Cahn-Hilliard

Lipson (1943, 1944), who had examined a copper-nickeMron ternary alloy. A few years ago, on an occasion in honour of Mats Hillert, Cahn (1991) mapped out in masterly fashion the history of the spinodal concept and its establishment as a widespread alternative mechanism to classical nucleation in phase transformations, specially of the solid-solid variety. An excellent, up-to-date account of the present status of the theory of spinodal decomposition and its relation to experiment and to other branches of physics is by Binder (1991). The Hillert/Cahn/Hilliard theory has also proved particularly useful to modern polymer physicists concerned with structure control in polymer blends, since that theory was first applied to these materials in 1979 (see outline by Kyu 1993). 

Non-Random Systems. As pointed out by Cahn and Hilliard(10,11), phase separation in the thermodynamically unstable region may lead to a non-random morphology via spinodal decomposition. This model is especially convenient for discussing the development of phase separating systems. In the linearized Cahn-Hilliard approach, the free energy of an inhomogeneous binary mixture is taken as ... 

Bates, P.W., and Fife, P.C., 1993, The dynamics of nucleation for the Cahn-Hilliard equation, SIAMJ. Appl. Math. 53 90. 

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. 

Processing conditions are known to play a critical role in establishment of morphology and final properties of the materials. Balazs and coworkers [245, 246] designed a multiscale method (coarse-grained Cahn-Hilliard approach and Brownian dynamics) and found that addition of solid particles significantly... 

The Cahn-Hilliard equation applies to conserved order-parameter kinetics. For the binary A-B alloy treated in Section 18.1, the quantity in Eq. 18.22 is the change in homogeneous and gradient energy due to a change of the local concentration cB and is related to flux by... 

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... 

Figure 18.4 Example of numerical solution for the Cahn-Hilliard equation, Eq. 18.25,...

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

These conclusions were later supported by time-resolved SAXS experiments by Stiihn et al. (1994) who studied the ordering of a PS-PI diblock with /PS = 0.44 following quenches from the disordered phase into the lamellar phase. They found that the relaxation times of the structure factor were wavevector dependent, and consistent with the Cahn-Hilliard from (Cahn and Hilliard 1958)... 

Keywords Cahn-Hilliard model Diffusion Nonlinear dynamics Pattern selection Polymer blends Soret effect Spinodal decomposition Thermal diffusion... 

A modified Cahn-Hilliard (CH) model [114] is used for the theoretical analysis of the impact of thermal diffusion on phase separation by taking into account an inhomogeneous temperature distribution, which couples to a concentration variation via the Soret effect. The Flory-Huggins model is used for the free energy of binary polymer-mixtures. The composition is naturally measured in terms of volume fraction 0 of a component A, which can be related to the weight fraction c by... 

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. 

For the description of phase separation we choose again the generic Cahn-Hilliard model in one spatial dimension [124, 125]... 

Theoretically we have employed a generalized Cahn-Hilliard model to describe the effects of stationary and traveling spatially periodic temperature-modulations... 

---