436 CHAPTER 18: SPINODAL AND ORDER-DISORDER TRANSFORMATIONS order parameters, such as for A2 -+ B2* ((qeq = 0) -+ (q = kqeq)), there is no bias to form one ordered B2 variant over another (the two equivalent variants are indicated by B2*; see Fig. 17.4). The two equivalent variants emerge at random locations, and interfaces develop as one impinges upon the other. For conserved order parameters, such as composition, interfaces between phases on phase-diagram tie-lines necessarily appear. In the absence of interfaces, a linear kinetic theory could be developed where the transformation driving force derives from decreases in homogeneous molar free energy as derived in Eqs. 17.28 and 17.29 for the conserved and nonconserved cases. However, at the onset of a continuous phase transition, the system is virtually all interface between new phases or variants. For example, when equivalent variants emerge in adjacent regions during ordering, gradients in the order parameter are generated; these constitute emerging diffuse antiphase boundaries. Neglecting the contribution of these interfaces leads to ill-posed linearized kinetics, as indicated by the negative interdiffusivity in Eq. 18.9. The theory for the free energy of inhomogeneous systems incorporates contribu- tions from interfacial free energy through the diffuse interface method [3]. Interfaces are defined by the locations where order parameters change and can be located by the regions with significant order-parameter gradients. Interfacial energy appears in the diffuse-interface methods because order-parameter gradients contribute extra energy. 18.2.1 Let J(F) represent either a conserved or a nonconserved order parameter, such as CB(F) or q(F). Also, let the field f(F) = f(J(F),VJ(F)) be the free-energy density (energy/volume) at position F. The homogeneous free-energy density, f = f (J, VJ = 0), is the free-energy density in the absence of gradients and is related to molar free energies, F(J) = N,(R) fhom(J), used to construct phase diagrams such as Figs. 17.5 and 17.7. Expanding the free-energy density about its homogeneous value in powers of gradient^,^ Free Energy of an inhomogeneous System f (e, VC) = f(J, 0) + 2 * VJ + VJ . K VJ + . . . (18.10) where (18.11) is a vector evaluated at zero gradient, and K is a tensor property known as the gradient-energy coefficient with components (18.12) The free-energy density should not depend on the choice of coordinate system [i.e., f (J, 05) should not depend on the gradient's direction] and therefore 2 = 0 and K will be a symmetric ten~or.~ Furthermore, if the homogeneous material is isotropic 4There are expansions that contain higher-order spatial derivatives, but the resulting free energy is the same as that derived here [I, 41. 51f the homogeneous material has an inversion center (center of symmetry), 2 is automatically zero. 18.2: DIFFUSE INTERFACE THEORY 437 or cubic, K will be a diagonal tensor with equal components K. The free-energy density will be, to second order, f (E, VE) = fhO"(E) + KVE ' VE = fhO"(E) + KIVEI2 (18.13) The free-energy density is thus approximated as the first two terms in a series expansion in order-parameter gradients: the first term is related to homogeneous molar free energy and the second is proportional to the gradient squared. In the expansion that leads to Eq. 18.13, it is assumed that the free energy varies smoothly from its homogeneous value as the magnitude of the order-parameter gradient increases from zero. This assumption is usually correct, but there may be cases that include a lower-order term proportional to lVSl if the free-energy density has a cusp at zero gradient. Such cusps appear in the interfacial free energy at a faceting orientation; they are also present at small tilt-misorientation grain-boundary energies [5], Models with crystallographic orientation as an order parameter incorporate gradient magnitudes, lV(l, into the inhomogeneous free- energy density [6]. 18.2.2 There are two energetic contributions to interfaces in systems that undergo decom- position and ordering transformations such as illustrated in Figs. 17.5 and 17.7. One is due to the gradient-energy term in Eq. 18.13; this contribution tends to spread the interface region and thereby reduce the gradient as the order parameter changes between its stable values in adjacent phases. A second contribution derives from the increased homogeneous free-energy density associated with the "hump" in Fig. 18.1, and this term tends to narrow the interface region. Thus, systems modeled with Eq. 18.13 contain diffuse interfaces where the order parameter varies smoothly as in Fig. 18.2. Equilibrium order-parameter profiles and energies can be determined by minimizing F, the volume integral of Eq. 18.13 [l, 41. Figure 18.2a shows a planar interface between two equilibrium phases possessing different conserved order parameters corresponding to local free-energy density min- ima in their order parameters as in Fig. 17.7a. Figure 18.2b shows a corresponding profile of the distribution of order between two identical ordered domains possessing different nonconserved order parameters corresponding to local free-energy density Structure and Energy of Diffuse Interfaces Ii I Figure 18.1: which has the maximum value A f,!,;:. Properties of diffuse interfaces expressed in ternis of the function Afho"(<), 438 CHAPTER 18 SPINODAL AND ORDERDISORDER TRANSFORMATIONS Figure 18.2: (a) Composition and (b) order variations across diffuse. planar interfaces. The profiles c(z) and q(z) are continuous. In (a). the grayscale image represents the spatial variation of a conserved variable, and the quantities ca' and ca" are the equilibrium values in the bulk phases at large distances from the interface (see Fig. 17.7). In (b). the drawing below the profile illustrates the spatial variation of a nonconserved variable such as local magnetization in the region around a domain wall. minima. Both kinds of interfaces can coexist, so that the variations of CB and 77 are coupled as in Fig. 18.3. In all cases, the distribution of the order parameter (or order parameters) minimizes the total free energy of the system. F. The coupled- parameter case can be treated as an extension to the theory so that the free energy is a function of both CE and 17. t 'I or c X- C Figure 18.3: antiphase boundary with segregation. Coupled system of order and concentration parameters representing mi 18.2: DIFFUSE INTERFACE THEORY 439 Minimizing 3 = f (<, VE)dV produces equilibrium interface profiles [(q. An equilibrated planar interface is characterized by its excess energy per unit area, y, ,=2l: ,/-,d[= ,/=A[ (18.14) and a characteristic width 6, (18.15) where A fhom is the increase in free-energy density relative to a homogeneous system at its equilibrium values of E (i.e., relative to the common-tangent line) and AfgT is the maximum value of Afhom (indicated in Fig. 18.1). y and 6 can be measured and their values uniquely determine the model parameters, AfzT and K. 18.2.3 Diffusion Potential for Transformation The local diffusion potential for a transformation, @(q, at a time t = to, can be determined from the rate of change of total free energy, 3, with respect to its current order-parameter field, [(F, to). At time t = to, the total free energy is 3(to) = s, [fhom(C(F, to)) + KV5 .V5] dV (18.16) which defines 3 as a functional of [(F‘, to).s If the order parameter is changing with local “velocity” [i.e., such that [(F, t) = c(F, to) + ((7, to)t], the rate of change of F can be summed from all the contributions to f([, V[) due to changes in the order-parameter field and its gradient, Using the relation Eq. 18.17 can be written ( 18.17) (18.18) Applying the divergence theorem to the second integral in Eq. 18.19, (18.20) d3 itO = s, ( :;(‘) - 2KV2[) 4 dV + 2K 1, iV[ d2 where aV is the oriented surface bounding the volume V. The boundary integral on the right-hand side of Eq. 18.20 is negligible. It vanishes identically if i(8V) = 0, ‘Some readers will recognize this development as the calculus of variations [7]. A functional is a function of a function; in this case, F takes the function [(F, to) and maps it to a scalar value that is numerically equal to the total free energy of the system. 440 CHAPTER 18 SPINODAL AND ORDER-DISORDER TRANSFORMATIONS which is the case if [(dV) has fixed boundary values (Dirichlet boundary condi- tions), or if the projections of the gradients onto the boundary vanish (Neumann boundary conditions), If neither Dirichlet or Neumann conditions apply, the bound- ary integral will usually be insignificant compared to the volume integral for large systems (e.g., if the volume-to-surface ratio is greater than any intrinsic length scale). Therefore, if the order parameter changes by a small amount 6[ = (dt, the change in total free energy is the sum of local changes: The quantity (18.2 1) (18.22) is the localized density of free-energy change due to a variation in the order- parameter field, 6[, and is therefore the potential to change [. Equation 18.22 is the starting point for the development of kinetic equations for conserved and nonconserved order-parameter fields. 18.3 EVOLUTION EQUATIONS FOR CONSERVED AND NON-CONSERVED ORDER PARAMETERS 18.3.1 Cahn-Hilliard Equation The Cahn-Hilliard equation applies to conserved order-parameter kinetics. For the binary A-B alloy treated in Section 18.1, the quantity in Eq. 18.22 is the change in homogeneous and gradient energy due to a change of the local concentration CB and is related to flux by (18.23) where the subscript is affixed to the gradient energy coefficient as a reminder that the homogeneous system is expanded in composition and its gradient. Therefore, the accumulation gives a kinetic equation for the concentration CB (T, t) in an A-B alloy: 2KCV2c~]} (18.24) 18.3: EVOLUTION EQUATIONS FOR ORDER PARAMETERS 441 which is the Cahn-Hilliard equation [3]. The Cahn-Hilliard equation is often lin- earized for concentration around the average value of the inherently positive kinetic coefficient M, = (M) = (5/[0(a2Pom )/(ax;)]), defined in Eq. 18.9: 1 2 horn ~ dCB = Ado [ ~V'CB - 2K,V4c~ at (18.25) The first term on the right-hand side in Eq. 18.25 is diffusive. The second term accounts for interfacial-energy penalties from concentration gradients. 18.3.2 Allen-Cahn Equation The Allen-Cahn equation applies to the kinetics of a diffuse-interface model for a nonconserved order parameter-for example, the order-disorder parameter ~(7, t) that characterizes the A2 t B2' phase transformation treated in Section 17.1.2. The increase in local free-energy density, @(F) from Eq. 18.22, does not require any macroscopic flux.7 In a linear model, the local rate of change is proportional to its energy-density decrease, (18.26) where Mq is a positive kinetic coefficient related to the microscopic rearrangement kinetics. According to the Allen-Cahn equation, Eq. 18.26, 77 will be attracted to the local minima of fhorn. Depending on initial variations in 77, a system may seek out multiple minima at a rate controlled by Mq. The second term on the right-hand side in Eq. 18.26 will govern the profile of 77 at the antiphase boundary and will cause interfaces to move toward their centers of curvature [8]. 18.3.3 Numerical models of conserved order-parameter evolution and of nonconserved order-parameter evolution produce simulations that capture many aspects of ob- served microstructural evolution. These equations, as derived from variational prin- ciples, constitute the phase-field method [9]. The phase-field method depends on models for the homogeneous free-energy density for one or more order parameters, kinetic assumptions for each order-parameter field (i.e., conserved order parameters leading to a Cahn-Hilliard kinetic equation), model parameters for the gradient- energy coefficients, subsidiary equations for any other fields such as heat flow, and trustworthy numerical implementation. The phase-field simulations reproduce a wide range of microstructural phenom- ena such as dendrite formation in supercooled fixed-stoichiometry systems [lo], dendrite formation and segregation patterns in constitutionally supercooled alloy systems [ll], elastic interactions between precipitates [12], and polycrystalline so- lidification, impingement, and grain growth [6]. Numerical Simulation and the Phase-Field Method 'This ordering transition occurs at constant composition and is accomplished by microscopic re-arrangement of atoms into two sublattices. 442 CHAPTER 18: SPINODAL AND ORDER-DISORDER TRANSFORMATIONS The simple two-dimensional phase-field simulations in Figs. 18.4 and 18.5 were obtained by numerically solving the Cahn-Hilliard (Eq. 18.25) and the Allen-Cahn equations (Eq. 18.26). Each simulation’s initial conditions consisted of unstable order-parameter values from the “top of the hump” in Fig. 18.1 with a small spatial Figure 18.4: Example of numerical solution for the Cahn-Hilliard equation, Eq. 18.25. demonstrating the kinetics of spinodal decomposition. The system is initially near an unstable concentration, (a), and initially decomposes into two distinct phases with compositions ca (black) and cB (white) with a characteristic length scale, (c) and (d). Subsequent evolution coarsens the length scale while maintaining fixed phase fractions. The effective time interval between images increases from (a)-(f). Figure 18.5: Example of numerical solution for the Allen-Cahn equation, Eq. 18.26, for an order-disorder transition such as A2 + B2*. Initial data are near the disordered state. 7 = 0 (gray) in (a). The system evolves into two types of domains (shown in black and white) with antiphase boundaries (APBs) separating them. The phase fractions are not fixed. The local rate of antiphase boundary migration is proportional to interface curvature [8. 131. The effective time interval between images increases from (a)-(f). 18.4: DECOMPOSITION AND ORDER-DISORDER: INITIAL STAGES 443 variation. In each simulation, the magnitude of the order parameter is indicated by grayscale. Initial medium gray values correspond to the unstable initial conditions. The characteristics of the initial evolution during spinodal decomposition or order-disorder transformations can be predicted by the perturbation analyses pre- sented in the following section. 18.4 INITIAL STAGES OF DECOMPOSITION AND ORDER-DISORDER TRANSFORMATIONS 18.4.1 A homogeneous free-energy density function fhom(cg) that has a phase diagram similar to Fig. 17.7b has the form Cahn-Hilliard: Critical and Kinetic Wavelengths l6f%F [(cg - C")(Cg - CP)] 2 fhoycg) = (CP - ca)4 (18.27) with stable (common-tangent) concentrations located at its minima C" and cP and a maximum of height fkaT at cg = co E (c" + cp)/2. Suppose that an initially uniform solution at CB = co is perturbed with a small one-dimensional concentra- tion wave, cg(z,t) = co + e(t)sinPz, where /3 = 2n/X. Substituting cg(T;t) into Eq. 18.25 and keeping the lowest-order terms in e(t) yields - [lSf&: - 2KcP2(cP - e(t) (18.28) de(t) - MOP2 - dt (cP - c")~ so that (18.29) where the sign of the amplification factor R(P) indicates whether a fluctuation will grow or not [i.e., only composition fluctuations wit.h wavelengths that satisfy I R(P) > 01, or d fkY 2Kc rr x > Xcrit = -(cP - c") - 2 (18.30) will have dcldt > 0 and will grow. Taking the derivative of the amplification factor in Eq. 18.28 with respect to p and setting it equal to zero, the fastest-growing (18.31) The characteristic length scale in the early stage of spinodal decomposition will correspond approximately to this wavelength.8 sReaders may recognize an analogy to the critical and fastest-growing wavelengths derived for surface diffusion and illustrated in Fig. 14.5. Both the surface diffusion equation and the Cahn- Hilliard equation are fourth-order partial-differential equations. The Allen-Cahn equation has analogies to the vapor transport equation. These analogies can be formalized with variational methods [14]. 444 CHAPTER 18: SPINODAL AND ORDER-DISORDER TRANSFORMATIONS 18.4.2 Allen-Cahn: Critical Wavelength A homogeneous free-energy density function f ho"(r]) that has an order-disorder transition similar to Fig. 18.6b has the form (18.32) with local minima at r] = f1 and a local maximum at r] = 0. Suppose that the system is initially uniform with an unstable disordered struc- ture (i.e., r] = 0). For instance, the system may have been quenched from a high- temperature, disordered state. r] = fl represents the two equivalent equilibrium ordered variants. If the system is perturbed a small amount by a one-dimensional perturbation in the z-direction, r](q = b(t) sin(pz). Substituting this ordering per- turbation into Eq. 18.26 and keeping the lowest-order terms in the amplification factor, b(t), (18.33) (18.35) which is about four times larger than the interface width given by Eq. 18.15. Note that the amplification factor is a weakly increasing function of wavelength (asymptotically approaching 4M, fk; at long wavelengths). This predicts that the longest wavelengths should dominate the morphology. However, the probability of finding a long-wavelength perturbation is a decreasing function of wavelength, and this also has an effect on the kinetics and morphology. Figure 18.6: (a) Free energy vs. nonconserved order parameter, q, at point TO,^) where the ordered phttse is stable. (b) Corresponding phase diagram. The ciirve is the locus of order-disorder transition temperatures above which qcq becomes zero. The equilibrium values of the order parameters, A$q, are the values that would be achieved at equilibrium in two equivalent variants lying on different sublattices and separated by an antiphase boundary as in Fig. 18.7~. 18 5: COHERENCY-STRAIN EFFECTS 445 It is instructive to contrast the nature of the evolving early-stage morphologies predicted by Eq. 18.25 (for spinodal decomposition) and Eq. 18.26 (for ordering) and illustrated by the simulations in Figs. 18.4 and 18.5. In spinodal decomposition, the solution to the diffusion equation gives rise to a composition wave of wavelength A,,, given by Eq. 18.31. The decomposed microstructure is a mixture of two phases with different compositions separated by diffuse interphase boundaries (see Fig. 18.7b). In continuous ordering, the solution to the diffusion equation gives rise to a wave of constant composition in which the order parameter varies. The theory does not predict that the order wave will have a “fastest-growing” wavelength-rather, it indicates that the longer the wavelength, the faster the wave should develop. The evolving structure will consist of coexisting antiphase domains, one with positive 71 and one with negative q, separated by diffuse antiphase boundaries (see Fig. 18.7~). The crystal symmetry changes that accompany order-disorder transitions, dis- cussed in Section 17.1.2, give rise to diffraction phenomena that allow the transitions to be studied quantitatively. In particular, the loss of symmetry is accompanied by the appearance of additional Bragg peaks, called superlattice reflections, and their intensities can be used to measure the evolution of order parameters. (a) Random < Diffuse 2 interfaces Spinodal Ordered APBs -L)( Figure 18.7: Interfaces resulting from two types of continnous transformatioii. (a) Initial structure consisting of ratidoirily mixed alloy. (b) After spinodal decomposition. Regions of R-rich and B-lean pliaves separated by diffuse interfaces formed as a result of long-range diffusion. (c) After an ordering transforniatiori. Equivalent ordering variants (domains) separated by two antiphase boundaries (APBs). The APBs result from A and B atomic rearrangement onto different sublattices in each domain. 18.5 COHERENCY-STRAIN EFFECTS The driving force for transformation, @ in Eq. 18.22, was derived from the to- tal Helmholtz free energy, and it was assumed that molar volume is independent [...]... to the coherency stresses, 23 23 using the equations of linear isotropic elasticity, l o e z p €tot zZ 1-v cospz The elastic strains, e:?, and €tot - tot xx - eyy - exy - tot - tot - eyr - E tot - ,, -0 (18.37) required to satisfy Eq 18.37 are 2v cos p z 1-u - - Q c 6c cos p z - €elas - cy,sc- €elas - elas €elas xy - elas - €elas - Eyz - Z X zZ xx - fyy (18.38) =o where u is Poisson's ratio The corresponding... entropy of mixing, s, is ideal; that is, + s = -nk[clnc ( 1- c) ln( 1- c)] cm-3 (number of atoms per unit volume) x fi = [c(l - c)]f”~~e-Q/(NokT) Q = 104 kJ mol-’ K = 1.6 x J cm-l b = -2 .6 x cm2 s-l (at 338 K and 22 at % Zn) f” = -1 .17 kJ cm-3 (at 338 K and 22 at % Zn) n =6 Solution We will need an expression for f ” ( T ) Because f = e - T s , a f / a T = -s and a(f”)/aT = -st’ Also, for ideal entropy of. .. generalized Cahn-Allen models J Stat Phys., 77( 1-2 ):17 3-1 84, 1994 14 W.C Carter, J.E Taylor, and J W Cahn Variational methods for microstructuralevolution theories JOM, 49:3 0-3 6, 1997 15 W.C Carter and W.C Johnson, editors The Selected Works of John W Cahn The Minerals, Metals and Materials Society, Warrendale, PA, 1998 16 J.W Cahn On spinodal decomposition Acta Metall., 9(10):79 5-8 01, 1961 17 W.C Carter. .. 28(8) :114 3115 3, 1980 22 S.M Allen Phase separation of Fe-A1 alloys with Fe3Al order Phil Mag., 36(1):1 8119 2, 1977 23 S.S Brenner, P.P Camus, M.K Miller, and W.A Soffa Phase separation and coarsening in Fe-Cr-Co alloys Acta Metall., 32(8):121 7-1 227, 1984 24 K.B Rundman and J.E Hilliard Early stages of spinodal decomposition in an aluminum-zinc alloy Acta Metall., 15(6) :102 5-1 033, 1967 25 H Baker, editor A S M Handbook:... Dislocations models of grain boundaries In Imperfections in Nearly Perfect Crystals John Wiley & Sons, New York, 1952 6 J.A Warren, R.Kobayashi, A.E Lobovsky, and W.C Carter Extending phase field models of solidification to polycrystalline materials Acta Muter., 51(20):603 5-6 058, 2003 452 CHAPTER 18 SPINODAL AND ORDER-DISORDER TRANSFORMATIONS 7 I.M Gelfand and S.V Fomin Calculus of Variations Prentice-Hall, Englewood... reported an atom-probe field-ion microscope study of decomposition in an Fe-Cr-Co alloy (see Fig 18 .11) [23] The atom probe allows direct compositional analysis of the peaks and valleys of the composition waves It is probably the best tool for verifying a spinodal mechanism in metals, because the growth in amplitude of the composition waves can be studied a a s function of aging time, with near-atomic resolution... center of Fig 1 8 7 ~ Only two sublattices are present in the structure Show that the long-range order parameter for domain 1 is the negative of the long-range order parameter of domain 2 Solution Using Eqs 17.7 and 17.8, 2vl = [Xgll - [Xg], (domain 1) 2q2 = [Xg1 2- [Xg], (domain 2) (18.80) But because the variants are equivalent, (18.81) Combining Eqs 18.80 and 18.81, 2112 = [xg] ,- [XEll 112 = -7 71... 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( @- p a ) 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... Examples of observations of spinodal microstructures include: 0 Kubo and Wayman made TEM observations of an aligned (100) spinodal decomposition product in thin foils of long-range ordered P-brass [20] (Interestingly, bulk material did not decompose, while thin foils with [OOl] foil normals did The difference was attributed to a relaxation of elastic constraint in the thin foil.) - 220 - 0 200 -_ 220... 3.9, the fluxes of components A and (local C-frame) can be written where L A and L g are intrinsic mobilities The flux o f J T = B B in a local crystal frame in the V-frame is then JT + C B ? ? ~ (18.50) where 5 is the velocity of the local C-frame in the V-frame as measured by the motion ; of an embedded inert marker a t the origin of the C-frame Using Eqs 3.15, 3.23, and A.10, = -[ RAJ~RByg] and therefore . possessing different conserved order parameters corresponding to local free-energy density min- ima in their order parameters as in Fig. 17.7a. Figure 18.2b shows a corresponding profile. profile of the distribution of order between two identical ordered domains possessing different nonconserved order parameters corresponding to local free-energy density Structure and Energy of Diffuse. (18.38) €elas - elas - €elas where u is Poisson&apos ;s ratio. The corresponding elastic stresses are given by u = CEelas . where C is the fourth-rank stiffness tensor: uzz = 0