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Register a new userSharp interface limits of phase-field models
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The use of continuum phase-field models to describe the motion of well-defined interfaces is discussed for a class of phenomena, that includes order/disorder transitions, spinodal decomposition and Ostwald ripening, dendritic growth, and the solidification of eutectic alloys. The projection operator method is used to extract the ``sharp interface limit from phase field models which have interfaces that are diffuse on a length scale $xi$. In particular,phase-field equations are mapped onto sharp interface equations in the limits $xi kappa ll 1$ and $xi v/D ll 1$, where $kappa$ and $v$ are respectively the interface curvature and velocity and $D$ is the diffusion constant in the bulk. The calculations provide one general set of sharp interface equations that incorporate the Gibbs-Thomson condition, the Allen-Cahn equation and the Kardar-Parisi-Zhang equation.
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In this paper, we propose and analyze a diffuse interface model for inductionless magnetohydrodynamic fluids. The model couples a convective Cahn-Hilliard equation for the evolution of the interface, the Navier-Stokes system for fluid flow and the possion quation for electrostatics. The model is derived from Onsagers variational principle and conservation laws systematically. We perform formally matched asymptotic expansions and develop several sharp interface models in the limit when the interfacial thickness tends to zero. It is shown that the sharp interface limit of the models are the standard incompressible inductionless magnetohydrodynamic equations coupled with several different interface conditions for different choice of the mobilities. Numerical results verify the convergence of the diffuse interface model with different mobilitiess.
We formulate a general shape and topology optimization problem in structural optimization by using a phase field approach. This problem is considered in view of well-posedness and we derive optimality conditions. We relate the diffuse interface problem to a perimeter penalized sharp interface shape optimization problem in the sense of $Gamma$-convergence of the reduced objective functional. Additionally, convergence of the equations of the first variation can be shown. The limit equations can also be derived directly from the problem in the sharp interface setting. Numerical computations demonstrate that the approach can be applied for complex structural optimization problems.
We consider sharp interface asymptotics for a phase field model of two phase near spherical biomembranes involving a coupling between the local mean curvature and the local composition proposed by the first and second authors. The model is motivated by lipid raft formation. We introduce a reduced diffuse interface energy depending only on the membrane composition and derive the $Gamma-$limit. We demonstrate 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.
Motivated by the drying pattern experiment by Yamazaki and Mizuguchi[J. Phys. Soc. Jpn. {bf 69} (2000) 2387], we propose the dynamics of sweeping interface, in which material distributed over a region is swept by a moving interface. A model based on a phase field is constructed and results of numerical simulations are presented for one and two dimensions. Relevance of the present model to the drying experiment is discussed.
Minimal models of self-propelled particles with short-range volume exclusion interactions have been shown to exhibit signatures of phase separation. Here I show that the observed interfacial stability and fluctuations in motility-induced phase separations (MIPS) can be explained by modeling the microscopic dynamics of the active particles in the interfacial region. In addition, I demonstrate the validity of the Gibbs-Thomson relation in MIPS, which provides a functional relationship between the size of a condensed drop and its surrounding vapor concentration. As a result, the coarsening dynamics of MIPS at vanishing supersaturation follows the classic Lifshitz-Slyozov scaling law at the late stage.
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