6 research outputs found
Convergence of phase-field approximations to the Gibbs-Thomson law
We prove the convergence of phase-field approximations of the Gibbs-Thomson
law. This establishes a relation between the first variation of the
Van-der-Waals-Cahn-Hilliard energy and the first variation of the area
functional. We allow for folding of diffuse interfaces in the limit and the
occurrence of higher-multiplicities of the limit energy measures. We show that
the multiplicity does not affect the Gibbs-Thomson law and that the mean
curvature vanishes where diffuse interfaces have collided.
We apply our results to prove the convergence of stationary points of the
Cahn-Hilliard equation to constant mean curvature surfaces and the convergence
of stationary points of an energy functional that was proposed by Ohta-Kawasaki
as a model for micro-phase separation in block-copolymers.Comment: 25 page
Small Volume Fraction Limit of the Diblock Copolymer Problem: I. Sharp Interface Functional
We present the first of two articles on the small volume fraction limit of a
nonlocal Cahn-Hilliard functional introduced to model microphase separation of
diblock copolymers. Here we focus attention on the sharp-interface version of
the functional and consider a limit in which the volume fraction tends to zero
but the number of minority phases (called particles) remains O(1). Using the
language of Gamma-convergence, we focus on two levels of this convergence, and
derive first and second order effective energies, whose energy landscapes are
simpler and more transparent. These limiting energies are only finite on
weighted sums of delta functions, corresponding to the concentration of mass
into `point particles'. At the highest level, the effective energy is entirely
local and contains information about the structure of each particle but no
information about their spatial distribution. At the next level we encounter a
Coulomb-like interaction between the particles, which is responsible for the
pattern formation. We present the results here in both three and two
dimensions.Comment: 37 pages, 1 figur
Copolymer-homopolymer blends: global energy minimisation and global energy bounds
We study a variational model for a diblock-copolymer/homopolymer blend. The
energy functional is a sharp-interface limit of a generalisation of the
Ohta-Kawasaki energy. In one dimension, on the real line and on the torus, we
prove existence of minimisers of this functional and we describe in complete
detail the structure and energy of stationary points. Furthermore we
characterise the conditions under which the minimisers may be non-unique.
In higher dimensions we construct lower and upper bounds on the energy of
minimisers, and explicitly compute the energy of spherically symmetric
configurations.Comment: 31 pages, 6 Postscript figures; to be published in Calc. Var. Partial
Differential Equations. Version history: Changes in v2 w.r.t v1 only concern
metadata. V3 contains some minor revisions and additions w.r.t. v2. V4
corrects a confusing typo in one of the formulas of the appendix. V5 is the
definitive version that will appear in prin
Stability of monolayers and bilayers in a copolymer-homopolymer blend model
We study the stability of layered structures in a variational model for
diblock copolymer-homopolymer blends. The main step consists of calculating the
first and second derivative of a sharp-interface Ohta-Kawasaki energy for
straight mono- and bilayers. By developing the interface perturbations in a
Fourier series we fully characterise the stability of the structures in terms
of the energy parameters.
In the course of our computations we also give the Green's function for the
Laplacian on a periodic strip and explain the heuristic method by which we
found it.Comment: 40 pages, 34 Postscript figures; second version has some minor
corrections; to appear in "Interfaces and Free Boundaries
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model
We present a computer-assisted proof of heteroclinic connections in the
one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a
fourth-order parabolic partial differential equation subject to homogeneous
Neumann boundary conditions, which contains as a special case the celebrated
Cahn-Hilliard equation. While the attractor structure of the latter model is
completely understood for one-dimensional domains, the diblock copolymer
extension exhibits considerably richer long-term dynamical behavior, which
includes a high level of multistability. In this paper, we establish the
existence of certain heteroclinic connections between the homogeneous
equilibrium state, which represents a perfect copolymer mixture, and all local
and global energy minimizers. In this way, we show that not every solution
originating near the homogeneous state will converge to the global energy
minimizer, but rather is trapped by a stable state with higher energy. This
phenomenon can not be observed in the one-dimensional Cahn-Hillard equation,
where generic solutions are attracted by a global minimizer
EXISTENCE AND STABILITY OF SPHERICALLY LAYERED SOLUTIONS OF THE DIBLOCK COPOLYMER EQUATION ∗
Abstract. The relatively simple Ohta–Kawasaki density functional theory for diblock copolymer melts allows us to construct and analyze exact solutions to the Euler–Lagrange equation by singular perturbation techniques. First, we consider a solution of a single sphere pattern that models a cell in the spherical morphology. We show the existence of the sphere pattern and find a stability threshold, so that if the sphere is larger than the threshold value, the sphere pattern becomes unstable. Next we study a spherical lamellar pattern, which may be regarded as a defective lamellar pattern. We reduce the existence and the stability problems to some finite dimensional problems which are accurately solved with the help of a computer. We find two thresholds. Only when the size of the sample is larger than the first threshold value does a spherical lamellar pattern exist. This patten is stable only when the size of the sample is less than the second threshold value. As the stability of the spherical lamellar pattern changes at the second threshold, a bifurcating branch with a pattern of wriggled spherical interfaces appears. The free energy of the latter pattern is lower than that of the first pattern. A similar bifurcation phenomenon also occurs in the single sphere pattern at its stability threshold. Key words. Ohta–Kawasaki diblock copolymer theory, sphere pattern, optimal size, spherical lamellar pattern, existence threshold, stability threshold, bifurcation, wriggled sphere pattern, wriggled spherical lamellar pattern AMS subject classifications. 34E05, 82D60 DOI. 10.1137/04061877