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PAGE 1 1 STUDY OF THERMAL CONDUCTIVITY IN BULK CRYSTAL AND KAPITZA CONDUCTANCE AT GRAIN BOUNDARIES OF STRONTIUM TITANATE BY M OLECULAR D YNAMICS SIMULATION By ZEXI ZHENG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR TH E DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013 PAGE 2 2 2013 Zexi Zheng PAGE 3 3 To my father and mother, for their support PAGE 4 4 ACKNOWLEDGMENTS I would like t o express sincere gratitude to my advisor Dr Youping Chen first, for her support and encouragement Then, I d like to thank my committee member: Dr. Curtis Taylor, for the willingness to review my work. Special thanks are given to Dr. Liming Xiong, for his advice and valuable comments. Also, I want to express my appreciation to my fellows in the lab: Xiang Chen, Shengfeng Yang, Ning Zhang, Shikai Wang, Chen Zhang, and Rui Che. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 LIST OF ABBREVIATIONS ................................ ................................ ........................... 10 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 13 2 BACKGROUND AND RELATED WORK ................................ ................................ 15 Strontium Tita nate ................................ ................................ ................................ ... 15 Brief Introduction to Molecular Dynamics Simulation ................................ .............. 16 Several Methods for Phonon Mediated Thermal Transport Investigati ons by Simulation ................................ ................................ ................................ ............ 17 3 MEASUREMENT OF THERMAL CONDUCTIVITY OF STRONTIUM TITANATE BULK CRYSTAL ................................ ................................ ................................ ..... 19 Modeling of Strontium Titanate Bulk Crystal ................................ ........................... 19 Molecular Dynamics Simulation of Thermal Conductivity Measurement for the Single Crystal System ................................ ................................ .......................... 20 Results and D iscussion ................................ ................................ ........................... 22 Effect of Heat Current Density Applied ................................ ............................. 22 Dependence of Thermal Conductivity on Cross Sectional Area ....................... 23 Dependence of Thermal Conductivity on Temperature ................................ .... 23 Finite Size Effect ................................ ................................ .............................. 24 4 STUDY OF KAPITZA CONDUCTANCE OF TILT GRAIN BOUNDARIES IN STRONTIUM TITANATE ................................ ................................ ........................ 30 Overview ................................ ................................ ................................ ................. 30 Kapitza Resistance and Conductance ................................ .............................. 31 Coincident Site Lattice Theory ................................ ................................ .......... 33 Modeling of SrTiO 3 Bicrystal Systems with Tilt Grain Boundaries ........................... 34 Molecular Dynamics Simulation of Kapitza Conductance at Grain Boundaries ...... 36 Results and Discussion ................................ ................................ ........................... 38 5 CONCLUSIONS ................................ ................................ ................................ ..... 51 PAGE 6 6 LIST OF REFERENCES ................................ ................................ ............................... 53 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 55 PAGE 7 7 LIST OF TABLES Ta ble page 3 1 Thermal conductivity values measured from systems with different sizes at T=600K. ................................ ................................ ................................ .............. 24 4 1 Measured tem perature discontinuities and the corresponding Kapitza conductance for the four tilt grain b oundaries at T=600K. ................................ .. 48 4 2 Rigid body translations of one grain with respect to the other for the four systems a nd the corresponding GB energy. ................................ ....................... 49 PAGE 8 8 LIST OF FIGURES Figure page 2 1 Structure of the 5 atom unit cell of SrTiO 3 ................................ ......................... 18 3 1 Three dimensional periodic simulation cell for measuring the bulk crystal thermal conductivity. ................................ ................................ ........................... 25 3 2 Three dimensional bulk crystal simulation model, with a total of 4080 atoms. .... 25 3 3 Evolution of the mean temperature of the whole system from 0 ~ 70,000 t ime step. ................................ ................................ ................................ .................... 25 3 4 Time averaged temperature distribution along z d irection. ................................ 26 3 5 Final temperature profile of the whole system along z direction, time averaged from 500,000 to 20,00,000 MD step. ................................ .................. 26 3 6 Temperature profile lin ear fitting. ................................ ................................ ........ 26 3 7 Results of thermal conductivity values of SrTiO 3 by others. ............................... 27 3 8 Effect of changing heat current applied to the simulation system, at an average temperature of 600K. ................................ ................................ ............ 27 3 9 Comparison of the rmal conductivity values measured under systems with different cross section sizes. ................................ ................................ ............... 28 3 10 Temperature profiles for systems of different cross section sizes. Each case was averaged through step 500,000 to step 2,000,000. ................................ ..... 28 3 11 The dependence of thermal conductivity on environmental temperatures for the system of the size 4*4*51 unit cells. ................................ ............................. 29 3 12 The dependence of thermal conductivity values on system sizes, under the ambient temperature of 600 K. ................................ ................................ ............ 29 4 1 Schematic representation of a 53( ) tilt grain boundary in cu bic crystal lattice. ................................ ................................ ................................ ................. 43 4 2 Schematic representation of a 36.9 ( ) tilt grain boundary in cu bic crystal lattice. ................................ ................................ ................................ ................. 43 4 3 SrTiO 3 bicrystal modeling pr ocedure. ................................ ................................ 44 4 4 SrTiO 3 bicrystal modeling procedure, conti nued. ................................ ................ 44 4 5 SrTiO 3 bicrystal modeling procedure, continued ................................ ................ 45 PAGE 9 9 4 6 Cutting out the fi nal model. ................................ ................................ ................. 45 4 7 Four SrTiO 3 bicrystal models (partial view, only left side of the model is shown) with different GB configurations. ................................ ............................ 46 4 8 Schematic representation of the three dimensional perio dic simulation cell. ...... 47 4 9 Temperature profiles of four bicrystal systems with a steady thermal current of 1.7910e+011 W/m 2 applied along the z direction ................................ .......... 47 4 10 Data analysis of the left part of the temperature profile ................................ ..... 48 4 11 Data analy sis of the right part of the temperature profile ................................ ... 48 4 12 Snap shot of the right grain boundary region, after the system is dynamically stable. ................................ ................................ ................................ ................. 49 4 13 Comparison of left and right grain boundary s tructure at the original 3 10 GB model. ................................ ................................ ................................ ................. 50 PAGE 10 10 LIST OF ABBREVIATIONS AMM Acoustic mismatch model CSL Coincidence site lattice DMM Diffuse mismatch model GB Grain boundary MD Molecular dynamics TE Thermoelectric PAGE 11 11 Abstract of Thes is Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requi rements for the Degree of Master of Science STUDY OF THERMAL CONDUCTIVITY IN BULK CRYSTAL AND KAPITZA CONDUCTANCE AT GRAIN BOUNDARIES OF ST RONTIUM TITANATE BY M OLECULAR D YNAMICS SIMULATION By Zexi Zheng May 2013 Chair: Youping Chen Major: Mechanical Engineering Perovskite type SrTiO 3 as a potential thermoelectric material has attracted great attention. Although it shows a high factor in th e figure of merit, the high thermal conductivity value is still not satisfactory. Nanostructuring approaches reveal the ability to modify the overall thermal transport performance; but before that, we need to know which type of grain boundary and what kind of structure has the highest thermal resistance. Therefore, studies on thermal properties at SrTiO 3 GBs will have great significance. We first measure th e thermal conductivity of SrTiO 3 bulk crystal using t he nonequilibrium molecular dynamics simulation a nd systematically explore the related issues coming with the met hod T he results are extrapolated and compared with those from experiments and other simulations which show s a reasonable agreement indicating that the method works well with the material st udied here Then, four bicrystal systems with different symmetric tilt grain boundaries ( 5(310), 5(210), 13(510) stable and 13(510) metastable) are modeled and their Kapitza conductanc e at grain boundaries are evaluate d by simulations using the same well tested technique. W e PAGE 12 12 compare the results and find the Kapitza conductance depends on the amount of disorder at grain boundary region most but show direct relationship with the G B energy W e also find the voids and gaps may impede the thermal transport and their density along the grain boundary plays an important role. Finally, we notice that temperature discontinuities measured at the two grain boundaries are slightly different in each case. The two GBs have exactly the same structure but f ace towards opposite directions. That means the side from which the incident heat current comes also matters, which suggests that the pattern of an incident plane may be a factor to influence the thermal transport. However, further investigations on larger sy stems and studies about phonon wave dynamics will be necessary in order to elucidate this. PAGE 13 13 CHAPTER 1 INTRODUCTION Thermoelectric (TE) materials have attracted great attention since they can interconvert a temperature difference and an electric potential directly and be applied to a variety of areas. The perovskite type SrTiO 3 has been recognized as a prospective good n typ e thermoelectric material, since it shows a high power factor of 1 for in the di mensionless figure of merit where is the Seebe ck coefficient, the electrical conductivity, the thermal conductivity and the absolute temperature. However, the value is still too low for practical applications due to the relative high intrinsic thermal conductivity value of about at room temperature 2 Recent studi es have found that nano grained structures show much lower thermal conductance since enhanced phonon scattering occurs at grain boundaries 3 23 which will greatly reduce the value. Thus, n anostructuring appro aches reveal the lar ge potential to modify the overall thermal transport performance without deteriorating But before that, we need to know which type of grain boundary and what kind of structure has the highest thermal resistance. Therefore, stu dies on thermal properties at SrTiO 3 grain boundaries will have great significance The objective of this work is to measure and compare the Kapitza conductance at SrTiO 3 tilt grain boundaries. A proper simulation method is first chosen and evaluated throu gh the measurement of thermal conductivity values of SrTiO 3 bulk crystal s Then the well tested and defined techniques are applied to study the Kapitza conductance at four SrTiO 3 tilt grain boundaries. T he following work is organized as follows: In Chapter 2, back ground and related works are introduced; In Chapter 3, measurement of thermal PAGE 14 14 conductivity of bulk crystal systems will be carried out and related issues with the method will be systematically explo red and discussed; Chapter 4, which is the main pa rt, will employ the techniques evaluated and further determined in Chapter 3 to perform simulations on SrTiO 3 bicrystal systems and compare the Kapitza conductance resulting at different tilt grain boundaries. PAGE 15 15 CHAPTER 2 BACKGROUND AND RELATED WOR K Stront ium Titanate Strontium titanate ( SrTiO 3 ) or STO is a perovskite type oxide of strontium and titanium, one of the most common ceramics. It has a crystallographic cubic structure with the space group number 221 and a lattice constant of 3.9049 angstroms, whi ch is similar to a diamond. The 5 atom unit cell structure is shown in Figure 2 1. Recent studies on strontium titanate have shown its great potential for practical applications such as : the usage in fuel cells, steam electrolysis and hydrogen gas sensors 4 6 because of its high photonic conductivity at high temperature; the usage as substrates for the fabrication of high Tc super conducting films and devices for its favorable dielectric constant 277 and the good lattice match; and employment in power genera tion through recovered waste heat or electronic refrigeration for electronic devices 7 due to its good thermoelectric properties. Since substantial achievements have been made on using thermoelectric materials to interconvert the temperature gradient and t he electrical potential the promising technology attracts much attention recently. STO presents a high power factor in figure of merit which indicates it s a good potential TE material. But its ther mal conductivity valu e is still relatively too high ; m ore efforts will be needed before it can be put into practical applications. Therefore, researchers made some attempts to try to reduce the value: Wang and his co workers 3 ex amined a series of nanograined dense SrTiO 3 ceramics and found the decreased with decreasing the average grain size, which is mainly due to the increase of interfaces as heat barriers. Chernatynskiy and his co workers 8 analyzed the thermal transport properties of the Ruddlesden PAGE 16 16 Popper phase, formed by interleaving perovskite layers of SrTiO 3 with strontium oxide rocksalt layers and found a difference between the thermal properties measured parallel to the structural layering and those measured perpendicular to the layering. Muta and his co workers 9 by contrast, studied the reduced and La doped single crystalline SrTiO 3 Their results show that both electrical conductivity and thermal conductivity are lower in reduced sample and o xygen vacancies will cause strong electron and phonon scattering. Brief Introduction to Molecular Dynamics Simulation The molecular dynamics method (MD) was first introduced by Alder and Wainwright in the late 1950 s. The purpose of the method was to study th e interactions of hard spheres B ut later, insights concerning behaviors of simple liquids were developed. Since MD simulations deal with and generate information only at t he atomistic level, knowledge on statistical mechanics is required to convert inf ormation from microscopic level to macroscopic level or vice versa. The MD simulation method is based on Newton s second law or the equation of motion which means it s purely classical. T his may result in some problems, beca use systems at the atomistic lev el obey quantum laws rather than classical laws. Furthermore, quantum e ffects become critical ly important when system temperature is below the Debye temperature. To avoid such issues, our models are genera ted as large as pos sible and simulated over the Debye temperature, i.e. 600K for SrTiO 3 Remarkably, various interaction types are supported in MD but we have to provide our own parameters. The potential function we use here is a combination of Buckingham and Coulombic for ce, which has a relative low computational efficiency, since a long cut off distance will include a vast number of atoms, resulting in a long computing ti me. We will employ the parameter s fit by Thomas et al. 10 PAGE 17 17 Several Methods for Phonon Mediated Thermal T ransport Investigations by Simulation W e have several different simulation based approaches that can be used to study the thermal conductance of a material. Among them, the most remarkable one should be p honon wave packet dynamics which was developed to p rovide a way to gain detailed insight into the mechanism of phonon scattering at interfaces. The method first creates phonon wave packets that are superposition of normal modes at one end of a specimen, then lets the wave packets propagate towards another end using molecular dynamics simulation. After they reach the interface, some of them will pass through and some will be reflected back. The energy transmission coefficient can thus be determined for every polarization and wave vector. Although it s a powe rful and comprehensive way to learn detailed information on how different phonon waves interact with different interface configurations, it doesn t seem to be easy when dealing with complex systems, such like the cases considered here. The Green Kubo metho d, a representation of equilibrium MD approach, can be employed to measure the thermal conductivity of a homogeneous system, as mentioned previously. Nevertheless, it is not appropriate to apply it into an inhomogeneous system especially when there involve s localized features such as GBs studied here, for the Green Kubo formalism was designed to handle situations being homogeneous only. Another approach, the nonequilibrium MD simulation method, which has already been successfully employed to calculate the t hermal conductivity in the bulk crystal Si systems, will also be introduced here. Since it s a method analogous to experiments, it is much more generous. As a result, it is capable of dealing with both homogeneous and inhomogeneous situations. Although the direct method has some limitation in the PAGE 18 18 ability of extracting detailed information regarding the phonon waves contacting with boundaries, it has another big advantage: being able to determine the Kapitza conductance within a s ingle simulation. Figure 2 1. Structure of the 5 atom unit cell of SrTiO 3 Blue: Strontium; Black: Titanium; Red: Oxygen. PAGE 19 19 CHAPTER 3 MEASUREMENT OF THERMAL CONDUCTIVITY OF STRONTIUM TITANATE BULK CRYSTAL We have already known that there are two common techniques for computing thermal conductivity by atomic level simulation, direct method and Green Kubo method. Since the Green Kubo method requires the simulation process to be very long in order to achieve convergence, it s not suitable to be employed her e for the ionic bonded material simulation, which is quite expensive in computational resource. Thus, the direct method will be chosen to perform our simulations In this chapter, we will systematically explore the issues related to the measurement of ther mal conductivity when using direct method. For the purpose of getting a relative good result that comparable to those from experiments, the simulation time necessary for the system to achieve dynamic equilibrium, the heat current loaded, and the temperatur e dependence, these aspects will be examined. And lastly, the system size effect, which has a significant influence on the on the measured thermal conductivity value will be considered. Modeling of Strontium Titanate Bulk Crystal The crystal structure of the perovskite type SrTiO 3 is cubic. This greatly simplifies the modeling procedure of a single crystal. As we can see from Figure 3 2, the final three dimensional rectangular model can be obtained by extending the cubic unit cell in three perpendicular directions, namely, the x, y and z directions. As an example here, if we want to gener a te a model with its dimension 4 by 4 by 51 unit cells, simply duplicate the 5 atom unit cell 4 times along x and y direction and 51 times along z direction. M ake sure the two adjacent cells are end to end and to keep a lattice constant distance between th em. After the model is finished, the spatial coordinates of each atom along PAGE 20 20 with their atom type and charge value must be printed into a data file that can be recognized by L am m ps as an input file. M olecular D ynamics Simulation of Thermal Conductivity Meas urement for the Single Crystal System In this section we demonstrate how the thermal conductivity value for a single crystal system is measured by direct method. Figure 3 1 shows the three dimensional periodic simulation cell for measuring the bulk crystal thermal conductivity. It has the exact same size as that of t he simulation model we created since periodic boundary conditions will be applied to all six borders in order to make the results as accurate as possible. At each side of the cell, there s a reg ion in the middle, where atom velocities are rescaled in every MD step. As we can see in the figure, some amount of energy is added into the region on the left by rescaling the particle velocities; meanwhile, the same amount of energy is subtracted from the region on the right. This will result in a thermal current flowing from the hot end (heat source) to the cool end (heat sink), as indicated in the figure. Due to the periodicity of the left and right two boundaries, the energy flow will split into two same parts, one going towards right a nd the other going towards left. The part to the left will leave the left border first and then come back from right border again finally reach the heat sink. The whole procedure is made up of four major steps : initiating velocity, environmental temperature control (0 ~ 40,000 MD step) microcanonical relaxation (40,000 ~ 70,000 MD step) and applying the heat flux (70,000 ~ 2,000,000 MD step) The first step is to give an ensemble of velocities based on the aim ed temperature designated to all the particles of the system Since the initial temperature measured according to the initial velocities will drop to about a half in the following steps, we may PAGE 21 21 set it to be twice higher. A s it can be seen from Figure 3 3A, t he initial temperature was firstly set to be 900 K, then it went down to about 600 K The second step is to apply the canonical ensemble control (NVT) to adjust and maintain the system mean temperature at the desired ambient temperature. As we can see from Figure 3 3B, the average temperature of the whole system is kept around 600K. However, the NVT algorithm can t be that precise to adjus t the temperature to exact 600K, which is also not necessary. Figure 3 4 shows the necessity of apply ing the third step microcanonical ensemble control (NVE) also known as relaxation F rom the diagram, we see the temperature distribution along z direction is quite uneven although the mean temperature is 600 K if only NVT control is applied B ut after 3 0,000 steps of relaxation, it s much smoother; what s more, the temperature peak originally in the middle is vanished. Lastly, a steady heat flux is applied to the system by rescaling the particle velocities at the two slabs. The final temperature profile is shown in Figure 3 5, which is obtained by averaging all data gathered through step 500,000 to step 2,000,000. It is found that the system may be ab le to reach a steady state after 200,000 steps of heat flux is applied and a span of 1,500,000 steps shoul d be long enough to produce a smooth and nice t emperature profile for gradient measurement. Figure 3 6 shows how we measure the temperature gradient from the temperature distribution curve. We simply make a linear fit for the two regions far away from the heat source and the heat sink, who display the strong nonlinearity of temperature distribution. For the region in the middle, it s fairly straightforward, just set the interval, and do the linear fit; but for the region made up with the two parts adjacent to the simulation box we need to cut them out first, and then PAGE 22 22 make a perfect match, since there are periodic boundary conditions at the simulation box borders. After they are spliced together, a linear fit can then be made. Results and Discussion The temp erature gradient values measured for the two linear regions of the system at the mean temperature of 600K, are 1.3543 K/ and 1.3732 K/ respectively Then take the average and apply Fourier s law: ( 3 1 ) we get the thermal conductivity of the system to be: 6.5664 W/mK For here, the heat flux q applied is: It s not difficult to measure the thermal conductivity value for a given system, but it seems not easy to answer how valid and how good the result is ; some issues related closely to the direct method need to be fu rther examined. Also, the finite size effect must be taken into consideration. Effect of Heat Current Density Applied The issue is also known as the effect of deviation from Fourier s law. This may occur when the heat current applied is too large that the system shows nonlinear response. Figure 3 8 shows the effect of changing the heat current applied to the system. The diagram is plotted as thermal conductivity value versus heat current intensity which should be a horizontal straight line if there s no d eviation from Fourier s law, i.e the value remains constant. However, we find thermal conductivity goes up with increase of heat flux, especially when the applied heat flux exceeds where ev (corresponding to q=0.597e11 W/m^2). Thus, to comply with Fourier s law, we may use any value between and since in this range, least PAGE 23 23 deviation is observed. And due to the l ow effi ciency of the computational process it s necessary to apply the stimulation as strong as possible to save response time. W e will thereby choose the value of for the remaining studies. Dependence of Thermal Conductivity on Cross Sectional Area Although periodic boundary conditions are applied along x and y directions and heat current travels only along z direction, there is still some size effect that will affect our results. As we have already known from the relationship: ( 3 2 ) that the summation can be different for systems with small or irregu lar cross sections, we must establish the bot tom line below which the values measured are invalid. The results are shown in Figure 3 9 and 3 10, where the former reveals the influence of cross sectional size on values and the la tter shows how the temperature profile of each case looks like. We see the thermal conductivity values measured for the systems of size 3*3 and 4*4 match well while those measured for systems with size s 1*1 and 2*2 deviate a lot. This can be understood by inspecting the four temperature profiles T he temperature distribution curves of the two cases 1*1 and 2*2, are not even stable which may be caused by the insufficient number of atoms in a slice that statistical averaging is based on. The results suggest that in order to get a smooth and nice temperature profile for the further measurement of thermal conductivity, a system with the cross section size at least 3 b y 3 unit cells will be needed Dependence of Thermal Conductivity on Temperature Figure 3 11 s hows the dependence of thermal conductivity on environmental temperatures for the system of size 4*4*51 unit cells. We see the value decreases PAGE 24 24 with increasing the ambient temperature which is a typical property for phonon media ted thermal transport. The values and trend can be compared with those results from experimen ts (Figure 3 7A, by Yamanaka et al.) and simulations (Seetawan et al.) and they are in good agreement. Finite Size Effect Finally, the finite size effect must be taken into account since it s quite important. F inite size effects arise when the length of the simulation cell (Lz) is not significantly longer than the phonon mean free path. Strictly speaking, the results we got above cannot be compared to those from ex periments since specimens used for investigations in real life are far much longer. But the good news is, the effect can be reliably eliminated by doing multiple simulations of different size systems and making an extrapolation of the results to an infinit e size. The results are shown in Table 3 1 and Figure 3 12. The thermal conductivity extrapolated for the infinite sized system at the ambient temperature of 600K is 9.9119 W/mK, which is larger than the results from experiments, but in good agreement with those reported by Chernatynskiy et al. 11 Table 3 1. Thermal conductivity values measured from systems with different sizes at T=600K. Lz(nm) 1/Lz(1/nm) (W/mK) (mK/W) 40 0.025 8.1569 0.1226 28.506 5 0.03508 8.06 0.1241 16.7915 0.05955 6.4129 0.1559 14.4485 0.06921 6.1228 0.1633 10.1527 0.0985 5.4429 0.1837 PAGE 25 25 Figure 3 1. Three dimensional periodic simulation cell for measuring the bulk crystal thermal conductivity. Figure 3 2. Three dimens ional bulk crystal simulation model, with a total of 4080 atoms. The size of the model is unit cells. A B Figure 3 3. Evolution of the mean temperature of the whole system from 0 ~ 70,000 time step A) 0 ~ 40,000 time step, B) 45,000 ~ 70,000 time step. PAGE 26 26 A B Figure 3 4. Time averaged temperature distribution along z direction A) averaged through 0 ~40,000 time step, B) averaged through 40,000 ~70,000 time step. Figure 3 5. Final temperature profile of the whole system alo ng z directi on, time averaged from 500,000 to 20,00,000 MD step. A B Figure 3 6. Temperature profile linear fitting. A) region from 30 to 30 B) region from 70 to 70 ( The two regions; 70 ~ right boundary and left boundary ~ 70 are joined toge ther due to the periodicity of the boundary conditions. ) PAGE 27 27 A B Figure 3 7. Results of thermal conductivity values of SrTiO 3 by others. A) from Yamanaka et al. 12 by experiment, B) from S eetawan et al. 13 by MD simulation. Figure 3 8. Effect of changin g heat current applied to the simulation system, at an average temperature of 600K. PAGE 28 28 Figure 3 9. Comparison of thermal conductivity values measured under systems with different cross section sizes. A B C D Figure 3 10. Tem perature profiles for sys tems with different cross section sizes. Each case was averaged through step 500,000 to step 2,000,000. A) 4 by 4 unit cells, B) 3 by 3 unit cells, C) 2 by 2 unit cells, D) 1 by 1 unit cells. PAGE 29 29 Figure 3 11. The dependence of thermal conductivit y on envir onmental temperatures for the system of the size 4*4*51 unit cells. Figure 3 12. The dep endence of thermal conductivity values on system sizes, under the ambient temperature of 600K. A linear fit is made to determine the thermal conductivity obtained by taking the reciprocal of the value at of the infinite system. PAGE 30 30 CHAPTER 4 STUDY OF KAPITZA CONDUCTANCE OF TILT GRAIN BOUNDARIES IN STRONTIUM TITANATE Overview It s known that semiconducting SrTiO 3 p olycrystals are broadly used as varistors, for they show good nonlinear current voltage (I V) characteristics 14,15 This suggests that investigations on defects and grain boundari es (GB) are indispensable. Also, some application of SrTiO 3 crystals (STO) su ch as the use of bicrystalline STO substrates as templates for Josephson junctions fabrication demands the understanding of GB structure, 16 18 which further confirms the importance of grain boundaries and defects study ing Among the numerous properties of this ionically bonded material, those related with thermal transport have attracted significant interest as mentioned previously However, the m ajority of the current works were done by experiments, only very limited number of researches were through molec ular dynamics simulation, even fewer dealt with bicrystal involving grain boundaries. So far several kinds of twin boundaries and tilt boundaries in SrTiO 3 have been observed and determined. And furthermore, investigations on atomic structure, electronic structure, and defect energetic s in selected SrTiO 3 symmetric tilt grain boundaries were being made by a first prin ciples projector augmented wave ( PAW ) calculation 19 20 which revealed the relationship s among the atomic structure, GB energy, and GB defect energetic In this chapter three types of tilt grain boundaries: [001](310) 5, [001](210) 5 and [001](510) 13 will be employe d; The 13 type GB contains two different cases, with different rigid body translations of one grain with respect to the PAGE 31 31 other. T he two cases stand for GB s with two energy states: stable and metastable, respectively. So, totally four different SrTiO 3 tilt grain boundaries will be modeled and their Kapitza conductance will be measured. Kapitza Resistance and Conductance Interfacial thermal resistance, also known as thermal boundary resistance or Kapitza resistance is a measure of an interface s resistance to thermal flow. It has been recognized that between two dissimilar materials there exists a temperature discontinuity at the interface if heat flux is applied across it 21 22 namely, there is an interfacial ther mal resistance. The Kapitza resistance is defined as follow: ( 4 1 ) where is the temperature discontinuity at the interface (for here, the grain boundary), and is the thermal current. The Kapitza conductance, i s simply the reciprocal of the Kapitza resistance and is thus defined as: ( 4 2 ) The relationships among the Kapitza conductance, resistance and thermal conductivity are as follows: ( 4 3 ) ( 4 4 ) PAGE 32 32 where is the thickness or length of the specimen along which direction heat current is applied. In phonon mediated thermal transport, the scattering of phonons at interfaces will lead to the rise of Kapitza res istance. 23 There are usually two theoretical frameworks that can be employed to study this interfacial resistance: the acoustic mismatch model (AMM) and the diffuse mismatch model (DMM). 23 24 The AMM says if the acoustic impedances of a material on two sid es are different, there will be phonon scattering occurring at the interface. The acoustic impedance is defined as: 25 26 ( 4 5 ) where is the density and is the speed of sound. For that the AMM is derived within continuum acoustics, it thereby has a limitation; It ignor es the structure of the interfaces, which contain important information on how phonon scattering actually takes place. The DMM, by contrast, takes into account the effect of interface scattering. It is assumed that incident phonon waves will be scattered w hen they go through the interfaces. However, both two models cannot always provide results be in good agreement with those from the experiments. 27 Fortunately, we have several different simulation based approaches that can be used to study the thermal con ductance of a material. Among them, the most remar kable one should be p honon wave packet dynamics 28 which was developed to provide a way to gain detailed insight into the mechanism of phonon scattering at interfaces. However, it doesn t seem to be easy wh en dealing with complex systems, such like the cases considered here. The Green Kubo method, a representation of equilibrium MD approach, can be employed to measure the thermal conductivity of a homogeneous PAGE 33 33 system, as mentioned previously. Nevertheless, it is not appropriate to apply it into an inhomogeneous system especially when there involves localized features such as GBs studied here, for the Green Kubo formalism was designed to handle situations being homogeneous only. Another approach, the nonequilib rium MD simulation method which has already been successfully employed to calculate the thermal conductivity in the bulk crystal systems, will also be introduced here. Since it s a method analogous to experiments, it is much more generous. A s a result, it is capable of deal ing with both homogeneous and inhomogeneous situations. Although the direct method has some limitation in the ability of extracting detailed information regarding the phonon waves contact ing with boundarie s, it has another big advantage: being able to determine the Kapitza conductance within a single simulation. Coincident Site Lattice Theory The coincident site lattice (CSL) provides a relatively easy way to study and understand the structure of grain boundari es with special orientations between them. A coincident site lattice can be defined when a finite fraction of lattice sites on two lattices coincide with each other. Usually, we use the notation followed by a number to express t he CSL type grain boundaries. The sigma value is defined as the ratio between the area enclosed by a unit cell of the coincidence sites and that enclosed by standard ones. For example, Figure 4 1 shows a 53 ( ) tilt grain boundar y in cubic crystal lattice. If the area of the smaller square in dotted line (enclosed by efgh) is defined as 1, the area of the larger one (enclosed by abcd) should be 5. Thus, the ratio between the two areas is 5 which leads to the sigma value to be 5, according to the definition. PAGE 34 34 It s clear that for every specific rotation angle, if exists, one and only one sigma value can be found. However, the rule doesn t apply when a sigma value is given. In other words, the number of the possible angles correspondi ng to a sigma value is not limited to only one. As in the example shown in Fi gure 4 2, we may easily find that for a sigma value of 5, not only a rotation angle of 53 but also 36.9 exists. The tilt grain boundary shown in Figure 4 2 is very similar to t hat in Figure 4 1. The only difference is the relative rotation angle s between two grains. W e should notice that, though the area enclosed by abcd is 10 times that enclosed by efgh, it doesn t mean we can draw the conclusion that the sigma value is 10. How ever, we should count in the coincidence lattice site s located in the center of each cell, and connect them with the nearest sites as presented in Figure 4 2 (the dotted square agdi ). T hen the true sigma value can be obtained by calculating the ratio o f a rea agdi which represents the smallest CSL cell to area efgh. From the above, we can see that for cubic cases, whenever an even sigma value appears, there should always be another coincidence site located in the center of the cell which will lead the tru e value to be half of the apparent quantity. Modeling of SrTiO 3 Bicrystal System s with Tilt Grain Boundaries As the same technique, namely the direct method, will be employed to study the thermal conductance of SrTiO 3 tilt grain boundaries here, the proced ure should be similar in spirit to that of the measurement of thermal conductivity of SrTiO 3 bulk crystal. However, there are still some issues we need to address first. The first, and the most significant one is how we design a simulation model that keeps both accuracy and efficiency. For the bulk crystal system, due to the fact that PAGE 35 35 along any direction it s isotropic, simply repeating the unit cell in a specific dimension may satisfy any size demanded. But in the bicrystal case, there exists an additional inhomogeneous region, the grain boundary, which needs further transformation and adjustment in order to match well with both two crystals. Furthermore, it involves rigid body translation of one crystal with respect to the other, which makes it even harder to apply the perio dic boundaries to the simulation box. Firstly, we need to determine the approximate size of the system. Based on the previous experience, a system of unit cells should be ok in order to get comparable results. Then, a single crystal block about twice as large as the approximate size is generated. It s necessary to obtain a relatively big blank at first since it make s sure when further manipulations are done, the part we want is still available. After that, we r otate the model chunk by an angle around the designated axis, according to the grain boundary type. For here, our goal is to create a bicrystal model with [001](310) 5 grain boundaries which means the rotation axis ma y be any of the three coordinate axes. We simply use the y axis as shown in Figure 4 3. And the rotation angle should be 18.435 degree The next step is to cut out the middle portion which is a single crystal, of the simulation system. In order to make sure the structure is stable and keeps c harge neutrality, we have to find out two characteristic atoms located on the two grain boundary plane s on each side of the central crystal, respectively, and set up two borders. In this case, the characteristic atom type is chosen to be Titanium, since it will appear on the boundary planes. Then, all atoms stay within the two borders will be kept and everything else outside will be cut off. The resulting geometry is shown in Figure 4 4A. Next, we divide the remaining central part into two sections, the lef t and PAGE 36 36 the right. The left section is further made a mirror copy with respect to the left border plane and right section also makes a mirror copy with respect to the right border. The resulting geometry is shown in Figure 4 4B. Although the model looks like it has three pieces actually it contains only two single crystals. Since periodic conditions will be applied to both left and right boundaries, the two parts adjacent to the two boundaries will be connected and matched as a single crystal. Before cutting out the final model we still have another important step need to do first. According to the study on GB energy of SrTiO 3 by Imaeda et al 19 a rigid body translation of one grain with respect to the other is necessary in order to minimize the grain bounda ry energy and stabilize the structure Thus, the grain in the middle stays and the remaining parts are subject to a rigid body translation to the proper positions as shown in Figure 4 5. For the case here, the corresponding translation vector is ( 4.5, 0. 5, 0.5) (in Angstrom). Figure 4 6 shows the final model which is obtained by cutting off the excess part of the roughcast. M olecular D ynamics Simulation of Kapitza Conductance at Grain Boundaries The results the from the previous chapter on measurement of thermal conductivity of bulk crystal show that the nonequilibrium MD simulation method is suitable to be employed to evaluate the thermal properties in this ionic bonded perovskite type material. Thus, the same technique is used to simulate the bicrystal systems with tilt grain boundaries. We make use of the modeling procedure developed in the last section and generate total ly four models of different GB configurations: [001](310) 5, [001](210) 5, [001](510) 13 stable and [001](510) 13 metastable, as shown in Figure 4 7 Due to the excess ive length along z direction s only a half of the model is presented for each PAGE 37 37 case. The fig ures on the left are models viewed along y direction and those on the right are m odels viewed along x direction. The schematic representation of the three dimensional simulation cell is shown in Figure 4 8. Each model is bounded in a simulation box has the exact matching size, with periodic border conditions in all three directions applied. That means t he boundaries along x and y directions are infinitely extended and the heat current will go along z direction only, which is very similar to the technique us ed in finite element analysis And because of the periodicity in the z direction, the system can thus be regarded as a simulation cell containing two crystals joined by two crystallographically identical but symmetric tilt grain boundaries. The heat curren t along the negative z direction will leave the left border and go back from the right border again forming a loop. At each MD time step (0.55fs) we rescale the particle velocities at two thin slabs, one located in the middle of the simulation cell and th e other near the right border. Each of the tw o slabs keeps a same distance from both of the two grain boundaries in order to get a symmetric temperature profile. The thickness of the slabs doesn t matter much as is already discussed, we thus simply choose it to be 6 according to the length of the whole system. To create a heat current, we add some energy at each MD step to one slab and remove the same amount of energy from the other slab, which alwa ys conserves the total energy of the whole system. The heat current can thus be calculated through the equation: ( 4 6 ) where is the transversal area of the cross section (lies in xy plane), is the MD time step, 1/2 means the amount of energy added into the heat source is split into two PAGE 38 38 equal parts, going through two ways as indicated in Figure 4 8, we only need to consider one way when doing calculation. Since we have known the heat current flowing in the system, the thing we care about left is the temperature discontinuity at the two grain boundaries which may be measured from the temperature profile when the whole system achieves a steady state. The whole simulation procedure consists of four steps : initiating velocity environmental tempe rature control, microcanonical relaxation and applying heat flux, which is basically the same as that in thermal conductivity measurement. It has already been known in the previous chapter that 200,000 to 300,000 steps after constant heat flux is applied s hould be long enough for the whole system to achieve the dynamic steady state. We thus extract out the temperature data at each point along z direction with a n interval of 4 angstrom s for the time steps from 500,000 to 2,000,000 and do the average. Then a smooth temperature profile will be seen and the post processing can be done. Results and Discussion The final temperature profiles of the four bicrystal systems are shown in Figure 4 9. Comparing with the results getting from the silicon bicrystal syste ms, we may easily find that the temperature discontinuities are very sharp here although drops are not that steep like those happen ing at silicon grain boundaries. The large temperature variation in the GB region indicate s there exists a high thermal resis tance, or equivalently a low thermal conductance. However, It is not easy to quantitative ly evaluate the Kapitza conductance here since the temperature doesn t come in with a sudden jump but with a transitional curve with finite gradient. This is because of the high computational cost of ionic bonded materials, our models c annot be created in large sizes like those in silicon. PAGE 39 39 Thus, it s necessary to develop a method to quantitatively determine the temperature discontinuities in our cases Figure 4 10 sho ws how the issue is addressed. W e break the data into two parts, the left part and the right part and inspect the left first. W e need to perform a function fitting for the raw data first since they come in with discrete form In order to keep as much inf ormation as possible from the original data, we choose the polynomial model of rank 12 which has the ability to mimic complex curves. We then take the derivative of the fitted function twice and find out the stationary points for the temperature gradient curve which can obtained by taking the derivative of the temperature distribution curve once, as shown in Figure 4 10B, blue dashed curve. The positions of the stationary points are where temperat ure gradient alters its trend of changing which in other w ords are potential boundaries of the area whose Kapitza conductance is affected by the presence of the GB Because of the fluctuations of the temperature distribution curve due to the existence of inherent errors, we may find several stationary points in e ach curve, as we can see from Figure 4 10 and Figure 4 11. However, only two of them are real or meaningful. Fortunately, they can be easily distinguished from the figure. As an example, in Figure 4 10B, the two meaning ful points are the two adjacent to the centered point which corresponds to the extreme value in the 1 st derivative curve. After the two points are located, their positions along z direction are known, and the corresponding two temperature values can be obtained from the temperature distrib ution curve. The difference of these two values, i.e. will be used to evaluate the Kapitza conductance. Table 4 1 summarizes the measured temperature discontinuities at each grain boundary, with same heat current applied (1.79 1e11 W/m 2 ) in each case Since in every PAGE 40 40 step same amount of energy is added to the heat source a nd equivalent amount of energy is sub tracted from the heat sink, and NVE control maintains at the same time the total energy is always conserv ed which makes t he mean temperature of the whole system stay at 600K. After the temperature discontinuities at both GBs in each system are measured, we take the average and apply Equation 4 2, to get the Kapitza conductance. The calculated results are also included in Tab le 4 1. As we can see, the (510) 13 metastable GB has the largest value while (310 ) 5 possesses the lowest thermal conductance. Interestingly, the same GB type with same misorientation angle but with different rigid body translation could show very diverse res ults. If we look at the values of 13 grain boundaries we may find that Kapitza conductance of the stable boundary is about 25% lower than that of the metastable boundary, which is unexpected sin ce the disparity is even greater than those between different GB types. However, we have to admit there may exist some relationship between the thermal conductance and the grain boundary energy, since the GB energy values of the two configurations are not close as well. Table 4 2 lists t he GB energy for each type of the grain boundary and their rigid body translation vectors. The data are from the study by M. Imaeda and co workers 19 and the study by H. S. Lee and co workers. 20 From the table, we may find the 13 stable structure has the lowest GB energy among the four while the 13 metastable s tructure has the largest. However, the trend of the correlation between the GB energy and value we got here is opposite to that reported by S chelling and co workers 23 for Silicon boundaries. While Schelling found the high energy, disordered grain boundary has a smaller value of if compared to the PAGE 41 41 lower energy, more ordered grain boundary, we observe d that the metasta ble grain boundary with a h igh er energy shows a stronger ability to pass the heat current, i.e., a higher Kapitza conductance, while the stable grain boundary wi th much lower GB energy shows a rather smaller value for the 13 grain boundaries Then we try to gain more information from the geometric construction s of the grain boundaries Figure 4 12 are the snap shots of the grain boundary structures of the four systems after they are dynamically stabilized We can find that except for the (210) 5 case, all other three models show big voids at the boundary. Especially for (310) 5 boundary, the boundary line is almost made up with continuous cavities, with only few number of atoms connect ing the two grains which make s it no t difficult to understand why this type of grain boundary possesses a low value (lowest among the four). By contrast, t he 13 stable structure show n in Figure 4 12C has a higher thermal conductance value than the (310) 5 case a lthough it also has comparable sized voids at the boundary. This is because there is a higher density of atoms joining the two grains here which makes the bridge to c onnect the two region s wider A s a result more phonons will be able to pass through. Com paring A, B and C three cases in Figure 4 12 and their corresponding values we may easily find that voids at boundaries may impede the thermal transport, but what s more, the am ount of atoms connecting the left and right part p lays a more important role Ho wever, we cannot make a judgement by only looking at the clues found above There are other factors that may be also critical. We should have noticed that the 13 metastable grain boundary (Figure 4 12D) is kind of special. Tho ugh its structure is very alike to that 13 stable structure as shown in PAGE 42 42 Figure 4 12 C, it has a much higher Kapitza conductance, ever higher than the (210) 5 boundary, which shows no obvious big voids. But if we inspect the atom alignments in the GB regio n, we may find the pattern in metastable structure is much more ordered if compared with that in stable structure, where disorders such as dislocations are observed. This suggests that the amount of disorder at grain boundary can have significant effect. F urthermore, it is noticed that temperature discontinuities measured at the two GBs are slightly different in each case, as we can see in Table 4 1, though same conditions and periodic borders are applied. T here must be some oth er reasons since the variatio n is already beyond the range of statistical errors. We make a comparison between the left GB structure and the right GB structure, a s shown in Figure 4 13. T o make life easier we simply rotate the le ft GB by 180 degrees to make it comparable to the right boundary. We see the two incident planes are different, or more pre cisely, they are symmetric. This is a potential factor that may chang e the behavior of phonon propagation. However, it s beyond my scope, investigations on phonon wave dynamics will be nee ded in order to elucidate this PAGE 43 43 Figure 4 1. Schematic representation of a 53 ( ) tilt grain boundary in cubic crystal lattice The ratio of the area enclosed by abcd to that enclosed by efgh is 5. Figure 4 2. Schematic rep resentation of a 36.9 ( ) tilt grain boundary in cubic crystal lattice. The ratio of the area enclosed by agdi to that enclosed by efgh is 5. PAGE 44 44 A B Figure 4 3. SrTiO 3 b icrystal modeling procedure A) Generating a single crysta l with size about twice as large as the approximate final one. B) Rotat ing the model by an angle according to the grain boundary type. In this case, [001](310) 5 grain boundary, the angle should be 18.435 A B Figure 4 4. SrTiO 3 bicrystal modeling proce dure continued A) Cutting out the middle part (the first single crystal) of the bicrystal system. B) Generating the second single crystal based on the first part by mirror copy. PAGE 45 45 A B Figure 4 5. SrTiO 3 bicrystal modeling procedure, continued. A) before rigid body translation B) after rigid body translation A B Figure 4 6. Cutting out the final model. A) view along y direction B) view along x direction. PAGE 46 46 A B C D Figure 4 7. Four SrTiO 3 bicrystal models ( partial view, only left side of the model is shown) with different GB configurations. A) B) [001]( 2 C) [001]( 5 13 stable state, D) [001]( 5 13 metastable state. PAGE 47 47 Figure 4 8. Schematic representation of the three dimensional periodic simulation cell. A B C D Figure 4 9. Temperature profiles of four bicrystal systems with a steady thermal current of 1.7910e+011 W/m 2 applied along the z direction averaged over 5 00,000 to 2,000,000 time steps. The background temperature is controlled at 600 K. A) [001] B) [001]( 2 C) [001]( 5 13 stable state, D) [001]( 5 13 metastable state. PAGE 48 48 A B Figure 4 10. Data analysis of the left part of the temperature profile A) Fit the temperature profile using order 12 polynomial, B) Find positions where tre nds shift, using the 2 nd order derivative. A B Figure 4 11 Data analysis of the right part of the temperature profile A) Fit the temperature profile using order 12 polynomial, B) Find positions where trends shift, using the 2 nd order derivative. Table 4 1. Measured t emperatur e discontinuities and the corresponding Kapitza conductance for the four tilt grain boundaries at T=600K. The heat current applied is 1.7910e+011 W/m 2 GB type at left boundary at right boundary averaged 116.2 0 130.6 0 123.4 0 1.45 [001]( 2 91.7 0 99 .0 0 95.35 1.88 [001]( 5 13 stable 107.5 0 116.2 0 111.85 1.60 [001]( 5 13 metastable 78.4 0 89 .0 0 83.7 0 2. 14 PAGE 49 49 Table 4 2. Rigid body translations of one grain with respect to the other for the four systems and the corresponding GB energy. Data from M. Imaeda and co workers and H. S. Lee and co workers 19,20 GB type Rigid body translation x( ) Rigid body tr anslation y( ) Rigid body translation z( ) GB energy (J/m 2 ) 4.5 0.5 0.5 1.02 [001]( 2 0.0 1.5 0.5 0.98 [001]( 5 13 stable 4.5 0.0 1.0 0.93 [001]( 5 13 metastable 10.5 0.0 1.0 1.37 A B C D Figure 4 12. Snap shot of the right grain boundary region after the system is dynamically stable. A) B) [001]( 2 C) [001]( 5 13 stable D) [001]( 5 13 metastable PAGE 50 50 A B F i gure 4 13. Comparison of left and right grain boundary structure at the origina l 310 GB model. A) GB structure on the right, the positive z direction points to th e right; B) GB structure on the left, the positive z direction points to the left (rotated by 180 degrees). The direction in which heat flows is indicated in red arrows. PAGE 51 51 CHAPTER 5 CONCLUSIONS We measure d the thermal conductivity of SrTiO 3 bulk cr ystal using the nonequilibrium molecular dynamics simulation first and systematically explore d the related issues coming with the method. It is found that a heat current density value of should be suitable for the simulation since the stimulation is strong enough for the system to quickly achieve a smooth temperature profile and the resulting thermal conductivity values show almost no deviation from Fourier s law. The study on the dependence of values on cross sectional area shows the system with a cross section size at least 3 by 3 unit cells will be required to obtain a reliable result. The final values are extrapolated and compared with those from experiments and other simulations, which shows a reasonable agreement, indicating that the method works well with the material studied here. Then, four bicrystal systems with different symmetric tilt grain bou metastable) are modeled and their Kapitza conductance at grain boundaries are evaluated by simulations using the same, well tested technique. We compare the results and find the Kapitza conductance d epends on the amount of disorder at grain energy among the four shows an intermediate thermal conductance while the metastable one with the highest GB energy but an ordered structure shows the strongest ability to pass the heat flux through. We also find the voids and gaps may impede the thermal transport and their density along the grain boundary plays an important role. Finally, we notice that temperature discontinuities measured at the two PAGE 52 52 grain boundaries are slightly different in each case. The two GBs have exactly the same structure but face towards opposite directions, which means fr om which side the incident heat current comes also matters. 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After he finished his high school studies at No.2 Secondary School Attached to East China Norm al University, he took the college entrance examination and was admitted to the School of Mechanical and Vehicular Engineering at Beijing Institute of Technology (BIT), Beijing, China, in 2007. He earned his Bachelor of Engineering d egree from BIT with a major in a utomotive e ngineering in 2011. Then he got his admission to the Department of Mechanical and Aerospace Engineering at the University of Florida and came to Gainesville, Florida in August, 2011. He worked with Dr. Youping Che n in the Atomistic and Multiscale Mechanics Group s ince 201 1 He received his M.S. in mechanical engineering from the University of Florida in the spring of 2013. 