## Abstract

In the presence of sufficiently strong surface energy anisotropy, the equilibrium shape of an isothermal crystal may include corners or edges. Models of edges have, to date, involved the regularization of the corresponding free-boundary problem resulting in equilibrium shapes with smoothed out edges. In this paper, we take a new approach and consider how a phase-field model, which provides a diffuse description of an interface, can be extended to the consideration of edges by an appropriate regularization of the underlying mathematical model. Using the method of matched asymptotic expansions, we develop an approximate solution which corresponds to a smoothed out edge from which we are able to determine the associated edge energy.

## 1. Introduction

The effect of anisotropic surface energy on determining the equilibrium shape of a solid crystal in contact with its own liquid phase is a classical problem in material science. The anisotropy of the surface energy of the interface is an expression of the underlying symmetry of the atomic lattice in the crystal. This, in turn, affects the macroscopic shape of the equilibrium interface. This shape is a circle (or sphere in three dimensions) in the absence of anisotropy. The presence of anisotropy changes the shape and, for strong anisotropy, it may be energetically favourable for the interface to exclude a range orientations (so-called ‘missing orientations’) that correspond to higher surface energies, resulting in the formation of corners, edges or facets in the interface shape.

This situation has been studied for over a century, dating back to Gibbs (1878) and Curie (1885). The first solution was given by Wulff (1901) using the well known Wulff construction which employs the polar plot of the surface energy. The Wulff theory was reviewed and refined by Herring (1951) and subsequently further interpretations have been developed by Burton *et al*. (1951), Frank (1963), Hoffmann & Cahn (1972), Cahn & Hoffmann (1974) and Andreev (1981). In two dimensions, the surface energy, , may be expressed as dependent on the polar angle of the normal vector to the interface, *θ*. The equilibrium theory shows that interfaces include corners when the so-called ‘surface stiffness’, , is negative.

The above developments are based on macroscopic models in which the interface is represented as a surface with no thickness, so-called ‘sharp interface’ models. The notion that an interface has an intrinsic thickness dates back to Lord Rayleigh (1892) and van der Waals (1893) in the context of an interface between two fluids. Since then this notion has been generalized to a wide range of interfaces and phase transitions (see Rowlinson & Widon (1989) for a comprehensive review). In the last two decades, these ideas have been applied to solid–liquid interfaces using so-called phase-field models. These models introduce an order parameter, *ϕ*, known as the phase field. The phase field takes a constant prescribed value in each bulk phase. Interfaces are represented as regions in which the value of the phase field changes between the bulk phase values. The contours of the phase field describe the shape and location of the solid–liquid interface. These models date back to Caginalp (1985, 1986) and Langer (1986). They can be considered as a mathematical device that allows a reformulation of the free-boundary problems associated with the sharp interface formulation. They have been successfully developed to describe a wide range of phase transitions, including eutectic and peritectic alloys (Karma 1994; Wheeler *et al*. 1996; Nestler & Wheeler 1998). They have also been used to model numerically a wide range of microstructure including dendritic growth (Kobayashi 1992; Wheeler *et al*. 1993; Karma & Rappel 1997) and Ostwald ripening (Warren & Murray 1996).

Surface energy anisotropy was first included in phase-field models by Kobayashi (1992). Subsequently, Wheeler & McFadden (1996, 1997) made a connection with the Cahn–Hoffman *ξ*-vector which provides a powerful mechanism to understand anisotropic interfaces in the context of a phase-field model.

The dynamic evolution of an interface with anisotropic surface energy is a problem that has been extensively studied. A particularly interesting situation is the evolution of a planar interface in a direction for which the surface stiffness is negative, and corresponds to a missing orientation. In this case, the underlying mathematical problem is a backward parabolic partial differential equation, resulting in an ill-posed problem so that small wavelength disturbances grow preferentially causing ‘blow up’ of the solution on the smallest length scale. This occurs in both the free-boundary formulation (Di Carlo *et al*. 1992) and the phase-field model (Wheeler 1999).

To date, the focus has been on the free-boundary formulation which involves applying a regularization by including higher terms to the surface energy (see Stewart & Goldenfeld 1992; Liu & Metiu 1993; Golovin *et al*. 1998, 1999*a*,*b*, 2001; Savina *et al*. 2003). This may involve adding surface diffusion or evaporation or more simply allowing the surface energy to depend also on curvature so that(1.1)where is the curvature of the interface and *β* is a positive constant. This can be expected to prevent the formation of corners by making high curvature of the interface energetically unfavourable thus smoothing out the corners in the equilibrium shape and damping the growth of short wavelength disturbances in the evolution of an interface. Recently, Spencer (2004) found asymptotic solutions that describe the shape of a smoothed corner in the equilibrium shape in the limit .

We consider the regularization of the phase-field model to investigate the smoothing of corners. Numerical computations of regularized phase-field models of substrate systems in thin film growth with strong anisotropy have been conducted by Wise *et al* (2005), Ratz *et al*. (2006) and Du *et al*. (2005).

In this paper, we confine our attention to two-dimensional interfaces and build on the anisotropic phase-field model that employs the generalized *ξ*-vector of Wheeler and McFadden (1996, 1997). We show that in the limit of the coefficient of the regularizing term going to zero a thin edge region is established within the interface, which connects the two adjacent interfaces. Within the edge region the interface orientation varies smoothly through the range of missing orientations predicted by both the free-boundary and phase-field models in the absence of any regularization. We use the method of matched asymptotic expansions to find an asymptotic solution in the five different adjacent regions comprising the two bulk phases, the two adjoining interfaces and the edge region. In the edge region itself we derive, at leading order, a nonlinear fourth-order partial differential equation for the interface orientation. Despite the nonlinearity, we are able reduce its solution to a single numerical quadrature. The edge problem has the character to a phase transition of an interface and is governed by a steady Allen–Cahn equation where the double-well nature of the associated potential is related to the form of . This analogy of the edge to a phase transition at an interface was first made by Cabrera (1963, 1964) and later by Stewart & Goldenfeld (1992) in the context of the sharp interface model. We go on to define and evaluate the edge energy associated with the presence of the edge region. In addition, we show that there is an underlying stress tensor associated with the regularized phase-field equation, which is a natural extension of the stress tensor for the phase-field model in the absence of regularization (Wheeler & McFadden 1997).

In §2, we set out the basic elements of the Cahn–Hoffman *ξ*-vector theory developed for the sharp interface model. We also describe the phase-field model and how it can be extended to the anisotropic case by the introduction of a generalized *ξ*-vector. In §3, we formulate the regularization of this model appropriate to the consideration of edges. In §4, we employ the method of matched asymptotic expansions to analyse corners and compute the corner energy.

## 2. Theory of interfaces with anisotropic surface energy

### (a) Sharp interface theory: the *ξ*-vector

Here, we consider the isothermal equilibrium shape of a solid in contact with its liquid phase when the surface energy of the solid–liquid interface, *γ*, depends on the local orientation of the interface. We restrict our consideration to a cylindrical solid crystal. Specifically, the interface, , is considered to be a cylindrical surface whose cross-sectional shape is represented as a curve, , in the -plane (see figure 1). An edge in is represented by a straight line parallel to the *z*-axis emanating from a corner in at which its gradient is discontinuous.

In the simple case where the surface energy is isotropic (i.e. *independent* of local interface orientation), the equilibrium shape, , is a circle, whose radius is related to the temperature of the system, , through the Gibbs–Thomson equation(2.1)Here, is the value of the surface energy, , is the curvature of the interface, is the free energy density difference between the solid and liquid phases, is the latent heat per unit volume and is the equilibrium melting temperature of the material. The Gibbs–Thomson equation results from minimizing the total free energy of the system, comprising the contributions from the two bulk phases and the interface through its surface energy, subject to the constraint of constant volume of solid.

When the surface energy depends on the local orientation of the interface, we represent it by(2.2)where describes the dependence of the surface energy on the orientation of the interface through the vector , the outward unit normal to the solid/liquid interface. In this situation, Cahn & Hoffman (1974) showed that the Gibbs–Thomson equation (2.1) becomes(2.3)Here, is the so-called Cahn–Hoffman *ξ*-vector and is the surface divergence operator on the interfacial surface . Taylor *et al*. (1992) provided the following convenient definition of the *ξ*-vector: first extend the domain of the argument of from unit to general vectors by making a homogeneous degree one extension,(2.4)for all non-zero vectors . With this definition, the Cahn–Hoffman *ξ*-vector is defined by(2.5)where subscript *i* represents the *i*th component of a vector. It follows from this definition that the *ξ*-vector has the properties(2.6)The above definition (2.5) for in fact holds more generally for fully three-dimensional interfaces. However, in the cylindrical case, if we express the anisotropy through the dependence of *γ* on the angle of the unit normal, , to (say) the *x*-axis, denoted by *θ*, then can be expressed as(2.7)where and are the unit vectors associated with the cylindrical polar coordinates *r* and *θ*. It follows that the modified Gibbs–Thomson equation (2.3) is(2.8)

There is geometrical interpretation of the *ξ*-vector which follows from its definition, given by equation (2.5), in which it is described as a vector perpendicular to the inverse *γ* polar plot: .

Equation (2.3) is satisfied by and thus the interfacial shape, , is given by itself. Specifically, the *ξ*-vector plot, defined by the locus of as a function of *θ*, describes . Edges form in the interface shape when the inverse gamma plot becomes non-convex. This corresponds to becoming negative for a range of interfacial orientations, *θ*. In this situation, the *ξ*-vector plot becomes multi-valued, developing ‘wings’. This is shown in figure 2 for the case . The equilibrium shape comprises the inner envelope with the orientations corresponding to a wing absent resulting in a point of discontinuity of the gradient and hence a corner in . At the corner, the *ξ*-vector (and hence ) is nevertheless continuous. We will denote at the corner by ; it is determined by a common tangent construction on the inverse gamma plot.

For an isotropic surface energy, the minimization of the total energy associated with the interface is accomplished by reducing its overall length, which results in a surface tension force acting within the local tangent plane of the interface. The presence of surface energy anisotropy means that the minimization of the surface energy of an interfacial element can be achieved by rotation *as well as* length contraction. Thus, the surface tension force no longer remains confined locally to the tangent plane of the interface. Cahn and Hoffman showed that the surface tension force, , acting on an interfacial element is given by(2.9)where represents a line element in . At an edge, the balance of forces from the two adjoining portions of the interface results in(2.10)where and are the *ξ*-vectors of each interface evaluated at the corner and is the tangent vector to the edge, being the unit vector in the *z*-direction. The continuity of at an edge is an expression of the local force balance there. In figure 3, we show the configuration of an edge. Using (2.7), we can express the common value of the *ξ*-vector, , in terms of at each of the two adjacent interfaces in terms of the basis unit vectors and shown in figure 3. Thus,(2.11)(2.12)Considering the orientation of and , we have that which gives that(2.13)(2.14)The force balance (2.10) may then be expressed as(2.15)Eliminating and between (2.13) and (2.15) gives the identity(2.16)where . This provides a relation between and as a result of the force balance at the edge.

### (b) Phase-field model

#### (i) Isotropic surface energy

We introduce a continuous phase-field function whose value denotes the state of the system at each point in space: *ϕ*=1 is liquid, *ϕ*=0 solid and intermediate values occur through interfaces. The dimensional Helmholtz free energy functional is assumed to be given by(2.17)where the free energy density, , is defined by(2.18)Here, is a constant, is defined by (2.4) and is the bulk free energy density. The governing equation is then given by a simple rate equation, which ensures that Helmholtz free energy of the system decreases with time(2.19)where is a homogeneous degree-one function that is related to the kinetic anisotropy of the interface. The dimensionless phase-field equation may then be expressed as(2.20)where , , . The dimensionless constants can be determined by the physical properties of the interface by relating the one-dimensional travelling wave solution of (2.20) to the physical properties of the interface (see Wheeler & McFadden (1996)),where a hat denotes a dimensional quantity. Here, the surface energy is given by , the kinetic parameter by , where is the outward unit normal to the interface, the interface thickness by , the macroscopic length scale of the interface by , the time-scale by and the latent heat per unit volume by . The precise choice of time-scale is not important to the discussion, provided that the dimensionless kinetic coefficient, *M*, is order one. The dimensionless interfacial temperature is given by(2.21)

The parameter *ϵ* may be re-expressed as and represents the ratio of the interface thickness to the macroscopic length scale and is therefore a small quantity. We observe from equation (2.20) that the length scale of the interface where *ϕ* varies between 0 and 1 scales with *ϵ*, i.e. on the short length scale associated with the interface thickness. Thus, we introduce a further rescaling of length so that the governing equation is appropriate to the description of the interfacial region. We also take the opportunity to scale time so that the resulting equation is independent of *γ* and *M*. We put and , in which case the phase-field equation may be expressed as(2.22)where the subscript stars have been omitted.

In (2.22) the *ξ*-vector, , naturally appears being again given by the definition (2.5), but with an argument of . This is the so-called generalized *ξ*-vector and, unlike in the sharp interface theory, it is defined throughout the whole domain and not confined to the surface of the interface, . It may be shown that, providing the inverse *γ*-polar plot is convex, the equation (2.22) is a forward parabolic partial differential equation. However, should it be non-convex then for those orientations of that correspond to non-convexity of the surface the equation locally becomes a *backward* parabolic partial differential equation. This corresponds to negative surface stiffness.

There is a one-dimensional travelling wave solution of (2.22) which represents a planar interface moving with speed *V* in a direction is given by(2.23)(2.24)when .

Using the above solution, Wheeler & McFadden (1996) showed that in the sharp interface limit, , in which the thickness of the interface, compared to the macroscopic length scale, goes to zero the anisotropic form of the Gibbs-Thomson equation (2.3) is recovered. Wheeler & McFadden (1997) subsequently investigated the structure of edges. As has been noted above in the sharp interface theory, at an edge the *ξ*-vector is continuous, denoted here by . They showed that in the phase-field theory, edges are represented as weak shocks at a straight line aligned with the direction of , across which the phase field is continuous but its gradient is not. They also showed that is continuous across the shock. This, along with continuity of *ϕ*, implies that the free energy density, , is also continuous across the shock.

Wheeler and McFadden also demonstrated that there existed a so-called *ξ*-tensor defined by(2.25)which satisfies . Using this, they were able to recover the classical force balance (2.10) across the edge.

## 3. Diffuse edge theory

### (a) Governing equation

As described above, an edge in the phase-field theory is represented by a weak shock involving a discontinuity in . Here, we seek to regularize this discontinuity by introducing a generalized free energy of the form(3.1)where we have extended the original functional associated with the phase-field model (2.17) by adding a term involving the square of the Laplacian of the phase field. Here, is a phenomenological constant. A similar approach to regularization has been adopted in the context of highly anisotropic quantum dot formation by Wise *et al*. (2005). The additional Laplacian term in the free energy functional corresponds to a higher-order approximation that accounts for longer-range atomic forces. We will show that in this model an edge region is present through which the interfacial orientation continuously varies through the range of missing orientations, thus smoothing out the corner in . The parameter, , characterizes the thickness of the edge region and is also related to the so-called ‘edge energy’ which is the excess free energy that, in this new model, is associated with the presence of the edge.

We assume that the dynamics are given by (2.19). Thus, the dimensionless phase-field equation associated with this generalized model is an extension of (2.22). It is given by(3.2)where *δ* is a non-dimensional representation of the coefficient of the new square Laplacian contribution to the free energy density. It is related to the dimensional parameters by(3.3)

### (b) Equilibrium edge structure: limit ,

We now consider the internal structure of an isothermal edge, at equilibrium and at a temperature below the freezing temperature ( is a negative constant). In order to make contact with the classical physics, represented by the sharp interface formulation, we consider the effect of the regularization on the sharp interface limit. Thus, we consider the limit, , followed by the limit . As we show below, *δ* characterizes the magnitude of the corner energy and thus this approach provides a way to determine experimentally the magnitude of both *δ* and *ϵ* and thus test whether the assumption associated with this particular limit, namely , is justified.

We consider the situation for a cylindrical interface. In the conventional sharp interface representation, the edge resides at the corner formed from the intersection of the two adjoining interfaces represented as curved portions of the cross-section . However, in the generalized diffuse edge model the interfaces and corner will all have a small thickness with a corresponding internal structure. We assume that the system is stationary and the interface regions meet in the inner edge region, at a contact angle given by the equilibrium value of the classical force balance at the edge. The detailed configuration in the vicinity of the edge is shown in figure 4.

The edge has a five region structure: two outer regions representing the bulk phases, where *ϕ* is zero or unity; two intermediate regions representing the interfacial layers away from the edge in which *ϕ* makes a transition between its bulk values; the inner edge region which represents the confluence of the intermediate regions and within which *ϕ* and vary continuously between their values in the four adjoining regions.

The two intermediate solutions will be denoted regions I and II, exterior to each side of the inner edge region itself. In each intermediate region, we set-up Cartesian coordinate systems and which are aligned to the orientation of the interfaces. Given the equivalence of the two outer regions, we will only discuss intermediate region I in detail. In the inner edge region, we establish another Cartesian coordinate system which is aligned to the common value of the *ξ*-vector at the edge, . We locate the common origin of the coordinate systems to correspond to in the inner edge region. Simple trigonometry gives the relation between the two coordinate systems aswhere represents the orientation of the interface as shown in figure 4.

The dimensionless governing equation (3.2) is represented on the length scale associated with the intermediate regions. The inner edge region connects the two intermediate regions within which the gradient of the phase-field variable varies rapidly, rather than undergoing a discontinuity as in the diffuse interface model.

We now go on to consider the intermediate and inner edge regions in turn. We develop the solution in each region using the method of matched asymptotic expansion to match them together.

### (c) Intermediate region I

We first take the limit . At leading order the governing equation (3.2) is(3.4)We develop its solution as a regular perturbation series in *δ*. It is straightforward to show that this is given bywhere is defined by (2.24) and is an undetermined constant which represents a translation of the interface.

Consideration of the development of the solution at first order in *ϵ* involves a similar secondary expansion as a regular perturbation series in *δ* as the leading order problem above. The solution of the leading order problem in this expansion at is identical to the analysis of Wheeler & McFadden (1996) and we deduce that the temperature is related to the curvature of the interface by(3.5)where is the non-dimensional temperature defined by (2.21) and is the curvature of interface *I* at the edge (non-dimensionalized with respect to *R*).

### (d) Inner edge region

Again we first take the limit . At leading order in *ϵ*, we recover the governing equation (3.4) and consider its solution in the subsequent limit . In this case, we are unable to find a solution as a simple regular perturbation series, as was the case in the interfacial layers. Thus, we seek a singular perturbation solution in which the biharmonic terms play a role at leading order and allow the inner edge region to support a continuous transition of the gradient of the phase field, and hence interface orientation, between its values in the interfacial regions, with interface orientations of and . In order to accomplish this, we introduce a thin inner edge region. To achieve an appropriate balance of terms in (3.4) where the biharmonic terms balance the interfacial gradient energy terms it is necessary to scale the lateral coordinate *s* by *δ*. We put(3.6)The inner edge region is therefore of lateral thickness . We will confine our attention to the solution of the leading order problem in *ϵ*, given by (3.4), as . We denote the phase field in the inner edge region by and express it in the form(3.7)The expansion of (3.4) in terms of the representation (3.7) is given in appendix A.

The inner expansion, appropriate to the edge region, of the intermediate solution is found from expressing it in terms of the inner edge region coordinates *r* and *σ* and taking the limit . This gives(3.8)

### (e) Leading order edge solution

The leading order problem, at , isThe matching condition is derived from the leading order representation of the intermediate solution given by (3.8). ThusWe deduce that(3.9)where matching it with Intermediate Region I requires(3.10)Matching the phase-field with the Intermediate Region II requires thatand hence(3.11)However, this relationship is automatically satisfied as we have assumed the system is in equilibrium and therefore subject to the classical force balance at the edge which requires the same relation (2.16) to hold. We note that we may conveniently write equation (3.10) as(3.12)

### (f) First-order edge solution

From the expansion of equation (3.4) at given in appendix (A 8), we find that(3.13)where(3.14)To simplify the notation, we writeand hence (3.13) may be written as(3.15)where(3.16)This may be integrated once to give(3.17)Now(3.18)as in the interfacial layers. Thus, the constant above is zero and we may integrate again to obtain(3.19)and, hence,(3.20)where is the common value of required by the matching across the edge region.

We define as the angle that makes with which is given by(3.21)Thus, and hence and we find that(3.22)where we have used the property that is a first-order homogeneous function and written . Hence(3.23)Integrating equation (3.21) with respect to *σ* gives the first-order correction to the phase field in the edge region as(3.24)where is a, as yet undetermined, function of *r*, and(3.25)To determine , we match the inner and intermediate solution to . The intermediate expansion of *Φ* is given by(3.26)We have re-expressed , given by (3.27), as(3.27)for *σ*>0, where(3.28)A comparison of the terms in (3.8) and (3.26) shows that the terms proportional to *σ* automatically match and that(3.29)We deduce that is a constant. However, we have co-located the common origin of the coordinate systems with in the inner edge region and thus we require that and(3.30)Thus, the interfaces adjoining the edge are translated by an extent with respect to the edge itself.

From equations (3.9) and (3.24), the phase field in the inner edge region is given as(3.31)The contours of , and hence the shape of the interface in the inner edge region, are approximated by

We have integrated (3.23) and (3.27) numerically for the case to determine and and hence the phase field in the inner edge region. In figure 5, we show the interface shape given by the contour and in figure 6 we plot .

## 4. Edge energy

In this section, we consider the excess energy associated with the edge in this generalized phase-field model. We define the edge energy, , in an analogous way to the surface energy in the phase-field model, i.e. the difference between the free energy of the system with and without the square Laplacian terms in the underlying free energy functional:(4.1)Using the asymptotic expressions for *ϕ* and in the inner edge region, derived above, it can be shown that(4.2)(4.3)and(4.4)Inserting these forms into (4.1) gives that(4.5)The first integral evaluates to and so the edge energy, at leading order, is given by(4.6)We may express this integral in terms of a quadrature with respect to as(4.7)

For the case , we have computed the edge energy, , as a function of *α* which represents the strength of the anisotropy. The computed dependence of the dimensionless edge energy on the degree of anisotropy, *α*, is shown in figure 7. For this fourfold symmetry an edge is present only when the surface stiffness is negative, which corresponds to .

## 5. The -tensor

In this section, we show that there is a -tensor for the time independent generalized phase-field model. The corresponding Euler–Lagrange equation is(5.1)We consider(5.2)which, using the Euler–Lagrange equation (5.1), may be expressed as(5.3)(5.4)(5.5)Using the identity(5.6)equation (5.5) may be expressed as(5.7)As is independent of position, we can write this as the divergence of a tensor :(5.8)where(5.9)where is the unit tensor. Evaluating the various terms gives the following (dimensionless) form for the tensor:(5.10)The last terms with coefficient represent the new contributions that are specific to the generalized phase-field model.

## 6. Discussion

In this paper, we have investigated the internal structure of edges in a generalized phase-field model. It can be shown that the leading equation in the inner edge region governing the orientation of the interface (3.17) may be expressed as(6.1)where and . This is an Allen–Cahn equation. The potential has a double-well form when the surface stiffness is negative for some orientations, i.e. corresponding to the formation of edges. Thus, we may interpret the leading-order inner edge solution as satisfying a common tangent construction on , spanning the values of *q* at the minima of each well, denoted and .

The edge energy can be expressed as(6.2)and is therefore proportional to the area bounded by the common tangent construction and the double-well in as well as the thickness of the edge region.

The common tangent construction on was established by Cabrera (1963, 1964) who made the analogy with a phase transition. The theory presented here provides a diffuse edge version of Cabrera's ideas, by developing a governing equation for the interface orientation that satisfies the same common tangent construction. Thus, the generalized phase-field model (3.2) contains within it a diffuse description of two types of phase transition involving *ϕ* and

The solid–liquid phase transition across a planar interface in which the value of

*ϕ*varies continuously and automatically satisfies a common tangent construction on the bulk free energy function .The edge phase transition in which the orientation of , and hence the interface, represented by

*q*, automatically satisfies a common tangent construction on .

We have developed a model based on a diffuse interface description of an interface and have shown that it describes smoothed edges, albeit in the limit . We can reasonably expect that smooth edges will also be present when *δ* is an order-one quantity. In this case, the thicknesses of the edge region and the adjoining interfaces would by comparable and this regime may well be useful in the numerical phase-field simulations in which edges form.

We have not ascertained the precise effect of the regularization on the relation of the interface temperature to the curvature of the interface, beyond identifying its effect is (see equation (3.5)). It remains to be discovered whether this would replicate the standard generalization of the sharp interface theory to high-curvature interfaces in which its often assumed that the surface energy depends on the square of the interfacial curvature (see (1.1)). This, in turn, would allow a comparison between the diffuse interface formulation of edges presented here to be compared to the sharp interface model of corners, developed by Spencer (2004), in the regime where the corner region described by Spencer is of the scale of the underlying dimensions of the interface itself. It might also allow the parameter *δ* to be estimated by relating it to a measurable physical macroscopic parameter, such as the coefficient *β* in (1.1).

In this work, we have not included the effect of stress. However, we have derived a ‘stress tensor’ which is a natural generalization of that for the standard phase-field model. In that situation, it may be shown that constitutes the reversible part of the stress tensor in a treatment that allows for motion of the two phases (see Anderson *et al*. 1998). This requires further work but may suggest that the regularization may affect the force balance and hence the equilibrium angle between the adjoining interfaces at an edge once bulk stress effects are explicitly included. It has been shown by Siegel *et al*. (2004) that when bulk stress effects are included in a sharp interface model the equilibrium angle is indeed affected.

## 7. Summary

To date investigations of corners and edges in interfaces in the presence of surface anisotropy have focussed on the regularization of sharp interface models. In this paper, we have considered this from the different perspective of a diffuse interface description of the interface. Specifically, we have extended the anisotropic phase-field equation, based on a generalized *ξ*-vector, by introducing a natural regularization. This model provides a diffuse description of edges as well as interfaces. We considered the model in the singular limit, , and were able to determine the interior structure of the edge region in which the edge is smoothed out and the interface orientations transition through the span of missing orientations associated with the edge in the Wulff shape. We are able to develop the analogy of the edge region as representing a phase transition first identified by Cabrera.

We have determined the associated edge energy and shown that it scales with the regularization coefficient, *δ*. Lastly, we have derived a stress tensor associated with the generalized phase-field equation.

## Acknowledgements

The author is very grateful to Dr G. B. McFadden for stimulating conversations during the conduct of this work.

## Footnotes

- Received January 2, 2006.
- Accepted March 21, 2006.

- © 2006 The Royal Society