Allen-Cahn Type Research Articles

Travelling-wave amplitudes as solutions of the phase-field crystal equation.

The dynamics of the diffuse interface between liquid and solid states is analysed. The diffuse interface is considered as an envelope of atomic density amplitudes as predicted by the phase-field crystal model (Elder et al. 2004 Phys. Rev. E70, 051605 (doi:10.1103/PhysRevE.70.051605); Elder et al. 2007 Phys. Rev. B75, 064107 (doi:10.1103/PhysRevB.75.064107)). The propagation of crystalline amplitudes into metastable liquid is described by the hyperbolic equation of an extended Allen-Cahn type (Galenko & Jou 2005 Phys. Rev. E71, 046125 (doi:10.1103/PhysRevE.71.046125)) for which the complete set of analytical travelling-wave solutions is obtained by the [Formula: see text] method (Malfliet & Hereman 1996 Phys. Scr.15, 563-568 (doi:10.1088/0031-8949/54/6/003); Wazwaz 2004 Appl. Math. Comput.154, 713-723 (doi:10.1016/S0096-3003(03)00745-8)). The general [Formula: see text] solution of travelling waves is based on the function of hyperbolic tangent. Together with its set of particular solutions, the general [Formula: see text] solution is analysed within an example of specific task about the crystal front invading metastable liquid (Galenko et al. 2015 Phys. D308, 1-10 (doi:10.1016/j.physd.2015.06.002)). The influence of the driving force on the phase-field profile, amplitude velocity and correlation length is investigated for various relaxation times of the gradient flow.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

An Allen–Cahn approach to the remodelling of fibre-reinforced anisotropic materials

We propose a theory of remodelling in fibre-reinforced biological tissues, in which the fibre orientation follows a given probability density. The latter is characterised by variance and mean angle. We claim that the fibres may change their orientation in time, thereby triggering a remodelling process that can be described by the spatiotemporal evolution of the mean angle. This is determined by solving a balance of external and internal generalised forces. We assign the latter ones by establishing a constitutive theory capable of resolving the spatial variability of the fibre mean angle and featuring a free energy density of the Allen–Cahn type. Through numerical simulations, we compare the predictions of our model with the results of another model available in the literature. Finally, we interpret the evolution of the mean angle as the consequence of a symmetry breaking that occurs in the tissue both spontaneously and due to the coupling between remodelling and deformation.

Néel walls with prescribed winding number and how a nonlocal term can change the energy landscape

We study a nonlocal Allen–Cahn type problem for vector fields of unit length, arising from a model for domain walls (called Néel walls) in ferromagnetism. We show that the nonlocal term gives rise to new features in the energy landscape; in particular, we prove existence of energy minimisers with prescribed winding number that would be prohibited in a local model.

A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions

In this paper, we consider a class of coupled systems of PDEs, denoted by (ACE)ε for ε≥0. For each ε≥0, the system (ACE)ε consists of an Allen–Cahn type equation in a bounded spacial domain Ω, and another Allen–Cahn type equation on the smooth boundary Γ:=∂Ω, and besides, these coupled equations are transmitted via the dynamic boundary conditions. In particular, the equation in Ω is derived from the non-smooth energy proposed by Visintin in his monography “Models of phase transitions”: hence, the diffusion in Ω is provided by a quasilinear form with singularity. The objective of this paper is to build a mathematical method to obtain meaningful L2-based solutions to our systems, and to see some robustness of (ACE)ε with respect to ε≥0. On this basis, we will prove two Main Theorems 1 and 2, which will be concerned with the well-posedness of (ACE)ε for each ε≥0, and the continuous dependence of solutions to (ACE)ε for the variations of ε≥0, respectively.

General set of traveling-wave solutions for amplitude equations in the phase field crystal model

Fronts dynamics of periodic crystalline state, which invades the homogeneous state (liquid phase), are analysed. These fronts are considered as traveling waves of atomic density amplitudes. The propagation of amplitudes is described by the hyperbolic equation of an extended Allen-Cahn type for which the complete set of analytical traveling-wave solutions are obtained by tanh-method. The set of solutions includes previously known traveling waves for the parabolic Allen-Cahn equation of both extended and standard form.

Numerical Approximations for Allen-Cahn Type Phase Field Model of Two-Phase Incompressible Fluids with Moving Contact Lines

AbstractIn this paper, we present some efficient numerical schemes to solve a two-phase hydrodynamics coupled phase field model with moving contact line boundary conditions. The model is a nonlinear coupling system, which consists the Navier-Stokes equations with the general Navier Boundary conditions or degenerated Navier Boundary conditions, and the Allen-Cahn type phase field equations with dynamical contact line boundary condition or static contact line boundary condition. The proposed schemes are linear and unconditionally energy stable, where the energy stabilities are proved rigorously. Various numerical tests are performed to show the accuracy and efficiency thereafter.

On the motion law of fronts for scalar reaction-diffusion equations with equal depth multiple-well potentials

Slow motion for scalar Allen-Cahn type equation is a well-known phenomenon, precise motion law for the dynamics of fronts having been established first using the socalled geometric approach inspired from central manifold theory (see the results of Carr and Pego in 1989). In this paper, the authors present an alternate approach to recover the motion law, and extend it to the case of multiple wells. This method is based on the localized energy identity, and is therefore, at least conceptually, simpler to implement. It also allows to handle collisions and rough initial data.

Spectral analysis for travelling waves in compressible two-phase fluids of Navier–Stokes–Allen–Cahn type

This is the first part of two papers whose purpose is to investigate stability of travelling wave solutions to the so-called Navier–Stokes–Allen–Cahn system. This set of equations is a combination of the Navier–Stokes equations for compressible fluids supplemented with a phase field description of Allen–Cahn type. The main part of this work deals with studying the problem obtained by linearizing the NSAC system around so-called standing waves. The main results are (1) local well-posedness of the linearized equations and (2) a detailed description of the point and essential spectrum. As a by-product, we obtain analyticity of the associated semigroup.

Experimental and numerical investigation of drop evaporation depending on the shape of the liquid/gas interface

We present experimental and numerical lifetime investigations of evaporating water drops on solid surfaces depending on the shape and size of their liquid/gas interface. The simulations are based on an Allen–Cahn type phase-field model, where the liquid–gas phase transition is driven by a concentration gradient. The experiments are conducted under defined initial and constant conditions. Throughout the whole experiment, contact angle, temperature, relative air humidity and drop weight are tracked continuously. The numerical and experimental results are compared with analytical predictions. In our study, we confirm that the lifetime of a drop is directly linked to its surface size and shape. We compare lifetimes of evaporating drops on smooth solid surfaces with different contact angles and find an increase in lifetime by 50% as the contact angle increases from 20° to 90°, both in experiments and in simulations. Furthermore, drops placed in different geometric set-ups such as a wedge or a corner show a lifetime behavior that is in accordance with analytical predictions. The presented computational approach captures experimentally measured drop lifetimes, for various set-ups, very well. These findings are especially relevant for industrial questions such as how fast complex components dry after cleaning or how long it takes till lacquer layers are cured on structured surfaces.

Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions

Our aim in this paper is to prove the existence and uniqueness of solutions to Cahn-Hilliard and Allen-Cahn type equations based on a modification of the Ginzburg-Landau free energy proposed in [12] (see also [16]) which takes into account strong anisotropy effects. In particular, the free energy contains a regularization term, called Willmore regularization.

Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian

We study entire solutions in $R$ of the nonlocal system $(-\Delta)^{s}U+\nabla W(U)=(0,0)$ where $W:R^{2}\rightarrow R$ is a double well potential. We seek solutions $U$ which are heteroclinic in the sense that they connect at infinity a pair of global minima of $W$ and are also global minimizers. Under some symmetric assumptions on potential $W$, we prove the existence of such solutions for $s>\frac{1}{2}$, and give asymptotic behavior as $x\rightarrow\pm\infty$.

Existence and uniqueness of monotone nodal solutions of a semilinear Neumann problem

In this paper, we study monotone radially symmetric solutions of semilinear equations with Allen–Cahn type nonlinearities by the bifurcation method. Under suitable conditions imposed on the nonlinearities, we show that the structure of the monotone nodal solutions consists of a continuous U-shaped curve bifurcating from the trivial solution at the third eigenvalue of the Laplacian. The upper branch consists of a decreasing solution and the lower branch consists of an increasing solution. In particular, we show the following equation Δu+λ(u−u∣u∣p−1)=0in B,∂u∂ν=0on ∂B has exactly two monotone radial nodal solutions, one is decreasing and the other is increasing. Here B is the unit ball in Rn, p>1 and λ>0.

An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions

We study nonlinear elliptic equations for operators corresponding to non-stable Lévy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of non-stable (i.e., non self-similar) Lévy processes. We establish the regularity of solutions, as well as sharp energy estimates. As a consequence, we prove a 1-D symmetry result for monotone solutions to Allen–Cahn type equations with a non-stable Lévy diffusion. These operators may still be realized as local operators using a system of PDEs — in the spirit of the extension problem of Caffarelli and Silvestre.

Front propagation for nonlinear diffusion equations on the hyperbolic space

We study the Cauchy problem in the hyperbolic space \mathbb{H}^n (n\ge2) for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space \mathbb R^n new phenomena arise, which depend on the properties of the diffusion process in \mathbb{H}^n . We also investigate a family of travelling wave solutions, named horospheric waves, which have properties similar to those of plane waves in \mathbb R^n .

Existence of Heteroclinic Solutions for a Pseudo-relativistic Allen-Cahn Type Equation

Abstract We study the existence and uniqueness of heteroclinic solutions to non-linear Allen-Cahn equation where G is a double-well potential. We investigate such a problem using variational methods after transforming the problem to an elliptic equation with a nonlinear Neumann boundary conditions.

A second order operator splitting method for Allen–Cahn type equations with nonlinear source terms

Allen–Cahn (AC) type equations with nonlinear source terms have been applied to a wide range of problems, for example, the vector-valued AC equation for phase separation and the phase-field equation for dendritic crystal growth. In contrast to the well developed first and second order methods for the AC equation, not many second order methods are suggested for the AC type equations with nonlinear source terms due to the difficulties in dealing with the nonlinear source term numerically. In this paper, we propose a simple and stable second order operator splitting method. A core idea of the method is to decompose the original equation into three subequations with the free-energy evolution term, the heat evolution term, and a nonlinear source term, respectively. It is important to combine these three subequations in proper order to achieve the second order accuracy and stability. We propose a method with a half-time free-energy evolution solver, a half-time heat evolution solver, a full-time midpoint solver for the nonlinear source term, and a half-time heat evolution solver followed by a final half-time free-energy evolution solver. We numerically demonstrate the second order accuracy of the new numerical method through the simulations of the phase separation and the dendritic crystal growth.

On the second inner variations of Allen–Cahn type energies and applications to local minimizers

In this paper, we obtain an explicit formula for the discrepancy between the limit of the second inner variations of p-Laplace Allen–Cahn energies and the second inner variation of their Γ-limit which is the area functional. Our analysis explains the mysterious discrepancy term found in our previous paper [8] in the case p=2. The discrepancy term turns out to be related to the convergence of certain 4-tensors which are absent in the usual Allen–Cahn functional. These (hidden) 4-tensors suggest that, in the complex-valued Ginzburg–Landau setting, we should expect a different discrepancy term which we are able to identify. Along the way, we partially answer a question of Kohn and Sternberg [6] by giving a relation between the limit of second variations of the Allen–Cahn functional and the second inner variation of the area functional at local minimizers. Moreover, our analysis reveals an interesting identity connecting second inner variation and Poincaré inequality for area-minimizing surfaces with volume constraint in the work of Sternberg and Zumbrun [16].

Front Propagation in Geometric and Phase Field Models of Stratified Media

We study front propagation problems for forced mean curvature flows and their phase field variants that take place in stratified media, i.e., heterogeneous media whose characteristics do not vary in one direction. We consider phase change fronts in infinite cylinders whose axis coincides with the symmetry axis of the medium. Using the recently developed variational approaches, we provide a convergence result relating asymptotic in time front propagation in the diffuse interface case to that in the sharp interface case, for suitably balanced nonlinearities of Allen-Cahn type. The result is established by using arguments in the spirit of $\Gamma$-convergence, to obtain a correspondence between the minimizers of an exponentially weighted Ginzburg-Landau type functional and the minimizers of an exponentially weighted area type functional. These minimizers yield the fastest traveling waves invading a given stable equilibrium in the respective models and determine the asymptotic propagation speeds for front-like initial data. We further show that generically these fronts are the exponentially stable global attractors for this kind of initial data and give sufficient conditions under which complete phase change occurs via the formation of the considered fronts.

A Phase-Field Approach for Wetting Phenomena of Multiphase Droplets on Solid Surfaces

We study the equilibrium wetting behavior of immiscible multiphase systems on a flat, solid substrate. We present numerical computations which are based on a vector-valued multiphase-field model of Allen-Cahn type, with a new boundary condition, based on appropriately designed surface energy contributions in order to ensure the right contact angles at multiphase junctions. Experimental investigations are carried out to validate the method and to support the numerical results.

Well-posedness and long time behavior of an Allen-Cahn type equation

The aim of this article is to study the existence and uniqueness of solutions for an equation of Allen-Cahn type and to prove the existence of the finite-dimensional global attractor as well as the existence of exponential attractors.