Constitutive Equation for Nematic Liquid Crystals under Weak Velocity Gradient Derived from a Molecular Kinetic Equation

Kuzuu, Nobu; Doi, Masao

doi:10.1143/jpsj.52.3486

Cited by 290 publications

(210 citation statements)

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“…Hence, it is very important to establish the relationship between two theories. Kuzuu and Doi [9] formally derive the Ericksen-Leslie equation from the Doi-Onsager equation (1.1), and determine the Leslie coefficients. However, the Ericksen stress is missed in the homogeneous case.…”

Section: 3mentioning

confidence: 99%

“…When h = h η i ,n (i = 1, 2), the problem becomes more complicated. Kuzzu and Doi [9] conjectured that all the eigenvalues of G h are non-positive, and Ker…”

Section: Spectral Analysis Of the Linearized Operatormentioning

confidence: 99%

“…Parodi's relation [19] gives a constraint for Leslie coefficients: α 2 + α 3 = α 6 − α 5 . While, σ E is the elastic (Ericksen) stress 9) where E F = E F (n, ∇n) is the Frank energy. The dynamic equation for the director field is given by 10) where γ 1 = α 3 − α 2 , γ 2 = α 6 − α 5 , and h is the molecular field…”

mentioning

confidence: 99%

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The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Wang

Zhang

2014

Comm Pure Appl Math

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Abstract. We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the EricksenLeslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized Doi-Onsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the Maier-Saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the Maier-Saupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional. We use x ∈ Ω ⊆ R 3 to denote the material point and f (x, m, t) to represent the number density for the number of molecules whose orientation is parallel to m at point x and time t. where De is the Deborah number, R is the rotational gradient operator(see Section 3), κ is a constant velocity gradient, and U is the mean-field interaction potential. Onsager [18]

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Section: 3mentioning

confidence: 99%

“…When h = h η i ,n (i = 1, 2), the problem becomes more complicated. Kuzzu and Doi [9] conjectured that all the eigenvalues of G h are non-positive, and Ker…”

Section: Spectral Analysis Of the Linearized Operatormentioning

confidence: 99%

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The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Wang

Zhang

2014

Comm Pure Appl Math

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“…6,15 Note that in order to arrive at the DEK equation, P ͑û Ј, t͒ in Eq. ͑7͒ has been put equal to the pdf P͑û Ј, t͒ of the central rod.…”

Section: ͑8͒mentioning

confidence: 99%

Brownian dynamics simulations of the self- and collective rotational diffusion coefficients of rigid long thin rods

Tao

Otter

Padding

et al. 2005

The Journal of Chemical Physics

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Mega-electron-volt proton irradiation on supported and suspended graphene: A Raman spectroscopic layer dependent study J. Appl. Phys. 110, 084309 (2011) Gel formation and aging in weakly attractive nanocolloid suspensions at intermediate concentrations J. Chem. Phys. 135, 154903 (2011) Waiting time dependence of T2 of protons in water suspensions of iron-oxide nanoparticles: Measurements and simulations J. Appl. Phys. 110, 073917 (2011) The effects of shape and flexibility on bio-engineered fd-virus suspensions J. Chem. Phys. 135, 144106 (2011) Quantitative measurement of scattering and extinction spectra of nanoparticles by darkfield microscopy Appl. Phys. Lett. 99, 131113 (2011) Additional information on J. Chem. Phys. Here this theory is put to the test by comparing it against computer simulations. A Brownian dynamics simulation program was developed to follow the dynamics of the rods, with a length over a diameter ratio of 60, on the Smoluchowski time scale. The model accounts for excluded volume interactions between rods, but neglects hydrodynamic interactions. The self-rotational diffusion coefficients D r ͑ ͒ of the rods were calculated by standard methods and by a new, more efficient method based on calculating average restoring torques. Collective decay of orientational order was calculated by means of equilibrium and nonequilibrium simulations. Our results show that, for the currently accessible volume fractions, the decay times in both cases are virtually identical. Moreover, the observed decay of diffusion coefficients with volume fraction is much quicker than predicted by the theory, which is attributed to an oversimplification of dynamic correlations in the theory.

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“…This formula is derived using the method introduced in paper of Kuzuu and Doi [16]. The angle dependance of this microscopic stress tensor is described by the orientation of the long molecular axis with respect to the direction n which is determined by the polar (6) and azimuthal [<p) angles, (4.6) a = n cos 6 + e sin 6, e x = cos (j), e y = sin </ >, e z = 0.…”

Section: 00mentioning

confidence: 99%

Static and Hydrodynamic Properties of Nematic Liquid Crystals

Chrzanowska

Sokalski

1992

Zeitschrift Für Naturforschung A

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The static properties of nematic liquid crystals are summarized. The mean field potential emerging from the static distribution function has been used to the hydrodynamic theory. Rotational viscosity coefficients have been investigated. The Parodi relation has been shown to be completely satisfied. Static and hydrodynamic properties have been predicted on the basis of one intermolecular potential.

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Constitutive Equation for Nematic Liquid Crystals under Weak Velocity Gradient Derived from a Molecular Kinetic Equation

Cited by 290 publications

References 11 publications

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Brownian dynamics simulations of the self- and collective rotational diffusion coefficients of rigid long thin rods

Static and Hydrodynamic Properties of Nematic Liquid Crystals

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