University of Southern California Dissertations and Theses

Convection irregularities during solution crystal growth and relation to crystal-defect formation.

(USC Thesis Other)

Convection irregularities during solution crystal growth and relation to crystal-defect formation.

PDF

Download

Open document

Flip pages

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content CONVECTION IRREGULARITIES DURING SOLUTION CRYSTAL GROWTH AND RELATION TO CRYSTAL- DEFECT FORMATION by Pei-Shiun Chen A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Chemical Engineering) September 1977 UMI Number: DP22093 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Dissertation Publ *-h *n g UMI DP22093 Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 U N IV E R S IT Y O F S O U T H E R N C A L IF O R N IA T H E G R A D U A T E S C H O O L U N IV E R S IT Y P A R K -p. LO S A N G E L E S , C A L IF O R N IA 9 0 0 0 7 * x - U s CK 17 § C 5 \g This dissertation, written by . . . & L - S L i u n . . . c U n .............. under the direction of lu .f..... Dissertation C o m mittee, and approved by a ll its members, has been presented to and accepted by The Graduate School, in p artia l fu lfillm e n t of requirements of the degree of D O C T O R O F P H I L O S O P H Y Dean DISSERTATION COMMITTEE Chairman ACKNOWLEDGMENTS I wish to gratefully acknowledge my advisor, Dr. William R. Wilcox, for his advice, guidance and encour agement throughout this work and also my committee members, Dr. C. J. Rebert and Dr. G. V. Chilingar, for their interest and suggestions in helping me to prepare my final manuscript. I also wish to acknowledge my deep appreciation to Dr. P. J. Shlichta for his valuable technical guidance and assistance with apparatus design and to Dr. R. A. Lefever for his constructive criticisms. Additionally, I am grateful to Mr. Frank Lum for permitting me to use his diamond wire saw. The research was financially supported by the National Aeronautics and Space Administration via a sub-contract from the Jet Propulsion Laboratory on Contract NAS8-29847. Finally, I am particularly grateful to my wife, Angela, for her understanding by assuming part of the financial responsibility and for helping with the English. I would also like to thank my mother for taking care of our baby during the period of this work. ii TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF FIGURES LIST OF TABLES ABSTRACT Chapter I. INTRODUCTION II. LITERATURE SURVEY (A) The Nature of Convection During Crystal Growth (B) Dimensional Analysis of the Equations of Change for Natural Convection Heat Transfer (C) Analogy Between Heat and Mass Transfer (D) Natural Convection from a Finite-Size Horizontal Plate (E) Natural Convection from Spheres 1. Natural convection in the laminar boundary layer for Pr_>l 2. Natural convection for Pr<<l 3. Natural convection in the turbulent boundary layer for Pr j> 1 page ii vii xv i xvii 1 6 6 8 13 15 24 26 - 32 32 Chapter 4. Natural convection for low Grashof Page number 34 (F) Crystal Growth from Solution (G) The Starvation Theory of Inclusion 35 Formation 38 III. EXPERIMENTAL METHOD (A) General Principles of the Schlieren 41 Method 41 (B) Experimental Apparatus 48 1. Mirror Schlieren system 48 2. Time lapse cinematography 3. Sample cell and temperature 51 control system 53 4. Vibration reduction set up 56 5. Impact producer 59 (C) Material Preparation 59 1. Preparation of solutions 59 2. Preparation of seed crystals 70 (D) Experimental Procedures 71 IV. EXPERIMENTAL RESULTS 76 (A) Hydrodynamics 77 1, Three types of convection patterns 2. The effect of crystal size on 77 convection irregularities 83 iv Chapter Page 3, The effect of crystal size on plume and eddy formation 89 4. The effect of supersaturation on convection irregularities 98 5. The importance of Grashof number to convective irregularity during solution growth 98 6. The effect of seed crystal depth under the free surface on the convection regularity 114 CB) Crystal Imperfections 116 1. Surface irregularities 116 2. Inclusions 134 (C) Influence of Mechanical Perturbation on Convection Irregularity and Crystal Defects 140 V. DISCUSSION AND THEORY 158 (A) Grashof Number for Irregularity 158 (B) The Effect of Surface Concentration on the Sherwood Number 162 (C) Surface Irregularities and Convection Patterns 166 1. Bottom face 169 2. Side surfaces 170 V Chapter Page 3. Top surface 172 (D) Bubble Generation and Inclusion Formation 175 VI. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RELATED WORK 185 NOMENCLATURE 188 REFERENCES 192 APPENDIX A Example Calculations of Volume Fraction of Solute, Grashof Number and Sherwood Number fora 30°C Saturated NaClO^ Solution with Growth at 25°C 201 APPENDIX B Example Calculation of the Change of the Relative Supersaturation During NaClO^ Crystal Growth 207 APPENDIX C The Calculation of the Schmidt Number for a 30°C Saturated NaClO^ Solution with Growth at 25°C 209 APPENDIX D Calculation of the Distribution Coefficient of Ag in NaClO^ Crystals from Atomic Absorption Data 211 APPENDIX E Example of the Weight Change of a NaC103 Crystal Grown with a Stable Plume 213 vi LIST OF FIGURES Figure 2-1 (a) Free convection along a heated plate facing downwards (b) Free convection along a heated plate facing upwards 2-2 Theoretical average Nusselt number for a horizontal heated plate facing downwards(37) 2-3 Local heat flux density for surface at different positive inclinations (43) 2-4 Local heat transfer data for surface at different negative inclinations (43) 2-5 The coordinate system for a sphere 2-6 Calculated local Nusselt number for spheres C48) 2-7 Local melting rate of benzene sphere (53) 3-1 Diagram of Schlieren optics (89) 3-2 Displaced image of Schlieren source at the knife edge when Qf is not intercepted (89) 3-3 Displaced image of Schlieren source at the knife edge when part of Qf is intercepted 3-4 Application of focused-image optics to mirror Schlieren system Page 16 16 21 23 25 27 29 31 43 45 47 49 vii Figure Page 3-5 Crystallization cell 54 3-6 Constant temperature water flow system 57 3-7 Vibration-reduction set up 58 3-8 Impact producer 60 3-9 The solubility curve of NaClO^ in water 63 3-10 The density of saturated NaClO^ solutions with the density measured at the saturation temperatures 65 3-11 The density of a supersaturated NaClO^ solution which was saturated at 30°C 66 3-12 The viscosity of saturated NaClO^ solutions with the viscosity measured at the saturation temperatures 68 3-13 The viscosity of a supersaturated NaClO^ solution which was saturated at 30°C 69 4-1 Laminar plume above a growing NaClO^ crystal 79 4-2 The definitions of crystal three dimensions and L 80 4-3 Early transition stage with irregular plume formation 81 4-4 Late transition stage with more irregular eddy formation 82 4-5 Convection irregularities in Rochelle salt growth 84 viii Figure Page 4-6 Convection irregularities with NaClO^ crystal mounted on a stiff 1.6 mm diameter copper wire 85 4-7 Influence of crystal dimensions on convection behavior with AT=5°C (30°C^25°C) 86 4-8 Transition from stable state to irregular state with same NaClO^ crystal 88 4-9 Influence of crystal size on eddy : formation and the width of the plume 90 4-10 The influence of Gr on the ratio of the plume neck width W to the crystal perimeter P 91 4-11 The number of convection, streamlines in the plume vs Gr for growing NaClO^ crystals 93 4-12 The number of irregular eddies in the plume vs Gr for growing NaClO^ crystals 94 4-13 Average period of formation of irregularities for growing NaClO^ crystals 95 4-14 The velocity v of irregularities at the center of the field of view over crystal: 97 4-15 The velocity of irregularities at the center of the field of view as a function of y and AT 99 ix Figure 4-16 4-17 4-18 4-19 4-20 4-21 4-22 4-23 4-24 4-25 4-26 4-27 4-28 Convection behaviors around growing NaClO^ crystals Influence of Gr and h on convection behavior Irregular convection currents above a short NaClO^ crystal Regular laminar plumes above a NaClO^ crystal The NaClO^ crystal growth rate as a function of Gr and AT Plot of the Sherwood number Sh against the Grashof number Gr for NaClO^ growth The average horizontal linear growth rate of NaClO^ crystals against Gr The average vertical linear growth rate of NaClO^ crystals against Gr The ratio of horizontal growth rate to vertical growth rate for NaClO^ crystals against Gr Plot of the Reynolds number of the plume against Gr Flaws on the top surface of a NaClO^ crystal Irregular convection currents above a NaClO^ crystal (shown in Fig. 4-26) which formed flaws on its top surface Diagonally patterned flaws on the square top surface of a NaClO^ crystal Page 100 102 103 104 105 107 111 112 113 115 118 119 120 x Figure Page 4-29 Flow patterns for horizontal plate in the laminar region (82) 121 4-30 Flow patterns above horizontal plates in the early turbulent region (82) 123 4-31 Terrace irregularities on the top surface of a NaClO^ crystal 124 4-32 Inclusions near the top surface of a NaClO^ crystal grown with irregular convection 126 4-33 Growth streaks on the side face of a NaClO^ crystal with irregular convection 127 4-34 A close view of growth streaks on a NaClO^ crystal side face with irregular convection 128 4-35 Slightly concave upper center part of a NaClOg crystal side surface with regular convection 129 4-36 Illustration of various types of concavities observed on the side faces of NaClO^ crystals 131 4-37 Needle-like growth steps formed on the side surface of a NaClO^ crystal with irregular convection 132 4-38 Layer, elongated channels and isolated spots of inclusions near a NaClO^ crystal bottom face with irregular convection 135 xi Figure 4-39 4-40 ! t i i 4-41 4-42 4-43 i ! ! 4-44 4-45 4-46 4-47 Inclusions in NaClO^ crystals.' Spheres with dark outlines are bubbles Inclusions near the upper part of the side surface of a NaClO^ crystal grown with steady convection Inclusion near the side surface of a NaClO^ crystal grown with irregular plume Inclusions around a concavity formed at the upper center part of a NaClO^ crystal side surface grown with regular convection The minimum amount of impact energy needed to induce transient convection irregularities lasting 3 sec or longer for NaClO^ crystals The duration time of irregularities vs. impact energy Convection patterns around a NaClO^ crystal tapped by 0.1 joule of impact energy Convection pattern around a NaClO^ crystal tapped by 0.7 joule of impact energy Inclusion near the top surface of a NaClO^ crystal grown with periodic tapping with Page 137 138 139 141 143 ! i i 144 145 147 xii Figure Page an impact energy of 0.035 joule and a period of 18 sec 148 4-48 The counter number (relative amount) of Ag in a NaClOg crystal grown with a I regular plume without tapping 150 i I ! 4-49 The microprobe tracing spots on NaClO^ j crystals 1 5 1 i 4-50 The counter number of Ag in a NaClO^ i crystal grown with five different conditions 152 j I j 4-51 The convection pattern around a NaClOg ! j crystal 2 sec after tapping by an impact I ! j energy of 0.062 joule I 5 4 ! I i 4-52 The counter number of Ag in a NaClO^ crystal grown with five different 1 1 conditions 155 ! 4-53 The convection pattern around a NaClOg j crystal 2 sec after tapping by an impact energy of 0.062 joule 157 5-1 Drag coefficients for a flat plate of finite thickness normal to the flow (83) 160 5-2 Flow pattern for a fluid impinging on a flat plate (83) 161 5-3 The Plot of E (Vc cal-Vc actual)2 vs. a for NaClO^ crystals 165 xiii Figure Page 5-4 Sketch of initial castellated stage in the growth of NaCl cubes immersed in flowing supersaturated brine (85) 168 5-5 Schematic diagrams illustrating (a) the formation of open channels on crystal side surface (b) sealing an open channel by a new growth layer and eddies, 173 5-6 Inclusions formed at an early stage on a side surface of a NaClO^ crystal grown with slightly irregular convection due to tapping (0.065 joule) 176 5-7 Inclusions formed on NaClO^ crystals with irregular plume (a) good seed crystal (b) bad seed crystal 179 5-8 Inclusions in a NaClOg crystal grown with irregular plume 180 5-9 Inclusions in a NaClO^ crystal grown with regular plume 181 5-10 Many inclusions formed at the interface between original seed crystal and newly grown part with irregular plume 183 A- 1 Mean molar volume vs. mole fraction of NaClOg solution 205 A-2 Mean molar volume vs. mole fraction of xiv Figure Page NaClO^ solution at 18°C (92) 206 E-l Example of the weight change of a NaClO^ crystal grown with a regular plume 213 xv LIST OF TABLES Table Page 2-1 Average Nusselt or Sherwood numbers for laminar convection about a sphere 33 4-1 Summary of least squares analysis 109 A-1 Density of aqueous NaClO^ solution (92) 204 xvi ABSTRACT An experimental study of the regularity of natural convection currents which arise from a crystal during solution growth is presented. In particular, the contri butions of hydrodynamic effects to the quality of the resulting crystal were investigated. The convection patterns and irregularity phenomena were observed with a Schlieren apparatus being developed for space processing studies. The convection plume above growing NaClOj crystals became irregular and exhibited strong oscillations when the Grashof number Gr of the system exceeded a critical value. That is, as either the crystal size or the super- saturation increased, there was a continuous and spon taneous transition from the "steady laminar" state to the "unsteady transition" state. The critical Gr was found to be dependent on the geometry of the crystal. The oscillations were not associated with perceptible motion of the crystal or of the free surface of the solution. The convection velocity of the irregularity was a function of Gr. The growth rate of the crystal was dependent both on Gr and supercooling AT. The oscillation frequency of the irregularity increased with increasing crystal growth xvii rate. The surface irregularities and growth inclusions appeared to be associated with prolonged oscillations of the convection. The surface irregularities were diagonal flaws and terraces on the top surface of the crystal. Streaks and growth steps appeared on the side surfaces. A hollow was often observed on the center of the bottom 1 surface. Solution and bubble inclusions were often found in the upper part of the crystal. The effect of mechanically-induced transient con vection irregularities on the distribution of Ag impurity and on the final crystal quality were also studied. The role of induced transient convection irregularities on Ag distribution is uncertain due to insufficient data. However, the results showed that this kind of forced con vection can reduce inclusion formation and lead to a better crystal. xviii CHAPTER I INTRODUCTION Of all the methods of crystal growth, room temperature solution growth is perhaps the oldest and cheapest. Many kinds of crystals have been grown this, way (1). In recent years, the growth of large clear crystals as perfect as possible has been stimulated by the need for them in the investigation of physical properties and for electronic and optical applications. Two well-known examples of growth from aqueous solution are ADP crystals (ammonium dihydrogen phosphate) for use in submarine - detection devices and EDT crystals (ethylene-diamine tartrate) for use in telephone carrier systems (2). Solution grown crystals suffer from a number of possible defects (3). The most obvious is the occurrence of inclusions. The term "inclusion" is given to any for eign body - either solid, liquid or gas - enclosed within a solid. The following specific terms are used in the literature to describe various forms of inclusions (4). (a) Bubbles - bubble-shaped cavities of various sizes filled with vapor or solution. (b) Negative crystals - faceted cavities. 1 ! (c) Veils- lines or thin sheets of small solvent inclus ions. (d) Ghosts- an envelope of planar veils each one of i ! which is parallel to some possible crystal face, i usually corresponding to the original shape of the ; seed crystal. : (e) Clouds- aggregates of fine bubbles or cavities. ; The presence of inclusions has many deleterious effects j |It has been found that the presence of water frequently i !results in a considerable error in precise chemical work j land in atomic weight determinations (5). Many physical | ■properties of .single crystal are degraded by the presence i 'of inclusions which cause light scattering, strain effects I i jand shorting of semiconductor junctions (6). It has also ibeen found that liquid inclusions can seep, releasing ; solution to the surfaces of the crystals (7). i ■ Several possible causes of the formation of inclusions ,during solution growth have been observed. Brooks et al. i j(8) found that the formation of inclusions in ADP and jNaClO^ was connected with a growth rate fluctuation. iDenbigh and White (9) showed that there is a critical I jsize for inclusion formation in hexamine crystals below which inclusions are not formed. The critical size was a function of growth rate. Published ideas on the origin of inclusions tend to be somewhat speculative since they were usually based on observations made after the crystal was grown. Moreover, many observations of growth were confined to microscopic crystals within thin horizontal films of solution (10, 11, 12, 13, 14). Convection was therefore highly suppressed or even completely eliminated. Similarly, there have been few if any quantitative measure ments of the free convection around a growing macroscopic crystal. In order to grow crystals from solution, people relied principally on experience. Both Buckley (2) and Holden (15) suggested that good crystals can be obtained simply by setting the solution aside to cool in a quiet spot without disturbance. No reason was given. Prior to the present study insufficient information was known about the effect of convection irregularities on formation of crystal defects. It was originally proposed that the present research would consist of direct observation, by Schlieren, of the role of convection in crystal growth. Photographic records would be made of convection patterns and growth rates. During post-experiment evaluation, these data would be correlated with the defect structure of the crystal. Therefore the primary purposes of this research were to: (a) Study free convection around crystals growing 3 from solutions in order to (1) Determine if there is a continuous transition from "steady convection" to nunsteady con vection" and (2) Observe transient convection instabilities due to vibrations. (b) Determine the effects of residual and transient convection irregularities on the defect structure of the crystallized sample and to correlate these defect structures with their convection history. It has been shown in this study that convection irregularities along with bubbles may cause the formation of defects, such as fluid inclusions and surface irregu larities, in the growing crystal. In contrast, convection in "low-gravity-simulation" systems (16), which crystallize at extremely low Grashof numbers, remains stable even when large perturbations are applied. This suggests that the low level of convection irregularities in a spaceflight could advantageously be utilized as an environment for solution crystal growth. In the second chapter of this dissertation the related literature is reviewed, including analytical and experi mental results on thermal and mass free convection from spheres and horizontal plates. A simple but useful model - the stagnant film model - is then reviewed. This model 4 enables us to predict the crystal growth rate in solution growth when the solution supersaturation and the convection situation are known. Finally, starvation theory is dis cussed. It can be used to explain some commonly observed phenomena on the formation of inclusions. Chapter III describes the detailed experimental methods and materials employed for the research. Chapter IV shows the experi mental results and observations. Interpretation of experi mental results is given in Chapter V. Some possible extensions of the present work are recommended in the last chapter. CHAPTER II LITERATURE SURVEY (A) The Nature of Convection During Crystal Growth In general, convection in fluids may be induced by four factors - a temperature gradient, a concentration gradient, a surface tension gradient and an external mechanical force. Since heat transfer and mass transfer are important processes during crystal growth, both temper ature and concentration gradients must exist in the fluid. The gradients of temperature and concentration required to drive transport processes also produce variations in fluid density. Under the influence of the gravitational field, a density gradient causes fluid motion - a downward flow of more dense fluid and an upward flow of the less dense fluid. The motion of the fluid caused by differences in density resulting from temperature change or concentration change is called ’’buoyancy-driven natural convection" or simply "natural convection." Therefore, on earth, the process of crystallization from solutions without stirring is dominated by natural convection (17). In a space environment, the gravity level is reduced to about 1 0 (where gE = 9.8 m/sec^ is earth’s gravity). 6 Thus buoyancy-driven natural convection is grealy di minished. However, if a fluid-fluid interface is present with a temperature gradient or concentration gradient along it, there will be surface tension driven convection, often called nMarangoni flow.” This flow is sometimes significant, in crystal growth but it has not been considered since gravity driven convection has been thought to dominate on earth. On the Apollo 14 mission, several experiments did show vigorous convection in a layer of heated liquid (18). Chang and Wilcox (19) found by numerical calculations that Marangoni flow in floating zone melting of silicon was large. It is therefore possible that surface tension- driven flow will often be the most significant type of convection in low gravity. In most crystal-growing systems the rate of growth depends directly on the rate of heat and mass transfer of the system. These transport rates are often accelerated through the use of mechanically induced fluid motion. This is called ’’forced convection.” An important example is the mounting of crystals on a rotating ’’tree” in solution growth (20) . Stirring is also effective in maintaining inclusion-free faceted growth (21) but care must be taken to assure that each face benefits more or less equally from the stirring. For crystals mounted on rotating trees, this requires frequent reversal of the direction of ro tation . _______________________________________________________________________ 7 (B) Dimensional Analysis of the Equations of Change for Natural Convection Heat Transfer For non-isothermal one component fluids, the equations of change describe the change of velocity and temperature with respect to time and position in the system. They contain three partial differential equations and are de rived from conservation laws (22). For simplicity, we restrict ourselves to systems with constant physical properties. With this restriction we may write the following equations for natural convection heat transfer: (1) Equation of continuity V•v = 0 (2-1) Where v is the velocity of the fluid. (2) Equation of motion = yV2v - PBg(T - Too) (2-2) Where p,t,y,j3, g and T represent density, time, viscosity, thermal expansion coefficient, gravity acceleration and temperature, respectively. (3) Equation of energy pCpgT = Kv2T (2-3) Where Cp is the heat capacity and % is thermal conduc tivity. Note that we have omitted the viscous dissipation 8 term in the equation of energy since that term is generally unimportant for free convection. For many flow systems one can select a characteristic length L and define the following dimensionless variables and differential operations: By writing the equations of change for natural convection in terms of these dimensionless variables we obtain, on rearrangement, v* = XPJi = dimensionless velocity ] i 7 t ui = dimensionless time p L T* = - j— = dimensionless temperature x*,y*,z*, = = dimensionless coordinates D Dt * (continuity) V* • v* = 0 (2-4) Dv* ? (motion) --- = V* v* - Gr-T* Dt* (2-5) 9 DT* 1 , o (energy) ---- = — v*^T* (2-6) Dt* Pr We see then that for free convection only the two dimension less groups, the Prandtl number, C u Pr = -P , and the Grashof number, gp2e(T0-TJE3 Gr = ------- 2------- > ■ appear in the equations of change. y If in two geometrically similar systems the Prandtl and Grashof numbers are the same for both, then both systems are described by identical dimensionless differential equations. If, in addition, the dimensionless initial and boundary conditions are the same, then the two systems are mathematically identical; that is, the dimensionless velocity distribution v* (x*,y*,z*,t*) and the dimensionless temperature distribution T*(x*,y*,z*,t*) are the same in each. Such systems are said to be "dynamically similar." The Prandtl number-is very important in all heat transfer problems involving fluid flow (23). Roughly speaking, this number is the ratio of the diffusivity of momentum to the diffusivity of heat. Since it only con tains thermophysical properties of the fluid, we may consider it as a property of the fluid. The Grashof number roughly represents the ratio of buoyancy force to the viscous force. The product of the Prandtl number and the 10 Grashof number is the Rayleigh number, Ra = Pr.Gr = gp2@(To - Too)L3cp/UK. This parameter is also often used to characterize free convection. In the case of a fluid situated between two infinite solid horizontal planes with the lower plane at a higher temperature, the onset of convection occurs at Ra >1710 (24). If the upper surface is free, but still at a constant temperature, then convection occurs for Ra >1100. The parameters characterizing free convection mass transfer are the compositional Grashof number coefficient and D is the molecular diffusivity. The product of the Schmidt number and the compositional Grashof number is the compositional Rayleigh number = p2ga (C» - C0 )L3/,a2 and the Schmidt number Sc - y/pD where is the compositional densification 11 Another possible way to write the Grashof number for either heat or mass transfer is Gr = g pApL3/]!2 where Ap is the density difference. It still remains to specify the characteristic length L which appears in Gr and Ra. In a typical solution growth system, we can simply treat the crystal as a thick horizontal plate. Several different definitions for L have been employed in the case of a horizontal plate (25, 26, 27, 28, 29). L may be taken as: (1) The short side of a rectangular plate, the side of a square plate, or the diameter of a circular plate (one dimensional viewpoint). A (2) L = p (two dimensional viewpoint) where A is the surface area and P is the perimeter. 1 1 1 (3) y- = y- ------ + y--------- -(three dimensional horizontal vertical viewpoint) The experimental results (27, 28) suggest that a common correlation for horizontal plates of different shapes can be attained if appropriate characteristic lengths are used in the Sherwood number Sh and compositional Grashof number Gr . The Sherwood number is the dimension- m less composition gradient 12 Sh = . C-C . o 0 0 y=0 where y is the distance from crystal surface. Goldstein (27) and Lloyd (28) investigated experimentally mass free convection from unshrouded circles, squares, rectangulars, equilateral triangles and non-symmetric right triangles. They concluded that only when the characteristic length L is defined as the ratio of the surface area to the encom passing perimeter do all horizontal forms exhibit a common Sherwood-Rayleigh correlation. However, none of these studies discussed the effect of vertical dimension on the convection patterns and mass transfer. Therefore, I am going to use characteristic length L-^ and treat the heights of the crystals as a second parameter. (C) Analogy Between Heat and Mass Transfer While the equations for heat and mass transfer are similar, there are several differences between heat and mass transfer (32). For most transfer problems, the solid interface is often chosen as the origin of the coordinates and is regarded as stationary. In crystal growth there must then be a flow of material to the interface since the crystal is growing. The flow created at the solid interface due to this mass transfer process may be called the "interfacial flow" (30). On the other hand, such a surface velocity is generally not present in corresponding ________________________________1 3 J heat transfer problems. In heat transfer mechanical energy is converted into heat by the action of viscosity. This phenomenon is called "viscous dissipation" (31). In addition there are differences in the variation of prop erties (viscosity, diffusivity, density, etc.) with temperature and composition. Finally, there is no mech anism in mass transfer corresponding to radiation in heat transfer. If one assumes that the properties of the fluid are constant, and that interfacial flow, viscous dissipation and radiation can be neglected, then heat and mass transfer are entirely analogous (33). When these are reasonable assumptions, theoretical and experimental results for heat transfer can be employed for mass transfer by a simple substitution of parameters. For example, substituting the Sherwood number for the Nusselt number, the Schmidt number for the Prandtl number and concentration for temper ature. The analogy between heat and mass transfer can apply not only for laminar flow but also for turbulent flow. This is particularly useful for empirical corre lations since heat transfer has been more widely investi gated. All important dimensionless parameters for natural convection are already defined in the last section except for the Nusselt number Nu. The Nusselt number is defined 14 as Nu T T o y=0 and is a dimensionless temperature gradient where L = characteristic length y = distance into fluid from solid surface T, T , = temperature of fluid at y, at surface (D) Natural Convection from a Finite-Size Horizontal Fluid motions produced.by buoyancy forces with a finite - thickness horizontal plate should be very similar to those arising in solution crystal growth. Two kinds of flow situations are considered - a heated plate facing downwards and a heated plate facing upwards, as shown in Figure 2-la and Figure 2-lb, respectively. Referring to Figure 2-la, as the fluid near the surface is heated by the wall it becomes less dense than the surrounding fluid. This lighter fluid flows across the plate under the influence of a hydrostatic pressure gradient and off the edges. Thus, under steady-state conditions a natural convection boundary layer is estab lished that has a maximum thickness at the plate center and decreases to a minimum (non-zero) thickness at the edges (34,35,37). The region of rapidly changing velocity (y=0) and in bulk fluid (y=°°) Plate 15 / / / / / / / / / / / / / / / / / Z / / / / To x '00 Ta>T 00 Fig. 2-la: Free convection along a heated plate facing downwards. Q To >Too oo / \ Fig. 2-lb: Free convection along a heated plate facing upwards. 16 is called the boundary layer. It extends from the solid boundary to the point in the fluid at which the velocity gradient becomes nearly zero. We note that the boundary layer is thin compared with the significant dimension L of the body for high Ra, i.e. the effects of fluid viscosity and of the no-slip surface condition are important only in a very narrow region adjacent to the surface. For a heated plate facing upwards, Rotem and Claassen (36) showed by the Schlieren method that a boundary layer also existed on the upper side of heated plates held in air. Fluid flowed from the edges to the center and finally flowed upwards. The lateral extent of the boundary layers was dependent on Ra. When Ra was sufficiently small, boundary layers starting at each of the leading edges ultimately met near the center-line of a plate, where the flow turned and formed a thermal plume above the plate. For larger values of Ra, however, the boundary layer became unstable well before reaching the vicinity of the axis of symmetry. Theoretical studies for both problems utilized the boundary-layer approximation (37,38). Since the Navier- Stokes and energy equations (i.e. Equations 2-2, 2-3) are too complicated to solve exactly, they are typically reduced to much simpler forms by an order-of-magnitude analysis (39), using the fact that the boundary-layer is thin. The simplified equations are called "boundary-layer ___________________17 equations.” For two-dimensional steady-state free convection, boundary layer flow along a horizontal plate can thus be described by the following equations: continuity 8 v 8 v S5T + 3y^ = 0 C2‘7) x-momentum ~ - B 2 V = - |f + u( (2-8) 3x 3yZ y-momentum = - P&[l-B(T-T0)]g (2-9) energy t r3T\ _ K r3 2T-, mi vx ^ + vy C5y} ' pc^ CV (2~10) The boundary conditions are at y=0, v =v =0 and T=T = constant (2-11) 7 ’ x y o at y-«o, vx^0, T+T*,, ^ +0, fy +0 and at x = 0, v (y) = 0 (2-12) A These boundary layer equations can then be solved either by similarity solutions (39) or by integral methods (37). Here, we do not attempt to go through any detailed 18 discussion of these methods but only the concepts and results. Roughly speaking, for a similarity solution a similarity variable n and a generalized stream function T are introduced into the boundary-layer equations. We let These are substituted into the original boundary layer In the integral method, the boundary layer equations are integrated from y=0 to y=6 for the momentum equation and from y=0 to y=fi^ for the energy equation, where 6 is the hydrodynamic boundary layer thickness and 6^ is the thermal boundary layer thickness. Some functional relationships are assumed between the dimensionless velocity distribu tion and y/6, and the dimensionless temperature distribu tion and y/6-t- The constants in the functions are then determined by the boundary-layer conditions. Finally the equations are solved for < 5 and 6-^. In this way the temperature distribution and velocity distribution are obtained approximately. In laminar boundary layer region, Clifton and Chapman (37) obtained thereby the following theoretical result for a heated plate facing downward. n ( x , y ) = y b ( x ) , y ( x , y ) = c ( x ) / ( n ) T-T (2-13) equations and functions b and c are sought such that T and f depend only on p. Then they are solved for T and f. 19 Nu = C Ra1^5 (2-14) Where constant C was found to be a function of Pr as shown in Fig. 2-2. The one-fifth power of Ra in expression 2-14 agrees with other theoretical results (34, 40, 41). These other theoretical predictions for the constant C are 0.46 for Pr = 0.7 by Singh and Birkebak (34), 0.5 for Pr >_1 by Wagner (40) and 0.66 for Pr by Singh et al. (41). Rotem and Claassen (36) treated theoretically two dimensional free convection flow from a horizontal heated plate facing upwards and found that the average Nusselt number Nu also obeyed the form of equation 2-14, except that the rate of heat transfer is somewhat higher than that of a plate facing downwards. For a heated plate facing upwards, eddies are formed due to the instability of the boundary layer and therefore more vigorous transport is expected. For very small Pr, the average Nusselt number was found to be Nu = 0.961 Ra1,/5 Pr7^15 (2-15) Suriano and Yang (42) used a numerical (finite difference) method to study the problem of two-dimensional, free-convection flow about a finite thickness horizontal plate heated on both sides with relatively small Rayleigh numbers (Ra < 300). They found that the Nusselt number of 20 I Fig. 2- 10° 5 6 10-2 lO 'l io O iol 1 0 2 P r | 2: Theoretical average Nusselt number for a horizontal heated plate facing downward (37). N e g le c t in g m e r f lc t e r m s Including Inertia terms 21 the lower surface was always larger than that of the upper surface, the difference being larger as the Rayleigh number increases. The fluid arriving in the neighborhood of the upper surface comes mostly from the lower surface, and has | therefore already been heated to some extent. In view of I the assumption that the temperature of both surfaces i | remains the same, the effective temperature difference between the boundary layer flow and the surface is smaller for the upper surface than for the lower surface. Conse quently, a higher transfer rate is expected for the lower surface. Hassan and Mohamed (43) measured the local heat transfer i along a horizontal heated plate facing upwards and found that the maximum flux density is at the edges. A dip in j the flux density occurs near the edges and increases again | i in the center of plate, as shown in Fig. 2-3. The number of degrees shown in the figure is the angle of inclination of the surface to the vertical. Mass transfer measurements ■ have been made by Lloyd and Moran (28). They suggested the following empirical relations. Sh = 0.54 Ra ^ (2.2xl04<Ra <8xl06) (2-16) m v m J K in the laminar region and SE = 0.15 Ra - 1/3 (8xl06<Ra <1.6xl09) (2-17) m v m J v J 22 Hect flu* density,g?'! Kcol/h 225 T e m p e r o f u r e d i f f e r e n c e - 3 8 2 ± I O ° C A i r f e m p e r o f o r e 2 7 2 ± 0 - 6 * C 200 E K a = * 1 5 * w i t h o 2 0 0 m m o d i a b o t i c at- 6 0 ® l e o d in g s e c t i o n 30® (75 E50 9 0 7 5 ' 6 0 125 100 lO O1 200 D i s t a n c e f r o m le o d i n g e d g e , jt , 300 "40i 500 mm Fig. 2-3: Local heat flux density for surface at different positive inclinations (43) 0° vertical plate 90° horizontal heated plate facing upwards. 23 in the turbulent region. In the very low Ra^ region (Ram _< 200), Goldstein et al. (37) suggested Sh = 0.96 R a ^ 6 (2-18) 'It is worth noting that the empirical 1/6 exponent on the i !Rayleigh number deviates from the 1/5 predicted via boundary-layer theoretical treatments. For a heated plate facing downwards, the instability sets in at a much higher Ra (44), since the buoyant fluid now cannot rise indefinitely but remains confined to a layer near the surface. Under steady-state conditions a natural convection boundary layer is established with a maximum depth at the center of the plate (35). Hassan and Mohamed’s (43) local heat transfer data also proved this, as shown in Fig. 2-4. Parker et al. (46) suggested the following correlation for a horizontal heated plate facing downwards (E) Natural Convection from Spheres The problem of natural convection from a sphere is relevant to solution crystal growth and a great deal of work has been done. Nu = 0.27 Ra4 (3xl05 <Ra< 3xl010) (2-19) 24 H e o f flux d e n s it y , q '\ k c o l / h 35 T e m p e r a t u r e - d i f f e r e n c e ~ 8 0 5 ± 0 - 2 ° C A i r t e m p e r a t u r e 2 8 - 4 ± 0 - 8 “ C 30 E O* 20 3 0 0 IOO 200 D i s t a n c e f r o m leading edge, x. 4 0 0 5 0 0 mm Fig. 2-4: Local heat transfer data for surface at different negative inclinations (43) 0° vertical plate -90° horizontal heated plate facing downwards. 25 1. Natural convection in the laminar boundary layer for Pr >. 1 Schmidt and Beckman (47) suggested that this problem could be solved analytically by means of boundary-layer ■ theory. For this method to be applicable all viscous and I I temperature (or concentration) effects must be restricted i |to a thin layer close to the surface, which requires that I 4 Ra > 10 . The coordinate system for a sphere is shown in Fig. 2-5, in which x is the distance coordinate along the body surface and is measured from the lower stagnation point, y is the normal distance from the surface, r is the distance from the surface to the axis of symmetry, and R is the sphere radius. Note that r = R sin x/R. Merk and Prins (48) showed that the appropriate boundary-layer equations for steady, nondissipative, con stant property laminar free convection flow are: continuity 8 (rv ) ■ - + 8 (rv ) — 2d = o (2- 20) 9y momentum v. x 3v — + v X 7 26 y Fig. 2-5: The coordinate system for a sphere x = the distance coordinate measured from the lower stagnation point along body surface y = normal distance from the body surface R = the sphere radius r = the distance from sphere surface to the symmetrical axis. 27 energy 3T , 3T K 32T v — + v — = --- o (2-22) x 3x y 3y pC 3y The boundary conditions are at I ! y = 0 v = v = 0 T=T i 7 x y o (2-23) y oo v 0 T->T J X c The boundary-layer equations then can be solved either by Saunder's approximation method (49) or by the Squire-Eckert integral approximation method (50). The final solution for both the average and the local Nusselt number was predicted 1 to obey an equation of the form ^ Nu = C x (Gr*Pr)^ (Pr>>l) (2-24) until boundary layer separation occurs. Merk and Prins (48) found by an integral method Nu = 0. 558 Ra^ (Pr+°°) (2-25) For the local Nusselt number, an approximate technique was used to estimate C as a function of position, yielding the result shown in Fig. 2-6. Chiang, Ossin and Tien (51) solved the boundary layer equations numerically for Pr = 0.72 for constant surface 28 Fig. 2-6: Calculated local Nusselt number for spheres (48) . 29 ___ temperature to obtain Nu 0.4576 - 0.03402 fjGr^ ) (2-26) Schutz (52) used limiting-current measurements to obtain the mass transfer rate for a sphere at high Ra which he claimed was in the transition region Kranse and Schenk (53) obtained the results shown in Figure 2-7 by measuring the local rate of melting of benzene spheres. For the average mass transfer rate, they also showed that equation 2-27 fits the experimental data of many other authors (54, 55, 56). The value of 2 in the equation comes from the theoretical result for steady state diffusion or conduction in the absence of convection (Ra=0). The flow pattern, which was observed by Schlieren method, showed that pure laminar flow was present between the stagnation point and the equator. Between 100° and the separation point at 140° ^ 150° some irregularities occurred, but real turbulence existed only in the plume. Yuge (57) obtained the following correlation for his experimental heat transfer data with spheres in air: SF = 2 + 0.59 Ram^ (2-27) for 2.3 x 108< Ram <1.5 x 1010 Nu = 2 + 0.392 Gr3 * (Gr <105) (2-28) 30 180 120 60 2-7: Local melting rate of benzene sphere (diameter 51.5 mm) (53). 31 Amato and Tien (58) investigated heat transfer from spheres to water and found Nu = 2 + 0.5 Ra*® (for 3xlOS<Ra<8xl08) (2- 29) j .Table 2-1 shows a comparison of the various analytical and |experimental results for laminar free convection heat or I jmass transfer from a sphere. All of these equations con taining the theoretical 1/4 power on the Rayleigh number 4 9 are expected to apply for Ra from 10 to about 10 , since turbulence has been found experimentally to arise in the 9 boundary layer at about Ra = 10 . j 2. Natural convection for Pr<<! (or Sc<<!) The formulas given above are inapplicable to free | convection in liquid metal for which Pr is much less than 1J i i For example, at 93°C Pr is 0.016 for Hg and 0.011 for Na (67). While molecular transfer is now thought to exert a I decisive effect on transport, one must also recognize the fact that the velocity field depends primarily on the inertia forces, while viscous effects are negligible. Kutateladze (68) concluded that Nu should be better correla ted by both Ra and Pr, i.e. Nu = 0.53Pr^Ra^ (103<Ra<107) (2-30) 5. Natural convection in the turbulent boundary layer for Pr _> 1 32. Table 2-1 Average Nusselt or Sherwood numbers for laminar convection about a sphere Investigator Nu or Sh Conditions Schutz (52) Kranse and Schenk (53) Yuge (57) Amato and Tien (58) Kyte, Madden and Piret (59) Ranz and Marshall (60) Schenkels and Schenk (61) Garner and Keey (54) Garner and Hoffman (55) Van der Burgh (62) Schenk and Schenkels (63) Jakob and Linke (64) Boberg and Starrett (65) Merk and Prins (48) Raithby and Hollands (66) 2+0.59 Ra"4 2+0.59 Ra®4 2+0.43 Ra^ 2+0.5 Ra®* 2+0.399 Ra*4 2+0.6 Pr1 Ra^ 2+0.59 Ra*4 23+0.585 Ra®* 5.4+0.44 Ra®4 0.525 Ra*4 0.56 Ra®* 0.555 Ra®4 0.51 Ra®4 0.558 Ra®4 2+0.56/ Pr ^Ra*4 \0.861+Pr/ Mass transfer-electrochemical Melting-benzene Heat transfer-air Heat transfer-water Heat transfer-air, He, Ar Evaporation-drops Dissolution-organic spheres Mass transfer-experimental Mass transfer-experimental Melting-benzene Melting-ice spheres Heat transfer-experimental Heat transfer-experimental Boundary-layer analytical Theoretical g For turbulent free convection (Ra >10 ), heat transfer data is usually correlated by an equation of the type (69) Nu = c’Ra1//3 (2-31) Only few experimental data are available. Bayley (70) I studied turbulent free convection heat transfer in two- 1 dimensional flow and showed that i i Nu = 0.1 Ra1 for 2xl09<Ra<1012 (2-32) Boberg and Starrett (65) showed that c* = 0.13 for a sphere. 4. Natural convection for low Grashof number i 4 I If the Grashof number is below about 10 , the convection flow regions are too thick for the boundary-layer approxi- ' mations to be valid. Hossain and Gebhart (71) considered theoretically this problem for a sphere and found: j Nu = 2 + Gr+Gr2(0.139-0.4S19Pr + 1.1902Pr2) (2-33) for Gr<1.0, and Mikheyev’s empirical formula, Nu = 1.18 Ra1^8 (10'3<Ra<5xl02) (2-34) is recommended by Kutateladze (68) for correlating experi mental heat transfer data at low Ra. Mathers, Madden and Piret (72) investigated experimen tally simultaneous heat and mass transfer for a sphere and obtained Sh = 2 + 0.282 Ra0-37 (Ra< 102) (2-35) Note that the empirical exponent deviates from 1/4 at low Ra. _______________________________________________________________________34 (F) Crystal Growth from Solution Now we consider a crystal growing from an isothermal solution. This problem differs from previous work for plates and spheres in two ways. These are (1) Shape - the geometry for a growing crystal is polyhedral. (2) Flux along surface of a growing crystal must be constant for it to remain planar, whereas spheres (or plates) have constant temperature T or con centration C. Because the geometry of a crystal is too complicated for boundary layer theory, it is frequently regarded in terms of the stagnant-film model. In this model it is imagined that a completely stagnant film of thickness 6- . resides at the solid surface. Within the stagnant film transport is by molecular processes only (no convection parallel to surface), while outside the film mixing is complete and the fluid is homogeneous. It is emphasized that this model does not correspond to reality. Indeed some authors call it the "fictitious film” model. Nevertheless it does give reasonably good predictions for the influence of inter facial flow and chemical reactions on the transport rate. It gives erroneous predictions for the influence of diffusion coefficient D, and no prediction for film thick ness 6S, which must be found by comparison with experiment or theory. 35 Steady state binary mass transfer in such a film would be governed by where D is diffusivity and vc£ is the .interfacial flow. This equation is valid for planar portions of the crystal except within a few stagnant film thicknesses of a corner. For constant fluid density Wilcox (30) showed that In order to solve for the growth rate of crystal v , it is necessary to specify boundary conditions. Outside of the stagnant film, the fluid is assumed to be completely mixed. At the outer edge of the stagnant film, y = <5s (by definition) At the crystal surface (regarding the surface as stationary), the solid growth flux equals the diffusive flux plus the convective flow of solute in the fluid to the surface. These must equal the rate of incorporation of solute into the crystal, which for linear interface (2-36) v c where v^ = the crystal growth rate p = density of fluid Pc = density of crystal. C = C CO (2-37) 36 kinetics is given by K, (C -C ). Here K. is the inter- & J 1 v o eq^ i facial kinetic constant. C is the concentration in fluid o at the crystal surface and C is the equilibrium solubi lity of the crystal material. Therefore, at the crystal surface, y=0 v ' l C = d6£) + V -C =K. (C -C' ) (2-38) efkpc / c Vdy/ q cf o iv o eqJ . 1 where C is the concentration of solute in the solid pc° (Cc= -jjr’when no solvent is incorporated, where M is the molecular weight of the solute). From equation 2-36 and the two boundary conditions, equations 2-37 and 2-38, we obtain an expression for the crystallization flow vc^ in terms of known concentrations Ceq - (X- ^ K C y ] exp(-^4£i) = Cra-Cc ^ f2-39^ i For slow growth and rapid interface kinetics, vcf ( 5svc ^ , 6svcf «1, exp ---x— )* 1-----p- K • 1 r 6v c£/D<0.1), and vc is given by V = D £_ f°_ ~Cea------- (2-40) c s pc c r-e-Vc c^pc/ eq Note that the C in the denominator arises from the eq crystallization flow'and must not be arbitrarily neglected. 37 The stagnant film thickness 8^ for forced convection is related to Sh by the following equation when no interfacial’ flow is present 5s = Sir |where L is the characteristic length of the system. Also ;note that in theoretical developments it is implicitly |assumed that 5 ^ f(vcf)• Experimental observations (2) j have also confirmed the general conclusions predicted by 1 the equations just given. i i i (G) The Starvation Theory of Inclusions i ! The existence of convection in crystal growth processes !has been known for a long time, but the importance of 1 convection to crystal growth has only been recognized in i i |recent years. Carlson (74) was the first to demonstrate ithe necessity of stirring for the growth of large inclusion- jfree crystals of ammonium dihydrogen phosphate from aqueous >solution. His theory is based on a simple fluid dynamic i jmodel and can be used to predict the maximum crystal size jwithout inclusions under some specified conditions. The results are in general agreement with the experiment, j Therefore, this theory is reviewed in detail. He treated the crystal growing from an agitated 38 j solution as a flat plate into which solute is being trans- ; ferred at a uniform rate. The flat plate has its leading edge perpendicular to the direction of oncoming flow. For simplicity, the crystallization flow (or interfacial flow) was neglected. Based on the result of Fage and Falkner ^ i (75), he found the concentration distribution along the surface to be C = C - b (2-42) O o o ^ J where C = solute concentration at the surface o C = bulk solution concentration oo b = constant which depends on the particular condition of the^experiment x = distance downstream from the leading edge The crystal growth flux Pcvc could be expressed as p v = 0.463 Db (Pr)1/3 (vp/vO*5 (2-43) where p = crystal density i c i p = solution density i vc = linear crystal growth rate | v = bulk fluid velocity D = diffusivity U = viscosity Pr = Prandtl number From equations 2-42 and 2-43, the downstream distance at which the surface concentration CQ decreases to the equilibrium solubility C is given by should be observed when x>xg , hypothesized to result in inclusion formation. Furthermore, for a given value of crystal growth rate v , the maximum value of xg will increase (according to equation 2-44) with the solution flow rate v. While Carlson's theory predicts general effects which are observed, it is an oversimplification and cannot be correct in its details. For example, it predicts that growth ceases when x = x r this would lead to a very large s step, rather than inclusion formation. Inclusions occur before this point and indeed a facet cannot be produced if the growth rate is so large as to yield such total depletion. eq 0.214vD(C -C ) oo eq 2 x (2-44) s Since growth requires that CQ>C^^, a "starvation” effect 40 CHAPTER III EXPERIMENTAL METHOD (A) General Principles of the Schlieren Method Schlieren is one of the fundamental methods of observing phenomena associated with fluid flow (76) . In our experiments on free convection in solution growth of crystals, rising convection columns of lower density fluid were revealed by the Schlieren image. As a means of flow visualization it was used to measure plume velocities and behavior. A sharp silhouette image also results which was useful for measuring changes of size in the crystals. The Schlieren method depends on refraction of a light beam by gradients in the refractive index of the fluid through which the light beam passes. Thus, an image is formed wherein the variations in brightness depend on differences in the gradients of refractive index in the light’s path. For most types of liquids, it has been shown (77) that the refractive index n of a liquid is related to its density p by the empirical equation D - n2-1 (3-1) c(n^O.4) where c is a constant. This equation has been found to be valid within ^ 3% over an extended temperature range. 41 Thus changes in density, are accompanied by changes of refractive index which may be observed by optical tech niques. Since density depends on temperature and compo sition, Schlieren can provide a quantitative measurement of either a concentration gradient or a temperature gradient. The formation of a Schlieren image depends on two superimposed optical systems. One provides general illumination of the field and forms a silhouette image of objects. The other produces variations in light intensity within the subject area depending on how the light is refracted by gradients in the index of refraction in the Schlieren field. Figure 3-1 illustrates how this double image is produced. The Schlieren-head lens M forms an image of the light source P at Q, passing through the Schlieren field S. The objective lens N then illuminates the screen T. At the same time, the objective lens N forms a real image of the point A at A r, the planes S and T being conjugate about lens N. Most important is the fact that all rays of light passing through any point in plane S form a real image at T, irrespective of the angle at which the rays pass through plane S. Thus, any object in the Schlieren field, plane S, produces a real image at plane T. Because the presence of a gradient in refractive index at plane S causes the rays of light to be refracted, 12 M S N T M S N T Fig. 3-1: Diagram of Schlieren optics (89) (P) point light source (M) Schlieren head lens (S) Schlieren field (N) objectives lens (T) Screen - £ = ■ say upward as at A, the image which they form at Q r is displaced from the normal position at Q. This displace ment y: . depends on the angle 0 , known as the "angular deviation" through which the rays are bent. Thus, all rays passing through A intersect the plane T at A' to form a real image and A also receives light from all points of the source P so that light passing through A produces an image of the source. The two independent modes of formation of the Schlieren image, therefore, are (1) displacement of the image of source by the angular deviation produced by a refractive-index gradient in the Schlieren field and (2) formation of a real image of the Schlieren field at the film plane. The average intensity of illumination of the image field is fixed by the position of the knife-edge. If a knife-edge is placed along the image Q in such a way that it intercepts a fraction of the light rays which have no angular deviation, the illumination of their corresponding points in the Schlieren image plane are uniformly decreased* Rays which do have an angular deviation produce a different illumination of the plane T at the point where they inter sect the plane, the intensity of illumination depending only on the angular deviation. Figure 3-2 shows the image Q ’ displaced from the image Q by a vertical distance y. Values of y are taken to be positive for displacements away from the knife edge 44 -I" I r Q a < T •1-tf I J_L * » * G , * //////. • • • Q................ y y / G Fig. 3-2: Displaced image of Schlieren source at the' knife edge GG when Q 1 is not intercepted (89) . 45 1 and negative for those which are toward the knife edge. edge is indicated by line G-G. If the knife edge is so placed that only a fraction a of the total luminous flux through Q passes it and if the image Q 1 is not intercepted !by it, then the conditions imposed on values of y and the ; ratio of the intensity of illumination I' at A' to the i ' intensity of illumination I at all other points in the ,plane T are given by 1 If y is sufficiently negative, Q' is totally intercepted by the knife edge. This condition and the corresponding j illumination ratio are given by | For intermediate values of y where - aS?<y<(1-a)6', as - shown in Figure 3-3, only fraction b of the total luminous j flux forming Q* passes the knife edge, where I The width of the images is 6 ’ and the position of the knife y > (l-a)<5 ’ II = I I a * (3-2) y <- a6’ (3-3) a<f< y < ( l-a)(T Fig. 3-3: Displaced image of Schlieren source at the knife edge when part of Qf is intercepted. -< i (B) Experimental Apparatus The observations we desired to make during crystal growth were (a) the average and local mass transfer rates, as measured by the change of size and shape of the crystal as a function of time, (b) the qualitative form of the convection pattern around the crystal and (c) the convec1 - tive velocity of the plume. As described in the last section (III-A), we expected that these requirements could be met by the technique of time-lapse Schlieren photo graphy. Therefore the experimental apparatus designed to study crystal growth from solution consisted of several parts: 1. A mirror Schlieren system, 2. Time-lapse cinematography attachments, 3. Sample cell and temperature control systems, 4. A vibration reduction system, and 5. An impact producer. These are now described in detail. 1. Mirror Schlieren system In practice, in order to minimize chromatic and spheri cal aberration in Schlieren systems, concave mirrors are often used in place of lenses, as we have done. A schematic diagram of our mirror Schlieren apparatus is shown in Fig. 3-4. The point light source A was a Sylvania concentrated-arc lamp (Model C-10) in which a tiny (0.4mm 48 IR Lamp Mirror C Mirror Lamp 4 3 * ’ - Crystal R Lamp o tK Lens Zoom Film i- Lens SCHLIEREN LIGHT PATH CRYSTAL IMAGE LIGHT PATH Fig. 3-4: Application of focused-image optics to mirror Schlieren system. The infrared lamps were used for heating the cell. mean diameter) brilliant (47 candles/mm) arc was maintained between a pointed rod and a concentric ring of sheet zirconium. After careful positioning of the lamp, we were able to focus this light onto a 4-inch diameter, 36-inch focal length concave spherical mirror B. This reflected a parallel collimated beam through the test area to a second spherical mirror C which refocused the undeviated light at point D. Here a knife edge cut off part of the focused beam, thereby uniformly dimming the background of the image in which the brightness at each point corresponded to the magnitude of the index of refraction gradient. In order to obtain good resolution pictures of growing crystals without loss of the Schlieren image, two convex lenses E and F were introduced between knife-edge D and camera G. It was important that the focal length of lens E, f£, be long enough so that the Schlieren image was large enough to fill the camera lens. Under these conditions, the Schlieren or shadowgraph image filled the film frame of the camera and the crystal image was also sharply focused on the film. For a detailed discussion of the optical design problem see reference (78). Any change in the zoom-lens setting required readjustment of the distance FG and refocusing on the crystal by readjusting distance CG. Finally, with the edge removed, a sharp silhouette 50 image formed. This "shadowgraph" model of the apparatus was used for measuring changes of size in the crystal. 2. Time-lapse cinematography Schlieren or shadowgraph pictures were taken with movie camera G which was mounted on a movable track so that sharp crystal images could be photographed. In order to obtain normal movie pictures when they were shown on the screen, the camera was mounted upside down. A Braun Nizo S480 8mm movie camera* was used. This camera included a Schneider variable-focus zoom lens, a through-the-lens viewer and automatic exposure control, and a built-in intervalometer with intervals ranging from 1/4 second to 200 seconds. The zoom lens was a very useful and convenient tool which provided both the view of the over-all fluid environment and a closeup view of the growth interface. It was found experimentally that the Schlieren image could fill the film frame of the camera only when the light cone converged to a focal point somewhere inside the zoom lens. For example, the zoom lens could have been placed between C and D. However, this position did not permit the insertion of the Schlieren knife edge. *Braun North America, Photo Products Division, Cambridge, MA 02142. 51 Therefore, an auxiliary convex lens F was inserted in front of the zoom lens. With regard to the Schlieren system, the camera position CG and the auxiliary lens distance FG were critically dependent on the zoom lens setting. However, if we inserted another convex lens at E, a parallel beam could be generated between E and F. Under these conditions the Schlieren image could fill the film frame and was independent of camera position CG. In practice, the focal length fg of lens £ had to be selected so that the parallel beam was about the same diameter as the zoom lens. The sharp crystal image could easily be obtained by adjusting only distance CG. The apparatus described above reached the following goals with complete satisfaction: (a) Clear Schlieren image of the fluid around the the crystal. (b) Sharply focused high-resolution imaging of the crystal profile. (c) Observation of both the over-all convection pattern and the closeup view of the growth interface. (d) Photographic records by automatic exposure control time-lapse camera. Two kinds of super 8 mm Kodak color movie film were used. These were Ektachrome 160 and Kodachrome 40. Only 52 the latter provided good pictures. The former always yielded over-exposed pictures, even when the knife edge was used. 5. Sample cell and temperature control system Cells were constructed from 6.4 mm thick plate glass which were ground to size and cemented with Dow Corning Silicone Adhesive. These cells, when inserted into the schlieren system, showed a few isolated schlieren striae but were adequate for the present use. The size of the cell was 15.2 cm x 12.7 cm x 2.5 cm or about 500 ml. A diagram of the crystallization cell is shown in Figure 3-5. It consisted of a glass solution cell, onto which was put a plastic lid. At the center of the lid, a large 0-ring seal permitted the insertion of a plastic plug through which was mounted a copper wire from which was suspended the seed crystal. In order to measure the size of the crystal two perpendicular numbered scales with lines 1 mm apart were made on the cell wall. At one end of the lid was mounted a small glass tube through which was inserted the thermocouple leads for temperature measurement or thermistor probe for temperature control. All of the experiments were supposed to have been carried out under isothermal conditions. However, by monitoring the ambient laboratory temperature, I found that there were diurnal temperature variations of up to 7°C. The chief difficulty in controlling the temperature ______________________________________________________________________5 . 3 _ Thermocouple Plug Leads Glass Tube Lid nn I lit illllilnl ill iillmi Ihilim Seed Crystal Scale 12.7 cm Fig. 3-5: Crystallization cell. 1 5 .2 cm of such a system was that any ordinary method of heating the cell, such as a heating wire around the cell, would heat by conduction inward from the walls of the cell. This would be expected to have caused considerable convection in the solution, the Schlieren from which would interfere with our observations. In the early experiments, infrared radia tion was tried with the hope that it might heat the solution more uniformly by being absorbed throughout the solution. Two infrared heat lamps were mounted above the concave mirrors of the Schlieren system and focused on the solution cell (see Fig. 3-4). The intensity of the lamps was regu lated by a voltage-proportioning control which was regulated, by a small thermistor probe immersed in the solution. In a preliminary test, monitored by the thermocouple immersed in the solution, it was found that the solution temperature could be stabilized thereby to within ±0.2°C. This system proved adequate for future experiments, but occasionally Schlieren were still observed in the solution next to the cell walls. It was then thought that the temperature regulation could be improved by using water jacketed cell windows to minimize heat losses from the cell to the air. A water jacket was made with 6.4 mm thick "Schlieren free" plate glasses. Its size was 15.6 cm x 15.2 cm x 8.9 cm, which was large enough to contain the crystallization cell. An insulated 38-liter constant temperature bath ______________________________________________ 55 was located 40.6 cm above the water jacket (see Fig. 3-6). The temperature of the bath was precisely controlled to ±0.01°C by a Bayley temperature controller* (Model-124). Temperature-controlled water smoothly flowed from the bath through 6mmID tygon ; “ tubing into the jacket under the static hydraulic pressure. There was an adjustable clamp on the tygon tubing which was used to adjust the water flow rate to the jacket. The water was then flowed to a well-insula ted 57-liter reservoir which was 81.3 cm below the cell. Finally, the water was pumped back to the constant tempera ture bath at a rate of 700 ml/min. The temperature of the cell could be controlled very accurately without causing any convection Schlieren. 4. Vibration-reduction set-up The most successful way to eliminate vibrations was to mount the crystallization cell with its water jacket on a 3.8 cm thick wood block which rested on a 7.6 cm thick polyether sponge immersed in glycerol (see Fig. 3-7). This set-up reduced the background vibration level to about 2x - 3 2' gg (gE=9:. 8m/sec ) .** When a 77 Kg person walked through the laboratory, the vibration level measured on the table on which the entire optical apparatus was mounted was about - 2 - 3 6.1x10 gE , while it was only 4x10 g^ on this set-up. *Bayley Instrument Co., Danville, CA 94526. **A11 acceleration values were measured by an Endevco Accelerometer Model 2219, Endevco,Co., Pasadena, CA. _________ 56 Constant Temperature B a t h ------------- insulated by 3 cm \ thick polyether / sponge 77777777777777777 cell W ater Jacket ✓ _ Insulated Reservior (insulated by 3 cm thick polyether) sponge Pump Fig. 3-6: Constant temperature water flow system. 8 1 .3 c m 40.6 cm Cell Water Jacket Cooling Water Woo 00 Polyether Sponge' Container Glycerol Fig. 3-7: Vibration-reduction set-up. 58 5. Impact Producer In order to study the effects of transient vibration on crystal growth, an impact producer was built and was mounted on the table (see Fig. 3-8). A steel shaft was coupled at one end to a d.c. motor and was supported by a bearing. At its other end, a circular aluminum cam was screwed. Four screws were set equal distance along the cam periphery. The screws were replaceable. Several different screw lengths were prepared in order to enable production of different amplitudes of impact. The rotary motion of the shaft was converted into an impact motion by the cam which lifted the hammer to a certain height and then released it to hit the cell wall. Some extra weight, such as metal washers, could be added onto the hammer to increase the impact strength. The strength of the impact was, therefore, varied by the screw length and the hammer weight. The shaft was driven by a Minarik* gear motor type KC1 2 2 RC. The frequency of the impact was varied by regulating the supply voltage to the d.c. motor or by decreasing the number of screws on the cam. (C) Material Preparation 1. Preparation of solutions Two constant temperature baths were assembled for the preparation and storage of saturated solutions. In all *Minarik Electric Co., Los Angeles, CA 90013. 59 Steel Shaft connected to dc motor c> Circular Cam ^ Screw Cell Hammer Fig. 3-8: Impact producer. 60 experiments described below, the term "saturated at 30°C" implies that the solution was initially stirred with excess solid solute for 1 hour at 40°C, then equilibrated with stirring in the 30°C bath for at least 48 hours, filtered and finally stored at 35°C until used. Subsequent experiments showed that continuous stirring was necessary for the equilibration procedure since free convection was not appreciable. Each such batch consisted of 7.6 liters of NaClO^ solution saturated at 30°C. The solubility, viscosity and density of the NaClO^ saturated solution were measured as functions of tempera ture. Measurements of the solubility of NaClO^ in water were made by slowly adding successive small portions of NaClOg (5g each time) to 50g of deionited water. This was stirred constantly and kept in a constant temperature bath whose temperature had been adjusted to that at which the solubility was to be determined. It was assumed that equilibrium was reached when undissolved solute remained after 4 hours of vigorous stirring. The magnetic stirrer was then stopped. When all undissolved solid had settled, a portion of the saturated solution was filtered and poured into a 250 ml beaker and the weight of the solution determined. After that the saturated solution was evaporated to dryness in a vacuum furnace at 50°C. The weight of solid which had been dissolved in a known weight of solution was thus determined, and from this the weight of solid dissolved in lOOg of water. From these data, we plotted the solubility curve of NaClC>3 in water shown in Fig. 3-9. Some results from the reference (79) are also shown in the figure. Agreement was within about 3%. Since the change of density with temperature is important in calculating the Grashof number, we made density measurements by using a capped 25 ml Weld pycno- meter carefully equilibrated in a water bath. The glass cap was fitted over the end of the pycnometer to prevent evaporation of solution. The solution was injected slowly into the pycnometer by a hypodermic syringe to avoid trapping air bubbles. All air bubbles trapped around the wall were removed by tapping or tilting the pycnometer. We determined (a) the mass of the empty pycnometer }(b) the mass of the pycnometer filled with deionized water Mr, (c) the mass of the pycnometer filled with solution Mx . The solution density px was then calculated by _ ( . MX-lMo-) Pr Px = (Mr-Mo) (3-5) Where pr was the density of deionized water. One possible error arose from the absorption of an uncertain amount of moisture on the glass. It was necessary to wipe the pycnometer with a dry cloth and allow it to stand in the balance case for 1/2 minute before weighing. Figure 3-10 _______________________________________________________________________&2L 50 u> 90 20 30 TEM PERA TU R E ( ° C ) 40 50 60 Fig. 3-9: The solubility curve of NaCIO, in H,0 . O my work ■ from reference (79). O' O) shows the density of saturated NaClOg solutions, with the density measured at the saturation temperatures. Figure 3-11 shows the temperature dependence of the density of a 30°C - saturated NaClOg solution. The viscosity of NaClOg solutions was measured by a U-tube capillary viscometer. A definite amount of liquid was put in the vessel, which was placed in a constant temperature bath. The liquid level was raised above the higher mark in the capillary and released. The time it took to fall from the upper mark to the lower mark was measured. When such an instrument is used, the kinetic energy correction is negligibly small, H. = ct (3 - 6 ) P where c is a constant and t is time. If a suitable liquid for which viscosity yr and density pr are known is available, the viscosity of the test solution px can be obtained relative to that of the reference liquid (deionized water) in the same viscometer from the relationship Em = ty Pv t3'7) |i p T ^ P p The viscosity of saturated NaClO^ solutions at various temperatures, with the viscosity measured at the saturation temperatures is shown in Figure 3-12. The viscosity of a 30°C - saturated solution at temperatures below 30°C is , _______________________________________________________________________64 DENSITY (g/cm3 ) 1.48 .46 1.44 .42 .40 1 . 38, 40 30 20 TE M P E R A TU R E (°C) Fig. 3-10: The density of saturated NaClO^ solu tions with the density measured at the saturation temperatures. DENSITY ( g / c m 1.452 .448 22 24 TEMPERATURE O Fig. 3-11: The density of a supersaturated NaClO^ solution which was saturated at 30°C. shown in Figure 3-13. In some cases, the NaClO^ solutions were doped with Ag by adding AgClO^. These solutions were then used to study the effects of transient vibration on crystal growth. For all experiments, the atomic ratio of Ag to Na in the solution was about 1/500. This was prepared by adding 1.06g AgClO^ to 400 cc of 30°C - saturated NaClO^ solution. Two precautions were observed with these doped solutions. First, metal wire (for example Nichrome wire) with a higher oxidation potential than Ag was not used to suspend the seed crystal, so as to avoid the following displacement reaction: Ag + + M M+ + Agf. Therefore, 0.015" Teflon wires were used to suspend seed crystals in the doped solutions. Second, the AgClO^ was added to the solution just before use. The NaClO^ solution became slightly cloudy when AgClOg was added. If the solution was stored for a long time after doping, the color of the doped solution always became brown and some brown particles were often found on the bottom of the cell. This indicated precipitation of AgCl due to probable: presence of Cl~ in NaClO^ and AgClO^. After contact with light, the AgCl particles might then convert, at least partially, to Ag and settle on the bottom of the cell. In these experiments 99% pure analytical reagent grade NaClO^ (Mallinckrodt, Inc.) and 9 5.99% pure AgClOg (K.§ K. Laboratories, Inc.) were used. 67 _____________________________________________________________________________________________; _____________________________________ I VISCOSITY ( C P ) F ig . 3 3,00 2.00 .60 40 20 TEMPERATURE O -12: The viscosity of saturated NaClO^ solutions with the viscosity measured at the saturation temperatures. 68 VISCOSITY ( C P ) 1 I 2.40 20 24 26 28 TEMPERATURE (°C) Fig. 3-13: The viscosity of a supersaturated NaClO^ solution which was saturated at 30°C. CT\ 2. Preparation of seed crystals The seed crystals were prepared by pouring a 25°C - saturated solution into an uncovered 250 ml beaker and lo cating it in an undisturbed place at about 22°C. As the solution evaporated, a few crystals usually began to grow on the bottom. The good seeds were transparent and with out any apparent defects to the naked eye. They were picked up with tweezers and placed onto a paper tissue for drying. The best way to dry the seeds was to wrap them in soft tissues several times. The size of the seeds obtained by this method was about 6.0x5.0x2.5 mm.* Their shapes were thin square or rectangular plates. In some cases, the seeds were cut by a 0.2 mm wire diamond saw to the desired size. Then, a 0.46 mm hole was drilled at the center of each crystal for the suspension wire. The drill worked much better and faster when wet with water, than when dry. The damage caused by the wire saw was removed by polishing the cut part of the crystal on a filter paper which was dampened with a 70% saturated NaClO^ solution. The damage caused by the drill was removed by placing the crystal into the 95% saturated NaClO^ solution for 1 minute with agitation. I also developed a technique for growing highly perfect seed crystals directly on a suspension wire. There were several disadvantages for seeds prepared by * In all dimensions given here, the last is the height. 70 previously-described way; (1) It was difficult to grow a large seed crystal (length or width is larger than 8 mm) without a hollow on the center of bottom face. (2) It took a long time to obtain a seed crystal with a height larger than 5 mm. (3) The damage caused by the diamond wire saw and the drill was inevitable. This new method was accomplished by dipping a 0.00 5” nichrome wire into a saturated solution for a few seconds, withdrawing, and then letting the wire dry in air so as to form many tiny crystals on it. The wire was then put into a supersaturated solution, causing the tiny crystals to grow to about 0.5 mm cubes. After that, the wire was removed to a 95% saturated solution in which a single crystal was selected and the others removed by tweezers. Finally, this single crystal was placed in another super saturated solution for further growth. (D) Experimental Procedures The size and weight of the seed crystal was measured before each experiment. The weighing of the crystal was carried out with an analytical balance with an accuracy of 10"3g. The size was measured with a caliper scale _ o with an accuracy of 10 cm. A NaClO^ solution saturated at 30°C was heated to 33°C and was poured into a preheated 71 cell. The nichrome wire (D=0.013cm) for suspending the seed was inserted through the 0-ring seal which allowed the crystal position to be adjusted very easily without lifting the lid or disturbing the solution. The crystallization cell was then covered by the lid. Since evaporation of water was prevented thereby, the solution and growing crystal could be considered as a closed system. At this moment, the seed crystal was placed in the cell at a position above the solution and allowed to reach a state of thermal equilibrium. At the same time, the solution temperature was still kept about 3°C above the saturation temperature for 10 minutes or longer. This presumably dissolved any tiny crystal which might have fallen from the seed into the solution. The cell with water jacket was then placed in the Schlieren apparatus and the knife was adjusted until a uniform gray back ground was achieved. To keep the temperature of the solution in the cell constant, the water in the jacket was recycled at a rate of 700 ml/min by a pump from the constant temperature _ ? bath, which was accurately controlled within 10 °C. Once a desired temperature had been reached, the seed crystal was carefully lowered down into the solution. The movie camera was turned on to take pictures contin uously, thereby recording the flow pattern around the growing crystal. The solutions were kept isothermal 7.2 during growth. Sometimes, mechanical impacts were intro duced during the experiments. At the completion of the growth cycle, the crystal was slowly withdrawn from the solution and was dried immediately with a paper tissue. The dry crystal was then examined by an optical microscope to observe features such as hillocks, hoppers and inclu sions. The size and weight of the crystal after growth were measured again by caliper scales and balance . In order to study the effects of transient vibration on impurity distribution, l«-06g of AgClO^ was added to 400 cc of NaClOj solution saturated at 30°C. The ratio of Na to Ag in the solution was about 500. During the growth period, mechanical impacts were introduced by the impact producer with several different frequencies and amplitudes. At the end of the run, a thin platelet of crystal was cut from the grown seed by a diamond wire saw. The platelet was polished by rubbing on filter paper containing a 95% alcohol (in water) solution. The slice was then coated with a thin layer of carbon for electron microprobe analysis of Ag. A 2ym of diameter of electron beam was focused onto a spot of the specimen, where the resulting excitation of the electrons of atom in the material produced characteristic x-radiation which was * monitored by detectors and recorded on counters. The counter output was periodically printed onto tape for data correction and evaluation. The possible errors by this method, unfortunately, were as high as 60%. The movies, after developing, were used for measuring the irregularity velocities. The movie films were projec- ' ted by a movie projector outfitted with stop motion, so as to give a resultant magnification of 10 times. This methodj 'allowed measurement of 0.1 mm quite accurately. The | i fposition of an irregularity was measured frame by frame. ' ■ From this information we calculated the instantaneous velocities of many irregularities. | Average over-all mass transfer coefficients k were i i :calculated directly from the weight increment Am of the I 1 :crystal m time interval At. 1 ] Since 1 N = v C = (3-8) : A MA c c 1-F v J where = mole flux of solute incorporated into the crystal 2 (mole/sec*cm ) vc = crystal growth rate (cm/sec) 3 Cc = molar concentration of crystal (mole/cm ) k = over-all average mass transfer coefficient (cm/sec) F = volume fraction of solutefin solution at interface m = mass of crystal (g) 2 : A = surface area of crystal (cm ) i i t = time (sec) ' M = molecular weight of solute (g/mole) 3 C = concentration of solute in solution (mole/cm ). i _____________ _74J Please note that Eq. 3-8 is not the same as Eq. 2-40, because for Eq. 2-40 we assumed constant density in the solution, while for this equation constant molar volumes i |are assumed. For NaClO^ solutions at 18 ‘°C, the relative i change of molar volume with respect to concentration is lower than the relative change of density with respect to i concentration (see Table A-l). Rearranging Eq. 3-8 we obtain k = U'F) ’ (Am/At) AC-M-A i The Sherwood number and the stagnant film thickness §s are given by Sh =? C-C . oo „ 8 (c - _ c) o «> Hi) kL D 6s y=0 The detailed calculations of F and Sh are given in Appendix A. CHAPTER IV EXPERIMENTAL RESULTS Many preliminary experiments were conducted to deter mine the parameters which affect the onset of convection irregularities during solution crystal growth in the absence of stirring. The term "convection irregularities" is taken to imply the existence of eddy motions in the convection plume leaving the growing crystal. There was a velocity component normal to the flow axis of the plume, and some degree of mixing of the fluid was observed. This was characterized either by a "mushroom" type or a "random eddy1 type of flow pattern. Our interest was principally in: (.a) The horizontal characteristic dimension L of the crystal, where L is the ratio of the surface area to the perimeter. (b) The vertical dimension (height h) of the crystal. (c) The depth H of the top of the crystal below the top* of the solution. (d) The supercooling AT of the solution. (e) The presence or absence of a free surface. ,(f) Rigid or flexible holding of crystal. 76 The preliminary observations showed that the convection irregularity of the plume was primarily dependent on the horizontal crystal size and the'degree of supersaturation. That is, the convection irregularity became more vigorous as crystal size increased and as supersaturation increased. The formation of growth defects and irregularities on the top face, bottom face and upper parts of side faces were associated with prolonged irregularity. A systematic experimental study was then planned to determine the effect of these two factors on crystal quality. Many phenomena which help in understanding of formation of inclusions were observed during the experi ments. The hydrodynamic results for NaC103 solution growth are presented first. The crystal defect structures from microscopic observation are presented next. The influence of mechanical perturbations on crystal growth is given at the end of this chapter. (A) Hydrodynamics 1. Three types of convection patterns The natural convection currents around sodium chlorate (NaC103) and Rochelle salt (KNaC4H406 * 4^0) crystals were observed by the Schlieren method. The pattern of this convection current was found to be significantly dependent on the crystal size and degree of supersaturation. Three 77 kinds of convection patterns were observed. Figure 4-1 displays laminar convection around a growing NaC103 crystal. The size of the crystal was 7.0 mm long x 6.9 mm wide x 6.3 mm high. (The definition of length, width and height of a growing crystal are shown in Figure 4-2. Here the last dimension always signifies the height of a crystal.) The supercooling AT was 5°C and Gr was 176* Some very well-defined laminar streamlines rose from all of the top of the crystal. The streamlines were smooth and regular. There was no structure or mixing within the convection columns in the plume and the entire system appeared stationary. Figure 4-3 shows another type of flow characterized by ’’mushroom" formation at same spots at regular time inter vals along the convection column. This behavior repre sented the early transition stage of the convection, with eddies carried up at fairly constant intervals in the mainly laminar stream. There was similarity between the behavior shown in Figure 4-3 and that of Figure 4-1 in that the convection was still largely laminar in this early transition stage. As the crystal size or supersaturation increased further, turbulence was more apparent. Streamlines were distorted and fluctuated. Irregular eddies formed more vigorously. Figure 4-4 shows early turbulent flow behavior. * Note. All Grashof numbers given here were calculated by assuming equilibrium at the crystal surface. Fig. 4-1: Laminar plume above a growing NaClO^ crystal (7.0mmx6.9mmx6.3mm) AT = 5 ° C (30°C^25°C) Gr = 176. (Assuming p=p at crystal surface) 79 Width (w) VIEWING DIRECTION _o> <D Fig. 4-2: The definitions of the three dimensions of crystals and L. L = (wxl)/2x(w+1). 80 Fig. 4-3: Early transition stage with irregular plume formation (8.Immx7.4mmx8.Omm NaC103) AT = 5 ° C (30°O25°C) Gr = 243. 81 Fig. 4-4: Late transition stage with more irregular eddies (8.8mmx8.Immx7.6mm NaClO^) AT = 5 ° C (30°O25°C) Gr = 315. 82 These results were reproducible. Similar phenomena were also found in Rochelle salt growth, as shown in Fig. 4-5. Figure 4-6 shows that the convection oscillations were still observed, even when the NaClO^ crystal was mounted on a rigid 1.6 mm diameter metal wire. We have therefore discounted the significance of rigid or flexible holding of crystals to convection irregularities. 2. The effect of crystal size on convection irregularities The purpose of these studies was to understand the relationship between crystal size and convection irregular ities. It was mentioned in the last section that there were three kinds of convection patterns around growing crystals. Figure 4-7 summarizes the Schlieren observations of the convection around growing NaC103 crystals in terms of crystal height h and horizontal characteristic dimension A L. The characteristic dimension L is defined as L=— , where A is the area of the top surface (in our studies it 2 2 ranged from 7.8 mm to 134 mm ) and p is the perimeter of the top surface (ranged from 18 mm to 49 mm). All experiments were run at the same supercooling, AT=5°C (30°C ^ 25°C). Crystals with sizes above the band shown were always found to have convection irregularities. This region was dominated by vigorous eddy convection. The flow was always 83 Fig. 4-5: Convection irregularities in Rochelle salt growth crystal size = 12.5mmxl6.8mmx22mm AT = 1°C ( 30 ° Cv2 9 °C) . 84 Fig. 4-6: Convection irregularities with NaClO^ crystal mounted on the stiff 1.6 mm diameter copper wire; crystal size = 9.7mmx8.4mmx5.4mm AT = 5 ° C (30°C^25°C) Gr = 383. 85 CHARACTERISTIC LENGTH ( m m ) THE HEIGHT OF SEED CRYSTALS (mm) Fig. 4-7: Influence of crystal dimensions on convection behavior with AT = 5°C (30 °C^25 °C). O regular convection plume 0 early irregular plume ■ irregular plume. stable laminar if the crystal size was below the band. The band area indicates the region of intermittent stable and irregular convection. Sometimes, continuous stable or irregular convection was observed in this band area. This was probably due to background vibration, as will become more clear later. It was found that the horizontal dimen sions of the crystal has a much stronger effect on the type of convection than did its height. As the crystal size increased the flow became less regular. The observations of the transition from regular to irregular convection with increasing crystal size appear to be in qualitative agreement with Wragg and Loomba's electrolytic mass trans fer results (26). They found that the electrode diameter was an important factor in determining the convection behavior. Figure 4-8 shows this kind of transition with the same NaClO^ crystal throughout at a fixed supercooling. When the crystal was small, the convection was very stable. As the crystal size increased, the stable condition broke down (Fig. 4-8b and Fig. 4-8c). During the whole four-hour growth period in 400 ml of solution, 0.53 g of solute was incorporated in the crystal, causing only a 4.5% decrease in the relative supersaturation. (For detailed calculation see Appendix B). The crystal height was also found to influence the convection regularity, although to a lesser degree. The 87 (a) Laminar convection (b) transitional convection (c) irregular convection size = 9.3mmx8.0mmx6.9mm size = 9.8mmx8.5mmx7.3mm size = 10.5mmx9.2mmx7.7mm Gr = 258 Gr = 306 Gr = 383 Fig. 4-8: Transition from stable state to irregular state with same NaClO^ crystal. AT = 4°C (30°C^26°C). oo oo result in Fig. 4-7 indicates that a crystal with the same horizontal surface area, but with larger height, would be more stable. This result is contrary to our intuition. In Chapter V I will explain why this ought to be so. 3. The effect of crystal size on plume and eddy formation Crystal size had a strong effect on the convection behavior. It is interesting to study this in more detail. Figure 4-9 revealed a clear tendency for both the convec tion column width and the number of eddies to increase with the crystal size, finally becoming more and more turbulent. Figure 4-10 shows a plot of the ratio of neck width W of the plume to the crystal perimeter P versus Grashof number Gr at AT=8°C (30°C^22°C). The neck width was defined as the width of the narrowest part of the convec tion column. The data seems to be represented well by a straight line, During the growth period, the width of the plume was not constant. The plume width steadily increased with time when convection was regular. The size of the plume sometimes fluctuated when convection was irregular. When eddies passed through the plume, its size sometimes increased temporarily. The neck width W, the number of convection streamlines ns, and the number of irregular eddies ne, were measured from the movie frames. As shown by Fig. 4-9, the number of convection ! (a) size=2. 0mmx8 . 4minx5 . 0 Gr=28 (b) size=4.5mmx8.7mmx5. Oram Gr=172 (c) size=13.6mmx8.9mmx5.Omm Gr=1024 (d) size=15.7mmx8.5mmx5.Omn Gr=1102 Fig. 4-9: Influence of crystal size on eddy formation and the width W of the plume. NaCIOAT = 8°C (30°C^22°C). UD O d/M 0.20 o O 0 . 1 0 1400 1600 Gr Fig. 4-10: The influence of Gr on the ratio of the plume neck width W to perimeter P. Height h = 5.0mm. AT = 8°C (30°O22°C) . streamlines and the number of irregular eddies in the plume both increased with the crystal size. The observed results are plotted in Fig. 4-11 and Fig. 4-12 as a function of Gr at AT=8°C. The irregular eddies started to occur at Gr = 240 (or L = 1.65, AT = 8°C). As Gr increased further to about 1460 (L = 3.03, AT = 8°C), the turbulence became more vigorous and the irregular eddies were observed in almost every streamline. All of the crystals used here had the same height, h = 5.00 mm. Also note that, from Fig. 4-10 to Fig. 4-12, L was the only parameter varied in 3 Gr, and those figures actually show the influence of L on W/p, ns or ne. The frequency of generation of irregulari ties at a given spot was relatively constant. The period icities of the appearance of eddies in the plume were measured for a range of operating conditions. This infor mation for NaClO^ is brought together in Fig. 4-13, wherein the period is plotted as a function of the growth rate. The periods shown here were values averaged over the total crystal. The growth rates were varied by varying both AT and L independently. The average period decreased markedly as growth rate increased, that is the rate of generation of irregularities increased as the growth rate increased. The upward rate of movement of irregularities in the plume was also measured from the movie frames. It was found that eddy velocity differed from point to point. An eddy rising 92 NUMBER O F STREAMLINES 20 Gr Fig. 4-11: The number of convection streamlines in the plume vs. Gr for growing NaClO^ crystals. h=5.00mm, AT = 8°C (30°Cr L22°C) . LKI Number of Irregular Eddies 1200 1400 1600 800 Gr Fig. 4-12: The number of irregular eddies in the plume vs. Gr for growing NaClO^ crystals. h=5.00mm. AT = 8°C (30°Cv22°C). o CD cn “O o Ql > CL 0,25 Growth Rate (g/hr) Fig. 4-13: Average period of formation of irregularities for growing NaClO^ crystals. (AT and L both were varied.) Q - AT=5 °C 0 ~ AT = 4°C ■ - AT = 3°C All solutions saturated at 30°C. 95 from the top crystal face at the center of the field of view accelerated for a short distance after which rose at a nearly constant velocity. Figure 4-14 shows the eddy velocity v, expressed as Reynold’s number Re, as a function of y/L for several sizes of NaClO^ crystals at AT = 5°C. The Reynold’s number is defined as Re = > and y is the vertical distance from the top crystal face. The fluid properties (density p and viscosity y) were evaluated as the arithmetic mean of those in the bulk solution and for a saturated solution at the growth temperature. The velocity of irregularities above the smaller crystal was lower than those above the larger crystal, since the irregularities emitted at the upper surface of the crystal mostly came from solution that had traversed the lower and side sur faces and hence had already been depleted of solute to some extent. If all other growth conditions were the same, the density of spent solution of larger crystals would be lower because the solution becomes more depleted in solute before emission. The density difference between the depleted solution and bulk solution is larger for larger crystals. Consequently, higher irregularity velocity was expected for larger crystal. 96 Reynolds Number Re A Fig. 4-14: The velocity v of irregularities in the plume at the center of the field of view AT= 5 ° C (30°C^25°C). Size of NaC103 crystals (A) 15.Immx7.8mmx4.4mm,(B) 7.4mmx7.lmmxl.1mm (C) 5.3mmx6.7mmxl.8mm,(y is the distance above the top of the crystal). 97 4. The effect of supersaturation on convection irregularities In all experiments, the supersaturation was produced by reducing the temperature. It was found that the convection currents around a crystal became Irregular at large supercooling AT. The average frequency of generation of irregularities over the crystal and the irregularity velocity at the center of field of view both increased with AT, as shown in Fig. 4-13 and Fig. 4-15 respectively. In the next section we use further experimental results to show that the effect of supersaturation can be contained in Gr along with L. 5. The importance of Grashof number to convective irregularities during solution growth As discussed in Chapter II the Grashof number of mass transfer is Gr = - ——P , which contains the two most P2 important experimentally varied parameters, crystal size L and density difference Ap. Figure 4-16(a) shows a small NaClO^ crystal (with size = 3.1 mm x 8.9 mm x 5.0 mm) which grew with laminar convection, even though the solu tion was at high supercooling AT = 8°C. Since L was small in this case, Gr was only 80. Figure 4-16(b) shows a large NaClO^ crystal (18.6 mm x 9.0 mm x 5.0 mm) which grew with irregular flow when AT was 3°C, but Gr was 560. 98 CD cn o y / L Fig. 4-15: The velocity of irregularities at the center of the field of view as a function of y and AT. (A) AT=8°C, Gr=893 (B) AT=3°C, Gr=341. Size of NaClO^ crystal = 9.2mm x 11.6mm x 5.0mm. Solution saturated at 30°C. 99 100 (a) (b) Crystal size=3.Immx8.9mmx5.Omm Crystal size=18.6mmx9.0mmx5.0mm AT=8°C (30°C'v22°C) AT=3°C (30°C'v27°C) Gr=80 Gr= 560 Fig. 4-16: Convection behavior around growing NaCIO, crystals. Experimentally, observations of convection irregulari ties around NaClO^ crystals for various sizes and values of AT are summarized in Figure 4-17 in terms of the Grashof number Gr and crystal height h. All crystals were suspended in 1% to 7% supersaturated NaC103 solutions. In the region above the top curve, the plume was always irregular, while in the region below the bottom curve, the plume was always stable. The band between these curves indicates the region of transition, in which irregular convection was still sometimes observed, probably due to background vibrations. As evidenced by the figure, the Grashof number above which irregularities were observed, varied with the crystal height h. The shorter the crystal the smaller the value of Gr required to produce irregularities. Figure 4-18 shows a short NaClC^ crystal which grew at AT=5°C (30°C v 25°C) with irregular convection currents. The crystal size was 7.4 mm x 7.1 mm x 1.1 mm and the corresponding Gr was 200. Figure 4-19 shows another NaClO^ crystal which grew exactly at the same conditions as the former one, but had a regular laminar plume. The size of the crystal was 8.0 mm x 6.9 mm x 10.0 mm and the Grashof number Gr = 213. Although the latter crystal had a higher Gr it still grew with a regular plume since its height was larger. The crystal growth rate in g/cm^hr vs. Gr is shown in Fig. 4-20. Gr was varied by varying both AT and L 101 500 ° 400 <D £ z o sz U) o <3 300 . rre <jular region 200 h —1 o K) The Height of Seed Crystal (mm) Fig. 4-17: Influence of Gr and h on convection behavior. AT varied from 1°C to 8°C, L varied from 0.44 mm to 3.03 mm. Solution saturated at 30°C ■ irregular plume Q regular plume Q transition (occasional irregularities) Fig . 4-18: Irregular convection currents above a short NaClO^ crystal. The size of the crystal = 7.4mmx7.lmmxl.1mm AT = 5 ° C (30°O25°C) Gr = 200. 103 Fig. 4-19: Regular laminar plume above a NaClO^ crystal. The size of the crystal = 8.0mmx6.9mmxl0.Omm. AT = 5°C (30°C^25°C) Gr = 213. 104 Growth Rate ( g / h r - c m 2) A T=8°C 700 400 Gr Fig. 4-20: The NaClO^ crystal growth rate as a function of Gr and AT. All solutions were saturated at 30°C. O Cn independently, and the crystal height h was varied from 0.8 mm to 14.5 mm. The growth rate decreased with increasing Gr (increasing L) but increased with increasing AT. These results are in agreement with Wragg's electro lytic mass transfer data for CuSO^ (80) . The reason for this is that for typical laminar free convection mass v v transfer from a suspended body (81), Sh=A+B Gr4Sc4 or 3/4 k L = a+b L , where a, b, A, B, are constants for a fixed AT, Ap, y , p, g, D etc. (only L varies). From this 3 b we see that k = j - + — so we expect k to decrease with L increasing L, or growth rate to decrease with increasing Gr (at fixed AT). Figure 4-21 presents the mass transfer data for NaClO^ crystal growth. The Sherwood number*is plotted as a func tion of the Grashof number on logarithmic coordinates. The shapes of the symbols reflect various growth tempera tures, as indicated in the figure’s caption. L was varied from 0.44 mm to 2.81 mm and h was varied from 0.8 mm to 14.5 mm. Within the scatter (approximately ±10%) the data could be correlated well by a straight line in the range 3 of Gr extending from 1 to 10 . A least squares fit through the points yielded the relation Sh = 1 . 563 Gr° ’ 252 (4-1) * As in calculating Gr, equilibrium at the crystal surface was assumed in calculating Sh. The effect of finite interface kinetics on the results is discussed in Appendix A. 106 8 — JZ CO 2 — Gr Fig. 4-21: Plot of the Sherwood number Sh against the Grashof number Gr for NaClO^ growth. All solutions were saturated at 30°C. Growth temperatures X22°C, O 25°c> #26°C, ■ 27°C, □ 28°C, 01 29°C, h=0.8 to 14.5mm, L = 0.44 to 2.81 mm. O 'J The summation of the squares of the difference between the calculated Sherwood number Sh and the experimental car Sherwood number Sh was 14.58155. I also completed least exp 1 squares fits on Sh = A + B Gr1 * (4-2) and Sh = 2 + A'GrB '. (4-3) The results are summarized in Table 4-1. It seems to me that Eq. 4-1 can fit the experimental data best, so I used this equation as a correlation equation for my results. Since the Schmidt number for NaClO^ solutions at 25°C is 3 around 10 (see Appendix C), the corresponding Rayleigh 3 6 number Ra = Gr*Sc ranged from 10 to 10 . Equation 4-1 could thus be rewritten as 0 2 5 2 Sh = 0. 274 Ra * (4-4) Forcing a slope of %, as has been reported in the litera ture for laminar free convection mass transfer (27,28), one obtains the relation Sh = 0.276 Ra0’25 (4-5) The data of Goldstein et al. (27) and Lloyd and Moran (28) 108 Table 4-1 Summary of least squares analysis Equation form 52 7 V (Sh - Sh ) 2 - * K cal exp' i = l 0 2 5 7 Sh=l.56 3Gr % Sh=-0.029+1.59Gr Sh=2+0.4Gr0,43 14.581550 14.585470 16.166620 109 can be compared directly to the present work since exactly the same definition was used for the characteristic length L in the Rayleigh number. Their constants in the correla ting equation were 0.59 for Goldstein et al. and 0.54 for Lloyd and Moran. In Figure 4-21, the height of the crystal used was varied between 0.8 mm and 14.5 mm. L was from 0.44 mm to 2.81 mm. Although the crystal height h was found important to convection irregularities of the plume above crystals, it was not an important factor on deter mining the mass transfer rate as seen in the figure. Due to limitations of the apparatus, we only obtained the average linear growth rates on the horizontal direction and the vertical direction. This was done by caliper measurements at the end of each set of experiments, to yield the results shown in Figure 4-22 and Figure 4-23 respectively. Within the scatter of the data, approximately ±15%, the average horizontal linear growth rate (cm/hr) had the same tendency as Fig. 4-20. The growth rate increased with decreasing Gr and increasing AT. Although the vertical growth rate, which scattered about 20%, increased with increasing AT, it didn’t have clear tendency with Gr. More accurate data is needed. Figure 4-24 shows the ratio of the horizontal growth rate to the vertical growth rate 3 against Gr (or L ). The data scattered so widely that only some qualitative trends can be discerned. The horizontal 110 Horizontal Linear Growth Rate (cm/hr) 0.08 0.06 0.02 100 200 400 500 600 300 700 Gr Fig. 4-22: The average horizontal linear growth rate of NaClO^ crystals against Gr. All solutions saturated at 30°C. (■) AT=8°C, (£) AT=5°C, (X)AT=4°C (Q)AT= 30 C, OAT=2°C, ®J)AT=1°C, h=0.8 to 14.5mm, L = 0.44 to 2.81 mm. Vertical Lineal Growth Rate(cm/hr) 0.08 0.06 0.04 0.02 700 Fig. 4-23: The average vertical linear growth rate of NaClO^ crystals against Gr. All solutions saturated at 30°C. J||)AT=80C, (jJ)AT=5°C, (X)AT=40C, 0)AT=3oC, O AT= 20C, U) AT=10C. h=0.8 to 14.5 mm, L=0.44 to 2.81mm. bO crystals against Gr. All solutions saturated at 30°C, (§f AT= AT=5°C, ( X ) AT=4°C, ( Q ) AT=3°C, O AT=2°C, ( f l | AT=1°C. £TI H* OP Horizontal Linear Growth Rate Vertical Linear Growth Rate ro -P» •-3 p" C D P P c+ H- O o Hi Ld O H H* t S I o p c+ P Crq H O s; P J P P c+ C D r+ O c C D P c+ H- n p G r q P O 3 r+ P J P P r+ C D O P n i —* o CkI CD rr growth rate was higher than the vertical growth rate. The ratio of horizontal growth rate to vertical growth rate decreased with increasing AT and Gr. Figure 4-25 shows that the terminal velocity vt of the irregularities in the plume at the center of the field of view was a function of Gr. The data are well represented by a straight line over a range of Gr extending from 0 to 3 10 . The equation of the line obtained by least squares is Re = 0.82 + 0.0071 x Gr. (4-6) 6. The effect of the depth of the crystal under the free surface on the convection regularity The influence of the depth of the seed crystal under the free surface H on the convection oscillations was studied by placing the same seed crystal in one saturated solution and varying its position. The convection patterns were observed by the Schlieren method. The results showed that this effect was important only when the crystal was near the solution surface, for example, when H <_ 2 . 4 cm for a NaClO^ crystal size of 7.5 mm x 6.4 mm x 7.1 mm at AT = 5°C (30°C^25°C). If a crystal was placed in a position very close to the free surface, some tiny transi tional convection irregularities were often observed at 114 6 4 2 0 0 200 1000 400 600 800 Gr Fig. 4-25: Plot of the Reynolds number of the plume against Gr. All NaClO^ solutions saturated at 30 ° C, (£) AT= 3 ° C , <0) AT=4°C, (Q AT=5°C (Q) AT=8°C. h-=0.8mm to 8.2mm, L=1.30 to 2.81 mm. 115 the center of the field of view over the crystal. When the same crystal was placed in a lower position, the convection remained steady. It occurred to us that these background vibrations at free surface might trigger the transition from steady laminar plumes to irregular plumes. Therefore in all experiments, the crystals were placed far below the free surface. For a 12 cm deep solution, the crystal was placed at a position 6 cm below the surface. (B) Crystal Imperfections The grown NaClO^ crystals were examined under a polarizing microscope. Surface irregularities and inclu sions were among the major defects. Crystal imperfections were always found when the crystals grew with irregular plumes. Imperfections were sometimes still observed even when crystals grew with steady plumes. This means that perfect crystals could be obtained with regular plumes, but not always. 1. Surface irregularities Several kinds of surface irregularities were found in NaClO^ crystals. These were: (a) Flaws or terraces on the top surface. 116 (b) Striations and growth steps on the upper part of side surfaces (c) Hollows on the center of bottom face. Figure 4-26 shows the flaws on the top surface of a NaClO^ crystal which grew at AT = 5°C(30°Cv25°C) with irregular convection currents (see Figure 4-27). The initial size of the crystal was 15.1 mm x 7.8 mm x 4.4 mm. Final size was 16 mm x 8.5 mm x 5.1 mm. The picture was taken by an optical microscope with reflected light. Therefore, most features seen on the picture are surface flaws. After careful examination, there seemed to be unsealed inclusions on the surface. As seen in Fig. 4-26, the traces of flaws at each corner of the rectangular crystal nearly coincided with the bisectors of the angles of the crystal. Beyond the four bisector lines, the orientation of the flaws was parallel to the direction of the convection. For a square crystal, diagonally patterned flaws were often found as shown in Fig. 4-28. This picture was taken by an optical microscope with transmitted light. Interior solution inclusions with bubbles were found just underneath the flaws. Interestingly, our flaw patterns were very similar to the convection patterns observed by Husar and Sparrow (82) for a heated horizontal plate facing upward. Figure 4-29 (a) and Fig. 4-29(b) show the laminar flow field for a Fig . 4-26: Flaws on the top surface of a NaClC>3 crystal. Incident light.AT=5°C (30°C^25°C). Size = 16mmx 8.5mmx 5.1mm Gr = 718. 118 Fig. 4-27: Irregular convection currents above a NaClO^ crystal (shown in Fig. 4-26) which formed flaws on its top surface. Size = 16mmx8.5mmx5.1mm AT = 5 ° C (30°O25°C) Gr = 718. 119 Fig. 4-28: Diagonally patterned flaws on the square top surface of a NaClO^ crystal. Size = 6.5mmx6.0mmx5.8mm AT = 5 ° C (30°O25°C) Gr = 127.5 120 121 Fig. 4-29: (a) (b) Flow patterns for horizontal plates in the laminar region (82). (a) rectangular plate (b) square plate Ra = 7.4xl04 Ra = 3.1xl04 size = 5.lcmxlO.2cmx0.05cm size = 8.9cmx8.9cmx0.05cm rectangular plate and a square plate. Figure 4-30 (a) and Fig. 4-30(b) represent the convection behavior in the early turbulent region. Comparing Fig. 4-26 to Fig. 4-30(a) and Fig. 4-28 to Fig. 4-29(b),we see that defect patterns similar to the flow irregularities were also found on the top surface of my growing crystals. From these photographs, it is seen that a basic characteristic of free convection flows adjacent to plates is the partitioning of the flow field. There was no flow across a partition line. Fluid moved inward from the edges along paths that were more or less straight and perpendicular to the edges. Upon reaching the neighborhood of the partition lines, the flow moved upward. Each partition line, therefore, was a cen tral element of the vertically-ascending buoyant plume. Since the partition lines were created by the interaction of different flow directions from the separate edges, we would expect these partition lines to be the most unstable regions. Terraces, as shown in Fig. 4-31, often formed on the top surface of a crystal when the convection was very irregular. The terraces at the center part of the crystal were higher than those at the periphery. When we examined the crystal with transmitted light, many interior solution inclusions with bubbles were found near the top surface as 122 (a) (b) Fig. 4-30: Flow patterns above horizontal plates in the early turbulent region (82). (a) rectangular heated plate (b) square heated plate size = 5.lcmxlO.2cmx0.0 5cm size = 8.9cmx8.9cmx0.05cm Ra = 1.5xl06 Ra = 2.3xl06 124 shown in Fig. 4-32. It seems likely that the same growth irregularities that gave rise to an irregular surface also gave rise to inclusions, and that surface irregularities are the precursors of inclusions. Growth streaks on the upper part of side surfaces were a common type of surface irregularity. They were actually many tiny unsealed inclusions. Figure 4-33 shows these streaks near the top of a NaClO^ crystal side face grown with irregular convection. They began some distance from the bottom face and grew in an upward direction, more or less parallel to each other. The streaks at the center of the face usually started at a lower position. Figure 4-34 shows a higher magnification photo (about 25x) of the streaks. They consisted of growth steps and open channels. We also saw many sealed solution inclusions next to the surface. Since inclusions cause light scattering, the upper part of the crystal in Fig. 4-33 was opaque while the lower part was transparent. The seepage of solution out of these streaks of inclusions sometimes caused various sizes of white pox of NaClO^ powder formed on the top surface, the side surface and especially at the base of the crystal suspension wire. It was often found that the upper center part of the side surface was slightly concave as shown in Fig. 4-35. 125 Fig. 4-32: Inclusions near the top surface of a NaClO^ crystal grown with irregular convection. AT = 4°C (30°C^26°C), size = 10.5mmx9.2mmx7.7 mm, G r = 3 79. 4-33: Growth streaks on the side face of c crystal with irregular convection. Size = 9mmx8.6mmx8.2mm AT = 5 ° C (30°O25°C) Gr = 357.0. NaC103 127 Fig. 4-34: A close view of growth streaks on a NaClO^ crystal side face with irregular convection. Size = 8.Immx7.5mmx7.2mm AT = 5 ° C (30°C^25°C) Gr = 248. 128 Fig. 4-35: Slightly concave upper center part of a NaClO^ crystal side surface with regular convection. Size = 7.4mmx6.9mmx6.5mm. AT = 5 ° C (30°C^25°C) Gr = 191. 129 This was another major type of surface irregularity. Different shapes of concavities formed on NaClO^ crystals are sketched in Fig. 4-36. U-shaped concavities like type (a) and type (b) were the most frequent ones. The top side of the concavity usually was wider than the bottom side. Inside the concavity, terraces sometimes could be found with height decreasing in the direction of the convection upward. When convection irregularities were vigorous, needle-like growth steps, as shown in Fig. 4-37, sometimes formed on the lower side of the concavity. The grooves between the growth steps seemed to connect to the flaws and inclusions. It seem likely that the grooves could have given rise to inclusions and flaws. Thus, sealed over grooves would be inclusions, while unsealed grooves left on the surfaces were flaws. Occasionally, Y-shape (type c) and rectangular shape (type d) concavities were also found. For Y-shaped concavities the growth of the crystal was retarded at the center of the face, leaving an open channel on it. It extended from the bottom face to the top face and became wider at the top, or became wider as one moved up. For rectangular shaped concavities, the growth of the crystal was retarded everywhere except at the edges. Quite often the shape of a concavity was not simple, but was the combination of two or more of the above shapes. For example, type (e) concavities shown in Fig. 4-36 were 130 7 (a) (b) (c) (d) (e) Fig. 4-36: Illustration of the various types of conca vities observed on the side faces of NaClO^ crystals (a) small U-shape (b) large U-shape (c) Y-shape (d) rectangular shape (e) combination of U shape and Y shape. 131 Fig . 4-37: Needle-like growth steps formed on the side surface of a NaClO^ crystal with irregular convection. Size = 8.0mmx7.8mmx8.Omm AT = 5 °C (30°C^25°C) Gr = 269. 132 a combination of U-shaped and Y-shaped. The height of crystal within the Y-shaped portion was smaller than that between U and Y. The "starvation theory" and "diffusion theory" can be used to explain the formation of concave faces. The starvation theory (74) predicted that at some distance from the leading edge, the solution would deplete to the equili brium concentration Ceq* At that point growth would stop and a "starvation" effect should be observed. The solution arriving at the side faces came principally from the bottom face and hence had already been partially depleted in solute. After the solution boundary layer reached the side faces, it flowed along the surfaces and gradually decreased in solute concentration. While supersaturation is expected to decrease with height up the crystal, it is also expected to be greater at the corners and edges of the crystal than at the center of a face (10,15). If the edges grew faster than the center, they would tend to shield the original face from fresh solution and so would further exaggerate the difference in supersaturation between the face and the edge. This finally would lead to a depression in the center of a face. From the foregoing we expect the conca vity would develop in the upper center part of a side face. It is not surprising that a hollow often forms at the center of a bottom face when the crystal rests on the bottom of a crystallization cell. However, we still often 133 obtained a concavity or a hollow at the center of a bottom face of our suspended crystals. This probably could be explained by Clifton and Chapman’s theoretical results for a hot plate facing downwards (37). They showed that the boundary layer had a maximum depth at the plate's center, at which there was a stagnation point with zero mass flow rate. This would be expected to lead to a depression in the center of the face until finally the hollow became deep enough to be noticeable. 2. Inclusions Examining the irregular surface parts under the microscope in monochromatic transmitted light, many inclu sions were found within the crystals. Inclusions contained both mother liquor and gas bubbles. As seen in Fig. 4-38, homogeneous inclusions of mother liquor and heterogeneous inclusions containing bubbles together with mother liquor were the most common types. Inclusions of bubbles alone without mother liquor occurred occasionally. The area photographed was at the center part of the bottom face. The sizes of the inclusions varied from 20 ym up to 3 mm. There were three classes of shapes of inclusions found in NaClO^ crystals: (a) Relatively large layers of solution whose lateral size substantially exceeded the thickness. Part 134 Fig. 4-38: Layer, elongated channels and isolated spots of inclusions near a NaClO^ crystal bottom face with irregular convection. Size = 8.7mmx7.9mmx8.Omm AT = 5 ° C (30°O25°C) of such a layer is seen at the right part of Fig. 4- 38 . (b) Thin, continuous elongated channels or chains of separate inclusions (c) Distinct small oval inclusions. In Fig. 4-39, inclusions with light outlines are mother liquor and the dark ones are bubbles. When the right hand side of the crystal was tilted up, the darkly outlined spherical inclusion moved to the right, proving it was a bubble and not a drop. We showed earlier that sometimes flaws formed diago nally across the top surface of a crystal, even when grown with a steady plume (see Fig. 4-28). Underneath these flaws were mother liquor inclusion with bubbles. Otherwise most other parts of the crystal were perfect. The appear ance of the bottom face and the side faces was good to the naked eye. With microscopic examination, we only found a few mother liquor inclusions in the upper part of the side surfaces, as shown in Fig. 4-40. When convection became vigorous, the terrace type of irregularities were formed on the top surface. The crystal contained many inclusions between each terrace boundary (see Fig. 4-32). The inclusions were not oriented and contained many bubbles. Figure 4-41 showed the tubular inclusions inside the side surface under the microscope. 136 137 (a) (b) Fig. 4-39: Inclusions in NaClO^ crystals. Spheres with dark outlines are bubbles. Note movement of the bubble from a to b as crystal was tilted. Fig. 4-40: Inclusions near the upper part of side surface of a NaC103 crystal grown with steady convec tion . Size = 6.5mmx6.0mmx5.8mm AT = 5 ° C (30°Cv25°C) Gr = 127.5. 138 Fig. 9 m rn 4-41: Inclusions near the side surface of a NaClO^ crystal grown with irregular plume . Size = 9.9mmx9.3mmx8.1mm AT = 3 ° C (30°C^27°C) Gr = 277. 139 The lower part of the crystal was perfect and no inclusions were found. At some distance from the bottom face, the first inclusions appeared on the side surfaces. They were tiny individual ovals of solution mostly without bubbles. As one moved up, the first inclusions were about 20 ym long They increased in size with height and contained more bubbles. Longer channels of separate inclusions were found half way up the crystal. At the upper part of the crystal, not only the diameter but also the length of the inclusions increased further. Sometimes they were connected together and formed large inclusions about 2 to 3 mm long. All inclusions were elongated vertically in the flow direction and more or less parallel to one another. Figure 4-42 shows a picture taken under the microscope of a U-shaped concavity on the upper center of the side surface. The inclusions outside the bottom edge of the concavity were distinct oval ones which were elongated perpendicular to the flow direction, as shown in the bottom of the picture. The inclusions on the corners of the concavity were thin and winding. (C) Influence of Mechanical Perturbations on Convection Irregularity and Crystal Defects Experiments described earlier demonstrated that the convection around a crystal growing at 1 g undergoes a 140 ! Fig. 4-42: Inclusions around a concavity formed at the upper center part of a NaClO^ crystal side surface grown with regular convection. Size = 7.4mmx6.9mmx6.5mm AT = 5 ° C ( 30 ° Cv2 5 °C) Gr = 191. 141 transition with increasing crystal size or supersaturation from steady laminar to irregular fluctuating flow. It was also shown that the convection irregularities are related to the formation of defects, such as inclusions and surface irregularities on the growing crystal. Below a critical Gr, convection was steady and laminar. However, accelcra- tional transients may induce temporary convection irregular ities. Figure 4-43 shows that the minimum amount of impact energy needed to cause transient convection irregularities lasting for about 3 seconds decreased with increasing Gr. The duration of the induced temporary convection irregular ities for a given accelerational intensity increased as the threshold Gr ffor spontaneous irregularities) was approached. For example, if the impact energy was 0.4 joule then the duration time was 24 sec for Gr=65 and 40 sec for Gr=15 0. For a given Gr, the duration time increased with impact strength as shown in Fig. 4-44. The size of the NaClO^ crystal was 7.0 mm x 6.9 mm x 6.3 mm and the corresponding Gr for the system was 176 at AT=5°C (30°C v 25°C). The threshold Gr for this system was 300. Figure 4-45 (a) shows the convection pattern 2 sec after impact around the crystal (h=6.3 mm, Gr=176) with 0.1 joules of impact energy. In this figure we could see an arc shaped fluid rising just above the crystal. This layer of depleted 142 IMPACT ENERGY (J O U L E ) 0.05 Continuous Spontaneous Irregularities 0.04 0.03 Temporary Irregularities Due To Transient Accelerations 0.02 Regular Convection 0.01 250 50 00 200 150 GRASHOF NUMBER Gr Fig. 4-43: The minimum amount of impact energy needed to induce transient convec tion irregularities lasting 3 sec or longer for NaCIO? crystals. Height of crystals = 3.5mm, 0-AT=S°C, ^-AT = 9°C. All solutions i i saturated at 30°C. - L * TRANSIENT TIME O F IRREGULAR PLUME(SEC) 60 45 30 0.| 0.70 0.60 0.30 0.50 0 0-80 0-20 0.40 0 IMPACT ENERGY (JOULE) Fig. 4-44: The duration time of irregularities vs. impact energy,size of NaClO^ crystal = 7.0mmx6.9mmx6.3mm. AT = 5°C (30°Cv25°C), Gr = 176. ) —1 - p* . 145 (a) (b) Fig. 4-45: Convection pattern around a NaClO^ crystal tapped by 0.1 joule of impact energy (a) 2 secs after tapping (b) 4 secs after tapping. Size = 7.0mmx6.9mmx6.3mm, AT = 5°C (30°C^25°C), Gr = 176. fluid rose up irregularly 4 sec after tapping as shown in Fig. 4-45(b). The time interval between the two pictures was 2 seconds. It seemed that the velocity of the irregu larity at the center of the field of view was faster than the outside part. Figure 4-46 shows the system when dis turbed by a large impact energy (about 0.7 joule). The convection current 2 sec after impact became very irregular. The quality of crystals grown with impacts applied periodically was better than those grown with spontaneous convection irregularities under otherwise identical conditions. Figure 4-47 shows the center of the top surface of a NaClO^ crystal grown with 0.035 joule impacts applied every 18 sec at AT=5°C. The size of the crystal was 7.1 mm x 6.0 mm x 6.4 mm and Gr = 144.3. Before tapping, the convection of the system was regular. Inclusions on the diagonal positions were observed. If we compare with Fig. 4-32, we find that both the number and the sizes of solution inclusions were greatly reduced. But what was more interesting was that most bubbles contained in inclu sions were very small, about 40 pm. Perhaps '’tapping” forced the bubbles originally attached on the crystal surface to leave, as was demonstrated on the sides of a glass vessel containing water. The distribution of Ag in NaClO^ crystals was deter mined by atomic absorption data. A standard solution of 146 Fig. 4-46: Convection pattern around a NaClC>3 crystal tapped by 0.7 joule of impact energy. Size = 7.0mmx6.9mmx6.3mm AT = 5 °C (30°Cv25°C) Gr = 176 147 Fig. 4-47: Inclusion near the top surface of a NaClO^ crystal grown with periodic tapping with an impact energy of 0.035 joule and a period of 18 sec. Size = 7.Immx6.0mmx6.4mm, AT = 5 ° C (30°O25°C), Gr = 144.3. 1 148 AgNO^ was prepared by dissolving 0.063g of reagent grade AgNO^ (Mallinckrodt Inc.) in 50 ml of deionized water and diluting to 10 liter solution. The concentration of Ag in the standard solution was about 4 pg/ml. A sample solution was prepared by dissolving 0.088 g of NaClO^, which had precipitated from a supersaturated solution (AT=5°C and growth temperature = 25°C) with Na/Ag = 100, in 250 ml of deionized water. The concentration of Ag in the sample solution was then obtained directly from the atomic absorp tion spectrophotometer by comparing the absorbance of the sample solutions with that of the standard solutions. From these data, the distribution coefficient of Ag in NaClO^ crystals was found to be about 2 (for detailed calculation please see Appendix D). The effect of convection on the Ag distribution in NaClO^ crystals was studied by an electron microprobe. Figure 4-48 shows the microprobe data of Ag on the side surface of a NaClO^ crystal. The electron beam traced along a line parallel to the bottom face and was 1 mm above the bottom face, as shown in Fig. 4-49(a). The crystal grew at 25°C (AT=5°C) without any mechanical disturbance and the convection plume was very regular. The counter numbers of Ag in the crystal fluctuated about a value of 300. (300 counter numbers ^1.5 wt% of Ag in NaClO^)• Figure 4-50 shows the microprobe data of a crystal 149 Counter Number 400 350 300 250 200 0 100 200 300 400 500 600 Distance From Seed Crystal Interface (um) I n O Fig: 4-48: The counter number (relative amount) of Ag in a NaClO^ crystal grown with a regular plume without tapping. Size of the crystal = 4. 8mmx4. 7mmx3. 5mm, AT = 5°C (30°O25°C), Gr = 56. 1.5m m ^Suspension Wire -J u/ Part 4.8 mm 1 .5 mm Tod Surface E E io ro Microprobe Tracing Sports 4.7 mm (a) Corresponding To Figure 4~48 1 .5 mm 4.6 mm 6.3 mm 2£) mm (b) Corresponding To Figure 4 - 5 0 6.1 m m 7.4 mm (c)Corresponding To Figure 4 ~ 5 2 Fig. 4-49: The microprobe tracing spots on NaC103 crystals. 151 l<8m C D -Q e Z 3 z C D c Z5 o o 350 300 4 0 0 Distance From Seed Crystal Interface (um) Fig. 4-50: The counter number of Ag in a NaClO^ crystal grown with five different conditions, (I) without tapping, (II) single tapping, (III) without tapping (IV) group tapping (V) without tapping. Size = 6.3mmx4.6mmx4.2mm, AT = 50C (30°O25°C) , Gr = 79. DO grown under varied conditions. The crystal first grew quietly without any tapping for half hour. The convection was very regular. Then the crystal was tapped by 0.062 joule of impact energy once every 8 minutes for 1 hr. The induced irregularities were very tiny, as shown in Fig. 4-51, and the duration time of each transient irregularity was about 4 seconds. After this tapping period, the crystal was allowed to sit quietly for another half hour. The convection plume was again steady. After that, the crystal was disturbed by group tapping for half hour. In each group tapping the system was tapped 6 times in one minute and then quieted for 6 minutes. Finally, the crystal was set quietly for the last half hour. The micro probe data shows no distinct difference between each period of growth. There is no big difference from the results of Fig. 4-48, except the average Ag level in the crystals was lowered. The boundaries drawn between each period are not accurate (±20 ym). They were based on the assumption that the crystal had uniform growth rate throughout (see Appendix E). On the other hand, when the accelerational transient was big enough to cause long duration convection irregular ities, the Ag concentration in crystals was affected by tapping, as shown in Fig. 4-52. When this crystal was tapped by an impact energy of 0.062 joules, the convection 153 Fig. 4-51: The convection pattern around a NaClO^ crystal 2 sec after tapping by an impact energy of 0.06 2 joule. Size = 6.3mmx4.6mmx4.2mm, AT = 5 ° C (30°O 2 5 ° C) , Gr = 79. 154 Counter Number 450 350 250 150 50 100 200 300 400 Distance From Seed Crystal Interface (urn) 500 600 Fig. 4-52: The counter number of Ag in a NaClO^ crystal grown with five different conditions (I) without tapping (II) single tapping (0.062 joule) (III) without tapping (IV) group tapping (V) without tapping. Size = 7.4mm x M S 6.1mm x 4.3mm, &T = 5°C (30°C'v25°C) , Gr = 153. became very vigorous (see Fig. 4-53), and the Ag level decreased as compared with that for quiet periods. Convec tion irregularities were still present during the quiet periods, but less vigorous. In the single tapping period, the Ag concentration was almost down to the background level. The reason for this can perhaps be explained as follows: Each time the NaClO^ solution were doped with AgC10^> we always found many tiny white particles suspended in the solutions and solutions became cloudy. Probably these white particle were AgCl, due to the presence of Cl in the NaClO^ and AgClOj. The Ag incorporated in the NaClO^ crystals was most likely AgCl particles rather than Ag+. If this is true, then the result is quite interesting - that vibration can greatly reduce trapping of foreign particles. Some more work on this part is highly desirable. 156 Fig. 4-53: The convection pattern around a NaClO^ crystal 2 sec after tapping by an impact energy of 0.062 joule. Size = 7.4x6.1x4.3mm AT = 5 °C (30°O25°C) CHAPTER V DISCUSSION AND THEORY (A) Grashof Number for Irregularity The first occurrence of irregularities in the plume above a growing crystal was shown (Fig. 4-17) to occur when the Grashof number exceeded a critical value. The Grashof number was defined as Gr = g*L3(p -p _) — (5-1) ° c > eq 2 H y where L is the characteristic length of the crystal, p^ and pe^ represent the density of bulk supersaturated solution and density of saturated solution at the growth temperature respectively, g is the acceleration of gravity and y is viscosity. The quantity is a fluid property grouping which was evaluated at the average of the bulk solution and saturated solution concentrations at the growth temperature. As evidenced by Fig. 4-17 , the Grashof number above which irregularities occurred varied with the height of the crystals h, especially for short crystals. The effect of increasing crystal height was to stabilize the plume. That 158 is, a crystal with the same horizontal area/perimeter but with a larger height had more stable convection. When a crystal grows, the influence of increasing L is much more than that of increasing h, so that the plume gradually becomes more irregular for a given crystal. The above observation appears to be in agreement with similar fluid hydrodynamic results (83). Figure 5-1 shows values of the form drag coefficient for forced convec tion fluid flow normal to a finite thickness flat plate. Note that increases as the ratio x/y increases, where x is the length of the plate and y is the thickness of the plate. For constant thickness y, is proportional to x. For constant x, is inversely proportional to y. A reduction of the drag coefficient is an indication of a reduction in the turbulent wake behind the body (23). In other words, convection becomes more turbulent as x increases and y decreases. This result is consistent with mine. To explain the above observation, the separation, reattachment and redevelopment of flow on a blunt leading edge is analyzed. Figure 5-2 shows fluid flow past a flat plate placed normal to the direction of flow. A boundary layer forms on the plate and separates at the edges. It is found that flow reattachment occurs at some distance from the leading edge. If the plate is thin, the separated flows often meet each other behind the plate 159 Drag coefficients for a flat plate of finite thickness normal to the flow (83). Fig. 5-2: Flow pattern for a fluid impinging on a flat plate (83). Flow is from left to right. 161 and so form a turbulent wake (23). However, if the plate is thick enough, then the separated flow reattaches on the plate side surfaces and subsequently redevelops stably in the downstream direction (84). Even after changing their courses by 90° at the end of side surfaces, the flow is still more regular than the former case. Maybe this is the reason why the thick plate (or crystal) usually can stabilize the flow. (B) The Effect of Surface Concentration on the In the calculation of Sh, as shown in Fig. 4-21, equilibrium was assumed at the crystal surface, which is not true. Thus the values shown for Sh were somewhat lower than the true value. Crystal growth from solution results from the two processes, transport and transformation, acting in series. Interface kinetics must be considered. If we assume linear interface kinetics then we can write Sherwood Number k (C -C ) v OO O K.(C -C ) l v o eq^ (5-2) 1 - F 3 where C is the interface concentration (in mole/cm ), o J 9 162 C is the bulk concentration and C is the equilibrium 0 0 eq concentration. We obtain from Eq. 5-2 the interfacial concentration k C +K.C c , 1-F. . . co i eo , (5_3) ° J£- + K. 1-F i or V C = --- ^ (5-4) C C 1 + 1-F K- k 1 which is just the old ’’addition of resistances” result. kL Assuming that = a Gr4, then C - C V C = ----— — — — (5-5) c c 1 1-F Ki (aGr^r JL Thus we have a growth rate that depends on and a. Since both and a were unknown, as well as the true value of Gr and F, we used a trial and error method combined with successive approximations. The following procedure was employed. First, a value of ”a” was guessed, and then was calculated by using the following equation for all data points along with Gr and F calculated for equilibrium at the interface 163 K. = 1 1 1-F eq The average for all data points was taken and then of the estimate. This was repeated for several values of "a" in order to find the best one (that giving the minimum error). Second, this K. and a were used to estimate C and J 9 1 o PQ for each data point. Third, these values were used to calculate Gr and F and repeat the first and second steps. This procedure was iterated until the values of CQ changed no longer(with 4 significant figures). After the second iteration, the average value of (C -C )/(C -C ) was about ’ & v o eq v 00 eq 0.66. The values of F and Gr finally determined differed from values calculated initially (assuming equilibrium) by about 0.48% and 70% respectively. Figure 5-3 shows that the best value for a was about 6.82. That is £ ( v i " v «. 1 v c cal c actual 2 ) was calculated to evaluate the error Sh 6.82 Gr4 (5-6) for 1 < Gr < 10 3 or Sh = 1.2 2 Ra4 (5-7) for 103 < Ra < 106 . The average value of thus obtained, is about 6.49x10 -3 164 ,7689 CM CO .768 " O CM 7687 o o .7686 1,7685 6.86 6.84 6.78 6.80 6.82 a Fig. 5-3: The plot of Z(Vc cal-Vc actual)2 vs. a for NaClO^ crystals. 165 cm/sec. One thing should be mentioned here. I determined the average growth rate by weighing the crystal after growth (out of solution). However, the crystal weight included the weight of occluded solvent. This probably accounts for the fact that my value of "a" exceeds 0.5, which is well known to be the value for a sphere. (C) Surface Irregularities and Convection Patterns Each cubic NaClO^ crystal consists of six growing surfaces. According to their positions, we can classify each as the top surface, a side surface or the bottom surface. For a growing crystal, the free convection current on the bottom face is laminar and flows horizontally out from the center to the edges. After changing its course by 90° at the edges, the convection current flows upwards and parallel to the side surfaces. At the end of the side surfaces, the flow changes its course by 90° again and extends laterally a short distance. Finally, it separates from the crystal and forms a convection plume above it. Along the path of the free convection current, the concentration of the solution gradually diminishes due to the incorporation of solute in the crystal. We expect each type of surface to be affected by the convection current differently. It was also found that different kinds of surface irregularities were formed on different types of surfaces. For example, diagonal position flaws 16 6 and terraces were often found on the top surface, streaks and concavities on the upper part of side surfaces and hollows on the bottom surface. Therefore, it is interest ing to examine the relationship between the formation of surface irregularities and the convection patterns on each face. Before going into details, we would first like to see how the crystals start their growth. Sclar and Schwartz (85) used a microscope to observe the growth of a cubic NaCl crystal suspended in a circulating solution and growing virtually unrestricted in all directions. At low supersaturations, the faces remained planar and no problems were encountered. As the supersaturation of the bulk solution was increased, skeletal overgrowths on plane-faced seeds were obtained. Continuous microscopic observation of the growth of these skeletal forms revealed the follow ing growth mechanism at the early stage of growth. (a) A step appeared on each vertex of the cube and its adjoining edges in a few minutes so that the seeds had a castellated appearance, as shown in Fig. 5-4. (b) Each step grew away from its vertex in the form of three rectilinear beams growing along the corresponding three edges. (c) As each step approached the center of the edge it 167 Fig. 5 4: Sketch of initial castellated stage in the growth of NaCl cubes immersed in flowing supersaturated brine (85). 168 might join with the step advancing from the opposite corner. Sometimes they did not join together and left an open notch near the middle of the edge. (d) A new actively growing superimposed step might then form at the vertex again. At the early stage of growth, we assume each face follows the same growth mechanism as mentioned above. After that, the convection effect, the starvation effect and the bubble effect should all be considered important to the crystal growth. We now examine these face by face. 1. Bottom face The bottom face was usually without defects. At high Gr, a hollow sometimes formed at the center of the bottom face. The boundary layer thickness is greater at the center of the bottom face. At low Gr, it is much more likely that new steps are generated near the center at screw dislocations, and spread outward into regions of higher supersaturation, thus accelerating so that the step trains are stable. As Gr increases, the supersaturation at the edges and corners may become large enough to generate steps by two dimensional nucleation. Then steps move inward, decelerate, pile up to form waves and macro steps and finally form a depression at the center of the 169 face (86). The starvation effect is not important on this face since the solution on this face is fresh. Bubbles might play an important role on the formation of inclu sions. We discuss this in more detailed in a later section. 2. Side surfaces This situation is similar to natural convection heat transfer from a heated vertical plate, which has frequently been studied (33). A boundary-layer flow is established along the plate surface. Since solute is depleted due to growth, the solution concentration decreases in a direction from bottom to top along the plate. Carlson (74) hypo thesized that the surface depressions on the downstream part of side surfaces of ADP crystals under forced convec tion were due to the ’’starvation effect.” In my experiments, the lower portions of the side faces were nearly always perfect. Slightly concave depressions often formed in the upper center portions of the faces. This was probably due to the starvation effect and diffusion effect. For crystals grown with natural convec tion without stirring, the starvation effect should be more serious on the upper parts of the side surfaces. Besides, the boundary layer will be thinner at the corners, and one also has transport in from three directions rather 170 than one. These effects perhaps caused the center part of the upper side surface to grow slower, leading to a conca vity there. The formation of inclusions on the crystal side surfaces probably was due to the effects of starvation and bubbles. This could be explained as follows. As the solution boundary layer flows up the side surfaces, solute is depleted. It causes the growth steps to decelerate, and thereby makes waves and macrosteps (86) . At the same time the boundary layer is enriched in rejected gas (87) as it flows up, since the solubility of gases in solids is much less than in the corresponding liquids. (For example, Scholander et al. (87) found that the solubility of air in ice is considerably less than a thousandth of its solubility in water.) Kuo and Wilcox (90) showed that the supersaturation of dissolved gas is insufficient for homo geneous nucleation. Thus if a bubble nucleates it must do so heterogeneously, either on a foreign particle or on the crystal surface itself. If it nucleates on a foreign particle, it is likely to float away and unlikely to be trapped in an inclusion. If it nucleates on the surface, the most likely site would be a step (the energy of nucleation is lower). Thus the probable scenario is that solute depletion causes a tiny step to form, which then helps bubbles to nucleate. The bubble, once formed, causes 171 additional solute depletion. As the crystal continues to grow, more and more tiny bubbles form at the crystalliza tion front. They grow and sometimes coalesce to form bigger bubbles, until their sizes are large enough to retard the growth underneath them. Inclusions then start, since the growth rate at the bubble bases is lower than that nearby. Finally, this may lead to many open channels on the side surfaces. Figure 5-5 (a) shows this phenomenon with an exaggerated scale. Since the steps move from bottom to top, they propagate into regions of lower supersaturation. Thus they gradually slow down as they move up. A new actively growing super imposed step might form at the bottom edge before the earlier one has completed its travel to the end of the face. This leads to a thickening at the bottom, more pronounced starvation of the old layer, and thus to coalescence. Due to the overgrowth of the new layer, the open channels then become one-end-open tubes. At the open mouth of tubes, eddies occasionally may bring some super saturated solution into the tubes and cause growth in the lateral direction, as shown in Fig. 5-5(b). The open- mouthed tubes then may be sealed to form inclusions. 5. Top surface As we mentioned earlier, the patterns of flaws formed 172 ^ 7 \ Z 7 La (a) Front View (b)Side View g. 5-5: Schematic diagrams illustrating (a) the formation of open channels on crystal side surface (b) sealing an open channel by a new growth layer formed on the top surface of the crystal were very similar to the natural convection patterns above a horizontal hot plate facing upwards. For a square crystal, diagonally patterned flaws were found. For a rectangular crystal, angle-bisector-position flaws were found. According to Husar and Sparrow’s observation (82), we see that solution flows inwards from the four edges, with these four streams meeting each other either at diagonal lines for square crystals or at angle-bisector lines for rectangular crystals. After that they separate from the crystal. These lines are the last positions at which the solution contacts the crystal. Since the solu tion is gradually depleted in solute concentration along its path, the solutions meeting each other at these lines may have a lower concentration (thicker boundary layer) than nearby areas, causing slower growth and formation of depressions there. In addition, the steps will be formed at the edges and will spread into the center of the face. The steps at two adjacent edges are most likely to meet each other at these lines. If bubbles are nucleated at the fronts of the growth steps, they may be contained between steps. These two factors (starvation and bubbles) may even lead to open channels along these lines. The convection on the top surface was usually less stable, especially along these hypothesized lines, since they are 174 created by the interaction of two converging flows from adjacent edges. This may cause eddies. When convection currents above the crystal became vigorous, terrace type surface irregularities were found. Due to the limitation of the apparatus, I couldn’t obtain a top view of the convection cells with vigorous convection. It would have been nice to know whether the appearance of the terrace was truly similar to that of the convection patterns. (D) Bubble Generation and Inclusion Formation Microscopic observations of grown NaClO^ crystals revealed that many gas bubbles were trapped inside the crystals. As we mentioned earlier, bubbles may play an important role in the formation of inclusions during solution growth. Since bubbles are most likely to nucleate at macrosteps, the growth rate at bubble bases is lower than that nearby, and an inclusion may start. Figure 5-6 shows an early stage of inclusion formation on a NaClO^ side surface. Many tiny inclusions (about 20 ym) with bubbles were trapped behind the macrosteps. In related experiments, Kliia and Sokolova (91) found that two kinds of inclusions can be induced by oil drops in NaNH^HPO^* 411^0 crystals growing from aqueous solution. The oil 175 Fig. 5-6: Inclusions formed at an early stage on a side surface of a NaClO^ crystal grown with slightly irregular convection due to tapping (0.065 joules). Note many tiny inclusions with bubbles were trapped. Size = 8.3x8.7x6.5mm, AT = 2.2 ° C (30°Cv27.8°C), Gr = 134.1. drops didn’t wet the crystal surface. Under stable conditions of slow growth, inclusions of mother liquor were formed. With rapid nonuniform growth, heterogeneous inclusions of oil and mother liquor were formed. Both single phase and two phase inclusions were also found in the present research. The bubble-aided inclusion model may help explain many phenomena observed during our solution crystal growth experiments. (a) The lower partsof the side surfaces were mostly perfect without any inclusions. The reason is neither certain nor well understood. Probably in that part the starvation effect was unimportant. Besides, the dissolved gas concentration was too low to nucleate bubbles at the macrosteps. As the solution moved up, more and more dissolved gases accumulated and the bubbles would have nucleated at steps. Therefore, at some distance away from the leading edge, many tiny inclusions might start to form. On the upper part of a side surface large inclusions were formed. Perhaps they were induced by larger bubbles together with the starvation effect. (b) Bubbles might nucleate preferentially at growth steps. Rough surfaces usually contained more macrosteps (or surface irregularities) than 177 smooth ones. Therefore, the perfection of the crystal surface is an important factor in the formation of bubble inclusions. The experimental data revealed that the better the surface quality the less bubbles were trapped. Figure 5-7(a) shows a perfect seed crystal with an unstable plume. Generally we only observed a few inclu sions on the top of those side surfaces which contained a few tiny bubbles. On the other hand, Fig. 5-7(b) shows a contrary case with more bubbles occluded. (c) Comparing Figs. 5-8 and 5-9 (the two original seed crystals were both perfect to the naked eye), we see a big difference in inclusion formation between when the plume was regular and when it was irregular. The inclusions formed with an irregular plume usually were heterogeneous and contained a larger number of bubbles. When the plume was regular the growth rate was low, and crystals usually grew without formation of inclusions. However, sometimes inclusions with very few bubbles were still observed. This result agrees with Kliia and Sokolova's (91). A possible way to explain this phenomena is that when the plume was regular the growth rate was 178 179 (a) smooth surface Fig. 5-7: Inclusions formed on NaClO^ crystals crystal (b) bad seed growth. Size = 9.2x9.9x7.1 mm AT = 4 . 5 ° C (30°O25.5°C) Gr = 393 Size = 9.1x8.3x8.8mm AT = 4°C (30°O26°C) Gr = 261 (b) rough surface with irregular plume (a) good seed 180 (a) side Fig. 5-8: Inclusions in a NaClO^ crystal Size = 7.9x8.7x8.7mm. AT = 50C (30°C^25°C) Gr = 298 (b) top grown with irregular plume. 181 * (a) side (b) top Fig. 5-9: Inclusions in a NaClO^ crystal grown with regular plume Size = 8.9x9.2x7.8mm AT = 1. 20C (30°O28.8°C) Gr = 92.6 low, and not only was the gas concentration low but also the interface could usually push away the bubble without trapping (88). As the bubbles grow, they may become large enough to depress the growth rate at bubble bases. Kliia and Sokolova found that at low growth rate, sometimes this induced solvent inclusions without trapping the foreign material. When the plume was irregular, the growth rate was fast and perhaps fluctuated, and even small bubbles might easily be trapped by the crystal to form heterogeneous inclusions. (d) Sometimes there was a layer of inclusions formed at the interface between the original seed crystal and the newly grown part, as shown in Fig. 5-10, especially when a dry seed crystal was used. Possibly many tiny bubbles were left on the surface after it was introduced into the solution. This might account for a succession of ’'ghost" crystals sometimes found. Large crystals could be grown without inclusions, even with repeated growth episodes, if the wet crystal was promptly transfered into each new solution. (e) The mechanically-induced transient irregularities would have caused local supersaturation 182 Fig. 5-10: Many inclusions formed at the interface between original seed crystal and newly grown part with irregular convection. Size = 10.6x8.0x4.3mm AT = 5 ° C (30°C^25°C) Gr = 388.1 183 fluctuations and might therefore be expected to have had more inclusions. However, the number and size of inclusions decreased significantly with tapping. Probably tapping the crystals during growth jarred many bubbles free of the crystal so they floated away, especially the large ones. It also decreased starvation by the addi tional mechanical stirring effect, therefore causing a reduction in inclusion formation. 184 CHAPTER IV CONCLUSION AND RECOMMENDATION FOR FUTURE RELATED WORK An experimental study has been made concerning convection irregularities around a growing NaClO^ crystal and their relation to crystal-defect formation. The following conclusions were obtained: (a) There was a continuous transition from steady laminar convection to irregular flow when the Grashof number of the system was brought beyond a critical value, either by increasing the crystal size or the supersaturation. The critical Grashof number increased with increasing crystal height. (b) As the Grashof number increased, both the irregularities in the plume increased and the number of morphological defects in the crystal increased. The types of defects formed on the top surface and the upper part of side surface are likely to be strongly influenced by the nature of the plume. With a laminar plume , 185 diagonal grooves sometimes were formed on the top surface, and concavities on the upper part of side surface. The inclusions were homogeneous. With an irregular plume, terraces were found on the top surface and streaks on the side surface. The inclusions were heterogeneous. (c) Below a critical Grashof number, accelerational transients induced temporary convection irregular ities, whose magnitude and duration for a given intensity increased as the threshold Gr (for spontaneous irregular convection) approached . Tapping greatly reduced the number and size of bubbles containing inclusions. For the present experiments I also conclude that the best conditions for growing good crystals from solution are: (a) Start with good wet seed crystals formed by directly nucleating the seed on a suspension wire and then transfering it quickly to the growth solution. (b) Use low supersaturations. There are two advanta ges to a low supersaturation or low growth rate. First, convection is regular. Second, bubbles would be less likely to form, and those that are formed would be pushed more readily. 186 (c) Grow crystals in the absence of dissolved gas (i.e. in a vacuum). (d) Use forced convection, especially as crystal size increases. Some possible related future research are: (a) Grow crystals from solutions in space. The most likely surface defect is a hollow on the center of each face. (b) Grow under a vacuum and see if that does reduce the number of inclusions formed. (c) Set up a long focal length microscope to directly observe the crystal surface during solution growth to see the details of inclusion formation. (d) Try different methods of stirring and find out which method can still yield high quality of crystals with the highest AT. Forced convection is necessary when crystal size is large, but convection patterns also affect the formation of surface irregularities. I think it may be helpful to keep the crystals moving slowly, sometimes up and down, sometimes back and forth, and tapping them very gently. (e) We need some more work on the influence of vibra tion on the trapping of foreign particles. 187 NOMENCLATURE A Constant A 2 Area of crystal top surface,cm a cons tant B constant C 3 solute concentration , g-mole/cm c constant CT constant CD Drag coefficient cp Heat capacity at constant pressure, cai/g-°c D 2 Diffusion Coefficient, cm /sec F Volume fraction of solute in solution g 2 Acceleration due to gravity, cm/sec Gr 3 2 Grashof number, g*p*Ap*L /y §E Acceleration due to gravity on earth, 9 80 cm/sec^ H Depth of crystal below the solution free surface, h 2 Heat transfer coefficient,cal/cm -sec- °C h Height of crystal, cm I Intensity of illumination, candle power K Thermal conductivity, cal/cm-sec-°C k Mass transfer coefficient, cm/sec tt Overall average mass transfer coefficient, cm/sec cm 188 K. 1 Interfacial kinetic constant, cm/sec L Characteristic length = A/p, cm 1 length of crystal, cm M Molecular weight, g/mole m Mass of crystal, g Mo Mass of the empty pycometer, g Mr Mass of pycometer filled with deionized water, g M X Mass of pycometer filled with solution, g n Refractive index na 2 Mass flux, g-mole/cm -sec Nu Nusselt number = hL/K P Perimeter of crystal top surface, cm P Pressure, dyne/cm^ Pr Prandtl number, yC^/K R Sphere radius, cm r Distance from sphere surface to the symmetrical axis, cm Ra Rayleigh number for heat transfer = Gr*Pr Ra^ Rayleigh number for mass transfer = Gr-Sc Re Reynolds number = pvL/y Sc Schmidt number = y/pD Sh Sherwood number = kL/D = L/6s = T Temperature, °C t time, sec C-C 3 ( < r ^ '-'oo *-*0 y=0 189 AT Supercooling, °C V Mean molar volume, ml/mole V Partial molar volume, ml/mo V convection velocity, cm/se vc Crystal growth rate, cm/se vcf crystallization flow rate, W neck width of plume, cm w width of crystal, cm X Mole fraction X Coordinate, distance along y Coordinate, distance from z Coordinate, cm Greek symbols 1 3 o a Compositional densification coefficient, — p o C 1 cm “ Vg-mole 3 Thermal densification coefficient, — ? °C ^ p 3 1 c 6 Hydrodynamic boundary layer thickness, cm Ss Thickness of stagnant film, cm 6^ Thermal boundary layer thickness, cm S' Width of Schlieren image, cm 3 p Density of solution, g/cm 3 Pc Density of crystal, g/cm p Similarity variable, defined by Eq. 2-13 190 0 Angular deviation of light beam y Viscosity, g/cm*sec 2 v Kinematic viscosity, y/p , cm /sec ip Generalized stream function, defined by Eq. 2-13 Subscripts C Crystal m Mass transfer 0 Surface condition x x component y y component 00 Bulk condition eq Equilibrium solute concentration, i.e. solubility 1 solvent 2 solute Superscript * Dimensionless variable 191 REFERENCES 1. R. A. Laudise, The Growth of Single Crystals, Prentice Hall, Inc., New Jersey, p. 261 (1970). 2. H. E. Buckley, Crystal Growth, John Wiley § Sons, Inc. New York, p. 45, 55, 57 (1958). 3. H. E. C. Powers, Sugar Technol. Rev., 1_, 85 (1970). 4. W. R. Wilcox, Ind. Eng. Chem. 6^, 13 (1968). 5. M. L. Salutsky, Treatise on Analytical Chemistry, I. M. Kolthoff and P. J. Elving, Eds. Vol. 1, Ch. 8, part 1, Interscience, New York (1959). 6 . J. J. Rosenfeld, A. B. Chase, Am. J. Sci. 2 59, 519 (1961). 7. D. E. Garrett, Chem. Eng. Progr. 6_2, 74 (1966). 8. R. Brooks, A. T. Horton and J. L. Torgesen, J. Crystal Growth, 2_, 279 (1968). 9. K. G. Denbigh and E. T. White, Chem. Eng. Sci. 21, 739 (1966). 10. C. W. Bunn, Discuss. Farad. Soc., 5_, 132 (1949). 11. J. Boscher and S. Goldsztaub, Comptes Rendus Academie Sciences, 25 3, 774 (1961). 12. S. Goldsztaub and R. Itti, ibid, 26 2, 1291 (1966). 192 13. R. Itti, ibid, 263, 1143 (1966). 14. H. E. Buckley, Crystal Growth, John Wiley $ Sons, Inc. New York, p. 45 (1958). 15. A. Holden and P. Singer, Crystals and Crystal Growing, Doubleday % Company, Inc. New York, p. 73, 104 (1960). 16. P. J. Shlichta, W. R. Wilcox, and P. S. Chen, ’’Simulation of Crystal Growth from Solution in Low Gravity,” Bull. Amer. Phys. Soc. , Ser. 11, 2_1, 361 (1976) . 17. J. R. Carruthers, Thermal Convection Instabilities Relevant to Crystal Growth from Liquidsj Preparation and Properties of Solid-State Materialss Vol. 3, W. R. Wilcox and R. A. Lefever, (eds.), Marcel-Dekker, N.Y. (1977). 18. P. G. Grodzka and T. C. Bannister, ’’Heat Flow and Convection Demonstration Experiments Aboard Appollo 14,” Science 176, 506 (1972). 19. W. R. Wilcox, C. E. Chang, ’’Analytics of Crystal Growth in Space,” Annual Report (5 June 1973 - 4 June 1974) Contract No. NAS8-29847, p. 48. C. E. Chang and W. R. Wilcox, J. Crystal Growth, 2_8 , 8 (1975). 20. J. C. Brice, The Growth of Crystals from Liquids3 North-Holland Publishing Co. p. 291 (1973). 21. A. P. Kapustin and V. E. Kovalynnaite, Growth of CrystalsVol. 2, Consultants Bureau Inc. N.Y. p. 29 (1959) . 193 22. 23. 24. 25. 26. 27. 28. 29 . 30. 31. 32 . R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, John Wiley § Sons, Inc., N.Y. p. 148, 338 (1960). J. G. Knudsen and D. L. Katz, Fluid Dynamics and Heat Transfer, McGraw-Hill Book Co., N.Y. p. 303, 319, 359 (1958). L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, London p. 214 (1959). M. Mikheyer, Fundamentals of Heat Transfer, Peace Publishers, Moscow pp.77-78, (1968). A. A. Wragg and R. P. Loomba, Int. J. Heat and Mass Transfer Vi, 439 (1970). R. J. Goldstein, E. M. Sparrow and D. C. Jones, Int. J. Heat and Mass Transfer, 1_6, 1025 (1973). J. R. Lloyd, W. R. Moran, J. Heat Transfer 96C, 443 (1974) . W. J. King, Mechanic Eng. p. 347 (May, 1932). W. R. Wilcox, J. Crystal Growth, 1_2 93 (1972). A. J. Chapman, Heat Transfer, 2nd edition, The Macmillan Co./Collier-Macmillan Limited, London, p. 227 (1967). W. R. Wilcox, Preparation and Properties of Solid State Materials, Vol. 1, edited by R. A. Lefever, Dekker, N.Y. p. 37 (1971). 194 33. W. M. Rohsenow, H. Y. Choi, Heat, Mass and Momentum TTansfer3 Prentice-Hall Inc., New Jersey, p. 413 (1961) . 34. S. N. Singh and R. C. Birkebak, ZAMP 2_0, 454 (1969). 35. T. Aihara, Y. Yamada and S. Endo, Int. J. Heat Mass Transfer 15, 2535 (1972). 36. Z. Rotem and L. Claassen, J. Fluid Mech. , 3_8, 173 (1969). 37. J. V. Clifton and A.J.Chapman, Int. J. Heat Mass Transfer 12, 1573 (1969). 38. W. N. Gill, D. W. Zeh and E. D. Casal, ZAMP, 16_, 539 (1965) 39. B. Gebhart, Heat Transfer, 2nd edition, McGraw-Hill Book Co., Ch. 8, (1971). 40. C. Wagner, J. Appl. Mech. 2_3, 320 (1956). 41. S. N. Singh, R. C. Birkebak and R. M. Drake, Jr., Progress tn Heat and Mass Transfer, Vol. 2, Pergamon Press, Oxford, p. 87 (1969). 42. F. J. Suriano and K. T. Yang, Int. J. Heat Mass Transfer, Iff, 473 (1968). 43. K. E. Hassan and S. A. Mohamed, Int. J. Heat Mass Transfer, 1_3, 1873 (1970). 44. J. R. Lloyd and E. M. Sparrow, J. Fluid Mech. _42_, 465 (1970) . 195 45. T. Fujii and H. Imura, Int. J. Heat Mass Transfer 15., 755 (1972). 46. J. D. Parker, J. H. Boggs and E. F. Blick, Introduction to Fluid Mechanics and Heat Transfer^ Addison-Wesley Publishing Co., Inc. p. 292 (1970). 47. E. Schmidt and W. Beckman, Tech. Mech. U. Thermody, _1 , 391 (1936). 48. H. J. Merk, J. A. Prins, Appl. Sci. Res. A4, 11, 195, 207, 435 (1953-54). 49. 0. A. Saunders, Proc. Roy. Soc. London, A172, 55 (1939) . 50. H. B. Squire, E. R. G. Eckert, Modern Developments -in Fluid Dynamics„ Oxford University Press, England, (1938). 51. T. Chiang, A. Ossin and C. L. Tien, J. Heat Transfer, 86C, 537 (1964). 52. G. Schutz, Int. J. Heat Mass Transfer, (3, 873 (1963). 53. A. A. Kranse and J. Schenk, Appl. Sci. Res. A15, 397 (1966) . 54. F. H. Garner and R. B. Keey, Chem. Eng. Sci., 9_, 218 (1959) . 55. F. H. Garner and J. M. Hoffman, A.I.Ch.E.J., 7_, 148 (1961) . 56. D. C. T. Pei, W. H. Gauvin, A.I.Ch.E.J., 9_, 375 (1963) . 196 57. T. Yuge, J. Heat Transfer, 82C, 214 (1960). 58. W. Amato and C. Tien, Int. J. Heat Mass Transfer 15, 327 (1972). 59. J. R. Kyte, A. J. Madden and E. L. Piret, Chem. Eng. Prog. 4-9, 653 (1953) . 60. W. E. Ranz and W. R. Marshall, Chem. Eng. Prog. 48, 173 (1952). 61. F. A. M. Schenkels and J. Schenk, Chem. Eng. Sci., 24, 585 (1969). 62. J. Vander Burgh, Appl. Sci. Res. A9_, 293 (1960). 63. J. Schenk and F. A. M. Schenkels, Appl. Sci. Res. 19, 465 (1968). 64. M. Jakob and W. Linke, Forsch. Ing. Wesens, <4, 75 (1933) . 65. J. E. Boberg and P. S. Starrett, Ind. Eng. Chem. 50, 807 (1958). 66. G. D. Raithby and K. G. T. Hollands, Int. J. Heat Mass Transfer, 1_7_, 1620 (1974). 67. F. Kreith., Principles of Heat Transfer, 3rd edition, Intext Educational Publishers, New York, p. 640 (1973). 68. S. S. Kutateladze, Fundamentals of Heat Transfer, Academic Press Inc., New York, p. 294 (1963). 69. W. R. Rohsenow and J. P. Hartnett, Handbook of Heat Transfer, McGraw-Hill Book Co., N.Y. p. 6-15 (1973). 197 70. F. J. Bayley, Proc. Inst. Mech. Eng., 169, 361 (1955). 71. M. A. Hossain and B. Gebhart, Heat Transfer, Vol. IV, NCI.6 (1970)r Elsevier Publishing Co., Amsterdam. 72. W. G. Mathers, A. J. Madden, Jr., and E. L. Piret, Ind. Eng. Chem. 4_9_, 961 (1957). 73. J. C. Brice, J. Crystal Growth, _1, 161 (1967). 74. A. Carlson, Growth and Perfection of Crystals, edited by R. H. Doremus, B. W. Roberts and D. Turnbull, Wiley, New York, p. 421 (1958). 75. A. Fage and V. M. Falkner, "Aeronautical Research Committee Reports and Memoranda," No. 1408, H.M. Stationery Office (1931). 76. N. F. Barnes and S. L. Bellinger, J. Opt. Soc. Am. 35, 497 (1945). 77. R. H. Perry and C. H. Chilton, Chemical Engineers r Handbook, 5th edition, McGraw-Hill Book Co., New York, p. 3-240 (1973). 78. P. J. Shlichta, P. S. Chen and W. R. Wilcox, "Simulation of Solution Growth in a Space Environment,” Final Report on experimental portion (Sept. 1973-Sept. 1974) Contract No. NAS8-29847. 79. International Critical Tables, Vol. IV, McGraw-Hill Book Co., N.Y. p. 235 (1928). 80. A. A. Wragg, Electrochimica Acta, 1_3, 2159 (1968). 198 81. P. Bomio, J. R. Bourne and R. J. Davey, J. Crystal Growth, 3_0 , 77 (1975) . 82. R. B. Husar and E. M. Sparrow, Int. J. Heat Mass Transfer, 1_1, 1206 (1968). 83. R. C. Binder, Fluid Meohanios, Prentice-Hall Inc., Englewood Cliffs, N.J., p. 183 (1955). 84. T. Ota, N. Kon, J. Heat Transfer 9_6C, 459 (1974). 85. C. B. Sclar and C. M. Schwartz, '’Proceedings of an International Conference on Crystal Growth," Boston, 20-24 June, 1966, p. 399. 86. W. R. Wilcox, J. Crystal Growth, 3_8, 73 (1977). 87. P. F. Scholander, Walter Flagg, R. J. Hock, and Laurence Irving, J. Cellular Comparative Physiology, £2, Suppl. 1 (1953). 88. K. H. Chen, "Particle Pushing and Separation of Particulate Mixtures by Solidification," Ph.D. dissertation, Univ. of Southern California, (1976). 89. W. R. Keagy, H. H. Ellis and W. T. Reid, "Schlieren Techniques for the Quantitative Study of Gas Mixture," The Rand Corporation, CA (1949) pp.3-4. 90. W. R. Wilcox and H. S. Kuo, J. Crystal Growth, 19, 221 (1973). 91. M. 0. Kliia and I. G. Sokolova, Sov. Phys.-Crystallogr. 3, 217 (1958). 199 92. Intennatbonal Cvlt'toal Tables, Vol. Ill, McGraw-Hill Book Co., N.Y. p. 80, 105 (1928). 93. R. H. Perry and C. H. Chilton, Chembeal Engbneevs 1 Handbook, 5th edition, McGraw-Hill Book Co., N.Y. p. 3-235 (1973). 200 APPENDIX A EXAMPLE CALCULATIONS OF VOLUME FRACTION OF SOLUTE, GRASHOF NUMBER AND SHERWOOD NUMBER ON A 30°C - SATURATED NaC103 SOLUTION WITH GROWTH AT 25°C (A) Volume Fraction of Solute F (Experiment Run Number = 042776-1) Figure A-l shows the mean molar volume V vs. mole fraction of NaClO^ in the solution at 18°C (data from ref. 92. See Table A-l). Figure A-2 shows the same data in a large scale. The curve of V vs. X£ is nearly straight. Therefore the partial molar volume of NaClO^ V^ is nearly _ 3 constant. From Fig. A-l we obtained V2=40.2 cm /mole at At the growth temperature (25°C) we only had the information of density and concentration of a 25°C- saturated solution. If the curve of V vs. X2 at 25°C is still straight and has the same slope as the 18°C curve 18 0 C. _ 3 then we obtained V2=40.45 cm /mole at 25°C. If the solu tion is saturated at 30°C, we have (see Table A-l) 6 .94x10 -3 mole/cm 3 201 Ceq = ( m r j TT> ' frag) = 6.76 x 10-3 mole/cm3 AC = 1.8 x 10 ^ mole/cm3 F = C V. ~C V9 = 6.94 x 10‘3 x 40.45 = 0.281. o 2 oo 2 This approximate value of F is only good if the interface kinetics are very slow. In the other limit of very fast interface kinetics then F ~ C V0. The real value of F is eq 2 F = C V. = 6.89 x 10"3 x 40.45 = 0.279 o 2 which is 0.7% lower than 0.281. (B) Grashof Number Gr (Using C ) From Eq. 5-1, we have Gr = g • L3 • (p -p )-^- ® v^oo ^eq y p^ = 1.449 g/ml (from Fig. 3-11) p = 1.439 g/ml (from Fig. 3-10) eq P +P p = _2L— = 1.444 g/ml 2 y^ = 2.08 c.p. (from Fig. 3-13) yeq = 2.02 c.p. (from Fig. 3-12) y +y 00 eq 0 n r p = ------n = 2.05 c.p. 2 202 crystal size = 6.9 mm x 7.9 mm x 0.8 mm 0 . 69x0 .79 n i o / i = 0.184 cm 2(0.69+0.79) 3 1'444 Gr = 980 x 0.184 x(l.449-1.439)x ^ 05xl0~ ^)^ = 209.8 (C) Sherwood Number Sh crystal growth rate = = 1.739x10 ^ g/sec 2 total crystal area A = 1.327 cm From Eq. 3-8 we have 0.719x1.739xl0'5 „ ,„-4 , ------- 7^---------------- = 4.92x10 cm/sec 1 . 8 x l 0 x 106.45x1.327 - 5 2 D = diffusivity - 1.5x10 cm /see (see Appendix C) kL _ 4.92x10 ^x0.184 * Sh = - j r - - ------------- f---- = 6.04 u 1.5x10” 203 204 Table A-l Density of aqueous NaClO^ solution (92) T (°C) wtl mole % density P(g/ml) mean molar volume (ml/mole) v = I[Mlx1 +M2 x2] 18 0 0 0.99862 18.02 18 1 0.17 1.0053 18.05 18 6 1.07 1.0397 18.22 18 12 2.25 1.0827 18.46 18 16 3.12 1.1131 18.65 18 20 4. 06 1.1449 18.86 18 24 5.07 1.1782 19.08 18 28 6.17 1.2128 19.34 18 32 7.37 1.2491 19.63 18 34 8.01 1.2680 19.78 18 48.2* 13. 59 1.4266** 21.04 25 50.0* 14.46 1.439** 21.40 2 o * * * 50.98* 14.96 1.4454** 21.61 Saturated solution (from Fig. 3-9). Density measured at saturated temperature (from Fig. 3-10) *** The density of 30°C saturated solution measured at 25°C is 1.449 g/ml (from Fig. 3-11). molar volume V (ml/mole) 40 36 24 0 . 8 mole fraction X 2 Fig. A-l: Mean molar volume vs. mole fraction of NaClO^ in aqueous solution. O 18°C ^ 18°C (saturated solution) ■ 25°C (saturated solution). 205 molar volume V (cm^mole) 20.8 2 0 . 4 20.0 9.6 0 0.02 0 . 0 8 0 . 0 4 0 . 1 0 0 . 1 2 mole fraction X2 Fig. A-2: Mean molar volume vs. mole fraction of NaClO^ solution at 18°C (92). 206 APPENDIX B EXAMPLE CALCULATION OF THE CHANGE OF THE RELATIVE SUPERSATURATION DURING NaC103 CRYSTAL GROWTH Experiment Run Number = 022676-1 The amount of NaClO^ dissolving in solution at 30°C before growth = 293.56 g. The amount of water in solution = 282.27 g. The crystal growth temperature = 26°C. The solution saturated at 30°C mi - | - t - - i , . r t o p 293.56 104 g The solubility at 30 C = ------- =- 5--- 282.27 100 gH20 The solubility at 26°C = (from Fig. 3-9). 10 OgH^O After 4 hours of growth at 26°C, the amount of NaClO^ incorporated in the crystal was 0.529 g. The amount of NaClO^ remaining in the solution after growth = 293.56-0.529=293.03 g. The ratio of NaClO^ to H20 after growth 293.03 = 103.82 282.27 100 gH20 207 n • • - 1 1 4 - 104-100 . Original relative supersaturation = -------- = 4 100 I- i i 4.- 4 . 4.- 103.82- 100 - OO0 Final relative supersaturation = ------ = 3.82%. 100 Percent change of relative supersaturation = 208 o\<= APPENDIX C THE CALCULATION OF THE SCHMIDT NUMBER FOR A 30°C-SATURATED NaClO^ SOLUTION WITH GROWTH AT 2 5°C The Schmidt number Sc = U = 2.08 c p ^ an ± y P D p = 2.02 c p Heq ^ U +U y = — — 52. = 2.05 c p Pm = 1.449 g/ml p = 1.439 g/ml eq 6 p = , P” 2 ^ = 1-444 g/ml From ref. 93, we have the diffusivity of NaClO^ at infinite dilute solution at T °C D = 5.045 x 10'8x (T+273) 209 at 2 5° C D = 1.50 x 10 ^ cm "V sec Sc . Z.OSxltT2 . g46-4 . 1.444x1.50x10 210 APPENDIX D CALCULATION OF THE DISTRIBUTION COEFFICIENT OF Ag IN NaC103 CRYSTALS FROM ATOMIC ABSORPTION DATA (1) The amount of Ag/Na in the original solution 1 100 (in moles). (2) The amount of Ag in the standard solution = 4 yg/ml (used as a reference for the spectro photometer) . (3) The sample solution consisting of 0.088g of NaClO^ dissolved in 250 ml deionized water. The number of moles of Na in this solution = °-088 = 0.00082668 mole. 106.45 The amount of Ag in sample solution from atomic absorption data = 6.46 yg/ml. The number of moles of Ag in the total solution = 6-46x10--x25£ = 1-497x10-5 mole- !07.87 (4) The ratio of Ag to Na in the solid = 0 (5) The distribution coefficient = ^ ^^^^ 0 .01 01811 1. 811 212 weight increasement (g) Appendix E 0.150 0.125 0.100 0 .0 7 5 0 .0 5 0 0 .0 2 5 200 time Fig. E-l: The weight change of a NaClO^ crystal grown with regular convection (weighed while crystal immersed in solution). Initial size of crystal = 5.6x4.6x6.3mm, (Gr=53), final size of crystal = 6.3x5.3x7.Omm, (Gr=107), AT = 4.2°C. 213

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Computerized modules for hydrogen plant design

PDF

Removal And Separation Of Particles (In Solid Organic Chemicals And Metals) By Crystallization

PDF

Conversion limits of reversible reactions in continuous non-ideal flow chemical reactors

PDF

Investigations On The Flow Behavior Of Disperse Systems

PDF

The Separation Of Multicomponent Mixtures In Thermally-Coupled Distillation Systems

PDF

Application of electrokinetic phenomena in: dewatering, consolidation andstabilization of soils and weak rocks in civil and petroleum engineering,and augmenting reservoir energy during petroleum...

PDF

The Defect Structure And Tracer Self-Diffusion Of Cadmium-Tellurium

PDF

Mechanistic Study Of Air Pollution

PDF

Nucleation And Solubility Of Monosodium-Urate In Relation To Gouty Arthritis

PDF

Auditors' risk attitudes: A hierarchical levels study within various decision contexts

PDF

Study of the parameters of rotameters with buoyant floats, for application to intravenous therapy

PDF

As(III) and As(V) removal by adsorption from geothermal fluids

PDF

An experimental investigation of thixotropy and flow of thixotropic suspensions through tubes

PDF

Characterization of polymer networks by swelling

PDF

Study of the flow properties of a thixotropic suspension

PDF

Non-Darcy Flow Of Gas Through Porous Media In The Presence Of Surface Active Agents

PDF

A multiplicative and additive correction procedure for the iterative improvement of finite element solutions

PDF

Thermal Conductivity Of Liquids

PDF

A high temperature catalytic ceramic membrane reactor: Theory and applications

PDF

Anodic alumina films: Preparation, characterization, and investigation of reaction and transport properites

Asset Metadata

Creator Chen, Pei-Shiun (author)

Core Title Convection irregularities during solution crystal growth and relation to crystal-defect formation.

Degree Doctor of Philosophy

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, chemical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Wilcox, William R. (committee chair), Chilingar, George V. (committee member), Rebert, C.J. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c17-24862

Unique identifier UC11348380

Identifier DP22093.pdf (filename),usctheses-c17-24862 (legacy record id)

Legacy Identifier DP22093.pdf

Dmrecord 24862

Document Type Dissertation

Rights Chen, Pei-Shiun

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA