Cahn-Hilliard equations

The free energy profile during the phase separation in a inhomogeneous mixture is described by the Cahn-Hilliard equation [23,24,31,32]  

The local free energy was presented with variety of mathematical models in different studies, such as the double-well model. A well Flory-Huggjns equation can be adapted for a polymer blend the ternary Flory-Huggins equation is as follows [33]  

The Flory-Huggins-type of free energy is well known and has been studied extensively for polymer blends [34], although it was initially derived for small-molecular solutions. The primary drawback of the Flory-Huggins model for 

The gradient energy coefficient, k, determines the influence of the composition gradient to the total free energy of the domain. The value of k is difficult to measure experimentally. Although efforts have been made by Saxena and Caneba [35] to estimate the gradient energy coefficient in a ternary polymer system from experimental methods, few results have been reported for specific conditions. Initially, the value of k can be estimated by the interaction distance between molecules [36]  

The mobility is estimated from the diffusivity of the polymers. The mobility of polymer blends with long chains can be estimated by the equation as follows [37]  

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

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... 

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

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. 

We derive here the governing equations necessary to describe the structure factor, S (q, t), for a complex liquid mixture subject to flow. Since this observable is the Fourier transformation of the spatial correlation of concentration functions, it is first required to develop an equation of motion for 5c (r, t). The approach described here employs a modified Cahn-Hilliard equation and is described in greater detail in the book by Goldenfeld [91]. To describe the physical system, an order parameter, q/ (r, ), is introduced. In a complex mixture, this parameter would simply be /(r, 0 = c(r, r)-(c), where c(r, t) is the local concentration of one of the constituents and mean concentration. The order parameter has the property of being zero in a disordered, or on phase region, and non-zero in the ordered or two-phase region. The observed structure factor, which is the object of this calculation, is simply... 

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]... 

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... 

Caco-2 absorption, 5,102 Cahn-Hilliard equations, 6, 240 CAMK group of kinases, 1,186,1%... 

The first phenomenon has been analyzed theoretically by various workers [83-86]. Most of them start with the Cahn-Hilliard equation relating the flux J of the polymer species A ... 

Conserved Fields and the Cahn-Hilliard Equation. The Allen-Cahn equation told us something about the spatial distribution and temporal evolution of a nonconserved order parameter which characterizes the state of order within the material. From the materials science perspective, it is often necessary to describe situations in which a conserved field variable is allowed to evolve in space and time. In this context, one of the most celebrated evolution equations is the Cahn-Hilliard equation which describes the spatio-temporal evolution of conserved fields such as the concentration. 

Like its nonconservative counterpart seen in the Allen-Cahn equation, the Cahn-Hilliard equation is aimed at describing the evolution of field variables used to describe microstructures. In the present setting, a particularly fertile example (of which there are many) of the use of this equation occurs in the context of phase separation. Recall that the Cahn-Hilliard equation describes a system with a conserved field variable. What we have in mind is the type of two-phase microstructures described in chap. 10 where the phase diagram demands the coexistence of the host matrix material and some associated precipitates. The Cahn-Hilliard equation describes the temporal evolution of such microstructures. 

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. 

Fig. 12.6. Illustration of the spatio-temporal evolution of the order parameter field for a system described by the Cahn-Hilliard equation (courtesy of W. Craig Carter).

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. 

Phase separation in binary alloys Cahn-Hilliard equation... 

In the course of evolution governed by the Cahn-Hilliard equation (7), the system tends to an attracting stationary state which corresponds to a minimum of the Ginzburg-Landau functional. In order to distinguish between this stationary state and all other stationary solutions, it is necessary to investigate the stability of all the solutions listed in the previous paragraph. 

For two- and three-dimensional Cahn-Hilliard equations, there are no stable spatially periodic solutions as well, and the development of stable spatially nonuiuform patterns is impossible. The evolution leads to the complete separation of phases. The growth law of the domain size L is [29], [30]. This... 

Dynamic Contact Line Generalized Cahn-Hilliard Equation... 

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 ... 

As an example of application of the generalized Cahn-Hilliard equation, we consider the case when a droplet is set into slow motion due to either external forces or long-range interactions. We assume that the deviation from equihb-rium shape remains weak and can be treated as a small perturbation everywhere. The droplet mobility can be deduced then from integral conditions based on an equihbrium solution. This allows us to avoid solving dynantic equations exphcitiy and computing a perturbed shape. 

Asaro-Tiller-Grinfeld instability, 125 Asters, 285, 291-292 Bifurcation, 227-228, 237, 258 Cahn-Hilliard equation, 3-5, 8-9, 29, 184 Combustion, 247... 

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.). 

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