Crystal Growth from Melts

Crystal Growth from Melts Applications to Growth of Groups 1 and 2 Crystals Bimalendu Narayan Roy Sheffield City Polytechnic, UK JOHN WILEY & SONS Chichester New York Brisbane Toronto Singapore

Contents Preface Acknowledgements List of Symbols xv xvii xix 1 Introduction 1 1.1 A Brief History of Crystal Growth 1 1.2 High Temperature Crystal Growth 2 1.3 Other Methods 3 1.3.1 Growth from the pure melt 3 1.3.2 Growth from the solid state 3 1.3.3 Growth from aqueous solutions 4 1.3.4 Growth from solvents other than water 4 1.3.5 Growth from the vapour phase 4 1.3.6 The hydrothermal method 4 1.4 A Brief Review of the Advantages and Disadvantages of all of the Methods 5 References 7 2 Applications of Crystals and Methods of Crystal Growth 9 2.1 General Applications of Crystals Grown from High Temperature Solutions 9 2.2 Applications of Alkaline-earth Metal Crystals 10 2.3 Classification of Crystal Growth Methods 11 2.3.1 Growth from the melt 14 2.3.2 Growth by evaporation 14 2.3.3 Growth from seeds 15 2.3.4 Growth from the solid state 15 2.3.5 Growth from liquid metals : 16 2.3.6 Temperature-gradient zone melting 16 2.3.7 Growth from aqueous solution 16 2.3.8 Crystal growth from non-aqueous solvents 17 2.3.9 Hydrothermal growth 17 2.3.10 Vapour-liquid-solid growth 17

2.4 Methods of Crystal Growth 18 2.4.1 The Czochralski method 18 2.4.2 The Bridgman-Stockbarger method 25 2.4.3 Zone-melting methods 28 2.4.4 The floating zone technique 31 2.4.5 The gradient-freeze technique 33 2.4.6 The hydrothermal growth method 35 2.4.7 The flame fusion or Verneuil method 37 2.4.8 Crystal growth by osmotic dewatering 40 2.5 Preparation of Single Crystal Thin Films 41 2.5.1 Vapour phase epitaxy 45 2.5.2 Liquid phase epitaxy 46 2.5.3 Molecular beam epitaxy 50 2.5.4 Van der Waals epitaxy 52 2.5.5 Metal-organic chemical vapour deposition 53 2.5.6 Chemical beam epitaxy 55 2.5.7 Langmuir-Blodgett thin film deposition 57 References 59 The Thermodynamics of Crystal Growth 63 3.1 Introduction 63 3.2 Mass Transfer in the Interface 63 3.3 Chemical Equilibria in Crystal Growth 64 3.4 Thermochemical Calculations 67 3.5 Solubility and Phase Diagrams 68 3.5.1 The dissolution-extraction method 68 3.5.2 Quenching 69 3.5.3 Hot-stage microscopy 69 3.5.4 Differential thermal analysis 69 3.6 Determination of the Solubility of Alkaline-earth Metal Salts 70 3.7 Phase Diagrams of Alkaline-earth Metal Systems 70 3.7.1 Binary systems 71 3.7.2 Reciprocal ternary systems 71 3.7.3 Quasi-binary systems 72 3.7.4 Alkaline-earth metal salt-alkali metal chloride systems 73 3.7.5 Alkaline-earth metal salt-alkali metal fluoride, borate, ferrite and molybdate systems 77 3.7.6 Alkaline-earth metal salt-alkaline-earth metal fluoride, borate, ferrite and molybdate systems (with common cation) 78 3.7.7 Alkaline-earth metal salt-alkali metal salt systems (with common anion) 79

3.7.8 Alkaline-earth metal salt-alkaline-earth metal chloride systems (with common cation) 79 3.7.9 Solubilities of barium titanate in different solvents 81 References 85 4 The Physical Chemistry of Molten Salts 87 4.1 Introduction 87 4.2 Electrostatic Interaction and Structure 88 4.3 Molten Salt Mixtures 88 4.4 Diffusion in Salt Melts 90 4.5 The Atomic Structure of Solid-Liquid Interfaces 92 References 94 5 Solvents and Solutes 95 5.1 Introduction 95 5.2 High Temperature Solvents 95 5.3 Crystallisation Conditions 97 5.4 Solute-Solvent Interactions 99 5.4.1 Alkaline-earth metal oxide solutions in alkali metal chloride melts 99 5.4.2 Alkaline-earth metal silicate solutions in alkaline-earth metal chloride melts 100 5.4.3 Alkaline-earth metal titanate solutions in alkaline-earth metal chloride melts 100 5.4.4 Alkaline-earth metal molybdate solutions in alkali metal chloride melts 100 5.4.5 Alkaline-earth metal molybdate solutions in alkaline-earth metal chloride melts 100 5.4.6 Alkaline-earth metal tungstate solutions in alkali metal chloride 101 5.4.7 Alkaline-earth metal tungstate solutions in alkaline-earth metal chloride melts 101 5.4.8 Alkaline-earth metal tungstate solutions in sodium tungstate melts 102 References 104 6 Materials, Methods and Variables 105 6.1 Introduction 105 6.2 Furnace and Chemicals 105 6.3 Methods of Crystallisation 107 6.3.1 Crystallisation by hot-stage microscopy 107 6.3.2 Crystallisation in platinum crucibles (by differential thermal analysis) 107 IX

X 6.3.3 Crystallisation in alumina crucibles (in a high temperature furnace) 109 6.3.4 Direct measurement of the degree of crystallisation 112 6.4 Instrumental Analysis 115 6.4.1 Infrared spectroscopy 115 6.4.2 X-ray methods 116 6.5 Particle Size Measurement 116 6.6 Particle Size Distribution 120 6.7 Standard Deviation and Standard Error 124 6.8 Crystal Number (N) 124 References 125 7 Theory of Crystallisation from Supersaturated Solutions at Constant Temperatures 126 7.1 Introduction 126 7.2 Supersaturation and Crystallisation Affinity 127 7.3 The Thermodynamics of Nucleation 128 7.4 The Kinetics of Nucleation 130 7.5 Crystal Growth (and Growth Rates) 135 7.5.1 The overall mechanism 135 7.5.2 Growth processes 137 7.6 The Overall Kinetics of Growth 142 7.6.1 Heterogeneous nucleation 142 7.6.2 Homogeneous nucleation 143 7.6.3 Growth onto seeds 143 7.7 Kinetic Equations for Growth 143 7.7.1 Diffusion-controlled growth 144 7.7.2 Surface nucleation-controlled growth 145 References 146 8 Theory of Crystallisation with Continuous Development of Supersaturation 147 8.1 Introduction 147 8.2 Nucleation 148 8.3 Critical Time and Critical Temperature 149 8.4 Variation of N with S (or C) 150 8.5 Overall Kinetics of Crystal Growth 151 8.5.1 Systems with no induction period 151 8.5.2 Systems with a short induction period 152 8.5.3 Systems with a prolonged induction period 152 8.5.4 Kinetic relationships at later stages 152 References 153

9 Preliminary General Studies on Crystallisation 154 9.1 Introduction 154 9.2 Crystallisation and Analysis 155 9.3 Variation of Crystal Size and Number with Metal Salt Structure 156 9.3.1 Crystallisation from lithium chloride melts 156 9.3.2 Crystallisation from alkaline-earth metal chloride melts 156 9.3.3 Crystallisation from sodium tungstate melts... 156 9.4 Variation of Crystal Size and Number with Rate of Cooling 156 9.4.1 Crystallisation from lithium chloride melts 157 9.4.2 Crystallisation from alkaline-earth metal chloride melts 161 9.4.3 Crystallisation from sodium tungstate melts... 161 9.5 Variation of Crystal Size and Number with Initial Crystallisation Temperature (and Solute Solubility) 164 9.5.1 Crystallisation from lithium chloride melts 164 9.5.2 Crystallisation from alkaline-earth metal chloride melts 164 9.5.3 Crystallisation from sodium tungstate melts... 167 9.6 Crystal Form 168 9.7 Preliminary Differential Thermal Analysis Studies 168 References 168 10 Crystallisation at Constant Temperatures 170 10.1 Introduction 170 10.2 Crystallisation 170 10.3 Analysis and Characterisation of Crystals 173 10.4 Crystal Size and Shape 173 10.4.1 Crystal size 173 10.4.2 Crystal size distribution 173 10.4.3 Crystal form 174 10.4.4 Crystal anisometry 178 10.5 Alternative Methods 181 10.5.1 Hot-stage microscopy 181 10.5.2 Chemical analysis of the residual solution 181 10.5.3 Differential thermal analysis 184 10.6 Nucleation (Seed Number) 184 10.7 Growth 185 10.7.1 Degree of crystallisation (a t ) 185 10.7.2 Course of growth 186 10.7.3 Maximum length and width 187 10.8 Kinetics of Growth 187 10.9 Mechanism of Growth 192 XI

xii 10.9.1 Kinetic relationships at early stages 192 10.9.2 Kinetic relationships at later stages 193 10.10 Rate Constants (for Diffusion-controlled Crystal Growth) 196 10.10.1 Variation of rate constants with metal cation... 198 10.10.2 Variation of rate constants with temperature... 198 10.11 Diffusion Rate Constants and Diffusion Coefficients 198 References 199 11 Crystallisation by Continuous Cooling from Lithium Chloride Melts in Alumina Crucibles 201 11.1 Introduction 201 11.2 Crystallisation 202 11.2.1 Overall excess solute concentrations 202 11.2.2 Residual excess solute concentrations 204 11.2.3 Total crystal weights 204 11.2.4 Methods of crystallisation 205 11.3 Analysis and Characterisation of Crystals 207 11.4 Crystal Size and Shape 207 11.4.1 Crystal size 207 11.4.2 Crystal size distribution 208 11.4.3 Crystal form 210 11.4.4 Crystal anisometry 211 11.4.5 Shape factor (/ s ) 211 11.5 Kinetics at Early Stages 211 11.5.1 Degree of crystallisation 211 11.5.2 Crystal length vs. time plots 213 11.6 Nucleation 217 11.6.1 Nucleation mechanism and kinetics 217 11.6.2 Nucleation rate constants 219 11.6.3 Nucleus number 221 11.6.4 Induction periods 223 11.6.5 Relationship betweentv and / 225 11.6.6 Relationship between (N/t) and ДС 226 11.7 Growth Kinetics 226 11.7.1 Crystal length vs. growth time 226 11.7.2 Growth-rates 228 11.7.3 Mechanism of growth 233 11.8 Rate Constants 233 11.9 Rate Constants and Diffusion Coefficients 234 11.9.1 Growth along major axes 236 11.9.2 Growth along minor axes 238 11.10 Factors Affecting Diffusion of Materials 239 11.11 Non-ideal Diffusion-controlled Crystal Growth 239 11.12 Crystal Anisometry 240

11.13 Final Crystal Size 240 References 241 XIII 12 Crystallisation by Continuous Cooling from Sodium Tungstate Melts in Alumina Crucibles 242 12.1 Introduction 242 12.2 Crystallisation 242 12.2.1 Development of supersaturation 243 12.2.2 Methods of crystallisation 246 12.3 Crystal Size and Shape 246 12.3.1 Crystal size 246 12.3.2 Crystal form 247 12.3.3 Crystal anisometry 249 12.3.4 Shape factor (/ s ) 250 12.4 Kinetics at Early Stages 254 12.4.1 Degree of crystallisation (a t ) vs. time (t) plots.. 254 12.4.2 Crystal length vs. time plots 254 12.4.3 Crystal width 254 12.5 Nucleation 254 12.5.1 Variation with rate of cooling (RT) 255 12.5.2 Variation with initial crystallisation temperature (To) 255 12.5.3 Variation with metal cation 255 12.6 Growth Kinetics 255 12.6.1 Variation with initial crystallisation temperature (Го) 255 12.6.2 Variation with rate of cooling (R T ) 258 12.7 Mechanism of Growth 259 12.8 Rate Constants (for Diffusion-controlled Growth) 259 12.9 Rate Constants and Diffusion Coefficients 261 12.9.1 Growth along the major axes 262 12.9.2 Growth along the minor axes 263 12.10 Non-ideal Diffusion-controlled Crystal Growth 263 12.11 Crystal Anisometry 264 12.12 Final Crystal Size 264 References 264 13 Crystallisation by Differential Thermal Analysis 265 13.1 Introduction 265 13.2 Crystallisation 265 13.3 Development of Supersaturation 267 13.3.1 Overall excess solute concentrations 267 13.3.2 Residual excess solute concentrations 269 13.3.3 Total crystal weights 270

13.3.4 Critical time (t), critical temperature (Г) and critical excess solute concentration (AC) 270 13.4 Overall Kinetics 271 13.4.1 Degree of crystallisation (a t ) 271 13.4.2 Crystal length 276 13.5 Nucleation 283 13.5.1 Variation of induction period with rate of cooling (R T ) 288 13.5.2 Variation of induction period with initial crystallisation temperature (Г 0 ) 290 13.5.3 Variation of induction period with metal cation 290 13.6 Growth after the Induction Period 290 13.6.1 Initial growth rates 290 13.6.2 Average crystal length (l T ) 291 13.6.3 /r vs. г relationships 291 13.6.4 The final growth 297 13.7 Mechanism of Growth 299 13.8 Rate Constants 299 References 302 Kinetic Parameters and Thermodynamic Properties in Crystal Growth 303 14.1 Introduction 303 14.2 A Mathematical Model 304 14.3 The Proximity of a Diffusing Particle and its Host Immediately before a Successful Diffusion Encounter... 306 14.4 Numerical Calculations of Kinetic Parameters and Thermodynamic Properties 306 14.5 Crystallisation in Alumina Crucibles 307 14.5.1 Activation energy 307 14.5.2 The pre-exponential factor 308 14.5.3 Activation enthalpy 312 14.5.4 Free energy of activation 312 14.5.5 Activation entropy 312 14.6 Crystallisation in Platinum Crucibles 312 14.6.1 The pre-exponential factor 312 14.6.2 Activation energy 313 14.6.3 Activation enthalpy 317 14.6.4 Free energy of activation 317 14.6.5 Activation entropy 317 14.7 Proximity (du) between the Reacting Partners 317 References 318