Competition between the surface and the boundary layer energies in a Ginzburg-Landau model of a liquid crystal composite

Leonid V. Berlyand, Evgen Khruslov

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

We consider a nonlinear homogenization problem for a Ginzburg-Landau 3D model with a surface energy term, in a liquid crystalline medium with inclusions. We show that the presence of the inclusions can be accounted for by an effective potential that can be viewed as an effective external field. Our main objective is to compute the contribution of the surface and bulk energies into this potential. We introduce a small parameter ε such that the average distances between the inclusions are of the order of ε, the inclusions sizes are of the order of εα, α > 1, and the coefficient in front of the surface energy term, is of the order of εβ. We found that the parametric half-plane {(α, β): α > 1, -∞ < β + ∞} is partitioned into two parts by a polygonal line, consisting of two linear parts. We show that, on the first part, the surface energy dominates and, on the second part, the boundary layer energy takes over. We focus our attention on the junction (critical or transitional) point, where both the bulk energy of thin layers around the inclusions and the surface energy provide finite contributions into the effective potential We present explicit formulas for computing the effective potential on the polygonal line and at the critical point in terms of specific surface energy and specific boundary layer energy. We also show that in the domain to the right of the polygonal line, the potential is zero and discuss the homogenized limit in the remaining part of the half plane. Our proof is based on the quasisolutions method, which incorporates some classical ideas of Stokes in hydrodynamics, as well as variational energy techniques previously developed in the study of linear homogenization problems.

Original languageEnglish (US)
Pages (from-to)185-219
Number of pages35
JournalAsymptotic Analysis
Volume29
Issue number3-4
StatePublished - Mar 2002

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Ginzburg-Landau Model
Surface Energy
Interfacial energy
Liquid Crystal
Liquid crystals
Boundary Layer
Boundary layers
Inclusion
Composite
Effective Potential
Composite materials
Energy
Half-plane
Homogenization
Line
Average Distance
Thin Layer
Term
Stokes
3D Model

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

Cite this

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title = "Competition between the surface and the boundary layer energies in a Ginzburg-Landau model of a liquid crystal composite",
abstract = "We consider a nonlinear homogenization problem for a Ginzburg-Landau 3D model with a surface energy term, in a liquid crystalline medium with inclusions. We show that the presence of the inclusions can be accounted for by an effective potential that can be viewed as an effective external field. Our main objective is to compute the contribution of the surface and bulk energies into this potential. We introduce a small parameter ε such that the average distances between the inclusions are of the order of ε, the inclusions sizes are of the order of εα, α > 1, and the coefficient in front of the surface energy term, is of the order of εβ. We found that the parametric half-plane {(α, β): α > 1, -∞ < β + ∞} is partitioned into two parts by a polygonal line, consisting of two linear parts. We show that, on the first part, the surface energy dominates and, on the second part, the boundary layer energy takes over. We focus our attention on the junction (critical or transitional) point, where both the bulk energy of thin layers around the inclusions and the surface energy provide finite contributions into the effective potential We present explicit formulas for computing the effective potential on the polygonal line and at the critical point in terms of specific surface energy and specific boundary layer energy. We also show that in the domain to the right of the polygonal line, the potential is zero and discuss the homogenized limit in the remaining part of the half plane. Our proof is based on the quasisolutions method, which incorporates some classical ideas of Stokes in hydrodynamics, as well as variational energy techniques previously developed in the study of linear homogenization problems.",
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Competition between the surface and the boundary layer energies in a Ginzburg-Landau model of a liquid crystal composite. / Berlyand, Leonid V.; Khruslov, Evgen.

In: Asymptotic Analysis, Vol. 29, No. 3-4, 03.2002, p. 185-219.

Research output: Contribution to journalArticle

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N2 - We consider a nonlinear homogenization problem for a Ginzburg-Landau 3D model with a surface energy term, in a liquid crystalline medium with inclusions. We show that the presence of the inclusions can be accounted for by an effective potential that can be viewed as an effective external field. Our main objective is to compute the contribution of the surface and bulk energies into this potential. We introduce a small parameter ε such that the average distances between the inclusions are of the order of ε, the inclusions sizes are of the order of εα, α > 1, and the coefficient in front of the surface energy term, is of the order of εβ. We found that the parametric half-plane {(α, β): α > 1, -∞ < β + ∞} is partitioned into two parts by a polygonal line, consisting of two linear parts. We show that, on the first part, the surface energy dominates and, on the second part, the boundary layer energy takes over. We focus our attention on the junction (critical or transitional) point, where both the bulk energy of thin layers around the inclusions and the surface energy provide finite contributions into the effective potential We present explicit formulas for computing the effective potential on the polygonal line and at the critical point in terms of specific surface energy and specific boundary layer energy. We also show that in the domain to the right of the polygonal line, the potential is zero and discuss the homogenized limit in the remaining part of the half plane. Our proof is based on the quasisolutions method, which incorporates some classical ideas of Stokes in hydrodynamics, as well as variational energy techniques previously developed in the study of linear homogenization problems.

AB - We consider a nonlinear homogenization problem for a Ginzburg-Landau 3D model with a surface energy term, in a liquid crystalline medium with inclusions. We show that the presence of the inclusions can be accounted for by an effective potential that can be viewed as an effective external field. Our main objective is to compute the contribution of the surface and bulk energies into this potential. We introduce a small parameter ε such that the average distances between the inclusions are of the order of ε, the inclusions sizes are of the order of εα, α > 1, and the coefficient in front of the surface energy term, is of the order of εβ. We found that the parametric half-plane {(α, β): α > 1, -∞ < β + ∞} is partitioned into two parts by a polygonal line, consisting of two linear parts. We show that, on the first part, the surface energy dominates and, on the second part, the boundary layer energy takes over. We focus our attention on the junction (critical or transitional) point, where both the bulk energy of thin layers around the inclusions and the surface energy provide finite contributions into the effective potential We present explicit formulas for computing the effective potential on the polygonal line and at the critical point in terms of specific surface energy and specific boundary layer energy. We also show that in the domain to the right of the polygonal line, the potential is zero and discuss the homogenized limit in the remaining part of the half plane. Our proof is based on the quasisolutions method, which incorporates some classical ideas of Stokes in hydrodynamics, as well as variational energy techniques previously developed in the study of linear homogenization problems.

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