Grain-boundary metastability and its statistical properties

Acta Materialia 104 (2016) 259e273

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Grain-boundary metastability and its statistical properties Jian Han a, Vaclav Vitek a, David J. Srolovitz a, b, * a b

Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 6 August 2015 Received in revised form 17 November 2015 Accepted 19 November 2015 Available online xxx

Grain-boundary (GB) structure and properties are usually analyzed in terms of ground-state (minimumenergy) GB states. However, global equilibrium is rarely achieved in materials. In this paper, we investigate the nature of GB metastability and its impact on material properties. Higher-energy GB states can be the result of nonequilibrium processes or simply thermal excitations. While the existence of limited GB metastability is widely known for a few simple GBs, we demonstrate that the multiplicity of metastable GB states is, in general, very large. This conclusion is based upon extensive atomistic bicrystal simulations for both symmetric tilt GBs and twist GBs in three very different materials. The energies of these GB states are densely distributed so that the dependence of the GB energy on misorientation is better described as an energy band rather than as a single curve as in the traditional picture. Based upon the distribution of metastable GB states, we introduce a GB statistical-mechanics picture and apply it to predict finite-temperature equilibrium and nonequilibrium properties. When GB multiplicity exists, GB structures can be thought of as domains of different GB states separated by various classes of line defects. The existence of a large set of metastable GB states, very close in energy, suggests an analogy between the behaviors of GBs and glasses and implies the potential for GB engineering. © 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Grain boundaries Grain-boundary energy Atomistic simulation Grain-boundary properties

1. Introduction Most crystalline materials are polycrystalline e an ensemble of single crystal grains separated from one another by a network of grain boundaries (GBs). While many material properties depend on grain size and grain orientation distributions, many also depend directly on the structure and properties of the constituent GBs [1,2]. The relationship between GB structure and properties has largely been built on our understanding of GBs in their equilibrium (and more commonly zero-temperature equilibrium) states. For example, the variation of the equilibrium GB energy with the macroscopic bicrystal geometric degrees of freedom (DOFs) is often evoked to describe the static and dynamic anisotropic properties of microstructures (e.g., see Refs. [3,4]) and segregation of solute to GBs (e.g., see Ref. [5]). However, in most situations of practical interest GBs are rarely in equilibrium (aside from the obvious issue that GBs themselves are nonequilibrium defects). GBs are almost always constrained by other GBs in a polycrystalline GB network, and their state is strongly influenced by the materials processing

* Corresponding author. Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA. E-mail address: [email protected] (D.J. Srolovitz). http://dx.doi.org/10.1016/j.actamat.2015.11.035 1359-6454/© 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

that leads to microstructure formation (e.g., solidification and annealing). Finally, GB structures are commonly exposed to dynamic perturbations resulting from an external treatment (e.g., severe plastic deformation and radiation damage). The latter issues are particularly important in phenomena as diverse as GB sliding [6e8], GB migration [9,10], absorption/emission of point defects [11,12], adsorption/transmission of dislocations during deformation and/or recrystallization [13]. All of these observations point to the importance of describing nonequilibrium GBs, including both their structure and properties. Several experimental [14e17] and atomistic simulation [18,19,11,20e22] studies have shown the existence of metastable GB structures. The relationship between metastable GB structure and GB behavior (e.g., temperature-induced structural phase transitions [23,11], GB sliding [20], GB migration [24,10], and dislocation nucleation [24]) has received some attention in the literature. At this point, it is well-established that metastable GB structures do exist and can influence GB properties. However, each of these studies focused on isolated examples of specific bicrystal systems (i.e., bicrystals with specific macroscopic geometries in a specific material). The outstanding questions include: (i) how widespread are metastable GB structures beyond the small set of “special” and/or low-angle GBs and the small set of material systems previously examined? (ii) How can we understand the

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relationship between GB geometry and the distribution of metastable GB structure? (iii) Can we predict the relationship between the distribution of metastable GB structure and GB properties? In this paper, we report on a systematic set of atomistic simulations designed to explore the extent of metastable structures in a series of symmetric tilt GBs (STGBs) and twist GBs in materials with three different crystal structures and with two distinct classes of bonding (metallic and covalent). The completeness and diversity of the resultant data sets provides the opportunity to examine GB structural metastability (and properties) in considerably greater generality than has previously been done. We demonstrate that “GB metastability” is an intrinsic character of GBs. While the GB energy as a function of some geometric parameter (e.g., misorientation) is often the only input employed to explain/predict the properties of a GB network, we demonstrate that this alone is insufficient to understand both finite-temperature equilibrium and dynamic properties of GBs. Much of the missing information, we assert, is contained in the distribution of metastable GB states. We suggest a quantity that can be used to provide a measure of the GBstate statistics for particular GB geometries in specific materials. Several of the ideas employed are based upon developments in the fields of glasses, glass forming liquids and granular materials (e.g. Ref. [25]) and treatments of structural uncertainty that are largely missing in traditional GB models (e.g., dislocation models [26] and structural unit models [27]). We show that a new statistical mechanics of GB states naturally arises when GB metastability is considered. 2. Theoretical construction 2.1. Definition of GB state We first define the term “GB state” as used in this paper. The equilibrium state for an isolated system of atoms at zero temperature and zero pressure (or ground state) corresponds to a minimum of the internal energy with respect to a set of generalized coordinates (e.g., atomic coordinates). Metastable states exist when there is more than one internal-energy minimum. When constraints are applied, it is possible that only some of the local energy minima can be accessed; the accessible local energy minima correspond to the stable states consistent with these constraints. The main constraint adopted in the definition of GB state in this study is that the system is a bicrystal where each of the grains is of the same phase. A homophase bicrystal is formed by joining two perfect, semi-infinite grains that differ only in orientation along a flat plane. This constraint precludes many possible states including, of course, the global ground state (an infinite, perfect, single crystal). We describe this constraint based upon the five macroscopic DOFs of a bicrystal [1]. While many descriptions of these DOFs are possible, we specify a rotation axis o (2 DOFs), a rotation angle q (1 DOF), and normal of the flat GB plane n (2 DOFs). The flat-GB-plane constraint also requires that n is constant all along the GB plane; thus, an orthogonal, Cartesian coordinate system (e1,e2,e3) can be defined such that e3kn, and e1 and e2 lie in the GB plane. Additional constraints arise naturally in thermomechanical equilibrium. The independent intensive variables in the homogeneous regions of the two grains that determine GB equilibrium include temperature T, stresses s3i (i ¼ 1,2,3), strains εij (i,j ¼ 1,2), and diffusion potential MI for each diffusing species I [28e30]. We choose the constraints that T ¼ 0, s31 ¼ s32 ¼ s33 ¼ 0, ε11 ¼ ε12 ¼ ε22 ¼ 0, and the system contains only a single element and the two grains are defect-free (hence, we need not consider MI). Stable (equilibrium and metastable) GB states are those which satisfy that (VRU)constraints ¼ 0, where U is the internal energy of the

bicrystal and R represents the coordinates of all atoms. Since the excess internal energy per unit area (measured for a fixed number of atoms) is exactly the GB energy g under this set of constraints, we can alternatively write the stability conditions as (VRg)constraints ¼ 0. Below, each solution is denoted by R(i), corresponding to the configuration of the ith GB state. Among these states, we are particularly interested in those associated with homogeneous GB structures; we call these intrinsic GB states. For a coincidence-site-lattice (CSL) GB, homogeneity implies that the GB structure repeats periodically [1]. This requirement rules out the states that can be described as intrinsic but are decorated by defects (e.g., non-uniformly distributed vacancies or extrinsic dislocations) or mixtures of several intrinsic states. The role of “intrinsic GB states” is analogous to that of “phases” in bulk materials. 2.2. Identification of GB state It is tedious to search for local energy minima by walking through the high-dimensional R space. Thus, we focus on the important DOFs (order parameters) that effectively identify these intrinsic GB states. Since the atoms in the grains away from the GB are constrained by the macroscopic DOFs (o,q,n) and the underlying crystal structure, their relative coordinates are not important; the important DOFs should only relate to the character of the GB. We describe the microscopic DOFs of a GB state by the relative displacement of one grain with respect to the other. The relative displacement includes two components, t1 and t2, parallel to the GB plane and one component, t3, perpendicular to the plane (t3 represents expansion or contraction of the bicrystal). For a given t1 and t2, t3 is determined uniquely by energy minimization, leaving only t ≡ (t1,t2) as the effective GB DOFs. We classify t as belonging to a set of microscopic DOFs or variables for a GB since it cannot be controlled by means of any type of action applied far from the GB (cf. Ref. [1], page 20), such as an externally imposed shear stress. The other microscopic DOF that determines a GB state is the atomic fraction in the GB plane: f ≡ 1  (number of removed atoms)/(number of atoms per layer parallel to the GB)1 where 0 < f  1. Removing an entire layer of atoms from the GB region changes the position of the GB plane along e3 (and is equivalent to shifting one grain relative to the other). Aside from shifts in t, the GB structure exactly repeats when f changes by one [11]. Of course, this also implies that the process of adding atoms to the GB plane is equivalent to that of removing atoms from the GB plane. It has recently been shown that change of f may lead to GB reconstructions that are not achievable by relative displacement (t) alone [31,11]. We consider the GB states that correspond to the local energy minima with respect to t for any value of f (instead of the local minima with respect to both t and f). We can think of GB states in an order-parameter space spanned by t and f; each stable state R(i) is identified by (t(i);f(i)). In this way, we avoid the direct enumeration of states in the full R space. Although such identification may not be unique (a glass structure is an extreme example), it is adequate for the case of GBs that possess structural order. In this paper, we first study the GB states resulting from variations of t at f ¼ 1. We do this for two reasons. (1) Since change of f requires the creation, annihilation and/or transport of interstitials or vacancies, such change occurs on a significantly longer time scale (diffusional) than change of t and in many cases does not occur at all on the time scale of interest. (2) Sampling GB states with respect to both t and f

1 If there are m atoms in a crystal basis, the number of atoms in a “layer” is the number of repeat units in the GB plane multiplied by m.

J. Han et al. / Acta Materialia 104 (2016) 259e273

is computationally expensive. We do, however, examine the GB states in the full (t;f) space for a limited number of GBs in order to determine the effect of f. We refer to the cases where we only examine the t space as conservative and those for which we also allow f to change as non-conservative. 3. Simulation model The bicrystal simulation model used in this study is illustrated in Fig. 1(a). A STGB is created as follows. A perfect cubic lattice fills all of space with an orientation: [100]ke1, [010]ke2 and [001]ke3 e this serves as the median lattice. Then, two lattices are generated by rotating the median lattice by ±q/2 about the tilt axis ok[100] such that they are joined with their respective planes (0kl) and ð0klÞ at the mean boundary plane with normal nk[001] (all indices refer to the median lattice). A STGB is then constructed by associating the crystal bases to one lattice on one side of the mean boundary plane and to the other lattice on the other side of this plane. To create a twist GB, two lattices are generated by ±q/2 rotations from the median lattice about the twist axis ok[001] such that their respective planes (hk0) and ðhk0Þ match. Then, as before, crystal bases are associated with one lattice on one side of the mean boundary plane with normal n ¼ o and to the other lattice on the other side of this plane. In order to insure that our conclusions are broadly valid, we examine bicrystals constructed from three different perfect crystal structures (face-centered-cubic FCC, body-centered-cubic BCC and diamond cubic DC) and two distinct types of bonding (metallic and covalent). To this end, we consider three different interatomic potentials: embedded-atom-method Al [32] (FCC), Finnis-Sinclair W [33] (BCC), and Tersoff-3 Si [34] (DC). Although these potentials have been widely used in simulations and have been shown to capture many of the features of GBs in the materials they are designed to represent (e.g., [35,36,23]), here they are chosen to serve as models reflecting the effects of different crystal structures and bonding characters (rather than representing strictly the specific, real materials). Finnis-Sinclair W and embedded-atommethod Al represent models with different crystal structures but similar bonding (metallic, non-directional character), while Tersoff3 Si describes the case where directional bonding is strong (covalent). 110 CSL STGBs were constructed for each model material with tilt angles in the 0  q  90+ range and with reciprocal coincident site densities S  941 [1]. We also constructed 55 CSL twist GBs in

261

the 0  q < 45+ range and with S  941 for W. The simulation supercells were created as shown in Fig. 1(a). Periodic boundary conditions were applied along e1 and e2; the size and shape of the supercells were fixed to keep the lattice in the regions far from the GB undeformed (with ideal lattice constant a0). In this way, the constraint ε11 ¼ ε12 ¼ ε22 ¼ 0 was satisfied. Two free surfaces were introduced along e3 in order to ensure that s31 ¼ s32 ¼ s33 ¼ 0 far from the GB. The thickness of each grain along e3 was 60e80 nm; i.e., sufficient to insure that the interactions between the GB and the surfaces is negligible. The sizes of the supercells along e1 and e2 are summarized in Appendix A.

4. GB states in t space 4.1. Conservative sampling Conservative sampling was employed to explore the GB states associated with local energy minima with respect to all possible relative displacements (i.e., in t space). Before we describe this, we first review some basic concepts and definitions about the translational symmetry of GBs. Fig. 2 demonstrates (for the particular case of a S5[001] twist GB) that all possible relative displacements of one grain with respect to the other in t space can be represented by shifts in one unit cell of the lattice (see Fig. 2(c)). The area of this unit cell is ac. In other words, we can sample all possible non-identical states simply by applying the relative displacements t represented by vectors in the area ac. Any t outside of ac will produce the same GB structure as a smaller t within ac. It is important to note that the area of the unit cell in this space ac is less than or equal to the area of the repeat cell of the GB plane in CSL, Ac, shown in Fig. 2(a) or (b) (shaded in yellow). ac ¼ Ac for STGBs and ac ¼ Ac/S2 for twist GBs (of course, S > 1). For each bicrystal (fixed o,q,n), the conservative sampling algorithm is as follows: (i) Construct a perfect (unrelaxed) bicrystal configuration and set the relative displacement t ¼ 0. (ii) Rigidly move one grain relative to the other by t. Uniformly grid the area of ac in t space with array of points of density  2 94a2 0 (for Al and W) or 257a0 (for Si). Choose a relative displacement t corresponding to one of the points (see Fig. 1(c)).

(a) Bicrystal Vacuum

(c) t space

e3 GB

–θ/2 e1 Lower grain

l1p1

e2

……

+θ/2

e1

……

Upper grain t

(b) GB plane e2

e3

Ac

Ac

……

Ac

……

t2 R(2)

t1

α(2) p1

αc

R(1)

α(1)

R(4) α(4)

α(3) R(3)

p2 Vacuum

l2p2

Fig. 1. Simulation model and method. (a) Schematic of a STGB bicrystal; the twist GB is similar but with e3 as the rotation axis. (b) The GB plane in the supercell as viewed from the e3-direction; it is tiled by periodic GB cells of area Ac. (c) Schematic of one period of the relative displacement t space; the phase-space area ac is Ac for STGBs and Ac/S2 for twist GBs. GB states are sampled by varying t as indicated by the black dots. The g-surface is indicated by the GB-energy contour (gray curves). R(i) denotes the ith stable state (blue  symbols), and a(i) is the area of the basin of attraction for state R(i) and is delimited by dividing surfaces (black lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 2. Schematic illustration of the relation between GB geometry, coincident site lattice (CSL), displacement-shift-complete (DSC) lattice, and the geometric parameters Ac and ac for the special case of S5[001] twist GB in a simple cubic crystal. (a) The GB structure constructed by stacking one grain (black solid points) on top of the other grain (the white points with red edges) such that the atoms in the (001) plane in one grain below the GB are directly below the atoms in the other grain, and then rotating one grain with respect to the other by q ¼ 36.9+ about the rotation axis [001]. One out of five (hence, S5) of the black atoms are coincident with those in the white lattice. The coincident atoms form a periodic lattice (i.e., a CSL plane parallel to the GB plane); the area of its period is Ac (shaded in yellow). (b) If we displace the black lattice with respect to the white one in (a) by the blue vector “1” (labeled in (a)), the identical CSL plane (and entire GB structure) will be recreated (apart from a shift of origin). Note that, similarly, the same GB structure can be recreated if the relative displacement is along any of the blue vectors labeled in (a) (“1” to “8”). (c) We can form a new two-dimensional lattice based upon these eight blue vectors, as shown by the gray square symbols. For the twist GB, this two-dimensional gray lattice is also DSC lattice. Note that the relative displacement corresponding to any translation vector in the gray lattice leaves the GB structure unchanged. ac (shaded in gray) denotes the area of the unit cell of this gray lattice. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(iii) Perform a conjugate-gradient energy minimization of the bicrystal with respect to all the coordinates of all atoms, allowing the relative displacement t to evolve to obtain a stable (equilibrium or metastable) state R(i). (iv) Repeat Steps (ii)e(iv) until t sweeps out entire ac. After the energy is minimized, we identify the area of basin of attraction associated with the ith state a(i), as schematically shown in Fig. 1(c). We note that the procedure outlined here does not guarantee that we are able to identify all metastable states. There are two distinct issues here. First, the sampling could be too coarse so that some minima are missed. We examined this possibility for several GBs by increasing the sampling density until the number of GB states converges. The sampling density described in (ii) is sufficiently large that we satisfy this criterion in all of our test GBs. Second, it is theoretically possible that there may be two different GB states that correspond to exactly the same t and our conjugategradient energy minimization only finds one of these. We expect such cases to be rare since distinct states will almost always correspond to different t. In fact, a recent study [9] showed that such local atomic rearrangements (shuffles) usually lead to changes in t. We also performed a series of molecular dynamics simulations where a GB was repeatedly quenched to zero temperature from high temperature (73% of the bulk melting point) and never observed multiple states at the same t. Hence, while we cannot rule out that a small number of states were missed, compared with the large number of states usually observed, the fraction of the states that may be missed due to this potential issue can reasonably be ignored. 4.2. Distribution and degeneracy of GB states The distributions of GB states mapped in t space (conservative sampling) are plotted in Fig. 3 for several representative W and Si bicrystals. The black circles in these figures indicate the location of intrinsic GB states. Examination of the distribution of these states in t space shows the underlying symmetry of the GB structures. This symmetry implies that certain intrinsic GB states are equivalent to/ degenerate with other intrinsic ones; i.e., they correspond to exactly the same GB energy.

Depending on the bicrystal crystallography, the degenerate GB structures are related by the point-group symmetry of the holosymmetric bicrystal configuration (i.e., the bicrystal with the highest symmetry of all possible t) [37]. For example, the configurations R(2) and R(1) in Fig. 3(a) are equivalent; their structures are related by a two-fold rotation (with axis lying in the GB plane and perpendicular to o). The degree of degeneracy of each state can be predicted from the symmetry associated with this state and the symmetry of the corresponding holosymmetric configuration (see Appendix B for details). The degenerate GB states can also result from shifts in the GBplane location (in the direction of its normal). A GB-plane shift has the same effect as varying relative displacement. In order to accommodate the relative displacement, some states correspond through a shift in the GB-plane location while keeping the GB structure fixed, rather than transforming to a state with a different structure. For example, the R(2) state in Fig. 3(a) is structurally equivalent to R(3) except for a GB-plane shift of one atomic layer. The vector connecting R(2) to R(3) is a translation vector of the displacement-shift-complete (DSC) lattice [1]. We independently count GB states with different t(i) even though they may have the same energies, because such states are distinguishable. When such states coexist in a single GB, they will be separated by partial GB dislocations (if they are related by a point-group symmetry) or DSC dislocations (if they differ by a GBplane shift/translational symmetry). The vector connecting any two such states in Fig. 3 is the Burgers vector of the (DSC or partial) GB dislocation. In this simulation study, we count two GB states as different when the difference between their energies is Dg > 104 J/ m2 and the difference between their locations is jDtj > 0.1 Å (for pffiffiffiffiffi STGBs) or 102 ac (for twist GBs). In this way, we automatically account for all degeneracy. 4.3. GB-energy bands (g-bands) We now investigate the variation of GB energy as a function of misorientation for Al, Si, and W STGBs and W twist GBs for the case of conservative sampling, as described above. These results can be found in the top set of panels in Fig. 4, where we show the data for all GB states. The minimum GB energies gmin versus q are shown as blue curves in these panels. The gmin(q) curves for Al and W STGBs

J. Han et al. / Acta Materialia 104 (2016) 259e273

263

Fig. 3. The distribution of GB states in t space (black circles) for the cases of (a) W S5 [100](012) STGB, (b) W S5[100](013) STGB, (c) Si S5[100](012) STGB, (d) Si S5[100](013) STGB, and (e) W S109[001] twist GB. The “g-surfaces” obtained by relaxation perpendicular to the GB planes are shown as color maps (the colorbars are in J/m2; see Appendix B for details). The red solid lines, the red dashed lines, the red ellipses, and the red squares represent mirrors, glide planes, two-fold rotation axes, and four-fold rotation axes, respectively. The green lines frame the area of a period of t space (ac); the dark gray shadowed areas correspond to the irreducible region of sampling. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

feature a series of cusps at misorientations corresponding to low S (< 20) [38] while the gmin(q) curve for Si STGBs is highly serrated with particularly sharp cusps at the misorientation corresponding to the two smallest values of S (i.e., S5 GBs). The gmin(q) curve for W twist GBs has only a shallow cusp at S5. These observations are in agreement with previously reported results (e.g., [39e41]). Aside from gmin(q), Fig. 4 also shows all of the metastable GB states for each misorientation and material. Clearly, the number of metastable states can be very large. In contrast to the classical picture of the GB energy versus misorientation being a curve, here we see that it may be more appropriate to consider the GB energetics versus misorientation to be a band which commonly extends well beyond gmin. The band width is associated with the dispersion of energies of the multiple GB states. We will refer to this energy band as “g-band”.2 Many previous studies have developed algorithms to identify gmin(q) by avoiding the higher-energy GB states. In contrast, we assert that the g-bands contain the information that will be crucial in interpreting GB structure, and, in particular, their thermodynamics and physical properties. The information contained in the g-bands is expected to play an important role since it represents all possible states of a GB, each of which may occur with finite probability in equilibrium and/or during kinetic processes. In general, when a g-band is wide, the corresponding GB properties may undergo large changes during different processes. When a g-band is dense, transformations between nearby GB states can occur easily; in such a case, metastable GB states can be expected to have a particularly strong influence on GB properties. In order to understand the relationship between the g-bands and GB properties, we first look for quantitative measures of key g-band features. We characterize a g-band in terms of its band width and the number of GB states. For each, we identify the misorientationspecific quantity X(q), where X denotes the GB property of

2 This terminology implies an analogy between the GB-energy spectrum g(i)(q) and the electronic energy band εn(k), where n is the electron energy level and k is the wave vector.

interest, and the misorientation-averaged quantity X defined as

1 X≡ qmax  qmin

qmax Z

qmin

  XðqÞ dq

;

(1)

o;n

where qmin and qmax delimit a range of misorientations for a series of GBs with fixed (o,n). Such a definition implicitly assumes that X(q) is integrable; i.e., X(q) is finite. Since the geometric variable q is eliminated by this averaging procedure, it allows us to compare a GB property X between different materials (through their X). 4.3.1. Band width The g-band width for a specific misorientation q is d(q) ≡ gmax(q)  gmin(q), where gmax(q) is the maximum energy amongst all GB states for a bicrystal of misorientation q. The misorientationaveraged band width for a series of GBs with fixed (o,n) is d. To compare between different materials, we consider the misorientation-averaged relative band width ðd=gmin Þ; this approach factors out the effects such as bond strength to allow us to focus on the effect of GB structural confirmation. The values of d and ðd=gmin Þ for the three model materials and GB series considered here are listed in Table 1. Based on either quantity, the band width in Si STGBs is considerably larger than any of the other cases. Al STGBs exhibit smaller d than the other materials, but larger ðd=gmin Þ than W STGBs. These differences may stem from differences in bonding character amongst these materials. The directional (covalent) bonds in Si stabilize more metastable GB structures than the non-directional (metallic) bonds in the metals, leading to a larger band width. Although typical bond strength (or cohesive energy) in Si is weaker than W, the d value of Si is still higher than that of W (likely due to the much larger configurational multiplicity in Si GBs than in W GBs). This is consistent with the fact that it is much easier to form amorphous Si than its pure metal counterparts. The d value of W is higher than that of Al, which is as expected since W bonds tend to be considerably stronger than Al bonds (recall that the melting point of W is about four times that of Al). However, normalizing out this bond-

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Fig. 4. The results of conservative sampling for (a) Al [100] STGBs, (b) Si [100] STGBs, (c) W [100] STGBs, and (d) W [001] twist GBs. The misorientations with S  29 are labeled by the vertical lines. The top panels show the g-bands, i.e. the GB-energy spectra of all the GB states (black dots) for each misorientation. The blue solid, green dashed, and red solid lines represent the minimum GB energy gmin, the equilibrium ensemble-averaged GB energy 〈g〉eq (at half bulk melting point), and the nonequilibrium ensemble-averaged GB energy 〈g〉sq, respectively. The panels of the second row show the number of GB states M that are shaded in gray, along with the number of states with distinct energies Me that are shaded in blue. The panels of the third row show the density of states. For STGBs, since the density of states in the real space r is the same as that in the phase space ra, only r is plotted. For twist GBs (d3), r is plotted as black dots while ra is shaded in red. The bottom panels show the degree of uncertainty Sc/kB by black dots. With reference to Eq. (4), the red and blue lines correspond to the values of SI/kB and SII/kB, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

strength effect, we find that ðd=gmin Þ in Al is nearly twice that of W. We suspect that this is related to the less densely packed structure of W than that of Al. Twist GBs experience much smaller energy variations than do STGBs [42], as seen by comparison of Fig. 4(c1) and (d1). We can understand this result by comparing the sizes of the repeat periods in t space ac. In most cases, ac of a STGB is larger than that of a twist GB with the same misorientation. Therefore, sampling all of the states for a STGB involves larger relative displacements, and thus Table 1 The misorientation-averaged quantities that describe several characteristics of the g-bands. Bicrystal systems

d (J/m2)

ðd=gmin Þ

r (a2 0 )

e d (J/m2)

Al STGB Si STGB W STGB W twist GB

0.27 1.7 0.75 0.046

0.72 1.4 0.39 0.020

13.6 241 22.4 6.43 (1  106)a

0.056 0.80 0.12 0.027

a The value in parenthesis for r in the W twist GBs corresponds to ra . For the other bicrystal systems, ra ¼ r.

requires larger relaxation, than for a twist GB. Larger relaxation generally implies larger variation in energy [1]. 4.3.2. Number of states Another important characteristic of a g-band is the number of GB states M(q), light gray shading in the panels of the second row in Fig. 4. M corresponds to the number of states sampled in a period of t space with area ac and for a period of the GB structure with area Ac. In all cases, small values of M are located at misorientations corresponding to low S values. The M(q) curves show a strong similarity amongst the STGBs of the three model materials. This suggests the existence of a hidden geometric factor in the g-bands; we return to this later. Also note that the twist GBs have much larger values of M than do the STGBs (compare Fig. 4(c2) and (d2)). This can be understood based on the size of a repeat GB cell Ac. Ac of a twist GB is always larger than that of a STGB with the same misorientation. When Ac is larger, the pool of the atoms that potentially can relax will be larger. This implies that there will be more ways to relax, leading to more states.

J. Han et al. / Acta Materialia 104 (2016) 259e273

20

(a)

(b)

20

W twist GBs Fitted parabola

ρα (105 Å-2)

15

25 Al STGBs Si STGBs W STGBs

265

ρα (Å-2)

15

10

10

5

5

0

0 0

200

400

Σ

600

800

1000

0

200

400

Σ

600

800

1000

Fig. 5. Density of states ra in t space as function of reciprocal coincident site density S for (a) STGBs and (b) twist GBs.

The number of GB states with distinct energies Me(q) are shown in blue, along with M(q) (gray) in Fig. 4. The gap between M and Me indicates the significant contribution of degeneracy, as discussed in Section 4.2. It is appropriate to normalize the number of GB states for each misorientation by the corresponding cell area, r(q) ¼ M(q)/Ac(q), where r is the density of GB states in real space. We can also define the density of GB states in t space as ra(q) ¼ M(q)/ac(q). For STGBs, ra ¼ r since ac ¼ Ac; however, for twist GBs, ra ¼ rS2 since ac¼ Ac/ S2. These are shown in the panels of the third row in Fig. 4. We find that M and ra for twist GBs are not well defined, since they diverge with increasing S for the twist-GB case (no such problem exists for r). The values of r for the three model materials are listed in Table 1. r for the Si STGBs is much larger than those of Al and W STGBs, as seen by comparison between Fig. 3(a) and (c) (or between Fig. 3(b) and (d)). Such differences may originate from the different bonding characters and crystal structures of these materials. As discussed above, the directional bonds in Si help stabilize more metastable configurations. We also note that, since the crystal basis for DC structures is twice that of the other two cubic structures, the number of DOFs for the relaxation is largest for Si. Examination of ra in Table 1 shows that ra is generally extremely large for twist GBs. The S-dependence of ra, obtained from our simulations, is shown in Fig. 5. ra grows approximately as S2 for twist GBs (see Fig. 5(b)), while ra tends to an upper bound with increasing S for the STGBs (see Fig. 5(a)). This can be understood from the relationship M f Ac, as suggested by the simulation results. Such a relationship implies that for twist GBs ra f S2 and for STGBs ra will be constant. This may have important consequences for GB properties. That is, any property that depends on ra will be nearly constant (independent of misorientation) for STGBs (except for low-S boundaries), while the same property will vary dramatically with misorientation for twist GBs. Since the magnitude of the 1=2 shortest (DSC or partial) GB dislocations scales as ra , we expect that GB sliding should be much more difficult for general STGBs than for typical twist GBs. 4.4. Uncertainty Based on the information contained in the g-bands, we define a measure of GB metastability Sc3 which describes our uncertainty

3 We denote this quantity as Sc to emphasize its relationship with the configurational entropy that we discuss later.

associated with the microscopic DOFs of a GB (since the macroscopic DOFs are usually well controlled or explicitly measurable). Therefore, Sc is defined based on the distribution of available states {R(i)} and their energetics {g(i)} for each fixed set of macroscopic DOFs (o,q,n). 4.4.1. Probability distribution The probability that a GB state R(i) is accessed in an unbiased sampling in t space is

. pðiÞ ¼ aðiÞ ac ;

(2)

where, with reference to Fig. 1(c), a(i) is the area associated with R(i) in the sense that any initial configuration sampled in this area will relax to R(i) by energy minimization. a(i) is the “volume” of the ith basin in t space (of course, since the t space is twodimensional, this generalized “volume” is an area). In this definition of the probability, there is no requirement that a lowenergy state must occur with higher probability than a highenergy state. Our simulation results suggest that the probability distribution of states in, for example, Si is fairly evenly spread across its g-band. 4.4.2. Uncertainty function The measure of GB metastability can be defined in the form of an uncertainty function:

Sc ðqÞ ¼ kB

MðqÞ X

pðiÞ ðqÞlnpðiÞ ðqÞ;

(3)

i¼1

where kB is the Boltzmann constant. This definition is analogous to that of information (Shannon) entropy [43] and the entropy of granular materials [44]. It measures our ignorance of a GB structure purely due to the existence of multiple metastable states, or the structural uncertainty of a GB in the case of unbiased sampling. Sc is plotted as the black dots in the bottom set of panels in Fig. 4, based on the distribution of states from our simulations. To clarify the features of Sc(q), we rewrite Eq. (3) using Eq. (2) and the relationship between Ac and ac:

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8 ! ! MðqÞ > X > Ac ðqÞ aðiÞ ðqÞ > ðiÞ > kB ln p ðqÞln  kB > > > a20 a20 > i¼1 > < ! ! MðqÞ Sc ðqÞ ¼ X aðiÞ ðqÞSðqÞ2 > ðiÞ > kB ln Ac ðqÞ kB p ðqÞln > > > a20 a20 > i¼1 > |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > : SI ðqÞ

for STGBs :

(4)

for twist GBs

SII ðqÞ

For convenience, the first and the second terms are denoted SI and SII, respectively. SI(q)/kB is plotted as the red lines and SII(q)/kB as the blue lines in the bottom set of panels in Fig. 4. Examination of the expression for SI shows that this term is solely determined by the bicrystal geometry. SI shows variations with q that is very similar to Sc (apart from a scaling factor), except for the low-angle GBs in W; i.e., the correlation is strong for q > 30+ in the STGBs and for q > 15+ in the twist GBs in W. SII, on the other hand, appears to scale as lnðra20 Þ, as shown in Fig. 6. We can understand this correlation as follows. If we assume that p(i) ¼ M1 for all R(i) (or, equivalently, all a(i)'s were identical for a given q), then SII would be exactly equal to kB lnðra20 Þ. The deviation from linearity in this plot is a measure of the accuracy of the assumption that p(i) ¼ M1. If the variation of SII is assumed to be largely materialdependent and insensitive to misorientation, then Sc(q) may be assumed to be proportional to SI(q) plus a material-dependent constant. However, although the variation of SII with q is relatively small, it still does have a q dependence of the same form as SI (comparing the red and blue lines in the bottom set of panels in Fig. 4). This implies that we can rewrite Sc(q) as

Eq. (5) suggests that the structural uncertainty of a GB grows with lnAc. This is intuitively reasonable. For a GB with small Ac (e.g., the S3 twin boundary in FCC materials), the GB structure is easily predicted (low uncertainty). For a GB with large Ac, the number of metastable GB states rises as the number of atoms that are free to relax increases; for such large-Ac cases, it is usually quite difficult to accurately predict the structure. From the viewpoint of a GB dislocation model, a large-Ac GB can always be considered as a small-Ac GBs decorated by an array of secondary GB dislocations (SGBDs) [1,27,45]. In addition to the states associated with the small-Ac GB segments, the arrangement and the core structure of the SGBDs further contributes to the structural complexity. In this section, we focused on the microscopic DOFs of a bicrystal that are associated with the relative displacement of one grain with respect to the other (t) for a fixed atomic fraction in the GB plane (f ¼ 1). However, there is evidence that variations in f may create additional GB states [11,31]. In the next section, we examine the effect of f and show that the discussions above is still valid for STGBs.

Sc ðqÞzkSI ðqÞ þ b;

5. GB states in (t;f) space

(5)

where k (> 0) and b are material-dependent constants. This relation is verified in Fig. 7. The linear relationship between Sc/kB and SI/kB works quite well for Al and Si STGBs. For W STGBs, the linear relation exists for high-angle (> 30+) and low-angle (< 30+) regions separately. The linear relation also works for W twist GBs (although this fit is less ideal).

6

Al STGBs Si STGBs W STGBs W twist GBs

SII / kB

4

2

0

-2 -2

0

2

ln(ρa02)

4

6

Fig. 6. Correlation between the second term of the measure of GB metastability SII/kB (defined in Eq. (4)) and the quantity lnðra20 Þ, where r is the density of states in the real space. The data points represent all of the GB data from the simulations (note that q is an implicit variable).

5.1. Non-conservative sampling Non-conservative sampling was designed to investigate how the distribution of GB states (with respect to t) depends on the atomic fraction in the GB plane f (as done earlier by Frolov et al. [11] and Alfthan et al. [22]). The atomic fraction 0 < f  1, where f ≡ 1  (number of removed atoms)/N and N is the number of atoms in a “layer” which equals the number of repeat units in the GB plane in a supercell multiplied by the number of atoms in the crystal basis. We define a “GB region” of thickness 16 Å around the GB plane and two “crystal regions” outside of this. We can then describe the non-conservative sampling as follows: (i) Construct an unrelaxed bicrystal supercell for the chosen macroscopic DOFs (o,q,n). (ii) Rigidly move one grain relative to the other by t (same as Step (ii) of conservative sampling). (a) Remove one atom from the GB region (i.e., f / f  1/N). (b) Fix the atoms in the crystal regions and heat the GB region to 98% of the bulk melting point, hold for 50 ps, and then cool to 0 K at 4.2  1013 K/s. (c) Minimize the energy of the bicrystal with respect to the coordinates of all atoms in e3-direction only (hence, t is fixed). (d) Perform a conjugate-gradient energy minimization of the bicrystal with respect to all the coordinates of all atoms, allowing t to evolve to obtain a stable state.

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10

(a)

8

Al STGBs Si STGBs W STGBs

12

(b)

267

W twist GBs

9

Sc/kB

Sc / kB

6 4

6 3

2 0 0 -1

0

1

SI / kB

2

3

-3 0

2

4

SI / kB

6

8

Fig. 7. Correlation between the measure of GB metastability Sc (defined in Eq. (3)) and the quantity SI ≡ kB lnðAc =a20 Þ in the cases of (a) STGBs and (b) twist GBs. The dashed lines show the results of linear fitting. The open green circles in (a) correspond to the data of low-angle STGBs (q < 30+) in W; the open black circles in (b) correspond to the data of lowangle twist GBs (q < 15+) in W. The open circles are not included in any fitting. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(e) Repeat Steps (a)e(e) based on the configuration obtained in Step (c) until f ¼ 0. (iii) Repeat Steps (ii)e(iii) until t sweeps out entire ac. The simulation supercell is shown in Fig. 1 (see Appendix A for

details). The procedure above may result in many GB states for each misorientation q corresponding to different values of t. However, we recall that the definition of an intrinsic GB state is a GB state for which the structure is homogeneous (i.e., one that is periodic). This

Fig. 8. The results of non-conservative sampling for (a) S5[100](013) STGB, (b) S29[100](037) STGB, (c) S13[100](023) STGB, (d) S5[001] twist GB, and (e) S13[001] twist GB in W. The GB energies g of all stable states are plotted against the atomic fraction in the GB plane f. Corresponding to each f value, multiple states exist for different relative displacements (black dots). The colored  symbols indicate intrinsic GB states at values of f indicated by the vertical, colored dashed lines above which the number of intrinsic GB states at that f are written. Blue dashed lines and symbols indicate the intrinsic GB states that can be found by conservative sampling and red ones indicate the additional intrinsic GB states that can only be found through non-conserved sampling. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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rules out the states that are effectively intrinsic GB states decorated by vacancies. Because intrinsic GB states are periodic, when f is fractional, the period of the GB must be larger than Ac. The non-conservative sampling simulations were performed with supercells much larger than Ac ¼ p1  p2; the size of supercells is l1p1  l2p2 (l1 and l2 are integers; see Fig. 1(b) and Appendix A). 5.2. Non-conservative sampling results Typical non-conservative sampling results are shown in Fig. 8(a)e(c) for three different [100] STGBs (each plot is for one misorientation q) in W. For almost every value of f sampled, multiple stable states were found corresponding to different values of t and with a distinct GB energy. However, most of these correspond to the structure of intrinsic GB states plus a non-periodic set of vacancies. Just as we do not consider a BCC crystal with point defects to be a different phase than BCC, we do not consider such states to be intrinsic GB states. In some cases, we also see a combination of two different intrinsic GB states; this is akin to a twophase alloy, which together do not constitute a new phase either. Fig. 8 shows that, at some values of f, cusps exist. Careful examination of the states corresponding to these cusps shows that their structures are always periodic; i.e., they correspond to intrinsic GB states. There may be non-cusp states that are also intrinsic GB states; we return to this later. For some STGBs, e.g., the S5[100] (013) and S29[100] (037) STGBs (Fig. 8(a) and (b)), a cusp is clearly seen at fractional f (labeled by the red  symbols). On the other hand, for all STGBs, we find that cusps also exist at f ¼ 1 (labeled by the blue  symbols), as seen in Fig. 8(a)e(c). Interestingly, for the S13[100] (023) STGB (Fig. 8(c)), the cusp only occurs at f ¼ 1. Checking all of the STGBs in this study, we find that nonconservative sampling identifies at most one additional intrinsic GB state corresponding to the cusp position in f, and no intrinsic GB states was ever found corresponding to non-cusp position in f. The additional GB states (other than those at f ¼ 1) may have either higher or lower energy than those found using conservative sampling (f ¼ 1), as shown in Table 2. While such fractional-f GB states may correspond to the global energy minimum, their energies are always found to be only slightly lower than those found using conservative sampling. The results of non-conservative sampling for the S5 and S13 [001] twist GBs are shown in Fig. 8(d) and (e). There are two Table 2 The characteristics of the additional GB states that are found by non-conservative sampling but cannot be found by conservative sampling. Madd/M is the ratio of the number of additional states to the number of states found by conservative sampling. g/gmin is the ratio of the GB energy of the additional states to the minimum GB energy found by conservative sampling. f is the atomic fraction in the GB plane corresponding to the additional states; it is written in form of fraction, where the denominator is the number of atoms per layer parallel to the GB within the supercell (N) and the numerator is the number of atoms remained in the GB when the additional state is found. “e” indicates no data since no additional state is found. Bicrystal geometry

Madd/M

Periods

g/gmin (J/m2)

f

S13[100](015) STGB S17[100](014) STGB S5[100](013) STGB S29[100](025) STGB S29[100](037) STGB S5[100](012) STGB S17[100](035) STGB S13[100](023) STGB S5[001] twist GB

0/1 0/2 1/2 1/15 1/7 1/3 1/6 0/11 2/1

S13[001] twist GBa

30/5

e e 5p1  p2 5p1  4p2 5p1  p2 5p1  2p2 5p1  4p2 e p1  p2 p1  p2 p1  p2

e e 1.933/1.797(¼1.08) 2.163/2.130(¼1.02) 2.116/2.127(¼0.99) 1.922/2.000(¼0.96) 2.034/2.060(¼0.99) e 1.910/2.580(¼0.74) 2.342/2.580(¼0.91) 2.150/2.576(¼0.83)

e e 12/60(¼0.2) 24/40(¼0.6) 16/40(¼0.4) 6/30(¼0.2) 16/80(¼0.2) e 8/40(¼0.2) 16/40(¼0.4) 16/52(¼0.31)

a

There are 30 additional states for this GB. Only the characteristics of the one with the lowest GB energy is provided as an example.

remarkable differences between the results for STGBs and twist GBs. First, for twist GBs, the additional GB states found by nonconservative sampling can have much lower GB energy than those found by conservative sampling. For all the STGBs in this study, the GB energy can be lowered by, at most, 3.9% by changing f; however, the GB energies are lowered by 26% and 17% for the S5 and S13 twist GBs, respectively. This suggests that the additional states associated with change of f are highly energetically favorable for twist GBs. Second, twist GBs can have much more additional intrinsic states than do STGBs (for which there is at most only one additional state found amongst the STGBs studied), as shown in Table 2. For example, the S13 twist GB has 30 additional states while only five states were found by conservative sampling, and the S5 twist GB have two additional states although only one state was found by conservative sampling. 5.3. Impact of additional GB states It is particularly important to know the influence of additional intrinsic GB states on the picture of g-bands and GB metastability proposed in Section 4. For STGBs that are not special (i.e., not lowS), the additional states can be reasonably neglected. As noted above, non-conservative sampling never introduces more than one additional state for any STGB, and the GB energies of these additional states are very close to gmin. Since, for non-special STGBs, the number of intrinsic states found by conservative sampling is very large (see Fig. 4), the addition of only one more state is insignificant to the GB statistics. Since the energy of the additional state is not very different from those found with conservative sampling, non-conservative sampling will also lead to only a very small change in GB energy. However, for special (low-S) STGBs, the number of intrinsic GB states found by conservative sampling is small; hence, in such cases non-conservative sampling is potentially important. The additional intrinsic states obtained by non-conservative sampling may be important for twist GBs. For twist GBs, changing f will, in general, lead to a large increase in the number of states (e.g., from 5 to 35 for the S13 twist GB). In addition, the GB energies of the additional states can routinely be significantly smaller than those found for f ¼ 1 (see Table 2). Hence, non-conservative sampling is potentially very important for GB statistics and energetics for twist GBs. This implies that, to obtain a meaningful gband for a twist GB, non-conservative sampling is necessary. Unfortunately, the computational costs for obtaining such data is prohibitive. We have found that, when conservative sampling is employed, the different states are related to one another through various types of relative displacement and that domains consisting of different states are separated by line defects that are of (DSC or partial) GB dislocation character. When non-conservative sampling is employed, we found additional intrinsic GB states, domains of which must also be separated by line defects. The additional states found at fractional f can be thought of as related to those at f ¼ 1 in the same way that an ordered alloy is related to its base structure; e.g., CsCl to BCC. We can think of fractional f as something like composition (in this case, we can think of finite f as related to a finite vacancy concentration) for which there are stable structures at fixed stoichiometry. Converting from BCC to CsCl implies an increase in the size of the unit cell, and a loss of some translational symmetry elements. So too in going from unity-to fractional-f GB structures. The new type of defect that appears in ordered alloys (e.g., CsCl) compared with elemental crystals is an anti-phase boundary (APB). The new type of line defect separating fractional-f GB structures compared with unity-f GB structures is the equivalent to this in two dimension. Similar one-dimensional APBs

J. Han et al. / Acta Materialia 104 (2016) 259e273

269

have been observed for other two-dimensional systems, especially free surfaces [46].

experimental [50] studies on single-component systems which show increasing GB disorder with increasing temperature.

6. Statistical mechanics of GB states

6.2. Stillinger quench

The traditional approach to understanding GB properties is to focus on the GB structure that corresponds to a minimum in the free energy or, more commonly, the internal energy (i.e., the ground state). However, the results presented above demonstrate that multiple GB states are the rule (especially for non-special GBs) rather than exception. Therefore, macroscopic GB properties should be determined by an appropriate ensemble average over all states. This suggests that real GB structure and properties should be reconsidered in light of this extremely large diversity of states and their distribution under experimental situations of interest. We approach this on the basis of the statistical mechanics of GB states.

Next, we consider the case where the GB is far from equilibrium. Consider a “random” GB with fixed macroscopic DOFs; i.e., a GB with the microscopic DOFs randomly chosen in (t;f) space (we will simplify this discussion by focusing on the more limited t space). We then quench this random GB to zero temperature and determine which GB state it ends in following the quench. This process is repeated a large number of times for differently chosen random GBs and we measure the probability of ending in a particular state. Such quench simulations were introduced by Stillinger [51] in the context of determining the structure and properties of liquids e hence the term, Stillinger quench (sq), see also Ref. [44]. The Stillinger-quench probability is simply the fractional size of the basin of each state; therefore,

6.1. Equilibrium The first application will be to equilibrium. A GB equilibrated at finite temperature can be treated as a two-dimensional mosaic structure made up of patches of particular structures (states). The smallest such patches can be of size Ac (smaller patches make no sense in light of the nature of the GB structure represented here). Patches with different states are separated by line defects, such as (DSC or partial) GB dislocations or one-dimensional APBs. However, for now, we ignore the line defects such that  we can write the ðiÞ probability of being in the ith state as Peq ∝exp gðiÞ Ac =kB TÞ. Here, we have explicitly assumed that the basin of attraction of each state is of equal size (same fraction of the total phase space). Now, consider the case that each state has the same energy but the size of each basin is different. The fractional size of the ith basin is simply ðiÞ p(i) ¼ a(i)/ac, as discussed above. In this case, Peq ¼ pðiÞ . Finally, in the general case where each state has a unique energy and its basin is  ðiÞ not uniform, Peq ∝pðiÞ exp gðiÞ Ac =kB TÞ [47]. We can normalize this result by the partition function Q and introduce the line-defect energy contribution Eline [20]; hence, ði Þ

Peq

 h  i ¼ pðiÞ exp  gðiÞ Ac þ Eline kB T Q   zpðiÞ exp gðiÞ Ac =kB T =Q 0 ;

(6)

where, in the last relation, we assume that the line-defect energy is constant and can be canceled with a similar term in Q. We return to the discussion of the line-defect contribution to the free energy below. In applying Eq. (6), we recall that the index (i) refers to distinct states even though several distinct states may share the same energy (e.g. owing to the symmetry-related degeneracy); in this way we automatically account for any degeneracy in summation over i. We note that in the analysis above, we explicitly ignore the vibrational entropy. This is done by analogy with amorphous materials (a case very similar to ours in that both exhibit a large number of nearly degenerate states), where vibrational effects are usually omitted in consideration of thermodynamic properties because the vibrational entropy varies little amongst accessible states (see e.g. Ref. [48]). Eq. (6) implies that GBs increasingly disorder with increasing temperature. At low temperature, the difference in energy between ðiÞ distinct states dominates Peq ; thus, only the few low-energy states will appear, leading to a highly ordered GB structure. At high ðiÞ temperature, however, Peq is strongly influenced by p(i), implying that the higher-energy states can be accessed; this leads to “disordered” GB structure in the sense that the GB structure consists of domains with structures corresponding to distinct states. This is consistent with the results of molecular dynamics [49] and

ðiÞ

Psq ¼ pðiÞ :

(7)

Comparing with the equilibrium case (Eq. (6)), we see that the sq case corresponds to a situation where the energy differences between states play no role. Hence, we can think of the sq case as yielding the extreme nonequilibrium distribution of GB states. Such a case is expected to be closer to the distribution of GB states following rapid quenches from high temperature or other far-fromequilibrium cases (e.g., during irradiation damage, GB migration, GB sliding, superplastic deformation). In reality, we expect most nonequilibrium distributions to be neither purely sq-like nor purely equilibrium. 6.3. Ensemble-averaged GB energy Equipped with the probability function, we can determine the GB energy corresponding to different physical situations. The ensemble-averaged (i.e., experimentally measurable) GB energy is M ðqÞ D E X g ðqÞ≡ P ðiÞ ðqÞgðiÞ ðqÞ;

(8)

i¼1 ðiÞ

ðiÞ

where P(i) is either Peq or Psq (or, in general, any probability function for the situation of interest), and accordingly we denote the appropriate GB energy as 〈g〉eq or 〈g〉sq. The top set of panels in Fig. 4 show 〈g〉eq and 〈g〉sq as green dashed and red solid curves, respectively. In order to calculate 〈g〉eq for these figures, we explicitly choose a temperature of half the bulk melting point. Examination of these plots shows that 〈g〉eq is very similar to gmin for all three materials, indicating that in equilibrium the GB energy is dominated by the minimum-energy states. For Al and W, the 〈g〉sq(q) curves resemble the gmin(q) curves, albeit shifted to somewhat higher energy. For Si, on the other hand, the 〈g〉sq(q) is much higher than gmin(q). The extremely large number of cusps in gmin(q) in Si disappear in 〈g〉sq; 〈g〉sq is a smooth function of q in Si with significant cusps only at the S5 misorientations. The GB energy measured in experiments or simulations is sensitive to how a sample has been prepared [52,53]. gmin represents the equilibrium limit that applies when the GB is annealed at high temperature and then cooled to 0 K at an infinitesimally small rate. 〈g〉sq corresponds to the rapid-cooling (quick-quench) limit from very high temperature. We expect that typically the measured values will fall within gmin < 〈g〉 < 〈g〉sq. Hence, the effective band width can be defined as e d ≡ ðhg〉sq  gmin Þ (shown in Table 1). Consistent with the results presented in the top set of panels in

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Fig. 4, we see that the range of accessible states is larger for Si than for the other materials in this study. 6.4. GB configurational entropy Mathematically, entropy S is related to uncertainty through the probability that a system is in any particular state i, say p(i): P S ¼  pðiÞ lnpðiÞ . Any application of this to a physical situation i the definition of an appropriate probability distribution requires p(i). If p(i) is the probability distribution appropriate to a system in equilibrium, then S is the thermodynamic entropy e e.g., related to the variation of the free energy with respect to temperature. The same entropy definition can also describe a situation where p(i) describes a nonequilibrium distribution of states. Comparison of S with the definition Sc described above (see Eq. (3)) implies that p(i) ¼ p(i). This demonstrates that Sc indeed has the form of an entropy. ðiÞ Given our definition of the probability Psq ¼ pðiÞ, Sc is the entropy associated with the configuration produced by Stillinger quench (sq). Since sq corresponds to zero temperature in a classical system, there is no vibrational contribution to the entropy; Sc is the configurational entropy. Although sq is an idealized process, corresponding to extreme nonequilibrium, we assert that Sc is a reasonable approximation to the configurational entropies for real, far-from-equilibrium processes, such as high-rate GB sliding (e.g., in superplastic deformation). We defer the demonstration of this assertion to a later publication. Sc also plays an important role in thermal equilibrium. In equiP ðiÞ ðiÞ ðiÞ Peq lnPeq , where Peq is defined in librium, we can write Seq ¼ kB i

(6). Inserting Peq into this expression, we obtain P ðiÞ ðiÞ Seq ¼ kB i sc wðiÞ þ hg〉eq Ac =T þ kB lnQ 0 , where sc ≡  pðiÞ lnpðiÞ and (i) (i) w ≡ exp(g Ac/kBT)/Q'. As temperature rises, the latter two terms in this expression decrease and the first term converges to Sc. Therefore, in the limit of high temperature Seq/Sc. In other words, as temperature rises, the equilibrium properties of a GB will increasingly resemble those described by the sq process (we note that Seq here is approximate and hence we do not advocate its application very close to the bulk melting point). In Section 4 we demonstrated that Sc is a linear function of the geometric parameter lnAc (see Eq. (5) and Fig. 7). This implies that, in cases where the GB configurational entropy is ~Sc (e.g., Stillinger quench or high-temperature equilibrium), we can infer a relationship between GB geometry and GB properties. Eq.

7. Discussion The simulation results and analysis above demonstrate that general (high-S) GBs are characterized by a large number of metastable states that can be very close in energy (nearly degenerate). As discussed in the Introduction, a wide range of experimental and simulation results have clearly demonstrated the existence of one or a few metastable GB states for high-symmetry (low-S) GBs. Unfortunately, experimental observations that clearly show the existence of many GB states is less direct. While it should be possible to observe several different GB structures by performing a series of high-resolution electron microscopy studies on a series of nominally identical bicrystals, such studies have not been performed systematically for high-S GBs. On the other hand, less direct experimental evidence is available. For example, Herbig et al. [54] (see their Fig. 4) recently performed a systematic study of GB segregation for a large ensemble of GBs by combining transmission electron microscopy and atom probe tomography. Their results demonstrate that the solute excess on high-angle (and high-

S) GBs vary by more than 80%. While segregation may indeed change the GB structure, this is evidence of considerable structural variation in the GBs. Clearly, more extensive experimental verification of the findings here is necessary to firmly establish the gband picture. It is unlikely that, with such as large a number of available states as found above, an entire GB will ever be found in a single state at any instant in time. This implies the coexistence of multiple states (for a discussion of the equilibrium situation see Ref. [55]). Coexistence of domains in different states, in turn, implies the existence of line defects separating these. With so many states, it is difficult to characterize all such line defects. However, such line defects can be easily categorized. Fig. 9 shows such a categorization for an example of (t;f) phase space. Depending on the translational and point-group symmetry of t space for f ¼ 1, line defects may be DSC dislocations (i.e., dislocations with Burgers vectors that are translation vectors of the DSC lattice [1]) or partial GB dislocations with zero or finite stacking-fault energy. If neighboring domains are associated with fractional f, then these same types of line defects may have APB character as well. We can characterize the structure of a GB at finite temperature through the probability distribution of different states (e.g., through Eq. (6)) or by performing a dynamic simulation under the conditions of interest (e.g., in finite-temperature equilibrium or during superplastic deformation). Both approaches are fraught; the first because of the approximations necessary and the second because of the large computational time and simulation size required to achieve meaningful statistics. An alternative approach can be borrowed from statistical physics, where it is common to employ Ising-model descriptions. Such an Ising model could be a two-dimensional array of spins, where each represents the local state of a domain in the GB. Consider a discrete spin on each site 2i ¼ {1,2,/,M}, each characterized by an energy H(2i). Neighboring spins interact with each other through a nearest-neighbor bond that has an energy depending on the states of these spins J(2i,2j). The Hamiltonian of the resultant N-site model can be represented as:

H¼

N X i¼1

Hð2i Þ þ

N X  1X J 2 ;2 2 i¼1 j2nnðiÞ i j

(9)

where nn(i) denotes the nearest neighbors of the ith site. When applied to GBs, H(2i) can be determined from the GB-state energies (Fig. 4) and the J(2i,2j) from the line-defect energies (read from a matrix that has the form as Fig. 9(b)). The equilibrium for such a Hamiltonian may be explored by Metropolis Monte Carlo [56] or its dynamics through kinetic Monte Carlo approaches. The picture of g-bands can be applied to understand why crystal-like GB structures [50,57] exhibit glass-like behavior [6,58]. Although GBs exhibit a high degree of structural order, they constitute a dense spectrum of local states in which GBs can be trapped. Just as many glass properties are determined by transitions between nearly degenerate states [59], GBs should exhibit similar glass-like properties. Inspired by the analogy with glasses, it may be productive to try to bring the theory of glasses to bear on the study of GBs. A convenient framework for such an approach could be the AdamGibbs relation between dynamic relaxation times in glass-forming liquid and the configurational entropy [59,60]. As discussed above, the configurational entropy of a GB can be obtained from the knowledge of GB states and their probability distribution (in equilibrium or in nonequilibrium processes). This could, in turn, be used to predict such relaxation-time-dependent properties, such as GB viscosity associated with GB sliding [1,6] and GB point-defect sink efficiency under irradiation [1,61]. Earlier work [58] has demonstrated that GB mobility follows a Vogel-Fultcher (rather

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271

than Arrhenius) law. Since the Vogel-Fultcher law can be derived from the Adam-Gibbs relation, this suggests that the link between GB behavior and the behavior of glasses is indeed reasonable [59]. Finally, we consider the implications of the existence of g-bands on GB engineering. The effective width of a g-band e d provides a measure of the variability of GB structure/properties. Large variability suggests the potential which a material and/or class of GBs has for GB engineering. As is now well established [62], processing a material (e.g., through thermomechanical treatment) can be used to manipulate a microstructure to produce higher or lower densities of particular types of GBs (e.g., twins). However, it is also possible to manipulate the properties of individual GBs at fixed microstructure (e.g., through mechanical deformation [52,53]). Systems for which the GB-structure spectrum is wide will be particularly sensitive to this type of manipulation. An interesting parameter that captures this is e d=d, where d is the grain size. Systems for which it is large may be more amenable to GB engineering strategies than those for which it is small. 8. Conclusions

Fig. 9. An example of analysis of the line defects in a GB (with fixed macroscopic DOFs) based on the distribution of GB states. (a) Schematic of (t;f) space. The blue and red levels represent the t spaces (i.e. the cross sections of the (t;f) space) which are similar to Fig. 1(c). The blue level corresponds to the t space with f ¼ 1, where R(i) labels the ith GB state in this space. R(1) and R(2) are related by point-group symmetry; R(1) and R(3) by a DSC translation vector; R(3) and R(4) by point-group symmetry; R(2) and R(4) by a DSC translation vector; and R(5) and R(6) by point-group symmetry. The red level ðiÞ corresponds to the t space with f fixed at a fraction “F”, where Rf labels the ith GB ð1Þ ð2Þ ð1Þ ð3Þ state in this space. R f and Rf are related by a DSC translation vector; R f and Rf by ð2Þ ð4Þ ð3Þ ð4Þ point-group symmetry; R f and Rf by point-group symmetry; and Rf and Rf by a DSC translation vector. (b) Summary of the types of GB line defects between GB states pairs shown in (a). Note that, “APB” indicates that an APB may or may not be present.

GB states are defined as the states corresponding to the local minima of internal energy when a system is under the constraints of bicrystal geometry and the conditions of thermomechanical equilibrium. We searched the GB states of the symmetric tilt GBs in three model materials and the twist GBs in Finnis-Sinclair W by atomistic simulations with two different sampling approaches. The first approach is called conservative sampling, i.e., exploring the GB states by changing the relative displacement of two grains t. The second one is called non-conservative sampling, i.e., exploring the GB states by changing t for various atomic fraction in the GB plane f. From the conservative sampling, we obtained the distribution of GB states in the t space and the energetic spectra over the whole range of misorientation (“g-bands”) due to the existence of multiple GB states associated with each misorientation. We characterized the features of g-bands by the band width and the number of states, and found that they are affected by the bonding character of materials and the bicrystal geometry. We defined the measure of GB metastability Sc to reflect the uncertainty of microscopic GB states induced by the distribution of metastable states. The geometry-dependence of Sc was figured out based on statistics of the simulation data. From the non-conservative sampling, we showed that for symmetric tilt GBs the change of f would not influence the conclusions obtained from the conservative sampling; however, for twist GBs the change of f might be important since many additional states were found and they have much lower energies than those obtained by conservative sampling. The picture of g-bands and GB metastability alter the way we think of GB properties. We revisited the GB thermodynamic properties based on the statistical mechanics of GB states. When a probability function that is consistent with the process of our interest (equilibrium or nonequilibrium) is determined, we can obtain the GB energy as ensemble average. We also revealed that the measure of GB metastability Sc had the physical meaning of the configurational entropy of an idealized nonequilibrium process (Stillinger quench). Finally, we noticed that the existence of multiple GB states might imply that GBs, while possessing crystal-like structures, may display properties analogous to those of glasses, and discussed the possibility of predicting GB dynamic behaviors from the knowledge of GB states based on the theory of glassy state. Appendix A. Simulation supercell size For the case of conservative sampling, the simulation supercell contains only one copy of the GB structural period (Ac); referring to

272

J. Han et al. / Acta Materialia 104 (2016) 259e273

Fig. 1(b), this implies that l1 ¼ l2 ¼ 1. A larger supercell is unnecessary since we focus on homogeneous GBs and all possible relative displacements can be accounted for in such a cell. For the case of non-conservative sampling, The values of l1 and l2 we used are shown in Table A1. In principle, it would be appropriate to use a very large supercell to allow for the formation of periodicities that are large multiples of the GB structural period (Ac). However, the computational work scales with supercell size; hence, we limit the supercell size to keep the problem computationally tractable. We cannot guarantee that additional intrinsic GB states do not exist with even larger periodicity. However, large-scale periodicity implies atomic ordering over large distances, while possible such very long-range periodicity is rare in singlecomponent systems. We expect that the supercell size used in our simulations should capture most intrinsic GB states, with the possible exception of some twist GBs for which a small number of states may be missed. Given the large number of states typical of a general GB, the number of missing states should not be significant. Table A1 The bicrystal configurations used in the non-conservative sampling. With reference to Fig. 1(b), p1 and p2 are the periods of Ac; l1 and l2 are the multiples of these periods in a simulation supercell. Bicrystal geometry S13[100](015) STGB S17[100](014) STGB S5[100](013) STGB S29[100](025) STGB S29[100](037) STGB S5[100](012) STGB S17[100](035) STGB S13[100](023) STGB S5[001] twist GB S13[001] twist GB

p1  p2 (a0) pffiffiffiffiffiffi 1  p26 ffiffiffiffiffiffi 1  p17 ffiffiffiffiffiffi 1  p10 ffiffiffiffiffiffi 1  p29 ffiffiffiffiffiffi 1  p58 ffiffiffi 1  p5 ffiffiffiffiffiffi 1  p34 ffiffiffiffiffiffi 1  13ffiffiffi pffiffiffi p 5  5ffiffiffiffiffiffi pffiffiffiffiffiffi p 13  13

l1

l2

Displacement grid

15 15 15 10 10 15 10 15 4 2

2 2 2 4 2 2 4 2 4 2

4 4 4 4 4 4 4 4 5 5

         

21 17 13 22 31 9 24 15 5 5

Appendix B. Degeneracy due to point-group symmetry According to the principle of symmetry compensation [63], the degree of degeneracy of a bicrystal is equal to the order of the layer group of the corresponding holosymmetric bicrystal configuration divided by the order of the layer group of this bicrystal [37] (a layer group is a three-dimensional space group of a two-dimensional lattice). For example, the configuration of the holosymmetric S5 [100] (012) STGB in W belongs to layer group p2'mm' (the prime operation transforms the upper grain into the lower grain and vice versa); the order of this group is 4. The point R(1) in Fig. 3(a) corresponds to a structure with layer group p1m1, which is of order 2. Hence, the degree of degeneracy of the R(1) state is 2. The layer groups of holosymmetric configurations for the GBs in this study have been summarized in Table A2. The order of the layer group is the largest possible degree of degeneracy associated with the bicrystal symmetry. In order to show the symmetry of the distribution of GB states clearly, in Fig. 3 we colored the t space based on “g-surfaces”. The “g-surface” describes the dependence of the energy on the relative displacement t, where the energy is obtained by energy minimization along the direction perpendicular to the GB plane at each fixed t. The GB states (indicated by the black circles in this figure) are not necessarily located at the minima of the “g-surfaces”. This suggests that the “g-surfaces” obtained in this way do not represent the true GB g-surface as schematically shown in Fig. 1(c) [64,20] (this is unlike the situation for a stacking fault). However, the symmetry of the distribution of GB states does coincide with the symmetry of these “g-surfaces”; they share the plane group consistent with the layer group of the holosymmetric bicrystal configuration (see Table A2).

Table A2 Symmetry of the holosymmetric bicrystal structures and the “g-surfaces” obtained by relaxation perpendicular to the GB planes in conservative sampling. Crystal, bicrystal types

Rotation Indices of GB planesa axis

FCC, STGB FCC, STGB BCC, STGB BCC, STGB DC, STGB DC, STGB BCC, twist GB

[100] [100] [100] [100] [100] [100] [001]

ð0klÞ=ð0klÞ, k ð0klÞ=ð0klÞ, k ð0klÞ=ð0klÞ, k ð0klÞ=ð0klÞ, k ð0klÞ=ð0klÞ, k ð0klÞ=ð0klÞ, k (001)/(001)

þ þ þ þ þ þ

l l l l l l

is is is is is is

odd even odd even odd even

Holosymmetric Plane group bicrystal layer of group (order) “g-surface” c2'mm' (4) p2'mm' (4) p2'mm' (4) c2'mm' (4) p2'11 (2) p2'1bm' (4) p42'2' (8)

c2mm p2mm p2mm c2mm p2 p2mm p4mm

a Plane indices of both grains refer to the median lattice. k < l are positive coprime integers (the only integer that evenly divides both is 1).

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