Communications in Mathematical Sciences

Volume 17 (2019)

Number 8

Bubble assemblies in ternary systems with long range interaction

Pages: 2309 – 2324

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n8.a10

Authors

Chong Wang (Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada)

Xiaofeng Ren (Department of Mathematics, George Washington University, Washington, District of Columbia, U.S.A.)

Yanxiang Zhao (Department of Mathematics, George Washington University, Washington, District of Columbia, U.S.A.)

Abstract

A nonlocal diffuse interface model, based on the Nakazawa–Ohta density functional theory for triblock copolymers, is used to study bubble assemblies in ternary systems. The model has three parameters weighing three types of long-range interaction and two parameters that fix the total area of each constituent. As the parameters vary, a large number of morphological phases appear as stable stationary states. One open question related to the polarity direction of double bubble assemblies is answered numerically. Moreover, it is shown that the average size of bubbles in a single bubble assembly depends on the sum of the minority constituent areas and the long-range interaction coefficients. One further identifies the ranges for area fractions and the long-range interaction coefficients for double bubble assemblies.

Keywords

diffuse interface model, Nakazawa–Ohta density functional, triblock copolymers, morphological phases, long-range interaction

2010 Mathematics Subject Classification

35R35, 70E55, 74N20, 82D60, 92C15

Full Text (PDF format)

X.R. is supported by National Science Foundation, DMS-1714371. Y.Z. is supported by a grant from the Simons Foundation through Grant No. 357963.

Received 16 September 2018

Accepted 3 September 2019

Published 3 February 2020