Cahn-Hilliard equation numerical simulations

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. 

Subsequently, a numerical model was introduced to simulate self-assembly by the phase separation of polymer blends on a heterogeneously functionalized substrate patterns. From thermodynamic principles, when polymer blends are quenched into the spinodal region in the phase diagram, phase separation can be initiated from small composition fluctuations in the blend. The Cahn-Hilliard equation is used to describe the energy profile in the domain with varying... 

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. 

This paper deals with kinetics of phase separation in ternary alloys, especiaUy asymptotic behavior of a minor element such as Mo in the above alloy associated with the decomposition of a major element. A theory on as rmptotic behavior of the minor element in Fe-based ternary alloy was proposed. And then Numerical simulation models based on the Cahn-Hilliard equation[l, 2) have been applied to the investigation of phase separation in Fe-Cr-Mo ternary alloys. Simulated asymptotic behavior of Mo or Cr asscociated with decomposition of Cr or Mo is compared with that predicted by the theory. 

Although experimental verifications are left as future problem, numerical simulations of phase separation in Fe-Cr-Cu and Fe-Cr-Mo ternary alloys based on the Cahn-Hilliard equation demonstrated the validity of the present analyses (8, 9. ... 

Numerical simulations based on the Cahn-Hilliard Equation were performed for Fe-Cr-Mo ternary alloys. Table 1 shows the conditions used for simulation. 

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. 

Research into the modeling of polymer self-assembly resides in two areas. The first is the formation of accurate and stable numerical methods. Among many studies in the numerical method for the Cahn-Hilliard Equation [5, 6], an elegant approach was reported by David J. Eyre [7] and L. Q. Chen etc [11]. The application of a semi-implicit Fourier-spectral method was extensively studied in this issue. This scheme is shown to be unconditionally gradient stable and solvable for all time steps. The semi-implicit Fourier-spectral method has been employed to model the phase separation as it s more efficient thus allowing us to simulate large systems for a longer time [2, 3]. 

Numerical models of conserved order-parameter evolution and of nonconserved order-parameter evolution produce simulations that capture many aspects of observed microstructural evolution. These equations, as derived from variational principles, constitute the phase-field method [9]. The phase-field method depends on models for the homogeneous free-energy density for one or more order parameters, kinetic assumptions for each order-parameter field (i.e., conserved order parameters leading to a Cahn-Hilliard kinetic equation), model parameters for the gradient-energy coefficients, subsidiary equations for any other fields such as heat flow, and trustworthy numerical implementation. 

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... 

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