NUCLEATI ON-From L I T E R A T U R E CITED
Amsler, J., H e h . Phys. Acta, 15, 699-732 (1942). Dehlinger and Wertz, Ann. Physik, 39, 226-40 (1941). Gopal, R., J . Indian Chem. SOC.,20, 183 (1943). Ibid., 21, 103-8, 145-7 (1944). Ibid., 24,279-84 (1947). Hersh, R. E., Fry, E. M., and Fenske, M. R., IND.ENG.CHEM., 30, 363 (1938). Heverly, J. R., Trans. Am. Geophys. Union, 30, 205-10 (1949). International Critical Tables, Vol. 3, p. 43, New York, McGrawHill Book Co., 1928. Ibid., p. 106. Ibid., Vol. 4, p. 239. Mellor, J. W., “Comprehensive Treatise on Inorganic and Theoretical Chemistry,” Vols. 7, 10, 11, London, Longmans, Green and Co., 1922. Ibid., Vol. 10, p. 786, Vol. 11, pp. 2-40 ff., 55. Neumann, K., and Miess, A., Ann. Physik, 41, 319-23 (1942). Perry, J. H., “Chemical Engineers’ Handbook,” 3rd ed., pp. 246, 1052, New York, McGraw-Hill Book Co., 1940. Rau, Schriften deut. Akad. Luftfahrt, 8, 65 (1944).
Liquids
(16) Rhodin, T. N., Discussions Faraday Soc., 5, 215 (1949). (17) Shearman, R. W., and Menzies, A. W. C., J . A m . Chem. SOC., 59,185 (1937). (18) Smith-Johannsen, R., Science, 108, 652-4 (1948). (19) Stranski, I. N., and Kuleliew, K., 2. physik. Chem., 142, 467 (1929). (20) Stranski, I. N., and Mutaftschiew, Z. C., Ibid., 150, 135 (1930). (21) Turnbull, D., and Fisher, J. C., J . Chem. Phys., 17,71-3 (1949). (22) Van der Merwe, J. H., Discussions Faraday Soc., 5 , 2 0 1 (1949). (23) Volmer, M., “Thermodynamik der Phasenbildung,” Dresden and Leipzig, Theodor Steinkopff, 1939. (24) Volmer, M., and Weber, A., 2. physik. Chem., 119,295 (1926). (25) Vonnegut, B., Chem. Revs.,44,277-89 (1949); J . Applied Phys., 18,593-5 (1947). (26) West, D. C., The Frontier, 8, No. 2, 12 (1945). (27) Wyckoff, R. W. G., “Crystal Structures.” New York, Interscience Publishers, 1948.(28) Young, S.W., J . Am. Chem. Soc., 33, 148 (1911). (29) Young, S.W., and Cross, R. G., Ibid., 33, 1375 (1911). (30) Young, S. W., and Van Sicklen, W. J., Ibid., 35, 1067 (1913). RECEIVED for review January 14, 1951.
ACCEPTED4pril 1, 1952.
NUCLEATION FROM SOLIDS
Theory of Nucleation in Solids ROMAN S M O L U C H O W S K I CARNEGIE INSTITUTE OF TECHNOLOGY, PITTSBURGH, P A ,
T h e basic problem of a theory of nucleation in solids is outlined and t h e complications caused by a departure from t h e conditions for applicability of Volmer’s theory are described. T h e various theories and their comparisons w i t h experimental data are summarized. In particular t h e nucleation theory of recrystallization and i t s successful interpretation of observations is discussed. It is pointed o u t t h a t t h e qualitative agreement w i t h experim e n t confirms general ideas about nucleation, while t h e absence of quantitative agreem e n t does n o t permit a critical appraisal of t h e specific assumptions.
A
T PRESENT there is no satisfactory theory of nucleation in
solids. This embarrassing situation is due t o the difficulty of experimental investigations of that phenomenon and also to the lack of theoretical estimates of the numerous factors which influence formation of nuclei in solids. Observations are made at relatively late stages of reactions when nucleation has been followed by appreciable growth and thus other unknown factors are added to the already complex picture. I n a few instances, as in recrystallization which is discussed here, it has been possible to extrapolate back to the process of nucleation without making too drastic assumptions about the rate of growth. The theories give at best a qualitative dependence of nucleation on time, temperature, composition, and other factors. A comparison with experiment sometimes leads to reasonable values for the adjustable constants. T H E BASIC PROBLEM
A comparison of nucleation in a solid, as for instance during precipitation in a supersaturated solid solution, with nucleation in a vapor phase during condensation illustrates well the various additional factors which have to be considered. The fair agreement between the various variants of Volmer’s theory ($00)of condensation and experiments indicates that the basic notions about the roles of free energy, surface energy, fluctuations of density, critical radius of the nucleus, etc., are in that particular case sound and convenient. How do these relatively simple concepts apply t o precipitation from a supersaturated solid solution? Considering first the free energy per mole of the precipitating compound, the first difJune 1952
ficulty is encountered: Information about its composition or its crystal structure is not certain, even if this information about the final precipitate obtained a t equilibrium were available. In fact there are many instances where it is definitely known that there are several intermediate stages in the development of a preand thus an unambiguous extrapolation to zero time cipitate (8), is not possible. Here the inherent limitations of the finest x-ray techniques become apparent: Usually only the later stages of the formation of a nucleus corresponding t o the development of its crystalline structure can be observed. Only in a few particularly favorable cases the presence and shape of clusters of (ultimately precipitating) atoms within the crystal lattice of the original matrix has been determined (9). Thus the notion of free energy per mole of the early nuclei becomes vague and it would be perhaps better to avoid that concept altogether and consider the phenomenon on a purely atomic basis. Clearly this will not be a n easy task. Next there is the surface energy. Does a nucleus in a solid have definite surface? There is little doubt that embryos-i.e., unstable nuclei corresponding, in the case of condensation, t o nuclei smaller than the critical size-are coherent with the matrix, and both their composition and atomic configuration gradually merge with the surrounding lattice. It could be assumed that a certain layer of atoms in this transition zone is the surface of the nucleus, but the ambiguity of t h a t concept is obvious. Thus the surface energy cannot be as uniquely defined as for condensation nuclei. For the same reason the critical radius or size of the nucleus in a solid is not rigorous concept. This is particularly true since solid nuclei are seldom spherical.
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Probably the strongest depaituie from the ideal case of condensation is the role of fluctuations. Instead of the fairly simple density fluctuation, fluctuations of chemical composition and of atomic configuration are present in a solid ( I ? ) . The two are, of course, not independent of each other. For instance, the probability of the occurrence of a particular crystalline lattice may depend not only on the presence of the proper ratio of the various atoms in a certain small volume-Le., local electron concentration-but also on their arrangement-Le., tendency to form a compound. These examples of the troublesome interactions between the nucleus and the matrix can sometimes be interpreted in ternis of strains of the nucleus and of the matrix. The most evident strain is due to the difference of the volume per atom in the matrix and in the aggregate. This strain is smallest for a diskshaped aggregate (14), whatever its crystal structure. The elastic anisotropy of the matrix promotes certain particular orientations of this disk with respect to the crystallographic axes of the matrix (15, 18). Another kind of strain is due to the tendency to match the positions of the atoms in the matrix with those in the aggregate across the boundary with the resulting coherency strains. The strains caused by volume change persist and increase with the size of the nucleus; the coherency strains increase until the shear strength in a direction tangential t o the boundary is reached and then, upon further growth of the nucleus, a rupture occurs, accompanied by a loss of coherency. While in condensation the process of supplying atoms is well described by kinetic theory of gases, in a solid the atoms move by the relatively complicated process of solid diffusion. The theory of the latter, although rather satisfactory in simple large scale applications, fails when high local concentration gradients and strain gradients are present. Both kinds of gradients depend upon the shape and size of the nucleus, which in this manner influences the progress of its growth from the early embryonic stage. THEORIES OF NUCLEATION
large, and thus improbable, nuclei are stable, while at lowcr temperatures the diffusion is slow. The various assumptions and simplifications underlying this theory are evident and its general applicability would be hardly expected. Borelius (9)approached the problem from an essentially different angle. He considered the probability of the occurrence of various fluctuations of composition a t a given temperature and their tendency to grow or to disappear. The relevant quantities are correlated directly 13 ith the dependence of free energy, F , on composition, c. I t appears that as long as the quantity
is positive the fluctuations will grow until separation is complctccl, while for negative p they will tend to disappear. Thus, whenever a change in c or '2 changes the sign of Q, there should be a rapid change of the nucleation state. This is indeed confirmed by experiments on the gold-platinum syetem, the same for which Becker's theory was checked. Furthermore the rate of changr of p , for instance
should correlate with the rate of change of nucleation. This too seems to be the case ( 1 7 ) . Both theories assume, of course, highly idealized conditions. The main difference between them is that Becker's theory does not consider in detail the behavior of various fluctuations while in Borelius's theory no account is taken of such factors as surface energy, atomic configurations, etc. Hobstetter (11) has recently proposed a theory which, a t least formally, bridges this gap by using the notions of average free energy, f(c), and of average internal energy, u(c), per pair of atoms in an area of uniform concentration, c. Whatever the physical significance of these concepts, they lead to a n expression foi the free energy of a fluctuation: If the coordination number is Z and the numher of pairs across the surface of the nucleus is S , the change in free energv on formation of a fluctuation of composition b containing n atoms is given by
In view of the existence of several surveys of the field ( 1 7 ) only a brief mention of the older theories will be given here and more emphasis will be placed on recent developments. The success of the Volmer-Becker theory of nucleation in conc , dc + ; b u ( b , c ) - u ( b ) - u(c) densation prompted Becker ( 2 ) to apply it to reactions in solid AF = - 2 state. As it is often done, the free energy of a binary alloy can be where u(b,c) corresponds to pairs of atoms a t the surface. -4sformally described in terms of contributions from pairs of atoms suming that the solid solution is random and that again the total AB, A A , and BB, each pair contributing its characteristic bond energy can be represented as a sum of interactions of nearest energy: VAB,V A A , VBB. .4ctuallp, for metals, this approach has neighbors, the expression for AF' can be written as little theoretical justification and should be considered only as a convenient method of describing various ordering and clustering phenomena ( 1 7 ) . A particularly bad aspect of this assumption is that the energy of a pair is independent of its surroundings. By where C = c In c (1 - e ) In ( I - c), B = b In b (1 - b ) In comparing an actual equdibrium diagram of a n alloy system (1 - b ) , and T , is the temperature a t the maximum of the miswhich has complete miscibility a t high tempeiatures and a miscibilit,y gap. With this expression for AF the size of the nuclei cibility gap a t low temperatures with the idealized model of pair can be calculated a t various temperatures and concentrations. interaction, one can obtain KO critical comparisons x i t h experiments are available, although li' one may expect to obtain an agreement at, least as good as with V A B - '/2(17AA VBB) either Becker's or Borelius's theories. This allows an estimate of the energy, W , per unit volume of the The formation of martensite, which occurs by a shear ti,ansnucleus a t each temperature and the energy, u, per unit of its formation, has been t,he subject of much interest. In part>icular, surface in terms of pairs of atoms. Assuming that the role of Fisher, Hollomon, and Turnbull (7) made an interesting att,empt strains, etc., is negligible and that the nucleus is a cube, one obto interpret this reaction in ternis of nucleation. Although there tains for the rate of nucleation are 110 changes of chemical composition involved, and in that respect the problem is soniexvhat simpler than precipitation, the IV = C exp. ( - Q/icT) exp. ( - A / k T ) formation of martensite is a complicated process mainly because the lattice strains play a ver3- important role. A compariwhere A = 32d/W2 and the first exponent represents the inson with experiment leads to surface energy and lattice strain of fluence of temperature on diffusion, C being a constant. A reasonable order of magnitude, although as shown by Machlin surprisingly good agreement of the rate of nucleation as a funcand Cohen ( 1 3 ) some of the basic assumptions made in this nution of undercooling with experiment was obtained for a 30% goldcleation theory of martensite are uncertain or even contradicted platinum alloy. According to this theory the maximum rate of by experiment. nucleation is due to the fact that a t higher temperatures only
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44, No. 6
NUCLEATION-From Another example of a reaction in which no chemical changes take place is the formation of ferromagnetic domains. Here again strains play an important role, as Dijkstra ( 6 ) has shown. There have been several attempts made t o develop formulas in which more factors occurring in nucleation in solids would be explicitly taken into account than has been done in the theories here described. One of the more recent theories is that of Sirota ( 1 6 ) . There is, however, not enough known about its applicability to experimental data t o appraise the model and its approximations. If it is realized how few of the various factors mentioned in the previous section were taken into account in any of the theories, i t is not surprising that only a qualitative, or a t best semiquantitative, correlation with experiment is obtained. I n fact it is astonishing that the theories work out so well. Part of this is unquestionably due t o the fortunate availability of experimental material for which the very drastic theoretical assumptions are perhaps not unreasonable, and part is a result of fortuitous cancellation of omitted factors. The qualitative agreement with experiment confirms the general notions about nucleation, while t h e absence of quantitative agreement does not yet give sufficiently clear criteria for the numerous more specific assumptions. RECRYSTALLIZATION
Kucleation is usually understood t o occur in a homogeneous unstable medium in which all atoms have, a priori, the same chance t o become part of the starting aggregate, embryo, and finally of the stable nucleus. Surprisingly enough the process of recrystallization in which this condition is not fulfilled seems to be the best understood and analyzed, both from the experimental and theoretical point of view. Of course, it may be argued whether recrystallization starts as a true nucleation process, but nevertheless the formal aspect of it can be successfully described in terms of nucleation and growth as has been shown by Anderson and Mehl(1) in the case of aluminum. The most striking features of recrystallization are: 1. Pronounced time dependence of the rate of nucleation and i n particular the long incubation period during which no reaction Beems t o occur 2. Increased nucleation in highly deformed parts of the matrix 3. Close relationship between statistical orientation of the recrystallized grains and the orientation of the original grains and their strain patterns I n order t o explain the incubation period Turnbull (19)assumed that it is related t o the time necessary for diffusjon of atoms from the deformed matrix to a strain-free embryo until it becomes large enough to be stable. This point of view, although formally satisfactory, is subject to some objections, as pointed out elsewhere ( 1 7 ) . The other aspects of recrystallization, the influence of deformation, and the relationship between orientations seem to be in contradiction: The highly deformed areas may be expected not t o “remember” their original orientation. This situation led t o a whole range of assumptions-from nucleation of recrystallization in a small essentially strain-free volume t o nucleation in a volume of maximuq strain. It has been also often suspected that it is the strain gradient ( 1 7 ) which controls nucleation rather than the absolute value of the strain itself, and this seems t o be confirmed by the recent theory of Cahn (6). I n his theory Cahn considers the highly deformed part of crystalline lattice as an area of high concentration of dislocations. As shown by Guinier and Tennevin (10) and by Lacombe and Beaujard ( 1 2 ) these dislocations, at proper temperatures, migrate t o form regular arrays of dislocations which outline strainfree areas. This process, called polygonization, results in formation of blocks disoriented by no more than a few minutes of arc. Cahn assumes that in a particular area one of these blocks starts growing by absorbing its neighbors and ultimately becomes a recrystallized grain. I n this way by combining the old suggestion that recrystallization nuclei form in areas of high local curvature ( 4 ) with the polygonization mechanism, Cahn recon-
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Solids
ciled the preferential nucleation in areas of high deformation with the existence of orientation relationships. The latter follows from the regular character of the polygonization process. The incubation period in this theory is simply the time necessary for the dislocations t o arrange themselves in regular patterns-Le., the time for completion of polygonization. The simple assumption that this time is proportional t o the radius of local curvature leads to good agreement with experiment. The formula is
N ( t ) = N exp.[- Z(tm/t
- 1)21
tm/t2
where N is a constant and t, is time a t which the most frequent nuclei become active. Fitting the curve a t the maximum value of N ( t ) leads to a very good agreement at other values oft. The temperature dependence of nucleation follows from the dependence of the mobility of dislocations on temperature. It seems thus that recrystallization which is a crystallographic and not a chemical process can be reasonably well described by a process of nucleation. Other nucleation phenomena in solids still await their theoretical and experimental clarification. NOMENCLATURE
A h,c C,N F
=
3zU3/w2 = concentration = a constant = free energy of a system AF = change of free energy of a system .f, P(c) = averaize free e n e r u Der Dair of atoms k = Boltzman constant" n = number of atoms in a fluctuation N ( t ) = rate of nucleation P = second derivative of free energy with respect t o concentration & = activation energy for diffusion u = energy of a surface per unit area s = number of pairs across the surface of a nucleus T = absolute temperature T , = temperature a t the maximum of a miscibility gap t = time t,, = time a t which the most frequent nuclei become active u ( c ) = average internal energy per pair of atoms unit of energy change associated with an interchange of two unlike atoms in a binary solid solution interaction energy of neighboring atoms A and B in a BAR = solid energy of a solid er unit volume coordination numier
v
=
w = z =
LITERATURE C I T E D
(1) Anderson, W. A., and Mehl. R. F., Trans. A m . Inst. Mining
Met. Engrs., 161, 150 (1945). (2) Becker, R.,Ann. Physik. 32, 128 (1938). (3) Borelius, G.,Ibid., 28, 507 (1937);33, 517 (1938);Arkiv. Mat. Astron. Fysik, A32, (1) (1945). (4)Burgers, W.G., and Louwerse, P. C., 2.Physik, 67,605 (1931). (5) Cahn, R.W., Proc. Phys. Soc. (London),63, 322 (1950). (6) Dijkstra, L. J., “Thermodynamics in Physical Metallurgy,” Cleveland, Ohio, Am. SOC.Metals, 1950. (7) Fisher, J., Hollomon, J., and Turnbull, D., Trans. Am. Inst. Mining Met. Engrs., 185, 691 (1949). ( 8 ) Geisler, A. H., “Phase Transformations in Solids,” New York, John Wiley & Sons, 1951. (9) Guinier, A.,Physica, 15, 148 (1949). (10) Guinier, A., and Tennevin, J., Compt. rend., 226, 1530 (1948). (11)Hobstetter, J. N., Trans. A m . Inst. Mining Met. Engrs., 180, 121 (1949). (12)Lacombe, P.,and Beaujard, L., J . Inst. Metals, 74, 1 (1943). (13) Machlin, E.S., and Cohen, M., J . Metals, 3,746 (1951). (14) Nabarro, F. R. N., Proc. Phys. SOC.(London), 52, 90 (1940); Proc. Roy. SOC.(London),175,519 (1940). (15)Opinsky, A., and Smoluchowski, R., Phys. Rev., 74, 343 (1948). (16) Sirota, N. N., Doklady Akad. N a u k U.S.S.R., 50, 337, 343 (1945). (17) Smoluchowski, R.,“Phase Transformations in Solids,” New York, John Wiley & Sons, 1951. (18)Smoluchowski, R., Physica, 15, 179 (1949). (19) Turnbull, D., Trans. Am. Inst. Mining Met. Engrs., 175, 774 (1948). (20)Volmer, M., “Kinetik der Phaaenbildung,” Dresden, Steinkopff, 1939. RECEIVED for review December 21, 1951.
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