A Diffuse Interface Model of Grain Boundary Faceting. F. Abdeljawad, 1 D. L. Medlin, 2 J. A. Zimmerman, 2 K. Hattar, 1 and S. M.

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1 A Diffuse Interface Model of Grain Boundary Faceting F. Abdeljawad, 1 D. L. Medlin, 2 J. A. Zimmerman, 2 K. Hattar, 1 and S. M. Foiles 1 1) Sandia National Laboratories, Albuquerque, New Mexico 87185, USA 2) Sandia National Laboratories, Livermore, California 94551, USA (Dated: 18 May 2016) Interfaces, free or internal, greatly influence the physical properties and stability of materials microstructures. Of particular interest are the processes that occur due to anisotropic interfacial properties. In the case of grain boundaries (GBs) in metals, several experimental observations revealed that an initially flat GB may facet into hill-and-valley structures with well defined planes and corners/edges connecting them. Herein, we present a diffuse interface model that is capable of accounting for strongly anisotropic GB properties, and capturing the formation of hill-and-valley morphologies. The hallmark of our approach is the ability to independently examine the various factors affecting GB faceting and subsequent facet coarsening. More specifically, our formulation incorporates higher order expansions to account for the excess energy due to facet junctions and their non-local interactions. As a demonstration of the modeling capability, we consider the Σ5 001 tilt GB in body-centered-cubic iron, where faceting along the {210} and {310} planes was experimentally observed. Atomistic calculations were utilized to determine the inclination-dependent GB energy, which was then used as an input in our model. Linear stability analysis and simulation results highlight the role of junction energy and associated non-local interactions on the resulting facet length scales. Broadly speaking, our modeling approach provides a general framework to examine the microstructural stability of polycrystalline systems with highly anisotropic GBs. 1

2 I. INTRODUCTION In pure metals and at a given temperature, the dependence of the interface free energy on the interface geometric degrees of freedom (DOF) sets the strength of anisotropy. For free surfaces, the interface energy generally depends on its inclination, defined by the plane normal. On the other hand, anisotropy of grain boundaries (GBs) generally depends on all GB DOF, i.e., misorientation due to the crystallographic orientations of the abutting crystals and inclination due to GB plane normal. Thermodynamic treatments incorporating interface anisotropy date back to over a century ago 1 3. The classic Wulff construction provides the equilibrium shape of crystals by utilizing the polar plot of the interface energy 3. For strong anisotropy, it may be energetically favorable for the interface to exclude a range of orientations, termed missing orientations, leading to the formation of facets, corners and edges 4 8. Based on the Wulff construction, Herring 9 examined the thermodynamic stability of interfaces against the formation of hill-and-valley structures and provided conditions for interface faceting. More specifically, if a plane of a crystal interface does not coincide in inclination with any of the boundary planes of the equilibrium shape, then there will always be a hill-and-valley structure that has a lower free energy than a flat interface, despite the potential increase in the total surface area incurred by this process. Frank 4 showed a geometrical, yet equivalent approach of describing faceting by examining the so-called Frank diagram, which is the polar plot of the reciprocal of the interface energy, {nγ 1 }, where n is the outward unit normal to the interface and γ is the corresponding interface energy. Nonconvex regions of {nγ 1 } correspond to interface inclinations that are unstable/metastable against the formation of facets with corners/edges connecting them. Cabrera 10 examined interface phase diagrams, where the projected surface energy is a function of inclination defined by two slopes, in three dimensions, and drew an analogy between interface faceting and spinodal decomposition in bulk phases. Within this description, stable, unstable and metastable interface inclinations are defined in terms of their locations with respect to the spinodal lines of the corresponding interface phase diagram. The so-called ξ-capillarity vector, developed by Hoffman and Cahn 5,6, provides a graphical representation of equilibrium shapes in the presence of anisotropic interfaces and their 2

3 propensity to break up into facets. In two dimensions, the ξ-vector is written as ξ = γe r + γ θ e θ, (1) where e r and e θ are the radial and tangential unit vectors, respectively, in the polar coordinate system. Here, γ = γ(θ) and γ θ = γ/ θ, where θ is the angle that the interface normal makes with the reference axis. Graphically, ξ becomes multivalued, or develops ears, when the interface stiffness, γ + γ θθ, becomes negative, where γ θθ = 2 γ/ θ 2. In fact, ears in the ξ-vector correspond to non-convex regions of the Frank diagram. Mullins 11 examined curvature-driven motion of GBs, where the sharp interface kinetic law states that the normal velocity, V n, at any point on the GB is given by V n = M gb P, (2) where M gb is the GB mobility and P is the driving force. Herring 12 extended this treatment and showed that the driving force in anisotropic GBs (in the absence of contributions due to bulk thermodynamics, where the grains on either side of the interface have the same free energy) is written as P = (γ + γ θθ )K, (3) where K is the mean curvature and the classic Gibbs-Thomson condition is recovered when γ θθ = 0, i.e., isotropic systems 13. Combining Eqs. 2 and 3 leads to the general kinetic law for the microstructural evolution of anisotropic GBs, which is written as V n = M gb (γ + γ θθ )K, (4) where in general M gb = M gb (θ). One can arrive at a linearized form of Eq. 4 by considering the schematic shown in Fig. 1(a), which shows a gently varying GB profile separating two crystals. With no loss of generality, invariance under translation and the small slope assumption (i.e., y/ x << 1) for the GB profile are assumed and the linearization of the spatial gradients is done about an inclination θ o. The spatio-temporal evolution of the GB is then described via the height function y = y(x, t) and the sharp interface kinetic law is then written as 8,11 y t = M gb(γ + γ θθ ) o [ 1 + ( y/ x) 2 ] 2 y x M 2 y gb(γ + γ 2 θθ ) o x, (5) 2 where (γ + γ θθ ) o denotes evaluation at the inclination given by θ o. For strongly anisotropic interfaces, the interface stiffness, γ + γ θθ, can be negative, which according to Eq. 5 leads to a locally backward parabolic equation. The aforementioned thermodynamic conditions for interface faceting correspond to inclinations with negative interface stiffness. In such cases, it is favorable for an interface to exclude such inclinations corresponding to high interface 3

4 energies, leading to the formation of facets and corners/edges connecting them. The problem of interface evolution in highly anisotropic systems has been the subject of intense research efforts, where several forms for the regularization of the inherent governing equations have been proposed 8, Motivated by the work of Angenent and Gurtin 14, Di Carlo et al. 8 proposed a regularized sharp interface kinetic law, where a higher order term, dependent on the local curvature and its spatial derivatives, was introduced. In problems involving crystal surfaces, Stewart and Goldenfeld 15 and later Liu and Metiu 16 derived sharp interface laws to examine thermal faceting in highly anisotropic crystals, where surface diffusion and evaporation are the dominant transport mechanisms. In their work, the regularization arises from a higher order expansion in the surface energy density, which is related to the free energy to form a corner on the crystal surface. Within the context of diffuse interfaces, Taylor and Cahn 17 provided a mathematical description for the treatment of non-differentiable and/or non-convex gradient energy terms. Wise et al. 19 introduced a higher order term into a Cahn-Hilliard-like phase field model 20 to examine the evolution of highly anisotropic thin films and the formation of quantum dots. Wheeler 7 utilized a regularization term similar to the one in Ref. 19 to investigate solidification in the presence of highly anisotropic interface energies. Motivated by the work of Rätz et al. 21, Torabi et al. 22 utilized a regularization term, different from the one in Refs. 7,19, in a phase field model for the evolution, through surface diffusion, of highly anisotropic crystals. Others proposed convexifying the anisotropic energy profile, thus avoiding regions with negative interface stiffness 23,24 ; a framework useful in phase transformation problems but that suppresses the nucleation of facets when an existing interface has an orientation with negative stiffness 22. Several theoretical and computational studies examined thermal faceting in GBs. Cahn 25 examined GB phase change mechanisms including faceting, where several analogies to bulk phase diagrams were drawn and the role of symmetry was examined. Brokman and Marchenko 26 proposed a Landau theory of GB phase transitions, which highlighted the role of GB facet junctions in interface reconstruction processes. Motivated by the treatment outlined in Ref. 27 and starting with a faceted GB geometry, Hamilton et al. 28 utilized a continuum elasticity treatment to examine the relevant energetics of a faceted twin GB in aluminum and concluded that the interface stress is too small to stabilize a configuration with fine-scale facets. A simple analytical model for the faceting of phase boundaries 4

5 y b = b x F (a) (b) FIG. 1. (Color online). (a) A GB between two adjoining grains, where the GB profile is defined via the curve y = y(x, t). (b) A schematic representation of a faceted GB, where the discontinuity in the interface stress, σ, leads to line forces F at facet junctions. Also, dislocation-like defects at hills and valleys of facet junctions are assumed to have Burgers vectors, b, that are equal in magnitude and opposite in direction. between immiscible constituents has been developed 29, which relies on calculating interface energies and torques at discrete numerical mesh points, but is unable to examine the kinetics associated with such a process. While the aforementioned studies provide useful insights on the evolution of highly anisotropic interfaces, they mainly focus on frameworks for the regularization of the inherent differential equations or treat free surfaces and solid-liquid interfaces. For GBs on the other hand and in addition to interface anisotropy, the faceting behavior is influenced by several factors, such as facet junction energetics, and the kinetics of both GB, through the GB mobility parameter, and facet junctions. Several studies of microstructure evolution and interface phase transitions have highlighted the role of line defects, triple and facet junctions 30,31. Most treatments of GB faceting are analytical in nature, which assume simple geometries and/or GB types. While able to capture some of the details associated with GB faceting, these treatments do not provide a general computational framework to examine the dependence of GB faceting and associated length scales on the relative energetics and interaction strengths, and can not be extended to more complex microstructural evolution problems. Herein, we present a diffuse interface model capable of capturing GB faceting instability due to an anisotropic dependence of energy on inclination, and accounting for the excess energy of facet junctions and their non-local interactions. As a demonstration of the modeling capability, we examine through linear stability analysis and numerical simulations the faceting behavior of Σ5 001 tilt GBs in body-centered-cubic 5

6 (BCC) iron, where facets along the {210} and {310} planes were experimentally observed. For this GB, the inclination-dependent interface energy was determined via atomistic calculations and used as an input in our model. This formulation allows us to deeply explore the influence of facet junction energies and interactions on the stability and evolution of faceted GB morphologies. The rest of the manuscript is organized as follows: In Sec. II, we present the theoretical modeling framework used in this work. Parametrization of the model to the Σ5 001 tilt GB in BCC Fe is described in Sec. III. Linear stability analysis and quantitative numerical results are presented and discussed in Sec. IV. Finally, concluding remarks are presented in Sec. V. II. THEORETICAL MODEL As described in the introduction, this study is mainly focused on GB faceting dynamics; a process driven by the boundary plane normal DOF and facet junction energetics. Therefore, in the treatment presented here we fix the GB misorientation and only consider the dependence on inclination (i.e., fixing the crystallographic orientations of adjoining grains and varying the GB plane normal). To examine the behavior of polycrystalline microstructures with multiple GB types, the model presented here could be extended to also incorporate the dependence of GB energy on the misorientation DOF, such as in the treatment presented by Kobayashi et al , which recovers the classic Read-Shockley description in the limit of a small misorientation angle. Figure 1(b) is a schematic illustration of a faceted GB between two adjoining grains with prescribed crystallographic orientations. The sum of surface energies of all individual facets constitutes the total surface energy of this structure. Due to the incompatibility of the translation states between two GB facets meeting at a junction, a dislocation-like defect exists at each of the these junctions 28,35. In general, the orientation of the junction dislocation Burgers vector depends on the nature of the incompatibility between the rigid body translation states of the adjacent facets with each having three DOF. In the configuration depicted in Fig. 1(b), the simplest case is shown, for which translation components normal to the facets are the same for each facet with no components parallel to the facets. Therefore, junction dislocations at hills and valleys have Burgers vectors that are parallel with the average inclination of the interface and are equal in magnitude and opposite in direction. 6

7 The presence of these junction dislocations and their associated energetics is a key difference between faceting in GBs, where the solid material exists at both sides of the interface, and in free surfaces. Moreover, when two GB facets with different inclinations meet to form a facet junction, the discontinuity in the interface stress, σ, leads to a line force F; the sign of which depends on whether the junction is at a hill or valley 27,28. Therefore, two energetic contributions arise due to facet junctions. The first is a local excess energy related to the defect content at these junctions and the second is long range junction-junction interactions due to their defect content/line forces, F. For a pair of facet junctions located at r i and r j, the interaction is inversely proportional to r i r j 15,26,27,36. Now, we turn our attention to our diffuse interface framework, where we start by introducing a dimensionless structural order parameter, ϕ(r, t), that describes the crystallographic orientation of two adjoining grains in a crystalline solid. Next, an additive decomposition is assumed for the total coarse-grained free energy, which incorporates anisotropic GBs, excess energy due to facet junctions and non-local interactions between these junctions, and is written as [ F tot = dr A H f(ϕ) + ϵ2 (θ) ( 2 ϕ 2 + Γ o 2 ϕ ) ] 2 + Γ1 dr 2 ϕ(r) 2 ϕ(r ). (6) r r >R c r r The first term on the right hand side (RHS) of Eq. 6 is the homogeneous bulk energy, where A H is a parameter that sets the energy density scale. f(ϕ) = (ϕ 2 1) 2 is a Landau polynomial with minima at ϕ = ±1 corresponding to two adjoining grains with distinct crystallographic orientations. gradient energy term. The second term on the RHS of Eq. 6 is the phase field In terms of (ϕ x, ϕ y ) = ( ϕ/ x, ϕ/ y), ϵ(θ) = ϵ(ϕ x, ϕ y ) sets the energy of the anisotropic GB, where again θ = tan 1 (ϕ y /ϕ x ) is the angle that the interface normal makes with the reference x-axis. Here, we note that we are examining the dependence of the energy on boundary inclination, while preserving the remaining GB DOF. The local facet junction energy is accounted for via the third term on the RHS of Eq. 6, where the model parameter Γ o controls this energy contribution [cf. Fig. 1(b)]. In fact, this term arises from a higher order expansion of the interface energy in terms of the local mean curvature. Following Herring 12, and Liu and Metiu 16 one expands the interface free energy in terms of the mean curvature, K, as γ = γ o (θ) + γ 2 (θ)k 2 + γ 4 (θ)k 4 +, (7) where γ o is the homogeneous part. γ 2 (θ) and γ 4 (θ) are the inclination-dependent coefficients of the higher order gradients. Within the phase field framework, the mean curvature is 7

8 defined as 13 K = n = ( ) ϕ 2 ϕ, (8) ϕ where n is the normal vector and the long wavelength approximation is assumed in the last step 15. Therefore, the third term on the RHS of Eq. 6 represents a higher order expansion of the interface energy in Eq. 7, where we only retained the leading gradient term and γ 2 is assumed constant 16. The last term on the RHS of Eq. 6 is a non-local symmetric kernel introduced to mimic long range interactions between GB facet junctions, where R c is a cutoff distance and Γ 1 controls the strength of this interaction. Stewart and Goldenfeld 15, in their sharp interface treatment, introduced a non-local kernel in terms of the curvature in order to account for the interface stress, which was found to always depress the faceting instability in crystal surfaces. It is worth mentioning that the non-local kernel employed in this work bears some similarity to the discrete Ewald lattice summation used in calculating long range electrostatic interactions 37. Within this framework, the GB anisotropy sets the preferential planes for the formation of hill-and-valley morphologies, while the model parameter Γ o controls the excess energy associated with junctions and leads to facet coarsening (a decrease in the number of facet junctions accompanied by an increase in the average facet length scale). On the other hand, Γ 1 phenomenologically accounts for non-local junction interactions and, depending on its sign, introduces a repulsive or an attractive effect. The spatio-temporal evolution of ϕ(r, t) follows from the non-conserved Allen-Cahn equation 38 ϕ t = L ( δftot δϕ ), (9) where L is a kinetic parameter related to GB mobility. δf tot /δϕ is the driving force for the microstructural evolution of GBs and is given by δf tot δϕ = A f H ϕ (ϵ2 ϕ) + x (ϵϵ θ y ϕ) y (ϵϵ θ x ϕ) + 2Γ o 4 ϕ + 2Γ 1 Ψ, (10) where ϵ θ = ϵ/ θ, f/ ϕ = 4(ϕ 3 ϕ) and ( x, y ) = ( / x, / y). Here, we introduced the auxiliary field, Ψ, as Ψ(r) = dr 2 ϕ(r ) r r >R c r r. (11) 3 When Γ o = Γ 1 = 0 and following the treatment of Cahn and Hilliard 20, and McFadden 8

9 et al. 39, the orientation dependent GB energy, γ gb, and width, δ gb, are given as γ gb = ϵ(θ) A H, (12) δ gb 11 3 ϵ(θ). (13) 2 AH With the aid of the asymptotic analysis by Wheeler 7 and the treatment of GB junctions outlined in Ref. 27, one can relate the model parameters defining the local junction energy to its defect content as Γ o ϵ αµb 2. Here, elastic isotropy of the abutting grains is assumed, where µ is the shear modulus, b is the magnitude of Burgers vector, and α is a constant that depends on the nature of junction dislocation and its length along the out-of-plane direction. Similarly, the model parameter describing non-local interactions scales as Γ 1 αµb 2 βσ 2 /µ, where σ is the interface stress and β is another material constant. The kinetics of GB faceting are captured via the GB mobility parameter, L, which can also be anisotropic. In threedimensional systems and using spherical coordinates, our modeling framework is capable of capturing faceting by defining the GB energy ϵ = ϵ(φ 1, Φ 2 ) and mobility L = L(Φ 1, Φ 2 ), where Φ 1 and Φ 2 are the polar and azimuthal angles, respectively. In this work, we seek to independently examine the relative role of each of the GB anisotropy, junction energy and non-local interactions on the faceting instability and subsequent facet coarsening. III. A MODEL GB SYSTEM In this section, we apply the theoretical framework to explore its implications for the influence of the junction energetics and interactions on GB faceting behavior. We focus on the specific case of the Σ5 001 tilt GB in BCC Fe, since our experimental observations have shown that this system can develop facets. Figure 2 shows an atomic resolution experimental image, which was collected by high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM), of one such boundary 40. Here, although the boundary has an average inclination that is roughly parallel with the {110} planes in the lower crystal, it has developed a hill-and-valley morphology forming nano-meter scale facets on {310} (blue) and {210} (red) planes. Motivated by this example, we choose this boundary as a model system to investigate GB faceting instability, subsequent facet coarsening, and the associated length scales. It is worth noting that we have conducted experimental and atomistic studies 9

10 to examine the impact of deviations in the GB misorientation from the exact Σ relation on the atomic structure of and defect content at GB facet junctions, and the results will be reported in a separate publication. The modeling framework is developed in general form, where the inclination-dependent energy of any GB can be used as an input parameter. 2 nm FIG. 2. (Color online). HAADF-STEM image of a Σ5 001 tilt GB in BCC Fe. The boundary has formed a hill-and-valley morphology consisting of facets on {310} (blue) and {210} (red) boundary planes. The first step in our analysis is the atomistic calculation of the energies of Σ5 001 tilt GBs in BCC Fe as a function of the inclination angle. GBs are constructed using the method developed by Tschopp and McDowell 41,42, wherein a fully periodic atomic system is created from two half-crystals, each of which is rotated such that a resulting planar boundary between the halves has a misorientation angle of from the [001] tilt axis (characteristic of the Σ5 {310} boundary), and the inclination angle varied over a range of 90. With this prescribed relative orientation, a sequence of relative displacements between the upper and lower half-crystals are used in conjunction with atom deletions and conjugate gradient energy minimizations to identify a low energy GB configuration. The dimensions of the resulting systems vary in order to accommodate an integer number of unit cells necessary to model each specific inclination. During energy minimizations, the dimensions within the GB plane are held fixed, but the one perpendicular to the GB plane is allowed to relax to achieve zero stress in that direction. After the initial construction and relaxation of each bicrystal system, the system is subjected to a simulated annealing process by which it is heated quickly, 0.1 nano-second (ns), from a temperature of 10K up to 85% of the anticipated melt temperature of 1800K, i.e., 1530K. The system is then held at this high temperature for 1 ns, and then slowly cooled back down to 10K over the span of 10 ns. This entire annealing process is also performed at zero pressure, allowing the system to expand or contract to some optimum combination of dimensions. After annealing and another 1 ns equilibration at low temperature, the system is again minimized in energy using a conjugate gradient approach. 10

11 The entire construction-minimization-annealing process was replicated for two Embedded Atom Method-type inter-atomic potentials: the one by Mendelev et al. 43 and the one by Chamati et al. 44 The Mendelev potential is commonly used in the literature, exhibits good physical properties representative of Fe, and has been previously used to examine GBs in Fe 45. The Chamati potential displays similar attributes, and in particular exhibits good qualitative agreement with density functional theory (DFT) calculations for the shape of the Peierls barrier to screw dislocation motion, as recently noted by Hale et al. 46 (a) (b) FIG. 3. (Color online). For the Σ5 001 tilt GB in BCC Fe: (a) Atomistic data using the potential by Mendelev et al. 43, which depict the variation of the GB energy with the inclination angle, along with the closed-form fit using Eq. 14. (b) The GB energy-inclination diagram using the inter-atomic potentials by Mendelev et al. 43 and Chamati et al. 44 Figure 3(a) is a plot of the GB energy as a function of the inclination angle using the potential by Mendelev et al. 43, where depressions, shallow cusps, in the boundary energy are observed at boundary inclinations { π/8, π/8, 3π/8} corresponding to {310}, {210} and {310} planes, respectively. Next, to incorporate this Wulff plot of the GB energy into the phase field model, we fit a closed-form function to the atomistic data. Motivated by the work of Kobayashi on anisotropic interfaces 47 and the fact that these cusps are π/4 apart, we select the following form for the inclination-dependent GB energy γ gb = γ nom [1 + δ cos(8θ)], (14) where γ nom and δ are fitting parameters corresponding to the nominal interface energy and degree of anisotropy, respectively. A least-squares fit to the atomistic data yields γ nom = 1.13 J m 2 and δ = 0.05 and this fit is shown in Fig. 3(a). Other closed-form functions can be used to fit the atomistic data, albeit with more fitting parameters. Recent atomistic studies examined GB energy variations in the five-space of GB macroscopic geometric DOF for a wide range of GB types and in several face- and body-centered cubic metals The form in Eq. 14 is chosen because of the smoothness of the function and the ability to get the location of energy depressions with a minimal number of fitting parameters. It is worth noting that the atomistic data for the GB energy can vary depending on the inter-atomic 11

12 potential used. This is evident in Fig. 3(b), which is a plot of the inclination-dependent interface energy using the potential by Mendelev et al. 43, used in this work, and the one by Chamati et al. 44 Locations of the cusps in the energy are captured by both potentials but the energy values differ by up to 20%. Based on this, the use of a closed-form function that mainly gets both the cusp locations and nominal energy values with the least number of fitting parameters is justified. With the aid of Eq. 14, negative interface stiffness, i.e., γ + γ θθ < 0, corresponds to δ > 1/ The least-squares fit to the atomistic data yields δ = 0.05, which indicates the existence of a range of inclinations with negative interface stiffness, and therefore are unstable against faceting. To demonstrate this graphically, Fig. 4 is a polar plot of the GB energy and the ξ-capillarity vector, Eq. 1, as a function of the inclination angle. As expected from the thermodynamic theory, ears (i.e., multivaluedness) in the ξ-vector correspond to inclinations with negative interface stiffness, and thus unstable/metastable against faceting. Regions with these inclinations will break into hill-and-valley structures corresponding to the neighboring low energy inclinations. FIG. 4. (Color online). Using the fit given by Eq. 14, a polar plot of the GB energy and the corresponding ξ-capillarity vector, Eq. 1. Values of interface stiffness are used to color the GB energy curve, where red (blue) denote positive (negative) stiffness. IV. RESULTS AND DISCUSSION In this section, we examine, through numerical examples, the faceting behavior of the Σ5 001 tilt GB in BCC Fe and the associated facet length scales. Based on the fit to the atomistic data detailed in Sec. III [cf. Fig. 3(a)], the phase field gradient energy coefficient is written as ϵ(θ) = λ [1 + δ cos(8θ)], (15) 12

13 where again λ and δ define the nominal GB energy and degree of anisotropy, respectively, both within the phase field framework. Next, the evolutionary equation for ϕ(r, t), i.e., Eq. 9, is made non-dimensional by introducing dimensionless space and time as r = r/d o, and t = t/τ, respectively, where d o and τ denote the characteristic length (pixel edge length) and time scales, respectively. For convenience, we introduce the non-dimensional quantities à H = A H /E ref, λ 2 = λ 2 /(E ref d 2 o), Γ o = Γ o /(E ref d 4 o), Γ 1 = Γ 1 /(E ref d 2 o), where E ref is a reference energy density. In the case of isotropic systems, δ = 0, the model parameters L and λ are related to GB mobility, M gb, and energy, γ gb, through 51 M gb γ gb = Lλ 2, which with the aid of Eqs. 9, 10 and 12 lead to the following for the characteristic time τ 3 4 λ d o. (16) 2 AH E ref M gb The resulting non-dimensional governing equation for ϕ(r, t) has the same form as its dimensional counterpart, i.e., Eqs. 9, but with the mobility parameter, L, now absorbed into the characteristic time, τ. Therefore, physical quantities are expressed in terms of the reference energy density and characteristic time scale, which are associated with GB energy and mobility, respectively. A quantitative description of GB faceting can be made by obtaining the inclination-dependent GB mobility and junction energies via atomistic calculations. Such detailed studies have been the subject of intense research efforts in recent years 48,52. For the remainder of this paper, we drop the tildes on all parameters for notational convenience. A. Linear Stability We begin our exploration of the model by examining the linear stability of a planar anisotropic boundary that is aligned along the reference y-direction with a plane normal along the x-direction. This GB profile is inclined from the {310} plane by 22.5, which is in close agreement with the experimentally observed GB [cf. Fig. 2], and is characterized by a negative interface stiffness [cf. Fig. 4]. Moreover, we assume a unit thickness in the out-of-plane z-direction, which reduces the system to a twodimensional one. This is equivalent to re-expressing the total coarse-grained energy given by Eq. 6 as one per unit thickness. Given an initially perturbed boundary profile, linear stability analysis provides insights on the range of growing and decaying wave vectors along with the fastest growing ones and their corresponding growth rates. 13

14 Following the sharp interface treatment by Stewart and Goldenfeld 15, we combine Eqs. 9 and 10 in a compact form to give ϕ t = A f H ϕ + ϕ 2 σ 2 σ 2 σ xx + 2ϕ ϕ 2 xy + ϕ yy 2Γ x ϕ x ϕ y ϕ 2 o 4 ϕ 2Γ 1 Ψ, (17) y where ϕ xx = 2 ϕ/ x 2 and similar expressions for ϕ yy and ϕ xy. σ = (1/2)ϵ 2 (θ) ϕ 2 is the anisotropic interfacial term. Ψ is the non-local field given by Eq. 11. Now, we consider a small perturbation of the form ϕ = ˆδ ϕ exp [ik x + ω(k)t], (18) where ˆδ ϕ is a constant. k = [k x k y ], x = [x y] and ω(k) are the two-dimensional wave and position vectors, and the amplification factor, respectively. It is shown in Appendix A that the dispersion relation (to the zeroth order for the non-local term) is written as ( ) ( ) ω(k) = 4A H λ 2 4πΓ 1 (1 + δ)[ 1 + δ k 2 4πΓ 1 λ 2 x δ k (1 + δ)r c λ 2 y 2 (1 + δ)r c ] (19) 2Γ o ( ) + k 2 λ 2 (1 + δ) x + ky 2 2, where positive (negative) values of ω indicate growing (decaying) modes. For the linear stability analysis presented in this section, we set (λ, A H, δ, R c ) = (2.0, 0.25, 0.05, 2.0), which yields nominal GB energy and anisotropy representative of the Σ5 001 tilt GB in BCC Fe. FIG. 5. (Color online). For Γ o = Γ 1 = 0, a plot depicting the neutral line (ω = 0) along with regions of unstable and stable perturbations. Here, we set (λ, A H, δ, R c ) = (2.0, 0.25, 0.05, 2.0). The asymptote of the neutral line is given by k y = (1 + δ)/ 1 63δ k x for δ > 1/ We start the linear stability analysis by examining the neutral line, which delineates regions with unstable (ω > 0) from ones with stable (ω < 0) perturbations and is obtained by setting ω = 0 in Eq. 19. First, when Γ o = Γ 1 = 0, i.e., no energy cost to having facet junctions and their non-local interactions, it is seen from Eq. 19 that for anisotropies with 1 63δ < 0 (i.e., δ 0.016) there exist growing perturbations along the y-direction; a condition that is consistent with negative interface stiffness for an interface energy of the form in Eq. 15. In this case, i.e., δ 0.016, the neutral line is a hyperbola with eccentricity e = 1 + [(1 + δ)/ 1 63δ ] and an asymptote given by k y = (1 + δ)/ 1 63δ k x. For the Σ5 001 tilt GB in BCC Fe, where the fit to atomistic data yielded δ = 0.05, Fig. 5 is a 14

15 plot depicting the neutral line along with regions of unstable and stable perturbations. For the model parameters chosen here, it can be seen that the GB is linearly unstable against any perturbations with wavelengths l y = 2π/k y and k x 1.0. Numerically, Fig. 5 indicates that if the governing equation is solved on a grid, then any perturbation leads to faceting, where the facet length scale would be dependent on the inherent numerical grid used. FIG. 6. (Color online). Dependence of the amplification factor, ω(k), on the junction energy model parameter, Γ o. A plot of the neutral line (ω = 0) for Γ o = (0.01, 0.05, 0.5, 2.5). Note that the shaded regions correspond to unstable perturbations, ω > 0. Here, we set (Γ 1, λ, A H, δ, R c ) = (0.0, 2.0, 0.25, 0.05, 2.0). Next, we examine the dependence of the amplification factor, ω(k), on the junction energy model parameter, Γ o, while setting Γ 1 = 0 (i.e., no junction-junction interactions). Figure 6 depicts plots of the neutral line (ω = 0) for Γ o = (0.01, 0.05, 0.5, 2.5). We note that the shaded regions shown in Fig. 6 correspond to unstable perturbations with ω > 0. It can be seen that for small values of Γ o, there exists a wide range of unstable perturbations, which lead to a faceted GB structure. On the other hand, large values of Γ o limit the unstable perturbations to ones with small wave vectors [cf. Fig. 6]; an indication of a GB profile with large wavelengths for the facets. Note that Γ o controls the excess energy due to facet junctions; therefore, beyond the linear stability regime the faceted GB structure minimizes its energy via facet coarsening. We now turn our attention to the role of Γ 1, which phenomenologically accounts for nonlocal interactions due to both the defect content at junctions and elastic relaxations due to the interface stress. It is worth noting that when elastic relaxations at facet junctions are dominant, Γ 1 is negative, which reflects a decrease in the system s free energy due to these relaxations 15,26,27. Figure 7 depicts plots of the neutral line for Γ 1 = ( 1.0, 0.5, 0.5, 1.0), where we set Γ o = 0.05 in all these cases. Again, the shaded regions in Fig. 7 correspond to unstable perturbations, ω > 0. For a given value of the local junction energy model parameter, Γ o, decreasing Γ 1 leads to a reduced range of unstable perturbations. For large values of Γ 1, the GB plane is linearly unstable against a wide range of initial perturbations. This is an indication that negative values of Γ 1 play a stabilizing role. As can be seen from 15

16 Eq. 19, when Γ 1 = 0.0 perturbations along the y-direction are unstable when the anisotropy parameter δ On the other hand, negative Γ 1 values shift the instability in the anisotropy to: δ > π Γ λ 2 R c, for Γ 1 < 0. (20) Therefore, negative values of Γ 1 depress the instability by increasing the threshold value of the anisotropy parameter, δ, needed to cause unstable perturbations. FIG. 7. (Color online). Dependence of the amplification factor, ω(k), on the junction-junction interaction model parameter, Γ 1. A plot of the neutral line (ω = 0) for Γ 1 = ( 1.0, 0.5, 0.5, 1.0). Note that the shaded regions correspond to unstable perturbations, ω > 0. Here, we set (Γ o, λ, A H, δ, R c ) = (0.05, 2.0, 0.25, 0.05, 2.0). In addition to establishing regions of unstable/stable perturbations, the critical wave vector, (k c x, k c y), corresponding to the maximum growth rate, ω max, which describes how fast an unstable GB facets, is of interest. These are found by setting ω/ k x = ω/ k y = 0. For the GB configuration used in this linear stability study and with the aid of Eq. 19, the wave vectors corresponding to the maximum growth rate are given by kx c = 0, (21a) ky c = 1 4πΓ 1 (1 + δ)(1 63δ)λ 2 R c. 2 Γ o R c (21b) Therefore, given a degree of anisotropy, δ, and facet junction energetics, Γ o and Γ 1, the critical wave vector represents the fastest growing Fourier mode, whose wavelength is 2π/k c y, and is physically related to the distance between facet junctions. Figure 8(a) is a contour plot of k c y as a function of Γ o and Γ 1, where it can be seen that the fastest growing wave vector increases with increasing Γ 1 and decreasing Γ o. Inserting the critical wave vectors given by Eq. 21 in Eq. 19 yields the following for the max growth rate, ω max [ ] ω max = 4A H + 2π2 Γ 2 1 (1 + δ)(1 63δ)λ2 + 8πΓ Γ o Rc λ 2 R c (1 + δ)(1 63δ). (22) 8Γ o R c Figure 8(b) is a contour plot of the max growth rate, ω max, for various values of Γ o and Γ 1. It can be seen that small values of Γ o and large (positive) values of Γ 1 yield unstable modes 16

17 with the fastest growth rates. (a) (b) FIG. 8. (Color online). For various values of Γo and Γ1, a contour plot of the (a) critical wave vector kyc, and (b) corresponding max growth rate, ωmax. In both panels, we set (λ, AH, δ, Rc ) = (2.0, 0.25, 0.05, 2.0). B. Numerical Results In this section, we numerically examine the spatio-temporal evolution of microstruc- tures with unstable GBs against faceting, and its dependence on various model parameters. Namely, we investigate the role of facet junction energy, Γo, and junction-junction interaction, Γ1, model parameters on the resulting facet length scales. To this end, a two-grain slab geometry is utilized where the planar GBs are aligned with the reference y-axis with a plane normal along the x-axis [cf. Fig. 9(a)]. Representative systems are discretized to produce a simulation box of Nx Ny pixels, where the pixel size (edge length) is x = y = 1.0 along the x- and y-directions, respectively. The phase field equation for ϕ, i.e., Eq. 9, is solved using explicit Euler method for the time derivative and central finite differencing with the Moore stencil for the spatial gradients. Periodic boundary conditions along all spatial dimensions for all fields are used. To induce the faceting instability, the phase field, ϕ(r, t = 0), was initially perturbed by adding a noise term, ξ(r), where ξ [ 0.1, 0.1] is a uniform random number. GB mobility can vary by orders of magnitude depending on GB type, temperature and impurity content48. In this study, we assume the GB mobility Mgb m4 J 1 s 1, characteristic length scale do = 1.0 nm and reference energy density Eref = J m 3. These parameters along with the fit to the atomistic data for the Σ5 001 tilt GB in BCC Fe [cf. Fig. 3] yield (λ, AH, δ) = (2.0, 0.25, 0.05) and a characteristic time scale τ 0.25s. Systems used in this study were spatially resolved with Nx = Ny = 120 and a numerical time step of t = was used. We start by simulating the GB microstructure depicted in Fig. 9(a) with various values for the junction energy model parameter, Γo. Figures 9(b)-9(d) are contours of the phase 17

18 (a) (b) (c) (d) FIG. 9. (Color online). The role of junction energy model parameter, Γ o, on the faceting behavior. Contour plots of the phase field, ϕ, depicting the configuration of the two-grain slab geometry in (a) the initial state (t=0), and at a simulation time of τ with Γ o of (b) 0.1, (c) 10.0, and (d) In all panels, black (red) denote grain 1 (2) and we set (Γ 1, λ, A H, δ, R c ) = (0.0, 2.0, 0.25, 0.05, 2.0). field, ϕ, at a simulation time of τ for Γ o values of (b) 0.1, (c) 10.0 and (d) 40.0, where black (ϕ > 0.85) and red (ϕ < 0.85) correspond to bulk grains 1 and 2, respectively. Two observations can be made here. First, the facets in these microstructures are at inclinations corresponding to low energy planes, which according to Fig. 4 are located at ±π/8 from the reference x-axis (i.e., {310} and {210} planes). Second, the average facet length scale increases with Γ o [cf. Figs. 9(b)-9(d)]. This behavior is mainly attributed to the fact that these systems initially minimize their free energy by faceting to the low energy planes, thus eliminating inclinations with negative stiffness. Then, further minimization of the energy occurs via the reduction of the total number of facet junctions, due to their excess energy, which in turn leads to larger facet length scales. This effect can be quantified by counting the number of facet junctions as a function of (simulation) time and calculating the average facet length scale, which in this work is defined as the total length of the faceted boundary divided by the number of junctions. Figure 10(a) displays the temporal evolution of the total number of facet junctions for various values of the junction energy parameter, Γ o. Small values of Γ o yield systems with the largest number of facet junctions. At large values of Γ o, the systems initially facet with a large number of junctions then subsequent facet coarsening leads to a reduction in the number of these junctions. Figure 10(b) on the other hand depicts the temporal evolution of the average facet length scale for various values of Γ o. Given a faceted microstructure, the reduction in the total number of facet junctions [cf. Fig. 10(a)] is accompanied by an increase in the average facet length scale [cf. Fig. 10(b)]. Moreover, except at early times facet coarsening was not observed in systems with small values of Γ o. It is worth noting that no change in the results presented in Fig. 10 was observed when we 18

19 ran the simulations to an order of magnitude longer in time. (a) (b) FIG. 10. (Color online). Temporal evolution of the (a) number of facet junctions, and (b) average facet length scale for various values of the junction energy parameter, Γ o. Finally, starting with the same initial microstructure used in the previous study [cf. Fig. 9(a)], we examine the role of non-local junction-junction interactions via the model parameter, Γ 1, while setting Γ o = 1.0. Here, we note that negative values of Γ 1 reflect reductions in the free energy due to dominant interface stress elastic relaxations. Contours of the phase field, ϕ, at a simulations time of τ are shown in Figs. 11(a)-11(c) for Γ 1 values of (a) -1.0, (b) 0.0, and (c) 1.0. It can be seen that negative values of Γ 1 [cf. Fig. 11(a)] play a stabilizing role, where fine-scale facets are observed, even at late times, compared to systems with positive Γ 1. Given a degree of anisotropy (i.e., fixing the value of δ), a close examination of the dispersion relation, i.e., Eq. 19, reveals that positive values of Γ 1 increase the propensity of the anisotropic GB to facet. This can be seen by examining Fig. 8(b) for the max growth rate, ω max. For a given Γ o value, increasing Γ 1 leads to larger growth rates, thus faster faceting dynamics. This trend is also observed in Fig. 11, where large values of Γ 1 [cf. Fig. 11(c)] lead to faster dynamics compared to small or negative ones [cf. Fig. 11(a)]. Again, the results depicted in Fig. 11 can be quantified by examining the evolution of the number of facet junctions [cf. Fig. 12(a)] and average facet length scale [cf. Fig. 12(b)] for various values of Γ 1. After a short initial transient that is associated with the time needed for the facets to develop, facet coarsening is observed in systems with positive Γ 1, while the ones with negative Γ 1 are characterized by fine-scale facets that are stationary, at least for the simulation times attained in this study. These results indicate that negative Γ 1 plays a stabilizing role, while positive ones increase the faceting dynamics. The results presented in this section indicate that details of the faceted GB microstructure, i.e., number of junctions and average facet length scale, are highly influenced by both facet junction excess energy and non-local junction-junction interactions. 19

20 (a) (b) (c) FIG. 11. (Color online). The role of junction-junction interaction model parameter, Γ 1, on the faceting behavior. Contour plots of the phase field, ϕ, depicting the configuration of the two-grain slab geometry at a simulation time of τ with Γ 1 of (a) -1.0, (b) 0.0, and (c) In all panels, black (red) denote grain 1 (2) and we set (Γ o, λ, A H, δ, R c ) = (1.0, 2.0, 0.25, 0.05, 2.0). (a) (b) FIG. 12. (Color online). Temporal evolution of the (a) number of facet junctions, and (b) average facet length for various values of the junction-junction interaction parameter, Γ 1. V. CONCLUDING REMARKS In this work, we presented a mesoscale framework for the microstructural evolution of highly anisotropic GBs and the formation of hill-and-valley structures. In addition to incorporating an inclination-dependent GB energy, higher order spatial gradients of the order parameter were introduced in our modeling framework to account for the energetics of facet junctions and their non-local interactions. These energetic contributions were found to highly influence the faceting dynamics. To demonstrate the modeling capability, we examined the Σ5 001 tilt GB in BCC iron, where faceting along the {310} and {210} planes was experimentally observed. The inclination-energy phase diagram of this GB was determined via atomistic calculations and used in our mesoscale framework as an input to examine the faceting dynamics of this GB. Linear stability analysis and numerical results revealed regimes of instability against faceting that are consistent with thermodynamic treatments of interface faceting. The roles of facet junction energy and junction-junction interactions were examined, where it was revealed that the average facet length scale increased with the junction energy. Furthermore, negative values for the junction-junction interactions played a stabilizing role, while positive ones increased the faceting dynamics. On the whole, our treatment provides a general framework to examine the stability of 20

21 GBs against the formation of hill-and-valley structures and subsequent facet coarsening. We anticipate that this framework can be further extended by incorporating an improved understanding of the relevant kinetic factors. For instance, the GB mobility, in general, can depend on boundary inclination and will thus affect the faceting dynamics. Furthermore, the kinetic behavior of the facet junctions themselves may affect the overall faceting behavior. Addressing these issues, which ultimately depend on the defect content and local structure of the junctions, will require detailed input from atomistic studies. ACKNOWLEDGMENTS We thank Dr. François Léonard at Sandia National Laboratories and professor Peter W. Voorhees at Northwestern University for useful discussions regarding the thermodynamic aspects of interface faceting. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy s National Nuclear Security Administration under contract DE-AC04-94AL Appendix A: The Dispersion Relation In this section, we derive the dispersion relationship given by Eq. 19. Assume without loss of generality that the linearization is about a planar interface, slab, aligned along the y-direction with a plane normal along the x-direction. This configuration corresponds to a critical quench, where the boundary has the highest interface energy [cf. Fig. 4]. Moreover, we assume a unit thickness in the out of plane direction, which reduces the system to a two-dimensional one. According to Eq. 15, one writes σ as σ = 1 2 ϵ2 (θ) ϕ 2 = 1 ( [ ) 2 λ2 ϕ 2 x + ϕ 2 y (1 7δ) δ ( ϕ 8 x + ϕ 8 y + 14ϕ 4 xϕ 4 )] 2 y. (A1) 1 7δ (ϕ 2 x + ϕ 2 y) 4 With the aid of Eq. A1, one expands the derivatives of σ about (ϕ xo, ϕ yo ) = (1, 0) in a 21

22 power series as 2 σ = 2 σ ϕ 2 x ϕ 2 x o 2 σ = ϕ x ϕ y + 3 σ ϕ ϕ 3 x + 3 σ ϕ x o ϕ 2 y + = λ 2 (1 + δ) 2 + O(ϕ 2 x o ϕ y, ϕ 2 x), yo 2 σ + 2 σ ϕ ϕ xo ϕ yo ϕ 2 x + x o ϕ yo 3 σ ϕ xo ϕ 2 y o ϕ y + = 0 + O(ϕ 2 y, ϕ 3 x), 2 σ = 2 σ + 3 σ ϕ ϕ 2 y ϕ 2 y o ϕ 2 x + 3 σ ϕ y o ϕ xo ϕ 3 y + = λ 2 (1 + δ) (1 63δ) + O(ϕ 2 y, ϕ 2 x). y o Using the convolution theorem, the Fourier transform, F, of Eq. 11 is given by F [Ψ(r)] = F [ 2 ϕ ] F [ 1/ r 3], where F [1/ r 3 ] corresponds to a zero order Hankel transform of the function 1/( r 2 +R 2 c) 3/2 (A2) (A3) and we only consider the zeroth-order approximation for the dependence on k. Finally, with the aid of Eqs. A2 and A3, we substitute the ansatz given by Eq. 18 into Eq. 17 and retain only linear terms in ˆδ ϕ, which yields the dispersion relation given by Eq. 19. REFERENCES 1 J. W. Gibbs, Trans. Conn. Acad. 3, 343 (1878). 2 P. Curie, Bull. Min. France 8, 145 (1885). 3 G. Wulff, Z. Kristallogr 34, 449 (1901). 4 F. C. Frank, in Metal surfaces: Structure, energetics and kinetics (Amer. Soc. Metals, Metal Park, OH, 1963). 5 D. Hoffman and J. W. Cahn, Surf. Sci. 31, 368 (1972). 6 J. W. Cahn and D. Hoffman, Acta Metall. 22, 1205 (1974). 7 A. Wheeler, Proc. Roy. Soc. A 462, 3363 (2006). 8 A. Di Carlo, M. Gurtin, and P. Podio-Guidugli, SIAM J. Appl. Math. 52, 1111 (1992). 9 C. Herring, Phys. Rev. 82, 87 (1951). 10 N. Cabrera, Surf. Sci. 2, 320 (1964). 11 W. W. Mullins, J. App. Phys. 27, 900 (1956). 12 C. Herring, in The physics of powder metallurgy, edited by W. E. Kingston (McGraw-Hill, NY, 1951). 13 R. W. Ballufi, S. M. Allen, and W. C. Carter, Kinetics of Materials (Wiley, 2005) p S. Angenent and M. Gurtin, Arch. Rational Mech. Anal. 108, 323 (1989). 15 J. Stewart and N. Goldenfeld, Phys. Rev. A 46, 6505 (1992). 16 F. Liu and H. Metiu, Phys. Rev. B 48, 5808 (1993). 22

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25 Grain 1 GB dy dt V n y x dy ds dx Grain 2

26 y x b = b F b = b F b = b F

27 [010] [100] 2 nm [010] [100]

28 {310} {210} {310} 1.20 GB Energy [J/m 2 ] Atomistic data Phase field fit π/8 0 π/8 2π/8 3π/8 Inclination Angle [Rad.]

29 {310} {210} 1.25 GB Energy [J/m 2 ] Mendelev potential Chamati potential π/8 -π/16 0 π/16 π/8 Inclination Angle [Rad.]

30

31

32 Γ = Γ = Γ = Γ =

33 Γ = - Γ = - Γ = Γ =

34

35

36 Grain 2 Grain 1 Grain 2 y x

37