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

Abstract

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...

Geometric percolation of hard nanorods: The interplay of spontaneous and externally induced uniaxial particle alignment

“…This phenomenon of re-entrance percolation has recently been found theoretically in the absence of orienting fields, and is caused by an increase in surface-to-surface distance as the particles transition into a more (orientationally) ordered state, [73] similarly to the disentanglement of rods in elongational flow fields. [87] What changes for field strengths K > 0 is that the percolation region becomes narrower and ultimately vanishes. This is because the additional alignment induced by the external field enhances the surface-to-surface distance of particles, thereby working against network formation.…”

Continuum percolation of polydisperse rods in quadrupole fields: Theory and simulations

“…Because the orientational distribution function ψ(ϑ) is independent of the azimuthal angle ϕ of the rods, the ϕ-average only requires the integration of |sin γ(u, u ′ )|. Expanding the integral kernel in Legendre polynomials P 2n and using the addition theorem for spherical harmonics [66,67], we obtain (10) with the coefficients d 0 = π 4 and, for n > 0,…”

Connectedness Percolation of Elongated Hard Particles in an External Field

“…(2), which, for q ! 0, reduces to gðuÞ ¼ hf Another simplification follows from expressing this integral in terms of a sum of the Legendre polynomials P 2n ðcos#Þ by invoking the addition theorem [15]. Because of cylindrical symmetry, we write c ð#Þ ¼ ð2Þ À1 ½a 0 þ a 2 P 2 ðcos#Þ þ a 4 P 4 ðcos#Þ [20], where a 0 ¼ 1=2 because c is normalized.…”

“…1. This kind of reentrance behavior is caused by the interaction-induced enhancement of the alignment of the particles and is not unlike the disentanglement of rodlike particles in elongational flow fields [15]. For weak fields, the densities at which this happens are preempted by the transition to the uniaxial nematic phase.…”