A Note on the Onsager Model of Nematic Phase Transitions

Fatkullin, Ibrahim; Slastikov, Valeriy

doi:10.4310/cms.2005.v3.n1.a2

Cited by 29 publications

(45 citation statements)

References 9 publications

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“…The principal reason for employing this particular methodology is recent improvement of analytical techniques addressing (spatiallyhomogeneous) Onsager-type theories. For example, a complete classification of all critical points in various models of this type has been recently established, see e.g., [3,10,9,14,23]. Combining these ideas with a Dirichlet energy estimate for S 1 -maps due to Sandier [20] allows us to achieve a complete rigorous understanding of patterns arising in the suggested model.…”

Section: Introductionmentioning

confidence: 99%

Vortices in two-dimensional nematics

Fatkullin

Slastikov

2009

Communications in Mathematical Sciences

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Abstract. We study a two-dimensional model describing spatial variations of orientational ordering in nematic liquid crystals. In particular, we show that the spatially extended OnsagerMaier-Saupe free energy may be decomposed into Landau-de Gennes-type and relative entropy-type contributions. We then prove that in the high concentration limit the states of the system display characteristic vortex-like patterns and derive an asymptotic expansion for the free energy of the system.

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Section: Introductionmentioning

confidence: 99%

Vortices in two-dimensional nematics

Fatkullin

Slastikov

2009

Communications in Mathematical Sciences

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“…Theorem 2 Type I equations (1,6,12), with constitutive equations (2, 3, 7, 8, 10, 11), with F = 0, σ (2) = 0 and (28), have a Lyapunov functional…”

Section: Energetics For Type I Equationsmentioning

confidence: 99%

“…The study of time asymptotics is also in its early development stage. High intensity asymptotics for uncoupled Smoluchowski equations have been studied in ( [2], [6], [15], [16]). The long time effects of shear in Doi-Smoluchowski equations have been investigated in ( [7], [8]).…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Fokker-Planck Navier-Stokes systems

Constantin¹

2005

Communications in Mathematical Sciences

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We consider Navier-Stokes equations coupled to nonlinear FokkerPlanck equations describing the probability distribution of particles interacting with fluids. We describe relations determining the coefficients of the stresses added in the fluid by the particles. These relations link the added stresses to the kinematic effect of the fluid's velocity on particles and to the inter-particle interaction potential. In equations of type I, where the added stresses depend linearly on the particle distribution density, energy balance requires a response potential. In equations of type II, where the added stresses depend quadratically on the particle distribution, energy balance can be achieved without a dynamic response potential. In unforced energetically balanced equations, all the steady solutions have fluid at rest and particle distributions obeying an uncoupled Onsager equation. Systems of equations of type II have global smooth solutions if inertia is neglected.

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“…[14,15,26,77] the authors conducted a detailed study of the 2-D model where the rod orientation is confined to a unit circle. In Ref.…”

Section: Mathematical Studies Of Pure Nematicsmentioning

confidence: 99%

Mathematical Studies and Simulations of Nematic Liquid Crystal Polymers and Nanocomposites

Zhou

Forest²,

Wang

2010

Jnl of Comp & Theo Nano

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Nematic liquid crystal polymers and nanocomposites have wide-ranging applications in modern technology including display devices, ultra-fast switches, high strength fibers, and materials with enhanced multi-functional properties including thermal, dielectric, electrical, and barrier properties. In this review article we provide an overview of selected mathematical issues and numerical simulations of nematic liquid crystal polymers and rigid rod nanocomposites. Some open questions will be addressed.

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A Note on the Onsager Model of Nematic Phase Transitions

Cited by 29 publications

References 9 publications

Vortices in two-dimensional nematics

Vortices in two-dimensional nematics

Nonlinear Fokker-Planck Navier-Stokes systems

Mathematical Studies and Simulations of Nematic Liquid Crystal Polymers and Nanocomposites

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