Equilibrium and stability of triple junctions in anisotropic systems

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Acta Materialia 60 (2012) 349–358 www.elsevier.com/locate/actamat

Equilibrium and stability of triple junctions in anisotropic systems R.A. Marks ⇑, A.M. Glaeser Department of Materials Science and Engineering, 210 Hearst Memorial Mining Building, University of California, Berkeley, CA 94720, USA Received 31 July 2011; received in revised form 23 September 2011; accepted 25 September 2011 Available online 4 November 2011

Abstract When a homophase or heterophase interface involves a crystalline solid, the interfacial energy is expected to depend upon the interface–plane orientation. In this paper, the equations governing equilibrium triple-junction configurations in anisotropic systems are reviewed. These equilibrium conditions were originally derived by considering the differential change in triple-junction interfacial energy associated with a differential change in triple-junction configuration and equating it to zero. However, the derived conditions do not distinguish between a triple-junction configuration that satisfies the equilibrium conditions by residing at a local energy minimum, a local energy maximum or at a saddle point. The present paper develops stability criteria for triple junctions with and without interfaces which have anisotropic energy, which can be used to determine whether triple-junction equilibrium conditions correspond to local energy minima (stable), maxima (unstable) or saddle points. For isotropic systems, there is a single solution to the triple-junction equilibrium conditions, and it is necessarily stable. For anisotropic systems, there are multiple solutions to the triple-junction equilibrium conditions, some of which may be stable, unstable or correspond to saddle points in triple-junction interfacial energy. Microstructure features and interface lengths/areas are expected to play a role in dictating the relative energies of stable configurations. Ó 2011 Published by Elsevier Ltd. on behalf of Acta Materialia Inc. Keywords: Interface energy; Surface energy (anisotropy); Thermodynamic stability; Grain boundary junctions; Torque terms

1. Introduction In his landmark 1948 paper, C.S. Smith [1] stated: “The art of metallography is mature and the forms in which various micro-constituents appear are well known. Investigations almost without end have disclosed the importance of the exact manner of distribution of phases on the physical properties and usefulness of an alloy. Surprisingly, however, relatively little attention has been paid to the forces that are responsible for the particular and varied spatial arrangements of grains and phases that are observed. Like the anatomist, the metallurgist has been more concerned with form and function than with origins.”

⇑ Corresponding author. Present address: Santa Clara University, Department of Mechanical Engineering, 500 El Camino Real, Santa Clara, CA 95053, USA. E-mail address: [email protected] (R.A. Marks).

Smith proceeded to discuss the role of surface and interfacial energetics in microstructures and properties. When materials of different physical form (solid, liquid or gas) or that differ in crystal structure, bulk composition or crystallographic orientation are in contact, interfaces (or surfaces) result. The intersection of three such interfaces occurs along a line termed a triple line or triple junction. Provided that the contacting phases or grains are in mutual chemical equilibrium, the interfaces will seek to adopt a configuration which minimizes the interfacial energy of the material system. The configuration depends on the energies of the three interfaces and, when crystalline phases are involved, the orientation dependencies of the energies. The configuration is described by specifying the angles between the intersecting interfaces and, when crystalline phases are involved, the crystallographic character of the interface plane. King [2] showed how the equilibrium configuration also depends on the triple-junction line energy. This implies that quadruple-junction point energies ultimately have an impact (albeit slight) on the equilibrium

1359-6454/$36.00 Ó 2011 Published by Elsevier Ltd. on behalf of Acta Materialia Inc. doi:10.1016/j.actamat.2011.09.043

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interfacial inclinations at triple junctions as well. However, both these contributions are beyond the scope of this paper. Interfacial energies and even triple-junction configurations have an important impact on several physical phenomena and properties, including the wetting of a solid surface by a liquid, the work of adhesion in wetting and fracture (including the flaw geometries at mono- or bi-material interfaces), the break-up or stabilization of thin films, the densification and coarsening of powder compacts, the geometry of grain-boundary networks, grain-boundary grooving, nucleation energetics at grain boundaries and surfaces, and several mechanical properties in metals, where these properties are determined by the population of special grain boundaries [3]. More recently, the re-examination of grain-growth phenomena has led to the incorporation of a triple-junction drag component [4–7] in the conventional grain-boundary-mobility model [8,9]. The desire to determine relative interfacial energies from triple-junction configurations has led to many studies in which isotropic interfacial energies have been assumed (see Refs. [10–18] as well as references therein), and relative interfacial energies were determined by balancing forces at the junction line. In some cases, additional constraints are assumed which further simplify the analysis. An example is the well-known Young equation, which is used to analyze the results of sessile-drop or wetting experiments [13–18]. It is assumed that the surface of the solid remains flat as the liquid drop spreads or recedes, since mass transport of the solid phase may be expected to be very slow in comparison with liquid-phase redistribution. Thus, a constrained equilibrium involving a force balance in one direction is treated. Although this may be reasonable at low temperatures, the same is not necessarily true at high temperatures [19]. Additional interpretation uncertainties arise when the droplet perimeter intersects multiple grains [20]. Sixty years ago, Herring [21] considered the effects of interfacial/surface-energy anisotropy on the equilibrium configuration of triple junctions. Herring’s result has seen relatively limited use in the interpretation of experimental results because its application requires detailed information regarding the interfacial energies that are involved; in general, there are more parameters or variables than can be determined from the measurement of the triple-junction configuration alone. The complications that arise due to interfacial/surface-energy anisotropy also affect modeling efforts. The resulting “torque” terms which emerge in the description complicate the mathematics, and simplifying assumptions regarding equilibrium geometries have been required in order to derive analytical expressions. Although the importance of anisotropy in determining triple-junction configurations is well established, and Herring provides an equilibrium condition, it is noteworthy that stability conditions have not been developed. In 1963, Mullins [22] pointed out that Herring’s equilibrium condition was a necessary but not sufficient condition for stable equilibrium. The topic resurfaced in the work of Adams et al.

[23] where, in discussing the stability of equilibrium triple-junction configurations, the authors noted: “A final technical point is that Herring’s equations say nothing about stability; they describe both stable and unstable equilibrium of intersecting interfaces. The latter, however, would not be expected to be observable. Further, a curvature driven evolving system with the Herring relations satisfied on triple lines is dissipative and its stationary configurations have stability properties.” In this paper, triple-junction stability criteria are derived, and the possible existence of unstable triple-junction configurations is discussed. 2. Background The equilibrium triple-junction configuration is often determined by executing a force balance along two directions perpendicular to the triple-junction line. The equilibrium conditions can be derived by considering the interfacial-energy change associated with the displacement of the triple-junction line in two perpendicular directions. For an equilibrium configuration, an infinitesimal displacement produces no energy change. This is essentially the approach used by Herring. The ensuing derivation serves as the starting point for subsequent extensions in this paper. Fig. 1a schematically illustrates a triple junction with three interfaces (black lines) drawn from the origin of the x and y axes. The triple-junction line is perpendicular to the plane of the page, and the triple-junction configuration is described by three positive angles, which are measured from the positive x axis to the appropriate interface. If a displacement dx of the triple junction occurs (from point O to point X in Fig. 1a), the angles for each interface will change as illustrated by the gray lines, where the interfaces are assumed to be fixed in space at some arbitrary distance from the triple junction. The interfaces may be expected to be pinned at some distant location, perhaps due to another triple junction or, alternatively, it may be expected that only a certain length of each interface is mobile as the triple junction moves; consequently, a change in inclination occurs with a change in triple-junction location. The interfacial-free-energy change for this displacement (dx) is equivalent to the change in length of each line multiplied by the length of triple-junction line considered (an arbitrary constant L) and the appropriate interfacial energy per unit area of interface ci. To incorporate effects of interfacial-energy anisotropy, terms representing the energy changes due to changes in interfacial inclination must be added. The length change of each interface can be determined by applying the law of sines to a triangle bounded by each pair of black and gray lines and the x axis in Fig. 1a. Consequently, the length of each triangle face can be expressed

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351

sin h1 sin½p  ðh1 þ dh1 Þ sinðdh1 Þ ¼ ¼ dx XA OA

ð1Þ

Using the small-angle approximations sin (dh1)  dh1 and cos(dh1)  1 (which are appropriate in the infinitesimal limit) as well as other trigonometric identities, Eq. (1) simplifies to sin h1 sinðh1 þ dh1 Þ sin h1 þ dh1 cos h1 dh1 ¼ ¼ ¼ dx XA OA OA

ð2Þ

The change in length of interface 1 is given by XA  OA ¼

sin h1 sin h1 þ dh1 cos h1 dx  dx dh1 dh1

¼ ðcos h1 Þdx

ð3Þ

Thus, the contribution of the length change of interface 1 to the total change in triple-junction interfacial free energy dF is Lc1(cos h1)dx. A similar analysis for the other triangles formed with the x axis yields a similar term for each interface. If a displacement in the x direction is considered, the same results are obtained. The contribution due to interfacial-energy anisotropy for a single interface can be expressed as a product of the rate of interfacial-energy change with orientation, oci/ohi, the change in orientation, dhi, the in-plane length of the interface that is reoriented ðXA; XB; XC; YA; YB or YCÞ, and the out-of-plane triple-junction length L. Considering an x-direction triple-junction displacement, and using Eq. (2) one obtains     @c @c1 L XA 1 dh1 ¼ L sin h1 dx ð4Þ @h @h When all terms of types derived in Eqs. (3) and (4) are combined for all three interfaces, the total interfacial-energy change takes the form  @c1 @c @c dF ¼ L sin h1 þ 2 sin h2 þ 3 sin h3  c1 cos h1 @h1 @h2 @h3  c2 cos h2  c3 cos h3 Þdx

ð5Þ

and the partial derivative of F with respect to x is  @F @c1 @c @c Fx ¼ ¼L sin h1 þ 2 sin h2 þ 3 sin h3  c1 cos h1 @x @h1 @h2 @h3  c2 cos h2  c3 cos h3 Þ Fig. 1. Schematic illustration of how the length and orientation of interfaces changes as the triple junction is displaced along (a) the x axis and (b) the y axis. The black lines represent the original position of the interfaces, and the gray lines represent their positions after an infinitesimal displacement in the positive x and y directions.

At equilibrium, the free-energy change for the infinitesimal displacement must be zero, which implies that c1 cos h1 þ c2 cos h2 þ c3 cos h3  

in terms of the described angles and the line-segment length OX , which is equivalent to the triple-junction displacement dx. This enables the change in interfacial lengths to be expressed in terms of dx as well. For example, applying the law of sines to interface 1 yields:

ð6Þ

@c2 @c sin h2  3 sin h3 ¼ 0 @h2 @h3

@c1 sin h1 @h1 ð7Þ

The other equilibrium condition is obtained by displacing the triple-junction line in the y direction, as illustrated in Fig. 1b, and applying the same construction, only forming triangles with the y axis. The law of sines applied to the triangle associated with interface 1 yields

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sin½ðp=2Þ þ ðh1 þ dh1 Þ sinððp=2Þ  h1 Þ sinðdh1 Þ ¼ ¼ dy OA YA cosðh1 þ dh1 Þ cos h1  dh1 sin h1 cos h1 dh1 ¼ ¼ ¼ dy OA OA YA

ð8Þ ð9Þ

Consequently, the change in length of the line that represents interface 1 is YA  OA ¼ 

cos h1 cos h1  dh1 sin h1 dy þ dy dh1 dh1

¼ ðsin h1 Þdy

ð10Þ

Again, expressions can be developed for the contribution due to interfacial-energy anisotropy for a y displacement of the triple junction, and the y displacement analogs to Eqs. (4)–(7) are given as Eqs. (11)–(14), respectively.     @c @c1 L YA 1 dh1 ¼ L cos h1 dy ð11Þ @h1 @h1  @c1 @c @c dF ¼ L cos h1 þ 2 cos h2 þ 3 cos h3 þ c1 sin h1 @h1 @h2 @h3  þc2 sin h2 þ c3 sin h3 dy ð12Þ Fy ¼

 @F @c1 @c @c ¼ L cos h1 þ 2 cos h2 þ 3 cos h3 @y @h1 @h2 @h3  þc1 sin h1 þ c2 sin h2 þ c3 sin h3

c1 sin h1 þ c2 sin h2 þ c3 sin h3 þ þ

@c3 cos h3 ¼ 0 @h3

ð13Þ

@c1 @c cos h1 þ 2 cos h2 @h1 @h2 ð14Þ

The result derived by Herring [21] and those presented here in Eqs. (7) and (14) are equivalent and can be expressed in the following common form:  3  X @c ci^li þ i ^ ni ¼ 0 ð15Þ @hi i¼1 where ^li is a unit vector parallel to interface i, but perpendicular to the triple-junction line, and ^ ni is a unit vector perpendicular to interface i. The vectors must be chosen such that the cross product of ^li and ^ ni for each interface results in the same unit vector parallel to the triple-junction line; i.e., the sense vector of the triple-junction line must be maintained. For the case where all three interfaces are isotropic ð@ci =@hi ¼ 0Þ, Eq. (15) simplifies to 3 X

ci^li ¼ 0

ð16Þ

i¼1

Eq. (16) represents the conventional picture where forces with magnitude proportional to each interfacial free energy and directed parallel to each interface (away from the triple junction) are resolved along at least two directions. At equilibrium, the net force along each direction must be

zero. This is equivalent to constructing a closed triangle by connecting the tail of an interfacial-energy vector (perpendicular to the interface and having a magnitude equal to the interfacial energy) to the head of another [2]. The inclination of these vectors can be adjusted until a closed triangle is formed, thus giving the equilibrium angles between interfaces when balance is achieved. Since the x axis can be arbitrarily chosen, equilibrium is satisfied for any rigid rotation of this equilibrium configuration. A consequence of this degree of freedom is that only two of the interfacial energies can be determined in terms of the third by measuring equilibrium interfacial orientations alone; i.e., relative values may be determined, but absolute values cannot be determined. The triple-junction equilibrium conditions in the anisotropic case involve at least three independent variables; in addition to the relative angles between the three interfaces (two angles), the crystallographic orientation of one interface (one absolute angular position) is required in order to completely describe the triple-junction configuration. Although the angular position of the x axis is arbitrary, the functions ci(hi) must be adjusted appropriately for the chosen reference-axis position. In addition to the three angular positions of the interfaces, the torque terms (oci/ ohi) are usually not known. Experimentally, one is limited to measuring the angular positions of the interfaces and hoping that enough information about the interfacial energies can be measured independently [24]. As a result, simplifying assumptions, such as those based on symmetries among the three phases [25], have been invoked when using Eq. (15); however, as will be pointed out shortly, even these systems do not present a completely soluble case. If the interfacial energies and torque terms are known, one can use a triangle construction similar to that for the isotropic-energy case; however, the n-vector developed by Hoffman and Cahn [26] must be used. As in the isotropic case, the n-vector has a component with magnitude and direction dictated by the interfacial free energy and angular position, respectively. However, the n-vector also includes a component perpendicular to the angular-position component and of magnitude equal to the appropriate torque term for that interface. Forming a closed triangle with nvectors yields an equilibrium triple-junction configuration. However, multiple interfacial orientations may have the same n-vector (e.g., at certain orientations on both sides of an interfacial-free-energy minima or cusp, or along a section of the Wulff plot that is described by a sphere intersecting the origin [27]). Alternatively, a given orientation may be represented by multiple n-vectors (i.e., at a cusp in the interfacial energy). Consequently, multiple equilibrium configurations may exist. This is also expected, since there are more unknowns than relationships among them implied by Eq. (15). King [2] has recently developed various constructions for finding equilibrium triple-junction configurations in anisotropic systems, and showed that there are ranges of allowed and forbidden angular posi-

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tions of two interfaces when the third is assumed to be fixed at an orientation described by a cusp in its interfacial-free-energy plot. Recently, Rohrer and co-workers [28–30] applied an iterative numerical method developed by Morawiec [31], which reconstructs the grain-boundary-energy distribution from the crystallographic and geometric characteristics of thousands of triple junctions. The method has been applied to MgO [28,29] and Ni [30]. In these systems, grain boundaries of lower energy have a relatively higher frequency of occurrence. 3. Derivation of stability conditions In the previous section, equilibrium conditions for triple-junction configurations were derived. It was assumed that the system is at equilibrium if the interfacial freeenergy change of the system is zero for an infinitesimal triple-junction-line displacement. However, this approach does not determine the stability of triple-junction configurations satisfying the equilibrium conditions, and nothing precludes the previously derived equilibrium conditions from being satisfied at configurations of local maximum energy. If the configurations predicted by Eqs. (7) and (14) are to be stable, it is necessary to show that the triple-junction interfacial free energy is, in fact, a minimum with respect to any triple-junction-line displacement. Since displacements with both x and y components are possible, this implies that the interfacial energy of the system varies in two dimensions, and thus, it is not adequate simply to show that the second derivatives of the triple-junction interfacial energy with respect to x and y are positive. Instead, a derivative test for locating extrema in multivariable functions, which incorporates all possible second derivatives, must be performed. If the equilibrium configurations determined by Eqs. (7) and (14) are stable, F xx F yy  F 2xy > 0 and F xx > 0 must be satisfied. If F xx F yy  F 2xy > 0, but F xx < 0, the configuration is unstable. If F xx F yy  F 2xy < 0, the configuration is at a saddle point in energy and is unstable with respect to some displacements, but stable with respect to others. If F xx F yy  F 2xy ¼ 0, higher-order derivatives are required in order to determine the nature of the equilibrium. The first partials of F, Fx and Fy, are provided by Eqs. (6) and (13), but are expressed in terms of the variables h1, h2 and h3. Using the chain rule, the second derivatives of interest may be expressed as F xx ¼

@ 2 F @F x @F x @h1 @F x @h2 @F x @h3 ¼ þ þ ¼ @x2 @x @h1 @x @h2 @x @h3 @x

ð17Þ

2

F yy ¼

@ F @F y @F y @h1 @F y @h2 @F y @h3 ¼ þ þ ¼ @y 2 @y @h1 @y @h2 @y @h3 @y

ð18Þ

F xy ¼

@2F @F x @F x @h1 @F x @h2 @F x @h3 ¼ ¼ þ þ @x@y @y @h1 @y @h2 @y @h3 @y

ð19Þ

From Eqs. (6) and (13), it follows that

353

! @F x @ 2 ci @ci @ci ¼L sin hi þ cos hi  cos hi þ ci sin hi @hi @hi @hi @h2i ! @ 2 ci ¼L þ ci sin hi ð20Þ @h2i ! @F y @ 2 ci @ci @ci ¼ L cos hi  sin hi þ sin hi þ ci cos hi @hi @hi @hi @h2i ! @ 2 ci ¼ L þ ci cos hi ð21Þ @h2i The law-of-sines constructions, which result in Eqs. (2) and (9), can be used to provide the remaining terms. Eq. (2) gives the variation of h1 with x as dh1 sin h1 sin h1 þ dh1 cos h1 ¼ ¼ ð22Þ dx XA OA which may be written as @h1 sin h1 sin h1 ¼ ¼ @x XA OA

ð23Þ

since XA approaches OA in the differential limit. In Fig. 1, the distances OA, OB and OC are arbitrarily chosen to be the distances d1, d2 and d3 from the origin of the figure. Applying the law of sines to the other triangles in Fig. 1a and considering the differential limit yields the following general expression for the partial derivative of hi with respect to x. @hi sin hi ¼ ð24Þ @x di The partial derivative of hi with respect to y is developed using Eq. (9). @hi  cos hi ¼ ð25Þ @y di Using Eqs. (20), (21), (24), and (25), the desired second partials, Eqs. (17)–(19), take the form " F xx ¼ L

@ 2 c1 þ c1 @h21

!

sin2 h1 @ 2 c2 þ þ c2 d1 @h22

!

sin2 h2 þ d2

@ 2 c3 þ c3 @h23

!

sin2 h3 d3

#

ð26Þ " F yy ¼ L

!

!

!

@ 2 c1 cos2 h1 @ 2 c2 cos2 h2 @ 2 c3 cos2 h3 þ c1 þ þ c2 þ þ c3 d1 d2 d3 @h21 @h22 @h23

#

ð27Þ " F xy ¼ L

2

!

2

!

2

!

@ c1 sinh1 cosh1 @ c2 sinh2 cosh2 @ c3 sinh3 cosh3 þ c1 þ þ c2 þ þ c3 d1 d2 d3 @h21 @h22 @h23

#

ð28Þ

Inserting Eqs. (26)–(28) into the stability criterion F xx F yy  F 2xy and simplifying yields  2  2 9 8 2 sin ðh1 h2 Þ @ c1 @ c2 > > 2 þc1 2 þc2 > > d1d2 @h @h > > > > = <  21  22  2 @ c3 sin ðh2 h3 Þ @ c2 2 2 ð29Þ F xx F yy F xy ¼L þ @h2 þc2 @h2 þc3 d2d3 > > 2 3 > >     > > 2 > > @ 2 c3 :þ @ 2 c1 þc 1 Þ; þc3 sin dðh3 3dh 1 @h2 @h2 1 2

3

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Recalling that stable equilibrium configurations require that Fxx > 0 and F xx F yy  F 2xy > 0, stable equilibrium is achieved if ! ! @c1 sin2 h1 @ 2 c2 sin2 h2 þ c þ þ c 1 2 d1 d2 @h21 @h22 ! @ 2 c3 sin2 h3 þ þ c 3 d3 @h23 ð30Þ

>0 and  2  2 9 8 2 sin ðh1 h2 Þ @ c1 @ c2 > > þ c þ c > > 1 2 d1d2 @h21 @h22 > > > > = < 2  2  2 sin ðh2 h3 Þ @ c3 @ c2 þ @h2 þ c2 @h2 þ c3 >0 d d 2 3 > > 2 > >  2 3  2 > > 2 > > @ c3 : þ @ c1 þ c 1Þ ; þ c3 sin dðh3 3dh 1 @h2 @h2 1 1

ð31Þ

3

are satisfied. If any or all @ 2 ci =@h2i are sufficiently negative, the equilibrium triple-junction configuration predicted by Eq. (15) may be unstable. It remains to show whether or not negative values of @ 2 ci =@h2i of this magnitude exist, and if so, whether or not equilibrium triple-junction configurations involving interfaces of such orientations are observable.  When all three interfaces are isotropic @ 2 ci =@h2i ¼ 0 , the stability conditions simplify to c1 c c sin2 h1 þ 2 sin2 h2 þ 3 sin2 h3 > 0 ð32Þ d1 d2 d3 and c1 c2 cc cc sin2 ðh1  h2 Þ þ 2 3 sin2 ðh3  h3 Þ þ 3 1 d 1d 2 d 2d 3 d 3d 1  sin2 ðh3  h1 Þ >0

ð33Þ

As a result, any equilibrium triple-junction configuration in an isotropic system is stable, since all sine terms are squared, and all interfacial-free-energy terms are necessarily positive. Furthermore, the unique solution to the equilibrium conditions necessarily represents the lowest-energy triple-junction configuration. However, the analysis of the anisotropic case in the following section will show that stable equilibrium is not inherent. 4. Discussion A comparison of the equilibrium triple-junction configurations in systems with exclusively isotropic interfaces and those involving anisotropic interfaces reveals several key differences. When all three interfaces are isotropic, Eq. (16) uniquely defines the equilibrium angles h1, h2 and h3. Equilibrium stability conditions for isotropic systems (Eqs. (32) and (33)) show that this unique solution is necessarily stable. The left-hand side of Eq. (33) hinges on the relative angles of the equilibrium angles and is thus independent of the placement of the coordinate system;

the equilibrium angles are unaffected by a rigid rotation of the set of interface planes. In this sense, the number of solutions to Eq. (16) is infinite, and the solution can be thought of as infinitely degenerate. The introduction of interfaces of anisotropic energy is accompanied by the appearance of torque terms in the equilibrium condition(s) (Eq. (15)). Owing to the torque terms, the equilibrium condition acquires a more complicated dependence on the angular positions of the interfaces, the nature of which is specific to each anisotropic triple-junction system. If triple-junction configurations in anisotropic systems are measured, but the effects of torque terms are neglected, incorrect relative interfacial energies result. An example in which torque terms are particularly important involves surface intersections of twins in Cu [25]. Rather than always forming a grain-boundary groove as would be expected in isotropic systems, the twin can also form a ridge, and the resolved components of the interfacial tensions are all directed into the solid. The potential magnitude of the errors hinges on the relative magnitude of the torque terms and interfacial energies. Some insight can be derived by applying a simple model which assumes that the surface energy scales with the nearest-neighbor bond energy and the number of missing nearest neighbors; effects of lattice relaxations, surface reconstructions or impurity segregation are neglected. If a vector of length c is used to represent the magnitude of the surface energy, and two angles and / define the direction normal to the surface of interest, the resulting c–h–/ (Wulff) plot for a simple cubic system consists of eight intersecting spheres which pass through the origin, with minimum surface energies along h1 0 0i directions. The orientation dependence of interfacial energy between three adjacent cusps in a Wulff plot is described by a sphere that passes through the three cusps and the origin of the plot [32]. For a triple junction, only a two-dimensional c–h slice of the interfacial Wulff plot needs to be considered, and the two-dimensional analog of the described spherical model is that each segment of the relevant interfacial Wulff plot will be described by a circle connecting two cusps and passing through the origin. Table 1 gives the range of values of torque term as a multiple of the interfacial energy for an increasing number of symmetric cusps in the two-dimensional Wulff plot. Clearly, for a low number of cusps, which may be expected at elevated temperatures, the torque terms are comparable with the interfacial energy. In fact, for up to six cusps, one may argue that torque terms are more important than interfacial energy terms, since the torque term multiples are in terms of plus or minus c. If more than six cusps exist, one expects that additional cusps will be at a different value of the interfacial energy and correspond to distinct planes. If one maintains the assumption that the interconnecting segments are circles passing through the origin, the torque terms will be greater than indicated in Table 1 for the set of cusps at lower energy, and less than those indicated in Table 1 for the set of cusps at higher energy. Experimental observations, to be

R.A. Marks, A.M. Glaeser / Acta Materialia 60 (2012) 349–358 Table 1 Range of values of torque terms against an increasing number of symmetric cusps in an interfacial Wulff plot in which neighboring cusps are connected by a circular segment passing through the origin of the plot. No. of cusps

Torque term (±ccusp)

Torque term (±cavg)

4 6 8 12 24

1.000 0.577 0.414 0.268 0.132

0.785 0.524 0.393 0.262 0.131

discussed shortly, indicate that the actual values of torque terms in some systems may exceed those in Table 1. Triple junctions in anisotropic systems are expected to exhibit multiple or even an infinite number of solutions to the equilibrium conditions [33]; however, the level of degeneracy seen in isotropic systems is not expected. The ranges of interfacial positions satisfying the anisotropic equilibrium conditions, as described by King [2], arise because the torque term of an interface lying at the cusp can assume a range of values. Thus, the terms for the other interfaces can satisfy the equilibrium conditions at various orientations (angles), values of interfacial free energy and/ or values of the torque term. The freedom to alter the relative angles between interfaces and the orientation of individual interfaces changes one or more terms in the stability criteria (Eqs. (30) and (31)). Thus, aside from systems exhibiting certain symmetries [33], it appears unlikely that multiple solutions with identical properties (degenerate) will exist in anisotropic systems. However, systems exhibiting special symmetries may be required to determine anisotropic interfacial energies using Herring’s equations. As mentioned earlier, such an attempt was made by Mykura [25] involving the intersection of pairs of twin boundaries in a metal with a free surface. Owing to the symmetry of the twin, Mykura concluded that the interfacial energies as well as the two torque terms for the two surfaces flanking the twin could be equated. Thus, a problem with six unknowns and two equations is reduced to one with four unknowns and two equations. Mykura further assumed that the torque term associated with the twin could be neglected, owing to the twin’s low energy. In light of the prior discussion, it is suggested that it would be just as valid to neglect the twin-boundary energy term instead or, alternatively, to neglect both the twin-boundary energy and twin torque term. In the latter case, the problem is treated as that of the decomposition of a free surface into a hilland-valley structure, where the equations governing this phenomenon have been extracted from Herring’s triplejunction equilibrium conditions by Gjostein [34]. In addition to there being multiple solutions to the anisotropic triple-junction equilibrium conditions, it has been shown that, in principle, these solutions may be stable, metastable, unstable or coincide with local saddle points in total triple-junction interfacial energy. In contrast with isotropic systems, the stability criteria in anisotropic systems include the second derivative of the interfacial-energy

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functions. When Wulff plots consist of circular segments that intersect the origin, it can easily be shown that, for any such circular segment, @ 2 c=@h2 ¼ c. Thus, it may be concluded that, for any triple junction consisting solely of interfaces obeying this circular-segment model, the stability of the triple-junction configurations is undetermined. (However, if at least one interface resides exactly at a cusp orientation, one may suppose that the equilibrium is stable.) This further implies that any point of a Wulff plot that makes an inner tangent to a circle passing through the origin (see circles centered in the upper two quadrants of Fig. 2) may be classified as an orientation satisfying @ 2 c=@h2 < c. Furthermore, if the segment connecting any two points along a smooth portion of the interfacial Wulff plot lies outside a circle connecting the two points and passing through the origin, a portion of this segment must satisfy @ 2 c=@h2 < c (see circles centered in the lower two quadrants of Fig. 2). Having shown the characteristics of an interfacial Wulff plot that are necessary for unstable triple-junction configurations to exist, it remains to: (1) show that at least some real materials systems have Wulff plots exhibiting these features and (2) assess whether or not orientations associated with @ 2 c=@h2 < c are expected to persist at a triple junction. The circle centered in the lower-right quadrant of Fig. 2 is intentionally chosen such that the interfacialenergy curve makes two outside tangents with the circle and can be related to a construction originally developed by Herring [32] and later illustrated by Winterbottom [35] for predicting the relative stability of planar surfaces and

Fig. 2. Example of a Wulff plot containing regions in which @ 2 c=@h2 < c (bold). The function 2 þ sin2 ð2hÞ was used to generate the plot. The circles centered in the upper two quadrants illustrate cases where the plot forms an inner tangent, and the tangent point is consistent with an orientation that satisfies @ 2 c=@h2 < c. The plot forms outer tangents with the circles centered in the lower quadrants, and it is clear that @ 2 c=@h2 < c is not satisfied at these points of tangency.

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a hill-and-valley structure. According to this construction, an interface of any orientation between those defined by the two tangent points will succumb to a hill-and-valley morphology. Since the criterion for a hill-and-valley structure concurrently guarantees that the Wulff plot contains a region in which @ 2 c=@h2 < c, and planar surfaces have been observed to decompose into hill-and-valley structures [36,37], this implies that at least some materials exhibit Wulff plots in which @ 2 c=@h2 < c for certain orientations. Instabilities in triple junctions satisfying Herring’s equilibrium conditions have been predicted for systems with as little as ±10% variation in interfacial free energy using a function of the same form as used in Fig. 2 [33]. However, the extent of negative curvature is ultimately responsible for making unstable equilibria apparent, and the prior statement coupled with the preceding discussion shows that extensive interfacial-energy variation is not required for portions of an interfacial Wulff plot to have sufficiently negative curvature. Conversely, as discussed more fully later, a triple junction may still be stable, while one or even two interface(s) reside(s) at an orientation where @ 2 c=@h2 < c, depending upon the magnitudes of the other parameters in the stability criteria (Eqs. (30) and (31)). However, it has just been shown that any interface satisfying @ 2 c=@h2 < c would decompose into a hill-and-valley morphology.1 Thus, one has two relevant stability criteria: one for the local balance at the triple junction, and another for the more global stability of the interfaces involved in the junction. Ultimately, one expects interfaces to migrate and transform in a fashion to reduce the total free energy of the system rather than just satisfying a condition at the triple junctions. The choice of which, if any, of the many possible solutions to the triple-junction equilibrium conditions a system adopts must ultimately be linked to which configuration minimizes the total interfacial energy. Although one does not expect unstable configurations to exist at equilibrium, there may be a few situations under which unstable configurations that do indeed satisfy the equilibrium conditions may be observed. For example, at elevated temperatures, it is possible that certain stable configurations may form, but upon cooling and perhaps the emergence of more cusps (or minima) in an interfacialfree-energy Wulff plot, these initially stable configurations may become unstable in a manner analogous to spinodal decomposition [38]. At elevated temperatures, the interface(s) may have resided near a cusp orientation (compared with the next-nearest cusp), but may be relatively far from adjacent cusps as more form at low temperatures. The present authors suspect that at least some of such “frozen-in” configurations would satisfy Herring’s equations at the 1

One can consider triple lines that intersect either the undecomposed unstable interface or one that has decomposed into a hill-and-valley structure. In the latter case, in assessing stability of the triple junction one would only consider the stable terminating facet that intersects the triple line.

lower temperatures, owing to the multiplicity of possible solutions. The lengths of the interfaces, di, do not appear in the triple-junction-equilibrium conditions (Eqs. (15) and (16)). The equilibrium hinges solely on the local geometry and associated energetics at the triple junction, and all configurations that preserve the angular relationships and the interface plane(s) in anisotropic systems are in this sense equivalent. It is not possible to assess the relative energies of states that could be connected by continuous shifts of the anchoring points, and thus no guidance on the allowed direction of change emerges. A rigid shift of the triple junction and the pinning-point positions does not affect the equilibrium, but would alter the interfacial lengths. It is noteworthy that the interface lengths are prominent in the stability criteria of triple junctions with isotropic interfaces (Eqs. (32) and (33)) and of triple junctions that include interfaces of anisotropic energy (Eqs. (30) and (31)). The stability criteria include terms of the form ci þ @ 2 ci =@h2i , which can, in principle, be positive, zero or negative, and these terms are scaled by coefficients that are inherently positive, but whose magnitudes depend inversely on di. One can systematically adjust the signs and magnitudes of the three ci þ @ 2 ci =@h2i terms and assess the values of Fxx and F xx F yy  F 2xy ; the results are summarized in Table 2. A few trends and features merit emphasis. When ci þ @ 2 ci =@h2i > 0 for all three interfaces, and each individual interface is stable relative to a hill-and-valley decomposition, Fxx > 0 and F xx F yy  F 2xy > 0, and the equilibrium is stable regardless of the lengths of the interfaces. When ci þ @ 2 ci =@h2i > 0 for two interfaces and ci þ @ 2 ci =@h2i ¼ 0 for the third, the equilibrium is also stable. As before, no ability to assess the relative energies of stable configurations emerges, and no direction of evolution is apparent. At the other extreme, when ci þ @ 2 ci =@h2i < 0 for all three interfaces, and each individual interface is unstable relative to a hill-and-valley decomposition, F xx < 0; F xx F yy  F 2xy > 0, and the equilibrium of the undecomposed interface is unstable regardless of the lengths of the interfaces. When ci þ @ 2 ci =@h2i < 0 for two interfaces and ci þ @ 2 ci =@h2i ¼ 0 for the third, the equilibrium is also unstable. When ci þ @ 2 ci =@h2i ¼ 0 for two or more interfaces, F xx F yy  F 2xy ¼ 0 and the stability of the equilibrium cannot be assessed with the described level of analysis. A configuration in which ci þ @ 2 ci =@h2i > 0 for one interface, ci þ @ 2 ci =@h2i ¼ 0 for a second, and ci þ @ 2 ci =@h2i < 0 for the third invariably leads to a saddle point since F xx F yy  F 2xy < 0 regardless of the lengths of the interfaces. Finally, two situation arise in which the signs of Fxx and F xx F yy  F 2xy depend upon the specific values of di. Since the combination ci þ @ 2 ci =@h2i > 0 for two interfaces and ci þ @ 2 ci =@h2i ¼ 0 for the third leads to a stable equilibrium, it follows that there must be some range of values of di for which the combination ci þ @ 2 ci =@h2i > 0 for two interfaces and ci þ @ 2 ci =@h2i < 0 for the third still yields Fxx > 0 and F xx F yy  F 2xy > 0, and a stable triple-junction configuration. The same situation arises when

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Table 2 Classes of triple-junctions based on the energy characteristics and stability of the participating interfaces in an anisotropic system; the stability criteria involving Fxx and F xx F yy  F 2xy are detailed in Eqs. (30) and (31). c1 þ @ 2 c1 =@h21

c2 þ @ 2 c2 =@h22

c3 þ @ 3 c3 =@h23

Fxx

F xx F yy  F 2xy

Nature of equilibrium

>0 0 0 0 <0 <0 <0 <0 <0 <0

>0 >0 0 0 0 <0 <0 0 >0 <0

>0 >0 >0 0 0 0 <0 >0 >0 >0

>0 >0 >0 0 <0 <0 <0 f(di) f(di) f(di)

>0 >0 0 0 0 >0 >0 <0 f(di) f(di)

Stable for all di Stable for all di Undetermined Undetermined Undetermined Unstable Unstable Saddle point Stable, saddle pt, unstable, undetermined Stable, saddle pt, unstable, undetermined

ci þ @ 2 ci =@h2i < 0 for two interfaces and ci þ @ 2 ci =@h2i > 0 for the third. For these two situations, it is necessary to consider the coupled changes in the lengths of all three interfaces to assess the effect on stability, and their effects on the coefficients that multiply the ci þ @ 2 ci =@h2i terms. In principle, for given interface energetics, triple-junction configurations could arise that are stable, lie at a saddle point, are unstable or whose stability cannot be assessed. Certain configurations would thus be avoided in an evolving system, saddle point configurations could be identified, but the relative energies of stable configurations would not be evident. The preceding discussion begs the question of the most relevant interfacial length to consider when evaluating triple-junction stability. When deriving the triple-junction equilibrium conditions, it is possible to equate the length of all three interfaces, progressively reduce this length, and thereby consider segments of equal infinitesimal length emanating from the triple junction. This approach was used in prior work [33] to weigh all interfaces equally. For a given d1:d2:d3 ratio, multiplying each length by a constant factor k does not change the sign of the terms in Eqs. (30) and (31), and thus has no influence on the nature of the stability. However, changing the d1:d2:d3 ratio can alter the sign of the sums and thus the stability. In real materials, the microstructure may define the relevant lengths. In such cases, length terms and phase fractions must be incorporated in an assessment of triple-junction and microstructural stability. In principle, the triple-junction equilibrium and stability criteria explored in this paper could be arrived at by differentiating a more global thermodynamic function. Such a function would help identify the lowest-energy triple-junction configuration in systems with multiple solutions to the equilibrium conditions (Eqs. (7) and (14)). Prior efforts to construct such a function based on the derivatives in this paper (Eqs. (6), (13), (24), and (25)) suggest that the integration of the total differential of this function would be path dependent [33], which is inconsistent with a true thermodynamic potential. The result is not surprising, since the anchor points of the interfaces can change as the microstructure evolves. Furthermore, not all possible triple-junction configurations are accessible by solely x- and y-directed translations for a given set of initial anchor points. Thus the

treatment used in deriving the equilibrium conditions is only valid for infinitesimal displacements of the triple-junction line. The multiplicity of solutions to the equilibrium conditions impairs quantitative measurements of interfacial energies. Even the combination of the equilibrium and stability conditions provides little guidance on assessing the relative energies of possible solutions. These issues are explored and the results of Herring’s equilibrium condition with those obtained from considering the total interfacial energy of a system are compared in a companion paper [39]. The comparison suggests a viable method for determining interfacial energies and their dependence on orientation. 5. Summary A review of the equilibrium conditions for triple-junction configurations in isotropic and anisotropic systems has been provided. The derivation follows an approach similar to that of Herring [21], which involves considering the total interfacial energy change associated with displacements of the triple-junction-line position. The derived equilibrium conditions do not address the issue of equilibrium stability. Stability conditions have been derived which show that it is indeed possible for the equilibrium conditions to be satisfied by unstable configurations if @ 2 c=@h2 < c for any or all of the interfaces involved in the triple junction. Furthermore, it is plausible that many interfacial Wulff plots have this characteristic. However, the resulting configurations are not expected to persist unless stabilized for kinetic reasons. Otherwise, any orientation satisfying @ 2 c=@h2 < c would decompose into a hill-and-valley structure, and the interfaces intersecting at a triple junction would always contribute to a stable triple junction configuration, as previously suggested by Adams et al. [23]. Unfortunately, the combination of the equilibrium triple-junction conditions and stability criteria do not identify which, if any, of the potentially multiple stable equilibrium configurations is preferred or lowest in energy. The present authors believe that changing the microstructure may change the preferred configuration. An exploration of this possibility, based on an alternative route to examining the relative stability of competing triple junction configurations, is the subject of a separate contribution.

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Acknowledgments A.M.G. acknowledges the support of the Division of Materials Research of the National Science Foundation, through Grant No. DMR-0606121. The authors wish to acknowledge and thank W. Craig Carter and Greg Rohrer for their careful reading of and insightful comments on an earlier version of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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