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UBC Theses and Dissertations

Growth of lithium triborate crystals Parfeniuk, Christopher Luke 1994

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GROWTH OF LITHIUM TRIBORATE CRYSTALSByChristopher Luke ParfeniukB. A. Sc., The University of British Columbia, 1988M. A. Sc., The University of British Columbia, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILosOPHYinTHE FACULTY OF GRADUATE STUDIESMETALS AND MATERIALS ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMay 1994©Christopher Luke Parfeniuk, 1994In presenting this thesis inpartial fulfilment of the requirements foran advanceddegree a.tthe University of BritishColumbia. I agree that the Library shallmake it freely availablefor reference and study.I further agree that permission for extensivecopying of thisthesis for scholarly purposesmay be granted by the headof my department or by hisor her representatives. It isunderstood that copyingor publication of this thesisforfinancial gain shall not be allowedwithout my written permission.Metals and Materials EngineeringThe University of BritishColumbia2075 Weshrook PlaceVancouver, CanadaV6T 1W5Date:AbstractLithium Triborate (LiB3O5)is a nonlinear opticalcrystal used to produce short wavelength radiation from a longer wavelength source. Large lithiumtriborate crystals ofgood quality are difficult to grow. The present investigation wasundertaken to examinethe parameters influencing the growth process, andthe growth process itself. for crystalspulled from an LiB3O5melt containing the fluxMOO3.The pseudo phase diagram of the LiB3O5 — MoO3 system wasestablished and theeutectic concentration shown to be 61.5 weight percent MoO3.The viscosity of theLiB3O5 system was measured as a function of temperature andMoO3 concentration. Itwas shown that the viscosity decreases with increasingMOO3 content. As the crystalgrows MoO3 is rejected by the crysta.l at the interface.A major factor in growing largercrystals is the movement of the rejected flux away from the interface,which depends onthe fluid flow in the melt. The fluid flow in turn is dependent on buoyancyforces due totemperature gradients, as well as crystal and cruciblerotations.Calculations were carried out using a mathematical modelfor heat and fluid flow inthe melt to establish the temperature distribution, fluidflow velocity and flow directionin the melt as a function of crystal and crucible rotation. Temperaturemeasurementswere then made in the melt in a crystal growerwith a simulated crystal over a rangeof crucible rotation rotation rates, and the results comparedto the mathematical modelpredictions. The boundary conditions usedin the model were determined from temperature measurements in the melt. Comparing thecalculated radia.l and axial temperaturegradients iii the melt with the measured values showedgood agreement between thecalculated and measured temperatures.UThe flow patterns in the melt predicted bythe model were also compared totheobserved flow in a physical, modelusing glycerine as the melt and ink as a tracer, forthe same size crucible and crystal usedin crystal growth. The observed flow patternwas consistent with the model predictions.The results of both the mathematical andphysical models clearly showed that most of themixing in the liquid is associated withthe crucible rotation and very little frombuoyancy forces. From these results it wasconcluded that maximum crucible rotation should beused during growth to move theconcentrated MoO3 away from the advancing int.erface as rapidly aspossible. MaximumMoO3 concentrations should also be used, consistent withother constraints, since theviscosity of the liquid decrea.ses with increasingMoO3 concentration. It was shown thatincreasing the size of the crucible increased the flow velocityin the melt. As a resultlarger crucibles were used in the crystal growth experiments.The length of good quality crystal which can be grown, is limitedby the formation ofinclusions in the crystal at the interfa.ce. The inclusions are shownto he primarily MoO3and are considered to form when the concentration of the MoO3reaches the eutecticconcentration of 61.5 wt% at the interface. Calculationsof the diffusion of MOO3 throughthe boundary layer away from the advancing interface, show thatgrowth must he slowwith strong liquid mixing below the interface, to produce a crystal1 cm in length.A series of crystals of LBO were growii in a commercial crystal grower selectingtheMOO3 concentration, crystal and crucible rotations, and the pullingrate from the optimum values of the growth parameters given by the model predictions.Using these growthparameters, larger aild better quality crystals were produced.Facets were observed onthe crystal surfaces for the [001] growth direction which resultedin stagnant areas onthe interface and the earlier appearance of MOO3 inclusions.It was also observed thatthe crystal cracked readily under therma,l stresses during cooling.To prevent cracking.crystals have been cooled slowly after growth in a uniform thermalgradient.UITable of ContentsAbstractiiList of FiguresixList of TablesxxivAcknowledgementxxvi1 Introduction12 Literature Review42.1 Growth of Borate Crystals . .42.1.1 LBO Crystal Growth42.1.2 Barium Metahorate Crystal Growth12.2 Physical Properties of LBO112.3 Growth Defects142.3.1 Flux Inclusions/Interface Breakdown1.52.3.2 Voids172.4 Fluid Flow192.4.1 General Concepts192.4.2 Standard Growth Practices202.4.3 Accelerated Crucible Rotation212.5 Mass Transfer .27iv3 Objectives334 Experimental354.1 Growth Process354.1.1 Growth Furnace3.54.1.2 LBO Seed364.1.3 Growth Procedure414.2 Temperature Measurements in the Melt474.2.1 Initial Temperature Measurements with NoCrucible Rotation 484.3 Temperature Measurements With and WithoutCrucible with Rotation 484.4 Physical Model of The Crystal GrowthProcess.554.5 Physical Properties5.54.5.1 Chemical Analysis554.5.2 LBO/Mo03Phase Diagram.574.5.3 Viscosity584.6 Crystal Quality605 Experimental Results615.1 Phase Diagram for the LBO/Mo03System615.2 Viscosity6.55.3 Preliminary LBO Crystal Growth Runs .666 Physical Model of the LBO Crystal GrowthProcess 766.1 Observed Fluid Flow Patterns776.2 Physical Explanation of Fluid Flow Patterns797 Temperature Measurements907.1 Initial Temperature Measurements with NoCrucible Rotation 90V7.2 Temperature Measurements Withand \Vithout Crucible Rotation7.2.1 Boundary Temperature Results7.2.2 Melt Temperature Results7.3 Thermal Gradients in the Crystal During Cooling8 Mathematical Model For LBO Crystal Growth8.1 Scope of Model and Assumptions8.2 Idealized Domain and Description ofCalculations8.3 Steady State Axisymmetric Fluid Flow Model8.3.1 Equations of Fluid Flow - Lagrangian Coordinates8.3.2 Equations of Fluid Flow - Euler Coordinates8.3.3 Temperature Boundary Conditions8.3.4 Velocity Boundary Conditions8.3.5 Solution Procedure9 Sensitivity Analysis of the Fluid Flow Model9.1 Natural Convection9.2 Forced Convection9.2.1 Crystal Rotation9.2.2 Crucible Rotation9.3 Mesh Size9.4 Fluid Viscosity9.5 Conductivity10 Modeling Results17010.1 Crystal Rotation17410.2 Crucible Rotation175919195114120120123124124126128129131132132136136142150150161vi10.3 Comparison of Crystalto Crucible Rotation17910.4 Comparison of the FlowFields in Small and Large Crucibles18810.5 Iso and Counter rotationof the Crystal and Crucible18811 Comparison of Temperature Measurementswith Model Results20211.1 Small Crucible (6.6 cm diameter) Results20211.1.1 Results Assuming no Thermocouple/MeltInteraction (Small Crucible)20711.1.2 Results assuming Thermocouple/MeltInteraction (Small Crucible)20911.2 Large Crucible (8.8 cm diameter)Results21411.2.1 Results Assuming no Thermocouple/MeltInteraction (Large Crucible)21411.2.2 Results assuming Thermocouple/MeltInteraction (Large Crucible) 22011.3 Summary of the TemperatureComparisons22412 Mass Transfer Calculations22812.1 Procedure for Estimating the Equivalent.Crystal Rotation Rate22912.2 Mass Transfer Behavior of MoO3below the Crystal23312.3 Maximum Growth Rates and GrowthTimes for LBO 23613 Application of Process Engineering Principlesto Crystal Growth 24213.1 LBO 1724313.2 LBO 18. 24513.3 LBO 1924613.4 LBO 20. 24713.5 LBO 21. 24813.6 LBO 22 and 23 .. 249vii13.7 LBO 2425013.8 LBO 2525114 Summary and Conclusions27315 Recommendations for Fiture Work280Bibliography282A Estimation of the Thermal Conductivity and theGas Temperature 286A. 1 High Temperature Thermal Couductivity Evaluation287A.2 Ambient Gas Temperature Approximation288ViiiList of Figures2.1 Phase Diagram of the Li20-B03system [7]72.2 Derived phase diagram of theLBO - MoO3 system. The compositionbetween C1 and C2 is the region wherethe direct crystal growth of LBOis possible92.3 TCA analysis showing the effectsof water vapour on the stability of LBOunder dry and wet nitrogen[61 92.4 Top view of BBO—Na20melt showing radialconvective cell boundariesand central cold spot [13]112.5 Viscosity versus l/T for moltenB2O3 [4]122.6 Habit shape of grown LBO crystal[5] 142.7 Transverse dark field view througha BBO crystal grown at a very highgrowth rate, orientation [001] parallelto growth direction. Many scattering centres are observed throughout, and concentra.tedin the core regionbeneath the seed [13]162.8 BBO crystal. Breakdown of growing interfacebreakdown at A-A [13] . . 162.9 An entrapment mechanismof gas—bubbles in crystals taking account offluid flow modes associated withcrystal rotation [25]2.10 Incorporation of gas—bubbles in Pb5Ce3O11as a. function of crystal rotation and rate audi crystal diameter [25]192.11 Theoretical Taylor-Proudrnancell shapes for counter rotation[28] . . . 222.12 The direction of fluid motion predictedin the Taylor—Proudma.n cells [28]23ixFluid motion due tonatural convection [29]. 24Predicted flow withcounter rotation large enough that forcedconvectiondominates [32] . Flow shownon right half and temperatureshown on lefthalf252.15 Predicted flow with counter rotationsmall enough that natural convectiondominates [32] . Flow shownon right half and temperatureshown on lefthalf2.16 General rotational fluid flow (shearing)due to ACRT [43] . (a) Top viewof circular tube filled with twodistinguishable fluids. Tube andcontentsare in uniform rotation. (b)Final shape of fluid after tubeand contentshave come to rest. Spiral shearing distortionis evident2.17 General fluid flow in the axial andra.dia.l direction due to ACRT[43] .2.18 Model prediction of fluid flowdue to ACRT [41] . Crucible rotation10to 30 rpm. Crystal rotation 40to SO rpm. Time period of acceleration15seconds2.19 Assumptions used in Burton,Prim and Slichter [45] calculationof theconcentration in the momentumboundary layer4.20 LBO crystal growth furnace, A andCpower and control box4.21 Schematic of LBO growth chamber4.22 Crucible rotation device4.23 Seed rotation device4.24 Platinum paddle for mixingthe melt4.25 Cleavage plane of an LBOCrystal4.26 Goniometer used for orienting LBOcrystals2.132.142628293032are the crystal puller, andB is the37383939404142x4.27 LBO Seed. 434.28 LBO Seed attached to platinumseed holder444.29 Appearance of LBO melt surfaceafter the seed ha.s been dipped464.30 Positions of thermocouples and simulatedcrystal for melt temperaturemeasurements494.31 Platinum cap used to simulate the crystal504.32 Thermocouple probe holder. A - guidetrack for radial movement of TCprobe. B - rack and pinion gear for axial movement.C - thermocoupleholder. D - micrometer and dial for positioningof the thermocouples. .524.33 Apparatus for moving thermocouples attachedto the crystal puller. . .534.34 Plexiglass crucible and crystal usedfor the physical model564.35 Platinum paddle for measuring viscosity ofLBO/Mo03595.36 Temperature difference versus temperature for 45Wt% MoO3 (sample 1)determined by DTA635.37 Phase Diagram of the LBO — MoO3 system645.38 Dependence of viscosity on temperature andMOO3 concentration685.39 Variation of viscosity with MOO3 at the temperaturesindicated 695.40 Liquidus temperature and viscosity as afunction of MoO3 705.41 Surface of an LBO crystal with inclusions.Magnified 30 times 735.42 Normal molybdenum inclusionin a.n LBO crystal magnified 400 times.(a)Backscatter image. (b) WDS molybdenumdot map 745.43 Molybdenum line inclusionin LBO crystal magnified 2,200 times.(a)Backscatter image. (b) WDS molybdenumline scan756.44 Initial dye tracer pattern in theglycerine. a.) Top view. b) Side view.Crucible rotated at 45 rpm82xi6.45 Dye tracer pattern fri the glycerinewhen the blue and red tracers reachthe centre of the fluid. a) Top view.h) Side view. Crucible rotated at 45rpm836.46 Dye tracer pattern in the glycerine whenthe dye reaches the bottom of thecrucible. a) Side view. b) View under thecrystal. Crucible rotated at 45rpm846.47 Dye tracer pattern near the bottom of thecrucible. a) Side view. b) Viewunder the crystal showing the red and bluedie moving up the side walls ofthe crucible. Crucible rotated at 45 rpm856.48 Dye tracer pattern at different crucible rotationrates. a) Crucible rotationrate of 45 rpm. b) Transition flow for a cruciblerotation rate between 45and 78 rpm. c) Crucible rotationrate of 78 rpm866.49 Top view of the dye tracer patternsat different crucible rotation rates. a)Crucible rotation rate of 45 rpm. h) Crucible rotation rateof 78 rpm. . 876.50 Interface curvature due to solid body rotation886.51 Crucible with portion of the upper surface constrainedto zero 897.52 Temperature variation with axial positionin a 55 Wt% MoO3solution. Thethree radial locations are r 0 mm,r 16 mm, and r = 32 mm. Thereis no crucible rotation during the temperature measurements.Cruciblediameter is 6.6 cm927.53 Temperature boundary conditions used in thefluid flow model for the sensitivity analysis and the examinationof the operating parameters. Cruciblediameter is 6.6 cm937.54 Melt temperature 0.2 cm from the crucible wallwith and without thesimulated crystal. Crucible diameter is6.6 cm96xii7.55 Melt temperature 0.47from the crucible bottom with and withoutthesimulated crystal. Crucible diameteris 6.6 cm977.56 Temperature boundary conditionsused in the mathematical model of meltwith the simulated crystal.Crucible diameter is 6.6 cm987.57 Melt temperature 0.2 cm from thecrucible wall with and without thesimulated crystal. Crucible diameter is8.8 cm997.58 Melt temperature 0.36 from the crucible bottomwith and without thesimulated crystal. Crucible diameter is8.8 cm 1007.59 Temperature boundary conditions used in the mathematicalmodel of meltwith the simulated crystal. Crucible diameteris 8.8 cm1017.60 Temperature distribution at r = 1.0 cm forcrucible rotations of 0, 15, 20,25 and 30 rpm. Crucible diameter is6.6 cm. Simulated crystal present. 1037.61 Temperature distribution at r =1.5 cm for crucible rotations of 0, 15, 20,25 and 30 rpm. Crucible diameter is6.6 cm. Simulated crystal present. 1047.62 Temperature distribution at r 2.6cm for crucible rotations of0, 15, 20,25 and 30 rpm. Crucible diameter is 6.6 cm. Simulated crystalpresent. 1057.63 Temperature distribution at. r = 3.1 cm for cruciblerotations of 0, 15, 20,25 and 30 rpm. Crucible diameter is6.6 cm. Simulated crystal present 1067.64 Change in liquid isotherms with cruciblerotation. (a) No crucible rotation.(h) Large crucible rotation1077.65 Temperature distribution at r = 0.4 cm for cruciblerotations of 0, 10, 20and 30 rpm. Crucible diameter is8.8 cm. Simulated crystal present. . .1097.66 Temperature distribution at r =0.9 cm for crucible rotations of 0. 10, 20and 30 rpm. Crucible diameter is8.8 cm. Simulated crystal present1107.67 Temperature distribution a.t r2.8 cm for crucible rotations of0. 10, 20and 30 rpm. Crucible diameter is 8.8cm. Simulated crystal present. . .111xlii7.68 Temperature distribution atr = 3.3 cm for crucible rotations of 0,10, 20and 30 rpm. Crucible diameteris 8.8 cm. Simulated crystal present. . .. 1127.69 Temperature distribution at r3.8 cm for crucible rotations of 0, 10, 20and 30 rpm. Crucible diameter is8.8 cm. Simulated crystal present. . . . 1137.70 Alumina aggregate used for determining thethermal gradients in the furnace.1167.71 The axial positions of the alumina modelcrystal used when measuringthethermal gradients1177.72 Temperature gradients measuredfor the different furnace configurationsasindicated1188.73 Schematic representation of domain examinedwith the model1259.74 Temperature boundary conditions usedin the sensitivity analysis1339.75 Vector plot of fluid velocity due to naturalconvection1379.76 Temperature contours tha.t occurin the LBO/Mo03melt1389.77 Axial velocity due to natural convection1399.78 Radial velocity due to natural convection1409.79 Tangential crystal surface velocity dueto natural convection 1419.80 Vector plot of fluid velocity due tocrysta.l rotation 1439.81 Rotational velocity plot of the LBO/Mo03melt with crystal rotation. 1449.82 Temperature cont.ours that occurin the LBO/Mo03melt with crystalrotation1459.83 Axia.1 velocity due to crystal rotation1469.84 Radial velocity due to crystalrotation1479.85 Tangential crystal surface velocitydue to crystal rotation 1489.86 Vector plot of fluid velocity due to cruciblerotation 1519.87 Rotational velocity plot of the LBO/Mo03melt with crucible rotation. 152xiv9.88 Temperature contours that occurin the LBO/Mo03melt with cruciblerotation1539.89 Axial velocity due to crucible rota.tion1549.90 Radial velocity due to crucible rotation1559.91 Tangential crystal surface velocity due tocrucible rotation1569.92 Two mesh densities used to examinethe models sensitivity. (a) Regularmesh density, approximately 1795 nodes. (b) Coarsemesh density, approximately 585 nodes1579.93 Axial velocity for different mesh densities1589.94 Radial velocity for different mesh densities1599.95 Tangential crystal surface velocity for differentmesh densities 1609.96 Axial velocity for different viscosities1629.97 Radial velocity for different viscosities1639.98 Tangential crystal surface velocity for different viscosities1649.99 Axial temperature profiles for different conductivities1669.100 Axial velocity for different conductivities1679.101 Radial velocity for different conductivities1689.102 Tangentia.l crystal surface velocity for differentconductivities 16910.103 Small crucible temperature boundaryconditions used in the results analysis.17210.104 Large crucible temperature boundary conditionsused in the results analysis.17310.105 Axial velocities a.t 0.5 of the fluid height.40.9 Wt% MoO3 present in thefluid. Crystal rotated a.t0, 10, and 20 rpm. Crucible is stationary. (a)Shape of the solid/liquid interface.(b) Axial Velocities176xv10.106 Radial velocities at0.5 of the crucible radius. 40.9 Wt% MoO3 presentinthe fluid. Crystal rotated at0, 10 and 20 rpm. Crucible is stationary.(a)Shape of the solid/liquid interface. (b) Radial velocities17710.107 Velocities tangential to the crystalsurface 0.5 of the crystal radius.40.9Wt% MoO3 present in the fluid. Crystal rota.tedat 0, 10 and 20 rpm. Crucible is stationary. (a) Shape of the solid/liquidinterface. (h) Tangentialvelocities17810.108 Axial velocities at 0.5 of the fluid height. 40.9Wt% MOO3 present in thefluid. Crucible rotated at0, 20, 40, and 60 rpm. Crystal is stationary. (a)Shape of the solid/liquid interface. (b) Axial velocities18010.109 Radial velocities at 0.5 of the crucible radius.40.9 Wt% MoO3 present inthe fluid. Crucible rotated at 0, 20, 40, and60 rpm. Crystal is stationary.(a) Shape of the solid/liquid interface. (b) Radialvelocities 18110.110 Velocities tangential to the crystal surface0.5 of the crystal radius. 40.9Wt% MoO3 present Crucible rota.ted at 0, 20, 40, and60 rpm. Crystal isstationary. (a) Shape of the solid/liquidinterface. (b) Tangential velocities.18210.111 The magnitude of the axial velocities alonga horizontal line at 0.5 of thefluid height. The two conditions examinedare crucible rotated at 20 rpmwith a stationary crystal and a stationarycrystal with a crucible rotatingat 20 rpm. (a’) Shape of the solid/liquid interface.(h) Axial velocities. . . 18410.112 The magnitude of the radial velocitiesalong a vertical line at 0.5of thecrucible radius. The two conditions examinedare crucible rotated at 20rpm with a stationary crystal and a stationarycrystal with a cruciblerotating at 20 rpm. (a) Shape of thesolid/liquid interface. (b) Radialvelocities185xvi10.113 The magnitude of the velocitiestangential to the crystal surface at0.5 ofthe crystal radius. The two conditionsexamined are crucible rotatedat20 rpm with a stationary crystal anda. stationary crystal with a cruciblerotating at 20 rpm. (a) Shape of the solid/liquidinterface. (h) Tangentialvelocities18610.114 Magnitude of the tangential velocity0.5 cm from the liquid/crystal interface for different crystal and crucible rotationrates. The calculations arefor a rotating crucible with a. stationary crystaland a stationary crucibleand rotating crystal18710.115 Axial velocities at 0.5 of the fluidheight. Large and small crucible shown.40.9 Wt% MOO3 present in the fluid. Crucible rotatedat 60 rpm. Crystalis stationary. (a) Shape of the solid/liquid interface.(b) Axial velocities. 18910.116 Radial velocities at 0.5 of the crucible radius.Large and small crucibleshown. 40.9 Wt% MoO3 present in thefluid. Crucible rotated at 60 rpm.Crystal is stationary. (a) Shape of the solid/liquidinterface. (b) Radialvelocities19010.117 Velocities ta.ngentia.l to the crystalsurface at 0.5 of the crystal radius.40.9Wt% MoO3 present. Large and small crucible shown. Cruciblerotated at60 rpm. Crystal is stationary. (a) Shape of the solid/liquidinterface. (b)Tangentia.l velocities19110.118 Axial velocities a.t 0.5 of the fluid height..40.9 Wt% MOO3 present in thefluid. Crysta.l rota.ted at± 10 rpm and crucible rotated at 20 rpm. (a)Shape of the solid/liquid interface.(b) Axial velocities19310.119 Radial velocities at 0.5 of the crucible radius.40.9 Wt% MoO3 present inthe fluid. Crystal rotated a.t± 10 rpm and crucible rotated at 20 rpm. (a)Shape of the solid/liquid interface. (b)Radial velocities194xvii10.120 Velocities tangentialto the crystal surface at 0.5 of the crystalradius. 40.9Wt% MoO3 present in the fluid. Crystal rotatedat + 10 rpm and cruciblerotated at 20 rpm. (a)Shape of the solid/liquid interface. (b) Tangentialvelocities19710.121 Fluid velocity tangentialto the solid/liquid interface as a. functionof crystalrotation rate. The velocity is 1mm from the crystal/melt interface andat 0.5 of the crystal radius. 40.9Wt% MoO3 present in the fluid. Thecrucible is rotated at 20 rpm19810.122 Vector plot of fluid velocityat a. crucible rotation rate of 20rpm and acrystal rotation rate of 0 rpm.Point B19910.123 Vector plot of fluid velocityat a crucible rotation rate of20 rpm and acrystal rotation rate of—23.5 rpm. Near point C20010.124 Vector plot of fluid velocity ata crucible rotation rate of 20rpm and acrystal rotation rate of—35 rpm. Point D20111.125 Temperature boundary conditionsused for t.he small crucible model. .. 20311.126 Temperature boundaryconditions used for the large cruciblemodel. . . 20411.127 Theta velocity boundary conditionsused in the model20511.128 Theta velocity boundary conditionsused in the model to account forthethermocouples in the melt20611.129 Experimental and calculated temperaturesa.s a function of axial height.Small (6.6 cm diameter) crucible.Crucible rotation 0 rpm20811.130 Experimental and calculated temperaturesas a function of axial heightat the radial locations indicated.Small (6.6 cm diameter) crucible. Crucible rotation = 1.5 rpm. Standardvelocity boundary conditions are used(Figure 11.127)210xviii11.131 Experimental and calculated temperatures as a function of axial heightatthe radial locations indicated, diameter) crucible. Crucible rotation = 20rpm. Standard velocity boundary conditions are used (Figure 11.127). .21111.132 Experimental and calculated temperatures as a function of axial heightat the radial locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 25 rpm. Standard velocity boundary conditions are used(Figure 11.127)21211.133 Experimental and calculated temperatures as a function of axial heightat the radial locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 30 rpm. Standard velocity boundary conditions are used(Figure 11.127)21311.134 Experimenta.1 and calculated temperatures as a function of axial heightat the radial locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 15 rpm. Modified velocity boundary conditions are used.(Figure 11.128)21511.135 Experimental and calculated temperatures as a functioii of axial heightat the radial locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 20 rpm. Modified velocity boundary conditions are used.(Figure 11.128)21611.136 Experimental and calculated temperatures as a function of axial heightat the radia.l locations indicated. Small (6.6 cm diameter) crucible. Crucible rotation = 25 rpm. Modified velocity boundary conditions are used.(Figure 11.128)217xix11.137 Experimental and calculated temperatures as a function of axialheightat the radial locations indicated. Small (6.6 cm diameter) crucible.Crucible rotation = 30 rpm. IViodifled velocity boundary conditions are used.(Figure 11.128)218.11.138 Experimental and calculated temperatures as a function of axial locationalong various vertical lines. Large (8.8 cm diameter) crucible. Zerocruciblerotation21911.139 Experimental and calculated temperatures as a function of axial heightat the radial locations indicated. La.rge(8.8 cm diameter) crucible. Crucible rotation = 10 rpm. Standard velocity boundary conditions are used(Figure 11.127)22111.140 Experimental and calculated temperatures as a function of axial heightat the radial locations indicated. Large (8.8 cm diameter) crucible. Crucible rotation = 20 rpm. Standard velocity boundary conditions are used(Figure 11.127)22211.141 Experimental and calculated temperatures as a. function of axial heightat the radial locations indicated. Large (8.8 cm diameter) crucible.Crucible rotation = 30 rpm. Standard velocity boundary conditions areused(Figure 11.127)22311.142 Experimental and calculated temperatures as a function of axial heightat the radial locations indicated. Large (8.8 cm diameter) crucible. Crucible rotation = 10 rpm. Modified velocity boundary conditions are used.(Figure 11.128)223xx11.143 Experimental and calculated temperaturesas a function of axial heightat the radial locations indicated. Large (8.8 cm diameter) crucible.Crucible rotation = 20 rpm. Modified velocity boundary conditions areused.(Figure 11.12S)22611.144 Experimental and calculated temperatures as a functionof axial heightat the radial locations indicated. Large (8.8 cm diameter) crucible.Crucible rotation = 30 rpm. Modified velocity boundary conditions areused.(Figure 11.128)22712.145 determination of the radius used in the analytical solution. Theradius, rais equivalent for both the finite element analysis and the analytical analysis.23112.146 Calculated and analytical velocity values. The calculated tangential velocity values are a.t at 0.5 of the crystal radius. The analytical solution (flowpast a rotating disk) for the radial velocity is at a radial location thatisequivalent to the surface length of the calculatedsolution 23212.147 Cross section of crystal grown a.t a crucible rotation rate of 60rpm. Theradius of the crystal is approximately 2 cm and the height at thecentreline is 0.85 cm23912.148 The concentration of MoO3 next to the growing interface as a functionoftime for the growth rates(.1)indicated for a crucible rotation of 60rpm. . 24013.149LB0 17 crystal frozen in the melt25213.l50Uncracked portions of LBO 17 crystal25313.151 Top view of LBO 18 crystal25413.152 Bottom view of LBO 18 crystal25513.l53Uncracked portions of LBO 18 crystal25613.154 Side view of LBO 19 crystal257xxi13.155 Cross section view of LBO19 crystal. The crosses are regions were sampleorientation was determined25813.156 Area in LBO 19 were interface breakdown/eutectic growthstarted. Magnified 20 times. (a) SEM photo. (1)) Map of molybdenumconcentration.The bright regions correspond to a high molybdenumconcentration. . . 25913.157 View of an molybdenum inclusion magnified200 times. (a) SEM photo.(b) Map of molybdenum concentration. The brightregions correspond toa high molybdenum concentration26013.158 Top view of LBO 20 crystal26213.159 Cross section view of LBO 20 crvsta.l26313.16OLBO 21 Crystal26413.161 Top view of LBO 23 crystal26.513.162 Bottom view of LBO 23 crystal26613.163 Interface appearance for planes were MoO3 was stuck to the surface.(a)Photo of underside of crystal. The area. A is a region ofMoO3 build up.(b) Schematic of surface along line B — B . .26713.164 Pieces of LBO 23 that were uncra.cked26813.16.5 Top view of LBO 24 crystal26913.166 Bottom view of LBO 24 crystal27013.167 Back lit view of LBO 24 Crystal27113.168 Schematic and photo of the MoO3 on the LBO 25 crystal272A.169 Temperature boundary conditions used to approximate the conductivityofthe LBO/Mo03melt289A.170 Fluid speed (u+ in the melt. The value of the conductivity used inthe model is 0.1 \V/cm K290xxiiA.171 Difference between the model and experimental temperature valuesas afunction of the conductivity291A.172 Comparison with the experimental temperature data andthe model resultsfor the best ambient gas temperature values293xxiiiList of Tables2.1 Reports on LBO crystal growth. SOT = seedon temperature; CR =cooling rate; PR = pull rate; atm = atmosphere,CDR = post growthcooling rate, SROT = seed rotation rateS2.2 Physical properties of Li20, B203,MoO3 andMo203 135.3 MoO3 and LBO concentrations used in determining thephase diagram. 625.4 The solidus and liquidus temperatures of theMOO3 — LBO samples . . 655.5 MoO3 and LBO concentration of the samples used in the viscositymeasurements665.6 Viscosity versus temperature for samples 7 to 10675.7 Viscosity versus temperature for sample 11717.8 MoO3 and LBO concentrations in the melt that wereused for the temperature measurements with and without crUcil)le rotation947.9 Temperature gradients measured for the different furnaceconfigurations asindicated1198.10 Assumptions used in fluid flow model228.11 Values used in the nondimensional analysisof the LBO/Mo03system . 1228.12 Non Dimensional Numbers for Gallium andLBO 239.13 Standard thermophysical propertiesused in the sensitivity analysis. . .1349.14 Parameters examined for sensitivity analysis134xxiv10.15 Thermophysical propertiesused in the results analysis17110.16 Parameters examined for themathematical model analysis17411.17 Thermophysical properties and rotation valuesused in the model for comparison with experimenta.1 temperaturemeasurements 20712.18 Numerical solutions for a rotating disk[46] 23012.19 Thermophysical properties used in the Analyticalsolution for flow belowa rotating disk23012.20 Values of the variables used in the determinationof the diffusion coefficient.23512.21 Diffusion coefficient of some liquids[57] 23512.22 The concentration of MoO3 next to the growinginterface as a function oftime. Diffusion coefficient is2.38 x10—8cm2/s. Growth rate(f)is 0.698mm/day24113.23 Growth conditions used for the crystalgrowth experiments25113.24 Growth conditions used for the crysta.l growthexperiments, continued. . . 261A.25 Thermophysical properties used in the modelto determine the conductivityof the melt and the ambient gas temperature287A.26 Difference between the model and experimentaltemperature values as afunction of the conductivity288xxvAcknowledgementI would like to thank Dr Indira Samerasa.kera,Dr Fred Weinberg, Jeff Edel, Dr KimFjeldsted and Brian Lent for their interest, suggestionsand help over the course of thisproject. The assistance of the professional and technicalstaff at Crystar and UBCinparticular, Jeff Clavdon, Mark Wa.ddington, Tim Elwel,Don Freschi, Fran Steeds, DaveWebb, Ernie Minkowitz and Mary Mager is greatlyappreciated.I especially thank my parents, Walter and RuthParfeniuk, for their love, encouragement and understanding that they have givenme over the years. I would also like tothank Tresca Batten for her friendship andfor putting up with me and my work scheduleduring my thesis. The conversation andfriendship of the ladies from the 6:30 and 9:00fitness class is also much appreciated.Thanks are also extended to iv fellow graduate students,especially Bernardo HernandezMorales, Cohn Edie, Dave Tripp, Ed Chong and BarryWiskel for their help during thisproject. I would like to express my gratitudeto the Natural Sciences and Engineering Research Council and the BritishColumbia Science Council for financial support andJohnson Matthey Electronics (Crysta.r Research)for materia.l support dun ng my studies.xxviChapter 1IntroductionLithium triborate LiB3O5,refered to as LBO, is a recently developednonlinear opticalcrystal [1]. The nonlinear optica.l behaviour of crystals is described bythe electromagnetic equations, as they relate to optics. The most important characteristicin the presentcase is the ability of the nonlinear optical crystal to generate higheroptical harmonics. Ifa monochromatic light beam, from a laser source, is passed througha nonlinear opticalcrystal, higher harmonic light is generated. Specificaly if infrared light froma Nd:YAGlaser is used as a source, ultra violet light will he generated in the crystal. Inmany applications the shorter shorter wavelength of ultra violet light has important advantages overvisible and infrared light. The specific advantages of LBO over othernonlinear opticalcrystals are [2, 3]: large transparency range (170 nm to 2.6urn),the largest effectivesecond harmonic generation conversion coefficient, high surface damage threshold(2.5GW/cm2for 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, inhigh powered lasersfor military applications, and in optical parametric amplifiers and oscillators.Other nonlinear optica.l crystals of the borate family include ,6—BaB2O4and1KB506(OH)4. 2H20.Lithium triborate crystals are difficult to grow due to properties that are intrinsicto the crystal and the growth process. One difficulty results from the high viscosityof LBO. The material contains 87.5 weight percent B203 which has aviscosity of 630poise at 727°C [4]. The high viscosity sharply reduces fluid flow a.nd masstransfer in themelt during crysta,l growth. A second difficulty is that LBO solidifiesin an incongruent1Chapter 1. Introduction2manner. Unlike GaAswhich solidifies congruently, LBO formsfrom a peritectic reaction(Li4B10O17+Liquid —* LiB3O5 at834°C). Although it is possible to produceLBOcrystals by peritectic growth,the process is very slow andonly small crystals can begrown. To overcome this difficultyadditional quantities of B203or the compound MoO3is added to the LBO which enablesLBO to be grown directly from theliquid without aperitectic reaction. This is normallytermed flux or solution growth.In this study MoO3was used as a flux.Adding MoO3 to LBO introducesnew factors in the crystal growthprocess. Theamount of MoO3 added must be established.The viscosity of the melt decreaseswithincreasing MoO3 concentrationwhich ca.n result in increasedfluid flow in the melt. Thisis a significant factor since theMoO3 is rejected 1w the solid atthe advancing solid/liquidinterface and must moveaway from the interface for growthto continue. However, thehigher the bulk concentration ofMoO3 in the melt the higher the concentrationbuild UJ)immediately ahead of the interfacefor a given rate of crystal growth. If theconcentrationof MoO3 at the solid/liquid interface reachesthe eutectic then eutectic growth willoccurwhich results in the formationof MoO3 rich phases in the solid.This reduces the opticalquality of the crystal and will causethe crystal to crack as it coolssince LBO has adifferent thermal expansion coefficient thanthe MoO3 phase.Crystal growth from solution, in other systems,has been examined previously. Mostof the work has been experimental, givingempirical correlations between growthparameters and theciuality of the grown crystal. LBO has been known to exist since1926.Research into the solution growthof LBO wa.s initiated in 1987[5] following the discovery of its special nonlinear optical properties.Reports on the growth of LBO crystalstothe present, do not include details of thegrowth process and do notrelate the growthparameters to defects generated in the crysta.lduring growth.LBO crystals are normally grown usingthe Top Seeded Solution Growth(ModifiedChapter 1. Introduction3Czochralski) Process in whicha rotating, oriented, seed crystal is dipped ina counterrotating bath at a temperature just abovethe L130—Mo03liquidus. The furnacetemperature is then slowly cooled to allow thecrystal to grow in the radial and axial directions.\Vhen the crystal has grown to a sufficient diameter,it is slowiy raised resulting in crystalgrowth in the axial direction. The controlof thermal gradients in the system and thecontrol of the mass transfer of the MoO3 at thesolid/liquid interface, is critical to thegrowth of high quality crystals. The mass transferof MoO3 is strongly dependent ontheviscosity of the melt, the crystal andcrucible rotations, and the diffusion rate of MoO3in the melt.In the present investigation direct temperature measurementsof the melt were madeduring simulated crystal growth. The viscosityand other physical parameters of theliquid were also measured as function of MoO3 concentrations.A mathematical model ofthe system is developed, using the finite element method,to characterize both the thermalfield and the fluid flow in the system during crystal growth.A simple mass transfer modelis used, in conjunction with the fluid flow model, to examinethe concentration of MoO3ahead of an advancing solid/liquid interface duringgrowth. The models are employed toselect the optimum MoO3 concentration, crystal and cruciblerotation rates, temperaturefields and growth rates to produce large diameterLBO crystals having minimal defects.Crystals of LBO were grown using a range of growthparameters, and the size of crystal,and visible crystal defects, related to the model predictions.Chapter 2Literature Review2.1 Growth of Borate CrystalsThis review will consider the growth of Lithium Trihorate(LBO) and Barium Metaborate (BBO) crystals. Barium Metaborate is reviewedto complement the very limitedpublished information available for LBO. Both crystalsare similar, having B203 as amajor component, and both crystals are grown in thesame manner.2.1.1 LBO Crystal GrowthThe Lithium Trihorate phase was initially reportedin 1926. The phase diagram ofLi20/B03system was reported in 1958 [7] and is shownin Figure 2.1. The LiB3O5.LBO, phase in this complex system is shown by the arrowat 87.5 wt% B203. Oncooling the melt at this borate concentration, there isa peritectic transformation at834 ± 4°C. There is a eutectoid transformation at595 ± 20°C; however, LBO was foundto be stable with no eutectoid transformation occurringon cooling below 595°C [5]. Thesuccessful growth of small crystals of LBO wasreported in 1978 [8] and 198019]usinga solid sta.te reaction process in which a B203 glass was coveredwith LiF powder andreacted at 750°C for 10 hours. The LBOcrystal structure was analyzed and found tobe orthorombic, symmetry class rnm2 and havinga. space group of P21. The strongnon—linear optical properties of LBO, reported in1987, resulted in the development ofthe solution growth of these crystals.4Chapter 2. LiteratureReview5Solution growth consistsof adding fluxes to L130 which modifiesthe phase diagramand allows LBO to he growndirectly as a solid from the liquid. Fluxmaterials used areMo03[2], LiF [5] or B203 as aself fluxing agent. The flux must havea high solubility inthe melt over a large temperaturerange and no solubility in the solidLBO. A derivedphase diagram for the LBO —MoO3system is shown in Figure 2.2. It wasconstructed byextrapolating the Li20/B03phase diagramfrom 87.5 wt% B203 towards100% MoO3.Crystals can he grown directly from themelt in the MoO3 concentrationrange betweenC1 and C2. Reports of the successful growth of LBOcrystals using various fluxes andstarting materials are listed in Table 2.1. No detailsof the growth process are given,norspecifics concerning the quality of the crystalsproduced. The post growth coolingrateis important as the thermal stresses duringcooling readily cause the crystal to fracture.LBO crystals are grown by slow coolingand may he combined with pullingthe crystalsvertically from the melt. The melt isheld in platinum crucibles, 50 mmin diameter andheight [1, 2], platinum being requiredbecause of the corrosive nature ofB203. Thecrucible is heated with nickel/chromium resistanceheating elements in a vertical furnacesystem. The insulation consists of aluminabased ceramics. The starting materialsaremade up of a combinations of LiOH, Li2B4O7,LiBO2,B203,H3B0 or similar compoundswhich contain extra hydrogen, carbon oroxygen components, the final mixture containing87.5 wt% B203.The amount of flux added can vary, near55 wt% MoO3,or enough toproduce a bulk composition of at least.90.3 wt% for B203 [2]. The amount of LiF addedas flux is not specified in tile relevant reports.The charge, including the LBO componentsa.nd flux, is heated to950°C to ensurethat all the components melt, and heldfor 5 hours, to allow the melt to homogenizeandH20 and CO2 vapour to form [1, 2,5]. Following homogenization, the melt is cooled.and at a suitable melt temperature. a. seedcrysta.] is clipped into the melt.The crystalis grown by slowly cooling the furnacea.t. a fixed ra.te. In some cases, the seedis slowlyGhapter 2. Literature Review6pulled from the melt to allowfor additional growth in the vertical direction.The seedorientation is generally[001] parallel to the growth direction. Duringgrowth the crystalis rotated as a constant rate.When growth is complete the crystalis lifted from themelt and slowly cooled in thefurnace to room temperature.The crystals are sensitiveto thermal strains and can fractureduring cooling. Details of thegrowth procedure aredependent on the fluxing agent, as outlinedbelow, for B203 self fluxing andMoO3.Crystal growth, with a B203 self fluxingsystem, is carried out nearthe liquidustemperature of 834°C for a. 90wt% B203 melt concentration. The procedureis to coolthe melt to 848°C, dip the seedinto the melt, hold for30 minutes, then rapidly coolto 833°C. The melt is thensiowly cooled between 0.5 to 2°C/da.y during whichtimethe crystal slowly growsin the melt. The seed is not pulled verticallyand the crystalis not rotated during growth[2]. Growth is terminated when the crystal hasreachedthe specified dimensions. SinceB203 has a high viscosity, the self fluxing processresultsin a melt having a high viscosity. Thismarkedly reduces fluid flow in the melt duetobouyancy forces and reduces mass transferin the melt adjacent to the growingcrystalinterface. This results ina high level of growth defects in the crystal[5].With an MOO3 flux, the homogenizedmelt is cooled to 673°C and held for5 hours.The seed crystal is then dipped intothe melt, held for 30 minutes, andthe melt thencooled at 5°C/day as the crystal grows.During growth the seed is rotated at 30rpm andpulled from the melt at a rate of1 mm/day [5].After crystal growth, the seedis separated from the melt and cooled toroom temperature at rates between 40 to100°C/hour. Stress in the crystalduring cooling can causethe crystal to fracture, dueto anisotropy of the expansion coefficientsof the crystal andthe solidified flux on the crystal surface[5].It was found that LEO decomposesat high temperatures when exposed to watervapour in an ambient atmosphere[6]. The decomposed material is Li3B7O12.AdryChapter 2. Literature Review7950900-‘ 850008007507006508090100.—Li20COMPOSITION (WT%) B203Figure 2.1: Phase Diagram of the Li20-B03system[7].nitrogen atmosphere was used to stop the decomposition. Figure2.3 shows the differencein the weight loss of an LBO sample in a dry and wet nitrogenatmosphere.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 beinclusion free,however no photographs of the LBO crystals were shown.2.1.2 Barium Metaborate Crystal GrowthThe successful growth of Barium Metaborate crystals (BBO), fl—BaB2O4,was reportedin 1985 [11} using Top Seeded Solution Growth (TSSG). The crystals were grownwithB203 flux, the melt containing 43wt% B203. This is appreciably lower than theB203content in LBO growth(87.5 wt%)which results in a much lower liquid viscosity inthe BBO melt. Na20 was examined as an alternative flux to B203 and was shown toproduce better quality crystals [12]. Na20 has a solubility range of 22 to30 wt% in BBOin the temperature range of 755 to 925°C [13].Chapter 2. Literature Review8Year Starting MaterialsFlux Growth ParametersCrystal Size1989 [1] Li20,H3B0, ?wt% B203 SOT = 833°C30 x 30 x 15 mm3CR = 0.5°C/dayPR = not givenatm = not givenCDR = 40°C/hourSROT = not given1989 [2] Li2CO3,H3BO, 90 wt% B203SOT = 848°C 18 x 20 x 6 mm3CR = 0.5°C/dayPR = not givenatm = not givenCDR = 40°C/hourSROT 0 rpm1989 [2] Li2CO3,H3B0, 55 wt%MoO3 SOT = 673°C 20 x35 x 9 mm3CR = 5°C/dayPR. = not givenatm = not givenCDR = 100°C/hourSROT = 30 rpm1989 [10] Li2CO3.H3B0. 93 wt% B2O3. SOT = 778°C No values listedCR. = 1.3°C/dayPR. = not givenatm = airCDR = not givenSROT = not given1990 [5] LiCO3,Li20, > 90 wt% B203, SOT = 834°C Size: 35 x 30 x 15 mm3L1OH, Li2B4O7 ? wt% LiF CR = 0.2-2°C/dayLiBO2,B2O3, PR = 1mm/day113B0,etcCDR = not givenTable 2.1: Reports on LBO crystal growth. SOT =seed on temperature: CR = coolingrate; PR=pull rate; atm = atmosphere, CDR = post growthcooling rate, SROT’ =seed rotation rate.Chapter 2. Literature Review9°°4__.LiO2BO3±LiquidE—Li2O5B203+ LiquidLiquid800c-)700LBO + LiquidEutectic600LBO + MoO3ci C2MoO3Concentration (Wt %)Figure 2.2: Derived phase diagram of the LBO - MoO3system. The composition betweenC1 and C2 is the region where the direct crystal growth of LBO is possible.TG8OOC1.2O4::ThrTime (hi)Figure 2.3: TGA analysis showing the effects of water vapour on the stability of LBOunder dry and wet nitrogen [6].Chapter 2. Literature Review10The procedure for growingBBO crystals, and evaluation of their quality, is givenbyFeigelson et al. [13]. Two types of furnaces wereused, a low gradient wire wound furnaceand a high gradient SiC heated furnace. Theplatinum crucibles containing the meltwere 55 mm in diameter and 55mm in height. Themelt composition used was8Oat%BBO—2Oat%Na20.The axial and radial gradients were approximately 50°C and30°C forthe high gradient furnace and 20°C and 10°C for the low gradient furnace.Temperaturefluctuations due to thermal convection was measured tobe +8°C.To grow a crystal, the charge was melted and then cooledto the temperature at whichthe seed crystal was dipped into the melt. The dipping temperature wasestablished byimmersing a platinum wire into the melt and coolinguntil a small amount of BBOsolidified on the wire. The melt was maintained at this temperature for12 hours. Next,the wire was removed and the melt temperature increased by several degreesafter whichthe BBO seed was immersed into the liquid. During growth the seedwas rotated at ratesbetween 2 and 16 rpm. There was no crucible rotation. Fluid flow occurredin the melt,due to natural convection which could be detect.ed by the presence ofradial convectiveboundaries at the top surface, Figure 2.4.Two seed crystal orientations were examined the c direction [001] andb direction[010] aligned parallel to the growth direction. The c orientation proved to he thebetterof the two, since the b direction crystals exhibited more cracking duringcooling. Theinterface shape during growth was more concave towards the melt for the corientation,because of the higher thermal conductivity of the crystal in this direction; oneorder ofmagnitude larger than in the b direction. Crystal quality improved athigher crystalrotation rates due to the increase in the fluid flow velocity atthe interface. At the highestrotation rates the interface inverted from concave to the liquid to concaveto the solid.During crystal growth, the cooling rate of the melt was maintainedat 2°C/day and apulling rate of 0.5 to 1.0 mm/day was used. After approximately 12 mmof crystal hadChapter 2. Literature Review11Top Viewot CrucibleFigure 2.4: Top view of BBO—Na20melt showing radial convective cell boundaries andcentral cold spot [13].grown the planar interface breaks down, resulting in flux inclusions being incorporatedinto the crystal.Better crystal quality was obtained using the large gradient furnace. It wa.s alsofound that crystal quality was best at the outside region of the crystal and worst at thecentre. These correspond to high flow and stagnant regions in the melt below the growinginterface [13].2.2 Physical Properties of LBOAn extensive literature search was carried out to obtain the thermo—physical propertiesof 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 MoO3are well established [14, 15]. Experimental measurements of the viscosity of B203 [4] areavailable but no data was found for the other components of the LB 0/flux mixtures.Theviscosity of B203,Figure 2.5, is very high, 280 poise at 833°C and 692 poise at 727°C.Chapter 2. Literature Review12800C 750C 700C1uUu I I900 ci800 - - -700.600500-ri4‘I)400C.?cI)300I I I I I I I8.8 9.0 9.2 9.4 9.6 9.8 10.0 10.2 10.41O,000IT[KJFigure 2.5: Viscosity versus 1/T for moltenB203 [4]The thermal conductivity of11203 [16] and LBO [17] are low as would be expected foroxide materials. The expansion coefficient of LBO [17] and Mo203 [16]are low whencompared to metals such as liquid bismuth which has a expansioncoefficient of 1.26x104 at 538°C. Since the LBO melt has a high viscosity and a small thermalexpansioncoefficient, bouyancy forces and thus fluid flow due tonatural convection in the melt willbe small. The LBO expansion coefficients are nonisotropicand the dimension change inthe c direction will be opposite to that of the b and a directions [18]. Thiswill increasethe stress levels that are present in the crystal during cooling. Evaporationcould be aproblem during crystal growth due to the vapour pressureof MoO3;1 atmosphere at1151°C.The crystal structure of LBO is orthorombic of classmm2 and has a space group ofPna2i. The cell dimensions for LBO are a = 7.379, b = 8.447 and c = 5.140 A. Growthdirections are generally parallel to the primary axesof a crystal which in this case couldbe a [100], b [010] or c [001]. We note, followingthe mm2 class designation, theChapter 2. Literature Review13Property Material Temperature Polynomial(units)Specific Heat Li20 298K < T < 500K 69.58+17.857 x 103T(J/g) —19.041 x 105T2B203 723K < T < 1400K 245.814— 145.511 x 103T—171.167 x 105T2+ 48.166 x 106T2MoO3 298K < T < 1068K 75.186+ 32.635 x 103T—8.786 x 105T2density Li20 - 2.013(g/cm3) B203 - 1.812LBO solid 2.474MoO3 - 4.692viscosity Li20 - -(poise) B203 - see figure 2.5MoO3 - -conductivity Li20 - -(W/cm 1<) B203 800 K 0.01LBO (a)f298 K 0.039LBO (b)f298 K 0.031MoO3 - -Expansion Li20 - -Coefficient B203 - -(K—’) LBO (c)f50°C - 400°C 11.6 x 10LBO (d) f[001] 30°C —88.0 x106LBO (e) ff010] 30°C 108.2 x106LBO (f) ff100] 30°C :33.6 x106MOO3 - -Mo203 395°C 5.35 x 1OVapour Li20 - -Pressure B203 - -(atm) MoO3 1151°C 1Table 2.2: Physical properties of Li20, B203.MoO3 and Mo203.LBO (a.) perpendicular to the c directionLBO (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 Review14Figure 2.6: Habit shape of grown LBO crystal[5](001) and (001) faces on a growing crystal will have differentfaceted surfaces. The otherprinciple axes a and b will have the same facetson the opposing faces of the cell. Afaceted LBO crystal is shown in Figure 2.6[5]. In orienting a seed crystal parallel to thegrowth direction care must be taken to determine whether the(001) or (001) is at thegrowing interface since the growing behaviour could he different.2.3 Growth DefectsGrowth defects found in single crystals include segregation, dislocations,flux entrapment,inclusions, cracks, and voids. In the case of LBO crystals the segregationcoefficient ofthe flux is zero, therefore no solid solution forms. While dislocationdensity is importantin semiconductor crystal growth it is not criticalin optical crystals. However, defectswhich reduce the optical quality of a crystal,such as voids, cracks, flux entrapment andrnciusions are a matter of great importance.The amount of published information ondefects in LBO is small [1, 2, 3, 5]. In the literature, similaritiesbetween LBO crystalszI xChapter 2. Literature Review15and other oxide crystals are examined.This is particularlr relevant when the oxidecrystals are of materials with high viscosities similarto B203 [20].2.3.1 Flux Inclusions/Interface BreakdownFlux inclusions in BBO crystals are commonly found at the centreof the crystal [13, 23].Figure 2.7 shows an increase in inclusion densityat the centre of a BBO crystal. Thesecore inclusions correspond to the region were fluid flow due to crystalrotation is minimal [13]. Core inclusions have been found in otheroxide crystals such as Bi4Ge3O12 [21,22]. It was concluded that the core inclusions formed as a resultof rapid changes inthe growth rate. The fast solidification produces a suddenincrease in concentrationof segregated materia,1 ahead of the interface. Inclusions will begenerated when theflux concentration ahead of the interfa.ce increasesmore rapidly than the rate at whichmaterial diffuses away from the interface. Flux inclusions havealso been found at theouter region of BBO crystals [23]. The inclusions at this a.rea areattributed to the largetemperature gradients and temperature fluctuations that occurin these regions.Interface breakdown can occur (luring BBO crystal growth. as shownin Figure 2.8,after the crystal has grown 12 mm [13]. The interface breakdownbegins at the centre ofthe growth interface and spreads to the edges a.s growth proceeds.This was correlatedwith the fluid flow under the crystal [13]. The interface breakdowninitiates at stagnantregions, such as the region below the centre of the crystal.Flux solidified on the outside of the crystal is responsible for crackingin BBO crystals [13]. Similarly, inclusions or solidlified flux at a surface of an LBO crystal maycausecracking due to the strong anisotropy of the flux and LBO crystals[5].Chapter 2. Literature Review16Figure 2.7: Transverse dark field view through a BBO crystal grown at a very highgrowth rate, orientation [001] parallel to growth direction. Many scattering centres areobserved throughout, and concentrated in the core region beneath the seed [13].I —110mmAAFigure 2.8: BBO crystal. Breakdown of growing interface breakdown at A-A [13].Chapter 2. LiteratureReview172.3.2 VoidsVoid formation in crystals may be due to the entrapmentof gas from the melt duringsolidification [28, 25,231.The gas may be from byproducts of the charge material,suchas C02,that are dissolved in the melt. The solubilityof dissolved gases in the liquidishigher than the solid. Thus, as the material solidifies,dissolved gas will be rejected bythe solid at the interface. Bubbles will nucleate when,at a given pressure, the dissolvedgas volume reaches a critical value. The bubbles trappedin the momentum boundarylayer ahead of the interface will likely be incorporatedin the crystal. The dissolved gasboundary layer will be thicker at the centre ofthe crystal, which contains a stagnantliquid region, and thinner at the edges of thecrystal. Figure 2.9 illustrates a possiblemechanism for the entrapment of gas bubbles in the crystal.The rotation of the crystal will cause two events to occurin the liquid. First, mixingwill increase in the liquid and reduce the thicknessof the dissolved gas boundary layerbelow the crystal. Second, the pressure below the rotating crystal willincrease proportionally with the centrifugal acceleration [25]. The increasein pressure at the growinginterface will delay or reduce bubble formation in thecrystal. A correlation betweenrotation rate, crystal diameter and void formationduring crystal growth has been reported [25, 36]. Figure 2.10 shows the extent of the incorporationof gas bubbles inPh5Ge3O11 as a function of crystal rotation rate and crystal diameter.At a fixed crystaldiameter a void free crystal could be grown at high rotationrates. At low crystal rotationrates natural convection dominates fluid flow. Increasingthe crystal rotation rate resultsin forced convection dominating the fluid flow andthe interface inverting from concaveto the liquid to concave to the solid! [25, 13].Chapter 2. LiteratureReview18(a) large convex______gs-bubb(esgas-sürptussmall convex:gas-bubblesgas—surpLusliquid(crack)Figure 2.9: An entrapment mechanism of gas—bubbles in crystals taking account of fluidflow modes associated with crystal rotation [25].._-(C) concaveChapter 2. Literature Review 1960 -bubble-free040oo\‘20bubble-in0 10 20Crystal diameter mmFigure 2.10: Incorporation of gas—hubbies in Pb5Ge3011 as a function of crystal rotationand rate and crystal diameter [25].2.4 Fluid FlowFluid flow in top seeded solution crystal growth is complex due to the rotation of both thecrystal and the crucible. Numerous fluid flow models have been developed to characterizeCzochralski (Cz) fluid flow [28, 29, 30, 31, 32, 33, 35, 38, 39].2.4.1 General ConceptsFluid flow in a crystal growth system in which the crucible is rotated in one direction, andthe 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 rotatingwith the liquid. The crystal and crucible rotating in opposite directions creates a numberof individual rotating fluid regions each separated from the other by a detached shearlayer. A shear layer is where the fluid velocity changes from one solid body rotation toanother. There is no mixing, with the exception of molecular diffusion, across a shearChapter 2. LiteratureReview20layer. The detached shear layers are shownin Figure 2.11 and 2.12 [28, 27].The crucible rotates a.t an angular velocity and thecrystal rotates at an angularvelocity of 1 The fluid adjacent to the cruciblewalls has solid body rotation with anangular velocity equal to crucible rotation rate.Directly below the crystal two individualcells, known as Taylor—Proudma.n cells, develop. The lowercell has fluid that rotates inthe same direction as the crucible but slower (2). Theupper cell rotates in the samedirection 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 atthe crystalsurface boundary layer and the crucible bottom boundarylayer are forced outward dueto the centrifugal force from the rotation. The upper cell has upward fluidflow to replacethe fluid that is moved due to the centrifugal force from the crystal and thelower cellhas some downward fluid flow to replace the fluid that is moving outward dueto thecentrifugal force of the crucible. In turn this fluid is replenished by themiddle shearlayer, the direction of its fluid motion is inward and up/down into the twocells.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 Figure2.13 Thetemperature in the liquid is highest a.t. the bottom a.nd lowest at the top.The lowerdensity liquid at the bottom rises up the wall arid cools, while the higher densityliquidat the top moves down the centre of the crucible.2.4.2 Standard Growth PracticesFluid flow during crystal growth results from both natural convectionand the forcedrotation predicted by the Taylor—Proudman analysis. Since the forced convectioncondition is wha.t was described with the Taylor—Proudman theorem we willbegin with theseresults shown in Figure 2.14. Two cells form as theoretically predicted.The outsidesolid body rotation is almost nonexistent with the lower Taylor-Proudmancell and itsChapter 2. Literature Review21detached shear layer goingto the crucible wall. The strong upward motionof the upperTaylor-Proudman cell cause thetemperature isotherms to flatten below thecrystal. Thecrystal interface will become concaveto the solid as the amount of forcedconvectionbecomes higher [32].If the crystal rotation rate is small,natural convection and/or crucible rotation dominates (Figure 2.15) and the uppercell disappears. The resulting shape of the isothermsindicate that there is more radial beat flow and theshape of the interface changestoconcave to the liquid. The drivingforce of many of the modeling papershas been topredict interface changes from concaveto convex [32, 33, 34, 36, 37].2.4.3 Accelerated Crucible RotationAs an alternative to crystal and crucible counterrotation during growth, the crucible(and crystal) can be accelerated and deceleratedalternately in clockwise and counterclockwise directions. The acceleratedcrucible rotation technique (ACRT) reduces thethickness 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 followthe change in crucible motion, whereasthe liquid in the centre will remain at rest dueto inertia. Thus shear rings in the liquid are producedby the difference in the outer andcentre fluid velocities. The number ofshear rings which develop increase with increasingacceleration. A top view of the shear rings producedis given in Figure 2.16. The increasein surface a.rea due to the shear rings results inan increase in diffusion. The resultingfluid flow in the radial and axial direction areshown in Figure 2.17. The rapid flowclose to the crucible bottom is referred toas Ekma.n-layer flow [43]. The rapid suctionof fluid close to the bottom is due to thepressure difference between the outside andinside of the crucible not being balanced by the centrifugalforces. High Ekrnan flowrates occur not only at the crucible bottom butalso on the crysta.l growth face that isChapter 2. Literature Review 22/CrystalUpperTaylor - ProudmanCellLayerUpper Transition______________IntermediateTransitionLayerDetachedStairnationShear_— SurfacesLayerc5Solid Body- RotationLowerILower TransitionTaylor - ProudmanLayerCellFigure 2.11: Theoretical Taylor-Proudman cell shapes for counter rotation [28].Chapter 2. Literature Review23+MELTI IFigure 2.12: The direction of fluid motion predicted in the Taylor—Proudman cells [28].Chapter 2. Literature Review24CentreLineI —-i//iui7Iii II\\III‘‘ ‘ ‘ \ ‘S —‘ ‘ “ \ “/Ji\\\\I\\ \I—\\\\N _,/‘ 1 IIN•— /--—--IICrucibleFigure 2.13: Fluid motion due to natural convection [29].Chapter 2. Literature Review25Crystala\ooCCrucibleFigure 2.14: Predicted flow with counter rotation large enough that forced convectiondominates [32]. Flow shown on right half and temperature shown on left half.Chapter 2. Literature Review 26Crystalo10.4—Q6-0.80.6-1(0CrucibleFigure 2.15: Predicted flow with counter rotation small enough that natural convectiondominates [32]. Flow shown on right half and temperature shown on left half.Chapter 2. Literature Review27perpendicular to the rotationaxis. Ekman-layer flow ceases as soon as uniformrotationis reached. A mathematical modelhas been used to examine the fluid flow thatresultsfrom combined forced and free convectionof accelerated crucible rotation Cz growthofmetals and semiconductors[40, 41]. The best mixing of the fluid is reportedto occurwhen the crystal and crucible are rotated/acceleratedin the same direction. The upperTaylor-Proudman cell is only present duringthe acceleration of the crucible (Figure2.1S).Thus the model predicts that the bulkfluid will mix best under these conditions. Thevariation of a solute boundary layer wasnot examined. The fluid properties of themodelare not close to those of LBO. This isespecially noticeable with the kinematic viscosityof 0.005 cm2/s. Using the density ofLBO, 2.474 g/cm3,the model viscosity is calculatedas 0.0123 poise. The viscosity ofan LBO solution for growth is nearly three ordersofmagnitude higher. It is difficult to predictif the same type of mixing would occur in ahigher viscosity liquid.2.5 Mass TransferThe most important aspect of mixing in the meltis tha.t which occurs in the boundarylayer under the crystal. In the past the difficultyof conducting comprehensive numericalintegration of the Navier-Stokes equations anddiffusion equations to determine the flow,the temperature, and solute distribution,has led to the development of boundary layertheory to provide simple models of key regionsof the flow in Cz growth [44]. Therehas also been no real attempt to verify the boundarylayer models or numerical modelswith measured solute distributionsahead of the crystal interface. Burton,Prim andSlichter [45] considered changesin the momentum boundary layer thickness and theresultant transport of solute, on the effective segregationof solute for steady state crystalgrowth in germanium. The domain wasassumed to be an infinite rotating disc onChapter 2. Literat tire Review28(b)Figure 2.16: General rotational fluid flow (shearing) due to ACRT [43]. (a) Top viewof circular tube filled with two distinguishable fluids. Tube and contents arein uniformrotation. (b) Final shape of fluid after tube and contents have come to rest.Spiralshearing distortion is evident.(a)CDChapter 2. Literature Review30:: ....-- ---—---lk7L7ffJ?M1’ff/it;qII11c./ 1....ii1t1._II1jIi44444t = 3.0 sv = 3.3 cm/smaxt 30.0 sv = 2.3cm/smaxt = 15.0 SV = 2.0 cm/s -44414444444144414444::‘t = 36.0 sVmax= 1.4cm/sFigure 2.18: Model prediction of fluid flow due to ACRT [41j. Crucible rotation 10 to30 rpm. Crystal rotation 40 to 80 rpm. Time period of acceleration 15 seconds.Chapter 2. Literature Review31the surface of a semi-infinitefluid. The boundary conditions used inthe mass transfercalculations are shown in Figure2.19. It is assumed that the bulk concentrationof thefluid is at the edge of the diffusion boundarylayer. The diffusion boundary layer thicknessis assumed to be constant over the radiusof the rotating disk. This is very relevantwhenlaminar flow and a thin diffusion layer arepresent [49]. The only velocitypresent in thesolute boundary layer is from the growing interface.The calculation of solute diffusionreverts to a simple steady state one dimensionaldiffusion calculation with aterm toaccount for the advancing interface.Chapter 2. Literature Review• —• —32Crystal, Growing at Velocity fConcentrationI66MomentumBoundaryLayerNO MIXINGBulk FluidCOMPLEThI1GFigure 2.19: Assumptions used in Burton, Prim and Slichter [45] calculation of theconcentration in the momentum boundary layer.Chapter 3ObjectivesThe objective of this investigation was to examine the crystal growth process using mathematical modeling, experimental measurements and crystal growth experiments in orderto establish procedures for growing larger and higher quality LBO crystals. The mathematical modeling quantitatively (lescribes the parameter interactions in the complexgrowth system. Experimental measurements were conducted to establish the requiredphysical 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 patterns in the melt. The crysta.l growth experiments were carried out to verify the modelpredictions. These experiments also demonstrated that better crystals can be producedby modifying the growth parameters in conformity with the model predictions. Theprincipal tasks undertaken to accomplish the objectives were as follows:1. The development of a mathematical model of heat transfer and fluid flow in theLBO crystal growth system.2. Application of the Burton, Prim. and Slichter analysis to examine the mass transferof MOO3 away from the solid/liquid interface.3. Use of a physical model to visualize the type of flow that occurs due to cruciblerotation.4. Measurement of the temperature distribution in the melt and a comparison withmodel predictions.33Chapter 3. ObJectives345. Temperature distributions for the mathematical models’ boundary conditions.6. Prediction of the effect of crucible rotation rate on the temperature distribution inthe melt.7. The phase diagram of the LBO/Mo03system, as determined by differential thermalanalysis.8. The dependence of viscosity of the melt as a function of temperature and MoO3content.9. The type, size and distribution of defects in the grown crystals.Chapter 4Experimental4.1 Growth Process4.1.1 Growth FurnaceThe system utilized in this investigation to grow LBO crystals is the modified NRCcrystal 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 LBOgrowth. The growth chamber A is 26.3 cm in diameter and 44.5 cm high. A schematicdiagram of the interior of the growth chamber is shown in Figure 4.21. The LBO/Mo03charge is contained in a crucible which sits on a rotating pedestal, thermally insulatedby a block of ceramic. The resistance heaters have Iron-chrome-aluminum elements. Theinsulation shown in Figure 4.21, consists of alurnina. A 6.35 cm diameter hole is locatedat the top of the chamber.The crucible is made of platinum. Two crucible sizes were used, a small crucible 66mm in diameter and height, and a larger crucible 88 mm in diameter and height. Theexpansion coefficient of LBO/Mo03is much la.rger tha.n that of platinum causing severecrucible distortion upon cooling. To ensure that the crucible remains firmly attached tothe insulation block during rotation, a cylinder of insulating material 5 mm in height isattached to the crucible and 3 pegs are attached to the cylinder which sit in holes in theinsulating block that is attached to the rotating pedestal. The pegs keep the cruciblefrom moving with respect to the block. The cylinder is attached to the crucible with35Chapter 4. Experimental36Cotronics Ultra—Temp 360 insulating tape and Cotronics904 Ceramic Adhesive. Theinsulating tape at the side of the crucible must be the samefor each consecutive crystalgrowth run. If not, the liquid temperature will change.The crystal growth furnace temperature is controlled witha Eurotherm 818 controller.This instrument is very accurate and is capable of maintainingramp rates as low as 0.01degrees per hour. The type K control thermocoupleis located adjacent to the furnacewall. The crucible rotation, seed rotation and seed lift wereall modified from the originalNRC equipment. The crucible rotation, Figure 4.22. consists of a Bodinemotor and a.pulley unit attached to the original rotation unit. A thermocouplegoes up the center ofthe rotation shaft for measuring the temperature at the base of the crucible.The gearingof the pulleys allows for crucible rotations between0 and 74 rpm. The seed is rotatedusing a Maxi-Pile gear motor a.nd planetary reduction gears to obtain a rotationof .5rpm, Figure 4.23.A stepper motor and indexer/driver package, was used for slow pulling of theseed.The seed lift velocity is 1.66 mm/day to 1.49 crn/miri. A platinum mixing paddle, shownin Figure 4.24 is used to homogenize the liquid. a.fterthe cha.rge has been melted. Theseed rod is made of hastelloy. The bottom of the seed rodis attached to either the paddleor to the platinum seed holder.4.1.2 LBO SeedLBO seeds were oriented such that the growth direction wa.s parallel tothe [001] direction.The orientation was determined by using the back reflectionLa.ue method on the cleavedsurface of the crystal, Figure 4.25. The unorient.edLBO crystals was attached to agoniometer, Figure 4.26, using wax. Samples were positioned suchthat the surface was3cm from film holder and were irradiated from a Cu targetfor 20 minutes at 30 k\7 and 20mA. After irradiation, the X-Ray film wa.s developedand the orientation of the crystalChapter 4. ExperimentalIi ‘$ ,B, ,.j ,37Figure 42O: LBO crystal growth furnace, A and C are the crystal puller, aid B is thepower arid control box.Chapter 4. Experimental38Base TC“rucib1e RotationFurnace TCFigure 4.21: Schematic of LBO growth chamberChapter 4. Experimental39Figure 4.22: Crucible rota.tion device.Figure 4.23: Seed rotation device.Chapter 4. Experimental 40IFigure 4.24: Platinum paddle for mixing the melt.was determined using a Greninger chart and Wulif net asdescribed in reference [19].On the basis of the established crystal orientationwith respect to the X-Ra.y beam, thegoniometer was adjusted such that the c axiswas parallel to the X-R.ay beam. Thisprocedure was repeated until the c axis of the sample wasparallel to the X-Ray beamwithin 0.2 degrees.Once the c axis was determined the (001) face was ground flat using a PM2A Logitechlapping and polisher machine with SiC grit as the lapping media. Thegoniomcter wasattached to the polishing instrument to ensure thatthe ground face was identical to themeasured orientation.The LBO crystal was repositioned on the goniometer by heating thewax. It wasreattacheci to the goniometer such that the C axis was perpendicular to the X-Raybeamdirection. The above procedure was then repeated to determinethe b axis. After thisaxis was determined and polished the procedure wa.s repeatedfor the a axis.Once the sample was oriented and the principle faces polished the LBO crystalwasChapter 4. Experimental41Figure 4.25: Cleavage plane of an LBO Crystalcut into smaller pieces using a. Buehler Isornet low speedsaw. The seed size is 5 x 2 x 10mm with the c axis being the longest dimension.Special notches are cut in the seedfor a.ttaching it to the platinum seed holder.The notches a.re 1 mm in depth. The finalseed shape is shown in Figure 4.27. The seed isattached to its holder by running aplatinum wire though the notches in the seed andholder. The wire is wound tightly onthe platinum holder to ensure that theseed is held in place. Figure 4.28 shown the seedand holder.4.1.3 Growth ProcedureThe LBO growth solution consist of lithium tetraborate(Li2B4O7),extra borate (B203)to obtain the correct stoichiometric amount ofLBO and molybdenum triborate (MoO3).Chapter 4. ExperimentalPignre 4.26: Coniorneter used for orienting LBO crystals42Chapter 4. Experimental43Figure 4.27: LBO SeedChap tei’ .1. Experiment a.]44Figure 4.28: LBO Seed attached to J)Iatinurn seedholderChapter 4. Experimental45The melting point of lithium tetral)otate is 930°C of borate is450°C, and molybdenumtrioxide is 795°C. All compounds were obtained from JohnsonMatthey and were 99.999%pure. The charge composition selected was 45 wt% MOO3(40 mol%). The compoundswere packed into a crucible with B203 at the bottom followed by the lithium tetrahorateand the top layer was MoO3. Dry nitrogen was introduced to the furnace at 12 cubicfeet per hour. This flow rate was used during the entire crystal growth run. The chargedcrucil)le is heated to 1050° C to allow the lithium tetraborate to completely melt at thistemperature for 48 hours. The liquid at this temperature has an opaque ambercolour.The mixing paddle is lowered into the liquid until it is completely submerged. Thecrucible 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 thenremoved from the furnace and replaced with the seed crystal which is attached to theseed 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 ofapproximately 930°C. The crucible rotation is increased to that used for growth whilethe furnace is cooling. The LBO seed is clipped by slowly lowering it into the liquid untilthe meniscus of the fluid is broken. The appearance of the fluid after the seed has beendipped is shown in Figure 4.29.If the insulation on the side of the crucible changes between crystal growth runsso will the furnace setting for the seed dip temperature. When this occurs the liquidtemperature is either below or above the liquidus temperature. If the seed is dipped andthe liquid is above the liquiclus temperature there will be a longer time before the crystalgrows out in size. The seed will not be melted awa since the LBO melting temperatureis approximately 150°C above the liquidus temperature. If the seed is dipped and theliquid ten3perature is below the liquiclus temperature spurious crystal will grow on theseed. The furnace temperature was increased 5°C every two hours until the spuriousChapter 4. Experimental46Platinum Crucible1LBO/Mo03ILiquidLBOIMoO31eniscus of LiquidSeedFigure 4.29: Appearanceof LBO melt surface afterthe seed has been (lippedChapter 4. Experimental47growth melted back to the seed. Growth is continued as per normal from thatpoint.The furnace is cooled at 0.1°C/hour after the seed had been dipped. Crystal growthis not apparent for up to 1 week after the start of the cooling. Once the crystal diameterhas reached 2/3 of the crucible diameter it is pulled at 1.66 mm/day and continued for5 days. Crystal growth is terminated by rapidly withdrawing the crystal from the meltat a rate of 2.1 x 1O cm/s. Once separated from the melt the crystal is slow cooled toroom temperature at 10°C/hour.The crucible is cleaned by heating it to 1000°C and decanting the liquid. Any residualmaterial is removed by soaking the inside of the crucible with reagent grade HCL. TheHCL fluid is mixed and heated for one day using a combination hot plate/magneticstirrer.4.2 Temperature Measurements in the MeltTemperature measurements in the melt are required to establish the temperature boundary conditions in the mathematical model, investigate the effect of crucible rotation onthe thermal field, and for comparison with results from the mathematical model. Thetemperature measurements were accomplished in three separate trials. The first set oftemperature measurements were undertaken to measure the temperature boundary conditions of the melt for use in a mathematical model that examines the crystal growthoperating parameters. A 6.6 cm diameter crucible with no rotation wa.s used to containthe melt for these measurements. The second and third set of temperature measurementswere conducted for determining the model boundary conditions to validate the model byexamining the effect of crucible rotation on the thermal field in the melt .A 6.6 cm diameter crucible was used to contain the melt for second set of temperature measurementsand an 8.8 cm diameter crucible contained the melt for the third set of measurements.Chapter 4. Experimental 484.2.1 Initial Temperature Measurements with No Crucible Rotation -Temperature measurements were carried out with a chromel/alumel thermocouple in aquartz sheathed tube connected to the seed rod. The thermocouple could be placed atthree different radial positions; at the center of the crucible, at the mid—radius of thecrucible and at the wall of the crucible. The axial temperature profile was determined byrecording the thermocouple output as it was slowly pulled out of the liquid. The furnacetemperature was set at 145°C above that normally used for the start of crystal growth toensure 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 diametercrucible with no rotation was used to contain the LBO—Mo03melt. No insulation wason the outside wall of the crucible.4.3 Temperature Measurements With and Without Crucible with RotationTwo sets of temperature measurements were carried out, one in a 6.6 cm diameter crucibleand 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 platinumsheathed chromel/alumel thermocouples were used for the temperature measurements.The position of the thermocouples were adjusted using a positioning device. A platinumsheet formed to simulate the crystal shape, as shown in Figure 4.31, was positionedto replace the crystal. The fluid flow conditions below the simulated crystal and itsthermophysical properties, are believed to be close to the actual conditions during LBOcrystal growth. As a result, the temperature measurements conducted with the simulatedcrystal could he applied to the actual growth system for model verification.Chapter 4. Experimental49iiFigure 4.30: Positions of thermocouples and simulated crystal for melt temperaturemeasurements.Chapter 4. Experimental 50I.iIWfl i.i CTb. D.... 1 bba a.4 Mfl ka%Figure 4.31: Platinum cap used to simulate the crystal.Chapter 4. Experimental51ApparatusThe thermocouple positioning was accurate andrepeatable to within 1 mm, using theapparatus shown in Figure 4.32; note that the thermocouplesare inside the furnace andthe positioning mechanism is outside the furnace.The apparatus consists of four keyparts which are a horizontal guide track for radial movement, a rackand pinion gearfor axial movement, a thermocouple holder and a micrometer/calibrateddial for spatialpositioning. The apparatus was positioned on the crystalpuller as shown in Figure 4.33.The thermocouple positioned closest to the wall is vertical and theinside thermocoupleis curved at the bottom to allow it to measure temperatures under thesimulated crystal.Figure 4.30 shows the thermocouple shapes and positions with respect tothe simulatedcrystal. Electronic cold junction compensation was used for the measurementsand a chartrecorder or computer based da.ta acquisition system was usedto record the temperatures.General ProcedureThe chromel/alurnel thermocouples used in the measurements were calibrated atthefreezing point of tin (231.9681°C). The temperature measurements weremade in anLBO/Mo03bath similar to that used for crystal growth. The furnace temperature wasset at 100°C above that normally used for the start of crystal growth to ensurethatno nucleation occurs in the melt or on the simulated crystal surface. The thermocouplepositioning systeni was set by determining the micrometer and dial settingswith thethermocouples touching the top of the melt. and the inside wall of thecrucible. In thesmall crucible temperatures were measured at axial and radial intervalsof 0.5 cm. Themelt temperature in the large crucible were measured at axial intervals of0.36 cm andradial intervals of 0.5 cm.Chapter 4. ExperimentalTOP52Figure 4.32: Thermocoupleprobe holder. A - guidetrack for radial movementof TCprobe. B - rack and piniongear for axial movement.C - thermocouple holder. Dmicrometer and dial for positioningof the thermocouples./ASIDE—Chapter 4. Experimental 53Figure 4.33: Apparatus for inoving thermocouples attached to the crystal puller.Cha.pter 4. Experimental54Procedure for Temperature Measurements in the SmallCrucibleInitial temperature measurements were made for one flux concentration withno cruciblerotation, without the simulated crystal. The measurements wererepeated with the simulated crystal added to the system. and finally wit.h the simulated crystal and cruciblerotations of 15, 20, 25 and 30 rpm. For the measurements, thermocoupleA was initiallypositioned 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 0.47from the bottom surface. The thermocouples were then moved vertically 0.18 cm thenat 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 of0.5 cmuntil thermocouple B was at the center of the crucible. The model crystal interfered withthe thermocouples limiting the horizontal movement to 1.55 cm above the bottom of thecrucible and thermocouple B to be at 1.0 cm from the center of the crucible. The radiallocation that the temperatures were measured at in the small crucible were 1.0, 1.5, 2.6and 3.1 cm.Procedure for Temperature Measurements in the Large CrucibleTemperature measurements were made for one flux concentration with no crucible rotation, without the simulated crystal. The measurements were repeated with the simulatedcrystal added to the system, and finally with the sinmiated crystal and crucible rotationsof 10, 15, and 30 rpm. For the measurements, thermocouple A wa.s initially positioned0.2 cm from the wall and 0.36 cm from the bottom surface. Thermocouple B, beingoffset 2.9 cm vertically from thermocouple A, was 2.9 cm from the wall and0.36 fromthe bottom surface. The thermocouples were moved vertically at intervals of0.36 cmChapter 4. Experimental55until they reached the maximum height of the fluid at 3.24 cm. The vertical temperature measurements were repeated at horizontal spacings of 0.5 cm until thermocouple Bwas close to the center of the crucible. The radial location that the temperatures weremeasured 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 ProcessA physical model was employed to visualize the general flow patterns that occur duringcrystal growth. The model consists of an 8.8 cm diameter plexiglass crucible, an 5.6 cmdiameter plexiglass crystal and a rotation device. The crystal had 0.2 cm diameter holesat 1/2 and 3/4 of the crystal diameter for the injection of a tracer fluid. Glycerine wasused to simulate the LBO/Mo03melt. Blue dye mixed with glycerin is used as a tracerto examine the flow patterns. Figure 4.34 shows the plexiglass crucible and crystal.4.5 Physical Properties4.5.1 Chemical AnalysisChemica.l analysis for Mo and Li was conducted using atomic absorption at an externallab. Samples from the phase diagram determination and the viscosity measurements wereanalyzed for two different reasons. The phase diagram samples were used to determinethe accuracy of the external lab’s results. The theoretical and actual composition of thesesamples will be very close since the samples that are sent for analysis spend no time atan elevated temperature. The viscosity samples must be analyzed since the compositionchanges with extended time at temperature due to the vapour pressure of MoO3.Chap ter 4. Experimental 56Figure 4.34: Plexiglass crucible and crystal used for the physical model.Chapter 4. Experimental574.5.2 LBO/Mo03Phase DiagramA phase diagram is necessary in crystal growth for selecting the liquid temperatureatwhich the seed is dipped. It is also important in mass transfer studies ofthe growing crystal. Of particular importance is the eut.ectic concentration, at which interfacebreakdownoccurs. The LBO/MoOa phase diagram was determined using differential thermalanalysis (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 astandard since it does not undergo a. phase change in the temperature rangeexamined.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 undergoesa phase change, the heat released or absorbed will be indicated by a positive or negativedeviation from the temperature of the alumina, sample. The peaks and valleyson the- furnace temperature curve thus correspond to phase changes in the LBO/Mo03sample.The equipment used consisted of a. Dupont 1090 Thermal Analyzer and a Dupont 910differential scanning calorimeter fitted with a. 1200°C DTA furnace. Special platinumcrucibles were used for containing the LBO/Mo03samples. The initial sample was55.4 wt% LBO and 44.6 wt% MoO3. Samples were prepared by combining lithiumtetraborate, borate and molybdenum trioxide. The lithium tetraborate and molybdenumtrioxide were in powder form and were easily mixed. The borate came in large blocksand 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. Thesamples were placed in a plastic bottle with alcohol and alumina halls which werethenplaced in the shaker and mixed for two 20 minutes periods. The samples were thenplaced in a drying furnace to evaporate the alcohol. The dried sample was remixedChapter 4. Experimental58using a mortar and pestle. A portion of the sample was kept to be sent for chemicalanalysis. Another portion of the sample wa.s used for the DTA analysis. The remainderof the sample was mixed with more MoO3 and used for the next DTA experiment. Thetheoretical LBO/Mo03wt% 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 colorfrom light green to blue as the MOO3 concentration increased.A sample was placed in the platinum crucible and placed in the DTA furnace. Thefurnace was rapidly heated to 150°C then slowly heated at 10°C/mm to a final temperature of 750°C. The differential temperature was recorded between 500°C and 730°C. Atthe same time the sample was examined for a. change in color associated with the phasechange.4.5.3 ViscosityThe viscosity of LBO containing MoO3 was measured, as a function of the MoO3 concentration, with a Brookfield Model RVTDV II viscometer, using a modified procedureof the ASTM standard C936-81. This procedure is used for measuring the viscosity ofglass above its softening point. The LBO/Mo03sample, in a platinum crucible, washeated 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 followingASTM specifications. The system, including the paddle, was calibrated by measuring theviscosity of a solution of glycerine and corn syrup which had a specified viscosity.Samples for the viscosity measurements were prepared using the same procedure aswas used for crystal growth. Depending on the test, either an air or dry argon atmospheres were used, the later wa.s to unsure that. no Li was lost from the melt due towater vapour. Once the LBO/Mo03melt was homogenized, a. sample was removed forquantitative chemical analysis of the molybdenum and lithium content. This wa.s doneChapter 4. Experimental59I IFigure 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. Thepaddle used for the viscosity measurements was then immersed in the melt together witha platinum sheathed chromel/alumel thermocouple. The viscometer was then turned on,the temperature stabilized at the specified test value and the thermocouple removed. Theviscosity of the melt as indicated by the viscometer was then recorded. The procedurewas then repeated at a lower set melt temperature, allowing approximately 10 minutesfor the melt to equilibrate at the new temperature. The viscosity measurements werecontinued until the viscosity of the solution was in excess of 100 poise.II SChapter 4. Experimental 604.6 Crystal QualityThe crystal quality of LBO was examined for MOO3 inclusions, cracking and bubblesusing optical microscopy and scanning electron microscopy (SEM). Quantitative chemical analysis for molybdenum was performed using wavelength dispersive spectroscopy(WDX). Samples were prepared by polishing the surface with a 1t diamond suspensionon a Buehler Ecomet IV. Polishing was continued until the surface of the sample wasflat and had no visible scratches. Optical examination was conducted using an ZeissStereomicroscope SV8. The samples for SEM/WDX analysis were sputter coated witha layer of amorphous carbon. SEM/WDS analysis was done using a Hitachi S-570 SEMwith Microspec WDX attachment. Measurement of the nonlinear optical properties wereconsidered beyond the scope of this project and not carried out. The variation in visible/electron image inclusion density was assumed to be a sufficient indication of crystalquality.Chapter 5Experimental Results5.1 Phase Diagram for the LBO/Mo03SystemThe concentration of the DTA samples examined are listed in Table 5.3. The chargeconcentrations listed were determined from the mass of the constituents used. The concentrations were measured by a commercial laboratory using atomic absorption. Themeasured values are given in weight percent Mo and Li. Equivalent MoO3 and LBOconcentrations were calculated from the measured data assuming that these were theonly species present in the melt.molecular weight of MoO3Mo03(equivalent) = Mo(measured) xmolecular weight of Momolecular weight of LBOLB 0 (equivalent) = Li(measu red) xmolecular weight of LiComparing the equivalent concentration of MoO3 and LBO to the chargeconcentrations,the former is observed to be approximately 5% below the later in all cases. The equivalentMoO3 concentration was normalized.Mo03(equivalent)Mo03(normalized) = .MoO3(equivalent) + LBO (equivalent)The normalized MOO3 concentration were close to the values of the chargeconcentration.These values were taken as the actual melt concentrations.The DTA results for 45 Wt% MoO3 (sample 1), are shown in Figure 5.36. Thecurve has four points of interest, the negative temperature difference at A (555°C) and61Chapter 5. Experimenta.l Results62SampleWeight PercentCharge ConcentrationMeasured EquivalentNormalizedMoO3 LBOMo Li MoO3LBO MoO31 44.5655.54 26.502.86 39.76 49.1944.762 49.6650.34 30.002.64 45.01 45.4049.783 59.5840.42 —— —— 604 64.4035.60 41.001.89 61.51 61.5165.425 69.1330.87 46.501.63 69.76 69.7671.346 73.7726.22 47.301.38 70.96 70.9674.94Table 5.3: MoO3and LBO concentrationsused in determiningthe pha.se diagram.D (575°C), thepositive temperaturedifference at B (682°C)and the change of slopeatC (610°C).During heatingthe sample a.ppearencewas noted. The samplewas initially auniform light greencolour until 555°C(point A) was reachedwhere a reactionoccurredand the sampleturned brown. Thesurface texture alsochanged frombeing uniform tohaving evenlydistibuted shiny regionson a dull surface.Between 575°C(point B) and610°C (point C)the sample turnedto a. transparent whitecolour and the surfacetexturechanged to being shinyin appearance.Between 610°C and682°C the samplecolourslowly changed to anamber colour. At. 682°C(point. D) the samplemelted.The liquidusof the sample was takenas the point wherethe temperaturedifferencestarted to increaseat point D and theslope betweenpoint C and D as thesolid—liquidphase region.The gradual changein slope aroundpoint C makes itdifficult to accuratelydetermine thesolidus.The temperaturesat which the phasetransformationsoccured are listedin Table 5.4and plotted inFigure 5.37. The hiquiduschanges from 682°Cat 44.76 Wt% MoO3to619°C at 60 Wt%MoO3 just beforethe eutectic concentration,61.5 Wt% MoO3.TheDTA analysisindicates that two otherphases occur abovethe eutectic compositionattemperature in excessof 610°C. Thestructure of these phaseswere not investigated.Chapter 5. Experimental Results63I I t f I I I I I I I I I I I I I I I3.12-0%0w04tIIIiIIItItIIItIIIIII500 522 540 560 580 800 622 640 660 882 700 720TemperQure(ec)DuPor 1090Figure 5.36: Temperature difference versus temperature for 45 Wt% MOO3 (sample 1)determined by DTA.Chapter 5. Experimental Results 64I I I680 -670 -5’660- Liquid650-640-E630- LBO + LiquidEutectic620 -CA+LiquidB+Liqaid-C-6101]LBO + MoO3600 I I I50 60 70MoO3Concentration (Wt %)Figure 5.37: Phase Diagram of the LBO — MoO3 system.Chapter 5. Experimental Results 65Sample Wt% MOO3 Solidus(s) Liquidus1 44 76 610°C 689°C2 49 78 612°C 661°C3 60 614°C 619°C4 65 4 614°C 618°C5 71 34 619°C, 618°C 630°C6 74.94 613°C, 618°C 632°CTable 5.4: The solidus and liquidus temperaturesof the MoO3 — LBO samples5.2 ViscosityThe viscosity of five samples (7-11)were measured as a function of temperature. Thecomposition of the samples are listed in Table 5.5.The viscosities are listed in Table 5.6and 5.7. The temperatures listed arethe average temperatures in the melt. The temperature difference across the sample was measuredto be 10°C. Two atmospheres wereused, air and dry argon. The laterwas used to ensure that no Li was lost from themeltdue to the presence of water vapourin the atmosphere.The viscosity is plotted on a logarithmic scale againstthe reciprocal of the test temperature, in Figure 5.38. In general, theviscosity is high, ranging from 4 poise to greaterthan 70 poise between 863°C and 661°C. Theviscosity decreases with increasing temperature and increasing MoO3 concentration. Theviscosity measurements are in goodagreement with each otherabove 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, 8and 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 Results66SampleWeight PercentCharge Con centra tion MeasuredEquivalent NormalizedMoO3 LBOMo Li MoO3LBO MoO37 34.865.2 17.30 3.SS25.96 61.5629.668 36.863.2 21.903.37 32.86 57.9636.189 39.360.7 26.403.3:3 39.61 57.9640.8910 43.057.0 20.602.85 30.91 49.0138.6711 50.050.0 21.203.14 31.81 54.0037.06Table 5.5: MoO3 andLBO concentrationof the samples usedin the viscosity measurements.The viscosity variationwith temperature forsample 10 (38.67 Wt%MoO3) and thefitted value for sampleS (36.18 Wt%MoO3)are similarbelow 702°C (10000/T[K}>10.25). Thus it isassumed that the lowtemperature viscositymeasurements, T <702°C(10000/T[K] > 10.25),for sample 11 areerroneous. Thevariation of viscositywithconcentration is plottedin figure 5.39.The viscosity decreasedwith increasing MoO3ata constant temperature.The viscosity of sample7, 8 and 9 attheir respective liquidus’swere determined. Theliquidus temperatureswere estimated byextrapolating the LBO/Mo03lic1uidus line onFigure 5.37 tothe composition ofthe samples. The liquidustemperatures wereusedwith the best fitviscosity equationsfor their respectivecompositions. Thevariation inviscosity and liquidustemperature is given inFigure .5.40. Theviscosities at the liquidusdecrease linearlywith increasing MoO3content.5.3 PreliminaryLBO Crystal GrowthRunsThis section examinesthe crystal growthparameters used andthe resulting crystalgrowth quality inruns conducted priorto the start of thisinvestigation. Mostof thepreliminary crystalgrowth runs used crystalrotation rates of approximately15 rpm, noChapter 5. Experimental Results67Sample Atmosphere Temperature 10,000/T[k] Viscosity (poise)7 Air 765°C 9.64 26.8Air 785°C 9.46 18.2Air 807°C 9.26 12.9Air 824°C 9.12 9.5Air 847°C 8.93 7.1Air 862°C 8.81 5.88 Air 736°C 9.91 25.3Air 754°C 9.74 18.2Air 779°C 9.51 12.1Air 800°C 9.32 8.4Air 817°C 9.17 6.6Air 847°C 8.85 4.7Air 863°C 8.80 4.2Air 914°C 8.43 3.79 Air 730°C 9.97 13.4Air 749°C 9.79 9.47Air 766°C 9.63 7.1Air 788°C 9.43 5.5Air 808°C 9.25 4.47Air 833°C 9.04 3.69Air 847°C 8.93 3.42Air 869°C 8.76 2.6310 Dry Argon 691°C 10.37 124.27Dry Argon 702.5°C 10.25 59.97Dry Argon 720°C 10.07 41.95Dry Argon 741°C 9.86 27.09Dry Argon 767°C 9.62 16.31Dry Argon 769°C 9.60 16.7Dry Argon 784°C 9.46 12.89Dry Argon 788°C 9.43 11.97Dry Argon S17°C 9.17 8.16Table 5.6: Viscosity versus temperature for samples 7 to 10.Chapter 5. ExperimentalResults68I I I<>k/0Q29.7 Wt% MoO1, o70.0- A 36.2 Wt%MoO3lc0‘V’40.9 Wt% MoO3— 0Wt% MoO3037.IWL%MoO -IcoY20.0-• —Cl)6.0 uI I I9.00 9.25 9.50 9.75 10.00 10.25 10.50 10.751O,000/T[K]Figure 5.38: Dependence of viscosity on temperatureand MoO3 concentration.Chapter5.Experimental Results69300T=72OC—-0T=77OC6T=820C CElI I28 30 32 34 36 38 40Wt% MoO3Figure 5.39: Variation of viscosity with Mo03 at the temperatures indicated.Chapter 5. Experimental Results7060 I I 750LiquidusD74050Viscosity A.—__________73() c• 40A720DI)7103070020I I I30 32 34 36 38 40Weight Percent MoO3Figure 5.40: Liquidus temperature and viscosityas a function of MoO3.Chapter 5. ExperimentalResults71Sample Atmosphere Temperature 10,000/T[k]Viscosity (poise)11 Dry Argon 653°C 10.80132.03Dry Argon 658°C 10.74 119.14Dry Argon 669°C 10.6296.13Dry Argon 676°C 10.54 78.90Dry Argon 687°C 10.4264.18Dry Argon 696°C 10.3250.50Dry Argon 734°C 9.9330.25Dry Argon 740°C 9.8721.83Dry Argon 755°C 9.7317.23Dry Argon 774°C 9.5513.15Dry Argon 775°C .54 12.10Dry Argon 787.5°C 9.4211.31Dry Argon 805°C 9.28 8.81Dry Argon 818°C 9.17 7.24Table 5.7: Viscosity versus temperature for sample11.crucible rotation and a cooling rate of 3°C/day. The initial charge concentration ofMoO3in the melt was 34 \1t%All the crystals cracked during the post growth cooling to room temperature.Theaverage size of the uncracked portions of thecrystal were 7 x 3 x 3 mm3. Opaquewhite/green colored inclusions were in the crystals having an averagesize of 3 x 3 x 3mm3.Samples of the grown crystals were examined using a scanning electron microscope(SEM) equipped with a wavelength dispersiveS1)eCtrometer(WDS). Figure .5.41 show thehackscatter images of the cross section of an LBO crystal. WDSanalysis was used toexamine the crystal defects as well as the defect free crystalmatrix. There was nomolybdenum present in the defect. free portions of thecrystal.The molybdenum inclusions on this sample formed as two types;line inclusions andnormal inclusions. Line inclusions run for distances that are greaterthan half the specimen size and the width is much smaller than the length. Normalinclusions have lengthsChapter 5. ExperimentalResults72that are comparable totheir widths. Backscatter images and thecorresponding dot mapsfor a normal inclusion areshown in Figure 5.42. The inclusion shape correspondsdirectlyto the molybdenum dot map.A backscatter image and its corresponding molybdenumline scan of a line inclusionis shown in Figure 5.43. The dot map and line scanof themolybdenum region corresponds t.o the inclusionsin the backscatter photos. The lineinclusions are parallel and perpendicularto the crystal growth axis.Flux inclusions in ceramic materialsare more significant than for other materials.Aninclusion with an expansion coefficientdifferent from the matrix is the most likely causeof the crystal fracturing during cooling toroom temperature. The expansion coefficientof LBO in the [001] direction andMo203 are —88.0 x 10/°C and 5.35 x10/°Crespectively (see Table 2.2).From these initial findings it was evident thatthe main crystal defects (inclusionsand cracking) are due to the MoO3 flux.It wa.s clear that the build up of MoO3,due tothe high viscosity of the melt, has a largeinfluence on the crystal quality. Mathematicaland physical modeling of the crystal growth processhas been employed to improve themixing in the melt thus reducing theMOO3 ahead of the growing interface.NormalInclusionsGrowth DirectionLine Inclusions73Chapter 5. Experirnenta.l ResultsFigure 5.41: Surface of an LBO crystal with inclusions. Magnified 30 timesChapter 5. Experimental Results74Figure 5.42: Normal molybdenum inclusion in an LBO crystal magnified400 times. (a)Backscatter image. (b) WDS molybdenum dotmapChapter 5. Experimental Results 75Figure 5.43: Molybdenum line inclusion in LBO crystal magnified 2,200 times. (a)Backscatter image. (b) WDS molybdenum line scanChapter 6Physical Model of the LBO Crystal Growth ProcessThe flow of liquid under a growing crystal was investigated by observing theflow directlyin a physical model of the growth system using a transparentliquid as the melt. Thephysical model gives a better visualization of the flow pa.tterns whilea mathematicalmodel will give semi—quantitative fluid velocityresults that will he superior to the physical model results. This type of flow has been previouslyexamined in detail for a rotatingcrystal in an infinite fluid [46]. Thus the physical model willonly examine the flow fieldsdue to a rotating crucible with a stationary crystal.The physical model consisted of a plexiglass crucible8.8 cm in diameter and a plexiglass crystal 5.6 cm in diameter. The melt level in the cruciblewas 2.5 cm in height.The fluid consisted of a glycerine solution having a viscosityof 7 poise. This correspondsto an 41.6 Wt% MOO3 melt at S20°C, which is 130°C abovethe liquidus temperature.The plexiglass model of the crystal was positioned such thatthe bottom surface of thecrystal was in contact with the melt. Blue dye was injected ina hole in the crystal at0.5 of the crystal radius and red dye in a hole at 0.75 of the crystals radius.The blueand red dye consisted of chart recorder ink mixed with glycerinein order to give it thesame buoyancy as the fluid. The crystal was stationary forall of the experiments and thecrucible was rotated at either 45 or 78 rpm in a clockwisedirection. The flow is discussedin terms of the fluids radial(Vr). axial (vi) and theta (swirl,v0)velocity components.76Chapter 6. PhysicalModel of the LBO Giystal Growth Process776.1 Observed FluidFlow PatternsFigure 6.44a showsthe initial dye pattern fromthe top of the crystalfor a cruciblerotation of 45 rpm. The injectionholes for the red and bluedye are shown in figure asA and B respectively. Similar injectionholes at the same radial locationbut rotated byone quarter of a circumferentialarc are shown as A’ andB’. The dye traces are movingin a semicircular arc andalso have a radial flow componentthat moves them towardsthe centre of the crystal. Thisis shown by the red dye trace.It is injected at a0.75of the crystal radius (PointA) and its position is at 0.5 of thecrystal radius (PointB’)after it travels one quarter of a circumferentialarc. The view of thedye from the side ofthe crucible is shown in Figure6.44b. The dyes remain veryclose to the model crystalafter they have been injected.It is noted that each picturein Figure 6.44 correspondsto different experimentalruns. Differences in the positionof the red and blue dyewithrespect to each other are dueto different injection start times.Thus initially the dyeshave a large swirl velocity asmaller radial velocity anda non existent axial velocity.The dye tracers patterns betweenthe injection locationsand near the centre of thecrystal are shown in Figure6.45. The view from the top of thecrystal, Figure 6.45a, showthat the dyes reach the centre ofthe crystal after they travel0.75 of a circumferentialarc. The view for the side of thecrucible, Figure 6.45b, showsthat the dye tracers startmoving toward the bottomof the crucible near point A. PointA is 0.75 of a circumferentialarc from the location where thedyes a.re injected. Itis evident that the axial velocitydownward increases as thedye traces get near the centerline of the melt.The flow of the dyes at the centreline of the crucible are shownin Figure 6.46. Theflow near the crucible, pointA, and at 0.5 of the fluid height,point B, are examined.The swirl in the 0 direction is largerat point A and small at pointB also the verticaltravel distance is large betweenboth points. This shows thatthe axial fluid velocity isChapter 6. PhysicalModel of the LBO Orvstai Growth Process78dominant at the centre of the cruciblenear the mid height of the fluid.The flow patterns of the dye tracers near the bottomof the crucible, Figure 6.47, aredifferent at the mid height of the fluid, pointA, the centre bottom of the crucible, pointB, and the outside bottom of the crucible, pointsC and D. The dye traces start movingin the radial directions and get a. larger swirla.s they approach the bottom of the crucible,points A to B. Once the dye traces have reachedthe bottom of the crucible their axialtravel distance decreases to near zero and there radialand swirl component increased asthey move outward along the bottom of the crucible.When the dye reaches the side ofthe crucible, point C and D, it is no longer a separate phasefrom the pure glycerine.This indicates that there is significant mixing occurringa.t the bottom of the crucible.Figure 6.48a shows the flow patterns after approximately3 minutes. The initial dye hasmixed to make the fluid a uniform red color except directlybelow the crystal were newdye is being injected into the liquid.The location of the maximum of the radia.I and axial flow componentsfor steady statecrucible rotation are at the following location. Maximum radialfluid flow occurs directlybelow the crystal and above the bottom of the crucible. Axialfluid flow is largest at thecentre of the crucible half way between the crystal and bottomof the crucible. This issimilar to what has been previousix’ l)een predicted[28, 32].The change in flow patterns a.t different crucible rotationrates are examined. Figure 6.48 shows the change in flow pattern between 45 and78 rpm. The sequence ofphotos with respect to the crucible rotation rates are, figure6.48a. is the flow patternat 45 rpm, figure 6.48b is the transient flow pattern andfigure 6.48c is the flow patternat 78 rpm. Point A, located at the mid height ofthe fluid, will be the location of thefluid that is examined. The location of the swirl at pointA with a crucible rotation rateof 45 rpm, Figure 6.48a., is large. The size of the swirlgets smaller a.s the flow adjuststo the higher crucible rotation rat.e of78 rpm, Point A Figure 6.48b. The steady stateChapter 6. Physical Model ofthe LBO Crystal Growth Process79flow pattern at 78 rpm, Figure6.4Sc, results in reducing the swirl of the dyetracer atpoint A to near zero. The dye tracerfollow a straight vertical line below the centreofthe crystal. This shows that the axial velocityat the centre line of the melt gets largerwith increasing crucible rotation rates.The change in flow patterns directly below thecrystal for crucible rotation ratesof45 and 78 rpm are shown in Figure 6.49.The dyes in the 45 rpm case reach the centre ofthe crystal in 0.75 of a. circumferential arc.A crucible rotation rate of 78 rpm causes thedyes to reach the centre of the crystal in0.5 of a circumferential arc. This shows thatthe radia.l velocity of the fluid increases withincreasing crucil)le rotation rates. BothFigures 6.48 and 6.49 clearly show tha.t axialand radial fluid velocities increase withincreasing crucible rotation rates.6.2 Physical Explanation of Fluid FlowPatternsThis type of flow has been previously examinedin detail for a rotating crystal andstationary crucible [46]. These results will be referredto in considering the forces actingon the fluid, and the resulting fluid motion, for thecase of crucible rotation with andwithout a stationary crystal in contactwith the top of the melt.When a crucible starts to rotate from rest withouta crystal present, liquid near thebottom of the crucible starts to move in the 0 directiondue to viscous drag. As thefluid rotates it gets a centripeta.l acceleration (—2r).The presence of the centripetalacceleration with no radial pressure gradient causesthe fluid to move outward in theradia.l direction. When the fluid reaches the crucible wallit is forced to move upward.Fluid a.t the center of the crucil)le fluid moves downwardto balance the outward flow inthe radial direction. As the flow progressesthe surface of the fluid rises at. the outside,and falls at the centre as shown in Figure6.50. The va.riation in fluid height with radiusChapter 6. Physical Modelof the LBO Crystal Growth Process80creates a radial pressure gradient which balance the centripetaiacceleration and stopsthe radial flow at the bottom of the crucible. When the radial flowstops the fluid movesas a solid body with the rotating crucible. Consider the radial forcesthat act on a controlvolume of fluid. In terms of Newtons law of motion the balanceof forces in the radialdirection is:F = marWhere Fr is the force acting in the radial direction, m is the mass andar is the accelerationiii the radial direction. The previous equation in terms of pressure is= —pa’.were P is pressure andp is density. With crucible rotation and no radial pressure gradientthe forces acting on the fluid at the bottom of the crucible is only due to the centripetalacceleration. 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 withthe centripetal acceleration giving:2=_p(_wr)For a stationary crystal present. it will be assumed that the crystal is inserted intothe melt after the fluid has become a solid body in the rotating crucible. It is alsoassumed that the bottom of the crystal is the same shape as the surface of the fluid. Thestationary boundary condition, v0 = 0, at the fluid surface with the crystal present isshown in Figure 6.51.The crucible is rotating and the fluid has solid body rotation. Constraining a portionof the fluid surface causes the fluid directly below it to have a lower 0 velocity than it hadChapter 6. Physical Modelof the LBO Ci stal Growth ProcessSiduring solid body rotation. Thus the centripetal accelerationis lower. Since, in this case,the radial pressure gradient is the same there is a netforce in the inward radial direction.Thus, fluid below the constrained surface will movein the inward radial direction untilit reaches the centre were it will move downward.The fluid moving downward will reachthe bottom of the crucible, then it will move radially outwardto the crucible wall, upwardalong the crucible side, and finally inward onceit reaches the surface of the fluid.Thisflow will continue since the centripetal acceleration willnot be balanced by the radialpressure gradient below the crystal. The flowpatterns as explained using fluid mechanicsare the same as has observed in the physical model.Chapter 6. Physical Model of the LBO Crystal GrowthProcess82Figure 6.44: Initial dye tracer pattern in the glycerine, a) Top view. b) Sideview.Cucible rotated at 45 rpm.Ohapter 6. Physical Model of the LBO Orysta.l Growth Process83Figure 6.45: Dye tracer pattern in the glycerine when the blue and red tracers reach thecentre of the fluid. a) Top view. b) Side view. Crucible rotated at 45 rpm.Chapter 6. Physical Model of the LBO Grystal Growth Process84Figure 6.46: Dye tracer pattern in the glycerine whenthe dye reaches the bottom of thecrucible. a) Side view b)View under the crystal. Crucible rotated at 45 rpm.Chapter 6. Physical Model of the LBO Crystal Growth Process 85Figure 6.47: Dye tracer pattern near the bottom of the crucible. a) Side view. b) Viewunder 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 Process86Figure 6.4$: Dye tracer pattern at different cruciblerotation rates. a) Crucible rotationrate of 45 rpm. h) Transition flow for a crucible rotation rate between45 and 78 rpm.c) Crucible rotation rate of 78 rpm.Ohapter 6. Physical Model of the LBO Crystal Growth Process 87Figure 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 Process88Centre LineMELTI_____CrucibleCrucible RotationFigure 6.50: Interface curvature due to solid body rotation.-7Chapter 6. Physical Model of theLBO Grysta,l Growth Process89Surface of meltconstrained to giveit a zero velocityMELTCentre LineCrucibleCrucible RotationFigure 6.51: Crucible with portion of the tipper surface constrainedto zero.Chapter 7Temperature MeasurementsTemperature measurements were conducted to establishthe boundary conditions in themodel, to investigate the effect of crucible rotation onthe thermal fields, to comparethe temperature fields measured with those predictedby the model and to examine thethermal gradients in the crystal as it is cools toroom temperature.7.1 Initial Temperature Measurements withNo Crucible RotationThe melt was a clear amber color. No convective cells were observedat the liquid surface.Temperature as a function of axial position at three radial locations;the center (r = 0cm), the mid position (r = 1.6 cm) and near the wall (r = 3.2 cm)of the crucible areshown in Figure 7.52The temperature oscillations at. r = 1.6 and 3.2 cm result fromthe furnace heatersturning on and ofF and are not significant. The oscillations were suppressedby employingelectrical shielding on the thermocouples. The melt temperatureswere assumed to bethe lowest temperature recorded in the oscillations. The melttemperatures range from760°C at the top of the melt to 868°C a.t the bottom.The temperature measurements were conducted with the furnaceset approximately145°C(T) higher than the conventional setting for crystal growth.It is assumed thatthe values of the measured temperatures change proportionally withthe furnace setting.Thus the boundary conditions for the mathematica.lmodel were obtained by decreasing90Gha.pter 7. TemperatureMeasurements91the measured temperatures bythe appropriate AT.T(boundary condition) = T(rneasured) — ATThe temperatures used as boundary conditions are assumedto he the temperature of theinside walls of the platinum crucible w’hich do not change.The temperature boundaryconditions used in the sensitivity analysis are givenin Figure 7.53.7.2 Temperature Measurements With and Without CrucibleRotationTemperature measurements were conducted to examine the influence ofcrucible rotationon the thermal fields for crucible diameters of 6.6 and8.8 cm. Rotation rates of 0, 15, 20.25 and 30 rpm were used in the 6.6 cm diameter crucible. Measurements wereconductedat crucible rotation rates of 0, 10, 20 a.nd 30 rpm in the8.8 cm diameter crucible.Table 7.8 gives the concentration of the melts that were investigated.In all cases theoutside of the crucible was thermally insulated. The bottom of thesmall crucible wasdeformed due to the expansion coefficient of LBO/Mo03being much larger thanthat ofplatinum. The bottom of the crucible was no longer flat, the center was0.5 cm lowerthan the side. Thermal boundary conditions were determined from the0 rpm cruciblerotation data. The measured thermal fields for different rotation rates werecompared tothe model predictions.7.2.1 Boundary Temperature ResultsSmall Crucible (6.6 cm diameter)Temperature measurements 0.2 cm from the vertical crucible wall, in a meltcontaining45.5 \Vt% MoO3,are shown in Figure 7.54. The dashed curve are the measurements without a simulated crystal present, and shows that. the temperature decreasesfrom 818°C.Chapter 7. Temperature I’Ieasurements 9288087G.860850840__830 -C)820Q1C.R=l6mm-800;ERrz32mm R=Ommci) II- I790I780 I0R1’5 2b30Height (mm)Figure 7.52: Temperature variation with axial position in a 5.5 Wt% MoO3solution. Thethree radial locations are r = 0 mm, r = 16 mm, and r = 32 mm. There is no cruciblerotation during the temperature measurements. Crucible diameter is 6.6 cm.Chapter 7.Temperature Measurements93Convective Heat Transfertttttz = 2.5(@•cMELTIIz=O.7an IIz = 0 ai_________________________r=3.3cmr=1.6anFixed TemperatureFigure 7.53: Temperature boundary conditions used in the fluid flow model for thesensitivity analysis and the examination of the operating parameters. Crucible diameteris 6.6 cm.Chapter 7. TemperatureMeasurements94CrucibleWeight PercentDiameter Charge Concent rationMeasured Equivalent NormalizedMoO3 LBO Mo LiMoO3 LBO MoO36.6 cm 50.05 49.95 29.00 3.0:343.51 52.11 45.508.8 cm 44.56 55.4433.30 3J3 48.46 53.83 47.38Table 7.8: MoO3and LBO concentrationsin the melt that were used for the temperaturemeasurements with and wj thout cm cil)lerotation.0.47 cm from the bottom, to805°C at 2.45 cm. With a simulated crystal positionedinthe melt the melt temperature increasedat the wall by approximately 12°C.The radialtemperature distribution in the melt 0.47 cmfrom the bottom of the crucible, is shownin Figure 7.55. The temperature remains essentiallyconstant at 807°C for 1 cm from thecentre then increases progressively toS1S°C at 0.2 cm from the crucible wall.In obtaining the temperature measurements shownin Figure 7.55 two thermocoupies were used, one for r < 1.6 cm and the secondfor r > 1.6 cm. The small increasein temperature at A may be associated with a.slight difference in behavior of the twothermocouples. The repeatability of the temperaturemeasurements were determined bymaking multiple measurements at points Band C. The difference in temperature was lessthan 1°C. Adding the sinnilated crysta.l increasesthe melt temperature, bitt does notsignificantly change the temperature gra.d ient. Followingthis observation, the boundarytemperatures were determined from the datawithout a crystal by adding 12°C to accountfor the temperature rise when a crystalis added, and extrapolating the measurementsto the outside surface of the platinum crucible.The resultant temperature boundaryconditions are given in Figure 7.56. Using theseboundary conditions, temperature profiles in the melt were calculated, using the mathematicalmodel, and compared to theexperimental measurements with no crucible rotationand with a simulated crystal. TheJChapter 7. Ternperat tire Measurements95boundary conditions were thenadjusted, and the calculations repeateduntil thecalculated temperatures fittedthe experimental values.The boundary conditionsdeterminedin this manner were then usedto calculate the thermalfield with crystal rotationandthe results were compared tothe corresponding measurements.Large Crucible (8.8 cm diameter)The melt temperature 0.2 cm fromthe wall of the crucibleis shown in Figure 7.57. Themelt temperature, without thecrystal, is essentially constantat 824.5°C between at0.36and 2 cm from the bottom of thecrucible. Above 2 cmthe temperature progressivelydecreases reaching807.3°C at 3.15 cm. Introducingthe simulated crystal increasesthemelt temperatureby approximately 5°C. The radialtemperature distribution0.36 cmfrom the bottom of the crucibleis shown in Figure 7.58. Thetemperature varies from806°C at 0.17 cm from the center of thecrucible to S24°C at 0.2cm from the wall ofthe crucible. The gradient does notappea.r to dramatically change with theinsertion ofthe simulated crystal. The temperatureboundary conditions used in themathematicalmodel are determined using theprocedure described in the previoussection. Figure 7.59shows the temperature boundaryconditions used in the model calculations.7.2.2 Melt Temperature ResultsSmall Crucible (6.6 cm diameter)Melt temperatures at cruciblerotation rates in the range0 and 30 rpm, with the simulatedcrystal present, were determined.The axial temperature profile 1cm from the centreof the crucible is given in Figure7.60. With 0 rpm crucible rotation,the temperaturedecreases with distance from thebottom, the curve being concaveupward. With 1.5rpm crucible rotation, the temperatureat 0.4 cm from the bottom decreasesand aboveChapter 7. Temperature Measurements 96BottomTop830825With’T1eSiThe Simulated Crystal,820E815-.. No Simulated Crystal810SSSS8oI IThO 0.5 1.01.5 2.0 2.5Axial Position (cm)Figure 7.54: Melt temperature 0.2 cm from t.he crucible wall with and without thesimulated crystal. Crucible diameter is 6.6 cm.Chapter 7. Temperature Measurements97CentreOutsideWith The S1edCsrn820W-8--815 -No Crystal Cr -810-80—— I I I0.5 1.0 1.5 2.0 2.53.0 3.5Radial Position (cm)Figure 7.55: Melt temperature 0.47 from the crucible bottom with and without thesimulated crystal. Crucible diameter is 6.6 cm.Chapter 7. TemperatureMeasurements98ConvectiveHeat TransferSIMULATFIDz=2.3cm812C10cm23cmMELTIHIcm- I35cmAz=-0115CmI05cmr=3615Cm8Crucible20C 0.115cm thicknessz -0.27cmr=1.6cmz=0.615cmr = 0 to 0.54 cmFixed TemperatureFigure 7.56: Temperature boundary conditions used in the mathematical model of me’twith the simulated crystal. Cruciblediameter is6.6 cm.Chapter 7. Temperature Measurements 99Bottom Top830 I IThe Simulated Crystal825 -,820‘SE815 -810No Simulated Crystal\b8% ‘.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Axial Position (cm)Figure 7.57: Melt temperature 0.2 cm from the crucible wall with and without thesimulated crystal. Crucible diameter is 8.8 cm.Chapter 7. Temperature Measurements 100CentreOutside825820No Crystal815AE 810 -‘I805,8000:5 1.0 1.52.0 2.5 3.0 3:5 4.0Radial Position (cm)Figure 7.58: Melt temperature 0.36 from the cruciblebottom with and without thesimulated crystal. Crucible diameter is 8.8 cm.Chapter 7. Temperature Measurements101Convective Heat Transferz = 3.3 cm—_—iz=2.5cm 1.0cmz = 1.7 cm 3.3 cmMELT IH____ _____I4.4 cm>1________________________________________Cruciblez=-0.ll5cm(,thickness0.115r 4.5 cmr=3.5cmr=OcmFixed TemperatureFigure 7.59: Temperature boundary conditions used in the mathematical model of meltwith the simulated crystal. Crucible diameter is 8.8 cm.Chapter 7. TemperatureMeasurements1020.5 cm increases. The shapeof the curve changes to concave downward.Increasingthe crucible rotation above15 rpm progressively raises the melt temperaturewith theaxial temperature distributionessentially remaining the same. The largesttemperatureincrease occurs in the 25 to30 rpm interval.The corresponding axial temperature distributionat a radial distance of 1.5 cm aregiven in Figure 7.61. In this caseat 0 rpm the temperature is observed to decreaselinearlywith distance from the bottom.With crucible rotation, the temperatureof the meltprogressively increa.ses, and the tern peraturedistribution curve is concave downward,asbefore. The axial temperature distributionat radial distances of 2.6 cm and 3.1cmfrom the centre are given in Figure 7.62and 7.63. At both distances at0 rpm, thetemperatures are higher than atr = 1.5 cm, and decrease linearly with distancefromthe bottom of the crucible. \Vit.h rotationrates between 15 and 25 thetemperaturedistribution becomes concave upward.The axial temperature profile becomes flatat arotation rate of 30 rpm.The movement of the fluid due to crucible rotationhas been qualitatively shown anddescribed in Chapter 6. Crucible rotation resultsin the movement of hot fluid up atthe crucible wall and cold fluid down underthe crystal. The movement of hot andcoldfluid causes the liquid isotherms tobecome more concave. The change in theliquidisotherm shape with the fluid motion is shownschematically in Figure 7.64. The axialtemperature profiles become flat at highcrucible rotation rates The experimentalresultsshown in Figures 7.60 through 7.63are in agreement with the physical model (qualitative)analysis.Large Crucible (8.8 cm diameter)Melt temperatures at crucible rotation rates between0 and 30 rpm, with the simulatedcrystal present, were determined. The axial temperatureprofile at 0.4 cm from theChapter 7. Temperature Measurements 103r-)CIEBottom Top828826-—D----—— 0RPM——--<>.-—— 15RPM20RPM——0——- 25RPM—••—••-X•—•--•’30RPM824-822-820-818 -816--I . . . .0.5 1.0 1.5Axial Position (cm)2.0Figure 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--—— 0RPM--—<>-——15RPM20RPM--0--- 25RPM——-X——30RPMBottom Top828826-824-822 - — — — — — — —‘0 ‘20Axial 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 Measurements105—Q————0 RPM------15RPM20RPM——0——- 25 RPM——-X—••—30RPMBottomTop828 .824.820818816-814oo0.5 1.0 1.52.0Axial 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]—-—— 0RPM---a--- 15RPM20RPM-_ 25RPM—— 30RPMBottom Top828826i:.820818816-814o• I I • I.0 0.5 1.0 1.5 2.0Axial 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 107iCrystalICrucibleFigure 7.64: Change in liquid isotherms with crucible rotation. (a) No crucible rotation.(b) Large crucible rotation.Chapter 7. Tempera.ttire Measurements108centre of the crucible is givenin Figure 7.65. The melt temperatureprofile with nocrucible rotation is concave downward.With rotation, the concavity remainsthe sameand the average melt temperatureincreases. The change in average melttemperaturewith crucible rotation is large between0 and 20 rpm and small between 20and 30 rpm.The axial temperature profile at r= 0.9 cm with no crucible rotation, Figure7.66,is slightly concave downward. A rotationrate of 10 rpm causes the melt temperatureto increase but the axial temperatureprofile remains concave upward. Higherrotationrates, 20 and 30 rpm, make the axia.ltemperature profile flat. The axia.l temperaturedistribution at radial distances of2.8 cm and 3.3cm from the centre are given inFigure 7.67and 7.68. At both distance the temperatureprofile at 0 rpm is flat. The interfacebecomes concave downward ata crucible rotation rate of 10 rpm. Between20 and 30rpm the axial temperature profiles becomesapproximately flat. For all of the previoustemperature profiles the largest changein melt temperature with crucible rotation rateoccurs between 0 and 20 rpm. Figure7.69 shows the axial temperature profile at3.8 cm.The axial temperature profile is flat withno crucible rotation. Rotation rates of0 to 30rpm do not change the shape of the temperatureprofile. The melt temperature changeslinearly with crucible rotation speed.Fluid motion due to crucible rotationhas been qualitatively examined in Chapter6.Crucible rotation causes the movement ofhot fluid up at the crucible wall and cold fluiddown under the crystal. This type of fluid motioncauses the liquid isotherms to becomemore concave (Figure 7.64). The axial temperatureprofiles become flat at high cruciblerotation ra.tes (Figure 7.64(b)) as shown experimentallyin Figures 7.65 and 7.66.Chapter 7. Temperature Measurements109—0--——0 RPM---c--- 10RPM20RPM——0-—- 30RPMBottom Top832 I830__——.:E--2-.828826- --“0,S 8240.5 1.0 1.5 2.0Axial Position (cm)Figure 7.65: Temperature distribution a.t r= 0.4 cm for crucible rotationsof 0, 10, 20and 30 rpm. Cruciblediameter is 8.8 cm. Simulated crystal present.Chapter 7. Temperature Measurements 110—D-—— 0RPM------•- 10RPM20RPM——0-—- 30 RPMBottomTop832 I8300-—-__.828826824820818816814oo 0.5 1.0 1.5 2.0Axial Position (cm)Figure 7.66: Temperature distribution at r = 0.9 cm for crucible rotations of 0, 10, 20and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present.Chapter 7. Temperature Measurements 111D-0 RPM---c--- 10RPMLi20RPM——0—-30 RPMBottomTop832 I•8300--2828‘—‘826E824E820818816814oo0.5 1.0 1.52.0Axial Position (cm)Figure 7.67: Temperature distribution at r = 2.8 cmfor crucible rotations of 0, 10, 20and 30 rpm. Crucible diameter is 8.8 cm. Simulatedcrystal present.Chapter 7. Temperature Measurements 1120RPM---0---10RPMLi 20RPM——0-—-30RPMBottomTop832I830__828- 0-‘—‘826E824C.820818816814oo0.5 1.01.5 2.0Axial Position (cm)Figure 7.68: Temperature distribution at r = 3.3 cm forcrucible rotations of 0, 10, 20and 30 rpm. Crucible diameter is 8.8 cm. Simulatedcrystal present.Chapter 7. Temperature Measurements 113—C---—0 RPM------ 10RPM20RPM——0—- 30RPMBottom Top832 I • I830828. 824822820818816814oo0.5k 1.0 1.5 2.0Axial Position (cm)Figure 7.69: Temperature distribution at r = 3.8 cm for crucible rotations of 0, 10, 20and 30 rpm. Crucible diameter is 8.8 cm. Simulated crystal present.chapter 7. Ternperattire Measurnts1147.3 Thermal Gradientsin the Crystal During CoolingHigh thermal stresses whichlead to cracking are dueto large temperature gradientsinthe crystal [51]. Thus, it isimportant to know the thermalgradients that occur in thecrystal during cooling for differentfurnace configurations. It is notpractical to measurethe thermal gradients in the crystalswith imbeddeci thermocouples.As an alternative asimulated crystal made of alumina.aggregat.e was used, having theconfiguration shown inFigure 7.70. Four thermocouples wereimbedded in the alumina in thepositions indicatedin the figure. The thermal conductivityof the ceramic alumina aggregateis comparableto tha.t of L130 which makesit a. suitable model material.Temperature measurements in the simulatedcrystal were made under equilibriumconditions with the furnaceset at 400°C and aft.er an intervalof five hours to allow thefurnace to reach equilibrium. Measurementswere made at two verticallocations in thefurnace, position A and B in Figure7.71, corresponding to the normal coolingpositions ofcrystals grown in the large andsmall crucibles respectively. Note thatthe bottom of thelarge crucible is positioned 1.1 cm abovethe bottom of the small crucible.Temperaturegradients were determined with nocrucible present, with a smallcrucible in the furnaceand with a large crucible in thefurnace with and without an insulatingblock at the topof the furnace as shown in Figure7.71.The crucible and sinmlated crystal positionsin the furnace for whichtemperaturemeasurements were made are shownin Figure 7.72. The valuesof the temperaturegradients are listed in Table 7.9 with thecorresponding measured temperaturegradientsacross the pairs of thermocouples indicated.no crucible in the furnace thegradients are larger for the simulated crystalinthe upper position A as comparedto B. The radial gradients a.re closefor 1)0th A and Ba.nd lower than the axial gradients withthe axial gradient at A beingappreciably largerChapter 7. TemperatureMeasurements115than the others.Adding the small crucible to thefurnace changes the temperature gradientsin thesimulated crystals. The radial gradientbetween thermocouples 1 and 2 is increasedwhen the crucible is added and the thesimulated crystal is above of the crucible.All ofthe other radial and axial gradients arelower with the crucible at positions Aand B. Inparticular there is a large decrease inthe axial gradients with the additionof the crucible.The decrease in gradients for positionB can be attributed to the presenceof the cruciblesurrounding the crystal in this position.The high conductivity of the platinum crucible,and its close proximity to the crystalreduces the gas temperature gradientadjacent tothe crystal which leads to lower giaclientsin the crystal. The higher radial gradients atposition at position A follow, since1)OSitiOfl A is above the top of the crucible.The radial gradient in the crystal with the large cruciblein the furnace is essentiallythe same as that observed with the crystalin the smaller crucible. (position B). The axialgradients however are effectively twicethat of the smaller crucible, and compared to thecondition of no crucible in the furnace.In effect the large crucible walls are sufficientlyfar away from crystal that they have little effect onthe gradients in the crystal. Addingan insulating brick over the top of the seed holesignificantly reduces the radial gra.dientsand to a lesser extend the axial gradients.Independent of crucible size, the thermalgia.clients in the crystal are most sensitiveto the addition of insulation over theseed hole in the furnace and the vertical positionofthe crystal. Moving the simulated crystaltowards the centre of the furnace reduces theaxial gradients and the insulated top reducesthe radial gradients. Using a. small cruciblefor crystal growth reduces both the a.xia.land ra.dia.l gradients provided that the crystalis surrounded by the walls of the crucibleduring the post growth cooling stage.Chapter 7. Temperature Measurements 1161 cm6cm1 cm1.65 cmTC PositionsTC1 TC2fltL:utiem1n’1TC3 TC40.4 cmFigure 7.70: Alumina aggregate used for determining the thermal gradients in the furnace.Chapter 7. Temperat tire MeasurementsInsulation CoverModel crystal for measuringtemperature gradient duringcrystal cooling117Furnace ElementFigure 7.71: The axial positions of the alumina model crystal used when measuring thethermal gradients.InsulationLarge CruciblePosn.A2.2 cmPosn.BSmall Cruc ,leBottom insulationChapter 7. Temperature Measurements 118No Crucible Small CrucibleLarge Crucible Large Crucible(66 mm diameter)(88 mm diameter) With Insulated Top________________I L‘‘ iLi’A‘jjGradients at A (C/cm): Gradients at A (C/cm)Gradients at A (C/cm): Gradients at A (C/cm):35 542.9fl0.93.6 4.22.3 1.6Gradients at B (C/cm): Gradients at B (C/cm):3.0>2.5>Figure 7.72: Temperature gradients measured for the different furnace configurations asindicated.Chapter 7. TemperatureMeasurements119Condition Temperature Gradient°C/crnRadial AxialTC1—TC2 TC3—TC4 TC1—TC3 TC2—TC41. No Crucible(a) crystal in upper position (A)3.5 3.6 7.3 7.2(h) crystal in lower position (B) 3.03.3 4.6 4.32. Small Crucible(a) crystal in upper position (A) 5.4 4.2 2.53.5(h) crystal in lower position (B) 2.5 2.6 2.6 2.53. Large Cruciblecrystal in upper position (A) 2.9 2.35.6 6.14. Large Cruciblecrystal in upper position (A)and seed hole insulated. 0.9 1.65.3 4.7Table 7.9: Temperature gradients measured for the different furnaceconfigurations asindicated.Chapter 8Mathematical Model For LBOCrystal Growth8.1 Scope of Model and AssumptionsThe objective of the model is to ({uantitativelvdetermine the temperature and velocity distributions in the melt during solidificationas a function of the growth variables.This requires establishing the thermal conditionsrelated to the furnace, calculating thethermal field, and fluid flow velocities, with and without crystaland crucible rotations.A fluid flow model, which incorporates hea.t transfer, hasbeen developed by FluidDynamics International (FIDAP) [52]. The FIDAP packagecan be used to determinethe velocities and temperatures in the melt duringcrystal growth. In growing a crystal,the melt temperature is adjusted t.o just above theliquidus temperature, a small seedcrysta.l is dipped into the melt, and the temperaturefurther lowered to allow the seed togrow. Crystal and crucible rotation is employedto enhance fluid flow in the melt. Thecrysta.l is slowly pulled awa.y from the melt, effectively understeady state conditions,once the required crystal diameter is reached by the seed. Themodel is used to predictthe fluid flow a.nd temperature distribution in themelt a.t steady state.A number of assumptions were used in the model, to simplifythe calculations. Theseare listed in Table 8.10, along with a. ratingof the validity of each assumption andcomments on the validity rating. The assumptionof an axisymmetric temperature fieldis necessa.ry to simplify the calculations. The crystalgrowth furnace is designed tohave an axisymmetric temperaturefield, and crystal and crucible rotation will tendto120Chapter 8. Mathematical ModelFor LBO Ciysta.l Growth121level out any local temperature variations in themelt. The assumption that thecrystalgrowth system is at steady stateis essential to the model, and is considered validonthe basis of the very slow cooling rate of5°C/day or withdrawal rate of 1.6 mm/dayfor LBO. In comparison the withdrawal ra.teof Ge is much faster at 100 mm/hr.Theassumption of laminar fluid flow is necessary for the modelcalculations. The assumptionis reasonable for LBO since the melt has ahigh viscosity (10 to 100 poise). For flowpast a. smooth plate, laminar flow breaks clown to turbulent flow whenthe Reynolds (Re)number exceeds 5 x iO.ReV L pItwhere V = flow velocity (cm/s), L is the characteristic length(cm), p = melt density(g/cm3)anditmelt viscosity (poise). For a. limiting case situation, assumeV = 1000cm/s, L = 3.2 cm,R= 12.3 poise (40.9% MOO3 at 730°C) andp = 3.26 g/cm3.Usingthese values gives Re = 848, which is very much lower than5 x i0, indicating that theflow is laminar.The air/melt interface is assumed to he flat, consistent with the high viscosity andlow fluid velocities in the melt. Heat. transfer a.t the air/melt and air/crystal interfaceisassumed to be by convective heat transfer only, which is consistentwith the low surfacemelt temperature (below 700°C). The model assumes that themelt is homogeneousduring crystal growth. As the crystal grows, MoO3 is rejectedat the interface resultingin an increase in the concentration of MOO3 in the melt ahead of the crystal.The LBO/Mo03thermophysicai properties and the dimensionless groups ofinterestare listed in Table 8.11 and Table 8.12 respectively. Thermophysical valueswhich arenot available in the literature are estimated.The Grashof, Prandtl and Schmidt numbers calculated for a LBO/Mo03liquid andfor liquid gallium are given in Table 8.12. The values for Galliumare typica.l for a liquidHCDz CDC CDC C C Co C CoCo,1Zr-t-CCoCoCoC-CD—.e-C-CDCD—.Se-CD•CD5..c-. Cd)CD—Co‘CoC.-,‘CD2.5CoC-CD—Cz CDCD Cl) Cd) CD CD -J C C CD Co C J) Co Co 0 CD C C C Cd) CM CD S(Th Sx I. 9C Co CD0C-)C C S C C- CDg)ji,X w-OO—‘---C-<c)CCCS----Ct’ -CoDq)CD<CD<c-q-CD—. CocCC-0-CCDO’ . c-_CChapter 8. Mathematical Model For LBO Crystal Growth123Group Definition Gallium LBO+MoO3(24.6 % Mo)Grashof Number (Cr) 2.4 x1060.0827Prandtl Number (Pr) 0.024 172kIISchmidt Number (Sc) fl— 2.4 x1024 x iOTable 8.12: Non Dimensional Numbers for Gallium and LBO.metal system. The small Crashof number for the LBO/Mo03liquid indicates buoyancyforces are smaller than viscous forces which implies that natural convection will be muchsmaller than forced convection. Gallium, in comparison, will have strong natural convection. 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 themomentum diffusion rate. The large Prandtl number for LBO/Mo03(172) indicates thatthe momentum diffusion rate is much larger than the energy diffusion rate. Thus for agiven boundary layer, the thermal boundar layer thickness is much smaller than the velocity boundary layer (6< 6). The Schmidt. number is the ratio of momentum and massdiffusivities. For convective mass transfer in laminar flow the Schmidt number is proportiona.l to the velocity and concentration boundary layers (Sc6/)The LBO/Mo03Schmidt number is large at 4 x i0 indicating that any concentration boundary layer willbe much thinner than the velocity boundary layer. In contrast liquid metals. as shown bygallium, have nondimensional parameters tha.t are very different to that of LBO/Mo03.8.2 Idealized Domain and Description of CalculationsThe 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 growthconditions. For example, the crucible walls are assumed to be smooth. The bottomChapter 8. Mathematica.lModel For LBO Ciystai Growth124of the crucible is assumed tobe flat for all calculations except when the temperaturemeasurements are compared to themodel calculations for the the small crucible.Therotation of the crucible and crysta.l are assumed notto he eccentric.Cylindrical polar coordinates are used inthe 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 andan axial velocity component(Vr, v6, vi).It is noted that the rotational component is treateddifferently from the axial and radialcomponents due to the assumption of axialsymmetry. The 0 component of the velocityand temperature variables are real, but they are constantand the spatial derivative iszero(g= o).The model is applied to small and large crucibles,66 mm in diameter and heightand 88 mm in diameter and height respectively.The sensitivity analysis, determination of the dominant parameters and model validation were conductedusing the smallcrucible. The comparison of the model predictions withthe experimental temperaturemeasurements for different crucible rotations were conductedon the small and large crucibles. A simulated crystal, formed from a platinum sheet,Figure 4.31, was used forthese measurements.8.3 Steady State Axisymmetric Fluid FlowModel8.3.1 Equations of Fluid Flow - LagrangianCoordinatesThe conservation of mass, momentum and energy governthe flow of a viscous fluid. Inorder to present them in their simplest form the equationsare formulated in Lagra.ngiancoordinates. In this case the reference frame is moving withthe fluid and the equationsare applied to a moving control volume of fluid. The conservationof mass in the control volume states tha.t the mass in the control volumeis constant with time, = 0.Chapter 8. Mathematical Model For LBO Crystal Growth 125CenterLineFluid(LBO + MoO)heightradius —>Platinum CruciblezFigure 8.73: Schematic representation of domain examined with the model.Ghapter 8. MathematicalModel For LBO Ciystai Growth126Expressing this equation in termsof the density of the fluid gives.d (p volume)dL=(8.1)The conservation of momentum followsfrom Newtons second law being appliedto thecontrol volume, F = m. Dividing bythe volume gives the force per unitvolume,changing the mass term to a density term.f= p (8.2)The energy equation describes the change in internalenergy 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 - EulerCoordinatesConverting the equations of fluid flow fromLagrangian coordinates to fixed (Euler) coordinates gives equations that are easily appliedto real situations. The following sectionwill give an overview of the final forms of the equations.Converting to Euler coordinatesrequires that the Material Derivative()be used on any property that changeswithtime.d D a -where the div term (V) in cylindrical coordinatesis(a ia aar r ãO ÔzThe fluid being examined is incompressible, thatis p is constant, which is used to simplify the equations. Using the material derivativeon the conservation of mass equation(Equation8.1) yields the following in Euler coordinates.pV = 0(8.4)Chapter 8. MathematicalModel For LBO Ciystal Growth127Likewise, the conservation momentum equationin Euler coordinates becomes:- -= .foa’y+.fsurjaceThe force term has been divided into a body and surfaceforce. The only body forces acting on the fluid are due to gravity. The body forceis p. The Boussinesq approximationis used to account for a density change with temperature.This is done by varying thedensity in the body force term usingPo(1 — /3AT), where ,8 is the expansion coefficient ofthe liquid, and AT is the temperature change. Thesurface forces consists of the externalstresses acting on the sides of the control volume.A length derivation of the stressesacting on the control volume gives the surface forces asfsurjace= where is thestress tensor. Manipulated, the equation further gives:fmrjace—Vp + V2()where p is pressure andIL Sviscosity. Substituting these into the conservationof momentum equation gives final form of the momentum equation:= Po(’— AT) Vp+V2 (8.5)The energy equation in Euler coordinates isDE DQ DWThe incompressible form of the energy equation isv (kv) (8.6)The equations are further simplified by using the penalty function approachfor thepressure variable. For this method the conservation of massequations is discarded andthe pressure is eliminated from the momentumequation usingp = —Vv. is theChapter 8. MathematicaiModel For LBO Crystal Growth128penalty parameter. The problemis solved for steady state; thus all time derivativesarezero. The resulting equations that are solved by FIDAPare: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 asSTRONGLY coupled. Thisterm is used when the full set of equations are solved simultaneously.Nine nodedquadratic elements are used in the model. Velocity and temperaturedegrees of freedom are present at each node.8.3.3 Temperature Boundary ConditionsThe temperature boundary conditions used in the model were experimentallymeasuredas discussed in Chapter 4. The location of the boundary conditions are differentfor eachset of model calculations. The temperature distribution along the wall and bottomofthe platinum crucible were considered as fixed temperature boundaries, assumed to helinear between points, and independent of time. The top of the fluid andcrystal weremodeled a.s a convective temperature boundary condition. The ambienttemperaturewas approximated using the experimental measurements. The convectiveheat transfercoefficient was taken as 0.006 W/cm2,which is an average value for a natura.l convectionheat transfer surface. The values of the temperature b&undary conditions used for eachset of calculations will be given prior to the presentation of the results.Sensitivity analysis and Parameter ExaminationThese calculations focus on the fluid flow in the melt. As a result the modelonly considersthe melt and the inside surface of the platinum crucible, which simplifiesthe calculations.Chapter 8. MathematicalModel For LBO iystal Growth129The temperature measurements wereconducted at a furnace setting higherthan thatused for crystal growth. The temperaturesused in the model are the measuredvaluesless the temperature difference betweenthe furnace setting at which the measurementswere made and the crystal growthfurnace setting which gives the thermalfield for acrystal that is approximately 2/3 of thecrucible diameter.Comparison with Experimental DataThese calculations examine the change in thermalfield with crucible rotation, with thesimulated crystal in the melt. The boundaryconditions are taken from the experimentalmeasurements tha.t have no crucible rotation withthe simulated crystal in the melt. Themelt temperatures are extrapolated to give the boundary conditiontemperatures whichare located at the outside of the crucible. Minor adjustments aredone to the boundarytemperatures by fitting the model predictions to the experimentalmelt temperatures.8.3.4 Velocity Boundary ConditionsFor the velocity boundary conditions it is assumed thatno slip occurs at the crucible andcrystal surfaces in contact with the melt, and that freeslip occurs at the air-melt surface.The azimuthal velocities are dependent. on the crystal and crucible rotationrate, L, and!Crespectively. The velocity boundary conditions areas follows:Bottom of Cruciblevz 0Vr = 01)9Il(rps) x 27r x r(cm)Chapter & MathematicalModel For LBO Crystal Growth130Side of Crucible= 0VT = 0=x 2ir x r(crn)Fluid/Air Interface= 0No Constraint= No ConstraintCrystal/Fluid Interface= 0= 0=x(rps) x 27r x r (cm)Fluid Center Line= No Constraint= 0V6 0Chapter 8.Mathematical Model For LBO Crystal Growth1318.3.5 Solution ProcedureSensitivity analysis and Parameter ExaminationThe calculations were carried out in sets of iterations.The initial shape of the crystal/melt interface wa.s taken as flat since the thermalfield is unknown prior to the firstcalculation. ‘With the solid-liquid interface positionand the boundary conditions defined.the model predicts the fluid velocities and temperaturefield using the successive substitutions iterative method. The new interface position,predicted by the model, is comparedto the position from the preceding model calculation. If theydiffer by more than 0.08mm the interface location is adjusted and the fluid velocities and temperaturefield arerecalculated. The adjustment of the interface and model calculations continuesuntil thedefined crystal/melt boundary location is within 0.08 mm of the calculatedvalue.Comparison with Experimental DataThe calculations with the simulated crystal were carried out in two steps.First, theboundary conditions were adjusted until the model accurately predicted the thermal fieldof the melt for the case of no crucible rotation. These boundary conditionswere thenused in the model which determines the thermal field as for different cruciblerotations.Chapter 9Sensitivity Analysis of the Fluid Flow ModelA sensitivity analysis was conducted to examining the flow due to natural and forcedconvection, and to establish how variations in the mesh density, viscosityand condllctivity values used in the model affect the calculations. The valuesof the mesh densities,conductivities and viscosities considered in the sensitivity ana.lysis are givenin Table 9.14.The mathematical model analysis was conducted for the small, 6.6 cm diameter,crucible with a fluid height of 2.5 cm. The temperature boundary conditions employed inthe sensitivity analysis are shown in Figure 9.74. These values were obtained by measurements, Figure 7.53, which were higher than usually employed for LBO crystal growth.The ambient gas temperatures, determined in Appendix A, were also approximately145°C higher. A correction of —145°C was applied to these measurements to bring theminto the correct range. The established values of the thermophysical propertiesare giveniii Table 9.13. The determination of the thermal conductivity used in the calculations isgiven in Appendix A. The MoO3 concentration for all of the calculations is 40.9Wt.%IVIoO3 with the exception of the viscosity analysis where all three MoO3 concentrationsare considered. It is noted that. the liquidus is assumed to be 650°C for all sensitivityanalysis calculations.9.1 Natural ConvectionThe 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 is132Chapter 9. Sensitivity Analysisof the Fluid Flow Model 133Convective HeatTransferT(C) =55+l87.8.r_23.7.r2tttz = 2.5(--)MELTIz=O.7an(z=Ocm1723Qr=3.3cm r=1.6anr=OcznFixed TemperatureFigure 9.74: Temperature boundary conditions used in the sensitivity analysis.‘—,-L.JC.Zt’.Z—. Cl)ci3(000 DODs.C)CC)CC)CD cc,2.—1-Cl)Cl)Cl)j©c9c- r.1‘—;I1 C)1 -cc,pP —:icc,0o©c, xx—‘—--91+++©0D èèc,)4)xxxI.c:DccCl)C)’xxx—---Ct, Cr) 0 0- Ct 0 Cl) 0 C)) Cl) 0-Cl) Ct C)) C)) C))-D.o-B0-Ci)C)Cl)<eCCC)c_ Cl)r c C — D , xc,) xcpDC)CCCiCl)0. l— x CC,,x (.%-Chapter 9. SensitivityAnalysis of the Fluid Flow Model135due to the high viscosity and low thermal expansion coefficient()of the LBO/Mo03melt. The flow pattern is counterclockwise moving up the wall and down belowthecrystal. There is a stagnant region, marked“S” in Figure 9.75, that the fluid rotatesaround. The flow regime shown in Figure9.75 results from the density gradient in thefluid, which is directly related to the temperaturedistribution. The density variationis proportional to the temperature difference themelt is above the liquidus, given byP = Po(1 — IBT). The temperature contours in the melt are shownin Figure 9.76.The hottest region of the fluid corresponds to the area werethe melts density is aminimum. In the LBO/Mo03melt the hottest region is locatednear the outside bottomof the crucible. This low density fluid rises upwardsand displaces the other fluid asit advances. The displaced fluid near the outside surface ofthe melt moves under thecrystal were it drops to the bottom of the crucible,then moves horizontally towards thehottest region to replace the less dense fluid tha.tis moving upward.The fluid velocities are examined at various locations in the melt. Theaxial velocitiesalong radia.l lines a.t 0.25, 0.5, and 0.75 of the heightof the fluid are shown in Figure 9.77.The flow is upward near the crucible wall and downwards under the crystalat velocitiesnear 3 x i0 and 3.2 x i0 cm/s respectively.The vertical flow is a. maximum a.t 0.5of the fluid height, and is a minimum, approaching zero, both belowthe centre of thecrystal, 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 crucibleradius are shown in Figure 9.78. The radial flow is negative belowthe crystal and positivenear the bottom of the crucible with velocities nea.r—2.4 x i0 cm/s and 2.5 x iOcm/s respectively. The largest ra.dia.l velocity occursat 0.5 of the crucible radius.The velocity tangential to the crysta.l surface at0.25, 0.5, and 0.75 of the ra.dius ofthe crystal is shown in Figure 9.79. The largest fluid velocitybelow the crystal occurs at0.75 of the crysta.1 radius. The smallest tangential velocity occursnear the centre of theChapter 9. SensitivityAnalysis of the Fluid Flow Model136crystal. The variation on tangentialfluid velocity is proportional to the amountof masstransfer that occurs ahead of the crystal.This is important since inclusion formation orinterface breakdown occurs when the mass transferof MoO3 away from interface is lowerthan the rate of MoO3 being rejected atthe growing crystal surface. Thus, inclusionor interface breakdown is less likely to occur atouter regions of the crystal where themagnitude of the tangential velocity is a maximum.9.2 Forced Convection9.2.1 Crystal RotationThe velocity and temperature fields are calculated fora crystal rotation rate of 12 rpmand no crucible rotation. The velocity vectors areshown in Figure 9.80. The magnitudeof the flow that occurs in the axial and radial directionsis approximately 5 x i0 crn/s,which is an order of magnitude higher than the flowthat occurs due to natural convection.There is also a rotational component of the fluid velocity. Figure9.81, which is large being2.6 cm/s near the edge of the crystal. Thus the flow that moves upward belowthe crystaland downward at the side of the crucible will also swirlin the 0 direction as it movesunder the crystal.This type of flow has been previously explained in Chapter6. The viscous drag of therotating crystal surface creates an upward swirling flow.The rotation of the crystal causesthe fluid near it to move in the0 direction. The imbalance between the fluids centripetalacceleration (—w2r) and pressure cause itto move outward in the radial direction. Fluidmoves upward below the crystal to balance the outward flowin the radial direction.The fluid moving in the radial direction reaches the cruciblewalls and them drops to thebottom of the crucible to take the place of the fluid that movesupward below the crystal.There is a stagnant region, marked i’S” in Figure9.80, that the fluid rotates around.Chapter 9. Sensitivity Analysis of theFluid Flow ModelCrystalO rpmCrucible 0 rpmReference Vector0.002 cm/s137— — _. —r — — —.-a-—— •••_._‘...\\kttt I-. \ \ .\I Ittt’I-4’‘ 44I ‘f ft ft( ‘I‘__HftftttI I III4,?4?lS4J/4IflI—— F, /I , \ \\\ ‘N\ ‘.\ \ \‘— - -I ‘I \ “ —-.— — — — -I I I*.S % — . * —— .—.asb S*•— — —-: I f.-:1.0 1.5 2.0 2.5 3.0Radial Position (cm)2.52.01.51.00.50.0 —0.0It’0.5Figure 9.75: Vector plot of fluid velocity due to natural convection.Chapter 9. Sensitivity Analysis of the Fluid Flow ModelICrystal 0 rpmCrucible 0 rpm0.02.52.01.51.00.50.0Radial Position (cm)1:380.5 1.0 1.5 2.0 2.5 3.0Figure 9.76: Temperature contours that occur in the LBO/Mo03melt.Chapter 9. Sensitivity Analysis of the Fluid Flow Model 139crystal Rotation 0 rpmCrucible Rotation 0 rpm40.9 Wt.% MoO3Liquid1.501.00B0.50A0 00 70.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)3.000.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)Figure 9.77: Axial velocity due to natural convection.Chapter 9. Sensitivity Analysis of the FluidFlow Model140Crystal Rotation 0 rpmCrucible Rotation 0 rpm40.9 Wt.% MoO3IC 1.00C0.00____________________________I .1 ,,..I.... I0.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)2.50 I2.00IERadial Velocity*i04 (cm/s)Figure 9.78: Radial velocity due to naturalconvection.Chapter 9. Sensitivity Analysis ofthe Fluid Flow Model 141Crystal Rotation 0 rpmCrucible Rotation 0 rpm40.9 Wt.% MoO3:1.50ABo 1.00Ct0.000.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)0.00C ,--- -0.50//\.BA1.001.5-3.0 -1.0Tangential Velocity*i04 (cm/s)Figure 9.79: Tangential crystal surfa.ce velocitydue to natural convection.Chapter 9. Sensitivity Analysisof the Fluid Flow Model142The temperature contours, Figure9.82, are similar to the natural convection temperature contours. The flow velocities a.re not largeenough to affect the shape of thecontours. The flow velocities in the axial directions at liquid heightsof 0.25, 0.5, and0.75 of the height of the fluid are shown in Figure 9.83. The liquid flows upwards at avelocity of approximately 6 x 103cm/s below the crystal and downwardat —3 x iOcm/s near the crucible wall. As with natura.l convection, the largest axia.l flow occursat0.5 of the fluid height and approaches zero 1)0th directly below the crystal and nearthe2.0 cm radial position.The flow velocity in the radia.l direction a,t positions 0.25, 0.5 and 0.74 of the crucibleradius are shown in Figure 9.84. The radial velocity is positive at approximately 4 x i0cm/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 thecrystal is shown if Figure 9.85. The fluid velocities coincide along the tangential linesat 0.5 and 0.75 of the crystal radius reaching 6 x 10 cm/s. The fluid velocity is lowertowards the center of the crystal, reaching a maximum of 3 x iO cm/s. The variationin the magnitude of the tangentia.l velocity with radius is similar to that obtained for thenatural convective flow.9.2.2 Crucible RotationThe 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 theaxial and radial flow is approximately 6 x10—2crn/s, which is two orders of magnitudehigher that the flow that occurs during natural convection. The rotational component ofthe flow is shown in Figure 9.87, with maximum flow near the crucible wall at 14 cm/sand minimum flow, approaching zero below the centre of the crystal. The flow resultingChapter 9. Sensitivity Analysis of the Fluid Flow Model 143Crystal 12 rpm Reference VectorCrucible 0 rpm0.02 cm/s2.5-— — — — ——_— — — — —— S S S S S- - — — — — — — ___r_r_._41’ e.e_ S S ‘. S S442 0 — — — — — — — ... — — .r’-’’’ -...———-.—. -- -‘s i; ; ;;o ;; ; ;;;..v -.I I I I I ‘/— -.- %5%\\ \ 4 I:1.51 ttt‘ ‘‘ t ‘ ‘5’.-, 111144 4444444’•‘• 10-““—/ / / / / /44444‘S\\4\i\\\ .—..I--._ ..— / / - - /4 5— —- ,,, ,0.5- ttt\\\\\.---.-- - .......t t145\ .. \\‘.-.i---.—*— — .— .— .— — — — — — — -.4 t %5. % 5.. — — — .— — — — —. — — — -, .t 1 S S S S ‘. S ‘.5. 5. 5.. 5..-..-..— .— - — - -I I I II I I I0.0 0.5 1.01.5 2.0 2.5 3.0Radial Position (cm)Figure 9.80: Vector plot of fluid velocity due to crystal rotation.Chapter 9. Sensitivity Analysisof the Fluid Flow Model 144Va (cm/s)2.52.744322.515632.0 2.286932.058241.829551.51.600851.372161.14347100.9147740.68608 10.4573870.5.:0.228694o.o I I Crystal 12rpm0.0 0.51.0 1.5 2.02.5 3.0Crucible 0 rpm40.9 Wt% MoO3Radial Position(cm)Figure 9.81: Rotational velocity plot of the LBO/Mo03meltwith crystal rotation.Chapter 9. Sensitivity Analysis ofthe Fluid Flow Model 145ICrystal 12rpmCrucible 0 rpm0.00.02.52.01.51.00.5Radial Position (cm)Figure 9.82: Temperature contours that occur in the LBO/Mo03melt with crysta’ ro0.5 1.0 1.5 2.0 2.5 3.0tation.chapter 9. Sensitivity Analysis of the Fluid Flow Model146crystal Rotation 12 rpmCrucible Rotation 0 rpm40.9 Wt.% MoO32.502.00C1.501.00B0.50A0 00_______________________________________________0.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)o 6.00.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)Figure 9.83: Axial velocity due to crystal rotation.Chapter 9. Sensitivity Analysis of the Fluid Flow ModelCrystal Rotation 12 rpmCrucible Rotation 0 rpm40.9 Wt.% MoO32.50o 1.00c—.4 0.500.000.147A B***3, .1 I,...,,30 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)2.502.00IE0.00-B--4.0 -2.0 0.0 2.0 4.0 6.0Radial Velocity*1 O (cm/s)Figure 9.84: Radial velocity due to crystal rotation.IC..BRadial Position (cm)3Tangential Velocity*10 (cmls)Chapter 9. Sensitivity Analysisof the Fluid Flow Model 148Crystal Rotation 12 rpmCrucible Rotation 0 rpm40.9 Wt.% MoO3I2.502.001.501.000.500.000. 0.50 1.00 1.50 2.00 2.50 3.00Figure 9.85: Tangential crystal surfacevelocity due to crystal rotation.Chapter 9. Sensitivity Analysis of the Fluid FlowModel149from the crucible rotation is similar to that ofcrystal rotation. except for theviscousdrag at the rotating crucible surface which resultsin an downward swirling flow. Thereis a stagnant region, marked “S” in Figure 9.86, around whichthe fluid rotates. Therotational or theta component of the velocity causesthe fluid to swirl as it moves in theradial and axial directions, which is important. The flow predictedby the mathematicalmodel is similar to what was shown by the physical model. Therotational and swirl ofthe fluid as determined using the physical model are shown in Figures6.44 through 6.47in Chapter 6.The high fluid flow velocities resulting from crucible rotation markedly affectthetemperature isotherms in the melt. The calculated isotherms withcrucible rotationare shown in Figures 9.88. These can be compared with the correspondingisothermsfor natural convection shown in Figure 9.76, which are markedly different.With largecrucible rotation, hot liquid moves upward adjacent to the crucible wall resultingin theupward movement of the isotherms in this region. The corresponding downwardflow ofthe cold liquid under the crystal moves the isotherm downward.The flow velocities in the axia.l direction atliciuiclheights of 0.25, 0.5 and 0.75 of theheight of the fluid are shown in Figure 9.89. The liquid flows upward ata velocity of6 x 102cm/s near the edge of the crucible and downward a.t at—6 x i0 cm/s underthe centre of the crystal. As with all the previous cases the largest axial flow occursat0.5 the fluid height. The axial flow approaches zero near the crystal as shownby the 0.75fluid height velocity tra.ce (line C on Figure 9.89) which goes to zeroas the line intersectsthe crystal.The radial velocity at positions 0.25, 0.5 and 0.75 of the crucibleradius are shown inFigure 9.90. The liquid flow directly below the crystal is inwardsat a value of —6 x102cm/s. The flow near the bottom of the crucible is outward and hasa maximum value of6 x102cm/s. The largest radial velocity occurs at 0.5 of the crucible radius.Chapter 9. SensitivityAnalysis of the Fluid Flow Model150The velocity tangentialto the crystal surface at 0.25, 0.5, and0.75 of the radius ofthe crystal is shown if Figure9.91. The fluid velocity decreases with decreasingradius,and changes from —7.6 x10—2cm/s at 0.75 of the crystal radius to —2x102cm/s at0.25 of the crystal radius.9.3 Mesh SizeIn order to determine the influence ofmesh size on the accuracy of the calculationsthe temperature and velocity fields were calculatedusing two node densities, 585 and1795, shown in Figure 9.92. The calculations were conductedfor a crystal rotation of6 rpm and a crucible rotation of —-18 rpm. The axial, radial andtangential velocitiesare examined at 0.5 of the fluid height,0.5 of the crucible radius and 0.5 of the crystalradius respectively. The axial velocity fields for the twoselected mesh sizes are shownin Figure 9.93 and the corresponding radial velocitiesin Figure 9.94. In both cases it isevident that the velocities are thesame for both mesh sizes. The calculated fluid velocitytangential to the crystal surface, Figure9..5. shows a small difference, less than 5 x iOcm/s, between the two mesh sizes. The results indicatethat a mesh density of 585 or1795 nodes ca.n be used in the calculations without introducingerrors.9.4 Fluid ViscosityThe effect of varying the viscosity in the calculations ofthe temperature and velocityfields was examined for melt concentrations of29.7, 36.2 and 40.9 Wt% MoO3. Thecorresponding viscosity of the melts is86, 30 and 12 poise at 730°C. For these calculationsthe crystal and crucible rotations were zero.The axial velocities for the three melts at0.5 of the fluid height are shown in Figure 9.96. Theaxial liquid velocity of the meltwith the lowest viscosity, (40.9Wt% MoO3)is small, approaching 3 x iO cm/s underChapter 9. Sensitivity Analysis of the Fluid Flow ModelCrystalO rpmCrucible 45 rpmReference Vector0.25 cm/s151- —-S— — -:—— # — - —— _— .- — —‘5 —-— —-5 ‘5 ‘5 ‘S—— S — _5_% ‘ ‘-————- ‘-“ ‘S\ \i-.5” \ ‘S ‘S-: - .— ——.5” “ ‘S ‘S ‘S4 1I]J 1/‘S ftft- riftfff\fffFfff.s’s’-w.‘yA’ft. -_wA,çç — -‘‘‘4 4 ‘ ,.5’‘S •‘*.— ‘I I S S S S S S •. %a..st’’— — — — —- - - .5-. .5 - — — -s* -. -. — - — - -. -‘I I . I I0.0 0.5 1.0 1.5 2.0 2.5 3.0Radial Position (cm)2.5 -I2.01.51.00.50.0‘5%‘5’‘5’‘ft‘ft‘ft‘ft‘ft‘ft‘ft‘ftttftftftftftftftftftIfttft4,4,1’iiI’ft.ft Ift.141.‘II.I,4,I,Figure 9.86: \/ector plot of fluid velocity due to crucible rotation.Va (cni/s)13.912612.753211.593810.43449.275078.115686.95635.796914.637533.478152.318761.15938Chapter 9. Sensitivity Analysis of the FluidFlow Model 152I!2.5 -2.0 -1.5 -1.0 -0.5 -0.00.0 0.5 1.0 1.5Ciystal 0 rpm2.0 2.5 3.0Crucible 45 rpmRadial Position (cm)40.9 Wt% MoO3Figure 9.87: Rotational velocity plot of the LBO/Mo03melt with crucible rotation.Chapter 9. Sensitivity Analysis of the FluidFlow Model 153CrystalO rpmCrucible 45 rpm2.5:__—6302.06501.5_7Z”1.00.50.00.0 0.5 1.01.5 2.0 2.5 3.0Radial Position (cm)Figure 9.88: Temperature contoursthat occur in the LBO/Mo03melt with cruciblerotation.Chapter 9. Sensitivity Analysis of the Fluid Flow Model 154Crystal Rotation 0 rpmCrucible Rotation 45 rpm40.9 Wt.% MoO32.50():1.501.000.500.00_____________________________________0.00BAI..I. .1.. .10.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)()*ISC.)C- -0.5-1.02.00 2.50Radial Position (cm)Figure 9.S9: Axial velocity due to crucible rotation.Chapter 9. SensitivityAnalysis of the Fluid Flow Model 155Crystal Rotation 0 rpmCrucible Rotation 45 rpm40.9 Wt.% MoO32.5O0 OC-,__________________________0.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)2.502.00Ao5oRadial Velocity*10 (cm/s)Figure 9.90: Radial velocity due to crucible rotation.Chapter 9. Sensitivity Analysis of the Fluid Flow Model 156Crystal Rotation 0 rpmCrucible Rotation 45 rpm40.9 Wt.% MoO3o 1.000.000.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)000 I - -Tangential Velocity*10 (cmls)Figure 9.91: Tangential crystal surface velocity due to crucible rotation./Chapter 9. Sensitivity Analysisof the Fluid Flow Model 157(a)_________(b)Figure 9.92: Two mesh densities used to examine the modelssensitivity. (a) Regularmesh density, approximately 1795 nodes. (h) Coarsemesh density, approximately 585nodesChapter 9. Sensitivity Analysis ofthe Fluid Flow ModelCrystal Rotation 6 rpmCrucible Rotation -48 rpm40.9 Wt.% MoO32.502.001.50C’)0C*0Ccc158iquidVelocity Sample1.000.500.1I . ,.I . .1 .1 I)0 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)1.00.50.0-0.5-1.0Radial Position (cm)Figure 9.93: Axial velocity for different mesh densities.Chapter 9. Sensitivity Analysis of the Fluid Flow Model 159Crystal Rotation 6 rpmCrucible Rotation -48 rpm40.9 Wt.% MoO3C.)C/)(IDC0.2—4>“-4Radial Position (cm)2.502.00C)O1.50..-(/D1.000.500.00Liquid2.502.001.501.000.500.000. 0.50 1.00 1.50 2.00 2.50 3.00-1.0 -0.5 0.00.5 1.0Radial Velocity*10 (cm/s)Figure 9.94: Radial velocity for different mesh densities.Chapter 9. Sensitivity Analysis of theFluid Flow ModelCryst2l Rotation 6 rpmCrucible Rotation -48 rpm40.9 Wt.% MoO31.000.50(1 flfl0.00CrystalVelocity Sample0.50 1.00 1.50 2.00 2.50 3.00160IRadial Position (cm)Tangential Velocity*10 (cm/s)0.02.50Liquid-1.0 -0.5Figure 9.95: Tangential crystal surface velocity for different mesh densities.Chapter 9. Sensitivity Analysisof the Fluid Flow Model 161the crystal. Increasing the viscosity of the melt from 12 poise (40.9Wt% MoO3)to 30poise (36.2 Wt% MoO3)decreases the magnitude of the axial velocity by afactor of 4.Increasing the viscosity to 86 poise (29.7 Wt% MoO3)effectively reducesthe velocity tozero. The corresponding radial and tangential calculated flow velocities for the viscositiesconsidered are shown in Figure 9.97 and 9.98. For both the radial and tangential velocitiesthe 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 inan LBO/Mo03melt.9.5 ConductivityThe 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. Theconductivities of thecrystal and the melt was assumed to he the same. Calculations werecarried out for a. crystal rotation of 6 rpm and crucible rotation of —12 rpm. The resultingcalculated 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 increasingconductivity. 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 Figures 9.100, 9.101, and 9.102 respectively. The axial velocity (Figure 9.100) is observed toincrease significantly with increasing conductivity. The change in axial velocity is due tothe change in crystal size. The crystal size increases in the axial direction with decreasingconductivity, moving the zero velocity boundary condition at. the crystal/fluid interfacecloser to the axial velocity sample location. The sample velocity will decrease the closerChapter 9. SensitivityAnalysis of the Fluid Flow Model 162crystal Rotation 0 rpmA = 29.7 Wt.% MoO3Crucible Rotation 0 rpmB = 36.2 Wt.% MoO3C = 40.9 WL% MoO31.501.00Velocity Samples0.500 00_______________________________________________0.00 0.50 1.00 1.50 2.00 2.503.00Radial Position (cm)3.00 /\2.0V1.0/*/‘_-X:tzzzzzzzz:/;•_C0.00 0.50 1.00 1.50 2.00 2.503.00Radial Position (cm)Figure 9.96: Axia.1 velocity for different viscosities.Chapter 9. Sensitivity Analysis of the Fluid FlowModelCxystal Rotation 0 rpmCrucible Rotation 0 rpmA=29.7Wt.%Mo03B = 36.2 Wt.% MoO3C = 40.9 Wi% MoO31632.502.00IE0.002.500.008>iws, .1..., .1 .1, I.,,,.)0 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)B AI I . . .-2.0 0.0 2.0 4.0Radial Velocity*i04 (cmls)Figure 9.97: Radial velocity for different viscosities.Chapter 9. SensitivityAnalysis of the FluidFlow Model 164Ciystal Rotation 0 rpmCrucible Rotation 0 rpmA = 29.7 Wt.% MoO3B = 36.2 Wt.% MoO3C=40.9Wt.% MoO3Velocity Samples2.50O 1.00.— 0.500.000.1I:)0 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)-4.0 -3.0 -2.0 -1.0 0.0Tangential Velocity*1 O (cm/s)Figure 9.98: Tangential crystal surface velocity for different viscosities.Chapter 9. SensitivityAnalysis of the Fluid Flow Model165the zero velocity boundary conditiongets to the sample location.The radial velocity, Figure 9.101, and tangential velocity,Figure 9.102. are also observed to increase with increasing conductivity.However, the change in velocityat theselocations is smaller than the change in the axialvelocity. The decrease in the radial andtangential velocity is believed, again, to be due tothe increase in crystal radius.The value of the conductivity has a. significant affect on the thermalfields in the meltand less of an effect in the liquid velocity fields.Chapter 9. Sensitivity Analysis of the Fluid Flow ModelCiystal Rotation 6 rpmCrucible Rotation -12 rpm40.9 WL% MoO3IA=0.04 W/cmKB = 0.05 W/cmKC=0.06W/cmK166I2.502.001.501.000.500.00Radial Position (cm)Temperature (C)Figure 9.99: Axial temperature profiles for different conductivities.Chapter 9. Sensitivity Analysis of the Fluid Flow ModelC)C*>-I4I.—001.000.50n neA =0.04 W/cmKB =0.05W/cmKC=0.06W/cmKVelocity Sample0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)Figure 9.100: Axial velocity for different conductivities.Crystal Rotation 6 rpmCrucible Rotation -12 rpm40.9 Wt.% MoO32.502.001.500.110.01675.00.0-5.0-10.0Radial Position (cm)Chapter 9. Sensitivity Analysis of the Fluid Flow ModelII2.50—Liquid002.502.001.501.000.500.000.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)Radial Velocity*i03 (cm/s)2.001.501.000.50168Crystal Rotation 6 rpmCrucible Rotation -12 rpm40.9 Wt.% MoO3A =0.04W/cmKB =0.05W/cmKC =0.06 W/cmK0.000.Figure 9.101: Radial velocity for different conductivities.Chapter 9. Sensitivity Analysis of the Fluid Flow ModelA=0.04W/cmKB =0.05W/cmKC = 0.06 W/cmKRadial Position (cm)169I0.000.501.001.50Tangential Velocity*i03 (cmls)Crystal Rotation 6 rpmCrucible Rotation -12 rpm40.9 Wt.% MoO32.50o 1.00\_\ \Velocity Sample0.000.00I . .10.50 1.00 1.50 2.00 2.503.00-8.0 -6.0 -4.0 -2.00.0Figure 9.102: Tangential crystal surface velocity for different conductivities.Chapter 10Modeling ResultsThe mathematical model was used to examine the effect of crystaland crucible rotationrates and crucible size on the interface shape andfluid velocity fields. The analysis inChapter 9 indicates that the fluid velocity decreases rapidlywith increasing melt viscosity.To obtain reasonable flow rates from the calculations, the minimummelt viscosity wasselected by using a MoO3concentration of 40.9wt% in the melt, based on the compositionat 698°C in Figure 5.40. The values of the thermopbysica.l propertiesused are given inTable 10.15, the same values used in the sensitivity analysis.A small crucible (6.6 cmdiameter) was used for all but one calculation. The large crucibledimensions were usedin comparing the flow fields in a. large and small crucible with the same crucible rotationrates. The temperature boundary conditions are the same a.s thoseused in the sensitivityanalysis and are given in Figure 10.10:3. The boundary conditionsused with the largecrucible are given in Figure 10.104. The crucible and crystal rotation ratesexamined aregiven in Table 10.16.Crystal rotation influences the shape of the solid/liquid interface. The calculationswith only cryst.a.l rotation were directed towards determimngthe rotation rate whichresults in the interface becoming concave to the liquid. clue to the upward flow ofhot liquidunder the crystal. Crucible rotation also influences the shape of the solid/Iic1uidinterface.Crucible rotation causes the cold liquid under the crysta.l to move downward.Thiswill result in the solid/liquid interface becoming convex to the liquid. The calculationswith only crucible rotation were to determine maximum rotationrate before the crystal170Chapter 10. Modeling Results171Property UnitsValuesSpecific Heat J/g K0.63Density g/cm33.26Viscosity for poise40.9 Wt% MoO3 exp (37.31 — 0.0784x T[C]+ 4.21 x 10 x (T[Cj)2)Conductivity W/cm K0.05Liquidus Celsius698Expansion K—’6 x106Coefficient (3)Table 10.15: Thermophysicai propertiesused in the results analysis.becomes extremely concave to the melt. Modelinga large rotating crucible (8.8 cmdiameter) with a. stationary crystal was to determinethe affect tha.t the crucible sizehas on the flow due to crucible rotation. Combined crysta.land crucible rotation willcreate different flow fields than what occurs with only crucible orcrystal rotation. Thus,combined crystal and crucible rotation rates wereexamined to determine the resultingflow fields that occurred. Both isorotation and counter rotationof the crucible and crystalare examined to determine their affect on the flowfields in the melt.Chapter 10. Modeling Results172Convective HeatTransferT(C) = 102 + 187.8r —23.7 r2ttttt= 2.5 cmCMELTIHIz=O.7cm Iz = 0 ii________________________________________________r=3.3cm r=1.6anr=OcmFixed TemperatureFigure 10.103: Small crucible temperatureboundary conditions usedin the results analysis.Chapter 10. ModelingResultsi 73Convective Heat TransferT(C) =628ttt- z =z=25cm70WIMELTIz = 1.7 anz =0 an701\7ççr=4.4cm r=3.5cznr=OcrFixed TemperatureFigure 10.104: Large crucible temperaturehoundar conditions usedin the resZ:s arysis.Chapter 10. Modeling Results174Mixing Conditions Crucible Size Units ValuesCrystal Rotation and no Crucible RotationCrystal Rotation small rpm 0, 10, 20Crucible Rotation rpm 0Crucible Rotation and no_Crystal_RotationCrystal Rotation small rpm 0Crucible Rotation rpm 0, 20, 40, 60Crystal Rotation large rpm 0Crucible Rotation rpm 60Isorotation of Crystal and CrucibleCrystal Rotation small rpm 10Crucible Rotation rpm 30Counter Rotation of Crystal and CrucibleCrystal Rotation small rpm 10Crucible Rotation rpm —30Table 10.16: Parameters examined for the mathematical model analysis.10.1 Crystal RotationThe effect of varying the crystal rotation rate on the temperature and velocity fields wasexamined. For these calculations the crystal rotation rates were 0, 10 and 20 rpm andthe crucible rotations were zero. The solid/liquid interface and the axial velocities for thethree rotation rates at 0.5 of the fluid height a.re shown ii Figure 10.105(b). The axialliquid velocity of the melt at zero crystal rotation is negative at —3 x i0 cm/s underthe crystal. Increasing the crystal rotation rates caiases the flow direction to change andmove upward under the crystal and down at the side of the crucible. At 10 rpm the fluidvelocity is 1.0 x10—2cm/s below the crystal and —4.0 x i0 cm/s near the cruciblewall. At 20 rpm the fluid velocity is 6.8 x10—2cm/s under the crystal and —2.4 x 10cm/s near the crucible wall. The solid/liquid interface, shown in Figure 10.105(a), movesupward and becomes flat with increasing crysta.l rotation, due to the increase in hot fluidmoving upward below the crystal at higher rotation rates. The model predicts that theChapter 10. Modeling Results175solid/liquid interface becomes concave to the melt at rotation rates larger than20 rpm.The flow velocities in the radial direction at 0.5 of the the crucible radius are shownin Figure 10.106(b). The radial velocities below the crystal are —3 x 10 cm/s, 8 x102cm/s and —4.4 x10—2cm/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 x10—2cm/s and —3.8 x102cm/s for crystal rotation rates of 0, 10 and 20 rpm. The magnitudeof the fluid velocity under the crystal is larger than the velocity near the crucible bottomfor a given rotation rate, the crystal rotation being the dominant mechanism for the fluidmotion.The velocity tangential to the crystal surface at 0.5 of the radius of the crystal isshown if Figure 10.107(b). The fluid velocity tangential to the crystal interface increaseswith increasing crystal rotation. The velocities are 3 x i0 cm/s, 0.9 x10_2cm/s and4.1 x10_2cm/s for crystal rotation rates of 0, 10 and 20 rpm.10.2 Crucible RotationThe thermal and fluid velocity fields were calculated with crucible rotation to determinethe effect of the rotation rate on the shape of the solid/liquid interface and the magnitudeof the fluid velocities. Crucible rotation rates of 0, 20, 40 and 60 rpm, with no crystalrotation, were examined. The axial fluid velocities along a radial line at 0.5 of the fluidheight are shown in Figure 10.108. The axia.l velocity directly under the crystal is nearconstant increasing from —3.2 x102,—2.2 x 10. and —3.3 x 10 cm/s for cruciblerotation rates of 20, 40 and 60 rpm. The smallest change occurs between crucible rotationrates of 0 and 20 rpm. The axial velocity upward near the crucible wall increases withincreasing crucible rotation being 4.4 x10_2,1.2 x 10 and 1.4 x 10 cm/s for rotationrates of 20, 40 and 60 rpm. The largest increase in axial velocity at this location occursChapter 10. Modeling Results 176Crystal Rotation: A =0 rpm, B 10 rpm, C = 20 rpmCrucible Rotation 0 rpm2.502.00B - - - -- Liquid1.50(a)o1 00Velocity Samples0.500.00.1 I.. .1,,.,00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)6.00C 4.0(b)2.00.0A-//N-2.00.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)Figure 10.105: Axial velocities at 0.5 of the fluid height. 40.9 Wt% MoO3 present in thefluid. Crystal rotated at 0, 10, and 20 rpm. Crucible is stationary. (a) Shape of thesolid/liquid interface. (h) Axial Velocities.Chapter 10. ModelingResults 177Crystal Rotation: A =0 rpm, B = 10 rpm,C =20 rpmCrucible Rotation 0 rpm(a)Liquid. 0.50 >0 00 ? 10.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)2.50 IC200B(b)><0.50 ‘I”0.00 I-2.0 0.0 2.0 4.0Radial Velocity* 102(cm/s)Figure 10.106: Radial velocities at 0.5 of the crucibleradius. 40.9 Wt% MoO3 present inthe fluid. Crystal rotated at 0, 10and 20 rpm. Crucible is stationary. (a) Shape of thesolid/liquid interface. (b) Radial velocities.Chapter 10. Modeling ResultsVelocity Samples are tangential0.50to the crystal interface at 1/2 ofthe crystal radius0.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)Figure 10.107: Velocities tangential to the crystal surface 0.5 of the crystal radius. 40.9Wt% MoO3 present in the fluid. Crystal rota.ted at 0, 10 and 20 rpm. Crucible isstationary. (a) Shape of the solid/liquid interface. (b) Tangential velocities.Crystal Rotation: A =0 rpm, B = 10 rpm, C =20 rpmCrucible Rotation 0 rpm1782.502.001.501.00ZLiquidI(a)(b)I. I. I0.0 1.0 2.0 3.0 4.0Tangential Velocity* 102(cm/s)Chapter 10. ModelingResults179between 40 and 60 rpm. The curvatureof the crystal increases with increased cruciblerotation, becoming more concave to the liquid. This is due tomovement of hot fluidupward at the crucible wall and downward belowthe crystal. Increasing the cruciblerotation rate causes the crystal to get smaller maintaining the sameinterface shape. Theradius of the crystal is 2.4, 2.3, 1.6 and 1.1 cm for crucible rotationrates of 0, 20, 40 and60 rpm respectively.The radial velocities along a horizontal line at 0.5 of the crucible radius is showninFigure 10.109. The radial fluid velocities near the bottom of thecrucible are 0.6 x 10,1.7 x10_iand 2.3 x 10 cm/s for crucible rotation rates of 20, 40 and60 rpm. The radialvelocities near the surface are —0.5 x 10, —1.5 x10_iand 2.2 x10_icm/s for cruciblerotations of 20, 40 and 60 rpm. The radial velocities near the crucible bottomare higherthan the velocities near the surface since crucible rotation producesthe fluid motion.The fluid velocity tangential to the crystal surface are shown in Figure 10.110(b).Thetangential velocities are —0.5 x lO’, —1.6 x lO and —2.1 x10_ifor crucible rotationrates of 20 40 and 60 rpm.10.3 Comparison of Crystal to Crucible RotationFluid 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 theaxialvelocities along a radia.l line at. 0.5 of the fluid height. are shownin Figure 10.111. Themagnitude of the a.xia.l fluid velocity below the crvsta.l is largestwith crystal rotationdecreasing from 6.8 x10_2cm/s at the centre line to zero at a radial position of 1.85cm. The flow below the crystal due to crucible rotation is a maximum at 1.2 cmfromthe centre of the crucible, with a. velocity of 4.8 x10_2cm/s. The axial flow velocitynear the crucible wall is 2.4 x102and 4.4 x10_2cm/s for crucible and crystal rotationChapter 10. Modeling Results180Czystal Rotation 0 rpmCrucible Rotation: A =0 rpm, B =20 rpm,C =40 rpm, D =60 rpm250CryStaljii2.001.50(aVelocity Samples0.500.00o.o 0.50 i.bô i.ô ZbÔ iÔ ibôRadial Position (cm)- 1.0C.)/,./\A,7/_--.\o 0.0 ‘I- - -,—•1 7/7*B //(b)-1.0__---/-2.0c,/-3.00.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)Figure 10.108: Axial velocities at 0.5 of the fluid height. 40.9 Wt% MoO3 present in thefluid. Crucible rotated at 0, 2O, 40, and 60 rpm. Crystal is stationary. (a) Shape of thesolid/liquid interface. (h) Axial velocities.Chapter 10. Modeling Results isiCrystal Rotation 0 rpmCrucible Rotation: A =0 rpm, B =20 rpm,C =40 rpm, D =60 rpm2 50•CrystalL___ II:Liquid(a) •C 1.000.500.00I .1,. .1. I..,.0.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)2.50I I2.00B’ A(b)o.soI I-2.0 -1.0 0.0 1.0 2.0Radial Velocity*10 (cmls)Figure 10.109: Radial velocities at 0.5 of the crucible radius. 40.9 Wt% MoO3 presentin the fluid. Crucible rotated at 0, 20, 40, and 60 rpm. Crystal is stationary. (a) Shapeof the solid/liquid interface. (h) Radial velocities.Chapter 10. Modeling Results 182Crystal Rotation 0 rpmCrucible Rotation: A 0 rpm, B =20rpm,C =40 rpm, D =60 rpm250cryStal I2.00zrZ_-Liquid1.50(a)L00Velocity Samples are tangential0.50to the crystal interface at 1/2 ofthe crystal radius.1... .1.... I0.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm). A AA______________________________________--> 0.50D, /UA(b)‘s.’1.00 ..-.---.-.5-..-.’ S.c5.) ..-o -.Cl)1.50-2.0 -1.5 -1.0 -0.5 0.0Tangential Velocity*10 (cm/s)Figure 10.110: Velocities tangential to the crysta.l surface 0.5 of the crystalradius. 40.9Wt% MoO3 present Crucible rotated at 0, 20, 40, and 60 rpm. Crystal isstationary. (a)Shape of the solid/liquid interface. (h) Tangentia.1 velocities.Chapter 10. Modeling Results183respectively. -The radial fluid velocities along a vertical line at 0.5 of the crucible radius are shownin Figure 10.112. The velocities near the crystal are approximately the same value foreach rotation method. The radial velocities being 4.3 x102and 4.7 x102cm/s forcrucible and crystal rotation respectively. The radial fluid velocity near the cruciblebottom are 3.6 x10_2and 5.9 xl0_2cm/s for crystal and crucible rotation respectively.The radial velocity due to crucible rotation is larger near the bottom of the crucibleand near the surface. This is due to the larger diameter of the crucible as compared tothe crystal diameter. The crucible being larger is able to rot.ate fluid at a higher thetavelocity (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 inFigure 10.113. The velocities are 4.0 x102and 5.2 x10_2cm/s for crystal and cruciblerotation respectively. As with the ra.dia.l velocity the larger tangential velocity is due tocrucible rotation.The magnitude of the largest fluid velocity tangential to the crystal interface as afunction of crystal or crucible rotation is shown in Figure 10.114. The tangential velocitydue to crystal rotation with a stationary crucible goes from near zero at no crystal rotationto 4 x10_2at 20 rpm. Higher crystal rotation ra.tes cannot be used due to melt backofthe crystal. The tangential velocity with crucible rotation and a. stationary crysta.l goesfrom near zero at no crucible rotation to approximately 20 x10_2cm/s at 60 rpm. Fora given rotation rate crucible rotation produces a. larger tangential velocity t.han crystalrotation. Much higher tangential velocities are thus attainable possible using cruciblerotation.Chapter 10. Modeling Results 184A: Crystal Rotation =20 rpm, Crucible Rotation =0 rpmB: Crystal Rotation =0 rpm, Crucible Rotation 20 ipm2.50-- LiquidQ2.00• 1.50(a)o1 00Velocity Samples0.500.00.1,., .1., .1,... I0.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)CtD6.0 AQ40 ‘“1.:::))—::—• — — — — — — — — —— ,j//\2.0,:‘0oô.ô i.bô i.ô ibô’ iô ibôRadial Position (cm)Figure 10.111: The magnitude of the axial velocities along a horizontal line at 0.5 ofthe fluid height. The two conditions examined are crucible rotated at 20 rpm with astationary crystal and a stationary crystal with a. crucible rotating at 20 rpm. (a) Shapeof the solid/liquid interface. (h) Axial velocities.Chapter 10. Modeling Results 185A: Crystal Rotation =20 rpm, CrucibleRotation =0 rpmB: Crystal Rotation =0 rpm, CrucibleRotation 20 ipm2.50--Liquid2.00o1.50II ‘majO 1.00—. 0.50>0.000.00 0.50 i.00 i.50 2.002.50Radial Position (cm)150-01.0 2.0 3.04.0 5.0 6.0RadialVeiocity* 102(cm/s)Figure 10.112: The magnitude of the radial velocities along a vertical line at 0.5 of thecrucible radius. The two conditions examined are crucible rotated at 20 rpm with astationary crystal and a stationary crystal with a crucible rotating at 20 rpm. (a) Shapeof the solid/liquid interface. (b) Radial velocities.I(b)186Chapter 10. Modeling ResultsA: Crystal Rotation=20 rpm, CrucibleRotation =0 rpmB: Crystal Rotation=0 rpm, Crucible Rotation20 rpm2.50CrystalA_-__4_I__(a):1.00Velocity Samples aretangential0.50to the crystalinterface at 1/2 ofthe crystal radius0.0c0.Liquid)0 0.50 1.00 1.502.00 2.50Radial Position (cm)3.000.03.0 4.05.0Tangential Velocity102(cm/s)Figure 10.113: The magnitude of the velocities tangential to the crystal surface at 0.5 ofthe crystal radius. The two conditions examined are crucible rotated at 20 rpm with astationary crystal and a stationary crystal with a crucible rotating at 20 rpm. (a) Shapeof the solid/liquid interface. (b) Tangential velocities.Chapter 10. Modeling Results 187I111111III II j— 20.0- -C)15.0-Ici)0Ci— 1 (\ rci) IU.Vc c.I,5.0-,‘ \*-0.11I I I I I I I I10 20 30 40 50 60Rotation Rate (rpm)Figure 10.114: Magnitude of the tangential velocity 0.5 cm from the liquid/crystal interface for different crystal and crucible rotation rates. The calculations are fora rotatingcrucible with a stationary crystal and a stationary crucible and rotating crystal.Chapter 10. Modeling Results18810.4 Comparison of the Flow Fields inSmall and Large CruciblesFluid flow velocities in a small and large crucible rotated a.t 20 rpm, withno crystalrotation are compared. The axial fluid velocities along a radial line at0.5 of the respectivefluid heights are shown in Figure 10.115. The variation of the axial velocity withradius issimilar for both crucibles. The large crucible has fluid velocities that are—5.3 x10_icm/sdownward under the crystal and 2.8 x10_iupward at the wall. These are approximatelydouble the velocities in the small crucible, which are —3.6 x lO cm/s downward belowthe crystal and 1.2 x10_icm/s upward near the wall. The radial velocities along avertica.l line at 0.5 of the crucible radius are shown in Figure 10.116. The radial velocitiesin the large crucible are 4.3 x10_Icm/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 crucibleare approximately half of the large crucible velocities, being 2.4 x10_icm/s outwardnear the bottom of the crucible and —2.1 x10_icm/s inward at the top of the melt.The melt velocity tangential to the crystal interface at 0.5 of the crystal radius is shownin Figure 10.117. The tangential fluid velocity is 5.0 x 10 cm/s and 2.1 x lO cm/sfor the large and small crucible respectively. Fluid velocity is proportional to cruciblesize for crucible rotation driven flows. The larger the crucible that faster the thetafluidvelocity near the outside of the melt. This in turn gives the fluid a largercentripetalacceleration which increases the overall flow velocity of the melt.10.5 Iso and Counter rotation of the Crystal and CrucibleThe axial, radial and tangential flow velocities are calculated for isorotation and counterrotation of the crysta.l and crucible. The rotation rates are +10 and 20 for the crystaland crucible respectively. The tangential velocity is calculated 1 mm from the crystalinterface for crucible rotation of 20 rpm and crystal rotation between —35 and 20 rpm.Chapter 10. Modeling Results 189A: Small crucible, crystal =0 rpm, crucible =60 rpmB: Large crucible, crystal = 0 rpm, crucible =60 rpm0C*>0C>.,-Velocity Samples3.50Crystal -3.00 - - - - -,_________Liquid2.502.00B:1.501.00A0.50°°8oo o.sö i.bo i.ö 2.0020 3..04.oRadial Position (cm)4.02.00.0(a)(b)Radial Position (cm)Figure 10.115: Axial velocities at 0.5 ofthe fluid height. Large a.nd small crucibleshown.40.9 Wt% MoO3 present in the fluid.Crucible rotated at 60 rpm. Crystal isstationary.(a) Shape of the solid/liquid interface.(b) Axial velocities.-2.0-4.0-6.01.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50Chapter 10. Modeling ResultsA: Small crucible, crystal 0 rpm, crucible 60 rpmB: Large crucible, crystal = 0 rpm, crucible =60 rpm1903.00C)C(IDC—..-3.002.502.000.—.—((IDC-1.000.500.00VrF)C.)0V>Radial Position (cm)3.50Crystal2.502.001.501.000.50°•°8o iö i.bLiquid2.00 2.50 3.00 3.50 4.00(a)(b)1.50Radial Velocity*10 (cmls)Figure 10.116: Radial velocities at 0.5 of the crucible radius. Large and smaii crucibleshown. 40.9 Wt% MoO3 present in the fluid. Crucible rotated at 60 rpm. Crys:al istationary. (a) Shape of the solid/liquid interface. (h) Radial velocities.Chapter 10. Modeling Results 191A: Small crucible, crystal 0rpm, crucible =60 rpmB: Large crucible, crystal = 0rpm, crucible =60 rpm3.50Crystal -3.00 --‘-- -- Liquid2.50 a2.00‘B1.50Velocity Samples1.000.50______________________________________‘‘I30 0.50 1.00 1.50 2.002.50 3.00 3.50 4.00Radial 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)0.000.501.001.-5.0 -4.0 -3.0-2.0 -1.0 0.0Chapter 10. Modeling Results192The axial fluid velocity along a vertica.l line at 0.5 of the fluid heightis shown inFigure 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 rotationrespectively. For both rotation cases the fluid velocity is downward under the crystalandupward near the crucible wall. This fluid motion occurs because the crucible rotation rateis higher and dominates the melt flow. For counter rotation (CR) larger fluid velocitiesare obtained than for isorotation (1). The axial velocities due to counter rotation (CR)are —5.2 x102cm/s downward under the crysta.l and 5.0 x102cm/s upward near thecrucible wall. The axial velocities due to isorotat.ion (I) are —3.2 x10_2cm/s downwardunder the crystal and 2.8 x102cm/s upward near the crucible wall. The solid liquidinterface is different for the two different rotation modes. Counter rotation causes thecrysta.l interface to be more convex to the melt and the radius smaller. This is due tothe 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 inFigure 10.119. The radial velocities are higher when the crystal and crucible are counterrotated. The radial velocities due to counter rotation are —5.2 x1O_2cm/s inward underthe crystal and 6.8 x10_2cm/s outward near the crucible bottom. The radial velocitiesdue to isorotation are —3.2 x10_2cm/s inward under the crystal and 4.0 x10_2cm/soutward nea.r the crucible bottom.The melt velocity tangentia.l to the crvsta.l surface at 0.5 of the crystal radius areshown in Figure 10.120. The maximum tangential velocities for iso and counter rotationare —3.5 x10_2cm/s and —5.8 x102cm/s respectively. The tangential fluid velocitydue to counter rotation is 60% higher than the isorotation case. The explanation of thedifference in the fluid velocity due to iso and counter rotation requires the calculation ofthe flow patterns as a function of crystal rotation rate for a fixed crucible rotation. Thefluid velocity at 0.5 of the crystal interface will he calculated for crystal rotation ratesChapter 10.Modeling Results 193A: Crystal Rotation =10 rpm, Crucible Rotation =20 rpmB: Crystal Rotation = -10 rpm, Crucible Rotation =20 rpm6.?joo0.50 1.00 1.50 2.00Radial Position (cm)Figure 10.118: Axial velocities at 0.5 of the fluid height. 40.9 Wt% MoO3 present in thefluid. Crystal rotated at ± 10 rpm and crucible rotated at 20 rpm. (a) Shape of thesolid/liquid interface. (h) Axial velocities.2.502.001.501.00Velocity Samples0.50(a)(b)Ic/)aC)*C)C)0 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)0.000.6.04.02.00.0-2.0-4.0-A2.50 3.00Chapter 10. ModelingResults 194A: Ciystal Rotation = 10 rpm, Crucible Rotation =20 rpmB: Crystal Rotation = -10 rpm, Crucible Rotation =20 rpm2.50(a)0 1.000.500.00 ç0.00 0.50 1.00 1.50 2.00 2.50 3.00Radial Position (cm)2.502.00(b)><0.50 -000’’’’’’-6.0 -4.0 -2.0 0.0 2.0 4.0 6.02Radial Velocity*10 (cm/s)Figure 10.119: Radial velocities at 0.5 of the crucible radius. 40.9 Wt% MoO3 presentin the fluid. Crystal rotated at ± 10 rpm a.nd crucible rotated at 20 rpm. (a) Shapeofthe solid/liquid interface. (b) Radial velocities.Chapter 10. Modeling Results195between —35 and 20 rpm at a cruciblerotation rate of 20 rpm. The tangentialvelocityat 1 mm from the crystal surfaceis used to examine the change in flow patternsand isshown in Figure 10.121. Four points, markedA to D, indicate positions where the flowpattern changes.At point A both the crystal and crucible arerotating at 20 rpm. With both rotatingat the same rate the fluid moves as a solid bodyin the theta (swirl) direction. Thisresult in natural convection being the only forcethat moves the fluid in the radial andaxia.l directions. The tangential fluid velocity is nearzero and is similar to the naturalconvection fluid velocity calculations of Chapter8. Decreasing the crystal rotation rates,moves the tangential velocity to point B, wherecrucible rotation dominates the flow.Fluid now moves U adjacent to the vall and down under thecrystal. The maximumflow in this direction occurs with the crystal rotationat zero, point D. The tangentialfluid velocity 1 mm from thecrvsta.l interface is —2.6 x102cm/s. A vector plot of theflow pattern is shown in Figure 10.122. The size ofthe reference vector in this plot andfor vector plots at the different rotation conditionsis the same, 0.1 cm/s. The flow withcrucible rotation of 20 rpm and crvst.a.l rotation of0 rpm goes up at the wall and downunder the crystal.The change in flow patterns between B and C is different than betweenA and B.Increasing the crystal counter rotation rate resultsin the fluid rotating in a directionopposite to that of the crucible. At pointC the crystal is rotating at a sufficiently highrate, that the centripeta.l acceleration of the fluid under thecrystal is large, causing thefluid to move outward. Thus, instead of thecrystal matching the crucibles rotation rateto reduce the flow, point A, thecrystal must be rotating a.t a sufficiently high rate tocause the fluid under the crystal to become stagnant. The vectorplot of the flow patternsnear point C is shown in Figure 10.123. Directly under thecrystal, marked UC, the flowis stagnant. IVloving from the crystal, the flow becomes significant,flowing up at the wallChapter 10. ModelingResults196and down under the crystal.Increasing the crystal rotation rate, pointC to D, causes the crystal to start todominate the flow patterns. This will result in the flow changing froma stagnant stateunder the crystal, point C, to flowing upward under the crysta.l anddownward at theedge of the crystal, point D. The tangential flow velocity 1 mm from the crystalinterfaceis —3.0 x10_2cm/s at a crystal rotation of -35 rpm and a crucible rotationof 20 rpm,point D. The vector flow pattern at point D is shown in Figure 10.124. The flowhastwo cells. The flow directly under the crystal is upward at the centre of the crucibleandoutward. The flow flow near the crucible bottom is outward and upward at the cruciblewall Both flows join and move outward and upward at the crucible wall and inwardanddownward at a 45 degree to the crucible bottom.Chapter 10. Modeling Results 197A: Crystal Rotation 10 rpm, CrucibleRotation =20 rpmB: Crystal Rotation = -10 rpm, CrucibleRotation =20 rpmRadial Position (cm)Tangential Velocity* 102(cm/s)Figure 10.120: Velocities tangential to the crystal surface at 0.5 of thecrystal radius.40.9 Wt% MoO3 present in the fluid.Crystal rotated at + 10 rpm and crucible rotatedat 20 rpm. (a) Shape of the solid/liquid interface. (b) Tangential velocities.1.501.00Velocity Samples0.50Anno.I(a)I(b))0 0.50 1.00 1.50 2.002.50 3.00-5.0 -4.0 -3.0 -2.0-1.0 0.0Chapter 10. Modeling ResultsCrystal Rotation Rate (rpm)198Figure 10.121: Fluid velocity tangential to the solid/liquid interface as a function ofcrystal rotation rate. The velocity is 1 mm from the crystal/melt interface and at 0.5 ofthe crystal radius. 40.9 Wt% MoO3 present in the fluid. The crucible is rotated at 20rpm.‘‘‘‘‘I’llDC3.02.01.00.0-1.0C.)©*.,-C)04-’HLAI--, I-30 -20 -100 10 20Chapter 10. Modeling Results 199Crystal 0 rpmReference VectorCrucible 20 rpm0.1 cm/s2.5 - .-— __a_ _____..%S ‘— — —.5%’’‘ ,- — — -5—.—.‘5’• ‘S‘ •L2.0 --\ ‘--J•..•,‘I’ htH•tH ttt• j/’’•,,ffftt1.0-• \\_.Ir,W//fffH’V—‘--“‘n’H,:- *“5-,4 q\, I4 4 ‘S\\ \‘ - ,I \ %-. . ._ — — —5’- -. .__-•_-.--..----*--,.--,. * - 1-- - ) •g- — — - •I I I I I I I I I I I I0.0 0.5 1.0 1.5 2.0 2.5 3.0Radial Position (cm)Figure 10.122: Vector plot of fluidvelocity at a crucible rotation rate of 20 rpm andacrystal rotation rate of 0 rpm. Point B.Chapter 10. Modeling Results 200Crystal -23.5 rpm ReferenceVectorCrucible 20 rpm0.1 cm/s2.5 -— — — — — — . ‘‘ , ‘‘a” 2.0-• S .• ;——• : -..-S -S . ;;—C:;;--;1.5 -ttCIDC-p0’i I / / /__.e__‘1A 4HL.U,..tfTjlIts. HH\‘ft1!!‘1 fit‘I.—,. ‘. i\V\\\ ‘% \ ‘t \ _‘*.‘•% \ ••••• * ‘ ‘ -. .. -. ..—... ‘a ‘a-* -a -‘a - -.. ‘a -a — - - -a - - -e -_ * — .+ *—a - -0.0I I I0.0 0.5 1.0 1.5 2.0 2.5 3.0Radial Position (cm)Figure 10.123: Vector plot of fluid velocity at a cruciblerotation rate of 20 rpm and acrystal rotation rate of —23.5 rpm. Near point C.Chapter 10. Modeling Results 201Crystal -35 rpmReference VectorCrucible 20 rpm0.1 cm/s2.5- ..---————— .;—- -—;;_-.S—r.U — f f ,.._.•.‘ ‘ ‘ ‘ ‘-t, .,__)r_,r_—”..I 4i ‘// ‘ - - .-.///_,__•_ S‘--- .•ot i‘‘S-1.5• V .. :0 ..V \ \\ \ ‘IV \ \ \ \\ \‘ t1.0\\ \t t t t t- . - - -- //i /// /1£‘—.‘y/f— “ - ‘ ‘ / ‘(.14 \ \\ “% fitSI..)- I I\ \ \ \ \ \\SS S. ‘S \ S‘ \I-.- .— , —-- -. -a-0.0I I II I0.0 0.51.0 1.5 2.0 2.5 3.0Radial Position (cm)Figure 10.124: Vector plot of fluidvelocity at a crucible rota.tion rateof 20 rpm and a.crystal rotation rate of —35 rpm.Point D.Chapter 11Comparison of Temperature Measurements withModel ResultsIn this section, the calculated temperature fieldsin melts contained in 6.6 cm and 8.8cm diameter crucibles determined from the model arecompared to the correspondingmeasured melt temperatures. The model boundary conditions weredetermined from themelt temperatures with no crucible rotation. The valuesused are given in Figure 11.12.5and Figure 11.126 for the small and large crucible respectively.The melt composition,material properties and crucible rotation ratesused for the calculations are given inTable 11.17.Melt temperatures were measured with two 3 mm outside diameter platinumsheathedthermoconples. The presence of the thermocouples in the rotatingmelt increases mixing and produces three dimensional flow. The axisvmrnetric modelused in this analysiscannot properly calculate this type of fluid flow. However,the increased mixing in themelt can be approximated by setting the theta velocity boundary condition tozero wherethe thermocouples enter the melt. The standard theta velocity boundarycondition foran axisymmetric melt are shown in Figure 11.127. The modifiedtheta velocity boundary conditions are shown in Figure 11.128. Both boundary conditions areused in thecalculations.11.1 Small Crucible (6.6 cm diameter) ResultsThe measured and calculated axial temperature profilesin the melt for the small nonrotating crucible are shown in Figure 11.129.The axia.l temperature profiles at r=1.O,202Chapter 11. Comparison ofTemperature Measurements with Model Results 203Convective Heat Transfer— T(gas) = 750 —>I.SIMIJLAT9D=23 cmCRYSTAL11.0cm23cmMELTIHI-z=0.7cm43.5 cm>IA05cmz= 0115cm /r=3 615 cmC 0.115 cm thicknessz= -0.27 cmr=l6cmz=-0.615cmr=OtoO.54cmFixed TemperatureFigure 11.125: Temperature boundary conditionsused for the small crucible model.Chapter 11. Comparison of Temperature Measurementswith Model Results 204Convective Heat TransferT(gas)75OHttItz=33cmASTALz=2.5cm1.0cmzI-.z=1.7cm 3.3cmH-eI‘1)ry4.4cm_____________________________Cruciblez-0.Il5cm________.:.::.::.:.thicknessQ.H5 czr = 4.5 cmr=3.5cmr=OcmFixed TemperatureFigure 11.126: Temperature boundary conditions used for the large crucib modei..Chapter 11. Comparison of Temperature Measurements with Model Results205theta velocity is unconstrained>tMYUs(theta velocity = 0I‘1)I1i1T’T m QolvariJ.i1I0 I .II II._0C0Io I- II..... ........................... ....... ..,.....,.................... ..........................•.!theta velocity = crucible rotation*2 it rFigure 11.127: Theta velocity boundary conditions used in the model.chapter 11. Comparison of Temperature Measurements with Model Results 206thermocoupletheta velocity isunconstrainedL9D/theta velocity 0I— IC.)MELT IzTiDC.)IIIC.)C.) —C II >> i. II .....:.:.:.:.:.;.:...:.:.:.:.:.:...........1theta velocity = crucible rotation*2 it rFigure 11.128: Theta velocity boundary conditions used in the model to account for thethermocouples in the melt.Chapter 11. Comparisonof Temperature Measurements with ModelResults 207PropertyUnit. small crucible largecrucibleAmount MoO3Wt% 45.5 47.4conductivity (1<) \V/cm K0.05 0.05Specific Heat (Cp) J/g K0.621 0.625density(p) g/cm3 3.14 3.19viscosity(it) poise 3.27 2.82Expansion Coefficient()K 6 x 10 6 x10crystal rotation rpm0 0crucible rotation rpm0, 15, 20, 25, 30 0, 10, 20, 30Table 11.17: Thermophysical propertiesand rotation values used in the modelfor comparison with experimental temperature measurements.1.5, 2.6 and 3.1 are shown. The calculatedtemperatures are within 1°Cof the measuredtemperatures at ra.dia.l locationsof 2.6 and 3.5 cm The calculated temperaturesnear thecentre of the crucible (r = 1.0 an(l1.5 cm) are within 2°C of the measured values.Thegood fit of the calculated and measured temperatureis expected since the experimentalvalues are used to determine the temperatureboundary conditions in the model.11.1.1 Results Assuming no Thermocouple/MeltInteraction (Small Crucible)This section assume that there is no interaction betweenthe melt and the thermocouple.The velocity boundary conditions are shown inFigure 11.127. The experimental andcalculated temperaturesfor a. crucible rotation rate of 15 rpm are shown in Figure11.130.The measured axial temperature profiles nearthe centre of the melt (r = 1.0 and 1.5 cm)are constant near 820°C between z = 0.4cm and 1.0 cm. Above 1.1 cm the temperaturedecreases with increasing height. Thecalculated temperature measurements atthe sameradial locations are within 2°C of the measured values.The measured temperatures at r = 2.6 and :3.1cm radius initially decrease withChapter 11. Compa.rison of TemperatureMeasurements with Model Results208IIE Calculated Measuredr= 1.0 cmr=1.5cmr=2.6cmr=3.lcm -----QI I II I I I . I I •0.0 0.51.0 1.52.0Axial Position (cm)Figure 11.129: Experimental and calculatedtemperatures as a function of axial height.Small (6.6 cm diameter) crucible. Crucible rotation = 0 rpm*Chapter 11. Comparison of Temperature Measurements withModel Results 209increasing z to 1.0 cm then appea.r to be constant. The calculatedresults at r = 2.6cm nearly coincides with three of the temperature measurementsand are higher at 1.0cm with a slight plateau. At r = 3.1 cm the calculated valuesare all higher than themeasured value, with no plateau.Figure 11.131 shows the measured and calculated axial temperatureprofiles at ahigher crucible rotation rate of 20 rpm. Increasing the rotationrate to 20 rpm from15 rpm does not significantly change the temperature distributions.The temperatureprofile at a crucible rotation rate of 25 rpm are shown in Figure 11.132. Themeasuredaxial temperature profiles a.t all locations become flatter and the temperaturedifferencebetween the different axial profiles decreases. The calculated axialprofiles near the centreof the melt (r=1.0 and 1.5 cm) l)ecome flatter and are within 2 to3°C of the measuredvalues. The calculated axial profiles near the cruciblewall (r=2.6 and 3.1 cm) do notchange 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 constantat823°C. The calculated temperature measurements at r 1.0 and 1.5 cm becomes flat at819°C and 820°C respectively. The calculated temperature at r = 2.6 and 3.1 cm do notchange significantly from the 25 rpm case.11.1.2 Results assuming Thermocouple/Melt Interaction (SmallCrucible)The model calculations in this section have the theta velocity boundary condition set tozero where the thermocouples enter the liquid (Figure 11.128). The experimental andcalculated temperature profiles at. 15 rpm are shown in Figure 11.134.The calculated axial temperaturel)ro1leshave a long plateau between z = 0.4 and1.2 cm a.t all radial locations. At. a rotation rate of 20 rpm, Figure11.135, the calculatedresults fit the measured results at z values less than 1.0 cm. At values greater thanthisChapter 11. Comparison of Temperature Measurements with Model Results 210I I I—-.- .—.—.—— 8- 820 - --‘------ .,—- .Calculated Measuredr=1.Ocm Cr1.5cmr=2.6cmr=3.lcm-•-.—.—.- Q800—.. ., •10.0 0.5 1.0 1.5 2.0Axial Position (cm)Figure 11.130: Experimental and calculated temperatures as a function of axial heightat 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________810r CalculatedMeasuredr1.Ocmr=1.5cmr=2.6cm————r3.1cm0800.1I0.0 0.51.0 1.52.0Axial Position(cm)Figure 11.13 1: Experimental and calculatedtemperaturesas a function of axial height atthe radial locations indicated, diameter) crucible. Cruciblerotation = 20 rpm. Standardvelocity boundary conditions are used (Figure 11.127).Chapter 11. Comparison of TemperatureMeasurements with Model Results 212—L. II I— — —0—820----a-810-Calculated Measuredr=1.Ocm Cr=1.Scmr=2.6cm ————r=3.lcm-•-•--•- Q800-i •• I•1 i •0.0 0.51.0 1.5 2.0Axial Position (cm)Figure 11.132: Experimental andcalculated temperatures as a. function ofaxial heightat the radial locations indicated. Small(6.6 cm diameter) crucible. Cruciblerotation =25 rpm. Standard velocity boundaryconditions are used (Figure 11.127).Chapter 11. Comparison of TemperatureMeasurements with Model Results213•L.II II§—820 - -4- - - -E810-Calculated Measuredr= 1.0 cmr=1.Scmr=2.6cm ————r=3.lcm----- Q800—p .I I •I •0.0 0.51.0 1.52.0Axial Position (cm)Figure 11.133: Experimental andcalculated temperatures as a function of axial heightat the radial locations indicated. Small (6.6 cmdiameter) crucible. Crucible rotation =30 rpm. Standard velocity boundaryconditions are used (Figure 11.127).Chapter 11. Comparison of Temperature Measurements with ModelResults 214the measured temperatures decreases faster than the model predictions.The calculatedresults a.t a crucible rotation rate of 25 rpm, Figure 11.136, are similar to the 20 rpmresults. The calculated axial temperature profiles near the crucible wall (r = 2.6 and 3.1cm) are in good agreement with the measured results. Figure 11.137 shows the measuredand calculated axial temperature profiles for 30 rpm crucible rotation. The calculatedresults have a good general fit with the measured values. The fit is not good near thecrucible wall (r =3.1 cm), and the centerline temperature values (r = 1.0 and 1.5 cm) nearz 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) ResultsThe measured and calculated temperatures for the large crucible (8.8 cm diameter) withno crucible rotation are shown in Figure 11.138. The calculated temperatures are in goodagreement with the measured temperatures close to the crucible wall (r greater than 2.8cm). 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 Crucible)Figure 11.139 shows the melt temperatures with the crucible rotating at 10 rpm. Thecalculated 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 cruciblerotation rates of 20 and 30 rpm are shown in Figure 11.140 and 11.141 respectively. Thecalculated results are not in agreement with the measured values with the axial temperature profile at r = 4.2 cm being in closest agreement. The measured and calculatedtemperature profiles a.t r = 0.4 cm behave similarly with with increasing CruCil)le rotation.Chapter 11. Comparison of Temperature Measurements with Model Results215•1 • I— —LE810Calculated Measuredr=1.Ocm Cr=1.5cmr=2.6cm ————r=3.lcm-.-.—.-.- Q800 — . •. I0.0 0.5 1.01.5 2.0Axial Position (cm)Figure 11.134: Experimental and calculated temperatures as a function of axial heightat 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 ModelResults 216I•I I I•ECalculated Measuredr= 1.0 cmr=1.Scmr=2.6cm ————r=3.lcm--•--- Q8 .• I . I . I .0.0 0.5 1.01.5 2.0Axial Position (cm)Figure 11.135: Experimental andcalculated temperatures as a function of axial heightat the radial locations indicated.Small (6.6 cm diameter) crucible. Crucible rotation =20 rpm. Modified velocity boundaryconditions are used. (Figure 11.128).Chapter 11. Comparison of TemperatureMeasurements with Model Results 217• ••—.:. . I • I • • II• IECalculated Measuredr=1.Ocm Cr=1.5cmr=2.6cm ————r=3.lcm----- Q8JJ —. II . . . I .0.0 0.5 1.01.5 2.0Axial Position (cm)Figure 11.136: Experimentaland calculated temperatures as afunction of axial heightat the radial locations indicated.Small (6.6 cm diameter) crucible. Cruciblerotation =25 rpm. Modified velocity boundaryconditions are used. (Figure 11.128).Chapter 11. Comparison ofTemperature Measurements with Model Results 218I I • II82O<<810Calculated Measuredr=1.Ocm Cr=1.5cmr=2.6cm ———— 0r=3.lcm----- Q800-. • . . .I . . . I0.0 0.5 1.01.5 2.0Axial Posifion (cm)Figure 11.137: Experimenta.1 andcalculated temperatures as a function of axial heightat the radial locations indicated.Small (6.6 cm diameter) crucible. Cruciblerotation =30 rpm. Modified velocity boundaryconditions are used. (Figure 11.128).Chapter 11. Comparison of TemperatureMeasurements with Model Results 219Calculated Measuredr=O.4cm Dr=O.9cmAr=2.8cm —r=3.3cmQr=3.8cmII•I • •• •jI830820810Axial Position (cm)Figure 11.138: Experimental and calculatedtemperatures as a function of axiallocationalong various vertical lines. Large(8.8 cm diameter) crucible. Zero crucible rotation.Chapter 11. comparisonof Temperature Measurements with ModelResults220Both the calculated and measuredtemperatures at z = 1.7 cm increase withincreasingcrucible rotation rate. The large difference betweenthe calculated and measured resultsshows that there is more mixing inthe melt than is predicted by the model. The resultsindicate that the thermocouples have a. large effecton mixing in the melt.11.2.2 Results assuming Thermocouple/MeltInteraction (Large Crucible)The calculated and measured axial temperatureprofiles at a crucible rotation rate of10 rpm are shown in Figure 11.142. The calculated axialtemperature profiles at r =2.8, 3.3 and 3.8 cm are approximately fiat and the temperaturedifference between thedifferent a.xia.l profiles is small.The measured temperature profiles at the same locationsare similar to the calculated values except that measuredtemperatures are higher thanthe calculated values. The measured and calculated temperatureprofiles at r = 0.9 cmdiffer from the profiles above 2.8 cm. The calculatedtemperatures are constant at 820°Cbetween z = 0.5 and 2.5 cm. The measured temperatures areapproximately constantat 826°C between z = 0.3 and 1.0 cm. The measured temperatureabove z = 1.0 cmdecreases progressively to 818°C at. z 1.6 cm.The calculated and measured axial temperature profiles at acrucible rotation rateof 20 rpm are shown in Figure 11.113 The difference betweenthe measured axial temperature profiles has decreased with increasing crucible rotation.The calculated axialtemperature profiles are similar to that for10 rpm. The measured axial temperatureprofiles are constant at 828°C for all radia.l positionsexcept r = 0.4 cm. At r = 0.4 cmthe measured temperature is constant. at 828°C betweenz = 0.4 and 1.0 cm. Above z =1.0 cm the temperature decreases progressively to 823°Cat z = 1.6 cm. The calculatedtemperatures are constant, nea.r824°C, for the axia.l temperature profiles at r= 3.8, 3.3and 2.8 cm. The calculated axial temperature profilesat r = 0.4 and 0.9 are constantnear 822°C.Chapter 11. Comparison of Temperature Measurements with Model Results 221Calculated Measuredr=O.4cmDr=O.9cmAr=2.8cmr=3.3cm—-- Qr=3.8cmI I • • .I • ••I83OI::zZ8J’I..Tho 0.5 1.0 1.5 2.0 2.5 3.0Axial Position (cm)Figure 11.139: Experimental and calculated temperatures as a function of axial heightat 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 222Calculated Measuredr=O.4cm Dr=O.9cm Ar= 2.8 cmr=3.3cm‘---“ Qr=3.8cmI I• I II830I:::80%d .Axial Position(cm)Figure 11.140: Experimental and calculated temperatures as a function of axial heightat 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 withModel Results 223CalculatedMeasuredr=O.4cmDr=O.9cmr=2.Scmr=3.3cm‘S--- Qr=3.8cniI I • UU I U II •I,830I:::1 I0.5 1.0 1.5 2.0 2.5 3.0Axial Position (cm)Figure 11.141: Experimental and calculatedtemperatures as a function of axia.l heightat the radia.l locations indicated. Large(8.8 cm diameter) crucible. Crucible rotation =30 rpm. Standard velocity boundary conditionsa.re used (Figure 11.127).Chapter 11. Comparison ofTemperature Measurements with Model Results224Figure 11.144 shows the measuredand calculated temperature profiles at a cruciblerotation rate of 30 rpm. All of themeasured temperature profiles are constant at830°C.The calculated values are constantbetween 822°C and 824°C. The calculated changein temperature profile with crucible rotation rateis similar to the change in measuredtemperature with crucible rotation.The most notable similarity is the decrease in temperature difference between the axial profiles nearthe crucible wall (r = 2.8 to 3.8 cm)with increasing crucible rotation. Another similarityis the slow increase in temperatureat the centre line of the crucible with increasing crucible rotationrate.11.3 Summary of the Temperature ComparisonsThe model can predict the temperatures within2 to 4°C of the measured temperatures ifthe correct boundary conditions and t.hermophysica.l properties areused. The accuracyof the model predictions at higher crucible rotation ra.tes indicatestha.t the calculatedmelt velocity must be near the actual melt velocity. The insertionof the thermocouple inthe melt increases mixing. This is not important when predictingthe melt temperaturesand melt velocity during crysta.l growth since no thermocouple ispresent.Chapter 11. Compa.rison of Temperature Measurements with Model Results225calculated Measuidr=O.4cm Dr=O.9cmr=2.8cmr=3.3cm--“ Qr=3.8cmI II••I,. • I III ••I.830I:::zZE0.5 1.0 1.5 2.0 2.5 3.0Axial Position (cm)Figure 11.142: Experimental and calculated temperatures as a function of axial heightat 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 226Calculated Measuredr=O.4cmDr=O.9cmr=2.8cmr=3.3cm --•“ 0r=3.8cmIII•I I830— ——-:=—r820E810.I I .. I I . I800oo 0.5 1.01.5 2.0 2.53.0Axial Position (cm)Figure 11.143: Experimenta.l and calculated temperatures as a function of axial heightat 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. Comparisonof Temperature Measurements with Model Results 227Calculated Measuredr=O.4cm Dr=O.9cm Ar=2.8cm ————r=3.3cm•—.—.—.-.- Qr=3.8cmIIII•I •-830820-810-800od0.5 1.01.5 2.02.5 3.0AxialPosition(cm)Figure 11.144: Experimental and calculated temperatures asfunction of axial heightat the radial locations indicated. Large (8.8 cmdiameter) crucible. Crucible rotation =30 rpm. Modified velocity boundaryconditions are used. (Figure 11.128).Chapter 12Mass Transfer CalculationsThe mass transfer calculations were conducted to elucidate the movement of MoO3 awayfrom the growing interface. This requires knowledge of the diffusion boundary layerthickness below the crystal, the growth velocity of the interface, the diffusion coefficientand the bulk concentration of MoO:3 in the melt. The boundary layer below a rotatingdisk in an infinite fluid has been determined analytically [45, 46]. The same solution hasbeen 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 thecrysta.l is not shaped like a fla.t disk, the fluid is not infinite and the crystal preferentiallygrows 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 crystalfor 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 thecrystal rotation rate tha.t gives the same flow velocity under the crystal. The equivalentcrystal rotation rate was used with analytical solution for mass transfer below a rotatingcrystal [45] to examine the change in interface concentration with growth time. Twocrystal growth runs (Chapter 13) provi(ledl data for this analysis.22$Chapter 12. Mass TransferCalculations22912.1 Procedure for Estimatingthe Equivalent Crystal Rotation RateThe calculated tangential velocity at0.5 of the crystal radius for an LBO crystal growingin an 8.8 cm diameter crucible rotating at 60rpm were compared with the analyticalsolution to determine a. rotation speed required foran infinite disc to produce the sameflow velocities. Once this was deterniined thecorresponding boundary layer thicknesswas calculated from Equation 12.11 and was assumedto apply to the growth of an LBOcrystal. The radial velocity below a rotating infinitedisk is given by.v;. = Va x w x F(z) (12.9)were4.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 axiallocation is given byz=zx(1x wwere z is the axial location, z is the dimensionless axial location,i is the viscosity,pis the density and w is the angular velocity. The radia.l location value(ra) used in theanalytical solution is equivalent to the surface length between thecentre and 0.5 of themodel crysta.l radius as shown in Figure 12.145.The results of the comparison of the ana.lytica.l solution to thecalculated solutionare given in Figure 12.146. The values of the thermophysicalproperties used for theanalytical and calculated solutions are given in Table12.19. The calculated fluid flowresults are for a crucible rotation of 60 rpm and are seen to be identicalto an infinite diskrotating at 40 rpm. Thus, the analytical solution with a infinitecrystal rotating at 40rpm (w = 4.19) can be used to determine themass transfer that occurs with an 8.8 cmdiameter crucible rotated at 60 rpm. It is assumed that thisratio of crucible to crystalrotation is independent of viscosity and density. as long as thesame values are used inboth calculations.Chapter 12. Mass Transferaiczi1ations230[z’ F(z)70.0 0.00.1 0.04620.2 0.08360.3 0.11330.4 0.13640.5 0.15360.6 0.16600.7 0.17420.8 0.17890.9 0.18071.0 0.1801Table 12.18: Numerical solutions for a rotating disk[46].Property Units Valuesdensity g/crn3 3.26viscosity for poise 21.3140.9 Wt% MOO3Table 12.19: Therrnophvsica.1 properties usedin the Analytical solution for flow below arotating disk.Chapter 12. Mass Transfer Calculations 231Analytical1<>1raCalculatedaFigure 12.145: determination of the radius used in the analytical solution. The radius,ra is equivalent for both the finite elementanalysis and the analytical analysis.Chapter 12. Mass Transfer Calculations 232• • • • • • • • • • • • •10.05-C.?AnalyticalC]Calculated C]0•10.05 0.10 0.15 0.20Velocity (cm/s)Figure 12.146: Calculated and analtical velocity values. The calculated tangential velocity values are at at 0.5 ofthe crystal radius. The analytica.lsolution (flow past a rotatingdisk) for the radial velocity is at a. radiallocation that is equivalent to the surface lengthof the calculated solution.Chapter 12. Mass TransferCalculations23312.2 Mass Transfer Behaviorof MoO3 below the CrystalThe concentration of MoO3 at the growing interfacecan he calculated [45]. The equationrelating the bulk concentration of MoO3 with theconcentration at the interface is:C0= CLX exp()(12.10)were C0 is the concentration of MoO3 at the growinginterface,CL is the concentrationof MoO3 in the bulk,•fis the growth velocity at the interface, 6 is thethickness ofthe diffusion boundary layer andD is the diffusion coefficient. It is assumed that themovement of the interface(.1)is the only flow within the diffusion boundarylayer. It isalso assumed that 6 is constant between the centreand edge of the crystal. The diffusionboundary layer thickness (6) for an infinite rotating disk is given as:/\1/66 = 1.6D113(12.11)were D is the diffusion coefficient of solute in the meltin cm2/s, t is the viscosity in poise(g/cm s),p isthe density in g/cm3 and is the rotationrate in rads/s. The growthvelocity of the LBO crystal a.nd the Mo03/LBOdiffusion coefficient is determined bycombiningEciua.tion12.11 and 12.10. The resulting equation is:/ / \1/6\3/2(fx1.6x{) \D = II (12.12)x in())All of the variables in the equation are readily available with the exceptionof the concentration of MoO3 next t.o the interface (C0). This value was determinedby growinga crysta.l a.t a. sufficiently high rate to cause eutectic formation. Whenthis occurs theconcentration of MoO3 next to the interface is61.5 Wt% MoO3 (Figure 5.37).A crystal growth experiment were conducted in which the crystal was slowcooledfor a short time then pulled at a rate of 1.66 mm/day. Eutectic growthoccurred uponChapter 12. MassTransfer Calculations234initiating the pulling sequence.Figures 13.155 and 13.156 show the interfacebreakdownthat occurred when the crystalpulling started. The upper portion ofthe crystal is clearand free of MoO3,the lowerportion has a eutectic structureof MoO3 rods in an LBOmatrix. The growth rate isa combination of the coolingrate of the furnace and thepull rate of the crystal. The growthrate due to programmed coolingof 2.4°C/day wascalculated to be 0.7 mm/day. This wasdetermined by dividing thefinal height of thecrystal by the total growth time for a.crystal grown only by melt cooling.Thus thetota.l growth rate(f)was 2.63 mm/day at eut.ectic formation.The rotation rate of thecrucible was 60 rpm which correspondsto a disk rotation rateof 40 rpm. Table 12.20gives the values of the variables used indetermining the diffusion coefficient.The bulkconcentration(CL) is assumed to be the initial theoretical concentration ofthe melt(44.7 Wt% MoO3). The viscosity was previouslymeasured (Figure 5.39). Thedensitywas calculated using weight fractions of theLBO and MOO3 with their respective densityvalues.The diffusion coefficient of MoO3 in an L130 melt wasestimated from Equation 12.12as 2.42 x108crn2/s. The same calculations were carriedout assuming the pull rateswithin +50% of 1.66 mm/day for comparisonreasons. The diffusion coefficients are3.73 x108and 1.24 x108cm2/s for pull rates of 2.49 and0.83 mm/day respectively.The variation in diffusion coefficientis minimal for the given changes in pullrate. Table 12.21 gives the values of diffusion coefficientsin other liquid systems. The diffusioncoefficients of liquid metals and liquid oxidesare of the order of iO and iOrespectively. The Mo03/LBO diffusion coefficientis lower than the otheroxide material diffusion coefficients. This ma be dueto the high temperatures of the oxidesgiven inTable 12.21.C) C) CriC’,CJD, CC)CCt’.)CI.CID C,C,,C, !“ C, CI) C Ct, C,, CI) C, -J C,C, C, C C C, IC,‘V CCt,J.XXXX.-NC-) CI)—,-•4Cfl<CC-)*CC,,Clc)Nr:-’C,C+—_.,Ct, C-CC,)c C),-J-IldlCCC-)C,,C-CC—C.)CI) Ct,-C•)C)—Ca)Chapter 12. Mass TransferCalculations23612.3 Maximum GrowthRates and Growth Times for LBOThe MoO3 concentration in thebulk of the melt and at the interface willchange asthe crystal grows. This is due to the rejectionof MoO3 by the solidifying crystalduringgrowth which increases the melt concentration.Equation 12.10 has been used to calculatethe concentrations at different times duringthe growth of the crystal. It was assumedthat the system wa.s at steady stat.e atthe slow growth rates used and tha.t thediffusionboundary layer thickness does not changeas the crystal grows. The crystal interfaceisassumed to be a cone growing into the melt andthat the ratio of radius to crystalheightis constant. The volume of the crystal is:Vctai= X Tr2wereVtai is the volume of the LBO crystal, ii. is the height of the crystal and risthe radius of the crystal. The relation ofthe radius to height was determinedfrom acrystal growth experiment where the interface wasquenched during the initial stages ofits growth (Figure 12.147). The relationship betweenthe radius and height. is:-=2.36The increase in crystal height withtime is assumed to be a function of the growth velocity:11= .1x Lwere t is the growth time,h is the crvsta.l height. and is the crysta.l radius. The volumeof the growing LBO crystal at anytime (luring the growth isv1 =(.rx x (2.36))The corresponding weight of the LBOcrystal is:=XPLBOChapter 12. MassTransfer Calculations237wereWxai is the weight of the LBO crystal andPLB0is the density of LBO(2.474 g/cm3).The bulk MoO3 concentrationof the fluid ca.n he calculated knowing theamount of LBOthat has been removed in growingthe crystal.c—_______________________________________‘L—l47tM0Q3+ (TI7tLBO — Wtxtai)wereCLis the bulk concentration of MOO3 iflthe melt, Wt03 is the initial chargeweight of MoO3 in the melt,T47tLBOis the initia.l charge weight of LBOin the melt andWtxtai is the weight of LBO in the crystal. Equation 12.10 canbe used to calculate theinterface concentration. The viscosityand density are assumed to changewith the bulkMoO3 concentration. The melt is assumedto be at 720°C for all of the calculationsandthe variation in viscosity withMoO3 concentration is taken from Figure.5.40.= erp (9.818 — Wt1700,x 0.169)p = Wi%jBo x 2.47 + lVt%1003 x 4.69The interface concentration was calculated forthree separate cases. The normalgrowth rate of 0.70 mm/day (2.4°C/day),a growth rate 50% higher than the normalvalue (1.05 mm/day, 3.75°C/day) anda. growth rate 50% lower than the normalvalue(0.35 mm/day, 1.2°C/day). The interface concentrationsas a function of time are givenin Figure 12.148. Table 12.22 gives the variation ininterface concentration with time forthe normal growth rate.The interface concentration at the normal growthrate (0.7 mm/day) increases ata constant rate from 48.9 Wt%MoO3 a.t da 0 to 49.39 Wt% MOO3 Oilday 10. Theinterface concentration continues toincrease until it reaches the eutectic concentration(61.5 Wt% MoO3)on day 27. Increasingthe growth rate by50% (1.05 mm/day) causesthe initial interface concentration to increaseto 51.3% MoO3. The rate of increa.seofthe interface concentration is larger and theeutectic concentration is reached after17.5Chapter 12. Mass TransferCalculations238days. Decreasing the growthrate by 50% causes the initial interfaceconcentration to be46.7%. At the slow growth ra.tes the eutecticis reached well after 30 days of growth. Thesecalculations are very approxi matedue t.o the number of assumptions that were employed.However, the trend in the rate of changein interface concentration with growth timecanbe used to determine the growthtime. Growth times should be limited to27 days orless. This will avoid growing in the regionwhere the increase in concentration withtimeis large.There are two factors tha.t have not. been accountedfor that will alter these predictions. Evaporation of MOO3 occursat the surface of the melt which reduces thebulkMoO3 concentration and in turn the interface concentration.Under these conditions itis possible to grow the crystal for a longertime due to the decreasing bulk MoO3concentration. The surface of the crystal is assumedto he flat. Any faceting of the crystalinterface, which is a. real possibility, villcrea.te stagnant areas and increase theMoO3concentration at the interface.Chapter 12. Mass Transfer Calculations 239Figure 12.147: Cross section of crystal grown at a crucible rotation rate of 60 rpm. Theradius of the crystal is approximately 2 cm and the height at the centre line is 0.85 cm.LBO CrystalFrozenMo03/LBOFlux LayerChapter 12. Mass Transfer CalculationsC240Figure 12.148: The concentration of MoO3 next to the growing interface as a function oftime for the growth rates(f)indicated for a crucible rotation of 60 rpm.65605550Time (days)30Jhapter 12. Mass TransferCalculations241‘lime Weight of Weight of Weight of ViscosityDensity C0(days) the LBO LBO in MoO3 in (poise)(g/brn3) (cm)Crysta.l the Melt the Melt(grams) (grams) (grams)012345689101112131415161718192021222324252627280.000.000.040.130.310.611.061.682.513.584.916.538.4810.7813.4716.5620.1024.1128.6233.6639.2645.4452.2559.7067.8476.6786.2596..59107.72256.33256.33256.29256.20256.02255.72255.27254.65253.82252.75251.42249.80247.85245.55242.86239.77236.23232.22227.71222.67217.07210.89204.08196.63188.49179.66170.08159.74148.6144.5744.5744.5744.5844.6044.6244.6744.7344.8144.9145.0445.204.5.4045.6345.9046.2246..5947.0247.5148.0648.7049.4250.2451.1752.2353.4254.7$56.3358.109.809.809.799.789.759.709.639.539.409.249.048.808.518.197.827.416.966.475.965.424.874.313.753.212.682.191J41.340.993.463.463.463.463.463.463.463.473.473.473.473.483.483.493.493.503.513.523.533.543.5.53.573.593.613.633.663.693.723.760.002570.002570.002570.002570.002570.002570.002560.002560.002550.002540.002530.002520.002510.002490.002470.002450.002420.002390.002360.002320.002280.002230.002180.002120.002050.001980.00 1910 .00 1820.0017348.9348.9348.9348.9448.9648.9849.0349.0849.1649.2649.3949.5449.7349.9550.2250.5250.8851.2951.7652.2952.9053.5954.3855.2756.2857.4258.7160.1961.87Table 12.22: The concentration of MoO3 next to the growinginterface as a function oftime. Diffusion coefficient is 2.38 x i0 cm2/s. Growthra.te(f)is 0.698 mm/day.Chapter 13Application of Process Engineering Principlesto Crystal GrowthThis section describes the Lithium Triborat.e crystal growthruns that were conductedwith parameters based on results from the mathematicalmodel, the physical model andexperimental measurements of temperature and viscositydescribed in Chapters 10,6.5 and 7 respectively. The experimental measurements and mathematicalmodel haveshown that crucible/crvstal rotation. cruciblesize and MoO3 content in the melt are theparameters that have the greatest influence onfluid flow below the crystal.In particular, the mathematical model established that cruciblerotation producesthe maximum fluid flow across the growing solid/liquid interface.Crystal rotation wasnot used in conjunction with crucible rotation since it would decreasethe fluid flow atthe growing crystal/liquid interface (Figure 10.121). The cruciblerotation rates testedwere 18.5, 30 and 60 rpm. The initial rotationrate (18.5 rpm) was similar to the crystalrotation used in the preliminary experiments (Section 5.3).An 8.8 cm diameter cruciblewas used for most of the crystal growth runs sincemelt velocity due to crucible rotationincreases with crucible size. A6.6 cm diameter crucible had to be used for the first twocrystal 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 increasingMoO3 content.Parameters for the mass transfer model were estimatedfrom the results of crystalgrowth runs LBO 19 and LBO 20 and wereused t.o elucidate the change in interfaceconcentration with growth time.The mass transfer analysis wa.s conducted for crystalsgrowing in the [001] direction. It was determinedthat the interface concentration initially242Chapter 13. Application of ProcessEngineering Principles to Crystal Growth243increases at a slow rate followedby a.n exponential increase. It was alsofound that thecrystal growth velocity in the[001] direction had to he slower thanthe minimum pullspeed of the crystal growth apparatus(1.66 mm/day) due to eutectic formationat thispull speed.Certain aspects of crystal growth werenot modeled or studied priorto the crystal growth runs. Two effects thatare likely to influence crysta.lgrowth that were notconsidered are:1. The influence of crysta.l orientationon the build up MoO3 at thegrowing interface and inclusion /eutecti c formation.Different crystal growth orientationscouldproduce faceting which would resultin higher inclusion formation/interfacebreakdown. The preferred growthdirection used in this investigation was[001] giventhat devices had to he fabricated tlia.t wereup to 1.5 cm in the [100] direction.Thefirst two runs were carried out in the [312] directionuntil enough seed material wasproduced in the preferred direction.2. The effect or growth atmosphere oncrystal quality was not investigated. Thefurnace atmosphere was changed fromair to dry nitrogen once it wasdeterminedthat LBO decomposed in the presenceof water vapour [6].Nine crystal growth runs were conductedduring this investigation (LBO 17 — LBO25). The complete list of the parametersused in crystal growth runs are givenin Tables 13.23 and 13.24. LBO 16 was completedbefore the initiation of this investigation(LBO 16) and is listed in Tables 13.23 and13.24 as a reference for comparison.13.1 LBO 17The first crysta.l (LBO 17) wasgrown using the 6.6 cm diameter cruciblein air. Thecrystal was grown in the [312] direction,the melt cooling rate for growthwas 2.4°C/day:Chapter 13. Applicationof Process Engineering Principles to Crystal Growth244which should yield a growthrate of 0.7 mm/day; the crucible rotation was18.5 rpmand the crystal was stationary. Theaim of this experiment was to determineif cruciblerotation and a higher MoO3 content wouldproduce a better crystal. The previousrun(LBO 16) was conducted with crystal rotationof 15 rpm, no crucible rotation and a 34.4Wt% MoO3.During growth the growing crystal/liquid interfaceat the surface of the liquid wasstrongly faceted. The crystal wa.s grownuntil its edges reached the outside of the crucible.After growth, the crysta.l was ra.ise(I fromthe melt before cooling to room temperatureat 10°C/hour. Unfortunately, the distance raised wasinsufficient, which resulted in itfreezing into the melt.Figure 13.149 confirms that LBO 17 froze intothe melt. The upper surface of thecrystal is a. white translucent color. Large cracks,marked A, run along the diagonals ofthe crystal. The edge marked B is due to a pieceof the crystal falling off while it wascooling to room temperature. Strong facetingis present at the edges of the crystal thatcorrespond to the liquid/solid interface(C). The rounded edge of the crystal (markedD) is due to the crystal growing close to the side ofthe crucible. The uncracked piecesof LBO 17 are shown in Figure 13.150. All of thepieces of the crystals were inclusionfree in the region that corresponded to the uppersurface (marked E on some of thecrystal pieces). The lower areas of the crystalshattered into a white powder when thesolid/liquid interface froze (marked F on some ofthe crystal pieces).Despite these problems there was a. significant increasein crysta.1 quality and yieldsize compared to preliminary growth experiments.Previously the average inclusion freeregion was approximately 3 x 3 x3 mm3 (Figure 5.41). LBO 17 was predominantlyinclusion free at the outer edgesof the crystal. and average size of the individual piecesof LBO crysta.l were 15 x 5 x 5 mm3.The increasedcr sta.l quality can be attributed tothe improved mixing in the melt due to crucible rotation,the higher MoO3 content (44.5Chapter 13. Application of ProcessEngineering Principles to Crystal Growth 245versus 34.3 Wt% MoO3)and theabsence of interface breakdown.13.2 LBO 18LBO 18 was grown in the [3T2] direction as per the previousrun. The melt coolingrate for growth was the same as in the previousrun (2.4°C/day). Two of the growthparameters, crucible rotation and crvsta.l pulling, were changed.The crucible rotationrate was increased to 30 rpm to improve mixing in themelt and the crysta.1 was pulleda.t 1.66 mm/day to increase its thickness. The mat.bematica.l modelpredicts that themagnitude of the tangential velocity beneath the crystal shouldincrease from 5 x10_2cm/s at 15 rpm to 10.5 x102cm/s at 30 rpm (Figure 10.114). Thepull rate of thecrystal was the minimum speed of the motor. The growth procedure consistedof slowcooling the furnace at 2.4°C/day until the crystal had grown toapproximately 2 cm indiameter. The crystal was then pulled for6 days at 1.66 mm/day to increase its axialthickness. After pulling, the furnace was slow cooled a.t 2.4°C/dayfor the remainder ofgrowth. 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 crystalhas two distinctareas, the upper portion (A), clue to a pull rate of 1.66 mm/day, anda lower portion (B)where only siow cooling was applied to the melt/crystal.The crystal grew along specificplanes. Major cracks were present in the crystal, running fromthe outer points of thecrystal inward to the centre. The bottomof the crystal was either clean (C) or coveredwith IVIoO3 flux (D) that wa.s white/green in color. The regionscovered with flux wereseverely cracked. The surface of the clear regions were predominantlyuncracked, howevercracks from the flux covered region propagated into the clear regions.The crystal wasfree of IVIoO3 inclusions with the exception of two locations,at the bottom surface ofthe crystal (D) and at the location where pulling was terminated (Betweenareas A a.ndChapter 13. Application of Process Engineering Principles to crystalGrowth246B). This would indicate that growth alongthe [312] direction could he as highas 1.66mm/day. Differences in growth rate andfaceting with orientation are expectedsince theLBO crystal structure is orthorombic andthe faceted shape has been previously shown(Figure 2.6). The crystal was moderately fragileand broke into pieces after a limitedforce was applied to it.The individual pieces of flux free portions of the crystal(A in Figure 13.151) are shownin Figure 13.153. The uncracked a.nd inclusion freepiece is approximately 20 x 15 x 6mm3.The decrease in inclusion density (increase incrystal quality) of LBO 17 and 18 canbe attributed to three factors; using crucible rotation to increasethe fluid velocity next tothe growing interface, increasing the MOO3 contentto decrease the viscosity (which alsoincreases the fluid velocity), and effects tha,t arepossibly due to the growth direction.Pulling the crystal at 1.66 mm/day did not result in interface breakdownwhich suggeststhat growth in the [312] direction can be large.13.3 LBO 19LBO 19 was grown in the [001] direction, which is the previousgrowth direction, in thelarge crucible (8.8 cm diameter). The crystal wasslow cooled until its diameter wasapproximately 6 cm after which it was also pulledat a rate of 1.66 mm/day. The crystalis shown in Figure 13.154. The seed is marked A, the slow cooledportion of the crystalis marked B, the slow cooled/pulled at 1.66 mm/day area of thecrystal is marked C andD. The area marked E is where a piece of the crystalhad broken off. The crystal wasvery fragile and broke easily. The lower areaof the crystal which grew while it was beingslow pulled (C and D) was a white/green color indicativeof interface breakdown. Thegrowth rate was sufficiently fast that thediameter of the crystal decreased to a value of1 cm (D).Chapter 13. Applicationof Process Engineering Principles to Crystal Growth247The polished cross section ofthe crystal is shown in Figure 13.155.The crossescorrespond to areas where X—Ray(hifraction was conducted to determine thecrystalorientation. The analysis was conductedat the seed, before the interface breakdownand at two points inside the white/greenportion of the crystal. Allof the X—Raydiffraction patterns were identical, andrevealed that the crystal consisted ofan LBOmatrix. SEM photos were taken in the areawere the interface break down occurred(AreaC on Figure 13.155). Wavelength dispersive spectroscopy(WDX) was also carried out atthe same locations to qualitatively determineif MoO3 wa.s present. Figure 13.156(a)isan SEM photo of the region whereinterface breakdown commenced. Thereare long rodlike inclusions in the LBO matrix, and the growthdirection is parallel to the rods. Thecorresponding molybdenum map of thesame area, Figure 13.156(h), shows that therodsconsist of molybdenum. A backscatter image andWDX image of an individual inclusionis shown in Figure 13.157. Unlike the previousrun, LBO 19 formed eutectic materialwhen the crystal was pulled at 1.66 mm/day. Twoparameters were changed relative togrowth of LBO 18; LBO iS was grownalong the [3T2] in a. small crucible while LBO19 was grown along the [001] directionin a. large crucible. Increasing the crUcil)le sizehas been demonstrated to increase the flowvelocity below the crystal, thus the eutecticformation in LBO 19 cannot be attributed t.othe change in crucible size. The occurrenceof the eutectic in LBO 19 must be due t.o the growthdirection, indicating that thatthe [001] direction readily forms eut.ecticat a. pull speed of 1.66 mm/day for cruciblerotations of 30 rpm or s’ower.13.4 LBO 20LBO 20 was grown with a. faster rotation rate(60 rpm) to improve the removal of MoO3from the interface. An increase in cruciblerotation from 30 to 60 rpm increases theChapter 13. Application ofProcess Engineering Principles to C’rystalGrowth 248magnitude of the tangential velocitybeneath the crystal from 10.5 x10—2cm/s to 21 x10_2cm/s (10.114). The crystals werecycled through a slow cooling/pulling period for1 dayand a slow cooling period of 1 day tofacilitate diffusion of MoO3 ahead ofthe interfaceand to prevent the rapid reductionin crysta.1 (liameter. The crystal, Figure13.158, is5 cm in diameter. The crystal was very fragileand broke into several pieces,as shownin the figure, when it was handled.The top of the crystal was covered witha thinwhite powder. The cross section ofthe crystal is shown in Figure 13.159.The areamarked A is part of the crystal that hadbroken off. The cross section of thecrystal,was similar to LBO 19 in tha.tthe upper (slow cooled) portion was clear andthe lowerportion had the eutectic structure. Thecrysta.l (liameter was reduced to approximately4 cm with the pulling/cooling cycle.The pulling mechanism on the crystal growthstation had a minimum pull rate of1.66 mm/day with no diameter controlmechanism.The previous two crystal growth runs, L13019 and 20, in conjunction with the masstransfer calculations, indicate that a pullrate of 1.66 mm/day will produce interfacebreakdown for [001] direction crystal growthfor a 8.8 cm diameter crucible rotated at60rpm. Larger crucible rotation rates werenot investigated since 60 rpm was close to themaximum rotation rate of the apparatus.13.5 LBO 21LBO 21 was grown using slow cooling without crystalpulling and with a dry N2 atmosphere. The dry N2 was used to prevent the uppersurface of the LBO from decomposingat an elevated temperature in the presence of water vapour[6]. Unfortunately LBO 21was slow cooled until a power outagefroze the crystal into the melt. The crystal, Figure 13.160, was severely cracked due to itbeing frozen into the melt. The white/greencolored material (A) is the LBO/fiux that has beenfrozen onto the outside of the crystalChapter 13. Applicationof Process Engineering Principles to CrystalGrowth 249due to the fast quench. The crystal was removedin this area to show the surfaceof thegrowing interface. Two type of surfaces are presentat the growing interface, rough andsmooth depending on the orientation of the grow’ingsurface. The crystal surface (B)is clear due to the dry N2 atmosphere. A portionof the surface of the crystal (C) isclear at the upper surface. The dry N2 prevented theupper surface of the crystal fromdecomposing is in the previous runs.13.6 LBO 22 and 23The subsequent crystal growth run, LBO 22, was conductedunder identical growthconditions as LBO 21. Unfortunately this run wasunsuccessful due to failure of themotor and the crystal accidentally froze with the melt during thepost growth coolingto room temperature. LBO 23 was grown usingthe identical conditions as LBO 21.Figure 13.161 and 13.162 shows the top and bottom of thecrystal respectively. Thebottom of the crystal has clear regions in additionto the portions that are covered withthe MoO3flux. The upper surface is clear with the crysta.l changingto a white color closeto the bottom of the crystal where the fluxis present. The white color corresponded toareas where the crystal shattered due to differencesin the thermal expansion coefficientsof MoO3 and LBO. The clea.r portion of crysta.l that grewa.s a flat surface was the 101plane. The flux covered portions grew as a. numberof irregula.r planes in a shingle likeconfiguration. A photo and schematic of the cross sectionof a plane that grew a.s a roughsurface is shown in Figure 13.163. The region marked B is werethe MoO3 builds up.The rough surface crea.tes stagnant areas wereMoO3 can accumulate. The areas of thecrystal where the flux was attached to the bottomsurface were fragile and broke apartwith moderate to little force. The area.s of thecrystal that were free of flux, Figure 13.164,were crack resistant. The largest. cr sta.l size was approximately20 x 10 x S mm3. Thechapter 13. Applicationof Process Engineering Principles to Crystal Growth250previous three crystal growthruns(LBO 21, 22 and 23) demonstrated that the 101planegrows as a fia.t surface while the other planes arefaceted. The facets result in a buildup of MoO3 at the solid/liquid interface which sticksto the surface of the crystal afterit has been separated from the melt. These crystalgrowth runs also confirm that LBOcrystals can be grown under these conditions without interface breakdownwhich is inaccordance with model predictions in Figure 12.148.13.7 LBO 24LBO 24 growth conditions were identical to LBO 21. The post growthcooling procedurewas modified to reduce the thermal gradients in thecrystal. Insulating bricks wereplaced over the top hole and separation of the crystal from the meltwas carried out bylowering the crucible rather than raising the crystal. was loweredinstead of the raisingcrystal to separate it from the melt. Measurements reportedin Chapter 5 suggest thatthe axial and radial temperature gradients in the crystalcan he reduced by movingthe crystal t.o the centre of the furnace and insulating the tophole while the crystal iscooling to room temperature. This procedure also ensured tha.tthe crystal was as closeas possible to the centre of the furnace elements thus reducing the thermal gradientin thecrystal. 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 previouscrystal growth runsthe crystal had a flux build up on the rough surfaces of the growing interface (markedAin Figure 13.166). The 101 family of planes (marked B in Figure13.166) were flat andthe surface was free of flux. Areas adjacent to the flux coveredsurface were cracked toa fine white powder. The crystal, while still havingmacro cracks, did not break when amoderate force was applied to it. Thus reducing the thermalgra.dient in the crystal didresult in improved strength. The largest crystal area that was uncracked betweentheChapter 13. Application ofProcess Engineering Principles to Grystal Growth251Growth Date CompositionSeed Crucible Rotation RateRun OrientationDiameter Crucible Crystal(mm/dd/yy) (Wt% MoO3) (cm)(rpm) (rpm)LBO 16 5/21/92 30.0 iion seeded6.6 0 6LBO 17 8/6/92 44.5 3126.6 18.5 0LBO 18 10/5/92 44.53 1 2 6.6 30.0 0LBO 19 11/26/92 44.50 0 1 8.8 30.0 0LBO 20 01/8/93 44.5 00 1 8.8 60.0 0LBO 21 03/24/93 44.5 0 01 8.8 60.0 0LBO 22 06/9/93 44.5 0 01 8.8 60.0 0LBO 23 07/21/93 44.5 00 1 8.8 60.0 0LBO 24 8/9/93 44.5 00 1 8.8 60.0 0LBO 25 12/7/93 44.5 00 1 8.8 60.0 0Table 13.23: Growth conditions used for the crystalgrowth experiments.top and growing interface was 20 x 10 x 5 mm3.The increased strengthof the crystal isa result of the lower thermal gradients (luring coolingto room temperature.13.8 LBO 25LBO 25 was grown to determine if the build upof flux at the interface was due to thelength of the growth runs. Growth conditionsidentical to LBO 24 were used exceptthat the crystal was grown for only 14 days. There was aMoO3 build up at the roughinterface. Figure 13.168 shows a portion of the crystal with theMoO3 build up. Thewhite regions correspond to flux attached to thecrevices between the growing planes. Asin the previous crystal growth runs, the 101 familyof planes grew as a flat surface with110 MOO3 build up. The presence of the MoO3at this small crystal size indicates thatflux build up occurs during all sta.ges of crystalgrowth.Chapter 13. Application of Process Engineering Principles to Crystal Growth 252Figure 13.149: LBO 17 crystal frozen in the melt.Chapter 13. Application of Process Engineering Principles to Ciysta.l Growth 253Figure 13.150: Uncracked portions of LBO 17 crystal.Chapter 13. Application of Process Engineering Principles to Crystal Growth 251Figure 13.151: Top view of LBO IS crystal.Chapter 13. Application of Process Engineering Principles to Crystal Growth 255Figure 13.152: Bottom view of LBO 18 crystal.Chapter 13. Application of Process Engineering Principles to Crystal Growth2.56JfIj IIIIIII1,1•11•3. I. 4S’anIesSs_IIFigure 13.153: Uncracked portions of L130 iS crystal.Ohapter 13. Application of Process Engineering Principles to cirystal Growth 257Figure 13.151: Side view of LBO 19 crystalChapter 13. Application of Process Engineering Principles to Grystal GrowthicIl,258Figure 13.155: Cross section view of LBO 19 crystal. The crosses areregions were sampleorientation was determined.LBO19Chapter 13. Application of Process Engineering Principles to Ci’vstal Growth 259Figure 13.156: Area in LBO 19 were interface breakdown/eutectic growth started. Magnified 20 times. (a) SEM photo. (b) Map of molybdenum concentration. The brightregions correspond to a high molybdenum concentration.Chapter 13. Application ofProcess Engineering Principles to Crvsta.l GrowthI,260Figure 13.157: View of an niolybdenum inc’usionmagnified 200 times. (a) SEM photo.(b) Map of molybdenum conccntraton.The bright regions correspond to a highmolybdenum concentration.aChapter 13. Applicationof Process Engineering Principles to CrystalGrowth261Growth GrowthII ate Growth Ti me AtmosphereRun cooling pulling cooling pulling(°C/day) (mm/day)(days) (days) (da.ys)LBO 16 3.8 1.66 223 AirLBO 17 2.4 027 0 AirLBO 18 2.4 0 296 AirLBO 19 2.4 1.6616 12 AirLBO 20 2.4 1.6630 5.9 AirLBO 21 2.4 017 0 N2LB022 2.4 0 210 N2LBO 23 2.4 0 270 N2LB024 2.4 0 32 0 N2LB025 2.4 09 0 N2Table 13.24: Growth conditions used for the crystal growthexperiments, continued.Chapter 13. Application of Process Engineering Principles to Crystai Growth 262Figure 13.158: Top view of LBO 20 crystalChapter 13. Application of Process EngineeringPrinciples to Ciysta1 Growth 263LBO20Figure 13.159: Cross section view of LBO 20crystal.Chapter 13. Application of Process Engineering Principles to Crystal Growth 264Smooth SurfaceFrozen MoO3/LBOFigure 13.160: LBO 21 Crystal.Flux LayerIChapter 13. Application of Process Engineering Principles toCistal Growth 265Figure 13.161: Top view of LBO 23 crystal.Chapter 13. Application of Process Engineering Principles to Grystal Growth 266Figure 13. I 62: Bottom view of LBO 23 crystal.Chapter 13. Application of ProcessEngineering Principles to ‘Cr stal Giowtii 267——BLiquidI.I/I.: CrystalIII.•IIII••••••••••••• .......••••. .. ._MoO3Build UpBFigure 13.163: Interface appearance for planes were MOO3was stuck to the surface. (a)Photo of underside of crystal. The area A is a. region of IVIoO3 buildup. (b) Schematicof surface along line B — B-Chapter 13. Application of Process Engineering Principles to Crystal GrowthFigure 13.164: Pieces of TJBO 23 that were iiiicracked.268ICthapter 13. Application ofProcess Engineering Principles to Crystal Growth 269Figure 13.l6Ti: Top view of LIlO 24 crystal.Chapter 13. Application of Process Engineering Principles to Crystal Growth 270Figure 13.166: Bottom view of LBO 24 crystal.Chapter 13. Application of Process Engineering Principles to Ciystal Growth 271Figure 13.167: flack lit view of LBO 24 Crystal.Ctha.pter 13. Application of Process EngineeringPrinciples to Crystal Growth 272• e• I• II I• a• a• aa a• a• a• I• I• I• I• aa aI •• aI I• II Ia. I• I• I• I• I• aI II Ia aI Ia a• aa a• a• II I• I• a• I• aa a• II II •a,• a:a1bMoO3ry:talI I• II IFigure 13.16$: Schematic and photo of the MoO3 onthe LBO 25 crystal.Chapter 14Summary and ConclusionsA study was undertaken to grow Lithium Triborate crystals by theTop Seeded SolutionGrowth (Czochralski) method. Lithium Triborate cannot begrown directly since itforms a.s a result of a peritectic transformation at834 ± 4°C. The addition of a MoO3flux modifies the phase diagram and allows LBO to be grown (lirectlya.s a solid from theliquid. The flux addition increases the complexity of the system,since the MoO3 that isrejected during growth builds up ahead of the growing interface which causesinterfacebreakdown. Poor mixing in the melt, due to the high viscosity ofB203.increases thepossibili tv of interface breakdown.The objective of this research wa.s to establish reasons for interface breakdownand tooptimize the growth process to produce large crystals of high optical quality.Mathematical and physical models have been employed to provide a. quantitative understandingofheat transfer, fluid flow and mass transfer during the growth process. Theseanalyseshave been complemented by experimenta.l measurements that were necessaryto quantifythermophysical properties and boundary conditions.The primary reason for interfacebreakdown is eutect.ic formation at the growing interface.Although the initial composition of the melt is different from the entectic, a,ccunmlation of 1\1003due t.o poor mixingallows the eutectic composition to be reached during growth rates.A summary of thekey findings a.re presented below:1. The phase diagram for the LTIO/Mo03system has been determined in the cornposition range of 44 and 74 weight percent MoO3.The liquidusof the LBO phase273Chapter 14. Summary andConclusions274decreases from 682°C to 619°C between 44.S and60 weight percent MoO3. Theeutectic 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 withASTM standard C936-S1 near 700°C. The viscosity of the melt containing 29.7and40.9 weight percent MoO3 wa.s found to be 234 and 21 poise respectively.The highviscosity retards mixing and inhibits mass transfer of the MoO3 flux awayfromthe growing interface. It has been found that interface breakdown occurs whenthecomposition of the liquid at the solid/liquid interface reaches the eutectic.The viscosity decreases with the decreasing liquidus temperature a.s shown in Figure5.40.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) andlarge (8.8 cm diameter) crucible as a function of crucible rotation. A simulatedcrystal fabricated from platinum sheet. was used create the type of mixing that occurred when an LBO crystal was present.. At zero crucible rotation, axial and radialgradients were present in the melt. With crucible rotation the average temperature of the melt increased. At a crucible rota.tion of 25 rpm the melt temperaturebecame more uniform in the axial and radial direction. In 1)0th crucihles, the difference in temperature of the melt. with and without crucible rotation(30 rpm) wassmall being approximately 8°C to 10°C. However, the shape of the axial and radialtemperature profiles changed significantly with crucible rotation.4. The physica.l model examined the mixing due to crucible rotation in a high viscositysolution. A large (8.8 cm diameter) plexiglass crucible, plexiglass crystal 5.6 cm indiameter, glycerine solution and blue glycerine dye were used. The dye was usedas a tracer to determine the flow patterns. It has been clearly demonstrated thatChapter 14. Summary andConclusions275the flow and mixing is significantfor crucibie rotations 45 rpm and higher.Nearthe crystal the fluid moves in a semicirculararc toward the centre line of themelt.The axial component of the flow increases andeventually dominates as thefluidmoves towards the centre of the crystal.This causes the fluid to move towardsthe bottom of the crucible. The fluid atthe centre line moves toward the bottomof the crUcil)le at which point it moves outward andthen upward at the cruciblewall. The fluid rotates in the theta direction whileit moves in the axial and radialdirections. The theta swirl is zero atthe centre line of the melt and a maximum atthe crucible walls. The fluid shears as it moves in thecrucible. This causes the dyeto mix into the bulk solution. The dye is completelydispersed in the melt after lessthan 5 minutes. Thus mixing of the melt due to cruciblerotation is significant.5. A mathematical model was used to determine the characteristicsof fluid flow due tonatural convection, crucible rotation,crystal rotation and crucible size. The modelcalculations without crystal and crucible rotationshown that the fluid velocitydue to natural convection is very small, approximately10 cm/s at 40.9 wt%MoO3(12 poise at 730°C). Decreasing the MoO3 contentincreases the viscosityof the solution which decreases the flow’ velocitiesin the melt. The flow due tonatural convection with 29.7 wt% MoO3 is al)proximately 10—6cm/s(86 poiseat 730°C). Thus the melt is essentially stagnantat low concentrations of MoO3.Inclusion formation and polyciystalline growthwould be predominant under theseconditions.Forced convection due to crystal a.nd cruciblerotation increases the amount ofmixing in the melt. The average flow velocity is approximately5 x10—2cm/s with20 rpm crystal rotation and no crucible rotation.The flow velocity is the same orderof magnitude for a crucible rotating at 20 rpm witha stationary crystal. HoweverChapter 14. Summaryand Conclusions276the direction of flow is reversed.With crystal rotation the fluidflows upwardunder the crystal and down a.t.the crucible walls. Crucible rotationcauses the fluidto move up at the crucible wallsand down under the crystal.Crucible rotationalso produces higher tangential flowvelocities at the crystal interfacethan crystalrotation. The fluid velocity tangentialto the crystal/liquid iHterface is a maximumat the outer crystal diameter. Thefluid velocity tangential to the crystal interfaceat 1/4 of the crystal radius is approximatelyone half of the tangential velocity at3/4 of the crystal radius.The model calculations show thatthe crystal interface will become concaveto theliquid and eventually melt back at highercrystal rotation rates (greaterthan 20rpm for the given thermal conditions).On the other hand, crucible rotation causesthe crystal interface to become convexto the melt because the direction ofthe flowis reversed. Higher crucible rotation ratescan be used to increase the flow velocityin the melt. The maximum calculated flow velocityat a crucible rotation of 60rpm is approximately 2 x10_Icm/s. The maximum calculatedfluid velocity at acrystal rotation rate of 20 rpm is 4 x102cm/s.The flow velocity due to cruciblerotation is dependent on the crucible size.Theaverage tangential flow velocity at0.5 of the crystal radius is —2 x10—1cm/s and5 x10—1cm/s for the 6.6 cm andS.S cm diameter crucibles rotating at 60 rpm.Fluid velocity is proportional to cruciblesize for crucible rotation driven flows. Thelarger the crucible the higher isthe rotational component of fluid velocitynear theoutside of the melt. This in turngives the fluid a larger centripetal accelerationwhich increases the overallmagnitude of the velocities of the melt.The effect of counter and isorotationof the crystal and crucible on fluid flow havebeen shown. Crystal and crucible rotationcause the fluid to move in oppositeChapter 14. Summaryand Conclusions277directions. Thus theycompete with each other in determiningthe final fluid motionand velocity. A rotatingcrucible with a stationary crystal willresult in a higherfluid velocity than a. rotatingcrysta.l with a stationary crucible.Introducing crystalrotation at a. given crucible rotationwill result in decreasing the averageflow rate.Increasing the crystal rotation toequal tha.t of the crucible will resultin the fluidbecoming stagnant. If thecrystal and crucible are rotatingin the same directionat the same rate all of the fluidis stagnant. If the crystal andcrucible are rotatingin opposite directions at the same rate. thenthe area under the crystal is stagnanta.nd there is some motion in the fluiddue to the shearing motion.If the crystalrotation is larger than the cruciblerotation then flow inducedby crystal rotationwill dominate.6. The model calculations were compared to temperaturemeasurements made in arotating crucible with a stationary simulatedcrystal. Measurements in thesmall(6.6 cm diameter) and large ($.S cm diameter)crucibles were employed for thecomparison. The model predicts temperaturesto within 2 to 4°C of the measuredtemperatures. The accnracof the model predictons at higher cruciblerotationrates indicates tha.t the calculatedmelt velocity must be similar to the actual meltvelocities.7. Mass transfer calculations havebeen conducted for an LBO crystal.conical inshape, growing in the [001] directionto determine the growth time requiredto reachthe eutectic concentrationat the solid liquid interface. It was determinedthat atthe norma.l growth rate (2.4°C/day0.7 mm/day) the interface concentrationreaches the LBO—Mo03eutectica.fter 27 days. The calculations indicatethatthe interface concentration increasesslowly at the start of growth andacceleratesat the end of growth. At a. growthvelocity of 0.35 mm/day results the eutecticChapter 14. Summary and Conclusions278concentration is not reached withinthe 30 day time period examined.An interfacegrowth velocity of 1.05 mm/dayresults in the interface concentration reachingtheeutectic in 10 days.8. The gradients in the crystal were measured todetermine the influence of varyingamounts of insulation. It wa.s determinedthat independent of cruciblesize, thethermal gradients in the crystal aremost sensitive to the addition of insulationoverthe seed hole in the furnace and the verticalposition of the crystal. Movingthesimulated crystal towards the centreof the furnace reduces the axial gradientsandthe insulated top reduces the ra.dia.l gradients.Using a small crucible for crystalgrowth reduces 1)0th the axial and radialgradients provided that the crystalissurrounded by the walls of the crucibleduring the post growth coolingstage.9. Crystals were grown with knowledge gained fromthe modeling and experimentalmeasurements. Employing crucible rotationrates of 60 rpm resulted in an increa.sein single crystal yield from 3 x3 x 3 mm3 to 20 x 10 x 8 mm3.Crystal growth ratesin excess of 1.66 mm/day were possible for growthin the [312] direction. The samegrowth rates caused interface breakdownwhen the crystal wa.s grownin the [001]direction. Growth in the [001] directioncaused faceting to occur at all locationsof the solid/liquid interface with exceptionof the 101 plane. The 101 plane grewas a flat surface and was free of MOO3flux. The crevices between the facetstrapthe MOO3 flux which sticks t.o the surfaceof the crystal after it. has been separatedfrom the melt. The MoO3 in theseregions ca.used cracking of the crystal becauseof the mismatch in therma.l expansion coefficientsbetween MOO3 and LBO.In conclusion the results of the presentinvestigation shows tha.t good qualitycrystalsof limited size can h)e grown froma. melt, of LBO containing MoO3 as aflux. The majorfa.ctor limiting the growth ra.teand crysta.l size is the concentrationof MoO3 at theChapter 14. Summary and Conclusions279advancing interface. SinceMOO3 IS not soluble inLBO, all of the MoO3 rejectedbythe solid is accumulated aheadof the interface. The MoO3moves into the liquidbydiffusion through a thinboundary layer adjacent to the interfaceand mixes in the bulkmelt. When the concentrationof MoO3at the interface reaches theeutectic concentrationof 61.5 wt% starting with a. melt concentrationof 44.5 wt%, eutectic willform, whichproduces phase regions richin MoO3,effectively terminatinggrowth of a good qualitycrystal. Accordingly, larger crystalscan be grown if the thicknessof the diffusion layeris decreased and the remainingMoO3 is distributed uniformlyin the liquid.Higher flow rates in the meltbelow the interface, and thinnerdiffusion boundarylayers are obtained with higher cruciblerotation rates. Highercrystal rotation ratescannot be used since they are limitedby remelting of the crystal.Buoyancy force flow isvery small, due to the high viscosityof the melt. Reducing the concentrationof MOO3in the melt would result in a decreasein the fluid flow in the melt due to anincreasein the viscosity. This would also resultin a higher MoO3 concentrationat the growinginterface due to the lower fluidvelocity. The size and qualityof LBO crystals whichcan be grown from the melt by thetop seeded solution growth (ModifiedCzochralski)process is limited by the nature of thematerials and process. Growth mustbe slow andgrowth conditions selected to movethe flux away from the interfaceas rapidly as possibleprimarily by crucible rotation.Some crysta.l orientations grow more readilythan othersdue to faceting, which affects the localmovement of the flux awayfrom the interface.The grown crystals can crack readilyunder therma.l stresses which requiresthe crystalsto be cooled in a. uniform thermal environment.Chapter 15Recommendations for Future WorkSevera.1 more sets of crystal grow’thruns should he carried out.In addition the masstransfer calculations should beimproved to more accurately predict theprocess.1. The size and quality of the L130 crystalsis dependent on the orientationof thecrystal. Crystals of [100], [010] and[101] directions should be grown toassesshow significant the orientationis in obtaining higher quality crystals.In particularthe extent of faceting of the interface,as a function of crystal orientationand thetemperature gradient in the melt,should be determined. The relationshipbetweenfaceting and the onset of theformation of MoO3 inclusions shouldhe established.On the basis of this information procedures shouldhe developed to reduce faceting.if faceting is found to instigate the formationof MoO3 inclusions.2. After growth the crysta.l nmst he cooledslowly in a. uniform thermal environmentto prevent cracking. This must be donein the crystal growth furnace.Accordinglythe furnace design should be modifiedto enable the crystal to cool at the centerofthe furnace where the thermal gradientsare low. This would require modificationof the lower portion of the furnaceto allow the crucible t.o be lowered outof thearea where the crystal will be cooledto room temperature.3. The model should be extendedto include important aspect.s of growth which werenot considered. During growth MoO3evaporates from the top surfaceof the melt..changing the melt composition. Thesurface of the solid liquid interfacein the280chapter 15. Recommendations for Future Work281present model is assumed to be smooth. In factit can he faceted which influencesthe model results and the onset of eutectic growth.The affect of faceting shouldbe incorporated into future mathematical models.4. The possibility of replacing the MOO3 flux with anothermaterial which producesmelts with much lower viscosities should be investigated.The flux would have tohe 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 CeramicSociety, 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 CrystalGrowth, 128 (1993) 933-937[7] B. Sastry and F. Hummel. Journal of the AmericanCeramic 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 andC. You, Scientia Sinica B, Vol. 28, No. 3, 235-243[12] A. Jiang, F. Cheng,Q.Lin, Z. Cheng and Y. Zheng, Journalof Crystal Growth, 79(1986) 963-969282Bibliography283[13] R..S. Feigelson, R.J. Raymakersand R.K. Route, Journal ofCrystal Growth, 97(1989) 352-366[14] I. Barin, 0. Knacke, and 0. hubaschewskiThermophysical Properties of InorganicSubstances, Springer-Verlag. Ben iii (1977).[15] D.D. Wagman et al., Selected Values of ThermodynamicProperties, National Bnreauof Standards Series 270, U.S. Department ofCommerce, Washington, (1968-1971).[16] Y.S. Touloukian, Series Editor, Thermophy.sicaiProperties of Matter, the TPRCdata series, lET/Plenum, NewYork, (1970-).[17] Cleveland Crystais, Inc. BBO and LBO InformationSheet, 19306 Redwood Avenue,Cleveland, Ohio 44110[18] Fujian Castech Crystals, Inc. LBO InformationSheet, 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 fromHigh-Temperature Solutions, Academic 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) 413-426Bibliography284[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 andEngineering, 12 (1977) 145-152[30] W. E. Langlois Journal of Crystal Growth. 42(1977) 386-399[31] N. Kobayashi and T. Arizumi, Journal ofCrystal 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 CrystalGrowth, 32 (1976) 89-94[37] V. Nikolov, K. Iliev and P. Peshev, Journal ofCrystal 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 GrystalGrowth, 97 (1989) 99-115[40] M. Mihelcic, C. Schroeck-Pauli. K. Wingerath, H. Wenzel,W. Uelhoff and A. VanDer Hart, Journal of Crystal Growth,53 (1981) 337-354[41] M. Mihelcic, C. Schroeck-Pauli, K. Wingerath,H. Wenzel, \‘V. Uelhoff and A. VanDer Hart, Journal of Crystal Growth,57 (1982) 300-317[42] H.J. Scheel Journal of Cry.stal Growth , 13/14(1972) 560-565Bibliography285[43] E.O. Schulz-Dubois Journal ofCrystal 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 Journalof 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, NewYork[50] Hewlett Packard, Practical Tern peralure Measurements ,Application Note 290[51] Chris Parfeniuk, M.A.Sc. Thesi.s , The University of BritishColumbia (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 MassTransfer, 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,WileyInterscience, New YorkAppendix AEstimation of the Thermal Conductivityand the Gas TemperatureThe accuracy of the mathematical modelresults depend on the how wellthe mathematica.l expressions describe the process being modeled,if the assumptions used to simplifythe model are correct, whether thecorrect values of the thermophysical propertiesareused and if the correct boundary conditionsare used. In thePresent case, the fluid flowcalculations are more reliable sincethe liquid being considered ha.sa high viscosity. Thethermophysical properties of the systemtha.t are required for the mathematicalmodelinclude, specific heat, density and thecoefficient of thermal expansion. Valuesfor theseproperties at high temperatures areavailable in the literature, a.s listedin Chapter 2.Viscosity values of the LBO/Mo03melts are required which are not available. Thesehave been determined inthe present investigation (Chapter 5). The lowtemperaturetherma.l conductivity of LBO is availablein the literature. High temperature values arenot available, and in the present caseare estimated as described below. Thefixed temperature boundary conditions at the crucible wallhave been estimated using temperaturemeasurements in the rrielt,Chapter 7. The ambient gas temperature abovethe melt willbe evaluated using the melt temperaturessho’’ii in Figure 7.52.The thermophysical properties of themelt are assumed to he the sum of the weightpercent average of the constituent properties.Cp(J/gK)= CpLBQx Wt% LBO+ CpA!00.x Wt% MoO3The mathematical model is appliedto a melt having a. charge concentration of5OWt%MoO3.The value of the melt viscosityis obtained by extrapolating the viscosity—concentration286Appendix A. Estimation of the ThermalConductivity and theGas Temperature 287Property Unit VahieCp J/g K 0.63p g/cm3 3.26i’ poise 2.29a’ K 6 x10_6Table A.25: Thermophysical propertiesused in the model to determine the conductivityof the melt and the ambient ga.stemperature.curves given in Figure 5.39. The averagemelt is taken to be 820°C. The viscosityasa function of concentration at 820°C extrapolatedto .50 Wt.% is 2.29 poise.The meltproperties used in the calculations are givenin Table A.25.A.1 High Temperature Thermal ConductivityEvaluationThe high temperature conductivity ofthe LBO/Mo03solution is approximatedusingthe temperature measurements givenin Figure 7..52 and the boundary conditions givenin Figure 7.53. The liquid is modeled withfixed temperatures boundary conditionsatthe l)ott.orn, side and top of themelt, Figure A.169. Model calculations with differentconductivities are compared withthe experimental results at distancesof 0.5 cm, 1.0cm, 1.5 cm and 2.0 cm fromthe bottom of crucible at both0 and 1.6 cm from thecenter of the crucible. Theconductivity that gives model resufts with the best fit ofthe experimental data is assumed to bethe correct value. The experimental results andmodel calculations are comparedlw the sum of the difference in the temperature valuesat the different locations.Difference =(Texpei.iientai — Tmodel)The model calculations using a conductivityof 0.1 W/cm K are examine. The fluid speedfrom natural convection, FigureA.170, is less than 0.02 cm/s due to thehigh viscosityAppendix A. Estimation of the Thermal conductivityand the GasTemperature 288Conductivity Difference(NV/cm K) (Celsius)1.0 19.80.1 13.60.075 10.30.05 4.70.025 -17.20.01 -95.4Table A.26: Difference between the model and experimentaltemperature values as afunction of the conductivity.of the melt. Thus, conduction is the dominant modeof heat tra.nsfer in the melt. Theapproximation of the conductivity is easier due tono natural convection occurringin themelt.Table A.26 and Figure A.171 gives the difference between the experimentalresults andmodel calculations for different conductivity values.An LBO/Mo03conductivity valueof 0.05 W/cm K used in the model givesthe best fit. to the experimental temperaturevalues. This is very close to the conductivity of LBOin the c direction, 0.039 W/cm K,as given in Table 2.2. It is assumed that the conductivit valueof the LBO/Mo03meltis 0.05 W/crn K a.nd it does not change with the stoichiornetryof the melt.A.2 Ambient Gas Temperature ApproximationThe ambient gas temperatures at the top of the meltare approximated by determiningwhich values give the best fit to the experimental(lata. Doing this allows the model tohave the best possible fit for thethermal field in the melt. The thermophysical propertiesused are given in Table A.25. The value of theconductivity is 0.05 as determined in theprevious section and the value of the convectiveheat transfer coefficient used is 0.006Appendix A. Estimation of the Thermal Conductivity and the Gas Temperature 289Fixed Temperaturer=32cmr=1.6an r=O.Ocmz = 2.5 cmMELTIIIz=O.7anjj...I__Iz=O czi__________________________________r3.2cmr=1.6crnr=O.OcmFixed TemperatureFigure A.169: Temperature boundary conditions used to approximate the conductivityof the LBO/Mo03melt.Appendix A. Estimation of the Thermal Conductivity and the Gas Temperature 290SPEED- CONTOUR PLOTLEGEND—— O3E—(2-— ()14E1J—— 0. I000E—Oi—— U.1182E—Ol—— O.1364E—O1—— O.1546E—O1—— O.1728E—O1MINIMUM0. 00000E+O0MAXIMUM0. 18187E—O1\\J_4i-”\NCentre\CrucibleFigure A.170: Fluid speed (u+ u)2in the melt. The value of the conductivity usedin the model is OJ W/cm K.Appendix A. Estimation of the Thermal Conductivityand the Gas Temperature 2912O110-0--10--20-I I0.00 0.25 0.50 0.75 1.00Conductivity (W/cm K)Figure A.171: Difference between the model and experimental temperature values asafunction of the conductivity.Appendix A. Estimation of the Therma.1Conductivity and theGas Temperature292W/cm2 K. The best fit for the ambientga.s temperature was foundto be:T = 200+ 1S7.Sx r 23.7 x r2(A.13)Where r is in centimeters and is the radialposition that the convective boundaryis incrucible and T is the ambient gas temperaturein degrees celsius. Figure A.172 showsthe measured temperature values andthe model calculations using thebest fit ambientgas temperature.Appendix A. Estimation of the Thermal Conductivityand the Gas Temperature293IIIliii!860-s•%ASSSS840- SSSSAc.)SSS‘82OSSASSSSS8O0lI-SAE_________SS780- Measured ModelCIr=O.Ocm C760- r=1.6cm AC0.0 0.5 1.01.5 2.0 2.5Axial Position(cm)Figure A.172: Comparison with the experimentaltemperature data and themodel resultsfor the best ambient gas temperaturevalues.

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Ashburn 3 0
Washington 2 0
Tokyo 1 0
Sunnyvale 1 0

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