Sharp Interface Energy Research Articles

On the sharp interface limit of a phase field model for near spherical two phase biomembranes

We consider sharp interface asymptotics for a phase field model motivated by lipid raft formation on near spherical biomembranes involving a coupling between the local mean curvature and the local composition. A reduced diffuse interface energy depending only on the membrane composition is introduced and a \Gamma -limit is derived. It is shown that the Euler–Lagrange equations for the limiting functional and the sharp interface energy coincide. Finally, we consider a system of gradient flow equations with conserved Allen–Cahn dynamics for the phase field model. Performing a formal asymptotic analysis, we obtain a system of gradient flow equations for the sharp interface energy coupling geodesic curvature flow for the phase interface to a fourth order PDE free boundary problem for the surface deformation.

Hybrid lattice Boltzmann finite difference model for simulation of phase change in a ternary fluid

In this paper, a hybrid lattice Boltzmann finite difference model based on the phase-field lattice Boltzmann and finite difference approaches is proposed to model phase-change phenomena in a ternary system. The system contains three immiscible incompressible fluids and the phase-change process happens at the interfaces of the fluids. Three distribution functions are used in the model; two of which are used to track the interfaces among three fluids and the other one is employed to recover the hydrodynamic properties (pressure and momentum). A sharp-interface energy equation is solved based on a finite difference approach and the net heat flux at the interface is considered as the driving force for the phase-change process. The proposed model is validated against available results and good agreement is found.

Phase-field lattice Boltzmann modeling of boiling using a sharp-interface energy solver.

The main objective of this paper is to extend an isothermal incompressible two-phase lattice Boltzmann equation method to model liquid-vapor phase change problems using a sharp-interface energy solver. Two discrete particle distribution functions, one for the continuity equation and the other for the pressure evolution and momentum equations, are considered in the current model. The sharp-interface macroscopic internal energy equation is discretized with an isotropic finite difference method to find temperature distribution in the system. The mass flow generated at liquid-vapor phase interface is embedded in the pressure evolution equation. The sharp-interface treatment of internal energy equation helps to find the interfacial mass flow rate accurately where no free parameter is needed in the calculations. The proposed model is verified against available theoretical solutions of the two-phase Stefan problem and the two-phase sucking interface problem, with which our simulation results are in good agreement. The liquid droplet evaporation in a superheated vapor, the vapor bubble growth in a superheated liquid, and the vapor bubble rising in a superheated liquid are analyzed and underlying physical characteristics are discussed in detail. The model is successfully tested for the liquid-vapor phase change with large density ratio up to 1000.

Existence, bifurcation, and geometric evolution of quasi-bilayers in the multicomponent functionalized Cahn–Hilliard equation

Multicomponent bilayer structures arise as the ubiquitous plasma membrane in cellular biology and as blends of amphiphilic copolymers used in electrolyte membranes, drug delivery, and emulsion stabilization within the context of synthetic chemistry. We present the multicomponent functionalized Cahn-Hilliard (mFCH) free energy as a model which allows competition between bilayers with distinct composition and between bilayers and higher codimensional structures, such as co-dimension two filaments and co-dimension three micelles. We construct symmetric and asymmetric homoclinic bilayer profiles via a billiard limit potential and show that co-dimensional bifurcation is driven by the experimentally observed layer-by-layer pearling mechanism. We investigate the stability and slow geometric evolution of multicomponent bilayer interfaces within the context of an [Formula: see text] gradient flow of the mFCH, addressing the impact of aspect ratio of the amphiphile (lipid or copolymer unit) on the intrinsic curvature and the codimensional bifurcation. In particular we derive a Canham-Helfrich sharp interface energy whose intrinsic curvature arises through a Melnikov parameter associated to amphiphile aspect ratio.

Domain Structure of Bulk Ferromagnetic Crystals in Applied Fields Near Saturation

We investigate the ground state of a uniaxial ferromagnetic plate with perpendicular easy axis and subject to an applied magnetic field normal to the plate. Our interest is the asymptotic behavior of the energy in macroscopically large samples near the saturation field. We establish the scaling of the critical value of the applied field strength below saturation at which the ground state changes from the uniform to a branched domain magnetization pattern and the leading order scaling behavior of the minimal energy. Furthermore, we derive a reduced sharp-interface energy giving the precise asymptotic behavior of the minimal energy in macroscopically large plates under a physically reasonable assumption of small deviations of the magnetization from the easy axis away from domain walls. On the basis of the reduced energy, and by a formal asymptotic analysis near the transition, we derive the precise asymptotic values of the critical field strength at which non-trivial minimizers (either local or global) emerge. The non-trivial minimal energy scaling is achieved by magnetization patterns consisting of long slender needle-like domains of magnetization opposing the applied field

A time-discrete model for dynamic fracture based on crack regularization

We propose a discrete time model for dynamic fracture based on crack regularization. The advantages of our approach are threefold: first, our regularization of the crack set has been rigorously shown to converge to the correct sharp-interface energy Ambrosio and Tortorelli (Comm. Pure Appl. Math., 43(8): 999–1036 (1990); Boll. Un. Mat. Ital. B (7), 6(1):105–123, 1992); second, our condition for crack growth, based on Griffith’s criterion, matches that of quasi-static settings Bourdin (Interfaces Free Bound 9(3): 411–430, 2007) where Griffith originally stated his criterion; third, solutions to our model converge, as the time-step tends to zero, to solutions of the correct continuous time model Larsen (Math Models Methods Appl Sci 20:1021–1048, 2010). Furthermore, in implementing this model, we naturally recover several features, such as the elastic wave speed as an upper bound on crack speed, and crack branching for sufficiently rapid boundary displacements. We conclude by comparing our approach to so-called “phase-field” ones. In particular, we explain why phase-field approaches are good for approximating free boundaries, but not the free discontinuity sets that model fracture.

Droplet Phases in Non-local Ginzburg-Landau Models with Coulomb Repulsion in Two Dimensions

We establish the behavior of the energy of minimizers of non-local Ginzburg-Landau energies with Coulomb repulsion in two space dimensions near the onset of multi-droplet patterns. Under suitable scaling of the background charge density with vanishing surface tension the non-local Ginzburg-Landau energy becomes asymptotically equivalent to a sharp interface energy with screened Coulomb interaction. Near the onset the minimizers of the sharp interface energy consist of nearly identical circular droplets of small size separated by large distances. In the limit the droplets become uniformly distributed throughout the domain. The precise asymptotic limits of the bifurcation threshold, the minimal energy, the droplet radii, and the droplet density are obtained.

Domain branching in uniaxial ferromagnets: asymptotic behavior of the energy

In this article we analyze the ground state energy of a ferromagnetic bulk sample with strong uniaxial anisotropy in a regime featuring domain branching. We derive the convergence of the micromagnetic energy towards a sharp interface energy in the spirit of Γ-limits. Then we use this convergence to rigorously justify the notion of a minimal energy per cross-section area. Compared to known results, the scaling bounds for the minimal energy are improved to an asymptotic equality.

Continuum limits of atomistic energies allowing smooth and sharp interfaces in 1D elasticity

We present two atomistic models for the energy of a one-dimensional elastic crystal. We assume that the macroscopic displacement equals the microscopic one. The energy of the first model is given by a two-body interaction potential, and we assume that the atoms follow a continuous and piecewise smooth macroscopic (continuum) deformation. We calculate the first terms of the Taylor expansion (with respect to the parameter representing the interatomic distance) of the atomistic energy, and deduce that the coefficients of that Taylor expansion represent, respectively, an elastic energy, a sharp-interface energy, and a smooth-interface energy. The second atomistic model is a variant of the first one, and its Taylor expansion predicts, in addition, a new term that accounts for the repulsion force between two sharp interfaces.

Deterministic equivalent for the Allen-Cahn energy of a scaling law in the Ising model

For the Allen-Cahn functional we study the following problem: for which prescribed amount m of volume is there the appearence of a droplet of one phase inside the other? Under a suitable assumption on the domain we show that the breaking of symmetry occurs at the same value of m as for the limit of the sharp interface energy. We also prove that there exists a threshold for m of order \(\varepsilon^\frac{n}{n+1}\) so that either there is the appearence of the droplet or there is no breaking of symmetry.

Grain boundary motion arising from the gradient flow of the Aviles–Giga functional

This paper considers the singular limit of the equation Θ t = − ϵ Δ 2 Θ + ϵ − 1 ∇ ⋅ ( [ | ∇ Θ | 2 − 1 ] ∇ Θ ) . Grain boundaries (limiting discontinuities in ∇ Θ ) form networks that coarsen over time. A matched asymptotic analysis is used to derive a free boundary problem consisting of curve motion coupled along hyperbolic characteristics and junction conditions. An intermediate boundary layer near extrema junctions is discovered, along with the relevant nonlocal junction conditions. The limiting dynamics can be viewed in the context of a gradient flow of the sharp interface energy on an attracting manifold. Dynamic scaling of the long-time coarsening process can be explained by dimensional analysis of the reduced problem.