Acta Materialia 51 (2003) 3687–3700 www.actamat-journals.com Solute segregation transition and drag force on grain boundaries N. Ma, S.A. Dregia, Y. Wang ∗ Department of Materials Science and Engineering, The Ohio State University, 2041 College Road, Columbus, OH 43210, USA Received 3 February 2003; received in revised form 14 March 2003; accepted 28 March 2003 Abstract We investigate solute segregation and transition at grain boundaries and the corresponding drag effect on grain boundary migration. A continuum model of grain boundary segregation based on gradient thermodynamics and its discrete counterpart (discrete lattice model) are formulated. The model differs from much previous work because it takes into account several physically distinctive terms, including concentration gradient, spatial variation of gradientenergy coefficient and concentration dependence of solute–grain boundary interactions. Their effects on the equilibrium and steady-state solute concentration profiles across the grain boundary, the segregation transition temperature and the corresponding drag forces are characterized for a prototype planar grain boundary in a regular solution. It is found that omission of these terms could result in a significant overestimate or underestimate (depending on the boundary velocity) of the enhancement of solute segregation and drag force for systems of a positive mixing energy. Without considering these terms, much higher transition temperatures are predicted and the critical point is displaced towards much higher bulk solute concentration and temperature. The model predicts a sharp transition of grain boundary mobility as a function of temperature, which is related to the sharp transition of solute concentration of grain boundary as a function of temperature. The transition temperatures obtained during heating and cooling are different from each other, leading to a hysteresis loop in both the concentration–temperature plot and the mobility–temperature plot. These predictions agree well with experimental observations.  2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Segregation; Solute drag; Grain boundary; Regular solution; Wetting transition 1. Introduction Most classical theories of interface migration are based on systems with isotropic and uniform boundary properties [1–3]. However, in a typical experimental microstructure one encounters a ∗ Corresponding author. E-mail address: wang.363@osu.edu (Y. Wang). population of grain boundaries where the thermodynamic and kinetic properties vary from one boundary to another [3,4]. In recent simulations of grain growth with anisotropic boundary properties based on the Phase Field [5–7] and Monte Carlo [7–9] methods, it was shown that anisotropy of boundary energy and mobility can have a profound effect on the morphology and kinetics of the microstructural evolution. In addition to crystallographic anisotropy, grain 1359-6454/03/$30.00  2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00184-8 3688 N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 boundaries in practical materials may exhibit different properties because of segregating defects such as dissolved impurities, second-phase particles, or inter-granular wetting films of second phases. For the case of solute atoms in metallic alloys, Aust and Rutter [10] showed that the rate of grain boundary migration could be reduced dramatically even by small average concentrations, but the effect was much less pronounced for certain high-angle boundaries with special structures. Subsequent experiments on a variety of bicrystalline and polycrystalline systems revealed further characteristics of solute drag, including its sensitivity to boundary speed and the dependence of boundary mobility on temperature and composition [4,11]. In heating-cooling experiments on doped Al bicrystals [12], for example, the boundary mobility exhibited a transition with a hysteresis that could not be rationalized on the basis of classical solutedrag models. In this paper we develop a theoretical model of solute segregation and solute drag at grain boundaries and investigate segregation profile, segregation transition, drag force and the corresponding effects on boundary migration. The present treatment follows the same formalism as Cahn’s solute-drag theory [13] but applies a more robust thermodynamic solution model. The original theories of solute drag are founded upon a simplified model of segregation in dilute, ideal solutions [13–16] under the influence of a potential well centered on the grain boundary. Here, to provide a basis for our model, we outline Cahn’s treatment, starting with the assumed form of solute chemical potential mB(x,c) ⫽ kTlnc(x) ⫹ EB(x) ⫹ const. (1) where x is the distance from the center of the grain boundary, c(x) is the solute atom fraction, and EB(x) is the solute-boundary interaction potential, which may translate with the boundary but is not otherwise altered in shape or amplitude. Thus, the influence of the defective structure of the boundary is represented by EB(x), and as described by Eq. (1), it is analogous to the effect of an external field imposed on an ideal solution. In a binary (A–B) substitutional system, and on the assumption that composition is varied by atomic exchanges, the equilibrium concentration profile obeys the following condition: ⌬m(x) ⫽ ⌬m(⬁) (2) where ⌬m = mB⫺mA is the exchange potential and “⬁” represents values in the bulk, far away from the boundary. Substituting Eq. (1) into (2) yields the equilibrium solute concentration at a grain boundary as a function of bulk concentration and temperature 冋 册 E(x) c⬁ c(x) exp ⫺ ⫽ 1⫺c(x) 1⫺c⬁ kT (3) and in this case E(x) ⫽ EB(x)⫺EA(x). (4) The segregation isotherm in the form of Eq. (3) is general, following directly from the conditions of chemical equilibrium. Cahn’s segregation profile can be obtained from Eq. (3) by assuming a dilute concentration throughout the ideal solution, i.e., c(x)≪1. The robustness of the segregation isotherm depends on the complexity of the chemical potential formulation. Equation (3) is similar to the McLean isotherm [17], but in this case the segregation is allowed to extend over the range of E(x), not localized to a single mathematical plane. Furthermore, if site exclusion is allowed, the quantity, [1⫺c(x)], in Eq. (3) is to be replaced by [c∗⫺ c(x)], where c∗ is the fraction of grain boundary sites available for segregated atoms at saturation. Using the chemical potential of Eq. (1) in a diffusion analysis, Cahn derived the steady-state composition profile across a boundary migrating with a constant speed. He showed that when E(x) is an even function of x, the drag force is associated with the asymmetry of the composition profile across the moving boundary and is insensitive to the sign of E(x). For certain combinations of solute concentration and temperature, the force–velocity relation becomes a multi-valued function, where two boundary velocities are possible at a given driving force, suggesting a jerky motion. The same phenomenon was also predicted by the analyses of Lücke and Detert [14] and Lücke and Stüwe [15]. The early ideal-solution models are convenient for illustrating basic characteristics of solute drag, N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 but more sophisticated models are needed for comparing theory with experiment. Hillert and Sundman (H–S) [18] and, more recently, Mendelev and Srolovitz (M–S) [19] developed more advanced models for solute drag by incorporating different elements of regular solution theory to account for atom–atom and atom–boundary interactions. The H–S model was applied to evaluate segregation and drag force over the entire composition range of a binary alloy, showing a non-monotonic variation of drag force with bulk concentration. The M–S model was applied to investigate the effect of the sign of E(x) on the segregation. It was shown that for attractive segregation, a positive mixing energy enhances the solute drag and a negative one reduces it. Such an effect becomes neglegible when E(x) is positive. Thus, in contrast to the prediction of Cahn’s ideal solution model where the drag force is independent of the sign of E(x), different drag forces were predicted for attractive and repulsive segregation. Even though both the H–S and M–S models are based on regular solutions, their treatments of segregation are significantly different from one another, as will be discussed further in section 2 below. More importantly, neither model considers the effects of steep composition gradients near the boundary. In addition, M–S model overlooks the possible coupling between composition and the solute–boundary interaction potential. When the level of segregation is high, steep composition gradients are present near the boundary. According to continuum (gradient-) thermodynamics of non-uniform systems [20] and its discrete counterparts [21,22], the contributions of concentration gradient to chemical potential must be included in the conditions for equilibrium and in calculating the driving forces for diffusion. In fact, independent to solute drag, there have been significant developments in the thermodynamics of solute segregation at static surfaces and interfaces based on discrete regular solution models [e.g. 23– 27] where all these contributions were accounted for automatically within the approximation of first nearest-neighbor interactions. In this paper, we develop continuum and discrete models for segregation and segregation transition at grain boundaries to obtain the steady-state concentration pro- 3689 files under static and dynamic conditions. The model is also applied to calculate the drag forces, segregation transition temperatures, and the transition of grain boundary mobility as a function of temperature. Using a prototype planar grain boundary in a regular solution, we illustrate the distinct terms that must be considered in grain boundary segregation, including concentration gradient, spatial variation of the gradient-energy coefficient and the coupling between concentration and structure in the solute–boundary interaction potential. 2. Segregation model In general, the chemical potential of solute in a chemically uniform but structurally non-uniform (incoherent or defective) system can be expressed as: m̄B ⫽ m̄0B(x) ⫹ kTlnc ⫹ m̄xs B (x,c) 0 B (5) xs B where m̄ is the standard-state value and m̄ is the part in excess of the contribution from configurational entropy of mixing. The variation of the standard-state value with position can be expressed as: m̄0B(x) ⫽ m̄0B(⬁) ⫹ EB(x) (6) and, correspondingly, the “excess” chemical potential is expressed as xs m̄xs B (x,c) ⫽ m̄B (⬁,c) ⫹ jB(x,c) (7) where jB(x,c), as in the H–S model, takes into account the position dependence of the enthalpy of mixing near a defect. The difference in solute chemical potentials near the boundary and in the bulk can be expressed as: m̄B⫺m̄⬁B ⫽ EB(x) ⫹ jB(x,c) ⫹ kTln(c / c⬁) xs ⫹ [m̄xs B (⬁,c)⫺m̄B (⬁,c⬁)]. (8) In gradient thermodynamics of non-uniform systems, the interfacial free energy is approximated by integrating contributions from local composition and from the composition gradient [20]. For a planar grain boundary, we express the interfacial energy g as follows g⫽ 冕冋 ⌬fo(x,c) ⫹ 冉 冊册 ␬ dc 2 dx 2 NV(x)dx (9) 3690 N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 where ⌬fo(x,c) is the free energy change (per atom) upon forming a uniform solution of composition c, at the location x, from constituents in the bulk reservoir of composition c⬁, ␬ is the gradientenergy coefficient, and Nv is the number of atomic sites per unit volume. Both ␬ and Nv are allowed to depend on position (or structure) but assumed to be independent of composition. By definition, ⌬fo(x,c) ⫽ c(m̄B⫺m̄⬁B ) ⫹ (1⫺c)(m̄A⫺m̄⬁A ). ⌬Fseg ⫽ E(x) ⫹ j(x,c) ⫹ ⌬m̄xs(⬁,c)⫺⌬m̄xs(⬁,c⬁) ⫺␬ 冉冊 d2c 1 d(NV␬) dc ⫺ . dx2 Nv dx dx The grain boundary energy in a chemically uniform system is given by: (11) ḡ ⫽ ⌬fs(x,c⬁)dx ⫽ gA ⫹ (gB⫺gA)c⬁ 冕 where 冕 xs A ⫹ (1⫺c)[m̄ (⬁,c)⫺m̄ (⬁,c⬁)] 冋 c 1⫺c ⫹ kT cln ⫹ (1⫺c)ln c⬁ 1⫺c⬁ 册 (12a) is the conventional chemical contribution [20] that arises wherever c ⫽ c⬁, and the remainder takes the following form ⌬fs(x,c) ⫽ EA(x) ⫹ cE(x) ⫹ jA(x,c) ⫹ cj(x,c) (12b) where j(x,c) = jB(x,c)⫺jA(x,c). Thus, ⌬fs(x,c) incorporates the structural contribution and would not vanish even in a chemically uniform system with c(x) = c⬁. Therefore, at a grain boundary, the structural incoherence drives the system away from composition uniformity. In Cahn’s critical point wetting theory [28], this structural contribution is assumed to be short-ranged, and the integration of ⌬fs(x,c) is replaced by a function depending only on c(0). The equilibrium composition profile minimizes g and obeys the following Euler–Lagrange equation: dc ∂⌬fo 1 d NV␬ ⫽ 0. ⫺ ∂c NVdx dx 冉 冊 (13) Equivalently, the equilibrium condition may be stated as a requirement of uniform exchange potential, analogous to Eq. (2) ⌬m ⫽ ⌬m̄(x,c)⫺␬ ⫽ ⌬m⬁ (16) 冕 ⫹ (1⫺c⬁) NVjAdx ⫹ c⬁ NVjBdx xs ⌬fc(c) ⫽ c[m̄xs B (⬁,c)⫺m̄B (⬁,c⬁)] xs A (15) (10) For convenience, we may also write ⌬f0(x,c) ⫽ ⌬fc(c) ⫹ ⌬fs(x,c) The segregation isotherm obtained from Eq. (14) has the same form as Eq. (3), but the free energy of segregation is given by: 冉冊 d2c 1 d(NV␬) dc ⫺ dx2 NV dx dx (14) where gA and gB are the grain boundary energies in pure A and B, respectively. In a chemically nonuniform system, 冕 冕 冉 冊册 g ⫽ gA ⫹ NVcEdx ⫹ NV[jA ⫹ cj]dx 冕冋 ⫹ NV ⌬fc ⫹ ␬ dc 2 dx (17) 2 dx. Note that the above analysis of segregation and boundary energy is quite general, not relying on any assumptions about a particular solution model. To apply the analysis to a given system, it is necessary to specify a particular model for the excess chemical potential, including structural contributions and the gradient-energy coefficient ␬. Following previous practice, we use a discrete, nearest-neighbor regular solution model to evaluate the requisite parameters. We treat a bicrystal as a stack of homogeneous atomic layers parallel to the grain boundary plane. Within each atomic layer the solute is randomly distributed, and the concentration is allowed to vary from layer to layer. Atoms in the grain boundary layers have a smaller coordination number than atoms in the bulk, but for any atom in a given layer the nearest-neighbors are distributed within the same layer and in the immediately adjacent layers. Thus, the total atomic coordination zi = zoi + zi+ ± + z⫺ is the number of nearest-neighbor i , where zi bonds in the adjacent layers below (⫺) and above N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 (+) the ith layer, and zoi is the in-layer coordination. Note that, owing to structural non-uniformity near + the boundary, z⫺ i and zi are unequal, which distinguishes the present treatment from previous models of interfaces, in which a fully coherent structure was assumed. However, by conservation of bonds between adjacent layers, the following relation holds niz+i ⫽ ni+1z⫺ i+1 (18) + ⫺2z⫺ i e(ci⫺1⫺ci)⫺2z i e(ci + 1⫺ci) in Eqs. (19) and (22) can be rewritten as ⫺ 冋 ⫺ 2e z⫺ i Ni⫺zi+1Ni+1 Ni 2 ⫹ ⌬Eseg ⫽ Ei ⫹ j(i,ci) ⫹ 2⍀(c⬁⫺ci) i and ⫺ 2z e(ci⫺1⫺ci)⫺2z e(ci+1⫺ci) where Ei ⫽ (zi⫺z⬁) eBB⫺eAA , 2 (20) j(i,c) ⫽ (zi⫺z⬁)e(1⫺2ci), ⍀=z⬁e is the regular solution parameter, e = eAB–1 / 2(eAA + eBB) and z⬁ is the total coordination number of atoms in the bulk. Ei and ji correspond to the E(x) and j(x,c) terms in Eq. (15), respectively. Using the ideal-mixing entropy employed of the regular solution model, we can express the chemical potential difference between the solute and solvent atoms as follows miB⫺miA ⫽ Ei ⫹ j(i,ci) ⫹ ⍀(1⫺2ci) + ⫺2z⫺ i e(ci⫺1⫺ci)⫺2z i e(ci+1⫺ci) ⫹ kTln (21) ci 1⫺ci Thus, the discrete-model segregation isotherm is given by ci c⬁ ⫽ exp 1⫺ci 1⫺c⬁ 冉 ⫺ + z⫺ i + zi . With the aid of expression 2 (23), the correspondence between continuum and discrete model parameters is summarized as follows where z̄i = ⌬m̄xs(⬁,c) / (⬁,c⬁) ⫽ / (c⬁⫺ci)⫺⌬m̄xs ⬁ 2⍀ + i (22) 冊 + Ei ⫹ j(i,ci) ⫹ 2⍀(c⬁⫺ci)⫺2z⫺ i e(ci⫺1⫺ci)⫺2zi e(ci+1⫺ci) . kT From the criterion of bond conservation, the terms (23) + z⫺ i ⫹ zi (ci+1 ⫹ ci⫺1⫺2ci) ⫺2e 2 ␬(x) ⫽ 2ed20z̄i (19) 册 z+i⫺1Ni⫺1⫺z+i Ni (ci⫺1⫺ci+1) 2 2 with ni denoting the number of atoms per unit area in the ith layer. The segregation energy is calculated by exchanging a solvent atom in the ith layer with a solute atom in the bulk, as implied by Eq. (15). For randomly distributed atoms with nearest-neighbor bond energies eAA, eBB and eAB, we obtain ⫺ i 3691 (24) (25) where d0 is the inter-layer spacing. Thus, the total segregation energy derived from continuum gradient thermodynamics can be viewed as a first approximation of the discrete counterpart with first derivatives approximating discrete differences, which becomes more accurate in the limit of small gradients. In contrast to previous treatments of grain boundary segregation and solute drag in regular solutions, the present treatment introduces selfconsistently several new terms. Not only are cond 2c centration gradients, ␬ 2, included, but owing to dx structural non-uniformity, the coupling between structure and composition is manifested in the prod␬dc duct, , and in j(c,x). Therefore, under the right dx dx assumptions, the present treatment can be reduced to previous results. For example, if we assume e = 0 (i.e., ideal solution), then ⍀, ␬, d␬/dx and j(x,c) all vanish and Eq. (15) reduces to Cahn’s ideal solution model or the McLean isotherm [17]. If we assume ␬ = 0 (d␬ / dx = 0) and j(x,c) = 0, but somehow keep ⍀ finite, then Eq. (15) reduces to the M–S model [19], or the Fowler–Guggenheim model [29]. If we assume ␬ = 0(d␬ / dx = 0), then Eq. (15) reduces to the H–S model [18]. 3692 N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 Below, we will investigate the effects of the new terms on the solute concentration profiles across the grain boundary, the corresponding drag forces, and the segregation transition temperature. 3. Diffusion and migration To simplify the diffusion analyses, we assume NV to be constant from now on. This should not alter the results qualitatively. For a moving boundary, the flux of solute atoms in a reference system moving with the grain boundary at a velocity, v, is given by: J ⫽ ⫺c(1⫺c)[(1⫺c)mA ⫹ cmB] ∂⌬m NV ∂x 冋 册 (26) Pi ⫽ 冕 kTNVv D ⬁ (c⫺c⬁)2 dx ⫺⬁ c(1⫺c) where D is the diffusivity of the impurity atoms. Hillert and Sundman [18] showed that the drag force calculated by Eq. (29) reduces to Cahn’s result under the ideal-solution assumption, but it is also appropriate for use in conjunction with the regular solution. In the current study, Eq. (29) is employed for the calculation of the drag force from the steady-state concentration profile. Under a small driving force (small velocity), boundary migration is dominated by solute drag, and Eq. (29) yields the following discrete form for the dependence of mobility on temperature and composition. M⫽ 冋冉 J ⫽ ⫺c(1⫺c)m ∂ ⌬m̄(x,c)⫺ ⫺␬ d␬dc dx dx (27) 冊 d2c / ∂x]NV⫺cvNV dx2 The evolution of solute concentration profile is obtained by solving the diffusion equation dc NV ⫽ ⫺divJ dt (28) using the finite difference method, with the flux given by Eq. (27). The steady state is defined by uniform flux in the grain boundary reference system. The steady-state concentration profile is obtained by evolving an assumed initial profile. Hillert [30] and Hillert and Sundman [18] provided a general method to evaluate the drag force based on the argument that the drag force derives from the free energy dissipation associated with solute diffusion during boundary migration. At the steady state and for equal atomic mobilities, the drag force (per unit area) may be expressed as 冉冘 D v ⫽ P kTNV ⫺cvNV where mB and mA are the atomic mobilities of solute and solvent atoms, respectively. To isolate the effects of segregation thermodynamics, we assume equal atomic mobilities. Thus, substituting Eq. (14) into Eq. (26), we obtain: (29) i (ci⫺c⬁)2 d ci(1⫺ci) 0 冊 ⫺1 (30) 4. Simulation results We apply the model to study segregation and segregation transition at both stationary and moving boundaries. The regular solution constant is chosen to be ⍀ = z ⬁e = 0.3 eV. Two independent functions are employed to describe the atomic coordination profile across the grain boundary, which, without loss of generality, are assumed to have the following Gaussian forms, zi ⫽ z⬁⫺2.0exp[⫺i2] (31a) ⫺ 2 z⫺ i ⫽ z⬁ ⫺1.0exp[⫺(i⫺0.5) ]. (31b) The bond conservation criterion for constant Nv gives z+i ⫽ z⫺ i+1 (32) and the in-layer coordination is given by + z0i ⫽ zi⫺z⫺ i ⫺z i (33) For illustrative purposes, we assume z⬁ = 12, z0⬁ = 4 and z⬁± = 4, which would be strictly correct for a twist boundary parallel to (001) in an f.c.c. 1 system. We also have chosen (eBB⫺eAA) = 0.1 2 eV. N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 4.1. Stationary boundary Based on the segregation model, the equilibrium solute concentration profile across a stationary boundary is given by Eq. (22) or equivalently by Eq. (3) with ⌬Eseg(x) given by Eq. (15). In this paper we solve Eq. (22) using the natural iteration method to find the equilibrium segregation profile. We started with a homogeneous system of ci = c⬁. Then we calculate the total energy of segregation by Eq. (19) and adjust the composition for each layer according to Eq. (22). This process is iterated till the composition profiles converge to a stationary value. Fig. 1 shows the comparison of equilibrium solute concentration profiles for attractive interaction (E(x) ⬍ 0) obtained from different models. The current, M–S and H–S models all predict an enhancement of solute segregation in the case of positive deviation of the regular solution from ideality. However, such an enhancement is overestimated significantly in both M–S and H–S models. For repulsive interaction, the solute enrichment in the system is so low that the difference among the three models becomes insignificant. For a system with positive mixing energy (e ⬎ 0), phase separation is expected when alloy composition and temperature are within the miscibility gap. If surfaces and interfaces exist, however, segregation transition at these surfaces and inter- 3693 faces have been predicted when the bulk alloy composition and temperature are outside the miscibility gap [26–28,31]. Below, we explore this transition at grain boundaries using the same approach and model system discussed above. A series of calculations were performed upon cooling and heating for a set of different bulk compositions. In the cooling cycle, we start with a homogeneous alloy of uniform solute concentration at a temperature much higher than the bulk miscibility gap temperature. The equilibrium solute concentration profile across the boundary is obtained by the iteration method. Then the same calculation is repeated at a lower temperature, using the equilibrium concentration profile of the previous temperature as the initial condition. The results obtained are shown in Fig. 2(a) by the solid circles, where the relative Gibbs excess of solute, ⌫B, is plotted against temperature. Note that the relative solute excess is defined by NVd0 (c ⫺c ) (34) ⌫B ⫽ 1⫺c⬁ i i ⬁ 冘 which is a meaningful measure of the grain boundary segregation defined through the Gibbs adsorption equation in conjunction with the Gibbs–Duhem relation. A clear transition from low to high segregation is observed for this case at T⫺ = 662.8 K. The same calculation procedures are repeated for a heating process, starting with the last equilibrium concentration profile obtained at the end of the cooling process. The results are given in Fig. 2(a) by the open triangles. A higher transition temperature, T + = 668.6 K, is predicted for the heating process as compared to the cooling process, resulting in a hysteresis loop. This indicates that the transition from low- to high-segregation or vice versa is a first-order phase transition. The transition temperature is obtained precisely by plotting the grain boundary energy in discrete version of Eq. (17): g⫺ gA ⫽ NVd0 冘 {⫺⍀(ci⫺c⬁)2 i Fig. 1. Equilibrium solute concentration profiles across a grain boundary obtained from different models for a system of attractive interaction between solute and grain boundary. The bulk composition is c ⬁ = 0.04 and the temperature is T = 1750 K. ci 1⫺ci ⫹ kT ciln ⫹ (1⫺ci)ln c⬁ 1⫺c⬁ 冋 册 ⫹ ciEi ⫹ (zi⫺z⬁)eci(1⫺ci) ⫹ ␬ (c ⫺c )2 2d20 i+1 i (35) 冎 3694 N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 Fig. 2. (a) Grain boundary segregation as a function of temperature during cooling (solid circles) and heating (open triangles) processes and, (b) grain boundary energy as a function of temperature for a system of bulk composition c ⬁ = 0.002. The segregation transition is indicated by the thick solid vertical line in (a). as a function of temperature for both the cooling and heating processes (Fig. 2(b)). The grain boundary energy increases with increasing temperature (less solute segregation) but with different slopes of the g–T curves for cooling and heating. At the transition temperature, grain boundaries with lowand high-segregations should have the same energy. Therefore, the temperature at which the two g–T curves intersect defines the transition temperature, and in this case, T0 = 665.0 K. By plotting the transition temperatures (marked by small symbols in Fig. 3) as a function of the bulk composition we obtain a stability boundary above the bulk miscibility gap for the segregation transition at grain boundaries. To investigate separd␬dc d2c terms, ately the effects of j(c,x),⫺␬ 2 and ⫺ dx dx dx we calculate the transition temperatures under various combinations of these three terms. The transition temperatures predicted by the H–S and M– S models are similar to each other but differ significantly from that of the current model (Fig. 3). The critical temperatures (defined by the temperature at which the sharp transition between low- and high-segregation ends, or in other words, the hysteresis disappears, and indicated by large symbols in Fig. 3) predicted by the M–S and current models differ from each other even more (~1000 K). The critical temperature predicted by the M–S model coincides with the critical temperature of bulk 3D system given by Fig. 3. Stability diagram for segregation transition at grain boundary. Different symbols represent results obtained under d2c various combinations of three terms: (1) ϕ(c,x); (2) ␬ 2; and dx d␬ dc . The M–S model excludes all three terms while the (3) dx dx H–S model includes only term (1). The current model includes all three terms. For each case, the critical temperature (Tgb c ) is indicated by the larger symbols. T3D c ⫽ ze 2k (36) which is indicated by the long-dashed horizontal line in Fig. 3, while the critical temperature predicted by the current model is closer to the critical temperature of an isolated 2D system, which is given by [26,31] N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 T2D c ⫽ z0e 2k (37) and is indicated by the short-dashed horizontal line in Fig. 3. It is interesting to note that the addition d␬dc terms, both of which are of j(c,x) and ⫺ dx dx related to the number of missing bonds of grain boundary atoms, lowers significantly the critical temperature, but has little effect on the transition d 2c temperatures, while the ⫺␬ 2 term lowers sigdx nificantly both the transition and the critical temperatures. Because of solute segregation, grain boundary energy is in general a function of both temperature and solute content. Such information should be very useful in materials process design. Fig. 4 summarizes the energy of the stationary boundary as a function of temperature and bulk impurity concentration for the model system considered. The results for ideal solution are also presented in Fig. 4 for comparison. At T = ⬁, the effect of segregation vanishes and grain boundary energy 3695 becomes a linear function of c (solid circles in Fig. 4) for ideal solution because all other terms in Eq. (35) vanish except ciEi. For regular solutions, the extra contribution from the parabolic term (zi⫺ z⬁)eci(1⫺ci) makes the g–c isotherm non-linear (solid squares in Fig. 4) even in the absence of segregation. For pure system, grain boundary energy usually decreases as temperature increases. For impure systems or alloys, however, segregation reduces grain boundary energy and as a consequence the grain boundary energy may increase as temperature increases at finite temperatures. More interestingly, the g–c curve becomes singular for regular solutions when the temperature is below the segregation transition temperature. An obvious horn structure is observed below the grain boundary critical temperature on the solvent rich side, which indicates the segregation transition. It is important to be aware of the segregation transition when one analyzes experimental data on g– c relations. On the other hand, careful examinations are necessary in experiments to reveal the singularity in the g–c plot because segregation transition occurs in a very narrow temperature or composition range. 4.2. Moving grain boundary Fig. 4. Relationship between grain boundary energy and bulk concentration under various conditions: (A) ideal solute at infinite temperature; (B) regular solution at infinite temperature; (C) regular solution at a T ⬎ Tc (bulk); (D) regular solution at T ⬍ Tc but T ⬎ Tgb c (critical temperature for grain boundary transition); (E) regular solution at T ⬍ Tgb c . For curve E, open symbols represent boundary energy obtained with decreasing bulk concentration while solid symbols represent values obtained with increasing bulk concentration. The inset illustrates the “horn” structure for curve E. ca and cb correspond to the miscibility gap at two different temperatures. First, we investigate the effect of solute drag at temperatures above the segregation transition temperature. The relationships between the drag forces, the relative Gibbs excess quantity of solutes and the boundary velocity are shown in Fig. 5. It can be seen that the drag force increases with increasing boundary velocity at small velocities and it decreases with increasing velocity at large velocities. The segregation at the grain boundary decreases gradually with increasing velocity. Qualitatively, these predictions are very similar to those predicted by Cahn’s ideal solution model and the M–S and H–S models. Quantitatively, however, the H–S model predicts a higher segregation as well as a higher drag force at both small and high velocities while M–S model predicts higher values at small velocities and lower values at high velocities (as shown in Fig. 6(a)). The comparison of solute concentration profiles of M–S and current 3696 N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 Fig. 5. Variation of drag force (a) and solute excess (b) with grain boundary velocity at a temperature T = 680 K, which is above the segregation transition temperature (T0). The bulk composition is c ⬁ = 0.002. Fig. 6. (a) Comparison of the drag forces predicted by the current model with those obtained from the M–S and H–S models at T = 2200 K which is above the transition temperature. (b) Steady state concentration profiles obtained from the current and M–S models at two different grain boundary velocities. The bulk composition is c ⬁ = 0.1. model under small and high velocities is shown in Fig. 6(b). Now we consider a boundary with an equilibrium solute concentration established at a temperature below the transition temperature where a phase transition has occurred, leading to an equilibrium solute concentration that is much higher than that established at temperatures above the transition temperature. Assuming that the boundary is now moving at a constant velocity at this temperature, we evaluate the steady-state concentration profiles and the drag force. Figure 7 shows the drag force and the relative Gibbs excess quantity of solute atoms of the boundary as functions of boundary velocity. An interesting observation in these simulations is that there exist two “breakaway” transitions. Similar to the previous case, the drag force increases and solute concentration decreases with increasing boundary velocity at small velocities, but the drag force is much higher because of the high solute concentration at the grain boundary. When the velocity reaches a critical value, a sharp N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 3697 Fig. 7. Variation of drag force (a) and solute excess (b) with grain boundary velocity at T = 660 K which is below the segregation transition temperature but above the bulk miscibility gap. The bulk composition is c ⬁ = 0.002. Solid circles represent values obtained with increasing velocity while open triangles represent values obtained with decreasing velocity. The inset in (a) shows the “low branch” of the drag force-velocity plot. transition from high- to low-segregation takes place, which leads to a similar transition for the drag force as well. Therefore, when the temperature is below the transition temperature, the system follows two different paths for segregation and drag force before and after the transition. For example, it follows the “high branches” when the velocity is low and switches to the “low branches” when the velocity is high. The low branches in the two plots are very similar to those predicted in the previous case when the temperature is above the transition temperature. If we start to decrease the boundary velocity after the transition, the inverse transition occurs at a lower critical velocity, leading to a hysteresis loop in both plots (Fig. 7). Therefore, as far as the grain boundary phase transition is concerned, changing velocity has a similar effect as changing temperature. The first “breakaway” is obviously associated with the grain boundary phase transition. It has a much stronger effect on the drag force and solute segregation at grain boundaries and hence should have a much stronger impact on boundary migration. 4.3. Mobility transition In the study of temperature dependence of grain boundary velocity [12], it was shown that the steady state migration velocity of tilt boundary in aluminum under a constant small driving force might change abruptly with temperature. Most interestingly, a hysteresis loop was observed during the heating–cooling cycle when the bulk concentration is below a critical value. In the literature, this phenomenon was explained as a consequence of the separation of solute atoms from the grain boundary, but this mechanism failed to explain the hysteresis and the relationship between the hysteresis and the bulk concentration. Below, we apply the segregation model developed in this study to analyze this phenomenon. The activation energy of solute atom diffusion is assumed to be 1.0 eV and all the other parameters are kept the same as previous calculations. The resultant plots of grain boundary mobility versus temperature for two alloys of different bulk composition are shown in Fig. 8. As can be readily seen, the model predicts successfully both a hysteresis and a critical alloy composition above which the hysteresis disappears. The result indicates that a phase transition underlies the mobility hysteresis. It is interesting to note that the high- and low-temperature branches have different slopes, indicating different apparent activation enthalpies for boundary migration. This is because the activation enthalpy contains the contribution from the segregation energy, which has a larger value at lower temperatures. 3698 N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 d␬dc , and the contribution from concentration dx dx dependence of the solute–grain boundary interacd2c tions, j(c,x). The addition of the ⫺␬ 2 term dx affects significantly the extent of the segregation, reducing the amplitude and increasing the width of the concentration peak across the grain boundary. This is not surprising because the gradient-energy term in the free energy is non-local and always suppresses any concentration inhomogeneity. As a result, it lowers the transition temperature and critical temperature and reduces the drag force. The d␬dc term is positive, contribution from the ⫺ dx dx which reduces the solute segregation. The addition of j(c,x) has no effect on the transition temperatures but lowers significantly the critical temperature. Physically, a system with j(c, x) only is equivalent to a system with ⍀ as a variable. Following the analysis of Wynblatt and Liu [26], it is easy to obtain the transition temperature and critical temperature of the ith layer: ⫺ Fig. 8. Prediction of grain boundary mobility transition as a function of temperature for different solute contents: c1⬁ = 0.005, c2⬁ = 0.001. The grain boundary mobility is normalized by its value at 1000K. The hysteresis loop disappears at higher solute concentration. Solid symbols are for the cooling process while open symbols are for the heating process. 5. Discussion Gradient thermodynamics and its microscopic counterpart have been used extensively to analyze segregation transition at surfaces and interfaces [26–28,31]. In this study, we employ a similar approach to study grain boundaries. In the framework of regular solution approximation, the model predicts several interesting phenomena, including the hysteresis for grain boundary segregation transition with respect to both temperature and velocity and two “breakaways” of solute atoms from a moving grain boundary. Similar behavior is observed for impurity segregation at dislocations [32]. In all the numerical calculations, the atomic volume is assumed to be constant. Such an assumption may not be valid, in particular, for a random large angle grain boundary. However, it should not alter the qualitative nature of the results. In contrast to previous grain boundary segregation models, three physically distinctive terms are introduced in the segregation energy in this study. They are the contribution from the concend2c tration gradient, ⫺␬ 2, the contribution from spadx tial variation of the gradient-energy coefficient, Ti0 ⫽ Ei⫺⍀⬁(1⫺2c⬁) , kln(c⬁ / 1⫺c⬁) ⍀i T ⫽ 2k (38) i c where ⍀i and ⍀⬁ are the regular solute constant at ith layer and bulk, respectively. These equations clearly show that the transition temperature, T0, is not sensitive to the structure change but the critical temperature, Tc, does. In his study of critical point wetting, Cahn [28] analyzed the phase transition behavior at a free surface of a binary alloy using gradient thermodynamics and variational analysis. In the grain boundary segregation model developed in this study, if we assume the coordination deficit, (zi⫺z⬁), is localized to a single atomic layer at x = 0, then our discrete model predicts quantitatively identical results as Cahn’s when the temperature is close to the bulk critical temperature, where the gradient of the concentration profile is small. At temperatures far below the critical point, the continuum variational analysis becomes invalid and fails to pre- N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 dict the phase transitions that are predicted by the discrete model. Note that the current model on segregation transition of a static boundary is similar to the surface segregation transition models of Wynblatt and coworkers [26,27]. The major difference is the number of layers considered in the formulation of the structure part (e.g z(x)) of the segregation energy. Furthermore, in the current study we have derived the connection between the continuous and discrete models and applied the models to moving boundaries for the drag effect and grain boundary mobility transition. It is interesting to note that the current model predicts a lower segregation and a smaller drag force at low velocity but a higher segregation and greater drag force at high velocity as compared to the M–S model. This is because the three extra terms introduced in the current model have different dependence on velocity and their interplay determines the overall behavior of the system. d2c According to the above analysis, both ⫺␬ 2 and dx d␬dc reduce the segregation. The contribution ⫺ dx dx from j(c,x) depends on the solute concentration: it enhances the segregation when solute concentration at the grain boundary, cgb ⬍ 0.5 and reduces it when cgb ⬎ 0.5. When the velocity of the grain d2c boundary is small, the contribution from ⫺␬ 2 dx d␬dc and ⫺ dominates. However, the absolute dx dx values of these two terms decrease monotonically with decreasing solute concentration in the grain boundary region but that of j(c,x) increases. At high enough velocity, j(c,x) becomes dominant, resulting in a stronger segregation and greater drag force. Note that only a single flat grain boundary is considered in the current paper. In polycrystalline materials, migration velocity of grain boundaries will depend on local curvatures. The segregation energy as well as grain boundary energy and mobility could be functions of misorientation and inclination of grain boundaries, as well as microstructure. Using the segregation model developed 3699 in this study, one could study directly the effects of segregation and phase transition at arbitrary grain boundaries on grain growth kinetics and microstructural evolution using the phase field method [33]. Corresponding work is underway. 6. Summary A self-consistent continuum model of grain boundary segregation and segregation transition is developed based on gradient thermodynamics and its relations to the discrete lattice model is derived. The model allows for an identification of several new physically distinctive terms that have been ignored in previous solute-drag models including concentration gradient, spatial variation of the gradient-energy coefficient and concentration dependence of solute–grain boundary interactions. The application of the model to a prototype planar grain boundary in a regular solution under various conditions provides considerable insight into the contributions of these terms to the total segregation free energy, the equilibrium segregation profile, the segregation transition and the corresponding drag force on grain boundary migration. Consideration of the contribution from the gradient-energy is critical in describing the behavior of solute segregation and drag force at grain boundaries. For example, its omission could result in a significant overestimate or underestimate (depending on the boundary velocity) of the enhancement of solute segregation and the corresponding drag force, and lead to much higher segregation transition temperature. The concentration dependence of the interaction energy between solute and grain boundary has little effect on the segregation transition temperature but lowers significantly the temperature and alloy composition of the critical point. The segregation transition (transition from lowto high-segregation) takes place with changing temperature, bulk composition or grain boundary velocity. The transition is first-order with a hysteresis. The segregation transition with respect to temperature is responsible for the observed sharp transition of grain boundary mobility with temperature and its dependence on bulk composition observed experimentally. 3700 N. Ma et al. / Acta Materialia 51 (2003) 3687–3700 Recent advances in computer simulations have made it possible to study the evolution of populations of boundaries in complex microstructures under various conditions. The model of grain boundary segregation developed in this study can be adopted easily into phase field modeling of interface migration. This will enable the study of solute segregation in a population of incoherent boundaries and characterization of possible effects on the kinetics of microstructural evolution. Acknowledgements We gratefully acknowledge the invaluable discussion with JW Cahn, DJ Srolovitz and MI Mendelev. This work is supported by the National Science Foundation under grant DMR-9905725 and the US Air Force Research Laboratory, Materials & Manufacturing Directorate under grant F33615-99-2-5215. References [1] Thompson CV. Solid State Physics 2001;55:269. [2] Weaire D, McMurry S. Solid State Physics 1997;50:1. [3] Sutton AP, Balluffi RW. Interfaces in crystalline material. New York: Oxford University Press. Inc., 1995. [4] Gottstein G, Shvindlerman LS. Grain boundary migration in metal. New York: CRC Press LLC, 1999. [5] Kazaryan A, Wang Y, Dregia SA, Patton BR. Phys Rev B 2000;61:14275. [6] Kazaryan A, Wang Y, Dregia SA, Patton BR. Acta Mater 2002;50:2491. [7] Upmanyn M, Hassold GN, Kazaryan A, Holm EA, Wang Y, Patton B, Srolovitz DJ. Interface Sci 2002;10:201. [8] Holm EA, Hassold GN, Miodownik MA. Acta Mater 2001;49:2981. [9] Rollett AD, Srolovitz DJ, Anderson MP. 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