466 CHAPTER 19: NUCLEATION Therefore, putting these relationships into Eq. 19.12 yields The lower limit of integration on the right-hand side can be replaced by oo without significant error, and carrying out the integration, Z,\i 3nN:kT (19.16) (1 9.17) ( 19.18) Equation 19.17 may be interpreted in a simple way. If the equilibrium concen- tration of critical clusters of size N, were present, and if every critical cluster that grew beyond size N, continued to grow without decaying back to a smaller size, the nucleation rate would be equal to J = PcN exp[-Ag,/(kT)]. However, the actual concentration of clusters of size N, is smaller than the equilibrium concentration, and many supercritical clusters decay back to smaller sizes. The actual nucleation rate is therefore smaller and is given by Eq. 19.17, where the first term (2) corrects for these effects. This dimensionless term is often called the Zeldovich factor and has a magnitude typically near 10-l. Non-Steady-State Nucleation: The Incubation Time. Although in principle, non- steady-state nucleation in single-component systems can be analyzed by solving the time-dependent nucleation equation (Eq. 19.10) under appropriate initial and boundary conditions, no exact solutions employing this approach have been ob- tained. Instead, various approximate solution have been derived, several of which have been reviewed by Christian [3]. Of particular interest is the incubation time described in Fig. 19.1. During this period, clusters will grow from some initial distribution, usually essentially free of nuclei, to a final steady-state distribution as illustrated in Fig. 19.5. Approximate solutions of the time-dependent nucleation equation discussed by Christian indicate that the time-dependent nucleation rate in Region I for a single- component system may be approximated by J(t) M Je-t/‘ ( 19.19) where J is the final quasi-steady-state rate and T is the incubation time [3]. As- suming that this is the case, a reasonably good estimate for the magnitude of T may be obtained using a physical argument introduced by Russell [4, 51. Here it is argued that the curve of AGN vs. N is essentially flat in the vicinity of N = Nc, as illustrated in Fig. 19.6, and that there is a range of cluster size, 6, over which the change in A~N is less than kT. Over this range A~N in Eq. 19.10 may be taken as constant, and this equation then becomes (19.20) 19.1: HOMOGENEOUS NUCLEATION 467 tI Figure 19.5: Cluster-size distribution during transient nucleation. which is of the form of the simple mass diffusion equation when only a concentration gradient is present. In this range, clusters will therefore grow (“move”) in cluster space by a random-walk process just as during the mass diffusion of particles. Well away from N,, drift arising from the force field of the potential (i.e., A~N) dom- inates. The transition from predominant random walking to predominant drifting occurs when the potential deviates from flatness by approximately kT on either side of N, (see Fig. 19.6). Because of drift, clusters of size Af < (Nc - 6/2) have a high probability of shrinking, whereas clusters of size n/ > (N, + 6/2) have a high probability of growing to stable nucleus size. The time required to form significant numbers of nuclei (i.e., the incubation time) will therefore be approximately the time required for clusters to random walk the distance 6 in cluster space, provided that the time required to reach size Af < (Nc -6/2) is shorter than the random-walk time. Other calculations indicate that this is indeed the case [3, 61. By analogy with the random walk for simple mass diffusion where, according to the one-dimensional form of Eq. 7.35, (R2) = 2Dt, 62 2PC 7%- (19.21) Figure 19.6: Variation of free energy with size of fluctuation in the nucleation regime. 468 CHAPTER 19: NUCLEATION Furthermore, it is shown in Exercise 19.4 that ( 19.22) and is therefore closely equal to the square of the Zeldovich factor given by Eq. 19.18. The results above are in reasonably good agreement with other estimates of r based on approximate analytic and numerical solutions of Eq. 19.10 [3, 61. 19.1.2 Classical Theory of Nucleation in a Two-Component System without Strain Energy Nuclei in two-component systems need not have the same composition as the parent phase. For example, B-rich p particles may precipitate from an A-rich a-phase matrix. The bulk free-energy change term in Eq. 19.1 is then given by (NIN,) AG,, (where the quantity AGc is shown in Fig. 17.6) rather than N(@ - pa). The rate of nucleation of the p phase can be determined by using a two-flux analysis where B atoms are added to a cluster by a two-step process consisting of a jump of a B atom onto the cluster from a nearest-neighbor matrix site followed by a replacement jump in the matrix in which a second B atom farther out in the matrix jumps into the site just evacuated by the first B atom [6]. The analysis for the steady-state nucleation rate is similar to that described previously, and the resulting expression for the rate is similar to Eq. 19.17. However, the p, frequency is replaced by an effective frequency that reduces to the smaller of either the frequency of the matrix+cluster jumping or the matrix+matrix replacement jumping. (Note that the controlling rate is always the slower rate in a two-step process.) The concentration of B atoms in the vicinity of the nucleus is expected to be close to its average concentration in the matrix. Further details are given by Russell [6]. 19.1.3 Effect of Elastic Strain Energy When clusters form in solids, an elastic-misfit strain energy is generally present because of volume and/or shape incompatibilities between the cluster and the ma- trix. This energy must be added to the bulk chemical free energy in the expression for AGN. Since the strain-energy term is always positive, it acts, along with the interfacial energy term, as a barrier to the nucleation. The magnitude of the elastic- energy term generally depends upon factors such as the cluster shape, the mismatch between the cluster and the matrix (see below), and whether the interface between the matrix and cluster is coherent, semicoherent, or incoherent, as described in Section B.6. The elastic energy of a p cluster in an a matrix can be calculated by carrying out the following four-stage process [7]: Assume the cluster and the matrix to be linearly elastic continua. Cut the cluster (modeled as an elastic inclusion) out of the a matrix, leaving a cavity behind, and relax all stresses in both the inclusion and matrix. The inclusion will then have a generally different shape than the cavity. The homogeneous strain required to transform the cavity shape to the inclusion shape is called the transformation strain, E:. 19 I HOMOGENEOUS NUCLEATION 469 (ii) Apply surface tractions to the inclusion so that it fits back into the cavity. The tractions necessary to accomplish this, -agnj, will be those required to produce the strains -E$. (iii) Insert the inclusion back into the cavity and join the inclusion and matrix along the inclusion/matrix interface in a manner that reproduces the type of interface (i.e., coherent, semicoherent, or incoherent) that existed initially between the p cluster and the matrix. (iv) Remove the applied tractions by applying equal and opposite tractions (i.e., a$nj). This step restores the system to its original state. The tractions ognj that act on the system at the a//? interface will give rise to ‘‘constrained” dis- placements w,C, and thus strains E:~, in both the inclusion and the matrix which can be computed using the strain-displacement relationships of elastic- ity theory. Corresponding stresses atj can then be computed from Hooke’s law. The final strains and stresses are then c,Cj and atj in the matrix and (&,Cj - &$) and (otj - a$) in the particle. Finally, the elastic energy can be calculated from a knowledge of these stresses and strains, since for any elastic body the elastic energy is given by 1/2 sv aij&ij dV. In problems of this type, the quantities that are given are the inclusion shape, the stress-free transformation strains E;, the elastic properties of the two phases, and the degree of coherence between the inclusion and the matrix. When the elastic properties of the inclusion and matrix are the same, the system is said to be elastically homogeneous. Otherwise, it is elastically inhomogeneous. The main difficulty is the calculation of the constrained strains, EC Having these, the calculation of the elastic strain energy in the inclusion and matrix is straightforward. The original reference to such calculations is Eshelby [7]. An overview is given by Christian [3]. Some of the main results are given below for simple shapes such as spheres, discs, and needles which can be derived from a general ellipsoid of revolution by varying the relative lengths of its semiaxes. Only the limiting cases when the alp interfaces are completely coherent or completely incoherent are included. Inclusions with semicoherent interfaces and interfaces where various patches possess different degrees of coherence will exhibit intermediate behavior which is much more com- plicated. Also, results for faceted interfaces are not included. In most cases, the energy of a faceted cluster can reasonably be approximated by using the result for a smoothly shaped cluster whose shape best approximates that of the faceted cluster. Incoherent Clusters. As described in Section B.l, for incoherent interfaces all of the lattice registry characteristic of the reference structure (usually taken as the crystal structure of the matrix in the case of phase transformations) is absent and the interface’s core structure consists of all “bad material.” It is generally assumed that any shear stresses applied across such an interface can then be quickly relaxed by interface sliding (see Section 16.2) and that such an interface can therefore sustain only normal stresses. Material inside an enclosed, truly incoherent inclusion therefore behaves like a fluid under hydrostatic pressure. Nabarro used isotropic elasticity to find the elastic strain energy of an incoherent inclusion as a function of its shape [8]. The transformation strain was taken to be purely. dilational, the particle was assumed incompressible, and the shape was generalized to that of an 470 CHAPTER 19: NUCLEATION ellipsoid of revolution with semiaxes a, a, c so that its shape was given by x2 y2 z2 -+-+-=I a2 a2 c2 (19.23) The shape could therefore be varied between that of a thin disc (c << a) and that of a needle (c >> a). The strain energy (per unit volume of inclusion) is expressed in the form (19.24) where E is the dilational transformation strain and E(c/a) is a dimensionless shape- dependent function that has the form sketched in Fig. 19.7. From this plot, and the dependence of AgE on E(c/a) given in Eq. 19.24, it is apparent that the elastic strain energy of an incoherent particle can be made arbitrarily small if the particle has the form of a thin disc. Of course, such a shape would have very large interfacial area and corresponding interfacial free energy. The preferred shape for the nucleation is therefore that which minimizes the sum of the strain and interfacial energies. AsE = 6pe2 E - (2 SDhere Figure 19.7: of aspect ratio cia. Elastic strain energy function E(c/a) for an incoherent ellipsoid inclusion Coherent Clusters. As described in Section B.6, for coherent interfaces all of the coherence (lattice registry) of the reference lattice is retained. For a + p phase transformations, the reference lattice is generally taken as the a-phase lattice, and the interface will contain an array of coherency dislocations as in Fig. B.8, which accounts for the surrounding stress field. A further example showing a spherical p cluster enclosed by a coherent interface is illustrated in Fig. 19.8~. As long as the a/@ interface remains coherent during the growth of a p cluster, any shear stresses across it will be unrelaxed, since no interface sliding is possible in complete contrast to the case of the incoherent interface discussed above. Eshelby treated systems that are both elastically homogeneous and elastically isotropic [7]. Some results for the ellipsoidal inclusion described by Eq. 19.23 are given below. Case 1. Pure dilational transformation strain with &Zz = E& = &T2. In an elastically homogeneous system, the elastic strain energy per unit vol- ume of the inclusion AgE is independent of inclusion shape and is given by (19.25) 19.1: HOMOGENEOUS NUCLEATION 471 (4 Figure 19.8: Interfacial structure for (a) coherent and (b) semicoherent interfaces between matrix phase Q and particle phase 0. The reference structure is the crystal lattice. Only coherency dislocations are present in (a); in (b), anticoherency dislocations relieve the elastic strain around the particle. where v is Poisson’s ratio and p is the shear mod~lus.~ Another feature of this case is that purely dilational strain centers do not interact elastically, so that the strain fields of preexisting inclusions do not affect the strain energy of new ones that form. This is sometimes referred to as the Bitter-Crum theorem [9]. Finally, there is the degree of accommodation-this refers to the fraction of the total elastic strain energy residing in the matrix. For this example, it can be shown that two-thirds of AgE always resides in the matri~.~ The case of a pure dilational transformation strain in an inhomogeneous elas- tically isotropic system has been treated by Barnett et al. [lo]. For this case, the elastic strain energy does depend on the shape of the inclusion. Results are shown in Fig. 19.9, which shows the ratio of Ag,(inhomo) for the inhomo- geneous problem to Ag,(homo) for the homogeneous case, vs. c/a. It is seen that when the inclusion is stiffer than the matrix, Ag,(inhomo) is a minimum 01234567 I cla + Needle -a 8 Disc SDhere Figure 9.9: Effect of elastic inhomogeneity on elastic strain energy of a coheren - ellipsoidal inclusion of aspect ratio c/a. Stress-free transformationstrains are E:~ = &rv = &Tz. From Barnett et al. [lo]. 31t is noted that Eqs. 19.24 and 19.25 do not agree exactly for the case of a sphere. Equation 19.25 correctly contains the factor (1 + v)/[3(1 - u)] % 2/3, introduced by Eshelby as an image term to make the surface of the matrix traction-free [7]. 4Further discussion of accommodation can be found in Christian’s text, p. 465 [3]. 472 CHAPTER 19: NUCLEATION for a spherical inclusion and, when the inclusion is less stiff than the matrix, it is a minimum for a disc. The elastic energy of inhomogeneous, anisotropic, ellipsoidal inclusions can be studied using Eshelby’s equivalent-inclusion method. Chang and Allen stud- ied coherent ellipsoidal inclusions in cubic crystals and determined energy- minimizing shapes under a variety of conditions, including the presence of applied uniaxial stresses [ 111. Case 2. Unequal dilational strains: €Zx = E~) €TV = E~) and €Tz = E,. Here (19.26) €2 + EP + 2V€,EV -[xc/(32a)][13 (€2 + E;) + 2 (16v - 1) E,E~ -8 (1 + 2~) (E~ + E~) E~ - 8~21 In this case the second and third terms become vanishingly small for a disc as it gets very thin, but the first term, which is independent of shape, remains. In addition, it may be seen that Eq. 19.25 is a special case of Eq. 19.26. Case 3. Pure shear transformation strain: €T3 = = 512; all other E; = 0. Here np2-u 2c ASE = S- 81-v a ( 19.27) Thus, for this case, AgE becomes vanishingly small for a disc as it gets very thin. Case 4. Invariant-plane strain with €T3 = &TI = S/2, ET, = E~, and all other An invariant-plane strain consists of a simple shear on a plane, plus a normal strain perpendicular to the plane of shear (see Section 24.1 and Fig. 24.1). This is a combination of Cases 2 and 3. The expression for Ag, then follows directly from Eqs. 19.26 and 19.27, with the result that AgE is proportional to cia. AgE is therefore minimized for a disc-shaped inclusion lying in the plane of shear. The term invariant-plane strain comes from the fact that the plane of shear in an invariant plane strain is both undistorted and unrotated. Hence the plane of shear is a plane of “exact” matching of the coherent inclusion and the matrix. In martensitic transformations, this matching is met closely on a macroscopic but not a microscopic scale (see Section 24.3). €$, = 0. Additional factors that should often be considered in the treatment of strain energies (although commonly ignored) are: elastic anisotropy, which can be consid- erable, even for cubic crystals; elastic inhomogeneity, which can be treated by the Eshelby equivalent-inclusion method [12] ; nonellipsoidal inclusion shapes; and elas- tic interactions between inclusions that can be significant, producing, for example, alignment of adjacent precipitates along elastically soft directions in anisotropic crystals [13]. 19.1: HOMOGENEOUS NUCLEATION 473 19.1.4 Both the interfacial energy and any strain energy associated with the formation of the critical nucleus act as barriers to homogeneous nucleation. Both energies are generally functions of the nucleus shape, and to find the nucleus of minimum energy, it is necessary to find the shape that minimizes the sum of these energies. As mentioned above, in the simple case where there is no strain energy, such as during solidification, the shape is given by the Wulff shape (described in Section C.3.1). However, in solid/solid transformations such as precipitation, where strain energy is generally present, the problem becomes considerably more complex. The many variables that play a role include the anisotropic interfacial energy, which will be affected by the degree of coherency, and the elastic strain energy variables, which include the transformation strain, the degree of coherency, and the elastic properties (including elastic anisotropy). No analytical treatments of this complex minimization problem therefore exist. However, it is generally anticipated that the interfacial energy will be the dominant factor in most cases. Because the strain energy is proportional to the nucleus volume while the interfacial energy is proportional to the nucleus area, the interfacial energy should tend to dominate at the large surface-to-volume ratio characteristic of the small critical nucleus. Both interfacial energy and strain energy have been incorporated in an analy- sis that gives some quantitative insight into the role that strain energy may play in determining the critical nucleus shape [14]. The nucleus is again taken to be ellipsoidal, so that the strain energy can be expressed as a function of c/a, as, for example, in Fig. 19.9. For simplicity, the interfacial energy is assumed to be isotropic. The free energy to form an ellipsoidal cluster may then be written Nucleus Shape of Minimum Energy where AgB is the bulk free-energy change per unit volume in the transformation, AgE is a function of t where t = c/a, and A(<) is a shape factor given by (2t2/4-) tanh-' 4- (C < 1) i (25/Jm) sin-' 4- (t > 1) A(t) = 2 (t = 1) (19.29) The energy of the critical nucleus is now found by minimizing AG with respect to a and 5. The first minimization produces the results and (19.30) (19.31) Equation 19.31 may be divided by the expression AG(1) = 16~~~/[3(Ag~)~], which is the form Eq. 19.31 would assume if the cluster were a sphere (5 = 1) and the strain energy were zero. Therefore, (19.32) 474 CHAPTER 19: NUCLEATION To find the effect of the strain energy on nucleus shape, the ratio AG(<)/AG(l) from Eq. 19.32 is now plotted vs. < for various fixed values of the energy ratio AgE(l)/AgB, where AgE(l) is the strain energy for the spherical nucleus (E = 1). Some results are shown in Fig. 19.10 for a coherent case corresponding to the lowest curve in Fig. 19.9, where the elastic energy decreased as the nucleus became disc-like. The minima in the curves correspond to the critical nuclei of minimum energy, and the critical nuclei remain spherical until the elastic energy is larger than about 85% of the absolute bulk free-energy change. E then decreases and the nucleus becomes progressively more disc-like. Similar results were found for other cases [14]. In general, the nucleus shape will not be strongly affected by the strain energy until lAgEI becomes comparable to IAgBI. But in most cases, AG(<) will be so large that no significant homogeneous nucleation is possible. Therefore, strain energy will not affect the nucleus shape significantly in most actual cases. However, there will be exceptional cases where the interfacial energy is particularly small, as in the case of coherent clusters with close lattice matching, where AG(l), and therefore AG(<), are small enough so that significant nucleation can occur in the presence of strain energies large enough to affect the nucleus shape. Lo.50 , , , , 1 " 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 f- Figure 19.10: Free energy to form ellipsoidal nucleus, Ag((), as a function of the aspect ratio ( = c/a for various fixed values of the ratio -Ag,(l)/AgB. Ag(() is normalized by AG(l), the value AG(<) would assume for a spherical nucleus (6 = 1) in the absence of any strain energy. Age(l) is the strain energy for a spherical nucleus. The elastic energy as a function of ( corresponds to the lowest curve in Fig. 19.9. After Lee et al. [14]. 19.1.5 More Complete Expressions for the Classical Nucleation Rate With the background above, more complete expressions for the classical nucleation rate can be explored. Single-Component System with Isotropic Interfaces and No Strain Energy. This rela- tively simple case could, for example, correspond to the nucleation of a pure solid in a liquid during solidification. For steady-state nucleation, Eq. 19.16 applies with AG, given by Eq. 19.4 and it is necessary only to develop an expression for Pc. In a condensed system, atoms generally must execute a thermally activated jump over a 19.1. HOMOGENEOUS NUCLEATION 475 local energy barrier in order to join the critical nucleus from the matrix. Therefore, ,& = z,Xs v, exp[-GF/(kT)] so that (19.33) Here, zcXg is the number of sites in the matrix from which atoms can jump onto the critical nucleus, vc is the effective vibrational frequency for such a jump, and GT is the free energy of activation for the jump. Two-Component System with Isotropic Interfaces and Strain Energy Present. An ex- ample of this case is the solid-state precipitation of a B-rich P phase in an A-rich a-phase matrix. For steady-state nucleation, Eq. 19.16 again applies. However, for a generalized ellipsoidal nucleus, the expression for AG will have the form of Eq. 19.28. Also, P must be replaced by an effective frequency, as discussed in Section 19.1.2. For nuclei that are coherent with the surrounding crystal, the lattice is continuous across the cr/P interface. The jumps controlling the Pc frequency factor will then be essentially matrix-crystal jumps and Pc will be equal to the product of the number of solute atoms surrounding the nucleus in the matrix, zcXS, and the solute atom jump rate, r, in the a crystal. The jump frequency can reasonably be approximated by r M *Dl/a2 (see Eq. 7.52, where *DI is the solute tracer diffusivity and a is the jump distance). Therefore, (19.34) For an incoherent nucleus, the jump rate across the cluster/matrix interface will be much faster than the lattice jump rate. Therefore, the pc frequency factor is controlled by the lattice-replacement jumping and Eq. 19.34 holds. In many cases, AGN may be affected by the presence of supersaturated lattice vacancies resulting from the rapid cooling necessary to induce the precipitation. Incoherent interfaces are generally efficient sources for vacancies (in contrast to the coherent interfaces considered above), and in cases where €Zx is positive, excess vacancies will annihilate themselves at the cluster/matrix interfaces and therefore eliminate the elastic strain energy that would otherwise have developed [6]. Fur- thermore, the excess vacancies may continue to annihilate beyond this point until the rate of buildup of elastic strain energy due to their annihilation is just equal to the rate at which energy is given up by the vacancy annihilation. In such a case, the excess vacancies provide a driving force aiding the nucleation and AGN takes the form AGN = flN(Ag~ + Agv) + 777fll3 (19.35) where SZ is the atomic volume and Agv is the free-energy change due to the vacancy annihilation. For an elastically homogeneous spherical cluster where the transfor- mation strain in the absence of any vacancy relaxation would be a uniform dilation, eTx, it may be shown (Exercise 19.5) that where E is Young’s modulus. On the other hand, when €rX is negative, Agv will be positive and excess vacancies will hinder the nucleation. [...]... in solids: The induction and steady-state effects Adv Colloid Interface Sci., 13( 3-4 ):20 5-3 18, 1980 7 J.D Eshelby On the determination of the elastic field of an ellipsoidal inclusion, and related problems Proc Roy SOC.A , 241(1226):37 6-3 96, 1957 8 F.R.N Nabarro The influence of elastic strain on the shape of particles segregating in an alloy Proc Phys SOC., 52(1):9 0-1 04, 1940 9 F Bitter On impurities... nucleation kinetics in binary metallic alloys Metall Tk-ans A , 23(7):191 5-1 945, 1992 485 EXERCISES 16 J.W Cahn and J.E Hilliard F'ree energy of a non-uniform system-111 Nucleation in a two-component incompressible fluid J Chem Phys., 31(3):68 8-6 99, 1959 17 H.I Aaronson and J.K Lee The Kinetic Equations of Solid-rSolid Nucleation Theory and Comparisons with Experimental Observations, pages 16 5-2 29 The... ~ ~ T x' / lo-'; z,Xz x 10'; v, x 1 013 s-1 ; N ) ~ 1023 cm-3; exp[-Gy/(kT)] x and J x 1 ~ m - ~ s - ' Therefore, AG, x 76kT and AG, must be no larger than approximately 76kT for observable rates of nucleation to occur The explosive onset of nucleation has made the experimental measurement of nucleation rates difficult, as measurable rates can be obtained only under a very limited range of experimental... volume 2, pages 8 9-1 40, Amsterdam, 1961 Nort h-Holland 13 A.J Ardell and R.B Nicholson On the modulated structure of aged Ni-A1 Acta Metall., 14(10):129 5-1 310, 1966 14 J.K Lee, D.M Barnett, and H.I Aaronson The elastic strain energy of coherent ellipsoidal precipitates in anisotropic crystalline solids Metall Trans A , 8(6):963970, 1977 15 H.I Aaronson and F.K LeGoues An assessment of studies on homogeneous... steady-state nucleation rate will be proportional t o exp[-AG,/(kT)] that at 800 K and X B = 0.3, lo6 = ~ ' e x p ( - 7 9 ) 50 we know (19.61) where the constant C' is equal t o NP.2 in the classical theory for steady-state nucleation We need t o find the critical nucleation barrier necessary t o achieve the nucleation rate of 10" and this will be or o6 -1 = loz1 In 1 0 - l ~= -7 9 exp( -7 9) exp[-AG,/(... h e n o d y - namically possible? (b) Below what temperature does coherent nucleation become t h e n o d y namically possible? (c) Which type of nucleation, coherent or incoherent, do you expect to occur at 510 K? Justify your answer Data -yc = 160 mJ m-2 7' = 800 mJ m-2 AgE= 2.6 x lo9 J m-3 AgB = 8 x lo6 (T - 900K)J m -3 Solution K-' (coherent interface) (incoherent interface) (coherent particle) (driving... interfacial equilibrium The volume of the nucleus is then the volume of the cone of height d plus the volume of the spherical cap of radius R and is given by The area of the cap, A , is given by 2rRh, where h is its height Therefore, A = 2rRh = 2rR2[ 1- COS(CY/~)] (19.90) The free energy t o form a nucleus as in Fig 19.22 is then + 2r~3 AG = -[ 1 - C O S ( C Y / ~ ) ]2rR2[ 1- C O S ( C Y / ~ ) ] ~ ~ ~ (19.91)... nucleus is Arect = bc c c = y l O / v + ylO/sub - yv/sub 2y10/" A94 &a (19.109) T h e two areas are equal when (yll/sub-yv/sub)2 = 2y10/v [ylO/wb -& - p u b + (Jz- l)yV/SUb] (19.110) 19.11 We wish t o prove by means of the Wulff construction (Section C.3.1) that the equilibrium shape of the grain boundary nucleus in Fig 19.12 is indeed composed of two spherical-cap-shaped interfaces The nucleus has cylindrical... book Kinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter Copyright 0 2005 John Wiley & Sons, Inc 501 502 20.1 CHAPTER 20: GROWTH OF PHASES IN CONCENTRATION AND THERMAL FIELDS G R O W T H OF PLANAR LAYERS We begin by analyzing the growth of planar layers when the growth rate is controlled by heat conduction, mass diffusion, or both simultaneously Growth under interface source-limited... The Minerals, Metals and Materials Society 2 D.T Wu Nucleation theory Solid State Phys., 50:3 7-1 87, 1997 3 J.W Christian The Theory of Transformations in Metals and Alloys Pergamon Press, Oxford, 1975 4 K.C Russell Linked flux analysis of nucleation in condensed phases Acta Metall., 16(5):76 1-7 69, 1968 5 K.C Russell Grain boundary nucleation kinetics Acta Metall., 17(8):112 3-1 131 , 1969 6 K.C Russell . lattice registry characteristic of the reference structure (usually taken as the crystal structure of the matrix in the case of phase transformations) is absent and the interface s core structure. side. Interactions with screw dislocations are generally considerably weaker, but can be important for transformation strains with a large shear component. Deter- minations of the various strain. and is therefore fur greater than the number of heterogeneous sites. The mechanism with the faster kinetics dominates. We shall consider two types of heterogeneous nucleation pro- cesses: nucleation