What we do not know about solidification theory

Materials Science and Engineering, 25 ( 1 9 7 6 ) 93 - 101

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© Elsevier S e q u o i a S.A., L a u s a n n e - - P r i n t e d in t h e N e t h e r l a n d s

What We Do Not Know About Solidification Theory

M. E. G L I C K S M A N

Materials Engineering Department, Rensselaer Polytechnic Institute, Troy, New York 12181 (U.S.A.)

1. I N T R O D U C T I O N

The title of this article, if pursued logically, purports that the writer has full perception of the boundaries of this field. Such great perception would be one of the key logical requirements to discuss what one does not know about a subject. Obviously, and more logically, the proper title should be: "What we should like to know about solidification theory". Even to attempt such a logical exercise in the limited format of these articles would be scientific hubris of an inexcusable nature. Instead, under the guise of a somewhat misleading and illogical title, I have selected just a few of the interesting problems in solidification theory which remain incompletely understood and about which we would like to know more. The selection of the four areas to be discussed is based solely on the author's choices of problems brought to his personal attention by colleagues and associates over the past few years. No attempt was made at achieving review-article status for any of these subjects, and the list discussed here is merely representative and by no means exhaustive. Indeed, from the outset, it is fair to say that whereas the pace of theoretical progress in this field is brisk, it nonetheless is barely keeping even with our growing perceptions of the physical and mathematical complexities which surround the theory of solid-liquid transformations.

2. C O U P L E D P R O C E S S E S

Most solidification and crystal growth processes involve the redistribution of heat and solute at a moving interface. The problems of mass and heat flow during solidification are intrinsically difficult, insofar as they generally involve solutions to linear

field equations with non-linear, strongly coupled, boundary conditions. Whereas major progress has been made recently in coping with the moving-boundary aspects using potential theoretic methods combined with high-speed computational techniques, substantial difficulties do remain. For interfaces with large curvatures, i.e., microscopic features, the Gibbs-Thomson relationship, or some variant, is usually required in formulating the temperature and solute concentration conditions at the moving solid-liquid interface. Fundamentally, the Gibbs-Thomson relationship provides a convenient means of ascribing local equilibrium to the interface with regard to all intensive field quantities, e.g., temperature, and solute concentrations. However, the applicability of the Gibbs-Thomson relationship to a moving interface is not immediately obvious, since the concept devolves from the thermodynamics of equilibrium systems and is usually applied without restriction to transforming systems. Indeed, Jones [ 1 ], in a recent paper, questions the utility of employing such a relationship for a moving solid-liquid interface, by claiming evidence for the fact that the solid-liquid interfacial free energy is a function of interface motion, and is a fundamental thermodynamic quantity only in the limited context of stationary interfaces which are at equilibrium. This view, while perhaps not widely held, must nonetheless be evaluated through critical experimentation since, if true, its implications would be profound. Results for the ice-water system obtained recently by Hardy [2] show, on the contrary, that a consistent value for the solid-liquid interfacial free energy obtains from both equilibrium and dynamic interfacial states. Although the velocity range originally investigated by Hardy and Coriell [3, 4] was limited to small values, their combined experiments show

94

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Fig. 1. Dendrite growth velocity v s . supercooling (adapted f r o m ref. 5). Steady-state t h e o r y predicts the correct slope of 2.6, which is i n d e p e n d e n t of the material, provided that local equilibrium and a c o n s t a n t interfacial energy occur. The vertical separation of the t h e o r y line and the data cannot be explained at present.

conclusively that the G i b b s - T h o m s o n relationship in relatively-pure ice-water systems appears consistent and, within their experimental precision, leads to constant values for the solid-liquid interfacial energy. In a recent series of experiments [5] conducted by the author and his colleagues R. J. Schaefer and J. D. Ayers, some definitive results were obtained which further support the notion of local equilibrium at moving solid-liquid interfaces. Dendritic growth velocities were measured in succinonitrile, a low entropy-of-fusion organic material. We found that over almost a onethousand-fold range of velocities, the dependence of the axial dendrite velocity on supercooling was again consistent with a constant value for the solid-liquid interfacial free energy. Our conclusion is based on the fact that the growth velocity-supercooling

relationship can be expressed theoretically [6 - 8] as V = {3GAO z s, so that In V is linear with In A0 with a slope of 2.6. Here V is the steady-state axial dendrite growth velocity, is a theoretical constant, G is a lumped material parameter, and A0 is the supercooling. Specifically, G = (~ASL/TC where is the thermal diffusivity of the melt, AS is the volumetric entropy of fusion, L is the heat of fusion, C is the specific heat of the melt, and T is the solid-liquid interfacial free energy. Figure 1 shows clearly that the correct slope of 2.6 is displayed by the logarithmic plot of V versus the ordinary (dimensional) supercooling, A T = AOL/C. Hence, it appears that in succinonitrile, at least, 7 remains constant over a wide range of interracial velocities, and local interracial equilibrium (which is responsible for the 2.6 slope) is well satisfied. The lack of vertical agreement of the theory line in Fig. 1 with the data remains puzzling. Insofar as both the theoretical constant, ~, and the materials constant, G, which appear above are well known, there remains little d o u b t that there is a serious defect in the steady-state theory of dendritic growth. Current thinking on this problem is that time-dependent processes, such as dendritic side-branching, must be considered in any complete kinetic theory of this type of solidification. Steady-state models are obviously lacking in this respect, although their utility is demonstrated by having justified certain aspects of the local equilibrium argument for non-facetted interfaces, as outlined above. Studies are needed on other pure systems, like succinonitrile, which do not form facetted interfaces during solidification, and on systems which do facet. Quantitative investigations are required to ascertain when and under what conditions interfacial equilibrium, as expressed by the G i b b s - T h o m s o n relationship, occurs at moving solid-liquid interfaces. Clearly, when a solid-liquid interface develops facets during freezing, then the moving interface must deviate from local equilibrium; however, it remains obscure as to whether the supercooling developed at such a facetted interface alters the value of the solid-liquid interracial free energy for the expressed crystallographic orientations of the facets. Answers to such questions will require critical studies of the

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coupled prc, cesses occurring at solid-liquid interfaces. Alloys pose additional challenges to our understanding of solidification. Again, in most instances where microscopic radii of curvature exist, e.g., when cells or dendrites are present, the temperature and compositions at the solid-liquid interface are related through constitutive equations, which are frequently based on considerations of local equilibrium. In the case of a binary alloy, the interfacial temperature can be expressed as = Te + MCz + FK, where T e is theAordinary melting point of the pure solvent, Cz is the solute concentration in the melt adjacent to the interface, and K is the local interfacial curvature (K < 0 when the interface is convex toward the liquid phase). The coefficients M and F are, in general, u n k n o w n functions of Cz and K, b u t are usually considered as constants for relatively dilute alloys with moderate curvatures. Specifically, M is taken as the slope of the liquidus, m --- ~ ~/~ ¢z, at zero curvature, and F is the usual GibbsThomson coefficient for the solvent, namely, ~/~ = 7 / A S , where again ~, is the solidliquid interfacial energy, and AS is the volumetric entropy of fusion. Similarly, the constitutive relationship for the interfacial composition of the solid can be expressed as ¢s = KCI -- ~2K, where both K and ~2 are functions of Cz and ~. ~2 is the FreundlichOstwald coefficient [ 9 ] , relating solid solubility and interfacial curvature, and K, in the case of dilute alloys, is usually approximated by k0, the well-known equilibrium segregation coefficient. Employing the constitutive relations relating temperature, solute concentration, and interfacial curvature, demands prior knowledge of the various coefficients described above and implies that local equilibrium obtains. The assumption of local equilibrium as mentioned earlier is becoming generally accepted when the solid-liquid interface advances at low velocities and retains an unfacetted morphology However, the transition from local equilibrium to appreciable non-equilibrium at a solidifying alloy interface remains purely a matter of conjecture for virtually all systems which do n o t form growth facets, because a definitive verification of the interfacial constitutive relationships at different solidification rates

has never been attempted. In the Z n - C d system, where a retrograde solid solubility occurs, appreciable deviation from local interfacial equilibrium was observed by Baker and Cahn [10] during rapid quenching. Their work served to show that the chemical rate theory of Jackson [11], and the irreversible thermodynamic theories of Borisov [ 12 ], Baralis [13], Jindal and Tiller [14], and Aptekar and Kamenetskaya [15] all led to inconsistencies in their respective predictions of deviations from local equilibria. Indeed, in a subsequent review of the thermodynamics of alloy solidification processes, Baker and Cahn [16] aptly summed up the status of this interesting area of solidification theory: " N e w principles are required in this field or tighter restrictions for applying the old principles are necessary, but, most of all, there is a need for experiments to study the deviations from local equilibrium and coupling effects.".

3. M O R P H O L O G I C A L STABILITY AND DEVELOPMENT

Of key importance in solidification theory is the prediction of the microscopic details of the shape, motion, and behavior of the solidliquid interface. The development of dynamical theories of morphological stability, over the decade beginning in 1964 has been reviewed recently by Sekerka [17] and b y Delves [ 1 8 ] , whose critical reviews contain excellent and exhaustive bibliographies of the field. The theories of morphological stability, to date, are linear theories, insofar as they describe the growth or decay of an infinitesimal perturbation of arbitrary wave vector. The linearization schemes employed in virtually all of the theoretical analyses of morphological stability require that the amplitude of the perturbation remains small when compared with the wavelength. The conditions for stability or instability are usually prescribed as an exponential rate of growth (or decay) o f the perturbation for a prescribed set of solidification conditions. These conditions involve the macroscopic growth rate of the unperturbed solid-liquid interface, the thermal gradient and solute content, as well as certain materials parameters such as the thermal conductivity,

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latent heat, chemical diffusivity, and solidliquid interfacial energy. Linear dynamic theory certainly provided a major advance in our theoretical understanding of solidification processes, although its limitations are still being clarified and much remains to be accomplished in this theoretical area. There is now abundant qualitative information [19] which supports the major findings of stability theory and lends credence to the approaches used. Recently, Hardy [2] has established firm quantitative proof that the value of the solid-liquid interfacial energy for the ice-water system at equilibrium agrees with the value derived earlier by Hardy and Coriell [3, 4] who studied the freezing of ice cylinders and analysed observed instabilities with linear theory. Curiously, in fact, it appears that the predictions from linear morphological stability theory remained satisfactory even when the perturbations grew well beyond what would be considered the linear regime, but the reasons for this axe not clear. Thus the bounds remain obscure beyond which an observable perturbation on a freezing solid-liquid interface no longer is well approximated by the theoretical notion of an infinitesimal perturbation. Morphological development during solidification involves much more than an initial description of the advent of instability. The development of complex cellular and dendritic crystal-growth forms cannot be described within the context of linear stability. To progress beyond this point new approaches are needed. A rather recent development which may be of use in exploring morphological development beyond initial instability is the formal, nonlinear, dynamic stability theory of Matkowsky [20, 21], who considered that a non-linear system became unstable when some critical parameter X* of the system was exceeded; that is, the stationary solution of the non-linear system was found to be stable for X < X* and unstable for k > X*. Here X* is the first bifurcation point of the related non-linear eigenvalue problem, the solutions of which are stationary solutions of the timedependent problem. The initial boundaryvalue problem for the non-linear system with k slightly greater than X* was considered, and an arbitrary perturbation was traced for all time until it evolved to a new stationary (or

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(b) Fig. 2(a). Calculated lamellar eutectic interface shapes (adapted from ref. 25). Shapes based on selfconsistent solutions, which permit local interfacial equilibrium and satisfaction of the diffusion equation in the liquid. Curves 1, 2 and 3 are for different ratios of the ~--~ and ~--liquid interfacial energies. A fiat interface occurs when 0l4 ffi 0. (b) Interfacial solute concentrations corresponding to the steadystate lamellar shapes in Fig. 2(a) (adapted from ref. 25). The broken curve is an analytical solution for the flat eutectic interface (c~4 = 0), and agrees well w i t h the computer-generated distributions based on Nash's potential theoretic approach. Accurate diffusion solutions are essential in advancing solidification theory. This method works well for a wide class of complex interface shapes for which orthogonal coordinate systems do not exist.

periodic) state. Matkowsky's theory provides a complete dynamical description of the nonlinear system for the case where X is close to X*, and therefore might be termed a theory of transitions between states. Such a theory could describe, potentially, such well-known

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morphological transitions as planar-to-cellular or cellular-to-dendritic. Nash [22] recently has shown that the first bifurcation point in the steady-state plane-front solidification problem is formally identical with the perturbed, marginally-stable state which occurs in Mullins and Sekerka's linear stability theory. This interesting correspondence of bifurcation (transition) theory and the dynamical stability theory may well point to the next major step in describing more complex solidification morphologies such as steepwalled cellular interfaces. An even more direct approach to improving our understanding of solidification employs time-dependent solutions to initial value problems, by so-called "marching" techniques. Here, one specifies the system at an initial state and permits progressive motion of the solid-liquid interface to occur in discrete steps. At each step the diffusion equations and the associated boundary conditions are solved at the new position of the interface. The problem of a solidifying sphere with nonlinear boundary conditions was solved in this manner several years ago by the author and R. J. Schaefer [23], and unexpected kinetic behavior at large supercoolings was found. Solution of solidification problems as initialvalue problems has the great advantage of not being limited to the regime of a small parameter, small amplitude, or small times. Indeed, a currently challenging problem at this time is to solve various morphological development problems by "marching", where the interface may assume an arbitrary form dictated only by the initial conditions, the boundary conditions, and the field equations. Solutions to these problems would permit detailed investigation of the influence of factors such as the crystal symmetry, interfacial curvature and orientation, molecular attachment mechanism, surface energy anisotropy, and defect content, on the morphological development and ultimate solidification structure. An essential step toward accomplishing this task has been taken by Nash [24], who recently has developed a set of time-dependent potential functions which provide solutions to the thermal and solute diffusion equations for arbitrary, moving interfaces. One recent application of this method was the development of a self-consistent theory of steady-

state lamellar eutectic solidification [ 25 ], which proved that Nash's potential-theoretic approach could indeed be used to solve the diffusion equation for relatively complex interface shapes. Figures 2(a) and (b) show the interface shapes and the solute concentration adjacent to these interfaces during steady-state eutectic growth. A comparison with the analytical results for a planar eutectic interface, obtained by Jackson and Hunt, is included in Fig. 2(b) for comparison. With the current ability to solve the associated heat and mass transport problems for an unconstrained solid-liquid interface, techniques could be developed soon to track the long-term morphological development of a solidifying particle. The growth process could then be followed in time through several transition stages until its asymptotic behavior as a dendrite was obtained. Thus, perhaps, many of the unknown aspects of morphological development can be explored by computer "experiments" which simulate solidification on a suitably microscopic level.

4. DENDRITIC INTERACTIONS AND SCALING LAWS

The behavior of the diffusion zones surrounding a moving interface is controlled by both the properties of the phases participating in the transformation and the rate of transformation. The well-known concept of a diffusion length expresses the fact that the diffusion gradient exists over a distance that is proportional to the speed of the diffusant and the time over which diffusion can occur. Thus, the more rapid the rate of solidJfication, the smaller is the diffusion length. For the case of a pure material, only a thermal diffusion length exists during solidification, which is the quantity a/V, where a is the thermal diffusivity and V is the velocity normal to the interface. Figures 3(a) (f) show the dendritic forms which occur in succinonitrile, the material for which the solidification kinetics are described in Fig. 1. The scaling of the solidification morphology with the thermal diffusion length shows the expected hundred-fold change as the velocity increases from 0.032 cm/s in Fig. 3(a) to 3.65 cm/s in Fig. 3(f). However, there is also an associated increase in the fraction of solid

98

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Fig. 3. Morphology of succinonitrile dendrites growing in the supercooled melt. ( F r o m ref. 5.) As the supercooling AT (°C) increases, both the number of primary dendrites and the density of side-branches increase. The kinetic data (Fig. 1) indicate that the thermal interactions may not be changing from (a) through (f), because of the scaling down of the morphology to the thermal diffusion length as the velocity V (cm/s) increases. (× 2.4)

which forms within the dendritic mass. This fraction is, in fact, equal to the dimensionless supercooling/xO = A T ( C / L ) and varies from approximately 0.05 in Fig. 3(a), to approximately 0.3 in Fig. 3(f), which is a six-fold increase in solid fraction corresponding to the six-fold increase in supercooling noted in the Figure. Now, since the morphological size scale varies inversely with the velocity, V, and the amount of solid per unit volume increases with the supercooling, A0, then the number density of dendritic arms should increase as V A O , or as A0 a.6 (since V cc A0 2.6). Factors

similar to those which set the density of branching in a dendritic mass of pure material also control the size scale of microsegregation in supercooled alloys, and a correspondence should be possible between the two cases. Flemings and his coworkers [26, 27] demonstrated empirical power-law correlations between the secondary-dendrite arm-spacing in directionally-solidified and furnace-cooled castings and the local solidification time, which is inversely related to the cooling rate and interface velocity. No fundamental approaches which might lead to

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a deeper understanding of dendritic branching and microsegregation have been suggested, and the similarities to the pure-material case might be gainfully exploited. In terms of the potential for improving the mechanical and corrosion properties of cast alloys, the subject of dendritic interactions is of great importance, and clearly one a b o u t which we desire to know much more.

5. INTERFACIAL MECHANICS

Several processes for producing technically advanced materials by solidification, e.g., crystals and eutectic in situ composites, require that certain "meniscus" conditions, which are n o t well understood, be maintained during crystal growth. For example, Fig. 4 shows that during crystal growth from the surface of a melt, three types of interfaces interact, viz., solid-liquid, solid-vapor, and liquid-vapor. The angular relationships among these interfaces are shown in Fig. 4, and have been discussed in detail recently by Surek and Chalmers [28]. For steady-state growth, the crystal diameter must remain stationary in time. This condition can be expressed

Fig. 5. Illustration of equilibrium at the ~-~-liquid triple point on a lamellar eutectic interface (from ref. 25). The interface shape, F(X), can adjust to alter the solute concentration field and the intersection angles 0~ and 0~. The role played by the crystallographic "torque" on the ff-l~ boundary in satisfying the force balance at the triple point is obscure, but such an effect may be responsible for the characteristic orientation relations which usually arise in steady-state lamellar euteetic solidification.

conveniently as 0 = 0 and ¢ = 5, and if these relationships are n o t satisfied then a tapering or expanding crystal, or perhaps one with an undulatory shape will result. Processes such as Czrochralski crystal growth, edge-defined film-fed growth (EFG), and the Stepanov technique of melt shaping all require precisely controlled steady-state conditions for o p t i m u m results. The value of the angle for steady-state growth had always been assumed to be zero [29, 30] until Antonov e t al. [31, 32] showed that ~ was actually in the range of 10 - 20 ° for silicon. Surek and Chalmers [28] have refined the measurements of ~ and found ¢ = 11 ° for (111} oriented silicon single crystals, and ~ = 8.5 ° for polycrystalline silicon specimens. Similar measurements for germanium showed that = 13 ° for single crystals and 8 ° for polycrystals. No theory exists at present for predicting the ¢ values which are critical for attaining steady-state crystal growth, although Bardsley e t al. [33] and Pogodin e t al. [34] have discussed the implications of ~ > 0 and attributed this fact to the partial wetting of the solid phase by its own melt. This, however, by no means provides a satisfactory

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explanation for the appearance of such special conditions for steady-state crystal growth. A related problem to that of the interfacial balance conditions during steady-state crystal growth, is explaining the nature of the microstresses within a crystal at the intersection of the liquid-vapor meniscus. One can imagine that the entire weight of the liquid in the meniscus is balanced by a force acting over a small region near the point 0 in Fig. 4, in the ¢ direction. This interfacial force, which may be several thousand dynes per centimeter of interfacial contact, acts over a thickness of only several atomic distances and must be balanced by an internal elastic stress in the crystal, which may be quite large in intensity. The implications for such a localised large stress in a crystal near its melting point are clear: dislocations might be generated, and residual stresses or lattice distortions may remain behind as permanent defects of the crystal. Clearly, the interfacial mechanics associated with interacting surfaces are poorly understood regarding their applications to crystal growth and perfection. Studies of the nature of such interfaces and their thermodynamic and mechanical interactions could shed considerable light on these questions. The final area to be discussed here under the subject of interfacial mechanics deals with the conditions required for steady-state lamellar eutectic growth. The problem of interfaciai mechanics germane to unidirectional, steady-state, polyphase solidification differs from that of single-phase crystal growth by the presence of solid-solid interfaces in the region of solidification which interact with the solid-liquid interfaces. Figure 5 shows the situation at a lamellar eutectic solid-liquid interface. The interfacial mechanics of this problem are complicated by the presence of interfacial "torques", which are generalized moments which arise from the angular derivative of the solid-solid boundary energy. Also, the fact that the a-liquid and /3-liquid interfacial energies differ, in general, leads to theoretical ambiguities in maintaining the a-/3-1iquid triple-point at equilibrium. One might consider that the conditions for steady-state lamellar eutectic solidification are satisfied when the solid-liquid surface tensions balance both in the Y-direction and X-direction through proper adjustment of the angles 0~ and 0~, but, in general, this is not

possible. The interfacial "torque" of the a-~ interface can be used to secure a balance of the unequal components of the forces in the X-direction, and the adjustment of the angles 0 ~ and 0 ~ can be employed independently to secure a balance of forces resolved along the Y-direction. The relationship of the precise nature of this interfacial equilibrium to the coupled diffusion problem associated with eutectic solidification has been discussed recently by Nash e t al. [35]. Rama Rao [36] has also reviewed some of the key problems remaining in obtaining a satisfactory theory of eutectic solidification, and Nash [25] has demonstrated techniques for solving the diffusion problem at coupled polyphase interfaces. Still in doubt, however, is the proper condition needed to select the operating state of a eutectic system from which the phase spacing, crystallography, and interface temperature may be predicted.

6. CONCLUSIONS

The field of solidification has undergone truly profound changes over the past two decades in terms of the sophistication and scope of its theoretical basis. New experimental findings will probably persist in outpacing the theoretical predictive capacity of this field, although occasionally theory has leapt ahead and remained there. In an article of this length major omissions are mandatory. For example, no mention was made of the impressive recent progress in statistical computer models of interface structure, or of the current activity in developing the finite element method for predicting solidification patterns, thermal gradients, and thermal stresses in ingots and shaped castings. Nor was mention made of the production of metastable phases using ultra-fast cooling techniques, such as splat cooling or spin quenching. This subject, and the allied area of metallic glasses, are highly active theoretically, in an attempt to keep up with the rapid pace of experimental progress. Thus, it is safe to say that the perimeter of knowledge in the field of solidification theory is expanding, albeit non-uniformly and fitfully. In terms of what we do n o t know about solidification theory -- clearly, a future author on that subject will have less to say.

101 REFERENCES 1 D. R. H. Jones, J. Mater. Sci., 9 (1974) 1. 2 S. C. Hardy, Nat. Bur. Stand., Washington, D. C., to be published. 3 S. C. Hardy and S. R. Coriell, J. Cryst. Growth, 5 {1969) 329. 4 S. C. Hardy and S. R. Coriell, J. Appl. Phys., 39 (1968) 3505. 5 M. E. Glicksman, R. J. Schaefer and J. D. Ayers, Metall. Trans., (1976) in press. 6 G. E. Nash and M. E. Glicksman, Rost Krist., 11 p. 278, Armenian Academy of Sciences, Erevan, USSR, 1975. 7 G. E. Nash and M. E. Giicksman, Acta Metall., 22 (1974) 1283. 8 G. E. Nash and M. E. Glicksman, Acta Metall., 22 (1974) 1291. 9 R. Defay, I. Prigogine, A. Bellemans and D. H. Everett, Surface Tension and Adsorption, Ch. XVI, Longmans, Green, London, 1966. 10 J. C. Baker and J. W. Cahn, Acta Metall., 17 (1969) 575. 11 K. A. Jackson, Can. J. Phys., 36 (1958) 683. 12 V. T. Borisov, Soy. Phys.-Dokl., 17 (1962) 50. 13 G. Baralis, J. Cryst. Growth, 3/4 (1968) 627. 14 B. K. Jindal and W. A. Tiller, J. Chem. Phys., 49 (1968) 4632. 15 I. L. Aptekar and D. S. Kamenetskaya, Fiz. Met. Metalloved., 14 (1962) 358. 16 J. C. Baker and J. W. Cahn, in Solidification, Ch. 2, American Society for Metals, Metals Park, Ohio, 1971. 17 R. F. Sekerka, in P. Hartman (ed.), Crystal Growth - - An Introduction, Ch. 15, North Holland Publ. Co., Amsterdam, 1973. 18 R. T. Delves, in Brian R. Pamplin (ed.), Crystal Growth, Vol. 1, Ch. 3, Pergamon Press, Oxford, 1975.

19 M. E. Glicksman, in Solidification, Ch. 6, American Society for Metals, Metals Park, Ohio, 1971. 20 B. J. Matkowsky, Bull. Am. Math. Soc., 76 (1970) 620. 21 B. J. Matkowsky, SIAM J. Appl. Math., 18 (1970) 872. 22 G. E. Nash, Naval Res. Lab., Washington, D. C., personal communication. 23 R. J. Schaefer and M. E. Glicksman, J. Cryst. Growth, 5 (1969) 44. 24 G. E. Nash, Ph.D. Thesis, George Washington University, 1974; Naval Res. Lab. Rep. 7679, May, 1974. 25 G. E. Nash, Naval Res. Lab. Rep. 7956, Feb, 1976. 26 M. C. Flemings, D. R. Poirer, R. V. Barone and H. D. Brody, J. Iron Steel Inst., 208 (1970) 371. 27 T. F Bower, H. D. Brody and M. C. Flemings, Trans. Metall. Soc. AIME, 236 (1966) 624. 28 T. Surek and B. Chalmers, J. Cryst. Growth, 29 (1975) 1. 29 R. J. Pohl, J. Appl. Phys., 25 (1954) 668. 30 G. K. Gaule and J. R Pastore, in R. O. Grubel (ed.), Metallurgy of Elemental and Compound Semiconductors, Interscience, New York, p. 201. 31 P. I. Antonov, Rost Krist., 6, Inst. Crystallography, Moscow, USSR, 1965, p. 158. 32 P. I. Antonov and A. V. Stepanov, Bull. Acad. Sci. USSR, Phys. Set., 33 (1969) 1805. 33 W. Bardsley, F. C. Frank, G. W. Green and D. T. J. Hurle, J. Cryst. Growth, 23 (1974) 341. 34 A. I. Pogodin, I. M. Tumin and A. M. Eidenzon, Bull. Acad. Sci. USSR, Phys. Ser., 37 (1973) 32. 35 G. E. Nash, J. D. Hunt, K. A. Jackson and M. E. Glicksman, Int. Conf. In-Situ Composites, Bolton Landing, New York, Sept. 1975. Unpublished discussion. 36 A. V. Rama Rao, P h . D . Thesis, Nagpur University, 1974.