Analysis of the operator splitting scheme for the Cahn‐Hilliard equation with a viscosity term

In this paper, we consider a second‐order fast explicit operator splitting method for the viscous Cahn‐Hilliard equation, which includes a viscosity term αΔu_t (α ∈ (0, 1)) described the influences of internal micro‐forces. The choice α = 0 corresponds to the classical Cahn‐Hilliard equation whilst the choice α = 1 recovers the nonlocal Allen‐Cahn equation. The fundamental idea of our method is to split the original problem into linear and nonlinear parts. The linear subproblem is numerically solved using a pseudo‐spectral method, and thus an ordinary differential equation is obtained. The nonlinear one is solved via TVD‐RK method. The stability and convergence are discussed in L²‐norm. Numerical experiments are performed to validate the accuracy and efficiency of the proposed method. Besides, a detailed comparison is made for the dynamics and the coarsening process of the metastable pattern for various values of α. Moreover, energy degradation and mass conservation are also verified.     

Slow dynamics for the hyperbolic Cahn‐Hilliard equation in one‐space dimension

The aim of this paper is to study the metastable properties of the solutions to a hyperbolic relaxation of the classic Cahn‐Hilliard equation in one‐space dimension, subject to either Neumann or Dirichlet boundary conditions. To perform this goal, we make use of an “energy approach," already proposed for various evolution PDEs, including the Allen‐Cahn and the Cahn‐Hilliard equations. In particular, we shall prove that certain solutions maintain a N‐transition layer structure for a very long time, thus proving their metastable dynamics. More precisely, we will show that, for an exponentially long time, such solutions are very close to piecewise constant functions assuming only the minimal points of the potential, with a finitely number of transition layers, which move with an exponentially small velocity.     

Phase Separation in the Advective Cahn–Hilliard Equation

The Cahn–Hilliard equation is a classic model of phase separation in binary mixtures that exhibits spontaneous coarsening of the phases. We study the Cahn–Hilliard equation with an imposed advection term in order to model the stirring and eventual mixing of the phases. The main result is that if the imposed advection is sufficiently mixing, then no phase separation occurs, and the solution instead converges exponentially to a homogeneous mixed state. The mixing effectiveness of the imposed drift is quantified in terms of the dissipation time of the associated advection–hyperdiffusion equation, and we produce examples of velocity fields with a small dissipation time. We also study the relationship between this quantity and the dissipation time of the standard advection–diffusion equation.

     

Nonlocal reaction--diffusion equations and nucleation

A nonlocal reaction-diffusion equation is presented and analysedusing matched asymptotic expansions and multiple timescales.The problem models a binary mixture undergoing phase separation.The particular form of the equation is motivated by argumentsfrom the calculus of variations, with the nonlocality arisingfrom an enforcement of conservation of mass. It is shown thatthe evolution of the solution can be characterized by trackingthe motion of fronts separating phases. The propagation of theinterfaces is found to be a coarsening process which dependsin a nonlocal fashion on mean curvature. Several special featuresof the equations of motion for the fronts are studied, and therelation of this evolution to Cahn-Hilliard theory and nucleationis discussed     

Coarsening dynamics of polymer droplets for a lubrication model with large slippage

We analyze the final stages of the dewetting process of nanoscopic thin polymer films on hydrophobized substrates using a lubrication model that captures the large slippage at the liquid-substrate interface. The final stages of this process are characterized by the slow-time coarsening dynamics of the remaining droplets. For this situation we derive a reduced system of equations from the lubrication model, using singular perturbation analysis. Such reduced models allow for an efficient numerical simulation of the coarsening process. The reduced model extends results of [2] for no-slip lubrication model. Apart from collapse and collision, we identify here some new coarsening dynamics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Metastable Dynamics and Spatially Inhomogeneous Equilibria in Dumbbell-Shaped Domains

The motion of internal layers for three singularly perturbed reaction diffusion problems, including the Allen–Cahn equation, is studied in a two-dimensional dumbbell-shaped domain. The channel region that connects the two attachments, or lobes, of the dumbbell is taken to be rectangular. The motion of straight-line internal layers in the channel region is analyzed by using an asymptotic projection method. It is shown that this motion is metastable and highly dependent on the local convexity properties of the boundary near the contact region between the ends of the channel and the two attachments. When the domain is nonconvex it is shown that the metastable internal layers dynamics in the channel tends, as t →∞, to a limiting, stable, spatially inhomogeneous equilibrium solution.     

A Cahn‐Hilliard–type equation with application to tumor growth dynamics

We consider a Cahn‐Hilliard–type equation with degenerate mobility and single‐well potential of Lennard‐Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn‐Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.     

Decoupled energy‐law preserving numerical schemes for the Cahn–Hilliard–Darcy system

We study two novel decoupled energy‐law preserving and mass‐conservative numerical schemes for solving the Cahn‐Hilliard‐Darcy system which models two‐phase flow in porous medium or in a Hele–Shaw cell. In the first scheme, the velocity in the Cahn–Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn–Hilliard equation. In the second scheme, an intermediate velocity is used in the Cahn–Hilliard equation which allows for the decoupling. We show that the first scheme preserves a discrete energy law with a time‐step constraint, while the second scheme satisfies an energy law without any constraint and is unconditionally stable. Ample numerical experiments are performed to gauge the efficiency and robustness of our scheme. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 936–954, 2016     

Efficient energy-stable schemes for the hydrodynamics coupled phase-field model

In this article, several efficient and energy-stable semi–implicit schemes are presented for the Cahn–Hilliard phase-field model of two-phase incompressible flows. A scalar auxiliary variable (SAV) approach is implemented to solve the Cahn–Hilliard equation, while a splitting method based on pressure stabilization is used to solve the Navier–Stokes equation. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of the phase-field variable, velocity, and pressure are totally decoupled. A finite-difference method on staggered grids is adopted to spatially discretize the proposed time-marching schemes. We rigorously prove the unconditional energy stability for the semi-implicit schemes and the fully discrete scheme. Numerical results in both two and three dimensions are obtained, which demonstrate the accuracy and effectiveness of the proposed schemes. Using our numerical schemes, we compare the SAV, invariant energy quadratization (IEQ), and stabilization approaches. Bubble rising dynamics and coarsening dynamics are also investigated in detail. The results demonstrate that the SAV approach is more accurate than the IEQ approach and that the stabilization approach is the least accurate among the three approaches. The energy stability of the SAV approach appears to be better than that of the other approaches at large time steps.     

Spectral analysis for periodic solutions of the Cahn�CHilliard equation on {\mathbb{R}}

We consider the spectrum associated with the linear operator obtained when the Cahn–Hilliard equation on \mathbbR{\mathbb{R}} is linearized about a stationary periodic solution. Our analysis is particularly motivated by the study of spinodal decomposition, a phenomenon in which the rapid cooling (quenching) of a homogeneously mixed binary alloy causes separation to occur, resolving the mixture into regions of different crystalline structure, separated by steep transition layers. In this context, a natural problem regards the evolution of solutions initialized by small, random (in some sense) perturbations of the pre-quenching homogeneous state. Solutions initialized in this way appear to evolve transiently toward certain unstable periodic solutions, with the rate of evolution described by the spectrum associated with these periodic solutions. In the current paper, we use Evans function methods and a perturbation argument to locate the spectrum associated with such periodic solutions. We also briefly discuss a heuristic method due to Langer for relating our spectral information to coarsening rates.     

On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects

Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn–Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell–cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. In the overall model an equation of Cahn–Hilliard type is coupled to the system of linear elasticity and a reaction–diffusion equation for a nutrient concentration. The highly non-linear coupling between a fourth-order Cahn–Hilliard equation and the quasi-static elasticity system lead to new challenges which cannot be dealt within a gradient flow setting which was the method of choice for other elastic Cahn–Hilliard systems. We show existence, uniqueness and regularity results. In addition, several continuous dependence results with respect to different topologies are shown. Some of these results give uniqueness for weak solutions and other results will be helpful for optimal control problems.     

Cahn-Hilliard inpainting and the Willmore functional

The Cahn-Hilliard equation has its origin in material sciences and serves as a model for phase separation and phase coarsening in binary alloys. A new approach in the class of fourth order inpainting algorithms is inpainting of binary images using the Cahn-Hilliard equation. Like solutions of the Cahn-Hilliard equation converging to two main values during the phase separation process, the grayvalues inside the missing part of the image are oriented towards the binary states black and white. We present stability/instability results for solutions of the Cahn-Hilliard equation and their connection to the Willmore functional. In particular we will consider the Willmore functional as a quantity to find the optimal scale of the inpainting result. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A phase field approach to solidification and solute separation in water solutions

We propose a phase field model for the solid–liquid phase transition in a water-salt (sodium chloride) solution in the absence of macroscopic motion, under possibly non-isothermal conditions. A thermodynamic approach based on a free energy functional is assumed. The model consists of three evolution equations: a time-dependent Ginzburg–Landau equation for the solid–liquid phase change, a diffusion equation of the Cahn–Hilliard kind for the solute dynamics and the heat equation for the temperature change. The proposed system is aimed to contribute to the modelling of the brine channels formation in the ice of the polar seas.     

Stability on Time-Dependent Domains

We explore the key differences in the stability picture between extended systems on time-fixed and time-dependent spatial domains. As a paradigm, we take the complex Swift–Hohenberg equation, which is the simplest nonlinear model with a finite critical wavenumber, and use it to study dynamic pattern formation and evolution on time-dependent spatial domains in translationally invariant systems, i.e., when dilution effects are absent. In particular, we discuss the effects of a time-dependent domain on the stability of spatially homogeneous and spatially periodic base states, and explore its effects on the Eckhaus instability of periodic states. New equations describing the nonlinear evolution of the pattern wavenumber on time-dependent domains are derived, and the results compared with those on fixed domains. Pattern coarsening on time-dependent domains is contrasted with that on fixed domains with the help of the Cahn–Hilliard equation extended here to time-dependent domains. Parallel results for the evolution of the Benjamin–Feir instability on time-dependent domains are also given.     

Equations de Cahn-Hilliard généralisées dans un milieu déform able

We consider in this Note models of generalized Cahn—Hilliard equations that take into account the effects of internal microforces and introduced by M. Gurtin in [6] coupled with the Navier equation of linear elasticity (under the small deformations assumption) for which we obtain the existence and uniqueness of weak solutions. When the deformations are infinitesimal and when the displacement gradient is small, in which case we can neglect the evolutive term in the Navier equation, we can furthermore prove the existence of finite-dimensional attractors by noting that the variational formulation can be uncoupled. It is important to note here that these results cannot in general be obtained for the coupled (classical) Cahn—Hilliard equations.     

Numerical and experimental investigations of phase segragation effects in binary brazing solders

Solder materials occupy many of fields for technical application (e.g. solder joints in automotive control units or in microelectronic packages). They are required to provide electrical and mechanical connections between different components. Due to their wide range of applications solder alloys are subject to a great variety of microstructural changes such as phase separation and coarsening processes. The micromorphological variations influence strength and life expectation of solder materials, in particular, in very small components such as solder joints in microelectronic packages. In order to analyze the microstructural evolution with a diffusion theory of heterogeneous solid mixtures we employ an extended Cahn-Hilliard phase field model. The diffusion equation under consideration constitutes a partial differential equation involving spatial derivatives of fourth order. Thus, the variational formulation of the problem requires approximation functions which are piecewise smooth and globally C1-continuous. In our contribution we fulfil the continuity requirement by means of rational B-spline finite element basis functions. To illustrate the versatility of this approach numerical simulations of phase decomposition and coarsening controlled by diffusion and by mechanical loading are discussed and compared with experimental results. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mathematical model for phase separation: A generalized Cahn–Hilliard equation

In this paper we present a mathematical model to describe the phenomenon of phase separation, which is modelled as space regions where an order parameter changes smoothly. The model proposed, including thermal and mixing effects, is deduced for an incompressible fluid, so the resulting differential system couples a generalized Cahn–Hilliard equation with the Navier–Stokes equation. Its consistency with the second law of thermodynamics in the classical Clausius–Duhem form is finally proved. Copyright © 2011 John Wiley & Sons, Ltd.     

Classical solvability of 1‐D Cahn–Hilliard equation coupled with elasticity

In this paper, we prove the classical solvability of a nonlinear 1‐D system of hyperbolic–parabolic type arising as a model of phase separation in deformable binary alloys. The system is governed by the nonstationary elasticity equation coupled with the Cahn–Hilliard equation. The existence proof is based on the application of the Leray–Schauder fixed point theorem and standard energy methods. Copyright © 2006 John Wiley & Sons, Ltd.     

Uniformly Distributed Circular Porous Pattern Generation on Surface for 3D Printing

We present an algorithm for uniformly distributed circular porous pattern generation on surface for three-dimensional (3D) printing using a phase-field model. The algorithm is based on the narrow band domain method for the nonlocal Cahn–Hilliard (CH) equation on surfaces. Surfaces are embedded in 3D grid and the narrow band domain is defined as the neighborhood of surface. It allows one can perform numerical computation using the standard discrete Laplacian in 3D instead of the discrete surface Laplacian. For complex surfaces, we reconstruct them from point cloud data and represent them as the zero-level set of their discrete signed distance functions. Using the proposed algorithm, we can generate uniformly distributed circular porous patterns on surfaces in 3D and print the resulting 3D models. Furthermore, we provide the test of accuracy and energy stability of the proposed method.