UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Growth of lithium triborate crystals Parfeniuk, Christopher Luke 1994

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
ubc_1994-894474.pdf [ 8.47MB ]
Metadata
JSON: 1.0078456.json
JSON-LD: 1.0078456+ld.json
RDF/XML (Pretty): 1.0078456.xml
RDF/JSON: 1.0078456+rdf.json
Turtle: 1.0078456+rdf-turtle.txt
N-Triples: 1.0078456+rdf-ntriples.txt
Original Record: 1.0078456 +original-record.json
Full Text
1.0078456.txt
Citation
1.0078456.ris

Full Text

GROWTH OF LITHIUM TRIBORATE CRYSTALS By Christopher Luke Parfeniuk B. A. Sc., The University of British Columbia, 1988 M. A. Sc., The University of British Columbia, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILosOPHY in THE FACULTY OF GRADUATE STUDIES METALS AND MATERIALS ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1994 © Christopher Luke Parfeniuk, 1994 In presenting this thesis in partial fulfilment of the requirements for an advanced degree a.t the University of British Columbia. I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Metals and Materials Engineering The University of British Columbia 2075 Weshrook Place Vancouver, Canada V6T 1W5 Date: Abstract Lithium Triborate (LiB3O5)is a nonlinear optical crystal used to produce short wave length radiation from a longer wavelength source. Large lithium triborate crystals of good quality are difficult to grow. The present investigation was undertaken to examine the parameters influencing the growth process, and the growth process itself. for crystals pulled from an LiB3O5melt containing the flux MOO3. The pseudo phase diagram of the LiB3O5 — MoO3 system was established and the eutectic concentration shown to be 61.5 weight percent MoO3. The viscosity of the LiB3O5 system was measured as a function of temperature and MoO3 concentration. It was shown that the viscosity decreases with increasing MOO3 content. As the crystal grows MoO3 is rejected by the crysta.l at the interface. A major factor in growing larger crystals is the movement of the rejected flux away from the interface, which depends on the fluid flow in the melt. The fluid flow in turn is dependent on buoyancy forces due to temperature gradients, as well as crystal and crucible rotations. Calculations were carried out using a mathematical model for heat and fluid flow in the melt to establish the temperature distribution, fluid flow velocity and flow direction in the melt as a function of crystal and crucible rotation. Temperature measurements were then made in the melt in a crystal grower with a simulated crystal over a range of crucible rotation rotation rates, and the results compared to the mathematical model predictions. The boundary conditions used in the model were determined from tempera ture measurements in the melt. Comparing the calculated radia.l and axial temperature gradients iii the melt with the measured values showed good agreement between the calculated and measured temperatures. U The flow patterns in the melt predicted by the model were also compared to the observed flow in a physical, model using glycerine as the melt and ink as a tracer, for the same size crucible and crystal used in crystal growth. The observed flow pattern was consistent with the model predictions. The results of both the mathematical and physical models clearly showed that most of the mixing in the liquid is associated with the crucible rotation and very little from buoyancy forces. From these results it was concluded that maximum crucible rotation should be used during growth to move the concentrated MoO3 away from the advancing int.erface as rapidly as possible. Maximum MoO3 concentrations should also be used, consistent with other constraints, since the viscosity of the liquid decrea.ses with increasing MoO3 concentration. It was shown that increasing the size of the crucible increased the flow velocity in the melt. As a result larger crucibles were used in the crystal growth experiments. The length of good quality crystal which can be grown, is limited by the formation of inclusions in the crystal at the interfa.ce. The inclusions are shown to he primarily MoO3 and are considered to form when the concentration of the MoO3 reaches the eutectic concentration of 61.5 wt% at the interface. Calculations of the diffusion of MOO3 through the boundary layer away from the advancing interface, show that growth must he slow with strong liquid mixing below the interface, to produce a crystal 1 cm in length. A series of crystals of LBO were growii in a commercial crystal grower selecting the MOO3 concentration, crystal and crucible rotations, and the pulling rate from the opti mum values of the growth parameters given by the model predictions. Using these growth parameters, larger aild better quality crystals were produced. Facets were observed on the crystal surfaces for the [001] growth direction which resulted in stagnant areas on the interface and the earlier appearance of MOO3 inclusions. It was also observed that the crystal cracked readily under therma,l stresses during cooling. To prevent cracking. crystals have been cooled slowly after growth in a uniform thermal gradient. UI Table of Contents Abstract ii List of Figures ix List of Tables xxiv Acknowledgement xxvi 1 Introduction 1 2 Literature Review 4 2.1 Growth of Borate Crystals . . 4 2.1.1 LBO Crystal Growth 4 2.1.2 Barium Metahorate Crystal Growth 1 2.2 Physical Properties of LBO 11 2.3 Growth Defects 14 2.3.1 Flux Inclusions/Interface Breakdown 1.5 2.3.2 Voids 17 2.4 Fluid Flow 19 2.4.1 General Concepts 19 2.4.2 Standard Growth Practices 20 2.4.3 Accelerated Crucible Rotation 21 2.5 Mass Transfer . 27 iv 3 Objectives 33 4 Experimental 35 4.1 Growth Process 35 4.1.1 Growth Furnace 3.5 4.1.2 LBO Seed 36 4.1.3 Growth Procedure 41 4.2 Temperature Measurements in the Melt 47 4.2.1 Initial Temperature Measurements with No Crucible Rotation 48 4.3 Temperature Measurements With and Without Crucible with Rotation 48 4.4 Physical Model of The Crystal Growth Process .55 4.5 Physical Properties 5.5 4.5.1 Chemical Analysis 55 4.5.2 LBO/Mo03Phase Diagram .57 4.5.3 Viscosity 58 4.6 Crystal Quality 60 5 Experimental Results 61 5.1 Phase Diagram for the LBO/Mo03System 61 5.2 Viscosity 6.5 5.3 Preliminary LBO Crystal Growth Runs . 66 6 Physical Model of the LBO Crystal Growth Process 76 6.1 Observed Fluid Flow Patterns 77 6.2 Physical Explanation of Fluid Flow Patterns 79 7 Temperature Measurements 90 7.1 Initial Temperature Measurements with No Crucible Rotation 90 V 7.2 Temperature Measurements With and \Vithout Crucible Rotation 7.2.1 Boundary Temperature Results 7.2.2 Melt Temperature Results 7.3 Thermal Gradients in the Crystal During Cooling 8 Mathematical Model For LBO Crystal Growth 8.1 Scope of Model and Assumptions 8.2 Idealized Domain and Description of Calculations 8.3 Steady State Axisymmetric Fluid Flow Model 8.3.1 Equations of Fluid Flow - Lagrangian Coordinates 8.3.2 Equations of Fluid Flow - Euler Coordinates 8.3.3 Temperature Boundary Conditions 8.3.4 Velocity Boundary Conditions 8.3.5 Solution Procedure 9 Sensitivity Analysis of the Fluid Flow Model 9.1 Natural Convection 9.2 Forced Convection 9.2.1 Crystal Rotation 9.2.2 Crucible Rotation 9.3 Mesh Size 9.4 Fluid Viscosity 9.5 Conductivity 10 Modeling Results 170 10.1 Crystal Rotation 174 10.2 Crucible Rotation 175 91 91 95 114 120 120 123 124 124 126 128 129 131 132 132 136 136 142 150 150 161 vi 10.3 Comparison of Crystal to Crucible Rotation 179 10.4 Comparison of the Flow Fields in Small and Large Crucibles 188 10.5 Iso and Counter rotation of the Crystal and Crucible 188 11 Comparison of Temperature Measurements with Model Results 202 11.1 Small Crucible (6.6 cm diameter) Results 202 11.1.1 Results Assuming no Thermocouple/Melt Interaction (Small Cru cible) 207 11.1.2 Results assuming Thermocouple/Melt Interaction (Small Crucible) 209 11.2 Large Crucible (8.8 cm diameter) Results 214 11.2.1 Results Assuming no Thermocouple/Melt Interaction (Large Cru cible) 214 11.2.2 Results assuming Thermocouple/Melt Interaction (Large Crucible) 220 11.3 Summary of the Temperature Comparisons 224 12 Mass Transfer Calculations 228 12.1 Procedure for Estimating the Equivalent. Crystal Rotation Rate 229 12.2 Mass Transfer Behavior of MoO3 below the Crystal 233 12.3 Maximum Growth Rates and Growth Times for LBO 236 13 Application of Process Engineering Principles to Crystal Growth 242 13.1 LBO 17 243 13.2 LBO 18 . 245 13.3 LBO 19 246 13.4 LBO 20 . 247 13.5 LBO 21 . 248 13.6 LBO 22 and 23 . . 249 vii 13.7 LBO 24 250 13.8 LBO 25 251 14 Summary and Conclusions 273 15 Recommendations for Fiture Work 280 Bibliography 282 A Estimation of the Thermal Conductivity and the Gas Temperature 286 A. 1 High Temperature Thermal Couductivity Evaluation 287 A.2 Ambient Gas Temperature Approximation 288 Viii List of Figures 2.1 Phase Diagram of the Li20-B3system [7] 7 2.2 Derived phase diagram of the LBO - MoO3 system. The composition between C1 and C2 is the region where the direct crystal growth of LBO is possible 9 2.3 TCA analysis showing the effects of water vapour on the stability of LBO under dry and wet nitrogen [61 9 2.4 Top view of BBO—Na20melt showing radial convective cell boundaries and central cold spot [13] 11 2.5 Viscosity versus l/T for molten B2O3 [4] 12 2.6 Habit shape of grown LBO crystal [5] 14 2.7 Transverse dark field view through a BBO crystal grown at a very high growth rate, orientation [001] parallel to growth direction. Many scatter ing centres are observed throughout, and concentra.ted in the core region beneath the seed [13] 16 2.8 BBO crystal. Breakdown of growing interface breakdown at A-A [13] . . 16 2.9 An entrapment mechanism of gas—bubbles in crystals taking account of fluid flow modes associated with crystal rotation [25] 2.10 Incorporation of gas—bubbles in Pb5Ce3O11 as a. function of crystal rota tion and rate audi crystal diameter [25] 19 2.11 Theoretical Taylor-Proudrnan cell shapes for counter rotation [28] . . . 22 2.12 The direction of fluid motion predicted in the Taylor—Proudma.n cells [28] 23 ix Fluid motion due to natural convection [29] . 24 Predicted flow with counter rotation large enough that forced convection dominates [32] . Flow shown on right half and temperature shown on left half 25 2.15 Predicted flow with counter rotation small enough that natural convection dominates [32] . Flow shown on right half and temperature shown on left half 2.16 General rotational fluid flow (shearing) due to ACRT [43] . (a) Top view of circular tube filled with two distinguishable fluids. Tube and contents are in uniform rotation. (b) Final shape of fluid after tube and contents have come to rest. Spiral shearing distortion is evident 2.17 General fluid flow in the axial and ra.dia.l direction due to ACRT [43] . 2.18 Model prediction of fluid flow due to ACRT [41] . Crucible rotation 10 to 30 rpm. Crystal rotation 40 to SO rpm. Time period of acceleration 15 seconds 2.19 Assumptions used in Burton, Prim and Slichter [45] calculation of the concentration in the momentum boundary layer 4.20 LBO crystal growth furnace, A and C power and control box 4.21 Schematic of LBO growth chamber 4.22 Crucible rotation device 4.23 Seed rotation device 4.24 Platinum paddle for mixing the melt 4.25 Cleavage plane of an LBO Crystal 4.26 Goniometer used for orienting LBO crystals 2.13 2.14 26 28 29 30 32 are the crystal puller, and B is the 37383939404142 x 4.27 LBO Seed . 43 4.28 LBO Seed attached to platinum seed holder 44 4.29 Appearance of LBO melt surface after the seed ha.s been dipped 46 4.30 Positions of thermocouples and simulated crystal for melt temperature measurements 49 4.31 Platinum cap used to simulate the crystal 50 4.32 Thermocouple probe holder. A - guide track for radial movement of TC probe. B - rack and pinion gear for axial movement. C - thermocouple holder. D - micrometer and dial for positioning of the thermocouples. . 52 4.33 Apparatus for moving thermocouples attached to the crystal puller. . . 53 4.34 Plexiglass crucible and crystal used for the physical model 56 4.35 Platinum paddle for measuring viscosity of LBO/Mo03 59 5.36 Temperature difference versus temperature for 45 Wt% MoO3 (sample 1) determined by DTA 63 5.37 Phase Diagram of the LBO — MoO3 system 64 5.38 Dependence of viscosity on temperature and MOO3 concentration 68 5.39 Variation of viscosity with MOO3 at the temperatures indicated 69 5.40 Liquidus temperature and viscosity as a function of MoO3 70 5.41 Surface of an LBO crystal with inclusions. Magnified 30 times 73 5.42 Normal molybdenum inclusion in a.n LBO crystal magnified 400 times. (a) Backscatter image. (b) WDS molybdenum dot map 74 5.43 Molybdenum line inclusion in LBO crystal magnified 2,200 times. (a) Backscatter image. (b) WDS molybdenum line scan 75 6.44 Initial dye tracer pattern in the glycerine. a.) Top view. b) Side view. Crucible rotated at 45 rpm 82 xi 6.45 Dye tracer pattern fri the glycerine when the blue and red tracers reach the centre of the fluid. a) Top view. h) Side view. Crucible rotated at 45 rpm 83 6.46 Dye tracer pattern in the glycerine when the dye reaches the bottom of the crucible. a) Side view. b) View under the crystal. Crucible rotated at 45 rpm 84 6.47 Dye tracer pattern near the bottom of the crucible. a) Side view. b) View under the crystal showing the red and blue die moving up the side walls of the crucible. Crucible rotated at 45 rpm 85 6.48 Dye tracer pattern at different crucible rotation rates. a) Crucible rotation rate of 45 rpm. b) Transition flow for a crucible rotation rate between 45 and 78 rpm. c) Crucible rotation rate of 78 rpm 86 6.49 Top view of the dye tracer patterns at different crucible rotation rates. a) Crucible rotation rate of 45 rpm. h) Crucible rotation rate of 78 rpm. . 87 6.50 Interface curvature due to solid body rotation 88 6.51 Crucible with portion of the upper surface constrained to zero 89 7.52 Temperature variation with axial position in a 55 Wt% MoO3 solution. The three radial locations are r 0 mm, r 16 mm, and r = 32 mm. There is no crucible rotation during the temperature measurements. Crucible diameter is 6.6 cm 92 7.53 Temperature boundary conditions used in the fluid flow model for the sen sitivity analysis and the examination of the operating parameters. Crucible diameter is 6.6 cm 93 7.54 Melt temperature 0.2 cm from the crucible wall with and without the simulated crystal. Crucible diameter is 6.6 cm 96 xii 7.55 Melt temperature 0.47 from the crucible bottom with and without the simulated crystal. Crucible diameter is 6.6 cm 97 7.56 Temperature boundary conditions used in the mathematical model of melt with the simulated crystal. Crucible diameter is 6.6 cm 98 7.57 Melt temperature 0.2 cm from the crucible wall with and without the simulated crystal. Crucible diameter is 8.8 cm 99 7.58 Melt temperature 0.36 from the crucible bottom with and without the simulated crystal. Crucible diameter is 8.8 cm 100 7.59 Temperature boundary conditions used in the mathematical model of melt with the simulated crystal. Crucible diameter is 8.8 cm 101 7.60 Temperature distribution at r = 1.0 cm for crucible rotations of 0, 15, 20, 25 and 30 rpm. Crucible diameter is 6.6 cm. Simulated crystal present. 103 7.61 Temperature distribution at r = 1.5 cm for crucible rotations of 0, 15, 20, 25 and 30 rpm. Crucible diameter is 6.6 cm. Simulated crystal present. 104 7.62 Temperature distribution at r 2.6 cm for crucible rotations of 0, 15, 20, 25 and 30 rpm. Crucible diameter is 6.6 cm. Simulated crystal present. 105 7.63 Temperature distribution at. r = 3.1 cm for crucible rotations of 0, 15, 20, 25 and 30 rpm. Crucible diameter is 6.6 cm. Simulated crystal present 106 7.64 Change in liquid isotherms with crucible rotation. (a) No crucible rotation. (h) Large crucible rotation 107 7.65 Temperature distribution at r = 0.4 cm for crucible rotations of 0, 10, 20 and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present. . . 109 7.66 Temperature distribution at r = 0.9 cm for crucible rotations of 0. 10, 20 and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present 110 7.67 Temperature distribution a.t r 2.8 cm for crucible rotations of 0. 10, 20 and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present. . . 111 xlii 7.68 Temperature distribution at r = 3.3 cm for crucible rotations of 0, 10, 20 and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present. . . . 112 7.69 Temperature distribution at r 3.8 cm for crucible rotations of 0, 10, 20 and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present. . . . 113 7.70 Alumina aggregate used for determining the thermal gradients in the furnace.116 7.71 The axial positions of the alumina model crystal used when measuring the thermal gradients 117 7.72 Temperature gradients measured for the different furnace configurations as indicated 118 8.73 Schematic representation of domain examined with the model 125 9.74 Temperature boundary conditions used in the sensitivity analysis 133 9.75 Vector plot of fluid velocity due to natural convection 137 9.76 Temperature contours tha.t occur in the LBO/Mo03melt 138 9.77 Axial velocity due to natural convection 139 9.78 Radial velocity due to natural convection 140 9.79 Tangential crystal surface velocity due to natural convection 141 9.80 Vector plot of fluid velocity due to crysta.l rotation 143 9.81 Rotational velocity plot of the LBO/Mo03melt with crystal rotation. 144 9.82 Temperature cont.ours that occur in the LBO/Mo03melt with crystal ro tation 145 9.83 Axia.1 velocity due to crystal rotation 146 9.84 Radial velocity due to crystal rotation 147 9.85 Tangential crystal surface velocity due to crystal rotation 148 9.86 Vector plot of fluid velocity due to crucible rotation 151 9.87 Rotational velocity plot of the LBO/Mo03melt with crucible rotation. 152 xiv 9.88 Temperature contours that occur in the LBO/Mo03 melt with crucible rotation 153 9.89 Axial velocity due to crucible rota.tion 154 9.90 Radial velocity due to crucible rotation 155 9.91 Tangential crystal surface velocity due to crucible rotation 156 9.92 Two mesh densities used to examine the models sensitivity. (a) Regular mesh density, approximately 1795 nodes. (b) Coarse mesh density, approx imately 585 nodes 157 9.93 Axial velocity for different mesh densities 158 9.94 Radial velocity for different mesh densities 159 9.95 Tangential crystal surface velocity for different mesh densities 160 9.96 Axial velocity for different viscosities 162 9.97 Radial velocity for different viscosities 163 9.98 Tangential crystal surface velocity for different viscosities 164 9.99 Axial temperature profiles for different conductivities 166 9.100 Axial velocity for different conductivities 167 9.101 Radial velocity for different conductivities 168 9.102 Tangentia.l crystal surface velocity for different conductivities 169 10.103 Small crucible temperature boundary conditions used in the results analysis. 172 10.104 Large crucible temperature boundary conditions used in the results analysis.173 10.105 Axial velocities a.t 0.5 of the fluid height. 40.9 Wt% MoO3 present in the fluid. Crystal rotated a.t 0, 10, and 20 rpm. Crucible is stationary. (a) Shape of the solid/liquid interface. (b) Axial Velocities 176 xv 10.106 Radial velocities at 0.5 of the crucible radius. 40.9 Wt% MoO3 present in the fluid. Crystal rotated at 0, 10 and 20 rpm. Crucible is stationary. (a) Shape of the solid/liquid interface. (b) Radial velocities 177 10.107 Velocities tangential to the crystal surface 0.5 of the crystal radius. 40.9 Wt% MoO3 present in the fluid. Crystal rota.ted at 0, 10 and 20 rpm. Cru cible is stationary. (a) Shape of the solid/liquid interface. (h) Tangential velocities 178 10.108 Axial velocities at 0.5 of the fluid height. 40.9 Wt% MOO3 present in the fluid. Crucible rotated at 0, 20, 40, and 60 rpm. Crystal is stationary. (a) Shape of the solid/liquid interface. (b) Axial velocities 180 10.109 Radial velocities at 0.5 of the crucible radius. 40.9 Wt% MoO3 present in the fluid. Crucible rotated at 0, 20, 40, and 60 rpm. Crystal is stationary. (a) Shape of the solid/liquid interface. (b) Radial velocities 181 10.110 Velocities tangential to the crystal surface 0.5 of the crystal radius. 40.9 Wt% MoO3 present Crucible rota.ted at 0, 20, 40, and 60 rpm. Crystal is stationary. (a) Shape of the solid/liquid interface. (b) Tangential velocities. 182 10.111 The magnitude of the axial velocities along a horizontal line at 0.5 of the fluid height. The two conditions examined are crucible rotated at 20 rpm with a stationary crystal and a stationary crystal with a crucible rotating at 20 rpm. (a’) Shape of the solid/liquid interface. (h) Axial velocities. . . 184 10.112 The magnitude of the radial velocities along a vertical line at 0.5 of the crucible radius. The two conditions examined are crucible rotated at 20 rpm with a stationary crystal and a stationary crystal with a crucible rotating at 20 rpm. (a) Shape of the solid/liquid interface. (b) Radial velocities 185 xvi 10.113 The magnitude of the velocities tangential to the crystal surface at 0.5 of the crystal radius. The two conditions examined are crucible rotated at 20 rpm with a stationary crystal and a. stationary crystal with a crucible rotating at 20 rpm. (a) Shape of the solid/liquid interface. (h) Tangential velocities 186 10.114 Magnitude of the tangential velocity 0.5 cm from the liquid/crystal inter face for different crystal and crucible rotation rates. The calculations are for a rotating crucible with a. stationary crystal and a stationary crucible and rotating crystal 187 10.115 Axial velocities at 0.5 of the fluid height. Large and small crucible shown. 40.9 Wt% MOO3 present in the fluid. Crucible rotated at 60 rpm. Crystal is stationary. (a) Shape of the solid/liquid interface. (b) Axial velocities. 189 10.116 Radial velocities at 0.5 of the crucible radius. Large and small crucible shown. 40.9 Wt% MoO3 present in the fluid. Crucible rotated at 60 rpm. Crystal is stationary. (a) Shape of the solid/liquid interface. (b) Radial velocities 190 10.117 Velocities ta.ngentia.l to the crystal surface at 0.5 of the crystal radius. 40.9 Wt% MoO3 present. Large and small crucible shown. Crucible rotated at 60 rpm. Crystal is stationary. (a) Shape of the solid/liquid interface. (b) Tangentia.l velocities 191 10.118 Axial velocities a.t 0.5 of the fluid height.. 40.9 Wt% MOO3 present in the fluid. Crysta.l rota.ted at ± 10 rpm and crucible rotated at 20 rpm. (a) Shape of the solid/liquid interface. (b) Axial velocities 193 10.119 Radial velocities at 0.5 of the crucible radius. 40.9 Wt% MoO3 present in the fluid. Crystal rotated a.t ± 10 rpm and crucible rotated at 20 rpm. (a) Shape of the solid/liquid interface. (b) Radial velocities 194 xvii 10.120 Velocities tangential to the crystal surface at 0.5 of the crystal radius. 40.9 Wt% MoO3 present in the fluid. Crystal rotated at + 10 rpm and crucible rotated at 20 rpm. (a) Shape of the solid/liquid interface. (b) Tangential velocities 197 10.121 Fluid velocity tangential to the solid/liquid interface as a. function of crystal rotation rate. The velocity is 1 mm from the crystal/melt interface and at 0.5 of the crystal radius. 40.9 Wt% MoO3 present in the fluid. The crucible is rotated at 20 rpm 198 10.122 Vector plot of fluid velocity at a. crucible rotation rate of 20 rpm and a crystal rotation rate of 0 rpm. Point B 199 10.123 Vector plot of fluid velocity at a crucible rotation rate of 20 rpm and a crystal rotation rate of —23.5 rpm. Near point C 200 10.124 Vector plot of fluid velocity at a crucible rotation rate of 20 rpm and a crystal rotation rate of —35 rpm. Point D 201 11.125 Temperature boundary conditions used for t.he small crucible model. . . 203 11.126 Temperature boundary conditions used for the large crucible model. . . 204 11.127 Theta velocity boundary conditions used in the model 205 11.128 Theta velocity boundary conditions used in the model to account for the thermocouples in the melt 206 11.129 Experimental and calculated temperatures a.s a function of axial height. Small (6.6 cm diameter) crucible. Crucible rotation 0 rpm 208 11.130 Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Cru cible rotation = 1.5 rpm. Standard velocity boundary conditions are used (Figure 11.127 ) 210 xviii 11.131 Experimental and calculated temperatures as a function of axial height at the radial locations indicated, diameter) crucible. Crucible rotation = 20 rpm. Standard velocity boundary conditions are used (Figure 11.127 ). . 211 11.132 Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Cru cible rotation = 25 rpm. Standard velocity boundary conditions are used (Figure 11.127 ) 212 11.133 Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Cru cible rotation = 30 rpm. Standard velocity boundary conditions are used (Figure 11.127 ) 213 11.134 Experimenta.1 and calculated temperatures as a function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Cru cible rotation = 15 rpm. Modified velocity boundary conditions are used. (Figure 11.128 ) 215 11.135 Experimental and calculated temperatures as a functioii of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Cru cible rotation = 20 rpm. Modified velocity boundary conditions are used. (Figure 11.128 ) 216 11.136 Experimental and calculated temperatures as a function of axial height at the radia.l locations indicated. Small (6.6 cm diameter) crucible. Cru cible rotation = 25 rpm. Modified velocity boundary conditions are used. (Figure 11.128 ) 217 xix 11.137 Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Cru cible rotation = 30 rpm. IViodifled velocity boundary conditions are used. (Figure 11.128 ) 218. 11.138 Experimental and calculated temperatures as a function of axial location along various vertical lines. Large (8.8 cm diameter) crucible. Zero crucible rotation 219 11.139 Experimental and calculated temperatures as a function of axial height at the radial locations indicated. La.rge (8.8 cm diameter) crucible. Cru cible rotation = 10 rpm. Standard velocity boundary conditions are used (Figure 11.127 ) 221 11.140 Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Large (8.8 cm diameter) crucible. Cru cible rotation = 20 rpm. Standard velocity boundary conditions are used (Figure 11.127 ) 222 11.141 Experimental and calculated temperatures as a. function of axial height at the radial locations indicated. Large (8.8 cm diameter) crucible. Cru cible rotation = 30 rpm. Standard velocity boundary conditions are used (Figure 11.127 ) 223 11.142 Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Large (8.8 cm diameter) crucible. Cru cible rotation = 10 rpm. Modified velocity boundary conditions are used. (Figure 11.128 ) 223 xx 11.143 Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Large (8.8 cm diameter) crucible. Cru cible rotation = 20 rpm. Modified velocity boundary conditions are used. (Figure 11.12S ) 226 11.144 Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Large (8.8 cm diameter) crucible. Cru cible rotation = 30 rpm. Modified velocity boundary conditions are used. (Figure 11.128 ) 227 12.145 determination of the radius used in the analytical solution. The radius, ra is equivalent for both the finite element analysis and the analytical analysis.231 12.146 Calculated and analytical velocity values. The calculated tangential veloc ity values are a.t at 0.5 of the crystal radius. The analytical solution (flow past a rotating disk) for the radial velocity is at a radial location that is equivalent to the surface length of the calculated solution 232 12.147 Cross section of crystal grown a.t a crucible rotation rate of 60 rpm. The radius of the crystal is approximately 2 cm and the height at the centre line is 0.85 cm 239 12.148 The concentration of MoO3 next to the growing interface as a function of time for the growth rates (.1) indicated for a crucible rotation of 60 rpm. . 240 13.149LB0 17 crystal frozen in the melt 252 13.l50Uncracked portions of LBO 17 crystal 253 13.151 Top view of LBO 18 crystal 254 13.152 Bottom view of LBO 18 crystal 255 13.l53Uncracked portions of LBO 18 crystal 256 13.154 Side view of LBO 19 crystal 257 xxi 13.155 Cross section view of LBO 19 crystal. The crosses are regions were sample orientation was determined 258 13.156 Area in LBO 19 were interface breakdown/eutectic growth started. Mag nified 20 times. (a) SEM photo. (1)) Map of molybdenum concentration. The bright regions correspond to a high molybdenum concentration. . . 259 13.157 View of an molybdenum inclusion magnified 200 times. (a) SEM photo. (b) Map of molybdenum concentration. The bright regions correspond to a high molybdenum concentration 260 13.158 Top view of LBO 20 crystal 262 13.159 Cross section view of LBO 20 crvsta.l 263 13.16OLBO 21 Crystal 264 13.161 Top view of LBO 23 crystal 26.5 13.162 Bottom view of LBO 23 crystal 266 13.163 Interface appearance for planes were MoO3 was stuck to the surface. (a) Photo of underside of crystal. The area. A is a region of MoO3 build up. (b) Schematic of surface along line B — B . . 267 13.164 Pieces of LBO 23 that were uncra.cked 268 13.16.5 Top view of LBO 24 crystal 269 13.166 Bottom view of LBO 24 crystal 270 13.167 Back lit view of LBO 24 Crystal 271 13.168 Schematic and photo of the MoO3 on the LBO 25 crystal 272 A.169 Temperature boundary conditions used to approximate the conductivity of the LBO/Mo03melt 289 A.170 Fluid speed (u + in the melt. The value of the conductivity used in the model is 0.1 \V/cm K 290 xxii A.171 Difference between the model and experimental temperature values as a function of the conductivity 291 A.172 Comparison with the experimental temperature data and the model results for the best ambient gas temperature values 293 xxiii List of Tables 2.1 Reports on LBO crystal growth. SOT = seed on temperature; CR = cooling rate; PR = pull rate; atm = atmosphere, CDR = post growth cooling rate, SROT = seed rotation rate S 2.2 Physical properties of Li20, B203,MoO3 and Mo203 13 5.3 MoO3 and LBO concentrations used in determining the phase diagram. 62 5.4 The solidus and liquidus temperatures of the MOO3 — LBO samples . . 65 5.5 MoO3 and LBO concentration of the samples used in the viscosity mea surements 66 5.6 Viscosity versus temperature for samples 7 to 10 67 5.7 Viscosity versus temperature for sample 11 71 7.8 MoO3 and LBO concentrations in the melt that were used for the temper ature measurements with and without crUcil)le rotation 94 7.9 Temperature gradients measured for the different furnace configurations as indicated 119 8.10 Assumptions used in fluid flow model 22 8.11 Values used in the nondimensional analysis of the LBO/Mo03system . 122 8.12 Non Dimensional Numbers for Gallium and LBO 23 9.13 Standard thermophysical properties used in the sensitivity analysis. . . 134 9.14 Parameters examined for sensitivity analysis 134 xxiv 10.15 Thermophysical properties used in the results analysis 171 10.16 Parameters examined for the mathematical model analysis 174 11.17 Thermophysical properties and rotation values used in the model for com parison with experimenta.1 temperature measurements 207 12.18 Numerical solutions for a rotating disk [46] 230 12.19 Thermophysical properties used in the Analytical solution for flow below a rotating disk 230 12.20 Values of the variables used in the determination of the diffusion coefficient.235 12.21 Diffusion coefficient of some liquids [57] 235 12.22 The concentration of MoO3 next to the growing interface as a function of time. Diffusion coefficient is 2.38 x 10—8 cm2/s. Growth rate (f) is 0.698 mm/day 241 13.23 Growth conditions used for the crystal growth experiments 251 13.24 Growth conditions used for the crysta.l growth experiments, continued. . . 261 A.25 Thermophysical properties used in the model to determine the conductivity of the melt and the ambient gas temperature 287 A.26 Difference between the model and experimental temperature values as a function of the conductivity 288 xxv Acknowledgement I would like to thank Dr Indira Samerasa.kera, Dr Fred Weinberg, Jeff Edel, Dr Kim Fjeldsted and Brian Lent for their interest, suggestions and help over the course of this project. The assistance of the professional and technical staff at Crystar and UBC in particular, Jeff Clavdon, Mark Wa.ddington, Tim Elwel, Don Freschi, Fran Steeds, Dave Webb, Ernie Minkowitz and Mary Mager is greatly appreciated. I especially thank my parents, Walter and Ruth Parfeniuk, for their love, encourage ment and understanding that they have given me over the years. I would also like to thank Tresca Batten for her friendship and for putting up with me and my work schedule during my thesis. The conversation and friendship of the ladies from the 6:30 and 9:00 fitness class is also much appreciated. Thanks are also extended to iv fellow graduate students, especially Bernardo Hernandez Morales, Cohn Edie, Dave Tripp, Ed Chong and Barry Wiskel for their help during this project. I would like to express my gratitude to the Natural Sciences and Engineer ing Research Council and the British Columbia Science Council for financial support and Johnson Matthey Electronics (Crysta.r Research) for materia.l support dun ng my studies. xxvi Chapter 1 Introduction Lithium triborate LiB3O5,refered to as LBO, is a recently developed nonlinear optical crystal [1]. The nonlinear optica.l behaviour of crystals is described by the electromag netic equations, as they relate to optics. The most important characteristic in the present case is the ability of the nonlinear optical crystal to generate higher optical harmonics. If a monochromatic light beam, from a laser source, is passed through a nonlinear optical crystal, higher harmonic light is generated. Specificaly if infrared light from a Nd:YAG laser is used as a source, ultra violet light will he generated in the crystal. In many appli cations the shorter shorter wavelength of ultra violet light has important advantages over visible and infrared light. The specific advantages of LBO over other nonlinear optical crystals are [2, 3]: large transparency range (170 nm to 2.6 urn), the largest effective second harmonic generation conversion coefficient, high surface damage threshold (2.5 GW/cm2 for a 0.1 nsec pulse at 1.064 tm), chemical stability and is non—hydroscopic. LBO crystals are used in medical and industrial Nd:YAC lasers, in high powered lasers for military applications, and in optical parametric amplifiers and oscillators. Other non linear optica.l crystals of the borate family include ,6—BaB2O4and 1KB506(OH)4. 2H0. Lithium triborate crystals are difficult to grow due to properties that are intrinsic to the crystal and the growth process. One difficulty results from the high viscosity of LBO. The material contains 87.5 weight percent B203 which has a viscosity of 630 poise at 727°C [4]. The high viscosity sharply reduces fluid flow a.nd mass transfer in the melt during crysta,l growth. A second difficulty is that LBO solidifies in an incongruent 1 Chapter 1. Introduction 2 manner. Unlike GaAs which solidifies congruently, LBO forms from a peritectic reaction (Li4B10O7 + Liquid —* LiB3O5 at 834°C). Although it is possible to produce LBO crystals by peritectic growth, the process is very slow and only small crystals can be grown. To overcome this difficulty additional quantities of B203 or the compound MoO3 is added to the LBO which enables LBO to be grown directly from the liquid without a peritectic reaction. This is normally termed flux or solution growth. In this study MoO3 was used as a flux. Adding MoO3 to LBO introduces new factors in the crystal growth process. The amount of MoO3 added must be established. The viscosity of the melt decreases with increasing MoO3 concentration which ca.n result in increased fluid flow in the melt. This is a significant factor since the MoO3 is rejected 1w the solid at the advancing solid/liquid interface and must move away from the interface for growth to continue. However, the higher the bulk concentration of MoO3 in the melt the higher the concentration build UJ) immediately ahead of the interface for a given rate of crystal growth. If the concentration of MoO3 at the solid/liquid interface reaches the eutectic then eutectic growth will occur which results in the formation of MoO3 rich phases in the solid. This reduces the optical quality of the crystal and will cause the crystal to crack as it cools since LBO has a different thermal expansion coefficient than the MoO3 phase. Crystal growth from solution, in other systems, has been examined previously. Most of the work has been experimental, giving empirical correlations between growth param eters and the ciuality of the grown crystal. LBO has been known to exist since 1926. Research into the solution growth of LBO wa.s initiated in 1987 [5] following the discov ery of its special nonlinear optical properties. Reports on the growth of LBO crystals to the present, do not include details of the growth process and do not relate the growth parameters to defects generated in the crysta.l during growth. LBO crystals are normally grown using the Top Seeded Solution Growth (Modified Chapter 1. Introduction 3 Czochralski) Process in which a rotating, oriented, seed crystal is dipped in a counter rotating bath at a temperature just above the L130—Mo0 liquidus. The furnace temper ature is then slowly cooled to allow the crystal to grow in the radial and axial directions. \Vhen the crystal has grown to a sufficient diameter, it is slowiy raised resulting in crystal growth in the axial direction. The control of thermal gradients in the system and the control of the mass transfer of the MoO3 at the solid/liquid interface, is critical to the growth of high quality crystals. The mass transfer of MoO3 is strongly dependent on the viscosity of the melt, the crystal and crucible rotations, and the diffusion rate of MoO3 in the melt. In the present investigation direct temperature measurements of the melt were made during simulated crystal growth. The viscosity and other physical parameters of the liquid were also measured as function of MoO3 concentrations. A mathematical model of the system is developed, using the finite element method, to characterize both the thermal field and the fluid flow in the system during crystal growth. A simple mass transfer model is used, in conjunction with the fluid flow model, to examine the concentration of MoO3 ahead of an advancing solid/liquid interface during growth. The models are employed to select the optimum MoO3 concentration, crystal and crucible rotation rates, temperature fields and growth rates to produce large diameter LBO crystals having minimal defects. Crystals of LBO were grown using a range of growth parameters, and the size of crystal, and visible crystal defects, related to the model predictions. Chapter 2 Literature Review 2.1 Growth of Borate Crystals This review will consider the growth of Lithium Trihorate (LBO) and Barium Metab orate (BBO) crystals. Barium Metaborate is reviewed to complement the very limited published information available for LBO. Both crystals are similar, having B203 as a major component, and both crystals are grown in the same manner. 2.1.1 LBO Crystal Growth The Lithium Trihorate phase was initially reported in 1926. The phase diagram of Li20/B3system was reported in 1958 [7] and is shown in Figure 2.1. The LiB3O5. LBO, phase in this complex system is shown by the arrow at 87.5 wt% B203. On cooling the melt at this borate concentration, there is a peritectic transformation at 834 ± 4°C. There is a eutectoid transformation at 595 ± 20°C; however, LBO was found to be stable with no eutectoid transformation occurring on cooling below 595°C [5]. The successful growth of small crystals of LBO was reported in 1978 [8] and 1980 19] using a solid sta.te reaction process in which a B203 glass was covered with LiF powder and reacted at 750°C for 10 hours. The LBO crystal structure was analyzed and found to be orthorombic, symmetry class rnm2 and having a. space group of P21. The strong non—linear optical properties of LBO, reported in 1987, resulted in the development of the solution growth of these crystals. 4 Chapter 2. Literature Review 5 Solution growth consists of adding fluxes to L130 which modifies the phase diagram and allows LBO to he grown directly as a solid from the liquid. Flux materials used are Mo03[2], LiF [5] or B203 as a self fluxing agent. The flux must have a high solubility in the melt over a large temperature range and no solubility in the solid LBO. A derived phase diagram for the LBO — MoO3 system is shown in Figure 2.2. It was constructed by extrapolating the Li20/B3phase diagram from 87.5 wt% B203 towards 100% MoO3. Crystals can he grown directly from the melt in the MoO3 concentration range between C1 and C2. Reports of the successful growth of LBO crystals using various fluxes and starting materials are listed in Table 2.1. No details of the growth process are given, nor specifics concerning the quality of the crystals produced. The post growth cooling rate is important as the thermal stresses during cooling readily cause the crystal to fracture. LBO crystals are grown by slow cooling and may he combined with pulling the crystals vertically from the melt. The melt is held in platinum crucibles, 50 mm in diameter and height [1, 2], platinum being required because of the corrosive nature of B203. The crucible is heated with nickel/chromium resistance heating elements in a vertical furnace system. The insulation consists of alumina based ceramics. The starting materials are made up of a combinations of LiOH, Li2B4O7LiBO2,B203,H3B0 or similar compounds which contain extra hydrogen, carbon or oxygen components, the final mixture containing 87.5 wt% B203. The amount of flux added can vary, near 55 wt% MoO3, or enough to produce a bulk composition of at least. 90.3 wt% for B203 [2]. The amount of LiF added as flux is not specified in tile relevant reports. The charge, including the LBO components a.nd flux, is heated to 950°C to ensure that all the components melt, and held for 5 hours, to allow the melt to homogenize and H20 and CO2 vapour to form [1, 2, 5]. Following homogenization, the melt is cooled. and at a suitable melt temperature. a. seed crysta.] is clipped into the melt. The crystal is grown by slowly cooling the furnace a.t. a fixed ra.te. In some cases, the seed is slowly Ghapter 2. Literature Review 6 pulled from the melt to allow for additional growth in the vertical direction. The seed orientation is generally [001] parallel to the growth direction. During growth the crystal is rotated as a constant rate. When growth is complete the crystal is lifted from the melt and slowly cooled in the furnace to room temperature. The crystals are sensitive to thermal strains and can fracture during cooling. Details of the growth procedure are dependent on the fluxing agent, as outlined below, for B203 self fluxing and MoO3. Crystal growth, with a B203 self fluxing system, is carried out near the liquidus temperature of 834°C for a. 90 wt% B203 melt concentration. The procedure is to cool the melt to 848°C, dip the seed into the melt, hold for 30 minutes, then rapidly cool to 833°C. The melt is then siowly cooled between 0.5 to 2°C/da.y during which time the crystal slowly grows in the melt. The seed is not pulled vertically and the crystal is not rotated during growth [2]. Growth is terminated when the crystal has reached the specified dimensions. Since B203 has a high viscosity, the self fluxing process results in a melt having a high viscosity. This markedly reduces fluid flow in the melt due to bouyancy forces and reduces mass transfer in the melt adjacent to the growing crystal interface. This results in a high level of growth defects in the crystal [5]. With an MOO3 flux, the homogenized melt is cooled to 673°C and held for 5 hours. The seed crystal is then dipped into the melt, held for 30 minutes, and the melt then cooled at 5°C/day as the crystal grows. During growth the seed is rotated at 30 rpm and pulled from the melt at a rate of 1 mm/day [5]. After crystal growth, the seed is separated from the melt and cooled to room temper ature at rates between 40 to 100°C/hour. Stress in the crystal during cooling can cause the crystal to fracture, due to anisotropy of the expansion coefficients of the crystal and the solidified flux on the crystal surface [5]. It was found that LEO decomposes at high temperatures when exposed to water vapour in an ambient atmosphere [6]. The decomposed material is Li3B7O12. A dry Chapter 2. Literature Review 7 950 900 -‘ 850 0 0 800 750 700 650 80 90 100 .—Li20 COMPOSITION (WT%) B203 Figure 2.1: Phase Diagram of the Li20-B3system [7]. nitrogen atmosphere was used to stop the decomposition. Figure 2.3 shows the difference in the weight loss of an LBO sample in a dry and wet nitrogen atmosphere. Crystal sizes grown were reported to be as large as 35 x 30 x 15 mm3 using MoO3 [5] and 30 x 30 x 15 mm3 using B203 [1]. The crystals were reported to be inclusion free, however no photographs of the LBO crystals were shown. 2.1.2 Barium Metaborate Crystal Growth The successful growth of Barium Metaborate crystals (BBO), fl—BaB2O4was reported in 1985 [11} using Top Seeded Solution Growth (TSSG). The crystals were grown with B203 flux, the melt containing 43wt% B203. This is appreciably lower than the B203 content in LBO growth ( 87.5 wt% ) which results in a much lower liquid viscosity in the BBO melt. Na20 was examined as an alternative flux to B203 and was shown to produce better quality crystals [12]. Na20 has a solubility range of 22 to 30 wt% in BBO in the temperature range of 755 to 925°C [13]. Chapter 2. Literature Review 8 Year Starting Materials Flux Growth Parameters Crystal Size 1989 [1] Li20,H3B0, ? wt% B203 SOT = 833°C 30 x 30 x 15 mm3 CR = 0.5°C/day PR = not given atm = not given CDR = 40°C/hour SROT = not given 1989 [2] Li2CO3,H3BO, 90 wt% B203 SOT = 848°C 18 x 20 x 6 mm3 CR = 0.5°C/day PR = not given atm = not given CDR = 40°C/hour SROT 0 rpm 1989 [2] Li2CO3,H3B0, 55 wt% MoO3 SOT = 673°C 20 x 35 x 9 mm3 CR = 5°C/day PR. = not given atm = not given CDR = 100°C/hour SROT = 30 rpm 1989 [10] Li2CO3.H3B0. 93 wt % B2O3. SOT = 778°C No values listed CR. = 1.3°C/day PR. = not given atm = air CDR = not given SROT = not given 1990 [5] LiCO3,Li20, > 90 wt % B203, SOT = 834°C Size: 35 x 30 x 15 mm3 L1OH, Li2B4O7 ? wt % LiF CR = 0.2-2°C/day LiBO2,B2O3, PR = 1mm/day 113B0,etc CDR = not given Table 2.1: Reports on LBO crystal growth. SOT = seed on temperature: CR = cooling rate; PR = pull rate; atm = atmosphere, CDR = post growth cooling rate, SROT’ = seed rotation rate. Chapter 2. Literature Review 9 °° 4__.LiO2BO3±Liquid E—Li2O5B203+ Liquid Liquid 800 c-) 700 LBO + Liquid Eutectic 600 LBO + MoO3 ci C2 MoO3 Concentration (Wt %) Figure 2.2: Derived phase diagram of the LBO - MoO3 system. The composition between C1 and C2 is the region where the direct crystal growth of LBO is possible. TG 8OOC 1.2O4::Thr Time (hi) Figure 2.3: TGA analysis showing the effects of water vapour on the stability of LBO under dry and wet nitrogen [6]. Chapter 2. Literature Review 10 The procedure for growing BBO crystals, and evaluation of their quality, is given by Feigelson et al. [13]. Two types of furnaces were used, a low gradient wire wound furnace and a high gradient SiC heated furnace. The platinum crucibles containing the melt were 55 mm in diameter and 55mm in height. The melt composition used was 8Oat% BBO—2Oat%Na20.The axial and radial gradients were approximately 50°C and 30°C for the high gradient furnace and 20°C and 10°C for the low gradient furnace. Temperature fluctuations due to thermal convection was measured to be +8°C. To grow a crystal, the charge was melted and then cooled to the temperature at which the seed crystal was dipped into the melt. The dipping temperature was established by immersing a platinum wire into the melt and cooling until a small amount of BBO solidified on the wire. The melt was maintained at this temperature for 12 hours. Next, the wire was removed and the melt temperature increased by several degrees after which the BBO seed was immersed into the liquid. During growth the seed was rotated at rates between 2 and 16 rpm. There was no crucible rotation. Fluid flow occurred in the melt, due to natural convection which could be detect.ed by the presence of radial convective boundaries at the top surface, Figure 2.4. Two seed crystal orientations were examined the c direction [001] and b direction [010] aligned parallel to the growth direction. The c orientation proved to he the better of the two, since the b direction crystals exhibited more cracking during cooling. The interface shape during growth was more concave towards the melt for the c orientation, because of the higher thermal conductivity of the crystal in this direction; one order of magnitude larger than in the b direction. Crystal quality improved at higher crystal rotation rates due to the increase in the fluid flow velocity at the interface. At the highest rotation rates the interface inverted from concave to the liquid to concave to the solid. During crystal growth, the cooling rate of the melt was maintained at 2°C/day and a pulling rate of 0.5 to 1.0 mm/day was used. After approximately 12 mm of crystal had Chapter 2. Literature Review 11 Top View ot Crucible Figure 2.4: Top view of BBO—Na20melt showing radial convective cell boundaries and central cold spot [13]. grown the planar interface breaks down, resulting in flux inclusions being incorporated into the crystal. Better crystal quality was obtained using the large gradient furnace. It wa.s also found that crystal quality was best at the outside region of the crystal and worst at the centre. These correspond to high flow and stagnant regions in the melt below the growing interface [13]. 2.2 Physical Properties of LBO An extensive literature search was carried out to obtain the thermo—physical properties of the components of LBO and MoO3, giving the values listed in Table 2.2 for LBO, Li20,B203,MoO3 and Mo203. The specific heat and density for Li20, B203 and MoO3 are well established [14, 15]. Experimental measurements of the viscosity of B203 [4] are available but no data was found for the other components of the LB 0/flux mixtures. The viscosity of B203, Figure 2.5, is very high, 280 poise at 833°C and 692 poise at 727°C. Chapter 2. Literature Review 12 800C 750C 700C 1uUu I I 900 ci 800 - - - 700 .600 500 -ri4 ‘I) 400 C.? cI) 300 I I I I I I I 8.8 9.0 9.2 9.4 9.6 9.8 10.0 10.2 10.4 1O,000IT[KJ Figure 2.5: Viscosity versus 1/T for molten B203 [4] The thermal conductivity of 11203 [16] and LBO [17] are low as would be expected for oxide materials. The expansion coefficient of LBO [17] and Mo203 [16] are low when compared to metals such as liquid bismuth which has a expansion coefficient of 1.26 x104 at 538°C. Since the LBO melt has a high viscosity and a small thermal expansion coefficient, bouyancy forces and thus fluid flow due to natural convection in the melt will be small. The LBO expansion coefficients are nonisotropic and the dimension change in the c direction will be opposite to that of the b and a directions [18]. This will increase the stress levels that are present in the crystal during cooling. Evaporation could be a problem during crystal growth due to the vapour pressure of MoO3; 1 atmosphere at 1151°C. The crystal structure of LBO is orthorombic of class mm2 and has a space group of Pna2i. The cell dimensions for LBO are a = 7.379, b = 8.447 and c = 5.140 A. Growth directions are generally parallel to the primary axes of a crystal which in this case could be a [100], b [010] or c [001]. We note, following the mm2 class designation, the Chapter 2. Literature Review 13 Property Material Temperature Polynomial (units) Specific Heat Li20 298K < T < 500K 69.58 + 17.857 x 103T (J/g) —19.041 x 105T2 B203 723K < T < 1400K 245.814— 145.511 x 103T —171.167 x 105T2+ 48.166 x 106T2 MoO3 298K < T < 1068K 75.186 + 32.635 x 103T —8.786 x 105T2 density Li20 - 2.013 (g/cm3) B203 - 1.812 LBO solid 2.474 MoO3 - 4.692 viscosity Li20 - - (poise) B203 - see figure 2.5 MoO3 - - conductivity Li20 - - (W/cm 1<) B203 800 K 0.01 LBO (a) f 298 K 0.039 LBO (b) f 298 K 0.031 MoO3 - - Expansion Li20 - - Coefficient B203 - - (K—’) LBO (c) f 50°C - 400°C 11.6 x 10 LBO (d) f[001] 30°C —88.0 x 106 LBO (e) ff010] 30°C 108.2 x 106 LBO (f) ff100] 30°C :33.6 x 106 MOO3 - - Mo203 395°C 5.35 x 1O Vapour Li20 - - Pressure B203 - - (atm) MoO3 1151°C 1 Table 2.2: Physical properties of Li20, B203.MoO3 and Mo203. LBO (a.) perpendicular to the c direction LBO (b) = parallel to the c direction. LBO (c) = LBO glass. LBO (d) = LBO crystalline material, [001] (lirection. LBO (e) = LBO crystalline material, [010] direction. LBO (f) = LBO crystalline material, [100] direction. Chapter 2. Literature Review 14 Figure 2.6: Habit shape of grown LBO crystal [5] (001) and (001) faces on a growing crystal will have different faceted surfaces. The other principle axes a and b will have the same facets on the opposing faces of the cell. A faceted LBO crystal is shown in Figure 2.6 [5]. In orienting a seed crystal parallel to the growth direction care must be taken to determine whether the (001) or (001) is at the growing interface since the growing behaviour could he different. 2.3 Growth Defects Growth defects found in single crystals include segregation, dislocations, flux entrapment, inclusions, cracks, and voids. In the case of LBO crystals the segregation coefficient of the flux is zero, therefore no solid solution forms. While dislocation density is important in semiconductor crystal growth it is not critical in optical crystals. However, defects which reduce the optical quality of a crystal, such as voids, cracks, flux entrapment and rnciusions are a matter of great importance. The amount of published information on defects in LBO is small [1, 2, 3, 5]. In the literature, similarities between LBO crystals z I x Chapter 2. Literature Review 15 and other oxide crystals are examined. This is particularlr relevant when the oxide crystals are of materials with high viscosities similar to B203 [20]. 2.3.1 Flux Inclusions/Interface Breakdown Flux inclusions in BBO crystals are commonly found at the centre of the crystal [13, 23]. Figure 2.7 shows an increase in inclusion density at the centre of a BBO crystal. These core inclusions correspond to the region were fluid flow due to crystal rotation is mini mal [13]. Core inclusions have been found in other oxide crystals such as Bi4Ge3O12 [21, 22]. It was concluded that the core inclusions formed as a result of rapid changes in the growth rate. The fast solidification produces a sudden increase in concentration of segregated materia,1 ahead of the interface. Inclusions will be generated when the flux concentration ahead of the interfa.ce increases more rapidly than the rate at which material diffuses away from the interface. Flux inclusions have also been found at the outer region of BBO crystals [23]. The inclusions at this a.rea are attributed to the large temperature gradients and temperature fluctuations that occur in these regions. Interface breakdown can occur (luring BBO crystal growth. as shown in Figure 2.8, after the crystal has grown 12 mm [13]. The interface breakdown begins at the centre of the growth interface and spreads to the edges a.s growth proceeds. This was correlated with the fluid flow under the crystal [13]. The interface breakdown initiates at stagnant regions, such as the region below the centre of the crystal. Flux solidified on the outside of the crystal is responsible for cracking in BBO crys tals [13]. Similarly, inclusions or solidlified flux at a surface of an LBO crystal may cause cracking due to the strong anisotropy of the flux and LBO crystals [5]. Chapter 2. Literature Review 16 Figure 2.7: Transverse dark field view through a BBO crystal grown at a very high growth rate, orientation [001] parallel to growth direction. Many scattering centres are observed throughout, and concentrated in the core region beneath the seed [13]. I —1 10mm A A Figure 2.8: BBO crystal. Breakdown of growing interface breakdown at A-A [13]. Chapter 2. Literature Review 17 2.3.2 Voids Void formation in crystals may be due to the entrapment of gas from the melt during solidification [28, 25, 231. The gas may be from byproducts of the charge material, such as C02, that are dissolved in the melt. The solubility of dissolved gases in the liquid is higher than the solid. Thus, as the material solidifies, dissolved gas will be rejected by the solid at the interface. Bubbles will nucleate when, at a given pressure, the dissolved gas volume reaches a critical value. The bubbles trapped in the momentum boundary layer ahead of the interface will likely be incorporated in the crystal. The dissolved gas boundary layer will be thicker at the centre of the crystal, which contains a stagnant liquid region, and thinner at the edges of the crystal. Figure 2.9 illustrates a possible mechanism for the entrapment of gas bubbles in the crystal. The rotation of the crystal will cause two events to occur in the liquid. First, mixing will increase in the liquid and reduce the thickness of the dissolved gas boundary layer below the crystal. Second, the pressure below the rotating crystal will increase propor tionally with the centrifugal acceleration [25]. The increase in pressure at the growing interface will delay or reduce bubble formation in the crystal. A correlation between rotation rate, crystal diameter and void formation during crystal growth has been re ported [25, 36]. Figure 2.10 shows the extent of the incorporation of gas bubbles in Ph5Ge3O11 as a function of crystal rotation rate and crystal diameter. At a fixed crystal diameter a void free crystal could be grown at high rotation rates. At low crystal rotation rates natural convection dominates fluid flow. Increasing the crystal rotation rate results in forced convection dominating the fluid flow and the interface inverting from concave to the liquid to concave to the solid! [25, 13]. Chapter 2. Literature Review 18 (a) large convex ______ gs-bubb(es gas-sürptus small convex: gas-bubbles gas—surpLus liquid (crack) Figure 2.9: An entrapment mechanism of gas—bubbles in crystals taking account of fluid flow modes associated with crystal rotation [25]. ._-(C) concave Chapter 2. Literature Review 19 60 - bubble-free 0 40 oo \ ‘20 bubble-in 0 10 20 Crystal diameter mm Figure 2.10: Incorporation of gas—hubbies in Pb5Ge3011 as a function of crystal rotation and rate and crystal diameter [25]. 2.4 Fluid Flow Fluid flow in top seeded solution crystal growth is complex due to the rotation of both the crystal and the crucible. Numerous fluid flow models have been developed to characterize Czochralski (Cz) fluid flow [28, 29, 30, 31, 32, 33, 35, 38, 39]. 2.4.1 General Concepts Fluid flow in a crystal growth system in which the crucible is rotated in one direction, and the crystal in the opposite direction was initially examined by Taylor and Proudman [28]. The solid body of rotating fluid is two dimensional with respect to coordinate axis rotating with the liquid. The crystal and crucible rotating in opposite directions creates a number of individual rotating fluid regions each separated from the other by a detached shear layer. A shear layer is where the fluid velocity changes from one solid body rotation to another. There is no mixing, with the exception of molecular diffusion, across a shear Chapter 2. Literature Review 20 layer. The detached shear layers are shown in Figure 2.11 and 2.12 [28, 27]. The crucible rotates a.t an angular velocity and the crystal rotates at an angular velocity of 1 The fluid adjacent to the crucible walls has solid body rotation with an angular velocity equal to crucible rotation rate. Directly below the crystal two individual cells, known as Taylor—Proudma.n cells, develop. The lower cell has fluid that rotates in the same direction as the crucible but slower (2). The upper cell rotates in the same direction as the crystal but again at a lower angular velocity (l). Figure 2.12 shows the axial and radial fluid motion that occur. The fluid at the crystal surface boundary layer and the crucible bottom boundary layer are forced outward due to the centrifugal force from the rotation. The upper cell has upward fluid flow to replace the fluid that is moved due to the centrifugal force from the crystal and the lower cell has some downward fluid flow to replace the fluid that is moving outward due to the centrifugal force of the crucible. In turn this fluid is replenished by the middle shear layer, the direction of its fluid motion is inward and up/down into the two cells. Natural convection due to buoyancy forces are also present during Cz crystal growth. The fluid flow for natural convection is shown in half the crucible in Figure 2.13 The temperature in the liquid is highest a.t. the bottom a.nd lowest at the top. The lower density liquid at the bottom rises up the wall arid cools, while the higher density liquid at the top moves down the centre of the crucible. 2.4.2 Standard Growth Practices Fluid flow during crystal growth results from both natural convection and the forced rotation predicted by the Taylor—Proudman analysis. Since the forced convection condi tion is wha.t was described with the Taylor—Proudman theorem we will begin with these results shown in Figure 2.14. Two cells form as theoretically predicted. The outside solid body rotation is almost nonexistent with the lower Taylor-Proudman cell and its Chapter 2. Literature Review 21 detached shear layer going to the crucible wall. The strong upward motion of the upper Taylor-Proudman cell cause the temperature isotherms to flatten below the crystal. The crystal interface will become concave to the solid as the amount of forced convection becomes higher [32]. If the crystal rotation rate is small, natural convection and/or crucible rotation dom inates (Figure 2.15) and the upper cell disappears. The resulting shape of the isotherms indicate that there is more radial beat flow and the shape of the interface changes to concave to the liquid. The driving force of many of the modeling papers has been to predict interface changes from concave to convex [32, 33, 34, 36, 37]. 2.4.3 Accelerated Crucible Rotation As an alternative to crystal and crucible counter rotation during growth, the crucible (and crystal) can be accelerated and decelerated alternately in clockwise and counter clockwise directions. The accelerated crucible rotation technique (A CRT) reduces the thickness of the stagnant boundary layer, thus allowing faster stable growth rates [42]. When a crucible is accelerated from rest, the liquid around the crucible wall will follow the change in crucible motion, whereas the liquid in the centre will remain at rest due to inertia. Thus shear rings in the liquid are produced by the difference in the outer and centre fluid velocities. The number of shear rings which develop increase with increasing acceleration. A top view of the shear rings produced is given in Figure 2.16. The increase in surface a.rea due to the shear rings results in an increase in diffusion. The resulting fluid flow in the radial and axial direction are shown in Figure 2.17. The rapid flow close to the crucible bottom is referred to as Ekma.n-layer flow [43]. The rapid suction of fluid close to the bottom is due to the pressure difference between the outside and inside of the crucible not being balanced by the centrifugal forces. High Ekrnan flow rates occur not only at the crucible bottom but also on the crysta.l growth face that is Chapter 2. Literature Review 22 / Crystal Upper Taylor - Proudman Cell Layer Upper Transition ______________ Intermediate Transition Layer Detached Stairnation Shear _— Surfaces Layer c5 Solid Body - Rotation Lower I Lower TransitionTaylor - Proudman Layer Cell Figure 2.11: Theoretical Taylor-Proudman cell shapes for counter rotation [28]. Chapter 2. Literature Review 23 + MELT I I Figure 2.12: The direction of fluid motion predicted in the Taylor—Proudman cells [28]. Chapter 2. Literature Review 24 Centre Line I —- i / /iui7 I ii II \ \ III ‘ ‘ ‘ ‘ \ ‘S — ‘ ‘ “ \ “ /J i \ \\ \ I \ \ \ I —\ \\ \ N _,/‘ 1 II N •— / --—-- I I Crucible Figure 2.13: Fluid motion due to natural convection [29]. Chapter 2. Literature Review 25 Crystal a\ooC Crucible Figure 2.14: Predicted flow with counter rotation large enough that forced convection dominates [32]. Flow shown on right half and temperature shown on left half. Chapter 2. Literature Review 26 Crystal o 1 0.4 —Q6 -0.8 0.6 -1(0 Crucible Figure 2.15: Predicted flow with counter rotation small enough that natural convection dominates [32]. Flow shown on right half and temperature shown on left half. Chapter 2. Literature Review 27 perpendicular to the rotation axis. Ekman-layer flow ceases as soon as uniform rotation is reached. A mathematical model has been used to examine the fluid flow that results from combined forced and free convection of accelerated crucible rotation Cz growth of metals and semiconductors [40, 41]. The best mixing of the fluid is reported to occur when the crystal and crucible are rotated/accelerated in the same direction. The upper Taylor-Proudman cell is only present during the acceleration of the crucible (Figure 2.1S). Thus the model predicts that the bulk fluid will mix best under these conditions. The variation of a solute boundary layer was not examined. The fluid properties of the model are not close to those of LBO. This is especially noticeable with the kinematic viscosity of 0.005 cm2/s. Using the density of LBO, 2.474 g/cm3, the model viscosity is calculated as 0.0123 poise. The viscosity of an LBO solution for growth is nearly three orders of magnitude higher. It is difficult to predict if the same type of mixing would occur in a higher viscosity liquid. 2.5 Mass Transfer The most important aspect of mixing in the melt is tha.t which occurs in the boundary layer under the crystal. In the past the difficulty of conducting comprehensive numerical integration of the Navier-Stokes equations and diffusion equations to determine the flow, the temperature, and solute distribution, has led to the development of boundary layer theory to provide simple models of key regions of the flow in Cz growth [44]. There has also been no real attempt to verify the boundary layer models or numerical models with measured solute distributions ahead of the crystal interface. Burton, Prim and Slichter [45] considered changes in the momentum boundary layer thickness and the resultant transport of solute, on the effective segregation of solute for steady state crystal growth in germanium. The domain was assumed to be an infinite rotating disc on Chapter 2. Literat tire Review 28 (b) Figure 2.16: General rotational fluid flow (shearing) due to ACRT [43]. (a) Top view of circular tube filled with two distinguishable fluids. Tube and contents are in uniform rotation. (b) Final shape of fluid after tube and contents have come to rest. Spiral shearing distortion is evident. (a) CD Chapter 2. Literature Review 30 :: ....- - ---—--- lk7L7ffJ?M1’ff / it;qII11c./ 1 ....ii1t1._II 1j Ii 44444 t = 3.0 s v = 3.3 cm/s max t 30.0 s v = 2.3cm/s max t = 15.0 S V = 2.0 cm/s - 4441 4444 44414 4414444 ::‘ t = 36.0 s Vmax = 1.4 cm/s Figure 2.18: Model prediction of fluid flow due to ACRT [41j. Crucible rotation 10 to 30 rpm. Crystal rotation 40 to 80 rpm. Time period of acceleration 15 seconds. Chapter 2. Literature Review 31 the surface of a semi-infinite fluid. The boundary conditions used in the mass transfer calculations are shown in Figure 2.19. It is assumed that the bulk concentration of the fluid is at the edge of the diffusion boundary layer. The diffusion boundary layer thickness is assumed to be constant over the radius of the rotating disk. This is very relevant when laminar flow and a thin diffusion layer are present [49]. The only velocity present in the solute boundary layer is from the growing interface. The calculation of solute diffusion reverts to a simple steady state one dimensional diffusion calculation with a term to account for the advancing interface. Chapter 2. Literature Review • — • — 32 Crystal, Growing at Velocity f Concentration I6 6 Momentum Boundary Layer NO MIXING Bulk Fluid COMPLEThI1G Figure 2.19: Assumptions used in Burton, Prim and Slichter [45] calculation of the concentration in the momentum boundary layer. Chapter 3 Objectives The objective of this investigation was to examine the crystal growth process using math ematical modeling, experimental measurements and crystal growth experiments in order to establish procedures for growing larger and higher quality LBO crystals. The math ematical modeling quantitatively (lescribes the parameter interactions in the complex growth system. Experimental measurements were conducted to establish the required physical parameters of the system and the boundary conditions required for the model. A physical model of the flow due to crucible rotation was used to visualize the flow pat terns in the melt. The crysta.l growth experiments were carried out to verify the model predictions. These experiments also demonstrated that better crystals can be produced by modifying the growth parameters in conformity with the model predictions. The principal tasks undertaken to accomplish the objectives were as follows: 1. The development of a mathematical model of heat transfer and fluid flow in the LBO crystal growth system. 2. Application of the Burton, Prim. and Slichter analysis to examine the mass transfer of MOO3 away from the solid/liquid interface. 3. Use of a physical model to visualize the type of flow that occurs due to crucible rotation. 4. Measurement of the temperature distribution in the melt and a comparison with model predictions. 33 Chapter 3. ObJectives 34 5. Temperature distributions for the mathematical models’ boundary conditions. 6. Prediction of the effect of crucible rotation rate on the temperature distribution in the melt. 7. The phase diagram of the LBO/Mo03system, as determined by differential thermal analysis. 8. The dependence of viscosity of the melt as a function of temperature and MoO3 content. 9. The type, size and distribution of defects in the grown crystals. Chapter 4 Experimental 4.1 Growth Process 4.1.1 Growth Furnace The system utilized in this investigation to grow LBO crystals is the modified NRC crystal puller shown in Figure 4.20. The crystals are grown in the chamber marked A, using the power and control system B. The upper chamber, C, is not significant for LBO growth. The growth chamber A is 26.3 cm in diameter and 44.5 cm high. A schematic diagram of the interior of the growth chamber is shown in Figure 4.21. The LBO/Mo03 charge is contained in a crucible which sits on a rotating pedestal, thermally insulated by a block of ceramic. The resistance heaters have Iron-chrome-aluminum elements. The insulation shown in Figure 4.21, consists of alurnina. A 6.35 cm diameter hole is located at the top of the chamber. The crucible is made of platinum. Two crucible sizes were used, a small crucible 66 mm in diameter and height, and a larger crucible 88 mm in diameter and height. The expansion coefficient of LBO/Mo03is much la.rger tha.n that of platinum causing severe crucible distortion upon cooling. To ensure that the crucible remains firmly attached to the insulation block during rotation, a cylinder of insulating material 5 mm in height is attached to the crucible and 3 pegs are attached to the cylinder which sit in holes in the insulating block that is attached to the rotating pedestal. The pegs keep the crucible from moving with respect to the block. The cylinder is attached to the crucible with 35 Chapter 4. Experimental 36 Cotronics Ultra—Temp 360 insulating tape and Cotronics 904 Ceramic Adhesive. The insulating tape at the side of the crucible must be the same for each consecutive crystal growth run. If not, the liquid temperature will change. The crystal growth furnace temperature is controlled with a Eurotherm 818 controller. This instrument is very accurate and is capable of maintaining ramp rates as low as 0.01 degrees per hour. The type K control thermocouple is located adjacent to the furnace wall. The crucible rotation, seed rotation and seed lift were all modified from the original NRC equipment. The crucible rotation, Figure 4.22. consists of a Bodine motor and a. pulley unit attached to the original rotation unit. A thermocouple goes up the center of the rotation shaft for measuring the temperature at the base of the crucible. The gearing of the pulleys allows for crucible rotations between 0 and 74 rpm. The seed is rotated using a Maxi-Pile gear motor a.nd planetary reduction gears to obtain a rotation of .5 rpm, Figure 4.23. A stepper motor and indexer/driver package, was used for slow pulling of the seed. The seed lift velocity is 1.66 mm/day to 1.49 crn/miri. A platinum mixing paddle, shown in Figure 4.24 is used to homogenize the liquid. a.fter the cha.rge has been melted. The seed rod is made of hastelloy. The bottom of the seed rod is attached to either the paddle or to the platinum seed holder. 4.1.2 LBO Seed LBO seeds were oriented such that the growth direction wa.s parallel to the [001] direction. The orientation was determined by using the back reflection La.ue method on the cleaved surface of the crystal, Figure 4.25. The unorient.ed LBO crystals was attached to a goniometer, Figure 4.26, using wax. Samples were positioned such that the surface was 3cm from film holder and were irradiated from a Cu target for 20 minutes at 30 k\7 and 20 mA. After irradiation, the X-Ray film wa.s developed and the orientation of the crystal Chapter 4. Experimental Ii ‘ $ , B , ,. j , 37 Figure 42O: LBO crystal growth furnace, A and C are the crystal puller, aid B is the power arid control box. Chapter 4. Experimental 38 Base TC “rucib1e Rotation Furnace TC Figure 4.21: Schematic of LBO growth chamber Chapter 4. Experimental 39 Figure 4.22: Crucible rota.tion device. Figure 4.23: Seed rotation device. Chapter 4. Experimental 40 I Figure 4.24: Platinum paddle for mixing the melt. was determined using a Greninger chart and Wulif net as described in reference [19]. On the basis of the established crystal orientation with respect to the X-Ra.y beam, the goniometer was adjusted such that the c axis was parallel to the X-R.ay beam. This procedure was repeated until the c axis of the sample was parallel to the X-Ray beam within 0.2 degrees. Once the c axis was determined the (001) face was ground flat using a PM2A Logitech lapping and polisher machine with SiC grit as the lapping media. The goniomcter was attached to the polishing instrument to ensure that the ground face was identical to the measured orientation. The LBO crystal was repositioned on the goniometer by heating the wax. It was reattacheci to the goniometer such that the C axis was perpendicular to the X-Ray beam direction. The above procedure was then repeated to determine the b axis. After this axis was determined and polished the procedure wa.s repeated for the a axis. Once the sample was oriented and the principle faces polished the LBO crystal was Chapter 4. Experimental 41 Figure 4.25: Cleavage plane of an LBO Crystal cut into smaller pieces using a. Buehler Isornet low speed saw. The seed size is 5 x 2 x 10 mm with the c axis being the longest dimension. Special notches are cut in the seed for a.ttaching it to the platinum seed holder. The notches a.re 1 mm in depth. The final seed shape is shown in Figure 4.27. The seed is attached to its holder by running a platinum wire though the notches in the seed and holder. The wire is wound tightly on the platinum holder to ensure that the seed is held in place. Figure 4.28 shown the seed and holder. 4.1.3 Growth Procedure The LBO growth solution consist of lithium tetraborate (Li2B4O7),extra borate (B203) to obtain the correct stoichiometric amount of LBO and molybdenum triborate (MoO3). Chapter 4. Experimental Pignre 4.26: Coniorneter used for orienting LBO crystals 42 Chapter 4. Experimental 43 Figure 4.27: LBO Seed Chap tei’ .1. Experiment a.] 44 Figure 4.28: LBO Seed attached to J)Iatinurn seed holder Chapter 4. Experimental 45 The melting point of lithium tetral)otate is 930°C of borate is 450°C, and molybdenum trioxide is 795°C. All compounds were obtained from Johnson Matthey and were 99.999% pure. The charge composition selected was 45 wt% MOO3 (40 mol%). The compounds were packed into a crucible with B203 at the bottom followed by the lithium tetrahorate and the top layer was MoO3. Dry nitrogen was introduced to the furnace at 12 cubic feet per hour. This flow rate was used during the entire crystal growth run. The charged crucil)le is heated to 1050° C to allow the lithium tetraborate to completely melt at this temperature for 48 hours. The liquid at this temperature has an opaque amber colour. The mixing paddle is lowered into the liquid until it is completely submerged. The crucible is then rotated at 30 rpm for 48 hours to ensure the liquid is completely mixed. After mixing is completed the liquid is a clear amber color. The stirring paddle is then removed from the furnace and replaced with the seed crystal which is attached to the seed rod and lowered to within 2.5 cm of the melt surface. The furnace temperature is then slowly cooled at 20°C/hr to the dip temperature of approximately 930°C. The crucible rotation is increased to that used for growth while the furnace is cooling. The LBO seed is clipped by slowly lowering it into the liquid until the meniscus of the fluid is broken. The appearance of the fluid after the seed has been dipped is shown in Figure 4.29. If the insulation on the side of the crucible changes between crystal growth runs so will the furnace setting for the seed dip temperature. When this occurs the liquid temperature is either below or above the liquidus temperature. If the seed is dipped and the liquid is above the liquiclus temperature there will be a longer time before the crystal grows out in size. The seed will not be melted awa since the LBO melting temperature is approximately 150°C above the liquidus temperature. If the seed is dipped and the liquid ten3perature is below the liquiclus temperature spurious crystal will grow on the seed. The furnace temperature was increased 5°C every two hours until the spurious Chapter 4. Experimental 46 Platinum Crucible 1LBO/Mo03 I Liquid LBOIMoO3 1eniscus of LiquidSeed Figure 4.29: Appearance of LBO melt surface after the seed has been (lipped Chapter 4. Experimental 47 growth melted back to the seed. Growth is continued as per normal from that point. The furnace is cooled at 0.1°C/hour after the seed had been dipped. Crystal growth is not apparent for up to 1 week after the start of the cooling. Once the crystal diameter has reached 2/3 of the crucible diameter it is pulled at 1.66 mm/day and continued for 5 days. Crystal growth is terminated by rapidly withdrawing the crystal from the melt at a rate of 2.1 x 1O cm/s. Once separated from the melt the crystal is slow cooled to room temperature at 10°C/hour. The crucible is cleaned by heating it to 1000°C and decanting the liquid. Any residual material is removed by soaking the inside of the crucible with reagent grade HCL. The HCL fluid is mixed and heated for one day using a combination hot plate/magnetic stirrer. 4.2 Temperature Measurements in the Melt Temperature measurements in the melt are required to establish the temperature bound ary conditions in the mathematical model, investigate the effect of crucible rotation on the thermal field, and for comparison with results from the mathematical model. The temperature measurements were accomplished in three separate trials. The first set of temperature measurements were undertaken to measure the temperature boundary con ditions of the melt for use in a mathematical model that examines the crystal growth operating parameters. A 6.6 cm diameter crucible with no rotation wa.s used to contain the melt for these measurements. The second and third set of temperature measurements were conducted for determining the model boundary conditions to validate the model by examining the effect of crucible rotation on the thermal field in the melt .A 6.6 cm diam eter crucible was used to contain the melt for second set of temperature measurements and an 8.8 cm diameter crucible contained the melt for the third set of measurements. Chapter 4. Experimental 48 4.2.1 Initial Temperature Measurements with No Crucible Rotation - Temperature measurements were carried out with a chromel/alumel thermocouple in a quartz sheathed tube connected to the seed rod. The thermocouple could be placed at three different radial positions; at the center of the crucible, at the mid—radius of the crucible and at the wall of the crucible. The axial temperature profile was determined by recording the thermocouple output as it was slowly pulled out of the liquid. The furnace temperature was set at 145°C above that normally used for the start of crystal growth to ensure that no nucleation occurs in the melt. The charge concentration used was 5OWt% MoO3. A sample of the melt wa.s not sent for chemical analysis. A 6.6 cm diameter crucible with no rotation was used to contain the LBO—Mo03melt. No insulation was on the outside wall of the crucible. 4.3 Temperature Measurements With and Without Crucible with Rotation Two sets of temperature measurements were carried out, one in a 6.6 cm diameter crucible and the other in a 8.8 cm diameter crucible. Both crucibles had insulation on the outside. Samples of the melt were sent for chemical analysis to an external lab. Two platinum sheathed chromel/alumel thermocouples were used for the temperature measurements. The position of the thermocouples were adjusted using a positioning device. A platinum sheet formed to simulate the crystal shape, as shown in Figure 4.31, was positioned to replace the crystal. The fluid flow conditions below the simulated crystal and its thermophysical properties, are believed to be close to the actual conditions during LBO crystal growth. As a result, the temperature measurements conducted with the simulated crystal could he applied to the actual growth system for model verification. Chapter 4. Experimental 49 ii Figure 4.30: Positions of thermocouples and simulated crystal for melt temperature measurements. Chapter 4. Experimental 50 I. i IWfl i.i C Tb. D.... 1 bba a.4 Mfl ka% Figure 4.31: Platinum cap used to simulate the crystal. Chapter 4. Experimental 51 Apparatus The thermocouple positioning was accurate and repeatable to within 1 mm, using the apparatus shown in Figure 4.32; note that the thermocouples are inside the furnace and the positioning mechanism is outside the furnace. The apparatus consists of four key parts which are a horizontal guide track for radial movement, a rack and pinion gear for axial movement, a thermocouple holder and a micrometer/calibrated dial for spatial positioning. The apparatus was positioned on the crystal puller as shown in Figure 4.33. The thermocouple positioned closest to the wall is vertical and the inside thermocouple is curved at the bottom to allow it to measure temperatures under the simulated crystal. Figure 4.30 shows the thermocouple shapes and positions with respect to the simulated crystal. Electronic cold junction compensation was used for the measurements and a chart recorder or computer based da.ta acquisition system was used to record the temperatures. General Procedure The chromel/alurnel thermocouples used in the measurements were calibrated at the freezing point of tin (231.9681°C). The temperature measurements were made in an LBO/Mo03bath similar to that used for crystal growth. The furnace temperature was set at 100°C above that normally used for the start of crystal growth to ensure that no nucleation occurs in the melt or on the simulated crystal surface. The thermocouple positioning systeni was set by determining the micrometer and dial settings with the thermocouples touching the top of the melt. and the inside wall of the crucible. In the small crucible temperatures were measured at axial and radial intervals of 0.5 cm. The melt temperature in the large crucible were measured at axial intervals of 0.36 cm and radial intervals of 0.5 cm. Chapter 4. Experimental TOP 52 Figure 4.32: Thermocouple probe holder. A - guide track for radial movement of TC probe. B - rack and pinion gear for axial movement. C - thermocouple holder. D micrometer and dial for positioning of the thermocouples. / A SIDE — Chapter 4. Experimental 53 Figure 4.33: Apparatus for inoving thermocouples attached to the crystal puller. Cha.pter 4. Experimental 54 Procedure for Temperature Measurements in the Small Crucible Initial temperature measurements were made for one flux concentration with no crucible rotation, without the simulated crystal. The measurements were repeated with the sim ulated crystal added to the system. and finally wit.h the simulated crystal and crucible rotations of 15, 20, 25 and 30 rpm. For the measurements, thermocouple A was initially positioned 0.2 cm from the wall and 0.47 cm from the bottom surface. Thermocouple B, being offset 1.6 cm vertically from thermocouple A, was 1.8 cm from the wall and 0.47 from the bottom surface. The thermocouples were then moved vertically 0.18 cm then at intervals of 0.45 cm until they reached the maximum height of the fluid at 2.45 cm. The vertical temperature measurements were repeated at horizontal spacings of 0.5 cm until thermocouple B was at the center of the crucible. The model crystal interfered with the thermocouples limiting the horizontal movement to 1.55 cm above the bottom of the crucible and thermocouple B to be at 1.0 cm from the center of the crucible. The radial location that the temperatures were measured at in the small crucible were 1.0, 1.5, 2.6 and 3.1 cm. Procedure for Temperature Measurements in the Large Crucible Temperature measurements were made for one flux concentration with no crucible rota tion, without the simulated crystal. The measurements were repeated with the simulated crystal added to the system, and finally with the sinmiated crystal and crucible rotations of 10, 15, and 30 rpm. For the measurements, thermocouple A wa.s initially positioned 0.2 cm from the wall and 0.36 cm from the bottom surface. Thermocouple B, being offset 2.9 cm vertically from thermocouple A, was 2.9 cm from the wall and 0.36 from the bottom surface. The thermocouples were moved vertically at intervals of 0.36 cm Chapter 4. Experimental 55 until they reached the maximum height of the fluid at 3.24 cm. The vertical tempera ture measurements were repeated at horizontal spacings of 0.5 cm until thermocouple B was close to the center of the crucible. The radial location that the temperatures were measured at in the large crucible were 0.4, 0.9, 2.8. 3.3 and 3.8 cm. 4.4 Physical Model of The Crystal Growth Process A physical model was employed to visualize the general flow patterns that occur during crystal growth. The model consists of an 8.8 cm diameter plexiglass crucible, an 5.6 cm diameter plexiglass crystal and a rotation device. The crystal had 0.2 cm diameter holes at 1/2 and 3/4 of the crystal diameter for the injection of a tracer fluid. Glycerine was used to simulate the LBO/Mo03melt. Blue dye mixed with glycerin is used as a tracer to examine the flow patterns. Figure 4.34 shows the plexiglass crucible and crystal. 4.5 Physical Properties 4.5.1 Chemical Analysis Chemica.l analysis for Mo and Li was conducted using atomic absorption at an external lab. Samples from the phase diagram determination and the viscosity measurements were analyzed for two different reasons. The phase diagram samples were used to determine the accuracy of the external lab’s results. The theoretical and actual composition of these samples will be very close since the samples that are sent for analysis spend no time at an elevated temperature. The viscosity samples must be analyzed since the composition changes with extended time at temperature due to the vapour pressure of MoO3. Chap ter 4. Experimental 56 Figure 4.34: Plexiglass crucible and crystal used for the physical model. Chapter 4. Experimental 57 4.5.2 LBO/Mo03Phase Diagram A phase diagram is necessary in crystal growth for selecting the liquid temperature at which the seed is dipped. It is also important in mass transfer studies of the growing crys tal. Of particular importance is the eut.ectic concentration, at which interface breakdown occurs. The LBO/MoOa phase diagram was determined using differential thermal anal ysis (DTA). This process consist of measuring the temperatures, of an alumina sample, and an LBO/Mo03sample simultaneously as they are heated. The alumina is used as a standard since it does not undergo a. phase change in the temperature range examined. The temperatures are plotted as the temperature difference between the two samples (ST) as a function of the furnace temperature. When the LBO/MoOa sample undergoes a phase change, the heat released or absorbed will be indicated by a positive or negative deviation from the temperature of the alumina, sample. The peaks and valleys on the - furnace temperature curve thus correspond to phase changes in the LBO/Mo03 sample. The equipment used consisted of a. Dupont 1090 Thermal Analyzer and a Dupont 910 differential scanning calorimeter fitted with a. 1200°C DTA furnace. Special platinum crucibles were used for containing the LBO/Mo03 samples. The initial sample was 55.4 wt% LBO and 44.6 wt% MoO3. Samples were prepared by combining lithium tetraborate, borate and molybdenum trioxide. The lithium tetraborate and molybdenum trioxide were in powder form and were easily mixed. The borate came in large blocks and was broken and ground to powder before it was mixed with the other constituents. Further mixing wa.s done using a. shaker with alumina. balls a.s the grinding media. The samples were placed in a plastic bottle with alcohol and alumina halls which were then placed in the shaker and mixed for two 20 minutes periods. The samples were then placed in a drying furnace to evaporate the alcohol. The dried sample was remixed Chapter 4. Experimental 58 using a mortar and pestle. A portion of the sample was kept to be sent for chemical analysis. Another portion of the sample wa.s used for the DTA analysis. The remainder of the sample was mixed with more MoO3 and used for the next DTA experiment. The theoretical LBO/Mo03 wt% composition ranges examined were 55.5/44.5, 50.3/49.7, 40.4/59.6, 35.6/64.4 30.9/69.1 and 26.2/73.8. The powdered samples ranged in color from light green to blue as the MOO3 concentration increased. A sample was placed in the platinum crucible and placed in the DTA furnace. The furnace was rapidly heated to 150°C then slowly heated at 10°C/mm to a final tempera ture of 750°C. The differential temperature was recorded between 500°C and 730°C. At the same time the sample was examined for a. change in color associated with the phase change. 4.5.3 Viscosity The viscosity of LBO containing MoO3 was measured, as a function of the MoO3 con centration, with a Brookfield Model RVTDV II viscometer, using a modified procedure of the ASTM standard C936-81. This procedure is used for measuring the viscosity of glass above its softening point. The LBO/Mo03 sample, in a platinum crucible, was heated in an electric furnace. The crucible dimensions are 66 mm diameter and height. A platinum paddle, shown in Figure 4.35 was fabricated for these measurements following ASTM specifications. The system, including the paddle, was calibrated by measuring the viscosity of a solution of glycerine and corn syrup which had a specified viscosity. Samples for the viscosity measurements were prepared using the same procedure as was used for crystal growth. Depending on the test, either an air or dry argon atmo spheres were used, the later wa.s to unsure that. no Li was lost from the melt due to water vapour. Once the LBO/Mo03melt was homogenized, a. sample was removed for quantitative chemical analysis of the molybdenum and lithium content. This wa.s done Chapter 4. Experimental 59 I I Figure 4.35: Platinum paddle for measuring viscosity of LBO/Mo03. with a quartz tube into which a sample from the high temperature melt was drawn. The paddle used for the viscosity measurements was then immersed in the melt together with a platinum sheathed chromel/alumel thermocouple. The viscometer was then turned on, the temperature stabilized at the specified test value and the thermocouple removed. The viscosity of the melt as indicated by the viscometer was then recorded. The procedure was then repeated at a lower set melt temperature, allowing approximately 10 minutes for the melt to equilibrate at the new temperature. The viscosity measurements were continued until the viscosity of the solution was in excess of 100 poise. II S Chapter 4. Experimental 60 4.6 Crystal Quality The crystal quality of LBO was examined for MOO3 inclusions, cracking and bubbles using optical microscopy and scanning electron microscopy (SEM). Quantitative chem ical analysis for molybdenum was performed using wavelength dispersive spectroscopy (WDX). Samples were prepared by polishing the surface with a 1t diamond suspension on a Buehler Ecomet IV. Polishing was continued until the surface of the sample was flat and had no visible scratches. Optical examination was conducted using an Zeiss Stereomicroscope SV8. The samples for SEM/WDX analysis were sputter coated with a layer of amorphous carbon. SEM/WDS analysis was done using a Hitachi S-570 SEM with Microspec WDX attachment. Measurement of the nonlinear optical properties were considered beyond the scope of this project and not carried out. The variation in visi ble/electron image inclusion density was assumed to be a sufficient indication of crystal quality. Chapter 5 Experimental Results 5.1 Phase Diagram for the LBO/Mo03System The concentration of the DTA samples examined are listed in Table 5.3. The charge concentrations listed were determined from the mass of the constituents used. The con centrations were measured by a commercial laboratory using atomic absorption. The measured values are given in weight percent Mo and Li. Equivalent MoO3 and LBO concentrations were calculated from the measured data assuming that these were the only species present in the melt. molecular weight of MoO3 Mo03(equivalent) = Mo(measured) x molecular weight of Mo molecular weight of LBO LB 0 (equivalent) = Li ( measu red) x molecular weight of Li Comparing the equivalent concentration of MoO3 and LBO to the charge concentrations, the former is observed to be approximately 5% below the later in all cases. The equivalent MoO3 concentration was normalized. Mo03(equivalent) Mo03(normalized) = . MoO3(equivalent) + LBO (equivalent) The normalized MOO3 concentration were close to the values of the charge concentration. These values were taken as the actual melt concentrations. The DTA results for 45 Wt% MoO3 (sample 1), are shown in Figure 5.36. The curve has four points of interest, the negative temperature difference at A (555°C) and 61 Chapter 5. Experimenta.l Results 62 Sample Weight Percent Charge Concentration Measured Equivalent Normalized MoO3 LBO Mo Li MoO3 LBO MoO3 1 44.56 55.54 26.50 2.86 39.76 49.19 44.76 2 49.66 50.34 30.00 2.64 45.01 45.40 49.78 3 59.58 40.42 — — — — 60 4 64.40 35.60 41.00 1.89 61.51 61.51 65.42 5 69.13 30.87 46.50 1.63 69.76 69.76 71.34 6 73.77 26.22 47.30 1.38 70.96 70.96 74.94 Table 5.3: MoO3 and LBO concentrations used in determining the pha.se diagram. D (575°C), the positive temperature difference at B (682°C) and the change of slope at C (610°C). During heating the sample a.ppearence was noted. The sample was initially a uniform light green colour until 555°C (point A) was reached where a reaction occurred and the sample turned brown. The surface texture also changed from being uniform to having evenly distibuted shiny regions on a dull surface. Between 575°C (point B) and 610°C (point C) the sample turned to a. transparent white colour and the surface texture changed to being shiny in appearance. Between 610°C and 682°C the sample colour slowly changed to an amber colour. At. 682°C (point. D) the sample melted. The liquidus of the sample was taken as the point where the temperature difference started to increase at point D and the slope between point C and D as the solid—liquid phase region. The gradual change in slope around point C makes it difficult to accurately determine the solidus. The temperatures at which the phase transformations occured are listed in Table 5.4 and plotted in Figure 5.37. The hiquidus changes from 682°C at 44.76 Wt% MoO3 to 619°C at 60 Wt% MoO3 just before the eutectic concentration, 61.5 Wt% MoO3. The DTA analysis indicates that two other phases occur above the eutectic composition at temperature in excess of 610°C. The structure of these phases were not investigated. Chapter 5. Experimental Results 63 I I t f I I I I I I I I I I I I I I I 3 .1 2 - 0 %0 w 0 4tIIIiIIItItIIItIIIIII 500 522 540 560 580 800 622 640 660 882 700 720 TemperQure (ec) DuPor 1090 Figure 5.36: Temperature difference versus temperature for 45 Wt% MOO3 (sample 1) determined by DTA. Chapter 5. Experimental Results 64 I I I 680 - 670 - 5’ 660 - Liquid 650- 640- E 630 - LBO + Liquid Eutectic 620 - C A+Liquid B+Liqaid -C-610 1] LBO + MoO3 600 I I I 50 60 70 MoO3 Concentration (Wt %) Figure 5.37: Phase Diagram of the LBO — MoO3 system. Chapter 5. Experimental Results 65 Sample Wt% MOO3 Solidus(s) Liquidus 1 44 76 610°C 689°C 2 49 78 612°C 661°C 3 60 614°C 619°C 4 65 4 614°C 618°C 5 71 34 619°C, 618°C 630°C 6 74.94 613°C, 618°C 632°C Table 5.4: The solidus and liquidus temperatures of the MoO3 — LBO samples 5.2 Viscosity The viscosity of five samples (7-11) were measured as a function of temperature. The composition of the samples are listed in Table 5.5. The viscosities are listed in Table 5.6 and 5.7. The temperatures listed are the average temperatures in the melt. The tem perature difference across the sample was measured to be 10°C. Two atmospheres were used, air and dry argon. The later was used to ensure that no Li was lost from the melt due to the presence of water vapour in the atmosphere. The viscosity is plotted on a logarithmic scale against the reciprocal of the test tem perature, in Figure 5.38. In general, the viscosity is high, ranging from 4 poise to greater than 70 poise between 863°C and 661°C. The viscosity decreases with increasing tem perature and increasing MoO3 concentration. The viscosity measurements are in good agreement with each other above 702°C (l0000/T[K] < 10.25). Sample 11(37.06 Wt% MoO3)deviates from sample 10 (38.67 Wt% MoO3)below 702°C (10000/T[K] > 10.25). The best fit lines for samples 7, 8 and 9 are plotted on the graph. 29.66Wt% MoO3: v(poise) = cxp (72.26 — 0.155 x T[C] + 8.51 x 10 x (T[Cj)2) 36.l8Wt% MoO3: v(poise) = e:rp (45.59 — 0.0946 x T[Cj + 5.04 x 10 x (T[C])2) 40.S9Wt% MoO3: v(poise) = cxp (37.31 — 0.0784 x T[C] + 4.21 x 10 x (T[Cj)2) Chapter 5. Experimental Results 66 Sample Weight Percent Charge Con centra t ion Measured Equivalent Normalized MoO3 LBO Mo Li MoO3 LBO MoO3 7 34.8 65.2 17.30 3.SS 25.96 61.56 29.66 8 36.8 63.2 21.90 3.37 32.86 57.96 36.18 9 39.3 60.7 26.40 3.3:3 39.61 57.96 40.89 10 43.0 57.0 20.60 2.85 30.91 49.01 38.67 11 50.0 50.0 21.20 3.14 31.81 54.00 37.06 Table 5.5: MoO3 and LBO concentration of the samples used in the viscosity measure ments. The viscosity variation with temperature for sample 10 (38.67 Wt% MoO3) and the fitted value for sample S (36.18 Wt% MoO3) are similar below 702°C (10000/T[K} > 10.25). Thus it is assumed that the low temperature viscosity measurements, T < 702°C (10000/T[K] > 10.25), for sample 11 are erroneous. The variation of viscosity with concentration is plotted in figure 5.39. The viscosity decreased with increasing MoO3 at a constant temperature. The viscosity of sample 7, 8 and 9 at their respective liquidus’s were determined. The liquidus temperatures were estimated by extrapolating the LBO/Mo03lic1uidus line on Figure 5.37 to the composition of the samples. The liquidus temperatures were used with the best fit viscosity equations for their respective compositions. The variation in viscosity and liquidus temperature is given in Figure .5.40. The viscosities at the liquidus decrease linearly with increasing MoO3 content. 5.3 Preliminary LBO Crystal Growth Runs This section examines the crystal growth parameters used and the resulting crystal growth quality in runs conducted prior to the start of this investigation. Most of the preliminary crystal growth runs used crystal rotation rates of approximately 15 rpm, no Chapter 5. Experimental Results 67 Sample Atmosphere Temperature 10,000/T[k] Viscosity (poise) 7 Air 765°C 9.64 26.8 Air 785°C 9.46 18.2 Air 807°C 9.26 12.9 Air 824°C 9.12 9.5 Air 847°C 8.93 7.1 Air 862°C 8.81 5.8 8 Air 736°C 9.91 25.3 Air 754°C 9.74 18.2 Air 779°C 9.51 12.1 Air 800°C 9.32 8.4 Air 817°C 9.17 6.6 Air 847°C 8.85 4.7 Air 863°C 8.80 4.2 Air 914°C 8.43 3.7 9 Air 730°C 9.97 13.4 Air 749°C 9.79 9.47 Air 766°C 9.63 7.1 Air 788°C 9.43 5.5 Air 808°C 9.25 4.47 Air 833°C 9.04 3.69 Air 847°C 8.93 3.42 Air 869°C 8.76 2.63 10 Dry Argon 691°C 10.37 124.27 Dry Argon 702.5°C 10.25 59.97 Dry Argon 720°C 10.07 41.95 Dry Argon 741°C 9.86 27.09 Dry Argon 767°C 9.62 16.31 Dry Argon 769°C 9.60 16.7 Dry Argon 784°C 9.46 12.89 Dry Argon 788°C 9.43 11.97 Dry Argon S17°C 9.17 8.16 Table 5.6: Viscosity versus temperature for samples 7 to 10. Chapter 5. Experimental Results 68 I I I <> k / 0Q 29.7 Wt% MoO1 , o70.0 - A 36.2 Wt% MoO3 lc 0 ‘V’40.9 Wt% MoO3 — 0 Wt% MoO3 037.IWL%MoO -I coY 20.0- • — Cl) 6.0 u I I I 9.00 9.25 9.50 9.75 10.00 10.25 10.50 10.75 1O,000/T[K] Figure 5.38: Dependence of viscosity on temperature and MoO3 concentration. Chapter 5. Experimental Results 69 300 T=72OC — - 0T=77OC 6 T=820C C El I I 28 30 32 34 36 38 40 Wt% MoO3 Figure 5.39: Variation of viscosity with Mo03 at the temperatures indicated. Chapter 5. Experimental Results 70 60 I I 750 Liquidus D 740 50 Viscosity A .— __________ 73() c • 40 A 720 D I) 71030 700 20 I I I 30 32 34 36 38 40 Weight Percent MoO3 Figure 5.40: Liquidus temperature and viscosity as a function of MoO3. Chapter 5. Experimental Results 71 Sample Atmosphere Temperature 10,000/T[k] Viscosity (poise) 11 Dry Argon 653°C 10.80 132.03 Dry Argon 658°C 10.74 119.14 Dry Argon 669°C 10.62 96.13 Dry Argon 676°C 10.54 78.90 Dry Argon 687°C 10.42 64.18 Dry Argon 696°C 10.32 50.50 Dry Argon 734°C 9.93 30.25 Dry Argon 740°C 9.87 21.83 Dry Argon 755°C 9.73 17.23 Dry Argon 774°C 9.55 13.15 Dry Argon 775°C . 54 12.10 Dry Argon 787.5°C 9.42 11.31 Dry Argon 805°C 9.28 8.81 Dry Argon 818°C 9.17 7.24 Table 5.7: Viscosity versus temperature for sample 11. crucible rotation and a cooling rate of 3°C/day. The initial charge concentration of MoO3 in the melt was 34 \1t% All the crystals cracked during the post growth cooling to room temperature. The average size of the uncracked portions of the crystal were 7 x 3 x 3 mm3. Opaque white/green colored inclusions were in the crystals having an average size of 3 x 3 x 3 mm3. Samples of the grown crystals were examined using a scanning electron microscope (SEM) equipped with a wavelength dispersive S1)eCtrometer (WDS). Figure .5.41 show the hackscatter images of the cross section of an LBO crystal. WDS analysis was used to examine the crystal defects as well as the defect free crystal matrix. There was no molybdenum present in the defect. free portions of the crystal. The molybdenum inclusions on this sample formed as two types; line inclusions and normal inclusions. Line inclusions run for distances that are greater than half the speci men size and the width is much smaller than the length. Normal inclusions have lengths Chapter 5. Experimental Results 72 that are comparable to their widths. Backscatter images and the corresponding dot maps for a normal inclusion are shown in Figure 5.42. The inclusion shape corresponds directly to the molybdenum dot map. A backscatter image and its corresponding molybdenum line scan of a line inclusion is shown in Figure 5.43. The dot map and line scan of the molybdenum region corresponds t.o the inclusions in the backscatter photos. The line inclusions are parallel and perpendicular to the crystal growth axis. Flux inclusions in ceramic materials are more significant than for other materials. An inclusion with an expansion coefficient different from the matrix is the most likely cause of the crystal fracturing during cooling to room temperature. The expansion coefficient of LBO in the [001] direction and Mo203 are —88.0 x 10/°C and 5.35 x 10/°C respectively (see Table 2.2). From these initial findings it was evident that the main crystal defects (inclusions and cracking) are due to the MoO3 flux. It wa.s clear that the build up of MoO3,due to the high viscosity of the melt, has a large influence on the crystal quality. Mathematical and physical modeling of the crystal growth process has been employed to improve the mixing in the melt thus reducing the MOO3 ahead of the growing interface. Normal Inclusions Growth Direction Line Inclusions 73Chapter 5. Experirnenta.l Results Figure 5.41: Surface of an LBO crystal with inclusions. Magnified 30 times Chapter 5. Experimental Results 74 Figure 5.42: Normal molybdenum inclusion in an LBO crystal magnified 400 times. (a) Backscatter image. (b) WDS molybdenum dot map Chapter 5. Experimental Results 75 Figure 5.43: Molybdenum line inclusion in LBO crystal magnified 2,200 times. (a) Backscatter image. (b) WDS molybdenum line scan Chapter 6 Physical Model of the LBO Crystal Growth Process The flow of liquid under a growing crystal was investigated by observing the flow directly in a physical model of the growth system using a transparent liquid as the melt. The physical model gives a better visualization of the flow pa.tterns while a mathematical model will give semi—quantitative fluid velocity results that will he superior to the phys ical model results. This type of flow has been previously examined in detail for a rotating crystal in an infinite fluid [46]. Thus the physical model will only examine the flow fields due to a rotating crucible with a stationary crystal. The physical model consisted of a plexiglass crucible 8.8 cm in diameter and a plex iglass crystal 5.6 cm in diameter. The melt level in the crucible was 2.5 cm in height. The fluid consisted of a glycerine solution having a viscosity of 7 poise. This corresponds to an 41.6 Wt% MOO3 melt at S20°C, which is 130°C above the liquidus temperature. The plexiglass model of the crystal was positioned such that the bottom surface of the crystal was in contact with the melt. Blue dye was injected in a hole in the crystal at 0.5 of the crystal radius and red dye in a hole at 0.75 of the crystals radius. The blue and red dye consisted of chart recorder ink mixed with glycerine in order to give it the same buoyancy as the fluid. The crystal was stationary for all of the experiments and the crucible was rotated at either 45 or 78 rpm in a clockwise direction. The flow is discussed in terms of the fluids radial (Vr). axial (vi) and theta (swirl,v0)velocity components. 76 Chapter 6. Physical Model of the LBO Giystal Growth Process 77 6.1 Observed Fluid Flow Patterns Figure 6.44a shows the initial dye pattern from the top of the crystal for a crucible rotation of 45 rpm. The injection holes for the red and blue dye are shown in figure as A and B respectively. Similar injection holes at the same radial location but rotated by one quarter of a circumferential arc are shown as A’ and B’. The dye traces are moving in a semicircular arc and also have a radial flow component that moves them towards the centre of the crystal. This is shown by the red dye trace. It is injected at a 0.75 of the crystal radius (Point A) and its position is at 0.5 of the crystal radius (Point B’) after it travels one quarter of a circumferential arc. The view of the dye from the side of the crucible is shown in Figure 6.44b. The dyes remain very close to the model crystal after they have been injected. It is noted that each picture in Figure 6.44 corresponds to different experimental runs. Differences in the position of the red and blue dye with respect to each other are due to different injection start times. Thus initially the dyes have a large swirl velocity a smaller radial velocity and a non existent axial velocity. The dye tracers patterns between the injection locations and near the centre of the crystal are shown in Figure 6.45. The view from the top of the crystal, Figure 6.45a, show that the dyes reach the centre of the crystal after they travel 0.75 of a circumferential arc. The view for the side of the crucible, Figure 6.45b, shows that the dye tracers start moving toward the bottom of the crucible near point A. Point A is 0.75 of a circumferential arc from the location where the dyes a.re injected. It is evident that the axial velocity downward increases as the dye traces get near the center line of the melt. The flow of the dyes at the centre line of the crucible are shown in Figure 6.46. The flow near the crucible, point A, and at 0.5 of the fluid height, point B, are examined. The swirl in the 0 direction is larger at point A and small at point B also the vertical travel distance is large between both points. This shows that the axial fluid velocity is Chapter 6. Physical Model of the LBO Orvstai Growth Process 78 dominant at the centre of the crucible near the mid height of the fluid. The flow patterns of the dye tracers near the bottom of the crucible, Figure 6.47, are different at the mid height of the fluid, point A, the centre bottom of the crucible, point B, and the outside bottom of the crucible, points C and D. The dye traces start moving in the radial directions and get a. larger swirl a.s they approach the bottom of the crucible, points A to B. Once the dye traces have reached the bottom of the crucible their axial travel distance decreases to near zero and there radial and swirl component increased as they move outward along the bottom of the crucible. When the dye reaches the side of the crucible, point C and D, it is no longer a separate phase from the pure glycerine. This indicates that there is significant mixing occurring a.t the bottom of the crucible. Figure 6.48a shows the flow patterns after approximately 3 minutes. The initial dye has mixed to make the fluid a uniform red color except directly below the crystal were new dye is being injected into the liquid. The location of the maximum of the radia.I and axial flow components for steady state crucible rotation are at the following location. Maximum radial fluid flow occurs directly below the crystal and above the bottom of the crucible. Axial fluid flow is largest at the centre of the crucible half way between the crystal and bottom of the crucible. This is similar to what has been previousix’ l)een predicted [28, 32]. The change in flow patterns a.t different crucible rotation rates are examined. Fig ure 6.48 shows the change in flow pattern between 45 and 78 rpm. The sequence of photos with respect to the crucible rotation rates are, figure 6.48a. is the flow pattern at 45 rpm, figure 6.48b is the transient flow pattern and figure 6.48c is the flow pattern at 78 rpm. Point A, located at the mid height of the fluid, will be the location of the fluid that is examined. The location of the swirl at point A with a crucible rotation rate of 45 rpm, Figure 6.48a., is large. The size of the swirl gets smaller a.s the flow adjusts to the higher crucible rotation rat.e of 78 rpm, Point A Figure 6.48b. The steady state Chapter 6. Physical Model of the LBO Crystal Growth Process 79 flow pattern at 78 rpm, Figure 6.4Sc, results in reducing the swirl of the dye tracer at point A to near zero. The dye tracer follow a straight vertical line below the centre of the crystal. This shows that the axial velocity at the centre line of the melt gets larger with increasing crucible rotation rates. The change in flow patterns directly below the crystal for crucible rotation rates of 45 and 78 rpm are shown in Figure 6.49. The dyes in the 45 rpm case reach the centre of the crystal in 0.75 of a. circumferential arc. A crucible rotation rate of 78 rpm causes the dyes to reach the centre of the crystal in 0.5 of a circumferential arc. This shows that the radia.l velocity of the fluid increases with increasing crucil)le rotation rates. Both Figures 6.48 and 6.49 clearly show tha.t axial and radial fluid velocities increase with increasing crucible rotation rates. 6.2 Physical Explanation of Fluid Flow Patterns This type of flow has been previously examined in detail for a rotating crystal and stationary crucible [46]. These results will be referred to in considering the forces acting on the fluid, and the resulting fluid motion, for the case of crucible rotation with and without a stationary crystal in contact with the top of the melt. When a crucible starts to rotate from rest without a crystal present, liquid near the bottom of the crucible starts to move in the 0 direction due to viscous drag. As the fluid rotates it gets a centripeta.l acceleration (—2r). The presence of the centripetal acceleration with no radial pressure gradient causes the fluid to move outward in the radia.l direction. When the fluid reaches the crucible wall it is forced to move upward. Fluid a.t the center of the crucil)le fluid moves downward to balance the outward flow in the radial direction. As the flow progresses the surface of the fluid rises at. the outside, and falls at the centre as shown in Figure 6.50. The va.riation in fluid height with radius Chapter 6. Physical Model of the LBO Crystal Growth Process 80 creates a radial pressure gradient which balance the centripetai acceleration and stops the radial flow at the bottom of the crucible. When the radial flow stops the fluid moves as a solid body with the rotating crucible. Consider the radial forces that act on a control volume of fluid. In terms of Newtons law of motion the balance of forces in the radial direction is: F = mar Where Fr is the force acting in the radial direction, m is the mass and ar is the acceleration iii the radial direction. The previous equation in terms of pressure is = —pa’. were P is pressure and p is density. With crucible rotation and no radial pressure gradient the forces acting on the fluid at the bottom of the crucible is only due to the centripetal acceleration. Thus there is a net force in the radial direction —p (_w2r) A radial pressure gradient is created with the change in fluid height, which balances with the centripetal acceleration giving: 2 =_p(_wr) For a stationary crystal present. it will be assumed that the crystal is inserted into the melt after the fluid has become a solid body in the rotating crucible. It is also assumed that the bottom of the crystal is the same shape as the surface of the fluid. The stationary boundary condition, v0 = 0, at the fluid surface with the crystal present is shown in Figure 6.51. The crucible is rotating and the fluid has solid body rotation. Constraining a portion of the fluid surface causes the fluid directly below it to have a lower 0 velocity than it had Chapter 6. Physical Model of the LBO Ci stal Growth Process Si during solid body rotation. Thus the centripetal acceleration is lower. Since, in this case, the radial pressure gradient is the same there is a net force in the inward radial direction. Thus, fluid below the constrained surface will move in the inward radial direction until it reaches the centre were it will move downward. The fluid moving downward will reach the bottom of the crucible, then it will move radially outward to the crucible wall, upward along the crucible side, and finally inward once it reaches the surface of the fluid. This flow will continue since the centripetal acceleration will not be balanced by the radial pressure gradient below the crystal. The flow patterns as explained using fluid mechanics are the same as has observed in the physical model. Chapter 6. Physical Model of the LBO Crystal Growth Process 82 Figure 6.44: Initial dye tracer pattern in the glycerine, a) Top view. b) Side view. Cucible rotated at 45 rpm. Ohapter 6. Physical Model of the LBO Orysta.l Growth Process 83 Figure 6.45: Dye tracer pattern in the glycerine when the blue and red tracers reach the centre of the fluid. a) Top view. b) Side view. Crucible rotated at 45 rpm. Chapter 6. Physical Model of the LBO Grystal Growth Process 84 Figure 6.46: Dye tracer pattern in the glycerine when the dye reaches the bottom of the crucible. a) Side view b) View under the crystal. Crucible rotated at 45 rpm. Chapter 6. Physical Model of the LBO Crystal Growth Process 85 Figure 6.47: Dye tracer pattern near the bottom of the crucible. a) Side view. b) View under the crystal showing the red and blue die moving up the side walls of the crucible. Crucible rotated at 45 rpm. Chapter 6. Physical Model of the LBO Grystal Growth Process 86 Figure 6.4$: Dye tracer pattern at different crucible rotation rates. a) Crucible rotation rate of 45 rpm. h) Transition flow for a crucible rotation rate between 45 and 78 rpm. c) Crucible rotation rate of 78 rpm. Ohapter 6. Physical Model of the LBO Crystal Growth Process 87 Figure 6.49: Top view of the dye tracer patterns at different crucible rotation rates. a) Crucible rotation rate of 45 rpm. 1)) Crucible rotation rate of 78 rpm. Chapter 6. Physical Model of the LBO Crystal Growth Process 88 Centre Line MELT I ___ __ Crucible Crucible Rotation Figure 6.50: Interface curvature due to solid body rotation. -7 Chapter 6. Physical Model of the LBO Grysta,l Growth Process 89 Surface of melt constrained to give it a zero velocity MELT Centre Line Crucible Crucible Rotation Figure 6.51: Crucible with portion of the tipper surface constrained to zero. Chapter 7 Temperature Measurements Temperature measurements were conducted to establish the boundary conditions in the model, to investigate the effect of crucible rotation on the thermal fields, to compare the temperature fields measured with those predicted by the model and to examine the thermal gradients in the crystal as it is cools to room temperature. 7.1 Initial Temperature Measurements with No Crucible Rotation The melt was a clear amber color. No convective cells were observed at the liquid surface. Temperature as a function of axial position at three radial locations; the center (r = 0 cm), the mid position (r = 1.6 cm) and near the wall (r = 3.2 cm) of the crucible are shown in Figure 7.52 The temperature oscillations at. r = 1.6 and 3.2 cm result from the furnace heaters turning on and ofF and are not significant. The oscillations were suppressed by employing electrical shielding on the thermocouples. The melt temperatures were assumed to be the lowest temperature recorded in the oscillations. The melt temperatures range from 760°C at the top of the melt to 868°C a.t the bottom. The temperature measurements were conducted with the furnace set approximately 145°C ( T) higher than the conventional setting for crystal growth. It is assumed that the values of the measured temperatures change proportionally with the furnace setting. Thus the boundary conditions for the mathematica.l model were obtained by decreasing 90 Gha.pter 7. Temperature Measurements 91 the measured temperatures by the appropriate AT. T(boundary condition) = T(rneasured) — AT The temperatures used as boundary conditions are assumed to he the temperature of the inside walls of the platinum crucible w’hich do not change. The temperature boundary conditions used in the sensitivity analysis are given in Figure 7.53. 7.2 Temperature Measurements With and Without Crucible Rotation Temperature measurements were conducted to examine the influence of crucible rotation on the thermal fields for crucible diameters of 6.6 and 8.8 cm. Rotation rates of 0, 15, 20. 25 and 30 rpm were used in the 6.6 cm diameter crucible. Measurements were conducted at crucible rotation rates of 0, 10, 20 a.nd 30 rpm in the 8.8 cm diameter crucible. Table 7.8 gives the concentration of the melts that were investigated. In all cases the outside of the crucible was thermally insulated. The bottom of the small crucible was deformed due to the expansion coefficient of LBO/Mo03being much larger than that of platinum. The bottom of the crucible was no longer flat, the center was 0.5 cm lower than the side. Thermal boundary conditions were determined from the 0 rpm crucible rotation data. The measured thermal fields for different rotation rates were compared to the model predictions. 7.2.1 Boundary Temperature Results Small Crucible (6.6 cm diameter) Temperature measurements 0.2 cm from the vertical crucible wall, in a melt containing 45.5 \Vt% MoO3,are shown in Figure 7.54. The dashed curve are the measurements with out a simulated crystal present, and shows that. the temperature decreases from 818°C. Chapter 7. Temperature I’Ieasurements 92 880 87G. 860 850 840 __830 - C) 820 Q1C . R=l6mm -800 ;E Rrz32mm R=Omm ci) I I- I 790 I 780 I 0 R 1’5 2b 30 Height (mm) Figure 7.52: Temperature variation with axial position in a 5.5 Wt% MoO3 solution. The three radial locations are r = 0 mm, r = 16 mm, and r = 32 mm. There is no crucible rotation during the temperature measurements. Crucible diameter is 6.6 cm. Chapter 7. Temperature Measurements 93 Convective Heat Transfer ttt tt z = 2.5 (@•c MELT I I z=O.7an I I z = 0 ai _________________________ r=3.3cm r=1.6an Fixed Temperature Figure 7.53: Temperature boundary conditions used in the fluid flow model for the sensitivity analysis and the examination of the operating parameters. Crucible diameter is 6.6 cm. Chapter 7. Temperature Measurements 94 Crucible Weight Percent Diameter Charge Concent ration Measured Equivalent Normalized MoO3 LBO Mo Li MoO3 LBO MoO3 6.6 cm 50.05 49.95 29.00 3.0:3 43.51 52.11 45.50 8.8 cm 44.56 55.44 33.30 3J3 48.46 53.83 47.38 Table 7.8: MoO3 and LBO concentrations in the melt that were used for the temperature measurements with and wj thout cm cil)le rotation. 0.47 cm from the bottom, to 805°C at 2.45 cm. With a simulated crystal positioned in the melt the melt temperature increased at the wall by approximately 12°C. The radial temperature distribution in the melt 0.47 cm from the bottom of the crucible, is shown in Figure 7.55. The temperature remains essentially constant at 807°C for 1 cm from the centre then increases progressively to S1S°C at 0.2 cm from the crucible wall. In obtaining the temperature measurements shown in Figure 7.55 two thermocou pies were used, one for r < 1.6 cm and the second for r > 1.6 cm. The small increase in temperature at A may be associated with a. slight difference in behavior of the two thermocouples. The repeatability of the temperature measurements were determined by making multiple measurements at points B and C. The difference in temperature was less than 1°C. Adding the sinnilated crysta.l increases the melt temperature, bitt does not significantly change the temperature gra.d ient. Following this observation, the boundary temperatures were determined from the data without a crystal by adding 12°C to account for the temperature rise when a crystal is added, and extrapolating the measurements to the outside surface of the platinum crucible. The resultant temperature boundary conditions are given in Figure 7.56. Using these boundary conditions, temperature pro files in the melt were calculated, using the mathematical model, and compared to the experimental measurements with no crucible rotation and with a simulated crystal. The J Chapter 7. Ternperat tire Measurements 95 boundary conditions were then adjusted, and the calculations repeated until the calcu lated temperatures fitted the experimental values. The boundary conditions determined in this manner were then used to calculate the thermal field with crystal rotation and the results were compared to the corresponding measurements. Large Crucible (8.8 cm diameter) The melt temperature 0.2 cm from the wall of the crucible is shown in Figure 7.57. The melt temperature, without the crystal, is essentially constant at 824.5°C between at 0.36 and 2 cm from the bottom of the crucible. Above 2 cm the temperature progressively decreases reaching 807.3°C at 3.15 cm. Introducing the simulated crystal increases the melt temperature by approximately 5°C. The radial temperature distribution 0.36 cm from the bottom of the crucible is shown in Figure 7.58. The temperature varies from 806°C at 0.17 cm from the center of the crucible to S24°C at 0.2 cm from the wall of the crucible. The gradient does not appea.r to dramatically change with the insertion of the simulated crystal. The temperature boundary conditions used in the mathematical model are determined using the procedure described in the previous section. Figure 7.59 shows the temperature boundary conditions used in the model calculations. 7.2.2 Melt Temperature Results Small Crucible (6.6 cm diameter) Melt temperatures at crucible rotation rates in the range 0 and 30 rpm, with the simulated crystal present, were determined. The axial temperature profile 1 cm from the centre of the crucible is given in Figure 7.60. With 0 rpm crucible rotation, the temperature decreases with distance from the bottom, the curve being concave upward. With 1.5 rpm crucible rotation, the temperature at 0.4 cm from the bottom decreases and above Chapter 7. Temperature Measurements 96 Bottom Top 830 825 With’T1eSih Simulated Crystal ,820 E815 -.. No Simulated Crystal 810 S S S S 8o I IThO 0.5 1.0 1.5 2.0 2.5 Axial Position (cm) Figure 7.54: Melt temperature 0.2 cm from t.he crucible wall with and without the simulated crystal. Crucible diameter is 6.6 cm. Chapter 7. Temperature Measurements 97 Centre Outside With The S1edCsrn 820W -8-- 815 - No Crystal Cr - 810- 80 —— I I I 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Radial Position (cm) Figure 7.55: Melt temperature 0.47 from the crucible bottom with and without the simulated crystal. Crucible diameter is 6.6 cm. Chapter 7. Temperature Measurements 98 Convective Heat Transfer SIMULATFID z=2.3cm 812C 10cm 23cm MELT I H I cm - I 35cm A z=-0115Cm I05cm r=3615Cm 8 Crucible20 C 0.115 cm thickness z -0.27cm r=1.6cm z=0.615cm r = 0 to 0.54 cm Fixed Temperature Figure 7.56: Temperature boundary conditions used in the mathematical model of me’t with the simulated crystal. Crucible diameter is 6.6 cm. Chapter 7. Temperature Measurements 99 Bottom Top 830 I I The Simulated Crystal 825 - ,820 ‘S E815 - 810 No Simulated Crystal \ b 8% ‘.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Axial Position (cm) Figure 7.57: Melt temperature 0.2 cm from the crucible wall with and without the simulated crystal. Crucible diameter is 8.8 cm. Chapter 7. Temperature Measurements 100 Centre Outside 825 820 No Crystal 815 A E 810 -‘ I 805, 800 0:5 1.0 1.5 2.0 2.5 3.0 3:5 4.0 Radial Position (cm) Figure 7.58: Melt temperature 0.36 from the crucible bottom with and without the simulated crystal. Crucible diameter is 8.8 cm. Chapter 7. Temperature Measurements 101 Convective Heat Transfer z = 3.3 cm —_—i z=2.5cm 1.0cm z = 1.7 cm 3.3 cm MELT I H ____ _____I 4.4 cm >1 ____ __ __ __ ____________ __ __ __ ____ Crucible z=-0.ll5cm ( ,thickness0.115 r 4.5 cm r=3.5cm r=Ocm Fixed Temperature Figure 7.59: Temperature boundary conditions used in the mathematical model of melt with the simulated crystal. Crucible diameter is 8.8 cm. Chapter 7. Temperature Measurements 102 0.5 cm increases. The shape of the curve changes to concave downward. Increasing the crucible rotation above 15 rpm progressively raises the melt temperature with the axial temperature distribution essentially remaining the same. The largest temperature increase occurs in the 25 to 30 rpm interval. The corresponding axial temperature distribution at a radial distance of 1.5 cm are given in Figure 7.61. In this case at 0 rpm the temperature is observed to decrease linearly with distance from the bottom. With crucible rotation, the temperature of the melt progressively increa.ses, and the tern perature distribution curve is concave downward, as before. The axial temperature distribution at radial distances of 2.6 cm and 3.1 cm from the centre are given in Figure 7.62 and 7.63. At both distances at 0 rpm, the temperatures are higher than at r = 1.5 cm, and decrease linearly with distance from the bottom of the crucible. \Vit.h rotation rates between 15 and 25 the temperature distribution becomes concave upward. The axial temperature profile becomes flat at a rotation rate of 30 rpm. The movement of the fluid due to crucible rotation has been qualitatively shown and described in Chapter 6. Crucible rotation results in the movement of hot fluid up at the crucible wall and cold fluid down under the crystal. The movement of hot and cold fluid causes the liquid isotherms to become more concave. The change in the liquid isotherm shape with the fluid motion is shown schematically in Figure 7.64. The axial temperature profiles become flat at high crucible rotation rates The experimental results shown in Figures 7.60 through 7.63 are in agreement with the physical model (qualitative) analysis. Large Crucible (8.8 cm diameter) Melt temperatures at crucible rotation rates between 0 and 30 rpm, with the simulated crystal present, were determined. The axial temperature profile at 0.4 cm from the Chapter 7. Temperature Measurements 103 r-) CI E Bottom Top 828 826- —D----—— 0 RPM ——--<>.-—— 15RPM 20RPM ——0——- 25RPM —••—••-X•—•--•’ 30RPM 824- 822- 820- 818 - 816- - I . . . . 0.5 1.0 1.5 Axial Position (cm) 2.0 Figure 7.60: Temperature distribution at r = 1.0 cm for crucible rotations of 0, 15, 20, 25 and 30 rpm. Crucible diameter is 6.6 cm. Simulated crystal present. Chapter 7. Temperature Measurements 104 —1J--—— 0 RPM --—<>-—— 15RPM 20RPM --0--- 25RPM ——-X—— 30RPM Bottom Top 828 826- 824- 822 - — — — — — — — ‘0 ‘20 Axial Position (cm) Figure 7.61: Temperature distribution at r = 1.5 cm for crucible rotations of 0, 15, 20, 25 and 30 rpm. Crucible diameter is 6.6 cm. Simulated crystal present. Chapter 7. Temperature Measurements 105 —Q———— 0 RPM ------ 15RPM 20RPM ——0——- 25 RPM ——-X—••— 30RPM Bottom Top 828 . 824 .820 818 816- 814oo 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 7.62: Temperature distribution at r = 2.6 cm for crucible rotations of 0, 15, 20, 25 and 30 rpm. Crucible diameter is 6.6 cm. Simulated crystal present. Chapter 7. Temperature Measurements 106 —I]—-—— 0 RPM ---a--- 15RPM 20RPM -_ 25RPM —— 30RPM Bottom Top 828 826 i : .820 818 816- 8 14o • I I • I .0 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 7.63: Temperature distribution at r = 3.1 cm for crucible rotations of 0, 15, 20, 25 and 30 rpm. Crucible diameter is 6.6 cm. Simulated crystal present. Chapter 7. Temperature Measurements 107 i Crystal I Crucible Figure 7.64: Change in liquid isotherms with crucible rotation. (a) No crucible rotation. (b) Large crucible rotation. Chapter 7. Tempera.t tire Measurements 108 centre of the crucible is given in Figure 7.65. The melt temperature profile with no crucible rotation is concave downward. With rotation, the concavity remains the same and the average melt temperature increases. The change in average melt temperature with crucible rotation is large between 0 and 20 rpm and small between 20 and 30 rpm. The axial temperature profile at r = 0.9 cm with no crucible rotation, Figure 7.66, is slightly concave downward. A rotation rate of 10 rpm causes the melt temperature to increase but the axial temperature profile remains concave upward. Higher rotation rates, 20 and 30 rpm, make the axia.l temperature profile flat. The axia.l temperature dis tribution at radial distances of 2.8 cm and 3.3cm from the centre are given in Figure 7.67 and 7.68. At both distance the temperature profile at 0 rpm is flat. The interface becomes concave downward at a crucible rotation rate of 10 rpm. Between 20 and 30 rpm the axial temperature profiles becomes approximately flat. For all of the previous temperature profiles the largest change in melt temperature with crucible rotation rate occurs between 0 and 20 rpm. Figure 7.69 shows the axial temperature profile at 3.8 cm. The axial temperature profile is flat with no crucible rotation. Rotation rates of 0 to 30 rpm do not change the shape of the temperature profile. The melt temperature changes linearly with crucible rotation speed. Fluid motion due to crucible rotation has been qualitatively examined in Chapter 6. Crucible rotation causes the movement of hot fluid up at the crucible wall and cold fluid down under the crystal. This type of fluid motion causes the liquid isotherms to become more concave (Figure 7.64). The axial temperature profiles become flat at high crucible rotation ra.tes (Figure 7.64(b)) as shown experimentally in Figures 7.65 and 7.66. Chapter 7. Temperature Measurements 109 —0--—— 0 RPM ---c--- 10RPM 20RPM ——0-—- 30RPM Bottom Top 832 I 830 __——.:E--2-.828 826- -- “0 ,S 824 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 7.65: Temperature distribution a.t r = 0.4 cm for crucible rotations of 0, 10, 20 and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present. Chapter 7. Temperature Measurements 110 —D-—— 0 RPM ------•- 10RPM 20RPM ——0-—- 30 RPM Bottom Top 832 I 830 0- —-__. 828 826 824 820 818 816 814oo 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 7.66: Temperature distribution at r = 0.9 cm for crucible rotations of 0, 10, 20 and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present. Chapter 7. Temperature Measurements 111 D- 0 RPM ---c--- 10RPM Li 20RPM ——0—- 30 RPM Bottom Top 832 I • 830 0- -2828 ‘—‘826 E824 E820 818 816 814oo 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 7.67: Temperature distribution at r = 2.8 cm for crucible rotations of 0, 10, 20 and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present. Chapter 7. Temperature Measurements 112 0RPM ---0--- 10RPM Li 20RPM ——0-—- 30RPM Bottom Top 832 I 830 __ 828 - 0- ‘—‘826 E824 C. 820 818 816 814oo 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 7.68: Temperature distribution at r = 3.3 cm for crucible rotations of 0, 10, 20 and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present. Chapter 7. Temperature Measurements 113 —C---— 0 RPM ------ 10RPM 20RPM ——0—- 30RPM Bottom Top 832 I • I 830 828 . 824 822 820 818 816 814oo 0.5k 1.0 1.5 2.0 Axial Position (cm) Figure 7.69: Temperature distribution at r = 3.8 cm for crucible rotations of 0, 10, 20 and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present. chapter 7. Ternperat tire Measurnts 114 7.3 Thermal Gradients in the Crystal During Cooling High thermal stresses which lead to cracking are due to large temperature gradients in the crystal [51]. Thus, it is important to know the thermal gradients that occur in the crystal during cooling for different furnace configurations. It is not practical to measure the thermal gradients in the crystals with imbeddeci thermocouples. As an alternative a simulated crystal made of alumina. aggregat.e was used, having the configuration shown in Figure 7.70. Four thermocouples were imbedded in the alumina in the positions indicated in the figure. The thermal conductivity of the ceramic alumina aggregate is comparable to tha.t of L130 which makes it a. suitable model material. Temperature measurements in the simulated crystal were made under equilibrium conditions with the furnace set at 400°C and aft.er an interval of five hours to allow the furnace to reach equilibrium. Measurements were made at two vertical locations in the furnace, position A and B in Figure 7.71, corresponding to the normal cooling positions of crystals grown in the large and small crucibles respectively. Note that the bottom of the large crucible is positioned 1.1 cm above the bottom of the small crucible. Temperature gradients were determined with no crucible present, with a small crucible in the furnace and with a large crucible in the furnace with and without an insulating block at the top of the furnace as shown in Figure 7.71. The crucible and sinmlated crystal positions in the furnace for which temperature measurements were made are shown in Figure 7.72. The values of the temperature gradients are listed in Table 7.9 with the corresponding measured temperature gradients across the pairs of thermocouples indicated. no crucible in the furnace the gradients are larger for the simulated crystal in the upper position A as compared to B. The radial gradients a.re close for 1)0th A and B a.nd lower than the axial gradients with the axial gradient at A being appreciably larger Chapter 7. Temperature Measurements 115 than the others. Adding the small crucible to the furnace changes the temperature gradients in the simulated crystals. The radial gradient between thermocouples 1 and 2 is increased when the crucible is added and the the simulated crystal is above of the crucible. All of the other radial and axial gradients are lower with the crucible at positions A and B. In particular there is a large decrease in the axial gradients with the addition of the crucible. The decrease in gradients for position B can be attributed to the presence of the crucible surrounding the crystal in this position. The high conductivity of the platinum crucible, and its close proximity to the crystal reduces the gas temperature gradient adjacent to the crystal which leads to lower giaclients in the crystal. The higher radial gradients at position at position A follow, since1)OSitiOfl A is above the top of the crucible. The radial gradient in the crystal with the large crucible in the furnace is essentially the same as that observed with the crystal in the smaller crucible. (position B). The axial gradients however are effectively twice that of the smaller crucible, and compared to the condition of no crucible in the furnace. In effect the large crucible walls are sufficiently far away from crystal that they have little effect on the gradients in the crystal. Adding an insulating brick over the top of the seed hole significantly reduces the radial gra.dients and to a lesser extend the axial gradients. Independent of crucible size, the thermal gia.clients in the crystal are most sensitive to the addition of insulation over the seed hole in the furnace and the vertical position of the crystal. Moving the simulated crystal towards the centre of the furnace reduces the axial gradients and the insulated top reduces the radial gradients. Using a. small crucible for crystal growth reduces both the a.xia.l and ra.dia.l gradients provided that the crystal is surrounded by the walls of the crucible during the post growth cooling stage. Chapter 7. Temperature Measurements 116 1 cm 6cm 1 cm 1.65 cm TC Positions TC1 TC2 fl tL:utiem 1n’1 TC3 TC4 0.4 cm Figure 7.70: Alumina aggregate used for determining the thermal gradients in the furnace. Chapter 7. Temperat tire Measurements Insulation Cover Model crystal for measuring temperature gradient during crystal cooling 117 Furnace Element Figure 7.71: The axial positions of the alumina model crystal used when measuring the thermal gradients. Insulation Large Crucible Posn. A 2.2 cm Posn.B Small Cruc ,le Bottom insulation Chapter 7. Temperature Measurements 118 No Crucible Small Crucible Large Crucible Large Crucible (66 mm diameter) (88 mm diameter) With Insulated Top ________________ I L ‘ ‘ i Li’ A ‘jj Gradients at A (C/cm): Gradients at A (C/cm) Gradients at A (C/cm): Gradients at A (C/cm): 35 54 2.9 fl 0.9 3.6 4.2 2.3 1.6 Gradients at B (C/cm): Gradients at B (C/cm): 3.0 > 2.5 > Figure 7.72: Temperature gradients measured for the different furnace configurations as indicated. Chapter 7. Temperature Measurements 119 Condition Temperature Gradient °C/crn Radial Axial TC1—TC2 TC3—TC4 TC1—TC3 TC2—TC4 1. No Crucible (a) crystal in upper position (A) 3.5 3.6 7.3 7.2 (h) crystal in lower position (B) 3.0 3.3 4.6 4.3 2. Small Crucible (a) crystal in upper position (A) 5.4 4.2 2.5 3.5 (h) crystal in lower position (B) 2.5 2.6 2.6 2.5 3. Large Crucible crystal in upper position (A) 2.9 2.3 5.6 6.1 4. Large Crucible crystal in upper position (A) and seed hole insulated. 0.9 1.6 5.3 4.7 Table 7.9: Temperature gradients measured for the different furnace configurations as indicated. Chapter 8 Mathematical Model For LBO Crystal Growth 8.1 Scope of Model and Assumptions The objective of the model is to ({uantitativelv determine the temperature and veloc ity distributions in the melt during solidification as a function of the growth variables. This requires establishing the thermal conditions related to the furnace, calculating the thermal field, and fluid flow velocities, with and without crystal and crucible rotations. A fluid flow model, which incorporates hea.t transfer, has been developed by Fluid Dynamics International (FIDAP) [52]. The FIDAP package can be used to determine the velocities and temperatures in the melt during crystal growth. In growing a crystal, the melt temperature is adjusted t.o just above the liquidus temperature, a small seed crysta.l is dipped into the melt, and the temperature further lowered to allow the seed to grow. Crystal and crucible rotation is employed to enhance fluid flow in the melt. The crysta.l is slowly pulled awa.y from the melt, effectively under steady state conditions, once the required crystal diameter is reached by the seed. The model is used to predict the fluid flow a.nd temperature distribution in the melt a.t steady state. A number of assumptions were used in the model, to simplify the calculations. These are listed in Table 8.10, along with a. rating of the validity of each assumption and comments on the validity rating. The assumption of an axisymmetric temperature field is necessa.ry to simplify the calculations. The crystal growth furnace is designed to have an axisymmetric temperature field, and crystal and crucible rotation will tend to 120 Chapter 8. Mathematical Model For LBO Ciysta.l Growth 121 level out any local temperature variations in the melt. The assumption that the crystal growth system is at steady state is essential to the model, and is considered valid on the basis of the very slow cooling rate of 5°C/day or withdrawal rate of 1.6 mm/day for LBO. In comparison the withdrawal ra.te of Ge is much faster at 100 mm/hr. The assumption of laminar fluid flow is necessary for the model calculations. The assumption is reasonable for LBO since the melt has a high viscosity (10 to 100 poise). For flow past a. smooth plate, laminar flow breaks clown to turbulent flow when the Reynolds (Re) number exceeds 5 x iO. Re V L p It where V = flow velocity (cm/s), L is the characteristic length (cm), p = melt density (g/cm3)and it melt viscosity (poise). For a. limiting case situation, assume V = 1000 cm/s, L = 3.2 cm, R = 12.3 poise (40.9% MOO3 at 730°C) and p = 3.26 g/cm3. Using these values gives Re = 848, which is very much lower than 5 x i0, indicating that the flow is laminar. The air/melt interface is assumed to he flat, consistent with the high viscosity and low fluid velocities in the melt. Heat. transfer a.t the air/melt and air/crystal interface is assumed to be by convective heat transfer only, which is consistent with the low surface melt temperature (below 700°C). The model assumes that the melt is homogeneous during crystal growth. As the crystal grows, MoO3 is rejected at the interface resulting in an increase in the concentration of MOO3 in the melt ahead of the crystal. The LBO/Mo03thermophysicai properties and the dimensionless groups of interest are listed in Table 8.11 and Table 8.12 respectively. Thermophysical values which are not available in the literature are estimated. The Grashof, Prandtl and Schmidt numbers calculated for a LBO/Mo03 liquid and for liquid gallium are given in Table 8.12. The values for Gallium are typica.l for a liquid H CD z CD C CD C C C Co C Co Co , 1 Z r- t- C Co Co Co C - CD — . e - C - C D CD — . S e - CD • CD 5 . . c -. Cd ) CD — Co ‘ Co C . - , ‘ CD 2 . 5 C o C - CD — C z CD CD Cl) Cd) CD CD -J C C CD Co C J) Co Co 0 CD C C C Cd) CM CD S (Th S x I. 9 C Co CD 0 C- ) C C S C C- CD g ) ji , X w - O O — ‘ - - - C - < c ) C C C S - - - - Ct’ - Co D q ) CD < CD < c- q- CD — . Co c C C - 0 - C C D O ’ . c- _ C Chapter 8. Mathematical Model For LBO Crystal Growth 123 Group Definition Gallium LBO + MoO3 ( 24.6 % Mo) Grashof Number (Cr) 2.4 x 106 0.0827 Prandtl Number (Pr) 0.024 172k IISchmidt Number (Sc) fl— 2.4 x 102 4 x iO Table 8.12: Non Dimensional Numbers for Gallium and LBO. metal system. The small Crashof number for the LBO/Mo03liquid indicates buoyancy forces are smaller than viscous forces which implies that natural convection will be much smaller than forced convection. Gallium, in comparison, will have strong natural con vection. A small Prandtl number (< 1) which is the ratio of thermal mass diffusivity, as present with gallium, indicates tha.t the energy diffusion rate is much larger than the momentum diffusion rate. The large Prandtl number for LBO/Mo03(172) indicates that the momentum diffusion rate is much larger than the energy diffusion rate. Thus for a given boundary layer, the thermal boundar layer thickness is much smaller than the ve locity boundary layer (6 < 6). The Schmidt. number is the ratio of momentum and mass diffusivities. For convective mass transfer in laminar flow the Schmidt number is propor tiona.l to the velocity and concentration boundary layers (Sc 6/) The LBO/Mo03 Schmidt number is large at 4 x i0 indicating that any concentration boundary layer will be much thinner than the velocity boundary layer. In contrast liquid metals. as shown by gallium, have nondimensional parameters tha.t are very different to that of LBO/Mo03. 8.2 Idealized Domain and Description of Calculations The crystal growth domain adopted for the mathematical model is shown in Figure 8.73. Note that the model assumptions make the domain a simplified version of actual growth conditions. For example, the crucible walls are assumed to be smooth. The bottom Chapter 8. Mathematica.l Model For LBO Ciystai Growth 124 of the crucible is assumed to be flat for all calculations except when the temperature measurements are compared to the model calculations for the the small crucible. The rotation of the crucible and crysta.l are assumed not to he eccentric. Cylindrical polar coordinates are used in the model. Thus there exists a radial, rotational (0) and axial component for all variables. In the case of the velocity vector, it consists of a radial velocity, theta. velocity and an axial velocity component (Vr, v6, vi). It is noted that the rotational component is treated differently from the axial and radial components due to the assumption of axial symmetry. The 0 component of the velocity and temperature variables are real, but they are constant and the spatial derivative is zero (g = o). The model is applied to small and large crucibles, 66 mm in diameter and height and 88 mm in diameter and height respectively. The sensitivity analysis, determina tion of the dominant parameters and model validation were conducted using the small crucible. The comparison of the model predictions with the experimental temperature measurements for different crucible rotations were conducted on the small and large cru cibles. A simulated crystal, formed from a platinum sheet, Figure 4.31, was used for these measurements. 8.3 Steady State Axisymmetric Fluid Flow Model 8.3.1 Equations of Fluid Flow - Lagrangian Coordinates The conservation of mass, momentum and energy govern the flow of a viscous fluid. In order to present them in their simplest form the equations are formulated in Lagra.ngian coordinates. In this case the reference frame is moving with the fluid and the equations are applied to a moving control volume of fluid. The conservation of mass in the con trol volume states tha.t the mass in the control volume is constant with time, = 0. Chapter 8. Mathematical Model For LBO Crystal Growth 125 Center Line Fluid ( LBO + MoO ) height radius —> Platinum Crucible z Figure 8.73: Schematic representation of domain examined with the model. Ghapter 8. Mathematical Model For LBO Ciystai Growth 126 Expressing this equation in terms of the density of the fluid gives. d (p volume) dL = (8.1) The conservation of momentum follows from Newtons second law being applied to the control volume, F = m. Dividing by the volume gives the force per unit volume, changing the mass term to a density term. f = p (8.2) The energy equation describes the change in internal energy of the control volume (dEe). It is equal to the sum of the change in heat added (dQ) and the work done (dW). dE1 = dQ + dW (8.3) 8.3.2 Equations of Fluid Flow - Euler Coordinates Converting the equations of fluid flow from Lagrangian coordinates to fixed (Euler) co ordinates gives equations that are easily applied to real situations. The following section will give an overview of the final forms of the equations. Converting to Euler coordinates requires that the Material Derivative () be used on any property that changes with time. d D a - where the div term (V) in cylindrical coordinates is (a ia a ar r ãO Ôz The fluid being examined is incompressible, that is p is constant, which is used to sim plify the equations. Using the material derivative on the conservation of mass equation (Equation8.1) yields the following in Euler coordinates. pV = 0 (8.4) Chapter 8. Mathematical Model For LBO Ciystal Growth 127 Likewise, the conservation momentum equation in Euler coordinates becomes: - - = .foa’y + .fsurjace The force term has been divided into a body and surface force. The only body forces act ing on the fluid are due to gravity. The body force is p. The Boussinesq approximation is used to account for a density change with temperature. This is done by varying the density in the body force term using Po (1 — /3AT), where ,8 is the expansion coefficient of the liquid, and AT is the temperature change. The surface forces consists of the external stresses acting on the sides of the control volume. A length derivation of the stresses acting on the control volume gives the surface forces as fsurjace = where is the stress tensor. Manipulated, the equation further gives: fmrjace —Vp + V2 () where p is pressure and IL S viscosity. Substituting these into the conservation of mo mentum equation gives final form of the momentum equation: = Po(’ — AT) Vp + V2 (8.5) The energy equation in Euler coordinates is DE DQ DW The incompressible form of the energy equation is v (kv) (8.6) The equations are further simplified by using the penalty function approach for the pressure variable. For this method the conservation of mass equations is discarded and the pressure is eliminated from the momentum equation using p = —Vv. is the Chapter 8. Mathematicai Model For LBO Crystal Growth 128 penalty parameter. The problem is solved for steady state; thus all time derivatives are zero. The resulting equations that are solved by FIDAP are: Po ( . v) = p(i — — V (v) + V2 (8.7) pc ( . v) = v (h.v) (8.8) The type of flow solved for this problem is described as STRONGLY coupled. This term is used when the full set of equations are solved simultaneously. Nine noded quadratic elements are used in the model. Velocity and temperature degrees of free dom are present at each node. 8.3.3 Temperature Boundary Conditions The temperature boundary conditions used in the model were experimentally measured as discussed in Chapter 4. The location of the boundary conditions are different for each set of model calculations. The temperature distribution along the wall and bottom of the platinum crucible were considered as fixed temperature boundaries, assumed to he linear between points, and independent of time. The top of the fluid and crystal were modeled a.s a convective temperature boundary condition. The ambient temperature was approximated using the experimental measurements. The convective heat transfer coefficient was taken as 0.006 W/cm2,which is an average value for a natura.l convection heat transfer surface. The values of the temperature b&undary conditions used for each set of calculations will be given prior to the presentation of the results. Sensitivity analysis and Parameter Examination These calculations focus on the fluid flow in the melt. As a result the model only considers the melt and the inside surface of the platinum crucible, which simplifies the calculations. Chapter 8. Mathematical Model For LBO iystal Growth 129 The temperature measurements were conducted at a furnace setting higher than that used for crystal growth. The temperatures used in the model are the measured values less the temperature difference between the furnace setting at which the measurements were made and the crystal growth furnace setting which gives the thermal field for a crystal that is approximately 2/3 of the crucible diameter. Comparison with Experimental Data These calculations examine the change in thermal field with crucible rotation, with the simulated crystal in the melt. The boundary conditions are taken from the experimental measurements tha.t have no crucible rotation with the simulated crystal in the melt. The melt temperatures are extrapolated to give the boundary condition temperatures which are located at the outside of the crucible. Minor adjustments are done to the boundary temperatures by fitting the model predictions to the experimental melt temperatures. 8.3.4 Velocity Boundary Conditions For the velocity boundary conditions it is assumed that no slip occurs at the crucible and crystal surfaces in contact with the melt, and that free slip occurs at the air-melt surface. The azimuthal velocities are dependent. on the crystal and crucible rotation rate, L, and !C respectively. The velocity boundary conditions are as follows: Bottom of Crucible vz 0 Vr = 0 1)9 Il(rps) x 27r x r(cm) Chapter & Mathematical Model For LBO Crystal Growth 130 Side of Crucible = 0 VT = 0 = x 2ir x r(crn) Fluid/Air Interface = 0 No Constraint = No Constraint Crystal/Fluid Interface = 0 = 0 = x (rps) x 27r x r (cm) Fluid Center Line = No Constraint = 0 V6 0 Chapter 8. Mathematical Model For LBO Crystal Growth 131 8.3.5 Solution Procedure Sensitivity analysis and Parameter Examination The calculations were carried out in sets of iterations. The initial shape of the crys tal/melt interface wa.s taken as flat since the thermal field is unknown prior to the first calculation. ‘With the solid-liquid interface position and the boundary conditions defined. the model predicts the fluid velocities and temperature field using the successive substitu tions iterative method. The new interface position, predicted by the model, is compared to the position from the preceding model calculation. If they differ by more than 0.08 mm the interface location is adjusted and the fluid velocities and temperature field are recalculated. The adjustment of the interface and model calculations continues until the defined crystal/melt boundary location is within 0.08 mm of the calculated value. Comparison with Experimental Data The calculations with the simulated crystal were carried out in two steps. First, the boundary conditions were adjusted until the model accurately predicted the thermal field of the melt for the case of no crucible rotation. These boundary conditions were then used in the model which determines the thermal field as for different crucible rotations. Chapter 9 Sensitivity Analysis of the Fluid Flow Model A sensitivity analysis was conducted to examining the flow due to natural and forced convection, and to establish how variations in the mesh density, viscosity and condllc tivity values used in the model affect the calculations. The values of the mesh densities, conductivities and viscosities considered in the sensitivity ana.lysis are given in Table 9.14. The mathematical model analysis was conducted for the small, 6.6 cm diameter, cru cible with a fluid height of 2.5 cm. The temperature boundary conditions employed in the sensitivity analysis are shown in Figure 9.74. These values were obtained by measure ments, Figure 7.53, which were higher than usually employed for LBO crystal growth. The ambient gas temperatures, determined in Appendix A, were also approximately 145°C higher. A correction of —145°C was applied to these measurements to bring them into the correct range. The established values of the thermophysical properties are given iii Table 9.13. The determination of the thermal conductivity used in the calculations is given in Appendix A. The MoO3 concentration for all of the calculations is 40.9 Wt.% IVIoO3 with the exception of the viscosity analysis where all three MoO3 concentrations are considered. It is noted that. the liquidus is assumed to be 650°C for all sensitivity analysis calculations. 9.1 Natural Convection The vector plot of the flow that occurs due to natural convection is shown in Figure 9.75. The average magnitude of the flow velocity is low a.t approximately i0 cm/s. This is 132 Chapter 9. Sensitivity Analysis of the Fluid Flow Model 133 Convective Heat Transfer T(C) =55+l87.8.r_23.7.r2 ttt z = 2.5 (--) MELT I z=O.7an( z=Ocm1723Q r=3.3cm r=1.6an r=Oczn Fixed Temperature Figure 9.74: Temperature boundary conditions used in the sensitivity analysis. ‘ — , - L .J C.Z t’. Z — . Cl) c i 3 ( 0 0 0 D O D s. C) CC ) CC ) C D c c , 2. — 1 - Cl) Cl) Cl) j © c 9 c - r . 1 ‘ — ; I 1 C ) 1 - cc , p P — :i cc , 0 o © c , x x — ‘ — - - 9 1 + + + © 0 D è è c , ) 4 ) x x x I . c : D c c Cl ) C)’ x x x — - - - Ct, Cr ) 0 0- Ct 0 Cl) 0 C)) Cl) 0- Cl) Ct C)) C)) C)) - D . o - B 0 - C i ) C) Cl) < e C CC) c _ C l ) r c C — D , x c, ) x c p D C ) C C C i Cl) 0 . l— x C C ,, x (. % - Chapter 9. Sensitivity Analysis of the Fluid Flow Model 135 due to the high viscosity and low thermal expansion coefficient () of the LBO/Mo03 melt. The flow pattern is counter clockwise moving up the wall and down below the crystal. There is a stagnant region, marked “S” in Figure 9.75, that the fluid rotates around. The flow regime shown in Figure 9.75 results from the density gradient in the fluid, which is directly related to the temperature distribution. The density variation is proportional to the temperature difference the melt is above the liquidus, given by P = Po (1 — IBT). The temperature contours in the melt are shown in Figure 9.76. The hottest region of the fluid corresponds to the area were the melts density is a minimum. In the LBO/Mo03melt the hottest region is located near the outside bottom of the crucible. This low density fluid rises upwards and displaces the other fluid as it advances. The displaced fluid near the outside surface of the melt moves under the crystal were it drops to the bottom of the crucible, then moves horizontally towards the hottest region to replace the less dense fluid tha.t is moving upward. The fluid velocities are examined at various locations in the melt. The axial velocities along radia.l lines a.t 0.25, 0.5, and 0.75 of the height of the fluid are shown in Figure 9.77. The flow is upward near the crucible wall and downwards under the crystal at velocities near 3 x i0 and 3.2 x i0 cm/s respectively. The vertical flow is a. maximum a.t 0.5 of the fluid height, and is a minimum, approaching zero, both below the centre of the crystal, and a.t approximately the 2.2 cm radia.l position. The radial velocity along vertica.l lines positioned at 0.25, 0.5 and 0.75 of the crucible radius are shown in Figure 9.78. The radial flow is negative below the crystal and positive near the bottom of the crucible with velocities nea.r —2.4 x i0 cm/s and 2.5 x iO cm/s respectively. The largest ra.dia.l velocity occurs at 0.5 of the crucible radius. The velocity tangential to the crysta.l surface at 0.25, 0.5, and 0.75 of the ra.dius of the crystal is shown in Figure 9.79. The largest fluid velocity below the crystal occurs at 0.75 of the crysta.1 radius. The smallest tangential velocity occurs near the centre of the Chapter 9. Sensitivity Analysis of the Fluid Flow Model 136 crystal. The variation on tangential fluid velocity is proportional to the amount of mass transfer that occurs ahead of the crystal. This is important since inclusion formation or interface breakdown occurs when the mass transfer of MoO3 away from interface is lower than the rate of MoO3 being rejected at the growing crystal surface. Thus, inclusion or interface breakdown is less likely to occur at outer regions of the crystal where the magnitude of the tangential velocity is a maximum. 9.2 Forced Convection 9.2.1 Crystal Rotation The velocity and temperature fields are calculated for a crystal rotation rate of 12 rpm and no crucible rotation. The velocity vectors are shown in Figure 9.80. The magnitude of the flow that occurs in the axial and radial directions is approximately 5 x i0 crn/s, which is an order of magnitude higher than the flow that occurs due to natural convection. There is also a rotational component of the fluid velocity. Figure 9.81, which is large being 2.6 cm/s near the edge of the crystal. Thus the flow that moves upward below the crystal and downward at the side of the crucible will also swirl in the 0 direction as it moves under the crystal. This type of flow has been previously explained in Chapter 6. The viscous drag of the rotating crystal surface creates an upward swirling flow. The rotation of the crystal causes the fluid near it to move in the 0 direction. The imbalance between the fluids centripetal acceleration (—w2r) and pressure cause it to move outward in the radial direction. Fluid moves upward below the crystal to balance the outward flow in the radial direction. The fluid moving in the radial direction reaches the crucible walls and them drops to the bottom of the crucible to take the place of the fluid that moves upward below the crystal. There is a stagnant region, marked i’S” in Figure 9.80, that the fluid rotates around. Chapter 9. Sensitivity Analysis of the Fluid Flow Model CrystalO rpm Crucible 0 rpm Reference Vector 0.002 cm/s 137 — — _. — r — — —. -a-— — •••_._‘...\\ktt t I -. \ \ . \ I I ttt’ I- 4’ ‘ 4 4 I ‘ f ft ft ( ‘I ‘ __Hftfttt I I III 4,? 4?l S4J / 4 I fl I — — F, / I , \ \\ \ ‘N \ ‘.\ \ \‘ — - - I ‘I \ “ — -. — — — — - I I I *. S % — . * — — .—.asb S*•— — — -: I f. -: 1.0 1.5 2.0 2.5 3.0 Radial Position (cm) 2.5 2.0 1.5 1.0 0.5 0.0 — 0.0 It’ 0.5 Figure 9.75: Vector plot of fluid velocity due to natural convection. Chapter 9. Sensitivity Analysis of the Fluid Flow Model I Crystal 0 rpm Crucible 0 rpm 0.0 2.5 2.0 1.5 1.0 0.5 0.0 Radial Position (cm) 1:38 0.5 1.0 1.5 2.0 2.5 3.0 Figure 9.76: Temperature contours that occur in the LBO/Mo03melt. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 139 crystal Rotation 0 rpm Crucible Rotation 0 rpm 40.9 Wt.% MoO3 Liquid 1.50 1.00 B 0.50 A 0 00 7 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 3.0 0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) Figure 9.77: Axial velocity due to natural convection. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 140 Crystal Rotation 0 rpm Crucible Rotation 0 rpm 40.9 Wt.% MoO3 I C 1.00 C 0.00 ____________________________ I .1 ,,..I.... I 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 2.50 I 2.00 I E Radial Velocity * i04 (cm/s) Figure 9.78: Radial velocity due to natural convection. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 141 Crystal Rotation 0 rpm Crucible Rotation 0 rpm 40.9 Wt.% MoO3 : 1.50 A B o 1.00 Ct 0.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 0.00 C ,-- - - 0.50 / / \. B A 1.00 1.5 -3.0 -1.0 Tangential Velocity * i04 (cm/s) Figure 9.79: Tangential crystal surfa.ce velocity due to natural convection. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 142 The temperature contours, Figure 9.82, are similar to the natural convection tem perature contours. The flow velocities a.re not large enough to affect the shape of the contours. The flow velocities in the axial directions at liquid heights of 0.25, 0.5, and 0.75 of the height of the fluid are shown in Figure 9.83. The liquid flows upwards at a velocity of approximately 6 x 103cm/s below the crystal and downward at —3 x iO cm/s near the crucible wall. As with natura.l convection, the largest axia.l flow occurs at 0.5 of the fluid height and approaches zero 1)0th directly below the crystal and near the 2.0 cm radial position. The flow velocity in the radia.l direction a,t positions 0.25, 0.5 and 0.74 of the crucible radius are shown in Figure 9.84. The radial velocity is positive at approximately 4 x i0 cm/s, below the crystal and negative, —4 x i0 crn/s, near the bottom of the crucible. The largest radial velocity occurs at 0.5 of the crucible radius. The velocity tangential to the crystal surface at 0.25, 0.5 and 0.75 the radius of the crystal is shown if Figure 9.85. The fluid velocities coincide along the tangential lines at 0.5 and 0.75 of the crystal radius reaching 6 x 10 cm/s. The fluid velocity is lower towards the center of the crystal, reaching a maximum of 3 x iO cm/s. The variation in the magnitude of the tangentia.l velocity with radius is similar to that obtained for the natural convective flow. 9.2.2 Crucible Rotation The flow is calculated for a crucible rotation rate of 4.5 rpm and no crystal rotation. The vector plot of the flow that occurs is shown in Figure 9.86. The magnitude of the axial and radial flow is approximately 6 x 10—2 crn/s, which is two orders of magnitude higher that the flow that occurs during natural convection. The rotational component of the flow is shown in Figure 9.87, with maximum flow near the crucible wall at 14 cm/s and minimum flow, approaching zero below the centre of the crystal. The flow resulting Chapter 9. Sensitivity Analysis of the Fluid Flow Model 143 Crystal 12 rpm Reference Vector Crucible 0 rpm 0.02 cm/s 2.5 - — — — — — — _ — — — — — — S S S S S - - — — — — — — ___r_r _._41’ e.e_ S S ‘. S S 44 2 0 — — — — — — — ... — — .r’-’’’ -...———-.—. - - -‘ s i ; ; ; ; o ;; ; ; ;; ..v -. I I I I I ‘/ — -. - % 5%\ \ \ 4 I : 1.5 1 ttt ‘ ‘ ‘ t ‘ ‘ 5’.-, 111144 4444444’• ‘• 10- ““ — / / / / / /44444 ‘S\ \4 \i\\\ .—..I--._ .. — / / - - / 4 5 — — - ,,, , 0.5 - tt t\\\\\.---.-- - ....... t t 14 5\ .. \ \‘.-.i---.—*— — .— .— .— — — — — — — - . 4 t % 5. % 5.. — — — .— — — — —. — — — - , . t 1 S S S S ‘. S ‘.5. 5. 5.. 5..-..-. .— .— - — - - I I I I I I I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Radial Position (cm) Figure 9.80: Vector plot of fluid velocity due to crystal rotation. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 144 Va (cm/s)2.5 2.74432 2.51563 2.0 2.28693 2.05824 1.82955 1.5 1.60085 1.37216 1.14347 10 0.914774 0.68608 1 0.457387 0.5 .: 0.228694 o.o I I Crystal 12 rpm 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Crucible 0 rpm 40.9 Wt% MoO3Radial Position (cm) Figure 9.81: Rotational velocity plot of the LBO/Mo03melt with crystal rotation. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 145 I Crystal 12rpm Crucible 0 rpm 0.0 0.0 2.5 2.0 1.5 1.0 0.5 Radial Position (cm) Figure 9.82: Temperature contours that occur in the LBO/Mo03melt with crysta’ ro 0.5 1.0 1.5 2.0 2.5 3.0 tation. chapter 9. Sensitivity Analysis of the Fluid Flow Model 146 crystal Rotation 12 rpm Crucible Rotation 0 rpm 40.9 Wt.% MoO3 2.50 2.00 C 1.50 1.00 B 0.50 A 0 00 _______________________________________________ 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) o 6.0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) Figure 9.83: Axial velocity due to crystal rotation. Chapter 9. Sensitivity Analysis of the Fluid Flow Model Crystal Rotation 12 rpm Crucible Rotation 0 rpm 40.9 Wt.% MoO3 2.50 o 1.00 c —.4 0.50 0.00 0. 147 A B ***3, .1 I,...,, 30 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 2.50 2.00 I E 0.00 - B - -4.0 -2.0 0.0 2.0 4.0 6.0 Radial Velocity * 1 O (cm/s) Figure 9.84: Radial velocity due to crystal rotation. I C..B Radial Position (cm) 3Tangential Velocity * 10 (cmls) Chapter 9. Sensitivity Analysis of the Fluid Flow Model 148 Crystal Rotation 12 rpm Crucible Rotation 0 rpm 40.9 Wt.% MoO3 I 2.50 2.00 1.50 1.00 0.50 0.00 0. 0.50 1.00 1.50 2.00 2.50 3.00 Figure 9.85: Tangential crystal surface velocity due to crystal rotation. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 149 from the crucible rotation is similar to that of crystal rotation. except for the viscous drag at the rotating crucible surface which results in an downward swirling flow. There is a stagnant region, marked “S” in Figure 9.86, around which the fluid rotates. The rotational or theta component of the velocity causes the fluid to swirl as it moves in the radial and axial directions, which is important. The flow predicted by the mathematical model is similar to what was shown by the physical model. The rotational and swirl of the fluid as determined using the physical model are shown in Figures 6.44 through 6.47 in Chapter 6. The high fluid flow velocities resulting from crucible rotation markedly affect the temperature isotherms in the melt. The calculated isotherms with crucible rotation are shown in Figures 9.88. These can be compared with the corresponding isotherms for natural convection shown in Figure 9.76, which are markedly different. With large crucible rotation, hot liquid moves upward adjacent to the crucible wall resulting in the upward movement of the isotherms in this region. The corresponding downward flow of the cold liquid under the crystal moves the isotherm downward. The flow velocities in the axia.l direction at liciuicl heights of 0.25, 0.5 and 0.75 of the height of the fluid are shown in Figure 9.89. The liquid flows upward at a velocity of 6 x 102cm/s near the edge of the crucible and downward a.t at —6 x i0 cm/s under the centre of the crystal. As with all the previous cases the largest axial flow occurs at 0.5 the fluid height. The axial flow approaches zero near the crystal as shown by the 0.75 fluid height velocity tra.ce (line C on Figure 9.89) which goes to zero as the line intersects the crystal. The radial velocity at positions 0.25, 0.5 and 0.75 of the crucible radius are shown in Figure 9.90. The liquid flow directly below the crystal is inwards at a value of —6 x 102 cm/s. The flow near the bottom of the crucible is outward and has a maximum value of 6 x 102 cm/s. The largest radial velocity occurs at 0.5 of the crucible radius. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 150 The velocity tangential to the crystal surface at 0.25, 0.5, and 0.75 of the radius of the crystal is shown if Figure 9.91. The fluid velocity decreases with decreasing radius, and changes from —7.6 x 10—2 cm/s at 0.75 of the crystal radius to —2 x 102 cm/s at 0.25 of the crystal radius. 9.3 Mesh Size In order to determine the influence of mesh size on the accuracy of the calculations the temperature and velocity fields were calculated using two node densities, 585 and 1795, shown in Figure 9.92. The calculations were conducted for a crystal rotation of 6 rpm and a crucible rotation of —-18 rpm. The axial, radial and tangential velocities are examined at 0.5 of the fluid height, 0.5 of the crucible radius and 0.5 of the crystal radius respectively. The axial velocity fields for the two selected mesh sizes are shown in Figure 9.93 and the corresponding radial velocities in Figure 9.94. In both cases it is evident that the velocities are the same for both mesh sizes. The calculated fluid velocity tangential to the crystal surface, Figure 9..5. shows a small difference, less than 5 x iO cm/s, between the two mesh sizes. The results indicate that a mesh density of 585 or 1795 nodes ca.n be used in the calculations without introducing errors. 9.4 Fluid Viscosity The effect of varying the viscosity in the calculations of the temperature and velocity fields was examined for melt concentrations of 29.7, 36.2 and 40.9 Wt% MoO3. The corresponding viscosity of the melts is 86, 30 and 12 poise at 730°C. For these calculations the crystal and crucible rotations were zero. The axial velocities for the three melts at 0.5 of the fluid height are shown in Figure 9.96. The axial liquid velocity of the melt with the lowest viscosity, (40.9 Wt% MoO3) is small, approaching 3 x iO cm/s under Chapter 9. Sensitivity Analysis of the Fluid Flow Model CrystalO rpm Crucible 45 rpm Reference Vector 0.25 cm/s 151 - —-S — — -: —— # — - — — _ — .- — — ‘5 — - — — -5 ‘5 ‘5 ‘S — — S — _5 _% ‘ ‘ -————- ‘- “ ‘S \ \ i -.5” \ ‘S ‘S -: - .— — —.5” “ ‘S ‘S ‘S 4 1I ]J 1/ ‘S ft ft - riftfff \fffFfff .s’s’ -w .‘y A’ ft. - _wA , ç ç — -‘‘‘ 4 4 ‘ ,. 5’ ‘S • ‘*. — ‘I I S S S S S S •. % a..st’’ — — — — — - - - .5-. .5 - — — -s * -. -. — - — - -. -‘ I I . I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Radial Position (cm) 2.5 - I 2.0 1.5 1.0 0.5 0.0 ‘5% ‘5’ ‘5’ ‘ft ‘ft ‘ft ‘ft ‘ft ‘ft ‘ft ‘ft tt ft ft ft ft ft ft ft ft ft I ft t ft 4, 4, 1’ ii I’ ft. ft I ft. 14 1. ‘I I. I, 4, I, Figure 9.86: \/ector plot of fluid velocity due to crucible rotation. Va (cni/s) 13.9126 12.7532 11.5938 10.4344 9.27507 8.11568 6.9563 5.79691 4.63753 3.47815 2.31876 1.15938 Chapter 9. Sensitivity Analysis of the Fluid Flow Model 152 I! 2.5 - 2.0 - 1.5 - 1.0 - 0.5 - 0.0 0.0 0.5 1.0 1.5 Ciystal 0 rpm 2.0 2.5 3.0 Crucible 45 rpm Radial Position (cm) 40.9 Wt% MoO3 Figure 9.87: Rotational velocity plot of the LBO/Mo03melt with crucible rotation. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 153 CrystalO rpm Crucible 45 rpm 2.5 :__— 630 2.0 650 1.5_7Z” 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Radial Position (cm) Figure 9.88: Temperature contours that occur in the LBO/Mo03 melt with crucible rotation. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 154 Crystal Rotation 0 rpm Crucible Rotation 45 rpm 40.9 Wt.% MoO3 2.50 () : 1.50 1.00 0.50 0.00 _____________________________________ 0.00 B A I ..I. .1.. .1 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) () * IS C.) C - -0.5 -1.0 2.00 2.50 Radial Position (cm) Figure 9.S9: Axial velocity due to crucible rotation. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 155 Crystal Rotation 0 rpm Crucible Rotation 45 rpm 40.9 Wt.% MoO3 2.5O 0 OC -, __________________________ 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 2.50 2.00 A o5o Radial Velocity * 10 (cm/s) Figure 9.90: Radial velocity due to crucible rotation. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 156 Crystal Rotation 0 rpm Crucible Rotation 45 rpm 40.9 Wt.% MoO3 o 1.00 0.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 000 I - - Tangential Velocity * 10 (cmls) Figure 9.91: Tangential crystal surface velocity due to crucible rotation. / Chapter 9. Sensitivity Analysis of the Fluid Flow Model 157 (a) _________ (b) Figure 9.92: Two mesh densities used to examine the models sensitivity. (a) Regular mesh density, approximately 1795 nodes. (h) Coarse mesh density, approximately 585 nodes Chapter 9. Sensitivity Analysis of the Fluid Flow Model Crystal Rotation 6 rpm Crucible Rotation -48 rpm 40.9 Wt.% MoO3 2.50 2.00 1.50 C’) 0 C * 0 C cc 158 iquid Velocity Sample1.00 0.50 0.1 I . ,.I . .1 .1 I )0 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 1.0 0.5 0.0 -0.5 -1.0 Radial Position (cm) Figure 9.93: Axial velocity for different mesh densities. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 159 Crystal Rotation 6 rpm Crucible Rotation -48 rpm 40.9 Wt.% MoO3 C.) C/) (ID C 0 .2 —4 > “-4 Radial Position (cm) 2.50 2.00C) O 1.50 ..- (/D 1.00 0.50 0.00 Liquid 2.50 2.00 1.50 1.00 0.50 0.00 0. 0.50 1.00 1.50 2.00 2.50 3.00 -1.0 -0.5 0.0 0.5 1.0 Radial Velocity * 10 (cm/s) Figure 9.94: Radial velocity for different mesh densities. Chapter 9. Sensitivity Analysis of the Fluid Flow Model Cryst2l Rotation 6 rpm Crucible Rotation -48 rpm 40.9 Wt.% MoO3 1.00 0.50 (1 flfl 0.00 Crystal Velocity Sample 0.50 1.00 1.50 2.00 2.50 3.00 160 I Radial Position (cm) Tangential Velocity * 10 (cm/s) 0.0 2.50 Liquid -1.0 -0.5 Figure 9.95: Tangential crystal surface velocity for different mesh densities. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 161 the crystal. Increasing the viscosity of the melt from 12 poise (40.9 Wt% MoO3) to 30 poise (36.2 Wt% MoO3) decreases the magnitude of the axial velocity by a factor of 4. Increasing the viscosity to 86 poise (29.7 Wt% MoO3) effectively reduces the velocity to zero. The corresponding radial and tangential calculated flow velocities for the viscosities considered are shown in Figure 9.97 and 9.98. For both the radial and tangential velocities the velocities are shown to be strongly reduced as the melt viscosity is increased. Clearly, the melt viscosity/MoO3concentration is a significant factor in calculating fluid flow in an LBO/Mo03melt. 9.5 Conductivity The effect of thermal conductivity on the melt temperatures and velocities was examined. Three melt thermal conductivities were considered, 0.04, 0.0.5 and 0.06 W/cm K. The conductivities of the crystal and the melt was assumed to he the same. Calculations were carried out for a. crystal rotation of 6 rpm and crucible rotation of —12 rpm. The resulting calculated temperature profiles at 0.5 of the crucible radius are shown in Figure 9.99. The axial temperature gradient is observed to decrease progressively with increasing conductivity. The interface position changes by 2 mm with a 20% change in conductivity. Changing the conductivity does not change the temperature a.t the bottom of the crucible, since it is a fixed temperature l)oundary condition. The corresponding axial, radial and tangential melt velocities are shown in Fig ures 9.100, 9.101, and 9.102 respectively. The axial velocity (Figure 9.100) is observed to increase significantly with increasing conductivity. The change in axial velocity is due to the change in crystal size. The crystal size increases in the axial direction with decreasing conductivity, moving the zero velocity boundary condition at. the crystal/fluid interface closer to the axial velocity sample location. The sample velocity will decrease the closer Chapter 9. Sensitivity Analysis of the Fluid Flow Model 162 crystal Rotation 0 rpm A = 29.7 Wt.% MoO3 Crucible Rotation 0 rpm B = 36.2 Wt.% MoO3 C = 40.9 WL% MoO3 1.50 1.00 Velocity Samples 0.50 0 00 _______________________________________________ 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 3.0 0 / \ 2.0 V 1.0 / * /‘_- X:tzzzzzzzz:/ ; •_C 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) Figure 9.96: Axia.1 velocity for different viscosities. Chapter 9. Sensitivity Analysis of the Fluid Flow Model Cxystal Rotation 0 rpm Crucible Rotation 0 rpm A=29.7Wt.%Mo03 B = 36.2 Wt.% MoO3 C = 40.9 Wi% MoO3 163 2.50 2.00 I E 0.00 2.50 0.00 8 > iws , .1..., .1 .1, I.,,,. )0 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) B A I I . . . -2.0 0.0 2.0 4.0 Radial Velocity * i04 (cmls) Figure 9.97: Radial velocity for different viscosities. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 164 Ciystal Rotation 0 rpm Crucible Rotation 0 rpm A = 29.7 Wt.% MoO3 B = 36.2 Wt.% MoO3 C=40.9Wt.% MoO3 Velocity Samples 2.50 O 1.00 .— 0.50 0.00 0.1 I: )0 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) -4.0 -3.0 -2.0 -1.0 0.0 Tangential Velocity * 1 O (cm/s) Figure 9.98: Tangential crystal surface velocity for different viscosities. Chapter 9. Sensitivity Analysis of the Fluid Flow Model 165 the zero velocity boundary condition gets to the sample location. The radial velocity, Figure 9.101, and tangential velocity, Figure 9.102. are also ob served to increase with increasing conductivity. However, the change in velocity at these locations is smaller than the change in the axial velocity. The decrease in the radial and tangential velocity is believed, again, to be due to the increase in crystal radius. The value of the conductivity has a. significant affect on the thermal fields in the melt and less of an effect in the liquid velocity fields. Chapter 9. Sensitivity Analysis of the Fluid Flow Model Ciystal Rotation 6 rpm Crucible Rotation -12 rpm 40.9 WL% MoO3 I A=0.04 W/cmK B = 0.05 W/cmK C=0.06W/cmK 166 I 2.50 2.00 1.50 1.00 0.50 0.00 Radial Position (cm) Temperature (C) Figure 9.99: Axial temperature profiles for different conductivities. Chapter 9. Sensitivity Analysis of the Fluid Flow Model C) C * >-I 4I .— 0 0 1.00 0.50 n ne A =0.04 W/cmK B =0.05W/cmK C=0.06W/cmK Velocity Sample 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) Figure 9.100: Axial velocity for different conductivities. Crystal Rotation 6 rpm Crucible Rotation -12 rpm 40.9 Wt.% MoO3 2.50 2.00 1.50 0.1 10.0 167 5.0 0.0 -5.0 -10.0 Radial Position (cm) Chapter 9. Sensitivity Analysis of the Fluid Flow Model I I 2.50 —Liquid 00 2.50 2.00 1.50 1.00 0.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) Radial Velocity * i03 (cm/s) 2.00 1.50 1.00 0.50 168 Crystal Rotation 6 rpm Crucible Rotation -12 rpm 40.9 Wt.% MoO3 A =0.04W/cmK B =0.05W/cmK C =0.06 W/cmK 0.00 0. Figure 9.101: Radial velocity for different conductivities. Chapter 9. Sensitivity Analysis of the Fluid Flow Model A=0.04W/cmK B =0.05W/cmK C = 0.06 W/cmK Radial Position (cm) 169 I 0.00 0.50 1.00 1.50 Tangential Velocity * i03 (cmls) Crystal Rotation 6 rpm Crucible Rotation -12 rpm 40.9 Wt.% MoO3 2.50 o 1.00 \ _\ \ Velocity Sample 0.00 0.00 I . .1 0.50 1.00 1.50 2.00 2.50 3.00 -8.0 -6.0 -4.0 -2.0 0.0 Figure 9.102: Tangential crystal surface velocity for different conductivities. Chapter 10 Modeling Results The mathematical model was used to examine the effect of crystal and crucible rotation rates and crucible size on the interface shape and fluid velocity fields. The analysis in Chapter 9 indicates that the fluid velocity decreases rapidly with increasing melt viscosity. To obtain reasonable flow rates from the calculations, the minimum melt viscosity was selected by using a MoO3 concentration of 40.9 wt% in the melt, based on the composition at 698°C in Figure 5.40. The values of the thermopbysica.l properties used are given in Table 10.15, the same values used in the sensitivity analysis. A small crucible (6.6 cm diameter) was used for all but one calculation. The large crucible dimensions were used in comparing the flow fields in a. large and small crucible with the same crucible rotation rates. The temperature boundary conditions are the same a.s those used in the sensitivity analysis and are given in Figure 10.10:3. The boundary conditions used with the large crucible are given in Figure 10.104. The crucible and crystal rotation rates examined are given in Table 10.16. Crystal rotation influences the shape of the solid/liquid interface. The calculations with only cryst.a.l rotation were directed towards determimng the rotation rate which results in the interface becoming concave to the liquid. clue to the upward flow of hot liquid under the crystal. Crucible rotation also influences the shape of the solid/Iic1u d interface. Crucible rotation causes the cold liquid under the crysta.l to move downward. This will result in the solid/liquid interface becoming convex to the liquid. The calculations with only crucible rotation were to determine maximum rotation rate before the crystal 170 Chapter 10. Modeling Results 171 Property Units Values Specific Heat J/g K 0.63 Density g/cm3 3.26 Viscosity for poise 40.9 Wt% MoO3 exp (37.31 — 0.0784 x T[C] + 4.21 x 10 x (T[Cj)2) Conductivity W/cm K 0.05 Liquidus Celsius 698 Expansion K—’ 6 x 106 Coefficient (3) Table 10.15: Thermophysicai properties used in the results analysis. becomes extremely concave to the melt. Modeling a large rotating crucible (8.8 cm diameter) with a. stationary crystal was to determine the affect tha.t the crucible size has on the flow due to crucible rotation. Combined crysta.l and crucible rotation will create different flow fields than what occurs with only crucible or crystal rotation. Thus, combined crystal and crucible rotation rates were examined to determine the resulting flow fields that occurred. Both isorotation and counter rotation of the crucible and crystal are examined to determine their affect on the flow fields in the melt. Chapter 10. Modeling Results 172 Convective Heat Transfer T(C) = 102 + 187.8 r —23.7 r2 ttt tt = 2.5 cm C MELT I H I z=O.7cm I z = 0 ii ________________________________________________ r=3.3cm r=1.6an r=Ocm Fixed Temperature Figure 10.103: Small crucible temperature boundary conditions used in the results anal ysis. Chapter 10. Modeling Results i 73 Convective Heat Transfer T(C) =628 ttt - z = z=25cm 70W I MELT I z = 1.7 an z =0 an 701 \7çç r=4.4cm r=3.5czn r=Ocr Fixed Temperature Figure 10.104: Large crucible temperature houndar conditions used in the resZ:s ar ysis. Chapter 10. Modeling Results 174 Mixing Conditions Crucible Size Units Values Crystal Rotation and no Crucible Rotation Crystal Rotation small rpm 0, 10, 20 Crucible Rotation rpm 0 Crucible Rotation and no_Crystal_Rotation Crystal Rotation small rpm 0 Crucible Rotation rpm 0, 20, 40, 60 Crystal Rotation large rpm 0 Crucible Rotation rpm 60 Isorotation of Crystal and Crucible Crystal Rotation small rpm 10 Crucible Rotation rpm 30 Counter Rotation of Crystal and Crucible Crystal Rotation small rpm 10 Crucible Rotation rpm —30 Table 10.16: Parameters examined for the mathematical model analysis. 10.1 Crystal Rotation The effect of varying the crystal rotation rate on the temperature and velocity fields was examined. For these calculations the crystal rotation rates were 0, 10 and 20 rpm and the crucible rotations were zero. The solid/liquid interface and the axial velocities for the three rotation rates at 0.5 of the fluid height a.re shown ii Figure 10.105(b). The axial liquid velocity of the melt at zero crystal rotation is negative at —3 x i0 cm/s under the crystal. Increasing the crystal rotation rates caiases the flow direction to change and move upward under the crystal and down at the side of the crucible. At 10 rpm the fluid velocity is 1.0 x 10—2 cm/s below the crystal and —4.0 x i0 cm/s near the crucible wall. At 20 rpm the fluid velocity is 6.8 x 10—2 cm/s under the crystal and —2.4 x 10 cm/s near the crucible wall. The solid/liquid interface, shown in Figure 10.105(a), moves upward and becomes flat with increasing crysta.l rotation, due to the increase in hot fluid moving upward below the crystal at higher rotation rates. The model predicts that the Chapter 10. Modeling Results 175 solid/liquid interface becomes concave to the melt at rotation rates larger than 20 rpm. The flow velocities in the radial direction at 0.5 of the the crucible radius are shown in Figure 10.106(b). The radial velocities below the crystal are —3 x 10 cm/s, 8 x 102 cm/s and —4.4 x 10—2 cm/s for crystal rotation rates of 0, 10 and 20 rpm respectively. The radial fluid velocities near the bottom of the crucible are 3 x iO cm/s, —7 x 10—2 cm/s and —3.8 x 102 cm/s for crystal rotation rates of 0, 10 and 20 rpm. The magnitude of the fluid velocity under the crystal is larger than the velocity near the crucible bottom for a given rotation rate, the crystal rotation being the dominant mechanism for the fluid motion. The velocity tangential to the crystal surface at 0.5 of the radius of the crystal is shown if Figure 10.107(b). The fluid velocity tangential to the crystal interface increases with increasing crystal rotation. The velocities are 3 x i0 cm/s, 0.9 x 10_2 cm/s and 4.1 x 10_2 cm/s for crystal rotation rates of 0, 10 and 20 rpm. 10.2 Crucible Rotation The thermal and fluid velocity fields were calculated with crucible rotation to determine the effect of the rotation rate on the shape of the solid/liquid interface and the magnitude of the fluid velocities. Crucible rotation rates of 0, 20, 40 and 60 rpm, with no crystal rotation, were examined. The axial fluid velocities along a radial line at 0.5 of the fluid height are shown in Figure 10.108. The axia.l velocity directly under the crystal is near constant increasing from —3.2 x 102, —2.2 x 10. and —3.3 x 10 cm/s for crucible rotation rates of 20, 40 and 60 rpm. The smallest change occurs between crucible rotation rates of 0 and 20 rpm. The axial velocity upward near the crucible wall increases with increasing crucible rotation being 4.4 x 10_2, 1.2 x 10 and 1.4 x 10 cm/s for rotation rates of 20, 40 and 60 rpm. The largest increase in axial velocity at this location occurs Chapter 10. Modeling Results 176 Crystal Rotation: A =0 rpm, B 10 rpm, C = 20 rpm Crucible Rotation 0 rpm 2.50 2.00 B - - - - - Liquid 1.50(a) o 1 00 Velocity Samples 0.50 0.00 .1 I.. .1,,., 00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 6.0 0 C 4.0 (b) 2.0 0.0 A - / / N -2.0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) Figure 10.105: Axial velocities at 0.5 of the fluid height. 40.9 Wt% MoO3 present in the fluid. Crystal rotated at 0, 10, and 20 rpm. Crucible is stationary. (a) Shape of the solid/liquid interface. (h) Axial Velocities. Chapter 10. Modeling Results 177 Crystal Rotation: A =0 rpm, B = 10 rpm,C =20 rpm Crucible Rotation 0 rpm (a) Liquid . 0.50 > 0 00 ? 1 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 2.50 I C 200 B (b) >< 0.50 ‘ I” 0.00 I -2.0 0.0 2.0 4.0 Radial Velocity * 102 (cm/s) Figure 10.106: Radial velocities at 0.5 of the crucible radius. 40.9 Wt% MoO3 present in the fluid. Crystal rotated at 0, 10 and 20 rpm. Crucible is stationary. (a) Shape of the solid/liquid interface. (b) Radial velocities. Chapter 10. Modeling Results Velocity Samples are tangential 0.50 to the crystal interface at 1/2 of the crystal radius 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) Figure 10.107: Velocities tangential to the crystal surface 0.5 of the crystal radius. 40.9 Wt% MoO3 present in the fluid. Crystal rota.ted at 0, 10 and 20 rpm. Crucible is stationary. (a) Shape of the solid/liquid interface. (b) Tangential velocities. Crystal Rotation: A =0 rpm, B = 10 rpm, C =20 rpm Crucible Rotation 0 rpm 178 2.50 2.00 1.50 1.00 ZLiquid I(a) (b) I . I. I 0.0 1.0 2.0 3.0 4.0 Tangential Velocity * 102 (cm/s) Chapter 10. Modeling Results 179 between 40 and 60 rpm. The curvature of the crystal increases with increased crucible rotation, becoming more concave to the liquid. This is due to movement of hot fluid upward at the crucible wall and downward below the crystal. Increasing the crucible rotation rate causes the crystal to get smaller maintaining the same interface shape. The radius of the crystal is 2.4, 2.3, 1.6 and 1.1 cm for crucible rotation rates of 0, 20, 40 and 60 rpm respectively. The radial velocities along a horizontal line at 0.5 of the crucible radius is shown in Figure 10.109. The radial fluid velocities near the bottom of the crucible are 0.6 x 10, 1.7 x 10_i and 2.3 x 10 cm/s for crucible rotation rates of 20, 40 and 60 rpm. The radial velocities near the surface are —0.5 x 10, —1.5 x 10_i and 2.2 x 10_i cm/s for crucible rotations of 20, 40 and 60 rpm. The radial velocities near the crucible bottom are higher than the velocities near the surface since crucible rotation produces the fluid motion. The fluid velocity tangential to the crystal surface are shown in Figure 10.110(b). The tangential velocities are —0.5 x lO’, —1.6 x lO and —2.1 x 10_i for crucible rotation rates of 20 40 and 60 rpm. 10.3 Comparison of Crystal to Crucible Rotation Fluid velocities are determined for a crvst.a.l rotating at 20 rpm with a stationary crucible. and a crucible rota.ting at 20 rpm with a stationary crystal. The magnitude of the axial velocities along a radia.l line at. 0.5 of the fluid height. are shown in Figure 10.111. The magnitude of the a.xia.l fluid velocity below the crvsta.l is largest with crystal rotation decreasing from 6.8 x 10_2 cm/s at the centre line to zero at a radial position of 1.85 cm. The flow below the crystal due to crucible rotation is a maximum at 1.2 cm from the centre of the crucible, with a. velocity of 4.8 x 10_2 cm/s. The axial flow velocity near the crucible wall is 2.4 x 102 and 4.4 x 10_2 cm/s for crucible and crystal rotation Chapter 10. Modeling Results 180 Czystal Rotation 0 rpm Crucible Rotation: A =0 rpm, B =20 rpm,C =40 rpm, D =60 rpm 250 CryStal j ii 2.00 1.50(a Velocity Samples 0.50 0.00 o.o 0.50 i.bô i.ô ZbÔ iÔ ibô Radial Position (cm) - 1.0 C.) /,./ \ A ,7/ _--.\o 0.0 ‘I - - - ,—•1 7/7 * B //(b) -1.0 __---/ -2.0 c ,/ -3.0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) Figure 10.108: Axial velocities at 0.5 of the fluid height. 40.9 Wt% MoO3 present in the fluid. Crucible rotated at 0, 2O, 40, and 60 rpm. Crystal is stationary. (a) Shape of the solid/liquid interface. (h) Axial velocities. Chapter 10. Modeling Results isi Crystal Rotation 0 rpm Crucible Rotation: A =0 rpm, B =20 rpm,C =40 rpm, D =60 rpm 2 50 •Crystal L___ I I : Liquid(a) • C 1.00 0.50 0.00 I .1,. .1. I..,. 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 2.50 I I 2.00 B’ A (b) o.so I I -2.0 -1.0 0.0 1.0 2.0 Radial Velocity * 10 (cmls) Figure 10.109: Radial velocities at 0.5 of the crucible radius. 40.9 Wt% MoO3 present in the fluid. Crucible rotated at 0, 20, 40, and 60 rpm. Crystal is stationary. (a) Shape of the solid/liquid interface. (h) Radial velocities. Chapter 10. Modeling Results 182 Crystal Rotation 0 rpm Crucible Rotation: A 0 rpm, B =20 rpm,C =40 rpm, D =60 rpm 250 cryStal I 2.00 zrZ_- Liquid 1.50(a) L00 Velocity Samples are tangential 0.50 to the crystal interface at 1/2 of the crystal radius .1... .1.... I 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) . A AA ______________________________________ -- > 0.50 D, / U A(b) ‘s.’ 1.00 ..-.---.- .5-..-.’ S. c5.) ..- o -. Cl) 1.50 -2.0 -1.5 -1.0 -0.5 0.0 Tangential Velocity * 10 (cm/s) Figure 10.110: Velocities tangential to the crysta.l surface 0.5 of the crystal radius. 40.9 Wt% MoO3 present Crucible rotated at 0, 20, 40, and 60 rpm. Crystal is stationary. (a) Shape of the solid/liquid interface. (h) Tangentia.1 velocities. Chapter 10. Modeling Results 183 respectively. - The radial fluid velocities along a vertical line at 0.5 of the crucible radius are shown in Figure 10.112. The velocities near the crystal are approximately the same value for each rotation method. The radial velocities being 4.3 x 102 and 4.7 x 102 cm/s for crucible and crystal rotation respectively. The radial fluid velocity near the crucible bottom are 3.6 x 10_2 and 5.9 x l0_2 cm/s for crystal and crucible rotation respectively. The radial velocity due to crucible rotation is larger near the bottom of the crucible and near the surface. This is due to the larger diameter of the crucible as compared to the crystal diameter. The crucible being larger is able to rot.ate fluid at a higher theta velocity (swirl) thus giving it a larger centripeta.l acceleration. The velocities tangential to the crvsta.l surface at. 0.5 of the crystal radius are shown in Figure 10.113. The velocities are 4.0 x 102 and 5.2 x 10_2 cm/s for crystal and crucible rotation respectively. As with the ra.dia.l velocity the larger tangential velocity is due to crucible rotation. The magnitude of the largest fluid velocity tangential to the crystal interface as a function of crystal or crucible rotation is shown in Figure 10.114. The tangential velocity due to crystal rotation with a stationary crucible goes from near zero at no crystal rotation to 4 x 10_2 at 20 rpm. Higher crystal rotation ra.tes cannot be used due to melt back of the crystal. The tangential velocity with crucible rotation and a. stationary crysta.l goes from near zero at no crucible rotation to approximately 20 x 10_2 cm/s at 60 rpm. For a given rotation rate crucible rotation produces a. larger tangential velocity t.han crystal rotation. Much higher tangential velocities are thus attainable possible using crucible rotation. Chapter 10. Modeling Results 184 A: Crystal Rotation =20 rpm, Crucible Rotation =0 rpm B: Crystal Rotation =0 rpm, Crucible Rotation 20 ipm 2.50 - - Liquid Q2.00 • 1.50(a) o 1 00 Velocity Samples 0.50 0.00 .1,., .1., .1,... I 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) CtD 6.0 A Q 40 ‘“1.:::)) —::—• — — — — — — — — — — ,j// \ 2.0 ,:‘ 0o ô.ô i.bô i.ô ibô’ iô ibô Radial Position (cm) Figure 10.111: The magnitude of the axial velocities along a horizontal line at 0.5 of the fluid height. The two conditions examined are crucible rotated at 20 rpm with a stationary crystal and a stationary crystal with a. crucible rotating at 20 rpm. (a) Shape of the solid/liquid interface. (h) Axial velocities. Chapter 10. Modeling Results 185 A: Crystal Rotation =20 rpm, Crucible Rotation =0 rpm B: Crystal Rotation =0 rpm, Crucible Rotation 20 ipm 2.50 - - Liquid 2.00 o 1.50 I I ‘m aj O 1.00 — . 0.50 > 0.00 0.00 0.50 i.00 i.50 2.00 2.50 Radial Position (cm) 150- 0 1.0 2.0 3.0 4.0 5.0 6.0 Radial Veiocity* 102 (cm/s) Figure 10.112: The magnitude of the radial velocities along a vertical line at 0.5 of the crucible radius. The two conditions examined are crucible rotated at 20 rpm with a stationary crystal and a stationary crystal with a crucible rotating at 20 rpm. (a) Shape of the solid/liquid interface. (b) Radial velocities. I (b) 186Chapter 10. Modeling Results A: Crystal Rotation =20 rpm, Crucible Rotation =0 rpm B: Crystal Rotation =0 rpm, Crucible Rotation 20 rpm 2.50 Crystal A _-__4_I__ (a) : 1.00 Velocity Samples are tangential 0.50 to the crystal interface at 1/2 of the crystal radius 0.0c 0. Liquid )0 0.50 1.00 1.50 2.00 2.50 Radial Position (cm) 3.00 0.0 3.0 4.0 5.0 Tangential Velocity 102 (cm/s) Figure 10.113: The magnitude of the velocities tangential to the crystal surface at 0.5 of the crystal radius. The two conditions examined are crucible rotated at 20 rpm with a stationary crystal and a stationary crystal with a crucible rotating at 20 rpm. (a) Shape of the solid/liquid interface. (b) Tangential velocities. Chapter 10. Modeling Results 187 I 111111 I I I I I j — 20.0- - C) 15.0- I ci) 0 C i— 1 (\ r ci) IU.V c c . I, 5.0- ,‘ \* - 0 .11I I I I I I I I10 20 30 40 50 60 Rotation Rate (rpm) Figure 10.114: Magnitude of the tangential velocity 0.5 cm from the liquid/crystal inter face for different crystal and crucible rotation rates. The calculations are for a rotating crucible with a stationary crystal and a stationary crucible and rotating crystal. Chapter 10. Modeling Results 188 10.4 Comparison of the Flow Fields in Small and Large Crucibles Fluid flow velocities in a small and large crucible rotated a.t 20 rpm, with no crystal rotation are compared. The axial fluid velocities along a radial line at 0.5 of the respective fluid heights are shown in Figure 10.115. The variation of the axial velocity with radius is similar for both crucibles. The large crucible has fluid velocities that are —5.3 x 10_i cm/s downward under the crystal and 2.8 x 10_i upward at the wall. These are approximately double the velocities in the small crucible, which are —3.6 x lO cm/s downward below the crystal and 1.2 x 10_i cm/s upward near the wall. The radial velocities along a vertica.l line at 0.5 of the crucible radius are shown in Figure 10.116. The radial velocities in the large crucible are 4.3 x 10_I cm/s outward nea.r the bottom of the crucible and —4.5x 10 cm/s inward at the top of the melt. The radia.l velocities in the small crucible are approximately half of the large crucible velocities, being 2.4 x 10_i cm/s outward near the bottom of the crucible and —2.1 x 10_i cm/s inward at the top of the melt. The melt velocity tangential to the crystal interface at 0.5 of the crystal radius is shown in Figure 10.117. The tangential fluid velocity is 5.0 x 10 cm/s and 2.1 x lO cm/s for the large and small crucible respectively. Fluid velocity is proportional to crucible size for crucible rotation driven flows. The larger the crucible that faster the theta fluid velocity near the outside of the melt. This in turn gives the fluid a larger centripetal acceleration which increases the overall flow velocity of the melt. 10.5 Iso and Counter rotation of the Crystal and Crucible The axial, radial and tangential flow velocities are calculated for isorotation and counter rotation of the crysta.l and crucible. The rotation rates are +10 and 20 for the crystal and crucible respectively. The tangential velocity is calculated 1 mm from the crystal interface for crucible rotation of 20 rpm and crystal rotation between —35 and 20 rpm. Chapter 10. Modeling Results 189 A: Small crucible, crystal =0 rpm, crucible =60 rpm B: Large crucible, crystal = 0 rpm, crucible =60 rpm 0 C * > 0 C > .,- Velocity Samples 3.50 Crystal - 3.00 - - - - -, _________ Liquid 2.50 2.00 B: 1.50 1.00 A 0.50 °°8oo o.sö i.bo i.ö 2.0020 3..04.o Radial Position (cm) 4.0 2.0 0.0 (a) (b) Radial Position (cm) Figure 10.115: Axial velocities at 0.5 of the fluid height. Large a.nd small crucible shown. 40.9 Wt% MoO3 present in the fluid. Crucible rotated at 60 rpm. Crystal is stationary. (a) Shape of the solid/liquid interface. (b) Axial velocities. -2.0 -4.0 -6.0 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 Chapter 10. Modeling Results A: Small crucible, crystal 0 rpm, crucible 60 rpm B: Large crucible, crystal = 0 rpm, crucible =60 rpm 190 3.00 C) C (ID C — ..- 3.00 2.50 2.00 0 .— .—( (ID C - 1.00 0.50 0.00 V rF) C.) 0 V > Radial Position (cm) 3.50 Crystal 2.50 2.00 1.50 1.00 0.50 °•°8o iö i.b Liquid 2.00 2.50 3.00 3.50 4.00 (a) (b) 1.50 Radial Velocity * 10 (cmls) Figure 10.116: Radial velocities at 0.5 of the crucible radius. Large and smaii crucible shown. 40.9 Wt% MoO3 present in the fluid. Crucible rotated at 60 rpm. Crys:al i stationary. (a) Shape of the solid/liquid interface. (h) Radial velocities. Chapter 10. Modeling Results 191 A: Small crucible, crystal 0 rpm, crucible =60 rpm B: Large crucible, crystal = 0 rpm, crucible =60 rpm 3.50 Crystal - 3.00 --‘ -- - - Liquid 2.50 a 2.00 ‘B 1.50 Velocity Samples 1.00 0.50 ______________________________________ ‘‘I 30 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 Radial Position (cm) Tangential Velocity * 10 (cmls) Figure 10.117: Velocities tangentia.l to the crystal surface at 0.5 of the crystal radius. 40.9 Wt% MoO3 present. Large and small crucible shown. Crucible rotated at 60 rpm. Crystal is stationary. (a) Shape of the solid/liquid interface. (1)) Tangential velocities. I (a) I (b) 0.00 0.50 1.00 1. -5.0 -4.0 -3.0 -2.0 -1.0 0.0 Chapter 10. Modeling Results 192 The axial fluid velocity along a vertica.l line at 0.5 of the fluid height is shown in Figure 10.118. The crystal and crucible rotation rates a.re ±10 and 20 rpm respectively. The curves in the Figures are marked I and CR. for isorotation and counter rotation respectively. For both rotation cases the fluid velocity is downward under the crystal and upward near the crucible wall. This fluid motion occurs because the crucible rotation rate is higher and dominates the melt flow. For counter rotation (CR) larger fluid velocities are obtained than for isorotation (1). The axial velocities due to counter rotation (CR) are —5.2 x 102 cm/s downward under the crysta.l and 5.0 x 102 cm/s upward near the crucible wall. The axial velocities due to isorotat.ion (I) are —3.2 x 10_2 cm/s downward under the crystal and 2.8 x 102 cm/s upward near the crucible wall. The solid liquid interface is different for the two different rotation modes. Counter rotation causes the crysta.l interface to be more convex to the melt and the radius smaller. This is due to the larger fluid velocities that occur with counter rotation. The radial fluid velocities at a vertical line at 0.5 of the crucible radius are shown in Figure 10.119. The radial velocities are higher when the crystal and crucible are counter rotated. The radial velocities due to counter rotation are —5.2 x 1O_2 cm/s inward under the crystal and 6.8 x 10_2 cm/s outward near the crucible bottom. The radial velocities due to isorotation are —3.2 x 10_2 cm/s inward under the crystal and 4.0 x 10_2 cm/s outward nea.r the crucible bottom. The melt velocity tangentia.l to the crvsta.l surface at 0.5 of the crystal radius are shown in Figure 10.120. The maximum tangential velocities for iso and counter rotation are —3.5 x 10_2 cm/s and —5.8 x 102 cm/s respectively. The tangential fluid velocity due to counter rotation is 60% higher than the isorotation case. The explanation of the difference in the fluid velocity due to iso and counter rotation requires the calculation of the flow patterns as a function of crystal rotation rate for a fixed crucible rotation. The fluid velocity at 0.5 of the crystal interface will he calculated for crystal rotation rates Chapter 10. Modeling Results 193 A: Crystal Rotation =10 rpm, Crucible Rotation =20 rpm B: Crystal Rotation = -10 rpm, Crucible Rotation =20 rpm 6.?joo 0.50 1.00 1.50 2.00 Radial Position (cm) Figure 10.118: Axial velocities at 0.5 of the fluid height. 40.9 Wt% MoO3 present in the fluid. Crystal rotated at ± 10 rpm and crucible rotated at 20 rpm. (a) Shape of the solid/liquid interface. (h) Axial velocities. 2.50 2.00 1.50 1.00 Velocity Samples 0.50 (a) (b) I c/) a C) * C) C )0 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 0.00 0. 6.0 4.0 2.0 0.0 -2.0 -4.0 - A 2.50 3.00 Chapter 10. Modeling Results 194 A: Ciystal Rotation = 10 rpm, Crucible Rotation =20 rpm B: Crystal Rotation = -10 rpm, Crucible Rotation =20 rpm 2.50 (a) 0 1.00 0.50 0.00 ç 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Radial Position (cm) 2.50 2.00 (b) >< 0.50 - 000’’’’’’ -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 2Radial Velocity * 10 (cm/s) Figure 10.119: Radial velocities at 0.5 of the crucible radius. 40.9 Wt% MoO3 present in the fluid. Crystal rotated at ± 10 rpm a.nd crucible rotated at 20 rpm. (a) Shape of the solid/liquid interface. (b) Radial velocities. Chapter 10. Modeling Results 195 between —35 and 20 rpm at a crucible rotation rate of 20 rpm. The tangential velocity at 1 mm from the crystal surface is used to examine the change in flow patterns and is shown in Figure 10.121. Four points, marked A to D, indicate positions where the flow pattern changes. At point A both the crystal and crucible are rotating at 20 rpm. With both rotating at the same rate the fluid moves as a solid body in the theta (swirl) direction. This result in natural convection being the only force that moves the fluid in the radial and axia.l directions. The tangential fluid velocity is near zero and is similar to the natural convection fluid velocity calculations of Chapter 8. Decreasing the crystal rotation rates, moves the tangential velocity to point B, where crucible rotation dominates the flow. Fluid now moves U adjacent to the vall and down under the crystal. The maximum flow in this direction occurs with the crystal rotation at zero, point D. The tangential fluid velocity 1 mm from the crvsta.l interface is —2.6 x 102 cm/s. A vector plot of the flow pattern is shown in Figure 10.122. The size of the reference vector in this plot and for vector plots at the different rotation conditions is the same, 0.1 cm/s. The flow with crucible rotation of 20 rpm and crvst.a.l rotation of 0 rpm goes up at the wall and down under the crystal. The change in flow patterns between B and C is different than between A and B. Increasing the crystal counter rotation rate results in the fluid rotating in a direction opposite to that of the crucible. At point C the crystal is rotating at a sufficiently high rate, that the centripeta.l acceleration of the fluid under the crystal is large, causing the fluid to move outward. Thus, instead of the crystal matching the crucibles rotation rate to reduce the flow, point A, the crystal must be rotating a.t a sufficiently high rate to cause the fluid under the crystal to become stagnant. The vector plot of the flow patterns near point C is shown in Figure 10.123. Directly under the crystal, marked UC, the flow is stagnant. IVloving from the crystal, the flow becomes significant, flowing up at the wall Chapter 10. Modeling Results 196 and down under the crystal. Increasing the crystal rotation rate, point C to D, causes the crystal to start to dominate the flow patterns. This will result in the flow changing from a stagnant state under the crystal, point C, to flowing upward under the crysta.l and downward at the edge of the crystal, point D. The tangential flow velocity 1 mm from the crystal interface is —3.0 x 10_2 cm/s at a crystal rotation of -35 rpm and a crucible rotation of 20 rpm, point D. The vector flow pattern at point D is shown in Figure 10.124. The flow has two cells. The flow directly under the crystal is upward at the centre of the crucible and outward. The flow flow near the crucible bottom is outward and upward at the crucible wall Both flows join and move outward and upward at the crucible wall and inward and downward at a 45 degree to the crucible bottom. Chapter 10. Modeling Results 197 A: Crystal Rotation 10 rpm, Crucible Rotation =20 rpm B: Crystal Rotation = -10 rpm, Crucible Rotation =20 rpm Radial Position (cm) Tangential Velocity * 102 (cm/s) Figure 10.120: Velocities tangential to the crystal surface at 0.5 of the crystal radius. 40.9 Wt% MoO3 present in the fluid. Crystal rotated at + 10 rpm and crucible rotated at 20 rpm. (a) Shape of the solid/liquid interface. (b) Tangential velocities. 1.50 1.00 Velocity Samples 0.50 Ann o. I (a) I (b) )0 0.50 1.00 1.50 2.00 2.50 3.00 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 Chapter 10. Modeling Results Crystal Rotation Rate (rpm) 198 Figure 10.121: Fluid velocity tangential to the solid/liquid interface as a function of crystal rotation rate. The velocity is 1 mm from the crystal/melt interface and at 0.5 of the crystal radius. 40.9 Wt% MoO3 present in the fluid. The crucible is rotated at 20 rpm. ‘‘‘‘‘I’ll D C 3.0 2.0 1.0 0.0 -1.0 C.) © * .,- C) 0 4-’ H L A I- -, I -30 -20 -10 0 10 20 Chapter 10. Modeling Results 199 Crystal 0 rpm Reference Vector Crucible 20 rpm 0.1 cm/s 2.5 - .- — _ _a_ _____..%S ‘ — — — .5%’’ ‘ , - — — - 5—.—. ‘5’• ‘S ‘ •L 2.0 - - \ ‘ -- J•..•, ‘I’ ht H • tH ttt • j/’’ •,,fff tt 1.0- • \\ _.Ir,W//fff H’ V —‘--“‘n’H,: - * “5-, 4 q \ , I 4 4 ‘S\\ \‘ - , I \ % -. . . _ — — — 5’ - -. .__-•_-.--..----*--,.--,. * - 1- - - ) • g- — — - • I I I I I I I I I I I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Radial Position (cm) Figure 10.122: Vector plot of fluid velocity at a crucible rotation rate of 20 rpm and a crystal rotation rate of 0 rpm. Point B. Chapter 10. Modeling Results 200 Crystal -23.5 rpm Reference Vector Crucible 20 rpm 0.1 cm/s 2.5 - — — — — — — . ‘ ‘ , ‘ ‘a” 2.0- • S . • ;—— • : -..- S - S . ;;— C : ;;--;1.5 - ttCID C -p0’i I / / / __.e__ ‘ 1A 4H L.U ,..tfTjl Its . HH \‘ ft 1!! ‘1 fit ‘I.—, . ‘. i \V\\ \ ‘% \ ‘t \ _‘*.‘• % \ •••• • * ‘ ‘ -. .. -. . . — . .. ‘a ‘a-* -a -‘a - - .. ‘a -a — - - -a - - -e -_ * — .+ * —a - - 0.0 I I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Radial Position (cm) Figure 10.123: Vector plot of fluid velocity at a crucible rotation rate of 20 rpm and a crystal rotation rate of —23.5 rpm. Near point C. Chapter 10. Modeling Results 201 Crystal -35 rpm Reference Vector Crucible 20 rpm 0.1 cm/s 2.5- . . ---— — — — — .;— - -—;;_-. S —r .U — f f , .._.•. ‘ ‘ ‘ ‘ ‘ - t , .,__)r_,r_—” .. I 4 i ‘ / / ‘ - - . - . ///_,__•_ S ‘--- . • o t i ‘‘S -1.5 • V . . : 0 . . V \ \ \ \ ‘I V \ \ \ \ \ \ ‘ t 1.0 \ \ \ t t t t t - . - - - - //i / / / / 1£ ‘—.‘y/f — “ - ‘ ‘ / ‘( .1 4 \ \ \ “% fit SI..) - I I \ \ \ \ \ \ \ S S S. ‘S \ S ‘ \ I-. - . — , — - - -. -a - 0.0 I I I I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Radial Position (cm) Figure 10.124: Vector plot of fluid velocity at a crucible rota.tion rate of 20 rpm and a. crystal rotation rate of —35 rpm. Point D. Chapter 11 Comparison of Temperature Measurements with Model Results In this section, the calculated temperature fields in melts contained in 6.6 cm and 8.8 cm diameter crucibles determined from the model are compared to the corresponding measured melt temperatures. The model boundary conditions were determined from the melt temperatures with no crucible rotation. The values used are given in Figure 11.12.5 and Figure 11.126 for the small and large crucible respectively. The melt composition, material properties and crucible rotation rates used for the calculations are given in Table 11.17. Melt temperatures were measured with two 3 mm outside diameter platinum sheathed thermoconples. The presence of the thermocouples in the rotating melt increases mix ing and produces three dimensional flow. The axisvmrnetric model used in this analysis cannot properly calculate this type of fluid flow. However, the increased mixing in the melt can be approximated by setting the theta velocity boundary condition to zero where the thermocouples enter the melt. The standard theta velocity boundary condition for an axisymmetric melt are shown in Figure 11.127. The modified theta velocity bound ary conditions are shown in Figure 11.128. Both boundary conditions are used in the calculations. 11.1 Small Crucible (6.6 cm diameter) Results The measured and calculated axial temperature profiles in the melt for the small non rotating crucible are shown in Figure 11.129. The axia.l temperature profiles at r=1.O, 202 Chapter 11. Comparison of Temperature Measurements with Model Results 203 Convective Heat Transfer — T(gas) = 750 —> I. SIMIJLAT9D =23 cm CRYSTAL1 1.0cm 23cm MELT I H I - z=0.7cm 4 3.5 cm >IA 05cm z= 0115cm / r=3 615 cm C 0.115 cm thickness z= -0.27 cm r=l6cm z=-0.615cm r=OtoO.54cm Fixed Temperature Figure 11.125: Temperature boundary conditions used for the small crucible model. Chapter 11. Comparison of Temperature Measurements with Model Results 204 Convective Heat Transfer T(gas)75O Htt It z=33cm ASTAL z=2.5cm 1.0cm z I-. z=1.7cm 3.3cm H -e I ‘1) ry 4.4cm _____________________________ Crucible z-0.Il5cm ________ .:.::.::.:. thicknessQ.H5 cz r = 4.5 cm r=3.5cm r=Ocm Fixed Temperature Figure 11.126: Temperature boundary conditions used for the large crucib modei.. Chapter 11. Comparison of Temperature Measurements with Model Results 205 theta velocity is unconstrained >tMYUs( theta velocity = 0 I ‘1) I1i1T’T m Q o lvariJ.i1 I 0 I . II I I. _ 0 C0 I o I - I I ..... ........................... ....... ..,.....,.................... .......................... •.! theta velocity = crucible rotation * 2 it r Figure 11.127: Theta velocity boundary conditions used in the model. chapter 11. Comparison of Temperature Measurements with Model Results 206 thermocouple theta velocity is unconstrained L9D / theta velocity 0 I — I C.) MELT I z TiDC.) II I C.) C.) — C I I > > i. I I .....:.:.:.:.:.;.:...:.:.:.:.:.:...........1 theta velocity = crucible rotation * 2 it r Figure 11.128: Theta velocity boundary conditions used in the model to account for the thermocouples in the melt. Chapter 11. Comparison of Temperature Measurements with Model Results 207 Property Unit. small crucible largecrucible Amount MoO3 Wt% 45.5 47.4 conductivity (1<) \V/cm K 0.05 0.05 Specific Heat (Cp) J/g K 0.621 0.625 density (p) g/cm3 3.14 3.19 viscosity (it) poise 3.27 2.82 Expansion Coefficient () K 6 x 10 6 x 10 crystal rotation rpm 0 0 crucible rotation rpm 0, 15, 20, 25, 30 0, 10, 20, 30 Table 11.17: Thermophysical properties and rotation values used in the model for com parison with experimental temperature measurements. 1.5, 2.6 and 3.1 are shown. The calculated temperatures are within 1°C of the measured temperatures at ra.dia.l locations of 2.6 and 3.5 cm The calculated temperatures near the centre of the crucible (r = 1.0 an(l 1.5 cm) are within 2°C of the measured values. The good fit of the calculated and measured temperature is expected since the experimental values are used to determine the temperature boundary conditions in the model. 11.1.1 Results Assuming no Thermocouple/Melt Interaction (Small Cru cible) This section assume that there is no interaction between the melt and the thermocouple. The velocity boundary conditions are shown in Figure 11.127. The experimental and calculated temperatures for a. crucible rotation rate of 15 rpm are shown in Figure 11.130. The measured axial temperature profiles near the centre of the melt (r = 1.0 and 1.5 cm) are constant near 820°C between z = 0.4 cm and 1.0 cm. Above 1.1 cm the temperature decreases with increasing height. The calculated temperature measurements at the same radial locations are within 2°C of the measured values. The measured temperatures at r = 2.6 and :3.1 cm radius initially decrease with Chapter 11. Compa.rison of Temperature Measurements with Model Results 208 I I E Calculated Measured r= 1.0 cm r=1.5cm r=2.6cm r=3.lcm ----- Q I I I I I I I . I I • 0.0 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 11.129: Experimental and calculated temperatures as a function of axial height. Small (6.6 cm diameter) crucible. Crucible rotation = 0 rpm* Chapter 11. Comparison of Temperature Measurements with Model Results 209 increasing z to 1.0 cm then appea.r to be constant. The calculated results at r = 2.6 cm nearly coincides with three of the temperature measurements and are higher at 1.0 cm with a slight plateau. At r = 3.1 cm the calculated values are all higher than the measured value, with no plateau. Figure 11.131 shows the measured and calculated axial temperature profiles at a higher crucible rotation rate of 20 rpm. Increasing the rotation rate to 20 rpm from 15 rpm does not significantly change the temperature distributions. The temperature profile at a crucible rotation rate of 25 rpm are shown in Figure 11.132. The measured axial temperature profiles a.t all locations become flatter and the temperature difference between the different axial profiles decreases. The calculated axial profiles near the centre of the melt (r=1.0 and 1.5 cm) l)ecome flatter and are within 2 to 3°C of the measured values. The calculated axial profiles near the crucible wall (r=2.6 and 3.1 cm) do not change significantly from the 20 rpm crucible rotation results. The temperature profiles at a crucible rotation rate of 30 rpm are shown Figure 11.133. The experimental temperatures at the four radial positions are essentially constant at 823°C. The calculated temperature measurements at r 1.0 and 1.5 cm becomes flat at 819°C and 820°C respectively. The calculated temperature at r = 2.6 and 3.1 cm do not change significantly from the 25 rpm case. 11.1.2 Results assuming Thermocouple/Melt Interaction (Small Crucible) The model calculations in this section have the theta velocity boundary condition set to zero where the thermocouples enter the liquid (Figure 11.128). The experimental and calculated temperature profiles at. 15 rpm are shown in Figure 11.134. The calculated axial temperature l)ro1les have a long plateau between z = 0.4 and 1.2 cm a.t all radial locations. At. a rotation rate of 20 rpm, Figure 11.135, the calculated results fit the measured results at z values less than 1.0 cm. At values greater than this Chapter 11. Comparison of Temperature Measurements with Model Results 210 I I I —-.- .—.—. — — 8 - 820 - --‘------ ., —- . Calculated Measured r=1.Ocm C r1.5cm r=2.6cm r=3.lcm -•-.—.—.- Q 800—. . ., •1 0.0 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 11.130: Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 15 rpm. Standard velocity boundary conditions are used (Figure 11.127). Chapter 11. Comparison of Temperature Measurements with Model Results 211 — — — — 820 _ _ _ _ _ _ _ _ 810 r Calculated Measured r1.Ocm r=1.5cm r=2.6cm — — — — r3.1cm 0 800 .1 I 0.0 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 11.13 1: Experimental and calculated temperatures as a function of axial height at the radial locations indicated, diameter) crucible. Crucible rotation = 20 rpm. Standard velocity boundary conditions are used (Figure 11.127). Chapter 11. Comparison of Temperature Measurements with Model Results 212 —L. I I I — — — 0 — 820 ----a- 810- Calculated Measured r=1.Ocm C r=1.Scm r=2.6cm ———— r=3.lcm -•-•--•- Q 800-i • • I •1 i • 0.0 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 11.132: Experimental and calculated temperatures as a. function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 25 rpm. Standard velocity boundary conditions are used (Figure 11.127). Chapter 11. Comparison of Temperature Measurements with Model Results 213 •L.I I I I § — 820 - - 4- - - - E 810- Calculated Measured r= 1.0 cm r=1.Scm r=2.6cm ———— r=3.lcm ----- Q 800—p . I I • I • 0.0 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 11.133: Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 30 rpm. Standard velocity boundary conditions are used (Figure 11.127). Chapter 11. Comparison of Temperature Measurements with Model Results 214 the measured temperatures decreases faster than the model predictions. The calculated results a.t a crucible rotation rate of 25 rpm, Figure 11.136, are similar to the 20 rpm results. The calculated axial temperature profiles near the crucible wall (r = 2.6 and 3.1 cm) are in good agreement with the measured results. Figure 11.137 shows the measured and calculated axial temperature profiles for 30 rpm crucible rotation. The calculated results have a good general fit with the measured values. The fit is not good near the crucible wall (r =3.1 cm), and the centerline temperature values (r = 1.0 and 1.5 cm) near z 1.5 cm. The calculated temperatures using the modified theta boundary condition, fit the measured temperatures reasonably well. 11.2 Large Crucible (8.8 cm diameter) Results The measured and calculated temperatures for the large crucible (8.8 cm diameter) with no crucible rotation are shown in Figure 11.138. The calculated temperatures are in good agreement with the measured temperatures close to the crucible wall (r greater than 2.8 cm). The calculated results at. r — 0.9 cm do not agree with the measured values. 11.2.1 Results Assuming no Thermocouple/Melt Interaction (Large Cru cible) Figure 11.139 shows the melt temperatures with the crucible rotating at 10 rpm. The calculated temperatures deviat.e markedly for the calculated values at all radial positions. with r = 4.2 cm showing the best fit. The calculated and measured results for crucible rotation rates of 20 and 30 rpm are shown in Figure 11.140 and 11.141 respectively. The calculated results are not in agreement with the measured values with the axial tem perature profile at r = 4.2 cm being in closest agreement. The measured and calculated temperature profiles a.t r = 0.4 cm behave similarly with with increasing CruCil)le rotation. Chapter 11. Comparison of Temperature Measurements with Model Results 215 •1 • I — — L E 810 Calculated Measured r=1.Ocm C r=1.5cm r=2.6cm ———— r=3.lcm -.-.—.-.- Q 800 — . • . I 0.0 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 11.134: Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 15 rpm. Modified velocity boundary conditions are used. (Figure 11.12$). Chapter 11. Comparison of Temperature Measurements with Model Results 216 I • I I I • E Calculated Measured r= 1.0 cm r=1.Scm r=2.6cm ———— r=3.lcm --•--- Q 8 . • I . I . I . 0.0 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 11.135: Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 20 rpm. Modified velocity boundary conditions are used. (Figure 11.128). Chapter 11. Comparison of Temperature Measurements with Model Results 217 • ••—.: . . I • I • • I I • I E Calculated Measured r=1.Ocm C r=1.5cm r=2.6cm ———— r=3.lcm ----- Q 8JJ — . I I . . . I . 0.0 0.5 1.0 1.5 2.0 Axial Position (cm) Figure 11.136: Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 25 rpm. Modified velocity boundary conditions are used. (Figure 11.128). Chapter 11. Comparison of Temperature Measurements with Model Results 218 I I • I I 82O<< 810 Calculated Measured r=1.Ocm C r=1.5cm r=2.6cm ———— 0 r=3.lcm ----- Q 800-. • . . . I . . . I 0.0 0.5 1.0 1.5 2.0 Axial Posifion (cm) Figure 11.137: Experimenta.1 and calculated temperatures as a function of axial height at the radial locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 30 rpm. Modified velocity boundary conditions are used. (Figure 11.128). Chapter 11. Comparison of Temperature Measurements with Model Results 219 Calculated Measured r=O.4cm D r=O.9cm A r=2.8cm — r=3.3cm Q r=3.8cm I I • I • • • • j I 830 820 810 Axial Position (cm) Figure 11.138: Experimental and calculated temperatures as a function of axial location along various vertical lines. Large (8.8 cm diameter) crucible. Zero crucible rotation. Chapter 11. comparison of Temperature Measurements with Model Results 220 Both the calculated and measured temperatures at z = 1.7 cm increase with increasing crucible rotation rate. The large difference between the calculated and measured results shows that there is more mixing in the melt than is predicted by the model. The results indicate that the thermocouples have a. large effect on mixing in the melt. 11.2.2 Results assuming Thermocouple/Melt Interaction (Large Crucible) The calculated and measured axial temperature profiles at a crucible rotation rate of 10 rpm are shown in Figure 11.142. The calculated axial temperature profiles at r = 2.8, 3.3 and 3.8 cm are approximately fiat and the temperature difference between the different a.xia.l profiles is small. The measured temperature profiles at the same locations are similar to the calculated values except that measured temperatures are higher than the calculated values. The measured and calculated temperature profiles at r = 0.9 cm differ from the profiles above 2.8 cm. The calculated temperatures are constant at 820°C between z = 0.5 and 2.5 cm. The measured temperatures are approximately constant at 826°C between z = 0.3 and 1.0 cm. The measured temperature above z = 1.0 cm decreases progressively to 818°C at. z 1.6 cm. The calculated and measured axial temperature profiles at a crucible rotation rate of 20 rpm are shown in Figure 11.113 The difference between the measured axial tem perature profiles has decreased with increasing crucible rotation. The calculated axial temperature profiles are similar to that for 10 rpm. The measured axial temperature profiles are constant at 828°C for all radia.l positions except r = 0.4 cm. At r = 0.4 cm the measured temperature is constant. at 828°C between z = 0.4 and 1.0 cm. Above z = 1.0 cm the temperature decreases progressively to 823°C at z = 1.6 cm. The calculated temperatures are constant, nea.r 824°C, for the axia.l temperature profiles at r = 3.8, 3.3 and 2.8 cm. The calculated axial temperature profiles at r = 0.4 and 0.9 are constant near 822°C. Chapter 11. Comparison of Temperature Measurements with Model Results 221 Calculated Measured r=O.4cm D r=O.9cm A r=2.8cm r=3.3cm —-- Q r=3.8cm I I • • . I • • • I 83O I :: zZ 8J’ I..Tho 0.5 1.0 1.5 2.0 2.5 3.0 Axial Position (cm) Figure 11.139: Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Large (8.8 cm diameter) crucible. Crucible rotation = 10 rpm. Standard velocity boundary conditions are used (Figure 11.127). Chapter 11. Comparison of Temperature Measurements with Model Results 222 Calculated Measured r=O.4cm D r=O.9cm A r= 2.8 cm r=3.3cm ‘---“ Q r=3.8cm I I • I I I 830 I ::: 80%d . Axial Position (cm) Figure 11.140: Experimental and calculated temperatures as a function of axial height at the radial locations indicated. La.rge (8.8 cm diameter) crucible. Crucible rotation = 20 rpm. Standard velocity boundary conditions are used (Figure 11.127). Chapter 11. Comparison of Temperature Measurements with Model Results 223 Calculated Measured r=O.4cm D r=O.9cm r=2.Scm r=3.3cm ‘S--- Q r=3.8cni I I • U U I U I I • I, 830 I ::: 1 I 0.5 1.0 1.5 2.0 2.5 3.0 Axial Position (cm) Figure 11.141: Experimental and calculated temperatures as a function of axia.l height at the radia.l locations indicated. Large (8.8 cm diameter) crucible. Crucible rotation = 30 rpm. Standard velocity boundary conditions a.re used (Figure 11.127). Chapter 11. Comparison of Temperature Measurements with Model Results 224 Figure 11.144 shows the measured and calculated temperature profiles at a crucible rotation rate of 30 rpm. All of the measured temperature profiles are constant at 830°C. The calculated values are constant between 822°C and 824°C. The calculated change in temperature profile with crucible rotation rate is similar to the change in measured temperature with crucible rotation. The most notable similarity is the decrease in tem perature difference between the axial profiles near the crucible wall (r = 2.8 to 3.8 cm) with increasing crucible rotation. Another similarity is the slow increase in temperature at the centre line of the crucible with increasing crucible rotation rate. 11.3 Summary of the Temperature Comparisons The model can predict the temperatures within 2 to 4°C of the measured temperatures if the correct boundary conditions and t.hermophysica.l properties are used. The accuracy of the model predictions at higher crucible rotation ra.tes indicates tha.t the calculated melt velocity must be near the actual melt velocity. The insertion of the thermocouple in the melt increases mixing. This is not important when predicting the melt temperatures and melt velocity during crysta.l growth since no thermocouple is present. Chapter 11. Compa.rison of Temperature Measurements with Model Results 225 calculated Measuid r=O.4cm D r=O.9cm r=2.8cm r=3.3cm --“ Q r=3.8cm I II••I,. • I I I I ••I. 830 I ::: zZE 0.5 1.0 1.5 2.0 2.5 3.0 Axial Position (cm) Figure 11.142: Experimental and calculated temperatures as a function of axial height at the radial locations indicated. Large (8.8 cm diameter) crucible. Crucible rotation = 10 rpm. Modified velocity boundary conditions are used. (Figure 11.128). Chapter 11. Comparison of Temperature Measurements with Model Results 226 Calculated Measured r=O.4cm D r=O.9cm r=2.8cm r=3.3cm --•“ 0 r=3.8cm I I I • I I 830 — — —-:= —r 820 E 810 . I I . . I I . I 800oo 0.5 1.0 1.5 2.0 2.5 3.0 Axial Position (cm) Figure 11.143: Experimenta.l and calculated temperatures as a function of axial height at the radial locations indicated. Large (8.8 cm diameter) crucible. Crucible rotation = 20 rpm. Modified velocity boundary conditions are used. (Figure 11.128). Chapter 11. Comparison of Temperature Measurements with Model Results 227 Calculated Measured r=O.4cm D r=O.9cm A r=2.8cm ———— r=3.3cm •—.—.—.-.- Q r=3.8cm I I I I • I • - 830 820- 810- 800od 0.5 1.0 1.5 2.0 2.5 3.0 Axial Position (cm) Figure 11.144: Experimental and calculated temperatures as function of axial height at the radial locations indicated. Large (8.8 cm diameter) crucible. Crucible rotation = 30 rpm. Modified velocity boundary conditions are used. (Figure 11.128). Chapter 12 Mass Transfer Calculations The mass transfer calculations were conducted to elucidate the movement of MoO3 away from the growing interface. This requires knowledge of the diffusion boundary layer thickness below the crystal, the growth velocity of the interface, the diffusion coefficient and the bulk concentration of MoO:3 in the melt. The boundary layer below a rotating disk in an infinite fluid has been determined analytically [45, 46]. The same solution has been extended to include a bottom rotating disk to account for crucible rotation [47, 48]. Unfortunately, the system examined in this research is very different. The bottom of the crysta.l is not shaped like a fla.t disk, the fluid is not infinite and the crystal preferentially grows in specific planes, and does not have a. uniform radius. Despite these differences. the analytical solution wa.s used in conjunction with the calculated fluid flow solution (Chapter 10) The mathematical model was used t.o calculate the flow velocity under the crystal for a given crucible rotation ra.te. The crystal was assumed to be growing in the [001] direction in an 8.8 cm diameter crucible .An analytical solution was used to determine the crystal rotation rate tha.t gives the same flow velocity under the crystal. The equivalent crystal rotation rate was used with analytical solution for mass transfer below a rotating crystal [45] to examine the change in interface concentration with growth time. Two crystal growth runs (Chapter 13) provi(ledl data for this analysis. 22$ Chapter 12. Mass Transfer Calculations 229 12.1 Procedure for Estimating the Equivalent Crystal Rotation Rate The calculated tangential velocity at 0.5 of the crystal radius for an LBO crystal growing in an 8.8 cm diameter crucible rotating at 60 rpm were compared with the analytical solution to determine a. rotation speed required for an infinite disc to produce the same flow velocities. Once this was deterniined the corresponding boundary layer thickness was calculated from Equation 12.11 and was assumed to apply to the growth of an LBO crystal. The radial velocity below a rotating infinite disk is given by. v;. = Va x w x F(z) (12.9) were 4. is the radial velocity. ra is the radia.l location. w is the angular velocity and F (z) is a dimensionless variable and is given on Table 12.18. The axial location is given by z=zx( 1 x w were z is the axial location, z is the dimensionless axial location, i is the viscosity, p is the density and w is the angular velocity. The radia.l location value (ra) used in the analytical solution is equivalent to the surface length between the centre and 0.5 of the model crysta.l radius as shown in Figure 12.145. The results of the comparison of the ana.lytica.l solution to the calculated solution are given in Figure 12.146. The values of the thermophysical properties used for the analytical and calculated solutions are given in Table 12.19. The calculated fluid flow results are for a crucible rotation of 60 rpm and are seen to be identical to an infinite disk rotating at 40 rpm. Thus, the analytical solution with a infinite crystal rotating at 40 rpm (w = 4.19) can be used to determine the mass transfer that occurs with an 8.8 cm diameter crucible rotated at 60 rpm. It is assumed that this ratio of crucible to crystal rotation is independent of viscosity and density. as long as the same values are used in both calculations. Chapter 12. Mass Transfer aiczi1ations 230 [ z’ F(z)7 0.0 0.0 0.1 0.0462 0.2 0.0836 0.3 0.1133 0.4 0.1364 0.5 0.1536 0.6 0.1660 0.7 0.1742 0.8 0.1789 0.9 0.1807 1.0 0.1801 Table 12.18: Numerical solutions for a rotating disk [46]. Property Units Values density g/crn3 3.26 viscosity for poise 21.31 40.9 Wt% MOO3 Table 12.19: Therrnophvsica.1 properties used in the Analytical solution for flow below a rotating disk. Chapter 12. Mass Transfer Calculations 231 Analytical 1< >1 ra Calculated a Figure 12.145: determination of the radius used in the analytical solution. The radius, ra is equivalent for both the finite element analysis and the analytical analysis. Chapter 12. Mass Transfer Calculations 232 • • • • • • • • • • • • • 10.05- C.? Analytical C] Calculated C] 0•1 0.05 0.10 0.15 0.20 Velocity (cm/s) Figure 12.146: Calculated and anal tical velocity values. The calculated tangential veloc ity values are at at 0.5 of the crystal radius. The analytica.l solution (flow past a rotating disk) for the radial velocity is at a. radial location that is equivalent to the surface length of the calculated solution. Chapter 12. Mass Transfer Calculations 233 12.2 Mass Transfer Behavior of MoO3 below the Crystal The concentration of MoO3 at the growing interface can he calculated [45]. The equation relating the bulk concentration of MoO3 with the concentration at the interface is: C0 = CL X exp ( ) (12.10) were C0 is the concentration of MoO3 at the growing interface, CL is the concentration of MoO3 in the bulk, •f is the growth velocity at the interface, 6 is the thickness of the diffusion boundary layer and D is the diffusion coefficient. It is assumed that the movement of the interface (.1) is the only flow within the diffusion boundary layer. It is also assumed that 6 is constant between the centre and edge of the crystal. The diffusion boundary layer thickness (6) for an infinite rotating disk is given as: / \1/6 6 = 1.6D13 (12.11) were D is the diffusion coefficient of solute in the melt in cm2/s, t is the viscosity in poise (g/cm s), p is the density in g/cm3 and is the rotation rate in rads/s. The growth velocity of the LBO crystal a.nd the Mo03/LBO diffusion coefficient is determined by combining Eciua.tion 12.11 and 12.10. The resulting equation is: / / \1/6\ 3/2(fx1.6x{) \ D = I I (12.12) x in () ) All of the variables in the equation are readily available with the exception of the con centration of MoO3 next t.o the interface (C0). This value was determined by growing a crysta.l a.t a. sufficiently high rate to cause eutectic formation. When this occurs the concentration of MoO3 next to the interface is 61.5 Wt% MoO3 (Figure 5.37). A crystal growth experiment were conducted in which the crystal was slow cooled for a short time then pulled at a rate of 1.66 mm/day. Eutectic growth occurred upon Chapter 12. Mass Transfer Calculations 234 initiating the pulling sequence. Figures 13.155 and 13.156 show the interface breakdown that occurred when the crystal pulling started. The upper portion of the crystal is clear and free of MoO3, the lower portion has a eutectic structure of MoO3 rods in an LBO matrix. The growth rate is a combination of the cooling rate of the furnace and the pull rate of the crystal. The growth rate due to programmed cooling of 2.4°C/day was calculated to be 0.7 mm/day. This was determined by dividing the final height of the crystal by the total growth time for a. crystal grown only by melt cooling. Thus the tota.l growth rate (f) was 2.63 mm/day at eut.ectic formation. The rotation rate of the crucible was 60 rpm which corresponds to a disk rotation rate of 40 rpm. Table 12.20 gives the values of the variables used in determining the diffusion coefficient. The bulk concentration (CL) is assumed to be the initial theoretical concentration of the melt (44.7 Wt% MoO3). The viscosity was previously measured (Figure 5.39). The density was calculated using weight fractions of the LBO and MOO3 with their respective density values. The diffusion coefficient of MoO3 in an L130 melt was estimated from Equation 12.12 as 2.42 x 108 crn2/s. The same calculations were carried out assuming the pull rates within +50% of 1.66 mm/day for comparison reasons. The diffusion coefficients are 3.73 x 108 and 1.24 x 108 cm2/s for pull rates of 2.49 and 0.83 mm/day respectively. The variation in diffusion coefficient is minimal for the given changes in pull rate. Ta ble 12.21 gives the values of diffusion coefficients in other liquid systems. The diffusion coefficients of liquid metals and liquid oxides are of the order of iO and iO respec tively. The Mo03/LBO diffusion coefficient is lower than the other oxide material dif fusion coefficients. This ma be due to the high temperatures of the oxides given in Table 12.21. C) C) Cri C’ , CJ D , C C) C C t’.) C I.CID C, C,, C , !“ C, CI) C Ct, C,, CI) C, -J C, C, C, C C C, I C , ‘ V C Ct, J. X X X X . - N C- ) CI ) — , - • 4 C f l< C C - ) * C C , , Cl c ) N r:- ’ C , C + — _ . , C t, C - C C,) c C ) , - J - I l d l C C C -) C , , C - C C — C.) CI ) Ct, - C•) C) — Ca ) Chapter 12. Mass Transfer Calculations 236 12.3 Maximum Growth Rates and Growth Times for LBO The MoO3 concentration in the bulk of the melt and at the interface will change as the crystal grows. This is due to the rejection of MoO3 by the solidifying crystal during growth which increases the melt concentration. Equation 12.10 has been used to calculate the concentrations at different times during the growth of the crystal. It was assumed that the system wa.s at steady stat.e at the slow growth rates used and tha.t the diffusion boundary layer thickness does not change as the crystal grows. The crystal interface is assumed to be a cone growing into the melt and that the ratio of radius to crystal height is constant. The volume of the crystal is: Vctai = X Tr2 were Vtai is the volume of the LBO crystal, ii. is the height of the crystal and r is the radius of the crystal. The relation of the radius to height was determined from a crystal growth experiment where the interface was quenched during the initial stages of its growth (Figure 12.147). The relationship between the radius and height. is: -=2.36 The increase in crystal height with time is assumed to be a function of the growth velocity: 11 = .1 x L were t is the growth time, h is the crvsta.l height. and is the crysta.l radius. The volume of the growing LBO crystal at any time (luring the growth is v1 = (.r x x (2.36)) The corresponding weight of the LBO crystal is: = X PLBO Chapter 12. Mass Transfer Calculations 237 were Wxai is the weight of the LBO crystal and PLB0 is the density of LBO (2.474 g/cm3). The bulk MoO3 concentration of the fluid ca.n he calculated knowing the amount of LBO that has been removed in growing the crystal. c — _______________________________________ ‘L —l47tM0Q3+ (TI7tLBO — Wtxtai) were CL is the bulk concentration of MOO3 ifl the melt, Wt03 is the initial charge weight of MoO3 in the melt, T47tLBO is the initia.l charge weight of LBO in the melt and Wtxtai is the weight of LBO in the crystal. Equation 12.10 can be used to calculate the interface concentration. The viscosity and density are assumed to change with the bulk MoO3 concentration. The melt is assumed to be at 720°C for all of the calculations and the variation in viscosity with MoO3 concentration is taken from Figure .5.40. = erp (9.818 — Wt1700, x 0.169) p = Wi%jBo x 2.47 + lVt%1003 x 4.69 The interface concentration was calculated for three separate cases. The normal growth rate of 0.70 mm/day (2.4°C/day), a growth rate 50% higher than the normal value (1.05 mm/day, 3.75°C/day) and a. growth rate 50% lower than the normal value (0.35 mm/day, 1.2°C/day). The interface concentrations as a function of time are given in Figure 12.148. Table 12.22 gives the variation in interface concentration with time for the normal growth rate. The interface concentration at the normal growth rate (0.7 mm/day) increases at a constant rate from 48.9 Wt% MoO3 a.t da 0 to 49.39 Wt% MOO3 Oil day 10. The interface concentration continues to increase until it reaches the eutectic concentration (61.5 Wt% MoO3) on day 27. Increasing the growth rate by 50% (1.05 mm/day) causes the initial interface concentration to increase to 51.3% MoO3. The rate of increa.se of the interface concentration is larger and the eutectic concentration is reached after 17.5 Chapter 12. Mass Transfer Calculations 238 days. Decreasing the growth rate by 50% causes the initial interface concentration to be 46.7%. At the slow growth ra.tes the eutectic is reached well after 30 days of growth. These calculations are very approxi mate due t.o the number of assumptions that were employed. However, the trend in the rate of change in interface concentration with growth time can be used to determine the growth time. Growth times should be limited to 27 days or less. This will avoid growing in the region where the increase in concentration with time is large. There are two factors tha.t have not. been accounted for that will alter these predic tions. Evaporation of MOO3 occurs at the surface of the melt which reduces the bulk MoO3 concentration and in turn the interface concentration. Under these conditions it is possible to grow the crystal for a longer time due to the decreasing bulk MoO3 con centration. The surface of the crystal is assumed to he flat. Any faceting of the crystal interface, which is a. real possibility, vill crea.te stagnant areas and increase the MoO3 concentration at the interface. Chapter 12. Mass Transfer Calculations 239 Figure 12.147: Cross section of crystal grown at a crucible rotation rate of 60 rpm. The radius of the crystal is approximately 2 cm and the height at the centre line is 0.85 cm. LBO Crystal Frozen Mo03/LBO Flux Layer Chapter 12. Mass Transfer Calculations C 240 Figure 12.148: The concentration of MoO3 next to the growing interface as a function of time for the growth rates (f) indicated for a crucible rotation of 60 rpm. 65 60 55 50 Time (days) 30 Jhapter 12. Mass Transfer Calculations 241 ‘lime Weight of Weight of Weight of Viscosity Density C0 (days) the LBO LBO in MoO3 in (poise) (g/brn3) (cm) Crysta.l the Melt the Melt (grams) (grams) (grams) 0 1 2 3 4 5 6 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 0.00 0.00 0.04 0.13 0.31 0.61 1.06 1.68 2.51 3.58 4.91 6.53 8.48 10.78 13.47 16.56 20.10 24.11 28.62 33.66 39.26 45.44 52.25 59.70 67.84 76.67 86.25 96..59 107.72 256.33 256.33 256.29 256.20 256.02 255.72 255.27 254.65 253.82 252.75 251.42 249.80 247.85 245.55 242.86 239.77 236.23 232.22 227.71 222.67 217.07 210.89 204.08 196.63 188.49 179.66 170.08 159.74 148.61 44.57 44.57 44.57 44.58 44.60 44.62 44.67 44.73 44.81 44.91 45.04 45.20 4.5.40 45.63 45.90 46.22 46..59 47.02 47.51 48.06 48.70 49.42 50.24 51.17 52.23 53.42 54.7$ 56.33 58.10 9.80 9.80 9.79 9.78 9.75 9.70 9.63 9.53 9.40 9.24 9.04 8.80 8.51 8.19 7.82 7.41 6.96 6.47 5.96 5.42 4.87 4.31 3.75 3.21 2.68 2.19 1J4 1.34 0.99 3.46 3.46 3.46 3.46 3.46 3.46 3.46 3.47 3.47 3.47 3.47 3.48 3.48 3.49 3.49 3.50 3.51 3.52 3.53 3.54 3.5.5 3.57 3.59 3.61 3.63 3.66 3.69 3.72 3.76 0.00257 0.00257 0.00257 0.00257 0.00257 0.00257 0.00256 0.00256 0.00255 0.00254 0.00253 0.00252 0.00251 0.00249 0.00247 0.00245 0.00242 0.00239 0.00236 0.00232 0.00228 0.00223 0.00218 0.00212 0.00205 0.00198 0.00 191 0 .00 182 0.00173 48.93 48.93 48.93 48.94 48.96 48.98 49.03 49.08 49.16 49.26 49.39 49.54 49.73 49.95 50.22 50.52 50.88 51.29 51.76 52.29 52.90 53.59 54.38 55.27 56.28 57.42 58.71 60.19 61.87 Table 12.22: The concentration of MoO3 next to the growing interface as a function of time. Diffusion coefficient is 2.38 x i0 cm2/s. Growth ra.te (f) is 0.698 mm/day. Chapter 13 Application of Process Engineering Principles to Crystal Growth This section describes the Lithium Triborat.e crystal growth runs that were conducted with parameters based on results from the mathematical model, the physical model and experimental measurements of temperature and viscosity described in Chapters 10, 6. 5 and 7 respectively. The experimental measurements and mathematical model have shown that crucible/crvstal rotation. crucible size and MoO3 content in the melt are the parameters that have the greatest influence on fluid flow below the crystal. In particular, the mathematical model established that crucible rotation produces the maximum fluid flow across the growing solid/liquid interface. Crystal rotation was not used in conjunction with crucible rotation since it would decrease the fluid flow at the growing crystal/liquid interface (Figure 10.121). The crucible rotation rates tested were 18.5, 30 and 60 rpm. The initial rotation rate (18.5 rpm) was similar to the crystal rotation used in the preliminary experiments (Section 5.3). An 8.8 cm diameter crucible was used for most of the crystal growth runs since melt velocity due to crucible rotation increases with crucible size. A 6.6 cm diameter crucible had to be used for the first two crystal growth runs since the 8.8 cm crucible was unavailable. A high MoO3 melt content (44.5 Wt%) was used since the viscosity decreases with increasing MoO3 content. Parameters for the mass transfer model were estimated from the results of crystal growth runs LBO 19 and LBO 20 and were used t.o elucidate the change in interface concentration with growth time. The mass transfer analysis wa.s conducted for crystals growing in the [001] direction. It was determined that the interface concentration initially 242 Chapter 13. Application of Process Engineering Principles to Crystal Growth 243 increases at a slow rate followed by a.n exponential increase. It was also found that the crystal growth velocity in the [001] direction had to he slower than the minimum pull speed of the crystal growth apparatus (1.66 mm/day) due to eutectic formation at this pull speed. Certain aspects of crystal growth were not modeled or studied prior to the crys tal growth runs. Two effects that are likely to influence crysta.l growth that were not considered are: 1. The influence of crysta.l orientation on the build up MoO3 at the growing inter face and inclusion /eutecti c formation. Different crystal growth orientations could produce faceting which would result in higher inclusion formation/interface break down. The preferred growth direction used in this investigation was [001] given that devices had to he fabricated tlia.t were up to 1.5 cm in the [100] direction. The first two runs were carried out in the [312] direction until enough seed material was produced in the preferred direction. 2. The effect or growth atmosphere on crystal quality was not investigated. The furnace atmosphere was changed from air to dry nitrogen once it was determined that LBO decomposed in the presence of water vapour [6]. Nine crystal growth runs were conducted during this investigation (LBO 17 — LBO 25). The complete list of the parameters used in crystal growth runs are given in Ta bles 13.23 and 13.24. LBO 16 was completed before the initiation of this investigation (LBO 16) and is listed in Tables 13.23 and 13.24 as a reference for comparison. 13.1 LBO 17 The first crysta.l (LBO 17) was grown using the 6.6 cm diameter crucible in air. The crystal was grown in the [312] direction, the melt cooling rate for growth was 2.4°C/day: Chapter 13. Application of Process Engineering Principles to Crystal Growth 244 which should yield a growth rate of 0.7 mm/day; the crucible rotation was 18.5 rpm and the crystal was stationary. The aim of this experiment was to determine if crucible rotation and a higher MoO3 content would produce a better crystal. The previous run (LBO 16) was conducted with crystal rotation of 15 rpm, no crucible rotation and a 34.4 Wt% MoO3. During growth the growing crystal/liquid interface at the surface of the liquid was strongly faceted. The crystal wa.s grown until its edges reached the outside of the crucible. After growth, the crysta.l was ra.ise(I from the melt before cooling to room temperature at 10°C/hour. Unfortunately, the distance raised was insufficient, which resulted in it freezing into the melt. Figure 13.149 confirms that LBO 17 froze into the melt. The upper surface of the crystal is a. white translucent color. Large cracks, marked A, run along the diagonals of the crystal. The edge marked B is due to a piece of the crystal falling off while it was cooling to room temperature. Strong faceting is present at the edges of the crystal that correspond to the liquid/solid interface (C). The rounded edge of the crystal (marked D) is due to the crystal growing close to the side of the crucible. The uncracked pieces of LBO 17 are shown in Figure 13.150. All of the pieces of the crystals were inclusion free in the region that corresponded to the upper surface (marked E on some of the crystal pieces). The lower areas of the crystal shattered into a white powder when the solid/liquid interface froze (marked F on some of the crystal pieces). Despite these problems there was a. significant increase in crysta.1 quality and yield size compared to preliminary growth experiments. Previously the average inclusion free region was approximately 3 x 3 x 3 mm3 (Figure 5.41). LBO 17 was predominantly inclusion free at the outer edges of the crystal. and average size of the individual pieces of LBO crysta.l were 15 x 5 x 5 mm3. The increased cr sta.l quality can be attributed to the improved mixing in the melt due to crucible rotation, the higher MoO3 content (44.5 Chapter 13. Application of Process Engineering Principles to Crystal Growth 245 versus 34.3 Wt% MoO3) and the absence of interface breakdown. 13.2 LBO 18 LBO 18 was grown in the [3T2] direction as per the previous run. The melt cooling rate for growth was the same as in the previous run (2.4°C/day). Two of the growth parameters, crucible rotation and crvsta.l pulling, were changed. The crucible rotation rate was increased to 30 rpm to improve mixing in the melt and the crysta.1 was pulled a.t 1.66 mm/day to increase its thickness. The mat.bematica.l model predicts that the magnitude of the tangential velocity beneath the crystal should increase from 5 x 10_2 cm/s at 15 rpm to 10.5 x 102 cm/s at 30 rpm (Figure 10.114). The pull rate of the crystal was the minimum speed of the motor. The growth procedure consisted of slow cooling the furnace at 2.4°C/day until the crystal had grown to approximately 2 cm in diameter. The crystal was then pulled for 6 days at 1.66 mm/day to increase its axial thickness. After pulling, the furnace was slow cooled a.t 2.4°C/day for the remainder of growth. During growth the crystal ha.d strong faceting at the solid/liquid interface. Crystal LBO 18 is shown in Figure 13.151 and 13.152. The crystal has two distinct areas, the upper portion (A), clue to a pull rate of 1.66 mm/day, and a lower portion (B) where only siow cooling was applied to the melt/crystal. The crystal grew along specific planes. Major cracks were present in the crystal, running from the outer points of the crystal inward to the centre. The bottom of the crystal was either clean (C) or covered with IVIoO3 flux (D) that wa.s white/green in color. The regions covered with flux were severely cracked. The surface of the clear regions were predominantly uncracked, however cracks from the flux covered region propagated into the clear regions. The crystal was free of IVIoO3 inclusions with the exception of two locations, at the bottom surface of the crystal (D) and at the location where pulling was terminated (Between areas A a.nd Chapter 13. Application of Process Engineering Principles to crystal Growth 246 B). This would indicate that growth along the [312] direction could he as high as 1.66 mm/day. Differences in growth rate and faceting with orientation are expected since the LBO crystal structure is orthorombic and the faceted shape has been previously shown (Figure 2.6). The crystal was moderately fragile and broke into pieces after a limited force was applied to it. The individual pieces of flux free portions of the crystal (A in Figure 13.151) are shown in Figure 13.153. The uncracked a.nd inclusion free piece is approximately 20 x 15 x 6 mm3. The decrease in inclusion density (increase in crystal quality) of LBO 17 and 18 can be attributed to three factors; using crucible rotation to increase the fluid velocity next to the growing interface, increasing the MOO3 content to decrease the viscosity (which also increases the fluid velocity), and effects tha,t are possibly due to the growth direction. Pulling the crystal at 1.66 mm/day did not result in interface breakdown which suggests that growth in the [312] direction can be large. 13.3 LBO 19 LBO 19 was grown in the [001] direction, which is the previous growth direction, in the large crucible (8.8 cm diameter). The crystal was slow cooled until its diameter was approximately 6 cm after which it was also pulled at a rate of 1.66 mm/day. The crystal is shown in Figure 13.154. The seed is marked A, the slow cooled portion of the crystal is marked B, the slow cooled/pulled at 1.66 mm/day area of the crystal is marked C and D. The area marked E is where a piece of the crystal had broken off. The crystal was very fragile and broke easily. The lower area of the crystal which grew while it was being slow pulled (C and D) was a white/green color indicative of interface breakdown. The growth rate was sufficiently fast that the diameter of the crystal decreased to a value of 1 cm (D). Chapter 13. Application of Process Engineering Principles to Crystal Growth 247 The polished cross section of the crystal is shown in Figure 13.155. The crosses correspond to areas where X—Ray (hifraction was conducted to determine the crystal orientation. The analysis was conducted at the seed, before the interface breakdown and at two points inside the white/green portion of the crystal. All of the X—Ray diffraction patterns were identical, and revealed that the crystal consisted of an LBO matrix. SEM photos were taken in the area were the interface break down occurred (Area C on Figure 13.155). Wavelength dispersive spectroscopy (WDX) was also carried out at the same locations to qualitatively determine if MoO3 wa.s present. Figure 13.156(a) is an SEM photo of the region where interface breakdown commenced. There are long rod like inclusions in the LBO matrix, and the growth direction is parallel to the rods. The corresponding molybdenum map of the same area, Figure 13.156(h), shows that the rods consist of molybdenum. A backscatter image and WDX image of an individual inclusion is shown in Figure 13.157. Unlike the previous run, LBO 19 formed eutectic material when the crystal was pulled at 1.66 mm/day. Two parameters were changed relative to growth of LBO 18; LBO iS was grown along the [3T2] in a. small crucible while LBO 19 was grown along the [001] direction in a. large crucible. Increasing the crUcil)le size has been demonstrated to increase the flow velocity below the crystal, thus the eutectic formation in LBO 19 cannot be attributed t.o the change in crucible size. The occurrence of the eutectic in LBO 19 must be due t.o the growth direction, indicating that that the [001] direction readily forms eut.ectic at a. pull speed of 1.66 mm/day for crucible rotations of 30 rpm or s’ower. 13.4 LBO 20 LBO 20 was grown with a. faster rotation rate (60 rpm) to improve the removal of MoO3 from the interface. An increase in crucible rotation from 30 to 60 rpm increases the Chapter 13. Application of Process Engineering Principles to C’rystal Growth 248 magnitude of the tangential velocity beneath the crystal from 10.5 x 10—2 cm/s to 21 x 10_2 cm/s (10.114). The crystals were cycled through a slow cooling/pulling period for 1 day and a slow cooling period of 1 day to facilitate diffusion of MoO3 ahead of the interface and to prevent the rapid reduction in crysta.1 (liameter. The crystal, Figure 13.158, is 5 cm in diameter. The crystal was very fragile and broke into several pieces, as shown in the figure, when it was handled. The top of the crystal was covered with a thin white powder. The cross section of the crystal is shown in Figure 13.159. The area marked A is part of the crystal that had broken off. The cross section of the crystal, was similar to LBO 19 in tha.t the upper (slow cooled) portion was clear and the lower portion had the eutectic structure. The crysta.l (liameter was reduced to approximately 4 cm with the pulling/cooling cycle. The pulling mechanism on the crystal growth station had a minimum pull rate of 1.66 mm/day with no diameter control mechanism. The previous two crystal growth runs, L130 19 and 20, in conjunction with the mass transfer calculations, indicate that a pull rate of 1.66 mm/day will produce interface breakdown for [001] direction crystal growth for a 8.8 cm diameter crucible rotated at 60 rpm. Larger crucible rotation rates were not investigated since 60 rpm was close to the maximum rotation rate of the apparatus. 13.5 LBO 21 LBO 21 was grown using slow cooling without crystal pulling and with a dry N2 atmo sphere. The dry N2 was used to prevent the upper surface of the LBO from decomposing at an elevated temperature in the presence of water vapour [6]. Unfortunately LBO 21 was slow cooled until a power outage froze the crystal into the melt. The crystal, Fig ure 13.160, was severely cracked due to it being frozen into the melt. The white/green colored material (A) is the LBO/fiux that has been frozen onto the outside of the crystal Chapter 13. Application of Process Engineering Principles to Crystal Growth 249 due to the fast quench. The crystal was removed in this area to show the surface of the growing interface. Two type of surfaces are present at the growing interface, rough and smooth depending on the orientation of the grow’ing surface. The crystal surface (B) is clear due to the dry N2 atmosphere. A portion of the surface of the crystal (C) is clear at the upper surface. The dry N2 prevented the upper surface of the crystal from decomposing is in the previous runs. 13.6 LBO 22 and 23 The subsequent crystal growth run, LBO 22, was conducted under identical growth conditions as LBO 21. Unfortunately this run was unsuccessful due to failure of the motor and the crystal accidentally froze with the melt during the post growth cooling to room temperature. LBO 23 was grown using the identical conditions as LBO 21. Figure 13.161 and 13.162 shows the top and bottom of the crystal respectively. The bottom of the crystal has clear regions in addition to the portions that are covered with the MoO3 flux. The upper surface is clear with the crysta.l changing to a white color close to the bottom of the crystal where the flux is present. The white color corresponded to areas where the crystal shattered due to differences in the thermal expansion coefficients of MoO3 and LBO. The clea.r portion of crysta.l that grew a.s a flat surface was the 101 plane. The flux covered portions grew as a. number of irregula.r planes in a shingle like configuration. A photo and schematic of the cross section of a plane that grew a.s a rough surface is shown in Figure 13.163. The region marked B is were the MoO3 builds up. The rough surface crea.tes stagnant areas were MoO3 can accumulate. The areas of the crystal where the flux was attached to the bottom surface were fragile and broke apart with moderate to little force. The area.s of the crystal that were free of flux, Figure 13.164, were crack resistant. The largest. cr sta.l size was approximately 20 x 10 x S mm3. The chapter 13. Application of Process Engineering Principles to Crystal Growth 250 previous three crystal growth runs(LBO 21, 22 and 23) demonstrated that the 101 plane grows as a fia.t surface while the other planes are faceted. The facets result in a build up of MoO3 at the solid/liquid interface which sticks to the surface of the crystal after it has been separated from the melt. These crystal growth runs also confirm that LBO crystals can be grown under these conditions without interface breakdown which is in accordance with model predictions in Figure 12.148. 13.7 LBO 24 LBO 24 growth conditions were identical to LBO 21. The post growth cooling procedure was modified to reduce the thermal gradients in the crystal. Insulating bricks were placed over the top hole and separation of the crystal from the melt was carried out by lowering the crucible rather than raising the crystal. was lowered instead of the raising crystal to separate it from the melt. Measurements reported in Chapter 5 suggest that the axial and radial temperature gradients in the crystal can he reduced by moving the crystal t.o the centre of the furnace and insulating the top hole while the crystal is cooling to room temperature. This procedure also ensured tha.t the crystal was as close as possible to the centre of the furnace elements thus reducing the thermal gradient in the crystal. Figure 13.165 and 13.166 shows the top and bottom of the crystal respectively. Figure 13.167 is a back light view of the crystal. As with the previous crystal growth runs the crystal had a flux build up on the rough surfaces of the growing interface (marked A in Figure 13.166). The 101 family of planes (marked B in Figure 13.166) were flat and the surface was free of flux. Areas adjacent to the flux covered surface were cracked to a fine white powder. The crystal, while still having macro cracks, did not break when a moderate force was applied to it. Thus reducing the thermal gra.dient in the crystal did result in improved strength. The largest crystal area that was uncracked between the Chapter 13. Application of Process Engineering Principles to Grystal Growth 251 Growth Date Composition Seed Crucible Rotation Rate Run Orientation Diameter Crucible Crystal (mm/dd/yy) (Wt% MoO3) (cm) (rpm) (rpm) LBO 16 5/21/92 30.0 iion seeded 6.6 0 6 LBO 17 8/6/92 44.5 312 6.6 18.5 0 LBO 18 10/5/92 44.5 3 1 2 6.6 30.0 0 LBO 19 11/26/92 44.5 0 0 1 8.8 30.0 0 LBO 20 01/8/93 44.5 0 0 1 8.8 60.0 0 LBO 21 03/24/93 44.5 0 0 1 8.8 60.0 0 LBO 22 06/9/93 44.5 0 0 1 8.8 60.0 0 LBO 23 07/21/93 44.5 0 0 1 8.8 60.0 0 LBO 24 8/9/93 44.5 0 0 1 8.8 60.0 0 LBO 25 12/7/93 44.5 0 0 1 8.8 60.0 0 Table 13.23: Growth conditions used for the crystal growth experiments. top and growing interface was 20 x 10 x 5 mm3. The increased strength of the crystal is a result of the lower thermal gradients (luring cooling to room temperature. 13.8 LBO 25 LBO 25 was grown to determine if the build up of flux at the interface was due to the length of the growth runs. Growth conditions identical to LBO 24 were used except that the crystal was grown for only 14 days. There was a MoO3 build up at the rough interface. Figure 13.168 shows a portion of the crystal with the MoO3 build up. The white regions correspond to flux attached to the crevices between the growing planes. As in the previous crystal growth runs, the 101 family of planes grew as a flat surface with 110 MOO3 build up. The presence of the MoO3 at this small crystal size indicates that flux build up occurs during all sta.ges of crystal growth. Chapter 13. Application of Process Engineering Principles to Crystal Growth 252 Figure 13.149: LBO 17 crystal frozen in the melt. Chapter 13. Application of Process Engineering Principles to Ciysta.l Growth 253 Figure 13.150: Uncracked portions of LBO 17 crystal. Chapter 13. Application of Process Engineering Principles to Crystal Growth 251 Figure 13.151: Top view of LBO IS crystal. Chapter 13. Application of Process Engineering Principles to Crystal Growth 255 Figure 13.152: Bottom view of LBO 18 crystal. Chapter 13. Application of Process Engineering Principles to Crystal Growth 2.56 JfIj I IIIIII 1,1•11• 3. I. 4 S’anIesS s_ I I Figure 13.153: Uncracked portions of L130 iS crystal. Ohapter 13. Application of Process Engineering Principles to cirystal Growth 257 Figure 13.151: Side view of LBO 19 crystal Chapter 13. Application of Process Engineering Principles to Grystal Growth icIl, 258 Figure 13.155: Cross section view of LBO 19 crystal. The crosses are regions were sample orientation was determined. LBO19 Chapter 13. Application of Process Engineering Principles to Ci’vstal Growth 259 Figure 13.156: Area in LBO 19 were interface breakdown/eutectic growth started. Mag nified 20 times. (a) SEM photo. (b) Map of molybdenum concentration. The bright regions correspond to a high molybdenum concentration. Chapter 13. Application of Process Engineering Principles to Crvsta.l Growth I, 260 Figure 13.157: View of an niolybdenum inc’usion magnified 200 times. (a) SEM photo. (b) Map of molybdenum conccntraton. The bright regions correspond to a high molyb denum concentration. a Chapter 13. Application of Process Engineering Principles to Crystal Growth 261 Growth Growth II ate Growth Ti me Atmosphere Run cooling pulling cooling pulling (°C/day) (mm/day) ( days) (days) (da.ys) LBO 16 3.8 1.66 22 3 Air LBO 17 2.4 0 27 0 Air LBO 18 2.4 0 29 6 Air LBO 19 2.4 1.66 16 12 Air LBO 20 2.4 1.66 30 5.9 Air LBO 21 2.4 0 17 0 N2 LB022 2.4 0 21 0 N2 LBO 23 2.4 0 27 0 N2 LB024 2.4 0 32 0 N2 LB025 2.4 0 9 0 N2 Table 13.24: Growth conditions used for the crystal growth experiments, continued. Chapter 13. Application of Process Engineering Principles to Crystai Growth 262 Figure 13.158: Top view of LBO 20 crystal Chapter 13. Application of Process Engineering Principles to Ciysta1 Growth 263 LBO 20 Figure 13.159: Cross section view of LBO 20 crystal. Chapter 13. Application of Process Engineering Principles to Crystal Growth 264 Smooth Surface Frozen MoO3 /LBO Figure 13.160: LBO 21 Crystal. Flux Layer I Chapter 13. Application of Process Engineering Principles to Cistal Growth 265 Figure 13.161: Top view of LBO 23 crystal. Chapter 13. Application of Process Engineering Principles to Grystal Growth 266 Figure 13. I 62: Bottom view of LBO 23 crystal. Chapter 13. Application of Process Engineering Principles to ‘Cr stal Giowtii 267 — — B Liquid I . I /I.: Crystal I I I . • I I I I ••••••••••••• .......•••• . .. . _ MoO3Build Up B Figure 13.163: Interface appearance for planes were MOO3 was stuck to the surface. (a) Photo of underside of crystal. The area A is a. region of IVIoO3 build up. (b) Schematic of surface along line B — B - Chapter 13. Application of Process Engineering Principles to Crystal Growth Figure 13.164: Pieces of TJBO 23 that were iiiicracked. 268 I Cthapter 13. Application of Process Engineering Principles to Crystal Growth 269 Figure 13.l6Ti: Top view of LIlO 24 crystal. Chapter 13. Application of Process Engineering Principles to Crystal Growth 270 Figure 13.166: Bottom view of LBO 24 crystal. Chapter 13. Application of Process Engineering Principles to Ciystal Growth 271 Figure 13.167: flack lit view of LBO 24 Crystal. Ctha.pter 13. Application of Process Engineering Principles to Crystal Growth 272 • e • I • I I I • a • a • a a a • a • a • I • I • I • I • a a a I • • a I I • I I I a. I • I • I • I • I • a I I I I a a I I a a • a a a • a • I I I • I • a • I • a a a • I I I I • a, • a : a1b MoO3 ry:tal I I • I I I Figure 13.16$: Schematic and photo of the MoO3 on the LBO 25 crystal. Chapter 14 Summary and Conclusions A study was undertaken to grow Lithium Triborate crystals by the Top Seeded Solution Growth (Czochralski) method. Lithium Triborate cannot be grown directly since it forms a.s a result of a peritectic transformation at 834 ± 4°C. The addition of a MoO3 flux modifies the phase diagram and allows LBO to be grown (lirectly a.s a solid from the liquid. The flux addition increases the complexity of the system, since the MoO3 that is rejected during growth builds up ahead of the growing interface which causes interface breakdown. Poor mixing in the melt, due to the high viscosity of B203. increases the possibili tv of interface breakdown. The objective of this research wa.s to establish reasons for interface breakdown and to optimize the growth process to produce large crystals of high optical quality. Mathemat ical and physical models have been employed to provide a. quantitative understanding of heat transfer, fluid flow and mass transfer during the growth process. These analyses have been complemented by experimenta.l measurements that were necessary to quantify thermophysical properties and boundary conditions. The primary reason for interface breakdown is eutect.ic formation at the growing interface. Although the initial composi tion of the melt is different from the entectic, a,ccunmlation of 1\1003 due t.o poor mixing allows the eutectic composition to be reached during growth rates. A summary of the key findings a.re presented below: 1. The phase diagram for the LTIO/Mo03 system has been determined in the corn position range of 44 and 74 weight percent MoO3. The liquidus of the LBO phase 273 Chapter 14. Summary and Conclusions 274 decreases from 682°C to 619°C between 44.S and 60 weight percent MoO3. The eutectic concentration has been estimated to be 61.5 weight percent MoO3. 2. The viscosity of a.n LBO/Mo03melt is high, and was measured in accordance with ASTM standard C936-S1 near 700°C. The viscosity of the melt containing 29.7 and 40.9 weight percent MoO3 wa.s found to be 234 and 21 poise respectively. The high viscosity retards mixing and inhibits mass transfer of the MoO3 flux away from the growing interface. It has been found that interface breakdown occurs when the composition of the liquid at the solid/liquid interface reaches the eutectic. The vis cosity decreases with the decreasing liquidus temperature a.s shown in Figure 5.40. Thus LBO crystal growth a.t higher MoO3 concentration levels is more desirable, provided the overall concentration of the melt is below the Mo03/LBO eutectic. 3. Temperature measurements have been conducted in a small (6.6 cm diameter) and large (8.8 cm diameter) crucible as a function of crucible rotation. A simulated crystal fabricated from platinum sheet. was used create the type of mixing that oc curred when an LBO crystal was present.. At zero crucible rotation, axial and radial gradients were present in the melt. With crucible rotation the average tempera ture of the melt increased. At a crucible rota.tion of 25 rpm the melt temperature became more uniform in the axial and radial direction. In 1)0th crucihles, the dif ference in temperature of the melt. with and without crucible rotation (30 rpm) was small being approximately 8°C to 10°C. However, the shape of the axial and radial temperature profiles changed significantly with crucible rotation. 4. The physica.l model examined the mixing due to crucible rotation in a high viscosity solution. A large (8.8 cm diameter) plexiglass crucible, plexiglass crystal 5.6 cm in diameter, glycerine solution and blue glycerine dye were used. The dye was used as a tracer to determine the flow patterns. It has been clearly demonstrated that Chapter 14. Summary and Conclusions 275 the flow and mixing is significant for crucibie rotations 45 rpm and higher. Near the crystal the fluid moves in a semicircular arc toward the centre line of the melt. The axial component of the flow increases and eventually dominates as the fluid moves towards the centre of the crystal. This causes the fluid to move towards the bottom of the crucible. The fluid at the centre line moves toward the bottom of the crUcil)le at which point it moves outward and then upward at the crucible wall. The fluid rotates in the theta direction while it moves in the axial and radial directions. The theta swirl is zero at the centre line of the melt and a maximum at the crucible walls. The fluid shears as it moves in the crucible. This causes the dye to mix into the bulk solution. The dye is completely dispersed in the melt after less than 5 minutes. Thus mixing of the melt due to crucible rotation is significant. 5. A mathematical model was used to determine the characteristics of fluid flow due to natural convection, crucible rotation, crystal rotation and crucible size. The model calculations without crystal and crucible rotation shown that the fluid velocity due to natural convection is very small, approximately 10 cm/s at 40.9 wt% MoO3 ( 12 poise at 730°C). Decreasing the MoO3 content increases the viscosity of the solution which decreases the flow’ velocities in the melt. The flow due to natural convection with 29.7 wt% MoO3 is al)proximately 10—6 cm/s ( 86 poise at 730°C). Thus the melt is essentially stagnant at low concentrations of MoO3. Inclusion formation and polyciystalline growth would be predominant under these conditions. Forced convection due to crystal a.nd crucible rotation increases the amount of mixing in the melt. The average flow velocity is approximately 5 x 10—2 cm/s with 20 rpm crystal rotation and no crucible rotation. The flow velocity is the same order of magnitude for a crucible rotating at 20 rpm with a stationary crystal. However Chapter 14. Summary and Conclusions 276 the direction of flow is reversed. With crystal rotation the fluid flows upward under the crystal and down a.t. the crucible walls. Crucible rotation causes the fluid to move up at the crucible walls and down under the crystal. Crucible rotation also produces higher tangential flow velocities at the crystal interface than crystal rotation. The fluid velocity tangential to the crystal/liquid iHterface is a maximum at the outer crystal diameter. The fluid velocity tangential to the crystal interface at 1/4 of the crystal radius is approximately one half of the tangential velocity at 3/4 of the crystal radius. The model calculations show that the crystal interface will become concave to the liquid and eventually melt back at higher crystal rotation rates (greater than 20 rpm for the given thermal conditions). On the other hand, crucible rotation causes the crystal interface to become convex to the melt because the direction of the flow is reversed. Higher crucible rotation rates can be used to increase the flow velocity in the melt. The maximum calculated flow velocity at a crucible rotation of 60 rpm is approximately 2 x 10_I cm/s. The maximum calculated fluid velocity at a crystal rotation rate of 20 rpm is 4 x 102 cm/s. The flow velocity due to crucible rotation is dependent on the crucible size. The average tangential flow velocity at 0.5 of the crystal radius is —2 x 10—1 cm/s and 5 x 10—1 cm/s for the 6.6 cm and S.S cm diameter crucibles rotating at 60 rpm. Fluid velocity is proportional to crucible size for crucible rotation driven flows. The larger the crucible the higher is the rotational component of fluid velocity near the outside of the melt. This in turn gives the fluid a larger centripetal acceleration which increases the overall magnitude of the velocities of the melt. The effect of counter and isorotation of the crystal and crucible on fluid flow have been shown. Crystal and crucible rotation cause the fluid to move in opposite Chapter 14. Summary and Conclusions 277 directions. Thus they compete with each other in determining the final fluid motion and velocity. A rotating crucible with a stationary crystal will result in a higher fluid velocity than a. rotating crysta.l with a stationary crucible. Introducing crystal rotation at a. given crucible rotation will result in decreasing the average flow rate. Increasing the crystal rotation to equal tha.t of the crucible will result in the fluid becoming stagnant. If the crystal and crucible are rotating in the same direction at the same rate all of the fluid is stagnant. If the crystal and crucible are rotating in opposite directions at the same rate. then the area under the crystal is stagnant a.nd there is some motion in the fluid due to the shearing motion. If the crystal rotation is larger than the crucible rotation then flow induced by crystal rotation will dominate. 6. The model calculations were compared to temperature measurements made in a rotating crucible with a stationary simulated crystal. Measurements in the small (6.6 cm diameter) and large ($.S cm diameter) crucibles were employed for the comparison. The model predicts temperatures to within 2 to 4°C of the measured temperatures. The accnrac of the model predictons at higher crucible rotation rates indicates tha.t the calculated melt velocity must be similar to the actual melt velocities. 7. Mass transfer calculations have been conducted for an LBO crystal. conical in shape, growing in the [001] direction to determine the growth time required to reach the eutectic concentration at the solid liquid interface. It was determined that at the norma.l growth rate (2.4°C/day 0.7 mm/day) the interface concentration reaches the LBO—Mo03 eutectic a.fter 27 days. The calculations indicate that the interface concentration increases slowly at the start of growth and accelerates at the end of growth. At a. growth velocity of 0.35 mm/day results the eutectic Chapter 14. Summary and Conclusions 278 concentration is not reached within the 30 day time period examined. An interface growth velocity of 1.05 mm/day results in the interface concentration reaching the eutectic in 10 days. 8. The gradients in the crystal were measured to determine the influence of varying amounts of insulation. It wa.s determined that independent of crucible size, the thermal gradients in the crystal are most sensitive to the addition of insulation over the seed hole in the furnace and the vertical position of the crystal. Moving the simulated crystal towards the centre of the furnace reduces the axial gradients and the insulated top reduces the ra.dia.l gradients. Using a small crucible for crystal growth reduces 1)0th the axial and radial gradients provided that the crystal is surrounded by the walls of the crucible during the post growth cooling stage. 9. Crystals were grown with knowledge gained from the modeling and experimental measurements. Employing crucible rotation rates of 60 rpm resulted in an increa.se in single crystal yield from 3 x 3 x 3 mm3 to 20 x 10 x 8 mm3. Crystal growth rates in excess of 1.66 mm/day were possible for growth in the [312] direction. The same growth rates caused interface breakdown when the crystal wa.s grown in the [001] direction. Growth in the [001] direction caused faceting to occur at all locations of the solid/liquid interface with exception of the 101 plane. The 101 plane grew as a flat surface and was free of MOO3 flux. The crevices between the facets trap the MOO3 flux which sticks t.o the surface of the crystal after it. has been separated from the melt. The MoO3 in these regions ca.used cracking of the crystal because of the mismatch in therma.l expansion coefficients between MOO3 and LBO. In conclusion the results of the present investigation shows tha.t good quality crystals of limited size can h)e grown from a. melt, of LBO containing MoO3 as a flux. The major fa.ctor limiting the growth ra.te and crysta.l size is the concentration of MoO3 at the Chapter 14. Summary and Conclusions 279 advancing interface. Since MOO3 IS not soluble in LBO, all of the MoO3 rejected by the solid is accumulated ahead of the interface. The MoO3 moves into the liquid by diffusion through a thin boundary layer adjacent to the interface and mixes in the bulk melt. When the concentration of MoO3 at the interface reaches the eutectic concentration of 61.5 wt% starting with a. melt concentration of 44.5 wt%, eutectic will form, which produces phase regions rich in MoO3, effectively terminating growth of a good quality crystal. Accordingly, larger crystals can be grown if the thickness of the diffusion layer is decreased and the remaining MoO3 is distributed uniformly in the liquid. Higher flow rates in the melt below the interface, and thinner diffusion boundary layers are obtained with higher crucible rotation rates. Higher crystal rotation rates cannot be used since they are limited by remelting of the crystal. Buoyancy force flow is very small, due to the high viscosity of the melt. Reducing the concentration of MOO3 in the melt would result in a decrease in the fluid flow in the melt due to an increase in the viscosity. This would also result in a higher MoO3 concentration at the growing interface due to the lower fluid velocity. The size and quality of LBO crystals which can be grown from the melt by the top seeded solution growth (Modified Czochralski) process is limited by the nature of the materials and process. Growth must be slow and growth conditions selected to move the flux away from the interface as rapidly as possible primarily by crucible rotation. Some crysta.l orientations grow more readily than others due to faceting, which affects the local movement of the flux away from the interface. The grown crystals can crack readily under therma.l stresses which requires the crystals to be cooled in a. uniform thermal environment. Chapter 15 Recommendations for Future Work Severa.1 more sets of crystal grow’th runs should he carried out. In addition the mass transfer calculations should be improved to more accurately predict the process. 1. The size and quality of the L130 crystals is dependent on the orientation of the crystal. Crystals of [100], [010] and [101] directions should be grown to assess how significant the orientation is in obtaining higher quality crystals. In particular the extent of faceting of the interface, as a function of crystal orientation and the temperature gradient in the melt, should be determined. The relationship between faceting and the onset of the formation of MoO3 inclusions should he established. On the basis of this information procedures should he developed to reduce faceting. if faceting is found to instigate the formation of MoO3 inclusions. 2. After growth the crysta.l nmst he cooled slowly in a. uniform thermal environment to prevent cracking. This must be done in the crystal growth furnace .Accordingly the furnace design should be modified to enable the crystal to cool at the center of the furnace where the thermal gradients are low. This would require modification of the lower portion of the furnace to allow the crucible t.o be lowered out of the area where the crystal will be cooled to room temperature. 3. The model should be extended to include important aspect.s of growth which were not considered. During growth MoO3 evaporates from the top surface of the melt.. changing the melt composition. The surface of the solid liquid interface in the 280 chapter 15. Recommendations for Future Work 281 present model is assumed to be smooth. In fact it can he faceted which influences the model results and the onset of eutectic growth. The affect of faceting should be incorporated into future mathematical models. 4. The possibility of replacing the MOO3 flux with another material which produces melts with much lower viscosities should be investigated. The flux would have to he insoluble in LBO. Bibliography [lj C. Chen, Y. Wu, A. Jiang, B. Wu, C. You, R. Li, and S. Lin, J. Opt. Soc. Am. B. Vol. 6, No. 4, 616 (1989). [2] C. Chen, Y. Wu, A. Jiang, B. Wu, C. You, R. Li, and S. Lin, United States Patent. Patent No. 4826283, Date: May 2, 1989 [3] C. Chen, Laser Focus World, November 1989 [4] A. Napolotano, P.B. Macedo and E.G. Hawkins. Journal of The American Ceramic Society, Vol. 48, No. 12, 613 (1965) [5] S. Zhao, C. Huang, H. Zhang, Journal of (rystal Growth, 99 (1990) 805-810 [6] E. Bruck, R.J. R.aymakers, R.1K. Route and R.S. Feigelson, Journal of Crystal Growth, 128 (1993) 933-937 [7] B. Sastry and F. Hummel. Journal of the American Ceramic Society, Vol. 41, No. 1, 7-17 [8] Von H. König and R.. Hoppe, Z. anorg. aug. (hem., 439 (1978) 71-79 [9] M. Iha.ra, M. Yuge and J. JKrogli-Moe, Yogyo-Kyokai-Shi, 88(4) (1980) 179-184 [10] C. L. Tang, Progress Report, Cornell University, fthica, NY [llj C. Chen, Y. Wu, A. Jiang and C. You, Scientia Sinica B, Vol. 28, No. 3, 235-243 [12] A. Jiang, F. Cheng, Q. Lin, Z. Cheng and Y. Zheng, Journal of Crystal Growth, 79 (1986) 963-969 282 Bibliography 283 [13] R..S. Feigelson, R.J. Raymakers and R.K. Route, Journal of Crystal Growth, 97 (1989) 352-366 [14] I. Barin, 0. Knacke, and 0. hubaschewski Thermophysical Properties of Inorganic Substances, Springer-Verlag. Ben iii (1977). [15] D.D. Wagman et al., Selected Values of Thermodynamic Properties, National Bnreau of Standards Series 270, U.S. Department of Commerce, Washington, (1968-1971). [16] Y.S. Touloukian, Series Editor, Thermophy.sicai Properties of Matter, the TPRC data series, lET/Plenum, New York, (1970-). [17] Cleveland Crystais, Inc. BBO and LBO Information Sheet, 19306 Redwood Avenue, Cleveland, Ohio 44110 [18] Fujian Castech Crystals, Inc. LBO Information Sheet, Get this address York [19] B.D. Cullity, Elements of X-Ray Diffraction, Addison-Wesley, Massachusetts, (1978) [20] D. Elweel and H.J. Scheel, Crystal Growth from High-Temperature Solutions, Aca demic Press, London [21] A. Horowitz, Jonrnai of Crystal Growth, 78 (1986) 121-128 [22] A. Horowitz, Jovrnai of Crystal Growth. 79 (1986) 296-302 [23] L. K. Cheng et al, Journal of Crystal Growth. 89 (1988) 553-559 [24] Feigelson, SPIE, 1104 (1989) [25] S. Miyazawa, Journal of Crystal Growth, 49 (1980) 515-521 [26] B. Cockayne, Journal of Crystal Growth. 42 (1977) 413-426 Bibliography 284 [27] J. R. Carruthers J. Electrochem. Soc., 114 (19) 959-962 [28] J. R. Carruthers and K. Nassau, Journal of Applied Physics, 39 (1968) 5205-5214 [29] W. E. Langlois arid C.C. Shir, Computer Methods in Applied Mechanics and Engi neering, 12 (1977) 145-152 [30] W. E. Langlois Journal of Crystal Growth. 42 (1977) 386-399 [31] N. Kobayashi and T. Arizumi, Journal of Crystal Growth, 49 (1980) 419-425 [32] N. Kobayashi, Journal of Crystal Growl/i, 52 (1981) 425-434 [33] N. Kobaya.shi, Journal of Crystal Growth, 55 (1981) 339-344 [34] R. Lamprecht, Journal of Crystal Growth. 65 (1983) 143-152 [35] M.J. Crochet and P.J. Wout.ers, Journal of Crystal Growth, 65 (1983) 153-165 [36] K. Takagi, T. Fukazawa and M. Ishi, Journal of Crystal Growth, 32 (1976) 89-94 [37] V. Nikolov, K. Iliev and P. Peshev, Journal of Crystal Growth, 89 (1988) 313-330 [38] A. Bottaro and A Zebib, Journal of Crystal Growth, 97 (1989) 50-58 [39] R. A. Brown, T.A. Kinney. P.A. Sackinger arid D.E. Bornside, Journal of Grystal Growth, 97 (1989) 99-115 [40] M. Mihelcic, C. Schroeck-Pauli. K. Wingerath, H. Wenzel, W. Uelhoff and A. Van Der Hart, Journal of Crystal Growth, 53 (1981) 337-354 [41] M. Mihelcic, C. Schroeck-Pauli, K. Wingerath, H. Wenzel, \‘V. Uelhoff and A. Van Der Hart, Journal of Crystal Growth, 57 (1982) 300-317 [42] H.J. Scheel Journal of Cry.stal Growth , 13/14 (1972) 560-565 Bibliography 285 [43] E.O. Schulz-Dubois Journal of Crystal Growth , 12 (1972) 81-87 [44] A. A. Wheeler, Journal of C’rystal Growth, 97 (1989) 64-75 [45] J. C. Burton, R. C. Prim, and W.P. Slichter, The Journal of Ghemical Physics, 21 (1953) 1987-1996 [46] F. XV. White, Viscous Fluid Flow, McGraw Full, New York [47] L. 0. Wilson, Journal of C’rysta.l Growth, 44 (1978) 371-376 [48] J. J. Favier and L. 0. \Vilson. Journal of Crystal Growth, 58 (1982) 103-110 [49] Klaus J. Vetter, Elect rochemical Kinetics. Academic Press, New York [50] Hewlett Packard, Practical Tern peralure Measurements , Application Note 290 [51] Chris Parfeniuk, M.A.Sc. Thesi.s , The University of British Columbia (1990) [52] Fidap Users Manual, Fluid Dynamics International, Evanston, Illinois 60201 (1991). [53] J.C. Brice, The Growth. of Crystals from Liquids , May 2, 1989 [54] F. Incropera, D. De Witt Fundamentals of Heat and Mass Transfer, John Wiley & Sons, New York [55] B. Pamplin Editor Grystal Growth, Pergamon Press, Oxford [56] F. Kreith and XV. Black, Basic heat Transfer, Harper and Row, New York [57] J. Szekely and J. Themelis, Rate phenomena n process metallurgy, Wiley Interscience, New York Appendix A Estimation of the Thermal Conductivity and the Gas Temperature The accuracy of the mathematical model results depend on the how well the mathemat ica.l expressions describe the process being modeled, if the assumptions used to simplify the model are correct, whether the correct values of the thermophysical properties are used and if the correct boundary conditions are used. In the Present case, the fluid flow calculations are more reliable since the liquid being considered ha.s a high viscosity. The thermophysical properties of the system tha.t are required for the mathematical model include, specific heat, density and the coefficient of thermal expansion. Values for these properties at high temperatures are available in the literature, a.s listed in Chapter 2. Viscosity values of the LBO/Mo03melts are required which are not available. These have been determined in the present investigation (Chapter 5). The low temperature therma.l conductivity of LBO is available in the literature. High temperature values are not available, and in the present case are estimated as described below. The fixed tem perature boundary conditions at the crucible wall have been estimated using temperature measurements in the rrielt, Chapter 7. The ambient gas temperature above the melt will be evaluated using the melt temperatures sho’’ii in Figure 7.52. The thermophysical properties of the melt are assumed to he the sum of the weight percent average of the constituent properties. Cp(J/gK) = CpLBQ x Wt% LBO + CpA!00. x Wt% MoO3 The mathematical model is applied to a melt having a. charge concentration of 5OWt% MoO3. The value of the melt viscosity is obtained by extrapolating the viscosity—concentration 286 Appendix A. Estimation of the Thermal Conductivity and the Gas Temperature 287 Property Unit Vahie Cp J/g K 0.63 p g/cm3 3.26 i’ poise 2.29 a’ K 6 x 10_6 Table A.25: Thermophysical properties used in the model to determine the conductivity of the melt and the ambient ga.s temperature. curves given in Figure 5.39. The average melt is taken to be 820°C. The viscosity as a function of concentration at 820°C extrapolated to .50 Wt.% is 2.29 poise. The melt properties used in the calculations are given in Table A.25. A.1 High Temperature Thermal Conductivity Evaluation The high temperature conductivity of the LBO/Mo03 solution is approximated using the temperature measurements given in Figure 7..52 and the boundary conditions given in Figure 7.53. The liquid is modeled with fixed temperatures boundary conditions at the l)ott.orn, side and top of the melt, Figure A.169. Model calculations with different conductivities are compared with the experimental results at distances of 0.5 cm, 1.0 cm, 1.5 cm and 2.0 cm from the bottom of crucible at both 0 and 1.6 cm from the center of the crucible. The conductivity that gives model resufts with the best fit of the experimental data is assumed to be the correct value. The experimental results and model calculations are compared lw the sum of the difference in the temperature values at the different locations. Difference = (Texpei.iientai — Tmodel) The model calculations using a conductivity of 0.1 W/cm K are examine. The fluid speed from natural convection, Figure A.170, is less than 0.02 cm/s due to the high viscosity Appendix A. Estimation of the Thermal conductivity and the Gas Temperature 288 Conductivity Difference (NV/cm K) (Celsius) 1.0 19.8 0.1 13.6 0.075 10.3 0.05 4.7 0.025 -17.2 0.01 -95.4 Table A.26: Difference between the model and experimental temperature values as a function of the conductivity. of the melt. Thus, conduction is the dominant mode of heat tra.nsfer in the melt. The approximation of the conductivity is easier due to no natural convection occurring in the melt. Table A.26 and Figure A.171 gives the difference between the experimental results and model calculations for different conductivity values. An LBO/Mo03conductivity value of 0.05 W/cm K used in the model gives the best fit. to the experimental temperature values. This is very close to the conductivity of LBO in the c direction, 0.039 W/cm K, as given in Table 2.2. It is assumed that the conductivit value of the LBO/Mo03melt is 0.05 W/crn K a.nd it does not change with the stoichiornetry of the melt. A.2 Ambient Gas Temperature Approximation The ambient gas temperatures at the top of the melt are approximated by determining which values give the best fit to the experimental (lata. Doing this allows the model to have the best possible fit for the thermal field in the melt. The thermophysical properties used are given in Table A.25. The value of the conductivity is 0.05 as determined in the previous section and the value of the convective heat transfer coefficient used is 0.006 Appendix A. Estimation of the Thermal Conductivity and the Gas Temperature 289 Fixed Temperature r=32cm r=1.6an r=O.Ocm z = 2.5 cm MELT I I I z=O.7anjj...I __ I z=O czi _ _______________ ________________ r3.2cm r=1.6crn r=O.Ocm Fixed Temperature Figure A.169: Temperature boundary conditions used to approximate the conductivity of the LBO/Mo03melt. Appendix A. Estimation of the Thermal Conductivity and the Gas Temperature 290 SPEED - CONTOUR PLOT LEGEND —— O3E—(2 -— ()14EJ —— 0. I000E—Oi —— U.1182E—Ol —— O.1364E—O1 —— O.1546E—O1 —— O.1728E—O1 MINIMUM 0. 00000E+O0 MAXIMUM 0. 18187E—O1 \\ J _ 4 i-”\N Centre \ Crucible Figure A.170: Fluid speed (u + u)2 in the melt. The value of the conductivity used in the model is OJ W/cm K. Appendix A. Estimation of the Thermal Conductivity and the Gas Temperature 291 2O1 10 - 0- - 10- -20- I I 0.00 0.25 0.50 0.75 1.00 Conductivity (W/cm K) Figure A.171: Difference between the model and experimental temperature values as a function of the conductivity. Appendix A. Estimation of the Therma.1 Conductivity and the Gas Temperature 292 W/cm2 K. The best fit for the ambient ga.s temperature was found to be: T = 200 + 1S7.S x r 23.7 x r2 (A.13) Where r is in centimeters and is the radial position that the convective boundary is in crucible and T is the ambient gas temperature in degrees celsius. Figure A.172 shows the measured temperature values and the model calculations using the best fit ambient gas temperature. Appendix A. Estimation of the Thermal Conductivity and the Gas Temperature 293 III liii! 860-s •% AS S S S840- S S S S Ac.) S S S‘ 82O S S A S S S S S8O0lI - S AE _ _ _ _ _ _ _ _ _ S S 780 - Measured Model CI r=O.Ocm C 760 - r=1.6cm A C 0.0 0.5 1.0 1.5 2.0 2.5 Axial Position (cm) Figure A.172: Comparison with the experimental temperature data and the model results for the best ambient gas temperature values.

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United States 6 0
China 5 2
Japan 3 0
France 2 0
Germany 2 8
City Views Downloads
Unknown 7 8
Beijing 5 2
Washington 2 0
Ashburn 2 0
Tokyo 1 0
Sunnyvale 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0078456/manifest

Comment

Related Items