Cahn-Hilliard model

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

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... 

Figure 2.8. Illustrating the meaning of symbols in the Cahn-Hilliard model.

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 ... 

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. 

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. 

While in the bulk the phases of the growing concentration waves are random, and also the directions of the wavevectors q are controlled by random fluctuations in the inital states, a surface creates a boundary condition, and working out adynamic extension [45,129,132,133,144,156] of the model in Sect. 2.1 Eqs. (7)-(10) one finds that under typical conditions wavevectors oriented perpendicular to the walls occur, with phases such that the maxima of the waves occur at the walls (Fig. 28). In terms of a normalized order parameter i /(Z, R, x) where x is a scaled time and Z, R, are scaled coordinates perpendicular and parallel to the walls, Z=z/2 b, R=q/2 b, V /=(( )-( )crit)/(( )coex-( )crit), this dynamic extension is the Cahn-Hilliard equation [291-294]... 

Besides the issue how t relates to there is the additional problem om that we have not tried to give an interpretation of this correlation length. Such an interpretation of would be possible using the theory of van der Waals or Cahn-Hilliard. Such models predict to increase with T, although there is some... 

For polymer liquids, the gradient approximation in conjunction with the lattice fluid model has been used to calculate surface tensions [24,25]. The Cahn-Hilliard relation for surface tension o, in terms of reduced variables, can be expressed as... 

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... 

Theory of dynamic critical phenomena by P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977) details many of the crucial ideas on order parameters and their dynamical evolution in the critical phenomena setting. Their Model A has been introduced here as the Cahn-Allen equation while their Model B we have referred to as the Cahn-Hilliard equation. 

With the form of free energy functional prescribed in equation (A3.3.52). equation (A3.3.43) and equation (A3.3. 48) respectively define the problem of kinetics in models A and B. The Langevin equation for model A is also referred to as the time-dependent Ginzburg-Landau equation (if the noise term is ignored) the model B equation is often referred to as the Cahn-Hilliard-Cook equation, and as the Cahn-Hilliard equation in the absence of the noise term. 

The dynamics of thin hlms or droplets bounded by a three-phase contact line is described by Eqs. (7), (8) with the potential p incorporating both interactions with the substrate, dehned by Eq. (40) with an appropriately chosen interaction model, and the effect of weak interfacial curvature according to Eq. (35). In the latter, the vapor density can be neglected, while the curvature expressed in lubrication approximation as k = —eV h the small parameter e due to a different scaling in the vertical and horizontal direction, cannot be excluded from the final form. One can also add here an external potential V(x), e.g. due to gravity. This leads to a generalized Cahn-Hilliard equation, appropriate for the case when the order parameter is conserved ... 

FIGURE 8.9 Spinodal decomposition process modeled using the Cahn-Hilliard equation. Time and length scales are dimensionless. Number of time steps following the quench are (a) 10,000, (b) 40,000, (c) 160,000, and (d) 640,000. Courtesy of Dr. Nigel Clarke (University of Durham, U.K.). 

EoS models can also be used in the frame of the gradient approximation, such as the Cahn-Hilliard theory [100] of inhomogeneous systems, for the description of surface properties. In the frame of this theory, the Helmholtz s free-energy density r in a one-component inhomogeneous system can be expressed as an expansion of density p and its derivatives ... 

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