Cahn-Hilliard diffusion equation

By taking the divergence of the above equation the general Cahn-Hilliard diffusion equation is obtained and this forms the basis of the interpretation of... 

To evaluate the demixing process under the nonisoquench depth condition, they carried out computer simulations of the time dependent concentration fluctuation using the Cahn-Hilliard nonlinear diffusion equation. 

Sj = Dj/D and D = (MkBTc b )/v are the Soret and the diffusion coefficient, respectively. In the absence of thermal diffusion, (49) reduces to the well known Cahn-Hilliard equation, which belongs to the universality class described by model B [3], In fact, (49) gives a universal description of a system in the vicinity of a critical point leading to spinodal decomposition. 

Keywords cell signaling lipid rafts BAR domains membrane curvature membrane elasticity PIP2 diffusion mean-field model coarse-grained theory Poisson-Boltzmann theory Cahn-Hilliard equations... 

The Kinetics of Spinodal Decomposition. Cahn s kinetic theory of spinodal decomposition (2) was based on the diffuse interface theory of Cahn and Hilliard (13). By considering the local free energy a function of both composition and composition gradients, Cahn arrived at the following modified linearized diffusion equation (Equation 3) to describe the early stages of phase separation within the unstable region. In this equation, 2 is an Onsager-type... 

Spinodal decomposition (SD) driven by the chemical reaction proceeds isothermally, but the quench depth AT (expressing the temperature difference between LCST and the reaction temperature), increases with time. This situation is quite different from the familiar SD under isoquench depth, where after a temperature jump (or drop) SD proceeds isothermally, and the AT is constant. However, the regular morphology is also obtained in the kinetically driven SD, as in the iso-quench SD. This observation was confirmed by the computer simulation using the Cahn-Hilliard non-linear diffusion equation [Ohnaga et al., 1994]. This should also be the case for the solution casting, described in preceding section and the shear-dependent decomposition in next section. 

A recent application of small angle X-ray scattering is for the determination of phase separation kinetics. Rundman and Hilliard (1967) noted the correspondence between the composition fluctuations corresponding to Cahn s solution to the diffusion equation and the small angle scattering intensity which results from these composition (i.e. electron density) fluctuations (Eq. 1.34). They obtain the relation,... 

A theoretical model based on a diffusion equation was developed by Cahn and Hilliard and introduces a diffusion equation that relates the interdilfusio-nal flux J of the two phases J = J = —Jj) to the gradient of chemical potential differences ... 

The kinetics of phase separation proper have been described by Cahn and Hilliard (37,38). For the early stages of decomposition, the diffusion equation is of the following form ... 

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 numerical modeling methods for polymer blends have been reviewed in this chapter, with different categories such as volume-of-fluid, molecular dynamics and diffusion-controlled methods being introduced. Use of the Cahn-Hilliard method was emphasized for binary and ternary polymer systems with no obvious mechanical flux, while specific factors such as elastic energy and functionalized substrate were considered for purposes of comparison. The diffusion-controlled model described, using the Cahn-Hilliard equation as the constitutive equation, can be used to depict the gradient of the interface as well as the composition profile of partially miscible blends hence, it is feasible to implement this equation in a polymer blend system. It should be noted that although these examples do not consider mechanical flux, additional constitutive equations (e.g., Navier-Stokes) can easily be added to this diffusion-controlled model. 

TDGL is a microscale method for simulating the structural evolution of phase-separation in polymer blends and block copolymers. It is based on the Cahn-Hilliard-Cook (CHC) nonlinear diffusion equation for a binary blend and falls under the more general phase-field and... 

The coefficient a varies from 0 to 1. When a is 0, the equation above is euquivalent to the Cahn-Hilliard with constant mobility, M. While a is 1, the bulk diffusion is severely decreased and the fluid is interface-diffiision-controlled. 7 is the noise term and defined as a function coordinates and time. 

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