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

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. Introduction A 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 stress has also been cr i t i~ized. '~ ,~ ' -~~ Doi and E d w a r d ~ ~ ~ l ~ ~ used the bare rotational friction coefficient, but others13*31-33 have opted for a renormalized one. This problem is unresolved. Lastly, the influence of 0024-9297/88/2221-3511$01.50/0 0 semiflexibility needs to be assessed. There is definitely an influence.34 However, present of the rotational diffusion of worms (with identical results) appear to conflict with most data except a t very high concentration26 or for a chain trapped in a fixed gel.3e It has been argued that the semiflexibility effect does show up clearly in equilibrium measurements.% Of these influences only the one pertaining to entanglementwB will be accounted for here. Entanglement Condition We consider a semidilute or concentrated solution of slender rods of length L and number density v. The solution is either isotropic or uniaxially ordered; the single-rod orientational distribution function f(u,t) depends on time and the unit vector u pointing along the axis of a test rod and defined with respect to some preferred axis. Doi and Edwards7bo have calculated the average number N(r) of rods intersecting a tube of radius r whose axis is aligned along a test rod. On averaging N(r) over all orientations of the probe we obtain ( N ( r ) ) = vrL2p (1)