Allen-Cahn equation

The Allen-Cahn equation applies to the kinetics of a diffuse-interface model for a nonconserved order parameter—for example, the order-disorder parameter r](f,t) 

The increase in local free-energy density, (r) from Eq. 18.22, does not require any macroscopic flux.7 In a linear model, the local rate of change is proportional to its energy-density decrease, 

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Toffoli and Margolus [tofF86] point out that what appears on the macroscopic scale is a good simulation of surface tension, in which the boundaries behave as though they are stretched membranes exerting a pull proportional to their curvature. Vichniac adds that the behavior of such twisted majority rules actually simulates the Allen-Cahn equation of surface tension rather accurately ... 

The simple two-dimensional phase-field simulations in Figs. 18.4 and 18.5 were obtained by numerically solving the Cahn-Hilliard (Eq. 18.25) and the Allen-Cahn equations (Eq. 18.26). Each simulation s initial conditions consisted of unstable order-parameter values from the top of the hump in Fig. 18.1 with a small spatial... 

Figure 18.5 Example of numerical solution for the Allen -Cahn equation, Eq. 18.26, for...

Generalizations of the Cahn-Hilliard and Allen-Cahn Equations... 

Nonconserved Fields and the Allen-Cahn Equation. Nonconserved order parameters (sueh as the state of order itself as introduced in eqn (12.22)) can have complex spatial distributions that evolve in time. The governing equation in this case is provided by the Allen-Cahn equation. To obtain this equation using the ideas introduced above, we hark back to the L2 norm. It is assumed that the instantaneous free energy of the system can be written down as a functional of the order parameter and its gradients. For example, in the context of the Allen-Cahn equation, the relevant free energy functional is... 

In the context of the Allen-Cahn equation, the claim is that it is the gradient as defined with respect to the L2 inner product which is of relevance. In concrete terms, what this means is that we may write... 

Fig. 12.5. Illustration of the spatio-temporal evolution of the order parameter field for a system described by the Allen-Cahn equation (courtesy of W. Craig Carter). The temporal sequence runs from left to right starting with the first row.

Conserved Fields and the Cahn-Hilliard Equation. The Allen-Cahn equation told us something about the spatial distribution and temporal evolution of a nonconserved order parameter which characterizes the state of order within the material. From the materials science perspective, it is often necessary to describe situations in which a conserved field variable is allowed to evolve in space and time. In this context, one of the most celebrated evolution equations is the Cahn-Hilliard equation which describes the spatio-temporal evolution of conserved fields such as the concentration. 

Like its nonconservative counterpart seen in the Allen-Cahn equation, the Cahn-Hilliard equation is aimed at describing the evolution of field variables used to describe microstructures. In the present setting, a particularly fertile example (of which there are many) of the use of this equation occurs in the context of phase separation. Recall that the Cahn-Hilliard equation describes a system with a conserved field variable. What we have in mind is the type of two-phase microstructures described in chap. 10 where the phase diagram demands the coexistence of the host matrix material and some associated precipitates. The Cahn-Hilliard equation describes the temporal evolution of such microstructures. 

An example of the type of results that can be obtained when the Cahn-Hilliard equation is solved numerically is shown in fig. 12.6. In this case, the homogeneous free energy is that of eqn (12.24). As with the numerical results for the Allen-Cahn equation shown in fig. 12.5, the initial conditions correspond to a small but random deviation from the homogeneous state. 

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