Cahn Hilliard Cook Equation Research Articles

A [formula omitted] weak Galerkin method for linear Cahn–Hilliard–Cook equation with random initial condition

This paper introduces a C0 weak Galerkin finite element method for a linear Cahn–Hilliard–Cook equation. The highlights of the proposed method are that the complexity of constructing the C1 finite element space for fourth order problem is avoided and the number of degree of freedom is apparently reduced compared to the fully discontinuous weak Galerkin finite element method. With the redefined discrete weak Laplace operator and the classical C0 Lagrange elements, the L2 optimal error estimates in spatial variable are obtained. In time, the classical Euler scheme is then used to do the numerical simulation. Finally, numerical experiments are presented to demonstrate the efficiency of the proposed numerical method.

Generalized non-classical nucleation model in binary alloys

In this study, we have developed the generalized non-classical nucleation model in binary alloys. The model implies application of Cahn–Hilliard or (and) Cahn–Hilliard–Cook (CHC) equations to determine kinetic coefficients and boundary conditions for master equation. Also, we have extracted properties of critical nuclei, i.e. size, average composition and nucleation barrier, from direct simulation by the CHC equation.

A direct meshless local collocation method for solving stochastic Cahn–Hilliard–Cook and stochastic Swift–Hohenberg equations

In this study, the direct meshless local Petrov–Galerkin (DMLPG) method has been employed to solve the stochastic Cahn–Hilliard–Cook and Swift–Hohenberg equations. First of all, we discretize the temporal direction by a finite difference scheme. In order to obtain a fully discrete scheme the direct meshless local collocation method is used to discretize the spatial variable and the Euler–Maruyama method is used for time discretization. The used method is a truly meshless technique. In order to illustrate the efficiency and accuracy of the explained numerical technique, we study two stochastic models with their applications in biology and engineering, i.e., the stochastic Cahn–Hilliard–Cook equation and a stochastic Swift–Hohenberg model.

An IMEX finite element method for a linearized Cahn–Hilliard–Cook equation driven by the space derivative of a space–time white noise

We consider a model initial- and Dirichlet boundary- value problem for a linearized Cahn–Hilliard–Cook equation, in one space dimension, forced by the space derivative of a space–time white noise. First, we introduce a canvas problem, the solution to which is a regular approximation of the mild solution to the problem and depends on a finite number of random variables. Then, fully discrete approximations of the solution to the canvas problem are constructed using, for discretization in space, a Galerkin finite element method based on $$H^2$$ piecewise polynomials, and, for time-stepping, an implicit/explicit method. Finally, we derive a strong a priori estimate of the error approximating the mild solution to the problem by the canvas problem solution, and of the numerical approximation error of the solution to the canvas problem.

Stochastic simulation of nucleation in binary alloys

In this study, we simulate nucleation in binary alloys with respect to thermal fluctuations of the alloy composition. The simulation is based on the Cahn–Hilliard–Cook equation. We have considered the influence of some fluctuation parameters (wave vector cutoff and noise amplitude) on the kinetics of nucleation and growth of minority phase precipitates. The obtained results are validated by the example of iron–chromium alloys.

Conforming finite element methods for the stochastic Cahn–Hilliard–Cook equation

This paper is concerned with the finite element approximation of the stochastic Cahn–Hilliard–Cook equation driven by an infinite dimensional Wiener type noise. The Argyris finite elements are used to discretize the spatial variables while the infinite dimensional (cylindrical) Wiener process is approximated by truncated stochastic series spanned by the spectral basis of the covariance operator. The optimal strong convergence order in L2 and H˙−2 norms is obtained. Unlike the mixed finite element method studied in the existing literature, our method allows the covariance operator of the Wiener process to have an infinite trace, including the space–time white noise is allowed in our model. Numerical experiments are presented to illustrate the theoretical analysis.

An attempt to handle the nanopatterning of materials created under ion beam mixing

ABSTRACTIn the past fifty years, experimental works based on TEM or grazing incidence X ray diffraction have clearly shown that alloys and ceramics exhibit a nano pattering under irradiation [1,2,3]. Many works were devoted to study the nano patterning induced by ion beam mixing in solids [17,18,19]. Understanding the nano patterning will provide scientific bases to tailor materials with well-defined microstructures at the nanometric scale. The slowing down of impinging particles in solids leads to a complex distribution of subcascades. Each subcascade will give rise to an athermal diffusion of atoms in the medium. In this work, we focused on this point. Based on the well-known Cahn Hilliard Cook (CHC) equation, we analytically calculate the structure factor describing the nano patterning within the mean field approximation. It has shown that this analytical structure factor mimics the structure factor extracted from direct numerical simulations of the time dependent CHC equation. It appears that this structure factor exhibits a universal feature under irradiation.

Finite-element approximation of the linearized Cahn-Hilliard-Cook equation

The linearized Cahn–Hilliard–Cook equation is discretized in the spatial variables by a standard finite-element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. Backward Euler time stepping is also studied. The analysis is set in a framework based on analytic semigroups. The main effort is spent on proving detailed error bounds for the corresponding deterministic Cahn–Hilliard equation. The results should be interpreted as results on the approximation of the stochastic convolution, which is a part of the mild solution of the nonlinear Cahn–Hilliard–Cook equation.

Finite Element Approximation of the Cahn–Hilliard–Cook Equation

We study the nonlinear stochastic Cahn-Hilliard equation perturbed by additive colored noise. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to $1$. We also prove strong convergence without known rate.

The phase dynamics and wetting layer formation mechanisms in two-step surface-directed spinodal decomposition

The two-step quench process of surface-directed spinodal decomposition is numerically investigated by coupling the Flory-Huggins-de Gennes equation with the Cahn-Hilliard-Cook equation. The phase dynamics and formation mechanisms of the wetting layer in two-step surface-directed spinodal decomposition have been concerned in detail. The results demonstrate that a parallel strip structure forms near the wetting layer and propagates into the bulk, when the first quench depth is very shallow and the bulk does not undergo phase separation, and the second quench depths are various points with deeper quench depths. In this case, the wetting layer turns to be unchangeable at the intermediate and later stages of the second quench process, compared to the growth with a time exponent 1/2 during the first quench process. When the first quench depth is deeper and phase separation occurs in the bulk during the first quench process, it is found that a deeper second quench depth can stimulate a more obvious secondary domain structure, and the formation mechanism of the wetting layer changes from logarithmic growth law to Lifshitz-Slyozov growth law.

Irregular semi-convex gradient systems perturbed by noise and application to the stochastic Cahn–Hilliard equation

We prove essential self-adjointness of Kolmogorov operators corresponding to gradient systems with potentials U such that DU is not square integrable with respect the invariant measure (irregular potentials). An application is given to the Cahn–Hilliard–Cook equation in dimension one. In this case the spectral gap is proved for the correspondig semigroup. We also obtain a log-Sobolev inequality.

Diffusion coefficient of the order parameter in the early stages of the time evolution of InGaN phase separation

We have carried out a study involving a model for the diffusion mechanism of group III atoms in an InGaN alloy semiconductor that uses a cell dynamical system (CDS) approach. In this CDS analysis, we focused on gathering information concerning the In-rich region with a dot-like structure that is formed at an early stage in the time evolution of phase separation of this alloy. A similar dot-like structure is known to appear in the active layer of InGaN-based multiquantum well laser diodes. Thus, we can determine the diffusion coefficient of group III atoms in InGaN by comparing the dot-like structures observed by cross-sectional transmission electron microscopy with those obtained by the CDS numerical approach. In order to perform this comparison quantitatively such that we could use it to calculate the diffusion coefficient, we derived a relationship between the rule of mapping used in CDS, and the discretization of the Cahn–Hilliard–Cook equation, which describes the dynamics of phase separation with a conserved order parameter. The value obtained for the diffusion coefficient is 4×10−16 cm2/s at a growth temperature of 800 °C. From this result, we can conclude that the diffusion of group III atoms arises from atomistic migration on the surface of an InGaN active layer during growth, instead of diffusion within the solid phase.

Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation

This paper gives theoretical results on spinodal decomposition for the stochastic Cahn–Hilliard–Cook equation, which is a Cahn–Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of a characteristic size.

Scaling properties of the morphological measures at the early and intermediate stages of the spinodal decomposition in homopolymer blends

The spinodal decomposition of the homopolymer blends has been studied by the numerical integration of the Cahn–Hilliard–Cook equation. We have investigated the time evolution of the morphological measures that characterize quantitatively the interface in the system. For symmetric blends we have found that the Euler characteristic of the interface is negative and increases with time as τ0.75 (connectivity of the domains decreases) regardless of the final quench temperature. The homogeneity index of the interface is constant in this case. This suggests that at the level of the integral geometry quantities (Minkowski functionals), the dynamic scaling hypothesis holds for the evolution of the interface morphology in quenched critical systems. The nonuniversal morphological evolution of the asymmetric blends have been studied. Also, we have shown that the thermal fluctuations can modify significantly the curvature distribution.

Spinodal decomposition in Fe-Cr alloys: Experimental study at the atomic level and comparison with computer models—II. Development of domain size and composition amplitude

The three-dimensional interconnected microstructures resulting from spinodal decomposition in a series of thermally aged Fe-Cr alloys have been analysed in terms of scale and composition amplitude. The development of the microstructure scale was found to fit a power law with a time exponent considerably smaller than that predicted by the LSW theory but in agreement with Monte Carlo simulations of the decomposition. Numerical solutions to the classical non-linear Cahn—Hilliard—Cook equation were found to fit the classical LSW theory. A model, based on the non-linear theory of spinodal decomposition by Langer et al. is used to quantify the composition amplitude at any stage of the phase separation. A detailed comparison between the atomic scale experimental results and computer simulations of spinodal decomposition is given.

Early stages of spinodal decomposition for the Cahn-Hilliard-Cook model of phase separation.

A computer simulation using discretized space and time is employed to examine the early stages of phase separation as described by the Cahn-Hilliard-Cook equation (CHCE). The relative simplicity of this simulation provides a direct test of various mathematical treatments of the nonlinear term in the CHCE. In particular, the numerical results are used to ascertain the validity of the approximations inherent in the perturbative expansions of Grant, San Miguel, Vinals, and Gunton, and of Langer, Bar-on, and Miller. The time of validity of all approximations is shown to be logarithmically related to the strength of the thermal fluctuations. In addition, the effect of the initial state on the dynamical evolution of the order parameter is examined.

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation