Dissertation/Thesis Abstract

A Geometric Variational Problem on a Periodic Domain
by Smpokos, Konstantinos, Ph.D., The George Washington University, 2020, 100; 27831110
Abstract (Summary)

A diblock copolymer melt is a soft material, characterized by fluid-like disorder on the molecular scale and a high degree of order at a longer length scale. A molecule in a diblock copolymer is a linear sub-chain of A-monomers grafted covalently to another sub-chain of B-monomers. The Ohta-Kawasaki density functional theory of diblock copolymer gives rise to a non local free boundary problem. We will work on a periodic lattice in C generated by two complex numbers and we will assume periodic boundary conditions. In this thesis we will find two stationary sets of the energy functional of the problem. The first set is a perturbation of a round disk in C. More specifically, we will perturb a round disk under the polar coordinates. The radius of this perturbed disk will be sufficiently small.

Also, we will minimize the energy of this stationary set with respect to the shape and size of the lattice. Additionally, we will show that for every K > = 2, K in N there exists a stationary set of the free energy functional that is the union of K disjoint perturbed disks in C. Later, we will assume that K = 2 and we will deal with the problem of finding the centers of these two perturbed disks. We will show that the centers of these disks are close to a global minimum of the Green's function of the problem. We will minimize the Green's function of the problem looking at some special cases of lattice structures. These lattices are the hexagonal lattice and a family of rectangular lattices.

Indexing (document details)
Advisor: Ren, Xiaofeng
Commitee: Baginski, Frank, Robinson, Robbie, Zhao, Yanxiang, Wu, Hao, Angoshtari, Arzhang
School: The George Washington University
Department: Mathematics
School Location: United States -- District of Columbia
Source: DAI-B 81/11(E), Dissertation Abstracts International
Source Type: DISSERTATION
Subjects: Mathematics
Keywords: Calculus of variations, Diblock copolymer, Geometric variational problems, Lattices, Perturbed disks, Stationary sets
Publication Number: 27831110
ISBN: 9798645445928
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