Rustum Choksi

On the Derivation of a Density Functional Theory for Microphase Separation of Diblock Copolymers

We consider here the problem of phase separation in copolymer melts. The Ohta–Kawasaki density functional theory gives rise to a nonlocal Cahn–Hilliard-like functional, promoting the use of ansatz-free mathematical tools for the investigation of minimizers. In this article we re-derive this functional as an offspring of the self-consistent mean field theory, connecting all parameters to the fundamental material parameters and clearly identifying all the approximations used. As a simple example of an ansatz-free investigation, we calculate the surface tension in the strong segregation limit, independent of any phase geometry.

Uniform energy distribution for an isoperimetric problem with long-range interactions

We study minimizers of a nonlocal variational problem. The problem is a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation induced by competing short- and long-range interactions. The short-range interaction is attractive and comes from an interfacial energy, and the long-range interaction is repulsive and comes from a nonlocal energy contribution. In particular, the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers. A natural conjecture is that in all space dimensions, minimizers are essentially periodic on an intrinsic scale. However, proving any periodicity result turns out to be a formidable task in dimensions larger than one. In this paper, we address a weaker statement concerning the distribution of energy for minimizers. We prove in any space dimension that each component of the energy (interfacial and nonlocal) of any minimizer is uniformly distributed on cubes which are suciently large with respect to the intrinsic length scale. Moreover, we also prove an L 1 bound on the optimal potential associated with the long-range interactions. This bound allows for an interesting interpretation: Note that the average volume fraction of the optimal pattern in a subsystem of size R fluctuates around the system average m. The bound on the potential yields a rate of decay of these fluctuations as R tends to +1. This rate of decay is stronger than the one for a random checkerboard pattern. In this sense, the optimal pattern has less large-scale variations of the average volume fraction than a pattern with a finite correlation

On the Phase Diagram for Microphase Separation of Diblock Copolymers: An Approach via a Nonlocal Cahn–Hilliard Functional

We consider analytical and numerical aspects of the phase diagram for microphase separation of diblock copolymers. Our approach is variational and is based upon a density functional theory which entails minimization of a nonlocal Cahn–Hilliard functional. Based upon two parameters which characterize the phase diagram, we give a preliminary analysis of the phase plane. That is, we divide the plane into regions wherein a combination of analysis and numerics is used to describe minimizers. In particular we identify a regime wherein the uniform (disordered state) is the unique global minimizer; a regime wherein the constant state is linearly unstable and where numerical simulations are currently the only tool for characterizing the phase geometry; and a regime of small volume fraction wherein we conjecture that small well-separated approximately spherical objects are the unique global minimizer. For this last regime, we present an asymptotic analysis from the point of view of the energetics which will be comp...

Scaling Laws in Microphase Separation of Diblock Copolymers

Summary. In this note, we study a nonlocal variational problem modeling microphase separation of diblock copolymers ([22], [3], [21]). We apply certain new tools developed in [5] to determine the principal part of the asymptotic expansion of the minimum free energy. That is, we prove a scaling law for the minimum energy and confirm that it is attained by a simple periodic lamellar structure. A previous result of Ohnishi et al. [23] was for one space dimension. Here, we obtain a similar result for the full three-dimensional problem.

On the first and second variations of a nonlocal isoperimetric problem

We consider a nonlocal perturbation of an isoperimetric variational problem. The problem may be viewed as a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation associated with competing short and long-range interactions. In particular, it arises as a Γ-limit of a model for microphase separation of diblock copolymers. In this article, we establish precise conditions for criticality and stability (i.e. we explicitly compute the first and second variations). We also present some applications.

Bounds on the micromagnetic energy of a uniaxial ferromagnet

We identify the optimal scaling law for a nonconvex, nonlocal variational problem representing the magnetic energy of a uniaxial ferromagnet. Our analysis is restricted to a certain parameter regime, in which the surface tension is sufficiently small relative to the other parameters of the problem.

Energy minimization and flux domain structure in the intermediate state of a type-I superconductor

Abstract The intermediate state of a type-I superconductor involves a fine-scale mixture of normal and superconducting domains. We take the viewpoint, due to Landau, that the realizable domain patterns are (local) minima of a nonconvex variational problem. We examine the scaling law of the minimum energy and the qualitative properties of domain patterns achieving that law. Our analysis is restricted to the simplest possible case: a superconducting plate in a transverse magnetic field. Our methods include explicit geometric constructions leading to upper bounds and ansatz-free inequalities leading to lower bounds. The problem is unexpectedly rich when the applied field is near-zero or near-critical. In these regimes there are two small parameters, and the ground state patterns depend on the relation between them.

Probability Distributions of Assets Inferred from Option Prices via the Principle of Maximum Entropy

This article revisits the maximum entropy algorithm in the context of recovering the probability distribution of an asset from the prices of finitely many associated European call options via partially finite convex programming. We are able to provide an effective characterization of the constraint qualification under which the problem reduces to optimizing an explicit function in finitely many variables. We also prove that the value (or objective) function is lower semicontinuous on its domain. Reference is given to a website which exploits these ideas for the efficient computation of the maximum entropy solution (MES).

Periodic phase separation: the periodic Cahn-Hilliard and isoperimetric problems

We consider two well known variational problems associated with the phenomenon of phase separation: the isoperimetric problem and minimization of the Cahn‐Hilliard energy. The two problems are related through a classical result in -convergence and we explore the behavior of global and local minimizers for these problems in the periodic setting. More precisely, we investigate these variational problems for competitors defined on the flat 2- or 3-torus. We view these two problems as prototypes for periodic phase separation. We give a complete analysis of stable critical points of the 2-d periodic isoperimetric problem and also obtain stable solutions to the 2-d and 3-d periodic Cahn‐Hilliard problem. We also discuss some intriguing open questions regarding triply periodic constant mean curvature surfaces in 3-d and possible counterparts in the Cahn‐Hilliard setting.

2D Phase Diagram for Minimizers of a Cahn-Hilliard Functional with Long-Range Interactions ∗

This paper presents a two-dimensional investigation of the phase diagram for global minimizers to a Cahn–Hilliard functional with long-range interactions. Based upon the