55 research outputs found
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
Minimality via second variation for a nonlocal isoperimetric problem
We discuss the local minimality of certain configurations for a nonlocal
isoperimetric problem used to model microphase separation in diblock copolymer
melts. We show that critical configurations with positive second variation are
local minimizers of the nonlocal area functional and, in fact, satisfy a
quantitative isoperimetric inequality with respect to sets that are
-close. The link with local minimizers for the diffuse-interface
Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative
estimate, we get new results concerning periodic local minimizers of the area
functional and a proof, via second variation, of the sharp quantitative
isoperimetric inequality in the standard Euclidean case. As a further
application, we address the global and local minimality of certain lamellar
configurations.Comment: 35 page
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
Optimal design of chemoepitaxial guideposts for directed self-assembly of block copolymer systems using an inexact-Newton algorithm
Directed self-assembly (DSA) of block-copolymers (BCPs) is one of the most
promising developments in the cost-effective production of nanoscale devices.
The process makes use of the natural tendency for BCP mixtures to form
nanoscale structures upon phase separation. The phase separation can be
directed through the use of chemically patterned substrates to promote the
formation of morphologies that are essential to the production of semiconductor
devices. Moreover, the design of substrate pattern can formulated as an
optimization problem for which we seek optimal substrate designs that
effectively produce given target morphologies.
In this paper, we adopt a phase field model given by a nonlocal
Cahn--Hilliard partial differential equation (PDE) based on the minimization of
the Ohta--Kawasaki free energy, and present an efficient PDE-constrained
optimization framework for the optimal design problem. The design variables are
the locations of circular- or strip-shaped guiding posts that are used to model
the substrate chemical pattern. To solve the ensuing optimization problem, we
propose a variant of an inexact Newton conjugate gradient algorithm tailored to
this problem. We demonstrate the effectiveness of our computational strategy on
numerical examples that span a range of target morphologies. Owing to our
second-order optimizer and fast state solver, the numerical results demonstrate
five orders of magnitude reduction in computational cost over previous work.
The efficiency of our framework and the fast convergence of our optimization
algorithm enable us to rapidly solve the optimal design problem in not only
two, but also three spatial dimensions.Comment: 35 Pages, 17 Figure
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