Physica A: Statistical Mechanics and its Applications
A second-order accurate non-linear difference scheme for the N -component Cahn–Hilliard system
Introduction
In this paper, we consider an efficient and accurate numerical method of a model for phase separation in a -component mixture. When a homogeneous system composed of -components, at high temperature, is rapidly cooled to a uniform temperature below a critical temperature, where it is unstable with respect to concentration fluctuations, spinodal decomposition [7] takes place: The system separates into spatial regions rich in one component and poor in the other components. It evolves into an equilibrium state with lower overall free energy [9].
Spinodal decomposition is of interest on two counts. First, it is one of the few solid-state transformations for which there is any plausible quantitative theory. Second, from a practical viewpoint, spinodal decomposition is of interest because it affords a means of producing a very finely dispersed structure that can enhance the properties of a material [22].
Most of the technologically important alloys are multi-component systems exhibiting multiple phases in their microstructures. Moreover, one or more of these phases are formed as a result of phase transformations induced during processing. Since the performance of these multi-component alloys depends crucially on the morphology of the phase, a fundamental understanding of the kinetics of phase transformations is important for controlling the microstructures of these multi-phase alloys [1].
Cahn [6] extended the van der Waals model [31] to time-dependent problems by approximating the interfacial diffusion as being proportional to chemical potential gradients. Generalization of the Cahn–Hilliard (CH) equations to multi-component systems appeared with de Fontaine [17] and Morral and Cahn [26]. Elliott and Luckhaus [15] gave a global existence result under constant mobility and specific assumptions on the form of the free energy. Elliott and Garcke [13] developed an existence theory for multi-component diffusion when the mobility matrix depends on the order parameters. Differences between binary and multi-component alloys were identified and the equilibrium and dynamic behavior of multi-component systems were studied by Eyre [16].
Although there are many numerical studies (see Refs. [2], [10], [11], [12], [14], [18], [19], [20], [29] and references therein) with binary CH equation and ternary CH equation [4], [9], [16], [25], [28], much less has been conducted on the quaternary CH system. In Ref. [24], vector-valued Allen-Cahn equations were considered. In Ref. [27], multi-component fluid mixtures were studied using molecular dynamics. In Ref. [25], a finite difference method is used for the constant mobility. In Ref. [3], a finite element approximation is used for the ternary CH system with a degenerate mobility matrix.
The purpose of this work is to consider a conservative second-order accurate nonlinear numerical method for the -component CH system with concentration dependent mobility for a -component mixture.
The contents of this paper are as follows. In Section 2 we briefly review governing equations for phase separation in a -component system which takes a concentration dependence of the mobility. In Section 3 we consider a fully discrete semi-implicit finite difference scheme and describe an efficient and accurate nonlinear multigrid V-cycle algorithm for the -component CH system. We present numerical experiments such as a second-order convergence test, comparison with a linear stability analysis of the equations, the evolution of triple junctions, and phase separation in a quaternary mixture in Section 4. Finally, in Section 5 we conclude.
Section snippets
Governing equations
We consider a system of a -component mixture. Let for be the mole fraction of the th component in the mixture as a function of space and time. Clearly the total mole fractions must sum to 1, i.e., so that, admissible states belong to the Gibbs -simplex
Let be a vector valued phase field. Without loss of generality, we choose a Helmholtz free energy functional of a generalized Ginzburg-Landau
Numerical solution
Since for -component systems, we only need to solve the equations with , and . Let and . In the following numerical scheme and solution algorithm, we restrict space dimensions to two for simplicity. The three-dimensional extension is straightforward.
Numerical experiments — the quaternary Cahn–Hilliard system
In this section, we perform numerical experiments such as a convergence test, a linear stability analysis, the evolution of triple junctions, and phase separation in a four component mixture. The extensions to higher systems than a quaternary system are algebraically complex but conceptually straightforward.
The composition of a quaternary mixture (A, B, C, and D) can be mapped onto an equilateral tetrahedron whose corners represent a 100% concentration of A, B, C or D as shown in Fig. 1(a).
Conclusions
We considered a fully discrete semi-implicit finite difference scheme for the -component CH system with a concentration dependent mobility and solved the resulting scheme by an efficient and accurate nonlinear multigrid method. We carried out numerical experiments such as a second-order convergence test, comparison with linear stability analysis, and evolution of triple junctions. We have also investigated phase separation via spinodal decomposition with a constant and degenerate concentration
Acknowledgments
This research was supported by the MKE (Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute for Information Technology Advancement) (IITA-2008- C1090-0801-0013). This work was also supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-C00225).
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