Numerical investigations of pattern formation in binary systems with inhibitory long-range interaction

Frank Baginski; Jiajun Lu; Frank Baginski; Jiajun Lu

doi:10.3934/era.2022081

Electronic Research Archive

2022, Volume 30, Issue 5: 1606-1631. doi: 10.3934/era.2022081

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Numerical investigations of pattern formation in binary systems with inhibitory long-range interaction

Frank Baginski ^{1
,
,},
Jiajun Lu ²

1.
The George Washington University, Washington, DC 20052, USA
2.
CCCC National Engineering Research Center of Dredging Technology and Equipment Co., LTD., Shanghai 201208, China

Academic Editor: Lei Wang

Received: 09 November 2021 Revised: 07 February 2022 Accepted: 07 February 2022 Published: 23 March 2022

We investigate pattern formation in a two-dimensional manifold using the Otha-Kawasaki model for micro-phase separation of diblock copolymers. In this model, the total energy includes a short-range and a long-range term. The short-range term is a Landau-type free energy that is common in phase separation problems and favors large domains with minimum perimeter. The inhibitory long-range interaction term is the Otha-Kawasaki functional derived from the theory of diblock copolymers and favors small domains. The balance of these terms leads to equilibrium states that exhibit a variety of patterns, including disk-like droplets, droplet assemblies, elongated droplets, dog-bone shaped droplets, stripes, annular rings, wriggled stripes and combinations thereof. For problems where analytical results are known, we compare our numerical results and find good agreement. Where analytical results are not available, our numerical methods allow us to explore the solution space revealing new stable patterns. We focus on the triaxial ellipsoid, but our methods are general and can be applied to higher genus surfaces and surfaces with boundaries.
- micro phase separation,
- diblock copolymer,
- pattern formation
Citation: Frank Baginski, Jiajun Lu. Numerical investigations of pattern formation in binary systems with inhibitory long-range interaction[J]. Electronic Research Archive, 2022, 30(5): 1606-1631. doi: 10.3934/era.2022081

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Abstract

We investigate pattern formation in a two-dimensional manifold using the Otha-Kawasaki model for micro-phase separation of diblock copolymers. In this model, the total energy includes a short-range and a long-range term. The short-range term is a Landau-type free energy that is common in phase separation problems and favors large domains with minimum perimeter. The inhibitory long-range interaction term is the Otha-Kawasaki functional derived from the theory of diblock copolymers and favors small domains. The balance of these terms leads to equilibrium states that exhibit a variety of patterns, including disk-like droplets, droplet assemblies, elongated droplets, dog-bone shaped droplets, stripes, annular rings, wriggled stripes and combinations thereof. For problems where analytical results are known, we compare our numerical results and find good agreement. Where analytical results are not available, our numerical methods allow us to explore the solution space revealing new stable patterns. We focus on the triaxial ellipsoid, but our methods are general and can be applied to higher genus surfaces and surfaces with boundaries.

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