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BIFURCATION AND MORPHOLOGICAL INSTABILITY By ARUNAN NADARAJAH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 f OMERS ITOF FLORIDA LIRAIES ACKNOWLEDGEMENTS In the course of the three and a half years of effort that went into this dissertation, I received help from numerous people in various ways that contributed to its completion. Enumerating them all would make this note be at odds with the spirit of conciseness of the rest of the document and so, reluctantly, I resolved to include only those who had made a direct contribution. Needless to say, I still remember with gratitude my debt to the rest. First mention should be made to my erstwhile mentor, Professor M.S. Ananth at the Indian Institute of Technology, Madras, who first kindled an enduring interest in theoretical transport phenomena and encouraged my proclivities toward graduate study. The greatest help came from my advisor Dr. R. Narayanan, who apart from getting me involved in morpho logical instability and giving many suggestions also imparted to me a solid background in mathematics and hydrodynamic stability theory, not to mention helping me in numerous other ways. Professor L.E. Johns, Jr., first introduced me to linear operator theory and gave suggestions too regarding my research, but more importantly, he was my "chemical engineering conscience," broadening my vision when I tended to specialize too much and keeping the objectives in perspective when I got wrapped up in abstract theoretical points. It is not an exaggeration to say that I probably would not have progressed this far academically without these three individuals. I would also like to thank Drs. S.R. Coriell and G.B. McFadden of the National Bureau of Standards for many discussions and suggestions regarding the subcritical nature of morphological instability; the members of my supervisory committee: Professor U.H. Kurzweg, Dr. G.K. Lyberatos, Dr. W.E. Lear, Jr., and Dr. S.A. Svoronos for their time and effort on my behalf; the Department of Chemical Engineering for provid ing a research assistantship during the first two years of my Ph.D. work and the Department of Mathematics for a lectureship during the last one and a half. My thanks also go to my uncle Dr. R.S. Perinbanayagam for being a role model and helping me evolve a "meaningful philosophy of life" and cope with the stress of graduate school. Finally, I wish to express my gratitude to my colleague S. Pushpa vanam for many "enlightening" discussions and to Debbie Hitt for doing a superb job of typing the manuscript and "correcting" my Queen's English spelling! TABLE OF CONTENTS ACKNOWLEDGEMENTS ......................................... ABSTRACT ................................................. CHAPTERS 1 DESCRIPTION OF THE PROBLEM ........................ 2 PREVIOUS WORK ON MORPHOLOGICAL INSTABILITY ........ 2.1 Early Work .................................. 2.2 Later Research .............................. 2.3 Inclusion of Other Effects .................. 2.4 Experiments in Morphological Instability .... 2.5 Limitations of Existing Models and Unaddressed Issues ........................ 3 A UNIFORM FORMULATION ............................. 3.1 The Formulation ............................. 3.2 The Linear Stability Problem ................ 3.3 The Adjoint Problem and Exchange of Stabilities ............................... 3.4 Finite Containers and the Most Dangerous Wavenumber ................................ 4 SUBCRITICAL BIFURCATION ........................... 4.1 Theory ..................................... 4.2 The Second Order Problem .................... 4.3 The Third Order Problem ..................... 4.4 Calculations and Comparisons ................ 5 COMPARISONS WITH RAYLEIGHMARANGONI CONVECTION .... 5.1 RayleighMarangoni Convection in Brief ...... 5. The Augmented Morphological Problem ......... 5.3 Comparison of Morphological Instability with RayleighMarangoni Convection ........ 6 BIFURCATION BREAKING IMPERFECTIONS ................ 6.1 Nature of Imperfections ..................... 6.2 Imperfection Due to Heat Loss ............... Page ii vi 1 8 8 10 12 14 15 17 17 23 30 35 38 38 41 46 49 61 61 65 70 74 74 76 6.3 The Outer Expansions ........................ 78 6.4 The Inner Expansions ........................ 83 6.5 Imperfection Due to Advection in the Melt ... 87 6.6 Nonexistence of the Planar State ............ 93 6.7 Asymptotic Solution ......................... 94 6.8 Controlling Imperfections ................... 99 7 NEW DIRECTIONS .................................... 102 7.1 Transition to Dendritic Growth .............. 102 7.2 Extension to Semiconductor Materials ........ 103 7.3 Inclusion of Microscopic Effects ............ 104 7.4 Numerical Methods ........................... 104 7.5 Experiments ................................. 105 APPENDICES A NOMENCLATURE .......................................107 B PHYSICAL PROPERTIES ................................113 REFERENCES ............................................... 115 BIOGRAPHICAL SKETCH ...................................... 120 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy BIFURCATION AND MORPHOLOGICAL INSTABILITY By Arunan Nadarajah August, 1988 Chairman: Dr. Ranganathan Narayanan Major Department: Chemical Engineering Morphological instability refers to the tendency towards spatial pattern formation on the liquidsolid interface when a dilute binary mixture is solidified or fused. The importance of this phenomenon is in the growth of metal alloy and semiconductor crystals from their melts, where it influences the solute or dopant concentration resulting in nonuniform physical and electrical properties. Previous formulations of morphological instability have involved several simplifying assumptions which restricted it to the study of a region immediately surrounding the interface. The models have limited validity and they require separate treatments for different situations like freezing and melting. In this study a new uniform approach is presented which considers the entire melt and crystal domain and is applicable to all situations. Earlier formulations are shown to be approximations of this and exchange of stabilities is proven asymptotically. The model is then used with a weakly nonlinear technique to predict the shape of the bifurcation diagram for various cell patterns. The subcritical nature of morphological instability is shown and regions of its prevalence are determined over the entire domain of experimental parameters. This was used to compare with experimental results and to determine optimal crystal growth regions. A comprehensive comparison of morphological instability with con vective instability was undertaken and this phenomenon was shown to resemble Marangoni convection in its mathematical and physical fea tures. This was done in order to introduce some of the multitudinous mathematical techniques employed in convective instabilities into mor phological instability and specifically was used here to complete the eigenspace of the linearized problem. Two imperfections which reside in the domain, heat loss at the container wall and advection in the melt, were considered and shown to be bifurcation breaking imperfections. Solutions to the problem were obtained in both cases by matched asymptotic expansions and based on these results a practical way of minimizing the effect of these imper fections was suggested. CHAPTER 1 DESCRIPTION OF THE PROBLEM Morphological instability refers to the process of spatial pattern formation at the liquidsolid interface when a binary mixture is solidified or fused. This is a problem of hydrodynamic instability and like all other problems of this nature, for this phenomenon too there is an onset point where the initially planar interface first begins to deform and forms cellular patterns. These grow into deeper fingerlike shapes and eventually forming side branches and treelike dendritic structures. The importance of this phenomenon is in the growth of metal alloy and semiconductor crystals from their melts. Morphological instability affects not just crystal shape but also the solute or dopant concentration, resulting in nonuniform physical and electrical properties in the crystal. This is especially perfidious in applications where superfine crystals with very consistent properties are required. Recently there have been some indications that crystal quality can be significantly improved by growing them in low gravity as this reduces other problems associated with crystal growth like natural convection, but unfortunately not morphological instability. Growing the crystal at very high temperature gradients or at very low growth rates avoids morphological instability but crystals grown at high temperature gradients are of poor quality due to thermal stresses and the very low growth rates makes the process very expensive. Hence this becomes a problem not only of avoiding crystal surface deformations but one of optimization of the process as well. There are several methods for growing crystals from the melt, distinguished by the hydrodynamics of growth. The three basic ones are Bridgman, Czochralski and float zone and most techniques are variations of these, like horizontal and vertical Bridgman. A typical Bridgman experiment is shown in Fig. 11. The material is usually in a quartz ampoule and is melted and then recrystallized in the Bridgman furnace. The upper part of the ampoule is maintained at a higher temperature than the lower and solidification proceeds upwards and the ampoule is pulled downwards at the same velocity v. The top of the melt is protected by a liquid encapsulant like B20 At the end of the process the ampoule is broken to retrieve the crystal. In the float zone technique shown in Fig. 12, the ampoule or the material itself is pulled through a circular furnace and the melting and recrystallization proceeds simultaneously. As this technique can be done even without a container it avoids the problem of impurities from the ampoule entering the crystal, but its more difficult to maintain a uniform temperature gradient. Figure 13 shows the Czochralski method where the crystal is rotated and pulled from a melt pool. As the emphasis here is on the liquidsolid interface, the modelling of the crystal growth process will be kept as general as possible but will resemble the Bridgman technique the most. The temperature and concentration profiles during typical crystal growth conditions are shown in Fig. 14. The temperature profiles in the liquid and solid are virtually straight lines and solute concentration in the solid is virtually constant. But the solute Heating Element Liquid Encapsulant Quartz Ampoule Melt Shield 'Crystal Pulling Rod Figure 11. Bridgman growth Heat Heat t Heat Figure 13. Czochralski Figure 12. Float zone concentration in the liquid CZ changes sharply near the interface because of solute rejection on solidification. This in turn has an effect on the freezing point depression in the liquid as shown in the figure. Figure 15 shows the same profiles but in a situation where the freezing point in the liquid TM now exceeds the actual liquid temperature (this change can be brought about by either reducing the liquid temperature gradient or by making the change in C near the interface even sharper by increasing the growth velocity). This is referred to as constitutional supercooling and the system responds to this unstable situation by interface deformation. Countering this is the interfacial tension which always acts to minimize the surface area which in this case is the planar surface. When this balance is upset, or in other words when the onset conditions are exceeded, the interface loses planarity and forms cellular patterns. Figure 16 shows the profiles for the fusion case, where we could have TM in the solid being less than the actual solid temperature and once again interfacial deformation is the system's response, balanced by interfacial tension. The only difference here is that since solid diffusivities tend to be several orders of magnitude lower than liquid diffusivities, the solute concentration profile in the solid near the interface will vary even more sharply resulting in lower onset conditions for morphological instability. In the paper of Trivedi and Somboonsuk (1984) there is a series of photographs from an experiment (see Fig. 1 in their paper) where succinonitrile/acetone crystals were grown. The first photograph shows the liquidsolid interface just after onset and has a discernible cellular pattern. Later ones show the cells becoming deeper, forming Temperature or Concentration Solid Figure 14. Concentration and temperature profiles during solidification Liquid Temperature or Concentration TMS Ts Solid *s Figure 15. TM Liquid C  Concentration and temperature profiles during solidification Temperature or Concentration STM Solid T, Figure 16. Concentration and temperature profiles during fusion TM Liquid C,  fingers and ending up as dendrites. The arrows in the first photograph mark the initial perturbations that eventually become dendrites. In this thesis we will concentrate only on the region near the onset point, shown by the first two photographs, though dendrites will be mentioned in discussions. In the papers by Morris and Winegard (1969) and Tiller and Rutter (1956) we see another aspect of morphological instability, the variety of cellular patterns, fingerlike shapes, hexagonal cells and variations of these. Other possible shapes are cylindrical rolls and rectangular cells but by far the commonly encountered pattern is the hexagonal one. The choice of the cell pattern is extremely important and factors determining this choice will be discussed later. Their figures also show that, unlike other forms of hydrodynamic instability, the number of cells on a single crystal is in the hundreds. Though it is customary to model morphological instability in terms of the temperature and concentration profiles, in reality this phenomenon seldom exists in isolation; it is usually coupled with fluid flow in the melt. There are two kinds of flows that occur. The first is buoyancy driven solutal convection which is caused by the sharp solute concentration gradients in the melt. The other is the result of density change during solidification. When solidification occurs there is a constant rate of volume decrease which causes the melt to move in to fill the vacated space. This motion is referred to as advection and the rigid side walls of the ampoule will cause closed streamlined flow in the melt as a result (see Fig. 62). In addition there will be flows in the melt in Czochralski growth due to rotation and other kinds of flows in special growth techniques. Apart from these, several other parameters affect this phenomenon, the most important of which are due to the fact that most crystals are faceted; that is, they have a crystal lattice structure. Hence whether the lattice axis is aligned or not with the growth direction is extremely important as can be seen from the experiments of Heslot and Libchaber (1985). Other important considerations are grain boundaries, wetting of the ampoule wall and the presence of impurities. Also in rapid solidification, the system will not be at thermodynamic equilibrium and kinetic undercooling of the melt becomes significant. CHAPTER 2 PREVIOUS WORK ON MORPHOLOGICAL INSTABILITY 2.1. Early Work The first successful attempt at explaining morphological instability qualitatively was by Rutter and Chalmers (1953). They coined the word "constitutional supercooling" to describe the existence of unstable melt regions near the interface where the freezing temperature can be higher than the liquid temperature itself and correctly identified this as the cause of interface deformation. Tiller, Rutter, Jackson and Chalmers (1953) quantified constitutional supercooling and for instability came up with the condition mG > GC (2.11) where m is the absolute value of the liquidus slope, G the liquid temperature gradient at the interface and G i the solute concentration gradient in the liquid at the interface. The negative sign is caused by Gc and Ge being in opposite directions (see Fig. 15). As can be seen these simple thermodynamic explanations do not take into account the stabilizing effect of interfacial tension. To do so would require casting the problem as one of hydrodynamic instability and obtaining the onset conditions from a linear stability analysis. This is exactly what Mullins and Sekerka (1963, 1964) did when they considered the problem with temperature and concentration equations in the liquid and solid and boundary conditions at the interface. Their criterion for instability to an infinitesimal disturbance was mG > Gh + aT X/L (2.12) where GT is the weighted temperature gradient GT = (k2.GI + kG )/(kz + k ) (2.13) Here ks and k are the solid and liquid thermal conductivities. G is a modified liquid concentration gradient given by G = G c(a v/2D )/(a (1/2 k)v/D ) (2.14) 2 2 c 1/2 a2 = (a2 + v /4D ) (2.15) where a is the wavenumber of the disturbance, k the solute distribution coefficient, TM the melting temperature of the pure solvent, Lh the latent heat of fusion and X the interfacial tension. This analysis laid the foundation for all further work in morphological instability. Following them Woodruff (1968) did the linear stability analysis along the same lines for the melting problem and came up with the same criterion as (2.12) but with G = G (na v/2D )/(nka + a. + (1 k)v/2D.) (2.16) 2 2 2 1/2 a = (a + v /4D2 ) (2.17) 5 S where Gso is the solid concentration gradient at the interface and n is D /D the ratio of solute diffusivities. 2.2. Later Research The next major contribution to the problem was made by Wollkind and Segel (1970) who proved "exchange of stabilities" for this problem for most parameter ranges. Proving exchange of stabilities is equivalent to showing the existence of the onset of steady state nonplanar solutions. They also considered the weakly nonlinear regime after the onset of instability and using the method of Stuart (1960) and Watson (1960) analyzed the problem for the case of twodimensional rolls, showing the existence of subcritical bifurcation at most growth velocities. The method of Stuart and Watson is essentially the theory of Landau (see Drazin and Reid (1981)) and follows the dominant mode of instability into the weakly nonlinear regime. In this form it is applicable only to disturbances of one cellular pattern at a time, but Segel and Stuart (1962) extended this theory to the prediction of the preferred pattern by considering the interaction of two specified modes of disturbance. Depending on the way these two modes were combined it was possible to obtain two dimensional rolls or hexagonal cells and they showed that the experimental parameters would determine the stability of these patterns. Sriranganathan, Wollkind and Oulton (1983) adopted this method for morphological instability and gave parameter ranges where each type of cell was stable. The limitation of the method is that it considers hexagonal and two dimensional roll patterns but not rectangular cells or cylindrical rolls and ignores the effect of container shape and size which have been shown to be important in wave pattern selection (see Koschmeider (1967)). Ungar and Brown (1984a) considered the highly nonlinear problem and after making several simplifications obtained solutions using the finite element method. Finite elements can handle highly nonlinear problems and give very accurate numerical solutions but are extremely time consuming. Solving the full morphological problem is a very expensive proposition by this method and hence Ungar and Brown simplified the problem by ignoring the latent heat and solid diffusivity and assuming that thermal conductivities in liquid and solid were equal. This allowed them to reduce the problem to a "onesided model" consisting of variables in the liquid region only, considerably simplifying the algebra and saving computer time. Such a model will have only limited validity in highly nonlinear regions and this was borne out when Ungar, Bennett and Brown (1985) solved the complete problem. But their most extensive calculations were done only for the one sided model and hence this is of chief interest. These were done only for the case of two dimensional roll disturbances and here they showed that contrary to that reported by Wollkind and Segel, there were multiple regions of supercritical and subcritical bifurcation. More importantly they showed that at large deformations of the interface secondary bifurcations occurred. Ungar and Brown (1985) also modelled the formation of deep cells in an attempt to follow the transition to dendritic growth. Nonlinear finite difference calculations were done in a more limited way by McFadden and Coriell (1984) for the two dimensional case. Later McFadden, Boisvert and Sekerka (1987) extended the calculations for the three dimensional patterns of hexagons and cross rolls. In both cases the enormous expenses involved restricted calculations to a few parameter values. 2.3. Inclusion of Other Effects While these workers were investigating the basic problem others were busy trying to incorporate various influences. The most important concern was the effect of fluid flow. Delves (1968, 1971 and 1974) in attempting to approximate the influence of advection and stirring in the melt, studied the influence of plane Couette flow on the problem. He showed that two dimensional roll disturbances in the flow direction were stabilized but there was no effect on disturbances perpendicular to the flow. Coriell, McFadden, Boisvert and Sekerka (1984) modelled Couette flow more systematically and came to the same conclusion. Recently McFadden, Coriell and Alexander (1988) examined the effect of plane stagnation flow on two dimensional disturbances perpendicular to the flow and here too the flow as found to be stabilizing. In another very important development Coriell, Cordes, Boettinger and Sekerka (1980) studied morphological instability with solutal convection. They showed that the two in.;abilities were essentially decoupled with the melt being unstable to convective disturbances of long wavelengths and the interface unstable to nonplanar disturbances of small wavelengths. Also at low growth rates the dominant instability was convective and the interface was not easily disturbed. At high growth rates the roles were reversed and at an intermediate velocity the two instabilities became comparable. It was only at this rate the two instabilities interacted and the result was the prevalence of oscillatory instabilities. Their conclusion was that, except at this particular growth rate, it is usually sufficient to study only the dominant instability near its onset. Following Coriell et al. several workers have looked at special aspects of these two instabilities and their work has been reviewed by Glicksman, Coriell and McFadden (1986). They all confirmed or refined the work of Coriell et al. but all the main conclusions mentioned above still hold. Several other influences apart from fluid flow have been incorporated into the model but only a few relevant ones will be considered here. Coriell and Sekerka (1972, 1973) tried to include the effect of grain boundaries on morphological instability by assuming that its only effect was to shift the onset conditions. They failed to observe that in the presence of grain boundaries there could be no planar solutions to the problem and that the interface will be nonplanar at all times. Ungar and Brown (1984b) obtained the solutions to this problem by matched asymptotic expansions for small grain angles and using finite elements for solutions of large grain angles. In rapid solidification kinetic undercooling of the melt is significant and Seidensticker (1967) included this and showed that it caused a shift in the onset conditions. The significance of this was shown by Hardy and Coriell (1968, 1969 and 1970) when they observed morphological instability in the growth of ice crystals. Constitutional supercooling was not a factor here and it was shown that kinetic undercooling was the primary cause. This dual cause for morphological instability is somewhat analogous to the situation in natural convection where we find that the variations of density and surface tension with temperature can both cause convective instability. 2.4. Experiments in Morphological Instability The early work on modelling morphological instability was prompted by experimental observations but beyond that very few quantitative experiments have been done near the onset conditions. This is an unfortunate state of affairs and experimental verifications of theoretical predictions are badly needed if further concrete progress on the theoretical front is to be made. The work of Morris and Winegard (1969), Trivedi and Somboonsuk (1984) and of Heslot and Libchaber (1985) have already been mentioned. Recently de Cheveigne, Guthman and Lebrun (1985, 1986) have attempted to verify the weakly nonlinear and strongly nonlinear theoretical predictions and one hopes that more work along these lines will follow. De Cheveigne et al. performed their experiments on succinonitrile/acetone and CBr4/Br2 systems. (These organic mixtures are much easier to work with than metal alloys as they are generally nonfaceted, transparent and require small temperature gradients and hence they have been very popular with experimentalists.) They found that the cell pattern formed and its dimensions were strongly dependent on geometry of the container. More importantly when they ran the experiments for two dimensional roll patterns, they found only subcritical instability. 2.5. Limitations of Existing Models and Unaddressed Issues In Chapter 1 the cause of constitutional supercooling was explained as being due to the sharp solute concentration gradient in the liquid near the interface, while elsewhere in the liquid and the solid the solute concentration was practically a constant. It would seem then that the only region of interest is the interface and a liquid "boundary layer" adjacent to it. This has prompted all previous workers in morphological instability to consider D /v as the characteristic length of the problem and to ignore solid diffusion. A typical value of D /v is 100 microns and this means that the far ends of the melt and crystal are infinitely far away and the domain of the problem is effectively confined to the liquid boundary layer mentioned above. For the melting problem a characteristic length of Ds/v is used and the domain becomes an even smaller boundary layer in the solid. These assumptions considerably simplify the algebra involved and hence their popularity. But they constrain the validity of the model in several ways. The most obvious one is that they necessitate the melting and solidification problems to be studied separately, even though they only differ in the direction of the growth velocity. Besides this assumption fails for very small growth velocities, as it introduces a singularity at v = 0. Later we will show that neglecting solid diffusion also introduces a singularity and makes the model fail in the nonlinear regime. Finally, any effect which resides in the entire domain, not merely the boundary layer, cannot easily be incorporated into the model, which is why all influences on morphological instability studied so far are either boundary layer effects (e.g., solutal convection) or interfacial effects (e.g., grain boundaries and kinetic undercooling). Phenomena that span the entire domain, like advection in the melt or imperfect insulation of the ampoule walls, have been either inadequately treated or ignored completely. Hence there is a need for a model that includes the entire liquid and solid domains which would be applicable for all growth velocities. This model should also dispense with the separate treatments accorded so far to the solidification and fusion problems with one uniform formulation. In Section 2.2 it was mentioned in connection with the work of Wollkind and Segel (1970) and of Ungar and Brown (1984a), that this problem oscillates between subcritical and supercritical instabilities for the case of two dimensional roll disturbances. They did not, however, compute the ranges of each type of instability for the experimental parameters involved. This is necessary in light of the experiments of de Cheveigne et al. (1986) who observed no supercritical instability. Also the extension of these predictions to three dimensional disturbances like hexagonal and rectangular cells is yet to be done. It would not be an exaggeration to state that the inspiration for all the theoretical work done so far in morphological instability has come from RayleighBenard convective instability. A comparison between the two problems would be invaluable as a source of continued inspiration and as a way to draw conclusions and conjectures about morphological instability from the vast published literature on RayleighBenard convection. Hurle (1985) has attempted this but his work can only be regarded as perfunctory and there exists a need for a more rigorous treatment of the issue. CHAPTER 3 A UNIFORM FORMULATION 3.1. The Formulation Since we are not making the assumption that the liquid and solid are very deep, the problem has to be formulated very carefully, especially with regard to the outer boundaries, if we are to avoid an intractable moving boundary problem. A typical crystal growth set up is shown in Fig. 31. The ampoule is heated by the heating coils surrounding it and they keep the melt region at a temperature T1 and the crystal at T2. The temperatures T1 and T2 are maintained constant by means of thermocouples located at z = s and z = X. The ampoule is pulled towards the cooler end at the same velocity V at which the crystal grows, thus keeping the interface stationary. The region near the interface is protected by an insulating shield and it is this region that becomes the domain in our model. So in this model the outer liquid and solid boundaries become fixed at z=1 and z=s respectively and the solid will be moving with a bulk velocity v and the liquid with a bulk velocity v/Y, where Y is the ratio of densities p /P In this section we will assume that Y=1 and consider the effects of Y not being unity in Chapter 6 as this would cause advection and fundamentally alter the basic problem. Also we will assume that the melt concentration at the outer liquid boundary is a constant C1. Thermocouple z = 0 So o o o o 0 0 0 0 0 0 1~~ v //1 77 7 7777 7 7f/. / f/ If/./If///7777 Melt z = I Crystal ///////1Shield o o o o o o o o o o o o z= S Heating Coil Figure 31. Experimental set up Thermocouple 19 The domain equations in the liquid melt are aT .+ at at az x 3z 2 z T at. De \ (3.11) (3.12) In the solid region the equations are aT s at ac a+ at aT 2 ac 2 v = D C 5z s s where T and C are the temperature and solute concentration, D diffusivity and thermal diffusivity, with the subscripts referring to the liquid and solid. The boundary conditions are Tz = T1, C C at z = 1 T =T at z=s s 2 (3.13) (3.14) and a the II and s (3.15) (3.16) At the liquidsolid interface we will use r to denote the departure from planarity and write the boundary conditions at z= T = T = T mC T H (3.17) s M I M Lh k VT n kT n = L(v + D ) (3.18) 2. 9. Ss h +Dt)(.18 kC = Cs (3.19) D VC* n (v + )C D VC n (v + )C (3.110) where TM is the melting point of the pure solvent, A the interfacial tension, Lh the latent heat, m the absolute value of the liquidus slope, ki and ks the liquid and solid thermal conductivities, k the distribution coefficient, n the normal at the interface directed into the solid and H is the curvature of the liquidsolid interface and in Cartesian coordinates is given by 2 2 2 H [ 'sC (1 + (i)2) 2 S + 1 (1 + (LC)2 )] S2 ax ax ay axay 2 ax dx yy [1 + 2 + (C))2 23/2 (3.111) ax Dy In cylindrical coordinates, if we assume circular symmetry, it becomes H [ 12 (1 + (,)2) a[1 + (,)2]3/2 (3.112) 2 Dar rar 3r ar It is assumed that the side walls are sufficiently far apart and well insulated to enable us to impose periodic boundary conditions in that direction. To convert these equations into the dimensionless form we will use the liquid depth Z as the characteristic length and the diffusive time D /v2 as the characteristic time. The temperature will be made dimensionless by T = (TTM)/G t with GT = (kG + ksG )/(ks+ k ) (3.113) where G, and Gs are the temperature gradients in the liquid and solid in the quiescent planar state. Similiarly the concentration will be made dimensionless by C = C/G a with c G = (D G + DG )/(D + D ) (3114) where G.o and Gso are the concentration gradients in the liquid and solid in the quiescent planar state. So in dimensionless form, if we neglect the Lewis numbers D /a in the temperature equations, we have 2i =0 (3.115) ac aC (3.116) + v = C at 3z V2T =0 (3.117) DC aC (3.118) +v r ni EC 5t zs s where v and = D /D The boundary conditions are Ti = (T TM)/GTL = T1 (3.119) C = 1 /GZ = C i 1 C at z=3s/=s, Ts = (T2 T )/G I T2 at the interface z= Ti = = SeC A f VT n BVT n = L( + ) Dt kCi = s VC n (v + )C = VC n (v + ) Dt Dt (3.120) (3.121) (3.122) (3.123) (3.124) where is the total derivative, a the ratio of thermal conductivities Dt k /ki and Se, A and L are the Sekerka, capillary and latent heat numbers respectively. mG Se = (3.125) T TM A M 2 (3.126) L hGT LhDQ L =k I (3.127) kiGT z at z= 1, At this point a discussion of the choice of a critical parameter becomes imperative. The experimentally variable parameters for this problem are Gt, C1 and v or their equivalents in this formulation GT, Go and v. Most previous workers have adopted G or C as their critical parameter but Hurle (1985) has proposed Se, by analogy with the RayleighMarangoni problem. Recently de Cheveigne et al. (1986) have advocated the use of v from an experimentalist's perspective. Although this is a valid choice the reason no one else has used it so far is probably because v occurs in the domain equations and will give rise to an infinite number of eigenvalues in the linearized problem. We second Hurle's suggestion and choose Se as this seems to be the naturally occurring coupling factor between temperature and concentration and the fundamental cause for morphological instability. Besides it includes both GT and Go But the suggestion of de Cheveigne et al. still remains a valid one. 3.2. The Linear Stability Problem From now onwards the for dimensionless variables will be dropped. The steady state planar solution to this problem occurs when c (x,y) or c (r) = 0 (3.21) The solution is then T c(z) = (T1 + SeC2 (o))z SeC2 (o) (3.22) T (z) = (T2 + SeCc (o))z/s SeC c(o) C (z)= C [ (1k)exp(vz) 1]/ (1k)exp(v) + 1] to 1kexp(v(1+s/n)) kexp(v(1+s/n)) C (z) C r (1k)exp(vz/n) + (1k)exp(v) + so1 kexp(v(1+s/n)) 1 kexp(v(1+s/n)) (3.23) (3.24) (3.25) To write the equations of the linear stability problem we will impose an infinitesimal disturbance on the steady state solution. T = Tto + T 2. ic 9& (3.26) with similar expansions for Ts, CV, Cs and . Considering the linear stability problem, in order to separate variables we will assume a horizontal cell pattern. This pattern for two dimensional rolls is given by 2irnx 2wn c (x) = Cos with wave number a = n L n L 2inx 2irny 2irn for cross rolls 0 (x,y) = Cos L + Cos a n L L n L (3.27) (3.28) for rectangular cells 0n(x,y) = Cos 2Ln Cos 2 an = 2rn(1/L2 + 1/L2) 1 n L1 L2 n 2 (3.29) for hexagonal cells 2wnx 2iny + ny C nay r (x,y) = 2Cos 23L Cos 3L + Cos L, an 4 n 73L 3L 3L n 3L for cylindrical rolls n (r) = J (a nr/R) n o n (3.210) (3.211) where an are the zeros of J1, and Jo and J1 are the zeroth order and first order Bessel's functions of the first kind. We can now write T (xy,z,t) = T1V(z) 41 (x,y)eot (3.212) with corresponding forms for the other variables, where a is the eigenvalue of the linear stability problem. The linear stability problem becomes in the domain 2 2 (D a )T = 0 2 2 oC2= (D vD a )C 2 2 (D a )T = 0 s1  C (3.213) (3.214) (3.215) (3.216) 2 vD 2  =(D a )Csl Here we have used D to denote  dzu The boundary conditions are T =0, C1 = 0 T = 0 si at z = 1 (3.217) at z = s (3.218) At the interface, z=0 T Ts + ;1 (G s) = 0 T + SeC + G (C + SeG c aA ) = DTl O DTsl = aL 1 K1 C + C 1 (kG c Gso) = 0 DCiM nDCs1 vC 1 + vCl1 = Gv(C 0 Cso c01 (3.219) (3.220) (3.221) (3.222) (3.223) Solving the system we obtain a general equation for morphological instability Ce + (1k)tanha stanha Ce k(na + v/2 tanha s)tanha + (a v/2 tanha )tanha s S S Xf C X* s = G + aA  aLtanhatanhas a(k tanhas + k tanha) A s a a a2 [2 v2 1/2 s n 2 a, = [a2 + + v2/4] 1/2 k G tanhas + k G tanha (3.224) k tanhas + k tanha II s (3.225) (3.226) (3.227) where STr G c(a v/2 tanha )tanha S + G (na v/2 tanha s)tanhaZ c (a v/2tanha )tanha s + k(na + v/2tanha s)tanhaZ R A 3 a S & (3.228) If we can show exchange of stabilities for this problem, then for neutral stability. GT + a2 A Se = (3.229) G Here we have already used the fact that the Se obtained from the neutral stability curve will be the same as Seo defined later in eqn. (4.22). In eqn. (3.229) if we let v become very large the critical wave number amin (for which Se0 is a minimum) also becomes very large and we can approximate all the tanh terms to unity. Further if we also neglect solid diffusion, n and G are zero and the equation reduces to the well known results obtained by Mullins and Sekerka (1964). G (a v) oC. (1k) 2 Seo 2 ] = 1 + a (3.230) a + v(k ) a + v(k ) a(k + k) For the case of neutral stability this becomes (1 + a2A )(a /v + k I) Se = (3.231) G(a/V 2 ) It must be pointed out that setting all the tanh terms to unity is equivalent to using the diffusive length as the characteristic length. This is a boundary layer approximation not unlike that used in the study of pipe flow at high Reynold's numbers. In our derivations we did not restrict v to be positive and hence eqn. (3.224) is also valid for negative velocities, that is the fusion problem. If we replace v with v and here too assume that v is very large and set the tanh terms to unity we obtain the result of Woodruff (1968). G c (a 1/2) C (1k) +  nka + a + v(1k) nka + a + v(1k) a(k + k ) (3.232) and for a = 0 (1 + a A)(nka + a + v(1k)) Se = ) 2 (3.233) G (na v) so a 2 To compare this model with the approximations of Mullins and Sekerka and that of Woodruff, Seomin was calculated for various growth velocities for the PbSn system and the results are shown in Fig. 32. As can be seen the approximations hold up very well for most growth velocities but begin to fail for small velocities. (The thermophysical data for the PbSn system were those of Coriell et al. (1980).) The ratio n for the PbSn system is of the order 105 but for systems with much smaller values of n like the CAustenite system (see Clyne and Kurz (1981) and Wolf, Clyne and Kurz (1982)) the approximations begin to fail at higher velocities. ig U) I s 's 0 > 4 8 Ow 00. 4 0 0C o o C qo 0C 0 0 a o o o ?* \ Io  a 0 O > a qt r' W ki 3.3. The Adjoint Problem and Exchange of Stabilities Exchange of stabilities refers to the nonexistence of time periodic infinitesimal perturbations. Timedependent infinitesimal perturbations about the planar state will generally have periodic and nonperiodic components, with a = or + iai. For some problems it is possible to show that a. is zero and this is called exchange of stabilities (see looss and Joseph (1980) for details). We still have to prove exchange of stabilities for this problem but before we can do that it is necessary to obtain the adjoint problem. To accomplish this we will define a column vector 9 and a matrix operator L T T s C C s (3.31) 0 Se 2 2  (D vDa a) 0 0 0 0 2 2 (D a )2 0 0 0 0 Se 2 2 (nD na oavD) x (3.32) and an inner product < *> = (T T + CC )dz + (TT + CC )dz 0 io ss se (1 3 (3.33) L = 2 2 (D a ) 0 0 0 where T denotes the complex conjugate of T, T* the adjoint function of T and x = (kNG0 Gsc)/(Ge G5) (3.34) Then the domain equations can be written as L*1 = 0 (3.35) In this inner product the adjoint problem becomes in the domain 2 2 ^"* (D a )T 2 2 ^* (D a )T = 0 2 2 ^ (D + vD a )Czi 2 vD 2 ^A* in sl a * = aC = c n s1 (3.36) (3.37) (3.38) (3.39) Subject to the boundary conditions T1 C 1 R1 L1 T = 0 s1 at z= 1 at z=s (3.310) (3.311) At the interface z=0 BT1 = T (3.312) nCt1 = c1 (3.313) kDCs1 DCl + (kG to Gs) = 0 (3.314) DT+l D1 + (G G) = 0 (3.315) DT + SeDC + (Gt + SeG aA) = T 51 Si (G SeG (G, G5) 11 GCCt (1k) (kG SeC (3.316) So far we have been unable to show exchange of stabilities for this problem directly. But Wollkind and Segel (1970) have proved it when the boundary layer approximation is valid and we will prove exchange of stabilities by performing an asymptotic analysis around the boundary layer solution, which corresponds to n=0 and Pe=0, where Pe is the Peclet number given by D ./v. ^ 00 ^10 01 T1 = T + nT1 + PeT1 + .... (3.317) 00 10 01 a = o + no + Pea + .... (3.318) The other variables are expanded similarly. If we use L to designate the operator L in (3.32) when n and Pe are zero, then the linear perturbed systems become L 10 =10 & L 00 f01 (3.319) & (3.320) 1 1 0 0 10000 a 0 With boundary conditions T10 ^10 s1 (3.321) & (3.322) at z=0 the equations are 00 ^10 ;10 10 B (0 ) = h 00 ^10 ^01 & B (0 1 1 l ) = 0 1 Sh (3.325) & (3.326) where B00 is the boundary operator defined by eqns. (3.219) (3.2.23) when n and Pe are zero and ^00 10  (G  ^00 10  (G + 10 "00 a LS 10 s 10 SeG ) to ^00 10 10  N(kG G ) 10 00 ^00 a C (1k)1 + to 1 00 10 ^ 00 a C (1k) toc 1 where f10 & f01 0 0 01 00 0 0 at z =  (3.323) at z= (3.324) 10 h = (3.327) h will have a similar form with 01 superscripts replacing the 10. It 00* ^00* we let L and 0 represent the adjoint operator and adjoint 00 ^00 function of L and #* respectively, we can use the solvability condition on the above system. 00 00 10 ^00* 10 < L > = < f > (3.328) 1 1 1 00*^ 00* ^10 Subtracting we get ^00* ^00* 10 <00* 01 J(41, ^\ h 0) = <0 f > (3.330) where the lefthand side of eqn (3.330) is the bilinear concomitant evaluated at z=0. ^00* ^00* 01)=0 <00 01 Similiarly J(01 h 0) = <* f > (3.331) ^00* ^00 We note that *, and ^o are real as they correspond to a state where 10 01 10 01 exchange of stabilities has been proved, while h10 h f and f are 10 01 also made up of real quantities except for o and o Hence we conclude from eqn (3.330) that a is real and from (3.331) that a is real; i.e. exchange of stabilities holds for the generalized morphological instability problem upto 0(n) and 0(Pe). So we are justified in using the neutral stability curve (3.229) to calculate Seo at least up to such order even though we often extend it further. Even when the boundary layer approximation is valid, exchange of stabilities does not hold for all growth conditions (see Coriell and Sekerka (1983)) and care must be exercised when such extensions are made. 3.4. Finite Containers and the Most Dangerous Wavenumber Appealing to the proof of Section 3.3, henceforth we shall only consider steady state solutions. A typical Seo versus a diagram is shown in Fig. 33 and Seomin is the minimum value of Seo and the wavenumber at which this occurs is amin. If we can maintain the growth conditions such that Se < Seomin we can at least say that the planar interface is stable to infinitesimal perturbations. As Seomin is the least value of Se at which the planar solution loses the stability, amin is the wavenumber of the disturbance which is most likely to occur. Hence this wavenumber is commonly regarded as the most dangerous and is the wavenumber at which morphological instability is usually studied. In the derivations and discussions that follow this is the wavenumber employed and it becomes our "operating wavenumber." The wavenumber is a 2w multiple of the reciprocal of the wavelength. If the domain being considered is regarded as being finite with periodic lateral boundary conditions, the wavelength R (or L depending on the wave pattern used) in its dimensionless form is now the aspect ratio, the ratio of the ampoule radius to the melt depth. For this situation Seo takes on different values depending on the number of cells formed on the crystal surface as shown in Fig. 34 (see also Rosenblat, Homsy and Davis (1982)). For each value of R there is a fixed number of cells which is the pattern that is most easily disturbed, except at certain values of R where two different patterns Unstable Stable Seo min  amin WAVENUMBER a Figure 33. Se vs. wavenumber diagram n=51 n=50 Seo min Figure 34. Se curves for various aspect ratios 0 Seo0 Seo+ 37 are equally dangerous. These "horizontal multiple points" are very important and will be discussed later. At other values of R the most dangerous number of cells is denoted by N and the corresponding wavenumber of each cell and the value of Seo are denoted as aN and SeoN respectively. For the analysis in Chapter 6 where the effect of a finite container on morphological instability is considered, R is chosen such that SeoN is as close as possible to Seomin so that the worst possible case can be examined. CHAPTER 4 SUBCRITICAL BIFURCATIONS 4.1. Theory The linear stability analysis will only give the onset conditions for morphological instability. Nonlinear calculations are necessary to determine the behavior beyond this point. Ideally numerical calculations should give the most amount of information by being applicable for small and large deformations, but as can be seen from the work of Ungar and Brown these calculations are very expensive to carry out and they were forced to simplify the nonlinear problem and perform calculations for very few experimental conditions. McFadden et al. (1987), even though they did not attempt calculations for larger deformations, were faced with the same restrictions. This then is the case for weaklynonlinear methods. They are generally valid only in a small region very close to the onset conditions but they can be used to predict the shape of the nonlinear curve for larger deformations and for several applications this information is sufficient. More importantly due to the analytical nature of the techniques they can be used to predict the weakly nonlinear behavior for all experimental conditions. A case in point is the work of McFadden et al., most of whose predictions could have been obtained more cheaply for all parameter values from weaklynonlinear theories. Probably the most useful information generated by these theories is the subcritical behavior of the nonlinear curve. Some typical bifurcation diagrams are shown in Figs. 41 to 44. The e = 0 axis corresponds to the planar solution and initially for small values of Se the planar solution is stable and usually the only possible solution. For Se Z SeoN the planar solution becomes unstable and a nonplanar solution bifurcatess" from the planar one. Figure 41 shows a symmetric bifurcation diagram where nonplanar solutions do not exist for Se < SeoN. For Se > SeoN, even an infinitesimal perturbation will make the solution jump from the unstable planar solution to the stable planar one while for Se < SeoN the planar solution is stable to all perturbations. This behavior is referred to as supercritical bifurcation and for these curves Se, = 0, Se2 > 0 (where Se, = dSe/de 2 2 and Se2 = d Se/de at e= 0). Figure 42 is nonsymmetric and as can be seen stable and unstable nonplanar solutions exist for Se < SeoN, making the planar solution stable to infinitesimal perturbations in this region, while a large perturbation can make it jump to the stable nonplanar branch. This is called a subcritical bifurcation diagram and is characterized by Sel 0. In this situation it is obvious that growing the crystal at Se < SeoN is no guarantee of avoiding morphological instability. Figures 41 and 42 display the behavior usually seen in most problems of hydrodynamic instability. Morphological instability is unusual in having nonlinear curves shown by Figs. 43 and 44 as well. These curves have been labelled "backward bending" to distinguish them from the usual "forward bending" curves. (Actually they are Januslike in appearance bending backwards and forwards.) From the point of view  Stable  Unstable Se Sei = 0 Se2 > 0 Figure 41. Forward bending, locally symmetric, supercritical bifurcation diagram Figure 42. Forward bending, unsymmetric sub critical bifur cation diagram S Stable  Unstable \Seo ; S I I I Sei = 0 Se2 > 0 Figure 43. Backward bending, locally symmetric, subcritical bifurcation diagram Figure 44. Se1 # 0 Backward bending, unsymmetric, sub critical bifurca tion diagram Se Sel # 0 of the crystal grower this is unfortunate as their subcritical nature increases the occurrence of subcritical bifurcation. Figure 43 is the symmetric case with Se1 0 and Se2 < 0 and Fig. 44 the nonsymmetric one with Se 0. The symmetry or nonsymmetry of the bifurcation diagram in hydrodynamic stability is dependent on the cell pattern (see Joseph (1976)). For morphological instability it will be shown that two dimensional rolls, rectangular cells and cross rolls produce locally symmetric bifurcation diagrams while cylindrical rolls and hexagonal cells produce nonsymmetric diagrams. It will also be shown that both backward bending and forward bending will occur for all cellular patterns depending on the experimental parameters used. Hence bifurcation can be subcritical or supercritical for two dimensional rolls, rectangular cells and cross rolls but for hexagonal cells and cylindrical cells bifurcation is always subcritical. 4.2. The Second Order Problem Here we begin our weakly nonlinear analysis and in this section we will calculate Sel, the first derivative of Se with respect to E. Considering the neutrally stable nonplanar solution near the bifurcation point Seo, we will expand the variables around the planar steady state solution. T T i + ET o+ E2T 3 ...... ........... (4.21) Se = Se + eSe + E2Se2 + ................ (4.22) 0 < where e = < 1 /> 2 c c (4.23) We have already obtained the linear perturbed solution in Section 3.2. Substituting these expansions into the steady state versions of eqns. (3.115) (3.124) and collecting the terms of order e2 we get the second order perturbed problem. If the first order perturbed variables are written, following eqn. (3.212), as T o(x,y,z,Se ) = T 01(zSe o (x,y)) (4.24) Then the solution to the second order problem becomes Tv = E Ttn(zSel, T O) )n(xy) (4.25) n=1l Substituting (4.24) and (4.25) into the second order problem and taking Fourier transforms horizontally, we get L11 0 (4.26) and T11 Call = 0 Till =0 B(1 C ) =h11 at z = 1 at z =s (4.27) (4.28) (4.29) at z=0 where L and B are the same as that used in (3.32) and (3.325) but with =0O and Se = Se and o o01(DT01 DTSo1 11 Coi(DTo + Se DC ) vG Se 2 o ]I Se (CO + C0 G 01 01 o 101' 2 to 0 01 11 1 +01 01 to a 01 01 T01 11 + 01(T101 BT01)12 C01( DCo DC ) v12 (kG G / n) 01 101 Sol 2 01 to so 11 a2c01 (C1l nC )1 + 01 (C01 C s)I12 (4.210) L 2 2 f o o 11 L= L 0 0 o o 3 (x,y)dxdy 2 (x,y)dxdy "2 "1 fx ay 12 o o (4.212) 12 L2 L 2 f (x,y)dxdy O O It can easily be seen that for two dimensional rolls, cross rolls and rectangular cells that Ill = 112 = 0. Hence from the solvability condition for these patterns Se1 will be zero; that is, the bifurcation will be locally symmetric. But for hexagonal cells and cylindrical rolls I,, and I12 will be nonzero and hence Se1 will also be nonzero and we have nonsymmetric bifurcation and the existence of subcritical instabilities. 11 (4.211) We will analyze the case of Se 0 by considering hexagonal cells as an example, but the results obtained are applicable to cylindrical rolls as well. It should be noted here that all the horizontal cell patterns we have been using are possible solutions to Helmholz's equation 12 a2 + a 2 = 0 (4.213) 2 2 ax ay and the solutions always come in pairs. For hexagonal cells the complementary solution to 11 given in (3.210) is i 1 = 2 Cos 3x Sin Jy Sin 41y (4.214) So the general form of eqn. (5.4) will be Tto T01( + Pi1) (4.215) To determine p the procedure outlined by Joseph (1976) will be used. We will proceed the same way as above but using (4.215) instead of (4.24) and multiplying by (1 and integrating horizontally. This will give the same set of eqns. (4.26) (4.210) but with L /3L 2 jf J (0 + p1) 21 dxdy S= 0 (1p)2 (4.216) f ( + p1 1dxdy 0 0 ,/3L 3x 3x S(0 1 + pi1)o1dxdy (1 + p "1 2]0 dxdy ay By 1 a 2 2 = 2 (1p (4.217) which can be used to calculate Se1. We then repeat the process with *1 and equate the two Se l's obtained. This will result in a cubic equation for p. p3 3p 0 (4.218) Se = Se (1p2) (4.219) Se1 G (k=G. G ) (a + v/2tanha )B (na v/2tanha s) 01 (k + ) tanha z ntanha s 11 (1+nB)tanha s (na + v/2tanha s) 12' 1 (4.220) B(G G ) a h s r 11 1 (B+A) tanha 2 a(GC + Seo G aA)(1/1) (G G )tanha I11 X> 3 (1+A)tanhas I + h (a+A) 12 2 Se I a(81)(kG G) h [ o 11 [os (BG + G /n)] 2 (k+B) (B+A)tanha 2c sc o I 12 IL o (4.221) (4.222) where B = (aI v/2tanha )tanha s/(na + v/2tanha s)tanhaZ (4.223) and A tanhas/tanha (4.224) There are three solutions 0, /3 and /3 for p, but it can be seen that Se1p for /3 and /3 coincide. It is also obvious that while any two of the solutions are independent the third is a linear combination of the other two. Here is an instance where the problem exhibits a multiplicity of solutions for the same eigenvalue, and could cause secondary bifurcations further along the bifurcation curve. The next obvious question is to ask if there is any point at which Sel in (4.220) goes to zero and hence causing Selp to go to zero. Calculations done for the CAustenite (i.e. steel) system did give such a curve (see Fig. 45) though for most growth conditions this curve lies far away from the critical (or most dangerous) wavenumber amin' intersecting it only at high velocities. 4.3. The Third Order Problem In Section 4.2, it was shown that for 2D rolls, cross rolls and rectangular cells Sel was zero, which implied a symmetric bifurcation diagram. But to learn more about the nature of this diagram it is necessary to go to the next order. If we repeat the procedure described for the second order problem for the terms of order E3 we will obtain the third order problem. (4.31) L 21 = 0 20 40 VELOCITY (pms1) Figure 45. Growth velocity vs. wavenumber chart for CAustenite system for hexagonal cells 3200 2400 1600 800 0 T121 9 C21 0 T21 B(21 21) = h 21 at z= 1 at z=s (4.33) (4.34) at z=0 2 a 2T  2 01 2 01 21 2 01 Z01 01 )121 + SeoC + 2 0 9.01 201Se0  (Gtot 101Gei)Se2  DC 1 2 3 SeoGC1 o01/ 6 01 0oI c A3 (3/2 a2 1 I ) 01 22 23  Cs01 + (k DC01 DC01n) + v1;(kGc G /n2) 6 01 io se 2  22  DCs01 21 O1 (DCo01  nDC01 22 s01l 22 (4.35) L L f i dxdy o o  21 L2 L1 fI f 22dxdy o o L2 L1 0 0 1 ax (4.36) + ( 2dxdy (4.37) 21 0 0 dxdy o o (4.32) h21 2 I[21 a 02(kC01 '( 2 ) o2  By ax y2 ay ax a7 2 2 21 axay 2 a u '1 2] + (ax dxdy ay L L fJ J 02dxdy 0o o0 (4.38) Once again using the solvability condition we obtain Se e = h + h + (I2 2 0 1 2 22 01 2 v(kG G s)(1+B)tanha s a op sc 5 2 21)Se 0 2 (k+B) (na +1/2vtanha s) s s (4.39) where 32 vG h 2 (3 2a +I )A 2 I Se v 1 a 22 23 21 0 2 (kG cG S)(a + v/2tanha )/tanha. 6(k+B) (4.310) I2 v(G c ) k(a + v/2 tanha ) 2 2 (k+B)2 tanhaz (na v/2 tanha s)B 2 So+ 121 n tanha s S Se v(kG9 G /n) (k so (B).311) 6(k+B) and B (a 1/2 vtanha )tanha s/(na + 1/2 vtanha s)tanha S& s s s s. L2 3 2 ax2 o ax 123 (4.312) 4.4. Calculations and Comparisons Using the results of the previous section, Se2 was calculated for various values of the experimental parameters and the results are shown in Figs. 46 to 49. Figure 46 was drawn in order to check the derivation with that of Ungar and Brown (1984a) and shows calculations done for 2D rolls with Ds 0 and 8 = 1 but is a more complete calculation for the experimental parameter ranges involved than theirs. The calculations agree with those of Ungar and Brown but it is an unexpected result nevertheless, showing multiple regions of subcritical and supercritical instability. It also seemed to contradict the earlier calculations of Wollkind and Segel (1970) who did not see any supercritical instability but this was resolved in the paper of Wollkind and Wang (1988) and hence Fig. 46 agrees with their calculations as well. As argued in Section 3.4 if we treat the amin curve as the experimental "operating line," the bifurcation is mostly supercritical which also seems to contradict the experimental results of de Cheveigne et al. (1986) who observed only subcritical bifurcation. In Fig. 47 calculations were done for more realistic values of Ds and 8 and the results are significantly different with only one region of supercritically and another of subcriticality. Ignoring solid diffusion introduces a singularity to the problem and this accounts for the distortions observed in Fig. 46. In Fig. 47 if we move along the operating line for a fixed liquid temperature gradient, initially for small growth velocities the bifurcation is supercritical, until at a critical velocity a transition point is reached and the bifurcation becomes subcritical. Hence for every imposed liquid temperature 16000 12000 8000 4000  0 Figure 46. 50 100 150 200 250 300 VELOCITY (gms'1) Velocity vs. wavenumber chart for the onesided approximations of Ungar and Brown for 2D rolls 350 20 40 60 80 VELOCITY (gms1) ONO Figure 47. Velocity vs. wavenumber chart for 2D rolls using the PbSn system gradient there will be a critical growth velocity below which the bifurcation for 2D roll disturbances is always supercritical. The really surprising result here is that when these transition points for various values of the gradient are joined, we get a straight line through the origin. We now have a clearly demarcated supercritical uppertriangular and subcritical lowertriangular one. It is well known that the onset condition Seo does not change much in the neighborhood of amin (see for example Coriell, McFadden and Sekerka (1985)) and so amin is more of an interval than a unique point. It can also be seen from Fig. 47 that the amin curve practically hugs the line of transition from sub to supercriticality and if we impose an interval for amin it would straddle the transition line. Thus for 2D rolls it is unlikely that we will ever see a sharp transition from subcritical to supercritical bifurcation in experiments. More likely we will observe subcritical behavior throughout as reported by de Cheveigne et al. Proceeding to the three dimensional patterns, we obtained almost identical results for square cells and cross rolls. Figure 48 is for square cells and we see quite a change with the supercritical region acquiring a characteristic balloon shape and having a sharp transition to subcritical bifurcation along the amin curve. But here too if these points of transition are joined, a straight line is the result, demarcating a supercritical uppertriangular operating region and a subcritical lower triangular one. Unlike the case of 2D rolls these should be visible to the experimentalist. So in order to avoid cross rolls and square cellular instabilities not only should the crystal be grown when Se < Seomin, but one should do so in the upper triangular region. Figure 49 demonstrates the universality of our result in being 0 2 4 6 8 10 12 14 VELOCITY (gms1) Figure 48. Velocity vs. wavenumber chart for square cells using the PbSn system 60 40 20 VELOCITY (gms1) 1 Figure 49. Velocity vs. wavenumber chart for the fusion problem for square cells using CAustenite system 8000 6000 4000 2000 0 applicable for the melting problem as well and repeating the derivation for this case separately as was done by Wollkind and Raissi (1974) is unnecessary. Finally, even though for hexagons Se1 was usually nonzero (and hence Se2 cannot easily be calculated), in Section 4.2 we saw that there were points at which Se1 did go to zero. If we attempt to evaluate Se2 for hexagons at these points I21, 122 and 123 become 21 15 (1 + p2) (4.41) 5 2 2 122 a (1+p) (4.42) 22 4 23 3 a (1+p2) (4.43) and as can be expected Se2 will be in the form Se = Se (1 + p2) (4.44) where Se2 is that corresponding to p = 0. The other possible value is when p is /3 or /3. Here Se2p can be calculated from (4.39) (4.3 12) and (4.41) (4.43). As mentioned in Section 4.2 these points are usually far away from amin and in Fig. 45 we found that along this curve Se2 is positive at low values of "a." As "a" increases, at a point above a=amin, Se2 becomes negative. Hence along this line the bifurcation diagram is forward bending at low "a's" (including amin) and becomes backward bending at high values of "a." Even though this is true only along the Se1=0 line, by analogy with other cell patterns we conjecture that this is valid when Se *0 as well. In other words we expect the bifurcation diagram to be forward bending along the ain line and below (as shown in Figs. 410 and 411) but a transition to backward bending along the amin line could occur at high velocities. Far above the amin line the bifurcation diagram should be backward bending (see Figs. 412 and 413). To confirm these conjectures we turned to the nonlinear calculations for hexagons of McFadden et al. (1987) but unfortunately they were unable to complete the bifurcation diagram as their attempts to compute the curve for <0O failed. Also their calculations were done only for the case of p=0 and not for p=/3. But they did confirm the existence of subcritical instability. To sum up, it has been shown that the Mullins and Sekerka and the Woodruff models of morphological instability are of limited validity. The uniform formulation is the more exact representation of the problem and it is applicable for all growth velocities and not just the relatively rapid solidification and fusion regions and provides a single formulation from which all the different models for various growth conditions can be obtained as limiting cases, eliminating the duplication of derivations for different cases; it incorporates the whole crystal and melt regions into the problem and not just a boundary layer region adjoining the interface facilitating the study of various domain effects like convection on morphological instability; Stable  Unstable Sei = Se2 > F4 , SeSO Se P = ^ p =+ Figure 410. Bifurcation diagram for hexagonal cells Figure 411. Bifurcation diagram for hexagonal cells N (K  Stable  Unstable Se e / Sei = 0 Se2 > 0O p =1= 0 P= Figure 412. Bifurcation diagram for hexagonal cells Se, 0 ( SP = 0 P=O Figure 413. Bifurcation diagram for hexagonal cells it avoids the incorrect predictions of subcritical bifurcation regions because of the singularities inherent in previous models. The principle of exchange of stabilities has been shown to be applicable to this model as well even though only in an asymptotic sense. When the weakly nonlinear technique of Malkus and Veronis (1959) is applied to this problem in a systematic way, it resulted in important information about the shape of the bifurcation diagram for various growth conditions. Some of these results are similar to those obtained for Marangoni instability (see Joseph (1976)) which leads us to assert that these results are valid for all hydrodynamic instability problems in which the nonlinearity lies only on the boundary. If rectangular cells, cross rolls or two dimensional rolls are the horizontal cell patterns then the bifurcation diagram will always be locally symmetric. For hexagonal cells or cylindrical rolls they are generally nonsymmetric and hence the bifurcation is subcritical. The problem can display a multiplicity of solutions for the same eigenvalue. Specifically for hexagons there are two possible solutions. Considering the morphological instability problem in particular the following were shown: for twodimensional rolls there are two operating regions, one subcritical (backward bending) and the other supercritical (forward bending), but since the demarcation is not sharp its probable that only subscritical bifurcation will be observed experimentally. 60 * For rectangular cells the forward bending region has a characteristic balloonlike shape and here too there is a straight line dividing the operating region into subcritical and supercritical zones, but here the transition is sharp and hence probably observable experimentally. * For hexagonal cells and cylindrical rolls the bifurcation diagram shows both backward and forward bending behavior but the exact regions of each can only be conjectured. CHAPTER 5 COMPARISONS WITH RAYLEIGHMARANGONI CONVECTION 5.1. RayleighMarangoni Convection in Brief When a horizontal layer of quiescent fluid is heated from below, on account of thermal expansion, the fluid at the bottom will be lighter than the fluid at the top. This unstable arrangement is maintained by the viscosity of the fluid which inhibits any flow and suppresses disturbances such that there will be a stable conduction profile in the fluid. But when the adverse temperature gradient exceeds a critical value, the viscous force is overcome and there will be cellular convection. This instability is known as RayleighBenard convective instability. There are several variations of this problem. Instead of an ad verse temperature gradient, there could be an adverse solute concentra tion gradient in the fluid causing once again an unstable topheavy arrangement. The convective instability arising from this is known as solutal convection or the solutal Rayleigh problem. Another way to cause convection is to have a very thin fluid layer heated from below, but the top surface of the fluid instead of being kept at a fixed lower temperature, is allowed to remain free. Here the thinness of the fluid layer makes buoyancy effects negligible but convection will be caused by surface tension variation on the free surface. This is known as Maran goni convection. Finally, there could be combinations of the above three types of convective instability. When convection is caused by thermal and concentration gradients it is known as doublediffusive convection. Combination of either thermal or solutal convection with surface tension driven flow is the RayleighMarangoni problem. In addition to these there are several other combinations possible like RayleighBenard convection with rotation or with a magnetic field but for purposes of comparison with morphological instability it will be seen that the three causes for convection mentioned above are the most relevant. In the discussion to follow it is desirable to consider the most general form of this problem. Despite there being several interesting features in the problem of doublediffusive convection, the causes for convection there, the temperature gradient and the solute concentration gradient are both bery similar and it is sufficient to look at the effect of one gradient. The manner in which the surface tension effects the problem is very different from the buoyancy effects and a general formulation should include both. Accordingly we will examine the Ray leighMarangoni problem with thermal convection. In the following sections several other reasons for looking at this problem will become apparent. The equations for the RayleighMarangoni problem are given by Sarma (1987) and we will reproduce them here and refer the interested reader to 'his paper for details. The steady state dimensionless Boussinesq equations in the domain are V*V = 0 (5.11) V p + a Tg V*VV = 0 Pr V2T PrV*VT 0 (5.12) (5.13) where p is the modified pressure, g the acceleration due to gravity, Ra the Rayleigh number and Pr the Prandtl number. The boundary conditions at the bottom of the fluid layer are at z = 0, T = T and w = 0, aw/3z = 0 0 (5.14) where w is the vertical component of velocity. The boundary conditions at the top are more complicated. Not only will there be surface tension variation across the surface but the surface is also free to deflect like the liquidsolid interface in crystal growth. at z = 1 + C, VTn = BiT Vn = 0  Bopn + Crn*[VV + VVT] = MaV HT + fin H (5.17) The dimensionless quantities are Bi the Biot number, Bo the Bond number, Cr the Crispation number, Ma the Marangoni number and H the surface curvature (see Scriven and Sternling (1964) for details). Initially there will be a quiescent, linear, stable, conducting solution to the problem with V = 0. At a critical value of the charac (5.15) (5.16) teristic parameter (Ma or Ra) this conducting solution becomes unstable to infinitesimal perturbations and we have a convective solution. Performing a linear stability analysis about the conduction state, separating variables and doing considerable manipulations we get in the domain (D a2 38 = a Ra6 2 2 3 2 (D a ) w = a Raw (5.18) (5.19) where "a" is the wavenumber, w component of velocity and 6 the the Fourier coefficient of the vertical Fourier coefficient of the temperature. At the boundary at z = 0, w = Dw = 8 = 0 At z = 1, w = 0 (5.110) (5.111) (5.112) BiD w = a MaD8 BiCr(D w 3a2Dw) = a2(Bo + a2)(De + Bi9) (5.113) When the density variation with temperature is negligible or in the absence of gravity, then Ra = 0 and we have the pure Marangoni problem with all the nonlinearities only in the boundary. Even with this effect in the boundary the important result of bifurcation, namely the fluid velocity, effects only the domain, the deflections in the boundary being only a secondary effect of convection. On the other hand, when the upper boundary is also kept at a fixed temperature we have, Ma = Bo = Cr = 0 and so at z = 1, w Dw = e 0 (5.114) This then is the pure RayleighBenard problem with all the nonlin earities only in the domain and the resulting nonquiescent solution also manifests itself in the domain as convection. The RayleighMarangoni problem described by eqns. (5.18) (5.113) is a mixed problem with nonlinearities in the domain and the boundary but the convective solu tion resulting from these nonlinearities shows up mainly in the domain. 5.2. The Augmented Morphological Problem As can be seen from Section 5.1, in the RayleighMarangoni problem there is a RaMa domainboundary duality which does not seem to exist in morphological instability. From the problem description in Chapter 3 it is easy to see that all the nonlinearities for this problem lie only in the liquidsolid interface. This is a limitation because by virtue of being on the boundary the Sekerka number is unique and hence also has a unique eigenfunction and is insufficient when solutions to inhomogeneous versions of the linearized morphological instability problem are needed, as in Chapter 6 where "imperfections" are considered. This difficulty also crops up in the pure Marangoni problem, but the RayleighMarangoni problem comes to the rescue, as there are countably many corresponding 66 values of Ra for each value of Ma and hence also countably many eigenfunctions forming a complete set (see Rosenblat, Homsy and Davis (1981)). The naturally occurring duality of Ma and Ra enables solutions to inhomogeneous problems to be obtained in a straightforward manner. In morphological instability there is no such obvious, naturally occurring boundarydomain duality and it is necessary to create one. To avoid confusion we will refer to the pure morphological problem of Chapter 3 as the Sekerka problem, and (by analogy with the Rayleigh Marangoni problem) set up an eigenvalue problem, with the eigenvalue in the domain, which we will call the "augmented morphological problem." liquid: V2p = MpL (5.21) 2 H S v 3q = (5.22) solid: Vp2 = Mp (5.23) 2 s nVq 3 = 0 (5.24) where M is the eigenvalue which we label as the morphological number. The boundary conditions are at z = 1, pt qt = 0 (5.25) at z = s, p = 0 (5.26) S pa pS + t(G Gs) =0 ( (5.27) at z = 0, pt + Seqe + t(G + SeG t) + A (5.28) ar I 8 a 0 (5.29) az az kqg qs + t(kG o G ) = 0 (5.210) n 3 vq + vq = 0 (5.211) Tz 'az Z with periodic conditions at the lateral boundary at r = R. We can now separate variables expressing the horizontal dependence as zeroth order Bessel's functions of the first kind Jo(air/R), where a, are the zeros of the first order Bessel's functions of the first kind J1. pt(r,z) = E Z P (z) J (a.r/R) (5.212) i=1 j=1 0 1 In addition if we take Fourier transforms in the horizontal direc tion and solve for q.., q .. and t the equations reduce to a system in pij and p ij. If we define a column vector tij Qij = (5.213) siJ and a matrix differential operator Li 2 2 (D a ) L. = (5.214) 2 2 BY.(D a.) 1 1 then the domain equations reduce to 1 ij ij 1Qj where Y =(G + Se aA )/(G + Se G a2 ) i o i s c i G = (G B G)/(B + k) (a v/2tanha )tanha .s 1 9_i s31 S (na + v/2tanha .s)tanha z Si 31 Ui 2 2 /4 1/2 ai =(a.i + v /4) 2 2 2 1/2 a = (a2 + v /4n ) The boundary conditions become at z = 1, at z = s, at z = 0, where p = 0 ~ij psij = 0 BiQij = 0 1 Bi D (5.215) (5.216) (5.217) (5.218) (5.219) (5.220) (5.221) (5.222) (5.223) (5.224) Defining an inner product 0 0 a  1 0 (5.225) where the "*" refers to the adjoint eigenfunction and "" the complex conjugate. It can easily be seen that the system described by eqns. (5.213)  (5.224) is selfadjoint in this inner product and so the eigenfunctions Qij are complete. Solving the system we get Qij A Sin X7 a7 (z + 1) Aij ij i A Sin/M. (s z) sij ij i (5.226) where Mij are solutions of the equation Se = ( GT + aA )/ Go T 1 (5.227) k G tan/M a S T k tanM a2S + Z ij i + k G tan/M . ai k G tan/M. a ss ij i Here A ,. and Asij can be determined from the normalizing condition xtij sij (5.229) where 6 is the Kronecker delta. 1J (5.228) 5.3. Comparison of Morphological Instability with RayleighMarangoni Convection The principal aim of this section is to relate the mathematical characteristics of the two problems so that we may introduce some of the extensive mathematical techniques used to study RayleighMarangoni convection to morphological instability, but we will make some physical comparisons as well. The augmented morphological problem described in Section 4.2 is similar to RayleighMarangoni convection. The augmented problem is self adjoint but the RayleighMarangoni problem is nonself adjoint. Both have an infinity of eigenvalues and corresponding eigen functions, but while completeness of the eigenspace is assured for the former, special theorems are required to show this for the latter (cf. Nadarajah and Narayanan (1987)). It should also be noted that while the RayleighMarangoni problem attempts to describe a realistic situation, the augmented morphological problem was artificially created in order to solve inhomogeneous versions of the Sekerka problem described in Sec tions 3.1 and 3.2. This brings us to the question whether there is a practical situa tion which is described by this mathematical concoction. The difficulty in coming up with one stems from another important difference between the two problems. In the pure RayleighBenard problem (where Ma, Bo and Cr are all zero) the nonlinearity is in the domain and the instability too manifests in the domain as convection. Even in the pure Marangoni problem (where Ra is zero) where the nonlinearity is in the boundary, the instability is still mainly in the domain. In contrast, in the Sekerka problem the nonlinearities and the resulting instability show up in the boundary. Though there are other boundary effects (like kinetic undercooling) which can cause morphological instability the only domain effect which could give M physical significance is a heat source term in the form MT or MeE/RT (see Joseph (1965)). We do not know of any experiment where morphological instability was observed as a result of a heat source in the melt or the crystal but if one does exist it will provide the true analogy to RayleighBenard convection. This is relevant as Hurle (1985) has attempted a comparison between the RayleighBenard problem and the Sekerka problem. It can now be seen that the Sekerka problem can only be compared to the pure Marangoni problem, with Se corresponding to Ma and A corresponding to the reci procal of Bo. Besides, for periodic lateral boundary conditions, the eigenvalues of both problems, Se and Ma, are unique. Based on this comparison we can make an important conjecture. Vrentas, Narayanan and Agrawal (1981) have shown that for the Marangoni and the RayleighMarangoni problems when the nonperiodic noslip condi tion for velocity is imposed at the sidewalls, the eigenvalue Ma is no longer unique and has countably many values. In other words when the walls are a finite distance apart Ma has many values, but as they are gradually moved apart we have "spectral crowding" and in the limit when they are sufficiently far apart to impose the periodic boundary condi tion of total slip, all the values of Ma coalesce into a unique num ber. Recently, following Coriell et al. (1980), several workers have looked at the coupled problem of morphological instability with solutal convection and all have assumed periodic boundary conditions. We sus pect that here too if the noslip condition for velocity at the side wall is imposed, the Sekerka number will no longer be unique. All this raises the question of completeness of the Marangoni and the Sekerka eigenspaces and its probable that generalized eigensolutions (see Nai mark (1967)) are needed when Ma and Se are chosen as eigenvalues. When these two problems are considered in a finite container we can see yet another difference. Both problems have simple eigenvalues except at certain aspect ratios of the container where two horizontal modes can coexist (cf. Rosenblat, Homsy and Davis (1982)). In a typical experiment of RayleighBenard convection we would expect to see a dozen or so convection cells (see for example Koschmieder (1967)) and increas ing or decreasing the number of cells by one can significantly effect the problem. Hence the multiple points in this problem are extremely important and have been the subject of study. But in morphological instability a single alloy crystal can contain hundreds of individual cells and the addition or loss of one has hardly a noticeable effect on the problem and consequently multiplicity of the lateral eigenfunctions loses its significance. In addition unlike the Rayleigh number it is well known that near the critical wave number aN, the critical value of the Sekerka number Seo hardly changes (see for example Coriell, McFadden and Sekerka (1985)) and the choice of aN has very little effect on SeoN. Conversely, the choice of the operating Se will have a tremendous impact on the resulting wavenumber (cf. Ramprasad and Brown (1987)). Other differences have been mentioned in Chapter 4. Both the Marangoni problem and the Sekerka problem have symmetric bifurcation diagrams near the bifurcation point for twodimensional rolls and rec tangular cells and nonsymmetric curves for hexagonal cells and cylindri cal rolls. But in morphological instability the curves can be "forward bending" or "backward bending" depending on the operating conditions, whereas in Marangoni convection the curves are forward bending every 73 where. Hence the occurrence of subcritical instability is more wide spread in morphological instability. CHAPTER 6 BIFURCATION BREAKING IMPERFECTIONS 6.1. Nature of Imperfections When the morphological instability problem was formulated in Chap ter 3, several effects were ignored and the resulting problem is an idealized or "perfect" one. Inclusion of these can alter the problem in several ways, for example kinetic undercooling of the melt becomes an important effect in rapid solidification, but all it does is alter the onset condition for morphological instability. In the parlance of bifurcation theory, an "imperfection" is an effect on the "perfect" problem which alters it in a specific way. Such an imperfection will cause the morphological instability problem not to have a planar solu tion at all even below the onset condition, and these are known as bifurcation breaking imperfections. The effect of a typical imperfection on the bifurcation diagram is shown in Fig. 61. The broken line is the solution in the presence of imperfection and it can be seen that the interface will be nonplanar for all nonzero values of Se. Obtaining solutions to the problem with imperfections is extremely difficult and we will only seek asymptotic solutions. Hence the problems to be considered should have very small imperfections. Under such conditions the method of matched asymptotic expansions of Matkowsky and Reiss (1977) can be employed and here it will be used in a way similar to the work of Tavantzis, Reiss and Mat kowsky (1978) for the RayleighBenard problem. ? I   Se =Seo N Figure 61. Imperfect bifurcation diagram showing inner and outer expansions The method.is fairly straightforward. The variables are expanded asymptotically with the imperfection parameter about the planar and the nonplanar solutions and two outer expansions 00O and 01 are obtained as shown in Fig. 61. At the bifurcation point SeoN these expansions break down and it is necessary to have inner expansions I1 and 12 near SeoN and to join the corresponding 00 and 01, matching conditions have to be specified. The imperfections that can be analyzed in this fashion must, of course, be small effects else a fullblown nonlinear solution will be needed. Two of the most important effects which are habitually ignored, namely imperfect insulation of the ampoule wall and advection in the melt, are such imperfections and readily lend themselves to this type of analysis. In Chapter 3 when the morphological instability problem was modelled by a uniform formulation, it was mentioned that one reason for this was to consider a finite crystal/melt region. This finiteness was only in the vertical direction and in the horizontal direction the imposition of periodic boundary conditions effectively meant that the ampoule side walls were infinitely far apart. The two imperfections that are to be considered are caused by nonperiodic lateral boundary conditions and another way of looking at the effect of these imperfections is to say that the container is now being considered to be finite in the lateral as well as the vertical direction. 6.2. Imperfection Due to Heat Loss As mentioned in the last section, it has been customary in this problem to assume periodic boundary conditions laterally, which is 77 equivalent to assuming that the walls of the ampoule are perfectly insulated or that they are so far apart that their effect can be ig nored. In practice neither of these is likely to be achieved and here we will examine the effect of a small amount of heat loss or heat gain from the wall on the problem. If we take the ampoule to be cylindrical with radius R, aT r = R, liquid: k2 6fD (z) (6.21) 3T solid: k s = 6f (z) (6.22) where f and f. are such that f (S) = fs () and fX(1) = f (s) = 0. If f and fs are positive it will mean heat loss and if they are nega tive, heat gain. If we make the transformations T (z,r) = 6 k r + 9 (z,r) (6.23) f (z) and T (z,r) = 6 k r + 9 (z,r) (6.24) s k s and substitute these in the steadystate versions of eqns. (3.115)  (3.124), the temperature equations in the domain become V2 = 6rD2f /k (6.25) V2 = 6rD2f /k (6.26) s 5 s The outer boundary conditions will remain unchanged but at the interface eqn. (3.121) becomes 6 = 6r(f /k f /k ) (6.27) 6 + SeC + AH = 6rf /k (6.28) and eqn. (3.122) converts to 7V Cn 8V6 *n = Lv dr(Df Df ) (6.29) In order to solve this system we will be treating the heat loss as an imperfection on the perfect problem (i.e. when 6 = 0). The perfect problem is of course the Sekerka problem of Section 3.1. 6.3. The Outer Expansions As this problem has been defined in a finite geometry, the number of cells are fixed by the container size and the growth conditions. We will choose these so that aN is very close to amin the wavelength corre sponding to Seomin the least value of SeoN. Another important decision is the selection of the wave pattern and our analysis is done for cylin drical rolls. An objection to this could be raised on the grounds that in most experiments it is the hexagonal pattern which is observed. We justify our assumption by once again making a comparison with Rayleigh Benard convection. It appears that in RayleighBenard convection the wave pattern selection is strongly influenced by the container size and shape, with the hexagonal pattern prevailing for all shapes in wide containers while in narrower ones the container shape determining the pattern (e.g., cylindrical rolls for circular containers. See Kosch mieder (1967)). Since the principal aim of this paper is to study a finite geometry effect on morphological instability, the cylindrical roll pattern would be the logical choice. This is especially valid for experiments such as those of Peteves (1986) where ampoules of radius 0.025 inches were used. In Chapter 4 it was shown that the bifurcation diagram is unsymme tric for cylindrical rolls and the form of the outer expansions 00 and 01 are shown schematically in Fig. 61. If we use superscript o to identify the problem with perfect insulation we can seek solutions by means of asymptotic expansions about the perfect problem. k = 6 (6.31) k=0 with similar expansions for s, CE, Cs and ;. Substituting these expan sions into the problem and collecting the terms of order 6 we get an inhomogeneous linear system. If we separate variables horizontally 1 1 9= 6 J (a r/R) (6.32) t 1t o i i=1 1 and eliminate C., C and C we get for the expansion about the planar jL s solution L 1 f= (6.33) 1 i i 1 S1 1 8s 1 1 6 s i f/k 1 2 I i f /k s s . = 2j rJ (a.r/R)dr/ rJ2(a.r/R)dr 0 0 1 O O The boundary conditions are at z = 1, at z = s, at z = 0, a =0 6 = 1 1 1ii 1 h = I f1 i / Df Y.f /k 1 s s Df s The eigenvalue problem of Section 5.2 has been shown to have a complete set of eigenfunctions and hence can be used to solve the above system. where (6.34) (6.35) (6.36) where (6.37) (6.38) (6.39) (6.310) 91 E= Q .J (a.r/R) (6.311) 