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Influence of Solidification Variables on the Microporosity Formation of Al-Cu(4.5wt%) Alloy with Axial Heat Processing

Permanent Link: http://ufdc.ufl.edu/UFE0022533/00001

Material Information

Title: Influence of Solidification Variables on the Microporosity Formation of Al-Cu(4.5wt%) Alloy with Axial Heat Processing
Physical Description: 1 online resource (335 p.)
Language: english
Creator: Kim, Joo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: aluminum, convection, copper, dendrite, hydrogen, porosity, solidification
Materials Science and Engineering -- Dissertations, Academic -- UF
Genre: Materials Science and Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The purpose of this study is to investigate the effect of solidification variables on porosity formation in Al-Cu (4.5 wt%) alloy. These variables include thermal gradient (from 19 to 25 K/cm), solidification rate (from 0.00007 to 0.15 cm/sec), initial hydrogen concentration (from 0.09 cc/100 g to 0.27 cc/100 g), and level of convection. The effect of the convection on porosity formation, which has not yet been studied, was investigated by comparing two different solidification techniques: the normal Vertical Bridgeman (DS) technique and the Axial Heat Processing (AHP) technique. The AHP technique makes use of a graphite disk (baffle) immersed in the melt. The baffle is made of high density graphite, which can conduct the heat axially and distribute it over the entire growing solid/liquid (= s/l) interface. The melt is separated into two zones when the baffle is brought close to the s/l interface. The small zone below the baffle reduces the melt height, leading to a reduction of convection. During the solidification, the solutes are rejected from the liquid due to their inherent lower solubility within the solid. These rejected solutes can be piled up locally in this small region, resulting in the change of solute concentration in the liquid and in the solid. The microstructures produced by the AHP and DS techniques do not show a noticeable difference according to the dendrite arm spacing measurement. However, samples produced by the AHP technique show a porosity 20 to 40% lower than those prepared by the DS technique, and its effect is more pronounced with decreasing cooling rates and increasing initial hydrogen concentration. Along the height of the AHP sample, the volume percent of microporosity exhibits a maximum in the early stage of solidification, and then drops considerably. In the DS samples, it is almost constant regardless of the sample height. Additionally, the pore size in AHP samples was 5 to 15% smaller than in DS samples. The level of convection has been estimated by Rayleigh number, where h is melt height: h (DS) = 8 cm and h (AHP) = 0.5 to 0.7 cm. The convection led to the composition change at the dendrite tip, which could be confirmed by Electron Micro Probe Analysis (EPMA). The liquid copper composition at the head of the tip in the AHP sample increased to 8.9 wt%, compared to 5.32 wt% for the DS samples when the solidification rate is 0.00007 cm/sec. Thus, the suppressed convection increased reduced the dendrite height in the AHP samples compared to the DS samples. Additionally, local buildup of hydrogen below the baffle in the AHP technique could be calculated by Tiller?s approach and confirmed by the Inductively Coupled Plasma (ICP) analysis. A hydrogen bubble can nucleate near the dendrite tip when the amount of hydrogen accumulates below the baffle exceeds its solubility. Accordingly, decreasing the dendrite height and increasing the hydrogen concentration near the dendrite tip are both helpful in removing gas bubbles from the dendrites. Thus, it is believed that the final gas porosity is lower in the AHP samples compared to the DS samples.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Joo Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Abbaschian, Reza J.
Local: Co-adviser: Fuchs, Gerhard E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022533:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022533/00001

Material Information

Title: Influence of Solidification Variables on the Microporosity Formation of Al-Cu(4.5wt%) Alloy with Axial Heat Processing
Physical Description: 1 online resource (335 p.)
Language: english
Creator: Kim, Joo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: aluminum, convection, copper, dendrite, hydrogen, porosity, solidification
Materials Science and Engineering -- Dissertations, Academic -- UF
Genre: Materials Science and Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The purpose of this study is to investigate the effect of solidification variables on porosity formation in Al-Cu (4.5 wt%) alloy. These variables include thermal gradient (from 19 to 25 K/cm), solidification rate (from 0.00007 to 0.15 cm/sec), initial hydrogen concentration (from 0.09 cc/100 g to 0.27 cc/100 g), and level of convection. The effect of the convection on porosity formation, which has not yet been studied, was investigated by comparing two different solidification techniques: the normal Vertical Bridgeman (DS) technique and the Axial Heat Processing (AHP) technique. The AHP technique makes use of a graphite disk (baffle) immersed in the melt. The baffle is made of high density graphite, which can conduct the heat axially and distribute it over the entire growing solid/liquid (= s/l) interface. The melt is separated into two zones when the baffle is brought close to the s/l interface. The small zone below the baffle reduces the melt height, leading to a reduction of convection. During the solidification, the solutes are rejected from the liquid due to their inherent lower solubility within the solid. These rejected solutes can be piled up locally in this small region, resulting in the change of solute concentration in the liquid and in the solid. The microstructures produced by the AHP and DS techniques do not show a noticeable difference according to the dendrite arm spacing measurement. However, samples produced by the AHP technique show a porosity 20 to 40% lower than those prepared by the DS technique, and its effect is more pronounced with decreasing cooling rates and increasing initial hydrogen concentration. Along the height of the AHP sample, the volume percent of microporosity exhibits a maximum in the early stage of solidification, and then drops considerably. In the DS samples, it is almost constant regardless of the sample height. Additionally, the pore size in AHP samples was 5 to 15% smaller than in DS samples. The level of convection has been estimated by Rayleigh number, where h is melt height: h (DS) = 8 cm and h (AHP) = 0.5 to 0.7 cm. The convection led to the composition change at the dendrite tip, which could be confirmed by Electron Micro Probe Analysis (EPMA). The liquid copper composition at the head of the tip in the AHP sample increased to 8.9 wt%, compared to 5.32 wt% for the DS samples when the solidification rate is 0.00007 cm/sec. Thus, the suppressed convection increased reduced the dendrite height in the AHP samples compared to the DS samples. Additionally, local buildup of hydrogen below the baffle in the AHP technique could be calculated by Tiller?s approach and confirmed by the Inductively Coupled Plasma (ICP) analysis. A hydrogen bubble can nucleate near the dendrite tip when the amount of hydrogen accumulates below the baffle exceeds its solubility. Accordingly, decreasing the dendrite height and increasing the hydrogen concentration near the dendrite tip are both helpful in removing gas bubbles from the dendrites. Thus, it is believed that the final gas porosity is lower in the AHP samples compared to the DS samples.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Joo Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Abbaschian, Reza J.
Local: Co-adviser: Fuchs, Gerhard E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022533:00001


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1 INFLUENCE OF SOLIDIFICATION VARIABLES ON THE MICROPOROSITY FORMATION OF Al-Cu (4.5 wt%) ALL OY WITH AXIAL HEAT PROCESSING By JOO RO KIM A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008

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2 2008 Joo Ro Kim

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3 To my family and my lab members

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4 ACKNOWLEDGMENTS Most of all, I m ust express my sincere gratitude to Dr. Reza Abbaschian, for the opportunity to work on such an interesting re search. His profound knowledge and advice have helped me to become a well-rounded and independent engineer and scientist. I will keep in mind his passion and profession. Add itionally, I would like to acknowledge the other committee, Dr. Gerhard Fuchs, Dr. Lusia A. Dempere, Dr. Mart in Glicksman, and Dr. Sadim Anghaie, for their support and guidance. I also show my apprecia tion to Wayne Acree for EPMA works and to Maria Collins for useful advice for the image analysis. Dr. Heesung Yoon and Takgun Oh deserve my gr atitude for their help to set up the instruments and the gas chromatography test. I am also grateful to Jinsoo Ahn, Taegun Kim, Sangsoo Ji, and, especially, Doh Won Jung for th eir support, encouragement, and willingness to help me. Finally, I cannot thank my wife enough. Her love and unwavering support for four years has been absolutely invaluable. I would also like to thank my family in Korea. In particular, I am appreciative to my parent for th eir wise and encouraging advise.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................................................................................................... 4LIST OF FIGURES .........................................................................................................................9ABSTRACT ...................................................................................................................... .............14 CHAP TER 1 INTRODUCTION .................................................................................................................. 162 BACKGROUND ....................................................................................................................18Thermodynamics in Solidification .........................................................................................18Equilibrium State ............................................................................................................. 18Melting Point Depression ................................................................................................20Kinetics in Solidification .................................................................................................... ....22Homogeneous Nucleation ............................................................................................... 22Heterogeneous Nucleation ...............................................................................................24Crystal Growth from the Melt ................................................................................................ 25Interface Stability and So lidification Variables .............................................................. 25Mass, Heat, and Momentum Conservation ..................................................................... 29Governing equations ................................................................................................29Convection in solidification ..................................................................................... 36Solute Redistribution ..............................................................................................................40Solidification in Planar Interface .....................................................................................41Equilibrium solidif ication model ............................................................................. 42Scheil model ............................................................................................................. 43Mixed model (I): Tille rs consideration ...................................................................44Mixed model (II) ...................................................................................................... 46Mixed model (III) ..................................................................................................... 47Diffusionless model ..................................................................................................48Zone melting ............................................................................................................ 48Faviers extension of mixed model (II) .................................................................... 50Solidification Models in De ndritic/Columnar Interface .................................................. 51Dendrite shape and curvature at the tip .................................................................... 53Fluid dynamics in dendritic solidification ................................................................58Scheils model in dendritic interface ........................................................................ 58Tip temperature depression (I): Burden and Hunts model ...................................... 59Tip temperature depression (II): Alexandorvs model ............................................. 60Crystal Growth Techniques from the Melt ............................................................................. 62Czochalski Method .......................................................................................................... 62Float Zone Method .......................................................................................................... 62Vertical Bridgman Method .............................................................................................. 63

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6 Axial Heat Processing ..................................................................................................... 64 Porosity Formation .................................................................................................................64Classification of Porosity Formation ............................................................................... 65Thermodynamic Consideration for Porosity Formation .................................................. 66Channel model ..........................................................................................................67Porous medium model .............................................................................................. 67Criteria model ...........................................................................................................72Prediction of Volume Percent Porosity and its Size ........................................................73Thermodynamic model ............................................................................................ 73Kinetic model ...........................................................................................................743 TECHNICAL APPROACH .................................................................................................112Equipment ..................................................................................................................... ........112Data Acquisition and Furnace Control ................................................................................. 115Experimental Procedures ...................................................................................................... 115Experimental Instruments Calibration ........................................................................... 115Sample and Crucible Preparation ..................................................................................116Initial Thermal Gradient in the Melt ............................................................................. 118Gas Atmosphere Control ............................................................................................... 119Crystal Growth ..............................................................................................................120Analysis ................................................................................................................................122Metallography ................................................................................................................ 122Image Analysis .............................................................................................................. 123Electron Probe Micro-Analysis .....................................................................................124Inductively Coupled Plasma At omic Emission Spectroscopy ...................................... 1244 RESULTS ....................................................................................................................... ......1315 DISCUSSION .................................................................................................................... ...168Porosity Formation: DS Samples.......................................................................................... 168Effect of Convection on Solute Redistribution .....................................................................173Zone Effect: AHP Technique ...............................................................................................178Porosity Formation: AHP Samples ....................................................................................... 1876 CONCLUSION .................................................................................................................... .206 APPENDIX A PARAMETER CALCULATION .........................................................................................208B ALEXANDROVS MODEL ................................................................................................209C MICROSTRUCTURES ........................................................................................................ 212D POROSITY DISTRIBUTION MEASUREMENT ..............................................................239

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7 E PORE RADIUS MEASUREMENT ..................................................................................... 279F THERMAL DATA ANALYSIS .......................................................................................... 319LIST OF REFERENCES .............................................................................................................326BIOGRAPHICAL SKETCH .......................................................................................................335

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8 LIST OF TABLES Table page 2-1 Solidification models accordi ng to the m ass transport method. ........................................ 892-2 Solidification condition with nondimen tional quantities by Fa viers criterion .................994-1 Physical and thermochemical pa rameters for Al-Cu (4.5 wt%) alloy .............................1444-2Solidification variab les and experimental conditions ...................................................... 1455-1Tip radius calculated for f our of solidification rates ........................................................199

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9 LIST OF FIGURES Figure page 2-1 Phase diagram of Al-Cu system ......................................................................................... 802-2 Thermal condition in columnar solidifica tion of pure metal substance at G > 0. .............. 812-3 Thermal condition at equiaxed solidifi cation of pure metal substance at G < 0. ............... 822-4 Binary phase diagram during solidificati on with solute composition profile ahead of the s/l in terface ............................................................................................................ ......832-5 Constitutional supercooling according to so lute buildup in front of s/l interface. ............. 842-6 Morphological instability diagram for Al-Cu (4.5 wt%) acco rding to initial composition of copper and the thermal gradient to solidification rate. ............................. 852-8 Vertical cylindrical container having isothermal temperature ........................................... 872-9 Solute redistribution behavior durin g solidification in a volume element ......................... 882-10 Solute redistribution in the equilibrium solidification model during solidification. ......... 902-12 Non-equilibrium phase diagram by Scheils equation .......................................................922-13 Concentration profile of the liquid at limited liquid diffusion of copper using Al-Cu (4.5wt%) sample.. ..............................................................................................................932-14 Solute redistribution profile versus the sample length using the Mixed model (I) and the copper compositional profile in the liquid ................................................................... 942-15 Comparison of composition profile ahead of the s/l interface using the Mixed model (I) and the Mixed model (II) ............................................................................................ 952-16 Zone melting technique and calculated co mpositional profile of copper in solid ............. 962-17 Solute distribution ahead of the s/l interface at l (zone length) > D/V(characteristic distance), (b) l < D/V ........................................................................................................ .972-18 Domains of solute distribution profile versus the sample length using, nondimential quantity Pe and GrSc ......................................................................................................... 982-19 Dendritic interface betw een solid and liquid ..................................................................1002-20 Compositional profile along the dendrite height and normal to dendrite height ............ 1012-21 Closed packed arrangements of primar y dendrite arms and growing dendrite in a volume element ................................................................................................................ 102

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10 2-22 Prediction of the tip temperature depr ession in Al-Cu (4.5wt%) alloy versus the solidification rate using Burden and Hunt m odel. ........................................................... 1032-23 Conventional crystal growth methods :C zochralski method, Float zone method, and Veridical Bridgeman method ........................................................................................... 1042-24 Axial Heat Processing technique ..................................................................................... 1052-25 Feeding mechanisms of th e melt during solidification. ................................................... 1062-26 Two main causes of microporosity: shri nkage porosity and gas porosity in simple pipe-like dendritic structure .............................................................................................1072-27 Arrnagement of primary dendrites: square arrangement, and interlocking arrangement. .................................................................................................................. ...1082-28 Calculated pressure drop versus interd endrtic channel size at G = 24 C/cm and V = 0.0008 cm/sec of Al-Cu(4.5 wt%) alloy. ......................................................................... 1092-29 Formation of isolated liquid pool during solidification by Fang and Grangers model: (a) comparison between upper and lowe r mushy zone, (b) pore at growing grain(solid line) and final grown grain(dash line) ...........................................................1102-30 The geometrical representation of the diffusion cell in Fang and Grnagers model (spherical shape) and the associated evolution of hydrogen concentrations in the local solid and liquid at two sequential times ..................................................................1113-1 Axial heat processing instruments ................................................................................... 1263-2 Details of the graphite crucible part in the axial heat processing: (1) graphite baffle, (2) graphite container, (3) heating element, (4) graphite pedestal, (5) Monel pedestal rod, and (6) quartz tube .................................................................................................... 1273-3 Thermal gradient in the liquid melt m easured by the thermocouple in the baffle (circle) and the directly immersed thermocouple to the melt (square) ............................ 1283-4 Gas chromatographic (GC) analysis for gas exited from the system prior to the solidification. ............................................................................................................... ....1293-5 Plot of hydrogen concentration measured by ICP sample and calculated by Sieverts law versus hydrogen partial pressure, PH2. ......................................................................1304-1 Temperature acquired from incorporated thermocouples in AHP samples according to time at the solidification rate = 0.0008 (cm/sec) and fu rnace temperature = 710 C. 1434-2 Temperature acquired from immersed thermocouples within the melt for DS samples versus time at the solidifi cation rate = 0.0005 (cm/sec) .................................................. 145

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11 4-3 Temperature acquired from incorporated therm ocouples (TC 3 and TC 4) for DS samples versus time at the solidif ication rate = 0.0005 (cm/sec) .................................... 1464-4 Photomicrography of Al-Cu (4.5 wt%) al loy solidified by the DS technique with different solidification conditions: c ooling rate = 0.0014C/sec (DS I 2) ,0.011C/sec ( DS II 5), 0.018C/sec (DS II 4), and 3.6C/sec (DS IV 1) ............. 1484-5 Photomicrography of Al-Cu (4.5 wt%) all oy solidified by the AHP technique with different solidification conditions: c ooling rate = 0.0015C/sec (AHP I 4), 0.012C/sec (AHP II 4), 0.018C/sec (AHP III 4), and 3.6C/sec (AHP IV 1) ..... 1494-6 The relation of primary and secondary dendrite arm spacing for Al-Cu (4.5 wt% 6 wt%) alloy with the cooling rate ...................................................................................... 1504-7 Microporosity observation w ith optical microscope with 500x magnification: at the eutectic (DS III-5), at grain boundary of cell structure (DS III-5), and between dendrite colony (grain) (DS IV-1) ................................................................................... 1514-8 Comparison of longitudinal porosity dist ribution between AHP and DS samples: at the bottom, 1.0 cm, 2.0 cm, 3.0 cm, 4.0 cm, and 5.0 cm ................................................. 1524-9 Average volume % porosity versus the sa mple height with various initial hydrogen concentrations for the DS I samples. ............................................................................... 1534-10 Average volume % porosity versus the sa mple height with various initial hydrogen concentrations for the DS II samples.. ............................................................................. 1544-11 Average volume % porosity versus the sa mple height with various initial hydrogen concentrations for the DS III samples. ............................................................................. 1554-12 Average volume % porosity versus the sa mple height with various initial hydrogen concentrations for the DS IV samples. ............................................................................ 1564-13 Average volume % porosity versus the sa mple height with various initial hydrogen concentrations for the AHP I samples.. ........................................................................... 1574-14 Average volume % porosity versus the sa mple height with various initial hydrogen concentrations for the AHP II samples. ........................................................................... 1584-15 Average volume % porosity versus the sa mple height with various initial hydrogen concentrations for the AHP III samples. .......................................................................... 1594-17 Comparison of average volume % porosity between AHP and DS samples according to the change of initial hydrogen concentration at different cooling rates: 0.0017 C/sec, 0.012 C/sec, 0.018 C/sec, and 3.6 C/sec ......................................................... 1614-18 Comparison of volume % porosity between AHP and DS samples versus the change of cooling rate at different initial hydrogen concentration .............................................. 162

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12 4-19 Average radius of pores according to th e change of initia l hydrogen concentration and cooling rate. ............................................................................................................. ..1634-20 Comparison of copper composition profile between AHP and DS samples within dendrite at cooling rate = 0.0017 C/sec .......................................................................... 1644-21 Comparison of copper composition profile between AHP and DS samples within dendrite at cooling rate = 0.018C/sec ........................................................................... 1654-22 Comparison of measured hydrogen concentrat ion with an increase of sample height for DS III 1 and AHP III 1 sample at cooling rate = 0.018 C/sec and [H]0= 0.27 cc/100g 1664-23 Comparison of hydrogen concentration afte r experiment with an increase of initial hydrogen concentration before experime nt for DS and AHP sample ............................ 1675-1 Calculated pressure drop ( P) and pressure against surface tension (P) versus cooling rate................................................................................................................... ....1905-2 Effect of copper content on the variati on of hydrogen solubility the variation of hydrogen pressure versus the solid frac tion at cooling rate = 0.018 C/sec. ................... 1915-3 Comparison of the gas pressure term w ith other pressure terms according to the change of initial hydrogen contents and cooling rates in DS samples and corresponding critical solid fraction accord ing to the initial hydrogen content. ............. 1925-4 Comparison of calculated and measured amount of porosity and radius in the DS samples using thermodynamic and Li and Changs model ............................................. 1935-5 Calculated permeability versus liquid frac tion using Piwonka and Flemings model, Murakami model, and Kozeny-Carman model. ............................................................... 1945-6 Domain of solute redistribution accordi ng to fluid characteristics by Favier et al: convective-diffusive boundary region, limited diffusion region, diffusive boundary region with fast diffusion, a nd diffusive boundary region with intermediate or slow diffusion. .................................................................................................................... ......1955-7 Comparison between the tip composition obtained by EPMA data and Burden and Hunts model and Alexandrovs model as a change of the solidification rate. ............... 1965-8 Mechanism of hydrogen buildup below the baffle .......................................................... 1975-9 Diffusion field accroding to the curved s/l interface and a pl anar s/l interface. ............. 1985-10 Concentration of hydrogen built up below th e baffle at 4 cm height versus initial hydrogen concentration, [H]0 of various cooling rates ................................................... 200

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13 5-11 Hydrogen concentration below and above the baf fle with an increase of sample height at cooling rate = 0.018 C/sec and [H]0= 0.09 cc/100 g ....................................... 2015-12 A calculated nucleation range according to the cooling rate: AHP sample at [H]0 = 0.09 cc/100 g and (b) DS samples for [H]0 = 0.27 cc/100 g. ........................................... 2025-13 Nucleation range on the dendrite fo r the DS sample and AHP samples ......................... 2035-14 Optical micrograph at the interface of the graphite baffle and metallic alloy. ................ 2045-15 Plots of calculated (Li and Changs m odel) and measured volume % porosity and pore size for AHP samples as a function of initial hydrogen concentration. ................... 205

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14 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INFLUENCE OF SOLIDIFICATION VARIABLES ON THE MICROPOROSITY FORMATION OF Al-Cu (4.5 wt%) ALL OY WITH AXIAL HEAT PROCESSING By Joo Ro Kim December 2008 Chair: Reza Abbaschian Cochair: Gerhard Fuchs Major: Materials Scie nce and Engineering The purpose of this study is to investigate the effect of solidification variables on porosity formation in Al-Cu (4.5 wt%) alloy. These variab les include thermal grad ient (from 19 to 25 K/cm), solidification rate (f rom 0.00007 to 0.15 cm/sec), initial hydrogen concentration (from 0.09 cc/100 g to 0.27 cc/100 g), and level of convecti on. The effect of the convection on porosity formation, which has not yet been studied, wa s investigated by comparing two different solidification techniques: the normal Vertical Bridgeman (DS) technique and the Axial Heat Processing (AHP) technique. The AHP technique makes use of a graphite disk (baffle) immersed in the melt. The baffle is made of high density graphite, which can c onduct the heat axially and distribute it over the entire growing solid/liquid (= s /l) interface. The melt is separated into two zones when the baffle is brought close to the s/l interface. The small zone below the baffle reduces the melt height, leading to a reduction of convecti on. During the solidification, the solutes are rejected from the liquid due to their inherent lower solubility within the solid. These rejected solutes can be piled up locally in this small region, resu lting in the change of solute c oncentration in the liquid and in the solid.

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15 The microstructures produced by the AHP a nd DS techniques do not show a noticeable difference according to the dendrite arm spaci ng measurement. However, samples produced by the AHP technique show a porosity 20 to 40% lo wer than those prepared by the DS technique, and its effect is more pronounced with decreasi ng cooling rates and in creasing initial hydrogen concentration. Along the height of the AHP sa mple, the volume per cent of microporosity exhibits a maximum in the early stage of solidif ication, and then drops co nsiderably. In the DS samples, it is almost constant regardless of th e sample height. Additionally, the pore size in AHP samples was 5 to15% smaller than in DS samples. The level of convection has been estimated by Rayleigh number, where h is melt height: h (DS) = 8 cm and h (AHP) = 0.5 to 0.7 cm. The c onvection led to the composition change at the dendrite tip, which could be c onfirmed by Electron Micro Probe Analysis (EPMA). The liquid copper composition at the head of the tip in the AHP sample increased to 8.9 wt%, compared to 5.32 wt% for the DS samples when the soli dification rate is 0.00007 cm/sec. Thus, the suppressed convection increased re duced the dendrite height in the AHP samples compared to the DS samples. Additionally, local buildup of hydrogen below the baffle in the AHP technique could be calculated by Tillers approach and c onfirmed by the Inductively Coupled Plasma (ICP) analysis. A hydrogen bubble can nuc leate near the dendrite tip when the amount of hydrogen accumulates below the baffle exceeds its solubility. Accordingly, decreasing the dendrite height and increasing the hydrogen concentration near the dendrite tip are both helpful in rem oving gas bubbles from the dendrites. Thus, it is believed that the final gas porosity is lower in the AHP samples compared to the DS samples.

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16 CHAPTER 1 INTRODUCTION The directional solidification t echnique of alum inum and its alloys in the automobile and aerospace industries has developed dramatically over the past two decades. So far, aluminum remains a superior engineering ma terial due to its i nherent light weight resulting in improved performance and fuel efficiency. Even with thes e advantages, defects in the microstructure can undermine the performance. One of the most detrimental defects is microporosity whose negative influence on mechanical properties ha s been well documented in the literature. Microporosity may stem from the relativel y fast casting speed during the directional solidification techni que, resulting in morphological instabil ities and the formation of complicated interface structures. Simply, this phenomenon is characterized by a change of the s/l planar interface to columnar-like or tree-like s/l interf ace (called dendritic structure) during solidification. In formation of the dendrite (solid), a shrunken region can be created due to a substantial density change between the liquid and solid. Shrinkage voids can occur when a molten metal fails to fill in the shrunken area in th e dendrites. In the case that solubility of the solid is less than that of the liquid, solutes are rejected and built up substantially between dendrites due to the lateral diffusi on of rejected solutes. In the cas ting of aluminum or its alloys, hydrogen has been a major concern due to the ten-fold decrease in solubility upon transfer from the liquid to solid. Thus, when the liquid cont ains small amount of hydrogen, it can easily become supersaturated between dendrites, so th at gas bubbles are able to nucleate. When the nucleated gas bubbles fail to escape to the free surface, they are engulfed by the growing solid within the dendrite, resulting in gas voids. This type of porosity is referred to as gas porosity. As dendritic structure is much complicated and more isolated, hydrogen atoms or nucleated gas

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17 bubbles are not able to escape out the bulk liquid. In this rega rd, the microporosity has been accepted as inevitable in norma l solidification rate. Through theoretical and experi mental investigations of microporosity resulting from the directional solidification technique over the pa st 50 years, thermodynamic and kinetic models have been developed and computational methods ha ve been used to pred ict volume of porosity and its size. The process parameters used in thes e models are very much the same as the actual solidification variables: thermal gradient, solid ification rate, convecti on, and initial solute concentration. However, the influence of c onvection, which has an important role in solidification, on porosity formation has been largely ignored. This is mainly due to difficulties associated with accurate contro l of the process parameters to reduce convection in the presence of gravity. Recently, however, an innovative so lidification technique called Axial Heat Processing (AHP) has been utili zed to control thermal buoyant c onvection near th e s/l interface by decreasing the melt height. Besides this, the AHP technique exhibits similar mechanisms as the fractional solidification techni que (zone melting) due to its small melt height and this is an important feature influencing hydrogen buildup and evolution. By isolating only a fractional zone, rejected solutes become more concentrated because they do not mix effectively with the bulk liquid owing to the presence of baffle. The present research is dedicated to inves tigating the influence of the AHP technique on porosity formation in Al-Cu (4.5 wt%) alloy. Thes e results have great importance, since the convection and zone melting concept have been ra rely applied to the dendritic s/l interfaces.

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18 CHAPTER 2 BACKGROUND This chapter presents fundam ental concepts re levant to the evaluati on of on the effect of solidification variables on microporosity formation. Due to the interdisciplinary nature of this investigation, this sect ion is composed of thermodynamics and the kinetics of solidification, crystal growth technique, solute redistribution, fluid dynamics, a nd porosity formation theories. Thermodynamics of Solidification During crystal growth from the melt, a so lid and liquid interf ace can be formed corresponding to the equilibrium state in the Gi bbs free energy. The shape of the interface and the formation of porosity are dependent on the Gibbs free energy. This section will provide the basic concept of thermodynamics re levant to solidification [1, 2]. Equilibrium State The equilibrium state is the most stable state. The energy of a system is defined by the free energy using the first and second laws of thermodynamics [1, 2]: TSPVUG [2-1] where G is the Gibbs free energy applie d at constant pressure cases. P is the pressure of the system, V is the molar volume of the system, T is the temperature of the system, S is its thermal entropy, and U is the internal en ergy per mole of the system. Here, enthalpy, H is defined as U + PV. The equilibrium state of one phase is satisfied when the Gibbs free energy, G is at the minimum. Phase diagrams can be plotted when two or more phases are in equilibrium and the phase diagram for Al-Cu system at one atm is show n in Fig. 2-1 [3]. From the phase diagram, the starting and finishing temperatur es of solidification (solidus and liquidus), stable phases at arbitrary temperatures and compositions, solidifi cation ranges, and equilibrium compositions can be obtained. A feature of the Al-Cu alloys phase diagram is a broad eutec tic range at about 0 ~

PAGE 19

19 33 (wt%) of copper. The melting point of pure al uminum is 660C, eutectic temperature is 548 C. At eutectic point, the maximu m solubility at the equilibriu m state is 5.65 (wt%). Another important equilibrium reaction in solidification occurs between the solid phase and the liquid. For example, gases such as hydrogen and oxygen di ssolve in the liquid and solid metal. In the gaseous state, the hydrogen and oxygen exist in diatomic form, H2 and O2. When dissolved, however, they are found in mona tomic form, H and O. In a chem ical reaction, a gas that in solution is indicated by H2 (g) = 2H in Al-Cu alloy solution. Th e equilibrium constant for this reaction is: )( 22gH Ha a K [2-2] where, K is the equilibrium constant, aH and aH2 the activity of hydrogen and hydrogen gas, respectively. Since, hydrogen gas at moderate pressures behaves as an ideal gas, its activity is just its pressure in atmospheres. The activity of materials dissolved as dilute solution (such as gases in metals) is usually defined in a differe nt way. Let us assume that hydrogen activity is linearly related to its concentr ation, not its pressure, in the metal. This assumption was reasonable based on numerous literature reports. The unit of concentration in the metal can be the volume of gaseous hydrogen dissolved in a given mass of Al-Cu: cubic centimeters of gas at standard temperature and pressure per 100g of Al-Cu. We can also arbitrarily pick a concentr ation of dissolved hydrogen equivalent to 1 cm3 per 100 g of Al-Cu as its standard st ate. With this standard state, aH = [H] and Eq. (2-2) can be transformed into Eq. (2-3) as following )( 22][gHP H K [2-3]

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20 where, [H] is the dissolved hydrogen volume pe r 100 g of Al-Cu. Thus, the amount of hydrogen dissolved in Al-Cu at a given te mperature is proportional to the square root of the hydrogen gas pressure. This relation is known as Sieverts law. Melting Point Depression Therm odynamics also provide deep insight conc erning the solidificati on of crystal growth. If a liquid exists under conditions where the solid is the equilibrium state, then, the driving force for melting/solidification is de pendent on the degree of deviation from the free energy at equilibrium. For the reaction of melting from solid to liquid (AL AS), the Gibbs free energy change is defined as follows )ln()ln(, pureL pures pureLaRT a a RTG [2-4] where, a is the activity which is defined mathematically as fxi where f is the activity coefficient and xi is the mole fraction of component i. as, pure is the activity of pure solid A and let the pure solid A be the most stable state at the arbitraril y chosen temperature at one atmosphere pressure. At the standard state, the ac tivity is defined as unity, so as, pure = 1. aL, pure is the activity of pure liquid. At melting point, solid and liquid can coexist and the two phases are thought as at the equilibrium and, hence, the Gibbs free energy change should be the same, G = 0 indicating the activity of liquid is also at unity. At temperatures below Tm, the melting temperature, while as, pure is still unity as the standard state, aL, pure is not at unity any longer. Accordingly, the change of Gibbs free energy is not zero due to deviation from the standard state, which can be written as m m m mSTLvSTHG [2-5]

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21 where Gm is the Gibbs free energy for the phase transformation from liquid to solid, Lv, the latent heat of fusion (J/g), repl aces the enthalpy change of melting, Hm, as a convention. The underbar of each term represents the term divi ded by mass to remove the consideration of mass of each component. In other words, G is Gibbs free energy per one mole. By assuming that the CP, or specific heat (J/gK), of the liquid and solid phases are the same, the Gibbs free energy change of melting become zero, Gm=0, at the melting point, [2-6] Thus, at an arbitrarily chosen temperature, T the change of Gibbs free energy can be rewritten by inserting Eq. (2-6) into Eq. (2-5), Pures PureL m m m ma a RT T TT Lv T T LvG, ,ln)()1( [2-7] In pure metal, the driving force for melting is determined by Tm-T, as shown in Eq. (2-7). From Eq. (2-7), it is apparent that the activity of pur e liquid A is greater than the case when the temperature, T, is below Tm resulting in G m > 0. This indicates that th e solid state is more stable and the phase transformation to liquid cannot occur. Note that the previous analysis is in regards to pure materials. In alloy systems, the Gibbs free energy change of mixing of the solutes needs to be considered and the melting temperat ure depression can be understood with similar treatment as the above. Now, let us assume that so lute, marked B, is introduced into the pure A system having ideal solution behavior in which the activity coefficient is one. Consider the dissolving of pure, liquid A in the liquid solution. For AL, pure = AL, solution, the Gibbs free energy change is written as [2-8] mSTLv pureL solutionLa a RTG, ,ln

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22 The dissolution of pure, solid A in the liqui d is the sum of the two processes above including the melting of pure A and the dissolution of pure liqui d A in the solution. The Gibbs free energy change is the su m of Eq. (2-7) and (2-8). pureL solutionL m ma a RT T TTLv G, ,ln )( [2-9] At G = 0 a new melting point by adding solutes can be calculated with Eq. (2-9). For example, in the case of Al-Cu ( 4.5 wt%), the activity of Al in solution is 0.98, assuming an ideal solution, and the Lv is 11116 J/mol and, thus, the melting temperature of Al-Cu(4.5wt%) is 649C, deviating from the melting te mperature of pure Al by 11C. Kinetics in Solidification The presence of a thermodynam ic driving force alone does not ensure that the solidification will initiate. The systems kinetic consideration determines if and when the solid will be formed from the melt. Microscopically, the solidification is a process of atomic rearrangement at the s/l in terface through mass transport by diffusion and convection. The microstructures such as equiaxed, cellular a nd dendritic types cannot be understood without kinetic factors, because the thermodynamics information from the phase diagram provides limited information. Also, the porosity formation takes place by nucleation and the porosity amount in solidification is affected by the pore growth. This chapter w ill review some basic theories of kinetics in order to understand the solidification and the porosity formation mechanism. Homogeneous Nucleation W ith undercooling/supercooling of liquid as a thermodynamic driving force, random fluctuations in local ar eas lead to the formation of small cl usters of solid phase atoms and the Gibbs free energy of these is give by [4, 5]:

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23 [2-10] where, VS is the volume of the solid sphere, VL is the volume of liquid, ASL is the solid/liquid interface area, GV S and GV L are the free energies per unit volume of the solid and the liquid respectively, and SL is the solid/liquid inte rface energy (dyne/cm). Without considering the solid and liquid interface, the energy of the system is given by; L VLSGVVG )(1 [2-11] The formation of solid therefore resu lts in a free energy change where: SLSLVSAGVGGG 12 [2-12] For the transformation from liquid to solid (solidification), m m VTTLvGG / can be substituted into Eq. (2-12). And, the critical ra dius of the cluster to initiate the nucleation(= herein, solidification) is determined by dG=0. It can be easily shown by differentiation of Eq. (212) that )( 1 ) 2 (*T Lv T rmSL [2-13] 2 2 3 *)( 1 ) 16 ( T Lv T GmSL [2-14] where, r* and *G are the critical radius and Gibbs free energy for initiation of the nucleation, respectively. The critical radius and critical free energy is inversely proportional to the supercooling, T as shown in Eq. (2-13) and (2-14). Regarding Eq. (2-12) and (2-13), the effect of surface energy term is only noticeable at small sizes of nuclei though it is excluded in thermodynamic consideration in Eq. (2-4). Acco rdingly, to initiate solid ification, excess energy from the equilibrium state is required and supercooling is a prerequisite condition for SLSL L VL S VSAGVGVG2

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24 solidification. The early research of Turnbull and Cech [6] showed that the required supercooling for Al-Cu (4.5 wt%) was about 0.18Tm (= 128C). Regarding pore formation, the system of pores is formed or expanded under one atmospheric pressure; thus, the Helmholz free energy is an appropriate term to attribute to the phenomena. Hirth et. al. [7] followed the same tr eatment from Eq. (2-10) and (2-14) and found the activation energy for the nuclea tion of gas pores at constant pr essure condition to be given by Eq. (2-15) and (2-16), 2 3 *)(3 16v LGF F [2-15] e r L m vP P V RT F ln [2-16] where, Fv is the Helmholz free energy, Pr is the gas pressure in the bubble, Pe is the equilibrium pressure of the gas over the metallic melt, L mVis the partial molar volume of the gas component over the metallic melt, GL is the surface tension between the ga s and liquid phases. Turnbull et al [8] revealed that about 3-5 times the equilibrium hydrogen pressure was required to nucleate the gas bubbles in Al-Cu alloy. Heterogeneous Nucleation Besides the difficulty in obt aining such a high supercool ing condition, in practice, hom ogeneous nucleation is a rare case mainly because a homogeneous liquid is rarely encountered. Impurities and container walls can play a role as preferential sites for nucleation by reduction of surface tension. The total free energy required to reach crit ical-size clusters on impurities, container surfaces, or a substrate, is much lower than that for the homogeneous nucleation by an amount, S( ): )(hom SGGhet [2-17]

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25 Herein, [2-18] )( S, a shape factor of nuclei, has numerical values less than unity and dependent on the wetting angle, Since the free energy required for heteroge neous nucleation is lower, the degree of critical supercooling required is also much less, usually by severa l orders of magni tude. In this study, the container material was ch osen to be high density graphite. The wetability of graphite by molten aluminum is known to be 148 at 800C [9]. Thus, the shape factor, )( S, is 0.983 and this is pretty close to homogeneous nucleation. In this regards, the heterogene ous nucleation of Al-Cu(4.5wt%) at the containe r surface may not expected. Crystal Growth from the Melt Crystal growth is a kind of phase transf orm ation estimated by aforementioned equations. However, with this section, we elaborate more on the process of solidif ication from the melt. Casting is a fabrication process whereby a totally molten metal is poured into and allowed to solidify in a mold having a desired shape. A large number of metal objects are also made by hot pressing and sintering alloy particles, which are produced vi a liquid-solid transformations. Casting is one of the most versat ile techniques that is able to approach the final desired shape and, furthermore, the casting method is known to be the most economical processing technique [10]. Since the properties of the end result are determined in large measure by the nature of the solidification process, the factor s involved in the transformation from liquid to solid are of the utmost practical importance. Interface Stability and Solidification Variables The feature of crystal growth is dete rmined mainly by heat transfer and mass transfer through which microstructure and physical/chemical properties of solidifie d ingot can be altered. 4/)cos1)(cos2()(2 S

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26 Processing variables of solidification, as a determ inant of mass/heat transport, are enumerated: (1) thermal gradient, G, (2) solidification rate, V and (3) convection. The vari ables are related to one another by cooling rate, GV, and local solidification time, tf, at which the solidification is completed. Morphological shape of the interf ace can be also understood by combining the relationships between aforementioned solidif ication variables. Th ere are two general classifications of the morphologi cal shape of the inte rface: planer interfac e and columnar (or cellular)/dendrite type interface. In alloy casting, the solidification variab les are affected by the compositional condition as well as the thermal cond itions near the interface, thus increasing the complexity, this will be disc ussed later in this section. In pure metal, planer interface type solidification is given by the heat flux, Tq, which increases in the y-direction when G is positive, G > 0 in Fig. 2-2. The interface will be found at the isotherm, denoted by Tm. If an interface experiences perturbations at the melting temperature, the temperature field of Tm becomes deformed as shown in Fi g. 2(b), so that the temperature gradient in the liquid at the tip along A-A increases while that in solid decreases. Accordingly, the heat flux will flow into the tip and less will flow out of it. Meanwhile, the reverse situation occurs in the depression along li ne B-B. As a result, the pert urbation tends to restore the equilibrium position and, finally, the interface shap e is once more smooth and planar. In the case where the temperature gradient is negative, G < 0, around the S/L interface, the heat need not be extracted to the mold wall because the liquid is undercooled below Tm and, therefore, the solid can be nucleated in the melt, not on the wall. In this case, equiaxed solidification is formed as described in Fig. 2-3. A perturbation will sense a higher gradient at its tip, leading to an increased heat flux, and a resultan t increase in the growth rate of the tip. The growth of the tip, finally, produces branch like structure and each branch is termed a dendrite Thus, the

PAGE 27

27 solidification producing this ty pe of S/L interface is called, dendritic solidification Note that the reason why the thermal gradient exists is due to th e radiation of the latent heat to the undercooled liquid [11, 12]. In pure metals, the interface morphology depends solely on thermal conditions. However, by taking into account the effect of compositional elements in all oy casting, the dendritic growth is possible under positiv e thermal gradients, G > 0 by adopting a important concept, named constitutional supercooling [13, 14]. In Fig. 2-4, a simple phase diagram is introduced and the initial composition of the solute befo re initiation of solidification is C0. As the solidification proceeds and the temperature decreases, the composition of solid, CS *, and liquid, CL at the interface corresponds to the phase diagram by assumi ng local equilibrium at the interface. Herein, the ratio of CS to CL is defined as the equilibr ium partitioning coefficient, **/LSCCk At the same time, the bulk composition of liquid is maintained at C0 and T0 by assuming the volume is large enough to neglect the local compositional ch anges. Due to solute buildup ahead of the s/l interface, the liquid composition can reach C0/k and it corresponds to the solid composition (= C0). These liquid and solid composition is hold c ontinuously just before the completion of the solidification. This constant com position region is defined as the st eady state region. It is easily determined that the liquid composition at the inte rface is much higher than that of the solid and the bulk liquid because the solutes are built up in front of the S/L interface. Since, as shown in the phase diagram, the temperature is a function of composition, the increase of solutes can lead to a local increase of the liquidus temperature at the in terface. In Fig. 2-5, the increase of the liquidus temperature is pl otted with the thermal gradient, G, which can be controlled. When the thermal gradient is less than liq uidus as shown A in Fig. 2-5, the liquid can be regarded as supercooled, even in G > 0. Therefore, the dendritic solidification can occur due to the instability

PAGE 28

28 of the interface and the di rectional solidification by extracting heat to desired direction of the mold wall is possible. The columnar structure is a unique structure de rived only by directional solidification with const itutional supercooling. The simple relation of the morphological instability in alloys with the constitutional supercooling is given as the following [13] dx dC m dx dT GL L L L [2-19] where, mL is the slope of the liquidus by simply assumi ng the slope is constant. Herein, the solute gradient in liquid is obtained by following Eq. (2-20) [15], )1(* 0kC D V dx dCL L x L [2-20] where V is the solidification rate and DL is diffusivity of solute in liquid. Combining Eqs. (2-19) and (2-20), and noting that kC* L = C* S, the general constitutional s upercooling criterion at which dendrite solidification can o ccur is shown as follows: [2-21] In the absence of convection, 0 *CCSand Eq. (2-21) is rewritten as Eq. (2-22) L L LkD kCm V G )1(0 [2-22] The right hand side term in Eq. (2-22) can be a constant in the same system because mL, DL, and k are fixed values during the entire solidificati on process. This indicat es that the ratio of thermal gradient, G to solidification rate, V is the governing factor in determining the shape of the interface. For the Al-Cu(4.5wt%) alloy, an inst ability diagram is plotted in Fig. 2-6 using mL=3.3 (C/wt%), DL= 4.5-5 (cm2/sec), and k =0.17 [16, 17]. The upper part of the line in Fig. L SL LkD kCm V G )1(*

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29 2-6 is a morphologically instable area and cellu lar/dendritic solidifica tion will occur. More accurate and developed analysis were performed a ccounting for the thermal field in front of the interface using fluctuation theory by Mullin s and Sekerka and elsewhere[18 21] Mass, Heat, and Momentum Conservation So far, the quiescent fluid flow of no conv ection has been applied whereby the diffusional process is th e dominant process for mass/therma l transport. To evaluate the influence of convection, a few simple fluid dynamics need to be reviewed. The quantitative assessment of fluid dynamics in solidification i nvolves the simultaneous solution of coupled partial differential equations, and typically, this is achieved numer ically. The various methods and mathematical algorithms used to produce a numerical solution comp rise a separate field of research, and they will not be reviewed here. Several texts and arti cles, applicable to crystal growth, cover this subject well [22-26]. Analytical solutions to the simplified fluid dynamic problems provide a good deal of physical insight into many phenomena. Accordingly, this sec tion is dedicated to presenting the general mathematical problem. It addresses many of the reasonable assumptions regarding the nature of the fluid in the present system as well. Note that only laminar flows and Newtonian, incompressible, dilute two-component fluids are considered. Several texts, which were consulted during the development of this section, provide extensive information on the subject [27-30]. Governing equations The governing equations that describe fluid m otion and interfacial transport come from three conservation principles rooted in physics. For any fluid system: (1 ) Mass is conserved, (2) Energy (heat) is conserved, and (3) Momentum is conserved. Conservation of mass and energy are themselves fundamental laws of physics. Conservation of momentum however, comes from Newtons second law that states th at the sum of the forces on a body (fluid) equals the rate of

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30 change in momentum (mass times acceleration). All three principles can de scribe the motion of a fluid assuming that it is a conti nuum, i.e., that density is an in tensive variable having a finite value at any point. The equation representi ng conservation of mass can be simplified by assuming that the fluid has a constant density at every point in the system and that linearized functions are applicable. With these, the fluid is termed as incompressible and expressed as a one-dimensional equation. Assuming no diffusion in the solid and keeping w ith the notation of th e previous section: )(LL LL LCDCv t C [2-23] Potential energy, kinetic energy, viscous dissi pation, mechanical work, and thermal energy all contribute to the overall energy balance in a fluid system. However, for solidification processes thermal energy outweighs all of th e others, and usually th ey are neglected. The corresponding thermal energy balance equation for th e present research then becomes Eq. (2-24) )(LL LL LT Tv t T [2-24] where L is the thermal conductivity of the fluid. Radiativ e heat transfer is also neglected in Eq. (2-24), so that Eq. (2-24) is simplified to yield a one-dimensi onal solution. If the only body force acting on the fluid is gravity, whic h is true in the present case, then the momentum balance is given by the Navier-Stokes equation gv Pvv t vLLL L LLL LL )( [2-25] where L, PL, and L are the density, pressure, and viscosity of the fluid, respectively, and g is the gravitational vector. Note th at the kinematic viscosity, is simply L/ L. The left-hand side of

PAGE 31

31 Eq. (2-25) represents the change in momentum a nd the right-hand side terms represent the forces due to pressure, viscous, and gravitat ional forces from left to right. Herein, the convective fo rce is expressed by the Lv term and conductive/diffusive motion in the melt is reflected by the ratio of DL to L. The convection results from the buoyancy force caused by temperature and soluta l gradients in the liquid, whic h are difficult to avoid during crystal growth. With the Oberbeck-Boussinesq ap proximation, variations of the coefficients of D and by the change of temperature, pressure, and composition is also ignored for further simplification. Also, the density of the fluid is assumed to be constant (i.e. an incompressible liquid) everywhere in the governi ng equations except in front of the gravity vector. Using linear approximations for the effects of temperature and composition on the fluid density, Eq. (2-25) can be transformed: gCCTT vPvv t vLL LL LLLL LLL L L )]()([* ** 2 [2-26] where, T* L and C* L are the reference temperature and com position for the constant density value L (these are typically taken at the S/L interface), and and are the thermal and solutal expansion coefficients of the liquid, respectiv ely. From Eq. (2-26), the buoyancy force for convection is caused by the density inversion du e to temperature and compositional gradients represented by the third term in ri ght-hand side of Eq. (2-26). In pr actice, the density gradient in response to the temperature and compositional grad ient can occur in any direction: vertically, laterally, and radially. Any direction can be a source for the convection. Boundary Conditions In order to solve the coupled Eq. (2-24) to (2-26), boundary conditions must be defined for the system. On the container walls, there are several assumed situations that are typically imposed. These include a no-slip condition for the fluid, imposed/measured temperatures or

PAGE 32

32 thermal fluxes, and zero species flux. In typical fluid dynamics problems, species, pressure, and thermal conditions are imposed at the inlet and outle ts of the domain to enable a solution to be determined. For directional solidification, the liquid is ofte n considered to enter the system domain at a chosen boundary/inlet. It is assumed to have a constant composition, velocity, and temperature profile. The s/l interface, how ever, is a more complicated boundary where the imposed conditions must account for the inte rfacial transport of energy and mass. The s/l interface is not only a source and sink for thermal energy and co mponents, but it moves as solidification takes place. Thus, the conditions at the s/l interface sh ould change in a complex manner. Specifically, they depend on conditions in the solid such that solid equivalents to Eq. (2-24) to (2-26) must be solved as well in solidification problems. However, the problem can be greatly simplified assuming that there is no species diffusion in the solid so that 0 v The only additional equation needed is then T t T2 [2-27] where is the thermal conductivity of the solid. General boundary cond itions at the s/l interface then become H S CmTTTL Al L LLmL 2* 0 [2-28] nvILvnTnTL LL )()()( [2-29] nvIkCnCDL L L ))(1()( [2-30] where mL<0 is the liquidus slope given by the two-component phase diagram; n is the unit normal vector to a point on the s/l interface; I is the velocity of the s/l interface in the lab frame

PAGE 33

33 at that point; and LV(<0) is the latent heat for the solid per unit volume, which is assumed identical for the solute. The inclusion of the liquid velocity at the s/l inte rface in Eq. (2-29) and (2-30) is necessary if the density of the solid and liquid are not e qual. The resulting shrinkage or expansion upon solidification will alter the velocity of the interface and a ffect the transport of components and heat, and Lv accounts for this. The absolute density change upon solidification is less than 2% for Al in the present system. Nondimensional Quantities The Eq. (2-24) (2-26) are expressed in terms of dimensional (also called primitive) variables. However, due to the vastly different length scales in most problems, it is often convenient to express the governing equations and boundary conditions using nondimensional quantities by scaling out using a reference state with the same units. It may be necessary to use nondimensional quantities to improve the stiffness of the problem and prevent possible sources of computational error. In fluid dynamics, there are several standard nondimensional numbers that give physical insight into the solidification processes and allow for comparison between systems. These quantities noted here are es sential to estimating the convection during solidification. Reynolds number Re compares the magnitudes of the in ertial forces to viscous forces on a fluid. The relationship is generally defined as chchVL Re [2-31] where Lch and Vch are a length and speed characteristic of the system. In solidification, these are typically taken as the diameter of the crucible and the growth rate of the solid. Practically, Re is used to estimate the type of flow in a system, i.e., laminar or tu rbulent flow. Laminar flow occurs in the system when viscous forces do minate as indicated by a low value of Re Generally, when

PAGE 34

34 Re < 104, the flow is regarded to be laminar. In th e Al-Cu (4.5 wt%) alloy, the crucible diameter is 38.1mm and the kinematic vi scosity of Al is 2.9-8 m2/sec, so turbulence during solidification of the alloy is not expect ed for a fluid veloc ity less than 3.5-8 m/sec. This is about 4 orders of magnitude gr eater than the withdrawing ra tes imposed in this study, as described later, so tur bulent flow is avoided. The thermal and solutal Rayleigh numbers, RaT and RaC, give an indication of the level of buoyancy forces relative to viscous forces in a system. They are defined as L chLL TLTTg Ra 3 ***))(( [2-32] L chLL CLCCg Ra 3 ****))(( [2-33] where, g is the magnitude of the acceleration due to gravity, is the thermal expansion coefficient of the liquid, ) (*** LLTT and ) (*** LLCC are temperature and solutal differences defined by values at two reference positions, an d **, which are usually th e s/l interface and the far-field liquid in directional solidifica tion, respectively. If the values of either RaT or RaC exceed a critical value, then convective instabilities (multiple convective cells/rolls) are predicted to occur. The characteristic length chosen for a ve rtical Bridgeman-type solidification is the melt height, h, the distance between the s/l interface and the top surface of melt. If the temperature and solutal axial gradients in the liquid are defined in terms of the melt height, the Rayleigh numbers can be rewritten as L L ThGg Ra 4 *))(( [2-34] L ChCg Ra 4 *))(( [2-35]

PAGE 35

35 Thus, low gradients and small melt heights can reduce the convection. While the Re, RaT, and RaC provide insight into the magnitude s of the different forces in a fluid system, the thermal and solutal Peclet numbers, PeT and PeC, compare different transport processes between convective tran sport and conducive (or diffusiv e) transport. These are given as L chch TVL Pe [2-36] L chch CD VL Pe [2-37] Note the similarity in the form of these quantities to Re The magnitudes of all three numbers Re, PeT, and PeC give indications of boundary layer ex tents in the fluid system. Herein, the physical meaning of boundary layer is an interface at whic h the component (temperature, composition, and momentum) changes abruptly. Several additional nondimensional quantities can be constructed from the previously defined numbers. Defined here are the Schmidt ( Sc ) Prandtl ( Pr ), Lewis ( Le ), and Grashof ( Gr ) numbers: L CD Pe Sc Re [2-38] L TPe Re Pr [2-39] L L T CD Le Re Re [2-40] PrT TRa Gr [2-41]

PAGE 36

36 PrC CRa Gr [2-42] These numbers are advantageously independent of the characteristic dimensions of the system and, consequently, they are simply nondime nsional material properties of the fluid. For example, metals and semiconductors typically have Sc and Le values much greater than one and Pr values much less than one. Thus, the ther mal transfer dominates the other transport mechanisms and the thermal condition may look cons tant during the entire experiment due to the overwhelming speed of heat transfer comp ared to solutal diffusion and convection. Convection in solidification The convection during solidification m ay be caused by rotation, natural convection, or the Marangoni effect [31-34]. If melts solidify vertically against the gravity without external sources of rotation or oscillation, the main factor for the convection is ther mally or compositionally driven natural convection. The na tural convection requires density inversion against the direction of gravity. Without this, ideally, the melt woul d be stable against th e convection. From the aspects of fluid dynamics, the friction force be tween the melt and the interface can produce an unphysical stagnant uns tirred fluid layer, at the interface [35, 36]. In this boundary layer, only diffusive transfer is possible. The diffusi on boundary layer consists of the solutal boundary layer, c, the momentum boundary layer, v and the thermal boundary layer, T. During the solidification of metals, the solutal boundary layer is important and the interaction of D with v and T is also considered for precise assessm ent of the solutal diffusion boundary layer. Physically, a strong convection can swipe more solutes from the solutal boundary layer. As a result, the amount of solute in the bulk liqui d will increase and the solutal diffusion boundary layer will be thinner. However, by assuming the local equilibrium at the interface, the

PAGE 37

37 concentration in the solutal diffusion boundary layer is higher than in bulk liquid. The level of convection can be described by the ratio of solu tes in the bulk liquid to the interface [37, 38]. k C CS/1 1 [2-43] LV V aDD [2-44] where, VD is the velocity at the edge of the diffusion boundary layer, a is a constant depending on the velocity profile in th e liquid (in linear approximation, a = 1/6), L is the interface length, and V is the interface moving velocity. Herein, D and VD are unknown parameters and difficult to measure. However, the bulk liquid velocity an d temperature in the bulk liquid can be easily measured, so that the soluta l boundary layer can be inferred by the relationship between D and v (or T). These relations are expressed in a variety of ways according to the fluid flow conditions during solidification and are well summarized by Ostrogorsk y and Mller [39]. First, the interaction between the solutal diffu sion layer and the velocity layer is linked by Rosenberger and Muller [40], 2/1Re PeC V [2-45] From Eq. (38), the term in the parentheses is equivalent to the Schmidt number and, for the Al-Cu (4.5wt%) alloy, this number is 420. The physical meaning of a large Sc, v>> C, is that the compositional profiles occurring in the melts change in a range near the phase boundary, where the velocities are already considerably redu ced. It should therefore be expected that the velocity profiles do not influence the concentra tion profile at the s/l interface. From this relationship, the C is obtained indirectly, so bulk liquid composition, C, can be estimated.

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38 Similarly, heat and mass transport results in a relationship between the thermal boundary layer, T, and the concentration boundary layer, C, in the absence of convection. n T C TPe Pe 1 3 1 n [2-46] The parentheses on the right-hand side of Eq. (2-46) is also represented by the Lewis number, which characterizes the ratio of th e diffusive heat and the solutal flux with Le = 680 for the Al-Cu (4.5 wt%) alloy. First, it indicates that the thermal boundary layer is extended much further by 680 times and, similar to the Sc number case, the compositional profiles near the phase boundary are not affected by the temperature profiles. From Eq. (2-45) and (2-46), the negligible effect of velocity and temperature on the compositional profil e near the interface can justify the local equilibrium condition during so lidification. Second, a large Le number indicates that the emitted latent heats cannot be built up in advance of the interface like solutes due to fast heat diffusion resulting in a temperature gradie nt remaining constant during the solidification process. Most of all, by using E q. (2-46), the level of convection, C can be estimated by measuring the temperature in the melt. Herein, by using the nondimensional quantitie s and its interactions, the solutal boundary layers could be estimated. From this, the le vel of convection can be found by the solutal diffusion boundary layer. Also, through Eq. (2-43) and (2-44), the composition of the bulk liquid can be calculated. In general, the amount of so lutes in the bulk liquid will increase with an increase of convection. When the Al-Cu alloy is solidified in a ve rtical direction against the gravity, the temperature is higher and the copper amount is lower at upper part of the sample. Herein, the dependence of density dist ribution along the height is as followed [41].

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39 )]()(1[* ** *0 LLLLLL LLCCTT [2-47] where, Lis the liquid thermal expa nsion coefficient, 2.74-5(/C), L is a constant corresponding to the change of copper concentration, 0.02 (/wt%), LT(=648C) and LC(= 4.5 wt%) is arbitrarily chosen as a reference state of temperature and composition, respectively. In Fig. 2-7, the density change is calculated in the case of Al-Cu (4.5 wt%) yielding no density inversion during solidification. Thus, this system should be convectively stable without another source of convection. In most cases, a small extent of radial heat extraction may be present and this amount can result in considerable amount of convection. In Fig. 2-8, the simple schematic representation for vertical directional so lidification, in which TA is the temperature at the bottom, TB is the temperature at the top, and TS is the temperature at the sidewall. Unless the sidewall is assumed to be insulating, the radial temperature gradie nt should exist horizonta lly. This is also represented by the radial Rayleigh number, RaW, L mS WdhTTg Ra /)(24 [2-48] where, h is the melt height and d is the diameter of th e melt container. Herein, TS can be simply chosen as an average of the top and bottom te mperatures. The radial temperature gradients, represented by Raw, can be as strong as a forced conv ection of 10 rpm of external rotation applied when the side wall temperature is 10 C above the cente r temperature [42, 43]. In sum, by considering the fluid flow, th e beginning for the practical mathematical solution for solute redistribution can be established based on the following assumptions: (1) local equilibrium condition is justified wi th the existence of stagnant solu te film in metals, (2) constant temperature gradient can be assumed regardless of the emission of latent heat from the s/l

PAGE 40

40 interface due to the fast heat diffusion. Thus, th e thermal condition during solidification can be fixed only by experimental parame ters. Finally, (3) thermally activ ated radial convection can be an established as the main source for melt mixing. Solute Redistribution During solid ification, the solutes are rejected to the liquid according to the inherent solubility difference between th e solid and the liquid, unless 2nd phase is formed in solid. The importance of the solute redistribution is that it determines the interface shape by changing the thermal condition near the interface and inhomogene ity of the solidified material when diffusion of the liquid and the solid is not fast enough. The compositional inhomogeneity is termed as s egregation For slow solid diffusion and fast solidification rates, the solute atoms may not have sufficient time to diffuse out into the entire sample. As a result, the compositional variation, which is marked on the solid part of Fig. 2-9, fo rms inside the sample. In the case of a negative liquidus slope ( mL) in Fig. 2-6, the equilibrium partitioning coefficient, k=CS/CL, becomes less than one. This leads that the solute is rejected in to the liquid. In most cases, the rejected solutes can be built up near the s/l interface or they co uld mix with the bulk immediately according to the mass transport conditions. He rein, it can be recognized that k diffusion, and convection are governing factors in determining solute redist ribution in the solid and the liquid, thermal inhomogeneity, and s/l interface morphology. Severa l solidification models were classified according to the mass transfer methods. Before introducing the solidification models, the principle of the mass balance near the interface needs to be explained first. Generally, solidification proceeds in a thermodyna mically closed system. In this respect, the mass balance should be satisfie d in any mathematical solution. 0 )()( LL SSCfdCfd [2-49]

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41 where, fS and fL are the volume fraction of solid and li quid during solidification. Eq. (2-49) is simple but important because it is the cornerstone of solidification classification. For example in Fig.2-9, a specific case of a solidification model is illustrated. To satisfy Eq. (2-49), area A2 should be the same with the summation of area A1 and A3, Herein, A2 represents the rejected solutes, while A3 is the solute dissipation into the solid, and A1 is the solute dissipation into the liquid by diffusion. Namely, Fig. 2-9 signifies that the amount of re jected solutes from the s/l interface is dissipated out to the solid and the li quid by diffusion. In this case, the diffusion of liquid is fast (resulting in a flat concentration profile in the liquid) and that of solid is limited (leading to an exponential decrease of concentr ation in the solid). By simplification of A1 as a triangle, the virtual mass balance at th e interface can be given by Eq. (2-50), *2 1 )1()(SDS LS SSLdC dCfLLdfCC [2-50] where, L is the ingot length and S,D, is the diffusion boundary layer of the solid and it is approximately equal to 2DS/V, as shown with the area A1. DS is the diffusion coefficient of the solid. If the liquid diffusion is not fast, the compositional profile in the liquid is not flat as shown in Fig 2-9 and Eq. 2-50 should be modifie d. Accordingly, it can be realized that the mass balance equation and following so lute redistribution can be determined by the mass transport method. Solidification in Planar Interface The planar interface solidification is used in practice to grow single crystals and refine materials, and results in a controlled uniform or non-uniform composition within materials. Mathematically, the solution can be easily obtai ned because of the simple geometry of the interface shape. As mentioned above solute redistribution is affected by the diffusion in the solid

PAGE 42

42 and the liquid, and convection in the liquid, leading to the differen t mathematical solutions which are summarized in Table 2-I. Each case will be reviewed briefly in this section. Equilibrium solidification model This type of solidification can occur when diffusion of solutes is extremely fast or the solidification time is extremely slow, i.e.fStDL 2 to ensure complete diffusion in the solid and the liquid, where L is the length of the growing crystal and tf is the solidification time. The mathematical solution of the mass balance gi ven by the simple level rule as following, 0CfCfCLLSS [2-51] where, fS and fL are the weight fractions of the solid and liquid, respectively, and CS and CL are the concentrations of the solid and liquid. When a sample having initial concentration, C0, is solidified through the vertical line of the phase diagram, in Fig. 2-4, the temperature and concentration of the solute are change d. The corresponding composition profile to TL, T and TS, is described in Fig. 2-10. At the liquidus temperature, TL, the solidification is initiated and the corresponding concentration is kC0. At T=T about 50% of the ingot was solidified and the liquid and solid concentrations are CL and CS, respectively. And, at T = TS, the solidification is completed with corresponding composition C0/k Due to long solidification times and fast diffusion rates in the solid and liquid, the composition of the interface and the bulk liquid remains constant along the length of the ingot. Physically, at the equilibrium solidification, segregation is not observed. This case is very rare, but, sometimes, it o ccurs in interstitial diffusion having a small solute size such as hydr ogen, whose concentration is a key component in porosity formation, diffusion in aluminum and carbon in steel.

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43 Scheil model The Scheil model was developed with tw o basic assumptions: no solid diffusion and complete liquid diffusion. Maximum segregation is produced due to the condition of no solid diffusion. However, this model [44] is the most versatile in all oy solidification be cause the solid diffusion is very limited in most cases. A quan titative expression is obt ained by equating (solute rejected from the solid) = (solute increase in the liquid) It is given by Eq. (2-52) LS SSLdCfdfCC )1()(* [2-52] By integrating from kC0 at fS = 0, 1 *)1(k S O SfkCC [2-53] The composition profile of the solid along the length of the solidified ingot can be calculated by Eq. (2-53). C* S is the composition of the solid onl y at the interface, not the entire solid unlike the aforementioned equilibrium solid ification model. Besides the assumptions in Table I, several additional assumptions are need ed in order to use th is model. First, the equilibrium partitioning coefficient, k, is constant for the entire solidification range. It indicates that the liquidus and solidus slopes are linear until the completion of solidification. Also, the local equilibrium is satisfied at the s/l interface, so that the composition of the solid and liquid can be simply obtained by the phase diagram. In the case of the Al-Cu (4.5 wt%) alloy, the compositional profile along the sample length is plotted in Fig. 2-11, in which the solid composition of copper at fS = 0 is 0.77 wt%, corresponding to C0 = 4.5 wt% and a fraction of eutectic iden tical with the fraction of the final liquid is found to be 0.091% by substituting C* S with the solubility lim it, 5.65 wt%, which is the copper concentration at the eutectic temperatur e. From these information and assumptions, the non-equilibrium phase diagram can be plotted in Fig. 2-12. The interfacial composition of the

PAGE 44

44 solid will follow the equilibrium phase diagram, i.e. the black-solid lines, and the average composition in the solid is found by the red-dotted line. As compared with Fig. 2-1, the eutectic cannot be formed based on the equilibrium phase diagram at Al-Cu (4.5 wt%), but from Fig. 212, the eutectic is expected due to slow solid di ffusion rate. Note that the Scheils equation is well applied in moderate solidification rates beca use, in these rates, the solute may not have sufficient time to diffuse in the solid. Mixed model (I): Tillers consideration In this model, solid diffusion is still ignored, but liquid diffu sion is limited without convective mixing. It leads to a solute enriched region ahead of the s /l interface due to the diffusion in the liquid not being fast enough to homogenize the rejected solutes with the bulk liquid. These enriched solutes ar e dissipated into the liquid by diffusion. Thus, solute enriched region is equal to the diffusion boundary layer. By considering the amount of solutes diffusing into unit area at x xdxdCD )/ ( and diffusing out at x+dx dxxdxdCD)/(, the net flow into a volume element is xdxCdD )/(2 2. If the s/l interface is consider ed and freezing is represented by moving the liquid distribution past it at the rate of the solidification rate, V the net flow out of the same volume element due to freezing is )/( dxdCV. The differential equation describing a stationary distribution is proposed by T iller, Jackson, Rutter and Chalmers [47] 02 2 dx dC V dx Cd D [2-54] The solution is found given with the following boundary conditions: x=0, CL=C0/k and at x = CL=C* L. ) 1 1()/( 0xDV LLe k k CC [2-55]

PAGE 45

45 Herein, DL/V is named as the characteristic di stance at which, mathematically, the CL C0 falls to 1/e of the C0/k C0. This length depends on the diffusivity of a material. The compositional profile in front of the s/l in terface is plotted in Fig. 2-13 in applying k = 0.17 in Al-Cu alloy and C0 = 4.5 wt%, when about ha lf of the ingot is solidified at a V = 0.00075 cm/sec. In Fig. 2-13, it is found that the characteristic length is as large as 0.25 cm in the absence of convection. The compositional profile of the enti re length scale of the solidified ingot is illustrated in Fig. 2-14 when the ingot size is assumed to be infinite. When the solid composition reaches the bulk liquid composition ( CS=C0) this region is defined as a steady state region. In the steady state region, the amount of enriched solutes reaches C0/k at the s/l interface and the concentration of the soli d and the bulk liquid are C0. Thus, the segregation disappears in this region. Before the steady state region, there is a region in which the solid concentration increases to C0 is called the initial tran sient region. When the diffusion boundary contacts the end of the ingot, the steady state cannot be maintained a ny longer and the composition increases abruptly. This is the final transient region. By using Eq. (2-54) and (2-55), the mathem atical expression from the mass balance for each region is different. x D V k k CCLexp 1 10 at steady state [2-55a] 1 exp1 10x D V k k CCL at initial transient [2-55b] 'exp 10x D V V D k k C at final transient [2-55c] The compositional profile in the initial transi ent region and the steady state is found easily with Eq. (2-55a) to (2-55b). In the final transient region, the analysis is complicated by the fact

PAGE 46

46 that the finite liquid region, in which the concentrat ion increases at all point s, must be considered, since the rejected solute s cannot be fully dissipated into the bulk liquid. Instead of the convenient relationship like Eq. (2-55a ) and (2-55b), the residua l solutes, which is not dissipated into the bulk liquid, is quantified by Eq. (2-55c) and repr esented by the grey area in Fig. 2-14. This residual solute may be back to the liquid, resulting in abrupt increase of the liquid and solid concentration. Herein, x is the distance from interface to th e end of specimen. As seen in Fig. 214, once the rejected solute is unable to be dissipated comple tely at the end of specimen, the overlap between the wall and the diffusion boundary layer begins and the solute may become concentrated in this region. Mixed model (II) To approach real cases of crystal growth, natural convecti on should be considered. This case was investigated by Burton, Prim, and Slic her [48], named as the BPS model. The convection can swipe out the solutes in the diffu sion boundary layer, resulting in the decrease of solute buildup ahead of the s/l interface, particularly when the diffusion layer extends far into the liquid. Thus, the swiped solutes are homogenized with the bulk liquid and the concentration of the bulk liquid increase ( C) to CL = C > C0. The influence of convectio n is taken into account by the parameter, D as seen in Eq. (2-43). Conceptua lly, the length of the boundary layer is extended to the maximum in th e quiescent condition, at which D = 4.6D/V [49]. This phenomena is shown in Fig. 2-15, in which the solid line indicates the composition profile by the Mixed model (II) and the dotted lines is the Mixed model (I). Note that the diffusion layer in the Mixed model (II) is not as extended as that of Mixed model (I) due to the convection. As a result, the concentration of the bulk liquid becomes homogenized at C (> C0). Thus, as the C is away from the C0, the level of convection is stronger. Here in, the effective par titioning coefficient, keff = CS/ C, is defined to quantify the level of convecti on and it was already di scussed in Eq. (2-43)

PAGE 47

47 and (2-44). Also, the mass balance equation is th e same as Eq. (2-54) with different boundary conditions: at x=0, CL=C0/k and this is the same as Eq. (2-54), but, at x= CL=C. And, the solution is given by Eq. (2-56): LDDV S SLe CC CC/ 0 ** [2-56] where, LC and SCare the interfacial compos itions of the liquid and th e solid, respectively, V is the solidification rate and DL is the diffusion coeffi cient in the liquid. Mixed model (III) Some experimental results were not matche d with aforementioned models, particularly when the diffusion of a solute is very fast [50] and the solidification rate is extremely slow [51, 52]. In the case of solidification including interstitial atoms such as C, H, B, and N, its inherent fast diffusion requires to consid er the diffusion in the solid. Bo wer, Brody and Flemings [16, 17] considered the solid state diffusion at the interf ace and the analytical expression is given by Eq. (2-57), * *)()1()(yy S S L SLdy dC D dt dC y dt dy CC [2-57] where, ingot length is l and the position of the s/l interface is y* The left side term represents the amount of solutes rejected from the inte rface by the moving boundary and the first term of the right hand side is the amount of solutes into liquids and the second term indicates solute back diffusion into the solid. Herein, the growth rate, dy/dt, can be assumed to be linear or parabolic for further simplification. k f kCCS S1 10 for linear growth [2-58] k k S Sfk kCC21 1 0 *)21(1 for parabolic growth [2-59]

PAGE 48

48 where, was defined in Eq. (2-60): 2L DfS [2-60] Here, DS is the solute diffusion coefficient in the solid, f is the solidification time, and L is the ingot length. Note that the extent of diffusion depends, not on solid ification time alone, but also on the dimensionless parameter, k ; for k <<1, Eq. (2-58) and Eq. (2-59) are approaching the Scheil model as show n in Eq. (2-53) ; for k >>1, composition of solid is close to uniformity. By oversimplification, as shown in Fig. 2-8 of the solid diffusion by using Zeners approach, however, mass balance breakdown occurs when solid state diffusion is fast due to high temperature and diffusion coeffici ent. Clyne and Kurz [53, 54] developed a modified constant, given by Eq. (2-61), to remove the oversimplification error. ) 2 1 exp( 2 1 ) 1 exp(1' [2-61] Diffusionless model This model [55, 56] is applicable when solidif ication occurs at high rate such as in fiber spinning, atomization, and melt spinning. Due to the high solidification rate, the atomic arrangement shows noncrystalline structure like liquids and glasses. In this case, thermal transport is only concern to manage the microstruc ture because the solutal transport is too slow at high solidification ra te. This is out of scope of this study. Zone melting Zone melting [57-61], a fractiona l crystallization, works beca use the solute concentration in the freezing solid differs from that in the liquid. Technically, a sm all melted zone slowly travels through the entire solid charge. As the zo ne travels, at its melting interface, solutes with designed composition enter to melted zone. And, at the trailing and freezing interface, solutes are

PAGE 49

49 frozen out to solid with its equilibrium partit ioning coefficient. It l eads to faster solute enrichment in zone and reaches to diffusion c ontrolled growth earlier, especially, in a large k When the solid diffusion is ignored and liquid di ffusion is fast like Scheils model, the solutes rejected will be homogenized with the bulk liquid instantly. In this case, the solutes distribution is obtained easily by simple mass balance, (sol ute entered from the melted interface) = (solute frozen into the trail inte rface). In the upper part of Fig. 2-16, the melted zone, l is the distance traveling through the ingot, L and x is the distance from the left side wall. In the small molten zone, solutes enter from the melted interface an d freeze out to the trail interface. Analytical solution was provided based on the mass balance, dxkCCdCL O)( [2-61] Assuming the two interfaces have uniform cross-sectional area, Eq. (2-61) can be transformed into Eq. (2-62) dx l s kCdsO)( [2-62] A solution from mathematical treatment is shown as: x l kek C C )1(10 [2-63] where, k is the equilibrium partiti oning coefficient. Using Eq. (2-63), composition profile of solute along the sample height ca n be calculated according to dist ance from the starting point. In the lower part of Fig. 2-16, the compositional va riation along the height for Al-Cu (4.5 wt%) is shown when l is chosen as 0.1 cm and 10 cm. The trend of Fig. 2-16 is understood easily because more solutes are enriched in the small melted zone at l = 0.1 cm and, in this case, diffusion controlled growth is possible.

PAGE 50

50 Provided limited liquid di ffusion, the zone length, l and characteristic length become important to obtain solute bu ildup in the small zone. At l > D/V as shown in Fig. 2-17(a), there is no solute built up ahead of the s/l interface. This leads to norm al solidification without zone effect. At l < D/V, some amounts of solutes, which are not fu lly discharged into bulk liquids, are built up in zone in Fig. 2-17(b), so that zone melt ing effect is predicted in this case [47]. })1)(1{(0kekCCx S [2-64] 1 )/(1 DLVe D V k [2-65] Solid composition can be calculated from Eq. (2 -64) and (2-65) proposed by Tillers et al. It can be easily understood that D and k can determine the effectiv eness of zone melting. For example, hydrogen in alloys will be more effectively stacked in the zone than copper due to two orders of magnitude difference of hydrogen a nd copper diffusion coefficient in aluminum (Dcu= 4.5-5 cm2/sec, DH = 3.1-3 cm2/sec at 648C). Faviers extension of mixed model (II) Favier et. al. [62-68] has extended the BPS model, which only considers the steady state of the distribution of the con centration, by using time dependent diffusion equation. In the BPS model, the diffusion boundary layer is found empiri cally because the level of convection depends on materials nature and solidifi cation variables. In th e absence of externa lly forced convection, the natural convection is determ ined by a few numbers of parameters such as the thermal and compositional profiles as well as solidification rate. Favier et. al. coupled the BPS model with aforementioned parameters to estimate the solute redistribution behavior a ccording to the natural convection. They also applied nondimensional quantities (Pe, Gr, Sc, and Pr) to generalize any solidification situations. With these, Favier et. al. produced the domain map with which the effect of solidification va riables on solute redistribution is able to be visualized at a glance. The

PAGE 51

51 domain is mapped for Al-Cu all oy in Fig. 2-18. Region (c) in Fig. 2-18 shows a similarity with the equilibrium solidific ation model. The solidification condi tion matched with Scheil's equation is depicted in region (a). The case that the vigorous convection destr oys the diffusion boundary layer is shown in region (b). The Mixed Model (I) case is in region (d) and the most practical case of the Mixed Model (II) co rresponds to the region (e). Each criterion domain can be classified with Table II and nondimensional quan tities were introduced in previous section. Solidification Models in Dendritic/Columnar Interface Dendritic/columnar shape a tree-like structur e stemmed from the planar interface. It requires definition of several terms, which are not needed in planar interf ace of solidification. In a simple and imaginary structure of a dendrite drawn in Fig. 2-19, dendrite trunk grown parallel to heat flux is defined as the primary dendrite and normal to heat flux are the secondary dendrite. The length between primary dendrites, as marked (a), is defined as the primary dendrite arm spacing and the dendrite height is al so marked as (b). At the tip of the dendrite, the solidification initiates and the temperature and compositi on can be regarded to be the liquidus, TL and kC0 predicted by the phase diagram. The solidificatio n is completed at the bottom of the dendrite. Thus, between the dendrite tip and bottom, th e compositional and temperature difference is present as shown at the bottom of Fig. 2-20(a) The primary/secondary dendrite arm spacing is determined by the local solidification time, tf. VG T tS f [2-66] Then, the primary and secondary dend rite arm spacing is expressed as n n fVGbtad)( [2-67] where, TS is temperature range of solidification, a, b, and n are constants, empirically. n value is between 1/3 and 1/2 for secondary dendrite ar m spacing and is very close to 1/2 for primary

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52 dendrite arm spacing. Generally, it has been shown that the final secondary arm spacing in fully solidified samples is much coarser than one that initially forms. This phenomenon is dendrite coarsening [70, 71] and is caused by the dendrite re-melting due to difference in surface energy. The surface energy difference leads to diffusion of solutes in smaller dendr ites to larger ones. The constants such as a, b, and n need to be measured empirically due to the dendrite coarsening. The differences between dendritic structure and planer interface structure can to be enumerated as following. First, the dendrite/co lumnar has a sharper edge and larger interface area relative to planar interface. It leads that re jected solute and discharged heat can dissipate easily into the bulk liquid. As a result, th e thermal undercooling and the constitutional supercooling are not as large relative to the planar front interface. Second, the sharp edge at the dendrite tip causes a Gibbs-Thomson effect, by which the composition a nd temperature of the dendrite tip can be altered by its in terfacial effect. Third, the capillar ity effect is present when the primary/secondary arm spacing is very small. Th en, the bulk liquid can experience the difficulty to penetrate through the interdendritic region. The capability to penetrate into the capillary pipe is expressed as the permeability ( K ). Finally, in most cases, the temperature and composition at the bottom of the dendrite corresponds to eutectic temperature (TE) and composition ( CE). The liquid composition becomes increasingly enriched near the dendrite bottom due to the lateral diffusion normal to the heat extraction directi on and the liquid composition can easily reach the eutectic composition even with very small amount of solute. For example, even in Al-Cu (1 wt%) alloy, a noticeable amount of eutectic was found [72, 73]. Besides aforementioned lateral diffusion betw een dendrites, the liquid diffusion from the dendrite bottom to the dendrite tip is also present through the interd endritic region and the radial diffusion at the dendrite tip plays an important role to dissipate th e rejected solutes into the liquid.

PAGE 53

53 Due to both diffusions, rejected so lutes between dendrites or from the dendrite tip can be built up ahead of the dendrite tip. It makes the tip composition, Ct, slightly higher than the bulk liquid composition, C. This is essential since Ct C0 ( > 0) is a supersatura tion for dendrite tip growth into the liquid. Besides the foregoing physical differences overwhelmingly complex and numerous factors compared to the simple plane interface impedes to obtain the mathematical solutions of the mass balance in dendritic interface. Howe ver, as shown in Fig. 2-20(a), a small volume element, in which the solid and the liquid coexis t, was chosen. The composition profile inside the small volume element can be calculated by the anal ytical solidification m odels used in planarfront interface, since the s/l interf ace is planar in this small volume element as shown in Fig. 220(b). Several assumptions were applied and they were reviewed in later section. With above background, this section will be dedicated to find analytical approach for dendrite shape and solute redistribution. Dendrite shape and curvature at the tip The Scheils equation in Eq. (2-53) can be transform ed to Eq. (2-68), )1/(1 0 0 k L LTT TT f [2-68] where, T0 is the melting temperatur e of pure aluminum, and TL is the liquidus temperature of AlCu alloy. As shown in Fig. 2-20, by connecting each s/l interface in each volume element (gray bar) along the dendrite height, h, it is possible to contour the dendrite shape according to the height. The dendrite is assumed to be closely packed and grown as a columnar structure in Fig. 2-21(a). In Fig. 2-21(b), a volume element is regard ed as a cylindrical disk shape having volume of dhdl2)2/(4 and the volume of solid in volume element is dhRTT 2 ')(4. Herein, RT=T is

PAGE 54

54 the radius of solid dendrite at arbitrarily chosen temperature, dl is the primary dendrite arm spacing, and h is the dendrite height. From the volume fraction of the solid in Fig. 2-21(b), the radius of the solid dendrite can be expressed with the solid fraction: )2/(5.0 'lSTTdfR [2-69] The dendrite height, h, is expressed as a function of te mperature and the thermal gradient, G TT hE TT [2-70] where, hT=T is the height at the arbitr arily chosen temperature (T ). By plugging Eq. (2-69) and (2-70) into Scheil equation expresse d as Eq. (2-53), the relation of R and h can be given by: )1/(1 0 0 2 2 '1 2 k L E l TTTT GhTTd R [2-71] The relationship between the curvature of the dendrite tip and growing conditions were developed by Ivansov [74] and modified by late r studies [75, 76]. From these studies, it was found that the curvature of the growing dendrite tip was approximately equal to the lowest wavelength by morphological instability theo ry [18, 19]. The minimum wave length, i, is given as followed 2 Ri [2-72] where, is the degree of supersaturation, is the coefficient of the Gibbs-Thomson effect. Even though the mathematical solution is complicated, the case of directional solidification with a columnar structure having 1 20C/cm and 10-4 10-5 cm/sec was well summarized by Kurz and Fisher [77] Rk mCkRkR DRGD V )1(4)1(2)1( 8 22 0 2 3 2 2 [2-73]

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55 The curvature of the growing dendrite tip is simply obtained with Eq. (2-73) by assuming a hemispherical growing tip. The increase of G and V make the dendrite tip shaper to accommodate the fast growth rate atmosphere because a needle-like structure has a better efficiency for solute dissipation into the bulk liqui d with a larger s/l interface and a more radial direction of solute diffusion. Thus by using Eq. (2-71) and (2-73), dendrite shape is able to be visualized. Diffusion boundary layer around growing sphere (Zeners approximation) The shape of dendrite tip can be simply assumed by a sphere. The diffusion boundary layer around the dendrite tip can be approximated by the diffusion so lution of a sphe re interface. When no tangential diffusion is occurring, the diffusi on equation can be written in terms of radial coordinates alone, [77 78] and its solution is shown as below, R CC dr dCR 0 [2-74] This shows that, to a first approximation, the thickness of the boundary layer around a growing sphere is equal to the radius (this is named as Zene rs approximation), which is the curvature at the dendrite tip. Thus, the di ffusion boundary layer around the dendrite tip is proportional to the dendr ite tip curvature. Permeability The permeability is the ability of the liqui d penetrating into a small pipe. From the experimental observation, DArcy found that at a sufficiently low rate of fluid flow through a packed bed, the flow rate is proportional to the pressure drop per unit length of bed given in Eq. (2-75): )(gP L Ak vL D L [2-75]

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56 where, vL is volume flow rate cm3/sec through the dendrite height, kD is permeability coefficient, cm4/dynesec, A is area of each bed, cm2, L is length of bed, is density of liquid, g/cm3, g is gravitational constant, cm/sec2 and P is pressure, dyne/cm2. The importance of DArcys law is applicable to the solidification when the stru cture of the packed bed is similar to the dendritic/columnar structure of di rectional solidificati on. If the volume flow rate can be obtained, the pressure drop, which is an im portant factor in porosity form ation, can be calculated. In Eq. (2-75), the value of kD in any fluid flow reflects the physical properties of the fluid and packing characteristics described by the distribution and shape of dendrites in the solidification process. The permeability, K, is defined as: DkK [2-76] where, is the viscosity having g/cmsec and the unit of the permeability is cm2. By combining Eq. (2-75) and (2-61), a familiar form of DAacys law for material scientists can be rewritten: )( gP K vL L [2-77] From Eq. (2-77), the permeability (cm2) can be regarded as an conductivity of fluid rate through a bed (or interdendritic region). In ot her words, a large perm eability indicates better fluidity through the interdendritic region. The first systematic measurement and theoretical approach was attempted by Piwonka and Flemings [79] by using partiall y solidified Al-Cu (4.5 wt%) in a steel tube by introducing molten lead. They found that the permeability is proportional to the square of the liquid fraction when the li quid fraction is greater than 0.3. The permeability of the dendritic network in Al-S i alloy was measured by Apelian et al. [80]. A strong relation of permeability with microstructure was revealed by Poirier and Ganesan [81] by adopting the Kozeny-Carman relation. Recently, Murakami et. al. [82] summarized th e permeability parallel

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57 and normal to dendritic growth direction by means of multilinear regression based on a large amount data: 2.3 7.1 2 2.2 1 13102.6L pgdd K [2-78] 2.3 4.2 2 3.1 1 19108.8L Ngdd K [2-79] where, Kp and KN are the permeabilities parallel and nor mal to columnar dendrites in m2, d1, and d2 are the primary and secondary dendrites arm spacing, respectively. The va lid ranges of liquid fraction, gL, to which the aforementioned equations can be applied, are 61.027.0 Lg for Eq. (78) and 66.026.0 Lg for Eq. (2-79). Piwonka and Flemings and D.R. Poirier et. al. [83, 84] introduced a permeability equation based on Hagen-Poiseuille law assuming a simple pipe-like structure, written by: 3 22 18 t gd KL P [2-80] where, t is the tortuosity factor, and gL is volume fraction of liquid. In a simple pipe-like structure, the theoretical value of permeability is proportional to square of liquid fraction which deviates from experimental values from Eq. (2-78) and (2-79). In addition, in a perfect pipe-like structure, t = 2 and in disordered equiaxed structure, t = 4. In our experiment, this value will be located between 2 and 4. The most well-known relation between perm eability and microstructure was developed, named as Kozeny-Carmen relation. 2 3 Sc L N PSk g K [2-81] where, kc is Kozeny-Carmen constant which was derive d according to various characteristics of bed structure (solidification microstructure). For example, in equiaxed Al-Cu(4.5wt%), kc =5. SS

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58 is the solid-liquid interfacial area in unit volume(/m3) and it depends on the microstructure of solidification measured by image analysis. Fluid dynamics in dendr itic solidification Volume flow rate through dendrites is an indicator of degr ee of convective motion [85,86]. In strong flow rate, the re jected solutes between dendrite can be delivered to the bulk liquid. Thus, the convection within the dendrites can alter the compositional variation within the dendrites and at the dendrite tip. Since the dendrite itself provides a resistan ce for fluid flow, the permeability should be account for to estimate the convection in dendrites [87-90]. The fluid flow can be characterized by Eq. (2-82) and (2-83) )(gPRa K vL m L [2-82] L mLKTg Ra* [2-83] where, Ram is the Rayleigh number of the dendrite, vL is the volume flow rate (cm3/sec), is the expansion coefficient by concentration change, g is the gravitational acceleration, L is the length of dendrite, T is the temperature variation within dendrite. Thus, the co nvective motion in dendrite can be determined with Ram along with K which also affects the solidification behavior of the dendrite. Scheils model in dendritic interface Between dendrites and far from the dendrite tip, the solute redistribution behavior is relatively well matched with the Scheil model if it has slow solid diffusion or equilibrium model if it has fast solid diffusion while liquid com position is somewhat homogeneous in the volume element due to the small size of the interdendritic region.

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59 In planar front interface, the compositional pr ofile is easily calculated by Eq. (2-53). To apply Scheils equation in the dendritic interface, the following assumptions are required: (1) there is negligible thermal undercooling for nucle ation of the solid: this is reasonable because latent heat can be dissipated relatively easily in the dendritic interface. By measuring the cooling curve at Al-Cu (4.5 wt%), the fact that the solidif ication initiates at just 1-3C below the melting temperature supports the assumption of negligible thermal undercooling [91]. (2) there is no net flow of solute into or out of the volume element However, at sufficiently steep thermal gradients and long solidification times, the liquid diffusion along the x direction in Fig. 2-20(a) cannot be ignored. Also, the convection can swipe solutes in inte rdendritic liquid. (3) there is complete liquid diffusion within volume element along the y direction in Fig. 2-20(b). The characteristic distance, DL/V within which diffusion dominates, can be compared with the half dendrite arm spacing. In Al-Cu (4.5 wt%), the charac teristic distance is 0.06 (cm) at DL = 4.5-5 cm2 /sec and V = 7.5-4 cm/sec which is larger than the ingot length. The half of primary dendrite arm spacing, = 0.0201 cm. Thus, this assumption may be reasonable due to a small length of the interdendritic region. (4) there is no diffusion in the solid and th ere is no radial diffusion on the dendrite tip. (5) a plate like de ndrite morphology is assumed to a pply the mathematical solution in planar s/l interface and this is reasonable when the volume element is sufficiently small. Generally, the assumptions were reasonable between dendrites and far from the dendrite tip. However, the situation is different near the dendrite tip Tip temperature depression (I): Burden and Hunts model The phenomenon near the dendrite tip is comp licated and complex, because the lateral, the radial diffusion and the convection are interplayed at the same time. In the absence of the convection, Burden and Hunt m odel [92, 93] considered well a bout radial diffusion and other factors. The liquid diffusion in th e x-direction in Fig. 2-20(a) a nd the radial diffusion ahead of

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60 the dendrite tip contribute to solute buildup ahead of dendrite tip. When th e rejected solutes are built up ahead of the dendrite tip, it can reduce the tip temperature by re-melting the dendrite tip. In addition, the curvature effect and kinetic eff ect give additive contribu tion to tip temperature depression by the Gibbs Thomson eff ect. By Burden and Hunt [92, 93], KRDTTTT [2-84] where, DT is the temperature depression due to the radial and lateral diffusion, RT is caused by the curvature of the dendrite tip, KT is the kinetic undercooling which was ignored because the low entropy of metal solidifi cation can make tip growing instant. Assuming that a tip can grow with minimum undercooling and k is constant, the analy tical solution is given: V GD kVC HD Tm TL LS LLsl 2/1 0)1( 8 [2-85] where, C0 is the initi al composition, TL is the liquidus temperature, sl is the curvature undercooling constant, H is the heat of fusion, mL is the liquid slope, V is the solidification rate, and G is the thermal gradient. In Fig. 2-22, the tip temperature dr op with various solidification rates is depicted for Al-Cu (4.5 wt%). At low solidification rates, the second term on the left hand side, representing the radial diffusion, beco me a governing factor for the tip temperature depression. In the high solidification rates, the ti p curvature mainly drops the tip temperature. The drawback of this model is that it ignores the inter action with convection and the compositional profile inside the de ndrite cannot be predicted. Tip temperature depression (II): Alexandorvs model The aforem entioned models were performed under stagnant flow of liquid flow (no convection) and, vL in Eq. (2-47) is ignored. The role of unavoidable convection based on the mass transfer and heat transfer was reviewed well by Worster et. al. [94] and Alexandrov et. al.

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61 [95 96]. In both approaches, the boundary cond itions were considered with the mass balance and heat balance equation at th e solid/mushy zone and liquid/mu shy zone. If the heat and mass balance of Eq. (2-23) and (2-24) as well as D Arcys equation of (2-75) is transformed to dimensionless forms by scaling out with refere nce state having same units (see appendix B) dz d N dz d dz d N dz d JH dz d 2 1))(( ))(( [2-86] dz d kc dz d D dz d dz dc Jc dz d ))(( ))1((0 [2-87] )(c dz dp RaJl m [2-88] where, m /( : temperature, m : liquidus slope, : initial concentration), c= /, J=v/V (v: volume flux of interdendritic fluid, V : solidification velocity), LLCCh /)( ( : volume fraction of solid, : density of interdendritic region, C : specific heat per unit volume, SS LLC CC )1 ( ) S /)( ( : thermal conductivity, S L )1 (), N1=LLLSCD / N2 = mCLLLV/ ( LV: latent heat), )/ /(0VgD ppL L m ** Vg Ra /0 (g: acceleration of gravity, : viscosity). Confusion can arise due to unfamiliar symbols and dimensionless forms between chemical engineers and material scientist. As shown by heat balance, Eq. (2-86) and mass ba lance, Eq. (2-87), the second term on the left hand side represents the mass tran sport by the convec tion. A variable, J representing ratio of convection and solidification veloci ty, is the indicator determining the strength of the fluid flow by convection. In the absence of co nvection, J should be unity since vL is zero and it increases as the convection increases. The effect of the convection on the tip com position is derived as Eq. (2-89) (see appendix b for details)

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62 1 )1/(0 Le JALe chz [2-89] where, cz=h is Ct/C, Le is Lewis number as seen in Eq. (2-40), and A is constant from the boundary condition. With Eq. (2-89), the tip composition can be obtained by consideration of convection. Crystal Growth Techniques from the Melt The m ost well-known techniques for crystal growth are the Czoc hralski method, Float zone method, and Bridgemans method, depicted in Fig. 2-23 (a), (b) and (c), respectively. Czochralski Method The pull rod is positioned over a crucible cont aining solid charge m aterial, which radial heaters melt to form a continuous source of liquid for solidification. In more modern technique, the entire apparatus is typically held within a chamber filled with an inert atmosphere to prevent oxygen contamination. To initiate growth, a portion of seed or cooling pad is lowered into the melt and created S/L interface is maintained by a te mperature gradient by the radial heaters. The rate and rotation of the pulling r od become a dominant factor to determine the microstructure. A main advantage of Czochralski method is that th e growing crystal is not in contact with the crucible wall. This helps to eliminate stress-induced defects due to mismatches in thermal expansion coefficients of the crystal and the crucib le material. A significant disadvantage of this method, however, is that the geometry of solidif ied crystal is not completely reproducible[97]. Additionally, it typically has radial temperature gr adients in the melt and a curved s/l interface, which increase inhomogeneity in the crystal. Float Zone Method The Float Z one (FZ) method is an alternative and crucible-free techni que typically used to grow Si, Te, and other materials [98, 99] by developing normal zone melting technique. As

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63 shown in Fig. 2-23 (b), radial h eaters are positioned around an ingot At the start of growth, the heaters create a molten zone, which is much smal ler than ingot length and held in place by surface tension. Then, the heaters are moved slow ly along the axis of the ingot, allowing the ingot material to replenish the zone at the top while the base of th e zone solidifies. Similar to the CZ method, an advantage of this technique is th at the growing crystal is not in contact with another material. Additionally, the Float Zone method can be used to grow particularly pure semiconductors because the melt is also free from c ontact with a crucible. The technique of zone refining can be used to eliminate impurities fr om a crystal as well. Unfortunately, the major drawback to this method is that the stability of the molten zone is often hard to maintain. The shape of the s/l interface, which strongly affects dopant segrega tion, is accordingly difficult to control. Vertical Bridgman Method The Vertical Bridgm an (VB) method, as depicted in Fig. 2-23(c), is a less complicated technique mechanically and ther e were several inventors and c ontributors to its development [100, 101]. It consists of a cruc ible surrounded by radial heater s and attached to a movable pedestal. The heaters are adjusted to create a hot and cold zone with temperatures above and below the melting point of the ma terial, respectively. Before grow th, a portion of the charge is brought into the hot zone and melted, establishi ng a s/l interface betw een the zones. The unmelted region could be a single seed crystal. Th en the crucible is slowly withdrawn into the cold zone. A stress-induced defect formation ma y occur because of differences in the thermal expansion properties of the crucible and crysta l materials. High level of axial and radial convection could also induce the macro-/microse gregations. Concave interface can occur due to radical heat flow.

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64 Axial Heat Processing Recently, an innovative crystal growth t echnique called Axial Heat Processing was developed in Russia to promote stable, planar crystal growth [102]. AHP is very similar technique to the VB method in that it utilizes a crucible, pedestal, and radial heaters. However, a feature of this technique in that it includes a baffle immersed inside the melt during growth, as shown in Fig. 2-24 at the desired height from the S/L interface. The baffle encloses a heater that provides heat axially to the S/L interface of the growing crystal. A significant advantage of AHP over the CZ, FZ, and VB methods has been verified [103-107] in planer crystal growth: These latter techniques only supply heat radially (from the outside) to the melt, leading to radial temperature gradients, increased convecti on, and curved s/l interfaces during growth. Similar with planer interface solidification, the axial heat provided by the baffle in AHP significantly reduces radial temperature gradients in front of dendrite ti ps and promotes more axially oriented dendrite structure. The AHP baffle also separates the melt into two large regions, above and below the baffle, linked by a channel between the baffle and the crucible wall. This effectively reduces the melt height above the interface and reduces the amount of natural convection in that region. The last feat ure of AHP technique is to limit the ingot volume similar with zone melting, so that the benefit of zone melting technique can be applied with same ways. The main disadvantage of AHP is that stre ss-induced defects are ofte n unavoidable, like in the VB method and the baffle should be coated wi th inert jacket to prevent the reaction with metallic melt. Accordingly, the AHP can affect the convection, as a main mass transport way, so that segregation and other structure re lated properties can be controlled. Porosity Formation The porosity for mation is classified according to size, feeding mechanism, and its cause. It was studied with thermodynamic and kinetic models. Theses will be introduced in this section.

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65 Classification of Porosity Formation W hen the solid is formed in the liquid, the volume shrinkage takes place instantly around the solid, because the solid density is smaller than the liquid, and, then, the liquid metal fills this shrunken empty space quickly. However, if the flow of the metallic melt cannot compensate this empty space properly, the shrinkage porosity [111-114] can occur. According to the pore size, it can be classified into the macroporosity and th e microporosity. The macroporosity has a large size pore having the magnitude of millimeter to ev en centimeter. However, it is known to be preventable by casting design and modeling and it is not our interest. The microporosity, having micron size pores, occurs between dendrites. Similar with the macroporosity, the feeding mechanism is also important. Compbell [108] classi fied with four different feeding mechanisms as shown in Fig. 2-25: liquid f eeding, mass feeding, interdendrit ic feeding, and solid feeding. The liquid feeding occurs prior to solidification. The second is the mass feeding at early stage of solidification, in which volume flow rate of the liquid is not affected by the presence of the solid. The third is the interden dritic feeding [109, 110] at which the fluid flow becomes slower due to the interaction with the solid at the later stage of solidification and this feeding mechanism is directly related to the microporosity formation. In the interdendritic feeding zone, the increase of the solid can reduce the channel size, through whic h the fluid can compensate the shrunken area, and, eventually, the capillarity effect can be seen, resulting in the pressure drop between dendrites. As shown in Fig. 2-26 (a), when the melt cannot reach the bottom of the dendrite, the shrinkage microporosity can be created. At k < 1, the gas element are rejected into the liquid during the solidification and enriched in the interdendritic region. When the gas concentration reaches its solubility, a gas pore can nucleate and grow as shown in Fig. 2-26 (b). In Al-Cu (4.5 wt%), it is reported that the interdendritic feeding zone has 70% of the mushy zone [110]. The last mechanism is the solid

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66 feeding and it occurs after the solidification by th e solute diffusion in the solid. The porosity can be formed at this stage at the grain boundary. This is called as the secondary porosity and it is frequently observed after the heat treatment process. The microporosity formation is by the nucleatio n/growth process and its Gibbs free energy is given: SLSL LGLG SGSG TAAAPVG ) ( [2-90] where, V is the volume of porosity, PT is the change of pressure terms, ASG, ALG, and ASL is the solid-gas, the liquid-gas, and the solid-liquid surface areas, and SG, LG, and SL the solid-gas, liquid-gas, and the solid-liquid interfacial energies, respectivel y. And, by converting the surface energies to the pressure, )(PPVGT [2-91] where, P is the pressure against surface tension. The porosity can be nucleated when the pressure terms on the right side of Eq. (2-91) become less than zero. Thermodynamic Consideration for Porosity Formation To understand the m icroporosity formation, the pressure terms affecting PT should be defined. During the dendritic solid ification, the pressure drop ( P ) and the gas pressure ( Pg) inside a gas pore equilibrate with the external pressures such as the atmospheric pressure ( Pa), the pressure generated by surface tension (P), and the metallostatic head pressure ( P). Pressure balance is formulated in Eq. (2-92) PPPPPag [2-92] If the left-handed side is gr eater than right-handed terms in Eq.(2-92), the porosity can form and grown until both sides are balanced. Various models were applied to calculate the P and the Pg. Three different theoretical models have been introduced to explain the porosity

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67 formation: (a) channel model, (b) porous medium model, and (c) thermal criteria. All models are developed based on the thermodynamics to judge wh ether the solidified mate rials have pores or not and where the pores are nuc leated during solidification. Channel model In the ch annel model, an attempt is made to balance all of the pressure terms in the interdendritic channel to determ ine where the flow of liquid metal ceases due to the pressure drop. Walther et. al. [114] suggested this model by assuming that feed metal would travel down a cylindrical channel with an ever-decreasing radi us in solidification lik e a centerline shrinkage. Ultimately, when the diameter of channel becomes small, the pressure drop would grow so large that the liquid would not compensate the shrinki ng area caused by solidification. As a result, pores can be formed. Mathematically, the pressu re drop along the channe l can be expressed by the following equation: r fL r L g P 32 1644 2'4 [2-93] where, P is the pressure drop al ong the length of the casting, is the viscosity of metal, is the heat-flow constant, =( SL)/ L, r is the channel radius, f is the friction force, and is solidification friction factor. This treatme nt did not account for the gas pressure ( Pg), so that it is only applicable to gas free system. Also, in the viewpoint of microstructu re, it considered only equiaxed structure. In this regar d, this is not useful for this study. Porous medium model The porous medium model considers the mu shy zone structure as a coherent porous medium. In this porous medium, a metallic melt is forced to wind its way between the dendrites. The shrinkage and gas evolution can take place in this porous medium. The difference from the channel model is to consider multi-channels a nd the contribution of gas components [122]. Thus,

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68 the porous medium model can describe the micros tructure which has the columnar or dendritic s/l interface. The concept of porous medium model was pione ered by P.S. Piwonka and M.C. Flemings based on a few assumptions [123-124]: (1) The size of the interdendritic liquid channel is regarded as the primary/secondary arm spacing. (2) Impurities or secondary phases are ignored. (3) The hydrogen concentration is assumed to be invariant during the entire solidification process. (4) Any effect of the alloying element on the porosity formation is neglected. (5) The pores can grow sufficiently fast to the its lim it. These oversimplified assumptions are modified for the better calculation below. (1) Pressure drop (P ) Piwonka and Flemings [124] suggested the idealized model to calculate the pressure drop in the interdendritic re gions based on the Darcys equation (Eq. 2-75) )( '322 2 4 22n R t r L P in equiaxed structure [2-94] where, P is the pressure drop along the length of the casting, is the viscosity of metal, is the heat-flow constant, = /(1) =( SL)/ L, r is the channel radius, L is the casting ingot length, n is the number of flow channel per unit area, and t is the tortuosity. In the case of the columnar or dendritic s/l interface, the solidified structure can be regarded as a simple pipe-like structure and the pressure drop is deri ved from Hagen-Poisseuille equation, t r VL P2'8 in columnar structure [2-95] where, V is the solidification rate. In Eq. (2-95), the tortuosity refl ects the complexity of channel shape. From Eq. (2-95), the pressure can dr op more with a decrease of the channel size r and an

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69 increase of channel length. Similarly, as the interdendritic region becomes smaller and the dendrite becomes more complex, the pressure drop is dropped more. A better calculation is possible by considering more practical dendritic structure. We can easily understand that the dendrite structure is si milar to the interlocki ng structure of Fig 2-27 (a), rather than the square structure. And, Fi g. 2-27 (b) shows how to assess the interdendritic channel size (). In the process of solidification, the dendrite (grey area) becomes thicker and interdendritic liquid (gL) becomes smaller, so that the in terdendritic channel diameter is expressed by =gld1/2 where d1 is the primary dendrite arm spacing and gL is the liquid fraction [125]. Sigworth and Wang [126] corrected a simple rectangular structure of the dendrites to the semi-spherical shape of the dendrite tip where th e pressure drop is not conspicuous. The fraction of semi-spherical region is defined as In other words, the interdendritic feeding starts below In the experimental range of this study, is about 0.3 [126]. With th ese, the pressure drop is calculated as following, sg Gd TV t P 1 ln8 )1(2 1 3 [2-96] Herein, is the geometrical portion of interdendritic feeding, gs is the volume fraction of solid during solidification, (= ( sL) / L) is the contraction factor, G is the thermal gradient, V is the solidification rate and t is the tortuosity of the dendrite. In Fig. 2-28, the pressure drop is depicted against dl, when G = 24C/cm and V = 0.0007 cm/sec. It is clear that the pressure drop becomes significant as the channel diameter is reduced. (2) Gas pressure ( Pg) Hydrogen is the only gaseous element with measurable so lubility in Al-Cu (4.5 wt%) alloy [127-129]. Thus, in this study, the gas pressu re indicates the hydrogen pr essure in a pore.

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70 Prior to the solidificat ion, the initial hydrogen co ncentration in the liquid is determined by the hydrogen partial pressure in the atmosphere surrounding the melt. When the liquid hydrogen concentration is in equilibrium with the hydroge n partial pressure in atmosphere, the initial hydrogen concentration can be obtained by Si everts law. Piwonka and Flemings [124] assumed that hydrogen concentration between dendrites ca n be obtained by the equilibrium solidification model. When the pore is nucleated, the hydrogen gas pressure inside the pore is also in equilibrium with the liquid hydrogen concentratio n. Thus, the hydrogen gas pressure in a pore is also easily calculated by combination of the e quilibrium solidification model and Sieverts law as given, 2]')'1([ kkfk V PLL i g [2-97] where, Vi is the volume of dissolved gas ini tially present in the molten metal and k =kS/kL. Poirier et.al. [130] transformed this equation to a familiar formula. )]1(1/[0 H S HHkgCC [2-98] where, CH is the concentration of hydrogen in the liquid, C0 H is the initial concentration of hydrogen in the liquid pr ior to solidification, kH is the distribution coefficient of hydrogen between solid and liquid. Eq. (2-98) can be transf ormed into the pressure term by Sieverts law, 2 2 S Cf PHH H [2-99] By substituting CH in Eq. (2-99) with Eq. (2-98) 2 2 0 2)1(1 1 H s HH HkgS Cf P [2-100] where PH 0 = (fHCH 0/S)2, and it indicates the in itial gas pressure prio r to the so lidification. fH is the activity coefficient of hydrogen, CH o is the initial concentr ation of hydrogen prior to

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71 solidification (cc/100 g), and S is the Sieverts constant (cm3atm1/2/100g). Eq. (2-100) is convenient because it can show the hydrogen gas pre ssure change in the process of solidification according to gs. However, it should be noted that the copper concentration also increases during solidification and it can l ead to hydrogen solubility change. So far, most researches have ignored this. Opie and Grant [131] determined the solubility constant of hydrogen in the Al-Cu alloy for 0
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72 =gld1/2 as shown in Fig. 2-27. Thus, the pressure generated by surface tension is shown as Eq. (2-103) 1/4 dgPl [2-103] Note that the surface tension is also able to be affected by the alloying elements. In Al-Cu (4.5 wt%), Poirier and Sperser [132] estimated the pressure against the surf ace tension in the AlCu alloy system: 20129.0721.0868Cu CuC C [2-104] where, Ccu is the concentration of copper (wt%) and the unit of surface tension is dyne/cm. In sum, by integrating Eq. (2-96), (2 -100), and (2-133) into Eq.( 2-92), [2-105] where, Pa is 1 atm and P is the Lgh, g is the gravitational acceleration constant, L is the liquid density and h is the sample height. With Eq. (2-105), th e critical solid fracti on at which the pore becomes stable can be obtained when the left-handed sides and the right-handed sides are balanced. However, the thermodynamic criteria are more or less limited following reasons. First, the equilibrium partitioning coefficient ( k ) is assumed to be constant at any location of phase diagram. Second, this model has no terms to account for kinetic factors such as supersaturation of porosity nucleation and growth. Criteria model The com plexity of the models and a few critical assumptions le d to a number of researchers to develop criteria functions to pr edict when and where there is a high probability of forming microporosity in a casting based on experimental observations. The well-known equation is Niyama criterion [134-136], in which temperature gradient and solidification time 2 0 2 3) )1(1 1 ( 1 ln8 )1( 4 kg P g Gd TR t dg PPs H s l ll a

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73 was rigorously calculated for mapping the solidi fication contour according to casting shape and mold to find final solidification area. Although th e application of this model [137-141] has been quite well-accorded with low-carbon steel castings, there are common limitations: (1) No consideration of presence of gas elements, (2) th ermal analysis in biased and little consideration in given to mass transport analysis, (3) usef ul only for normal freezing and no criteria for directional solidification. (4) No consideration of nucleation and growth, (5 ) it is not versatile for any materials due to limited tests on limited mate rials. Most of all, it is experimental based model, so it requires numerous data to obtain reliability. Prediction of Volume Percent Porosity and its Size Thermodynamic model Sigworth and Wang [126] formulated th e following equation to predict the volume percent porosity by assuming the ideal gas behavior, )1(1 )1( 1 273 (%)* 0 H S s H H LH Pkg g P kTC V [2-106] where, C0 H is the initial hydr ogen concentration, L is the liquid density, P* H and g* s are the critical hydrogen pressure in a pore and the critical solid fraction when pores become just stable thermodynamically as depicted in Eq. (2-105). is the kinetic factor. Eq. (2-106) assumes that (1) the pore grows instantly at the nucleated position, (2) all hydrogen exceeding its solubility is used for the nucleation and growth of a pore, (3) the hydrogen con centration is always constant even after the pore nucleation, (4) the supersatur ation is not required to nucleate a gas pore, and (5) no pore can escape from the dendrites. One other extreme is the homogeneous nucleation of a pore because it requires overcoming the highest energy barrier for the nucl eation. Hirth et. al. [141 ] derived an equation to calculate the activation energy for the homogeneous pore nucleation,

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74 2 3)(3 16vF F [2-107] And e r L m vP P V RT Fln [2-108] where, Fv is the Helmholz free energy, Pr is the gas pressure for th e homogeneous nucleated pore, Pe is the equilibrium pressure of the gas in the metallic melt, is the surface tension between gas and metal melt, and L MV is the partial molar volume of hydrogen in the melt. By obtaining the Fv, the Pr can be calculated. With this, the minimum concentration of gaseous elements for the homogeneous nucleation can be estimated. Turnbu ll et. al. [142, 143] found the activation energy for the homogeneous pore nucleation in metals, F*=60kT By putting this expression into Eq. (107) and (108), the Pr should exceed Pe by 5.13 times. Thus, the homogeneous nucleation is possible only when the hydrogen gas pressure is about 5 times greater than the hydrogen pressure in equilibrium with the hydrogen conc entration in the liquid. In Al-Cu (4.5 wt%) alloy, the required amount of hydrogen for hom ogeneous nucleation is 0.87 cc/100 g when Pe is in equilibrium with 0.38 cc/100 g. It will be revealed later how difficult to reach 0.87 cc/100 g. Kinetic model For the heterogeneous pore nuc leation, the nucleati on sites are necessary and potent pore nucleation sites are num erous such as the surface of primary/secondary dendrites, 2nd phases, and oxides [144]. After nucleation, the ki netic models assume that there is no supersaturation for the heterogeneous pore nucleation and the pore size is determined by its growth controlled by hydrogen diffusion into the pore. Most kinetic mode ls consider that a pore nucleates in the middle of the dendrite and it grows until it meets the eutectic front. Thus, the volume % porosity and its size depend on where a pore nucleates and how long it takes to meet the eutectic front.

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75 Fang and Granger [145, 146] stand the con cept of the pore growth. At early stage of mushy zone (upper mushy zone) as shown in Fig. 229 (a), the microstructure is coarse and open, resulting in less hydrogen buildup and more escaping of gas bubbles As solidification proceeds, the local structure become isolated (isolated pool) and the hydrogen co ncentration begins to increase greatly in the isolated pool, as depicted in Fig. 2-29 (b). This isolated liquid pool is defined as the diffusion cell, in which a pore can nucleate and grow by sucking the hydrogen rejected from the s/l interface. Th e structure of the isolated liqui d pool is simplified as a sphere and the compositional profile is demonstrated in Fig. 2-30. R0 is the size of the initial diffusion cell just when the isolated liquid pool is formed, r0 is the pore size at nucleation, C is the hydrogen concentration and subscr ipt L and S represents the li quid and the soli d, respectively. Once the hydrogen concentration in the isolated liquid pool exceeds the solubility limit, it is assumed that the hydrogen bubble is nucleated and detached immediatel y from the nucleation sites having a spherical shape. From the mass balance at the interface, amount of hydrogen ente red into pore is the same with amount of hydrogen reject ed from the s/l interface. ) )((4/) 3 4 (2 3SSLLCC dt dR Rdtnrd [2-109] where, r is the radius of pore, n is the density of nucleated bubble inside a diffusion cell, R is the radius of the instantaneous s/l interface and L and, S are liquid and solid density, respectively. By rewriting Eq. (2-92), PPPPnkTPa g [2-110] Here, k is the gas constant. And, by substituting n of Eq. (109) for Eq. (110), Eq. (111) can be transformed as given following relation:

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76 ) 3 2 ( 3 )(3)() ()()(2r P r dt dP TP dt dT rCCkT dt dR r R dt drg S SSLL [2-111] The first term in the blanket on the right-h and side of Eq. (2-111) accounts for hydrogen release contributing mainly to the pore growth. The second term relates to the surface tension, and the third term represents the pressure drop term. To solv e Eq. (2-111), Fang and Granger assumes that (1) the solubility is constant regardless of the temperature and solute concentration change within the mushy zone, (2) the exce ss hydrogen in liquid diffuses into the pore immediately, (3) there is no hydr ogen depletion near the hydrogen pore, and (4) the nucleated gas bubble is surrounded by liquid only. However, the temperature within the mushy z one is changed and the hydrogen solubility has exponential relation w ith temperature. Thus, the change of hydrogen solubility along the mushy zone height should be considered. Furthe rmore, the nucleated pore may contact the solid and liquid at the same time on the dendrite surface. These lead to apply a large correction factor when the calculation result of this model is compared with experimental data. Fang and Grangers model need to be reviewed becau se other models were developed based on it Atwood and Lee [147-149] accepted Fang and Grangers concept of the isolated pool. However, they consider that the nucleated gas po re touches the solid and the liquid at the same time and hydrogen depletion can occur around the gas pore. Thus, they assume that the gas pore growth is controlled by hydrogen diffusion within the isolated pool. A differential mass balance equation is shown in Eq. (2-112) 0)()(* *Rr H H P HPr C D dt dR CC [2-112] where, CP is the hydrogen concentration within a pore, C* H is the average hydrogen concentration at g/s and g/l interface, and RP is the pore radius, r is the distance from g/l interface

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77 into bulk liquid, and D* H is the effective hydrogen diffusion coe fficient when a pore is in contact with the solid and the liquid. The effective diffu sion coefficient of hydrogen is sensitive to the solid and liquid fraction due to several orders magnitude differ ence of the hydrogen diffusivity between the solid and the li quid [150]. Conceptually, the D* H decreases dramatically as the solidification proceeds. It indicates that the pore growth rate decreases with an increase of solid fraction. )]1()1([ )]1()1([* S S H S L H S S H S L H L H HgDgD gDgDD D [2-113] where, L HD and S HD is the hydrogen diffusivity in the liquid and the solid, respectively. In addition, the effect of temperature cha nge along the mushy zone height on the hydrogen solubility is also added. Most of assumptions used in Fang an d Grangers model are corrected properly. However, this model deals with the hydrogen diffusion only in a single pore. Thus, it was difficult to obtain the volume % porosity of the entire sample. In addition, this model ignored the pore nucleation. Recently, K.D. Li and E. Chang [151] modi fied the foregoing models with rigorous insight of thermodynamics and different diffusion solution. Although Li and Chang also used the effective diffusivity like the above model, Ti llers model is used to calculate the hydrogen concentration rather than the equilibrium solid ification model. The hydrogen supersaturation for pore nucleation is considered by using Lee and Hunts observation [152]. As for the diffusion solution, Li and Chang used the classic theory of diffusion-controlled precipitation in the single medium. With these concept and mathematical treat ment, Li and Chang succeeded to derive a practical model to estimate amount and size of a pore, by assuming periodic distribution of pores.

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78 273 )1( exp1)((%)* 0 g L n fS HSH PP T t tg CCV [2-114] where, tf is the local solidification time, 0 HC is the initial hydrogen concentration, C* H,S is the hydrogen concentration at the s/g in terface, which is equal to hydroge n solubility in the solid at given temperature, n is the constant depending on precipitate shape [153], and Sgis the critical solid fraction when the pore becomes stable and it can be obtained by Eq. (2-105). The relaxation time, t, can be given by using Hams solution [153]: )( ~ 3 )(* 0 3/2 0 23/1 SHH H egCCD CrC t [2-115] ) ( )1( 1 ~SsHL LH HDgkdg gkk D [2-116] where, gCis the concentration of hydrogen in the gas por e when the gas pore is in contact with the solid and liquid. Generally, the average hydr ogen solubility between the liquid and the solid is chosen, g SS g LLCfCf and re is the radius of the diffusion cell. Li and Chang assume the tetrahedron type of diffusion cell, so that by knowing final grain size (rG) in Fig. 2-31(b), the diffusion cell size can be obtained by re=1.225rG. D ~ is the effective hydrogen diffusion coefficient when the pore contacts the solid and li quid. By integrating Eq. (2-115) and Eq. (2116) to Eq.( 2-114), the volume per cent porosity can be predicted. Since (%) )3/4(3PppVnr in which np is the pore density per unit ( = diffusion cell) volume, the pore radius can be given simply by: 3/1(%)][PepVrr [2-117]

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79 Thermodynamic models and kinetic models have been developed as shown above. Due to complexity of pore formation mechanism and co mplicated dendritic structure, the models are still with intermediate confidence. However, a recent kineti c model present by Li and Chang is the most adequate to describe microporosity formation mechanism and its amount in this study, because Li and Chang successfully includes mo st proposed concepts for the prediction of porosity amount.

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80 Figure 2-1. Phase diagram of Al-Cu system

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81 Figure 2-2. Thermal condition in columnar solidification of pure metal substance at thermal gradient (G) > 0. x y Solid liquid A A B B x y Solid liquid A A B B Tm

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82 Figure 2-3. Thermal condition at equiaxed solidification of pure metal substance at thermal gradient (G) < 0. Solid liquid T m A B A A B B Solid liquid

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83 CS CL = C0 C0 CEC* L TS TL TE C* S = C0 C0C0 C* S CL=C0/k Solid Liquid DL/V C T Figure 2-4. Simple binary phase diagram duri ng solidification and solu te composition profile ahead of the s/l interface on upper-right corner

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84 Figure. 2-5. A schematic drawing of the constitutional supercooling according to solute buildup in front of s/l interface: line A and B indicates the imposed temperature gradient, G TS Temperature profile in liquid S/L interface TL Liquidus temperature B A

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85 Figure 2-6. Morphological inst ability diagram for Al-Cu (4.5 wt%) according to initial composition of copper and ratio of ther mal gradient to so lidification rate. 5.25.45.65.86.06.26.46.66.87.07.2 0 5 10 15 20 25 30 35 Cell/Dendrite COlog (G/V) Planar

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86 Figure 2-7. Density profile from the dendrit e bottom to the bulk liquid by composition and temperature change in Al-Cu (4.5wt%) alloy.

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87 Figure 2-8. Schematic representing the vertic al cylindrical container having isothermal temperature: top (TB), bottom (TA), and side wall (TS)

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88 Figure 2-9. Schematic illustration of the solute redistribution during so lidification in a volume element Solid Liquid A3 A2 A1 C0 C* S dC* S dy dCLC y

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89 Table 2-1. Solidification models acc ording to the mass transport method. Solidification Model Solid Liquid Remarks Equilibrium Model Complete diffusion Complete diffusion Super slow solidification Scheil Model No diffusion Complete diffusion Maximum segregation Mixed Model (I) No diffusion Limited diffusion + no convection Simple model Mixed Model (II) No diffusion Limited diffusion + convection Realistic model Mixed Model (III) Limited diffusion Limited diffusion Realistic model Diffusionless Model No diffusion No diffusion Rapid cooling

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90 Figure 2-10. Solute redist ribution in the equilibrium solidification model at TL T TS. CS=C0 CL=C0/k CS=kC0 S/L interface at T=TL S/L interface at T=T S/L interface at T=TS CS=C*S CL=C* LAt T = TL At T = T At T = TS x C

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91 Figure 2-11. Solute redistribution pr ofile according to solid fraction (fS) by Scheils equation for Al-Cu (4.5 wt%).

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92 Figure 2-12. Non-equilibrium phase diagram from Scheils equation CS CL = C0CE C* S=kC0CL TS TL TE C S at eutectic C*S= k CE

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93 Figure 2-13. Concentration profile in front of s/l interface when the solute diffusion in the liquid is limited at Al-Cu(4.5wt%) sample..

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94 Figure 2-14. Solute redistribution profile (sol id curve) along the sample length by using the Mixed model (I) and the compositional profil e ahead of s/l interface (dotted line)

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95 Figure 2-15. Comparison of composition profile ahead of the s/l interface by using the Mixed model (I) (solid line) and the Mi xed model (II) (dotted line).

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96 Figure 2-16. Schematic of zone melti ng technique (above) at zone length = l The calculated compositional profile (below) of copper in solid versus sample length, x, when x = 0.1 cm and 1 cm. An ingot material is Al-Cu (4.5 wt%) a nd total length of sample (L) = 10 cm. Heater l Solid Unmelted solid L Molten zone X=0 X 0246810 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 CSx (cm) l = 1(cm) l = 0.1 (cm)

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97 (a) (b) Figure 2-17. Solute distribution ahead of the s/l interface: (a) l (zone length) > D/V ( characteristic distance) (b) l < D/V

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98 Figure 2-18. Domains of solute distribution profile along the sa mple length on log Pe vs log GrSc plane

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99 Table 2-2. Solidification condition with nondi mentional quantities by Faviers criterion Regime a b c d e condition A: ratio of melt height to diameter of sample, h/d is the flow length sc ale over cell dimension (DS: 0.2, AHP: 0.159)

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100 Figure 2-19. Schematic drawing for dendritic st ructure: (a) the primary dendrite arm spacing, (b) the dendrite height (a) (b)

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101 (a) (b) Figure 2-20. Compositional profile versus the dendrite height (x direction) is shown in Fig. (a) and compositional profile versus the volum e element (y-direction) which is graycolored rectangular shape i ngot is shown in Fig. (b) solid liquid CE Ct CL C0 y x S L Concentration (wt%) Distance (y) CS *CL l

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102 (a) (b) Figure 2-21. (a) Closed packed arrangements of primary dendrite arms by upper view. (b) Schematic of growing dendrite (g ray area) having a radius RT=T in volume element 1d d l /2 solid liquid RT=T

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103 Figure 2-22. Calculation of tip temperature depression of Al-C u (4.5wt%) alloy versus the solidification rate by Bu rden and Hunt model. -5-4-3-2-101 520 540 560 580 600 620 640 660 Tip Temperature(C)log(V) (cm/sec)

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104 (a) (b) (c) Figure 2-23. (a) A schematic of a Czochralski method, (b) Float zone method, (c) Veridical Bridgeman method

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105 Figure 2-24. A schematic of Axia l Heat Processing technique B B a a f f f f l l e e w w i i t t h h i i m m m m e e r r s s e e d d h h e e a a t t e e r r

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106 Figure 2-25. Schematic representation of feeding mechanisms. Liquid feeding Mass feeding Interdendric feeding Solid feeding

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107 (a) (b) Figure 2-26. Two main cause of microporosity (a ) shrinkage porosity, (b) gas porosity in simple pipe-like dendritic structure P a PMZ Primary dendrite Pa P Pg Primary dendrite H H Cu Cu

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108 (a) (b) Figure 2-27. Schematic description for primary dendrite arms: (a) square arrangement (above), and interlocking arrangement (below), (b) dendritic grooves among three primary dendrite arms. 1d

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109 Figure 2-28. Pressure drop is calculated versus interdendritic channel size at thermal gradient = 24 C/cm and V = 0.0007 cm/sec of Al-Cu (4.5 wt%) alloy. 1.41.61.82.02.22.42.62.83.0 1 2 3 4 5 6 7 8 9 10 log(P) (Pa)log(dLm)

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110 (a) (b) Figure 2-29. Formation of isol ated liquid pool duri ng solidification: (a ) comparison between upper and lower mushy zone, (b) pore at growing grain (solid line) and final grown grain (dashed line)

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111 Figure 2-30. The geometrical representation of the diffusion cel l in Fang and Grangers model (spherical shape) and the associated evol ution of hydrogen con centrations in the local solid and liquid at tw o sequential times (t : dotted line and t: arbitrarily chosen time after the structure is isolated) C R or r r0 r CL (r=r) CL (r=r0) C0CS (r=r) CS (r=r0) Isolated Liquid Growing Pore Solid R0

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112 CHAPTER 3 TECHNICAL APPROACH Experim ental investigations of solidifica tion behavior are limited by the ability to accurately control and determine the conditions n ear the s/l interface during solidification. As mentioned, crystal growth techniques such as the Czochralski, Vertical Bridgman, and Float Zone methods supply heat radially to the melt, leading to large radial temperature gradients across the s/l interface. These in turn lead to convection and solute segregation, making it extremely difficult to know the thermal and solu tal conditions accurately during solidification. Axial Heat Processing (AHP), how ever, is designed to minimize radial gradients by using a conductive graphite baffle. Additionally, ther mocouples embedded in the baffle enable measurement of the thermal conditions near th e s/l interface without disturbing the growth process. Thus, AHP was the method employed in th e present investigation. The results are also compared to directional solidif ication technique by using the vertical Bridgeman method. The following subsections describe the AHP eq uipment used, the experimental procedures for Al-Cu crystals, and the charac terization methods utilized to examine the solidified crystals. Equipment The AHP unit was specifically constructed for the present inves tigation. Although its design was slightly different from convectiona l directional solidificat ion technique, it was possible to grow virtually identical solidifie d samples. Descriptions of the units and accompanying controlling and data acquisition modules follow. A tubular three-zone furnace built by the Mellen Corporation was vertically attached to a rigid steel frame, also constructed by Mellen. The furnace was 45.6 cm tall with a 25.4 cm outer diameter (OD) and a 8.4 cm inner diameter (ID ). It had two upper temperature zones (top furnace and middle furnace), 10.2 cm and 15.2 cm long, and one 10.2 cm lower zone (bottom furnace).

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113 Each zone was independently controllable a nd consisted of a coiled Kanthal wire heating element embedded in ceramic insulation. The maximum rated temperature of each zone was 1350C. The AHP experimental unit is schematically represented in Fig. 3-1. It is composed of the metallic bellows cap at the top and the metallic bottom cover, the baffle parts (baffle, graphite rod, and stainless steel rod), and graphite crucible. The metallic bellows cap was placed on the quartz tube. The bellows was applied to control the baffle position. The baffle position can be stationary when the sample is withdrawn to th e cold zone. The minimum height of the bellows was 3.5 cm and the extended height is 12 cm. It could be more extende d by applying external force to 16 cm. Six holes were made and clos ed with bolts on the metallic cap. When the thermocouples were incorporated in the baffle, these bolts we re unscrewed to insert the thermocouples and additional fittings were used between the hole and th ermocouple to prevent a leak. High temperature O-rings (OD: 3.17 mm and ID: 0.15 mm) were jointed in the top and bottom cover for better sealing. Endurable temperature is up to 275 C. In addition, three fixing screws were incorporated on the side of the bello ws cover, which were us ed to hold the stainless steel baffle rod. The feature of AHP was the use of the baffle parts. The baffle rods were composed of the stainless steel rod and the graphite rod. Graphite baffle was conn ected to the graphite rod. The stainless rod was connected with the metallic bell ows cap with three fixing bolts. The graphite baffle is also shown in the left-h anded side of Fig. 3-1, which is composed of the graphite baffle cover and body. Two holes were made inside the baffle body to incorporate the thermocouples in the graphite baffle. Thickness of the baffle botto m 3.8 mm and entire height of the baffle was 38.1 mm. The stainless steel gas pi pes were connected in metallic bottom cover and exited to the

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114 pipe connected to the metallic bellows cap The cooling water pipe made by copper wound around Monel pedestal rod and the t op parts of pipe contacts the bot tom of graphite pedestal to enhance the cooling efficiency. The characteristic feature of the AHP experime ntal unit, as shown sc hematically in Fig. 32, was the use of a 37.1 mm diameter graphite baffle (1) immersed in the melt near the s/l interface. The specimen was contained in a 210 mm height, 39.2 mm and 38.1 mm inner diameter graphite crucible (2). The annular gap be tween the baffle and the crucible is 0.5 mm or less. A 3-zone radially heating vertical, tubu lar furnace (background heate r) (3) supplied the heat to melt the entire charge. The heat is extracte d axially from the grow ing crystal through a graphite pedestal (4) and a Monel pedestal rod (5). Six calib rated K-type Inconel sheathed thermocouples (TC 1-TC 6) are positioned throughout the furnace, pedestal, crucible, and the baffle to measure and control the thermal profile near the s/l interface.TC1 was located at the center of the baffle and TC 2 was at the corner of the baffle. Both thermocouples were used to monitor the radial temperature difference inside the melt in AHP samples. TC 3 was placed 1cm from the bottom of the crucible inside the graphi te container. TC 4 was located 5 ~ 7 cm from the crucible bottom. TC 5 and 6 are embedded 1 and 2 cm beneath the crucible, and provide temperature gradient of solid. The error range fo r K-type thermocouples is reported as less than 0.7 C. LabTech software controls the background heat er zones with PIDs. A quartz tube (6), placed inside the background furnace, encases the aforementioned parts of the AHP unit. Each furnace zone thermocouple was hand-made by we lding 0.5 mm diameter chromel and alumel wire together. Hollow alumina sleeves electrically shielded the leads inside the furnace, and the tips were exposed at the ID of the furnace. All other thermocouples were 0.15 mm diameter

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115 grounded thermocouples, coated in magnesium oxide and protected by Inconel sheaths. The actual chromel and alumel wire diameters were 0.025 mm. These thermoc ouples were purchased from Omega Engineering, Inc., where they were calibrated to 0.4% of their reading above 275C. Below this temperature their uncertainty is .55C. The lower cap is attached to a vertical screw-driven translation system supplied by Mellen Inc. Data Acquisition and Furnace Control The thermocouples in each unit were conn ected to a PC equippe d with a CIO-DAS08 analog data acquisition card att ached to a CIO-EXP16 (or EXP-32) expander board. The cards were purchased from Omega. The expander boa rd received the voltage signals from the thermocouples in separate channels, amplified th e signals, and sent them to the data acquisition card. The expander board also supplied a cold junction compensation voltage, which converted to room temperature. The data acquisition car d interfaced with LabTech Notebook Pro software, which was used to convert the voltage signals to temperatures. These were acquired in real time during crystal growth at a rate of 10 Hz. Ten measurements we re averaged every second to produce an output of 1 Hz. The software also was used to record the temperature data in a file for later analysis. Experimental Procedures Experimental Instruments Calibration Before loading samples, the furnace was heated to the desired temperature with an empty graphite container surrounded by the quartz tube. A calibrated th ermocouple was inserted from the top and scanned the temperature along the furnace height to find the constant temperature zone (hot zone). With these, it was found th at 20 25 cm from the bottom of furnace had constant temperature region (i.e., less than 1C of fluctuation) and 16 28 cm had a temperature fluctuation less than 3C.

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116 The pulling speed was determined by controllin g the electrical current using a dial-type controller which flowed into the electrical moto r. Once the desired speed was determined, the quartz tube, on which five indicato rs was marked by every 0.5 cm, 1 cm, and 5 cm, were inserted and the pulling speeds to cold zone were meas ured. By measuring the travel time between indicators, the pulling speed could be measured and the average value from five measurements was used. Four different pulling speeds were de termined. In the case of intermediate pulling speed, the travel time was measured by 1 cm inte rval. For example, after setting the electrical current in dial-type controller, the travel times were measured: 1,346 seconds, 1,310 seconds, 1,416 seconds, 1,252 seconds, and 1,370 seconds and the average pulling speed was 0.000758 cm/sec. In other case, the travel time were 2,018 seconds, 1,857 seconds, 1,937 seconds, 1,812 seconds, and 1,887 seconds and the average pulling speed was determined as 0.000526 cm/sec. When the pulling speed was slower, the measur ed distance can be reduced to 0.5 cm for convenience. The travel times were 6,742 sec onds, 6,767 seconds, 7,031 seconds, 6,435 seconds, and 6,338 seconds and the average pulling rate was determined as 0.0000742 cm/sec. In the case of the fastest pulling speed, the travel times (33 seconds, 28 seconds, 36 seconds, 36 seconds, and 33 seconds) were measured using 5 cm interval. The average pulling speed was 0.153 cm/sec. By considering error range (= standard deviation), the pulling rates were defined as 0.00007 cm/sec, 0.0005 cm/sec, 0.0008 cm/sec, and 0.15 cm/sec. Sample and Crucible Preparation The master charges of Al-Cu (4.5 wt%) all oy were provided by Alcoa Inc. (OD: 3cm, and Height: 60cm). It was reported that the copper composition variation was at the range of 4.491 wt% and 4.508 wt% and the commer cial pure aluminum (99.8 wt% of Al) was used as a source materials. The master charge was sectione d by every 12.9 ~ 13.0 cm by the band saw. After sectioning, each charge was clean ed with methanol in sonicator to remove debris and lubricant

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117 for machining work. Then, the sectioned master charges were dried in a vacuum oven at 150C over 2 hours and were kept in vacuum oven un til experiment at room temperature. The container and the baffle were made by high-conductivity and hi gh-density graphite from Graphite Store Inc and the conductivity was reported as 190 J/sec m C. The inner surface of graphite crucibles was prepared with an ultrafine finish. The graphite baffle and inner wall of the graphite container was coat ed with a Zirconium oxide (ZrO2) spray purchased by ZYP coatings Inc. to prevent the reaction between graphite and th e Al-Cu melt, as well as the specimen release agent. Its thickness was 0.5 mm or less. The coated graphite containers were dried in air for 12 hours and baked in a vac uum oven for 3 hours at 200C depending on the company instruction. After inspecting for cracks and roughness of coating by the optical microscope, the coated graphite containers were stored in the vacuum oven until utilization at the room temperature. Once the master charge, graphite container, a nd the graphite baffle were prepared, the DS and AHP units should be assembled. In the DS experiment, the master charge was placed on the graphite crucible which, then, was attached to th e graphite pedestal rod. The graphite crucible and pedestal were placed within the furnace. The quartz tube was inse rted from the top and placed on the bottom cover. Then, the metallic bellows cover was placed on top of the quartz tube. Simultaneously, the top and bottom metallic covers were sealed with O-rings. Vacuum grease is slightly applied around Oring for better sealing. Due to relatively low temperature of stability of the O-ring and vacuum grease, the to p of furnace was covered with refractory brick, which also helped to stabilize the furnace temperature. For the AHP experiments, the master charge and the quartz tube was loaded in the graphite tube at the same way with the DS experiment. Before placing the metallic top on the quartz tube, the baffle parts were attached on

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118 the upper metallic cap in a careful manner. Then, the assembled top cover with the baffle parts was attached to the quartz tube and the baffle was lowered above the top of graphite crucible. After loading a sample and enclosing the system with quartz tube, the graphite container height was controlled to place at about 20 cm height from the furnace bottom, above which the furnace temperature is stable and below which a cons tant temperature gradient is present. Initial Thermal Gradient in the Melt Prior to the solidification, th e temperatures in the liquid were measured at the same thermal condition of experiment. In DS experiment s, the thermocouple was di rectly inserted into the melt until it touched the bottom and the temp erature was measured by 1 cm interval. For example, when the furnace temperatures we re set at top furnace = 720 C, middle furnace = 720 C, and bottom furnace = 300 C, result (square poi nts) is shown in Fig. 3-3. The liquidus temperature (648 C) is located at 0 ~ 1 cm heights based on the measured temperature and the thermal gradient of the melt is 24 C/cm to 3 cm of the melt. In AHP case, the melt temperature was measured using the thermocouple incorporated in the baffle. Similar with DS experiment, the baffle was driven to the bottom and temper ature was measured by every 1 cm. When the furnace temperature was set: top furnace = 725 C, middle furnace = 725 C, and bottom furnace = 300 C, the thermal profiles of DS and AHP experiments show the same trend. The bottom temperature was 642 C and the ther mal gradient in the liquid is 22 C at 0 ~ 2 cm. This method has been used to measure the temperature grad ient [105-107, 109]. The thermal gradient in the liquid can be varied slightly ac cording to furnace temperature a nd environmental factors. Thus, the more precise measurements were introduced in next chapter. In addition, it should be noted that the bottom at 0 cm may not indicate the crucible bottom, because a part of sample could be already solidified near the bottom of crucible in the experimental temperature setup. Thus, it is possible that the thermocouple or the baffle contacted the solidif ied portion of sample, not the

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119 crucible bottom. However, temperature measured in this condition is closer to the real temperature gradient of the liquid. Gas Atmosphere Control The hydrogen concentration in the melt was controlled by the hydrogen partial pressure. The Ar-H2 (5, 10%) mixture gases and ultra high purity Ar (99.999 %) gas were purchased from Praxiar and Air Gas Inc. and the partial pressure of hydrogen (0.1, 0.075, 0.05, 0.025, and 0 atm) was controlled by mixing both gases using calibra ted three ball flow meters. Two ball flow meters were attached in front of each pipe of pure Ar and mixture gas cylinder and both pipes were merged using Y shape connector. Another ba ll flow meter was used to check the total gas flow rate between Y shape connecter and the furn ace. The flow rate of the third ball flow meter must be the same as the summation of pure Ar a nd mixture gas flow rates. Each ball flow meter was calibrated by flowing gas bubbles through a bubble meter. In addition, the mixture gas exiting the system was analyzed with Gas Chromatography technique (GC) to ensure no other hydrogen gas sources such as moisture (H2O). The result is shown in Fig. 3.4 with Ar (95%) and H2 (5%) mixture gas, excluding the peak for argo n due to exceptionally high intensity. Although some impurity gases such as CH4, N2, and O2 were detected with ignorable level, it is noted that there was no moisture (H2O) as another hydrogen source exce pt controllable hydrogen gas. To confirm equilibrium between melt and mixture gas, three samples (produced under PH2 = 0.1 atm, 0.5 atm, and pure Ar gas) were rapidly cooled to room temperature after three hours holding at 750C and hydrogen an alysis was performed with IC P analysis (explained later). Results were summarized in Fig. 3-5, in wh ich the hydrogen concentrat ion measured shows good agreement with calculated va lues by Sieverts law.

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120 Crystal Growth After establishing the sample, the systems were slightly heated at 200C under an inert argon atmosphere for one hour to remove any ab sorbed moisture during loading the sample. Then, the systems were first evacuated to 10-1 Torr with mechanical pump. The systems were backfilled with argon and this evacuation/backfil ling process was repeated at least three times. After purging the system, argon atmosphere was substituted with the desired mixture gas (Ar+H2) as introduced above. The gas outlet line was conn ected to an empty flas k, followed by a flask partially filled with water. Th e argon or mixture gases exited through the water, creating a few bubbles, indicating a s light overpressure. Once a desired gas atmosphere was establishe d, cooling water was circulated through a Monel pedestal just after heating was initiated. Simultaneously, the data acquisition of all thermocouples began. Then, the furnace zones we re heated to the desi red temperatures (Top: 750C, Middle: 750C, Bottom: 500C) and held th e temperature for at least 3 hours to ensure complete melting of entire sample and equili brium between gases and molten alloy [155]. Also, by setting the bottom temperature at 500C, the experimental temperature could be reached quickly. Then, the furnace temperature was reduced to set TC 3 = 640 ~ 650C, which is close to the liquidus temperature (=648C) of Al-Cu (4.5 wt%) alloy. This can place the de ndrite tip at fixed height (1 cm) near the TC 3. With expe rience, this temperature was obtained when the furnace temperatures were reached: Top: 695C ~ 730C, Middle: 695C ~ 730C, Bottom: 300C. For the DS experiment, after ensuring the sy stems thermally stabili zed, the solidification was initiated by powering the translation unit, preset to the desired pulling speed (0.00007, 0.0005, 0.0008, and 0.15 cm/sec). During the translat ion, the crucible was lowered away from

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121 the hot zones of the furnace. Wh en solidification was complete, the furnace was turned off and the sample was allowed to be cool to room temperature in the furnace. For the AHP experiment, after the samples were completely melted at Top: 750C, Middle: 750C, Bottom: 500C, the baffle was slowly su bmerged towards the bo ttom of the graphite container and lifted until it cleared the tap of the melt. The purpose of the procedure was to remove any possible oxide films, which can hind er the melt flow between the baffle and wall of the graphite container as well as homogenization of the molten sample. After repeating this process three times, the baffle was positioned at 2 cm height from the bottom. Then, the furnace temperature was reduced to set TC 3 = 640 ~ 650 C. In AHP experiments, the baffle temperature was also targeted at about 675C (it was controlled at a bout 670 ~ 680) to obtain constant distance (about 1 cm) be tween the baffle and dendrite tip, because the initial thermal gradient was pre-measured as 22 ~ 24 C/cm. Furt her adjustments were required to establish the desired temperature gradients and initial melt height, which is the distance between the baffle and the dendrite tip. When the in itial temperature condition was set, the pedestal was translated down to the cool zone with preset pulling spee d as mentioned above. During this process, only the crucible was lowered away from the baffle, but the baffle was held at its original position. When the baffle came out of the melt and the liqui d alloy was allowed to solidify completely, the furnace was turned off and allowed to cool in th e furnace. Note that the preset mixture gases continued to flow until completion of solidification, regardless of DS and AHP experiments. After cooling, the unit was disassembled, and th e crucible tube contai ning the crystal was unscrewed from the pedestal. The solidified sample was then removed from the crucible by using a mechanical press, unless otherwise noted. The sa mple and recorded temperature data were then analyzed. The graphite and quartz tube were clea ned in between runs. In the case of the quartz

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122 tube, the graphite debris and impurities could be deposited on the quartz tube. Thus, after soft washing with a paper towel and flowing water, it was immersed in HF (5% + 95% Water) in a rubber boat, if necessary. The ZrO2 coating in the graphite tube and baffle could be easily removed with methanol with a brush, but it may re quire several repetitions to complete cleaning. Analysis Sample analysis consisted of metallo graphy, image analysis, Electron Probe MicroAnalysis (EPMA), and Inductively Coupled Plasma (ICP) analysis. The details are described below. Metallography Longitudinal surfaces of the solidified sample s were prepared by sectioning the samples with a saw followed by standard grinding and polis hing. SiC papers were used first, and the polishing direction was radial fo r 200 grit, longitudina l for 320 grit, radi al for 400 grit, and longitudinal for 600, 800, and 1200 grit papers. Hydr aulic suspensions of 5.0 m, 1.0 m and 0.3 m alpha alumina particles were then used on se parate Buehler Mastertex polishing cloths, with a figure-eight polishing motion. Each sample was cl eaned with an ultrasonic cleaner at each step of the normal grinding and polishing. Then, th e samples were observed with an optical microscope up to 200 x magnification to check for embedded alumina particles from the polishing slurry. In the case that embedded particles were observ ed, the polishing procedure was repeated starting with the 400 grit SiC paper. To reveal the microstructure, each polished sa mple was etched in Kellers reagent (2.5% Nitric acid+1.5% HCl+1.5% HF+95% H2O) for 15 seconds to observe the porosity. For porosity observation, the light etching removes any sm eared metal on the exposed hole and provides better visualization. In samples that were over-etched, eutectics and pores were not distinguished in the microphotograph. After etch ing, the samples were rinsed w ith distilled water and dried.

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123 Photographs of the samples are collected with a CCD camera in optical microscope with 50x magnification and in bright image mode. The CCD camera was essential for image analysis because it can provide uniform contrast and brightness on an entire photograph. Image Analysis Photographs of specimen surfaces were analy zed with an image an alyzer (Image Pro. Version 4.5). The image analysis software can digitize signal s of the photograph and compute size and length as well as provide a clean view of photograph by removing noise and emphasizing a specific signal. First, the length un it should be calibrated with a micro-ruler made by Nikon Inc. In this study, a micrometer ( m) was selected for the length scale. Images taken by the optical microscope were loaded and a calib rated unit was selected in natural image mode. Area and radius measurements were chosen from the measurement options in the software. With the help of light etching, the porosity was rec ognized as dark area and, technically by controlling brightness and exposure pr operly, the dark regions (pores) co uld be distinguished easily. In image analysis, the perimeters of the pores were described with lines. However, in low magnification (50x), it is diffi cult to judge the porosity area that was properly selected by software. Thus, the selected area was magnified di gitally in the image analysis software and by controlling signal (light) intensity manually, the poro sity area selected by the image analysis was effectively controlled to match with the por osity observed using by the optical microscope. When images were analyzed, the resultant data was automatically transferred a MS Excel work sheet. For the volume % porosity measurement, about 6 ~ 10 photographs were analyzed at 1 cm height intervals. The average va lue of each height was used to obtain the variation of the volume % porosity along the sample height and the porosity level of a sample is measured by averaging all measurements. To measure pore size, pores we re assumed to be a circle. The Image analysis

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124 software converted the area to ra dius, automatically. To check reliability of data, analysis of a few photos was repeated 5 times. From repeated an alysis, we found that the error is less than 5%. Electron Probe Micro-Analysis Electron Prove Micro-Analysis (EPMA), which measures the characteristic X-rays emitted from the sample surface, was used to observe the copper con centration distribution. Ultra-high purity aluminum (purity: 99.99%) and copper (pur ity: 99.99%) was used as a standard. Samples were obtained by reducing the dimensions of the pr imary solidified ingot to fit the sample holder following standard grinding and polishing to at least 1 m. To observe the copper distribution between the dendrite and interdendritc region, 1 m beam size was chosen to avoid the error caused by the interaction volume inside the samp le with electron beam. The beam is scanned inside the primary and secondary dendrites. The beam energy was selected as 20keV. The interaction volume with this beam condition (beam size and ener gy) in Al-Cu alloy is known as 5-8 m from previous works [156], which is small enough compared to the size of the dendritic region of this study. Generally, the error rang e of EPMA results is less than 5%, what was allowable in this study. Inductively Coupled Plasma At omic Emission Spectroscopy The Inductively Coupled Plasma Atomic Em ission Spectroscopy (ICP) was conducted to determine the amount of hydrogen in the sample. Briefly, the induced coupled argon plasma is an excitation source for atomic emission microsc opy. By building a plasma torch, a section of the sample in vaporized carried by carrier gas (A r) through the channel in th e plasma central axis. During this transit, the sample is atomized and the atoms are excited electrically. When the atoms return to the ground state in the cooler region, the characteristic wave lengths of the elements are emitted and detected. By analyzing a characteristic wavelengths and intensities, simultaneous qualitative and quant itative analysis is possible.

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125 Samples were obtained by cutting a specimen r oughly in the traverse direction by using the a band saw and, then, it was sliced by the diam ond cutter to reduce the dimension to about 5 mm x 5 mm x 2 mm. The sample wei ght was controlled to have about 0.2 g, since the maximum allowable sample weight is 0.5 g. Note that the eutectic of Al-C u alloy has lower hydrogen solubility relative to inside of dendrite. Thus, when eutectic amount in selected area for ICP analysis is higher than the av erage amount of eutectic, the measured hydrogen concentration can be underestimated. In this regard, a large ar ea (5 mm x 5 mm x 2 mm) wa s selected and three different locations in the same sample were chosen at the same sample height. Standard deviation of three measurements was 0.48 ppm and the error range is reported during the calibration as 0.5 ppm (=0.02 cc/100g). This error range was allowable considering that the measured hydrogen was in the range of 3 17 ppm.

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126 Figure 3-1. Axial heat processing instruments

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127 Figure 3-2. Details of the gra phite crucible part in the axial heat processing: (1) graphite baffle, (2) graphite container, (3) hea ting element, (4) graphite pedestal, (5) Monel pedestal rod, and (6) quartz tube TC 1 TC 2 TC 4 TC 3 TC 5 TC 6 (1) (2) (3) (4) (5) (6)

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128 Figure 3-3. Thermal gradient in the liquid me lt measured by the thermocouple in the baffle (circle) and the directly immersed thermocouple to the melt (square)

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129 Figure 3-4. Gas chromatographic (GC) analysis for gas exited from the system prior to the solidification.

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130 Figure 3-5. Plot of hydrogen concentration measured by IC P sample and calculated by Sieverts law versus hydr ogen partial pressure, PH2. 0.000.020.040.060.080.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Hydorgen concentration (cc/100g)PH2 (atm) calculated measured

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131 CHAPTER 4 RESULTS The inform ation about cooling rate ( GV), solidification rate ( V ), and thermal gradients (GL ) can be obtained by Fig. 4-1 in the AHP samples. At solidifi cation rate = 0.0008 cm/sec, the temperature change versus the solidification time is present in Fig. 4-1. After the temperature stabilizes up to 13,385 seconds, the specimen be gan to be withdrawn to the cold zone. The temperature of the baffle obtained by TC 1 and TC 2 is almost constant throughout the solidification process. When, however, the sample was withdrawn for 21,100 seconds, the temperature of TC 1 and 2 increases abruptly when the baffle leaves the melt. This increase occurs because the heat is conduc ted to the cold zone by the melt, and this heat extraction does not happen outside the melt. From this, we can also confirm the solidification rate. In this case, the sample height is 7.8 cm and the initial ba ffle height (distance between baffle bottom and crucible bottom) is 2 cm. Thus, the baffle should travel by 5.8 cm until it leaves the melt. By dividing the baffles travel di stance (= 5.8 cm) with the to tal travel time (= 21,110 13,385 seconds), the withdrawing rate can be calculated as 0.00075 cm /sec, which is reasonably accorded with pre-determined wit hdrawing rate (= 0.0008 cm/sec). To determine the thermal gradient ( GL), the initial crystal height, H0, must be found using the procedure outlined in [157, 158] (see Appendix I), )()( )(0 0 MBLPMS PMSTTkTTk TTdk H [4-1] where, kS and kL are the thermal conductivities of the so lid and the liquid of Al-Cu (4.5 wt%) alloy, respectively, d0 is the initial distance between the thermocouples sensing temperatures TB (= TC 1) and TP (= TC 5), and TM is the melting temperature of Al-Cu (4.5 wt%). Then, the thermal gradient can be given by GL =(TB TM)/(d0-H0) All necessary physic al parameters are

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132 summarized in Table 41. From Fig. 4-1, TB = 678 C, Tp = 623 C, and d0 = 3.4 cm is known from the apparatus design. Thus, H0 is equal to 2.2 cm and GL = 24 C/cm by using Eq. (4-1). This value is reasonable compared to the direct measurement of thermal gradient (= 22 C/cm) in previous chapter. The initial melt height (distanc e between baffle bottom to dendrite tip) is also calculated by h = d0 H0 0.4 cm (= thickness of baffle bottom) and it is 0.8 cm in Fig. 4-1. It is noteworthy that the above method in determining the cooling rate and the thermal gradient can be used up to pulling speed = 1 cm/sec or less as the response time of the thermocouples would not be fast enough to monitor realistically. In add ition, the temperature at the sides (TC 2) of the baffle is slightly higher than th e center (TC 1) by about 2 C. The temperature gradients can also be determ ined directly in the AHP technique, because the baffle incorporating the thermo couples is immersed in the melt. However, in the case of the DS samples, there are no immersed thermocoupl es. For this reason, a single experiment to measure the radial temperature difference and temp erature gradient of DS samples was designed. Two thermocouples were inserted into the me lt directly. One thermoc ouple was located just beside side wall of graphite crucible and the second thermocouple was placed at the center of crucible to measure the radial temperature grad ient. Both thermocouples were positioned at a 2 cm height, where the baffle is usually located in the AHP technique. The temperature from this experiment is presented in Fig. 4-2 and the temp erature collected from TC 3 and TC 4 in the DS experiment is shown for the comparison in Fig. 43. The tangent of the curve in Fig. 4-2 is the cooling rates. Between 9,000 and 13,000 seconds in Fig. 4-2, the cooling rate is obtained as 0.011C/sec and the corresponding thermal gradient is 22 C/cm attained by dividing the obtained cooling rate with the pr e-determined solidification rate (=0.0005 cm/sec). From Fig. 43, the cooling rate is obtaine d as 0.010 cm/sec between 11,000 seconds and 18,000 seconds and

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133 the thermal gradient is 20 C/cm. Strictly speaking, the cooling rate should be obtained by a thermocouple immersed in the melt. However, w ith this reasonable proximity, the temperature gradients can be collected from TC 3 in DS experiments. The solidification variables are summarized in Table 4-2 and hydrogen partial pressure, PH2, and calculated initial hydrogen concentration, [H]0, are also included. The microstructures obtained from the DS a nd AHP samples are shown in Fig. 4-4 and 45, respectively. The growth directions are mark ed by the arrows inserted in each photograph at 50 x magnification. The traces of primary dendrite stems can be seen with somewhat dark area along to the growth direction at the center of dendrite. The eutectics at the interdendritic region are shown by grey-colored areas due to reacti on with the etching solu tion. Some black spots which stand out in Fig. 4-4 (d) and Fig. 4-5 (d ) are the pores. Regardless of the DS and the AHP samples, the dendrite shows a somewhat traditiona l regular cell structure at the cooling rate = 0.0017 C/sec. The eutectic region is partially continuous and parallel to the growth direction. As the cooling rate increases to 0.012 0.018 C/sec, the secondary dendrites which grow normal to the growth direction become rec ognizable, and the contin uity of the eutectic region disappears. Instead, the eutectic region has mo stly ellipsoidal shape. This ki nd of structure is expressed as porous medium. The directionality of the primary and secondary dendrites relative to the growth direction is still rema ined. However, at about the coo ling rate = 1.7 ~ 3.6C/sec, the directionality of the dendrites no longer rema ins. Instead, the equiax ed-like structure is commonly seen in both the AHP and the DS samples. Heat is extracted to crucible wall as well as crucible bottom, because the sample is escaped from the hot zone and withdrawn to the cold region rapidly. On contrast, in slow solidification ra te, the heat is readily extracted to the crucible bottom, where the cooling water is circulating. Also, equiaxed-s tructure is caused by the strong

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134 constitutional and thermal supercooling in high withdrawing rate. This large supercooling can induce numerous heterogeneous nucleations on the crucible wall or even in the melt. This breakage of directionality in Al -Cu alloys tends to be severe with an increase of the solutes amount [159, 160]. The equiaxed stru cture indirectly indicates that the solidification is initiated after the baffle leaves the melt in AHP experiments. If crystals are nucleated at the side wall or in the melt, those impede the melt flow through th e gap between the baffle and crucible wall. Furthermore, it takes just 30 seconds until the baffle leaves the melt, at which most of the sample is still above the melting temperature as shown in Fig.F-7 (Appendix F). Thus, it can be inferred that the influence of AHP techni que at solidification rate = 1.5 C/sec or above is negligible compared to DS technique. In Fig. 4-6, the primary and secondary de ndrite arm spacings are pl otted versus cooling rates and compared with previous results, all pr oduced by directional soli dification technique are attained by averaging at least 10 measured dist ances between dendrite st ems. The primary and secondary dendrite arm spacing ca n be defined. Sarreal and A bbaschian [54] measured the primary/secondary dendrite arm spacing of Al-Cu (4 .9 wt%) and used coo ling curves to extract the solidification variables. Rong et al. [161] used Al-Cu (4.5 wt%) alloy as a material and measured GL by the furnace temperature gradient. McCa rtney and Hunt [162] investigated a comprehensive range of cooling ra tes and various aluminum alloys The data of Al-Cu (6 wt%) alloy was chosen because the cooling rate and co mposition are similar to this study. These were shown in Fig. 4-6. And, Kattamis and Flemings attained dendrite arm spacing for Al-Cu (4.5 wt%) with similar range of this study [163]. As expected in ob servation of the microstructure, the measured dendrite arm spacings of the DS and the AHP samples do not exhibit an appreciable difference. In additi on, the agreement of the DS samples with previous data can

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135 confirm the reliability of current experiments. For mathematical convenience, the relation of the primary dendrite arm spacing and the cooling rate was obtained by regression of data in Fig. 4-6 to be, 0057.2)log(33712.0)log( VG DAS [4-2] Fig. 4-7 shows the porosity sh ape observed with high magnifi cation (500 x). Fig. 4-7 (a) shows the interdendritic porosity. Th is type of interdendritic poro sities are found at cooling rate = 0.0017 ~ 0.018 C/sec. The shape of porosity linke d with eutectic region. The porosity shape is difficult to define because it is distorted by the local conditions, as shown vividly in Fig. 4-7 (a). This is understandable, because the eutectic region corresponds to the final liquid just before completion of solidification. Thus, the porosity size is usually smaller than eutectic size in most cases. Fig. 4-7 (b) shows a few pores are ra rely found on the grain boundary. This type of porosity is called the secondary porosity becau se it is nucleated af ter solidification by supersaturated hydrogen in the solid. The secondary porosity is frequent ly observed in heat treatment process and it has been studied by Talbot and Granger [164] Fig. 4-7 (c) shows another type of interdendritic porosity, having hi gh cooling rate = 1.5 ~ 3.6C/sec. As seen in Fig.4-4 (d) and 4-5 (d), porositi es tend to be located at the junction between dendrite colonies (grains). Porosity distributions of the AHP and the DS samples are compared at a specific height (0, 1.0, 2.0, 3.0, 4.0, and 5.0 cm) in Fig. 4-8. These examples are produced when the cooling rate is about 0.018 cm/sec and initial hydrogen concentration, [H]0, of 0.27 cc/100 g. All measurements are included in Appendix D. The relative distance is given with respect to zero at the center. Negative and positive values correspond to left and right center, respectively. It is observed that the volume % porosity in the AHP samples tends to be less than that in DS samples at all of the

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136 heights. Due to the scattering of volume % por osity, the radial trend of porosity amount is difficult to obtain. This irregular distribution of porosity may signify that the porosity formation and growth depend on a local solid ification condition rega rdless of the AHP and the DS samples. By averaging the porosity data of each height, the variation of the volume % porosity for the DS I samples with the sample height is sh own in Fig. 4-9 per the different initial hydrogen concentration. Error ba rs are added at [H]0 = 0.27 cc/100g, 0.19 cc/100g and 0.09 cc/100g. The error bar is produced using the st andard deviation of the data pe r each height (Appendix D). The change of porosity with the sample height is not observed in DS I sample series. Also, by considering error bar, the porosity level between [H]0 = 0.23 cc/100g and 0.19 cc/100g can be regarded as the same level of porosity. However, it is clear that the porosity level tends to increase with higher initial hydr ogen concentration. In Fig. 410, the average volume % porosity for DS II samples is present with the sample height. The error bars are also added at [H]0 = 0.27 cc/100g, 0.19 cc/100g and 0.09 cc/100g. Similar with Fig. 4-9, the variation of porosity level is almost the same with the change of the sample height. In this case, the porosity level at [H]0 = 0.19 cc/100g and 0.13 cc/100g is in the error range. However, th e increase of porosity can be verified with an increase of initi al hydrogen concentrati on in DS II sample seri es. In Fig. 4-11, the average volume % porosity of DS III samples is shown along the sample height. The porosity level is constant with the change of the sample height regardless of initial hydrogen concentrations. The porosity level di fference is not clear between [H]0 = 0.19 cc/100g and 0.13 cc/100g because most of the points at [H]0 = 0.13 cc/100g are in the error range of [H]0 = 0.19 cc/100g. Fig. 4-12 shows the porosity variation with the sample height for DS IV samples. It is also similar with others shown above. In this cas e, the samples at [H]0 = 0.27 cc/100g and 0.23

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137 cc/100g have the same level of porosity. In DS sa mples, the average volume % of porosity did not change along the sample height, but affect ed by the initial hydrogen concentrations. In Fig. 4-13, the average volume % of poros ity for AHP I samples are shown with the sample height. The error bars are added for [H]0 = 0.27 cc/100g and 0.09 cc/100g. Comparing to DS samples, the porosity level in AHP I samp les shows clear influence of sample height, particularly, at high initial hydrogen concentration. At [H]0 = 0.27 cc/100g and 0.23 cc/100g, the porosity level is greater at the lower part of samples (0 ~ 2 cm) by about 0.15 ~ 0.18% than the higher sample parts (2 ~ 5 cm). It is reasonabl e by considering the maximum standard deviation (= 0.11%). In the case of [H]0 = 0.19, 0.13, and 0.09 cc/100g, the average porosity of the first 2 cm height is about 0.21%, 0.18%, and 0.17%, respectively, and the por osity levels of each sample decrease to 0.15%, 0.14%, and 0.12% at 2 ~ 5 cm. The maximum standard deviation of samples at [H]0 = 0.19, 0.13, and 0.09 cc/100g are 0.08%, 0.06%, and 0.06%, respectively. Thus, it is difficult to say the increase of porosity at lo wer parts of the sample. However, this trend will be seen in other cases of AHP samples. After soli dification by 2 ~ 3 cm, the porosity levels of all samples become similar regardless of the init ial hydrogen concentration. Fig. 4-14 shows the variation of average volume % of porosity for AHP II samples. Similar with AHP I samples, the porosity amount is less sensitive to the initial hydrogen concentration, particularly above 2 ~.3 cm heights. At [H]0 = 0.27 cc/100g and 0.23 cc/100g, the differ ence of porosity amount between 0 cm and 2 cm is about 0.14% and 0.10%, respectively. The data at [H]0 = 0.23 cc/100g are overlapped with the error ra nge of the sample at [H]0 = 0.27 cc/100g in this case. The average porosities of the first 2 cm for the samples at [H]0 = 0.19, 0.13, and 0.09 cc/100g are 0.22%, 0.20%, and 0.13% respectively, which reduce to 0.17%, 0.15%, and 0.11%. At each case, the standard deviation ranges are 0.11%, 0.06%, and 0.05%. In Fig. 4-15, the average porosity

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138 variation of AHP III samples with the sample height is present. The porosity amount at the bottom is higher than 2 cm height by 0.12% at [H]0 = 0.27 cc/100g and 0.15% at [H]0 = 0.23 cc/100g. Also, the average porosities be tween 0 cm and 2 cm at at [H]0 = 0.19, 0.13, and 0.09 cc/100g are 0.22%, 0.14%, and 0.15%. The averag e porosities above 2 cm height reduce to 0.15%, 0.12%, and 0.08%. Fig. 4-16 shows the average porosity distribution along the sample height for AHP IV samples. In this case, the porosity variatio n along the sample height is no longer observed and the porosity di stribution is somewhat similar to that of DS samples. The effect of initial hydrogen concentration on porosit y amount is clearly seen even in considering the error range. In Fig. 4-17, the average volume % porosity of the entire sample is plotted versus the initial hydrogen concentration. Using the AHP technique, the average volume % porosity is suppressed by 20 ~ 40% at cooling rate = 0.001 7 ~ 0.018C/sec. The reduction of porosity amount in the AHP samples is obvious at high [H]0 and low the cooling rate. This advantage is negligible or disappears at cooling rate = 3.6 C/sec at any initial hydroge n concentration because the porosity measured between the AHP and the DS samples are within experimental uncertainty. In the case of AHP samples, the volume % porosity reaches a plateau after the [H]0 exceeds 0.19 cc/100 g. But, for the DS sample s, the volume % porosity shows a continuously increasing trend with an increase of the initial hydrogen concentration, [H]0. At [H]0 = 0.09 cc/100 g, the pore amount of DS and AHP samples is almost identical except at the cooling rate = 0.0017 C/sec. With increasing the initial hydrogen concentrations from 0.09 cc/100g to 0.27 cc/100g, the average porosity amounts increase by about 0.25 ~ 0.35% acco rding to the cooling rate. To confirm the reproducibility of a sample DS I-1, DS I-5, AHP III-1, and AHP III-5 samples were repeated five times and the error bars (as standard deviations) are marked as

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139 arrows. Other error bars without arrows were produced by the standard deviation from all porosity measurements as shown in Appendix D. In Fig. 4-18, the volume % porosity of the DS and the AHP samples are plotted versus the cooling rate for several initi al hydrogen concentrations. The e rror bars with the arrows also indicate repeated experiments and others are produced as the standa rd deviation from all porosity measurements. In the DS samples, the volume % porosity tends to deceas e as the cooling rate increases with the concomitant finer microstructure With the change of cooling rate, the porosity level is varied by 0.13 ~ 0.25%. The volume % porosity of the AHP samples is not as sensitive to the cooling rate as the DS samples at 0.0015 to 0.018C/sec. When the cooling rate exceeds 0.018C/sec, the volume % porosity in the AHP sa mples becomes as high as that in the DS samples, regardless of the initial hydrogen concen trations. As seen in Fi g. 4-17, the porosity amount is not suppressed at cooling rate = 3.6C/sec. From Fig. 4-17 and Fig. 4-18, the reduction of porosity is greater with the higher initial hydroge n concentrations and slower cooling rates when the AHP technique is used. Plotted average pore size obtained using image analysis versus the initial hydrogen concentration at various cooling rates is shown in Fig 4-19. A ll measurements are included in appendix E and the error bars are added using standard deviation based on them. In the DS samples, the average pore size tends to become larg er as the cooling rate decreases and the initial hydrogen concentration increases. At the cooling rate = 0.0015C/s ec, the average pore size is about 12 ~ 17 m depending on the initial hydrogen concentration and it is varied about 7 ~ 17 m at [H]0 = 0.27 cc/100g, according to the cooling ra te change. The effect of the initial hydrogen concentration on the porosity size is almost negligible at cooling rate = 0.012C/sec and 0.018C/sec. However, the pore size change can be realized by the change of cooling rate

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140 regardless of the initial hydr ogen concentration. Thus, th e effect of initial hydrogen concentration on pore size is observed only at th e cooling rate = 0.0017 C/sec and the pore size is more determined by the cooling rate. Herein, in the case of cooling rate = 3.6C/sec, the data of pore size is excluded in Fig. 4-19 due to a different microstruc ture (equiaxed structure) as shown in Fig. 4.4 (d) and Fig. 4-5 (d). In AHP samples, the pore sizes (= 10-12 m) are slightly smaller than the DS samples at cooling rate = 0.0017 C/sec and the pore size is almost identical with DS samples at other cooling rates. For AHP examples, the influence of initial hydrogen concentration on pore size is not as noticeable as for the DS samples, and the average pore size in the AHP samples is somewhat constant even when the initial hydrogen concentration changes. The average pore size commonly increases as cooli ng rate increases in the AHP samples as well as DS samples. Electron Probe Microanalysis was operated to investigate coring along the dendrite as shown in Fig. 4-20 and Fig. 4-21. Solidificatio n begins at the dendrite tip, where the copper composition is at the minimum from the phase diag ram as shown in Fig. 2-1. Note that the dendrite tip exists only during th e solidification and, after the solidification, only the dendrite trunk is observed as shown in Fig. 4-4 and Fig. 4-5. The dendrite tip may be placed near the center of dendrite trunk. Thus, the minimum com position at the dendrite ti p should be found near the dendrite center. For the copper compositi onal analysis, samples DS I-2 and AHP I-3 produced at cooling rate = 0.0017 C /sec are investigated and the re sults are shown in Fig. 4-20. At cooling rate = 0.018C/sec, DS III-2 a nd AHP III-3 are compar ed and their copper compositional profiles are plotted in Fig. 4-21. The electron probe scanned inside a dendrite and the scanning region is shown in Fig. 4-20 and Fi g 4-21. For the slower solidification rate, the primary dendrite trunk was investigated and, for the faster solidificati on rate, the secondary

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141 dendrite truck was scanned by the electron probe. Since, it was difficult to define the dendrite center with 2 dimensional photographs. Thus, five different locations were scanned in each DS and AHP sample produced at cooling rate = 0.0017 C/sec. The minimum composition was then used as the dendrite tip composition. Herein, the cen ter of scanned line is defined to be zero. The right side and left side ends of the scanned line are expressed by + 5 and 5, respectively. Fig. 421 shows the copper composition profile at higher cooling rate ( = 0.018C/sec), in which three locations were scanned by the electron probe The minimum copper compositions of the DS I-2 and the AHP I-3 sample in Fig. 4-20 are 1.74 wt % and 2.54 wt%, respectively. In the case of DS III-2 and the AHP III-3, the minimum values are 1.69 wt% and 1.82 wt%. There is a considerable difference of minimum copper co mposition between the AHP and the DS samples at cooling rate = 0.0017 C/sec. Results of the ICP analysis to investigate the hydrogen concentration are shown in Fig. 422. Herein, the measured hydrogen concentration is plotted versus the sample height for the samples (DS III-1 and AHP III-1) produced at th e cooling rate = 0.018 C/sec and the initial hydrogen concentration = 0.27 cc/100 g. The hydrogen concentration of the DS samples is almost constant (0.23 ~ 0.25 cc/100 g) with an increase of the sample hei ght. In this case, the standard deviation is not obtai ned depending on several data. However, during the calibration of ICP, it is known that the error range is 0.5 cc/100g and it was marked on Fig. 422. In the AHP sample, the hydrogen concentration decreases from 0.27 cc/ 100 g to 0.18 cc/100 g at the early stage of solidification and become constant for the rest of the sample height. The measured hydrogen concentration in the AHP sa mple is slightly higher than the DS sample at 1 cm height by 0.3 cc/100 g, but at 4 cm, the AHP sample has a lower [H] by about 0.6 cc/100g relative to DS sample. In Fig. 4-23, the hydrogen concen tration for the DS and the AHP samples are

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142 compared with an increase of the initial hydr ogen concentration, in which the samples were obtained at 4 cm height of the center and the error range, is also marked. The hydrogen concentration is higher in the DS samples than the AHP samples regardless of the initial hydrogen concentration. This tr end is more pronounced as the initial hydrogen concentration increases. In addition, the hydrogen concentration in the DS samples is almost equal to the initial hydrogen concentration and the difference from th e initial hydrogen concen tration is less than 0.02 cc/100 g.

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143 Figure 4-1. Temperature acquired from in corporated thermocouples in AHP samples according to time at the solidificatio n rate = 0.0008 (cm/sec) and furnace temperature = 710 C.

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144 Table 4-1. Physical and thermochemical parameters for Al-Cu (4.5 wt%) alloy l s ml[3] DL [54] DS [54] 2.42 (g/cm3) 2.59 (g/cm3) 3.3 (C/wt%) 4.5-5 (cm2/sec) 1.710-8 (cm2/sec) DH,L[152] DH,S[152] kl [3] ks[3] L[3] 3.010-3(cm2/sec) 5.2-4 (cm2/sec) 90 (J/sec m K) 174(J/sec m K) 0.321 (cm2/sec) *[3] [84] Cl [3] CS [3] 0 [132] 0.000027(1/K) 0.045 (poise) 1.18(J/g k) 0.9(J/g k) 847 (dyne/cm) TM [3] t [124] 0.018 (cm2/sec) 648C 2 l : liquid density, s: solid density, ml: liquidus slope in phase diagram of Al-Cu alloy, Cl : heat capability of liquid, CS : thermal capability of solid, kl: thermal conductivity of liquid, ks: heat conductivity of solid, DL: diffusivity of copper in liquid aluminum, DS: diffusivity of copper in solid aluminum, DH,L: diffusivity of hydrogen in liquid, DH,S: diffusivity of hydrogen in solid L: thermal diffusivity of liquid, : thermal expansion coefficient, : viscosity of liquid, 0 : interfacial energy of pure aluminum, : kinematic viscosity, t : tortuosity, TM: melting temperature of Al-Cu(4.5 wt%)

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145 Figure 4-2 Temperature acquired from immersed thermocouples within the melt for DS samples according to time at the soli dification rate = 0.0005 (cm/sec)

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146 Figure 4-3. Temperature acquired from incorporated thermocouples (TC 3 and TC 4) for DS samples according to time at the soli dification rate = 0.0005 (cm/sec)

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147 Table 4-2. Solidification variab les and experimental conditions V10-3 (cm/s) GL (C/cm) GV10-3 (C/sec) PH2 ( atm) [H]0 (cc/100g) h (cm) DS I-1 0.07 24 1.7 0.1 0.27 2 0.07 22 1.5 0.75 0.23 3 0.07 19 1.3 0.05 0.19 4 0.07 22 1.5 0.025 0.13 5 0.07 19 1.3 0 0.09 AHP I-1 0.07 23 0.8 1.6 0.1 0.27 2 0.07 23 0.8 1.6 0.75 0.23 3 0.07 24 0.7 1.7 0.05 0.19 4 0.07 23 0.8 1.6 0.025 0.13 5 0.07 22 0.6 1.5 0 0.09 DS II-1 0.5 24 12 0.1 0.27 2* 0.5 24 12 0.75 0.23 3* 0.5 24 12 0.05 0.19 4 0.5 19 10 0.025 0.13 5 0.5 19 10 0 0.09 AHP II-1 0.5 23 0.7 12 0.1 0.27 2 0.5 22 0.8 11 0.75 0.23 3 0.5 22 0.8 11 0.05 0.19 4 0.5 22 0.6 11 0.025 0.13 5 0.5 24 0.6 12 0 0.09 DS III-1 0.8 20 16 0.1 0.27 2 0.8 20 16 0.75 0.23 3* 0.8 24 15 0.05 0.19 4* 0.8 24 19 0.025 0.13 5 0.8 19 19 0 0.09 AHP III-1 0.8 21 0.9 17 0.1 0.27 2 0.8 22 0.7 18 0.75 0.23 3 0.8 20 0.6 16 0.05 0.19 4 0.8 22 0.8 18 0.025 0.13 5 0.8 22 0.9 18 0 0.09 DS IV-1 150 11 1650 0.1 0.27 2** 150 12 1650 0.75 0.23 3** 150 12 1650 0.05 0.19 4 150 12 1800 0.025 0.13 5 150 12 1800 0 0.09 AHP IV-1 150 24 0.7 3450 0.1 0.27 2 150 21 0.9 3600 0.75 0.23 3 150 24 0.6 2550 0.05 0.19 4 150 23 0.7 3300 0.025 0.13 5 150 21 1 3450 0 0.09 *: GL is obtained directly by insertin g a thermocouple into the melt. **: GL is obtained by averaging measured va lues of DS IV-1, 4, and 5.

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148 (a) (b) (c) (d) Figure 4-4. Photomicrography of Al-Cu (4.5 wt %) alloy solidified by the DS technique with different solidification conditions: (a) c ooling rate = 0.0017C/sec (DS I 2) (b) 0.012 C/sec ( DS II 5), (c) 0.018C/sec (DS II 4), and (d): 3.6C/sec (DS IV 1)

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149 (a) (b) (c) (d) Figure 4-5. Photomicrography of Al-Cu (4.5 wt %) alloy solidified by the AHP technique with different solidification conditions: (a) cooling rate = 0.0015C/sec (AHP I 4), (b) 0.010C/sec (AHP II 4), (c) 0.018C/sec (AHP III 4), and (d): 3.6C/sec (AHP IV 1)

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150 Figure 4-6. The relation of primary and sec ondary dendrite arm spacing for Al-Cu (4.5 wt% 6 wt%) alloy with the cooling rate

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151 (a) (b) (c) Figure 4-7. Microporosity obser vation with optical microscope with 500x magnification: (a) at the eutectic (DS III-5), (b) at grai n boundary of cell structure (DS III-5), (c) between dendrite colony (grain) (DS IV-1)

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152 (a) (b) (c) (d) (e) (f) Figure 4-8. Comparison of longitudinal poros ity distribution between AHP III-1 and DS III1 samples: (a) at the bottom, (b) = 1.0 cm, (c) = 2.0 (cm), (d) = 3.0 (cm), (e) = 4.0 (cm), and (f) = 5.0 (cm) -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume (%) of porosityRelative distance (cm) DS III 1 AHP III 1-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 DS III 1 AHP III 1 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 DS III 1 AHP III 1 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 DS III 1 AHP III 1 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 DS III 1 AHP III 1 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 DS III 1 AHP III 1 Volume (%) of porosityRelative distance (cm)

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153 Figure 4-9. Average volume % porosity versus the sample height with various initial hydrogen concentrations for the DS I samples.

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154 Figure 4-10. Average volume % porosity versus the sample height with various initial hydrogen concentrations for the DS II samples..

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155 Figure 4-11. Average volume % porosity versus the sample height with various initial hydrogen concentrations for the DS III samples.

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156 Figure 4-12. Average volume % porosity versus the sample height with various initial hydrogen concentrations for the DS IV samples

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157 Figure 4-13. Average volume % porosity versus the sample height with various initial hydrogen concentrations for the AHP I samples..

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158 Figure 4-14. Average volume % porosity versus the sample height with various initial hydrogen concentrations for the AHP II samples

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159 Figure 4-15. Average volume % porosity versus the sample height with various initial hydrogen concentrations for the AHP III samples.

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160 Figure 4-16. Average volume % porosity versus the sample height with various initial hydrogen concentrations for the AHP IV samples.

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161 Figure 4-17. Comparison of volume % porosity between AHP (solid lines) and DS (dotted lines) samples according to the change of initial hydrogen concentration at different level of cooling rates: 0.0017 (C/sec), 0.012 (C/sec), 0.018 (C/sec), and 3.6 (C/sec)

PAGE 162

162 Figure 4-18. Comparison of volume % porosity between AHP (solid lines) and DS (dotted lines) samples according to the change of cooling rate at different level of initial hydrogen concentration

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163 Figure 4-19. Average radius of pores according to the change of initial hydrogen concentration and cooling rate (= 0.0017, 0.012, 0.018C/sec).

PAGE 164

164 Figure 4-20. Comparison of copper composition profile between AHP and DS samples within dendrite at cooling rate = 0.0017C/sec (t he scanned region is marked in the photomicrograph as an arrow)

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165 Figure 4-21. Comparison of copper composition profile between AHP and DS samples within dendrite at cooling rate = 0.018C/sec (t he scanned region is marked in the photomicrograph as an arrow).

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166 Figure 4-22. Comparison of measured hydrogen concentration with an increase of sample height for DS III 1 (square) and AHP III 1 sample (circle) at cooling rate = 0.018C/sec and [H]0= 0.27 cc/100g

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167 Figure 4-23. Comparison of hydrogen concentration (after experiment) with an increase of initial hydrogen concentrat ion (before experiment) for DS (square) and AHP sample (circle).

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168 CHAPTER 5 DISCUSSION The experim ental and theoretical comparison between AHP and DS samples are discussed in this chapter. First, the porosity formati on mechanism of DS samples will be reviewed thermodynamically and kinetically. Subsequently the feature of the AHP technique will be explained based on solute redist ribution theories. Finally, the influence of the AHP technique on porosity formation is discussed. Porosity Formation: DS Samples As explained in Eq. (2-92), the porosity form ation is determined by the pressure balance between pore closing factors on the right-hand side of Eq. (2-92) and pore opening factor on the left-hand side. It should be noted that when the pressure terms on left side of Eq. (2-92) are greater than the right side, th e pore is thermodynamically stable To find the countermeasure for porosity formation, the dominant cause between shrinkage (by pressure drop) and gas porosity (by gas pressure) is needed to be precisely investigated. While pressure drop and gas pressure terms are sensitive to local solidification enviro nment such as solid fraction and initial gas concentration, the pressure te rms of atmospheric pressure an d hydrostatic head pressure are almost invariable except pressure against the surface tension. In Fig. 5-1, the influence of cooling rate on pressure drop, P and pressure against surface tension, P, is plotted according to the cooling ra te change by using Eq. (2-96) and Eq. (2-103), respectively. The parameters used for th e calculation are found in Table (4-1). As noted, small channel size (interdendritic region) w ith increase of cooling rate can increase P and P. The range of cooling rate of this study is also marked in the Fig. 5-1. The P and P are obtained by assuming a spherical pore at gL=0.1, arbitrarily, at which the P and P were maximized. It is understood that P is insignificant in the entire range of the experiment because it cannot

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169 overcome the surface tension, which is one of the pore closing terms on the right side of Eq. (292). In this regard, it is easily recognized that the porosity in Al-Cu (4.5 wt%) cannot be formed without the contribution of a nother pore opening factor, that in the gas pressure ( Pg). Thus, control of porosity is strongly depe ndent on the gas pressure term. In most previous research variation of temperature a nd copper concentration along the dendrite height has not been pr operly included or it has been i gnored. In the study, the precise pressure calculation is able to be perform ed with Eq. (2-100). In Eq. (2-100), hydrogen solubility is a function of copper composition, and its change af fects the exponential relation of solubility to temperature. C opper concentration is variable from 4.5 wt% to 33 wt% within dendrite height. In Fig. 5-2 (a), the solubility of hydrogen with and without consideration of copper composition is estimated with increase of the solid fracti on. In Fig. 5-2(b), the gas pressure is calculated and it is easily found that consideration of copper concentration can give a substantial change in gas pressure within dendrite s. Fig. 5-2 (b) shows influence of gas pressure, PH2, on gas formation is dominant due to a dramatic increase at the middle of the solid fraction. This overwhelming effect of the gas pressure is caused by an exponential relation of solubility with temperature as well as solid fraction. The effects of solid fraction on the gas pressure and other pressure terms in DS samples are summarized in Fig. 5-3 (a) for various init ial hydrogen concentrations and cooling rates used in this study. All pre ssure terms are calculated by Eq. (2 -96), Eq. (2-100) and Eq.(2-103). Parameters are in Table (4-1) and solidification conditions are summarized in Table (4-2). For convenience, as the variation of thermal gradie nts and solidification ra tes in each group is ignorable in calculation, 0.00007 cm/sec, 0. 0005 cm/sec, 0.0008 cm/sec, and 0.15 cm/sec are used as the solidification rates in each group and thermal gradient is used as 24 C. In Fig. 5-3

PAGE 170

170 (a), the critical solid fraction ( gs *) at which the nucleated pore b ecome stable thermodynamically can be obtained from the intersection point from Pa+P+PP and Pg curves. In thermodynamics aspect, the volume % of porosity tends to increase as g* s approaches zero. Because the solid fraction has a re lation with dendrite height as shown in Eq. (2-69) and Eq. (271), a small g* s indicates that pores can be stable earlier near the dendr ite tip. In other aspects, because the dendrite itself is the heterogeneous nucleation site for gas bubbles, decrease of g* s can signify the increase of nucleation sites [118][119]. Furthermore, a gas bubble formed earlier may have more time to grow [146]. Increase of the cooling rate can push the critical solid fraction back slightly by affecting the P. However, by controlling the initial hydrogen pressure, the g* s can be varied in the range of gs = 0.3 to gs = 0.8 in this study. Thus, initial hydrogen concentration should be contro lled carefully to determine g* s. In addition, the threshold hydrogen concentration at which gas porosity is a voided can be found when the gas pressure, Pg, does not exceed to Pa+P+PP at the eutectic fraction in Fig. 53 (a), marked by vertical line. For example, in the case of cooling rate = 3.6 C/sec, the pressure at the eutectic is 4.73 atm and the gas pressure cannot reach it when the initial hydrogen content is less than 0.03 cc/100g. When the cooling rate decreases to 0.0017 C/sec, the threshold hydrogen concen tration is 0.021 cc/100 g and the corresponding partial pressure of hydrogen is 1.710-3 atm by Sieverts law. In Fig. 5-3(b), the critical solid fraction ba sed on Fig. 5-3(a) is plotted versus initial hydrogen concentration [H]0. There is linear dependence of the critical solid fraction on the initial hydrogen concentration rega rdless of the cooling rate. Howeve r, the cooling rate has little influence on the critical solid fraction up to coo ling rate = 0.012 (C/sec). With information of the critical solid fraction and the gas pressure the amount of porosity can be predicted by thermodynamics and kinetics as discussed in th e following section using Eq. (2-106). By putting

PAGE 171

171 = 1, we can evaluate the porosity amount based purely on therm odynamic consideration without knowing the activation energies for nucleation and growth. Note that has been regarded as a kinetic factor a nd previous researchers arbitraril y chose this value to match the experimental data, because thermodynamic predictions usually overestimate the volume % porosity. For this reason, the thermodynamic pred iction was performed for prediction of the minimum amount of porosity, which can be obtaine d when pores can nuclea te at the eutectic front: T = 548 C, P* H = 2.1 atm at cooling rate = 0.018 C/sec. This case is depicted in Fig. 5-4 (a) as a black line. As expected, even the using minimum pred iction condition of thermodynamic model, the calculated volume % porosity (solid line) is much higher than the empirical values. Deviation between calculated valu e and empirical value in this study is greater than that in previous research [126], which cons idered solubility in pure aluminum. By solving Eq. (2-114), Eq. (2-115) and Eq. (2-116), volume % porosity can be estimated including kinetic considerations Simply, this model is able to assess the volume % porosity based on the diffusion controlled pore growth mo del, which say the size and volume % porosity increases when hydrogen diffusion into a gas bubbl e increases. Most of the parameters can be easily obtained from Tables (4-1) and (4-2). Howe ver, note that this model was derived based on the equiaxed microstructure and the assumption that hydrogen solutes accumulate around the single dendrite colony (grain) ra ther than between the dendrite s. Conceptually, this is understandable because at a high cooling rate, the dendrite height and arm spacing are very small, so that rejected solutes can surround the dendrite colony (gra in) itself. In this respect, Li and Chang [151] accepted the isolated pool (=di ffusion cell in Fang and Grangers model in chapter 2) and diffusion controlled growth theory concept from Fang and Granger. The diffusion cell size (re) was defined as 1.225 rG. Herein, rG is the size of a single dendrite (grain). However,

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172 in this study, the cooling rate is relatively slow and the microstructure is a cellular or dendritic structure with good directionality in the growth direction as shown in Fig. 4-4 and Fig. 4-5 (except the cooling rate = 3.6 C/sec). Also, the dendrite heights and primary/secondary arm spacing are sufficient to allow the porosity forma tion between dendrites.. T hus, simple structural modification for a diffusion cell is required to apply Li and Changs model to a columnar structure of this study. The diffusion cell size can be regarded as spherical shape, and re is defined as half the interdendritic spacing, dl/2 For continuous plotting, Eq. (4-2) is adopted to calculate dl and it is assumed that pores formed between dendrites cannot escape from the dendrite and pores are distributed uniformly else swhere. By adopting this diffusion cell and the local solidification time, tf = (TL-TS)/(GV), the volume % porosity proposed by Li and Changs model can be calculated. The calculated results are shown in Fig. 54 (a) and (b). They agree reasonably with experimental data, especially for the high initial hydrogen cont ent region. In the predictions shown in Fig. 5-4 (a), the e ffect of the cooling rate on th e volume % porosity is slightly underestimated. This may be due to lack of cons ideration for increase of surface tension with increase of copper concentrati on and decrease of temperature. The reaction kinetics for pore growth determined by the relaxation time, t, relative to the local solidification time, tf, as shown in Eq. (2-114). A longer local solidification time would allow more hydrogen to diffuse into the pores to achieve thermodynamic equilibrium, whereas larger values of the relaxation time, which is controlled mostly by re and C0, indicate slow kinetics for hydrogen. Thus, a small C0 would imply a small driving force for diffusion of hydrogen, which decreases the value of t, resulting in decrease of porosity amount. In Fig. 5-4 (b), the pore radius is co mpared with the result of Eq. (2-117), which shows reasonable agreement but devi ated slightly from th e experimental values.

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173 This may be caused by the simple assumption that only a single pore can nucleate and grow in a diffusion cell and no gas bubble can escape from th e dendrites. In practice, several pores can coexist at high initial hydrogen c oncentration [145, 146], and the in terdendritic region must not be completely isolated, so no pore in the di ffusion cell integrate into a larger pores. With above information, it can be realized that the porosities in DS samples were well accorded with the prediction of the kinetic m odel and reasonably understood with previously proposed models. However, we knew that AHP samples have smaller volume % porosity even under similar solidification conditi ons. Thos is possibl e only when the nucleation is suppressed or pore growth is restrained for other reason s. Among solidification variables introduced in chapter 2, the level of convec tion in the AHP technique will be suppressed due to a small solidification height and, physica lly, the solutes reject ed from the solid may also be confined below the baffle relative to the DS technique. These two characteristics of the AHP technique relating to porosity formation are examined below. Effect of Convection on Solute Redistribution In Al-Cu (4.5 wt%) solidification, ther e was no density inversion thermally or compositionally as shown in Fig. 2-7 in the dend rite or bulk liquid when an ingot grows in the opposite direction of gravity. Thus, the source of convection is the radial temperature gradient. The level of convection can be estimated by Raw for a bulk liquid and Ram inside the mushy zone. By previously mentioned equations, L mS WdhTTg Ra /)(24 [2-48] L mLKTg Ra* [2-83]

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174 where, g is the magnitude of the acceleration due to gravity, is the thermal expansion coefficient in liquid, d is the diameter of sample, Ts is the temperature of side wall, Tm is the melting temperature, L is the mushy zone length, is the kinematic viscosity, L is the thermal diffusivity, T is the maximum temperature difference in mushy zone, and K is the permeability. The Rayleigh number represents the ratio of the buoyancy force to the viscous force. An increase in the Ra values indicates more convective mixing in th e liquid. These values will allow to be for calculation of solute redistribut ion in convective motion in Al exandrovs model (described below). To calculate Ram, the permeability, K is needed. In Fig. 5.5, the permeability in Al-Cu(4.5 wt%) was calculated using models introduced by Hagen-Poiseuille model (Eq. 2-80), Murakami Model (Eq. 2-78), and Kozeny-Carmen relation (Eq. 2-81). Even though numerous models have been proposed, the regression models based on numerous data points (Murakami model) and empirical values ( Kozeny-Carmen model) for AlCu (4.5 wt%) may be more reliable. Also, as predicted in Pb-Sn experiment by Poirier [165], Piwonka and Flemings model in which K is proportional to square of gL (marked dotted line) is greatly overestimated relative to K based on empirical data points from solidification experiments of Al-Cu (4.5 wt%). The permeability increases abruptly to unity at some critical level of solid fraction, above which the solid can not affect the fluid flow. This behavior was revealed by the viscosity measurement in mushy zone. Thus, the permeability can be obtained only below this critical limit (s olid fraction). In the limit of each models reliability, K can be calculated between 3.86-9 cm2 for this study at the eutectic front (gL=0.091) and 1.910-8(cm2) at Murakamis limitation (gL=0.66). By selecting the highest value of K as 1.910-8 cm2, the maximum value of Ram is obvtained as 1.4110-6. This value is much less than the Rac = 0.161, below which the region is convectively stable [41]. In this respect due to very small

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175 permeability between dendrites, the dendrite forest itse lf can act as a solid wall as far as the flow in the liquid region is concerned. Thus, the convection effect in bul k liquid is important only near the dendrite tip and estimation of convectiv e fluid flow may be treated with only Raw similar with that in plane front interface. Convection in the bulk liquid is estimated by Raw in Eq. (2-43). From the radial temperature difference, Raw can be easily calculated: 40,832 for DS and 5 for AHP. The difference is caused mostly by solidification height, h (DS= 8 cm and AHP = 0.7 cm) and, weakly, by the radial temperature gradient ( T = 4 C for DS and 2 C for AHP sample). To determine that this suppression of convection can affect so lute distribution, Faviers criterion is used as shown in Fig. 2-18 and Table (2-2). It should be noted that alt hough, Faviers analysis was developed for the planar-front solidification, but it may be applicable in this study due to the small permeability within the dendrite. The n ecessary nondimensional quantities and parameters are obtained by Eq. (2-31) (2-42) in Al-Cu (4.5 wt%) alloy: Gr (DS) = 1.6106, Gr (AHP) = 202, Sc = 400. Pe number can be calculated as the solidification rate, V change. Pe (AHP) is 58, 38, 4.6 and Pe (DS) is 133, 88, and 10 as V decreases by 0.0008, 0.0005, and 0.00007 cm/sec For Pe calculations, the characteristic length Lch, is the minimum value betw een height and diameter of sample. Thus, in the case of DS sample, the sample height (= about 8 cm) and, for the AHP sample, the sample diameter (= 3.7 cm) can be us ed. Domain of solute redistribution in Al-Cu (4.5wt%) alloy is plotted in Fig. 5-6. For given G and V it is clearly confirmed that the DS sample is dominated by convective flow, even th ough there is no particular source for convection except the radial temperature grad ient, as predicted [166]. As fo r AHP samples, two points at V = 0.0008 and 0.0005 cm/sec are in region d (diffusion controlled growth regi on), but one point at V =0.00007 cm/sec is in the convective region. More over, the convection may be reduced, but not

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176 negligibly so, relativ e to the DS samples. To conf irm the suppression of convection experimentally, the composition of dendrite tip was examined. In the absence of convection, more solute can be built up ahead of the dendrite tip by radial di ffusion at the tip and by liquid diffusion from the interdendritic region. On contrast, the c onvection causes the accumulated solutes to be driven into the bul k liquid. Therefore, the absence of convection should result in an increase in copper concentration to the dendrite tip, correspondi ng to tip temperature depression (It should be noted that a very strong convection can carry heat to the dendrite tip. At this level of convection, the tip temperature can increase, so that the dendrite tip can be re-melted. In this study, this strong convection is not considered be cause this strong convec tion is able to occur particularly when density conversion is created during solidification). The minimum values of the copper concentra tion in dendrite are determined by EPMA as shown in Fig. 4-20 and Fig. 4-21. It should be noted that the mi nimum concentration of copper in EPMA is regarded as dendrite tip composition, but it is overe stimating to imitate the tip composition, since the diffusion in the solid c ontinues to occur during solidification and after freezing and it increases the tip concentration subs tantially relative to th at at the moment of freezing. The solid diffusion of Al-Cu (4.5 wt%) was already well treated by Bower and Flemings [17] who calculated th e solid diffusion by Ficks 2nd law with characteristic diffusion time, (=4 f/(dlg*)2), where dl/2 is the diffusion length and g* is a correction factor (=0.32 in Al-Cu (4.5 wt%) alloy). They plotted the tip composition change of copper as a function of increasing can be calculated easily since the real tip composition can be obtained from the plot at the moment of solidific ation by finding the composition at = 0 (Fig. 7 in [17]). At cooling rate = 0.0017 C/sec, is 3.4108 and the minimum compositi ons of copper from EPMA are 1.74 wt% for DS and 2.54 wt% for AHP, resp ectively, which should decease to 0.91(wt%)

PAGE 177

177 and 1.50(wt%) by tracking the plot of Fig. 7 [17] to =0. Corresponding tip temperature is 643 C for 0.91 wt% and 633 C for 1.50 wt% at the moment of solidifi cation. In the case of cooling rate = 0.018 C/sec, the is 1.5108 and tip composition of DS and AHP decreases to 0.91 wt% and 0.93 wt% from the minimum composition of 1.69 wt% and 1.82 wt% in EPMA analysis. In Fig. 5-7, the tip compositions measured at cooling rate = 0.0017 C/sec and cooling rate = 0.018 C/sec are compared to th eoretical calculations using m odels of Burden and Hunts model of Eq. (2-85) and Alex androv of Eq. (2-89) and para meters in Table (4-1). At V = 0.00007 cm/sec (cooling rate = 0.0017 C/s, it is observed that a substantial tip temperature depression relative to the liquidus temperature (= 648 C) occurs in th e AHP sample (=632 C) compared to a small temperature depression by 4 C in the DS sample. When the solidification rate increases to V = 0.0008 cm/sec (cooling rate = 0.018 C/sec), the tip temperatures measured are 643 and 641 C for DS and AHP samples, respectivel y. With allowable deviation from the aforementioned models, it is shown that tip te mperature of AHP samples obeys Burden and Hunts model and DS samples are more accorded with Alxandrovs model. The former model was derived assuming no convective mixing in the bulk liquid. A consider able tip temperature depression at V = 0.00007 cm/sec may be caused by enhanced radial diffusion at the dendrite tip, which causes more solutes accumulated ahead of th e dendrite tip, so that the tip temperature can be reduced. At V = 0.00007 and 0.0005 cm/sec, neither the radial diffusion nor Gibbs Thomson effect can reduce the tip temperat ure even in suppressed convecti on. Thus, at the intermediate solidification rates, it is expected that there should be no significant tip te mperature depression even in the absence of convection. Because the convection causes the mixing of accumulated solutes ahead of dendrite tip, there should be limited solute build-up near the dendrite tip, which leads to the bulk liquid with C0. This convection effect is obvious at V = 0.00007 cm/sec, where

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178 Burden and Hunts model predicts a large tip te mperature depression. Experimentally, a large tip temperature drop of the AHP sample is observed at V = 0.00007 cm/sec, but the temperature is still has about 10 C higher than the value obtained by Burden and Hunts model. This is caused by the presence of weak convection, as shown in Fig. 5-6, where the point at V= 0.00007 cm/sec is in the convective region. In the case of DS samples, all data are in good agreement with Alexandrovs model. In this regard, we can also conclude that the dendr ite height of the AHP sample should be smaller than that of the DS sample if the same temperature gradient is assessed. Zone Effect: AHP Technique So far, the copper redistribution at the dendr ite tip was examined with and without the presence of convective mixing. Prior to examinati on of the zone effect in the AHP technique, the different role of hydrogen and copper, as kinds of solutes, needs to be introduced. First, hydrogen does not affect liquidus an d solidus temperature or micros tructure of Al-Cu (4.5 wt%) [167]. Thus, regardless of hydroge n buildup, the tip temperature depression does not occur as mentioned previously. Furthermore, because kH < 1, there is no density inversion from Eq. (247). Thus, there is no driving force to produce th e convective motion except radial temperature variation. However, the diffusion coefficient of hydrogen is higher two orders of magnitude than that of copper in Al-Cu (4.5 wt%). Rapid diffusion is able to dissipate the hydrogen solutes quickly to the bulk liquid, so th at a significant amount of solute buildup head of the interface is not expected in normal solidification such as in the DS technique. However, in the AHP technique, the baffle is able to confine dissipation of hydroge n to the small zone below the baffle. As a result, the hydrogen solutes can be c oncentrated below the baffle. In Fig. 5-8, the phenomenon ahead of the dendrite tip can be depi cted as a combination concept of zone melting and final transition of planar-front solid ification. The dendrite tip composition (Ct) is relatively

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179 constant, as in steady state of planar front solidification. Wh en the diffusion field of hydrogen ahead of dendrite tip meets the baffle, more solute s are built up in the liquid as shown in Fig. 214 because some of rejected hydr ogen cannot be dissipated complete ly into the bulk liquid. The difference of the AHP technique from the final tr ansition region is the constant distance from the s/l interface to the baffle, which continues to decrease in the final transition region. To have a the small zone effect, Tiller et. al. [15] concluded that a high diffusion coefficient with an extended diff usion field is advantageous. In contrast, when the rejected solutes can be fully dissipated within a zone le ngth, solutes will not be accumulate in a small zone. For example, in the case of a planar interf ace, the characteristic length (D/V) of copper and hydrogen are 0.06 cm and 4.9 cm at cooling ra te = 0.0017 C/sec, respectively. Also, the approximation of the diffusion boundary layer (2D/V) [77] [168] is also 0.12 cm and 9.2 cm, respectively. As shown in Table (4-2), the baffle height is abou t 0.5 cm-0.7cm, which is greater than the diffusion boundary layer of copper and shorte r than that of hydrogen. In this regard, the solute accumulation below the baffle is expected only for hydrogen. In pract ice, the shape of the dendrite tip is not planar in de ndritic growth. When a spherical sh ape is assumed at the dendrite tip, the diffusion boundary layer is approximately equal to tip ra dius, R, from the diffusion solution as shown in Eq. (2-74). P hysically, this means that a sharp tip with a small tip radius is able to dissipate solutes effec tively, due to a larger interface and more radial direction of diffusion, resulting in a less extend ed diffusion field. This is described by in Fig. 5-9, which shows how, conceptually, we can understand that decease of diffusion layer in dendritic shape of interface. The tip radii ca lculated by using Eq. (2-73) are summ arized at Table (5-1) according to solidification rate. The rejected hydrogen solutes may be dissipated completely at solidification rate = 1.5 (cm/sec) regardless of DS and AHP tec hnique. Quantification of solute buildup below

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180 the baffle is obtained by modi fying Eq. (2-55c). Herein, the characteristic length (D/V ) is replaced by 2/R R xR k k CH H'2 exp 2 10 [5-1] where, kH is the equilibrium partit ioning constant of hydrogen, x is the distance from the dendrite tip to the baffle, and R is the curvature of the dendrite tip. Excess hydrogen built up under the baffle would be calculated by using Eq (5-1), in which the hydrogen buildup is affected by R and C0. In the case of the DS sample, hydroge n concentration can be obtained from the Tillers equation for the steady state case as shown in Eq. (2-55a ), since the hydrogen composition of the dendrite tip should be constant during normal freezing from the onset of solidification. By using Table 4-1, all parameters are contained for the calculation of Eq. (2-55b) and Eq. (5-1). Fig. 5-10 shows the concentration of accumulated hydrogen below the baffle (AHP technique) at 4 cm height plotted versus the change of initial hydrogen concentration at three cooling rates. In constructing Fig. 5-10, it was assumed that rejected hydrogen is not mixed with bulk liquid above the baffle and that gas bubbl es are not nucleated. Fig. 5-10 shows that hydrogen accumulation below the baffle is sensit ive to the cooling rate as well as initial hydrogen concentration, due to its interaction with diffusion field. For example, the accumulated hydrogen concentration is 1.1 cc/100g at [H]0=0.09 cc/100 g and it increases to 3.0 cc/100 g at [H]0 = 0.27 cc/100 g when the cooling rate is 0.0017 C/sec. At a faster cooling rate of 0.018 C/sec, accumulated hydrogen concentra tion is down to 0.36 cc/100 g for [H]0 =0.09 cc/100 g and 1.3 cc/100 g at [H]0=0.27 cc/100 g, respectively. However, it is unlikely that Fig. 5-10 accurately predicts the hydrogen concentration be low the baffle, because it is highly probable for gas pores to nucleate when hydr ogen concentration exceeds the solubility limit. Once gas pores

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181 are nucleated, substantial amount of hydrogen can be consumed by the pores, resulting in decease of hydrogen in liquid. The solubility limit is calculated by Eq. (2-101). The solubility of hydrogen between the dendrite tip and baffle is between 0.38 cc/100 g and 0.42 cc/100 g due to the temperature gradient between dendrite tip a nd baffle. Thus, in most cases of AHP samples, hydrogen concentrations below the baffle are grea ter than its own solubility limit. Thus, it is inferred that gas pores can be nucleated near th e dendrite tip and even on the crucible wall and the baffle bottom except with cooling rate s = 0.012 C/sec and 0.018 C/sec with [H]0=0.09 cc/100 g. To confirm the increase of hydrogen buildup, IC P analysis was performed as a function of height, and the results are plotte d in Fig. 5-11. The hydrogen con centration was measured 0.5 cm below the baffle and 1 cm above baffle. Here in, a designed experiment was performed by placing a pin at 4 cm height to ho ld the baffle at this desired height (thickness of baffle was 1 cm). The initial height of the dendrite before star ting withdrawal was about 1 ~ 1.5 cm and, as in the normal AHP experiment, the baffle is located about 0.5 ~ 1 cm above the s/l interface. After 1 cm of withdrawing crucible into cold zone, the baffle is stuck by pins at pre-set position and the dendrite tip contacts the baffle after about 2 cm of w ithdrawing. This is solidified at 0.018 C/sec and [H]0 = 0.09 cc/100 g. In this condition, the solute accumulation is the slowest below the baffle. In the case of higher [H]0 and slower cooling rate, the rate of solute buildup may be too fast to observe the hydroge n concentration profile along the height. Also, by allowing 2 cm of dendrite growth to the baffle, the amount of solutes built up will not reach the solubility limit at the dendrite tip, so that gas pore nucleati on may not occur on the de ndrite tip. If the gas bubbles are nucleated, the measur ed hydrogen concentration measur ed must be underestimated.

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182 In Fig. 5-11, a noticeable increase of hydroge n concentration is observed below the baffle, and it reaches 0.21 cc/100 g at 2 cm height. In co ntinuing solidification af ter the dendrite touched the baffle, the solidification situation become the same as in DS experiments, because the baffle was absent ahead of dendrite tip. Above the baff le, the hydrogen concentration is about 0.07 ~ 0.08 cc/100 g, which is almost the same with the equilibrium hydrogen concentration (= 0.09 cc/100g). This strongly supports the hydrogen solute buildup below the baffle. However, accumulated hydrogen concentration measured by ICP (dotted line) is less than the calculated value as shown by the dotted line in Fig. 5-11. This may be caused by (1) porosity formation within dendrite below the baffle and (2) dissipation of hydroge n into the bulk through the gap between the container wall and the baffle after the baffle is fixed by the pin. Additional information is needed for the prec ise analysis of pore nucleation in the AHP and DS samples, in particular, the supersat uration needed for pore nucleation. Recently, nucleation behavior of hydrogen bubbles during directi onal solidification was observed by an insitu x-ray temperature gradient stage (XTGS) technique for the Al-Cu(10wt%) alloy [169] and for Al-Si alloy [147], in which the location and temperature at the moment of nucleation could be observed directly as well as the specific regi on of the dendrite where the nucleation rate was fastest. With these results, the hydrogen supersat uration in the liquid is then calculated by the ratio of hydrogen concentration in po res in the interdendritic liquid to the solubility limit at that temperature. Results showed that the supersatur ation ratio in Al-Cu (10 wt%) was 1.1 ~ 3.1 at [H]0 = 0.25 cc/100 g [169] with 95% of the gas bubbles nucleated. This means that most gas bubbles were able to be formed when the hydr ogen concentration was 1.1 ~ 3.1 times the maximum solubility, (Fig. 3 in [147]). The por e nucleation rate was maximized at a specific dendrite area which satisfies the supersaturation ra tio and this area is defi ned as the nucleation

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183 range. It was explained that, once numerous gas pores have nucleated with considerable consumption of hydrogen, the available hydrogen d ecreases and, due to diffusion of remaining hydrogen into growing pores, additional hydroge n consumption occurs. This reduction of hydrogen causes a decrease of nuc leation rate as solidification progresses. Thus, most of the pores exist in this specific range, and the numbe r of pores decreases towards the bottom of the mushy zone. This work was confirmed by anot her supportive observati on of pore nucleation location [170, 171]. With the above supersaturation ratio, the nuc leation range can be calculated by the method previously used in the DS samples. By multip lying the supersaturation ratio (1.1~3.1) by the original solubili ty in Eq. (2-101), we can find a supersaturated solubility, which includes the supersaturation effect. Then, by in serting this supersaturated sol ubility to Eq. (2-100), a new gas pressure, which gives the supers aturation consideration, can be obtained. In case of AHP samples, the hydrogen concentration after buildup needs to be replaced with initial hydrogen concentration. Then, the pore nucleation range of this study can be obt ained by inserting the above values to Eq. (2-115). Fig. 5-12 (a) and (b) shows nucleation ranges when [H]0=0.09 cc/100 g for the AHP sample and when [H]0=0.27 cc/100 g for the DS sample. The ne w gas pressure lines were plotted by multiplying the supersaturation ratio; dotted line for the supersaturation ratio is for 1.1 and solid line is for 3.1. These are directly compared with Pa+P+PP according to the cooling rate (0.0017, 0.012, 0.018 C/sec) expressed on Fi g. 5-12 (a) and (b). The nucleation range is found between two intersect ion points of the new Pg and the Pa+P+PP covering by an arrow. Due to supersaturation, the possible nu cleation starting point of the DS samples is changed from gs = 0.32 when the supersaturation is not c onsidered as shown in Fig. 5-3 (a) to

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184 0.44 in Fig. 5-12 (b). For AHP samples, [H]0 = 0.09 cc/100 g was selected as a representative [H]0, corresponding to the nucleation range when the smallest solute concentration has built up below the baffle. However, in this case, th e pore will nucleate heavily in the range of gs=0.140.51, approximately regardless of co oling rate, which is closer to the dendrite tip. As the initial hydrogen concentration increases, th e nucleation range approaches th e dendrite tip and, in most AHP sample, the nucleation range is extended from the dendrite to the graphite container wall and the baffle between the dendrite tip and the baffle. The DS sample with [H]0 = 0.27 cc/100 g is compared to show the nucleation range at th e maximum initial hydrogen concentration. Here, the nucleation range is gs= 0.44-0.66, which is still within the confines of the dendrite. In Fig. 5-13 (a) and (b), four representative cases were se lected to demonstrate the trend of nucleation range according to the solidification variables. The schematic of the dendrite shape was calculated by the Scheil equation, as show n in Eq. (2-69) (2-71). Even at [H]0=0.23 cc/100 g for the DS sample, the nucl eation range is lower than [H]0= 0.09 cc/100 g of the AHP sample due to solute buildup. This means that nu cleation in the AHP sample occurs close to the dendrite tip relative to DS samples regardless of initial hydrogen concentration. As shown in Fig. 5-13 (b), the initial hydroge n concentration of the AHP sample increases to [H]0 =0.23 cc/100 g and, by contrast, it deceases to 0. 09 cc/100 g in the DS samples. It is clear that for DS samples the nucleation range approaches the dendrite bottom with a decrease of initial hydrogen concentration. In the case of AHP samples, th e nucleation can occur anywhere below baffle due to a large hydrogen buildup. The locations of nucleated gas bubbl es are important, since the gas pores are known to be formed because the ga s cannot escape from the dendrite due to low permeability of the interdendritic region. Thus, the gas bubbles formed on the tip of the dendrite are capable of escaping more easil y [172]. Compared with the stagnant interd endritic region near

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185 the dendrite tip, relatively strong fl uid flow helps gas bubbles to es cape to the free surface. In the case of a metallic material, the direct observation of an escaping pore is difficult due its inherent opaqueness. However, this is well visualized in organic materials [173, 174], where a pore nucleated on the dendrite tip show s either escape from dendrite or entrapment into the dendrite. The outcome is determined by a competitive relation between the buoyancy force of the pore and the interfacial energy between por e and base material. From observations of pores escaping from the dendrite tip, it was found that a flat the dendrit e tip and a slow growth rate are effective at releasing the pores into the liquid. When a nucl eated pore cannot escape from the dendrite and is trapped inside the dendrite, th e pores can grow along with the primary dendrite by attracting a considerable amount of hydrogen. This is call ed the bubble-solid duplex structure, which can occur frequently when bubbles cannot be re pulsed from the dendrite tip [175, 176]. To confirm the formation of gas bubbles on the dendrite tip and their escape as in organic materials, a modified experiment was performe d. In the AHP technique, escaping gas bubbles may be imprisoned below the baffle, since the gap between the baffle and the side wall of crucible is small and the melt flows down through this gap during solidification. To observe the region below the baffle, alumina pins were inserted in the side wall of co ntainer at 5 cm height and the baffle is positioned initially at 2 cm he ight. When the sample was withdrawn to cold zone by 3 cm, the baffle became stuck by the pins at a pre-set position (5 cm height). After about 3 cm solidification, the sample was withdrawn rapidly into the lower temperature region in 30 seconds and a traverse section of the interface region was inves tigated microscopically. As shown Fig. 5-14, unusually large pores of 100 200 m were observed below the baffle where the size of normal pores is typically 715 m. This visual result suppo rts the concept of escaping of pores and hydrogen buildup below the baffle.

PAGE 186

186 Lee et. al. [178] proposed mass concentrati on equation for hydrogen concentration in the presence of gas bubbles on the dendrite tip: T CP CgCgCVHg LHLSHL H, 0)1( [5-2] where, C0 H is the initial hydrogen concentration, and the subscripts of s, l, and v indicate the solid, liquid, vapor concentrati ons, respectively. Equation 5-2 in cludes an additional term based on the ideal gas law ( is the gas constant) to relate the concentration of hydrogen in the pores to mass balance equation. Due to the newly introduced gas term, the hydrogen concentration in the liquid can be neglected as bubbl e nucleation increases According to Lee et al. [178], the hydrogen pore nucleation is suppr essed by pre-nucleated pores. Ha n and Viswanathan [179] also proposed that the nucleation a nd the subsequent growth of pores can consume hydrogen and reduce the hydrogen concentration. At this time, the nucleation and growth rate cease because of reduced hydrogen concentration and begin again when hydrogen is built up. Han and Viswanathan supported their hypothe sis with a rapid water solid ification experiment done by Carte et al. [180]. In the water solidification experiment, gas (air) bubb les are nucleated on the dendrite and, after gas bubbles are escaped from the dendrite, new gas bubbl es are nucleated. In the AHP technique, the nucleated pores below the baffle on the dendrite tip are relatively stationary due to the difficult to escaping through the gap and they can play a role as a hydrogen sink, so that, in turn, the local hydrogen concentration of the liquid can be reduced. This was confirmed by ICP analysis as shown in Fig. 423, in which the hydrogen concentration in AHP samples become less than those in DS samples by 0.4 0.8 cc/100 g in spite of hydrogen buildup.

PAGE 187

187 Porosity Formation: AHP Samples It is known that both dendrite heights can decrease and hydrogen concentration can decrease as a result of gas bubble escape. In Fi g. 5-15 (a) and (b), volume % porosity and their sizes are predicted by Li and Ch angs model in AHP samples. To use this model in AHP cases, the hydrogen concentration measured by ICP substitu tes for the initial hydrogen concentration in Eq. (2-114) and the locally increas ed copper concentration ahead of the tip is considered from EPMA results. The ICP analysis was performed only at the cooling rate = 0.018 C/sec and the same measurement of [H] was assumed to hold fo r cooling rate = 0.0017 and 0.012 C/sec. Fig. 5-15 (a) and (b) show the predicted and observed porosities and pore size for the AHP samples and the results are in reasonable agreement, even though the calculated results are more sensitive to the hydrogen concentration. When the initial hydrogen concentr ation is 0.23cc/100 g and 0.27 cc/100 g at cooling rate = 0.018 C/sec, the hyd rogen concentration of the AHP sample is reduced to 0.16 and 0.18 cc/100 g, at which the volume % porosity is 0.19 % and 0.20 %, respectively. In comparing close case of sim ilar hydrogen concentrati on in DS samples, the volume % porosity of DS samples is 0.24 % at 0.19 cc/100 g. At this si milar initial hydrogen concentration, the volume % poros ity of the DS sample is still higher than that in the AHP sample by 0.4 0.5%. This difference, howev er, may be allowable by relatively low hydrogen concentration of AHP samples relative to th e DS sample (AHP: 0.16 cc/100g, DS: 0.19 cc/100g) and considering the error range of porosity measurement. This reasonable prediction of the AHP sample with the theoretical model and the DS samples may signify that there are two steps of nucleation ranges in the AHP samples. If most of gas bubbles nucleate on the dendrite tip and escape from dendrite, the volume % porosity with in dendrite may be much less than what was measured value. Also, in principle, the pred iction of volume % porosit y by Li and Changs model assumes that the gas bubbles nucleate within the dendrite and do not escape from dendrite.

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188 Thus, there is no reason that the volume % poros ities in the AHP samples should agree with the Li and Changs model or the volume % porositie s in the DS samples containing low initial hydrogen content. In this regard, the followi ng two steps could be proposed. (1) When the hydrogen solutes are being accumulate d in the early stage of solidif ication, the nucleation region may approach the tip of the dendrite until the solubility li mit is reached. The solute buildup in the early stage of solidification is observed in Fig. 4-23 and Fig. 5-11. Also indirectly, it can be confirmed that the volume % porosity is higher in early stage of solidification as shown in Fig. 413 ~ Fig. 4-15. The increase of porosity amount may be result of the increased hydrogen buildup in solidification progress. (2) Just after starting of gas bubble nucleation near the dendrite tip, the gas bubbles can begin to escape a nd grow soon by coalescence or attracting rejected hydrogen on the dendrite tip [176], because th e gas bubble growth is free above the dendrite relative to between dendrites. However, the gas bubble can not escape to the free surface easily with the presence of the baffle, so that the gas bubbles stay between dendrite tip and the baffle during solidification, as shown in Fig. 5-14. With the incr ease of stationary hydr ogen bubbles, the local hydrogen concentration ahead of the tip decreases as measured in the ICP analysis of Fig. 4-22 and 4-23. The decrease in hydrogen concentratio n may push back the nucl eation range from the dendrite tip towards the bottom of the dendrite as similar to the case of a low initial hydrogen concentration in the DS samples. When nucleati on resumes within the dendrite again, as in DS samples, the measured porosities is close to Li and Changs model. This two step process leads to a decrease in porosity in hi gh initial hydrogen concentrations. In addition, with above information, the re sults can to be understood. From Fig. 4-17 and Fig. 4-18, the advantage of the AHP technique in suppressing porosity amount is more obvious with a decrease in cooling rate and an increase in initial hydrogen concentration. These

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189 parameters correspond to more hydrogen buil dup below the baffle. The volume % porosity becomes flat with an increase of initial hydrogen concentration, because there is sufficient hydrogen buildup below the baffle, regardless of the initial hydrogen concen tration. Curiously, at [H]0 =0.09 cc/100 g, the porosity redu ction in the AHP sample is insignificant except when the cooling rate = 0.0015 C/sec, becau se the hydrogen buildup is favored at a slower cooling rate due to the larger hydrogen diffusi on field. In contrast, the AHP technique has a negligible effect on porosity at higher cooling rate because it takes time for the hydrogen buildup to nucleate at the tip. Thus, a significant number of pores can form until hydrogen buildup is enough to nucleate the gas bubbles near the dendrite tip, resu lting in increase of porosity in early stage of solidification, as shown in Fig. 4-15. The decrease in the dendrite height in AHP samples shown in Fig. 5-7 at cooling rate = 0.0017 C/sec would also be conducive to escape of gas bubbles due to short distance of travel inside the dendrite. In addition, decreasing the dendrite height can decrease the solubility limit, resulting in the nucleation range approaching the dendrite tip more rapidly. At cooling rate = 3.6 C/sec, there is no indication that the AHP technique has advantage over the DS method, because the solidification initiat ed after the baffle left. Even if the baffle is present, the hydrogen cannot be accumulated below the baffle in such high cooling rate as calculated in Table 5-1.

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190 Figure 5-1. Calculat ed pressure drop ( P) and pressure against surface tension (P) versus cooling rate

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191 0.00.20.40.60.81.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 S (cc/100g)gs pure Al Al-Cu0 00 20 40 60 8 0 1 2 3 4 5 PH2(atm)gs Pure Al Al-Cu (a) (b) Figure 5-2. (a) Effect of copper content on th e variation of hydrogen solubility versus solid fraction, (b): Effect of copper conten t on the variation of hydrogen pressure versus the solid fraction at cooling rate = 0.018 C/sec.

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192 (a) (b) Figure 5-3. (a) Comparison of the gas pressure term with other pressure terms according to the change of initial hydrogen contents ([H]0 = 0.09, 0.13, 0.19, 0.23, 0.27 cc/100 g) and cooling rates (cooling rate = 0.0017, 0.012, 0.018, 3.6 C/sec) when samples are solidified with DS technique (b) Corresponding critical solid fraction according to the initial hydrogen content. 0.090.120.150.180.210.240.27 0.0 0.2 0.4 0.6 0.8 1.0 g* s[H]0 (cc/100g) GV=0.0015 (oC/sec) 0.012 (oC/sec) 0.018 (oC/sec) 3.6 (oC/sec)

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193 (a) (b) Figure 5-4. (a) Comparison of calculated and m easured amount of porosity in the DS samples by using thermodynamic and Li and Changs model, (b) Comparison of measured radius with calculated pore size of porosity using Li and Changs model

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194 Figure 5-5. Calculated permeability vs gL based on Piwonka and Flemings model (solid line at t = 2 and dashed lines at t = 3), Murakami model (dashed-dotted line), and Kozeny-Carman model (black squares).

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195 Figure 5-6. Domain of solute redistribution according to fluid characteristics by Favier et al: (a) convective-diffusive boundary region, (b) limited diffusion region, (c) diffusive boundary region with fast di ffusion, and (d) diffusive boundary region with intermediate or slow diffusion.

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196 -4.5-4.0-3.5-3.0-2.5-2.0-1.5-1.0 620 625 630 635 640 645 650 655 660 Temperature (oC)log (V) (cm/sec) Burden and Hunt's model Alexandrov's model Exp. (DS) Exp. (AHP) Figure 5-7. Comparison between the tip temper ature obtained by EPMA data and Burden and Hunts model (solid line) and Alexandrovs model (dotted line) as a change of the solidification rate.

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197 Figure 5-8. Mechanism of hydrogen buildup below the baffle

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198 (a) (b) Figure 5-9. Simple schematic of a diffusion field (dotted line): (a) at the curved s/l interface at a dendrite tip and (b) at a planar s/l interface.

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199 Table 5-1. Tip radius calculated for four of solidification rates V4 (cm/sec) 0.7 5 8 150 R (cm) 2.7 1.4 1.3 0.011 D/V (cm) 49.0 6.1 4.0 0.2

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200 Figure 5-10. Concentration of hydrogen built up below the baffle at 4 cm height versus initial hydrogen concentration, [H]0 of various cooling rates (0.0017, 0.012, and 0.018C/sec). 0.090.120.150.180.210.240.27 0.5 1.0 1.5 2.0 2.5 3.0 [H] (cc/100g)[H]0 (cc/100g) cooling rate = 0.0017 (oC/sec) 0.012 (oC/sec) 0.018 (oC/sec)

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201 Figure 5-11. Hydrogen concentration below and above the baffle with an increase of sample height at cooling rate = 0.018 C/sec and [H]0 = 0.09 cc/100 g

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202 (a) (b) Figure 5-12. A calculated nucleation range according to the cooling rate: (a) AHP sample at [H]0 = 0.09 cc/100 g and (b) DS samples for [H]0 = 0.27 cc/100 g: dotted line represents the supersaturation rati o of 1.1 and solid line is 3.1.

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203 (a) (b) Figure 5-13. (a): Nucleation range on an dendrite with gs = 0.54-0.70 for the DS sample (blue area) produced at [H]0 = 0.23 cc/100 g and cooling rate = 0.018 C/sec and gs = 0.14-0.51 AHP samples (yellow area) produced at [H]0 = 0.09 cc/100 g and cooling rate = 0.018 C/sec, (b): Nucleation range is gs = 0.82-0.88 for DS samples produced at [H]0 = 0.09 cc/100 g and cooling rate = 0.018 C/sec and gs = 0.0-0.1 for AHP samples produced at [H]0 = 0.23 cc/100 g and cooling rate = 0.018 C/sec

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204 Figure 5-14. Optical micrograph of a transverse section of sample at the interface of the graphite baffle (right and dark) and metallic alloy (left and bright) when the baffle is held at 5 cm height. Graphite baffle Ni Al Ta Mo Alloy Normal gas pore Agglomerated gas pore at the baffle interface 100( m)

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205 (a) (b) Figure 5-15. Plots of calculated (Li and Cha ngs model) and measured (a) volume % porosity and (b) pore size for AHP samples as a function of initial hydrogen concentration.

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206 CHAPTER 6 CONCLUSION The influence of the Axial Heat Pro cessi ng (AHP) technique and the normal vertical Bridgeman technique (DS) on the mechanism of porosity formation was successfully compared in Al-Cu (4.5 wt%) alloy. It is confirmed th at the porosity formation in AHP samples were reduced by 20 ~ 40% compared to DS samples. Th e reduction of porosity would be helpful to enhance the product life and improve mechanical properties. Also, AHP technique could reduce the production cost by simplifying th e pre-treatment to minimize ga s elements before initiation of casting. The specimens were produced in a wide range of so lidification conditions: solidification rates of 0.00007 ~ 0.15 cm/sec, thermal gradients of 17 ~ 24C/cm, and initial hydrogen concentration of 0.09 cc/100g ~ 0.27 cc/ 100g, by equilibration with hydrogen partial pressure. The level of convection in AHP sample s was suppressed by three or four orders of magnitude compared with DS samples. Experime ntal observations and an alysis are summarized as followings. 1. The microstructures produced by the AHP and the DS techniques do not show a noticeable difference with regard to primary and secondary dendrite arm spacing. However, samples produced by the AHP technique show a por osity 20 ~ 40% lower than samples prepared by the DS technique, and this effect is more pronounced with decrea sing cooling rates and increasing initial hydrogen con centration. Along the height of the AHP sample, the volume percent of microporosity e xhibits a maximum at the early stage of solidification, and then drops considerably. In the DS samples, it is almost constant regardless of the sample height. Additionally, the pore size in the AHP samples was 5 ~ 10% smaller than in the DS samples. 2. A thermodynamic assessment for the porosity formation and its growth were precisely performed by considering the variation of c opper concentration and temperature along the

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207 dendrite height. Volume % poros ity and pore size of the DS samples were predicted and good agreement was observed with Li and Changs model. 3. The suppression of convection in the dendr ites and the bulk liqui d was calculated by Ram and Raw, respectively. Regardless of the AHP and the DS technique, the convection within the dendrite can be ignored due to its small permeability (~10-12 m2). Thus, the effect of convection on the tip composition is determined by the convection in the bulk liquid. As the Raw of the AHP samples were smaller by three or four orders relative to DS samples, the convective mixing of the AHP samples is negligible. As a resu lt, more solutes could be built up ahead of the dendrite tip as confirmed by EPMA. This incr ease of the copper composition can reduce the dendrite height, resulting in easy esca pe of gas bubbles from the dendrite. 4. Using the AHP technique, hydrogen buildup under the baffle was observed using ICP analysis Accumulated hydrogen concentrations were calculated by Tillers approach and they reach the hydrogen solubility limit at the dendr ite tip during solidification. This results in nucleation of gas bubbles on the dendrite tip and the escape of gas bubbles was observed by photomicrograph.

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208 APPENDIX A PARAMETER CALCULATION The center location of th e s/l interface can be determined at all times using a onedimensional heat balance, VL dz dT K dz dT KL L (A-1) Linearzing this equation produces: VL tztz TtT K tz tTT KL i B M BC L i pcM )()( )( )( )( (A-2) Where at given time, t, zi(t) is the position of the center of the interface and zB(t) is the baffle position. Likewise, TBC(t) is the temperature in th e center of the baffle and TPC(t) is the temperature in the center of the pedestal. All quantities in Eq. (A-2) ar e known experimentally except zi(t). After algebraic manipulation: 0)]([)()(])([)( )(][)]([2 tTTKtzVLtzTtTKtTTKtzVLtzPC M B BM BCL PC M i i (A-3) This equation is solved by the quadratic equation to give zi(t), also designated as the height of the s/l interface fromt the bottom, H, at a ny given time. Once this is determined, the melt height (h) and liquid and solid gradient are displayed simply: ) ()()( tztzthi B (A-4) )()( )( )( tztz TtT tGi B M BC L (A-5) )( )( )( tz tTT tGi PCM L (A-6)

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209 APPENDIX B ALEXANDROVS APPROACH G eneral approach for solute redistributi on is only based on mass transfer with no convective mixing. To consider the c onvection, the bulk liquid flow rate, vL cannot be ignored in Eq. (2-23), Eq. (2-24), and Eq. (2-25). The bounda ry conditions were derived from mass balance and heat balance equation at the interfaces( =solid/mush zone and liquid/mushy zone). By assuming one-dimensional steady state convective problem and continuity equation for fluid speed at solid/mush interface and liquid/much in terface, we can scale out the units by choosing references. By scaling with D/V, Eq. (2-23), Eq. (2-24), and D Arcys equation are transformed to dimensionless forms: dz d N dz d dz d N dz d JH dz d 2 1))(( ))(( [B-1] dz d kc dz d D dz d dz dc Jc dz d ))(( ))1((0 [B-2] )( c dz dp RaJl [B-3] where, m /( : temperature, m : liquidus slope, : initial concentration), c= /, J=U/V (U: volume flux of interdendritic fluid, V : solidification velocity), LLCCh /)( ( : volume fraction of solid, : density of interdendritic region, C : specific heat per unit volume, SS LLC CC )1 ( ) S /)( ( : thermal conductivity, S L )1 (), N1=LLLSCD / N2 = mCLLLV/ ( LV: latent heat), ) / /(0VgD ppL L m ** Vg Ra /0 (g: acceleration of gravity, : viscosity). Confusion can be arisen due to unfamiliar symbols between chemical engineers. He rein, a variable, J, represents the ratio of convective and solidification velocity. Thus, variation of J provides a different strength of the

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210 fluid flow. Boundary condition Boundary conditions were created from the continuity between solid/mush and liquid/mush: c 0 ( m /00) ; dz dc dz d [B-5] )1()(* 1 2 N N dz dc GS 0)1( dz dc ck at z = 0 [B-6] dz d dz dL dz d dz dL at z = h [B-7] Here, z indicates the height from the s/m interface. Thus, z = 0 means s/m interface and z = h is the location of l/m interface. *denotes the volume fraction of solid fraction. Using boundary conditions, Integrating Eq. (B-1) by usi ng Eq. (B-5) express solute concentration gradient in the mushy zone )( )))(((1 2 0 N ANchJ dz dc [B-8] where A is a constant. Substituting dc/dz into Eq. (B-2) and di viding this equation by dzd / the resultant equation is as given: 0])()( )1(2 1 kccMMJcc d d [B-9] )( ))(( )1()(1 20 1 N ANhJ M [B-10] )( ))(( )1()(1 0 2 N hJ M [B-11] As can be seen, this differential equation of the first kind determines function c( ). Integration gives the solute concentration as a function of solid fraction:

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211 0 1 2]))(exp( 1 )1[( )1()( ))(exp( )( dyykF dy dM c Le Le J J M kF chz [B-12] where 0 2)1()( )( JxxM dx F, Here, LLLLCDLe / and this is named as Lewis nu mber. Now, the composition profile within interdendritic equati on is derived Simply, by putting =0 into Eq. (B-12), we can get the composition of solute at the dendrite tip (cz=h) 1 )1/(0 Le JALe chz [B-13] Now, J is found from Eq. (B-3). A in Eq. (B13) should be found. This can be solved from another boundary condition at s/m interface. Usin g Eq. (B-5) and substituting Eq. (B-8) at z=0 into the second boundary condition of Eq.(B-6), the following relati ons are given: 0 *)( c [B-14] )( ))(( ))(1(* 1 2 ** 0 N AN hJ k [B-15] By solving above algebraic equation, *and A can be obtained and by putting these values into Eq. (13), the tip composition at the dendrite tip can be calculated.

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212 APPENDIX C MICROSTRUCTURES Micros tructures of each sample were attach ed here. The experimental conditions were summarized in table 4-2. In each experiment, 50 ~ 90 microstructures were used to obtain the volume percent porosity with magnification 50 x ~ 100 x. In this section, 8 pictures were selected at 1 ~ 2 cm height and another 8 pictures were selected at 3 ~ 4 cm height of samples, respectively. Figure C -1. Microstructure of DS I 1 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -2. Microstructure of DS I 1 at 3 ~ 4 cm height of the sample (the arrow indicates solidification direction) with 50 x magnification.

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213 Figure C -3. Microstructure of DS I 2 at 1 ~ 2 cm height of the sample (the arrow indicates solidification direction) with 50 x magnification. Figure C -4. Microstructure of DS I 2 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -5. Microstructure of DS I 3 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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214 Figure C -6. Microstructure of DS I 3 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -7. Microstructure of DS I 4 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -8. Microstructure of DS I 4 at 3 ~4 cm height of the sample (the arrow indicates the solidification directi on with 50 x magnification.

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215 Figure C -9. Microstructure of DS I 5 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -10. Microstructure of DS I 5 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -11. Microstructure of AHP I 1 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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216 Figure C -12. Microstructure of AHP I 1 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -13. Microstructure of AHP I 2 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -14. Microstructure of AHP I 2 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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217 Figure C -15. Microstructure of AHP I 3 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -16. Microstructure of AHP I 3 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -17. Microstructure of AHP I 4 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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218 Figure C -18. Microstructure of AHP I 4 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -19. Microstructure of AHP I 5 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -20. Microstructure of AHP I 5 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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219 Figure C -21. Microstructure of DS II 1 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -22. Microstructure of DS II 1 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -23. Microstructure of DS II 2 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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220 Figure C -24. Microstructure of DS II 2 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -25. Microstructure of DS II 3 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -26. Microstructure of DS II 3 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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221 Figure C -27. Microstructure of DS II 4 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) Figure C -28. Microstructure of DS II 4 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -29. Microstructure of DS II 5 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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222 Figure C -30. Microstructure of DS II 5 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -31. Microstructure of AHP II 1 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C -32. Microstructure of AHP II 1 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification.

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223 Figure C -33. Microstructure of AHP II 2 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -34. Microstructure of AHP II 2 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -35. Microstructure of AHP II 3 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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224 Figure C -36. Microstructure of AHP II 3 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -37. Microstructure of AHP II 4 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -38. Microstructure of AHP II 4 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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225 Figure C -39. Microstructure of AHP II 5 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 100 x magnification. Figure C -40. Microstructure of AHP II 5 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 100 x magnification. Figure C -41. Microstructure of DS III 1 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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226 Figure C -42. Microstructure of AHP II 1 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C -45. Microstructure of DS III 3 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -46. Microstructure of DS III 3 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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227 Figure C -47. Microstructure of DS III 4 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -48. Microstructure of DS III at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C -49. Microstructure of DS III 5 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 100 x magnification.

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228 Figure C -50. Microstructure of DS III 5 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 100 x magnification. Figure C 51. Microstructure of AHP III 1 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C 52. Microstructure of AHP III 1 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification.

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229 Figure C 53. Microstructure of AHP III 2 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C 54. Microstructure of AHP III 2 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C 55. Microstructure of AHP III 3 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification.

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230 Figure C 56. Microstructure of AHP III 3 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C 57. Microstructure of AHP III 4 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C 58. Microstructure of AHP III 4 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification.

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231 Figure C 59. Microstructure of AHP III 5 at 1 ~ 2 cm height of the sample (the arrow indicates the solidifi cation direction) with 100 x magnification. Figure C 60. Microstructure of AHP III 5 at 3 ~ 4 cm height of the sample (the arrow indicates the solidifi cation direction) with 100 x magnification. Figure C 61. Microstructure of DS IV 1 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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232 Figure C 62. Microstructure of DS IV 1 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C 63. Microstructure of DS IV 2 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C 64. Microstructure of DS IV 2 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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233 Figure C 65. Microstructure of DS IV 3 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C 66. Microstructure of DS IV 3 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C 67. Microstructure of DS IV 4 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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234 Figure C 68. Microstructure of DS IV 4 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C 69. Microstructure of DS IV 5 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification. Figure C 70. Microstructure of DS IV 5 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification direction) with 50 x magnification.

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235 Figure C 71. Microstructure of AHP IV 1 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C 72. Microstructure of AHP IV 1 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C 73. Microstructure of AHP IV 2 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification.

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236 Figure C 74. Microstructure of AHP IV 2 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C 75. Microstructure of AHP IV 3 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C 76. Microstructure of AHP IV 3 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification.

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237 Figure C 77. Microstructure of AHP IV 4 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C 78. Microstructure of AHP IV 4 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification. Figure C 79. Microstructure of AHP IV 5 at 1 ~ 2 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification.

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238 Figure C 80. Microstructure of AHP IV 5 at 3 ~ 4 cm height of the sample (the arrow indicates the solidification dire ction) with 50 x magnification.

PAGE 239

239 APPENDIX D POROSITY DISTRIBU TION MEASUREMENT (a) (b) (c) (d) (e) (f) Figure D. Porosity distribution of DS I 1 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of PorosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of PorosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of PorosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of PorosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of PorosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of PorosityRelative distance (cm)

PAGE 240

240 (a) (b) (c) (d) (e) (f) Figure D-2. Porosity distributi on of DS I 2 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Relative distance (cm) Volume (%) of porosity-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume % of porosityRelative distance (cm)

PAGE 241

241 (a) (b) (c) (d) (e) (f) Figure D-3. Porosity distribution of DS I 3 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume % of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)

PAGE 242

242 (a) (b) (c) (d) (e) (f) Figure D-4. Porosity distributi on of DS I 4 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Volume (%) of porosityRelative distance (cm)

PAGE 243

243 (a) (b) (c) (d) (e) (f) Figure D-5. Porosity distribution of DS I 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)

PAGE 244

244 (a) (b) (c) (d) (e) (f) Figure D-6. Porosity distribution of AHP I 1 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 245

245 (a) (b) (c) (d) (e) (f) Figure D-7. Porosity distribution of AHP I 2 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 246

246 (a) (b) (c) (d) (e) (f) Figure D-8. Porosity distribution of AHP I 3 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 247

247 (a) (b) (c) (d) (e) (f) Figure D9. Porosity distribution of AHP I 4 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosity Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 248

248 (a) (b) (c) (d) (e) (f) Figure D-10. Porosity distribution of AHP I 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosity Relative distance (cm)

PAGE 249

249 (a) (b) (c) (d) (e) (f) Figure D-11. Porosity distribution of DS II 1 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Volume (%) of porosityRelative distance (cm)

PAGE 250

250 (a) (b) (c) (d) (e) (f) Figure D12. Porosity distribution of DS II 2 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)

PAGE 251

251 (a) (b) (c) (d) (e) (f) Figure D-13. Porosity distribution of DS II 3 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)

PAGE 252

252 (a) (b) (c) (d) (e) (f) Figure D14. Porosity distribution of DS II 4 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 253

253 (a) (b) (c) (d) (e) (f) Figure D-15. Porosity distribution of DS I 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 254

254 (a) (b) (c) (d) (e) (f) Figure D-16. Porosity distribution of AHP II 1 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)

PAGE 255

255 (a) (b) (c) (d) (e) (f) Figure D-17. Porosity distribution of AHP II 2 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)

PAGE 256

256 (a) (b) (c) (d) (e) (f) Figure D-18. Porosity distribution of AHP II 3 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)

PAGE 257

257 (a) (b) (c) (d) (e) (f) Figure D-19. Porosity distribution of AHP II 4 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)

PAGE 258

258 (a) (b) (c) (d) (e) (f) Figure D-20. Porosity distribution of AHP II 5 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)

PAGE 259

259 (a) (b) (c) (d) (e) (f) Figure D-21. Porosity distribution of DS III 1 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume (%) of porosityRelative distance (cm)

PAGE 260

260 (a) (b) (c) (d) (e) (f) Figure D-22. Porosity distribution of DS III 1 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative dsitance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)

PAGE 261

261 (a) (b) (c) (d) (e) (f) Figure D-23. Porosity distribution of DS III 3 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 262

262 (a) (b) (c) (d) (e) (f) Figure D-24. Porosity distribution of DS III 4 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 263

263 (a) (b) (c) (d) (e) (f) Figure D-25. Porosity distribution of DS III 5 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 Volume (%) of porosityRelative distance (cm)

PAGE 264

264 (a) (b) (c) (d) (e) (f) Figure D-26. Porosity distribution of AHP III 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 265

265 (a) (b) (c) (d) (e) (f) Figure D-27. Porosity distribution of AHP III 2 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 266

266 (a) (b) (c) (d) (e) (f) Figure D-28. Porosity distribution of AHP III 3 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 267

267 (a) (b) (c) (d) (e) (f) Figure D-29. Porosity distribution of AHP III 4 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)

PAGE 268

268 (a) (b) (c) (d) (e) (f) Figure D-30. Porosity distribution of AHP III 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)

PAGE 269

269 (a) (b) (c) (d) (e) (f) Figure D-31. Porosity distribution of DS VI 1 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)

PAGE 270

270 (a) (b) (c) (d) (e) (f) Figure D-32. Porosity distribution of DS IV 2 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)

PAGE 271

271 (a) (b) (c) (d) (e) (f) Figure D-33. Porosity distribution of DS IV 3 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)

PAGE 272

272 (a) (b) (c) (d) (e) (f) Figure D-34. Porosity distribution of DS IV 4 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Volume (%) of porosityRelative distance (cm)

PAGE 273

273 (a) (b) (c) (d) (e) (f) Figure D-35. Porosity distribution of DS IV 5 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Volume (%) of porosityRelative distance (cm)

PAGE 274

274 (a) (b) (c) (d) (e) (f) Figure D-36. Porosity distribution of AHP IV 1 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 275

275 (a) (b) (c) (d) (e) (f) Figure D-37. Porosity distribution of AHP IV 2 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 276

276 (a) (b) (c) (d) (e) (f) Figure D-38. Porosity distribution of AHP IV 3 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Volume (%) of porosityRelative distance (cm)

PAGE 277

277 (a) (b) (c) (d) (e) (f) Figure D-39. Porosity distribution of AHP IV 4 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)

PAGE 278

278 (a) (b) (c) (d) (e) (f) Figure D-40. Porosity distribution of AHP IV 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 0.5 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0.0 0.1 0.2 0.3 0.4 Volume (%) of porosityRelative distance (cm)

PAGE 279

279 APPENDIX E PORE RADIUS MEASUREMENT (a) (b) (c) (d) (e) (f) Figure E-1. Measured pore radius of DS I 1 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)

PAGE 280

280 (a) (b) (c) (d) (e) (f) Figure E-2. Measured pore radius of DS I 2 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)

PAGE 281

281 (a) (b) (c) (d) (e) (f) Figure E-3. Measured pore radius of DS I 3 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 22 24 Radius of pore (m)Relative distance (cm)

PAGE 282

282 (a) (b) (c) (d) (e) (f) Figure E-4. Measured pore radius of DS I 4 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m) Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)

PAGE 283

283 (a) (b) (c) (d) (e) (f) Figure E-5. Measured pore radius of DS I 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)

PAGE 284

284 (a) (b) (c) (d) (e) (f) Figure E-6. Measured pore radius of AHP I 1 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)

PAGE 285

285 (a) (b) (c) (d) (e) (f) Figure E-7. Measured pore radius of AHP I 2 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)

PAGE 286

286 (a) (b) (c) (d) (e) (f) Figure E-8. Measured pore radius of AHP I 3 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)

PAGE 287

287 (a) (b) (c) (d) (e) (f) Figure E-9. Measured pore radius of AHP I 4 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)

PAGE 288

288 (a) (b) (c) (d) (e) (f) Figure E-10. Measured pore radius of AHP I 5 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 289

289 (a) (b) (c) (d) (e) (f) Figure E-11. Measured pore radius of DS II 1 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)

PAGE 290

290 (a) (b) (c) (d) (e) (f) Figure E-12. Measured pore radius of DS II 2 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 291

291 (a) (b) (c) (d) (e) (f) Figure E-13. Measured pore radius of DS II 3 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 292

292 (a) (b) (c) (d) (e) (f) Figure E-14. Measured pore radius of DS II 4 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 5 6 7 8 9 10 11 12 13 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 5 6 7 8 9 10 11 12 13 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 5 6 7 8 9 10 11 12 13 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 5 6 7 8 9 10 11 12 13 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 5 6 7 8 9 10 11 12 13 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 5 6 7 8 9 10 11 12 13 14 15 Radius of pore (m)Relative distance (cm)

PAGE 293

293 (a) (b) (c) (d) (e) (f) Figure E-15. Measured pore radius of DS II 5 at each height : (a) at th e bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)

PAGE 294

294 (a) (b) (c) (d) (e) (f) Figure E-16. Measured pore radius of AHP II 1 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 295

295 (a) (b) (c) (d) (e) (f) Figure E-17. Measured pore radius of AHP II 2 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 296

296 (a) (b) (c) (d) (e) (f) Figure E-18. Measured pore radius of AHP II 3 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 297

297 (a) (b) (c) (d) (e) (f) Figure E-19. Measured pore radius of AHP III 4 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 298

298 (a) (b) (c) (d) (e) (f) Figure E-20. Measured pore radius of AHP II 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)

PAGE 299

299 (a) (b) (c) (d) (e) (f) Figure E-21. Measured pore radius of DS III 1 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 6 8 10 12 14 16 18 20 Radius of pore (m)Relative distance (cm)

PAGE 300

300 (a) (b) (c) (d) (e) (f) Figure E-22. Measured pore radius of DS III 2 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 301

301 (a) (b) (c) (d) (e) (f) Figure E-23. Measured pore radius of DS III 3 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 302

302 (a) (b) (c) (d) (e) (f) Figure E-24. Measured pore radius of DS III 4 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 303

303 (a) (b) (c) (d) (e) (f) Figure E-25. Measured pore radius of DS III 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)

PAGE 304

304 (a) (b) (c) (d) (e) (f) Figure E-26. Measured pore radius of AHP III 1 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 305

305 (a) (b) (c) (d) (e) (f) Figure E-27. Measured pore radius of AHP III 2 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 306

306 (a) (b) (c) (d) (e) (f) Figure E-28. Measured pore radius of AHP III 3 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 307

307 (a) (b) (c) (d) (e) (f) Figure E-29. Measured pore radius of AHP III 4 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 308

308 (a) (b) (c) (d) (e) (f) Figure E-30. Measured pore radius of AHP III 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 Radius of pore (m)Relative distance (cm)

PAGE 309

309 (a) (b) (c) (d) (e) (f) Figure E-31. Measured pore radius of DS IV 1 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 310

310 (a) (b) (c) (d) (e) (f) Figure E-32. Measured pore radius of DS IV 2 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 311

311 (a) (b) (c) (d) (e) (f) Figure E-33. Measured pore radius of DS IV 3 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 312

312 (a) (b) (c) (d) (e) (f) Figure E-34. Measured pore radius of DS IV 4 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 313

313 (a) (b) (c) (d) (e) (f) Figure E-35. Measured pore radius of DS IV 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 314

314 (a) (b) (c) (d) (e) (f) Figure E-36. Measured pore radius of AHP IV 1 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 315

315 (a) (b) (c) (d) (e) (f) Figure E-37. Measured pore radius of AHP IV 2 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 316

316 (a) (b) (c) (d) (e) (f) Figure E-38. Measured pore radius of AHP IV 3 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 317

317 (a) (b) (c) (d) (e) (f) Figure E-39. Measured pore radius of AHP IV 4 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 318

318 (a) (b) (c) (d) (e) (f) Figure E-40. Measured pore radius of AHP IV 5 at each height : (a) at the bottom, (b) 1 cm, (c) 2 cm, (d) 3 cm, (e) 4 cm, and (f) 5 cm -2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)-2.0-1.5-1.0-0.50.00.51.01.52.0 0 2 4 6 8 10 12 14 Radius of pore (m)Relative distance (cm)

PAGE 319

319 APPENDIX F THERMAL DATA ANALYSIS Figure F-1. Temperature acquired from in corporated thermocouples in AHP samples according to the time at the solidification rate = 0.0005 (cm/sec) and furnace temperature = 715C.The sample height is 7. 7 cm and the initial baffle height is 2 cm. Thus, the baffle should travel by 5.7 cm in the melt. By dividing expected travel distance with measured trav el time (=22,243 13,005 seconds), the calculated withdrawing rate is 0.00055 cm /sec. It is reasonably accorded with measured withdrawing rate (= 0.0005 cm/sec) By using Eq. 4-1, the initial crystal height can be calculated as 2.1 cm and GL = 23C.

PAGE 320

320 Figure F-2. Temperature acquired from in corporated thermocouples in AHP samples according to time at the solidificatio n rate = 0.00008 (cm/sec) and furnace temperature = 700C.The sample height is 7. 1 cm and the initial baffle height is 2 cm. Thus, the baffle should travel by 5.1 cm in the melt. By dividing expected travel distance with measured trav el time (=83,395 14,830 seconds), the calculated withdrawing rate is 0.000075 cm/sec. It is reasonably accorded with measured withdrawing rate (= 0.00007 cm /sec). By using Eq. 4-1, the initial crystal height can be calculated as 2.3 cm and GL = 22 C.

PAGE 321

321 Figure F-3. Temperature acquired from in corporated thermocouples in AHP samples according to time at the solidificatio n rate = 0.15 (cm/sec) and furnace temperature = 730C.The sample height is 7. 5 cm and the initial baffle height is 2 cm. Thus, the baffle should travel by 5.5 cm in the melt. By dividing expected travel distance with measured trav el time (=13,095 13,067 seconds), the calculated withdrawing rate is 0.00008 cm /sec. It is reasonably accorded with measured withdrawing rate (= 0.00007 cm /sec). By using Eq. 4-1, the initial crystal height can be calculated as 2.2 cm and GL = 23C.

PAGE 322

322 Figure F-4. Temperature acquired from incorporated thermocouples in DS samples according to time at the solidificati on rate = 0.0008 (cm/sec) and furnace temperature = 720 C. Fig. F-4 shows the temperature varia tion versus time at the solidification rate = 0.0008 cm/sec. The baffle temperatur es (TC 1 and TC 2) are not present in DS samples. Arbitrarily, two points were selected to obtain the cooling rate and thermal gradient in the liquid as shown in Figure F-4. In this case, the cooling rate is 0.015 C/sec and GL is 20 C/cm.

PAGE 323

323 Figure F-5. Temperature acquired from incorporated thermocouples in DS samples according to time at the solidificati on rate = 0.0005 (cm/sec) and furnace temperature = 715 C. Fig. F-5 shows the temperature varia tion versus time at the solidification rate =0.0005 cm/sec. The baffle temperatur es (TC 1 and TC 2) are not present in DS samples. Arbitrarily, two points were selected to obtain the cooling rate and thermal gradient in the liquid as shown in Figure F-5. In this case, the cooling rate is 0.012 C/sec and GL is 24 C/cm.

PAGE 324

324 Figure F-6. Temperature acquired from incorporated thermocouples in DS samples according to time at the solidificati on rate = 0.00007 (cm/sec) and furnace temperature = 700 C. Fig. F-6 shows the temperature variation versus time at the solidification rate =0 .00007 cm/sec. The baffle temperatures (TC 1 and TC 2) are not present in DS samples. Arbitrarily, two points were selected to obtain the cooling rate and thermal gradient in the liquid as shown in Figure F-6. In this case, the cooling rate is 0.0018 C/sec and GL is 25 C/cm.

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325 Figure F-7. Temperature acquired from incorporated thermocouples in DS samples according to time at the solidification rate = 0. 15 (cm/sec) and furnace temperature = 705 C. Figure F-7 shows the temperature variation versus time at the solidification rate =0.15 cm/sec. The baffle temperatures (TC 1 and TC 2) are not present in DS samples. Arbitrarily, two points were selected to obtain th e cooling rate and thermal gradient in the liquid as shown in Figure F-7. In this case, the cooling rate is 1.6 C/sec and GL is 11 C/cm.

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335 BIOGRAPHICAL SKETCH Joo Ro Ki m was born in 1975 in Seoul, Korea (Republic of). He graduated from Dankuk High School in 1994 and, then, attended the Yonsei University to earn a BS degree in metallurgy in 1995. While pursuing the BS degree, he comp leted his military service for 26 months from May1998 to July 2000. He spent the fall semester in Centenary College in New Jersey, US. On graduating from Yonsei University, he advanced to graduate school at the Department of Materials Science and Engineering in Yonsei Un iversity and chose chemical metallurgy for his MS thesis project in 2002. After graduation, he worked for the Iron and Steel Institute of Korea as a research assistan t involving an innovative ir on and steel-making process, named FINEX. In August 2004, he joined the University of Flor ida and Department of Materials Science and Engineering to work in Dr. R eza Abbaschians solidification la boratory to pursue a doctoral degree. After passing his qualifying exam and defe nding his proposal of re search in 2007, he received his PhD in 2008.