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THERMAL TRANSPORT IN SELECTED CERAMIC MATERIALS FOR POTENTIAL APPLICATION AS INERT MATRIX FUEL OR THERMAL BARRIER COATING USING ATOMIC LEVEL SIMULATION By PRIYANK SHUKLA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 2010 PRIYANK SHUKLA With love to my whole family ACKNOWLEDGMENTS After reaching at this important milestone of my life, I would like to express my gratitude and thanks to all the people who made my PhD journey a wonderful and memorable one. First and foremost, I would like to thank and express my deepest gratitude to my mentor Prof. Phillpot. His passion for science, willingness to help; offer advice on anything, insightful remarks and a great sense of humor made my PhD a very enjoyable learning experience. He is my role model. Without his guidance and critical remarks and constant push to bring out the best of you, this work would not have seen the day. I cannot thank enough to my supervisory committee members Prof. Sinnott, Prof. Nino, Prof. Baney and Prof. Tulenko for their constant support, suggestions and guidance. I was fortunate enough to work with the great group of students and post docs of Computational Materials Science Focus Group. The group is very supportive and helpful. There are many, whose friendship and support, I cherished and enjoyed during these years. They know who they are and they have my gratitude. Special thanks to Dr. Rakesh Behera for his unconditional help, insightful remarks and delicious food. This journey wouldn't have been possible without the constant support, love and encouragement from my family. Although they are physically far away, weekly phone calls, yahoo messenger and emails have changed the distance to a few clicks away. I consider myself fortunate to have such a great family. In the end, I would like to say my special thanks to my wife, Monika. Thanks for bearing me for past three years, when I cannot tolerate myself for more than a few minutes. This work was funded by DOENERI Award DEFC07051D14647. TABLE OF CONTENTS page A C KNO W LEDG M ENTS ......... ............... ............................................. ............... LIST O F TA BLES .......... ..... ..... ............................................................. ........ 8 LIS T O F F IG U R E S .................................................................. 9 A BST RA CT ............... ... ..... ......................................................... ...... 13 CHAPTER 1 BACKGROUND ............................................................. ........ .......................... 15 1.1 Energy Sources.................................... .......... 15 1.2 Greenhouse Effect and Global Warming ................................. 17 1.2.1 Carbon Dioxide (CO2) Emission ........................ ............ 19 1.2.2 Global W arm ing Potential .......................... ....... ....... ...... .. 20 1.3 Carbon Em issionFree Energy..................................... ..... 20 1.4 Nuclear Energy .................. ....... .... ..................... 23 1.5 Thermal Conductivity: An Important Property ............................... 25 1.6 Motivation for This W ork .............................................................................. 25 1.7 Objective of This Work................................... ............... 26 1.8 Organization of Dissertation.......................... ...... ......... 28 2 THERMAL TRASNPORT AND SIMULATION METHODOLOGY......................... 29 2.1 Basics of Therm al Transport............................................... .................... 29 2 .2 M odes of H eat T transport .......................................................................... 32 2.3 Heat Transport in Nonmetallic Solids.................................. .. ......... 33 2.4 S im ulation M ethods ................................................................... 39 2.5 M olecular Dynam ics Basics .................................................................... 40 2 .6 Integration A lgo rithm s ........................................................................ .............. 4 0 2.7 Periodic Boundary C condition .................................................................... 43 2.8 Interatom ic Interactions................................. ............... 45 2.8.1 Longrange Interactions.......................... ..... ......... 45 2.8.2 Shortrange Interactions ....................... ......................................... 51 2 .9 E n s e m b le s .......................................................................................... 5 4 2.9.1 Pressure Control ......................... ........................ .................... 57 2.9.2 Tem perature Control ................................. ........................... ... .. 59 2.10 Thermal Expansion Simulation ................................. ...... 60 2.11 Thermal Conductivity Simulation ....... ......... ............... ....... 60 3 CRYSTALLOGRAPHY OF RELEVANT MATERIAL SYSTEMS............................ 65 3.1 G general Introduction ......... ........... .................. ............... ............... 65 3.2 MgO: Rocksalt Crystal Structure......................................... 66 3.3 Nd2Zr207: Pyrochlore Crystal Structure ................................... ...... ............ ... 68 3.4 MgAI204: Spinel Crystal Structure................ ......................... 71 3.5 Bi4Ti3012: Aurivillius Crystal Structure................ ........ ... .......... ........ 75 4 THERMAL TRANSPORT PROPERTIES OF MgONd2Zr207.............................. 84 4 .1 In tro d u c tio n ........................................................................ 8 4 4.2 Simulation Setups................................ ............... 85 4.2.1 Thermal Expansion............................. 85 4.2.2 Thermal Conductivity........................ ..... ............................. 85 4.3 Results ............... ..... .. ........... ....... ........... ... ..... ......... 86 4.3.1 Thermal Expansion............................. 86 4.3.2 Anharmonic Analysis ................. ...................... ................ 87 4.3.3 Thermal Conductivity........................ ..... ............................. 89 4.3.4 Finite Size Analysis ........................................................................ ........ 90 4.3.5 MgO: Thermal Conductivity ...... ...................... .............. 91 4.3.6 NDZ: Thermal Conductivity ............... ............... ................................. 92 4.4 MgONDZ: Polycrystals and Composite ............................. ............ 93 4.4.1 Polycrystal Structure............................. ............... 94 4.4.2 Com posite Structure..................... ........................... 96 4.4.3 Interfacial Effect on Thermal Conductivity ............... ....... ................ 98 4.4.4 Grain Size Dependence on Thermal Conductivity............................... 102 4.4.5 Thermal Conductivity of MgONDZ Composite ................................... 103 4.5 Discussion ........... .. .......... ........ ........................ 105 5 THERMAL TRASNPORT PROPERTIES OF MgAI204............................ 106 5.1 Introduction .... .............. ................................. ................. 106 5.2 Selection of Inverted Structure.............................. ............... 107 5.3 Effects of Inversion on Properties ............................... ............ 112 5 .3 .1 Lattice P a ra m ete r ............................................. ............... 1 12 5.3.2 A ntisite Form ation Energy ........... ............................. .................. 114 5.3.3 Bulk Modulus ......... ... .... ......... ..... .............. 118 5.3.4 Thermal Expansion........................................ 122 5.3.5 Therm al Conductivity....................... .................... 125 5.4 Conclusion ................. ........ ............ ............... 132 6 THERMAL TRANSPORT PROPERTIES OF Bi4Ti3O12 ..................................... 134 6.1 Introduction: Thermal Barrier Coating ....... .......... ......................... ....... 134 6.2 Selection of Potential Param eter .......... .............................. .................. 139 6.3 Thermal Expansion .............. ............... ..................... 140 6.4 Thermal Conductivity ............ ............... .................... 146 6 .5 C conclusion and Future S tudy................................................... .................... 155 7 GENERAL CONCLUSIONS ....................... ............. ......... ............... 156 6 7 .1 M g O N D Z ................................... ................................................... ............... 1 5 6 7.2 M gA 20 4 ........... ......... ..................................................... 157 7 .3 B i4T i3O 1 2 ......................................................................................................... 1 5 8 LIST O F REFERENC ES ............... ......... ...... ... ..................................... 160 BIOGRAPHICAL SKETCH ............... .... .. ............ .................... 175 LIST OF TABLES Table page 11 GWP (100 year timehorizon) of greenhouse gases ............... .................... 21 12 Types of commercial nuclear reactors............... ...... ....... ............... 24 21 The ZBL param eters ............................................... .............. 53 22 Potential parameters for the description of MgONd2Zr207 system .................. 53 23 Short range potential parameters for MgAI204 ....... ..... ......... ........................ 53 24 Short range potential parameters for Bi4Ti3O12 ......................... ................. 54 25 Summary of the eight statistical ensembles.............. ....................... 57 31 R ocksalt A X crystal structure ................................................... .. ................... 68 32 Pyrochlore A2B2X6X' crystal structure with origin at B cation.............................. 71 33 Fractional coordinates for atomic positions and vacancies for spinel ............. 73 34 Various Aurivillius structures for different n values................... ................... 75 35 Fractional coordinates for tetragonal structure (14/mmm) of BIT......................... 77 36 Fractional coordinates for monoclinic structure (Blal) of BIT.......................... 77 37 Historical preview for structures of BIT....................................... 78 41 Coefficient of thermal expansion for MgO and NDZ................................ ..... 87 4.2 Comparison of Debye temperature and C............................................. 88 51 Comparison of thermal expansion and bulk modulus............ ................ 126 61 Different forms of layered perovskites ..... .................................. 137 62 Short range potential parameters for Bi4Ti3012...................... ..................... 139 63 Comparison of interaction parameters for BIT............... .... ................ 140 64 Comparison of thermal expansion values.............................. 142 65 Thermal conductivity variation for all 4 systems ....... .............................. 154 LIST OF FIGURES Figure page 11 Energy demand and supply for the US ........................................ ...... ........... 15 12 Energy distribution among major sectors ................................. ......... ........ 16 13 Primary energy consumption by source and sector....................... ............... 16 14 Difference in average global temperature from 19611990 .............................. 17 15 US greenhouse gas em ission..................................... ............................ 18 16 CO2 concentration in the atmosphere for past 10000 years .............................. 19 17 Regions of restructured fuel pellet.................................. .. ............... 27 21 Moore's law for microprocessor..................... ..... ........................... 29 22 Power consumption of a microprocessor .................. ................................. 30 23 Schematic of TBC system ................................ .. .................. 31 24 Three modes of Heat Transfer.... .................................... 32 25 Threephonon processes ......... ................. .............................. ............... 34 26 N and U processes .............. .... .......... ....... ......... 34 27 Debye temperature and thermal conductivity (in W/cmK) at 300 K ............... 38 28 Temporal and Spatial resolution of simulation methods................. ........ 39 29 Schematic representation of a 2D periodic system. ............. ................ 44 210 Coulomb energy per ion as a function of cutoff in NaCI .............................. 47 211 Energy per ion against net system charge ......... ..................... ......... ...... 48 212 Schematic of dipolar shell of NaCI Bravais Lattice. ................ ......... ........... 48 213 Comparison of the Coulomb energy between charged and charge neutral....... 49 214 Comparison between the Madelung energy and the approximate ones........... 50 215 Schematic of flash diffusivity measurement and temperature rise curve............ 61 216 Schematics representation of simulation setup for direct method ................ 62 31 Conventional cubic unit cell of M gO ........... ...................................... ............ 66 32 Schematics showing three different symmetry operations.............................. 67 33 Pyrochlore sub unit (1/8) and Fluorite unit cell ............ ............................... 69 34 Two types of subcells, that forms the unit cell of pyrochlore........................... .. 70 35 Cubic spinel unit cell with Mg, Al and O atoms ........ ................ ............... 72 36 U nit cell of Bi4T i3O 12 ............................................................. ............... 76 37 Groupsubgroup graph connecting 14/mmm to Blal ................. ................ 80 38 Octahedra tilt of 6x6x1 supercell of BIT................. ......................... 82 41 Simulation setup to simulate thermal conductivity ...... .............................. 86 42 Lattice parameter variation with temperature for MgO and NDZ.................... 86 43 System size effect on thermal conductivity MgO and NDZ.............................. 91 44 Thermal conductivity data of MgO ................................ ............... 92 45 Therm al conductivity of Nd2Zr20 7............................................... ............ ...... 93 46 MgO polycrystal ~ 3 nm grain size .... .................. ........................ 94 47 Different connectivity patterns for twocomponent composite system ................ 96 48 MgONDZ composite (21 composite)............................................. .......... 97 49 Comparison of thermal conductivity of MgO ............................... ... ................ 99 410 Comparison of thermal conductivity of NDZ ............................. ... ................ 100 411 Temperature dependence of the interfacial conductance, G and Kapitza le ngth ................. ............... ......................................... 10 1 412 Predicted grainsize dependence of the normalized thermal conductivity of MgO and NDZ polycrystals compared with experimental results of YSZ.......... 103 413 Comparison of thermal conductivity of 6040 MgONDZ composite with predicted value .................. ............ ......... ............... .............. 104 51 Distribution of energy as a function of cation arrangement............................. 108 52 PDF for normal spinel and comparison of AIMg PDF ............................... 109 53 Comparison of PDFs for inverted structures............. ..... ................. 110 54 Decrease of lattice parameter with inversion............................ ... ................ 113 55 Structural energy as a function of inversion for MgAI204. .............................. 115 56 Cation antisite formation energy as a function of inversion. ........................... 116 57 Formation energy as a function of inversion site distance.............................. 118 58 Bulk modulus as a function of inversion ....... .... ............ ....................... 119 59 Comparison of normalized bulk modulus with MD study by Chartier.............. 120 510 Separating out various effects on bulk modulus ............... ...... ....... ............ 121 511 Variation of lattice parameter with temperature for i=0 ............................... 123 512 Coefficient of thermal expansion as a function of inversion .............................. 124 513 Dependence of the Gruneisen coefficient on inversion ................................ 124 514 Variation of thermal conductivity for MgAI204 with temperature ............ ....... 126 515 System size analysis for different inversion values for MgA204............ ....... 127 516 Dependence of thermal conductivity variation on inversion............................ 128 517 Density effect on thermal conductivity .......................... .... .......... ........ 130 518 Com binatoric analysis .............................. ... .... ............ ............... 131 61 Simple schematic view of turbofan gas turbine engine................................. 134 62 Idealized Brayton cycle ............................... ................... 135 63 Schematic of a TBC............................................. 136 64 6x6x1 super cell of BIT viewed along x and y axes..................... ............... 141 65 Coefficient of thermal expansion of BIT along x,y and z axes....................... 142 66 Temperature variation of tilt angles of ..... .................................. 144 67 Temperature profile of nearest TiTi distance in BIT structure........................ 145 68 Temperature profile of BiBi cation distance in Bi202 layer............................. 146 69 System size analysis for T = 300 K in all the direction................ ............... 147 610 Temperature and direction dependence of thermal conductivity for BIT........... 148 611 Thermal conductivity of BIT by experiment.............. ...................... 149 612 Schematic of method to investigate the density effect on thermal conductivity of B IT ........... ..... ... .. ......................................................... 152 613 System size and density dependence of thermal conductivity for BIT.............. 153 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THERMAL TRANSPORT IN SELECTED CERAMIC MATERIALS FOR THEIR POTENTIAL APPLICATION AS INERT MATRIX FUEL OR THERMAL BARRIER COATING BY ATOMIC LEVEL SIMULATION By PRIYANK SHUKLA August 2010 Chair: Simon R. Phillpot Major: Materials Science and Engineering Thermal management is a critical issue in nuclear reactor and thermal barrier coating applications. Current research for nextgen nuclear fuel is looking into materials to provide uraniumfree alternate with high thermal conductivity, whereas thermal barrier coating research is in need of low thermal conductivity materials. In this work, an understanding of various structural factors on thermal conductivity using atomic level simulation, is presented Thermal conductivity is a key parameter in the selection of nuclear fuel. Next gen nuclear fuel is suggested as to improve the efficiency of the nuclear fuel and make it environment friendly. Nextgen nuclear fuel is called as inert matrix fuel. Thermal conductivity of two potential materials (MgONd2Zr207:NDZ and MgAI204) is characterized using molecular dynamics simulation. In a composite system, interfaces play a dominant role in governing thermal conductivity of the system. To characterize the interface effects, very fine 2D textured grains (<10 nm) are build. Homogeneous interface simulations showed the dominance of interface effects on grains. Extension to bigger grain sizes showed good compatibility of the simulation data with the experimental dependence of grain size. In addition, MgAI204, a spinel system, has been characterized also for its potential application as an inert matrix. Since system undergoes cation antisite inverted state under irradiation, thermal conductivity as a function of inversion is characterized. It has been shown that, thermal conductivity remains invariant under inversion. In the process of characterizing thermal conductivity, thermal expansion, bulk modulus and elastic properties are also illustrated. Another key area for heat management is thermal barrier coating application. Thermal conductivity of a layered perovskite material, Bi4Ti3012 (BIT) belonging to Aurivillius family, is characterized. The effect of layers (Bi202 layer and perovskite layer) on the thermal transport of BIT is characterized by changing the mass of the layers under selected scenarios. The findings of the study suggested an effect of disparity between layers on the thermal conductivity of BIT in the through direction but limited role in the other directions. CHAPTER 1 BACKGROUND 1.1 Energy Sources Energy is the driving tool for the development of human civilization. The United States was self sufficient in energy until the 1950s. Since then, energy consumption has outpaced production, with the result that the deficit in energy continues to increase (Fig. 11). 120 20 Impdrts Exports is90 1o0 1970 180 19g 2 p ao Figure 11. Energy demand and supply for the US [1] Energy is given in the units of Btu, acronym for British thermal unit, which is a traditional energy unit. One Btu is the energy needed to heat up 1 pound of water by 1 degree Fahrenheit. It is approximately equivalent to 1.06 kJ. If the energy deficit continues, we will face an energy crisis in very near future. To overcome this deficit and regain energy independence, energy production needs to be increased and/or usage has to be reduced. Figure 12 shows the energy consumption among various sectors. As it clearly depicts, the industrial sector has leveled off in energy consumption but all other sectors are linearly increasing. This brings us to the question: how do different energy sources contribute towards meeting our demand of energy? F T pranspo rtatii n 10 On mmearcnal Residential 195I 1980 1970 19g0 1990 2QQQ Figure 12. Energy distribution among major sectors [1] The contribution from five different energy sources is shown in Fig. 13. It can be seen clearly that carbonemission free energy sources (renewable and nuclear) contribute only 16% of the total share. With coal, petroleum and natural gas as the leading contributors of the energy sources the biggest concern is: what is the effect on the environment of these energy sources? Transportation Industrial Petroleum, 37%O Natural Residential and Commercial Gas 24% Electric Power / Nuclear Renewable Electric Energy Power Petroleum N Natural Gas ECoal 7,' 9%0 Renewable Energy Nuclear Electric Power Total U.S. Energy = 99.3 Quadrillion Btu Source: Energy Information Administration, Annual Energy Review 2008, Tables 1.3, 2.1 b2.1f. Figure 13. Primary energy consumption by source and sector [1] 1.2 Greenhouse Effect and Global Warming Carbon dioxide (CO2) is the leading contributor to the greenhouse effect and global warming. The greenhouse effect is the absorption of infrared radiation in the Earth's environment. In the total absence of greenhouse effect, the Earth's surface temperature would be 255 K instead of 288 K [2]. The "greenhouse effect" of current concern is the further rise in temperature of Earth's surface from man's activities. 0,6 Global air temperature 0.4 2009 anomaly +0.44C 8(6th warmest on record) S0.2 0,II II l l l II' w 0.2 0.4 .6  1860 1880 1900 1920 1940 1960 1980 2000 Figure 14. Difference in average global temperature from 19611990 (best coverage period) [3] Figure 14 shows the trend in the Earth's surface temperature from 1850. Over 3000 stations all over the world recorded monthly temperature. The average is taken and the deviation from the period of 19611990 (best coverage period) is plotted here as the "Temperature anomaly". The increase over the past 30 years is the highest temperature rise seen in the past 10,000 years. To keep track of climate changes and these potential environmental and socioeconomic effects, The Intergovernmental Panel on Climate Change (IPCC) was formed by the United Nations Environment Program (UNEP) and the World Meteorological Organization (WMO). In its fourth Assessment Report, it is stated that the planet is warming rapidly due to the increase in greenhouse gases (GHGs) [4]. Gases which act in a similar fashion to C02 and absorb the infrared radiation from the earth's surface are collectively referred as GHGs. Although most of the GHGs are naturally present in the atmosphere, human activities have increased their levels manyfold over preindustrial era levels. This imbalance has caused the rise in global temperature and has given birth to today's biggest challenge: Global Warming. The GHGs can be classified as three types: 1. Natural: C02, methane (CH4), nitrous oxide (N20), water vapor and ozone (03). 2. Man made: Chlorofluorocarbons (CFCs), hydrochlorofluorocarbons (HCFCs), perfluorocarbons (PFCs), hydrofluorocarbons (HFCs), bromofluorocarbons (halons) and sulfur hexafluoride (SF6). 3. Indirect: Carbon monoxide (CO), nitrogen oxides (NOx), aerosols and non methane volatile organic compounds. They influence the formation of other forms of GHGs and contribute in global warming indirectly. HFC P FC S F6 NitrdOUS xi d MMethan4 mCrbon Dpede 8,000 7,030 6,127 6,089 6,192 6,312 6,413 6,489 6,7{  4,4,  5,000 5,000 3,000 2,000 1,000 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 Figure 15. US greenhouse gas emission [5] Even though contributions of these other GHGs are individually less than that of C02, their total collective contribution is of about the same magnitude. That is why their contribution in global warming is termed as "effective doubling". Figure 15 captures the contribution of various GHGs since 1990. 1.2.1 Carbon Dioxide (C02) Emission Being the largest contributor among GHGs, C02 becomes of utmost importance to understand the emission of C02. In the equilibrium carbon cycle there will be no buildup of C02 in the environment and thus no global temperature rise. Because of technical advancements, dependence on energy has increased and consequently the emission of C02 has risen. Figure 16 shows the concentration of C02 in the atmosphere for the past 10,000 years. IPCC 2007: WG1AR4 S3500 350 3o0 CI X 100 1900 2000 0 Year 0 300 Z 0 r 0 m 250  10000 5000 0 Time (before 2005) Figure 16. C02 concentration in the atmosphere for past 10000 years [6] As shown in Fig. 16, the concentration of C02 has increased markedly since the beginning of industrialization. The right axis label of Fig. 16: "Radiative Forcing" is a measure of the difference in net irradiance at the boundary of troposphere and the stratosphere. In simple terms, positive radiative forcing means a warming effect on the climate. By comparing Figs. 14 and 16, a linear relation between C02 emission and global temperature is found. All GHGs have different warming effects radiativee forcing) on the climate. A common metric system is used to measure the effect on climate by all these GHGs and is quantified as the global warming potential (GWP). 1.2.2 Global Warming Potential The GWP of any GHG is a quantified measure of its global time averaged relative radiative forcing impact. Conventionally the GWP of C02 is taken as 1. The GWP of any gas is defined as the equivalent mass of C02 to produce the same effect in global warming. A GWP is always specified over a timeperiod. A comparison between different GHGs is shown in Tablel1. 1.3 Carbon EmissionFree Energy In order to build a safer environment for coming generations, global warming needs to be eliminated. To achieve this goal, carbonfree energy is required. As discussed, carbonfree energy sources form a minor 16% share of the total energy resources. There are three ways to achieve the carbonfree environment: 1. Energy conservation by efficient usage 2. C02 separation and sequestration 3. Advancement of carbonfree energy sources (nuclear, solar, wind etc.) Efficient usage of energy does play an important role in contributing towards carbon free environment by lowering the gap between energy demand and supply. That, in effect, lowers the consumption of fossil fuels. But in the long run, it does not alleviate the global warming problem. Table 11. GWP (100 year timehorizon) of greenhouse gases [5] Gas Chemical formula GWP Atmospheric lifetime (Years) Carbon dioxide CO2 1 50200 Methane CH4 21 12 3 HFC152a C2H4F2 140 1.5 Nitrous Oxide N20 310 120 HFC32 CH2F2 650 5.6 HFC134a CH2FCF3 1,300 14.6 HFC4310mee CsH2F10 1,300 17.1 HFC125 C2HF5 2,800 32.6 HFC227ea C3HF7 2,900 36.5 HFC143a C2H3F3 3,800 48.3 HFC236fa C3H2F6 6,300 209 Perfluoromethane CF4 6,500 50,000 Perfluorobutane C4F10 7,000 2,600 Perfluorohexane C6F14 7,400 3,200 Perfluoroethane C2F6 9,200 10,000 HFC23 CHF3 11,700 264 Sulpher hexafluoride SF6 23,900 3,200 C02 separation and sequestration has the potential to be very efficient in abating global warming problem but has serious challenges. Turner [7] compared the global warming and C02 separation and sequestration solution to a smoker who is exercising. Even though the smoker is exercising, it will be of no effect if he continues to smoke. Similarly, C02 storage and sequestration cannot be the final solution. Besides, the sequestration process is very expensive and the risk is associated with the failure of C02 storage pose a great danger [7]. This limits the prospect of using C02 separation and sequestration technology as a solution to global warming. That makes the third alternative to be the most attractive. A very good review of the ways to solve the current energy problem is given by Lior [8]. The following steps are suggested for the future energy solution: 1. The immediate challenge is to meet the energy demand. Thus, fossil fuel will still be used until other energy sources become large shareholders in the energy resource family. 2. C02 separation and sequestration methods are needed to accompany the CO2 releasing energy sources. 3. Build more combined cycle power plants. 4. Where wind is economically available, wind power generation needs to be deployed. 5. Develop less expensive and more efficient solar cells. 6. Even with the issue of long term radioactive waste storage and safety issues, nuclear power plants need to be built and alternate nuclear fuels need to be developed. It is evident that all carbonfree energy sources will play a vital role in resolving the energy crisis/global warming problem. Nuclear energy is clearly recognized by scientists and technologists as environment friendly [9]. This view is now also shared by some environmentalists [10]. This view is further strengthened by the fact that according to ONUIPCC AR4 Synthesis Report, nuclear energy is considered as an effective GHG mitigation option [1]. 1.4 Nuclear Energy Nuclear energy is a way to generate energy from the splitting of nucleus. Heavier atoms release more energy in the process of the splitting of the nucleus. Controlled fission reaction forms the basis of nuclear energy. U02 is the most common nuclear fuel used in nuclear plants in USA. Naturally occurring uranium contains 99.3% 238U, 0.7% 235U, and < 0.01% 234U. 235U is fissionable and most of the current reactors use enriched uranium, which contains ~4% of 235U in the fuel. Fissile 235U and fertile 238U both are used in the fission process. Slow moving neutrons are captured by 235U to start fission reaction. Whereas, fast moving neutrons are absorbed by 238U and converts into 239Pu, which is fissionable by slow as well as fast moving neutrons. Various kinds of nuclear reactors are used worldwide to produce the nuclear energy. Some of them are shown in Table 12. As shown in Table 12, enriched U02 is the main nuclear fuel in US. The nuclear fuel cycle starts from the mining of uranium and ends at the burial of spent fuel. The whole process contributes towards radioactive waste. During mining and enrichment, waste of 238U, 226Ra and 222Rn and 230Th is generated. During the fission reaction, neutron capture and difference decay schemes leads to the formation of transuranic nuclides (atomic number >92), including 239Pu, 237Np, 241Am and 244Cm. In addition to the nuclear reactors, dismantling of nuclear warheads also increases the stockpile of Pu. The presence of these transuranic nuclides make the spent fuel radioactive and a period of more than 100,000 years is required to bring down the radioactivity level of this spent fuel to the level of mined uranium. Handling and keeping safe the Pu stockpile and transuranic nuclides poses a great environment challenge. An innovative fuel approach, termed inert matrix fuel (uranium free), has been suggested as a way to alleviate the problem by embedding Pu and minor actinides in a material matrix of non fissile or fertile material and incinerate in nuclear reactors. The stable matrix, after burning in a reactor will be directly disposed of in a once through fuel cycle concept. Table 12. Types of commercial nuclear reactors [11] Reactor type Main fuel Main countries Pressurized Water Reactor (PWR) Boiling Water Reactor (BWR) Pressuried Heavy Water Reactor 'CANDU' (PHWR) Gascooled Reactor (AGR & Magnox) Light Water Graphite Reactor (RBMK) Fast Neutron Reactor Enriched U02 Enriched U02 Natural U02 Natural U (metal), enriched U02 Enriched U02 Pu02 and U02 (MOX) US, France, Japan, Russia, China US, Japan, Sweden Canada UK Russia Japan, France, Russia Some key parameters in the selection of the inert matrix fuel (IMF) include high temperature stability, good irradiation behavior, high thermal conductivity, good mechanical properties, and low neutron capture and crosssections [12, 13]. Based on these criterions, MgAI204, ZrO2MgO and MgO Nd2Zr207 (NDZ), among others have been suggested as possible inert matrix materials [1416]. 1.5 Thermal Conductivity: An Important Property A nuclear fuel pellet experiences a temperature drop of ~ 800 K from the centerline of the fuel pellet to the edge. This large temperature difference is due to the low thermal conductivity of U02. At high temperature, center region of the pellet expands more than the rim. Because of the thermal stress thus formed, the microstructure of the grain changes and the fuel cracks and degrades. Efficient removal of heat from the fuel will maintain a more uniform temperature across the fuel and increase the fuel life time. This makes high thermal conductivity a desirable property for a nuclear fuel. The Thermal conductivity of materials depends on various parameters including but not limited to bulk modulus, density, and microstructure. The thermal conductivity is also an important parameter in the selection of thermal barrier coating (TBC) materials. In contrast to nuclear fuel, TBC applications demand low thermal conductivity materials. A detailed description of TBCs and their applications are given in Chapter6. Being important for both, nuclear fuel and TBC materials, thermal conductivity becomes an important material property for investigation. That raises an important question: What controls thermal conductivity? 1.6 Motivation for This Work A good understanding of the physics governing thermal conductivity in nonmetals is essential to answer the question, asked in Sec. 1.5. Phonons are the heat carriers in nonmetals/ceramic materials and are defined as lattice vibrations. Analysis of thermal transport is performed in phonon space (momentum space or frequency space). While phonon space is convenient for analysis, design and synthesis of the materials is performed in real space. It is difficult to move conceptually from phonon space to real space. For example, the position of the atoms in the unit cell and the relaxation times of the associated phonon modes are not related to each other I an obvious way. Molecular dynamics (MD) simulations are a suitable tool for the analysis of thermal conductivity at a finite temperature and for the bridging of real and phonon space analysis. Having total control over the system, simulation provides easy access to capture the effects of structural features affecting the thermal conductivity. In the following chapters, the effects of interface, structure and mass on material's thermal transport properties are presented and analyzed in detail. 1.7 Objective of This Work This work is primarily focused on understanding the thermoelastic properties of MgO, Nd2Zr207 (NDZ), MgAI204 and Bi4Ti3012 (BIT). The thermal properties of MgO NDZ and MgAI204 are studied in the light of their potential application as IMF systems, whereas BIT is studied for its potential application as TBC material. The effects of interfaces on the thermal properties of MgONDZ system are very important to understand. During burnup, large temperature and steeper temperature gradient causes the morphological changes of the nuclear fuel during power operation, as shown in Fig. 17. Although fuel pellet starts with all solid, a sizable void develops in the center. The void forms by the movement of the porosity to the center. Immediately adjacent to void region, columnar grains are formed. The radial boundaries of the columnar grains are formed by the trails of the pores and/or by the movement of fissiongas bubbles. Moving outward from the columnar region, a region of large textured grains is formed (Equiaxed grains) [17]. As fabricated Equiaxed grains Columnar grains VOID Figure 17. Regions of restructured fuel pellet [17] The outermost part of the pellet experiences more radiation damage and contains more fission damage than the bulk due to neutron resonance capture by 238U in this "rim" region to form fissile 239Pu. This leads to the formation of smaller grain sizes ( ~ 02 pm) from the asfabricated grain sizes of ~ 10 pm [18]. This effect is termed as "rim effect" and changes thermoelastic properties of the fuel. These morphological changes of the fuel, makes it essential to characterize the interface effects. Similar to experimental investigation of the interfacial effects on thermal transport for YSZ [19], MgONDZ composite system is investigated. To characterize the effect of interfaces for MgONDZ system, single crystal and polycrystals MgO and NDZ material system results are used. Thermal and elastic properties of another candidate inert matrix material, MgAI204 are also characterized. The ability of MgAI204 to resist radiation damage comes from its ability to withstand cation anti site disorder (inversion) without structural damage. Thus characterization of the thermal and elastic properties of MgAI204 as a function of inversion becomes very important for the potential application as an IMF matrix. In this work, effects of inversion on thermal as well as elastic properties are investigated. Another material, Bi4Ti3012 (BIT), is characterized for its potential application as thermal barrier coating (TBC) material. The anisotropic structure of BIT is determined; Intrinsic density difference between layers has been identified as the reason of its low thermal conductivity [20]. In this work, characterization of the effect of the density difference on thermal conductivity is extended. 1.8 Organization of Dissertation Chapter 2 lays out the foundation of thermal conductivity theory for ceramic materials and how it is implemented in molecular dynamics simulation. It covers important aspects of simulation and thermal conductivity. Chapter 3 contains crystallographic information of the materials discussed in this dissertation. Chapters 4, 5 and 6 focus on thermal transport properties of MgONd2Zr207, MgAI204 and Bi4Ti3012. Chapter 7 summarizes the results and the impact of this work. CHAPTER 2 THERMAL TRASNPORT AND SIMULATION METHODOLOGY 2.1 Basics of Thermal Transport In many aspects of technological developments, better management of heat is a key for improvement. In all sections of technology, heat flow is in the heart of many systems. Control over heat flow is necessary for the performance, life time and reliability of the devices. Performance, reliability and lifetime of electronic devices depend greatly on the efficiency of heat management. Intel cofounder Gordon Moore made the prediction in 1965 that number of transistors on a chip will double in every two years. Figure 21 captures the development in logic circuit industry. With each year, numbers of transistors are increasing while chip size decreases. This evolution corresponds to increasing energy density and the need for better heat management. transistors 10,000.000.000 DU~j~Iol htel' IjIanm ZP. Fio so MOORE'S LAW rir'ITni r ni penssr  licr Iaiunm' PiaLcZ*or Inler Pentiurm 4 Proe:L sor 1 InerPeoDqIIf,' IIPrOceKr irnel'P"wil "llP,'oessad'" 10,000,000 I0r4e .L .iiwrpocsn . 10,000.00 ie 8 6 r 100,000 fE,, .." 10,000 8004 ." 1.000 1970 1975 1980 1985 1990 1995 2000 2COS 2010 Figure 21. Moore's law for microprocessor [21]. The feature sizes in microelectronics is decreasing very fast and even outpacing Moore's law. Already by 2008, 35 nm Logic technology has been achieved [22]. With the increasing number of transistors on an integrated circuit and the decreasing size, power density increases. Figure 22 shows the implication of Moore's law on power consumption in microprocessor [23]. Figure 22 clearly suggests that one of the key issues is proper channeling of heat in the microprocessor. 100000 1000 Pentlum I S100 pfoces e " 10 286 46 .. 0.1 1 Bo^ .n ri" r 1971 1974 1978 1985 1992 20DD 2004 ) 82W Year Figure 22. Power consumption of a microprocessor, dashed line represents Moore's Law [23] Another area in which heat management is vital is nuclear energy. A nuclear fuel pellet experiences a temperature range of ~800 K from fuel pellet center to the rim within a distance of ~0.5 cm. Even though the heat generated during fission reaction inside the fuel is released to the coolant, this large temperature gradient often leads to structural deterioration of the fuel material. The high temperature and large temperature gradient causes physical and chemical changes in the fuel pin that often result in the deformation and cracking of the fuel pellet and possible interaction with the coolant [24]. To release heat from the fuel, higher thermal conductivity than current nuclear fuel (U02) is a desirable property of nuclear fuel. Therefore understanding heat transport mechanisms is important for the next generation nuclear fuel: IMF. Heat management is critical issue for the thermal barrier coatings (TBC) also. Research is primarily focused on finding low thermal conductivity materials for the application of TBC. TBCs act as a thermal barrier between hot gas stream and engine component thus reducing thermally activated oxidation and in effect delaying failure. TBC also increases the lifetime of the part by reducing thermal fatigue [25]. Figure 23, captures the temperature profile across the gas turbine engine and TBC. t oxidation resistant eatingg thermal barrier coating I TiAT I cowmporient I AT150K Distance Figure 23. Schematic of TBC system [25] All these examples points to the importance of heat management. A better understanding of thermal conductivity provides a greater control on materials. To capture the effects on thermal conductivity in ceramics, atomiclevel simulation is used. After gaining insight into the mechanisms of thermal transport, better selection of materials will be possible. Mechanisms of heat transfer in solids and the simulation methods used to capture the heat transport mechanisms will be discussed in detail in the following sections. 2.2 Modes of Heat Transport There are three modes of heat transfer relevant to materials: conduction, radiation and convection (Fig. 24). Conduction heat transfer involves heat carriers such as electrons, phonons (lattice vibrations); radiation involves electromagnetic waves and does not need any medium for heat transfer, convection involves movements of mass within fluids. Convection does not occur in solids, so this will not be discussed further. Conduction Convection : Radiation Radiation Figure 24. Three modes of Heat Transfer [26] In looking at heat transfer, it is useful to place solid materials in three distinct categories: metals, semiconductors and insulators. The major heat carriers in metals are electrons. Electrons in metals are very mobile and are not held tightly by the nucleus. Thus metals are generally described as having a communal sea of valence electrons. For metals the electrical conductivity (o) and thermal conductivity (K) are related by the famous WiedemannFranz Law [27]: = LT (21) a" T is temperature and L, a constant called the Lorentz number, is given by 72 (k 12 L =  = 2.44x10 WOK 2 (22) Where, kB is Boltzmann constant and e is the electron charge. This empirical law states that K/o has same value for all the metals at the same temperature. The proportionality of K/o on temperature was discovered by Danish mathematician and physicist Ludvig Lorenz in 1872. While this law generally holds, L is not exactly same for all the metals. Rosenberg stated that this law holds true at low and high temperatures but fails at intermediate temperatures [28]. In semiconductors and insulators, phonons (lattice vibrations) are the major heat carriers. This mechanism will be discussed in detail in this work. 2.3 Heat Transport in Nonmetallic Solids Phonons are the major heat carriers in nonmetals. The following discussion is based on standard presentations from various solid state texts [2932]. Heat transport in nonmetallic solids takes place through lattice vibrations. These are termed as phonons and are the major heat carriers in non metals. Harmonic potential treatment of phonons gives zero thermal expansion and infinite thermal conductivity of the material. Since materials do have finite value of thermal conductivity and nonzero thermal expansion, anharmonic terms are included in the discussion of phonons and heat transport. These anharmonic terms are labeled as threephonon (3rd order term in interaction potential) and fourphonon (4th order term in potential) processes. Other higher order terms are also possible but contribution to thermal conductivity is mostly dominated by threephonon processes. During heat transports two kinds of threephonon processes may occur, which are shown in Fig. 25 and are described below: k2 kk2 k3 A) k3B) Figure 25. Threephonon processes A) creation of one phonon from 2 phonons B) creation of two phonons from one phonon. 1. Annihilation of two phonons with the creation of one phonon (Fig. 25A) w(ki)+ w(k2)4(k3), kl+k2k3 2. Annihilation of one phone with the creation of two phonons (Fig. 25B) w(ki) w(k2)+w(k3), kl k2+k3 Both of the three phonons processes shown in Fig. 2.5 conserve momentum and are called normal (N) processes. In Nprocess, the total momentum is conserved and thus the phonon distribution is not changed. Another three phonon process is also possible in which momentum is not conserved, as shown in Fig. 26. NProcess UProcess k k, 1 / G "k .......... Figure 26. N and U processes, kl, k2, k3, and k3' represent phononmomenta and G reciprocal vector [33] Figure 26 shows both the N and U processes. G is a reciprocal lattice vector and based on the value of G, two processes are possible: normal (N) for G=0 and umklapp (U) for G/O. The shaded region in Fig. 26 is first Brillouin zone. In the Nprocess, as expected phonon moment lies in the first Brillouin zone and so no further operation is needed. Uprocesses do need an additional operation of the addition of reciprocal vector G to bring back the phonon moment to first Brillouin zone. The Nprocesses contribute to thermal conductivity indirectly as explained in detail by Callaway [34]. The dominant contribution to the heat resistance comes from Uprocesses, i.e. momentum nonconserving processes. There are a number of theoretical approaches to understand phonon conductivity in solids, including: 1. Relaxation time approach 2. Variational approach 3. Green's function approach (density matrix approach) The relaxation time and variation approaches assume that the thermal conductivity can be calculated from the linearized Boltzmann equation, whereas the Green's function approach takes the route of quantum statistics. A very good discussion of these methods is given by Srivastava [32]. From the kinetic theory of gases, the thermal conductivity of phonon gas can be expressed as /K= Cv2 (23) 3 where Cis the specific heat, v is group velocity and A is mean free path of the phonons. In a phonon gas, C can be approximated well by using the Debye model, v can be taken as the speed of sound; the real problem is in the calculation of 2. The specific heat is defined by the amount of heat required to raise the temperature by 1 K per unit mass. It is one of the easiest thermodynamic quantities to measure experimentally. Depending on which system parameter is kept constant, the specific heat can be of two types: Cp for constant pressure and Cv for constant volume. Since for a solid, it is experimentally unrealistic to keep volume constant while changing temperature, Cp is of more experimental relevance. There is a simple relationship between Cv and Cp for solids [35]: C = C + TV K (24) where a is coefficient of thermal expansion and KT is isothermal compressibility. At high temperatures, all solids obey DulongPetit result for the heat capacity, which gives the value as: C, = 3NAk (25) where NA is Avogadro's number and kB is the Boltzmann constant. This gives fairly good value at high temperatures but is qualitatively incorrect at lower temperatures; in particular T= 0 K, the specific heat value goes to zero. Einstein proposed a crude but simple approximation that all vibrations have the same frequency and are set to the value oE. The heat capacity then can be given as: mE 2 Xp~ E/kT) C = 3NZk, Bh [ exp(h9E /kBT) (26) kT [exp(hE,/kBT)1]2 Equation 26 is known as the Einstein model for heat capacity. It correctly converges to zero at 0 K but gives the wrong temperature dependence, decreasing faster than experiment near 0 K. Debye proposed a model in 1912 that overcame the problem of the Einstein model. Debye approximated a linear relation for phonon frequencies w = ck and treated phonons in a similar fashion as a particle in a box. Instead of infinite possible values of w, Debye suggested an upper limit on frequency as wD. Density of states under the Debye approximation is defined as: D(w)= 22v 3 D (27) 0 0 > OD where V is the volume of the solid. The maximum frequency WD is determined by the number density of atoms n and given as: OD = 2r2V3n (28) The specific heat in Debye model is given as: D.o ) n( / T3 \ C = 3N 3hK d n (29) At low temperatures, Eq. 29 become C =2NkB where D is Debye 5 OD) temperar, d d as ho ch 652 N , temperature, defined as O At high temperatures, it reaches the kB kB V DulongPetit limit of 3NAkB. The Debye temperature is the temperature above which material behaves classically and quantum effects to thermal conduction are negligible. As explained above, Debye model imposes a condition as maximum allowable phonon frequency to achieve finite number of modes in a solid (Eq. 28). That maximum frequency corresponds to the Debye temperature. Figure 27 shows the Debye temperature for various materials. The simplicity or complexity of Eq. 23 lies in the understanding of phonon mean free path. Based on the phonon model (Debye or Einstein), the temperature dependence of thermal conductivity can be easily described. At high temperatures, the specific heat of a solid is a constant, given by the DulongPetit law (Eq. 25). The group velocity of phonons in a solid has a weak temperature dependence [36]. Although the exact solution is not known, Peierls showed phonon mean free path has T dependence at high temperatures [37]. More sophisticated analysis has shown that temperature dependence is not T but I where n lies between 1 and 2 and confirmed by experiments [29, 38]. Be Detbye temperature and thermal conductivity B C N 0 F Ne 1440 Low temperature limit of Debye temperature in Kelvn ...2230... ... .. 75 2.00 Thermal conductivity at 300K, in W/cmK 0.27 1.29 Mg Al Si P S Cl Ar 400 428 645 ... ... 92 1.56 2 .37 148 Ca Sc Ti V Ci Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 230 3 420 630 410 470 445 0343 327 320 374 282 90 72 0.16 220.31 940.08 0.801.000.91 01 1.16 0 0.50002 Sr Y Nb Mo Tc R Rh Ag Cd n Sb Te I Xe 147 280 291 275 450 ... 480 274 5 209 200 211 153 64 0.1723 054 138 0.51 1.17 150 072 4 .97 0.82 0.67 0.24 0.02 Ba La H Ta Re Au Hg Pb i Po At Rn 110 142 252 240 4 430 5 420 24065 719 75 105 119 64 0.14 23 0.58 174 0.48 1.47 072317... 046 0.35 0.08 Pr Nd 0.13 0.16 Pa U 207 .. 0_28 Eu Gd 200 ... 200 ... 0.11 Am Cm ... 210 011 0.11 Bk Cf Ho Er Tm ... o... ... 0.16 0.14 0.17 Es Fm Md Yb Lu 120 210 0.35 0.16 No Lr ... Figure 27. Debye temperature and thermal conductivity (in W/cmK) at 300 K for selected elements [30] At low temperatures, the phonon mean free path becomes extremely long and thermal conductivity is limited by the system size, because scattering is dominated by the surfaces. In low temperature limit, the temperature dependence of thermal conductivity comes from the specific heat. Because the specific heat shows T3 dependence, the thermal conductivity does also. In the intermediate temperature range, Peierls showed K  expD OD where n, and b can be determined by fitting to experimental results. Typical values of nb lie in the range of 1535. 2.4 Simulation Methods There are number of ways to simulate materials. Based on the desired properties, an appropriate simulation method can be selected based on the length and time scale. Figure 28 captures different simulation methods based on time and length scales. Time Years Hours Minutes Seconds Microseconds Nanoseconds Picoseconds Femtoseconds Distance 1A 1 nm 1 p mm1 meters Figure 28. Temporal and Spatial resolution of simulation methods [39] For example, to investigate the electronic structure of a material, quantum mechanical simulation methods (density functional theory and HartreeFock) are appropriate [40]; to simulate a material of the size of mm, finite element methods are appropriate [41]. With current technological improvements, pm size length system have been simulated using molecular dynamics (MD) simulation with trillion atoms [42]. In this case the limiting factor is the huge amount of data and its analysis and the possible time devotion for the simulation. A 10 ps MD simulation of trillion atoms took a couple of days to finish and 40 TB hard disk space to store position, velocity and force data [42]. MD simulation is deterministic in nature and is a useful tool in the study of non equilibrium processes at the atomic level. Atoms are the basic units of MD simulation and are generally treated as having no internal electronic structure. Each atom has a specified mass and charge. It is classical in nature and every atom follows Newton's equation of motion (Eq. 210). d2 r F=m (210) dt2 The key physical input required for MD simulations is the interatomic potential. Since there is no interatomic potential that can exactly describe all the properties of any material, MD simulations can capture the properties of material only with limited fidelity. 2.5 Molecular Dynamics Basics As explained in the previous section, the trajectory of each atom is calculated in MD simulation using Newton's equation of motion. Based on the trajectory as a function of time, the average values of properties can be determined. For any atom i, the equation of motion can be written as follows: d2r, F, = m, = VU, (211) dt2 2.6 Integration Algorithms There are various integration algorithms [43, 44]. All the algorithms expand the position (r), velocity (v) and acceleration (a) into Taylor series around the time t: r(t + t)= r(t) + v(t)6t + r(t) 2 .. 2 v(t+ t)= v(t) + a(t)6t + v(t)t2 +... (212) a(t + t) = a(t) + a(t)t +... The Verlet Algorithm: The Verlet algorithm uses position and acceleration at time t and t6t to calculate position at t+ 5 t [45]. Because of its simplicity and low memory usage, it is a very common algorithm. According to Verlet algorithm, the position at t+ St is given by r(t + 8t)= 2r(t) r(t t)+ a(t)&t2 (213) The LeapFrog Algorithm: The leapfrog algorithm is a modified version of Verlet algorithm. It computes velocities at half timestep intervals and uses these velocities to calculate positions at time t+ S t r(t +t)= r(t)+v(t + t (214) The Beeman's Algorithm: The position prediction is similar to the Verlet algorithm but velocities are determined more precisely. r(t+ st) = r(t) + v(tt+ 2 a(t)t 2 _at t)t 3 6 (215) v(t + t) =v(t)+ a(t +t)t + a(t)t a(t t)st 3 6 6 The PredictorCorrector Algorithm: The Beeman's algorithm is modified into the predictorcorrector algorithm. i(t +9t)dp"re=ctd + i(t) I(t)dt + 2 (t)t 21 ( 5t)3t 2 3 6 (+ t)prd, = v(t)+ 3 (t )t {t St)at (216) 2 2 (t +t)rrctd = ()+ tt) + t + i(t 8(t t) 3 6 6 The 5th order Gear predictorcorrector method [44] is used in this work for numerical integration. The predicted new position of an atom is calculated using 5th order derivative of current position. S(t +St)= (t)+st(t)+ 1 ~t2 (t+ ISt3 (t)+ IS4 (t)+ 1S5 d(t) 2 6 24 120 SP(t + t) = (t) + 5tc(t) +1 St2b(t) + 3 (t)+ t4 t(t) 2 6 24 i P (t +,St)= d(t)+,t b(t)+ 1 t2 (t) 1 3 (t) (217) 2 dp t+ t)= (t) Where r, v, and care the position, velocity, acceleration of each atom respectively. b c and dare third, fourth and fifth derivatives of the position with time of each atom respectively. The superscript 'p' corresponds to the 'predicted' values. Using these predicted positions, forces on atoms are calculated thence the corrected accelerations. The difference between the corrected and predicted acceleration gives the error from the prediction step that is used to calculate the trajectory: Ac(t + St)= tc(t + St) tP (t +St) (218) c(t + 6 t) = P (t + 6t)+ c3Aa(t + 6 t) j (t + tt) = i(t + 5t)+ c, A (t + 5t) ac(t +t) = P(t +6t)+c2Ad(t +t) bc(t+6t)=bP(t+6t)+c3Ad(t+,t) ec(t +6't) = e (t + 6 t)+c4Ad~t + t) dc(t +t) = d(t +t)+cAa(t +t) The coefficients for a fifth order predictorcorrector are Co = 3/16, c, = 251/360, c2 = 1, C3 = 11/18, C4 = 1/6, and C5 = 1/60 [44, 46]. These corrected values are used as an input for the next simulation step. The procedure is repeated for all the atoms to determine their trajectories for the duration of the simulation. The timestep is an important factor in sampling the atom trajectory during simulation. The time step must be smaller than the fastest vibrational frequency in the system. Typical time steps for MD simulations are of the order of lfs (1015 s). This constraint comes from the fact that higher time step leads to errors in the trajectory and unreliable simulation results. Smaller time steps might capture more accurate trajectory but will limit the total simulation time. There are several ways developed to increase the time step and maintain the accuracy level, e.g. multiple time scale [47, 48] and stochastic dynamics [49]. 2.7 Periodic Boundary Condition The application of periodic boundary conditions (PBC) is a very useful technique to mimic large system by simulating smaller systems. Depending upon the requirements, PBC can be applied to replicate the system in one, two or three dimensions. When applied in three dimensions, the system is extended to mimic bulk material. A schematic of PBC is shown in Fig. 29. Original simulation cell (cell # 5) is in the center of the simulation system and other 8 cells surrounding it are the outcome of PBC. Arrows represent the movement of atoms. During simulation, all the image cells follow the actual cell. Therefore, during simulation if any atom moves in the actual cell, its periodic images follow its motion. For example, if atom moves from cell 5 to cell 6. All its images, also move to the next cell and in this process, atom from cell 4 enters cell 5. That way the total numbers of atoms in the cell are always conserved. 0 *r 0 4 ft 5 00 6 o0 S o0 0 r* 0 7 ; : 8* 9 Figure 29. Schematic representation of a 2D periodic system. Simulation is performed with the highlighted box. Arrows shows how atoms are affected under PBC [50] When PBC is applied, one needs to be careful as it can lead to unphysical interactions (self interaction: between atom and its image). All the interactions are taken to be zero after a specified distance (known as cutoff). Typically, to avoid self interactions, system size needs to be at least twice the cut off. 2.8 Interatomic Interactions Newton's equation of motion forms the basis of MD simulation and the interaction potential is the most important input for MD simulation. The quality of any simulation depends merely on the quality of the interatomic potential. In ionic systems, atoms are made of charged ions and interactions between the ions can be separated into a long range, Coulombic part, and a short range part. 2.8.1 Longrange Interactions The longrange interaction between two charged species is given by Coulomb's law: Ucou =q qj (220) r.i Where, qi and qj are the charges on atom i and j respectively, and rij is the distance between them. For a crystal structure with N ions, the total Coulombic energy is N1 N q ( Ucoul = (221) i1 j>7 r, The 1/r dependence of the energy makes the summation conditionally convergent. Conventionally, the Ewald summation is used to compute the longrange electrostatic interactions [51]. A more recent direct summation method is used to compute the summation [52]. Both methods are described below. Ewald Sum: The Ewald method is the standard technique to calculate electrostatic interaction in a periodic system. It was introduced by P. Ewald in 1921 [51]. In this technique, Eq. 219 is rewritten as two rapidly converging parts plus a constant term. UEwad Ur +Um +U0 (222) Where Ur is the short range term that sums quickly in real space, Um is a long range term that sums quickly in reciprocal (Fourier) space and Uo is the constant (self) term. These terms are given as: U 1 N erfc(ar ) Um=  q,qj 2 (223) 2u so m UO = 2 V is the volume of the simulation cell, m = (I,j,k) is a reciprocalspace vector. There are a few disadvantages of the Ewald sum. Firstly, it is not straightforward and fairly complicated to code, secondly it scales as O(N312) [53]. Direct Sum: A powerful alternative to the Ewald summation was developed by Wolf et al. [52] and is referred to as the direct summation. It is inspired by the fact that the Coulombic interactions in ionic crystals are short ranged, as demonstrated by both, theories [30, 54] and simulations [55, 56]. The only factor not allowing Coulomb sum to converge is the lack of charge neutrality over spherical crystallattice shell. This can be best illustrated with the help of an example. Consider a simple rocksalt structure of NaCI. The NaCI unit cell can be viewed as interpenetrating FCC lattices of anions and cations with half unit cell length displacement. The rocksalt structure will be discussed in detail in Chapter 3. As different cutoff distances are set for the potentials (cutoff distance is defined as the distance beyond which atoms do not interact Eq 226), the total system will be either cation deficient or anion deficient or rarely charge neutral. Figure 210 shows the variation of energy with cutoff radius. 0 0 0 MaCI lattice S10 o 0 S o O 0o S10 00 0o o o o 0 0 S20 o oo oo00 T 0 0 0 1 2 3 4 R [e] Figure 210. Coulomb energy per ion as a function of cutoff in NaCI [52] Figure 210 straightforwardly captures the nonconvergence of the Coulomb energy. The dashed line in the Fig. 210 is the actual value of the Coulomb energy for Rc oo of NaCI which is also known as the Madelung energy. The arrow in the Fig. 2 10 shows the points where the energy value converges to the actual value, however no relationship can be deduced between the cutoff (Re) and convergence. When the same energy graph is viewed against the net system charge (Fig. 211), a clear trend is captured for the convergence. Figure 211 renders the convergence criterion: charge neutral system. Fluctuations in Fig. 210 around the dashed line show that NaCI system never achieves charge neutrality when it is spherically terminated. To overcome this charge neutrality problem, initially a dipolar solution was suggested by Wolf [57]. A schematic is shown in Fig. 212, where a NaCI crystal is built of Na+Cl dipoles. In the evaluation of the Coulomb energy for a truncated crystal shell, counter dipoles are created at the surface for each dipole present in the shell and the expression for the Coulomb energy is given below: E = 7 N(r ) ( 5/cos43 [ 8Z J "<,^ ^ (224) 12 NaCI lattice o 8 0 h 4 0 4 S0 0 0 0 0o 8 0 8' = So 0 8 30 20 10 0 10 20 30 a qs[N(+)N()]q Figure 211. Energy per ion against net system charge [52] +q q Figure 212. Schematic of dipolar shell of NaCI Bravais Lattice. Adapted from Ref. [57] This dipole approach solves the problem of charge neutrality but leaves the system polarized. To overcome the polarization effect in the energy calculation, the polarization energy per unit volume is subtracted to achieve the correct Coulomb energy value. The whole problem of polarization energy can be avoided altogether, if the basic unit is taken without a dipole moment. For the NaCI crystal system, instead of Na+Cl as a basic unit, a subunit cell consisting of 4 Na+ and 4 Cl ions, is taken as a basis. Charge neutrality is achieved by adding countercharges at the terminated sphere. If Aq(Rc) is the net charge within the cutoff sphere of radius R, for ion i, the Coulomb energy can be written as: Ead (Rc q, qj q Aq(R,) (225) (r, r R , . 1 1m I2 . Scirhae neuta zef b m II iL/ *l a i ir f e [2 6 I P, o 10 e y  how :: i LI 6 i h ti I 210 3 D D direct t/r sum 30 iC1 tile 1 1.5 2 2.5 3 3.5 4 n [al Figure 213. Comparison of the Coulomb energy between charged (open circle) and charge neutral (filled circle) systems as a function of R,. Adapted from reference [52] A comparison between the charge neutral crystal shell and charged one is shown in Fig. 213. Figure 213 shows the effect of charge neutrality on the energy calculation and all the fluctuations of the charged system go away. Comparison with real Madelung energy is shown in Fig. 214 and it shows a damping effect in the energy calculation by the direct sum. Energy converges by 1/r in the direct sum as seen in Fig. 214. This 1/r damping effect on energy makes the potential practically very slow and useless for practical uses. That calls for some improvement in the potential. 1.0 1.5 2.0 2.5 R [al c 3.0 3,5 4,0 Figure 214. Comparison between the Madelung energy (open square) and the approximate ones (filled circle) as a function of R,. Adapted from reference [52] Before suggesting improvement to iron out the damping effect in energy, potential energy in itself needs to be rectified for the direct sum. By terminating energy calculation at R,, forces and stresses beyond R, become zero, which makes these otherwise physically continuous functions, discontinuous. To retain the continuity of energy, forces and stress, a shifted potential is used instead of the actual potential Form of this shifted potential is given in Eq. 226. V, (r,)= 'Lqj V~~r}^ lim q q (226) By introducing damping term in the shifted potential (Eq. 226), fluctuations in the Madelung energy calculation can be flattened out at smaller R,. Damping is introduced in the potential by error function because of its i) simplicity and ii) similarity to the Ewald summation method. After all these modifications, Eq. 222 is expressed as follows: E Mad ( NA=I q, qerfc(ar ) im qqerfc(ar ) (erfc(daa) R 2 o, (R) 2 m + Iq (227) 2 c 2 ,r r >Rc r2Rc Where a is a damping constant. It has been shown for NaCI that the Madelung energy is almost independent of a all the way up to 1.5/a for Rc=1.5a. Here a is the lattice parameter of NaCI [52]. This method also works very well with disordered systems including liquids and defects. 2.8.2 Shortrange Interactions The need for short range interaction comes from the 1/r dependence of the Coulomb energy. In the limit "r" approaches zero, the long range interactions diverge which certainly is not the case in a real system. Without any other interactions, two oppositely charged ions would fall into each other and the system energy will become infinite. Various functional forms of shortrange interactions have been developed for years, based on the material and properties under investigation. For example, to define characteristics of inert gases or neutral atoms or molecules LennardJones [58] potential is used, for semiconductors StillingerWeber [59] potential is used. The Buckingham potential [60] is used in this work, which is widely used to describe ionic materials. The functional form of the Buckingham potential is given by, VBuk (r)= exp ( r Pj) C / (228) where rij is the distance between ions i and j and Aij, Pij and Cij are ion pair (i,j) dependent parameters. The first term of Eq 228, embeds the quantum mechanical exchange interaction effect into classical description of interaction. Exchange interaction in simple terms, takes care of the fact that no two electrons can have the same four quantum numbers (n,l,mi,ms). Second term of Eq. 228 describes the Van der Waals attractive interaction due to dipole interaction. These empirical potentials are developed by fitting to the bulk material properties. That limits its use under extreme conditions such as high energy collisions. A more realistic model to capture the effect at very short range is described by Ziegler, Biersak and Littmark (ZBL) [61]. The ZBL interaction is a screened electrostatic potential of the nucleonnucleon interaction: V(r)= ZZ2e (r) (229) r where Z1 and Z2 are the atomic numbers of the interacting atoms, e is the electron charge and r is the interatomic distance. O(r) is the screening function, B(r)= AZ exp(r (230) i=1,4 V au where Ai and bi are predefined parameters and are given in Table 21 and au is: 0.8854 a = ao (231) Z0 r 23 +z 023 where ao is the Bohr radius, 0.539177 A. Table 21. The ZBL parameters i A b 1 0.1818 3.2000 2 0.5099 0.9423 3 0.2802 0.4028 4 0.02817 0.2016 The potential parameters for the MgONd2Zr207 system are summarized in Table 22. To incorporate the covalent nature of the crystal in case of MgO and Nd2Zr207, partial charges are assigned to ions and are shown as superscripts. Table 22. Potential parameters for the description of MgONd2Zr207 system [62, 63] Species A (eV) p(A) C (eV Ab) O1 O1 35686.18 0.20100 32.00 Nd+2.5O1.7 2148.14 0.32270 22.59 Zr+3.4_1.7 1402.57 0.33120 5.10 Mg+1.71.7 929.69 0.29909 0.00 Table 23 summarizes the interaction parameters for MgAI204. Every ion is assigned formal charges in this case. Table 23. Short range potential parameters for MgAI204 from Ball et al. [64] Species A (eV) p(A) C (eV Ab) Mg+2O2 1279.69 0.2997 0.00 O2O2 9547.96 0.2240 32.0 Al+3O2 1361.29 0.3013 0.00 Table 24 summarizes the interaction parameters for Bi4Ti3012. Every ion is assigned a formal charge in Bi4Ti3012. Three different TiO interaction parameters are investigated and labeled by their A parameter values. Table 24. Short range potential parameters for Bi4Ti3012 [6567] Species A (eV) p(A) C (eV Ab) Bi+3O2 49529.4 0.2223 0.00 O202 9547.96 0.2192 32.00 2131 2131.04 0.3038 0.00 Ti+4O2 760 760.47 0.3879 0.00 2549 2549.4 0.2989 0.00 Even though the interactions for Bi4Ti3012 were defined for coreshell model, the shell part is neglected interaction parameters for corecore only are used in this work. In a coreshell model, the electrons of an ion are divided into two parts: core and shell. Core electrons are combined with nucleus and assigned a +ve charge and the shell having all electrons, is given a ve charge. The charge on each ion is described by the sum of the respective core and shell charges. The core and shell of each ion are connected by either a harmonic or anharmonic spring. For more details, please refer to the article by Dick and Overhauser [68]. 2.9 Ensembles Statistical mechanics is a useful method to study a system with large number of particles. It converts microscopic properties of the collections of individual atoms (positions, velocities, etc.) or molecules to macroscopic bulk properties of materials that can be observed experimentally. The ensemble is simply a statistical equivalent of repeating the same experiment under the same macroscopic conditions but different microscopic ones and expects to observe a range of outcomes. A thermodynamic state of the system can be defined using various state variables: pressure (P), temperature (T), total number of atoms (N). Other thermodynamic properties (density, chemical potential, heat capacity etc.) may be derived using the equation of state and fundamental relations of thermodynamics. Even other microscopic properties, like diffusion coefficient, structure factor which clearly depicts microscopic structure and dynamics at the atomic level are dictated by the variables characterizing the thermodynamic state and not by microscopic properties. The atomic positions and moment are coordinates in the phase space (F). For a single particle, its phase space has 3 positions and 3 moment totaling 6 dimensions. Let us assume, the instantaneous value of a property c can be written as (F). Since system evolves with time and macroscopic property c is a time average of all these instantaneous value, and can be written as: 1 tobs obs < >tme = lim J (F(t))t (232) to,b tobs 0 As stated earlier, time evolution of the phase space is governed by the Newton's equation of motion. As the number of atoms increase in the system, solving Eq. 232 becomes a formidable task. Another concern is also the infinite time limit to perform the time average of Eq. 232. That can be resolved by taking a long enough time r and solving Newton's equation of motion in smaller time steps (St). Replacing infinite time by r and 6 t, Eq. 232 can be rewritten as: ob, Y,,E (ri)) (233) Equation 233 provides a simulated value of the macroscopic property c by taking its time average. Instead of time evolution of the bulk properties, Gibbs suggested replacing time averages by the ensemble average. The ensemble is regarded as a collection of points r in the phase space. The points are distributed according to a probability density p(F). This density function depends on the chosen fixed macroscopic parameters (NPT, NVE etc). These points are selected from i) Pair of extensive and intensive quantities including N (total no of particles) or p (chemical potential) ii) The volume (V) or pressure (P) iii) The energy (E) or related thermodynamic potential or temperature (T) In general, ensembles are categorized as microcanonical, canonical and grand canonical based on different macroscopic constraints [44]. The microcanonical ensemble is used to describe the properties of an isolated system with fixed N, V and E and is also referred to as the NVE ensemble. The canonical ensemble is a set of systems which can share its energy and maintains a constant temperature. They are in constant contact with a heat bath or reservoir and exchange energy to maintain a constant temperature. It is referred as NVT ensemble. The grand canonical ensemble is a collection of systems which are in thermal contact with a heat bath or reservoir and can exchange particles. It can be viewed as an extension of a canonical ensemble with the addition of particle exchange. The chemical potential p is introduced to define the fluctuations in the number of particles. These three ensembles are not sufficient to cover all the experimental constraints. A list of all the statistical ensembles is given in Table 25. Table 25. Summary of the eight statistical ensembles. For specific conditions the internal energy E is modified as the enthalpy (H = E + PV), the Hill energy (L = E tZiNi), and the Ray enthalpy (R = E +PV ZJiNi) [69]. Fixed Variable Ensemble NVE Microcanonical NVT Canonical NPT IsothermalIsobaric (Gibbs) NPH Isoenthalipclsobaric pVT Grandcanonical pVL Grandmicrocanonical pPR Grand IsothermalIsobaric pPT Generalized A generalized ensemble is not physical as its size is not specified [69].Since in the simulations, mostly NPT and NVE ensembles are applied, it becomes important to briefly discuss the implementation of constant temperature and pressure in the simulation. 2.9.1 Pressure Control The pressure control procedure developed by Andersen [70] and subsequently extended by ParrinelloRahman [71] is now a standard tool. In a constant pressure simulation, the volume of the system fluctuates. Atom coordinates are redefined as , = (234) V3 where, V is the box volume. Let us imagine the fluid to be simulated is in a container of variable volume V and is compressed by a piston. M is the mass of the piston and P is the pressure applied. Let us also assume that the volume change is isotropic or, equivalently an equal pressure is applied from all the directions (hydrostatic pressure). This scenario can be represented by a Lagrangian given as follows: 1 2 1 2 L = mVy p, P u +V IMV PV (235) 2 1,1 I The first two terms of Eq. 235 are the Lagrangian of the fluid in the container, third term corresponds to the kinetic energy of the piston and the last term refers to the potential energy of the piston. After constructing the Hamiltonian and forming equation of motions, the trajectory of each atom can be derived as follows: V P+3 2m (236) 1 ^ where, Pi is defined as / and nr is the conjugate momentum of p (atomic /V3 coordinates, as defined in Eq. 234). This method was extended by Parrinello and Rahman (PR) [71] to anisotropic stress. These methods are not good to capture first order phase transitions [72] for non periodic systems [73]. First order phase transition in PR scheme does not take place via nucleation and growth. Instead it takes place as a collective transformation which might result in an overestimation of the energy barrier for the phase transition. This can result in missing intermediate phases and lowers the predictive power of the PR scheme. To overcome these issues, a new method has been Md2 V, dt2 developed by Martoiak, Laio and Parrinello [74]. Another scheme has been developed for finite system sizes by Sun and Gong [73] 2.9.2 Temperature Control Most experiments are carried out at constant pressure and temperature. Usually, temperature is related to the average kinetic energy of the system as (KE) = Y = NkBT (237) 1 2 2 This issue is resolved by implementing a thermostat in MD. A simulated system is in constant contact with a heat bath or reservoir that acts as a thermostat. The first and simplest method to implement constant temperature in MD simulation is a momentumscaling algorithm proposed by Woodcock in 1971 [75]. In this method, at each simulation step, the velocity of each particle is rescaled to keep the average kinetic energy at the desired constant value calculated by Eq 237. It is also known as velocity rescaling method. Rescaled velocity is given by: v, rescaledd)= Fgtv, (238) Various other methods to implement constant temperature in MD simulation have been reviewed here [44, 76]. They are proposed by Nose [77], Hoover et a/.[78] and Evans and Morriss [79], Haile and Gupta [80] and an extension to the extended system method of Nose [76] by Hoover [81] that is known as NoseHoover algorithm. Because of its simplicity, velocity rescaling method is used for the simulations. 2.10 Thermal Expansion Simulation As a starting point in the simulation of thermal conductivity, coefficient of thermal expansion is computed using simulation. Thermal expansion is related to the lattice parameter to the first order, as follows: = 1+aT (239) a0 where "a" is lattice parameter at temperature T and ao is the lattice parameter of the system at T = 0 K and a is the linear thermal expansion. To the first order approximation a can be computed by plotting a/ao as a function of temperature. In the simulation, a cubic system is generated to calculate thermal expansion in all the directions. To compute thermal expansion, NPT ensemble is used and simulations are run at various temperatures to get a good profile between lattice parameter and temperature. 2.11 Thermal Conductivity Simulation Fourier law relates the heat flux to the temperature gradient on a material: ?T J, =K (240) where J,, is the /th component of heat flux (energy per unit area per unit time), Ky is an element of conductivity tensor and OT/x, is the temperature gradient along j direction. Experimentally, the thermal conductivity is measured using thermal diffusivity and the following relation K = apC, (241) where a is the thermal diffusivity, p is the density and Cp is the specific heat capacity at constant pressure. Laser flash is the most widely used method to calculate thermal diffusivity and is described in detail by Parker et al. [82]. In brief, A few millimeters thick thermally insulated sample is irradiated with a highintensity shortduration light pulse and temperature history of the rear surface is measured. The thermal diffusivity is measured from the shape of temperature evolution with time curve at the rear surface. A schematic of flash method is shown in Fig. 215. In simulation, it can be determined by two ways: Non equilibrium MD (NEMD: direct method) [63, 79, 8391] or equilibrium MD (EMD: GreenKubo method) [83, 9297]. AMPLIFIER S URNACE COMPUTER MONITOR POWER + S SUPPLY j DATA LM wIsmo PRINTER *I  FURNACi EST 0 500 1000 1500 2000 T~me/nms Figure 215. Schematic of flash diffusivity measurement and temperature rise curve. Adapted from the Ref [98]. The simulation setup to measure thermal conductivity for direct method is shown in Fig. 216. System in direct method is setup in a way that it is of cylindrical shape. LASER or FLASH SOURCE USUa 5UWLY 3 25 S15 o I 0 0.5 2500 ii This size constraint (longer in one direction: z compared to other two: x and y) transforms Eq. 241 into scalar form as given below: dT J = K (242) 0z Heat source Heat sink Z=L42 ZL4/4 Z0 Z=L.J4 Z=L_/2 I I I_ _ S/  \ Figure 216. Schematics representation of simulation setup for direct method (NEMD). Simulation cell has length Lz and heat source and sink both have thickness 6 and at a distance Lz/4 from the edge/center of the cell. As energy is added and removed at the source and sink respectively. Figure is adapted from reference [99]. To simulate the thermal conductivity, the system setup follows these steps 1. Coefficient of thermal expansion is calculated at the desired temperature (Section 210). 2. System is equilibrated at 0 K. 3. Expansion coefficient is applied to the relaxed structure and equilibrated at the set temperature 4. Once the system reaches equilibrium at the set temperature, the temperature gradient is generated by giving extra energy to the atoms at the heat source and removing the same amount of energy from the atoms at the heat sink region. 5. This sets up a temperature gradient and its average over time is used to compute the thermal conductivity using Eq. 242. The size of the simulation cell strongly affects the value of thermal conductivity determined, as illustrated in the case of Si [99]. When the simulation cell is smaller than the phonon mean free path, thermal conductivity is limited by system size. This is known as the Casimir limit. To account for this effect, the thermal conductivity is calculated as a function of system size and then extrapolated to infinite system size. Following is the physics behind this phenomena and methodology to overcome the issue. By applying the kinetic theory of gases, the thermal conductivity was shown to be equal to: S= cvl (243) 3 where, c is the specific heat, vis the group velocity and / is the effective mean free path of the phonons. Specific heat above the Debye temperature is given by the DulongPetit value for a noninteracting gas: c = kn (244) 2 where, kg is the Boltzmann constant and n is the number density of the ions. For a unit cell of volume V and total number of ions N, n is given as N/. Group velocity v can (vl, + 2v, ) be expressed as: v= where vL and v, are the longitudinal and transverse 3 sound velocities. For a simulation cell of length Lz, the phonon mean free path is Lz/4 (As shown in Fig. 216, heat source and heat sink are at Lz/4). If lo is the phonon mean free path for infinite system, then the effective mean free path for system size with Lz length is given by [99]: 1 = 1 + (245) The final expression for thermal conductivity is: I 3 BN vL + 2v, 1 4(246) r = kg + (246) 3 2 V 3 )l L, For a cubic system, the thermal conductivity and system size relation reduces to: 1 2a3 1 4 = + (247) K kNv Il L, Equation 247 shows a linear relation between the inverse of thermal conductivity and inverse of system size. The methodology to infer thermal conductivity for infinite system size can be simply achieved by extrapolating the thermal conductivity values at 1/Lz=0. This approach has been applied to Si and diamond single crystals [99] and will be used here. In case of thermal conductivity, the coefficient of thermal expansion and elastic properties are calculated first. "Anharmonic analysis" is used to correct simulated thermal conductivity values based on the fidelity of interaction parameters with experimental thermal expansion and bulk modulus values. Details of the analysis are presented in chapter4 with MgONd2Zr207 results. CHAPTER 3 CRYSTALLOGRAPHY OF RELEVANT MATERIAL SYSTEMS 3.1 General Introduction Crystallography is formed from two Greek words, crystallon meaning frozen drop/cold drop, and grapho meaning write. Crystallography refers to the science of determining the arrangement of atoms in solids. Crystallography also provides insight into material properties and plays a major role in many subjects, including mineralogy, physics, materials science, chemistry, geology, geophysics and medicine. In essence, it encompasses various areas of science and engineering. Materials with the same chemical composition exhibit different properties based upon their crystal structures. For example, graphite and diamond both are forms of carbon but differ in crystal structure and show disparity in almost all physical properties. Another example is PbTiO3, which is cubic at room temperature and paraelectric but as it changes to the tetragonal structure at 763 K shows ferroelectricity [100]. The structure of a material can be understood as a description of the arrangement of the components that make up the material. All the atoms/ions can be taken as building blocks of the materials, especially solids. The limiting factor here is the sheer number of atoms to describe a small piece of material. For example a tiny cube of salt (NaCI) has approximately 1020 atoms. If a material is perfectly ordered, then by exploiting periodicity, a smaller representative unit of NaCI can be used to describe NaCI for any length scale. The smallest representative unit of any structure is called its unit cell. Standard descriptions of all the space groups are given in The International Tables for Crystallography [101]. Both HermannMauguin and Schonflies notations are used to describe the symmetry of the materials investigated. Schonflies notation is adequate only to describe the point group and thus generally used in spectroscopy. HermannMauguin notation is generally used in crystallography because it includes space group symmetry also in the notation. 3.2 MgO: Rocksalt Crystal Structure The rocksalt structure is named after NaCI halitee). Almost every alkali halide, sulfide, selenide and telluride form rocksalt structure [102]. Almost one third of all MX type compounds crystallize to the rocksalt structure, where M is the cation and sits at (0, 0, 0) and X is an anion and sits at (0, 0, 1/2). MgO has the rocksalt crystal structure. The equal number of cations and anions in the rocksalt structure are arranged as interpenetrating FCC sublattices. The unit cell of rocksalt crystal structure with MgO as an example is shown in Fig. 31, with O as large ions and Mg as smaller ions. The conventional cubic unit cell contains 4 formula units of MgO (Z=4). O Mg Figure 31. Conventional cubic unit cell of MgO. Mg (small circle) and O (big circle) The rocksalt structure belongs to the Fm3m (o,, No. 225 in the international table) space group, where Fm3m is space group representation in HermannMaugin and o' is in Schonflies notation. o, represents octahedral symmetry (four 3fold axes along body diagonals with three 4fold axes along the faces) and mirror planes. In HermannMaugin notation, F represents a face center cubic structure and m3m, a short form for 3 2 point group represents operations along the primary <001>, secondary <111> and tertiary <110> direction of the cubic system. For MgO, the 4/m represents fourfold rotational symmetry at the corners and the cell center, and a mirror perpendicular to the rotational axis [100] (Fig. 32(a)). The3 symmetry represents the rotoinversion along the body diagonal [111] of the unit cell (Fig. 32(b)). The 2/m symmetry operation represents the twofold rotation along the edges of the unit cell, and a mirror plane perpendicular to the axis of rotation (Fig. 3 2(c)). a )B) Figure 32. Schematics showing three different symmetry operations represented by MgO A) A fourfold rotation symmetry along [100]. B) A threefold inversion symmetry along [111]. C) A twofold rotation symmetry along [110]. The lattice parameter of MgO unit cell is 4.191 A [103]. Table 31 lists the site symmetry, the Wyckoff position and coordinates of the atoms for rocksalt structure AX. Table 31. Rocksalt AX crystal structure Ion Wyckoff position Site symmetry Coordinates (0,0,0; 0,1/2,1/2; 1/2, 0, 1/2; 1/2, 1/2, 0)+ A 4a Oh 0,0,0 X 4b Oh 1/2,1/2,1/2 The first column of Table 31 lists the ions present in the system (for example, AX has cation A and anion X). The second column gives information about Wyckoff position that represents a total number of equivalent points. The letter in the Wyckoff position goes in alphabetic order from high to low symmetry, for example 'a' is assigned to the equipoint having the least number of equivalent points. The third column contains information about the point symmetry of the corresponding position. The last column gives the coordinates of the ions. The positions under the "coordinates" title are called the generators. All the positions of atoms/ions are calculated by adding the coordinates to the generators. For example, from Table 31, ion A has 4 multiplicity and all 4 positions are calculated by adding (0,0,0) to (0,0,0), (0,1/2,1/2), (1/2,0,1/2) and (1/2,1/2,0). Similarly, coordinates of X can be calculated by adding (1/2,1/2,1/2) to (0,0,0), (0,1/2,1/2), (1/2,0,1/2). 3.3 Nd2Zr207: Pyrochlore Crystal Structure Nd2Zr207 (NDZ) belongs to the pyrochlore family and has general formula A2B207 [104]. The space group for pyrochlore structure is Fd3m (F4y32/m, o No. 227 in the international table [101]). Similar to the rocksalt crystal symmetry, pyrochlore also has 3fold rotoinversion along (111) and 2fold rotation axis with perpendicular mirror along (110). Instead of 4fold rotation axis with perpendicular mirror plane, this has 41 screw axis with diamond glide. A screw axis (helical axis or twist axis) is a combination of rotation plus translation symmetry. 41 screw axis represents a rotation of 900 (3600/4) and translation of 14 of the lattice vector parallel to the rotation axis. The crystal structure of pyrochlore materials is closely related to the fluorite structure. There are two main differences between the fluorite and pyrochlore crystal structures: anions occupy all 8 tetrahedral sites in fluorite and only 7 in pyrochlore, and pyrochlores have approximately twice the lattice spacing of fluorite structures, i.e. one unit cell of pyrochlore structure can be divided into 8 sub units with each sub unit similar to a fluorite structure [105]. Figure 33 shows oneeighth of the pyrochlore structure and the fluorite structures side by side. Nd Zr U A) B) Figure 33. A) Pyrochlore sub unit (1/8) B) Fluorite unit cell Cations A and B are 8 and 6fold coordinated respectively, and can have either +3 and +4 or +2 and +5 oxidation states. Based on the charge and oxidation state, the ionic radius of cation A should lie within 0.0870.151 nm [105]. Similarly ionic radius of B should lie in the range of 0.0400078 nm [105]. Furthermore, the radius ratio of rA/rB should lie between 1.292.30 [105, 106]. Generally for the +3/+4 pyrochlores, lanthanides (3+) occupy A sites and 4+ cations take the B sites. As seen in Fig. 33, cations sit at the corner and face center positions; while anions occupy tetrahedral sites. In pyrochlore, the cations form face centered cubes and are ordered in alternate (110) and (110) rows (face diagonals). The anions are located in all of the tetragonal sites except the ones which are surrounded by the A cations only [104]. The unit cell of pyrochlore is composed of two types of subunit cells, arranged in alternate manner. Type I. The A cations form face diagonal at the upper righthand corner and missing anion is at the lower lefthand. (Fig. 34A) Type II. The A cations form face diagonal at the lower lefthand corner and missing anion is at the opposite corner. (Fig. 34B) ,4 Vacant Type I Cell Type II Cell Figure 34. Two types of subcells, that forms the unit cell of pyrochlore with alternate mixing [107] The structural difference of one less anion to complete the cube inside the cell identifies two types of cations in the system. They are denoted by X and X', where X' is the diagonally situated cation from the vacant site and rest are defined by X. The unit cell of pyrochlore contains equal number of TypeI and Typell subcells and is arranged in a manner that only different types of subcells become neighbors. Figure 34 shows one typical arrangement of both types of subcells. Due to this structural feature, The lattice parameter of NDZ is (a = 8.089 A) which is almost double that of MgO (a = 4.202 A). This difference becomes important when composite structures of MgO and NDZ are built for the simulations. The polycrystalline structures of MgO and NDZ are used to separate out the interface effects of each individual component of the proposed inert matrix system. Structural information for pyrochlore crystal structure including location, site symmetry and coordinates with origin fixed at Zr cation, are given in Table 32. Table 32. Pyrochlore A2B2X6X' crystal structure with origin at B cation [104] Ion Wyckoff position Site symmetry Coordinates (0,0,0; 0,1/2,1/2; 1/2, 0, 1/2; 1/2, 1/2, 0)+ A 16d D3d 1/2,1/2,1/2; 1/2,1/4,1/4; 1/4,1/2,1/4; 1/4,1/4,1/2 B 16c D3d 0,0,0; 0,1/4,1/4; 1/4,02,1/4; 1/4,1/4,0 X 48f C2v x,1/8,1/8; x,7/8,7/8; 1/4x,1/8,1/8; 3/4+x,7/8,7/8 1/8,x,1/8; 7/8,x,7/8; 1/8,1/4x,1/8; 7/8,3/4+x,7/8 1/8,1/8,x; 7/8,7/8,x; 1/8,1/8,1/4x; 7/8,7/8,3/4+x X' 8b Td 3/8,3/8,3/8; 5/8,5/8,5/8 x for regular tetrahedral = 5/16 (0.3125) x for regular cube = 3/8 (0.375) From Table 32, value of x ranges from 0.3125 to 0.375 and depend on the specific composition. Exact value of x can be determined by experimentally characterizing the sample using for example Xray diffraction, neutron diffraction. 3.4 MgAI204: Spinel Crystal Structure The general formula for the spinel compounds is AB204, where A is usually a divalent cation and B is a trivalent cation. In this thesis, A is Mg2+ and sits on 8a tetrahedral sites, and B is Al3+ and occupies the 16b octahedral sites. The cubic unit cell contains 8 formula units for a total of 24 cations (8 Mg ions and 16 Al ions) and 32 anions (all O ions). This unit cell can be considered as consisting of 8 cubic subunits, as shown in Fig. 3.5. Bragg [108] and Nishikawa [109] determined the crystal structure of spinel in the early 1900s. The majority of spinels belong to space group Fd3m (Oh7, No. 227 in the international table [101]). Figure 35. Cubic spinel unit cell with Mg (blue, medium sized), Al (green, small) and O (red, large) atoms. The unit cell can be considered as being divided into 8 subunits with Mg tetrahedral (blue) and Al octahedral (green) in alternating sub cells There are total 64 octahedral and 32 tetrahedral interstices between anions in the spinel unit cell. Among these, 8 tetrahedral interstices are occupied by A cation and 16 octahedral interstices by B cation in the normal spinel structure. The atomic positions in the spinel structure depend largely on the choice of origin in the Fd3m space group. Two different points with point symmetries 43m and 3m are possible choices for the origin. Table 33 lists the atomic positions for normal spinel structure with both origin choices. Table 33. Fractional coordinates for atomic positions and vacancies for spinel [110] Fractional coordinates Wyckoff Point (0,0,0; 0,1/2,1/2; 1/2, 0, 1/2; 1/2, 1/2, 0)+ Lattice Site Position Symmetry Origin 43m 3m Asite 8a/ 1/ 1/ 1/ 1/ 1/7/ 7/ 7/ Bsite X Tetrahedral vacancy Octahedral vacancy Tetrahedral vacancy 43m 3m 3m 78' 878' /8, /8 /8 2'/2'/2; 2'Y4'/4' Y4'/2'/4' 4'/4'/2' u,u,u;'u,u, 3/ 3] 3/. 8'/ 8'/8' + t/ t/ 5/ +0 3 1 1 81 3 1. 1 3. Y8, 8, 8; 8, Y8 Y8 Y8 Y8 Y8 The oxygen atoms sit at (u,u,u) where u is approximately YA. For each spinel structure, permissible values of u can be calculated based on the cations radii and charges. The value of u changes with inversion and that in effect changes the lattice 16d 32e 8b 16c 48f 5/ 5/ 5/.5 7/ 7/ (] /Q /( /+UO /O / 4 V4 V4X 7/ 5/ 7/.7 7/ 5 /8'/8'/8'/8'/8'/8' V4 V4 V4,K u,u,u;u,u,u;u,u,u;u,u,u; U] U UU UW UU +U)U Y4'00 OY4'0; 4''Y4; 3 ~~1 3 .3 3 _Y _Y 84' 8'0; 8' 8'0 8' 8' Y2'Y4'Y4' Y4'2'Y4' Y4'Y4'Y2 '4'Y4' Y4'0'Y4' 4'Y4''; 43m 3m mm , ;, /4'/4'/4 parameter of the structure. The relation between bond lengths and lattice parameter is given as [111]: 40dT + 8 33do2 +8d a= (31) where, a = unit cell edge or lattice parameter do = bond length of cationO at octahedral site dT = bond length of cationO at tetrahedral site Cation Arrangement: It has long been established that spinel crystals can accommodate significant amount of cation disorder or so called variatee atom equipoints" [112, 113]. The structure with cation disorder is termed as "inverse spinel", introduced by Verwey and Heilmann [114]. Inversion is traditionally denoted by i and is given as: No. of Al on tetrahedral sites i = (32) Total no. of Mg The value of i ranges from 0 to 1, where 0 has all the cations on their normal crystal sites (Mg at tetrahedral and Al on octahedral) and 1 refers to the structure with all the Mg on octahedral sites and Al equally divided on octahedral and tetrahedral sites. Based on inversion, disorder in the spinel structure is described by Q, which is defined as [115, 116]: Q =1 (33) 2 The value of Q, ranges from 1(i=0) to 0.5 (i=1). Q is 0 for randomly disordered structure (i=2/3). The modified structural formula is used to project cation disorder in the system [(2) + M()P [M +M2+ X4 where M(1)P refers to divalent and M(2)q to trivalent cations. The first bracket represents the cation arrangement on tetrahedral sites. Similarly the second bracket represents cation arrangement on octahedral sites. 3.5 Bi4Ti3012: Aurivillius Crystal Structure Bismuth layer compounds belong into the Aurivillius family of layered perovskite and are denoted by (Bi202)2+(An1Bn03n+1)2 [117]. The crystal structure can be visualized as alternate stacking of (AniBnO3n+l)2 perovskite layers and (Bi202)2+ fluorite like layers along the caxis where n represents the number of perovskite layers between Bi202 layers. Various compounds with different n values have been prepared and are presented in Table 34. Table 34. Various Aurivillius structures for different n values n value Material Reference 1 Bi2WO6 Rae et al. [118] 2 Bi2SrNb209 Ismunandar et al.[119, 120] 3 Bi4Ti3012 Aurivillius [117] 4 Bi5Ti3MnO15 Kumar et al.[121] 5 Bi2Ba2Ti5O18 Aurivillius and Fang [122] Bismuth titanate Bi4Ti3012 (BIT) belongs to Aurivillius structure family with n=3 value. The crystal structure of BIT in ferroelectric phase (T >950 K) is tetragonal with 14/mmm symmetry and has a~3.85 A and c ~ 32.832 A [118]. As temperature is lowered, the system undergoes structural changes and forms a lower symmetry structure with "a" and "b" lattice constants elongated by ~2 of their tetragonal structure values. 6' ' o ' . as alt7 Figure 36. Unit cell of Bi4Ti3012 for A) low temperature monoclinic structure B) high temperature tetragonal structure, Bi is violet, 0 is red and Ti is light blue Figure 36 compares the high temperature tetragonal structure with low temperature structure. The atomic positions for tetragonal as well as low temperature monoclinic structure are given in Tables 35 and 36, respectively. Low temperature lattice parameters are taken as a=5.450 A, b=5.4059 A, and c=32.832 A and all the angles 900 [123]. There is no consensus over the low temperature crystal structure of BIT and scientific community is divided between orthorhombic and monoclinic structures. A list of experimental investigations and their conclusion on the structure of BIT is shown in Table 37. Table 35. Fractional coordinates for tetragonal structure (14/mmm) of BIT [124] Atom Site Fractional coordinates X Y Z Bi 4a 0.5 0.5 + 0.5667 Bi 4a 0.5 0.5 + 0.0662 Ti 4a 0.0 0.0 0.0 Ti 4a 0.0 0.0 + 0.1259 0 4a 0.0 0.5 0.0 0 4a 0.5 0.0 0.0 0 4a 0.0 0.5 0.1141 0 4a 0.5 0.0 +0.1141 0 4a 0.0 0.0 +0.0593 0 4a 0.0 0.0 +0.1802 0 4a 0.0 0.5 0.25 0 4a 0.5 0.0 0.25 Table 36. Fractional coordinates for monoclinic structure (Blal) of BIT [125] Atom Site Fractional coordinates X Y Z Bi 4a 0.0407 0.5041 0.5667 Bi 4a 0.0427 0.4777 0.7104 Ti 4a 0.0849 0.0044 0.4969 Ti 4a 0.0948 0.0014 0.6279 0 4a 0.3401 0.2710 0.5086 0 4a 0.3036 0.2389 0.2473 0 4a 0.1375 0.0614 0.5585 0 4a 0.0879 0.0406 0.6815 0 4a 0.3212 0.2747 0.6118 0 4a 0.4071 0.2036 0.6257 Bi 4a 0.0429 0.4991 0.4332 Bi 4a 0.0386 0.5170 0.2878 Ti 4a 0.0884 0.0093 0.3716 0 4a 0.3884 0.2203 0.4944 0 4a 0.3213 0.2443 0.7480 0 4a 0.1215 0.0638 0.4402 0 4a 0.1039 0.0573 0.3187 0 4a 0.3418 0.2720 0.3890 0 4a 0.3899 0.2095 0.3775 Aurivillius in his Xray diffraction study conceived the structure as orthorhombic with space group Fmmm. Later neutron diffraction studies suggested the orthorhombic structure with B2cb space group (#41, in the international table [101]) [126, 127]. In other studies using single crystal Xray diffraction, Rae et al. [123] and Shimakawa et al. [128] reported the monoclinic structure with the Blal space group (#7, in the international table [101]). From the crystallography perspective, the only difference between Blal and B2cb space group is the absence or presence of bglide. Experimentally, it is very hard to differentiate between these two space groups. The controversy arises because all of these possible structures are subgroups of 14/mmm. Figure 37 represents a groupsubgroup graph connecting the space groups 14/mmm and Blal. Table 37 summarizes the experimental investigations and their results for the structure of BIT. Table 37. Historical preview for structures of BIT Year Method Structure Lattice Reference Parameters 1949 Single crystal Xray B2cb a=5.410A Aurivillius [117] (Orthorhombic) b=5.448 A c=32.800 A 1971 B2cb a=5.448A Dorrian et al. [127] (Orthorhombic) b=5.411 A c=32.83 A 1990 No b glide polarization Blal a=5.45A Rae et al. [123] along caxis (Monoclinic) b=5.4059 A c=32.832 A (3=900 Table 37 Cont Year Method 1999 Neutron power diffraction 2001 Neutron power diffraction 2003 Neutron power diffraction 2007 Rietvald refinement of powder neutron diffraction 2008 DFT 2008 Odd n:B2cb Even n:A21am Structure B2cb (Orthorhombic) Blal (Monoclinic) Blal (Monoclinic) B2cb (Orthorhombic) Blal (Monoclinic) Blal (Monoclinic) Lattice Parameters a=5.444A b=5.4086 A c=32.8425 A a=5.44521A b=5.41118 A c=32.8432 A a=5.4475A b=5.4082 A c=32.8066 A a=5.4432A b=5.4099 A c=32.821 A a=5.4289A b=5.4077 A c=32.8762 A a=5.411A b=5.448 A c=32.83 A Reference Hervoches and Lightfoot [126] Shimakawa al. [128] Kim et al. [129] Chakraborty et al. [130] Shrinagar et al. [131] Jardiel ey al. [132] Figure 37. Groupsubgroup graph connecting 14/mmm to Blal [133] The structural controversy can be resolved with the help of symmetry constraints on the physical properties. BIT showed a finite polarization along the c axis, which is unlikely in the B2cb structure because the bglide has its mirror plane perpendicular to the c axis and thus all the polarization vectors along the c axis cancel out. Following the study by Shrinagar et al. [131], the fractional coordinates for Blal crystal structure can be derived by the appropriate transformations to its original Bravais lattice P1nl. Point 1 80 of P1n1 (x,y,z) remains unchanged but point (x+1/2,y+1/2,z) changes to (x+1/2, y+1/2,z+1/2). An important feature of these layered perovskites is the tilting of the octahedra. Figure 37 captures the nature of octahedra tilting along different axes. The effect of tilting is associated with the onset of ferroelectric phase change in Aurivillius structures. A study by Newnham et al. [134] suggested that increasing the number of perovskite units in Aurivillius structure (Bi4Ti3012, SrBi4Ti4015, Sr2Bi4Ti50O8) reduces To (Curie temperature). Another study by Shimakawa et al. [128] related structural disorder to curie temperature. A tolerance factor can be obtained for the layered perovskite structures by considering the perovskite tolerance factor; t = (RR + Ro) (34) where RA, RB, and Ro are the ionic radii of A, B and oxygen ions, respectively. This can be applied to Aurivillius structures. Due to additional Bi202 layers present in Aurivillius structures, stability region becomes very narrow due to extra structural constraints. Most of the perovskite structures exhibit tolerance factor between 0.771.01 [135]. The t value for layered structures is suggested to lie between 0.810.89 and with further decreasing as perovskite layers increase [136]. A brief description about octahedron tilting and its notation is discussed below. Octahedron Tilting: Octahedron tilting reduces the symmetry of the structure. Tilting also has a prominent effect on lattice parameter but it is very difficult to describe it. Attempts are made to explain the effect by including the effect of octahedral into it [137]. However, there are multiple ways the octahedra can tilt and change the environment around the cation. A standard notation has been developed to describe octahedra tilting in perovskite by Glazer [138]. The notation describes the tilting of MX6 octahedra about any of the three orthogonal Cartesian axes. %z % %4 %4% *rt j rX WAW^We99c ^^ ^^^^^^'s^^ S^^V~i, '^S^^i^Kr^S .yWWWWW^ ^^A^^^y. ^0^ yOyS. Br rY'r^mv B) r Y Figure 38. Octahedra tilt of 6x6x1 supercell of BIT along A) Xaxis [100] B) Yaxis [010] C) Zaxis [001] The general notation for tilt is used as aboc, where a,b,c are the magnitude of the tilt corresponding to [100], [010], and [001] axes. Superscripts depend on the sense of the tilting of successive octahedra. A + (positive) superscript would denote the same direction (inphase) tilt of the successive octahedra and a (negative) superscript implies tilt in the opposite direction (out of phase). A superscript of 0 (zero) denotes no tilting about that axis. There are certain important results that can be inferred from the tilting notation and are given below: 1. Equality of tilts about two or more pseudocubic axes implies equality of axial lengths (a+a+a+a=b=c). 2. Two or more superscripts are '0' or'+', system has orthogonal axes. 3. If two of the superscripts are '', then respective axes are inclined to each other and third one is perpendicular to both of them. 4. If all there superscripts are '', then all axis are inclined to one another. Based on this brief description and Fig. 38, it can be inferred that BIT shows aob c octahedra tilting in Glazer notation. CHAPTER 4 THERMAL TRANSPORT PROPERTIES OF MgONd2Zr207 4.1 Introduction As discussed in the Chapter 1, next generation fuel is one of the solutions of today's energy crisis. In a study of diluents materials for Pu burning in nuclear reactors, MgO along with AIN and ZrN showed promising properties to be a candidate for nuclear reactor applications [139]. Neeft et al. [140] also suggested MgO as a potential inert matrix fuel (IMF) material and studied the effect of neutron irradiation in MgO dispersed IMF and modeled the crack propagation in the fuel. Further study by Chauvin et al. [141] showed a good behavior by MgO based IMF. Recognizing the importance of MgO, Medvedev et al. [15] proposed MgOZrO2 as a candidate IMF material to overcome hydration limitation of MgO. Detailed report on MgOZrO2 as an IMF, is presented in the dissertation of Medvedev [14]. Inspired by this, Yates et al. [16] suggested MgO Nd2Zr207 (NDZ) as a candidate material for IMF. The selection of this specific pyrochlore composition (NDZ) is based in part on insights obtained from atomiclevel simulations. In particular, Sickafus et al. [142] used atomiclevel simulation methods to characterize the radiation tolerance of a wide range of pyrochlores (A2B207). They found that the radiation tolerance increases significantly as the B ion radius increases. In another atomic level simulation study by Schelling et al. [143], it has been reported that the thermal conductivity increases as the B cation radius decreases. Combining these two results with the guidelines suggested by Matzke et al. [12], neodymium zirconate, NDZ, is identified as a suitable candidate pyrochlore material. Interface effects on the thermal properties of MgONDZ cercer is of primary concern for its potential application as IMF. How the microstructure, interfaces, volumetric composition governs the thermal transport of the composite is investigated in this chapter. Before investigating interface effects for the MgONDZ cercer composite, fidelity of interaction parameters is tested for both the system. MD simulations were run to compute elastic and thermal transport properties of MgO and NDZ. By comparing them with experiment and establishing the veracity of interaction parameters, interface effects are simulated for polycrystals. These polycrystal results are extended to composite system of MgONDZ. 4.2 Simulation Setups 4.2.1 Thermal Expansion The simulation setup to capture thermal expansion consists of cubic arrangement of 6x6x6 or 4x4x4 unit cells in the case of MgO and NDZ respectively. Having 8 atoms in the unit cell of MgO gives a total of 1278 atoms in the simulation supercell. Similarly for NDZ, unit cell having 88 atoms gives 5632 atoms in the simulation setup. Periodic boundary conditions are applied in all the spatial directions. 4.2.2 Thermal Conductivity As explained in Sec. 2.11, thermal conductivity is calculated by creating a temperature gradient across the system. After reaching steady state heat flow, thermal conductivity is calculated using Fourier's law (Eq. 240). The system can be compared with simple setup of creating a heat flow across a rod by putting one end at the flame and the other at the cold bath. Schematic of simulation setup is shown in Fig. 41 on the right and a simple cartoon representing heat flow is shown on the left. Simulation setup has non cubic arrangement with zaxis longer than x and y axes. The set up is chosen to have unidirectional heat flow in zdirection. 1560 'C Heat Figure 41. Simulation setup to simulate thermal conductivity 4.3 Results Heat 4.3.1 Thermal Expansion Simulation setups are ran at several temperatures under NPT ensemble (please refer, Sec. 29) to compute the coefficient of thermal expansion. After reaching equilibrium state at each temperature, an average of lattice parameter of the system is calculated. The lattice parameters calculated by the simulation are compared with experiment for MgO (Fig. 42A) and NDZ (Fig. 42B). The lattice parameters for both the system are in quite good agreement with the experiment [144, 145]. MgO ) 500 1000 T[K] *Sim OExp g 8 1500 2000 1500 2000 1.093 1.089 1.085 E1.081 m 1.077 1.073 1.069 * 1.065 0 B) NDZ cO00 400 800 T[K] Figure 42. Lattice parameter variation with temperature for A) MgO and B) NDZ Using these simulation values and Eq. 239, coefficient of thermal expansion (CTE) is computed for each of the material. CTE for MgO and NDZ from simulation and experiment is shown in Table 41. Fidelity of interaction parameters is tested by 0.45 0.44 0.43 0.42 0.41 A) *Sim oExp 0 O 1200 0 . 0 1600 comparing the simulated values with experiment. The comparison for MgO is within 5% and for NDZ it is 22% of the experimental value. Such levels of agreement are typical of empirical potentials for oxide materials as shown by other simulation studies [143, 146, 147]. Table 41. Coefficient of thermal expansion for MgO and NDZ Species Simulation Experiment NDZ 7.4x10/K 9.5x10/K [145] MgO 12.4x106/K 13.5x106/K [148] 4.3.2 Anharmonic Analysis The different values of thermal expansion for the simulated and experimental values provide valuable information on the nature of the interatomic interactions. As is well known, the thermal expansion arises from the asymmetry in the interaction potential, i.e. from the anharmonicity in the interatomic interactions. An elementary analysis gives the Gr0neisen relation for the thermal expansion [29]: C= yc ,B (41) where c, is the specific heat, B is the bulk modulus and y is the Grineisen parameter, which is a measure of the anharmonicity of the material. Thus, the reduced values of the simulated thermal expansions are attributed to the interatomic interactions being insufficiently anharmonic and/or the bulk modulus being too high. This conclusion has important implications for the thermal transport properties, because the thermal conductivity is also determined by the anharmonicity of the interactions. In particular, in most electricallyinsulating materials the dominant scattering mechanism is phononphonon scattering. Such scattering would be completely absent if all of the interactions were harmonic. Using a perturbation expansion of the interatomic potential to third order, thereby including the lowest anharmonic terms, Liebfried and Schloemann [149] showed that the thermal conductivity is inversely proportional to the square of the Grineisen parameter: c Mv (42) 10 72 h T where kB is the Boltzmann constant, h is the Planck constant, v and M are the volume and the mass per atom. The only two materials constants that enter into Eq. 42 are the Debye temperature, e, and the frequencyaveraged Grineisen parameter, y. From this relation, it is clear that the thermal conductivity is inversely proportional to y. Through their dependence on the Grineisen parameter, a simple empirical relationship between the thermal conductivity and thermal expansion is shown as [150]: a2 = oc, (43) B The constant contains all of the nonmaterials constants in Eqs. 42 and 43. Table 42 shows the values of OD and Cp for MgO and NDZ is determined using GULP (General Utility lattice Program) [151, 152] and compared with the experimental values. Table 4.2. Comparison of Debye temperature and Cp calculated using GULP with experiment. F=Gsim/Gexpt, where G=OD3Cp2 OD (K) Cp(J/molK) Material Sim. Exp. Sim. Exp. MgO 830 946 [153] 38.77 34.82 [153] 0.83 Nd2Zr207 549 489 [154] 207.54 232.70 [154] 1.13 Moreover, the experimental and calculated values of G=OD3C 2, analogous to the combination in Eq. 43 are within 20% for both materials. After merging Cv and OD with other constants Eq. 43 becomes K= (44) a2B where X' = X 3Cv2. From Eq. 44, a relationship between experimental and simulation thermal conductivity comes out to be: Ksim Kexpt ((Lexpt/ Csim)2(Bexp/Bs=m) (45) In Eq. 45, ((lexpt/ (,)2(Bexp/Bs ) ratio is termed as "correction factor". After dividing the simulated value with the correction factor, comparison with experimental value is done. The bulk modulus for MgO and NDZ from simulation for these potentials came out to be 170 GPa and 167 GPa respectively at 0 K. The corresponding experimental roomtemperature values for single crystals are 155 GPa and 159 GPa, which are quite consistent [155, 156]. From these data, Eq. 45 predicts Ksim = 1.02 Kexpt for MgO and Ksim= 1.61 Kexpt for NDZ; that is, simulations of the thermal conductivity of MgO should match the experimental results rather well, while the simulations should significantly overestimate the thermal conductivity of NDZ. 4.3.3 Thermal Conductivity As described in section 4.2.2, simulation is set up to generate unidirectional heat flow. By generating temperature gradient, conduction is simulated and then calculated by using Fourier's law, Eq. 46 dT J= K (46) dx The key determinant of thermal conductivity is the mean free path of the phonons before they scatter from each other or from other components of the microstructure. If the system size in the direction of heat flow is smaller than the phonon mean free path, then the intrinsic anharmonic properties of the system will not be probed completely. This regime is known as the Casimir limit. It is thus no surprise that previous studies have also shown that the accuracy of the simulations of the thermal conductivity depends significantly on the length of the simulation cell. In a previous study by Schelling et al. [157], a finite size scaling approach is used to determine the thermal conductivity of the material. 4.3.4 Finite Size Analysis This approach, which is based on the inverse additivity of contributions to the mean free path yields (for details, please refer Eq. 247): 1 1 4P  = + (47) K K L where K is the thermal conductivity determined from simulation, K is the thermal conductivity in the infinite systemsize limit (i.e., the best estimate of the 'true' thermal conductivity of the material), Lz is the length of the simulation cell and P is a constant. Equation 47 thus suggests that a plot of 1/K vs 1/Lz should be linear, and that the thermal conductivity of an infinite system can be obtained by extrapolating to 1/Lz = 0. Figure 43 shows such analysis for MgO and NDZ at 573 K. By using the above described system size analysis at various temperatures, the temperature dependence of thermal conductivity is extracted for both MgO and NDZ. 0.12 0.10 .......... 0.50 0.08 ..... 0.40.... ....... S 0.06 ....... ... 0.30 0 .... T573K T 573 K S0.04 0.20 0.02 MgO 0.10 NDZ 0.00 0.00 0.00 0.02 0.04 0.06 0.08 0.02 0.03 0.08 1/L, [nm1] A 1/L, [nm1] B Figure 43. System size effect on thermal conductivity A) MgO B) NDZ Finitesize scaling analysis can lead to a significant increase in the simulated value of thermal conductivity, as is illustrated in Fig. 43. If only a single simulation had been performed for the shortest simulation cell (Lz = 15.3 nm for MgO and Lz = 12.80 nm for NDZ), then the estimated thermal conductivities would be less than onefourth of the best estimated value. 4.3.5 MgO: Thermal Conductivity Figure 44 compares the best estimate of the thermal conductivity determined from simulation and finitesize scaling analysis, with the experimental data for MgO. The data of Cahill (open circles) [158] is for a highquality single crystal thin film, while the polycrystalline data (open squares) is for coarsegrained polycrystal [159]. The experimental polycrystalline data is systematically lower than the experimental data for a single crystal by a small amount. While physically this is reasonable since the polycrystals contain grain boundaries, and possibly other defects arising from processing, all of which provide additional scattering mechanisms, this difference could also simply be statistical in origin [31]. 60 50 *Simulation 0 G o Experiment Cahill 0 40 o Experiment Ronchi 00 S OO E i o 30  20 0 0 10 q> ] [ MgO 0 500 1000 1500 2000 T [K] Figure 44.Thermal conductivity data of MgO: experiment data for single crystal thin film (open circle) [158], experiment polycrystal data (open square [159]) and simulation data (solid circle). The simulated data (solid circles) agreed very well with the single crystal experimental data. Had a finite size analysis not been performed, these values would have been significantly lower and would not have agreed with the experimental values. That the experimental and simulated values are in such good agreement provides validation for the harmonic analysis described in the Sec. 4.3.2, which predicted that they should agree within 2%. 4.3.6 NDZ: Thermal Conductivity The anharmonicity analysis yielded a multiplicative correction factor of 1/1.61 = 0.62 for NDZ. Thermal conductivity values are computed using the simulation method as described in Sec. 2.11. The simulated values of thermal conductivity are modified using the correction factor. Figure 45 shows both the uncorrected and corrected values for the thermal conductivity of NDZ. While the uncorrected values are, as expected, consistently higher than the experimental values, the corrected values are much more consistent with the experimental results [160]. 5 o Single Crystal  uncorrected *Single Crystal  4 corrected 0 Experiment 0 0 0 0 3 NDZ 0 0 200 400 600 800 1000 1200 1400 1600 T [K] Figure 45. Thermal conductivity of Nd2Zr207: uncorrected single crystal simulation values (open circle), corrected single crystal values (solid circle) and experimental polycrystal values (open square) [160] Both the thermal conductivity results for MgO and NDZ are done on single crystals. The difference between experiment and simulation results, besides other reasons is because experimental system has defects. To illustrate the effect of grain boundaries (GBs), simulations on polycrystals are performed and analyzed. 4.4 MgONDZ: Polycrystals and Composite Defects are an essential part of any material. In nuclear reactor, fuel pellets have grain boundaries as one of the primary defects. The working temperature in the reactor ranges from 800 K1500 K. Grains in the nuclear fuel undergoes structural changes and that leads to changes in the thermal and mechanical properties of the fuel. It is important to know how these interfaces affect thermal transport in the fuel. That can be understood if the effects of interfaces are characterized in the fuel. Since at atomic level interface effects are enhanced, atomic level simulations provide an ideal vehicle to characterize the interface effects. To characterize the effect of grain boundaries the thermal transport properties of MgO and NDZ polycrystals and MgONDZ composites are simulated. Interface effects are enhanced at smaller grain size level. It is possible to extend the size into the tens of nanometers range but, due to intrinsic computational limitations, it is not possible to extend the range to microns and tens of microns grain size, present in typical ceramic materials. To extend the results for larger grains, a simple analysis model is used and results are compared with experiment results. The following section will outline the method used in this work to generate polycrystal and composite structures. 4.4.1 Polycrystal Structure y x z Figure 46. MgO polycrystal ~ 3 nm grain size; black arrows showing orientation of the grain Interface effects can be estimated using 2D textured polycrystals. They provide an efficient alternate to 3D grains to capture interface effects. A simple code has been developed to generate 2D polycrystals as shown in Fig. 46. The polycrystal is textured with all the grains being parallel to the [100] direction. For both materials, MgO and NDZ, 24 perfectly hexagonal grains are filled with perfect crystal with the [100] directions oriented in almost random directions with respect to the horizontal xaxis, with the stipulation that all grainboundary (GB) misorientations should be >150, thereby reducing the tendency for grain coalescence. The grainboundaries in these polycrystals are thus a reasonably random selection of <001> tilt boundaries. Periodic boundary conditions are applied such that there are no additional interfaces at the boundaries. The whole process of generating polycrystal can be summarized as 1. 24 hexagonal grains of equal size were created by specifying their centers but all of them are empty at this step. 2. Each grain was oriented in a way that no neighboring grains form low angle GB. 3. Each grain is filled with the unit cells of the material (MgO or NDZ). 4. Each grainatom is screened and are removed if they are closer to neighboring grain centers. 5. All the grains are combined to make the polycrystal. 6. Any two atoms, with a distance less than a specified cutoff distance are identified and termed as bad atoms. 7. Charge of each grain is calculated and maintained to zero by removing bad atoms from the grain. 8. Still if system is not neutral, bad atoms are removed to achieve charge neutrality. 9. If system is charge neutral but still have bad atoms, bad atoms are removed as a formula unit. For example, for each Zr of NDZ, two nearest oxygens are removed to keep the neutral charge of the system. 4.4.2 Composite Structure Depending upon the connectivity of constituent phase of the composite, it can be one, two, or threedimensionally continuous. A simple scheme to classify the connectivity for twocomponent system is presented by Newnham et al. [161]. All the possible connectivity of twocomponent composite system is shown in Fig. 47. Among the ten different connectivity patterns, 33 composites are extremely rare and 03 is very common for particulates in a matrix and 13 for aligned fibermatrix composites. 00 01 02 3 11 12 13 22 23 33 33 Different views of the same object Figure 47. Different connectivity patterns for twocomponent composite system [162]. To understand this notation, a simple method is used: 1. First number represents the connectivity of white 2. Second number represents the connectivity of black Black and white nomenclatures are used here to define two different materials because it is easier to follow Fig.47 with the convention. For example, 11 composite means, only 1dimension connectivity for both white and black blocks. As shown in Fig. 47, 11 composite has only vertical alignment for both white and black. Another example of 23 composite means, white is continuous in 2dimensions and black in all 3 dimensions. For the textured composite, the composite will exhibit at least 11 connectivity. Figure 48. MgO (blue)NDZ (red) composite (21 composite) A Simple schematic representation of composite structure is shown in Fig. 48. Composite structure can be either MgO connected and/or NDZ connected or not connected. Based on the experimental insight by Yates et al. [16], same volumetric composition of MgO and NDZ gives microstructural dependent temperature dependence of thermal conductivity. Similar to the procedure used to build polycrystals, composites are built. Slight changes are made because of the difference in lattice parameters of MgO and NDZ. The method to generate composite is as follows: 1. 24 hexagonal equal size grains were created by specifying their centers but all of them are empty at this step. 2. Each grain was oriented in a way that no neighboring grains form low angle GB. 3. Atomic coordinates of MgO atoms are changed according to NDZ lattice parameter. For example, (1, 0, 0) coordinate corresponds to ~ (0.39,0,0). 4. Based on the connectivity, grains are assigned to MgO or NDZ.. 5. Each grain is filled with the corresponding material unit cells. 6. Atoms with a smaller distance than a specified cutoff are termed as bad atoms for each grain 7. Grains are made charge neutral by removing bad atoms 8. Still if system is not neutral, bad atoms are removed to achieve charge neutrality. 9. Grains are combined and composite is built. Composite structures are equilibrated and high energy atoms are screened. These high energy atoms are removed from the system. Equilibrated composite structures are used to compute thermal conductivity and capture interface effects. 4.4.3 Interfacial Effect on Thermal Conductivity As described in section 4.4.1, polycrystals of MgO and NDZ structures are generated. Thermal transport properties are calculated using the same approach as used for the single crystals. The effect of system size is expected to be much less important for such finegrained polycrystals because the dominant mechanism should be phonon/grain boundary rather than phonon/phonon scattering. Simulations on the system shown in Fig. 46 and a system with the same microstructure and grain size, but twice the length (formed by simply putting two simulation cell shown in Fig. 46 end to end), gave thermal conductivity values within 5% of each other, which is within the statistical errors of the simulations. The anharmonicity analysis predicted the thermal conductivities of the single crystals from simulations would be overestimated by 2% for MgO and 62% for NDZ. Anharmonic correction factors are used in predicting the thermal conductivity of MgO and NDZ polycrystals. In this step, the thermal expansion and bulk modulus are taken as essentially microstructure independent. However, within the approximation made in separating the contributions in the polycrystal thermal conductivity from that of grain interior and gain boundary, the accuracy may not be sufficient to consider more careful analysis. As Figs. 49 and 410 show, the calculated thermal conductivities of the finegrained polycrystals of MgO and NDZ are considerably smaller than their singlecrystal counterparts. The thermal conductivity for polycrystalline NDZ shows a smaller reduction from the singlecrystal value than does polycrystalline MgO (note the difference in scales); this smaller reduction is attributed to the already much smaller mean free path of phonons in NDZ as compared to MgO. Also, for both systems, the effect is considerably larger at lower temperatures, at which the phonon mean free path is longer. 60 *Single Crystal 50 0 Polycrystal ,40 1 30 ju * 20 SMgO 10 0 0 0 0 0 0 0 0 200 400 600 800 1000120014001600 T [K] Figure 49. Comparison of thermal conductivity of MgO: single crystal (solid circle) and 3 nm polycrystal (open circle). 3.00 *Single Crystal  2.50 corrected * 2.00 Polycrystal  2.00 * 20 corrected 1.50  1.00 0 o o o o o 0.50 NDZ 0.00 0 200 400 600 800 1000 1200 1400 1600 T [K] Figure 410. Comparison of thermal conductivity of NDZ: single crystal (solid circle) and 3 nm polycrystal (open circle). Using a simple model in which the polycrystal is assumed to be describable in terms of a volume of bulk with thermal conductivity (KO) and a volume of grain boundary with interface conductance (G) (all grain boundaries being assumed to be physically the same), Yang et al. [19] derived a simple relation for the dependence of the thermal conductivity, K, of a polycrystal on grain size, d: K= K (48) / 1+o Gd Using Eq. 48, estimates for G are extracted; the result is shown in Fig. 411 for both materials, MgO and NDZ. As for other materials that have been studied experimentally and/or in simulation, G increases with temperature, reflecting the enhanced anharmonic coupling of weakly coupled phonon modes across the interfaces [163]. 100  * 0 I I 0 400 80 T [ 8 S 0 400 800 T [K] 35.0 * * S 30.0 25.0 20.0 ' 15.0 10.0 0 0 Mg 5.0 O O 0.0 0 1200 1600 K] 4.5 4.0 3.5 3.0 * 2.5 E 2.0 * 1.5 0 o 1.0 0.5 NDZ N0.0 1200 1600 Figure 411. Temperature dependence of the interfacial conductance, G (solid symbols) and Kapitza length, Ik (open circles) for A) 3 nm polycrystalline MgO and B) 3 nm polycrystalline NDZ. Because the thermal conductivity and G are in different units, it is not possible to compare them directly. To make a better comparison, a new variable named Kapitza length is defined as Ik =KO/G (49) 101 2.0 1.5 1.0 0.5 0.0 1.5 1.3 1.0 I 0.8  0.5 0.3 0.0 B) The Kapitza length (Ik) is a measure of the length of perfect crystal that offers the same resistance to heat transport as the interface i.e. the larger the value of Ik, the larger the thermal barrier offered by the interface. From Fig 411, Ik goes from 30 nm at 300 K to ~3 nm at 1473 K for MgO. The effect is less dramatic in NDZ, with Ik decreasing from 4 nm at 300 K to 1.75 nm at 1473 K. These decrease in MgO and NDZ are consistent with the trends seen previously for U02 [150] and diamond [164] in both of which, Ik decreases with temperature for both materials. 4.4.4 Grain Size Dependence on Thermal Conductivity Having determined G, Eq. 48 was used to predict the grainsize dependence of the thermal conductivity. Figure 412 shows the calculated grain size dependence of the thermal conductivity at 300 K and 573 K for MgO and NDZ, normalized to the corresponding single crystal values; thus in both cases, the large grain size limit is K/K0=1. In the absence of any experimental data on the thermal conductivity of MgO or NDZ as a function of grain size against which to compare these trends, Fig. 412 includes the experimental results of Yang et al. for yttriastabilized zirconia (YSZ) [19]. Despite the considerable difference in materials and microstructures (textured vs. random polycrystal), the simulation data is broadly consistent with the experimental data. In particular, the agreement between the simulation results for NDZ and the experiments for YSZ is remarkably good; this agreement probably arises from the fact that they both are low thermal conductivity materials with fluoriterelated crystal structures. After gaining insight at the effect of homogeneous interface on thermal transport, study was extended to include composites. 102 1.0 0.8 * A o0 0 0 0 0 .0.6 A o 0 o Experiment: YSZ: O 480K 0.4 o oMD: MgO:300K 0 *MD: MgO:573K 0.2 0.2 MD: NDZ:300K A MD: NDZ:573K 0.0 1 0 20 40 60 80 100 120 Grain size [nm] Figure 412. Predicted grainsize dependence of the normalized thermal conductivity of MgO (solid and open circle) and NDZ (solid and open triangle) polycrystals compared with experimental results of YSZ (solid square) [19] Polycrystal simulations provided information for MgOMgO and NDZNDZ interfaces. To understand the nature of MgONDZ interface, composite system with different connectivity is generated. Section 4.4.5 details the information on simulation studies done on MgONDZ composites. 4.4.5 Thermal Conductivity of MgONDZ Composite Based on experimental investigation by Yates et al. [16], a volumetric ratio of 60 40 (MgONDZ) was suggested for the composite for the simulation study. To achieve the similar ratio, 10 grains were selected as NDZ and remaining 14 were MgO grains. Polycrystal investigations gave us insight into interfacial effects of both the constituents separately. Interfacial effects were larger than grain size itself at this sub nano size level 103 and that led to the assumption that no additional interface effects can be seen in the composite. A simple model based on volumetric composition is used to predict the thermal conductivity of composite. The predicted value of thermal conductivity is given by: K = XNDZ KNDZ,poly + XMgO MgO,poly (49) where XNDZ gives the volemtric composition of NDZ (For example, in this case 10/24 is for NDZ) and Kpoly,NDZ is the thermal conductivity of the same size NDZ polycrystal. Similarly values are defined for MgO grain component. 4.0 3.5 3.0 2.5 S2.0 ,1.5 Calculated 1.0 OMD 0.5  0.0 I , 0 200 400 600 800 1000 1200 1400 1600 T (K) Figure 413. Comparison of thermal conductivity of 6040 MgONDZ composite with predicted value Based on the simulation results, no additional interface effects are observed in the composite. This can be attributed to the limitation of current status of simulation power. 104 4.5 Discussion The atomiclevel simulation method has provided an important insight into the MgONDZ system. The fidelity of interaction parameters for MgO and NDZ was tested separately. The interaction parameters for MgO have shown better fidelity with experiment than for NDZ. Finite size and anharmonic analysis tools are used to correct the simulated thermal conductivity values. The modified values after the analysis have shown a good comparison with the experiment. After establishing single crystal results, interfaces are introduced into the system. A simple program is written to generate 2D textured polycrystals. Very fine grain size enhanced the interface effect. Interfacial conductance for MgO and NDZ polycrystals showed correct temperature dependence. Calculation of Kapitza length has shown that interface effects are as big as grain itself for both the materials: MgO and NDZ. A simple analysis is used to extrapolate the effect of grain size and compared with the experiment study. Both the materials showed the independence of the thermal conductivity on the grain size beyond ~100 nm. That implies the limiting effect of interfaces for bigger grains. A simple analysis based on the volumetric composition of the individual grains is used to predict the thermal conductivity of the composite. Thermal conductivity values of the polycrystals are used to predict the composite thermal conductivity. Comparable results for predicted value and MD results showed that thermal conductivity depends only on the volumetric composition of the composite and not the microstructure at very fine grain size level. 105 CHAPTER 5 THERMAL TRASNPORT PROPERTIES OF MgAI204 5.1 Introduction Based on the guidelines for inert matrix materials (for details, please refer Section 1.5) proposed by Matzke et al. [12], magnesium aluminate spinel MgAI204 is among the potential inert matrix materials for minor actinide incineration. In particular, MgAI204 is very tolerant to neutron irradiation [165, 166]. Its ability to tolerate radiation damage is attributed to i) a very high recombination rate of Frenkel pairs [165] and, ii) the ability to tolerate significant intrinsic cation antisite disorder [167].The property of cations interchanging their sites in the spinel structure is called inversion. It has been well established that cation arrangement plays an important role in the material properties. For example, the elastic properties change with cation arrangement in spinel [168, 169], and the bulk modulus of MgTi205 changes with cation ordering [170]. It has also been found that with increasing temperature MgAI204 undergoes inversion [171173]. A deeper understanding of the effects of inversion on the thermoelastic properties of MgAI204 is highly desirable before it can be fully assessed as a potential IMF material. The experimental investigation of the effects of inversion on MgAI204 faces two big challenges: 1. It is hard to control inversion during the synthesis process. 2. It is impossible to separate out the effect of temperature from inversion. Both of these challenges are easily overcome in atomic level simulation. MD is chosen as an ideal vehicle to perform the simulations to capture the effect of inversion on MgAI204 properties. One of the key performance metrics of any IMF material is 106 thermal conductivity; therefore an important question here is: How does inversion affect the thermal conductivity of MgAI204? 5.2 Selection of Inverted Structure Conventional MD simulation methods and lattice dynamics are used to investigate the effect of inversion on mechanical and thermal transport properties of MgAI204. Interaction parameters are taken from Table 22. As discussed in Chapter3, each unit cell of MgAI204 contains 8 Mg, 16 Al and 32 0 ions. For the MD simulation study, a cubic arrangement of 4x4x4 unit cells for a total of 3584 atoms is formed. For the lattice dynamics calculations in GULP a 2x2x2 system with 448 atoms is formed. This large system size benefited us by giving us more choices for the degree of inversion. There are 2 simple ways to create an inverted structure in MgAI204: i) Direct interchange of Mg and Al; ii) Al can move to a vacant tetrahedral site and Mg to a vacant octahedral. A firstprinciples electronic structure study by Gupta [174] has shown that direct cation exchange is energetically more favorable. Therefore randomly selected Mg and Al atoms were simply exchanged to generate the inverted structures. Since there is no experimental information on possible ordering in inverted structures available, ~7000 structures were generated for each inversion value via random cation interchange. Three lowest energy structures were identified and used them for the subsequent simulations. Figure 51 captures the distribution of structural energy among the generated structures for the same inversion value. The xaxis of Fig. 51 covers the whole energy range spanned by generated structures for the inversion values i=0.0625 and i=1.0, and the yaxis shows the number of structures having that energy. The 107 energy of the structure increases from left to the right on the xaxis. Thus, structures having energy in the left extreme of the graph are picked. The large number of structures for each inversion maximizes the chances of identifying the energetically most favorable structure. 0.14 0.12 U, 0.1 0.08 0 S0.06 .2 0.04 0.02 0 A) 0.12 0.10 12487 Energy(eV) i=0.0625 12462 i= 1.0 12764 Energy(eV) 12664 Figure 51. Distribution of energy as a function of cation arrangement for the inversion value of A) 0.0625 B) 1.0 108 After selecting structures for each inversion, pair distribution function (PDF) for each cation is investigated. PDF provides structural insight with a very simple and easy analysis method. PDF for normal (i=0) is shown in Fig. 52A, and it shows that first Mg Al neighbor is beyond 3 A. As inversion is introduced in the system, the height of the peaks is decreasing. A comparison between the PDF of AIMg for i=0 and 0.125 is shown in Fig. 52B. They have very similar peaks but the actual values have gone down by more than 20% and peaks are more spread out for i=0.125. 2.00 4.00 6.00 f=0.0  MgMg  MgAI AIMg AIAI I Il  U I  10.00 12.00 14.00 16.00 18.00 r(A) I I i=0AIMg  i=0.125 AIMg I U ________________________________ 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 B) r(A) Figure 52. PDF for A) normal spinel B) comparison of AIMg PDF for i=0 and i=0.125 109 0.30 0.25 0.20 0.15 0.10 0.05 0.00  0.00 0.30 0.25 0.20 0.15 0.10 0.05 000  i 0.035 0.030 MgMg 0.025 0.125 0.020 S0.25 c'5 0.015 p0.5 0.75 0.010 I 1 o, 4 I 0.005 ' 0.000 " ;  ) 0.00 3.00 6.00 9.00 12.00 15.00 18.00 r(A) 0.035 0.030 MgAI 0.025 0.125 0.020 0.25 0.015 0.5 I I I 0.75 0.010  1 0.005 I I' 0.000 I 4 . 0.00 3.00 6.00 9.00 12.00 15.00 18.00 B) r(A) Figure 53. Comparison of PDFs for inverted structures for A) MgMg, B) MgAI, C) Al Mg and D) AlAl 110 Figure 53 Cont... Comparison of PDFs for inverted structures for A) MgMg, B) MgAI, C) AIMg and D) AIAI 0.035 0.030 AIMg 0.025 0.125 0.020 0.25 0.015 0.5 I 0.75 0.010 1 I I 0.005 , 0.000 , ,  C) 0.00 3.00 6.00 9.00 12.00 15.00 18.00 r(A) 0.025 0.020 AIAI 0.015 0.125 S1 0.25 0.5 0.010  0.75 0.005 I I 0.000,  0.00 3.00 6.00 9.00 12.00 15.00 18.00 D) r(A) 111 For better comparison between inverted structures, PDFs for each cationcation pair are plotted against each other for all the inverted structures except i=0 (Fig. 53). As seen in Fig. 53, with the increase in the value of i, the system shows disorder in the form of less sharp peaks (more spread out). Even though spreading of the peaks does not change significantly at i=1, the intensity of peaks gets higher. That corresponds to ordering at tetrahedral site in favor of Al but disorder on the octahedral sites (Fig. 53B). Similar results are shown for Al in Figs. 53C and D. 5.3 Effects of Inversion on Properties 5.3.1 Lattice Parameter The unit cell of MgAI204 has 8 Mg ions which limits the stepsize of inversion to 0.125 (1/8). Each MgAI interchange changes the inversion by 0.125. Although this large step size covers the whole range of inversion, it might fail to capture some important physics for intermediate inversion values. To overcome this, the system size is increased by 8fold, thus reducing the stepsize from 0.125 to 0.015625. This provided more inversion values for the simulation and better insight into the effects of inversion on material properties. MD and GULP simulations methods were used to equilibrate the structure at T = 0 K. As explained above, among the various possible cation arrangements for any particular i value, the lowest energy structure is chosen for the simulation. The effect of inversion is shown in Fig. 54 and is compared to the effect captured by Andreozzi et al. [175]. 112 8.12 8.10 8.08 8.06 8.04 8.02 8.00 0.00 1.0005 1.0000 0.9995 0.9990 S0.9985 0.9980 0.9975 0.9970 0.9965 8.0870 8,0860 8.0850 8.0840 8.0830 0.75 1.00 8.0820 0. B) 0.2 Inversion Figure 54. Decrease of lattice parameter with inversion A) this MD simulation study B) reproduced from experimental investigation by Andreozzi et al. [176] C) comparison between MD and experimental study Comparison between experiment and this MD simulation study gives an impression of a much stronger dependence of lattice parameter on inversion in MD but in reality that is not the case. Experimental dependence also involves the expansion due to temperature which is absent in simulation data. Even though the experiment in 54 B) does not span the full range of inversion available to simulation, both show a 113 0.25 0.50 Inversion a a = 80900 0.0242 x R'= 0.983 S: & Disordering 0 Ordering o SP3 0.20 0.24 x 0.28 0.32 O Exp * 0 0O 0 *cO ok decrease in lattice parameter with inversion. This increase in density with increasing inversion can be qualitatively understood in terms of a simple picture of the bond strength as a function of bond charge. In normal spinel (i=0), each tetrahedral site is occupied by Mg2+ with 4 02 neighbors and each octahedral site is occupied by A13+ with 6 02 neighbors. Each cation shares /2 e with each neighboring oxygen on both tetrahedral or octahedral sites. On inversion, Al3+ on a tetrahedral site with 4 neighboring oxygens, shares now % e with each neighboring oxygen, an addition of /4 e per bond. Similarly, Mg2+ on the octahedral site has 6 neighboring oxygen ions, thus sharing 1/3 e a loss of 1/6 e, compared to noninverted spinel. A simplified picture of bond strength being proportional to bond charge gives a larger change at inverted tetrahedral sites than at octahedral sites. In other words, the decrease in AIO bond length at tetrahedral sites is larger than the increase in MgO bonds at octahedral sites. That, in turn, should produce an overall densification of the spinel upon inversion as is observed. 5.3.2 Antisite Formation Energy To investigate the effects of inversion further, the energy is determined as a function of inversion. Consistent with the normal state being more stable, Fig. 55 shows an increase in energy with inversion. This appears to be linear up to i0.5, above which there is a change in slope. A further analysis is performed by calculating the formation energy of each additional cation interchange (Fig. 56). It is evident from Fig. 55 that up to i~0.375, the energy required to form new antisite defects is almost constant. Equivalently, it can be said that for these low inversion cases, existing defects make it neither easier nor harder to form additional antisite defects. Conversely, beyond i=0.375, it becomes comparatively 114 easier to form new defects with the defect formation energy decreasing by ~25% for the fully inverted system. This decrease in formation energy can be understood in terms of the effects of the antisite defects on the local strain fields. At low inversion, the local strain produced by interchanging Mg and Al cations is small, and is easily relieved by structural relaxations, thus not affecting the formation energy for the next antisite pair to form. As the inversion increases, however cation disordering increases and rearrangement of cations is not able to relieve the strain dipole. Such unrelieved strain dipoles lower the energy required to form strain dipoles of opposite orientation. As a result, the antisite formation energy for each additional disorder decreases. 23 0.2 230. = E 0 231.0 . 0.00 0.25 0.50 0.75 1.00 Inversion Figure 55. Structural energy as a function of inversion for MgAI204. 115 1.0 0.9 0.8 LU 0.6  0.5 0.00 0.25 0.50 0.75 1.00 Inversion Figure 56. Cation antisite formation energy as a function of inversion. A simple analysis provides further insight into the orderdisorder process and will help us understand Figure 56. The orderdisorder process can be described by a chemical exchange reaction: [177] TAI+MMg4MAI+TMg where, T is a tetrahedral site and M is an octahedral site. The forward reaction represents the restoration of the ordered structure, the backward reaction represents disordering. K1 and K2 are taken as the forward and backward reaction rate constants, respectively. The time rate of change of the Al concentration on a T site is given by  dX Al) = Ki (T Al)X(T Al (Mg)X(MMg) K2 TMg)X(TMg) (MAl)X(MAl) (51) dt where X is the fraction of T sites occupied by Al and D is analogous to an activity coefficient and is generally a function of temperature, composition and volume. Assuming ideal mixing in each site [178] and at equilibrium dX() is 0, applying dt these two conditions on Eq. 51, it becomes: K2 = X( Al )X(Mg Al (Mg) (52) K, X'TMg)X(MAl)'(TMg)T(MAl) 116 In MgAI204 spinel, X( Al)= x, X(TMg)= x, X(MMg)= x and X(MAl)= 1 . After substituting, Eq. 52 turns into K,2 x2 TAl) Mg) I f l4 (53) K (I xX2 x Mg TMAl) As already pointed out, K2 signifies the reaction rate for disordering and K1 for ordering. In the original analysis by Mueller, all 0 values are taken to be unity for the crystals with two nonequivalent sites [178]. K2 2 " X 2 (54) /K, (1 x)(2 x) (5) In that case for the Eq. 54, r >1 signifies the disordering of the structure being the dominant reaction. Solving the inequality equation for disordering as the preferential reaction, gives > 2. This simple analysis shows that, disordering becomes preferential after the system achieves ~66% inversion. Even though the threshold value for inversion is different, this simple model shows that each additional disorder in MgAI204 costs less energy after the threshold value. The analysis here is based on all the ( values being unity. The value of ( may change from unity, given high temperature, electrostatic charge analysis [179] favors inversion and disordered structure release strain by more disorder. Combining it with the work presented here, it is proposed that the disordered structure at low inversion values forms stress dipoles at inverted sites to release the strain, thereby lowering the energy of the system. This is also shown by the smallest possible inversion, exchange of 1 antisite pair. Figure 56 shows the variation of the formation energy for the inverted structure as a function of distance between inversion sites. 117 1.4 S1.2 > 1.0  = 0.8 0.6 o 0.4 o 0.2 0.0 0 2 4 6 8 10 12 Cation seperation(A) Figure 57. Formation energy as a function of inversion site distance. As seen in Fig. 57, inverted sites prefer to be in close proximity. Based on the result of 1 antisite pair energy and theoretical understanding, it can be hypothesized that, as disorder increases, tetrahedral sites are increasingly occupied by Al. Also the AIO bond length is decreased at tetrahedral sites. At 50% inversion, tetrahedral sites are equally occupied by Al and Mg. Combined with electrostatic analysis (favoring of Al on tetrahedral sites) and reaction rate analysis (less energy expense for each additional cation disorder after certain inversion value), favoring of disorder can be understood. This transition is seen at lower than 50% inversion, which can be attributed to the approximations and simple reaction rate analysis. The effect of inversion on the elastic properties is discussed in the next section. 5.3.3 Bulk Modulus For a typical solid, the bulk modulus increases as the system becomes denser due to the rapidly increasing interatomic repulsion. Based on this argument, the bulk modulus is expected to increase with inversion. As the crosses in Fig. 58 indicate, the 118 behavior is actually more complex: initially the bulk modulus decreases slightly,then levels off, and then increases above i=0.5. A similar strong dependence of the bulk modulus at high inversions in MgAI204 was also reported by Chartier et al. in their MD simulations [180] (Fig. 58 B)). The interatomic interaction in their study had Buckingham interaction with Morse potential as short range interaction which was fitted by Morooka et al. [181]. That potential is developed to capture the Frenkel pair accumulation under neutron irradiation. The system might also turn to an amorphous state. Since the system under investigation had no point defects, other than cation anti site disorder, potential parameters are chosen as described in Table 23. 204 300 204 x 202 200 290  198 Sx  196 280  Sx 194  m x x x x x C 192 270 0.00 0.25 0.50 0.75 1.00 A) 0.00 0.25 0.50 0.75 1.00 Inversion B) Inversion Figure 58. Bulk modulus as a function of inversion A) this work B) reproduced from the reference [180] Even though values of the bulk modulus are different in the two studies, qualitative effect of inversion on the bulk modulus is very similar in nature. To compare the qualitatitive behavior of bulk modulus, the bulk modulus is normalized to normal spinel (i=0) for both the simulation studies and compared in Fig. 59. The small differences are due to the different description of the interatomic interactions used in the two studies. The potential description used in the study by Chartier [180] and developed by Morooka et al.[181] and has the Buckingham form and a Morse potential. 119 The good agreement argues that the specific description of the interatomic forces are not essential to obtain qualitative and semiquantitative trends in behavior. The dependence of the bulk modulus on inversion can be understood in terms of the separate contributions of inversion to cation disordering alone and to the associated volume change. To dissect these effects further, each contribution was studied separately: such separation is not possible experimentally, but is straightforward in simulation. The separate effects on the bulk modulus are shown in Fig. 510. 1.08 *MD * = 1.6 SOEChartier * 1.04  v 1.02 E O o 1.00o 0.98 .. 0.00 0.25 0.50 0.75 1.00 Inversion Figure 59. Comparison of normalized bulk modulus with MD study by Chartier [180] First, the volume of the system was fixed to that of normal spinel and the bulk modulus was calculated as a function of inversion, thereby picking out the the effect of the cation arrangement on the bulk modulus (filled circles in Fig. 510). Initially the bulk modulus decreases with inversion, then levels off. It has been already shown (Sec. 5.3.1) that inversion causes shortening of AIO bonds and expansion of MgO bonds at inverted sites, respectively. A similar effect of decrease in bulk modulus with cation disordering was experimentally observed in MgTi205 [170]. 120 310 Cation disorder AVolumetric A Cation disorder + Volumetric A 300 x Simulation x A A cn 290 A A 0 280 A A A S270 * 260 0.00 0.25 0.50 0.75 1.00 Inversion Figure 510. Separating out various effects on bulk modulus In the case of single crystal MgTi20s, the decrease in bulk modulus was attributed to the different compressibility of the weaker MgO and stronger TiO bonds and the way in which the octahedra are arranged. Even though MgTi205 (orthorhombic dipyramidal Bbmm) and MgAI204 are structurally different, the behavior of cation disordering is analogous because they both display weaker MgO bonds under cation disordering. Since this bond weakening is primarily responsible for lowering bulk modulus in MgTi205 for low inversions, it is also most likely related to lowering of bulk modulus in MgAI204. The leveling off of bulk modulus with inversion suggests the limited effect of weak MgO bonds in this case, as compared to MgTi205 [170]. This also suggests that there is an increasing effect of AIO bonds with inversion. Since the lengthening of the AIO bond has precisely the opposite effect on elastic properties, a small increase in the bulk modulus at higher values of / is seen. Second, the bulk modulus for normal spinel was calculated as a function of volume. Using the relationship between the degree of inversion and volume, the volume dependence of the the bulk modulus was replotted as a function of the inversion (filled 121 triangles in Fig. 510). It is evident from Fig. 510 that the decrease in volume monotonically increase the bulk modulus, consistent with the usual behavior of crystals. To capture the combined effect of volume and cation disordering, the average of the two effects is taken as an estimate for the dependence of the bulk modulus on inversion. As the triangles in Fig. 510 shows, this prediction matches the actual simulation results reasonably well. In particular, it shows that the volumetric effect essentially negates the effect of cation disorder up to i0.375. For higher degrees of inversion, further cation disorder has little effect, while the volumetric contraction leads to an increase in the bulk modulus. 5.3.4 Thermal Expansion The behavior of bulk modulus was attributed to the relative strength between inverted MgO and AIO bonds and to the volume change. These effects are also manifested in the thermal expansion. To calculate the thermal expansion, a series of MD simulations were run at different temperatures for NPT ensembles (for details refer to Chapter 2). The coefficient of thermal expansion is computed for each inversion value following the procedure described in Sec. 2.10. The variation of lattice parameter for i=0 with temperature is shown in Fig. 511. Similar variation of lattice parameter with temperature is shown for all other inversion values. The thermal expansion coefficients for each inversion value are compiled in Fig. 512. The similarity in the effect of inversion on the bulk modulus and thermal expansion is expected. The bulk modulus and the thermal expansion can be related through the Gruneissen relation [30]: /3B (54) 122 where Cv is the specific heat and y is the Gineisen coefficient, which measures the anharmonicity of the interatomic interactions. In Eq. 54, if everything else is constant, the thermal expansion and bulk modulus are expected to show opposite behavior with inversion. As seen by Fig. 58, bulk modulus increases with inversion after reaching i=0.5 and from Fig. 512, thermal expansion decreases. 1.009 1.008 1.007 1.006 1.005 1.004 1.003 1.002 1.001 1.000 i=0.0 0 250 750 T [K] 1000 1250 1500 Figure 511. Variation of lattice parameter with temperature for i=0. 123 6.60 , 6.20 o 5.80 x 5.40 * . 5.00 I I. 0.00 0.25 0.50 0.75 1.00 Inversion Figure 512. Coefficient of thermal expansion as a function of inversion. Using simulated values of a, Cv and B; y is calculated using Eq. 54. The dependence of y on inversion is illustrated in Fig. 513 and suggests that y does not depend strongly on inversion. The average value of y~2 is consistent with the value of 2.0 experimentally reported by Chopelas and Hofmeister [182]. 2.1  2.0 > 1.9 1.8 1.7 I 0.00 0.25 0.50 Inversion 0.75 1.00 Figure 513. Dependence of the Gruneisen coefficient on inversion. 124 I I I I The Gruneisen coefficient is a measure of the anharmonicity of the system. The good agreement between MD results and the experimental findings shows that the interatomic potential yields the same level of anharmonicity as the experiment. As shown previously that the thermal expansion is related to the thermal conductivity through the anharmonicity (for details, please refer Ch. 2). Thus, the analysis suggests that the simulated thermal conductivity values should be at least semiquantitatively correct. 5.3.5 Thermal Conductivity As noted in the Sec 5.1, MgAI204 undergoes inversion under neutron irradiation. A similar simulation setup as described in Section 2.10 is used to compute the thermal conductivity. These thermal conductivity values are then corrected for system size effect and compared to the experiment (Fig. 514). Thermal conductivity results for normal (i=0) MgAI204 are shown as open circles in Fig. 514. The MD results are shown as open circles and "corrected" represents the thermal conductivity values after harmonic analysis is performed on these MD values. As explained in Chapter 2, harmonic analysis provides a correction factor based on the accuracy of interaction parameters in reproducing bulk modulus and thermal expansion values to the experimental values. Table 51 provides detailed information about the correction factor calculated by harmonic analysis. Based on the numbers from Table 51, the correction factor comes out to be equal to 1.52. That means, based on the fidelity of the interaction potential, thermal conductivity values will be 1.5 times higher than the experimental counterpart (For details, please refer Eq. 46). After dividing MD simulation thermal conductivity values by 1.52, they are plotted as open triangles in Fig. 513 and termed as 125 "corrected". The filled circles are taken from the experimental studies of Burghartz et al. [183]. Table 51. Comparison of thermal expansion and bulk modulus for harmonic analysis a (x10 /OK) Bulk Modulus (GPa) Experiment 8.89 [184] 197.4 [172] Simulation 6.1 275.74 14.0 02 OMD 12.0 SExp 10.0 0 AO Corrected 8.0 O 6.0 4.0  2.0 0.0 250 500 750 1000 1250 1500 T [K] Figure 514. Variation of thermal conductivity for MgAI204 with temperature, open symbols represent the simulation results and filled circles are points reproduced from the experimental study by Burghartz et al. [183] The similarity between experimental and simulated thermal conductivity values gave the confidence to use the simulation tool for further investigation. Since the working temperature for nuclear fuels is in the range of 800 K1500 K, investigation on thermal conductivity as a function of inversion is performed at 1250 K. Finite system size analysis (For details, please refer Eq. 246) is performed to take into account any artifacts produced by system size limitations. A system size analysis 126 for each inversion value is shown in Fig. 515. The circles at y=0 gives the estimate of inverse of thermal conductivity for infinite system size. These infinite system size thermal conductivity values are computed using a linear fit and taking the inverse of intercept at y=0 (Chapter 2). 0.24 0.22 . 0.20 i=0.2 0.18 i=0. , 0.16 i=1.0 0.14 i=0.0 0.12 0.10 0 0.002 0.004 0.006 0.008 0.01 0.012 1/L [A1] Figure 515. System size analysis for different inversion values for MgAI204 The extrapolated thermal conductivity values for different inversion values of MgAI204 system are plotted as open triangles against inversion in Fig. 516. The rather weak dependence of the thermal conductivity on inversion is quite surprising when viewed in terms of the thermal transport properties of defected ceramics materials. In particular both experiments and MD simulations of fluoritestructure materials (e.g., yttriadoped zirconia and hyperstoichiometic U02) have shown a strong decrease in thermal conductivity with even quite low defect concentrations [63, 185]. The key difference between the spinel case and the fluorite structured materials is that here disorder only involves antisite defects, while for the fluorites the disorder involved 127 oxygen vacancies (zirconia) and oxygen interstitials (urania), which presumably affect the phonon properties and hence thermal conductivity much more strongly. By recalling the relation between anharmonicity and thermal conductivity of the system, as defined by the Eq. 42, the inversion dependence in a quantitative manner can be understood. This representation of thermal conductivity is based on the Debye approximation which assumes all phonon modes present in the system are acoustic and contribute to the thermal conductivity. It works well for very simple systems but as the structural complexity increases, the actual thermal conductivity values deviate from the predicted values. For the system, even though Eq. 42 cannot be expected to quantitatively predict the thermal conductivity, it should still capture the qualitative picture of the variation of thermal conductivity with inversion. 10 8  El 6 E6 * 4 *MD 2 2 Eo Calculated 0 I 0.00 0.25 0.50 0.75 1.00 Inv Figure 516. Dependence of thermal conductivity variation on inversion The thermal conductivity calculated from MD does show a shallow minimum at i0.5, not predicted from the anharmonicity analysis alone. This minimum can be attributed to the degree of disorder in the system which, for the fluorites, led to a significant drop in the thermal conductivity [186188]. There are various factors that can 128 produce differences between thermal conductivity values calculated from MD and the predicted values by using Eq. 42. They are investigated one by one: Effect of density: As already shown, the volume decreases with inversion (Fig 5.2). This increase in density increases the sound of speed in the material. The effect of cation disorder and volume on the thermal conductivity is separated out with the help of these two simple cases: Case 1: Change the lattice parameter to i=0.0 value for i=0.5 and 1.0 The lattice parameter of the inverted structures (i=0.5 and 1.0) is increased to the value of i=0.0 (normal). By comparing the thermal conductivity values of the i=0.5 structure with two different lattice parameters, the effect of density on thermal conductivity is calculated. Similar study is done on i=1.0 structure also. The results are shown in Fig. 516 and showed a negligible effect (~1%) of the density on the thermal conductivity. Case2: Change the lattice parameter to i=1.0 for i=0.0 To extend the study of case 1 on perfectly ordered structure (i=0), the lattice parameter of normal MgAI204 is changed to the value of i=1.0. The results are also shown in Fig. 515 and again no significant effect of density on thermal conductivity is found. Arrangement of cations: This analysis is motivated by the fact that for each inversion, there could be thousands of possible ways to arrange the cations. Similar to the analysis mentioned in Sec. 5.2, was performed here. Instead of choosing the lowest energy structures (as was done for the thermal expansion), structures with lowest and highest energies for same i values were chosen for thermal conductivity simulations. A 129 difference of less than 1% in the thermal conductivity between these structures was found. This is in accord with the expectation. Since random distribution of cations does not introduce any new defect in the system, no change in thermal conductivity was expected. 7 i=0.0, volume of i=1.0 i=1.0, volume of i=1.0 6 i=0.5, volume of i=0.0 i=0.0, volume of i=0.0 + i=1.0, volume of i=0.0 4 E i=0.5, volume of i=0.5  2 1 0 0 i 0.00 0.25 0.50 0.75 1.00 Inversion Figure 517. Density effect on thermal conductivity Total scattering sites: A key effect of point defects is the number of possible scattering sites in the system produced by antisite defect creation. To capture the effect of point defects, a simple "combinatorics analysis" was used. With each inversion, the possible number of scattering sites is CMg .C A n, (55) where C"M denotes the number of possible ways to generate the structure for the inversion i, involving ni Mg and ni Al ions. Thus, the product is a measure of possible number of scattering sites for the inversion value (i). As an example, for i=0, Mg and Al occupy tetrahedral and octahedral positions, and there is only one way to arrange the 130 cations. For one antisite pair, in one unit cell system, 1 Mg can be in any of the 16 possible Al sites and also Mg can be chosen in 8 different ways. The multiplicity of these possible arrangements with inversion can be expected to have an impact on the thermal conductivity. Figure 517 captures the variation of the number of possible arrangements for each cation site for each inversion. *Combination 50 Mg *** * 40 AAI * E 30  .0 o 20 10  0.00 0.25 0.50 0.75 1.00 Inversion Figure 518. Combinatoric analysis: yaxis shows number of possible combinations for MgAI exchange possible in 2x2x2 unit cell system of MgAI204 Even though i=1 signifies the totally inverted structure, it does not have the most number of possible combinations, because all of the A sites are now totally occupied by Al. There is a maximum in the number of different defect arrangements at i=0.67, at which point the system would display the maximum number of different scattering environments. If there were no other effects, a minimum in the thermal conductivity at i=0.67 is expected. The fact that the thermal conductivity is actually a minimum at close to i=0.5 suggests that the disorder on the A sublattice actually affects the phonon scattering more strongly than the disorder on the B site. This can be understood in terms of local strain produced at inverted sites. As previously discussed, the TAI sites experience greater strain than the MMg sites and thus affect material properties to a 131 greater extent. As inversion increases, TAI experiences more chemical disorder; at i=0.5 half are occupiedby Mg and half by Al above i=0.5 they become more and more populated by Al ions. 5.4 Conclusion The MD study has provided significant insights into the effects of inversion on the thermoelastic properties of MgAI204. Ease of formation of more disorder in already inverted structures is shown by this study. The effect of inversion is extended on thermoelastic properties of MgAI204 by scanning the whole range of inversion (i=01). The effects of temperature and cation arrangement are also separated. The decrease in lattice parameter with inversion is shown to arise from a simple charge sharing between cationoxygen at tetrahedral and octahedral site. The effect of disorder already present in the structure on the formation of new cation arrangement is also presented. A simple theoretical analysis favors the formation of disordered structure after i0.67 and the simulation results presented the favoring of additional cation disorder after i~0.37. The difference from the theoretical model is explained in the light of simplified analysis and charge analysis favoring the inverted structure for MgAI204. The variation in elastic constants and bulk modulus with inversion are reported and compared with other DFT and MD studies. Contributions to the bulk modulus are analyzed in two parts: volumetric and cation disorder. This study has shown that weakening of MgO bonds at inverted octahedral sites essentially negates the volumetric effect on bulk modulus up to i~0.5. After i=0.5, stronger AIO bonds at inverted tetrahedral sites and volumetric effects dominate and control the bulk modulus of MgAI204. 132 The effect of thermal expansion as a function of inversion is also captured and explained. As expected from the bulk modulus results, thermal expansion remain almost unchanged up to i~0.5 and decreased afterwards. Anharmonicity of the system is calculated using bulk modulus and thermal expansion and compared to the experimental value. The thermal conductivity for the normal structure reproduces values similar to the experimental ones. The study is extended to include the effect of inversion on thermal conductivity, which has never been done before. The thermal conductivity results are compared with the theoretically computed values. Deviation from the theoretical model is reported and explained with the help of a "combinatorics analysis". 133 CHAPTER 6 THERMAL TRANSPORT PROPERTIES OF Bi4Ti3O12 6.1 Introduction: Thermal Barrier Coating Thermal barrier coatings (TBCs) are applied on materials operating at elevated temperatures to act as insulation. TBCs are widely used in turbine blades, combustor cans, gas turbines and aero engine parts. The most common place where to see a gas turbine engines is commercial jets and in electricity generating plants. In gas turbines a pressurized gas spins the turbine. Pressurized gas is generated by burning fuel in the engines. Possible fuels include propane, kerosene, jet fuel or natural gas. The main advantages of gas engines are that they are smaller in size than their counterparts and have a greater power to weight ratio. Their use is limited to power reactors and commercial jets or tanks because of the cost, high operating temperature and high spin speed of the turbine and limitations of constant load rather than fluctuating one. Jet engines can be classified as either turbofan or turbojet. Both work on the same principle. A schematic of a simple turbofan is shown in Fig. 61. They have three parts: The compressor compresses the incoming gas to about thirty times its original pressure. The combustion area burns the fuel and produces high pressure, high velocity gas. The turbine extracts power from the high velocity high pressure gas. Compressor Turbine Nozzle Combustion Shaft chamber Figure 61. Simple schematic view of turbofan gas turbine engine [189] 134 Thermodynamically the Brayton cycle is used to describe gas turbines. In the Baryton cycle, air is compressed isoentropically, combusts at constant pressure and expands again isoentropically at the initial pressure. An idealized Brayton cycle is shown in Fig. 62. Fuel Combustion P T 3 2 3 qin 2 3 Work4 1 utq ut 4 !out 4 2 Fresh Air Exhaust 1 qout gasses Pv Diagram v Ts Diagram s Figure 62. Idealized Brayton cycle [190] Similar to other heat cycles, the efficiency of the Brayton cycle is proportional to the combustion temperature. There is thus a strong incentive to increase the operating temperature. The limiting factor to elevate the operating temperature is the limitations of the performance of the materials that make up the engine and its parts to withstand heat and pressure. To elevate the temperature, thermal barrier coatings (TBCs) are applied to the turbine blade itself [191]. A TBC is primarily made of four layers (Fig. 63): 1. Ceramic topcoat: This layer provides the thermal insulation for the turbine blade. It has very low thermal conductivity and is designed to withstand a large number of thermal cycles. The most common TBC material is yttria stabilized zirconia (YSZ) which has a thermal conductivity around 2.3 W/mK at ~10000C and melting point around 27000C. These are typically 100 pm to 2 mm thick. 2. Thermally Grown oxide (TGO): This layer is created when the topcoat reacts with the bond coat layer. It is about 110 pm thick. 135 3. Bond coat: This layer is about 75150 pm thick and is below the TGO. The purpose of this layer is to hold the topcoat on the substrate. This layer is made of Ni and Pt and often consists of more than one layer of different compositions. 4. Substrate: The turbine itself is generally made of a superalloy and is cooled by hollow channels inside the turbine blade. .... . Superalloy C'11Iling Air Figure 63. Schematic of a TBC [192] The important considerations in the selection of a TBC material are its ability to withstand thermal stresses with repeated heating and cooling cycles; thermodynamic compatibility with oxides formation of bond coat; and low thermal conductivity. The current material of choice for TBCs is yttriastabilized zirconia (YSZ) in its metastable tetragonalphase; it will also remain the material of choice for the current operating temperatures [193]. However, YSZ has poor creep resistance at high temperatures 136 which results in crack and eventually spallation. YSZ also has high diffusivity of oxygen that results in oxidation of bondcoat layer. The limiting factors of YSZ and the need for higher operating temperature provides a motivation for developing new materials for TBC applications. Table 61. Different forms of layered perovskites Composition/Formula Layer orientation Name and example An+lBnX3n+1 or A2'[An1BnX3n+l] A'[An1BnX3n+l] {001}p RuddlesdenPopper Sr3Ti207 DionJacobson RbCa2Nb300o Bi202[An1BnX3n+l] AnBnX3n+2 An+1 BnX3n+3 {001}p {110}p {111}p Aurivillius Bi4Ti3012 Ca4Nb4014 Ba5Nb4015 Even though all the requirements are critical in the selection of TBC material, low thermal conductivity remains the primary criterion. Desirable properties of TBCs include large average atomic numbers, large unit cell and the possibility of bonding and structural disorder [194]. Aurivillius compounds possess many of the required properties for TBCs. These anisotropic layered materials have weakly bonded planes (Bi202 layers 137 {001}p or any other layer between perovskites) between layers of pseudoperovskites having rigid polyhedra. This intrinsic structural disorder acts as a barrier to phonons and reduces their mean free paths. Thus the weakly bonded planes in layered perovskite materials are useful in achieving lower thermal conductivity values without introducing any external defects or dislocations in the system. Anisotropic thermal expansion exhibited by these layered materials is useful in minimizing residual stresses. Based on the composition of interslabs and/or the orientation of the layer plane relative to aristotype cubic lattice, these layered perovskites are divided into five groups as shown in Table 61. In Table 61 for {001}p notation, p refers to pseudocubic. Among these layered perovskite types, Aurivillius structures possess many desirable properties for use as a TBC. As shown in Table 61, Aurivillius compounds are represented by the general formula (Bi202)2+[An1BnO3n+1]2, where n represents the number of perovskite layers separating Bi202 layers. A simple Aurivillius compound with n=3, Bi4Ti3012 {(Bi202)[Bi2Ti301o]} has many promising properties (large atomic number, large unit cell, anisotropic structure, layered structure) to be a potential TBC material. A recent study by Shen et al. has investigated the low value and anisotropic nature of the thermal conductivity of Bi4Ti3012 (BIT) and attributed the low value to the density difference between layers [20]. Further investigations pertaining to the effects of density difference between layers and other intrinsic factors on limiting thermal transport properties of BIT might be useful. The experimental studies, however, face two big challenges: 1. It is very hard to synthesize single crystals 2. It is impossible to tune the density of layers to the desired value. 138 Both of the experimental limitations are easily handled by atomiclevel simulation. Therefore, the experimental investigation is expended with the help of MD simulation. In this chapter, the effects and limitations of density and intrinsic structural factors on the thermal transport of single crystal BIT are illustrated. 6.2 Selection of Potential Parameter As discussed in Section 28, for the study of BIT, three sets of potential parameters were chosen to capture the shortrange interaction. They all had same BiO and 00 interactions but different TiO parameters. Table 24, has all the parameters. They are given again below for quick reference. Table 62. Short range potential parameters for Bi4Ti3012 [65, 67] (Repeat of Table 24) Species A (eV) p(A) C (eV Ab) Bi+3O2 49529.4 0.2223 0.00 O2O2 9547.96 0.2192 32.0 2131 2131.04 0.3038 0.00 Ti+4O2 760 760.47 0.3879 0.00 2549 2549.4 0.2989 0.00 These interactions are used to calculate to calculate structural properties of BIT. To check the quality of parameter sets, they are compared with experimental values as shown in Table 63. As explained in Chapter2, interaction potentials are the back bone of MD simulation. Interaction potentials are developed in order to reproduce material properties by simulation. Based on the Table 62, all interaction parameters overestimate the system volume. Interactions 760 and 2549, overestimated the volume 139 by ~25%.Whereas for interaction 2131, the simulated volume is within 0.05% of experimental value. This simple comparison of structural features to the experiment provides us an opportunity to choose the bestfit interaction parameters. Bearing this in mind, all the further simulations are performed using 2131 potential parameters. Table 63. Comparison of interaction parameters for BIT 2131 760 2549 Exp. [118] a(A) 5.42 5.53 5.48 5.45 b(A) 5.34 5.46 5.43 5.41 c(A) 33.41 33.43 33.35 32.83 a 90.00 90.00 90.00 90.00 (3 89.93 90.16 89.90 90.00 y 90.00 89.99 89.99 90.00 Vol. (A3) 968.29 1009.06 992.46 967.98 6.3 Thermal Expansion At low temperature (T < 900 K), BIT has monoclinic structure with all the crystallographic angles of 900. Confusion arises because of the conventional classification of the crystal systems. If all the angles are 900 and sides are not equal, it is classified as orthorhombic and not monoclinic. However, there are symmetry operations tied with each system. The orthorhombic B2cb space group has 2fold rotation, a 21 screw axis, mirror plane and glide plane symmetries whereas Blal has only mirror and glide as symmetry elements. So even though description of lattice features leads to the conclusion of BIT being orthorhombic, when combined with symmetry considerations it is clearly monoclinic. 140 The thermal expansion is simulated for the NPT ensemble. Since the cutoff distance is ~10A, the smallest system is generated by building a structure of 6x6x1 unit cells in the x, y and z directions. This supercell structure has dimension of ~30A in each direction, and is shown below (Fig. 64). Figure 64 clearly shows the difference in tilting behavior of TiO6 octahedra along x and y axes which can be represented as aobc in the Glazer notation (for details, please refer the Sec. 3.5) [138]. Z Srirw* i*n 1 06 W A111. %4%'%4% 0%. 4 @ 0 0 0 . rterrrrrrrrv '.wrarrrr X Y Figure 64. 6x6x1 super cell of BIT viewed along x and y axes The tilting of the TiO6 octahedra in the pseudoperovskite layer affects the Bi202 layer also. As seen in Fig. 64, the two layers are tilted in a similar fashion along the y axis. The effect of octahedron tilting on lattice parameter becomes clear when the coefficient of thermal expansion along each direction is computed. The normalized lattice parameters are compared for the different directions in Fig. 65. Table 64 compares the simulation values to the experimental studies. 141 Table 64. Comparison of thermal expansion values Axis Simulation (xl 0/K) Experiment (xl 0/K) X 5.14 10.1 [195] Y 3.09 4.45 [195] Z 3.77 0.20 [195], 16.12 [196] Average 4.00 4.91 [195], 11[197] 1.006 1.005 1.004 1.003 1.002 1.001 1.000 n QQQnnn 09. ~ A 0 o A Q DD s D 0 200 400 600 800 Temperature (K) 1000 1200 Figure 65. Coefficient of thermal expansion of BIT along x,y and z axes Very few experiments have reported the thermal transport properties of BIT. The discrepancy among the experiments is larger than that between the simulation results in this work and experiment [195197]. The experimental study by Hirata [195] was done to investigate the physics behind ferroelectric behavior of BIT near its Curie temperature. It reported that the aaxis (xaxis) displacement of Bi atom with respect to 142 U.3 TiO6 octahedra in the perovskite layer is mainly responsible for ferroelectricity of BIT. The experimental investigations by Subbarao [197] were done on polycrystalline samples and mainly focused on to the structural changes around the Curie temperature of BIT (6750C). Another experimental study by Hirata et al. [195] found an average expansion value (4.9x106/K) similar to the one reported earlier by Subbarao (10x106/K) [197]. Due to this large variation among experiments, expansion coefficients are compared to the average value. Similar to the experiment findings, MD simulation also captures the anisotropy in expansion coefficients. The strong difference between the x and y values of the thermal expansion is interesting and can be attributed to two structural features: 1. Octahedra tilting, 2. The difference in thermal expansion of the layers. A very good description of octahedron tilting on lattice parameter is given by Megaw [137]. To compute the octahedra tilting effect, structural analysis is used in Crystalmaker. Since there is no tilt along x, only y and z tilts need be computed. An analysis tool is required to calculate the tilting of the octahedra. Due to the lack of any sophisticated analysis tool, tilting is calculated using the angle tool 4 in Crystalmaker. As seen from Fig. 64, tilting of octahedra along Yaxis can be measured in two ways: insidetilt (inset of Fig. 66 A) and outsidetilt (inset of Fig. 66 B). Both are measured and shown in Fig. 66. 143 14 12  8 6 4 Insidetilt 2A 0 0 100 200 300 400 500 600 700 A) Temperature (K) 12 10  P 4 6 2 Outsidetilt SM 0 I I I I I I 0 100 200 300 400 500 600 700 B) Temperature (K) Figure 66. Temperature variation of tilt angles of A) Inside and B) Outside tilt of the octahedra along Yaxis (baxis). As seen in Fig. 66, there is negligible effect of temperature on octahedra tilting. The second structural feature of the layering is investigated to characterize the non uniformity of the thermal expansion. To measure expansion of different layers, the cationcation distance is measured as a function of temperature. It is not trivial to resolve the distances in different direction. 144 The minimum separation between same cations in each layer can be used to compute thermal expansion of each layer. Again Crystalmaker is used to calculate the distances between the same cations of each layer. The methodology to calculate change in same type of closest cations is described below: 4.0 TiTi AvgX 3.9 Avg Y 3.8 i AvgZ I I 3.8  3.6 3.5 100 0 100 200 300 400 500 600 700 Temperature (K) Figure 67. Temperature profile of nearest TiTi distance in BIT structure Pseudoperovskite layer: Ti is only present in pseudoperovskite layer, thus temperature profile of TiTi smallest distance is used to compute thermal expansion of the layer. The distance between TiTi is measured along both X and Y axes using Crystalmaker. Interatomic distance of closest TiTi is calculated in each pseudo perovskite layer. Average of the bond length at each temperature is taken and is shown in Fig. 68. Since there is only one layer to compute thermal expansion in Zdirection, no error bar is denoted for zaxis bond length. Bi202 layer: Since Bi is present in both the layers, BiBi is calculated separately for each layer. A system similar to that used for TiTi bond calculation is used Instead of masking Bi and 0, Ti and O are masked. BiBi temperature profile, shown in Fig. 68, is used to compute the thermal expansion of Bi202 layer. 145 3.9 3.8  3.7  S3.6 Avg X ,o Avg Y F 3.5  .,5 Avg Z 3.4  3.3 100 0 100 200 300 400 500 600 700 Temperature (K) Figure 68. Temperature profile of BiBi cation distance in Bi202 layer Analysis for both the layers provides some interesting insight into their response towards the temperature. But it is not trivial to suggest any effect of these layers on the overall thermal expansion. Besides the atomic disorder, density difference between layers might also play some role in the anisotropy of the heat transport properties of BIT. A detailed investigation to capture the effect of mass on thermal expansion gave no conclusive indication. The investigation is extended to observe the effect on thermal conductivity. A recent experimental investigation of thermal transport properties of BIT by Shen et al. reasoned the density difference between layers was responsible for its low thermal conductivity [20]. The experimental study and the effect of density on thermal transport are discussed in detail in the following section. 6.4 Thermal Conductivity BIT exhibits anisotropic thermal conductivity. As discussed in sec. 23, BIT can be indexed as orthorhombic (because of its unique nature of being monoclinic with orthorhombic axes). The thermal conductivity can be written in matrix form as 146 A1 1 0 0 K= 0 K22 0 0 0 K33 (61) 1.6 1.4 1.2 Z 1.0 E 0.8 S0.6 0.4 0.2 0.0 0 0.02 0.04 0.06 0.08 0.1 0.12 A) 1/Lz [nm1] 0.6 0.5 my S0.4 OX E 0.3 0.2 0.1 0.0 0 0.02 0.04 0.06 0.08 0.1 0.12 B) 1/L, [nm1] Figure 69. System size analysis for T = 300 K in all the direction A) x,y and z system size dependence B) x and y system size analysis only The offdiagonal terms are zero because all the angles are 900. The thermal conductivity is calculated in all three directions. As discussed in Chapter 2, it is necessary to perform system size analysis, from which the thermal conductivity values for the infinite system size are computed. As a representative example, the system size 147 analysis for T = 300 K is shown in Fig. 69. A similar system size analysis is performed at all the temperatures. The thermal conductivity for infinite system size systems are shown in Fig. 610. 1  0  100 A A A 300 500 T (K) 700 Figure 610. Temperature and direction dependence of thermal conductivity for BIT Anisotropy in the thermal conductivity is very well captured by the simulation study. Experimental investigation of the thermal conductivity of BIT is shown in Fig. 611 and simulation does produce similar qualitative dependence of the thermal conductivity on temperature. 148 III 2.U Random Y 2.5 E 2. .. ab plane 2.0 U ...  ... ... in  1.5 0 0.5 0.0 200 400 600 800 1000 Temperature (C) Figure 611. Thermal conductivity of BIT by experiment [20] The experimental results are shown on Fig. 611 and need some explanation as figure in itself is not sufficient to explain its results. Experimentally, it is very hard to synthesize single crystal BIT, so results shown are of textured polycrystalline sample. In the experiment, two types of samples were prepared: i) caxis textured sample and ii) random polycrystalline sample. The thermal diffusivity was measured using the thermal flash method [82] and thermal conductivity is calculated using Eq. 241. Reported experimental sample densities (textured: 7882 kg m3, random: 7590 kg m3) were lower than the theoretical value (8040 kg m3). The thermal conductivity from the random sample is shown by filled circles and the textured sample with filled triangles in Fig. 611. It has been assumed in the experiment that the thermal conductivity along a and b axes is the same and is shown by filled 149 squares (ab plane) in Fig. 611. Since the thermal conductivity along a and b axes is the same, Eq. 61 can be written as Kab 0 0 K= 0 Kab 0 LO0 O r L 0 oco (62) If random polycrystalline sample has thermal conductivity Kp, it can be written in the form of axial thermal conductivities as: = 1 (2, + cc) (63) 3 Since experimentally, Kp and Kcc is measured, Kab can be calculated as: IKa= (3 Kc) (64) 2 The calculated data for thermal conductivity along ab plane is shown as filled squares in Fig. 610. The thermal conductivity results from the experiment and simulation are in qualitative agreement. A theoretical discussion on thermal conductivity behavior over temperature gives the 1/Tn dependence. This temperature dependence corresponds to a decrease in thermal conductivity values with temperature for non metals. An increase in the thermal conductivity at high temperatures (T >800 K) is seen in Fig. 611. This is attributed to radiative component of the thermal transport. Contribution of this radiative component of heat transfer in BIT thermal conductivity increases the apparent thermal conductivity of the material beyond 800 K. Density Effect on Thermal Conductivity: As explained in Section 6.1, low thermal conductivity materials are needed for TBC applications. The motivation behind 150 this analysis is to understand the factors contributing towards lowering the thermal conductivity of BIT and then to exploit them to reduce the thermal conductivity value even lower. Clarke has described, a minimum thermal conductivity value expression based on material's elastic and structural properties [194]: m 3 6E2 (65) Kmm OC (65) MY Where, M is the molecular mass, m is the number of atoms per molecule. The expression of 62, changes different atomic masses to effective mass and that leads to absence of optical phonons in the material. Overall effects of this assumption are unknown but this underestimates thermal conductivity. As shown in experiment thermal conductivity (Fig. 611), even after the phase change in BIT, the thermal conductivity does not change much. This reflects the limited role of specific atomic arrangements on thermal conductivity in BIT. When the phonon mean free path is calculated in the limit of high temperature, it gives a value of 0.24 nm (half of interatomic spacing 0.46 nm) [20]. Since there is no ionic disorder present in the system and atomic disorder has a very limiting effect on thermal conductivity, the low thermal conductivity is attributed to the scattering of phonons from the superlattice of alternating pseudoperovskite and Bi202 [20]. The density of Bi202 fluoritelike block is 11,397 kg m3 and the thickness is 0.45 nm, whereas the perovskite block has a density of 6487 kg m3 with a thickness of 1.21 nm. This density difference becomes larger in low temperature structure. Since MD simulation gives total control over structure and its features, the density effect can be easily probed. The effect of the density difference is investigated in a systematic manner. 151 In the BIT structure, the PP layer is less dense than the BO layer. A simple method to probe the density effect on thermal conductivity of BIT is described below: 1. The PP layer is made heavier by increasing the mass of Ti (HT) 2. The PP layer is made heavier by increasing the mass of Bi only in PP layer (PPB) 3. The BO layer is made lighter by reducing the mass of Bi in BO layer only (LB). A simple schematic is shown below of the above methodology (Fig. 612). The density of BO and PP layers are denoted by dl and d2, respectively. In the case of HT, the masses of the Ti in PP layer are artificially increased and since there is no Ti in BO layer, this captures the effect of mass only. The other two procedures, LB and PPB, involve changing mass of Bi only in BO layer or in PP layers that introduces two different kinds of Bi in the system. Thus these two scenarios include the effect of extra scattering sites along with the mass effect. 4 dl>d2 4 * III : T Bo O BO C d1 d1 d? d1 \, t C d2 dl d2 dl PP SP Normal HT LB PPB Xtra scattering sites Figure 612. Schematic of method to investigate the density effect on thermal conductivity of BIT 152 At room temperature, all the above systems were compared for their respective values of thermal conductivity. Anisotropic nature of thermal conductivity allows us to choose between x, y or z axes thermal conductivity values to compare the effect of density on thermal conductivity. Thermal conductivity along the z axis is expected to reflect the effect of density more than x or y axis. Since dependence along x or y axis is expected to be same. The effect of the density is captured along the x and the z axis. The temperature is set at 300 K for the comparison. A system size analysis is performed at T=300 K for the thermal conductivity in x direction for all the systems and the comparisons are shown in Fig. 613. Figure 613 shows a systematic system size dependence on all the systems along the xaxis, but it does not become clear what the effect of density on thermal conductivity is. After extracting the infinite thermal conductivity for all the systems, the result is shown in Table 63. 0.35 0.30 0.25 o normal EL HT 0.20 SLB 0.15 \PPB 0.10 0.00 0.02 0.04 0.06 0.08 0.10 0.12 1/L [nm1 ] Figure 613. System size and density dependence of thermal conductivity for BIT 153 Table 65. Thermal conductivity variation for all 4 systems System K (W/mK) Normal 5.63 LB 5.86 HT 5.78 PPB 7.16 The effect along the zaxis is more prominent as shown in Fig. 614. As expected, density difference is responsible for the low value of the thermal conductivity of BIT. Both the cases, PPB and HT increase the density of PP layer to be equal to the BO layer. If density was the only reason for the low value of thermal conductivity, the thermal conductivity increase should have been same for PPB and HT scenarios. The increase in the thermal conductivity is larger for HT case than PPB. The PPB scenario has two types of Bi in the system and that reflects as point defect in the system. The density increases the thermal conductivity value but presence of point defects lower the overall increase in the thermal conductivity. Whereas in HT, mass of all the Ti increases, thus no point defects are created in the system. That is why, a larger increase is observed in the case of HT than PPB and this absolute increase can be attributed to the density effect. 154 1.4 1.2 1.0 F 0.8 0.6 0.4 0.2 0.0 Normal PPB HT Figure 614. Density effect on the thermal conductivity along the zaxis in BIT 6.5 Conclusion and Future Study Structural constraints and their effect on thermal transport properties are well documented and investigated by the MD simulation. A suitable potential parameter is chosen based on its performance to reproduce structural features of BIT. Using the potential, anisotropy in thermal expansion was shown. The octahedron tilting was found to be temperature independent. Anisotropy in thermal conductivity qualitatively agreed with experimental investigation. Density difference between layers is attributed to the low value of thermal conductivity. A systematic investigation revealed the contribution of density effect and point defects on the thermal conductivity along the zaxis. No density effect was observed along the x axis. It can be concluded that by the introduction of point defects in the system, thermal conductivity can be lowered. 155 CHAPTER 7 GENERAL CONCLUSIONS In this work, the molecular dynamics (MD) simulation method has been successfully used to investigate thermal transport in potential inert matrix fuel materials (MgONDZ and MgAI204) and thermal barrier coating material (Bi4Ti3012). 7.1 MgONDZ As a candidate of inert matrix fuel, interfaces play a dominant role on thermal conductivity in composite systems. In this dissertation, it was determined that volumetric composition governs the thermal transport at very fine (<10 nm) grain size level for MgONDZ composite. The composite can have MgONDZ, MgOMgO and NDZNDZ interfaces. Homogenous interfaces were investigated by simulating MgO and NDZ polycrystals separately. Using thermal conductivity values of polycrystals and single crystal, interface effects were computed for both materials. For very fine grain size, interface effects were comparable to the grain size, itself. Due to the limitation of MD simulations, grain sizes were limited in tens of angstroms. Even with limited computational power, polycrystal thermal conductivity results were extrapolated for larger grain sizes and gave similar grain size dependence in comparison with the experiment results for YSZ [19]. Fidelity with experiment provides an additional tool to rectify the simulation data. Anharmonic analysis was used to quantify the fidelity of the interatomic interaction parameters of the simulation. Based on the difference from experiment for thermal expansion and bulk modulus, a correction term was calculated. The correction term quantified the deviation in thermal conductivity results for the material. 156 The study provides a key insight into the interface effects on thermal conductivity of MgONDZ composite. However, the study was limited to <10 nm grain size compared to micrometer size grains observed experimentally. Therefore, a systematic study with larger grain size of MgO, NDZ and MgONDZ system will provide additional insight into the thermal properties. Defects generally act as a barrier to thermal conduction. It will be interesting to capture the coupling of defectgrain boundary interactions on thermal conductivity of MgONDZ cercer. 7.2 MgAI204 With relative ease of formation of inverted structure (cation antisite structure) in MgAI204 under neutron irradiation, effect on thermal and mechanical properties is an important issue. Experimentally temperature and inversion are interrelated, thus separating out one effect from another is extremely hard. Atomiclevel simulation provided an ideal tool to capture the individual effects in the MgAI204 system. Experimentally MgAI204 system has been characterized for i<0.3. Using MD simulation, the entire range of inversion (i=01) has been probed for thermal and mechanical properties. Thermal expansion has shown no variance for i< 0.5 and decreases afterwards. Similarly, analysis on bulk modulus has shown no change for i< 0.5 and increases for i>0.5. Using the above response of inversion on thermal expansion and bulk modulus, thermal conductivity was predicted. Thermal conductivity values were also calculated independently for each inversion using nonequilibrium MD (using heat source, heat sink). Critical analysis of the thermal conductivity values calculated by both methods showed the inability of the existing model in capturing the inversion effect. 157 "Combinatorics analysis" method has been used to incorporate the inversion effect on thermal properties. Additionally, experimental and simulation observations on lattice parameters of MgAI204 has shown a decrease with inversion. However, the effect has not been properly characterized. For the first time, this study explained the reason for decrease of the lattice parameter with inversion. Besides, the study also presented a possible cation arrangement for 1 cation antisite pair. Another interesting question is to check whether the existing inversion enhances or reduces further disordering of the system. For i< 0.5, no change in further disorder of the system has been reported, however for i>0.5, further disorder is facilitated. Low formation energy of additional antisites for i>0.5 has been reported and explained through the formation of strain dipoles at inverted sites and simple reaction rate analysis model. The present study has considered the arrangement of atoms in a spinel structure. However, defected rocksalt structure is also a possible arrangement of atoms for MgAI204. Therefore, for a complete understanding of the thermoelastic properties, it is essential to characterize and compare both spinel and defected rocksalt arrangements of MgAI204. Similar to MgONDZ system, effect of interfaces on thermoelastic properties can be characterized. 7.3 Bi4Ti3012 Bi4Ti3012 (BIT), a layered perovskite material, is a candidate material for thermal barrier coating (TBC) applications. The layering in BIT alternates between Bi202 and perovskitelike layers. Due to the presence of these layers, thermal conductivity results 158 in low values for BIT. This study focused on the fundamental understanding of the low value of the thermal conductivity of BIT. An experimental study reported that the difference in the density of the layers guides the overall thermal conductivity of BIT [20]. In order to investigate the effect of density, simulations were performed by varying the mass of the cations in an appropriate manner. Due to the layered structure of BIT, anisotropy in thermal properties is reported in the literature. The current calculations also captured the anisotropic behavior in thermal conductivity and expansion. Since BIT is a potential material for TBC applications mainly due to its low thermal conductivity, it is desired to further reduce its thermal conductivity. Characterization of the effects of intrinsic and extrinsic defects (like La [198]) on the thermal conductivity of BIT is essential. Diffusion of oxygen ion in BIT is also an important criterion for TBC application. Thus, characterization of ionic diffusion in BIT will provide a better understanding of the material for TBC application. Overall, molecular dynamics is a very useful tool in characterizing atomic level details of various materials. This study successfully characterized the thermal transport properties of various oxide materials for their potential application as inert matrix fuel (MgONDZ, MgAI204) and thermal barrier coating (Bi4Ti3012). 159 LIST OF REFERENCES [1] "Annual Energy Review 2008 Prepared by Energy Information Administration," Washington, D.C., 2009. [2] D. Bodansky, "Global Warming and NuclearPower," Nucl. 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It is a 5year degree program that combines the basic training of both engineering and physical sciences. He came to the United States for his higher studies and joined the graduate program in Physics Department at University of Florida (UF), Gainesville, in Fall 2002. He received his Master of Science in Physics from UF in May 2005. After completion, he joined the doctoral degree under Prof. Simon R. Phillpot at UF, Gainesville. He received his Ph.D. in materials science and engineering from the University of Florida in the summer of 2010. 175 PAGE 1 1 THERMAL TRANSPORT IN SELE CTED CERAMIC MATERIALS FOR POTENTIAL APPLICATION AS INERT MATRIX FU EL OR THERMAL BARRIER COATING USING ATOMIC LEVEL SIMULATION By PRIYANK SHUKLA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 PAGE 2 2 2010 PRIYANK SHUKLA PAGE 3 3 With love to my whole family PAGE 4 4 ACKNOWLEDGMENTS After reaching at this important milestone of my life, I would like to express my gratitude and thanks to all the people who made my PhD journey a wonderful and memorable one. First and foremost, I would like to thank and express my deepest gratitude to my mentor Prof Phillpot. His passion for science, willingness to help; offer advice on anything, insightful remarks and a great sense of humor made my PhD a very enjoyable learning experience. He is my role model. Without his guidance and critical remarks and constant push to bring out the best of you, this work would not have seen the day I cannot thank enough to my supervisory committee members Prof. Sinnott, Prof. Nino, Prof. Baney and Prof. Tulenko for their constant support suggestions and guidance. I was fortunate enough to work with the great group of students and post docs of Computational Materials Science Focus Group. The group is very supportive and helpful There are many, whose friendship and support, I cherished and enjoyed during these years They know who they are and they have my gratitude. Special thanks to Dr. Rakesh Behera for his unconditional help, insightful remarks and delicious food. This journey wouldnt have been possible without the constant support, love and encouragement from my family. Although they are physically far away, weekly phone calls, yahoo messenger and emails have changed the distance to a few clicks away. I consider myself fortunate to have such a great family In the end, I would like to say my special thanks to my wife, Monika. Thanks for bearing me for past three years, when I cannot tolerate myself for more than a few minutes. This work was funded by DOE NERI Award DE FC07 051D14647. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 8 LIST OF FIGURES .......................................................................................................... 9 ABSTRACT ................................................................................................................... 13 CHAPTER 1 BACKGROUND ...................................................................................................... 15 1.1 Energy Sources ................................................................................................. 15 1.2 Greenhouse Effect and Global Warming .......................................................... 17 1.2.1 Carbon Dioxide (CO2) Emission .............................................................. 19 1.2.2 Global Warming Potential ........................................................................ 20 1.3 Carbon EmissionFree Energy .......................................................................... 20 1.4 Nuclear Energy ................................................................................................. 23 1.5 Thermal Conductivity: An Important Property ................................................... 25 1.6 Motivation for This Work ................................................................................... 25 1.7 Objective of This Work ...................................................................................... 26 1.8 Organization of Dissertation .............................................................................. 28 2 THERMAL TRASNPORT AND SIMULATITON METHODOLOGY ......................... 29 2.1 Basics of Thermal Transport ............................................................................. 29 2.2 Modes of Heat Transport .................................................................................. 32 2.3 Heat Transport in Nonmetallic Solids ............................................................... 33 2.4 Simulation Methods .......................................................................................... 39 2.5 Molecular Dynamics Basics .............................................................................. 40 2.6 Integration Algorithms ....................................................................................... 40 2.7 Periodic Boundary Condition ............................................................................ 43 2.8 Interatomic Interactions ..................................................................................... 45 2.8.1 Long range Interactions ........................................................................... 45 2.8.2 Short range Interactions .......................................................................... 51 2.9 Ensembles ........................................................................................................ 54 2.9.1 Pressure Control ..................................................................................... 57 2.9. 2 Temperature Control ............................................................................... 59 2.10 Thermal Expansion Simulation ....................................................................... 60 2.11 Thermal Conductivity Simulation .................................................................... 60 3 CRYSTALLOGRAPHY OF RELEVANT MATERIAL SYSTEMS ............................. 65 3.1 General Introduction ......................................................................................... 65 PAGE 6 6 3.2 MgO: Rocksalt Crystal Structure ....................................................................... 66 3.3 Nd2Zr2O7: Pyrochlore Crystal Structure ............................................................ 68 3.4 MgAl2O4: Spinel Crystal Structure ..................................................................... 71 3.5 Bi4Ti3O12: Aurivillius Crystal Structure ............................................................... 75 4 THERMAL TRANSPORT PROPERTIES OF MgO Nd2Zr2O7 ................................. 84 4.1 Introduction ....................................................................................................... 84 4.2 Simulation Set ups ............................................................................................ 85 4.2.1 Thermal Expansion .................................................................................. 85 4.2.2 Thermal Conductivity ............................................................................... 85 4.3 Results .............................................................................................................. 86 4.3.1 Thermal Expansion .................................................................................. 86 4.3.2 Anharmonic Analysis ............................................................................... 87 4.3.3 Thermal Conductivity ............................................................................... 89 4.3.4 Finite Size Analysis ................................................................................. 90 4.3.5 MgO: Thermal Conductivity ..................................................................... 91 4.3.6 NDZ: Thermal Conductivity ..................................................................... 92 4.4 MgO NDZ: Polycrystals and Composite ........................................................... 93 4.4.1 Polycrystal Structure................................................................................ 94 4.4.2 Composite Structure ................................................................................ 96 4.4.3 Interfacial Effect on Thermal Conductivity ............................................... 98 4.4.4 Grain Size Dependence on Thermal Conductivity ................................. 102 4.4.5 Thermal Conductivity of MgO NDZ Composite ..................................... 103 4.5 Discussion ...................................................................................................... 105 5 THERMAL TRASNPORT PROPERTIES OF MgAl2O4 ......................................... 106 5.1 Introduction ..................................................................................................... 106 5.2 Selection of Inverted Structure ........................................................................ 107 5.3 Effects of Inversion on Properties ................................................................... 112 5.3.1 Lattice Parameter .................................................................................. 112 5.3.2 Anti site Formation Energy .................................................................... 114 5.3.3 Bulk Modulus ......................................................................................... 118 5.3.4 Thermal Expansion ................................................................................ 122 5.3.5 Thermal Conductivity ............................................................................. 125 5.4 Conclusion ...................................................................................................... 132 6 THERMAL TRANSPORT PROPERTIES OF Bi4Ti3O12 ........................................ 134 6.1 Introduction: Thermal Barrier Coating ............................................................. 134 6.2 Selection of Potential Parameter .................................................................... 139 6.3 Thermal Expansion ......................................................................................... 140 6.4 Thermal Conductivity ...................................................................................... 146 6.5 Co nclusion and Future Study .......................................................................... 155 7 GENERAL CONCLUSIONS ................................................................................. 156 PAGE 7 7 7.1 MgO NDZ ....................................................................................................... 156 7.2 MgAl2O4 .......................................................................................................... 157 7.3 Bi4Ti3O12 ......................................................................................................... 158 LIST OF REFERENCES ............................................................................................. 160 BIOGRAPHICAL SKETCH .......................................................................................... 175 PAGE 8 8 LIST OF TABLES Table page 1 1 GWP (100 year timehorizon) of greenhouse gases .......................................... 21 1 2 Type s of commercial nuclear reactors ................................................................ 24 2 1 The ZBL parameters ........................................................................................... 53 2 2 Potential parameters for the description of MgO Nd2Zr2O7 system .................... 53 2 3 Short range potential parameters for MgAl2O4 .................................................. 53 2 4 Short range potential parameters for Bi4Ti3O12 .................................................. 54 2 5 Summary of the eight statistical ensembles. ....................................................... 57 3 1 Rocksalt AX crystal structure .............................................................................. 68 3 2 Pyrochlore A2B2X6X crystal structure with origin at B cation .............................. 71 3 3 Fractional coordinates for atomic positions and vacancies for spinel ................. 73 3 4 Various Aurivillius structures for different n values ............................................. 75 3 5 Fractional coordinates for tetragonal structure (I4/mmm) of BIT ......................... 77 3 6 Fractional coordinates for monoclinic structure (B1a1) of BIT ............................ 77 3 7 Historical preview for structures of BIT ............................................................... 78 4 1 Coefficient of thermal expansion for MgO and NDZ ........................................... 87 4.2 Comparison of Debye temperature and Cp ......................................................... 88 5 1 Comparison of thermal expansion and bulk modulus ....................................... 126 6 1 Different forms of layered perovskites .............................................................. 137 6 2 Short range potential parameters for Bi4Ti3O12 ................................................. 139 6 3 Comparison of interaction parameters for BIT .................................................. 140 6 4 Comparison of thermal expansion values ......................................................... 142 6 5 Thermal conductivity variation for all 4 systems ............................................... 154 PAGE 9 9 LIST OF FIGURES Figure page 1 1 Energy demand and supply for the US .............................................................. 15 1 2 Energy dist ribution among major sectors ........................................................... 16 1 3 Primary energy consumption by source and sector ............................................ 16 1 4 Difference in average global temperature from 19611990 ................................ 17 1 5 US greenhouse gas emission ............................................................................. 18 1 6 CO2 concentration in the atmosphere for past 10000 years ............................... 19 1 7 Regions of restructured fuel pellet ...................................................................... 27 2 1 Moore's law for microprocessor .......................................................................... 29 2 2 Power consumption of a microprocessor ............................................................ 30 2 3 Schematic of TBC system .................................................................................. 31 2 4 Three modes of Heat Transfer ............................................................................ 32 2 5 Three phonon processes. ................................................................................... 34 2 6 N and U processes ............................................................................................. 34 2 7 Debye temperature and thermal conductivity (in W/cm K) at 300 K ................... 38 2 8 Temporal and Spatial resolution of simulation methods ..................................... 39 2 9 Schematic representation of a 2 D periodic system. .......................................... 44 2 10 Coulomb energy per ion as a function of cut off in NaCl .................................... 47 2 11 Energy per ion against net system charge ......................................................... 48 2 12 Schematic of dipolar shell of NaCl Bravais Lattice. ............................................ 48 2 13 Comparison of the Coulomb energy between charged and charge neutral ....... 49 2 14 Comparison between the Madelung energy and the approximate ones ............. 50 2 15 Schematic of flash diffusivity measurement and temperature rise curve. ........... 61 PAGE 10 10 2 16 Schematics representation of simulation set up for direct method ..................... 62 3 1 Conventional cubic unit cell of MgO. .................................................................. 66 3 2 Schematics showing three different symmetry operations. ................................. 67 3 3 Pyrochlore sub unit (1/8) and Fluorite unit cell ................................................... 69 3 4 Two types of subcells, that forms the unit cell of pyrochlore ............................... 70 3 5 Cubic spinel unit cell with Mg, Al and O atoms. .................................................. 72 3 6 Unit cell of Bi4Ti3O12 ........................................................................................... 76 3 7 Groupsubgroup graph connecting I4/mmm to B1a1 .......................................... 80 3 8 Octahedra tilt of 6x6x1 supercell of BIT .............................................................. 82 4 1 Simulation set up to simulate thermal conductivity ............................................. 86 4 2 Lattice parameter variation with temperature for MgO and NDZ ....................... 86 4 3 System size effect on thermal conductivity MgO and NDZ ................................. 91 4 4 The rmal conductivity data of MgO ...................................................................... 92 4 5 Thermal conductivity of Nd2Zr2O7 ....................................................................... 93 4 6 MgO polycrystal ~ 3 nm grain size ..................................................................... 94 4 7 Different connectivity patterns for twocomponent composite system ................ 96 4 8 MgO NDZ composite (21 composite) ................................................................ 97 4 9 Comparison of thermal conductivity of MgO ....................................................... 99 4 10 Comparison of thermal conductivity of NDZ ..................................................... 100 4 11 Temperature dependence of the interfacial conductance, G and Kapitza length ................................................................................................................ 101 4 12 Predicted grainsize dependence of the normalized thermal conductivity of MgO and NDZ polycrystals compared with experimental results of YSZ .......... 103 4 13 Com parison of thermal conductivity of 6040 MgO NDZ composite with predicted value ................................................................................................. 104 5 1 Distribution of energy as a function of cati on arrangement ............................... 108 PAGE 11 11 5 2 PDF for normal spinel and comparison of Al Mg PDF .................................... 109 5 3 Comparison of PDFs for inverted structures ..................................................... 110 5 4 Decrease of lattice parameter with inversion .................................................... 113 5 5 Structural energy as a function of inversion for MgAl2O4. ................................. 115 5 6 Cation antisite formation energy as a function of inversion. ............................. 116 5 7 Formation energy as a function of inversion site distance. ............................... 118 5 8 Bulk modulus as a function of inversion ........................................................... 119 5 9 Comparison of normalized bulk modulus with MD study by Chartier ................ 120 5 10 Separating out various effects on bulk modulus ............................................... 121 5 11 Variation of lattice parameter with temperature for i= 0. .................................... 123 5 12 Coefficient of thermal expanion as a func tion of inversion. ............................... 124 5 13 Dependence of the Grneisen coefficient on inversion. ................................... 124 5 14 Variation of thermal conductivity for MgAl2O4 with temperature ....................... 126 5 15 System size analysis for different inversion values for MgAl2O4 ....................... 127 5 16 Dependence of thermal conductivity variation on inversion .............................. 128 5 17 Density effect on thermal conductivity .............................................................. 130 5 18 Combinatoric analysis ...................................................................................... 131 6 1 Simple schematic view of turbofan gas turbine engine ..................................... 134 6 2 Idealized Brayton cycle .................................................................................... 135 6 3 Schematic of a TBC .......................................................................................... 136 6 4 6x6x1 super cell of BIT viewed along x and y axes .......................................... 141 6 5 Coefficient of thermal expansion of BIT along x,y and z axes .......................... 142 6 6 Temperature variation of tilt angles of .............................................................. 144 6 7 Temperature profile of nearest Ti Ti distance in BIT structure .......................... 145 6 8 Temperature profile of Bi Bi cation distance in Bi2O2 layer ............................... 146 PAGE 12 12 6 9 System size analysis for T = 300 K in all the direction ...................................... 147 6 10 Temperature and direction dependence of thermal conductivity for BIT ........... 148 6 11 Thermal conductivity of BIT by experiment ....................................................... 149 6 12 Schematic of method to investigate the density effect on thermal conductivity of BIT ................................................................................................................ 152 6 13 System size and density dependence of thermal conductivity for BIT .............. 153 PAGE 13 13 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THERMAL TRANSPORT IN SELECTED CERAMIC MATERIALS FOR THEIR POTENTIAL APPLICATION AS INERT MATRIX FUEL OR THERMAL BARRIER COATING BY ATOMIC LEVEL SIMULATION By PRIYANK SHUKLA August 2010 Chair: Simon R. Phillpot Major: Materials Science and Engineering Thermal management is a critical issue in nuclear reactor and thermal barrier coating applications. Current research for next gen nuclear fuel is looking into materials to provide uranium free alternate with high thermal conductivity whereas thermal barrier coating research is in need of low thermal conductivit y materials. In this work, an understanding of various structural factors on thermal conductivity using atomic level simulation, is presented Thermal conductivity is a key parameter in the selection of nuclear fuel. Next gen nuclear fuel is suggested as to improve the efficiency of the nuclear fuel and make it environment friendly. Next gen nuclear fuel is called as inert matrix fuel. Thermal conductivity of two potential materials (MgO Nd2Zr2O7:NDZ and MgAl2O4) is characterized using molecular dynamics simulation. In a composite system, interfaces play a dominant role in governing thermal conductivity of the system. To characterize the interface effects, very fine 2D textured grains (<10 nm) are build. Homogeneous interface simulations showed the domina nce PAGE 14 14 of interface effects on grains. Extension to bigger grain sizes showed good compatibility of the simulation data with the experimental dependence of grain size. In addition, MgAl2O4, a spinel system, has been characterized also for its potential applic ation as an inert matrix. Since system undergoes cation anti site inverted state under irradiation, thermal conductivity as a function of inversion is characterized. It has been shown that, thermal conductivity remains invariant under inversion. In the process of characterizing thermal conductivity, thermal expansion, bulk modulus and elastic properties are also illustrated. Another key area for heat management is thermal barrier coating application. Thermal conductivity of a layered perovskite material, B i4Ti3O12 (BIT) belonging to Aurivillius family, is characterized. The effect of layers (Bi2O2 layer and perovskite layer) on the thermal transport of BIT is characterized by changing the mass of the layers under selected scenarios. The findings of the study suggested an effect of disparity between layers on the thermal conductivity of BIT in the through direction but limited role in the other directions PAGE 15 15 CHAPTER 1 BACKGROUND 1 1 Energy Sources Energy is the driving tool for the development of human civilizati on. The United States was self sufficient in energy until the 1950s. Since then, energy consumption has outpaced production, with the result that the deficit in energy continues to increase (Fig.1 1). Figure 11 Energy demand and supply for the US [1] Energy is giv en in the units of Btu, acronym for British thermal unit, which is a traditional energy unit. One Btu is the energy needed to heat up 1 pound of water by 1 degree Fahrenheit. It is approximately equivalent to 1.06 kJ. If the energy deficit continues, we wi ll face an energy crisis in very near future. To overcome this deficit and regain energy independence, energy production needs to be increased and/or usage has to be reduced. Figure 12 shows the energy consumption among various sectors. As it clearly depicts, the industrial sector has leveled off in energy consumption but all other sectors are linearly increasing. This brings us to the question: h ow do different energy sources contribut e towards meeting our demand of energy? PAGE 16 16 Figure 12. Energy distribution among major sectors [1] The contribution from five different energy sources is shown in Fig. 13. It can be seen clearly that carbonemission free energy sources (renewable and nuclear) contribute only 16% of the total share. With coal, petroleum and natural gas as the leading contri butors of the energy sources the biggest concern is: what is the effect on the environment of these energy sources? Figure 13. Primary energy consumption by source and sector [1] PAGE 17 17 1.2 Greenhouse Effect and Global Warming Carbon dioxide (CO2) is the leading contributor to the greenhouse effect and global warming. The greenhouse effect is the absorption of infrared radiation in the Earths environment. In the total absence of greenhouse effect, the Earths surface temperature would be 255 K instead of 288 K [2] The greenhouse effect of current concern is the furth er rise in temperature of Earths surface from mans activities. Figure 14. Difference in average global temperature from 19611990 (best coverage period) [3] Figure 14 shows the trend in the Earths surface temperature from 1850. Over 3000 stations al l over the world recorded monthly temperature. The average is taken and the deviation from the period of 19611990 (best coverage period) is plotted here as the Temperature anomaly. The increase over the past 30 years is the highest temperature rise seen in the past 10,000 years. To keep track of climate changes and these potential environmental and socioeconomic effects, The Intergovernmental Panel on Climate Change (IPCC) was formed by the United Nations Environment Program (UNEP) and the World Meteorological Organization (WMO). In its fourth Assessment Report, it is stated that the planet is warming rapidly due to the increase in greenhouse PAGE 18 18 gases (GHGs) [4]. Gases which act in a simi lar fashion to CO2 and absorb the infrared radiation from the earths surface are collectively referred as GHGs. Although most of the GHGs are naturally present in the atmosphere, human activities have increased their levels manyfold over preindustrial er a levels. This imbalance has caused the rise in global temperature and has given birth to todays biggest challenge: Global Warming. The GHGs can be classified as three types: 1 Natural: CO2, methane (CH4), nitrous oxide (N2O), water vapor and ozone (O3). 2 Man made: Chlorofluorocarbons (CFCs), hydrochlorofluorocarbons (HCFCs), perfluorocarbons (PFCs), hydrofluorocarbons (HFCs), bromofluorocarbons (halons) and sulfur hexafluoride (SF6). 3 Indirect: Carbon monoxide (CO), nitrogen oxides (NOx), aerosols and nonm ethane volatile organic compounds. They influence the formation of other forms of GHGs and contribute in global warming indirectly. Figure 15. US greenhouse gas emission [5] PAGE 19 19 Even though contributions of these other GHGs are individually less than that of CO2, their total collective contribution is of about the same magnitude. That is why their contribution in global warming is termed as effective doubling. Figure 15 captures the contribution of various GHGs since 1990. 1.2.1 Carbon Dioxide (CO2) Emission Being the largest contributor among GHGs, CO2 becomes of utmost importance to understand the emission of CO2. In the equilibrium carbon cycle there will be no buildup of CO2 in the environment and thus no global temperature rise. Because of technical advancements, dependence on energy has increased and consequently the emission of CO2 has risen. Figure 16 shows the concentration of CO2 in the atmosphere for the past 10,000 years. Figure 16. CO2 concentration in the atmosphere for past 10000 years [6] As shown in Fig. 16, the concentration of CO2 has increased markedly since the beginning of industrialization. The right axis label of Fig. 16: Radiative Forcing is a measure of the difference in net irradiance at the boundary of troposphere and the PAGE 20 20 stratosphere. In simple terms, positive radiativ e forcing means a warming effect on the climate. By comparing Figs. 1 4 and 16, a linear relation between CO2 emission and global temperature is found. All GHGs have different warming effects (radiative forcing) on the climate. A common metric system is used to measure the effect on climate by all these GHGs and is quantified as the global warming potential (GWP). 1.2.2 Global Warming Potential The GWP of any GHG is a quantified measure of its global time averaged relative radiative forcing impact. Conventionally the GWP of CO2 is taken as 1. The GWP of any gas is defined as the equivalent mass of CO2 to produce the same effect in global warming A GWP is always specified over a timeperiod. A comparison between different GHGs is shown in Table11. 1.3 Carbon EmissionFree Energy In order to build a safer environment for coming generations, global warming needs to be eliminated. To achieve this goal, carbonfree energy is required. As discussed, carbon free energy sources form a minor 16% share of the total energy resources. There are three ways to achieve the carbonfree environment: 1 Energy conservation by efficient usage 2 CO2 separation and sequestration 3 Advancement of carbonfree energy sources (nuclear, solar, wind etc.) Efficient usage of energy does play an important role in contributing towards carbon free environment by lowering the gap between energy demand and supply. That, in effect, low ers the consumption of fossil fuels. But in the long run, it does not alleviate the global warming problem. PAGE 21 21 Table 11. GWP (100 year timehorizon) of greenhouse gases [5] Gas Chemical formula GWP Atmospheric lifetime (Years) Carbon dioxide CO 2 1 50 200 Methane CH 4 21 12 3 HFC 152a C 2 H 4 F 2 140 1.5 Nitrous Oxide N 2 O 310 120 HFC 32 CH 2 F 2 650 5.6 HFC 134a CH 2 FCF 3 1,300 14.6 HFC 43 10 mee C 5 H 2 F 10 1,300 17.1 HFC 125 C 2 HF 5 2,800 32.6 HFC 227ea C 3 HF 7 2,900 36.5 HFC 143a C 2 H 3 F 3 3,800 48.3 HFC 236fa C 3 H 2 F 6 6,300 209 Perfluoromethane CF 4 6,500 50,000 Perfluorobutane C 4 F 10 7,000 2,600 Perfluorohexane C 6 F 14 7,400 3,200 Perfluoroethane C 2 F 6 9,200 10,000 HFC 23 CHF 3 11,700 264 Sulphe r hexafluoride SF 6 23,900 3,200 CO2 separation and sequestration has the potential to be very efficient in abating global warming problem but has serious challenges. Turner [7] compared the global warmi ng and CO2 separation and sequestration solution to a smoker who is exercising. Even though the smoker is exercising, it will be of no effect if he continues to smoke. Similarly, CO2 storage and sequestration cannot be the final solution. Besides, the PAGE 22 22 sequ estration process is very expensive and the risk is associated with the failure of CO2 storage pose a great danger [7] This limits the prospect of using CO2 separation and sequestration technology as a solution to global warming. That makes the third alternative to be the most attractive. A very good review of the ways to solve the current energy problem is given by Lior [8]. The following steps are suggested for the future energy solution: 1 The immediate challenge is to meet the energy demand. Thus, fossi l fuel will still be used until other energy sources become large shareholders in the energy resource family. 2 CO2 separation and sequestration methods are needed to accompany the CO2 releasing energy sources. 3 Build more combined cycle power plants. 4 Where wind is economically available, wind power generation needs to be deployed. 5 Develop less expensive and more efficient solar cells. 6 Even with the issue of long term radioactive waste storage and safety issues, nuclear power plants need to be built and alternate nuclear fuels need to be developed. It is evident that all carbonfree energy sources will play a vital role in resolving the energy crisis/ global warming problem. Nuclear energy is clearly recognized by scientists and technologists as environment friendly [9]. This view is now also shared by some environmentalists [10] This view is further strengthened by the fact that according to PAGE 23 23 ONUIPCC AR4 Synthesis Report, nuclear energy is considered as an effective GHG mitigation option [1] 1.4 Nuclear Energy Nuclear energy is a way to generate energy from the splitting of nucleus. Heavier atoms release more energy in the process of the splitting of the nucleus. Controlled fission react ion forms the basis of nuclear energy. UO2 is the most common nuclear fuel used in nuclear plants in USA. Naturally occurring uranium contains 99.3% 238U, 0.7% 235U, and < 0.01% 234U. 235U is fissionable and most of the current reactors use enriched uraniu m, which contains ~4% of 235U in the fuel. Fissile 235U and fertile 238U both are used in the fission process. Slow moving neutrons are captured by 235U to start fission reaction. Whereas, fast moving neutrons are absorbed by 238U and converts into 239Pu, which is fissionable by slow as well as fast moving neutrons. Various kinds of nuclear reactors are used worldwide to produce the nuclear energy. Some of them are shown in Table 12. As shown in Table 12, enriched UO2 is the main nuclear fuel in US. The nuclear fuel cycle starts from the mining of uranium and ends at the burial of spent fuel. The whole process contributes towards radioactive waste. During mining and enrichment, waste of 238U, 226Ra and 222Rn and 230Th is generated. During the fission reaction, neutron capture and difference decay schemes leads to the formation of transuranic nuclides (atomic number >92), including 239Pu, 237Np, 241Am and 244Cm. In addition to the nuclear reactors, dismantling of nuclear warheads also increases the stoc kpile of Pu. The presence of these transuranic nuclides make the spent fuel radioactive and a period of more than 100,000 years is required to bring down the radioactivity level of this spent fuel to the level of mined uranium. Handling and keeping safe the Pu PAGE 24 24 stockpile and transuranic nuclides poses a great environment challenge. An innovative fuel approach, termed inert matrix fuel (uranium free), has been suggested as a way to alleviate the problem by embedding Pu and minor actinides in a material matrix of nonfissile or fertile material and incinerate in nuclear reactors. The stable matrix, after burning in a reactor will be directly disposed of in a once through fuel cycle concept. Table 12. Types of commercial nuclear reactors [11] Reactor type Main fuel Main countries Pressurized Water Reactor (PWR) Enriched UO 2 US, France, Japan, Russia, China Boiling Water Reactor (BWR) Enriched UO 2 US, Japan, Sweden Pressuried Heavy Water Reactor CANDU (PHWR) Natural UO 2 Canada Gas cooled Reactor (AGR & Magnox) Natural U (metal), enriched UO2 UK Light Water Graphite Reactor (RBMK) Enriched UO 2 Russia Fast Neutron Reactor PuO 2 and UO 2 (MOX) Japan, France, Russia Some key parameters in the selection of the inert matrix fuel (IMF) include high temperature stability, good irradiation behavior, high thermal conductivity, good PAGE 25 25 mechanical properties, and low neutron capture and cross sections [12, 13] Based on these criterions, MgAl2O4, ZrO2MgO and MgO Nd2Zr2O7 (NDZ), among others have been suggested as possible inert matrix materials [14 16] 1.5 Thermal Conductivity: An Important Property A nuclear fuel pellet experiences a temperature drop of ~ 800 K from the centerline of the fuel pellet to the edge. This large temperature difference is due to the low thermal conductivity of UO2. At high temperature, center region of the pellet expands more than the rim. Because of the thermal stress thus formed, the microstructure of the grain changes and the fuel cracks and degrades. Efficient removal of heat from the fuel will maintain a more uniform temperature across the fuel and increase the fuel life time. This makes high thermal conductivity a desirable property for a nuclear fuel. The Thermal conductivity of materials depends on various parameters including but not limited to bulk modulus, density, and microstructure. The thermal conductivity is also an important parameter in the selection of thermal barrier coating (TBC) materials. In contrast to nuclear fuel, TBC a pplications demand low thermal conductivity materials. A detailed description of TBCs and their applications are given in Chapter 6. Being important for both, nuclear fuel and TBC materials, thermal conductivity becomes an important material property for investigation. That raises an important question: What controls thermal conductivity ? 1.6 Motivation for T h is Work A good understanding of the physics governing thermal conductivity in nonmetals is essential to answer the question, asked in Sec. 1.5. P honons are the heat carriers in nonmetals/ceramic materials and are defined as lattice vibrations. Analysis of thermal PAGE 26 26 transport is performed in phonon space (momentum space or frequency space). While phonon space is convenient for analysis, design and sy nthesis of the materials is performed in real space. It is difficult to move conceptually from phonon space to real space. For example, the position of the atoms in the unit cell and the relaxation times of the associated phonon modes are not related to each other I an obvious way. Molecular dynamics (MD) simulations are a suitable tool for the analysis of thermal conductivity at a finite temperature and for the bridging of real and phonon space analysis. Having total control over the system, simulation pr ovides easy access to capture the effects of structural features affecting the thermal conductivity. In the following chapters, the effects of interface, structure and mass on materials thermal transport properties are presented and analyzed in detail. 1.7 Objective of This Work This work is primarily focused on understanding the thermoelastic properties of MgO, Nd2Zr2O7 (NDZ), MgAl2O4 and Bi4Ti3O12 (BIT). The thermal properties of MgO NDZ and MgAl2O4 are studied in the light of their potential application as IMF systems, whereas BIT is studied for its potential application as TBC material. The effects of interfaces on the thermal properties of MgO NDZ system are very important to understand. During burn up, large temperature and steeper temperature gradient causes the morphological changes of the nuclear fuel during power operation, as shown in Fig. 17. Although fuel pellet starts with all solid, a sizable void develops in the center. The void forms by the movement of the porosity to the center. Immediately adjacent to void region, columnar grains are formed. The radial boundaries of the columnar grains are formed by the trails of the pores and/or by the movement of fissiongas bubbles. Moving PAGE 27 27 outwar d from the columnar region, a region of large textured grains is formed (Equiaxed grains) [17] Figure 17. Regions of restructured fuel pellet [17] The outermost part of the pellet experiences more radiation damage and contains more fission damage than the bulk due to neutron resonance capture by 238U in this rim region to form fissile 239Pu. This leads to the formation of smaller grain sizes ( ~ 02 m) from the as fabricated grain sizes of ~ 10 m [18] This effect is termed as rim effect and changes thermoelastic properties of the fuel. These morphological changes of the fuel, makes it essential to characterize the interface effects. Similar to experimental investigation of the interfacial effects on thermal transport for YSZ [19] MgO NDZ composite system is investigated. To characterize the effect of interfaces for MgO NDZ system, single crystal and polycrystals MgO and NDZ material system results are used. Thermal and elastic properties of another candidate inert matrix material, MgAl2O4 are also characterized. The ability of MgAl2O4 to resist radiation damage comes from its As fabricated Equiaxed grains Columnar grains PAGE 28 28 ability to withstand cation anti site disorder (inversion) without structural damage. Thus characterization of the thermal and elastic properties of MgAl2O4 as a function of inversion becomes very important for the potential application as an IMF matrix. In this work, effects of inversion on thermal as well as elastic properties are investigated. Another material, Bi4Ti3O12 (BIT), is characterized for its potential application as thermal barrier coating (TBC) material. The anisotropic structure of BIT is determined; Intrinsic density difference between layers has been identified as the reason of its low thermal conductivity [20] In this work, characterization of the effect of the density difference on thermal conductivity is extended. 1.8 Organization of Dissertation Chapter 2 lays out the foundation of thermal conductivity theory for ceramic materials and how it is implemented in molecular dynamics simulation. It covers important aspects of simulation and thermal conductivity. Chapter 3 contains crystallographic information of the materials discussed in this dissertation. Chapters 4, 5 and 6 focus on thermal transport properties of MgO Nd2Zr2O7, MgAl2O4 and Bi4Ti3O12. Chapter 7 summarizes the results and the impact of this work. PAGE 29 29 CHAPTER 2 THERMAL TRASNPORT AND SIMULATITON M ETHODOLOGY 2.1 Basics of Thermal Transport In many aspects of technologi cal developments, better management of heat is a key for improvement. In all sections of technology, heat flow is in the heart of many systems. Control over heat flow is necessary for the performance, life time and reliability of the devices. Performance, reliability and lifetim e of electronic devices depend greatly on the efficiency of heat management. Intel cofounder Gordon Moore made the prediction in 1965 that number of transistors on a chip will double in ever y two years. Figure 2 1 captures the development in logic circuit industry. With each year, numbers of transistors are increasing while chip size decreases. This evolution corresponds to increasing energy density and the need for better heat management. Figure 21 Moore's law for microprocessor [21] PAGE 30 30 The feature sizes in microelectronics is decreasing very fast and even outpacing Moores law. Already by 2008, 35 nm Logic technology has been achieved [22] With the increas ing number of transistors on an integrated circuit and the decreasing size, p ower density increases. Figure 2 2 shows the implication of Mo ores law on power consumption i n microprocessor [23] Figure 22 clearly suggests that one of the key issues is proper channeling of heat in the microprocessor. Figure 22. Power consumption of a microprocessor dashed line represents Moore's Law [23] Another area in which heat management is vital is nuclear energy. A nuclear fuel pellet experiences a temperature range of ~800 K from fuel pellet center to the rim within a distance of ~0. 5 cm. Even though the heat generated during fission reaction inside the fuel is released to the coolant, this large temperature gradient often le ads to structural deterioration of the fuel material. The high temperature and large temperature gradient causes physical and chemical changes in the fuel pin that often result in the deformation and cracking of the fuel pellet and possible interaction with the coolant [24] To release heat from the fuel, higher thermal conductivity than current nuclear fuel (UO2) is a desirable property of nuclear fuel. Therefore understanding heat transport mechanisms is important for the next generation nuclear fuel: IMF PAGE 31 31 Heat management is critical issue for the thermal barrier coatings (TBC) also Research is primarily focused on finding low thermal conductivity materials for the application of TBC. TBCs act as a thermal barrier between hot gas stream and engine component thus reducing thermally activated oxidation and in effect delaying failure. TBC also increases the lifetime of the part by reducing thermal fatigue [25] Figure 23 captures the temperature profile across the gas turbine engine and TBC. Figure 23. Schematic of TBC system [25] All these examples points to the importance of heat manag ement. A better understanding of thermal conductivity provides a greater control on materials. To capture the effects on thermal conductivity in ceramics, atomic level simulation is used. After gaining insight into the mechanism s of thermal transport, bett er selection of materials will be possible. M echanisms of heat transfer in solids and the simulation methods used to capture the heat transport mechanisms will be discussed in detail in the following sections PAGE 32 32 2.2 Modes of Heat Transport There are three mo des of heat transfer relevant to materials: conduction, radiation and convection (Fig. 24). Conduction heat transfer involves heat carriers such as electrons, phonons (lattice vibrations); radiation involves electromagnetic waves and does not need any medium for heat transfer, convection involves movements of mass within fluids. Convection does not occur in solids, so this will not be discussed further. Figure 24 Three modes of Heat Transfer [26] In looking at heat transfer, it is useful to place solid materials in three distinct categories: metals, semiconductors and insulators. The major heat carriers in metals are electrons. Electrons in metals are very mobile and are not held tightl y by the nucleus. Thus metals are generally described as having a communal sea of valence electrons. For metals the electrical conductivity ( ) and thermal conductivity ( ) are related by the famous WiedemannFranz Law [27] : LT (2 1) PAGE 33 33 T is temperature and L, a constant called the Lorentz number, is given by 2 8 2 210 44 2 3 K W e k LB (2 2) W here, kB is Boltzmann constant and e is the electron charge. This empirical law states that / has same value for all the metals at the same temperature. The proportionality of / on temperature was discovered by Danish mathematician and physicist Ludvig Lorenz in 1872. While this law generally holds, L is not exactly same for all the metals. Rosenberg stated that this law holds true at low and high temperatures but fails at intermediate temperatures [28] In semiconductors and insulators, phonons (lattice vibrations) are the major heat carriers. This mechanism will be discussed in detail in this work. 2.3 Heat Transport in Nonmetallic Solids Phonons are the major heat carriers in nonmetals. The following discussion is based on standard presentations from various solid state texts [29 32] Heat transport in nonmeta llic solids takes place through lattice vibrations. These are termed as phonons and are the major heat carriers in non metals. H armonic potential treatment of phonons gives zer o thermal expansion and infinite thermal conductivity of the material. Since materials do have finite value of thermal conductivity and nonzero thermal expansion, anharmonic terms are included in the discussion of phonons and heat transport. These anharmonic terms are labeled as threeph onon (3rd order term in interaction potential) and four phonon (4th order term in potential) processes. Other higher order terms are also possible but contribution to thermal conductivity is mostly dominated by threephonon processes. During heat transport s PAGE 34 34 two kinds of threephonon process es may occur, which are shown in Fig. 2 5 and are described below: A) Figure 25. Threephonon processes A) creation of one phonon from 2 phonons B) creation of two phonons from one phonon. 1 Annihilation of two phonons with the creation of one phonon (Fig. 2 5A) 12) 3), k1+k2 k3 2 Annihilation of one phone with the creation of two phonons (Fig. 2 5B) 1) 23), k1 k2+k3 Both of the three phonons processes shown in Fig. 2.5 conserve momentum and are called normal (N) processes. In N process, the total momentum is conserved and thus the phonon distribution is not changed. Another three phonon process is also possible in which momentum is not conserved, as shown in Fig. 26. Figure 26. N and U processes, k1, k2, k3, and k3 represent phononmomenta and G reciprocal vector [33] k 1 k 2 k 3 k 1 k 2 k 3 B) PAGE 35 35 Figur e 2 6 shows both the N and U processes. G is a reciprocal lattice vector and based on the value of G, two processes are possible: normal (N) for G =0 and umklapp (U) for G The s haded region in Fig. 2 6 is first Brillouin zone. In the N process, as expect ed phonon momenta lies in the first Brillouin zone and so no further operation is needed. U process es do need an additional operation of the addition of reciprocal vector G to bring back the phonon momenta to first Brillouin zone. The N processes contribute to thermal conductivity indirectly as explained in detail by Callaway [34] The dominant contribution to the heat resistance comes from U processes, i.e. momentum nonconserving processes. There are a number of theoretical approaches to understand phonon conductivity in solids, including: 1 Relaxation time approach 2 Variation al approach 3 Greens function approach (density matrix approach) The relaxation time and variation approaches assume that the thermal conductivity can be calculated from the linearized Boltzmann equation, whereas the Greens function approach takes the route of quantum statistics. A very good discussion of these methods is given by Srivastava [32] From the kinetic theory of gases, the thermal conductivity of phonon gas can be expressed as Cv 3 1 (2 3) w here C is the specific heat, v is group velocity and is mean free path of the phonons. In a phonon gas, C can be approximated well by using the Debye model, v can be taken as the speed of sound; the real problem is in the calculation of PAGE 36 36 The specific heat is defined by the amount of heat required to raise the temperature by 1 K per unit mass. It is one of the easiest thermodynamic quantities to meas ure experimentally. Depending on which system parameter is kept constant the specific heat can be of two types: Cp for constant pressure and Cv for constant volume. Since for a solid, it is experimentally unrealistic to keep volume constant while changing temperature, Cp is of more experimental relevance. There is a simple relationship between Cv and Cp for solids [35] : T V PK TV C C2 (2 4) where is coefficient of thermal expansion and TK is isothermal compressibility. At high temperatures, all solids obey Dulong Petit result for the heat capacity, which gives the value as: B A Vk N C 3 (2 5) where NA is Avogadros number and kB is the Boltzmann constant. This gives fairly good value at high temperatures but is qualitatively incorrect at lower temperatures; in particular T= 0 K the specific heat value goes to zero. Einstein proposed a crude but simple approximation that all vibrations have the same frequency and are set to the valueE. The heat capacity then can be given as: 2 21 exp exp 3 T k T k T k NZk CB E B E B E B V (2 6) Equation 26 is known as the Einstein model for heat capacity. It correctly converges to z ero at 0 K but gives the wrong temperature dependence, decreasing faster than experiment near 0 K. Debye proposed a model in 1912 that overcame the problem of the Einstein model. Debye approximated a linear relati on for phonon PAGE 37 37 = ck and treated phonons in a similar fashion as a particle in a box. D. Density of states under the Debye approximation is defined as: D D DV 0 23 2 2 (2 7) w D is determined by the number density of atoms n and given as: nD 3 22 (2 8) The specific heat in Debye model is given as: d T T n NCD VD 3 33 0 (2 9) At low temperatures Eq. 2 9 become 3 45 12 D B VT Nk C where D is Debye temperature, defined as 3 1 26 V N k c kB B D D At high temperatures, it reaches the Dulong Petit limit of 3NAkB T he Debye temperature is the temperature above which material behaves classi cally and quantum effects to thermal conduction are negligible. As explained above, Debye model imposes a condition as maximum allowable phonon frequency to achieve finite number of modes in a solid (Eq. 28). That maximum frequency corresponds to the Debye temperature. Figure 27 shows the Debye temperature for various materials. The simplicity or complexity of Eq. 23 lies in the understanding of phonon mean free path. Based on the phonon model (Debye or Einstein) the PAGE 38 38 temperature dependence of thermal conductivity can be easily described. At high temperatures, the specific heat of a solid is a constant, given by the Dulong Petit law (Eq. 2 5). The group velocity of phonons in a solid has a weak temperature dependence [36] Although the exact solution is not known, Peierls showed phonon mean free path has T 1 dependence at high temperatures [37] More sophisticated analysis has shown that temper ature dependence is not T 1 but nT 1 where n lies between 1 and 2 and confirmed by experiments [29, 38] Figure 27. Debye temperature and thermal conductivity (in W/cm K) at 300 K for selected elements [30] At low temperatures, the phonon mean free path becomes extremely long and thermal conductivity is limited by the system size, because scattering is dominated by the surfaces. In low temperature limit, the temperature dependence of thermal conductivity comes from the specific heat B ecause the specific heat shows 3T PAGE 39 39 dependence, the thermal conductivity does also. In the intermediate temperature range, Peierls showed bT TD n Dexp ~ where n, and b can be determined by fitting to experimental results. Typical values of nb lie in the range of 1535. 2.4 Simulation Methods There are number of ways to simulate materials. Based on the desired properties an appropriate simulation method can be selected based on the length and time scale. Figure 28 captures different simulation methods based on time and length scale s. Figure 28. Temporal and Spatial resolution of simulation methods [39] For example, to investigate the electronic structure of a material, quantum mechanical simulation methods (density functio nal theory and HartreeFock) are appropriate [40] ; to simulate a material of the size of m m, finite element methods are appropriate [41] With current technological improvements, m size length system have been simulated using molecular dynamics (MD) simulation with trillion atoms [42] In this PAGE 40 40 case the limiting factor is the huge amount of data and its analysis a nd the possible time devotion for the simulation. A 10 ps MD simulation of trillion atoms took a couple of days to finish and 40 TB hard disk space to store position, velocity and force data [42] MD simulation is deterministic in nature and is a useful tool in the study of nonequilibrium processes at the atomic level. Atoms are the basic units of MD simulation and are generally treated as having no internal electronic structure. Each atom has a specified mass and charge. It is classical in nature and every atom follows Newtons equation of motion (Eq. 210). 2 2dt r d m F (2 10) The key physical input required for MD simulations is the interatomic potential. Since there is no interatomic potential that can exactly describe all the properties of any material, MD simulations can capture the properties of material only with limited f idelity. 2.5 Molecular Dynamics Basics As explained in the previous section, the trajectory of each atom is calculated in MD simulation using Newtons equation of motion. Based on the trajectory as a function of time, the average values of properties can be determined. For any atom i, the equation of motion can be written as follows: i i i iU dt r d m F 2 2 (2 11) 2.6 Integration Algorithms There are various integration algorithms [43, 44] All the algorithms expand the position (r), velocity (v) and acceleration (a) into Taylor series around t he time t: PAGE 41 41 ... ) ( ) ( ) ( ... ) ( 2 1 ) ( ) ( ) ( ... ) ( 2 1 ) ( ) ( ) (. 2 .. 2 .. t t a t a t t a t t v t t a t v t t v t t r t t v t r t t r (2 12) The Verlet Algorithm : The Verlet algorithm uses position and acceleration at time t and t t to calculate position at t+ t [45] Because of its simplicity and low memory usage, it is a very common algorithm. According to Verlet algorithm, the position at t+ t is given by 22 t t a t t r t r t t r (2 13) The LeapFrog Algorithm : The leapfrog algorithm is a modified version of Verlet algorithm. It computes velocities at half timestep intervals and uses these velocities to calculate positions at time t+ t t t t v t r t t r 2 (2 14) The Beemans Algorithm : The position prediction is similar to the Verlet algorithm but velocities are determined more precisely. t t t a t t a t t t a t v t t v t t t a t t a t t v t r t t r 6 1 6 5 3 1 6 1 3 22 2 (2 15) The Predictor Corrector Algorithm : The Beemans algorithm is modified into the predictor corrector algori thm. PAGE 42 42 t t t a t t a t t t a t v t t v t t t a tt a t v t t v t t t a t t a t t v t r t trcorrected predicted predicted 6 1 6 5 3 1 2 1 2 3 6 1 322 2 (2 16) T he 5th order Gear predictor corrector method [44] is used in this work f or numerical integration. The predicted new position of an atom is calculated using 5th order derivative of current position. t d t t d t d t t c t t c t d t t c t t b t t b t d t t c t t b t t a t t a t d t t c t t b t t a t t v t t v t d t t c t t b t t a t t v t t r t t rp p p p p p 2 3 2 4 3 2 5 4 3 22 1 6 1 2 1 24 1 6 1 2 1 120 1 24 1 6 1 2 1 (2 17) Where r v and a are the position, velocity, acceleration of each atom respectively. b c and d are third, fourth and fifth derivative s of the position with time of each atom respectively T he superscript p corresponds to the predicted values. Using these predicted positions, forces on atoms are calculated thence the corrected accelerations. The difference between the corrected and predicted acceleration gives the error from the prediction step that is used to calculate the trajectory : t t a t t a t t ap c (2 18) PAGE 43 43 t t a c t t d t t d t t a c t t c t t c tt a c t t b t t b t t a c t t a t t a t t ac t t v t t v t t a c t t r t t rp c p c p c p c p c p c 5 4 3 2 1 0 (2 19) The coefficients for a fifth order predictor corrector are c0 = 3 / 16 c1 = 251 / 3 6 0 c2 = 1, c3 = 11/ 18, c4 = 1/6 and c5 = 1/60 [44, 46] These corrected values are used as an input for the next simulation step. The procedure is repeated for all the atoms to determine their trajectories for the duration of the simulation. The time step is an important factor in sampling the atom trajectory during simulation. The tim e step must be smaller than the fastest vibrational frequency in the system. Typical time steps for MD simulations are of the order of 1fs (1015 s). This constraint comes from the fact that higher time step leads to errors in the trajectory and unreliable simulation results. Smaller time steps might capture more accurate trajectory but will limit the total simulation time. There are several ways developed to increase the time step and maintain the accuracy level, e.g. multiple time scale [47, 48] and stochastic dynamics [49] 2.7 Periodic Boundary Condition The application of periodic boundary conditions (PBC) is a very useful technique to mimic large system by simulating smaller system s. Depending upon the requirements, PBC can be applied to replicate the system in one, two or three dimensions. When applied in three dimensions, the system is extended to mimic bulk material. A schematic of PBC is shown in Fig. 2 9 Original simulation cell (cell # 5) is in the center of the PAGE 44 44 simulation system and other 8 cells surrounding it are the outcome of PBC. Arrows represent the movement of atom s. During simulation, all the image cells follow the actual cell. Therefore, during simulation if any atom moves in the actual cell, its periodic images follow its motion. For example, if atom moves from cell 5 to cell 6. All its images, also move to the next cell and in this process, atom from cell 4 enters cell 5. That way the total numbers of atoms in the c ell are always conserved. Figure 29. Schematic representation of a 2D periodic system. Simulation is performed with the highlighted box. Arrows shows how atoms are affected under PBC [50] When PBC is applied, one needs to be careful as it can lead to unphysical interactions (self int eraction: between atom and its image). A l l the interactions are taken to be zero after a specified distance (known as cut off). Typically, to avoid self interactions, system size needs to be at least twice the cut off. 1 2 3 4 5 6 7 8 9 PAGE 45 45 2.8 Interatomic Interaction s Newtons equation of motion forms the basis of MD simulation and the interaction potential is the most important input for MD simulation. The quality of any simulation depends merely on the quality of the interatomic potential. In ionic systems, atoms are made of charged ions and interactions between the ions can be separated into a long range, Coulombic part, and a short range part. 2.8.1 Longr ange Interactions The long range interaction between two charged species is given by Coulombs law: ij j i Coulrq q U (2 20) Where, qi and qj are the charges on atom i and j respectively, and rij is the distance between them. For a crystal structure with N ions, the total Coulombic energy is 1 1 N i N i j ij j i Coulr q q U (2 21) The 1/r dependence of the energy makes the summation conditionally convergent. Conventionally, the Ewald summation is used to compute the long range electrostatic interactions [51] A more recent direct summat ion method is used to compute the summation [52] Both methods are described below. Ewald Sum : The Ewald method is the standard technique to calculate electrostatic interaction in a periodic system. It was introduced by P. Ewald in 1921 [51] In this technique, Eq. 219 is rewritten as two rapidly converging parts plus a constant term. 0U U U Um r Ewald (2 22) PAGE 46 46 Where Ur is the short range term that sums quickly in real space, Um is a long range term that sums quickly in reciprocal (Fourier) space and U0 is the constant (self) term. These terms are given as: N i i N ji m j i j i m N j i ij ij j i rq U m r r m i m q q V U r rerfc q q U1 2 0 ', 0 2 2 ,2 exp 2 1 2 1 (2 23) V is the volume of the simulation cell, m = (l,j,k) is a reciprocalspace vector. There are a few disadvantages of the Ewald sum. First ly it is not straightforward and fairly complicated to code, secondly it scales as O(N3/2) [53] Direct Sum: A powerful alternative to the Ewald summation was developed by Wolf et al. [52] and is referred to as the direct summation. It is inspired by the fact that the Coulombic interactions in ionic crystals are short ranged, as demonstrated by both, theories [30, 54] and simulations [55, 56] The only factor not allowing Coulomb sum to converge is the lack of charge neutrality over spherical crystal lattice shell. This can be best illustrated with the help of an example. Consider a simple rocksalt structure of NaCl. The NaCl unit cell can be viewed as interpenetrating FCC lattices of anions and cations with half unit cell length displacement. The r ocksalt structure will be discussed in detail in C hapter 3 As different cut off distances are set for the potential s (cut off distance is defined as the distance beyond which atoms do not interact Eq 2 26), the total system will be either cation deficient or anion deficient or r arely charge neutral. Figure 210 shows the variation of energy with cut off radius. PAGE 47 47 Figure 210. Coulomb energy per ion as a function of cut off in NaCl [52] Figure 210 straightforwardly captures the nonconvergence of the Coulomb energy. The dashed line in the Fig. 210 is the actual value of the C oulomb energy for Rc of NaCl which is also known as the Madelung energy. The arrow in the Fig. 210 shows the points where the energy value converges to the actual value, however no relationship can be deduced between the cut off (Rc) and convergence. When the same energy graph is viewed against the net system charge (Fig. 211), a clear trend is captur ed for the convergence. Figure 211 renders the convergence criterion: charge neutral system. Fluctuations in Fig. 210 around the dashed line show that NaCl system never achieves charge neutrality when it is spherically terminated. To overcome this charg e neutrality problem, initially a dipolar solution was suggested by Wolf [57] A schematic is shown in Fig. 2 12, where a NaCl crystal is built of Na+Cldipoles. In the evaluation of the Coulomb energy for a truncated crystal shell, counter dipoles are created at the surface for each dipole present in the shell and the expression for the Coulomb energy is given below: PAGE 48 48 s sr s r sr b r N b q E4 5 2cos 5 1 8 7 (2 24) Figure 211. Energy per ion against net system charge [52 ] Figure 212. Schematic of dipolar shell of NaCl Bravais Lattice. Adapted from Ref. [57] This dipole approach solves the probl em of charge neutrality but leaves the system polarized. To overcome the polarization effect in the energy calculation, the polarization energy per unit volume is subtracted to achieve the correct Coulomb energy PAGE 49 49 value. The whole problem of polarization energy can be avoided altogether, if the basic unit is taken without a dipole moment. For the NaCl crystal system, instead of Na+Clas a basi c unit, a sub unit cell consisting of 4 Na+ and 4 Clions, is taken as a basis. Charge neutrality is achieved by adding countercharges at the terminated sphere. If q (Rc) is the net charge within the cutoff sphere of radius Rc for ion i the Coulomb energy can be written as: c c i R r j i ij j i c Mad iR R q q r q q R Ec ij (2 25) Figure 213. Comparison of the C oulom b energy between charged (open circle) and charge neutral (filled circle) systems as a function of Rc. Adapted from reference [52] A comparison between the charge neutral crystal shell and charged one is shown in Fig. 2 13. Figure 213 shows the effect of charge neutrality on the energy calculation and all the fluctuations of the charged system go away. Comparison with real Madelung energy is shown in Fig. 214 and it shows a damping effect in the energy calculation by PAGE 50 50 the direct sum. Energy converges by 1/r in the direct sum as seen in Fig. 214. This 1/r damping effect on energy makes the potential practically very slow and useless for practical uses. That calls for some improvement in the potential. Figure 214. Comparison between the Madelung energy (open square) and the approximate ones (filled circle) as a function of Rc. Adapted from referenc e [52] Before suggesting improvement to iron out the damping effect in energy, potential energy in itself needs to be rectified for the direct sum. By terminating energy calculation at Rc, forces and stress es beyond Rc become zero which makes these otherwise physically continuous functions discontinuous. To retain the continuity of energy, forces and stress, a shifted potential is used instead of the actual potential Form of this shifte d potential is given in Eq. 226 ij j i R r ij j i ij C shr q q r q q r Vc ijlim (2 26) PAGE 51 51 By introducing damping term in the shifted potential (Eq. 226), fluctuations in the Madelung energy calculation can be flattened out at smaller Rc. Damping is intr oduced in the potential by error function because of its i) simplicity and ii) similarity to the Ewald summation method. After all these modifications, Eq. 222 is expressed as follows: N i i c c R r j i ij ij j i R r ij ij j i N i c Mad toiq R R erfc r r erfc q q r r erfc q q R Ec ij c ij2 12 lim 2 1 (2 27) Where is a damping constant. It has been shown for NaCl that the Madelung energy is almost independent of all the way up to 1.5/a for Rc=1.5a. Here a is the lattice parameter of NaCl [52] This method also works very well with disordered systems including liquids and defects. 2.8.2 Short range Interactions The need for short range interaction co mes from the 1/r dependence of the C oulomb energy. In the limit r approaches zero, the long range interactions diver ge which certainly is not the case in a real system. Without any other interactions, two oppositely charged ions would fall into each other and the system energy will become infinite. Various functional forms of short range interactions have been developed for years, based on the material and properties under investigation. For example, to define characteristics of inert gases or neutral atoms or molecules LennardJones [58] potential is used, for semiconductors Stillinger Weber [59] potential is used The Buckingham potential [60] is used in this work which is widely used to describe ionic materials. The functional form of the Buckingham potential is given by, 6expij ij ij ij ij ij Buckr C r A r V (2 28) PAGE 52 52 w here rij is the distance between ions i and j and Aij, ij and Cij are ion pair (i,j) dependent param eters. The first term of Eq 2 28, embeds the quantum mechanical exchange interaction effect into classical description of interaction. Exchange interaction in simple terms, takes care of the fact that no two electrons can have th e same four quantum numbers (n,l,ml,ms). Second term of Eq. 228 describes the Van der Waals attractive interaction due to dipole interaction. These empirical potentials are developed by fitting to the bulk material properties. That limits its use under ex treme conditions such as high energy collisions. A more realistic model to capture the effect at very short range is described by Ziegler, Biersak and Littmark (ZBL) [61] The ZBL interaction is a screened electrostatic potential of the nucleonnucleon i nteraction: r r e Z Z r V2 1) ( (2 29) w here Z1 and Z2 are the atomic numbers of the interacting atoms, e is the electron charge and r is the interatomic distance. r is the screening function, 4 1expi u i ia r b A r (2 30) w here Ai and bi are pre defined parameters and are given in Table 2 1 and au is : o ua Z Z a23 0 2 23 0 18854 0 (2 31) w here a0 is the Bohr radius, 0.539177 PAGE 53 53 Table 21. The ZBL parameters i A b 1 0.1818 3.2000 2 0.5099 0.9423 3 0.2802 0.4028 4 0.02817 0.2016 The p otential parameters for the MgO Nd2Zr2O7 system are summarized in Table 2 2. To incorporate the covalent nature of the crystal in case of MgO and Nd2Zr2O7, partial charges are assigned to ions and are shown as superscripts Table 22. Potential parameters for the description of MgO Nd2Zr2O7 system [62, 63] Species A (eV) ( ) C (eV 6 ) O 1.7 O 1.7 35686.18 0.201 00 32.0 0 Nd +2.55 O 1.7 2148.14 0 3227 0 22.59 Zr +3.4 O 1.7 1402.57 0.3312 0 5.1 0 Mg +1.7 O 1.7 929.69 0.29909 0.0 0 Table 23 summarizes the interaction parameters for MgAl2O4. Every ion is assigned formal charges in this case. Table 23. Short range potential parameters for MgAl2O4 from Ball et al. [64] Species A (eV) ( ) C (eV 6 ) Mg+2O2 1279.69 0.2997 0.00 O2O2 9547.96 0.2240 32.0 Al+3O2 1361.29 0.3013 0.00 PAGE 54 54 Table 2 4 summarizes the interaction parameters for Bi4Ti3O12. E very ion is assigned a formal charge in Bi4Ti3O12. T hree different Ti O interaction parameters are investigated and labeled by their A parameter values. Table 24. Short range potential parameters f or Bi4Ti3O12 [65 67] Species A (eV) ( ) C (eV 6 ) Bi+3O2 49529.4 0.2223 0.00 O2O2 9547.96 0.2192 32.0 0 Ti+4O2 2131 2131.04 0.3038 0.00 760 760.47 0.3879 0.00 2549 2549.4 0.2989 0.00 Even though the interactions for Bi4Ti3O12 were defined for coreshell model the shell part is neglected interaction para meters for core core only are used in this work In a coreshell model, the electrons of an ion are divided into two parts: core and shell. Core electrons are combined with nucleus and assigned a +ve charge and the shell having all electrons is given a ve charge. The charge on each ion is described by the sum of the r espective core and shell charges. The core and shell of each ion are connected by either a harmonic or anharmonic spring. For more details, please refer to the article by Dick and Overhauser [68] 2.9 E nsembles Statistical mechanics is a useful method to study a system with large number of particles. It converts microscopic properties of the collections of individual atoms PAGE 55 55 (positions, velocities etc.) or molecules to macroscopic bulk properties of mater ials that can be observed experimentally The ensemble is simply a statistical equivalent of repeating the same experiment under the same macroscopic conditions but different microscopic ones and expects to observe a range of outcomes. A thermodynamic stat e of the system can be defined using various state variables: pressure (P), temperature (T), total number of atoms (N). Other thermodynamic properties (density, chemical potential, heat capacity etc.) may be derived using the equation of state and fundamental relations of thermodynamics. Even other microscopic properties, like diffusion coefficient, structure factor which clearly depicts microscopic structure and dynamics at the atomic level are dictated by the variables characterizing the thermodynamic state and not by microscopic properties. The atomic positions and momenta are coordinates in the phase space ( ). For a single particle, its phase space has 3 positions and 3 momenta totaling 6 dimensions. Let us assume, the instantaneous value of a property can be written as l these instantaneous value, and can be written as: dt t tobs onbt obs t time obs 01 lim (2 32) As stated earlier, time evolution of the phase space is g overned by the Newtons equation of motion. As the number of atoms increase in the system, solving Eq. 2 32 becomes a formidable task. Another concern is also the infinite time limit to perform the time average of Eq. 232. That can be resolved by taking a long enough time and PAGE 56 56 solving Newtons equation of motion in smaller time steps ( t ) Replacing infinite time by and t Eq. 2 32 can be rewritten as: t i time obsi 11 (2 33) Equation 233 provides a simulated value of the macroscopic property ti me average. Instead of time evolution of the bulk properties, Gibbs suggested replacing time averages by the ensemble average. The ensemble is regarded as a collection of points in the phase space. The points are distributed according to a probability density parameters (NPT, NVE etc). These points are selected from i ) Pair of extensive and intensive quantities including N (total no of particles) or ii) The volume (V) or pressure (P) iii) The energy (E) or related thermodynamic potential or temperature (T) In general, ensembles are categorized as microcanonical, canonical and grandcanonical based on different macroscopic constraints [44] The microcanonical ensemble is used to describe the properties of an isolated system with fixed N, V and E and is also referred to as the NVE ensemble. The canonical ensemble is a set of systems which can share its energy and maintains a constant temperature. They are in constant contact with a heat bath or reservoir and exchange energy to maintain a constant temperature. It is referred as NVT ensemble. The grand canonical ensemble is a collection of systems which are in thermal contact with a heat bath or reservoir and can exchange particles. It can be viewed as an PAGE 57 57 extension of a canonical ensemble with the addition of particle exchange. The chemical potential is introduced to define the fluctuations in the number of particles. These three ensembles are not sufficient to cover all the experimental constraints. A list of all the statistica l ens embles is given in Table 25 Table 25 Summary of the eight statistical ensembles. For specific conditions the internal energy E is modified as the enthalpy (H = E + PV), the Hill energy (L = E iNi) and the Ray enthalpy (R = E +PV iNi) [69] Fixed Variable Ensemble NVE Microcanonical NVT Canonical NPT Isothermal Isobaric (Gibbs) NPH Isoenthalipc Isobaric Grand canonical Grand microcanonical Grand Isothermal Isobaric Generalized A g eneralized ensemble is not physical as its size is not specified [69] .Since in the simulations, mostly NPT and NVE ensembles are applied it becomes important to briefly discuss the implementation of constant temperature and pressure in the simulation. 2.9.1 Pressure C ontrol The pressure control procedure developed by Andersen [70] and subsequently extended by ParrinelloRahman [71] is now a standard tool. In a constant pressure simulation, the volume of the system fluctuates. Atom coordinates are redefined as PAGE 58 58 3 1V ri i (2 34) w here, V is the box volume. Let us imagine the fluid to be simulated is in a container of variable volume V and is compressed by a piston. M is the mass of the piston and P is the pressure applied. Let us also assume that the volume change is isotropic or, equivalently an equal pressure is applied from all the directions (hydrostatic pressure). This scenario can be represented by a Lagrangian given as follows: N i N j i ij i iPV V M V u mV L1 1 2 3 1 3 22 1 2 1 (2 35) The first two terms of Eq. 2 35 are the Lagrangian of the fluid in the container, third term corresponds to the kinetic energy of the piston and the last term refers to the potential energy of the piston. After constructing the Hamiltonian and forming equation of motions, the trajectory of each atom can be derived as follows: N i N j i ijij i i ir u r m p p V P dt V Md1 1 ^ 2 2' 3 1 2 3 2 1 (2 36) w here, pi is defined as 3 1Vi and is the conjugate momentum of (atomic coordinates, as defined in Eq. 234) This method was extended by Parrinello and Rahman (PR) [71] to anisotropic stress. These methods are not good to capture first order phase transitions [72] for non periodic systems [73] Firs t order phase transition in PR scheme does not take place via nucleation and growth. Instead it takes place as a collective transformation which might result in an overestimation of the energy barrier for the phase transition. This can result in missing intermediate phases and lowers the predictive power of the PR scheme. To overcom e these issues, a new method has been PAGE 59 59 developed by Martok, Laio and Parrinello [74] Another scheme has been developed for finite system sizes by Sun and Gong [73] 2.9.2 Temperature C ontrol Most experiments are carried out at constant pressure and temperature. Usually, temperature is related to the average kinetic energy of the system as T Nk mv KEB i i2 3 22 (2 37) This issue is resolved by implementing a thermostat in MD. A simulated s ystem is in constant contact with a heat bath or reservoir that acts as a thermostat. The first and simplest method to implement constant temperature in MD simulation is a momentum scaling algorithm proposed by Woodcock in 1971 [75] In this method, at each simulation step, the velocity of each particle is rescaled to keep the average kinetic energy at the desired constant value calculated by Eq 237. It is also known as velocity rescaling method. Rescaled v elocity is given by: i et t iv T T rescaled varg (2 38) Various other methods to implement constant temperature in MD simulation have been reviewed here [44, 76] They are propose d by Nos [77] Hoover et al [78] and Evans and Morriss [79] Haile and Gupta [80] and an extension to the extended system method of Nos [76] by Hoover [81] that is known as Nos Hoover algorithm. Because of its simplicity, velocity rescaling method is us ed for the simulations. PAGE 60 60 2.10 Thermal Expansion Simulation As a startin g point in the simulation of thermal conductivity, coefficient of thermal expansion is computed using simulation. Thermal expansion is related to the lattice parameter to the first order, as follows: T aa 10 (2 39) w here a is la ttice parameter at temperature T and a0 is the lattice parameter of the system at T = 0 K and is the linear thermal expansion. To the first order approximation can be computed by plotting a/a0 as a function of temperature. In the simulation, a cubic system is generated to calculate thermal expansion in all the directions. To compute thermal expansion, NPT ensemble is used and simulations are run at various temperatures to get a good profile between lattice parameter and temperature. 2.11 Thermal Conductivity Simulation Fourier law relates the heat flux to the temperature gradient on a material: j ij ix T J (2 40) w here iJ is the ith component of heat flux (energy per unit area per unit time), ij is an element of conductivity tensor and jx T is the temperature gradient along j direction. Experimentally, the thermal conductivity is measured using thermal diffusivity and the following relation pC (2 41) w here is the thermal diffusivity, is the density and Cp is the specific heat capacity at constant pressure. Laser flash is the most widely used method to calculate thermal PAGE 61 61 diffusivity and is described in detail by Parker et al. [82] In brief, A few millimeters thick thermally insulated sample i s irradiated with a highintensity short duration light pulse and temperature history of the rear surface is measured. The thermal diffusivity is measured from the shape of temperature evolution with time curve at the rear surface. A schematic of fl ash met hod is shown in Fig. 215. In simulation, it can be determined by two ways: Non equilibrium MD (NEMD: direct method) [63, 79, 8391] or equilibrium MD (EMD: Green Kubo method) [83, 9297] Figure 215. Schematic of flash diffusivity measurement and temperature rise curve. Adapted from the Ref [98] The simulation set up to measure thermal conductivity for direct method is shown in Fig. 2 1 6 System in direct method is set up in a way that it is of cylindrical shape. PAGE 62 62 This size const raint (longer in one direction: z compared to other two: x and y) transforms Eq. 241 into scalar form as given below: z T J (2 42) Figure 216. Schematics representation of simulation set up for direct method (NEMD). Simulation cell has length Lz and heat source and sink both have thickness and at a distance Lz/4 from the edge/center of the cell. energy is added and removed at the source and sink respectively. Figure is adapted from reference [99] To simul ate the thermal conductivity, the system set up follows these steps 1 Coefficient of thermal expansion is calculated at the desired temperature (Section 210). 2 System is equilibrated at 0 K. 3 Expansion coefficient is applied to the relaxed structure and equil ibrated at the set temperature 4 Once the system reaches equilibrium at the set temperature, the temperature gradient is generated by giving extra energy to the atoms at the heat source and removing the same amount of energy from the atoms at the heat sink r egion. Heat source Heat sink PAGE 63 63 5 This sets up a temperature gradient and its average over time is used to compute the thermal conductivity using Eq. 242. The size of the simulation cell strongly affects the value of thermal conductivity determined, as illustrated in the case of Si [99] When the simulation cell is smaller than the phonon mean fre e path, thermal conductivity is limited by system size. This is known as the Casimir limit. To account for this effect, the thermal conductivity is calculated as a function of system size and then extrapolated to infinite system size. Following is t he phys ics behind this phenomena and methodology to over come the issue By applying the kinetic theory of gases, the thermal conductivity was shown to be equal to: cvl 3 1 (2 43) w here, c is the specific heat, v is the group velocity and l is the effective mean free path of the phonons. Specific heat above the Debye temperature is given by the Dulong Petit value for a noninteracting gas: n k cB2 3 ( 2 44) w here, Bk is the Bol t zmann constant and n is the number density of the ions. For a unit cell of volume V and total number of ions N n is given as V N Group velocity v can be expressed as: 3 2T Lv v v where Lv and Tv are the longitudinal an d transverse sound velocities. For a simulation cell of length Lz, the phonon mean free path is Lz/4 (As shown in Fig. 216 heat source and heat sink are at Lz/4). If l is the phonon mean PAGE 64 64 free path for infinite system, then the effective mean free path for system size with Lz length is given by [99] : z effL l l 41 1 (2 45) The final expression for t hermal c onductivity is : 14 1 3 2 2 3 3 1 z T L BL l v v V N k (2 46) For a cubic system, the thermal conductivity and system size relation reduces to: z BL l Nv k a 4 1 2 13 (2 47) Equation 247 shows a linear relation between the inverse of thermal conductivity and inverse of system size. The methodology to infer thermal conductivity for infinite system size can be simply achieved by extrapolating the thermal conductivity values at 1/Lz=0. This approach has been applied to Si and diamond single crystals [99] and will be used here. In case of thermal conductivity, the coefficient of thermal expansion and elastic properties are calculated first A nharmonic analysis is used to correct simulated thermal conductivity values based on the fidelity of interaction parameters with experimental thermal expansion and bulk modulus values. Details of the analysis are presented in chapter 4 with MgO Nd2Zr2O7 results. PAGE 65 65 CHAPTER 3 CRYSTALLOGRAPHY OF RELEVANT MATERIAL SYS TEMS 3.1 General Introduction Crystal lography is formed from two Greek words, crystallon meaning frozen drop/cold drop, and grapho meaning write. Crystallography refers to the science of determining the arrangement of atoms in solids. Crystallography also provides insight into material proper ties and plays a major role in many subjects, including mineralogy, physics, materials science, chemistry, geology, geophysics and medicine. In essence, it encompasses various areas of science and engineering. Materials with the same ch emical composition exhibit different properties based upon their crystal structures. For example, graphite and diamond both are forms of carbon but diff er in crystal structure and show disparity in almost all physical properties. Another example is PbTiO3, which is cubic at r oom temperature and paraelectric but as it changes t o the tetragonal structure at 763 K shows ferroelectricity [100] The structure of a material can be understood as a description of the arrangement of the components that make up the material. All the atoms/ions can be taken as building blocks of the materials, especially solids. The limiting factor here is the sheer number of atoms to describe a small piece of material. For example a tiny cube of salt (NaCl) has approximately 1020 atoms. If a material is perfectly ordered, then by exploiting periodicity, a smaller representative unit of NaCl can be used to describe NaCl for any length scale. The smallest representative unit of any structure is called its unit cell. Standard descriptions of all the space groups are given in The International Tables for Crystallography [101] Both HermannMauguin and Sch nflies notations are used t o describe the symmetry of the material s investigated. Sch nflies notation is PAGE 66 66 adequate only to describe the point group and thus generally used in spectroscopy. HermannMauguin notation is generally used in crystallography because it includes space group symmetry also in the notation. 3.2 MgO: Rocksalt Crystal Structure The rocksalt structure is named after NaCl (hal ite). Almost every alkali halide, sulfide, selenide and telluride form rocksalt structure [102] Almo st one third of all MX type compounds crystallize to the rock salt structure, where M is the cation and sits at (0, 0, 0) and X is an anion and sits at (0, 0, 1/2). MgO has the rocksalt crystal structure. The equal number of cations and anions in the rocks alt structure are arranged as interpenetrating FCC sublattices. The unit cell of rocksalt crystal structure with MgO as an example is shown in Fig. 31, with O as large ions and Mg as smaller ions. The conventional cubic unit cell contains 4 formula units of MgO (Z=4). Figure 31. Conventional cubic unit cell of MgO. Mg (small circle) and O (big circle) Mg O PAGE 67 67 The rocksalt structure belongs to the m Fm_3 ( 5 hO No. 225 in the international table) space group, where m Fm_3 is space group representation in HermannMaugin and 5 hO is in Sch nflies notation. 5 hO represents octahedral symmetry (four 3fold axes along body diagonals with three 4fold axes along the faces) and mirror planes. In HermannMaugin notation, F represents a face center cubic structure and m m_3 a short form for _2 3 4 m m point group represents operations along the primary <001>, secondary <111> and tertiary <110> direction of the cubic system. For MgO, the 4/m represents four fold rotational symm etry at the corners and the cell center and a mirror perpendicular to th e rotational axis [100] (Fig. 3 2(a)). The 3 symmetry represents the rotoinve rsion along the body diagonal [111] of the unit cell (Fig. 3 2(b)). The 2/m symmetry operation represents the twofold rotation along the edges of the unit cell and a mirror plane perpendicular to the axis of rotation (Fig. 3 2(c)). Figure 32. Schematics showing three different symmet ry operations represented by MgO A ) A four fold rotation symmetry along [100]. B ) A three fold inversion symmetry along [111]. C ) A two fold rotation symmetry along [110]. a A) a a B) C) PAGE 68 68 The lattice p arameter of MgO unit cell is 4.191 [103] Table 31 lists the site symmetry, the Wyckoff position and coordinates of the atoms for rocksalt structure AX. Table 31. Rocksalt AX crystal structure Ion Wyckoff position Site symmetry Coordinates (0,0,0; 0,1/2,1/2; 1/2, 0, 1/2; 1/2, 1/2, 0)+ A 4a O h 0,0,0 X 4b O h 1/2,1/2,1/2 The f irst column of Table 3 1 lists the ion s present in the system (for example, AX has cation A and anion X) The s econd column give s information about Wyckoff position that represents a total number of equivalent points The letter in the Wyckoff position goes in alphabetic order from high to low symmetry, f or example a is assigned to the equipoint having the least number of equivalent points. The third column contains information about the point symmetry of the corresponding position. The last column gives the coordinates of the ions. The positions under the coordinates title are called the generators. A ll the positions of atom s/ions are calculated by adding the coordinates to the generators. For example, from Table 31, ion A has 4 multiplicity and all 4 positions are calculated by adding (0,0,0) to (0, 0,0), (0,1/2,1/2), (1/2,0,1/2) and (1/2,1/2,0). Similarly, coordinates of X can be calculated by ad ding (1/2,1/2,1/2) to (0,0,0), (0,1/2,1/2), (1/2,0,1/2). 3.3 Nd2Zr2O7: Pyrochlore Crystal Structure Nd2Zr2O7 ( NDZ ) belongs to the pyrochlore family and has general formula A2B2O7 [104] The space group for pyrochlore structure is m Fd_3 ( m d F 23 4_ 1, 7 hO No. 227 in the international table [101] ). Similar to the rocksalt crystal symmetry, pyrochlore also has 3fold rotoinversion along (111) and 2fold rotation axis with perpendicular PAGE 69 69 mirror along (110). Instead of 4fo ld rotation axis with perpendicular mirror plane, this has 41 screw axis with diamond glide. A screw axis (helical axis or twist axis) is a combination of rotation plus translation symmetry. 41 screw axis represents a rotation of 900 (3600/4) and translati on of of the lattice vector parallel to the rotation axis. The crystal structure of pyrochlore materials is closely related to the fluorite structure. There are two main differences between the fluorite and pyrochlore crystal structures: anions occupy all 8 tetrahedral sites in fluorite and only 7 in pyrochlore, and pyrochlores have approximately twice the lattice spacing of fluorite structures, i.e. one unit cell of pyrochlore structure can be divided into 8 sub units with each sub unit similar to a fluorite structure [105] Figure 33 shows oneeighth of the pyrochlore structure and the fluorite structures side by side. Figure 33. A) Pyrochlore sub unit (1/8) B) Fluorite unit cell Cations A and B are 8and 6fold coordinated respectively, and can have either +3 and +4 or +2 and +5 oxidation states. Based on the charge and oxidation state, the ionic radius of cation A should lie within 0.087 0.151 nm [105] Similarly ionic radius of B should lie in the range of 0.0400078 nm [105] Furthermore, the radius ratio of B Ar r A) B) Nd Zr O O U PAGE 70 70 sh ould lie between 1.292.30 [ 105, 106] Generally for the +3/+4 pyrochlores lanthanides (3+) occupy A sites and 4+ cations take the B sites. As seen in Fig. 33 cations sit at the corn er and face center positions; while anions occupy tetrahedral sites. In pyrochlore, the cations form face centered cubes and are ordered in alternate (110) and ( _1 10) rows (face diagonals) The anions are located in all of the tetragonal sites except the ones which are surrounded by the A cations only [104] The unit cell of pyrochlore is composed of two types of subunit cells, arranged in alternate manner. Type I. Th e A cations form face diagonal at the upper right hand corner and missing anion is at the lower left hand. (Fig. 34A) Type II. The A cations form face diagonal at the lower left hand corner and missing anion is at the opposite corner. (Fig. 3 4B) Figure 34. Two types of subcells, that forms the unit cell of pyrochlore with alternate mixing [107] The structural difference of one less anion to complete the cube inside the cell identifies two types of cations in the system. They are denot ed by X and X, where X is the diagonally situated cation from the vacant site and rest are defined by X. The unit cell of pyrochlore contains equal number of TypeI and TypeII subcells and is arranged PAGE 71 71 in a manner that only different types of subcells become neighbors. Figure 34 shows one typical arrangement of both types of subcells. Due to this structural feature, The lattice parameter of NDZ is (a = 8.089 ) which is almost double that of MgO (a = 4.202 ). This difference becomes imp ortant when composite structures of MgO and NDZ are built for the simulations. The polycrystalline structures of MgO and NDZ are used to separate out the interface effects of each individual component of the proposed inert matrix system. Structural information for pyr ochlore crystal structure including location, site symmetry and coordinates with origin fixed at Zr cation, are given in Table 32. Table 32. Pyrochlore A2B2X6X crystal structure with origin at B cation [104] Ion Wyckoff position Site symmetry Coordinates (0,0,0; 0, 1/2,1/2; 1/2, 0, 1/2; 1/2, 1/2, 0)+ A 16d D 3 d 1/2,1/2,1/2; 1/2,1/4,1/4; 1/4,1/2,1/4; 1/4,1/4,1/2 B 16c D 3 d 0,0,0; 0,1/4,1/4; 1/4,02,1/4; 1/4,1/4,0 X 48f C 2 v x,1/8,1/8; x,7/8,7/8; 1/4 x,1/8,1/8; 3/4+x,7/8,7/8 1/8,x,1/8; 7/8,x,7/8; 1/8,1/4x,1/8; 7/8,3/4+x,7/8 1/8,1/8,x; 7/8,7/8, x; 1/8,1/8,1/4 x; 7/8,7/8,3/4+x X 8b T d 3/8,3/8,3/8; 5/8,5/8,5/8 x for regular tetrahedral = 5/16 (0.3125) x for regular cube = 3/8 (0.375) From Table 32, value of x ranges from 0.3125 to 0.375 and depend on the specific composition. Exact value of x can be determined by experimentally characteriz ing the sample using for example X ray di f f raction neutron diffraction. 3.4 MgAl2O4: Spinel Crys tal Structure The general formula for the spinel compounds is AB2O4, where A is usually a divalent cation and B is a trivalent cation. In this thesis A is Mg2+ and sits on 8a tetrahedral sites, and B is Al3+ and occupies the 16b octahedral sites. The cubic unit cell contains 8 formula units for a total of 24 cations (8 Mg ions and 16 Al ions) and 32 PAGE 72 72 anions (all O ions) This unit cell can be considered as consisting of 8 cubic subunits, as shown in Fig. 3. 5 Bragg [108] and Nishikawa [109] determined the crystal structure of spinel in the early 1900s. The majority of s pinels belong to space group 3 m Fd (Oh 7, No. 227 in the international table [101] ). Figure 35 Cubic spinel unit cell with Mg (blue, medium sized), Al (green, small) and O (re d, large) atoms. The unit cell can be considered as being divided into 8 subunits with Mg tetrahedral (blue) and Al octahedral (green) in alternating sub cells There are total 64 octahedral and 32 tetrahedral interstices between anions in the spinel unit cell. Among these, 8 tetrahedral interstices are occupied by A cation and 16 octahedral interstices by B cation in the normal spinel structure. The atomic positions in the spinel structure depend largely on the choice of origin in the m Fd_3 space group. Two different points with point symmetries m 3 4_ and m_3 are possible choices for the origin. Table 33 lists the atomic positions for normal spinel structure with both origin choices PAGE 73 73 Table 33. Fractional coordinates for atomic positions and vacancies for spinel [110] The oxygen atoms sit at u u u where u is approximately For each spinel structure, permissible values of u can be calculated based on the cations radii and charges. The value of u changes with inversion and that in effect changes the lattice Lattice Site Wyckoff Position Point Symmetry Fractional coordinates (0,0,0; 0,1/2,1/2; 1/2, 0, 1/2; 1/2, 1/2, 0)+ Origin m 3 4_ m_3 A site 8a m 3 4_ 4 1 4 1 4 1 ; 0 0 0 8 7 8 7 8 7 ; 8 1 8 1 8 1 B site 16d m_3 ; 8 5 8 7 8 7 ; 8 7 8 5 8 7 ; 8 7 8 7 8 5 ; 8 5 8 5 8 5 ; 2 1 4 1 4 1 ; 4 1 2 1 4 1 ; 4 1 4 1 2 1 ; 2 1 2 1 2 1 X 32e m 3 u u u u u u u u u u u u u u u u u u u u u u u u 4 1 4 1 4 1 ; 4 1 4 1 4 1 ; 4 1 4 1 4 1 ; 4 1 4 1 4 1 ; ; ; ; ,_ _ ; 4 3 4 3 ; 4 3 4 3 ; 4 3 4 3 ; 4 1 4 1 ; 4 1 4 1 ; 4 1 4 1 ; ,_ _u u u u u u u u u u u u u u u u u u u u u u u Tetrahedral vacancy 8b m 3 4_ 4 3 4 3 4 3 ; 2 1 2 1 2 1 8 5 8 5 8 5 ; 8 3 8 3 8 3 Octahedral vacancy 16c m_3 ; 8 1 8 3 8 3 ; 8 3 8 1 8 3 ; 8 3 8 3 8 1 ; 8 1 8 1 8 1 0 4 1 4 1 ; 4 1 0 4 1 ; 4 1 4 1 0 ; 0 0 0 Tetrahedral vacancy 48f mm ; 0 4 1 4 1 ; 4 1 0 4 1 ; 4 1 4 1 0 ; 2 1 4 1 4 1 ; 4 1 2 1 4 1 ; 4 1 4 1 2 1 ; 4 1 0 0 ; 0 4 1 0 ; 0 0 4 1 ; 4 1 0 0 ; 0 4 1 0 ; 0 0 4 1 8 1 8 7 8 7 ; 8 7 8 1 8 7 ; 8 7 8 7 8 1 ; 8 1 8 1 8 1 ; 8 1 8 1 8 1 ; 8 1 8 1 8 1 ; 8 3 8 7 8 7 ; 8 7 8 3 8 7 ; 8 7 8 7 8 3 ; 8 3 8 1 8 1 ; 8 1 8 3 8 1 ; 8 1 8 1 8 3 PAGE 74 74 parameter of the structure. The relation between bond lengths and lattice parameter is given as [111] : 3 11 8 33 8 402 2 T O Td d d a (3 1) w here, a = unit cell edge or lattice parameter dO = bond length of cationO at octahedral site dT = bond length of cationO at tetrahedral site Cation Arrangement: It has lo ng been established that spinel crystals can accommodate significant amount of cation disorder or so called variate atom equipoints [112, 113] The structure with cation disorder is termed as inverse spinel, introduced by Verwey and Heilmann [114] Inversion is traditionally denoted by i and is given as : Mg of no Total sites l tetrahedra on Al of No i (3 2) The value of i ranges from 0 to 1, where 0 has all the cations on their normal crystal sites (Mg at tetrahedral and Al on octahedral) and 1 refers to the structure with all the Mg on octahedral sites and Al equally divided on octahedral and tetrahedral sites. Based on inversion, disorder in the spinel structure is described by Q which is defined as [115, 116] : 2 3 1 i Q (3 3) The value of Q ranges from 1( i =0) to 0.5 ( i =1). Q is 0 for randomly disordered structure ( i =2/3). The modified structural formula is used to project cation disorder in the PAGE 75 75 system 4 2 2 1 2 12 1 1 2 X M M M MVI IVB q i p i A p i q i where pM 1 refers to divalent and qM 2 to trivalent cations. The first bracket represents the cation arrangem ent on tetrahedral sites. Similarly the second bracket represents cation arrangement on octahedral sites. 3.5 Bi4Ti3O12: Aurivillius Crystal Structure Bismuth layer compounds belong into the Aurivillius family of layered perovskite and are denoted by (Bi2O2)2+(An 1BnO3n+1)2 [117] The crystal structure can be visualized as alternate stacking of (An 1BnO3n+1)2 perovskite layers and (Bi2O2)2+ fluorite like layers along the c axis where n represents the number of perovs kite layers between Bi2O2 layers. Various compounds with different n v alues have been prepared and are presented in Table 34. Table 34. Various Aurivillius structures for different n values n value Material Reference 1 Bi 2 WO 6 Rae et al. [118] 2 Bi 2 SrNb 2 O 9 Ismunandar et al. [119, 120] 3 Bi 4 Ti 3 O 12 Aurivillius [117] 4 Bi 5 Ti 3 MnO 15 Kumar et al. [121] 5 Bi 2 Ba 2 Ti 5 O 18 Aurivillius and Fang [122] Bismuth titanate Bi4Ti3O12 (BIT) belongs to Aurivillius structure family with n=3 value. The crystal structure of BIT in ferroelectric phase (T > 950 K ) is tetragonal with I4/mmm symmetry and has a~3.85 and c ~ 32.832 [118] As temperature is lowered, the system undergoes structural changes and forms a lower symmetry PAGE 76 76 st ructure with a and b lattice constants elongated by ~ 2 of their tetragonal structure values. Figure 36 Unit cell of Bi4Ti3O12 for A) low temperature monoclinic structure B) high temperature tetr agonal structure, Bi is violet, O is red and Ti is light blue Figure 36 compares the high temperature tetragonal structure with low temperature structure. The atomic positions for tetragonal as well as low temperature monoclinic structure are given in Tables 3 5 and 36, respectively. Low temperature lattice parameters are taken as a=5.450 b=5.4059 and c=32.832 and all the angles 900 [123] There is no consensus over the low temperature crystal structure of BIT and scientific community is divided between orthorhombic and monoclinic PAGE 77 77 structures. A list of experimental investigations and their conclusion on the structure of BIT is shown in Table 3 7. Table 35. Fractional coordinates for t etragonal structure (I4/mmm) of BIT [124] Atom Site Fractional coordinates X Y Z Bi 4a 0.5 0.5 0.5667 Bi 4a 0.5 0.5 0.0662 Ti 4a 0.0 0.0 0.0 Ti 4a 0.0 0.0 0.1259 O 4a 0.0 0.5 0.0 O 4a 0.5 0.0 0.0 O 4a 0.0 0.5 0.1141 O 4a 0.5 0.0 0.1141 O 4a 0.0 0.0 0.0593 O 4a 0.0 0.0 0.1802 O 4a 0.0 0.5 0.25 O 4a 0.5 0.0 0.25 Table 36. Fractional coordinates for monoclinic structure (B1a1) of BIT [125] Atom Site Fractional coordinates X Y Z Bi 4a 0.0407 0.5041 0.5667 Bi 4a 0.0427 0.4777 0.7104 Ti 4a 0.0849 0.0044 0.4969 Ti 4a 0.0948 0.0014 0.6279 O 4a 0.3401 0.2710 0.5086 O 4a 0.3036 0.2389 0.2473 O 4a 0.1375 0.0614 0.5585 O 4a 0.0879 0.0406 0.6815 O 4a 0.3212 0.2747 0.6118 O 4a 0.4071 0.2036 0.6257 Bi 4a 0.0429 0.4991 0.4332 Bi 4a 0.0386 0.5170 0.2878 Ti 4a 0.0884 0.0093 0.3716 O 4a 0.3884 0.2203 0.4944 O 4a 0.3213 0.2443 0.7480 O 4a 0.1215 0.0638 0.4402 O 4a 0.1039 0.0573 0.3187 O 4a 0.3418 0.2720 0.3890 O 4a 0.3899 0.2095 0.3775 Aurivillius in his X ray diffraction study conceived the structure as orthorhombic with space group Fmmm Later neutron diffraction studies suggested the orthorhombic PAGE 78 78 structure with B2cb space group (#41, in the international table [101] ) [126, 127] In other studies using single crystal X ray diff raction, Rae et al. [123] and Shimakawa et al. [128] reported the monoclinic structure with the B1a1 space group (#7, in the international table [101] ). From the crystallography perspective, the only difference between B1a1 and B2cb space group is the absence or presence of bglide. Experimentally, it is very hard to differentiate between these two space groups. The controversy arises because all of these possible structures are subgroups of I4/mmm. Fi gure 37 represents a groupsubgroup graph connecting the space groups I4/mmm and B1a1. Table 37 summarizes the experimental investigations and their results for the structure of BIT. Table 37. Historical preview for structures of BIT Year Method Structu re Lattice Parameters Reference 1949 Single crystal X ray B2cb (Orthorhombic) a=5.410 b=5.448 c=32.800 Aurivillius [117] 1971 B2cb (Orthorhombic) a=5.448 b=5.411 c=32.83 Dorrian et al. [127] 1990 No b glide polarization along c axis B1a1 (Monoclinic) a=5.45 b=5.4059 c=32.832 0 Rae et al. [123] PAGE 79 79 Table 37 Cont Year Method Structure Lattice Parameters Reference 1999 Neutron power diffraction B2cb (Orthorhombic) a=5.444 b=5.4086 c=32.8425 Hervoches and Lightfoot [126] 2001 Neutron power diffraction B1a1 (Monoclinic) a=5.44521 b=5.41118 c=32.8432 Shimakawa et al. [128] 2003 Neutron power diffraction B1a1 (Monoclinic) a=5.4475 b=5.4082 c=32.8066 Kim et al [129] 2007 Rietvald refinement of powder neutron diffraction B2cb (Orthorhombic) a=5.4432 b=5.4099 c=32.821 Chakraborty et al. [130] 2008 DFT B1a1 (Monoclinic) a=5.4289 b=5.4077 c=32.8762 Shrinagar et al. [131] 2008 Odd n:B2cb Even n:A21am B1a1 (Monoclinic) a=5.411 b=5.448 c=32.83 Jardiel ey al. [132] PAGE 80 80 Figure 37 Group subgroup graph connecting I4/mmm to B1a1 [133] The structural controversy can be resolved with the help of symmetry constraints on the physical properties. BIT showed a finite polarization along the c axis, which is unlikely in the B2cb structure because the bglide has its mirror plane perpendicular to the c axis and thus all the polarization vectors along the c axis cancel out. Foll owing the study by Shrinagar et al. [131] the fractional coordinates for B1a1 crystal structure can be derived by the appropriate transformations to its original Bravais lattice P1n1. Point 1 I4/mmm (130) I4/mmm (107) I _4 2m (121) Fmmm (69) Fmm2 (42) F2mm (42) Bbab (68) Bmab (64) Bbam (64) Bmam (67) I12/m1 (12) I1m1 (8) C2cb (41) B2cb (41) C2mb (39) B2cm (39) P12/n1 (13) P12 1 /n1 (14) B1a1 (7) PAGE 81 81 of P1n1 (x,y,z) remains unchanged but point (x+1/2, y+1/2,z) changes to (x+1/2, y+1/2,z+1/2). An important feature of these layered perovskites is the tilting of the octahedra. Figure 37 captures the nature of octahedra tilting along different axes. The effect of tilting is associated with the onset of ferroelectric phase change in Aurivillius structures. A study by Newnham et al. [134] suggested that increasing the number of perovskite units in Aurivillius structure (Bi4Ti3O1 2, SrBi4Ti4O15, Sr2Bi4Ti5O18) reduces Tc (Curie temperature) Another study by Shimakawa et al. [128] related structural disorder to curie temperature. A tolerance factor can be obtained for the layered perovskite structures by considering the perovskite tolerance factor; O B O AR R R R t 2 (3 4) where RA, RB, and RO are the ionic radii of A, B and oxygen ions, respectively. This can be applied to Aurivillius structures. Due to additional Bi2O2 layers present in Aurivillius structures, stability region becomes very narrow due to extra structural constraints. Most of t he perovskite structures exhibit tolerance factor between 0.771.01 [135] The t value for layered structures is suggested to lie between 0.810.89 and with further decreasing as perovskite layers increase [136] A brief description about octahedron tilting and its notation is discussed below. Octahedron Tilting: Octahedron tilting reduces the symmetry of the structure. Tilting also has a prominent effect on lattice parameter but it is very difficult to describe it. Attempts are made to explain the effect by including the effect of octahedral into it [137] However, there are multiple ways the octahedra can tilt and change the environment around the cation. A standard notation has been developed to describe PAGE 82 82 octahedra tilting in perovskite by Glazer [138] The notation describes the tilting of MX6 octahedra about any of the three orthogonal Cartesian axes. Figure 38 Octahedra tilt of 6x6x1 supercell of BIT along A) X axis [100] B) Y axis [010] C) Z axis [001] The general notation for tilt is used as a+b0c, where a,b,c are the magnitude of the tilt corresponding to [100], [010], and [001] axes. Superscripts depend on the sense of X Y Z A) B) C) PAGE 83 83 the tiltin g of successive octahedra. A + (positive) superscript would denote the same direction (inphase) tilt of the successive octahedra and a (negative) superscript implies tilt in the opposite direction (out of phase). A superscript of 0 (zero) denotes no tilting about that axis. There are certain important results that can be inferred from the tilting notation and are given below: 1 Equality of tilts about two or more pseudocubic axes im plies equality of axial lengths (a+a+a+ a=b=c) 2 Two or more superscripts ar e 0 or +, system has orthogonal axes. 3 If two of the superscripts are then respective axes are inclined to each other and third one is perpendicular to both of them. 4 If all there superscripts are then all axis are inclined to one another. Base d on this brief description and Fig. 3 8 it can be inferred that BIT shows a0bcoctahedra tilting in Glazer notation. PAGE 84 84 CHAPTER 4 THERMAL TRANSPORT PROPERTIES OF MgO Nd2Zr2O7 4.1 Introduction As discussed in the C hapter 1 next generation fuel is one of the solutions of todays energy crisis. In a study of diluents materials f or Pu burning in nuclear reactors MgO along with AlN and ZrN showed promising properties to be a candidate for nuclear reactor applications [139] Neeft et al. [140] also suggested MgO as a potential inert matrix fuel (IMF) material and studied the effect of neutron irradiation in MgO dispersed IMF and mod eled the crack propagation in the fuel Further study by Chauvin et al. [141] showed a good behavior by MgO based IMF Recognizing the importance of MgO, Medvedev et al. [15] proposed MgO ZrO2 as a candidate IMF material to overcome hydration limitation of MgO Detailed report on MgO ZrO2 as an IMF, is presented in the dissertation of Medvedev [14] Inspired by this, Yates et al. [16] suggested MgO Nd2Zr2O7 (NDZ) as a candidate material for IMF The selection of this specific pyrochlore composition (NDZ) is based in part on insights obtained from atomic level simulations. In particular, Sickafus et al. [142] used atomic level simulation methods to characterize the radiation tolerance of a wide range of py rochlores (A2B2O7) They found that the radiation tolerance increases significantly as the B ion radius increases In another atomic level simulation study by Schelling et al [143] it has been reported that the thermal conductivity increases as the B cation radius decreases Combining these two results with the guidel ines suggested by Matzke et al. [12] neodymium zirconate, NDZ is identified as a suitable candidate pyrochlore material Interface effects on the thermal properties of MgO NDZ cercer is of primary concern for its potential application as IMF. How the microstructure, interfaces, PAGE 85 85 volumetric composition governs the thermal transport of the composite is investigated in this chapter. Before investigating interface effects for the MgO NDZ cercer composite, fidelity of interaction parameters is tested for both the system. MD simulations were run to compute elastic and thermal transport properties of MgO and NDZ. By comparing them with experiment and establishing the veracity of interaction parameters, interface effects are simulated for polycryst als. These polycrystal results are extended to composite system of MgO NDZ. 4.2 Simulation Set ups 4.2.1 Thermal Expansion The simulation set up to capture thermal expansion consists of cubic arrangement of 6x6x6 or 4x4x4 unit cells in the case of MgO and NDZ respectively. Having 8 atoms in the unit cell of MgO gives a total of 1278 atoms in the simulation supercell. Similarly for NDZ, unit cell having 88 atoms gives 5632 atoms in the simulation set up Periodic boundary conditions are applied in all the spatial directions. 4.2.2 Thermal Conductivity As explained in Sec. 2.11, thermal conductivity is calculated by creating a temperature gradient across the system. After reaching steady state heat flow, thermal conductivity is calculated using Fouriers law (Eq. 240). The system can be compared with simple set up of creating a heat flow across a rod by putting one end at the flame and the other at the cold bath. Schematic of simulation set up is shown in Fig. 4 1 on the right and a simple cartoon representing heat flow is shown on the left. Simulation set up has non cubic arrangement with z axis longer than x and y axes. The set up is chosen to have unidirectional heat flow in z direction. PAGE 86 86 Figure 41. Simulation set up to simulate thermal conductivity 4.3 Results 4.3.1 Thermal E xpansion Simulation set ups are ran at several temperatures under NPT ensemble (please refer, Sec 2 9) to compute the coefficient of thermal expansion. After reaching equilibrium state at each temperature, an average of lattice parameter of the system is calculated. The lattice parameters calculated by the simulation are compared with experiment for MgO ( Fig. 4 2A) and NDZ (Fig. 4 2B). T he lattice parameters for both the system are in quite good agreement with the experiment [144, 1 45] Figure 42. Lattice parameter variation with temperature for A) MgO and B) NDZ Using these simulation values and Eq. 239, coefficient of thermal expansion (CTE) is computed for each of the material. CTE for MgO and NDZ from simulation and experiment is shown in Table 41. Fidelity of interaction parameters is tested by 0.41 0.42 0.43 0.44 0.45 0 500 1000 1500 2000a [nm]T [K] Sim Exp MgO 1.065 1.069 1.073 1.077 1.081 1.085 1.089 1.093 0 400 800 1200 1600a [nm]T [K] Sim Exp NDZ Heat sour ce Heat sink [00 1] A) B) PAGE 87 87 comparing the simulated values with experiment. The comparison for MgO is within 5% and for NDZ it is 22% of the experimental value. Such levels of agreement are typical of empirical potentials for oxide materials as shown by other simulation studies [143, 146, 147] Table 41. Coefficient of thermal expansion for MgO and NDZ Species Simulation Experiment NDZ 7.4x10 6 /K 9.5x10 6 /K [145] MgO 12.4x10 6 /K 13.5x10 6 /K [148] 4.3.2 Anharmonic A nalysis The different values of thermal expansion for the simulated and experimental values provide valuable information on the nature of the interatomic interactions. As is well known, the thermal expansion arises from the asymmetry in the interaction potential, i.e. from the anharmonicity in the interatomi c interactions. An elementary analysis gives the Grneisen relation for the thermal expansion [29] : B 3 cv ( 4 1 ) where cv is the specific heat, B is the bulk modulus and is the Grneisen parameter, which is a measure of the anharmonicity of the material. T hus the reduced values of the simulated thermal expansions are attribute d to the interatomic interactions being insufficiently anharmonic and/or the bulk modulus being too high. This conclusion has important implications for the thermal transport properties, because the thermal conductivity is also determined by the anharmonicity of the interactions. In particular, in most electrically insulating materials the dominant PAGE 88 88 scattering mechanism is phononphonon scattering. Such scattering would be completely absent if all of the interactions were harmonic. Using a perturbation expansion of the interatomic potential to third order, thereby including the lowest anharmonic terms, Lieb f ried and Schloemann [149] showed that the thermal conductivity is inversely proportional to the square of the Grneisen parameter : T Mv h kB 3 3 24 10 24 ~ (4 2) where kB is the Boltzmann constant, h is the Planck constant, v and M are the volume and the mass per atom. The only two materials constants that enter into Eq. 4 2 are the Debye temperature, and the frequency averaged Grneisen parameter From this relation, it is clear that the thermal conductivity is inversely proportional to Through their dependence on the Grneisen parameter, a simple empirical relationship between the thermal conductivity and thermal expansion is shown as [150] : B Cv 2 3 2 (4 3) The constant contains all of the nonmaterials constants in Eq s. 4 2 and 4 3 Table 42 shows the values of D and Cp for MgO and NDZ is determined using GULP (General Utility lattice Program) [151, 152] and compared with the experimental values. Table 4.2. Comparison of Debye temperature and Cp calculated using GULP with experiment. =Gsim/Gexpt, where G= D 3Cp 2 Material D (K) C p (J/mol K) Sim. Exp. Sim. Exp. MgO 830 946 [153] 38.77 34.82 [153] 0.83 Nd 2 Zr 2 O 7 549 489 [154] 207.54 232.70 [154] 1.13 PAGE 89 89 Moreover, the experimental and calculated values of G= D 3Cp 2, analogous to the combinat ion in Eq. 4 3 are within 20% for both materials. After merging Cv and D with other constants Eq. 4 3 becomes B2' (4 4) where = 3Cv 2. From Eq. 4 4 a relationship between experimental and simulation thermal conductivity comes out to be: ) ( ) (B B sim exp 2 sim expt expt sim (4 5) In Eq. 4 5, ) ( ) (B B sim exp 2 sim expt ratio is termed as correction factor. After dividing the simulated value with the correction factor, comparison with experimental value is done. The bulk modulus for MgO and NDZ from simulation for these potentials came out to be 170 GPa and 167 GPa res pectively at 0 K. The corresponding experimental room temperature values for single crystals are 155 GPa and 159 GPa, which are quite consistent [155, 156] From these data, Eq. 4 5 sim expt simexpt for NDZ ; that is simulations of the thermal conductivity of MgO should match the experimental results rather well, while the simulations should significantly overestimate the thermal conductivity of NDZ. 4.3.3 Thermal C onductivity As described in section 4.2.2, simulation is set up to generate unidirectional heat flow. By generating temperature gradient, conduction is simulated and then calculated by using F ouriers law, Eq. 46 dx dT J (4 6) PAGE 90 90 The key determinant of thermal conductivity is the mean free path of the phonons before they scatter from each other or from other components of the microstruc ture. If the system size in the direction of heat flow is smaller than the phonon mean free path, then the intrinsic anharmonic properties of the system will not be probed completely. This regime is known as the Casimir limit. It is thus no surprise that previous studies have also shown that the accuracy of the simulations of the the rmal conductivity depends significantly on the length of the simulation cell. In a previous study by Schelling et al. [157] a finite size scaling approach is used to determine the thermal conductivity of the material 4.3.4 Finite Size Analysis Thi s approach, which is based on the inverse additivity of cont ributions to the mean free path yields (for details, please refer Eq. 247) : zL P 4 1 1 (4 7) where is the thermal conductivity determined from simulation, is the thermal conductivity in the infinite system size limit ( i.e., the best estimate of the true thermal conductivity of the material), Lz is the length of the simulation cell and P is a constant. Equation 47 th z should be linear, and that the thermal conductivity of an infinite system can be obtained by extrapolating to 1/Lz = 0. Figure 4 3 shows such analysis for MgO and NDZ at 573 K By using the above described system size analysis at various temperatures, the temperature dependence of thermal conductivity is extracted for both MgO and NDZ. PAGE 91 91 Figure 43. System size effect on thermal conductivity A) MgO B) NDZ Finite size scaling analysis can lead to a significant increase in the simulated value of thermal conductivity, as is illustrated in Fig. 4 3 If only a single simulation had been performed for the shortest simulation cell (Lz = 15.3 nm for MgO and Lz = 12.80 nm for NDZ), then the estimated thermal conductivities would be less than onefourth of the best estimated value. 4.3.5 MgO: Thermal Conductivity Figure 44 compares the best estimate of the thermal conductivity determined from simulation and finitesize scaling analysis, with the experiment al data for MgO. The data of Cahill (open circles) [158] is for a highquality single crystal thin film, while the polyc rystalline data (open squares ) is for coarse grained polycrystal [159] The experimental polycrystalline data is systematically lower than the experimental data for a single crystal by a small amount. While physically this is reasonable since the polycrystals contain grain boundaries, and possibly other defects arising from processing, all of which provide additional scattering mechanisms, this difference could also simply be statistical in origin [31] 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.00 0.02 0.04 0.06 0.081/k [mK/W]1/Lz[nm1] T~573K MgO 0.00 0.10 0.20 0.30 0.40 0.50 0.02 0.03 0.081/k [mK/W]1/Lz[nm1] NDZ T ~ 573 K A B PAGE 92 92 Figure 44 Thermal conductivity data of MgO: experiment data for single crys tal thin film (open circle) [158] experiment polycrystal data (open square [159] ) and simulation data (solid circle). T he simulated data (solid circles) agree d ve ry well with the single crystal e xperimental data. Had a finite size analysis not been performed, these values would have been significantl y lower and would not have agreed with the experimental values. That the experimental and simulated values are in such good agreement provides validation for the harmonic analysis described in the Sec. 4.3.2, which predicted that they should agree within 2%. 4.3.6 NDZ: Thermal Conductivity The anharmonicity analysis yielded a multiplicative correction factor of 1/1.61 = 0.62 for NDZ. Thermal conductivity values are computed using the simulation method as described in Sec. 2.11. The s imulated values of the rmal conductivity are modified using the correction factor. Figure 45 shows both the uncorrected and corrected values for the thermal conductivity of NDZ. While the uncorrected values are, as expect ed 0 10 20 30 40 50 60 0 500 1000 1500 2000k [W/m K]T [K] Simulation Experiment Cahill Experiment Ronchi MgO PAGE 93 93 consistently higher than the experimental values, the corrected values are much more consistent with the experimental results [160 ] Figure 45. Thermal conductivity of Nd2Zr2O7: uncorr ected single crystal simulation values (open circle), corrected single crystal values (solid circle) and experimental polycrystal values (open square) [160] Both the thermal conductivity results for MgO and NDZ are done on single crystals. The difference between experiment and simulation results besides other reasons is because experimental system has defects. To illustrate the effect of grain boundaries (GBs), simulations on polycrystals are performed and analyzed. 4.4 MgO NDZ: Polycrystals and Composite Defects are an essential part of any material. In nuclear reactor, fuel pellets have grain boundaries as one of the primary defects. The working temperature in the reactor ranges from 800 K 1500 K Grains in the nuclear fuel undergoes structural changes and that leads to changes in the thermal and mechanical properties of the fuel. It is 0 1 2 3 4 5 0 200 400 600 800 1000 1200 1400 1600k [W/m K]T [K] Single Crystal uncorrected Single Crystal corrected Experiment NDZ PAGE 94 94 important to know how these interfaces affect thermal transport in the fuel. That can be understood if the effects of interfaces are characterized in the fuel. Since at atomic level inter face effects are enhanced, atomic level simulations provide an ideal vehicle to characterize the interface effects. To characterize the effect of grain boundaries the thermal transport properties of MgO and NDZ polycrystals and MgO NDZ composites are simul ated Interface effects are enhanced at smaller grain size level It is possible to extend the size into the tens of nanometers range but, due to intrinsic computational limitations, it is not possible to extend the range to microns and tens of microns grain size present in typical ceramic materials. To extend the results for larger grains, a simple analysis model is used and results are compared with experiment results. The fo llowing section will outline the method used in this work to generate polycr ystal and composite structures. 4.4.1 Polycrystal S tructure Figure 46. MgO polycrystal ~ 3 nm grain size; black arrows showing orientation of the grain x z y PAGE 95 95 Interface effects can be estimated using 2D textured polycrystals. They provide an efficient alternate to 3D grains to capture interface effects. A simple code has been developed to generate 2D polycrystals as shown in Fig. 46. The polycrystal is tex tured with all the grains being parallel to the [100] direction. For both materials MgO and NDZ, 24 perfectly hexagonal grains are filled with perfect crystal with the [100] directions oriented in almost random direc tions with respect to the horizontal x axis, with the stipulation that all grainboundary (GB) misorientations should be >15, thereby reducing the tendency for grain coalescence The grain boundaries in these polycrystals are thus a reasonably random selection of <001> tilt boundaries. Periodic boundary conditions are applied such that there are no additional interfaces at the boundaries The whole process of generating polycrystal can be summarized as 1 24 hexagonal grains of equal size were created by specifying their centers but all of them are empty at this step. 2 Each grain was oriented in a way that no neighboring grains form low angle GB. 3 Each grain is filled with the unit cells of the material (MgO or NDZ). 4 Each grainatom is screened and are removed if they are closer to neighboring grain centers. 5 All the grains are combined to make the polycrystal. 6 Any two atoms, with a distance less than a specified cut off distance are identified and termed as bad atoms. 7 Charge of each grain is calculated and maintained to zero by removing bad ato ms from the grain. 8 Still if system is not neutral, bad atoms are removed to achieve charge neutrality. 9 If system is charge neutral but still have bad atoms, bad atoms are removed as a formula unit. For example, for each Zr of NDZ two nearest oxygens are r emoved to keep the neutral charge of the system. PAGE 96 96 4.4.2 Composite S tructure Depending upon the connectivity of constituent phase of the composite, it can be one, two or threedimensionally continuous. A simple scheme to classify the connectivity for twocomponent system is presented by Newnham et al. [161] All the possible connectivity of twoco mponent composite system is shown in Fig. 47. Among the ten different connectivity patterns, 33 composites are extremely rare and 03 is very common for particulates in a matrix and 13 for aligned fiber matrix composites. Figure 47. Different connec tivity patterns for two component composite system [162] To understand this notation, a simple method is used: 1 First number represents the connectivity of white 2 Second number represents the connectivity of black B lack and white nomenclatures are used here to define two different materials because it is easier to follow Fig.4 7 with the convention. For example, 11 composite means, only 1dimension connectivity for both white and black blocks. As shown in Fig. PAGE 97 97 4 7, 1 1 composite has only vertical alignment for both white and black. Another example of 23 composite means, white is continuous in 2dimensions and black in all 3 dimensions. For the textured composite, the composite will exhibit at least 11 connectivity. Figure 48 M gO (blue) NDZ (red) composite (21 composite) A Simple schematic representation of composite structure is shown in Fig. 48. Composite structure can be either MgO connected and/ or NDZ connected or not connected. Based on the experimental insight by Yates et al. [16] same v olumetric composi tion of MgO and NDZ gives microstructural dependent temperature dependence of thermal conductivity Similar to the procedure used to build polycrystals, composites are built Slight changes are made because of the difference in lattice parameters of MgO and NDZ. The method to generate composite is as follows: 1 24 hexagonal equal size grains were created by specifying their centers but all of them are empty at this step. PAGE 98 98 2 Each grain was oriented in a way that no neighboring grains form low angle GB. 3 Atomic coordinates of MgO atoms are changed according to NDZ lattice parameter. For example, (1, 0, 0 ) coordinate corresponds to ~ (0.39,0,0). 4 Based on the connectivity, grains are assigned to MgO or NDZ. 5 Each grain is filled with the corresponding material unit cells. 6 Atoms with a smaller distance than a specified cut off are termed as bad atoms for each grain 7 Grains are made charge neutral by removing bad atoms 8 Still if system is not neutral, bad atoms are removed to achieve charge neutralit y. 9 Grains are combined and composite is built Composite structures are equilibrated and high energy atoms are screened. These high energy atoms are removed from the system. Equilibrated composite structures are used to compute thermal conductivity and capture interface effects. 4.4.3 Interfacial Effect on Thermal Conductivity As described in section 4.4.1, polycrystals of MgO and NDZ structures are generated. Thermal transport properties are calc ulated using the same approach as used for the single crys tals. The effect of system size is expected to be much less important for such fine grained polycrystals because the dominant mechanism should be phonon/grain boundary rather than phonon/phonon scattering. S imulations on the system shown in Fig. 4 6 and a system with the same microstructure and grain size, but twice the length (formed by simply putting two simulation cell shown in Fig. 4 6 end to end), gave thermal conductivity values within 5% of each other, which is within the statistical errors of the si mulations. T he anharmonicity analysis predicted the thermal conductivities of the single crystals from si mulations would be overestimated by 2% for MgO and 62% for NDZ. Anharmonic correction factors are used i n predicting the thermal conductivity of MgO and NDZ PAGE 99 99 polycrystals In this step, the thermal expansion and bulk modulus are taken as essentially microstructure independent. However, within the approximation made in separating the contributions in the polycrystal thermal conductivity from that of grain interior and gain boundary, the accuracy may not be sufficient to consider more careful analysis. As Figs. 4 9 and 410 show, the calculated thermal conductivities of the fine grained polycrystals of MgO and NDZ are considerably smaller than their single crystal counterparts. The thermal conductivity for polycrystalline NDZ shows a smaller reduction from the singlecrystal value than does polycrystalline MgO (note the difference in scales); this smaller reduction is attributed to the already much sm aller mean free path of phonons in NDZ as compared to MgO. Also, for both systems, the effect is considerably larger at lower temperatures, at which the phonon mean free path is longer. Figure 49 Comparison of thermal conductivity of MgO: single crystal (solid circle) and 3 nm polycrystal (open circle). 0 10 20 30 40 50 60 0 200 400 600 800 1000 1200 1400 1600k [W/mK]T [K] Single Crystal PolycrystalMgO PAGE 100 100 Figure 410. Comparis on of thermal conductivity of NDZ : single crystal (solid circle) and 3 nm polycrystal (open circle). Using a simple model in which the polycrystal is assumed to be describable in t erms of a volume of bulk with thermal conductivity ( 0) and a volume of grain boundary with interface conductance ( G ) (all grain boundaries being assumed to be physically the same), Yang et al. [19] derived a simple relation for the dependence of the thermal conductivity, of a polycrystal on grain size, d: Gd 10 0 (4 8) Using Eq. 4 8, estimates for G are extracted; the result is shown in Fig. 4 11 for both materials, MgO and NDZ As for other materials that have been studied experimentally and/or in simulation, G increases with temperature, reflecting the enhanced anharmonic coupling of weakly coupled pho non modes across the interfaces [163] 0.00 0.50 1.00 1.50 2.00 2.50 3.00 0 200 400 600 800 1000 1200 1400 1600k [W/m K]T [K] Single Crystal corrected Polycrystal correctedNDZ PAGE 101 101 Figure 411. Temperature dependence of the interfacial conductance, G (solid symbols) and Kapitza length, lk (open circles) for A) 3 nm polycrystalline MgO and B ) 3 nm polycrystalline NDZ. Because the thermal conductivity and G are in different units, it is not possible to compare them directly. To make a better comparison, a new variable named Kapitza length is defined as lk = 0/G (4 9 ) 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 0.0 0.5 1.0 1.5 2.0 2.5 0 400 800 1200 1600lk[nm] G [GW/m2K]T [K]MgO 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.3 0.5 0.8 1.0 1.3 1.5 0 400 800 1200 1600lk[nm] G [GW/m2K]T [K] NDZ B) A) PAGE 102 102 The Kapitza length (lk) is a measure of the length of perfect crystal that o ffers the same resistance to heat transport as the interface i.e. t he larger the value of lk, the larger the thermal barrier offered by the interface. From Fig 4 11, lk goes from 30 nm at 300 K to ~3 nm at 1473 K for MgO The effect is less dramatic in NDZ, with lk decreasing from 4 nm at 30 0 K to 1.75 nm at 1473 K T hese decrease in MgO and NDZ are consistent with the trends seen previously for UO2 [150] and diamond [164] in both of which, lk decreases with temperature for both materials. 4.4.4 Grain Size Dependence on Thermal Conductivity Having determined G, Eq. 4 8 was used to predict the grainsize dependence of the t hermal conductivity. Figure 412 shows the calculated grain size dependence of the thermal conductivity at 30 0 K and 573 K for MgO and NDZ normalized to the corresponding single crystal values; thus in both cases, the large grain size limit is / 0=1. In the absence of any experimental data on the thermal conductivity of MgO or NDZ as a function of grain size against which to compare these trends, Fig. 4 1 2 includes the experimental results of Yang et al for yttria stabilized zirconia (YSZ) [19] Despite the considerable difference in materials and microstructures (textured vs. random polycrystal), the simulation data is broadly consistent with the experimental data In particular, the agreement between the simulation results for NDZ and the experiments for YSZ is remarkably good; this agreement probably arises from the fact that they both are low thermal conductivity materials with fluoriterelated crystal structures. After gaining insight at the effect of homogeneous interface on thermal transport, study was extended to include composites. PAGE 103 103 Figure 412. Predicted grain size dependence of the normalized thermal conductivity of MgO (solid and open circle) and NDZ (solid and open triangle) polycrystals compared with experimental results of YSZ (solid square) [19] Polycrystal simulations pr ovided information for MgO MgO and NDZ NDZ interfaces To understand the nature of MgO NDZ interface, composite system with different connectivity is generated. Section 4.4.5 details the information on simulation studies done on MgO NDZ composites. 4.4.5 T hermal Conductivity of MgO NDZ C omposite Based on experimental investigation by Yates et al [16] a volumetric ratio of 6040 (MgO NDZ) was suggested for the composite for the simulation study. To achieve the similar ratio, 10 grains were selected as NDZ and remaining 14 were MgO grains. Polycrystal investigations gave us insight into interfacial effects of bot h the constituents separately. Interfacial effects were larger than grain size itself at this sub nano size level 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 20 40 60 80 100 120k/k0Grain size [nm] Experiment: YSZ: 480K MD: MgO:300K MD: MgO:573K MD: NDZ:300K MD: NDZ:573K PAGE 104 104 and that led to the assumption that no additional interface effects can be seen in the composite. A simple model based on volumetric composition is used to predict the ther mal conductivity of composite. The predicted value of thermal conductivity is given by: poly MgO MgO poly NDZ NDZx x, (4 9) where xNDZ gives the volemtric composition of NDZ (For example, in this case 10/24 is for NDZ) and poly,NDZ is the thermal conductivity of the same size NDZ polycrystal. Similarly values are defined for MgO grain com ponen t. Figure 413. Comparison of thermal conductivity of 6040 MgO NDZ composite with predicted value Based on the simulation results no additional interface effects are observed in the composite. This can be attributed to the limitation of current s tatus of simulation power. PAGE 105 105 4.5 Discussion The atomic level simulation method has provided an important i nsight into the MgO NDZ system. The fidelity of interact ion parameters for MgO and NDZ was tested separately. The interaction parameters for MgO have shown better fidelity with experiment than for NDZ. Finite size and anharmonic analysis tools are used to correct the simulated thermal conductivity values. The modified values after the analysis have shown a good comparison with the experiment. After est ablishing single crystal results, interfaces are introduced into the system. A simple pr ogram is written to generate 2D textured polycrystals. Very fine grain size enhanced the interface effect. Interfacial conductance for MgO and NDZ polycrystals showed correct temperature dependence. Calculation of Kapitza length has shown that interface effects are as big as grain itself for both the materials: MgO and NDZ. A simple analysis is used to extrapolate the effect of grain size and compared with the experim ent study. Both the materials showed the independence of the thermal conductivity on the grain size beyond ~100 nm. That implies the limiting effect of interfaces for bigger grains. A simple analysis based on the volumetric composition of the individual grains is used to predict the thermal conductivity of the composite. Thermal conductivity values of the polycrystals are used to predict the composite thermal conductivity. Comparable results for predicted value and MD results showed that thermal conductiv ity depends only on the volumetric composition of the composite and not the microstructure at very fine grain size level. PAGE 106 106 CHAPTER 5 THERMAL TRASNPORT PROPERTIES OF Mg A l2O4 5.1 Introduction Based on the guidelines for inert matrix materials (for details, please refer Section 1.5) proposed by Matzke et al. [12] magnesium aluminate spinel M gAl2O4 is among the potential inert matrix materials for minor actinide incineration. In particular, MgAl2O4 is very tolerant to neutron irradiation [165, 166] Its ability to tolerate radiation damage is attributed to i) a ver y high recombination rate of Frenkel pairs [165] and, ii) the ability to tolerate significant intrinsic cation antisite disorder [167] .The property of cations interchanging their sites in the spinel structure is called inversion. It has been well established that cation arrangement plays an important role in the material properties. For example, the elastic properties change with cation arrangement in spinel [168, 169] and the bulk modulus of MgTi2O5 changes with cation ordering [170] It has also been found that with increasing temperature MgAl2O4 undergoes inversion [171 173] A deeper understanding of the effects of inversion on the thermoelastic properties of MgAl2O4 is highly desi rable before it can be fully assessed as a potential IMF material. The experimental investigation of the effects of inversion on MgAl2O4 faces two big challenges: 1 It is hard to control inversi on during the synthesis process. 2 It is impossible to separate out the effect of temperature from inversion. Both of these challenges are easily overcome in atomic level simulation. MD is chosen as an ideal vehicle to perform the simulations to capture the effect of inversion on MgAl2O4 properties. One of the key perfor mance metrics of any IMF material is PAGE 107 107 thermal conductivity; therefore an important question here is: How does inversion affect the thermal conductivity of MgAl2O4? 5.2 Selection of Inverted Structure Conventional MD simulation methods and lattice dynamics are used to investigate the effect of inversion on mechanical and thermal transport properties of MgAl2O4 Interaction parameters are taken from Table 22. As discussed in Chapter 3, each unit cell of MgAl2O4 contains 8 Mg, 16 Al and 32 O ions. For the MD simulation study, a cubic arrangement of 4x4x4 unit cells for a total of 3584 atoms is formed. For the lattice dynamics calculations in GULP a 2x2x2 system with 448 atoms is formed. This large system size benefit ed us by giving us more choices for the degree of inversion. There are 2 simple ways to create an inverted structure in MgAl2O4: i ) Direct interchange of Mg and Al; ii) Al can move to a vacant tetrahedral site and Mg to a vacant octahedral. A first principles electronic structure study by Gupta [174] has shown that direct cation exchange is energetically more favorable. T herefore randomly selected Mg and Al atoms were simply exchanged to generate the inverted structures. Since there is no experimental information on possible ordering in inverted structures available, ~7000 structures were generated for each inversion value via random cation interchange. T hree lowest energy structures were identified and used them for the subsequent simulations. Figure 51 captures the distribution of structural energy among the generated structures for the same inversion value. The x axis of Fig. 5 1 covers the whole energy range spanned by generated structures for the inversion values i= 0.0625 and i= 1.0, and the y axis shows the number of structu res having that energy. The PAGE 108 108 e nergy of the structure increases from left to the right on the x axis. Thus structures having energy in the left extreme of the graph are picked. The large number of structures for each inversion maximizes the chances of identifying the energetically most favorable structure. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 1 3 5 7 9 11 13 15 17 19 21 23 25 faction of total structures i =0.0625 12487 Energy (eV) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Fraction of total structures i =1.0 Energy (eV) 12764 12664 Figure 51 Distribution of energy as a function of cation arrangement for the inversion value of A) 0.0625 B) 1.0 12462 B) A) PAGE 109 109 After selecting structures for each inversion, pair distribution function (PDF) for each cation is investigated. PDF provides structural insight with a very simple and easy analysis method. PDF for normal ( i= 0) is shown in Fig. 5 2A, and it shows that first Mg Al neighbor is beyond 3 As inversion is introduced in the system, the height of the peaks is decreasing. A comparison between the PDF of Al Mg f or i= 0 and 0.125 is shown in Fig. 52B. They have very similar peaks but the actual values have gone down by more than 20% a n d peaks are more spread out for i= 0 125 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 g(r) r() Mg Mg Mg Al Al Mg Al Al i =0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 g(r) r () i=0 Al Mg i=0.125 Al Mg Figure 52 PDF for A) normal spinel B) comparison of Al Mg PDF for i=0 and i=0.125 B) A) PAGE 110 110 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.00 3.00 6.00 9.00 12.00 15.00 18.00 g(r) r() 0.125 0.25 0.5 0.75 1 Mg Mg 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.00 3.00 6.00 9.00 12.00 15.00 18.00 g(r) r() 0.125 0.25 0.5 0.75 1 Mg Al Figure 53 Comparison of PDFs for inverted structures for A) Mg Mg, B) Mg Al, C) AlMg and D) Al Al A) B) PAGE 111 111 Figure 53 Cont.. Comparison of PDFs for inverted structures for A) MgMg, B) Mg Al, C) AlMg and D) Al Al 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.00 3.00 6.00 9.00 12.00 15.00 18.00 g(r) r() 0.125 0.25 0.5 0.75 1 Al Mg 0.000 0.005 0.010 0.015 0.020 0.025 0.00 3.00 6.00 9.00 12.00 15.00 18.00 g(r) r() 0.125 0.25 0.5 0.75 1 Al Al C) D) PAGE 112 112 For better comparison between inverted structures, PDFs for each cationcation pair are plotted against each other for all the inverted structures except i =0 (Fig. 5 3). As seen in Fig. 53, with the increase in the value of i the system shows disorder in the form of less sharp peaks (more spread out). Even though spreading of the peaks does not change significantly at i= 1, the intensity of peaks gets higher. That corresponds to ordering at tetrahedral site in favor of Al but disorder on the octahedral sit es (Fig. 5 3B ) Similar resu lts are shown for Al in Fig s. 5 3 C and D 5.3 Effects of Inversion on Properties 5.3.1 Lattice Parameter The unit cell of MgAl2O4 has 8 Mg ions which limits the stepsize of inversion to 0.125 (1/8). Each Mg Al interchange changes the inversion by 0.125. Although this large step size covers the whole range of inversion, it might fail to capture some important physics for intermediat e inversion values. To overcome this, the system size is increased by 8fold, thus reducing the stepsize from 0.125 to 0.015625. This provided more inver sion values for the simulation and better insight into the effects of inversion on material properties MD and GULP simulations methods were used to equilibrate the structure at T = 0 K. As explained above, among the various possible cation arrangements for any particular i value, the lowest energy structure is chosen for the simulation. The effect of inv ersion is shown in Fig. 5 4 and is compared to the effect captured by Andreozzi et al. [175] PAGE 113 113 0.9965 0.9970 0.9975 0.9980 0.9985 0.9990 0.9995 1.0000 1.0005 0 0.1 0.2 0.3 0.4 a/a 0 Inversion Exp Figure 54 Decrease of lattice parameter with inversion A) this MD simulation study B) reproduced from experimental investigation by Andreozzi et al. [176] C) comparison between MD and experimental study Compa rison between experiment and this MD simulation study gives an impression of a much stronger dependence of lattice parameter on inversion in MD but in reality that is not the case. Experimental dependence also involves the ex p ansion due to temperature which is absent in simulation data Even though the experiment in 5 4 B) does not span the full range of inversion available to simulation both show a 8.00 8.02 8.04 8.06 8.08 8.10 8.12 0.00 0.25 0.50 0.75 1.00a ()Inversion C) A) B) PAGE 114 114 decrease in lattice parameter with inversion. This increase in density with increasing inversion can be qualitat ively understood in terms of a simple picture of the bond strength as a function of bond charge. In normal spinel ( i =0), each tetrahedral site is occupied by Mg2+ with 4 O2 neighbors and each octahedral site is occupied by Al3+ with 6 O2 neighbors. Each cation shares ewith e ach neighboring oxygen on both tetrahedral or octahedral sites. On inversion, Al3+ on a tetrahedral site with 4 neighboring oxygens, shares now ewith each neighboring oxygen, an addition of eper bond. Simil arly, Mg2+ on the octahedral site has 6 neighboring oxygen ions thus sharing 1/3 e, a loss of 1/6 e, compared to noninverted spinel. A simplified picture of bond strength being proportional to bond charge gives a larger change at inverted tetrahedral sites than at octahedral sites. In other words, the decrease in Al O bond length at tetrahedral sites is larger than the increase in Mg O bonds at octahedral sites. That, in turn, should produce an overall densification of the spinel upon inversion as is o bserved. 5.3.2 Anti site Formation Energy To inves tigate the effects of inversion further, the energy is determined as a function of inversion. Consistent with the normal s tate being more stable, Fig. 55 shows an increase in energy with inversion. This appears to be linear up to i ~0.5 above which there is a change in slope. A further analysis is performed by calculating the formation energy of each additional cation interchange (Fig. 56). It is evident from Fig. 5 5 that up to i ~0.375, the energy required to form new antisite defects is almost constant. Equivalently, it can be said that for these low inversion cases, existing defects make it neither easier nor harder to form additional antisite defects. Conversely, beyond i= 0.375, it becomes comparatively PAGE 115 115 easier to form new defects with the defect formation energy decreasing by ~25% for the fully inverted system This decrease in formation energy can be understood in terms of the effects of the antisite defects on the local strain fields. At low inversion, the local strain produced by interchanging Mg and Al cations is small, and is easily r elieved by structural relaxations, thus not affecting the formation energy for the next antisite pair to form. As the inversion increases, how ever cation disordering increases and rearrangement of cations is not able to relieve the strain dipole. Such unrelieved strain dipoles lower the energy required to form strain dipoles of opposite orientation. As a result, the antisite formation energy for each additional disorder decreases. Figure 55 Structural energy as a function of inversion for MgAl2O4 PAGE 116 116 0.5 0.6 0.7 0.8 0.9 1.0 0.00 0.25 0.50 0.75 1.00 E/ inversion Inversion Figure 56 Cation a ntisite formation energy as a function of inversion. A simple analysis provides further insight into the order disorder process and wi ll help us understand Figure 56 The order disorder process can be described by a chemical exchange reaction: [177] TAl+MMg MAl+TMg w here, T is a tetrahedral site and M is an octahedral site. The f orward reaction represents the restoration of the ordered structure, the backward reaction rep resents disordering. K1 and K2 are taken as the forward and backward reaction rate constants respectively. The time rate of change of the Al concentration on a T site is given by Al X Al Mg X Mg K Mg X Mg Al X Al K dt Al dXM M T T M M T T T 2 1 (5 1) w here X is the fraction of T sites occupied by Al and is analogous to an activity coefficient and is generally a functi on of temperature, composition and volume. Assuming ideal mixing in each site [178] and at equilibrium dt Al dXT is 0, a pplying these two conditio ns on Eq. 5 1, it becomes: Al Mg Al X Mg X Mg Al Mg X Al X K KM T M T M T M T 1 2 ( 5 2 ) PAGE 117 117 In MgAl2O4 spinel, 2 1 x Mg X x Mg X x Al XM T T and 2 1 x Al XM After substituting, Eq. 5 2 turns into Al Mg Mg Al x x x K KM T M T 2 12 1 2 (5 3 ) As already pointed out, K2 signifies the reaction rate for disordering and K1 for ordering. In the original analysis by Mueller, all values are taken to be unity for the crystals with two nonequivalent sites [178] x x x K K 2 12 1 2 (5 4) In that case for the Eq. 5 4, >1 signifies the disordering of the structure being the dominant reaction. Solving the inequality equation for disordering as the preferential reaction, gives 3 2 x This simple analysis shows that, disordering becomes preferential after the system ach ie ves ~66% inversion. Even though the threshold value f or inversion is diff erent, t his simple model shows that each additional disorder in MgAl2O4 costs less energy after the threshold value. The analysis here is based on all the val ues being unity. The value of may change from unity, given high temperature, electrostatic charge analysis [179] favors inversion and disordered structure release strain by more disorder Combining it with the work presented here, it is proposed that the disordered st ructure at low inversion values forms stress dipoles at inver te d sites to release the strain, thereby lowering the energy of the system This is also shown by the smallest possible inversion, exchange of 1 antisite pair. Figure 56 shows the variation of the formation energy for the inverted structure as a function of distance between inversion sites. PAGE 118 118 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 2 4 6 8 10 12 Formation energy (eV) Cation seperation() Figure 57 F ormation energy as a function of inversion site distance. As se en in Fig. 57 inverted sites prefer to be in close proximity. Based on the result of 1 antisite pair energy and theoretical understanding, it can be hypothesized that, a s disorder increases, tetrahedral sites are increasingly occupied by Al A lso the Al O bond length is decreased at tetrahedral sites. At 50% inversion, t etrahedral sites are equally occupied by Al and M g. Combined with electrostatic analysis (favoring of Al on tetrahedral sites) and reaction rate analysis (less energy expens e for each additional cation disorder after certain inversion value) favoring of disorder can be und erstood. T his transition is see n at lower than 50% inversion, which can be attributed to the approximations and simple reaction ra te analysis. The effect of inversion on the elastic properties is discussed in the next section 5.3.3 Bulk Modulus For a typical solid, the bulk modulus increases as the system becomes denser due to the rapidly increasing interatomic repulsion. Based on this argument, the bulk m odulus is expected to increase with inversion. As the crosses in Fig. 5 8 indicate, the PAGE 119 119 behavior is actually more complex: initially the bulk modulus decreases slightly then levels off, and then increases above i= 0.5. A similar strong dependence of the bulk modulus at high inversions in MgAl2O4 was also reported by Chartier et al in their MD simulations [180] (Fig. 5 8 B)). The interatomic interaction in their study had Buckingham interaction with Morse potential as short range in t eraction which was fitted by Morooka et al [181] That potential is developed to capture the Frenkel pair accumu lation under neutron irradiation. The s ystem migh t also turn to an amorphous state. Since the system under investigation had no point defects other than cation anti site disorder, potential parameters are chosen as described in Table 23. 270 280 290 300 0.00 0.25 0.50 0.75 1.00 Bulk Modulus (GPa) Inversion Figure 58 Bulk modulus as a function of inversion A) this work B) reproduced from the reference [180] Even though values of the bulk modulus are different in the two studies, qualitative effect of inversion on the bulk modulus is very similar in nature. To compare the qualitatitive behavior of bulk modulus, the bulk modulus is normalized to normal spinel ( i= 0) for both the simulation studies and compared in Fig. 59 The small differences are due to the different description of the interatomic interactions used in the two studies. The potential description used in the study by Chartier [180] and developed by Morooka et al. [181] and has the Buckingham form and a Morse potential. 192 194 196 198 200 202 204 0.00 0.25 0.50 0.75 1.00Bulk Modulus (GPa)Inversion A) B ) PAGE 120 120 The good agreement argues that the specific description of the interatomic forces are not essential to obtain qualitative and semi quantitative trends in behavior. The dependence of the bulk modulus on inversion can be unders tood in terms of the separate contributions of inversion to cation dis ordering alone and to the associated volume change. To dissect these effects further, each contribution was studied separately: such separation is not possible experimentally, but is str a igh t forward in simulat ion. The s eparate effects on the bulk modulus are shown in Fig. 510. Figure 59 Comparison of normalized bulk modulus with MD study by Chartier [180] First, the volume of the system was fixed t o that of normal spinel and the bulk modulus was calculated as a function of inversion, thereby picking out the the effec t of the cation arrangement on the bulk modulus (filled circles in Fig. 5 10). Initially the bulk modulus decreases with inversion, the n levels off. It has been already shown (Sec. 5.3.1) that inversion causes shortening of Al O bonds and expansion of MgO bonds at inverted sites, respectively. A similar effect of decrease in bulk modulus with cation disordering was experimentally observed in MgTi2O5 [170 ] PAGE 121 121 260 270 280 290 300 310 0.00 0.25 0.50 0.75 1.00 Bulk Modulus (GPa) Inversion Cation disorder Volumetric Cation disorder + Volumetric Simulation Figure 510. Separating out various effects on bulk modulus I n the case of single crystal MgTi2O5, the decrease in bulk modulus was attributed to the different compressibility of the weaker MgO and stronger Ti O bonds and the way in which the octahedra are arranged. Even though MgTi2O5 (orthorhombic dipyramidal Bbmm) and MgAl2O4 are structurally different, the behavior of cation disordering is analogous because they both display weaker Mg O bonds under cation disordering. Since this bond weakening is prima rily responsible for lowering bulk modulus in MgTi2O5 for low inversions, it is also most likely related to lowering of bulk modulus in MgAl2O4. The leveling off of bulk modulus with inversion suggests the limited effect of weak Mg O bonds in this case, as compared to MgTi2O5 [170] This also suggests that there is an increasing effect of Al O bonds with inversion. Since the lengthening of the Al O bond has precisely the opposite effec t on elastic properties, a small increase in the bulk modulus at higher values of I is seen. Second, the bulk modulus for norm al spinel was calculated as a function of volume. Using the relationship between the d egree of inversion and volume, the volume d ependence of the the bulk modulus w as replotted as a function of the inversion (filled PAGE 122 122 triangles in Fig. 5 10). It is evident from Fig. 5 10 that the decreas e in volume monotonically increase the bulk m odulus, consistent with the usu al behavior of crystal s. To capture the combined effect of volume and cation dis ordering, the average of the two effects is taken as an estimate for the dependence of the bulk modulus on inversion. As the triangles in Fig. 510 show s, this prediction matches the actual simulati on results reasonably well. In particular, it shows that the volumetric effect essentially negates the effect of cation disorder up to i ~0.375. For higher degrees of inversion, further cation disorder has little effect, while the volumetric contracti on lea ds to an increase in the bulk modulus 5.3.4 Thermal Expansion The behavior of bulk modulus was attributed to the relative strength between inverted Mg O and Al O bonds and to the volume change. These effects are also manifested in the thermal expansion. To calculate the thermal expansion, a series of MD simulations were run at different temperatures for NPT ensembles (for details refer to Chapter 2). The coefficient of thermal expansion is computed for each inversion value following the procedure descibed in Sec. 2. 10 The variation of lattice parameter for i= 0 with t emperature is shown in Fig. 511. Similar variation of lattice parameter with temperature is shown for all other inversion values. T he thermal expansion coefficients for each inversion val ue a re compiled in Fig. 5 12. The similarity in the effect of inversion on the bulk modulus and thermal expansion is expected. The bulk modulus and the thermal expansion can be related through the Grneissen relation [30] : B Cv3 (5 4 ) PAGE 123 123 where Cv is the neisen coefficient, which measures the anharmonicity of the interatomic interactions. In Eq. 5 4, if everything else is constant, the thermal expansion and bulk modulus are expected to show opposite behavior with inversion. As seen by Fig 5 8 bulk modulus increases with inversion after reaching i= 0.5 and from Fig. 51 2 thermal expansion decreases. Figure 511. Variation of lattice parameter with temperature for i= 0. 1.000 1.001 1.002 1.003 1.004 1.005 1.006 1.007 1.008 1.009 0 250 500 750 1000 1250 1500ratio (a/a0)T [K]i =0.0 PAGE 124 124 5.00 5.40 5.80 6.20 6.60 0.00 0.25 0.50 0.75 1.00 x10 6 ) [/K] Inversion Figure 512. Coefficient of thermal expanion as a function of inversion. v and B; calculated using Eq. 54 The 13 and does not depend strongly on inversion 2.0 experimentally reported by Chopelas and Hofmeister [182] Figure 51 3 Dependence of the Grneisen coefficient on inversion. 1.7 1.8 1.9 2.0 2.1 0.00 0.25 0.50 0.75 1.00Inversion PAGE 125 125 The G rneisen coefficient is a measure of the anharmonicity of the syst em. The good agreement between MD result s and the experimental find i ngs shows that the interatomic potential yields the same level of anharmonicity as the experiment. As shown previously that the thermal expansion is related to the thermal conductivit y through the anharmonicity (for details, please refer Ch. 2). Thus, the analysis suggests that the simulated thermal conductivity values should be at least semi quantitatively correct. 5.3.5 Thermal Conductivity As noted in the Sec 5.1 MgAl2O4 undergoes inversion under neutron irr adiation. A similar simulation set up as described in Section 2.10 is used to compute the thermal conductivity. These thermal conductivity values are then corrected for system size effect and compared to the experiment (Fig. 514). Thermal conductivity res ults for normal ( i= 0) MgAl2O4 are shown as open circles in Fig. 514. The MD results are shown as open circles and corrected represents the thermal conductivity values after harmonic analysis is performed on these MD values As explained in Chapter 2, harmonic analysis provides a correct ion factor based on the accuracy of interaction parameters in reproducing bulk modulus and thermal expansion values to the experiment al values. Table 51 provides detailed information about the correction factor calculated by harmonic analysis. Based on the numbers from Table 51, the correction factor comes out to be equal to 1.52. That means, based on the fidelity of the interaction potential, thermal conductivity values will be 1.5 times higher than the experiment al coun terpart (For details, please refer Eq. 46). After dividing MD simulation thermal conductivity values by 1.52, they are plotted as open triangles in Fig. 513 and termed as PAGE 126 126 corrected. The filled circles are taken f rom the experimental studies of B u rghart z et al [183] Table 5 1. Comparison of thermal expansion and bulk modulus for harmonic analysis (x10 6 / 0 K) Bulk Modulus (GPa) Experiment 8.89 [184] 197.4 [172] Simulation 6.1 275.74 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 250 500 750 1000 1250 1500 k [W/m K] T [K] MD Exp Corrected Figure 514. Variation of thermal conductivity for MgAl2O4 with temperatur e, open symbols represent the simulation results and filled circles are points reproduced from the experimental study by Burghartz et al. [183] The similarity between experimental and simulated thermal conductivity values gave the confidence to use the simulation tool for further investigation. Since the working temperature for nuclear fuels is in the range of 800 K 1500 K, investigation on thermal conductivity as a function of inversion is performed at 1250 K. Finite system size analysis (For details, please refer Eq. 246) is p erformed to take into account any artifacts produced by system size limitations. A system size analysis PAGE 127 127 for each inver sion value is shown in Fig. 515 The circles at y=0 gives the estimate of inverse of thermal conductivity for infinite system size. These infinite system size thermal conductivity values are computed using a linear fit and taking the inverse of intercept at y=0 (Chapter 2). Figure 515. System size analysis for different inversion values for MgAl2O4 The extrapolated thermal conductivity values for different inversion values of MgAl2O4 system are plotted as open triangle s against inversion in Fig. 516. The rather weak dependence of the thermal conductivity on inversion is quite surprising when viewed in t erms of the thermal transport properties of defected ceramics materials. In particular both experiments and MD simulations of fluoritestructure materials (e.g., yttria doped zirconia and hyperstoichiometic UO2) have shown a strong decrease in thermal cond uctivity with even quite low defect concentrations [63, 185] The key difference between the spinel case and the fluorite structured materials is that here disorder only involves antisite defects, while for the fluorites the disorder involved 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0 0.002 0.004 0.006 0.008 0.01 0.012 mK/W ]1/Lz1] i =0.0 i =0.25 i =1.0 i =0.75 i =0.50 PAGE 128 128 oxygen vacancies (zirconia) and oxygen interstitials (ur ania), which presumably affect the phonon properties and hence thermal conductivity much more strongly. B y recalling the relation between anharmonicity and thermal conductivity of the system, as defined by the Eq. 4 2 the inversion dependence in a quanti t a tive manner can be understood. This representation of thermal conductivity is based on the Debye approximation which assumes all phonon modes present in the system are acoustic and cont ribute to the thermal conductivity. It works well for very simple sys tems but as the structural complexity increases, the actual thermal conductivity values deviate from the predict ed values. For the system, even though Eq. 4 2 cannot be expected to quantitatively predict the thermal conductivity, it should still capture the qualitative picture of the variation of thermal conductivity with inversion. 0 2 4 6 8 10 0.00 0.25 0.50 0.75 1.00 k (W/m K) Inv MD Calculated Figure 516. Dependence of thermal conductivity variation on inversion The thermal conductivity calculated from MD does show a shallow minimum at i ~0.5, not predicted from the anharmonicity analysis alone. This minimum can be attributed to the degree of disorder in the system which, for the fluorites, led to a significant drop in the thermal conductivity [186188] There are various factors that can PAGE 129 129 produce differences between thermal conductivit y values calculated from MD and the predicted values by using Eq. 42 They are investigated one by one: Effect of density: As already shown, t he volume decreases with inversion (Fig 5.2) This increase in density increases the sound of speed in the material. The effect of cation disorder and volume on the thermal conductivity is sepa rated out with the help of these two simple cases: Case 1: Change the lattice parameter to i= 0.0 value for i= 0.5 and 1.0 The lattice parameter of the inverted structures ( i= 0.5 and 1.0) is increased to the value of i= 0.0 (normal). By comparing the thermal conductivity values of the i= 0.5 structure with two different lattice parameters, the effect of density on thermal conductivity is calculated. Similar study is done on i= 1.0 structure also. The results are shown in Fi g. 5 16 and showed a negligible effect (~1%) of the density on the thermal conductivity. Case2: Change the lattice parameter to i= 1.0 for i= 0.0 To extend the study of case 1 on perfectly ordered structure ( i= 0), the lattice parameter of normal MgAl2O4 is changed to the value of i= 1.0. The results are also shown in Fig. 515 and again no significant effect of density on thermal conductivity is found. Arrangement of cations : This analysis is motivated by the fact that for each inversion, there could be thousands of possible ways to arrange the cations. Similar to the analysis mentioned in Sec. 5.2, was performed here. Instead of choosing the lowest energy structures (as was done for the thermal expansion) structures with lowest and highest energies for same i values we re chosen for thermal conductivity simulations. A PAGE 130 130 difference of less than 1% in the thermal conductivity between these structures was found. This is in accord with the expectation. Since random distribution of cations does not introduce any new defect in the system, no change in thermal conductivity was expected. Figure 51 7 Density effect on thermal conductivity Total scattering sites: A key effect of point defects is the number of possible scattering site s in the system produced by antisite defect creation. To capture the effe ct of point defects, a simple combinatorics analysis was used. W ith each inversion, the possible number of scattering sites is Al n Mg ni iC C (5 5) w here Mg niC denotes the number of possible ways to generate the structure for the inversion i involving ni Mg and ni Al ions. Thus the product is a measure of possible number of scattering sites for the inversion value ( i ) As an example, for i =0, Mg and Al occupy tetrahedral and octahedral positions and there is only one way to arrange the 0 1 2 3 4 5 6 7 0.00 0.25 0.50 0.75 1.00 (W/m K)Inversion i =0.0, volume of i =0.0 i =0.0, volume of i =1.0 i =0.5, volume of i =0.5 i =0.5, volume of i =0.0 i =1.0, volume of i =0.0 i =1.0, volume of i =1.0 PAGE 131 131 cations. For one anti site pair, in one unit cell system, 1 Mg can be in any of the 16 possible Al sites and also Mg can be chosen in 8 different ways. The multiplicity of these possible arrangements with inversion can be expected to have an impact on the thermal conductivity. Figure 517 captures the variation of the number of possible arrangement s for each cation site for each inversion. 0 10 20 30 40 50 60 0.00 0.25 0.50 0.75 1.00 Log (combination) Inversion Combination Mg Al Figure 51 8 Combinatoric analysis: y axis shows number of possible combinations for Mg Al exchange possible in 2x2x2 unit cell system of MgAl2O4 Even though i= 1 signifies the totally inverted structure, it does not have the most number of possible combinations, because all of the A sites are now totally occupied by Al. T here is a maximum in the number of different defect arrangements at i =0.67, at which point the system would display the maximum number of different scattering environments. If there were no other effects, a minimum in the thermal conductivity at i= 0.67 is expected. The fact t hat the thermal conductivity is actually a minimum at close to i =0.5 sugge sts that the disorde r on the A sublattice actually a ffects the phonon scattering more strongly than the disorder on the B site. This can be understood in terms of local strain produced at inverted sites. As previously discussed, the TAl sites experience gr eater strain than the MMg sites and thus affect material properties to a PAGE 132 132 greater extent. As inversion increases, TAl experiences more chemical disorder; at i= 0.5 half are occupiedby Mg and hal f by Al above i =0.5 they become more and more populated by Al ions. 5.4 Conclusion The MD study has provided significant insights into the eff ects of inversion on the thermo elastic properties of MgAl2O4. Ease of formation of more disorder in already inverted structures is shown by this study. The effect of inversion is extended on thermo elastic properties of MgAl2O4 by scanning the whole range of inversion ( i= 0 1). The effects of temperature and cation arrangement are also separated. The decrease in lattice parameter w ith inversion is shown to arise from a simple charge sharing between cationoxygen at tetrahedral and octahedral site. The effect of disorder already present in the structure on the formation of new cation arrangement is also presented. A simple theoretical analysis favors the formation of disordered structure after i~ 0.67 and the simulation results presented the favoring of additional cation disorder after i ~0.37. The difference from the theoretical model is explained in the light of simplified analysis an d charge analysis favoring the inverted structure for MgAl2O4. The variation in elastic constants and bulk modulus with inversion are reported and compared with other DFT and MD studies. Contributions to the bulk modulus are analyzed in two parts: volumet ric and cation disorder. This study has shown that weakening of Mg O bonds at inverted octahedral sites essentially negates the volumetric effect on bulk modulus up to i ~0.5. After i= 0.5, stronger Al O bonds at inverted tetrahedral sites and volumetric eff ects dominate and control the bulk modulus of MgAl2O4. PAGE 133 133 The effect of thermal expansion as a function of inversion is also captured and explained. As expected from the bulk modulus results, thermal expansion remain almost unchanged up to i ~0.5 and decreased afterwards. Anharmonicity of the system is calculated using bulk modulus and thermal expansion and compared to the experimental value. The thermal conductivity for the normal structure reproduces values similar to the experimental ones. The study is exte nded to include the effect of inversion on thermal conductivity, which has never been done before. The thermal conductivity results are compared with the theoretically computed values. Deviation from the theoretical model is reported and explained with the help of a combinatorics analysis. PAGE 134 134 CHAPTER 6 THERMAL TRANSPORT PROPERTIES OF B i4T i3O12 6.1 Introduction: Thermal Barrier Coating Thermal barrier coatings (TBCs) are applied on materials operating at elevated temperatures to act as insulation. TBCs are widely used in turbine blades, combustor cans, gas turbines and aero engine parts. The most common place where to see a gas turbi ne engines is commercial jets and in electricity generating plants In gas turbines a pressurized gas spins the turbine. Press urized gas is generated by burning fuel in the engines. Possible fuels include propane, kerosene, jet fuel or natural gas. The main advantages of gas engines are that they are smaller in size than their counterparts and have a greater power to weight rati o. Their use is limited to power reactors and commercial jets or tanks because of the cost, high operating temperature and high spin speed of the turbine and limitations of constant load rather than fluctuating one. Jet engines can be classified as either turbofan or turbojet. Both work on the same principle. A schematic of a simple turbofan is shown in Fig. 61. They have three parts: The c ompressor compresses the incoming gas to about thirty times its original pressure. The c ombustion area burns the fuel and produces high pressure, high velocity gas. The t urbine extracts power from the high velocity high pressure gas. Figure 61. Simple schematic view of turbofan gas turbine engine [189] PAGE 135 135 Thermodynamically the Brayton cycle is used to describe gas turbines. In the Baryton cycle, air is compressed isoentropically, combusts at constant pressure and expands again isoentropically at the initial pressure. An idealized Brayton cycle is shown in Fig. 62. Figure 62. Idealized Brayton cycle [190] Similar to other heat c ycles, the efficiency of the Brayton cycle is proportional to the combustion temperature. There is thus a strong incentive to increase the operating temperature. The limiting factor to elevate the operating temperature is the limitations of the performance of the materials that make up the engine and its parts to withstand heat and pressure. To elevate the temperature, thermal barrier coatings (TBCs) are applied to the turbine blade itself [191] A TBC is primarily made of four layers (Fig. 63): 1 Ceramic topcoat: This layer provides the thermal insulation for the turbine blade. It has very low thermal conductivity and is designed to withstand a large number of thermal cycles. The most common TBC material is yttria stabilized zirconia (YSZ) which has a thermal conductivity around 2.3 W/mK at ~10000C and melting point around 27000 ck. 2 Thermally Grown oxide (TGO): This layer is created when the topcoat reacts with the bond coat layer. It is about 1 PAGE 136 136 3 Bond coat: This layer is about 75purpose of this layer is to hold the topcoat on t he substrate. This layer is made of Ni and Pt and often consists of more than one layer of different compositions. 4 Substrate: The turbine itself is generally made of a superalloy and is cooled by hollow channels inside the turbine blade. Fig ure 63. S chematic of a TBC [192] The important considerations in the selection of a TBC material are its ability to withstand thermal stresses with repeated heating and cooling cycles; thermodynamic compatibility with oxides formation of bond coat; and low thermal conductivity. The current material of choice for TBCs is yttria stabilized zirconia (YSZ) in its metastable tetragonal phase; it will also remain the material of choice for the current oper ating temperatures [193] However, YSZ has poor creep resistance at high temperatures PAGE 137 137 which results in crack and eventually spallation. YSZ also has high diffusivity of oxygen that results in oxidation of bond coat layer. The limiting factors of YSZ and the need for higher operating temperature provides a motivation for developing new materials for TBC applications. Table 61. Different forms of layered perovskites Composition/Formula Layer orientation N ame and example A n+1 B n X 3n+1 or A2[An 1BnX3n+1] {001} p Ruddlesden Popper Sr3Ti2O7 A[A n 1 B n X 3n+1 ] {001} p Dion Jacobson RbCa2Nb3O10 Bi 2 O 2 [A n 1 B n X 3n+1 ] {001} p Aurivillius Bi4Ti3O12 A n B n X 3n+2 {110} p Ca 4 Nb 4 O 14 A n+1 B n X 3n+3 {111} p Ba 5 Nb 4 O 15 Even though all the requirements are critical in the selection of TBC material, low thermal conductivity remains the primary criterion. Desirable properties of TBCs include large average atomic numbers, large unit cell and the possibility of bonding and st ructural disorder [194 ] Aurivillius compounds posses s many of the required properties for TBCs. These anisotropic layered materials have weakly bonded planes (Bi2O2 layers PAGE 138 138 or any other layer between perovskites) between layers of pseudoperovskites having rigid polyhedra. T his intrinsic structural disorder acts as a barrier to phonons and reduces their mean free paths. Thus the weakly bonded planes in layered perovskite materials are useful in achieving lower thermal conductivity values without introducing any external defec ts or dislocations in the system. Anisotropic thermal expansion exhibited by these layered materials is useful in minimizing residual stresses. Based on the composition of inter slabs and/or the orientation of the layer plane relative to aristotype cubic l attice, these layered perovskites are divided into five groups as shown in Table 61. In Table 61 for {001}p notation, p refers to pseudocubic. Among these layered perovskite types, Aurivillius structures possess many desirable properties for use as a TB C. As shown in Table 61, Aurivillius compounds are represented by the general formula (Bi2O2)2+[An 1BnO3n+1]2 , where n represents the number of perovskite layers separating Bi2O2 layers. A simple Aurivillius compound with n=3, Bi4Ti3O12 {(Bi2O2)[Bi2Ti3O10]} has many promising properties (large atomic number, large unit cell, anisotropic structure, layered structure) to be a potential TBC material. A recent study by Shen et al. has investigated the low value and anisotropic nature of the thermal conductiv ity of Bi4Ti3O12 (BIT) and attributed the low value to the density difference between layers [20] Further investigations pertaining to the effects of density difference between layers and other intrinsic factors on limiting thermal transport properties of BIT might be useful. The experimental studies however, face two big challenges: 1 It is very hard to synthesize single crystals 2 It is impossible to tune the density of layers to the desired value. PAGE 139 139 Both of the experimental limitations are easily handled by atomic level simulation. Therefore, the experimental investigation is expended with the help of MD simu lation. In this chapter, the effects and limitations of density and intrinsic structural factors on the thermal transport of single crystal BIT are illustrated. 6.2 Selection of Potential Parameter As discussed in Section 28, for the study of BIT, th ree sets of potential parameters were chosen to capture the short range interaction. They all had same Bi O and O O interactions but different Ti O parameters. Table 24, has all the parameters. They are given again below for quick reference. Table 62 Sh ort range potential parameters for Bi4Ti3O12 [65, 67] (Repeat of Table 24) Species A (eV) ( ) C (eV 6 ) Bi+3O2 49529.4 0.2223 0.00 O2O2 9547.96 0.2192 32.0 Ti+4O2 2131 2131.04 0.3038 0.00 760 760.47 0.3879 0.00 2549 2549.4 0.2989 0.00 These interactions are used to calculate to calculate structural properties of BIT. To check the quality of parameter sets, they are compared with experimental v alues as shown in Table 6 3 As explained in Chapter 2, interaction potentials are the back bone of MD simulation. Interaction potentials are developed in order to reproduce material properties by simulation. Based on the Table 62, all interaction parameters overestimate the system volume. Interactions 760 and 2549, overestimated the volume PAGE 140 140 by ~2 5%.Whereas for interaction 2131, the simulated volume is within 0.05% of experimental value. This simple c omparison of structural features to the experiment provides us an opportunity to choose the best fit interaction parameters. Bear ing this in mind, all the further simulations are performed using 2131 potential parameters. Table 63 Comparison of interacti on parameters for BIT 2131 760 2549 Exp. [118] a() 5.42 5.53 5.48 5.4 5 b() 5.34 5.46 5.43 5.41 c() 33.41 33.43 33.35 32.8 3 90.00 90.00 90.00 90.00 89.93 90.16 89.90 90.00 90.00 89.99 89.99 90.00 Vol. ( 3 ) 968.29 1009.06 992.46 967. 98 6.3 Thermal Expansion At low temperature (T < 900 K), BIT has monoclinic structure with all the crystallographic angles of 900. Confusion arises because of the conventional classification of the crystal systems. If all the angles are 900 and sides are not equal, it is classified as ortho rhombic and not monoclinic. However, there are symmetry oper ations tied with each system. The orthorhombic B2cb space group has 2fold rotation, a 21 screw axis, mirror plane and glide plane symmetries whereas B1a1 has only mirror and glide as symmetry elements. So even though description of lattice features leads to the conclusion of BIT being orthorhombic, when combined with symmetry considerations it is clearly monoclinic PAGE 141 141 The thermal expansion is simulated for the NPT ensemble. Since the cut off distance is ~10, the smallest sys tem is generated by building a structure of 6x6x1 unit cells in the x, y and z directions. This supercell structure has dimension of ~30 in each direction, and is shown below (Fig. 64). Figure 64 clearly shows the difference in tilting behavior of TiO6 octahedra along x and y axes which can be represented as a0bcin the Glazer notation (for details, please refer the Sec. 3.5) [138] Figure 64. 6x6x1 super cell of BIT viewed along x and y axes The tilting of the TiO6 octahedra in the pseudoperovskite layer affects the Bi2O2 layer also. As seen in Fig. 64, the two layers are tilted in a sim ilar fashion along the y axis. The e ffect of octahedron tilting on lattice parameter becomes clear when the coefficient of thermal expansion along each direction is computed The normalized lattice parameters are compared for the different directions in Fig. 6 5. Table 64 compares the simulation values to the experimental studies X Y Z PAGE 142 142 Table 64 Comparison of thermal expansion values Axis Simulation (x10 6 /K) Experiment (x10 6 /K) X 5.14 10.1 [195] Y 3.09 4.45 [195] Z 3.77 0.20 [195] 16.12 [196] Average 4.00 4.91 [195] ,11 [197] 0.999 1.000 1.001 1.002 1.003 1.004 1.005 1.006 0 200 400 600 800 1000 1200 a T /a 0 Temperature (K) X Y Z Figure 65. Coefficient of thermal expansion of BIT along x,y and z axes V ery few experiments have reported the thermal transport properties of BIT. The discrepancy among the experiments is larger than that between the simulation results in this work and experiment [195197] The experimental study by Hirata [195] was done to investigate the physics behind ferroelectric behavior of BIT near its Curie temperature. I t reported that the a axis (x axis) displacement of Bi atom with respect to PAGE 143 143 TiO6 octahedra in the perovskite layer is mainly responsible for ferroelectricity of BIT. The experimental investigations by Subbarao [197] were done on polycrystalline samples and mainly focused on to the structural changes around the C urie temperature of BIT (6750C) Another experiment al stud y by Hirata et al. [195] found an average expansion value (4.9x106/K) similar to the one reported earlier by Subbarao (10x106/K) [197] Due to this large variation among experiments, expansion coefficients are compared to the average value. Similar to the experiment findings, MD simulation also captures the anisotropy in expansion coefficients. The strong difference between the x and y values of the thermal expansion is interesting and can be attributed to two structural features: 1 Octahedra tilting, 2 The difference in thermal expansion of the layers. A very good description of octahedron tilting on lattice parameter is given by Megaw [137] To compute the octahedra tilting effect, structural analysis is used in Crystalmaker. Since there is no til t along x, only y and z tilts need be compute d An analysis tool is required to calculate the tilting of the octahedra. Due to the lack of any sophisticated analysis tool tilting is calculated using the angle tool in Crystalmaker. As seen from Fig. 64, tilti ng of octahedra along Y axis can be measured in two ways: inside tilt (inset of Fig. 66 A) and outsidetilt (inset of Fig. 66 B) Both are measured and shown in Fig. 66. PAGE 144 144 0 2 4 6 8 10 12 14 0 100 200 300 400 500 600 700 Tilt Angle ( 0 ) Temperature (K) Inside tilt 0 2 4 6 8 10 12 0 100 200 300 400 500 600 700 Tilt Angle ( 0 ) Temperature (K) Outside tilt Figure 66. Temperature variation of tilt angles of A) Inside and B) Outside tilt of the octahedra along Y axis (b axis). As seen in Fig. 6 6, there is negligible effect of temperature on octahedra tilting. The second structural feature of the layering is investigated to characterize the nonuniformity of the thermal expansion. To meas u re expansion of different layers, the cation cation distance is measured as a function of temperature. It is not trivial to resolve the distances in different direction. B) A) PAGE 145 145 Th e minimum s eparation between same cations in each layer can be used to compute thermal expansion of each layer. Again Crystalmaker is used to calculate the distances between the same cations of each layer The methodology to calculate change in same type o f closest cations is described below: 3.5 3.6 3.7 3.8 3.9 4.0 100 0 100 200 300 400 500 600 700 Ti Temperature (K) Avg X Avg Y Avg Z Ti Ti Figure 67 Temperature profile of nearest Ti Ti distance in BIT structure Pseudoperovskite layer: Ti is only present in pseudoperovskite layer thus temperature profile of Ti Ti smalle st distance is used to compute thermal expansion of the layer. The distance between Ti Ti is measured along both X and Y axes using Crystalmaker. Interatomic distance of closest Ti Ti is calculated in each pseudoperovskite layer. Average of the bond length at each temperature is taken and is shown in Fig. 6 8. Since there is only one layer to compute thermal expansion in Z direction, no error bar is denoted for z axis bond length. Bi2O2 layer: Since Bi is present in both the layers, Bi Bi is calculated separately for each layer. A system similar to that used for Ti Ti bond calculation is used Instead of masking Bi and O, Ti and O are masked. Bi Bi temperature profile, shown in Fig. 68 is used to compute the thermal expansion of Bi2O2 layer. PAGE 146 146 3.3 3.4 3.5 3.6 3.7 3.8 3.9 100 0 100 200 300 400 500 600 700 Bi Bi distance () Temperature (K) Avg X Avg Y Avg Z Figure 68 Temperature profile of Bi Bi cation distance in Bi2O2 layer Analysis for both the layers provides some interesting insight into their response towards the temperature. But it is not trivial to suggest any effect of these layers on t he overall thermal expansion. Besides the atomic disorder, density difference between layers might also play some role in the anisotropy of the heat transport properties of BIT. A detailed investigation to capture the effect of mass on thermal expansion gave no conclusive indication. The investigation is extended to observe the effect on thermal conductivity. A recent experimental investigation of thermal transport properties of BIT by Shen et al. reasoned the density difference between layers was responsible for its low thermal conductivity [20] The experimental study and the effect of density on thermal transport are discussed in detail in the following section. 6.4 Thermal Conductivity BIT exhibit s anisotropic thermal conductivity. As discussed in sec. 23, BIT can be indexed as orthorhombic (because of its unique nature of being m onoclinic with orthorhombic axes). The thermal conductivity can be written in matrix form as PAGE 147 147 33 22 110 0 00 0 0 (6 1) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0 0.02 0.04 0.06 0.08 0.1 0.12 1/ [mW/K] 1/L z [nm 1 ] Z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.02 0.04 0.06 0.08 0.1 0.12 1/ 1/L z 1 ] Y X Figure 69 System size analysis for T = 300 K in all the direction A) x,y and z system size dependence B) x and y system size analysis only The off diagonal terms are zero because all the angles are 900. The thermal conductivity is calculated in all three directions. As discussed in Chapter 2, it is necessary to perform system size analysis, from which the thermal conductivity values for the infinite system size are computed. As a representative example, the system size A) B) PAGE 148 148 analysis for T = 300 K is shown in Fig. 6 9 A similar system size analysis is performed at all the temperatures. The thermal conductivity for infinite system siz e systems are shown in Fig. 610. 0 1 2 3 4 5 100 300 500 700 (W/mK) T (K) X Y Z Figure 610. Temperature and direction dependence of thermal conductivity for BIT Anisotropy in the thermal conductivity is very well captured by the simulation study. Experimental investigation of the thermal conductiv ity of BIT is shown in Fig. 6 11 and simulation does produce similar qualitative dependence of the thermal conductivity on temperature. PAGE 149 149 Figure 61 1 Thermal conductivity of BIT by experiment [20] The experimental results are shown on Fig. 61 1 and need some explanation as figure in itself is not sufficient to explain its results. Experimentally it is very hard to synthesize single crystal BIT, so results shown are of textured polycrystalline sample. In the experiment, two types of samples were prepared: i) c axis textured sample and ii) random polycrystalline sample. The thermal diffusivity was measured using the thermal flash method [82] and thermal conductivity is calculated using Eq. 241. Reported experimental sample densities (textured: 7882 kg m3, random: 7590 kg m3) were lower than the theoretical value (8040 kg m3). The the rmal conductivity from the random sample i s shown by fi lled circles and the textured sample with filled triangles in Fig. 61 1 It has been assumed in the experiment that the thermal conductivity along a and b axes is the same and is shown by filled PAGE 150 150 squares (ab plane) in Fig. 61 1 Since the thermal conductiv ity along a and b axes is the same, Eq. 6 1 can be written as cc ab ab 0 0 0 00 0 (6 2) p it can be written in the form of axial thermal conductivities as: cc ab p 2 3 1 (6 3) p cc ab can be calculated as: cc p ab 3 2 1 (6 4) The calculated data for thermal conductivity along ab plane is shown as fi lled squares in Fig. 61 0 The t hermal conductivit y results from the experiment and simulation are in qualitative agreement. A theoretical discussion on thermal conductivity behavior over temperature gives the 1/Tn dependence. This temperature dependence corresponds to a decrease in thermal conductivity v alues with temperature for non metals. A n increase in the thermal conductivity at high temperatures (T >800 K) is seen in Fig. 6 1 1 This is attributed to radiative comp onent of the thermal transport Contribution of this radiative component of heat trans fe r in BIT thermal conductivity increases the apparent thermal conductivity of the material beyond 800 K. Density Effect on Thermal Conductivity : As explained in Section 6.1, low thermal conductivity material s are needed for TBC applications. The motivation behind PAGE 151 151 this analysis is to understand the factors contributing towards lowering the thermal conductivity of BIT and then to exploit them to reduce the thermal conductivity value even lower. Clarke has described, a minimum thermal conductivity value expression based on materials elastic and structural properties [194] : 3 2 2 1 6 1 3 2 minM E m (6 5 ) Where, M is the molecular mass, m is the number of atoms per molecule. The expression of 62, changes different atomic masses to effective mass and that leads to absence of optical phonons in the material. Overall effects of this assumption are unknown but this underestimates thermal conductivity. As shown in experiment thermal conductivity (Fig. 611), even after the phase change in BIT, the the rm al conduct ivity does not change much. This reflects the limited role of specific atomic arrangement s on thermal conductivity in BIT. When the phonon mean free path is calculated in the limit o f high temperature, it gives a value of 0.24 nm (half of interat omic spacing 0.46 nm) [20] Since there is no ionic disorder present in the system and atomic disor der has a very limiting effect on thermal conductivity, the low thermal conductivity is attributed to the scattering of phonons from the superlatt i ce of alternating pseudoperovskite and Bi2O2 [20] The density of Bi2O2 fl uoritelike block is 11,397 kg m3 and the thickness is 0.45 nm whereas the perovskite block has a density of 6487 kg m3 with a th ickness of 1.21 nm. This density difference becomes larger in low temperature structure. Since MD simulation gives total control over structure and its features, the density effect can be easily probed. T he effect of the density difference is investigated in a systematic manner. PAGE 152 152 In the BIT structure, the PP layer is less dense than the BO layer. A simple method to probe the density effect on thermal conductivity of BIT is described below: 1 The PP layer is made heavier by increasing the mass of Ti (HT) 2 The P P layer is made heavier by increasing the mass of Bi only in PP layer (PPB) 3 The BO la yer is made lighter by reducing the mass of Bi in BO layer only (LB ) A simple schematic is shown below of the above methodology (Fig. 612). The density of BO and PP layers are denoted by d1 and d2, respectively. In the case of HT, the masses of the Ti in PP layer are artificially increased and since there is no Ti in BO layer, this captures the effect of mass only. The other two procedures, LB and PPB, involve changing mass of Bi only in BO layer or in PP layers that introduces two different kinds of Bi in the system. Thus these two scenarios include the effect of extra scattering sites along with the mass effect. Figure 612. Schematic of method to investigate the density effect on thermal conductivity of BIT B O P P d1 d2 d1 d1 d2 d2 Normal HT d1>d2 LB Xtra scattering sites d1 d1 PPB BO PP PAGE 153 153 At room temperature, all the above systems were compared for their respective values of thermal conductivity. Anisotropi c nature of thermal conductivity allows us to choose between x, y or z axes thermal conductivity values to compare the effect of density on thermal conductivity. Thermal conductivity along the z axis is expected to reflect the effect of density more than x or y axis. Since dependence along x or y axis is expected to be same. The effect of the density is captured along the x and the z axis. T he temperature is set at 300 K for the comparison. A s ystem size analysis is performed at T=300 K for the thermal cond uctivity in x direction for all the systems and the comparisons are shown i n Fig. 6 13. Figure 613 shows a systematic system size dependence on all the systems along the x axis, but it does not become clear what the effect of density on thermal conducti vity is After extracting the infinite thermal conductivity for all the systems, the result is shown in Table 63. 0.10 0.15 0.20 0.25 0.30 0.35 0.00 0.02 0.04 0.06 0.08 0.10 0.12 1/ [ m K/W] 1/L z [nm 1 ] normal HT LB PPB Figure 613. System size and density dependence of thermal conductivity for BIT PAGE 154 154 Table 65 Thermal conductivity variation for all 4 systems System K) Normal 5.63 LB 5.86 HT 5.78 PPB 7.1 6 The effect along the z axis is more prominent as shown in Fig. 614. As expected, density difference is responsible for the low value of the thermal conductivity of BIT. Both the cases, PPB and HT increase the density of PP layer to be equal to the BO layer. If density was the only reason for the low value of thermal conductivity, the thermal conductivity increase should have been same for PPB and HT scenarios. The increase in the thermal conductivity is larger for HT case than PPB. The PPB scenario has two types of B i in the system and that reflects as point defect in the syst em. The density increases the thermal conductivity value but presence of point defects lower the overall increase in the thermal conductivity. Whereas in HT mass of all the Ti increases, thus no point defects are created in the system. That is why, a larger increase is observed in the case of HT than PPB and this absolute increase can be attributed to the density effect PAGE 155 155 Fig ure 614. Density effect on the thermal conductivity along the z axis in BIT 6.5 Conclusion and Future Study Structural constraints and their effect on thermal transport properties are well documented and investigated by the MD simulation. A suitable potential parameter is chosen based on its performance to reproduce structural features of BIT. Using the potent ial, anisotropy in thermal expansion was shown. The octahedron tilting was found to be temperature independent. Anisotropy in thermal conductivity qualitatively agreed with experimental investigation. Density difference between layers is attributed to the low val ue of thermal conductivity. A systematic investigation revealed the contribution of density effect and point defects on the thermal conductivity along the z axis. No density effect was observed along the x axis. It can be concluded that by the intro duction of point defects in the system, thermal conductivity can be lowered. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Normal PPB HT (W/mK) PAGE 156 156 CHAPTER 7 GENERAL CONCLUSIONS In this work, the molecular dynamics (MD) simulation method has been successfully used to investigate thermal transport in potential inert matrix f uel materials (MgO NDZ and MgAl2O4) and thermal barrier coating material (Bi4Ti3O12). 7.1 MgO NDZ As a candidate of inert matrix fuel, interfaces play a dominant role on thermal conductivity in composite system s. In this dissertation, it was determined tha t volumetric composition governs the thermal transport at very fine (<10 nm) grain size level for MgO NDZ composite. The composite can have M gO NDZ, MgO MgO and NDZ NDZ interfaces Homogenous interfaces were investigated by simulating MgO and NDZ polycrystals separately. Using thermal conductivity values of polycrystals and single crystal, interface effects we re computed for both materials. For very fine grain size, interface effects were comparable to the grain size, itself. Due to the limitat ion of MD simulations grain sizes were limited in tens of angstroms. Even with limited computational power, polycrystal thermal conductivity results were extrapolated for larger grain sizes and gave similar grain size dependence in comparison with the experiment results for YSZ [19] Fidelity with experiment provides an additional tool to rectify the simulation data. Anharmonic analysis w a s used to quantify the fidelity of the interatomic interaction parameters of the simulation. Based on the difference from experiment for thermal expansion and bulk modulus, a correction term wa s calculated. The c orrection term quantified the deviation in thermal conductivity results for the material. PAGE 157 157 The study provides a key insight into the interface effects on thermal conductivity of MgO NDZ composite. However, the study wa s limited to <10 nm grain size compared to micrometer size grains observed experi mentally. Therefore, a systematic study with larger grain size of MgO, NDZ and MgO NDZ system will provide additional insight into the thermal properties. Defects generally act as a barrier to thermal conduction. It will be interesting to capture the coupl ing of defect grain boundary interactions on thermal conductivity of MgO NDZ cercer. 7 .2 MgAl2O4 With relative ease of formation of inverted structure (cation anti site structure) in MgAl2O4 under neutron irradiation, effect on thermal and mechanical properties is an important issue. Experimentally temperature and inversion are interrelated, thus separating out one effect from another is extremely hard. Atomic level simulation provided an ideal tool to capture the individual effects in the MgAl2O4 system. Experimentally MgAl2O4 system has been characterized for i< 0.3. Using MD simulation, the entire range of inversion ( i= 0 1) has been probed for thermal and mechanical properties. Thermal expansion has shown no variance for i< 0.5 and decreases afterwards. Similarly analysis on bulk modulus has shown no change for i< 0.5 and increases for i> 0.5. Using the above response of inversion on thermal expansion and bulk modulus, therm al conductivity wa s predicted. Thermal conductivity values were also calculated i ndependently for each inversion using non equilibrium MD (using heat source, heat sink). Critical analysis of the thermal conductivity values calculated by both methods showed the inability of the existing model in capturing the inversion effect. PAGE 158 158 Combinat orics analysis method has been used to incorporate the inversion effect on thermal properties. Additionally, experimental and simulation observations on lattice parameters of MgAl2O4 has shown a decrease with inversion. However, the effect has not been pr operly characterized. For the first time, this study explained the reason for decrease of the lattice parameter with inversion. Besides, the study also presented a possible cation arrangement for 1 cation anti site pair. Another interesting question is to check whether the existing inversion enhances or reduces further disordering of the system. For i< 0.5 no change in further disorder of the system has been reported, however for i >0.5, further disorder is facilitated. L ow formation energy of additional anti sites for i >0.5 has been reported and explained through the formation of strain dipoles at inverted sites and simple reaction rate analysis model. The present study has considered the arrangement of atoms in a spinel structure. However, defected rocks alt structure is also a possible arrangement of atoms for MgAl2O4. Therefore, for a complete understanding of the thermoelastic properties, it is essential to characterize and compare both spinel and defected rocksalt arrangement s of MgAl2O4. Similar to M gO NDZ system, effect of interfaces on thermoelastic properties can be characterized. 7 .3 Bi4Ti3O12 Bi4Ti3O12 (BIT), a layered perovskite material, is a candidate material for t hermal barrier coating (TBC) applications. The layering in BIT alternates between Bi2O2 and perovskitelike layers. Due to the presence of these layers, thermal conductivity results PAGE 159 159 in low values for BIT. This study focused on the fundamental understanding of the low value of the thermal conductivity of BIT. An e xperimental study reported that the difference in the density of the layers guides the overall thermal conductivity of BIT [20] In order to investigate the effect of density, simulations were performed by varying the mass of the cations in an appropriate manner. Due to the layered structure of BIT, anisotropy in thermal properties is reported in the literature. The current calculations also captured the anisotropic behavior in thermal conductivity and expansion. Since BIT is a potential material for TBC applications mainly due to its low thermal conductivity, it is desired to further re duce its thermal conductivity. Characterization of the e ffects of intrinsic and extrinsic defects (like La [198] ) on the thermal conductivity of BIT is essential Diffusion of oxygen ion in BIT is also an important criterion for TBC application. Thus, characterization of ionic diffusion in BIT will provide a better und erstanding of the material for TBC application. Overall, molecular dynamics is a very useful tool in characterizing atomic level details of various materials. 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Jo, "Lanthanum Substituted Bismuth Titanate for Use in NonVolatile Memories," Nature vol. 401, pp. 682684, 1999. PAGE 175 175 BIOGRAPHICAL SKETCH Priyank Shukla was born in Kanpur, UP India. He received his Integrated Masters of Science in physics from Indian Institute of Technology (c ommonly known as IIT ), Kanpur, India in 2002 It is a 5year degree program that combines the basic training of both engineering and physical sciences. He came to the United States for his higher studies and joined the graduate program in Physics Department at University of Florida ( U F), Gainesville, in Fall 2002. He received his Maste r of Science in P hysics from UF in May 2005. After completion, he joined the doctoral degree under Prof. Simon R. Phillpot at UF Gainesville. He received his Ph.D. in materials science and engineering fr om the University of Florida in the summer of 2010. 