Cahn-Hilliard equation numerical results

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. 

In this chapter, attention will be focused on applications of the Cahn-Hilliard equation on the numerical simulation of an inhomogeneous polymer blend. The numerical model of a binary polymer system and a polymer-polymer-solvent system will be reviewed as examples to illustrate the application of such modeling methodologies. Attention will be paid in particular to a diffusion-controlled system with no mechanical flow, and the effects of substrate patterning will be taken into consideration to highlight the influences of external attraction during the phase separation of polymer blends. The results of the numerical simulation will then be verified using realistic experimental results, on a quantitative basis. 

The evolution of polymer composition in the spatial domain can be derived using the Cahn-Hilliard equation. In numerical simulations, the fourth-order nonlinear parabolic partial differential equations are solved using Fourier-spectral methods, while the partial differential equations are transferred by the discrete cosine transform into ordinary partial equations. The result is then transformed back with the inverse cosine transform to the ordinary space. 

Asymptotic behavior of a minor element Y in a Fe-X-Y ternary system associated with phase decomposition of the major element, X, was investigated by using a model based on the Cahn Hillirad equation for multicomponent s)rstems. Numerical simulations of phase separation in Fe-Cr-Mo ternary alloys were performed with use of the Cahn-Hilliard equation. The following results are obtained. 

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