The weak shear kinetic phase diagram for nematic polymers

Forest, M. Gregory; Wang, Qi; Zhou, Ruhai

doi:10.1007/s00397-003-0317-8

Cited by 85 publications

(106 citation statements)

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“…These simple closure approximations respect the traceless property of the orientational dynamic equation, and have been shown to yield a good approximation of kinetic theory in the dynamics of monodomains at the nematic concentrations of interest here [10,15,16]. These closures are exact when the molecules are aligned perfectly.…”

Section: Continuity Equationmentioning

confidence: 85%

“…The model has been benchmarked in the longwave, monodomain regime with resolved simulations of the Doi kinetic theory [15,16]. Indeed, the motivation for an analytical study of structure properties is to provide guidance for structure simulations of mesoscopic [17] and kinetic [15,16] models, where the parameter space is too large to assimilate any kind of collapse of the numerical data through scaling laws.…”

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confidence: 99%

“…First, we consider a more general physical model to admit anisotropic distortional elasticity (unequal bend, splay, twist elasticity constants). The second-moment orientation model is derived from a recent generalization of the Doi-Hess-Marrucci-Greco kinetic theory [33] and guides our numerical studies [15,16], for which there are no preceding numerical or analytical results. Second, the asymptotic analysis is extended from plate-driven Couette cell properties to pressure-driven Poiseuille flows.…”

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confidence: 99%

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On Weak Plane Couette and Poiseuille Flows of Rigid Rod and Platelet Ensembles

Cui¹,

Forest²,

Wang³

et al. 2006

SIAM J. Appl. Math.

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On weak plane Couette and Poiseuille flows of rigid rod and platelet ensembles (with Z. Cui, M.G. Forest and Q. Wang), SIAM J. Appl. Math. , 66 (4), 2006 Abstract. Films and molds of nematic polymer materials are notorious for heterogeneity in the orientational distribution of the rigid rod or platelet macromolecules. Predictive tools for structure length scales generated by shear-dominated processing are vitally important: both during processing because of flow feedback phenomena such as shear thinning or thickening, and postprocessing since gradients in the rod or platelet ensemble translate to nonuniform composite properties and to residual stresses in the material. These issues motivate our analysis of two prototypes for planar shear processing: drag-driven Couette and pressure-driven Poiseuille flows. Hydrodynamic theories for high aspect ratio rod and platelet macromolecules in viscous solvents are well developed, which we apply in this paper to model the coupling between short-range excluded volume interactions, anisotropic distortional elasticity (unequal elasticity constants), wall anchoring conditions, and hydrodynamics. The goal of this paper is to generalize scaling properties of steady flow molecular structures in slow Couette flows with equal elasticity constants [M. G. Forest et al., J. Rheol., 48 (2004), pp. 175-192] in several ways: to contrast isotropic and anisotropic elasticity; to compare Couette versus Poiseuille flow; and to consider dynamics and stability of these steady states within the asymptotic model equations.

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Section: Continuity Equationmentioning

confidence: 85%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On Weak Plane Couette and Poiseuille Flows of Rigid Rod and Platelet Ensembles

Cui¹,

Forest²,

Wang³

et al. 2006

SIAM J. Appl. Math.

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“…First we set a nematic concentration N = 6, for which stable quiescent equilibria satisfying F (Q) = 0 are uniaxial (β = 0), ordered phases with Flory order parameter s given below, equation (7). We select a = 0.8, or aspect ratio r = 3, which for the Doi closure gives excellent agreement with the full kinetic theory in the longwave limit of sheared monodomains [10,11,12]. Finally we set α = 2.0, µ 1 = 2.3867 × 10 −4 , µ 2 = 3.1667 × 10 −3 , µ 3 = 3.5 × 10 −3 , consistent with our previous papers [1,9] so that we identify anchoring distortion effects; similarly, we use a constant rotational diffusivity.…”

Section: Hong Zhou and M Gregory Forestmentioning

confidence: 94%

Anchoring distortions coupled with plane Couette & Poiseuille flows of nematic polymers in viscous solvents: Morphology in molecular orientation, stress & flow

Zhou

Forest²

2005

DCDS-B

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Abstract. The aim of this work is to model and simulate processing-induced heterogeneity in rigid, rod-like nematic polymers in viscous solvents. We employ a mesoscopic orientation tensor model due to Doi, Marrucci and Greco which extends the small molecule, liquid crystal theory of Leslie-Ericksen-Frank to nematic polymers. The morphology has various physical realizations, all coupled through the model equations: the orientational distribution of the ensemble of rods, anisotropic viscoelastic stresses, and flow feedback. Previous studies in plane Couette & Poiseuille flow (with the exception of [7]) have focused on the coupling between hydrodynamics and the orientational distribution of rigid rod polymers with identical anchoring conditions at solid boundaries; without flow, the equilibrium structure is homogeneous. Here we explore steady structures that emerge with mismatch anchoring conditions at the walls, which couple an equilibrium elastic distortion across the channel, short and long range elasticity potentials, and hydrodynamics. In plane Couette (where moving plates drive the flow) and Poiseuille flow (where a pressure gradient drives the flow), in the regime of weak flow and strong distortional elasticity, asymptotic analysis yields closed-form steady solutions and scaling laws with identical wall conditions. We focus simulations in this regime to expose the effects due to wall anchoring conflicts, and illustrate the induced morphology of the orientational distribution, stored viscoelastic stresses, and non-Newtonian flow. A remarkably simple diagnostic emerges in this physical parameter regime, in which salient morphology features are controlled by the amplitude and sign of the difference in plate anchoring angles of the director field at the two plates.

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“…In the past, it has been perceived as colossal and complicated-hardly accessible to theoretical analysis. Given the rising interest in kinetic theory in the mathematics community these days, various attempts have been made to analyze the properties of the partial differential equations in the kinetic theories and obtain their solutions semianalytically and numerically [23,11,12,13,14,15,16,17,18,5,6,7,10,23,34,35,26,40,38,20]. A recent review of the state of the art in the mathematical and numerical analysis of multi-scale models of complex fluids is given by Li and Zhang [25].…”

Section: Introductionmentioning

confidence: 99%

Steady states and their stability of homogeneous, rigid, extended nematic polymers under imposed magnetic fields

Wang²,

Zhang³

et al. 2007

Communications in Mathematical Sciences

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Abstract. We study the steady state phase behavior of homogeneous, rigid, extended (polar) nematic polymers or nematic dispersions under imposed magnetic (or electric fields), in which the intermolecular dipole-dipole and excluded volume interaction as well as external field contribution are accounted for. We completely characterize the phase diagram for polar nematics with and without permanent magnetic moments (or dipoles). For nematics without magnetic moments, the steady state is either purely nematic or polar depending on the strength of the excluded volume and the dipole-dipole interaction, in which the nonzero polarity vector (the first moment vector) is coaxial with the second moment tensor; thereby the steady state pdf is determined essentially by up to three scalar order parameters and a rotational group of SO(2) transverse to the imposed field direction. For nematics with permanent magnetic moments (or dipoles), the steady states are polar and the polarity vector is parallel to the external field direction when a necessary condition for parameters is met. When the condition is violated, only stable steady states have their polarity vector parallel to the external field direction and there are thermodynamically unstable nonparallel states. The stability of the steady states is inferred from the minimum of the free energy density.Key words. nematic liquid crystal, kinetic theory, polymers, magnetic field, electric field, bifurcation, steady state, phase transition, polarity, stability AMS subject classifications. 45K05, 70K50, 82B26, 82B27 IntroductionKinetic theory is a useful tool in modeling soft matter and complex fluids [2,4,8,24,9]. In the past, it has been perceived as colossal and complicated-hardly accessible to theoretical analysis. Given the rising interest in kinetic theory in the mathematics community these days, various attempts have been made to analyze the properties of the partial differential equations in the kinetic theories and obtain their solutions semianalytically and numerically [23,11,12,13,14,15,16,17,18,5,6,7,10,23,34,35,26,40,38,20]. A recent review of the state of the art in the mathematical and numerical analysis of multi-scale models of complex fluids is given by Li and Zhang [25].

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The weak shear kinetic phase diagram for nematic polymers

Cited by 85 publications

References 50 publications

On Weak Plane Couette and Poiseuille Flows of Rigid Rod and Platelet Ensembles

On Weak Plane Couette and Poiseuille Flows of Rigid Rod and Platelet Ensembles

Anchoring distortions coupled with plane Couette & Poiseuille flows of nematic polymers in viscous solvents: Morphology in molecular orientation, stress & flow

Steady states and their stability of homogeneous, rigid, extended nematic polymers under imposed magnetic fields

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