Creator:David Turnbull Date Created:1950 Place Created:Schenectady, New York Keywords:thermodynamics in physical metallurgy Context:article from American Society for Metals ************************************************** RES. LAB. REPRINT 1715 GENERAL $11 ELECTRIC PRINCIPLES OF SOLIDIFICATION BY D. TURNBULL I , Reprinted from "Thermodynamics in Physical Metallurgy," Pages 282-306 Copyright, 1950, American Society for Metals, Cleveland, Ohio SCHENECTADY, NEW YORK PRINCIPLES OF SOLIDIFICATION By David Turnbull BROADLY, "solidification" implies the separation of a crystalline phase from gaseous or liquid solution. However, the present discussion is limited to the kinetics of formation of crystals from uniary systems of gases or liquids, and the latter process is to be discussed in more detail. Rates of solidification may be interpreted in terms of the nuclea-tion of crystals and their growth to macroscopic dimensions. The general theory of nucleation has been developed in a preceding paper (l)1 of this seminar. We shall examine critically how it applies to the nucleation of crystals, particularly in liquids. Experimental results on the rate of crystal nucleation in liquids appear to be confusing and contradictory. However, many of the apparent contradictions can be resolved if the role of extraneous influences (such as heterogeneities and container walls), often overlooked or not clearly understood, is properly considered. The experimental conditions most favorable for homogeneous nucleation (i.e., nucleation in the absence of extraneous influences) can be defined and in large part fulfilled. The evidence now at hand indicates that if these conditions are established, consistent and highly reproducible results are obtained. These results are particularly interesting because of the valuable clues that they furnish to the solution of the problem of interfacial energies involving solid phases. Such data also should be valuable in evaluating theories dealing with the relation between the structure of liquids and solids. The kinetics of growth of crystals to macroscopic size is probably better understood than their nucleation. It is generally agreed that kinetic factors determine the form of large crystals, and some important generalizations relating these kinetic factors to crystal structure have been made. Nevertheless there are many phenomena involved in crystal growth that have not yet been explained satisfactorily in terms of our present models of rate theory and of structure. The mechanism for the marked changes in growth rates brought about by adsorbed impurities is imperfectly understood. Although 1Thc figures appearing in parentheses pertain to the references appended to this paper. The author, David Turnbull, is research associate, General Electric Co., Schenectady, N. Y. 282 PRINCIPLES OF SOLIDIFICATION 283 there appears to be good evidence for the periodic discontinuities in growth rate of crystals predicted by structural and kinetic theory, the growth rates observed experimentally are larger by several orders of magnitude than predicted by the theory. These problems and others would be clarified considerably if there were a more satisfactory picture of the structure of the surface layer of a pure solid in equilibrium with its vapor or its melt. Great progress in setting up such models for solid surfaces should become possible as reliable values for interfacial energies involving solid phases become available. Nucleation of Crystals Vapor —> Crystal According to the theory reviewed in a preceding paper (1), an approximate expression for the rate of homogeneous nucleation of crystals in vapor can be written as follows: I = n(p/kT) (AV2)(-AFTV./3m) * exp. [-K crystal reactions. However, in liquid metals that have been significantly subcooled the growth rate of crystals is very rapid, one might say cataclysmic, so that most of the time elapsed in the transformation is that required for forming the first crystal nucleus. Measurement of the over-all rate of transformation dilatometrically, by heat evolution, etc., thus gives the desired information on the rate of nucleation. Experiments of this general character in which a relatively small number of nuclei are developed in the course of the transformation are particularly susceptible to the influence of nucleation catalysts as already explained (1). Since as little as one part in 1016 of the right kind of heterogeneity can catalyze the formation of one nucleus, the magnitude of the problem of eliminating such extraneous influences in these experiments can be appreciated. Clearly, the normal standards of chemical purity are insignificant in nucleation experiments. 286THERMODYNAMICS IN PHYSICAL METALLURGY In fact there is good experimental evidence (1, 7) that nucleation in practically all such experiments occurs heterogeneously. The hypothesis that the well-known effect of thermal history upon nucleation kinetics is but another manifestation of catalytic influences has been justified by many experiments (7, 8). Mechanical vibrations are another important disturbing influence on nucleation in liquid —» crystal reactions. It is well known that comparatively mild mechanical vibrations can induce nucleation of crystals in a subcooled liquid at a temperature several degrees higher than that at which it would normally occur. For example, the author has observed that gallium samples which would remain liquid indefinitely at AT_ = 45 °C under static conditions could be made to crystallize immediately at AT_ = 30 °C by vibrations brought about manually. It has been maintained (9) that liquids can be prevented from subcooling perceptibly by employing mechanical vibrations of sufficient intensity. Vonnegut (10) has suggested that the effect is due to cavitation at the container wall, or elsewhere, induced by the vibrations. At the instant the cavity is closed, the liquid rushing back into position has acquired an extremely high velocity so that a wave of rather high alternating positive and negative pressures will be set up. In a liquid which contracts upon solidification, the free energy of the liquid —»crystal transformation is decreased in regions of positive pressure so that the effect of this positive pressure upon the rate of crystal nucleation will be roughly equivalent to lowering the temperature by an amount A calculable by the Clausius equation. The magnitude of local pressures that can be built up in this manner can easily account for the shift in AT_ due to mechanical vibration. Because the magnitudes of these pressures are unpredictable, it is evidently necessary that mechanical vibrations be minimized in experiments purporting to measure the rate of homogeneous nucleation in liquid —» crystal transformations. How then can the influence of heterogeneities and vibrations be eliminated and the actual rate of homogeneous nucleation in liquid —» crystal reactions be measured? The answer to this question is suggested by the work of Volmer and Flood (3). That is to select experimental conditions such that a very large number of nuclei (i.e., large in comparison with the number of possible heterogeneities present) form in the interior of the sample. When working with large continuous specimens this solution of the problem is practicable only for substances whose crystals grow relatively slowly at tem- PRINCIPLES OF SOLIDIFICATION 287 peratures where homogeneous nucleation becomes measurable. Under these circumstances, a very large number of crystals can form before the liquid is entirely transformed. These conditions are often realized in subcooled organic liquids having high viscosity, but they can never be realized in large liquid metal samples that have been significantly subcooled because of the tremendous rate of crystal growth. In order to assure the formation of a very large number of nuclei in liquid metal specimens, it is necessary to break the liquid up into very small parts that are prevented somehow from intercommunicating. Thus, as the author has pointed out (11), the nucleation catalysts are isolated in individual parts of the specimen and are capable of promoting transformation only in that particular part in which they are localized. If the parts are small enough the vast majority of them will contain no catalyst whatever and the nucleation must be homogeneous. As an example, suppose that a liquid metal contains 106 catalytic centers/cm3 for the formation of crystals. The probability of finding 1 cm3 of this liquid free of centers is practically nil and it would be observed that such a sample would crystallize at a very small amount of subcooling, AT__If 1 cm3 of the liquid is entirely broken up into isolated droplets 10 microns in diameter, the probability of finding a catalytic center in a given droplet is only 1 in 2000. The entire sample could then be subcooled to a very much greater extent, in fact to the characteristic temperature of homogeneous nucleation. The heterogeneous component of the reaction would not be detected at all excepting with very sensitive experimental procedures. If, instead of looking at the behavior of the whole assembly of droplets, individual droplets were studied, the probability would be very large (1999/2000) that a given individual would subcool to the maximum extent. It has been known for at least sixty years that comparatively small particles of liquid gold subcool farther than do large samples in crucibles which solidify nearly at the melting temperature. However, no quantitative information regarding the amount of subcooling or the effect of time upon it seems to be available. The first more or less systematic experiments on the subcooling of small metal particles seem to have been made by Mendenhall and Ingersoll (12). They watched the solidification of some high melting metal particles 50 to 100 microns in size that were melted by heating on a Nernst glower. The maximum subcooling obtained was quite large and 288THERMODYNAMICS IN PHYSICAL METALLURGY seemed to be proportional to the melting point of the particular element. However, the only results given were the maximum subcooling for platinum and rhodium, 370 °C in both cases. The full significance of these results and of the earlier qualitative observations was not realized and greater significance was attached to the results from subcooling experiments on comparatively large melts contained in crucibles. Vonnegut (13) seems to have been the first to study the solidification behavior of large aggregates of small liquid particles in his investigation of the kinetics of the liquid —» crystal transformation in tin. The particles of liquid tin were kept apart by an oxide film surrounding each particle, and the rate of transformation was followed dilatometrically. About the same time Cwilong (14) and Schaefer (15) established independently the maximum subcooling required for the formation of snow crystals in aggregates of small water droplets suspended in air. In these experiments, as in the earlier small-particle experiments, the maximum subcooling was very much larger than ever attained on large samples. The author (11) suggested that this could be attributed to the isolation of nucleation catalysts effected by breaking up the sample, and with his co-workers (16, 17) extended the small-particle technique to the kinetics of liquid - crystal reactions in a large number of pure metals. The experimental details and complete results of these experiments are to be published elsewhere (16, 17) and only a summary will be presented. Table I summarizes the results of the subcooling experiments of the author and co-workers as well as those of other investigators whose results are known to have been obtained by the small-particle technique. In nearly all cases the maximum subcooling, AT_, observed for small particles is very much larger than that for large continuous samples. The maximum subcooling recorded in the table was chosen arbitrarily in each case to correspond to a nucleation rate of the order of 1 nucleus/particle in 10 seconds. The average particle size in most of the experiments was of the order of 50 microns. Since AT_ is time-dependent and because nucleation rates are not closely specified, one may question whether a comparison of AT_ values is significant. Actually, the nucleation rate changes so sharply with temperature that AT_ is practically a characteristic property of the substance. The marked dependence of nucleation rate on temperature has been PRINCIPLES OF SOLIDIFICATION 289 demonstrated by many experiments. For example, the author (11) found that mercury samples could be held for 1 hour at AT_ = 43 °C without detectable reaction, but completely solidified within a minute at AT. = 46 °C. From this, it may be inferred that the rate changes by a factor of at least 104 in 3 degrees. Similar phenomena were found in microscopic observations of solidification (16, 17). Thus, a AT_ value can be bracketed within ±2% for most subcooled liquid metals such that at AT_/1.02 the nucleation rate, I, is too slow to measure in a reasonable period and at AT_/0.98 is too rapid to measure. An analogous problem was met with and solved similarly by Volmer and Flood (4) in their experiments on the nucleation of liquids in vapors. They had no way of measuring the nucleation rate quantitatively, but it changed so markedly with small changes of the supersaturation ratio p/p0 that it sufficed to ascribe some arbitrary finite value to the nucleation rate at the minimum value of p/p0 at which nuclei were observed to form. Even so, the agreement that they found between interfacial tensions calculated from the results of nucleation experiments and values of these tensions measured by other methods was excellent. For most of the metals the largest fraction of the particles crystallized at or very close to the maximum subcooling reported. This fraction was 9/10 in the case of mercury, for example. In some instances, however, a large proportion of the particles crystallized before the maximum AT_ was reached and only a small but significant fraction crystallized at this temperature. Some interesting observations indicating the possibility of finding a "6 spectrum" were made on metals exhibiting this behavior. For example, with lead particles it was observed that the temperature of crystallization for individual particles was sharply defined and reproducible throughout several successive liquefaction/solidification cycles. Particle "A" might be observed to crystallize consistently at a temperature very close to T0, particle "B" at about y2 the maximum subcooling and particle "C" at the maximum. Thus, each particle might be said to have a characteristic AT_ and one could construct a spectrum of such values from the observations on a large number of particles. It is evident, however, that this behavior is symptomatic of the influence of heterogeneities in the particle. The characteristic AT_ value observed for a given particle can be related to the contact angle 0 made by the crystal with the effective heterogeneity contained in 290THERMODYNAMICS IN PHYSICAL METALLURGY the particle. Thus, a 6 spectrum might be constructed from the characteristic AT_ values. Such a spectrum could be continuous or it might be more or less discrete if a large number of particles contain some specific heterogeneity. An arbitrary representation of this possibility is shown by Fig. 1. According to the theory of homogeneous nucleation, the rate of nucleus formation in small particles Iv is related to their volume, V, by the following relation Iv = IV Equation VI where I is the nucleation rate per unit volume. In interpreting the Possible ©"Spectrum" in J*! o q. o QJ JD £ 3 Z 0 „ 180 e Fig. 1—Distribution of Small Particles in an Aggregate as a Function of 6. small particle results it is necessary to know the relative importance of volume changes and temperature changes upon the nucleation rate. One alternative explanation of the small-particle results that has been advanced is that realizable volume changes are very important in their effect upon nucleation rate relative to the total amount of subcooling obtained. This explanation is untenable for several reasons. It is tantamount to the requirement that the temperature coefficient of nucleation rate be very much smaller than it actually is. For example, in the author's experiments on mercury, a mass, approximately 1 centimeter in linear dimension, that did not significantly subcool was broken up to particles approximately 50 microns in linear dimensions that subcooled 46 °C. The volume change was of the order of 107, but the rate changed by a factor of at least 10* in 3 degrees so that the total effect of the volume change per se is equivalent to changing the te'mperature 6 degrees or less. Further PRINCIPLES OF SOLIDIFICATION 291 evidence for the relative unimportance of volume change is obtained in microscopic observations of solidification. Generally, small particles are less likely to contain heterogeneities than large particles and thus subcool the maximum amount much more frequently. Occasionally, however, a 10-micron particle is observed to solidify consistently at a temperature only a little less than T0, while an adjacent 50-micron particle consistently exhibits the maximum subcooling. Also, it should be noted that the maximum subcooling obtained in large continuous samples sometimes approaches that observed for small particles. For example, Johannsen (18) subcooled 1-cc water samples 36 degrees, while the small-particle result is 39. Turnbull (11) subcooled. 1-gram gallium samples 55 degrees compared with 76 degrees for small particles. Bardenheuer and Bleckman (21) report that they were able to subcool a 150-gram sample of iron, entirely coated with a layer of liquid slag, 258 degrees before solidification. This number is of the same order of magnitude as one would anticipate from the small-particle technique. Some interesting correlations and consequences emerge from the data on maximum subcooling. One is that the ratio AT_/T0 is about the same (see Table I) for many of the substances and falls within the range 0.13 to 0.25 for all. For metals crystallizing in a close-packed structure, AT_/T0 is of the order of 0.18 excepting for lead. Such a correlation might be expected on the basis of homogeneous nucleation theory, provided the liquid-solid interfacial energy a were proportional to the heats of fusion AHt and, in addition, the entropies of fusion were constant for various substances. Since entropies of fusion are not constant, a better correlation should be obtained by comparing o directly with AHf. Having the approximate value of the nucleation rate at one temperature, the equation of Turnbull and Fisher (5) can be used to calculate numerical values of a. Fisher, Hollomon and Turnbull (22) first made this calculation for water. The author (16) has extended this calculation to all of the substances listed in Table I, with the results shown. In addition to the assumption made in deriving the equation, the following were used in making the calculations : 1. a is isotropic so that K = 16"V3 corresponding to a spherically-shaped nucleus. 2. AF., = 0. A value of (1/10) sec-1 per particle was assigned to I, and Equation Ila used to calculate AFV. 292 THERMODYNAMICS IN PHYSICAL METALLURGY Although there is no clear theoretical basis for assuming that the interfacial energy between a metal crystal and its melt is isotropic, there are certain lines of experimental evidence which indicate that this may be a fair approximation for metals having cubical crystal structures. Thus, the experiments of C. S. Smith (23) indicate that the interfacial energy between a metal crystal and the liquid phase of a different metallic element (e.g., solid copper — liquid lead) is nearly isotropic. This does not prove that the interfacial energy between a metal crystal and its own melt is isotropic, but it is strong evidence in favor of such a hypothesis. Also, C. G. Dunn and F. Lionetti (24) have shown that the interfacial energy between differently oriented crystals of silicon ferrite is practically independent of the orientation difference, A, when this exceeds a relatively small value. These measurements offer further indirect support for the assumption under discussion. At present there is no direct measurement of the magnitude of AFa, but the evidence indicates that it must be quite small in comparison with AF* in metal liquid —» crystal reactions. In all of the metals that the author and his co-workers have studied, the growth rate of crystals into their melts, when significantly subcooled, was too rapid for accurate estimation. This was also true of metals such as gallium, germanium, and bismuth, having relatively complex crystal structures. Further, it is well known that the energies of activation for viscous flow, Q, are abnormally small for liquid metals, and it is probable that AFa is of the same order of magnitude as Q. For example, if Q is equated to AFa in the case of mercury AFA«*1.6kT while AF* « 75 kT at —85 °C, the temperature of rapid crystallization. If AFa is of this order of magnitude, equating it to zero introduces an error of less than 1% in the calculated value of the interfacial energy. Calculation of AFV by assuming a constant entropy of fusion, is only a fair approximation in many cases and could be improved considerably. Improved calculations of AFt are currently being carried out in this laboratory. The validity of the assumptions made in deriving Equation IV have already been discussed. However, it is worthwhile to note that if the quantity Ih/nkT in Equation IV were in error by a factor of 1010 due to experimental errors in measuring I and the approximations made in deriving nkT/h, the calculated value of a would be only 10% in error. PRINCIPLES OF SOLIDIFICATION 293 It is reasonable to correlate (25, 16) the gram-atomic interfacial energy, oK, rather than a in ergs/cm2 with the gram-atomic heat of fusion aH(. oe is defined as crystal transformation in many of these compounds. It would be very desirable to reinvestigate the kinetics of many solid —» solid transformations by the small-particle technique. Some evidence that there is an effect is furnished by the results of several ®This suggestion was made by Dr. R. A. Oriani of this laboratory. 296THERMODYNAMICS IN PHYSICAL METALLURGY investigators (26, 27) who observed that the y —» a transformation in small iron particles precipitated from copper was very much slower than in large iron samples. It is possible that the interfacial energies between solid phases involved in transformations may bear a relation to the heat of transformation that is analogous to that discovered in liquid —> crystal reactions. Kinetics of Crystal Growth General Principles It might be expected that small crystal nuclei and embryos that appear most frequently should have a form that minimizes their surface energies. It has been supposed at times that the form of large crystals also is of necessity determined by the condition of minimum surface energy. However, there is abundant evidence which indicates that the form of large crystals is almost always determined by the effects of orientation upon growth rate rather than by purely thermodynamic factors. The insignificant contribution of surface energy to the free energy of large crystals can easily be shown. Consider a metal crystal having linear dimensions of the order of 1 centimeter and having two possible sets of crystal faces that differ in surface energy by 1000 erg/cm2. The vapor pressure of a crystal having the higher surface energy faces exposed is larger by a factor of about 1 -f- 10~7 than that for a crystal having the other set of faces exposed. A temperature fluctuation of the order of 10~6 °C is sufficient to change the vapor pressure by this factor. Although many processes in nature take place when the driving force is this small, the influences of minute temperature fluctuations during crystal growth are sufficient to obliterate the importance of pure thermodynamic factors in determining crystal form. It is readily demonstrated (28) that crystal planes perpendicular to the direction of most rapid growth grow out of existence while the planes that finally appear are those perpendicular to the direction of slowest growth. A generalization due to Bravais (29) is that the most closely packed planes develop during crystal growth. Thus, the most closely packed planes are normal to the direction of slowest growth and contain the directions of most rapid growth. This generalization can be explained as follows: an atom added to a close-packed plane bonds with nearest neighbors with which its interaction is greatest, while an atom added in a direction normal to a most closely packed PRINCIPLES OF SOLIDIFICATION 297 plane must bond with nonnearest neighbors with which its interaction is weaker. The former event is more probable than the latter so that, after a closely packed plane is formed, growth will tend to progress in directions contained within it rather than in directions normal to it. There are many exceptions to the Bravais generalization. Some of them are attributed to external disturbing influences such as the effect of other components in the system. Many examples are known in which the crystal form is completely altered when developed in the presence of compounds strongly adsorbed on the crystal faces. Other exceptions cannot be explained in this way, but Donnay and Harker (30) have shown that many of these can be rationalized in terms of an extended law which they formulate, that includes the law of Bravais as a special case. Their law reduces to that of Bravais if the lattice is of the hexahedral mode and the space group is devoid of screw axes and glide planes. Growth of Crystals in Supersaturated Vapor According to kinetic theory, the number of atoms from the vapor added to 1 cm2 of crystal surface per second is: dn/dt = a (p — p0)/(2irmkT)* Equation XI where a — fraction of impinging molecules retained p = pressure of supersaturated vapor p0 = pressure of vapor in equilibrium with the crystal This relationship has been verified for large crystals, although values of a obtained in different investigations have not been consistent. There is good evidence, however, that a is greatly diminished by adsorbed surface films of foreign substances. Volmer and Estermann (31) have measured a values of the order of 0.9 for surfaces of mercury crystals free of adsorbed films. It is possible, therefore, that a is close to unity for most clean, solid surfaces. This might be considered surprising, since in general an atom must impinge on the surface of a growing crystal at a point where its binding energy is considerably less than average. The fact that atoms do stick most of the time seems to imply that atoms immediately adsorbed on their crystal surface are very mobile and in addition have a mean lifetime on the surface that is large relative to the surface jump period. In fact, there is a good experimental basis for the existence of a mobile layer of atoms on solid surfaces. Volmer and Estermann (32) performed the classic experiment in demonstration of this. 298 THERMODYNAMICS IN PHYSICAL METALLURGY After permitting mercury vapor to impinge upon the surface of a growing crystal for a definite period, they found that the crystal was 1000 times greater in its largest dimension, and only 1/10 as large in thickness, than could be accounted for on the basis of the collision frequency, adopting the concept that each atom struck at the point of impingement. They rejected the implausible hypothesis that the crystal plastically deformed during the course of its growth and proposed instead that atoms attached themselves to the growing lattice only after having migrated great distances within an adsorbed layer of mobile mercury atoms. Fig. 3—Schematic Representation of Periodic _ Discontinuities in Crystal Growth Rates with Addition of Lattice Planes. A large number of subsequent investigations have provided evidence for the existence of a mobile layer on the surface of many types of solids. One of the most striking of these was that of Kowarski (33) in which Brownian movement of a 1-micron liquid droplet on a solid surface of P-toluidine was observed. There is reason to believe that a mobile "self-adsorbed" layer exists on clean surfaces of all solids, at least at temperatures not too far removed from their melting points. The Problem of Two-Dimensional Nucleation Consider the attachment of atoms in the initiation of a new layer On a perfect crystal plane. It is evident that the energy decrease in the attachment of the first few atoms is much smaller than that of the atoms which finally perfect the new layer. Thus, there should be a period of slow attachment of atoms while a two-dimensional nucleus is formed followed by a period of rapid growth of this nucleus to a complete crystal plane. The resulting fluctuations in growth rate, di/dt, with the addition of successive crystal layers is shown PRINCIPLES OF SOLIDIFICATION 299 schematically in Fig. 3. Kossel (34) and Stranski (35) have developed the theory of two-dimensional nucleation and growth. There is experimental evidence for the existence of such periodic discontinuities in crystal growth rates. Marcelin (36) first observed the phenomenon in his experiments on the crystallization of P-toluidine from solution. Thickening of the crystal could be followed by the change in color of interference tints that accompanied it. These tints were observed to deepen in a stepwise manner— each step sweeping rapidly over the crystal surface. From the change in wave length accompanying a step, Marcelin calculated that some of the steps were indeed of the order of 1 molecular layer in thickness. Observations similar to Marcelin's have been made on the crystal growth of many other substances. For example, Volmer (37) has reported on similar phenomena in the electrodeposition of cadmium and tin. Qualitatively these experiments seem to confirm the Kossel-Stranski mechanism. However, the lengths of the two-dimensional nucleation periods are very much shorter than have been calculated on the basis of the idealized Kossel-Stranski theory. In fact, calculations based on the theory (38) indicate that a crystal ought not to grow at a measurable rate for a vapor supersaturation of a few per cent, while experience proves that crystals do grow quite rapidly under such conditions. Probably the solution of this dilemma is that the crystal is not perfect to its outermost boundary. Thomson (38) has advanced an explanation in these terms that is quite plausible. He points out that the growth of crystal layers takes place not by direct attachment of atoms from the vapor but by attachment from the mobile self-adsorbed layer which can be assumed to have a relatively constant configuration with respect to the vapor. The energy et required for attaching an atom to a two-dimensional nucleus from this self-adsorbed layer must be much less than the energy eT of attaching it directly from the vapor. It is reasonable to suppose that the ratio Ei/eT is of the order of the ratio of the liquid - crystal interfacial energy to the liquid-vapor interfacial energy. The surface energy opposing the initial nucleation of the crystal is the interfacial energy between the vapor and the self-adsorbed layer, oT, plus the interfacial energy between the self-adsorbed layer and the crystal, Oi. If the mobile layer is relatively dense, aT, which should remain nearly constant during crystal growth, predominates over