The Oseen–Frank Limit of Onsager’s Molecular Theory for Liquid Crystals

Liu, Yuning; Wang, Wei

doi:10.1007/s00205-017-1180-6

Cited by 11 publications

(17 citation statements)

References 34 publications

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“…Actually, sincek(ξ) = e − π 2 |ξ| 2 a , it is not difficult to see that (2.5) holds with c 0 ≤ π 2 a . We also remark that our choice of k here weaken the assumptions in our previous work [26] on the static problem.…”

Section: The Maier-saupe Energymentioning

confidence: 90%

“…Furthermore, we assume (1.21). The first assumption in (1.21) is crucial to deduce the Oseen-Frank energy with bounded coefficients, see [26,33]. On the other hand, we deduce from it the following condition which will be employed in the proof of Theorem 1.1 in the last section:…”

Section: The Maier-saupe Energymentioning

confidence: 90%

“…On the other hand, Wang, Wang and Zhang [31] justified the limit from the Q-tensor flow to (1.9) in the framework of weak solutions, where the key ingredient is to establish some monotonicity formulas. In [26], the authors considered the asymptotic limit of ǫ for critical points and minimizers of the energy functional (1.2)-(1.4), and the one-constant approximation of Oseen-Frank energy is derived in the limit. See also [30] for a Γ-convergence approach where a more general energy than (1.8) is obtained in the limit.…”

Section: )mentioning

confidence: 99%

“…The proof of (5.3) was motivated by [1,26]. First, we infer from the assumption (2.5) for the kernel function k(x) that…”

Section: Compactness Of the Second Momentsmentioning

confidence: 99%

“…The following two lemmas are concerned with the properties of T ǫ and will be employed in the rest of the work. Though the proof can be found in [26] (except (5.13)), we present them here for completeness.…”

Section: Compactness Of the Second Momentsmentioning

confidence: 99%

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The small Deborah number limit of the Doi–Onsager equation without hydrodynamics

Liu

Wang

2018

Journal of Functional Analysis

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We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals without hydrodynamics. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position x ∈ R d (d = 2, 3) and orientation vector m ∈ S 2 (the unit sphere). We prove that, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into S 2 . This flow is a special case of the gradient flow to the Oseen-Frank energy functional for nematic liquid crystals. The key ingredient is to show the strong compactness of the family of number density functions and the proof relies on the strong compactness of the corresponding second moment (or the Q-tensor), a spectral decomposition of the linearized operator near the limit local equilibrium distribution, as well as the energy dissipation estimate.Date: September 17, 2018. 1 We refer to the book by de Gennes-Prost [5] for physics of this theory.1.2. From microscopic theories to macroscopic theories for liquid crystals. Exploring the connections between different theories for liquid crystal flow is a fundamental issue in liquid crystal studies. first derived the Ericksen-Leslie equations and determined the Leslie coefficients from the Doi-Onsager equation under the small Deborah number limit. However, the Ericksen stress was missing. E-Zhang [9] extended Kuzuu and Doi's formal derivation to the inhomogeneous case and the Ericksen stress was obtained from an extra introduced body force. Roughly speaking, E and Zhang showed that the solution (f, v) of (1.5)-(1.6) with De = ǫ has a formal expansionIt can be verified using |n(x, t)| ≡ 1 that, if a weak solution n(x, t) is smooth, then it fulfills (∂ t n − Λ∆n) ∧ n = 0 and this is equivalent to (1.9).Remark 1.2. The first part of Theorem 1.1 is concerned with the wellposedness of (1.10), which is proved in the beginning of Section 4. Although these issues can be discussed under much more relaxed assumptions on the interaction potential (1.3) as well as the initial data, for the sake of investigating the scaling limit, we restrict ourselves to the inhomogeneous Maier-Saupe potential defined by (1.11) and initial data near the local equilibria, which include local equilibrium distributions as especial cases. More precisely, if n ǫ (x) : R d → S 2 fulfills n ǫ − e 0 H 1 (R d ) ≤ C for some C independent of ǫ and for some e 0 ∈ S 2 , then f in ǫ (m, x) = 1 Z e η(m·nǫ(x)) 2 satisfies (1.22). Remark 1.3. We will give a more detailed discussion on assumptions (1.21) in Section 2.1.

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Section: The Maier-saupe Energymentioning

confidence: 90%

Section: The Maier-saupe Energymentioning

confidence: 90%

Section: )mentioning

confidence: 99%

“…The proof of (5.3) was motivated by [1,26]. First, we infer from the assumption (2.5) for the kernel function k(x) that…”

Section: Compactness Of the Second Momentsmentioning

confidence: 99%

Section: Compactness Of the Second Momentsmentioning

confidence: 99%

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The small Deborah number limit of the Doi–Onsager equation without hydrodynamics

Liu

Wang

2018

Journal of Functional Analysis

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Modelling and computation of liquid crystals

Wang¹,

Zhang

2021

Acta Numerica

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Liquid crystals are a type of soft matter that is intermediate between crystalline solids and isotropic fluids. The study of liquid crystals has made tremendous progress over the past four decades, which is of great importance for fundamental scientific research and has widespread applications in industry. In this paper we review the mathematical models and their connections to liquid crystals, and survey the developments of numerical methods for finding rich configurations of liquid crystals.

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Γ-convergence of a mean-field model of a chiral doped nematic liquid crystal to the Oseen–Frank description of cholesterics

Taylor

2020

Nonlinearity

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Systems of elongated molecules, doped with small amounts of molecules lacking mirror symmetry can form macroscopically twisted cholesteric liquid crystal phases. The aim of this work is to rigorously derive the Oseen-Frank model of cholesterics from a more fundamental model concerned with pairwise molecular interactions. A non-local mean-field model of the two-species nematic host/chiral dopant mixture is proposed, and it is shown that Oseen-Frank's elastic free energy for cholesteric liquid crystals can be obtained in a simultaneously large-domain and dilute-dopant asymptotic regime. By techniques of Γ-convergence, it is shown that in the asymptotic limit, dopantdopant interactions are negligable, the Frank constants and nematic host order parameter are unperturbed by the presence of dopant, but the mirror asymmetry of the dopant-host interaction leads to a macroscopically twisted ground state. The constant of proportionality between the helical wavenumber and dopant concentration, the helical twisting power (HTP), can be explicitly found through such an analysis, with a nonlinear temperature dependence. Depending on the relative strengths of the host-host and host-dopant interactions, it is shown that HTP may increase or decrease with temperature.

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The Oseen–Frank Limit of Onsager’s Molecular Theory for Liquid Crystals

Cited by 11 publications

References 34 publications

The small Deborah number limit of the Doi–Onsager equation without hydrodynamics

The small Deborah number limit of the Doi–Onsager equation without hydrodynamics

Modelling and computation of liquid crystals

Γ-convergence of a mean-field model of a chiral doped nematic liquid crystal to the Oseen–Frank description of cholesterics

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