An extended Cahn-Hilliard model for interfaces with cubic anisotropy

Abstract

For studying systems with a cubic anisotropy in interfacial energy ¼, we extend the Cahn±Hilliard model by including in it a fourth-rank term, namely, ®ijlm‰q 2 c=…qxi qx j †Š‰q 2 c=…qxl qx m †Š. This term leads to an additional linear term in the evolution equation for the composition parameter ®eld. It also leads to an orientation-dependent eOE ective fourth-rank coe cient ®hhkli in the governing equation for the one-dimensional composition pro®le across a planar interface. The main eOE ect of a non-negative ®hhkli is to increase both ¼ and interfacial width w, each of which, upon suitable scaling, is related to ®hhkli through a universal scaling function. In this model, ¼ is a diOE erentiable function of interface orientationn n , and does not exhibit cusps; therefore, the equilibrium particle shapes (WulOEshapes) do not contain planar facets. However, the anisotropy in the interfacial energy can be large enough to give rise to corners in the WulOE shapes in two dimensions. In particles of ®nite sizes, the corners become rounded, and their shapes tend towards the WulOEshape with increasing particle size.

Phase‐field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy

“…To formulate the phase-field model for strongly anisotropic fracture, we have combined the classical variational phase-field model of brittle fracture [27] with the ECH framework [40,41], proposed in the context of phase-field models of crystal growth. The result is a fourth-order model, because the energy functional involves the Hessian of the phase-field.…”

“…From these definitions and because the order of differentiation can be exchanged for sufficiently smooth functions, these tensors possess various symmetries. Following similar arguments as before, the number of fourth-rank tensors can be reduced from five to three [40], and hence, the local free energy density f can be written as…”

“…Because higher-rank tensors can produce more general anisotropy, we extend the Taylor series expansion of local free energy density to higher order, here up to fourth order. This leads to ECH-type equations [40][41][42].…”

“…A different and natural way to take anisotropy into account is to follow the approach presented in the original work by Cahn and Hilliard [39] and write the Taylor series expansion of the free energy including higher-order terms. In the context of crystal growth, this idea has been shown to provide a satisfactory way of describing systems with anisotropic surface energy and is intrinsically regularized [40,41]. However, as shown later, this method imposes constraints on the kinds of orientation dependence of G c .n/.…”

On the incorporation of cubic and hexagonal interfacial energy anisotropy in phase field models using higher order tensor terms

“…The origins of anisotropy could be energetic (such as anisotropies in interfacial, elastic or magnetic energies) and/or kinetic (such as anisotropies in the attachment kinetics). Hence, a large number of phase field models have been developed to account for these anisotropies: even though it is not possible to list all the phase field studies that deal with anisotropies within the purview of this article, the following listing is fairly representative: see, for interfacial anisotropy [6][7][8][9][10][11][12][13][14][15][16][17][18][19] ; for elastic anisotropy [20][21][22][23][24][25] , for magnetocrystalline anisotropy 26 , and, for anisotropy in attachment kinetics 27 . In a typical phase field model, the microstructure is described by order parameters and the thermodynamic quantities (free energy or entropy) are represented as functionals in these order parameters.…”