Theory of lyotropic polymer liquid crystals

Odijk, Theo

doi:10.1021/ma00163a001

Cited by 534 publications

(604 citation statements)

References 21 publications

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“…Near this point at concentration c = 6 + u (IV. 4) we argue that the integrodifferential equation has the following solution obtained by combining eq A.9,10, and 15. Having thus ensured that the nematic state is feasible, we next turn to a numerical analysis of its properties.…”

Section: Bifurcation Analysismentioning

confidence: 84%

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Induced chain rigidity, splay modulus and other properties of nematic polymer liquid crystals

Vroege¹,

Odijk²

1988

Macromolecules

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We present a numerical analysis of the induced chain rigidity or global persistence length, the order parameter, the splay modulus, and other properties of a polymer nematic. The macromolecules are viewed as long slender wormlike cylinders interacting via hard-core repulsions in the second virial approximation. We calculate the orientational distribution function from the nonlinear integrodifferential equation first formulated by Khokhlov and Semenov. A bifurcation analysis of this equation is also given. Exact expressions for the susceptibility and the global persistence length are derived in terms of the distribution function. Analytical estimates of thew quantities based on the usual methods we extremely poor approximations to thwe determined numerically. We also discuss the splay modulus which is directly related to the susceptibility and the global persistence length.

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Section: Bifurcation Analysismentioning

confidence: 84%

“…Minimizing U d e f with respect to g yields 9,1988 If their orientations are described by polar angles (8,4) and (e' , 49 defined with respect to the director, then cos the solution of chains is effectively a solution of rods of length g because we want to count "end" defects (see Figure 2). In view of the identity lpl = Ppp = gps we obtain…”

Section: T Is Simplymentioning

confidence: 99%

Induced chain rigidity, splay modulus and other properties of nematic polymer liquid crystals

Vroege¹,

Odijk²

1988

Macromolecules

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“…An alternative route is to choose a (normalized) trial function with one or more variational parameters and then minimize the free energy with respect to these parameters. Here we shall use a Gaussian distribution function [72]:…”

Section: -Calculationsmentioning

confidence: 99%

“…Using the Gaussian distribution one obtains to leading order (large a) [72] Substitution of eqs. (22) and (23) in the Helmholtz free energy (6) and the free-volume fraction (17) leads to an analytic expression for the grand potential (3) which depends on a single variational parameter and hence is easily minimized.…”

Section: -Calculationsmentioning

confidence: 99%

Phase behaviour of rod-like colloid+flexible polymer mixtures

1994

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Summary. -The effect of non-adsorbing, flexible polymer on the isotropic-nematic transition in dispersions of rod-like colloids is investigated. A widening of the biphasic gap is observed, in combination with a marked polymer partitioning between the coexisting phases. Under certain conditions, areas of isotropicisotropic-nematic or isotropic-nematic-nematic three-phase coexistence appear in the phase diagram of rod-polymer mixtures.

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“…Even with additional interactions, the terms analogous to the first two in eq 36 will be negligible for very high 6. Thus we obtainI5 (39) where EH i= Dcp-'J2 is of the order of the average distance between the rods.…”

Section: Elongational Stress Pression7mentioning

confidence: 98%

Disentanglement of rods in semidilute and liquid-crystalline solutions in elongational flow

Odijk¹

1988

Macromolecules

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An analytical theory is presented of the disentanglement of rods in both semidilute and liquid-crystalline solutions within the context of the preaveraged Doi equation. The excluded-volume effect is accounted for in the second virial approximation. It is assumed that the degree of orientational order is high at all times. The diffusion equation and the stress are solved to leading order. IntroductionA mere glimpse at the experimental literature on the rheology of polymer liquid crystals reveals that many phenomena are not well understood.'" Nevertheless, the theory advanced by Doi6J does rationalize several remarkable effects like the sharp decrease in the viscosity when the concentration is increased beyond the isotropic-nematic transition. Moreover, steady-state rheological properties are surprisingly well described by the Doi equations provided two parameters are adjusted to conform to one or two e~periments.~ The relative success of the reptation theory for liquid crystals has caused a flurry of theoretical activity.&13 Most previous analyses have concentrated on weak flow. Kuzuu and Doi14 have analyzed the influence of weak and strong flows on a solution of entangled rods but only for very low volume fractions. Here, we show that it is straightforward to extend their calculations for elongational flow even when the excluded-volume effect is nonnegligible. An asymptotic time-dependent solution to the preaveraged Doi equation6i7 is obtained for a high enough degree of orientational order. The latter is a nontrivial function of the elongational rate and the excluded-volume effect. Since the number of rods enveloping a test rod decreases with increasing order there is a distinct possibility of the rods disentangling with strong enough flow. Thus, beyond this critical rate the rotational diffusion should be close to ideal. At very high elongational rates the rods should more or less align along the lines of flow. In that case, the stress is determined mainly by hydrodynamic friction so that Batchelor's limit a~p1ies.l~ Several workers16-'* have attempted to study the isotropienematic transition for solutions of rodlike particles in steady elongational flow by adding a term of Kramers' typelg to the usual free energy. Here, we point out that these analyses contradict the integral equation for the orientational distribution function arising from the Doi equation. In effect, ref 16-18 neglect the effect of entanglement altogether.Finally, let us recall some of the criticisms that can be leveled at the Doi theory. Arguing that a rigid tube ccnstraint may be too severe, Fixman20,21 proposed an alternative model in which the mean-square torque on a test rod is calculated by kinetic arguments. Nevertheless, the rotational diffusion coefficient would still increase with orientational order though less rapidly than in the Doi theory. Next, computer simulations20-28 show that entanglement starts developing at much higher concentrations than was originally s u r m i~e d .~~~~~The formulation of the hydrodynamic s...

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Theory of lyotropic polymer liquid crystals

Cited by 534 publications

References 21 publications

Induced chain rigidity, splay modulus and other properties of nematic polymer liquid crystals

Induced chain rigidity, splay modulus and other properties of nematic polymer liquid crystals

Phase behaviour of rod-like colloid+flexible polymer mixtures

Disentanglement of rods in semidilute and liquid-crystalline solutions in elongational flow

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