Sharp Interface Limit Research Articles

Existence of weak solutions for a sharp interface model for phase separation on biological membranes

<p style='text-indent:20px;'>We prove existence of weak solutions of a Mullins-Sekerka equation on a surface that is coupled to diffusion equations in a bulk domain and on the boundary. This model arises as a sharp interface limit of a phase field model to describe the formation of liqid rafts on a cell membrane. The solutions are constructed with the aid of an implicit time discretization and tools from geometric measure theory to pass to the limit.

Stochastic Cahn\u2013Hilliard equation in higher space dimensions: the motion of bubbles

We study the stochastic motion of a droplet in a stochastic Cahn–Hilliard equation in the sharp interface limit for sufficiently small noise. The key ingredient in the proof is a deterministic slow manifold, where we show its stability for long times under small stochastic perturbations. We also give a rigorous stochastic differential equation for the motion of the center of the droplet.

Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

We establish a quantitative rigidity estimate for two-well frame-indifferent nonlinear energies, in the case in which the two wells have exactly one rank-one connection. Building upon this novel rigidity result, we then analyze solid–solid phase transitions in arbitrary space dimensions, under a suitable anisotropic penalization of second variations. By means of $$\Gamma $$-convergence, we show that, as the size of transition layers tends to zero, singularly perturbed two-well problems approach an effective sharp-interface model. The limiting energy is finite only for deformations which have the structure of a laminate. In this case, it is proportional to the total length of the interfaces between the two phases.

A Cahn--Hilliard Model for Cell Motility

We introduce and study a diffuse interface model describing cell motility. We provide a detailed rigorous analysis of the model in dimension 1 and formally derive the sharp interface limit in any d...

Phase field modelling of surfactants in multi-phase flow

A diffuse interface model for surfactants in multi-phase flow with three or more fluids is derived. A system of Cahn–Hilliard equations is coupled with a Navier-Stokes system and an advection-diffusion equation for the surfactant ensuring thermodynamic consistency. By an asymptotic analysis the model can be related to a moving boundary problem in the sharp interface limit, which is derived from first principles. Results from numerical simulations support the theoretical findings. The main novelties are centred around the conditions in the triple junctions where three fluids meet. Specifically the case of local chemical equilibrium with respect to the surfactant is considered, which allows for interfacial surfactant flow through the triple junctions.

Phase field modeling of rapid resolidification of Al-Cu thin films

Abstract A binary alloy multi-order parameter phase field model is used to study rapid solidification in Al-Cu under conditions corresponding to recent dynamic transmission electron microscopy (DTEM) experiments. The phase field model’s sharp interface limit is set through a recent matched asymptotic analysis to follow the solute trapping and interface undercooling kinetics of the Continuous Growth Model (CGM). The phase field model convergence to the CGM sharp interface model is investigated, and based on this an optimal interface width is chosen to simulate the DTEM experimental conditions. The temperature distribution used in the phase field simulations is taken from an analytic expression extracted from experiments. Simulated solidification structures are compared to experiments, including time-resolved DTEM images and post-mortem TEM-based image quality and orientation maps. We find that the large scale morphological features of the simulated microstructures are in good agreement with the experiments, and the corresponding concentration profiles that emerge are in qualitative agreement with experiments. These results show that phase field simulations, informed with DTEM experiments, provides a promising framework to investigate rapidly solidified microstructure evolution and solute segregation, and to calibrate hard-to-determine solidification parameters.

Convergence of the Allen–Cahn equation to the mean curvature flow with 90°-contact angle in 2D

We consider the sharp interface limit of the Allen–Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain \Omega , in the situation where an interface has developed and intersects \partial \Omega . Here a parameter \varepsilon &gt; 0 in the equation, which is related to the thickness of the diffuse interface, is sent to zero. The limit problem is given by mean curvature flow with a 90°-contact angle condition and convergence using strong norms is shown for small times. Here we assume that a smooth solution to this limit problem exists on [0, T] for some T &gt; 0 and that it can be parametrized suitably. With the aid of asymptotic expansions we construct an approximate solution for the Allen–Cahn equation. In order to estimate the difference of the exact and approximate solution we use a spectral estimate for the linearized Allen–Cahn operator.

The role of degenerate mobilities in Cahn‐Hilliard models

AbstractThe present work provides an insight into the role of degenerate mobilities in phase field models and in particular their influence on the evolution of the interface. The equation of interest is the Cahn‐Hilliard equation (in two space dimensions) with a polynomial double well free energy and different order‐parameter dependent, degenerate mobilities. According to the preliminary work [1], considering an anisotropic version of the equation, the asymptotic sharp interface limit subtly dependents on the degeneracy of the mobility. Whilst a quadratic degenerate mobility leads to a sharp interface model where bulk diffusion is present at the same asymptotic order as surface diffusion, a bi‐quadratic degenerate mobility leads to a sharp interface model where bulk diffusion is subdominant. The present study continues the preliminary work and shows that the type of the degeneracy has a qualitative impact on the evolution: Considering numerical simulations of the dewetting process of a thin solid film with four‐fold anisotropic surface energy, the evolution with bi‐quadratic degenerate mobility turns out to be more effective for film pinch‐off than with quadratic degenerate mobility. An extension to the isotropic Cahn‐Hilliard equation ensures that the characteristic difference in the pinch‐off behavior is in fact mobility‐dependent.

Self-similar dynamics of two-phase flows injected into a confined porous layer

We study the dynamics of two-phase flows injected into a confined porous layer. A model is derived to describe the evolution of the fluid–fluid interface, where the effective saturation of the injected fluid is zero. The flow is driven by pressure gradients due to injection, the buoyancy due to density contrasts and the interfacial tension between the injected and ambient fluids. The saturation field is then computed after the interface evolution is obtained. The results demonstrate that the flow behaviour evolves from early-time unconfined to late-time confined behaviours. In particular, at early times, the influence of capillary forces drives fluid flow and produces a new self-similar spreading behaviour in the unconfined limit that is distinct from the gravity current solution. At late times, we obtain two new similarity solutions, a modified shock solution and a compound wave solution, in addition to the rarefaction and shock solutions in the sharp-interface limit. A schematic regime diagram is also provided, which summarises all possible similarity solutions and the time transitions between them for the partially saturating flows resulting from fluid injection into a confined porous layer. Three dimensionless control parameters are identified and their influence on the fluid flow is also discussed, including the viscosity ratio, the pore-size distribution and the relative contributions of capillary and buoyancy forces. To underline the relevance of our results, we also briefly describe the implications of the two-phase flow model to the geological storage of $\text{CO}_{2}$, using representative geological parameters from the Sleipner and In Salah sites.

Motion by Mean Curvature from Glauber–Kawasaki Dynamics

We study the hydrodynamic scaling limit for the Glauber-Kawasaki dynamics. It is known that, if the Kawasaki part is speeded up in a diffusive space-time scaling, one can derive the Allen-Cahn equation which is a kind of the reaction-diffusion equation in the limit. This paper concerns the scaling that the Glauber part, which governs the creation and annihilation of particles, is also speeded up but slower than the Kawasaki part. Under such scaling, we derive directly from the particle system the motion by mean curvature for the interfaces separating sparse and dense regions of particles as a combination of the hydrodynamic and sharp interface limits.

Droplet phase in a nonlocal isoperimetric problem under confinement

<p style='text-indent:20px;'>We address small volume-fraction asymptotic properties of a nonlocal isoperimetric functional with a confinement term, derived as the sharp interface limit of a variational model for self-assembly of diblock copolymers under confinement by nanoparticle inclusion. We introduce a small parameter <inline-formula><tex-math id="M1">\begin{document}$ \eta $\end{document}</tex-math></inline-formula> to represent the size of the domains of the minority phase, and study the resulting droplet regime as <inline-formula><tex-math id="M2">\begin{document}$ \eta\to 0 $\end{document}</tex-math></inline-formula>. By considering confinement densities which are spatially variable and attain a unique nondegenerate maximum, we present a two-scale asymptotic analysis wherein a separation of length scales is captured due to competition between the nonlocal repulsive and confining attractive effects in the energy. A key role is played by a parameter <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> which gives the total volume of the droplets at order <inline-formula><tex-math id="M4">\begin{document}$ \eta^3 $\end{document}</tex-math></inline-formula> and its relation to existence and non-existence of Gamow's Liquid Drop model on <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula>. For large values of <inline-formula><tex-math id="M6">\begin{document}$ M $\end{document}</tex-math></inline-formula>, the minority phase splits into several droplets at an intermediate scale <inline-formula><tex-math id="M7">\begin{document}$ \eta^{1/3} $\end{document}</tex-math></inline-formula>, while for small <inline-formula><tex-math id="M8">\begin{document}$ M $\end{document}</tex-math></inline-formula> minimizers form a single droplet converging to the maximum of the confinement density.

Sharp interface model for elastic motile cells.

In order to study the effect of cell elastic properties on the behavior of assemblies of motile cells, this paper describes an alternative to the cell phase field (CPF) we have previously proposed. The CPF is a multi-scale approach to simulating many cells which tracked individual cells and allowed for large deformations. Though results were largely in agreement with experiment that focus on the migration of a soft cancer cell in a confluent layer of normal cells, simulations required large computing resources, making a more detailed study unfeasible. In this work we derive a sharp interface limit of CPF, including all interactions and parameters. This new model scales linearly with both system and cell size, compared to our original CPF implementation, which is quadratic in cell size, this gives rise to a considerable speedup, which we discuss in the article. We demonstrate that this model captures a similar behavior and allows us to obtain new results that were previously intractable. We obtain the full velocity distribution for a large range of degrees of confluence, [Formula: see text], and show regimes where its tail is heavier and lighter than a normal distribution. Furthermore, we fully characterize the velocity distribution with a single parameter, and its dependence on [Formula: see text] is fully determined. Finally, cell motility is shown to linearly decrease with increasing [Formula: see text], consistent with previous theoretical results.

Phase-field simulation of core-annular pipe flow

Phase-field methods have long been used to model the flow of immiscible fluids. Their ability to naturally capture interface topological changes is widely recognized, but their accuracy in simulating flows of real fluids in practical geometries is not established. We here quantitatively investigate the convergence of the phase-field method to the sharp-interface limit with simulations of two-phase pipe flow. We focus on core-annular flows, in which a highly viscous fluid is lubricated by a less viscous fluid, and validate our simulations with an analytic laminar solution, a formal linear stability analysis and also in the fully nonlinear regime. We demonstrate the ability of the phase-field method to accurately deal with non-rectangular geometry, strong advection, unsteady fluctuations and large viscosity contrast. We argue that phase-field methods are very promising for quantitatively studying moderately turbulent flows, especially at high concentrations of the disperse phase.

A Phase-Field Approach to Eulerian Interfacial Energies

We analyze a phase-field approximation of a sharp-interface model for two- phase materials proposed by M. Silhavy [32, 33]. The distinguishing trait of the model resides in the fact that the interfacial term is Eulerian in nature, for it is defined on the deformed configuration. We discuss a functional frame allowing for existence of phase- field minimizers and {\Gamma}-convergence to the sharp-interface limit. As a by-product, we provide additional detail on the admissible sharp-interface configurations with respect to the analysis in [32, 33].

Space–time discontinuous Galerkin methods for the ε-dependent stochastic Allen–Cahn equation with mild noise

Abstract We consider the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild space–time noise posed on a bounded domain of $\mathbb{R}^2$. The positive parameter $\varepsilon $ is a measure for the inner layers width that are generated during evolution. This equation, when the noise depends only on time, has been proposed by Funaki (1999, Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sin., 15, 407–438). The noise, although smooth, becomes white on the sharp interface limit $\varepsilon \rightarrow 0^+$. We construct a nonlinear discontinous Galerkin scheme with space–time finite elements of general type that are discontinuous in time. Existence of a unique discrete solution is proven by application of Brouwer’s Theorem. We first derive abstract error estimates and then, for the case of piecewise polynomial finite elements, we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter $\varepsilon $. Finally, we present a linear approximation of the nonlinear scheme, for which we prove existence of solution and optimal error in expectation in piecewise linear finite element spaces. The novelty of this work is based on the use of a finite element formulation in space and in time in $2+1$-dimensional subdomains for a nonlinear parabolic problem. In addition this problem involves noise. These types of schemes avoid any Runge–Kutta-type discretization for the evolutionary variable, and seem to be very effective when applied to equations of such a difficulty.

On Large Deviations of Interface Motions for Statistical Mechanics Models

We discuss the sharp interface limit of the action functional associated to either the Glauber dynamics for Ising systems with Kac potentials or the Glauber+Kawasaki process. The corresponding limiting functionals, for which we provide explicit formulae of the mobility and transport coefficients, describe the large deviations asymptotics with respect to the mean curvature flow.

Asymptotic behavior of Allen–Cahn-type energies and Neumann eigenvalues via inner variations

We use the notion of first and second inner variations as a bridge allowing one to pass to the limit of first and second Gateaux variations for the Allen–Cahn, Cahn–Hilliard and Ohta–Kawasaki energies. Under suitable assumptions, this allows us to show that stability passes to the sharp interface limit, including boundary terms, by considering noncompactly supported velocity and acceleration fields in our variations. This complements the results of Tonegawa, and Tonegawa and Wickramasekera, where interior stability is shown to pass to the limit. As a further application, we prove an asymptotic upper bound on the $$k\mathrm{th}$$ Neumann eigenvalue of the linearization of the Allen–Cahn operator, relating it to the $$k\mathrm{th}$$ Robin eigenvalue of the Jacobi operator, taken with respect to the minimal surface arising as the asymptotic location of the zero set of the Allen–Cahn critical points. We also prove analogous results for eigenvalues of the linearized operators arising in the Cahn–Hilliard and Ohta–Kawasaki settings. These complement the earlier result of the first author where such an asymptotic upper bound is achieved for Dirichlet eigenvalues for the linearized Allen–Cahn operator. Our asymptotic upper bound on Allen–Cahn Neumann eigenvalues extends, in one direction, the asymptotic equivalence of these eigenvalues established in the work of Kowalczyk in the two-dimensional case where the minimal surface is a line segment and specific Allen–Cahn critical points are suitably constructed.

Validity of Formal Asymptotic Expansions for Singularly Perturbed Competition-Diffusion Systems

We consider a two-species competition-diffusion system involving a small parameter $\varepsilon>0$ and discuss the validity of formal asymptotic expansions of solutions near the sharp interface limit $\varepsilon\approx0$. We assume that the corresponding ODE system has two stable equilibria. As in the scalar Allen--Cahn equation, it is known that the motion of the sharp interfaces of such systems is governed by the mean curvature flow with a driving force. The formal expansion also suggests that the profile of the transition layers converges to that of a traveling wave solution as $\varepsilon\rightarrow0$. In this paper, we rigorously verify this latter ansatz for a large class of initial data. The proof relies on a rescaling argument, the super--subsolution method, and a Liouville type theorem for eternal solutions of parabolic systems. Roughly speaking, the Liouville type theorem states that any eternal solution that lies between two traveling waves is itself a traveling wave. The same Liouville type theorem was established for the scalar Allen--Cahn equation by Berestycki and Hamel. In view of their importance, we prove the Liouville type theorems in a rather general framework, not only for two-species competition-diffusion systems but also for $m$-species cooperation-diffusion systems possibly with time periodic or spatially periodic coefficients.

Uniqueness and traveling waves in a cell motility model

<p style='text-indent:20px;'>We study a non-linear and non-local evolution equation for curves obtained as the sharp interface limit of a phase-field model for crawling motion of eukaryotic cells on a substrate. We establish uniqueness of solutions to the sharp interface limit equation in the subcritical parameter regime. The proof relies on a Grönwall estimate for a specially chosen weighted <inline-formula><tex-math id="M1">\begin{document}$L^2$\end{document}</tex-math></inline-formula> norm. <p style='text-indent:20px;'>As persistent motion of crawling cells is of central interest to biologists, we next study the existence of traveling wave solutions. We prove that traveling wave solutions exist in the supercritical parameter regime provided the non-linearity of the sharp interface limit equation possesses certain asymmetry (related, e.g., to myosin contractility). <p style='text-indent:20px;'>Finally, we numerically investigate traveling wave solutions and simulate their dynamics. Due to non-uniqueness of solutions of the sharp interface limit equation we numerically solve a related, singularly perturbed PDE system which is uniquely solvable. Our simulations predict instability of traveling wave solutions and capture both bipedal wandering cell motion as well as rotating cell motion; these behaviors qualitatively agree with recent experimental and theoretical findings.

Sharp interface limit for stochastically perturbed mass conserving Allen–Cahn equation

This paper studies the sharp interface limit for a mass conserving Allen–Cahn equation, added an external noise and derives a stochastically perturbed mass conserving mean curvature flow in the limit. The stochastic term destroys the precise conservation law, instead the total mass changes like a Brownian motion in time. For our equation, the comparison argument does not work, so that to study the limit we adopt the asymptotic expansion method, which extends that for deterministic equations used originally in de Mottoni and Schatzman [Interfaces Free Bound. 12 (2010) 527–549] for the nonconservative case and then in Chen et al. [Trans. Amer. Math. Soc. 347 (1995) 1533–1589] for the conservative case. Differently from the deterministic case, each term except the leading term appearing in the expansion of the solution in a small parameter $\varepsilon$ diverges as $\varepsilon$ tends to $0$, since our equation contains the noise which converges to a white noise and the products or the powers of the white noise diverge. To derive the error estimate for our asymptotic expansion, we need to establish the Schauder estimate for a diffusion operator with coefficients determined from higher order derivatives of the noise and their powers. We show that one can choose the noise sufficiently mild in such a manner that it converges to the white noise and at the same time its diverging speed is slow enough for establishing a necessary error estimate.