16.4 Viscosity and Alignment in Nematics 335
viscous behavior of liquid crystals can also be analyzed by computer simulations.
Non-Equilibrium Molecular Dynamics (NEMD) simulations were first performed
for perfectly oriented nematics in [147]. Nematics composed of particles interact-
ing via a Gay-Berne potential were studied both in NEMD simulations, cf. [152],
and in equilibrium Molecular Dynamics (MD) calculations, e.g. see [153]. In MD
simulations, time correlation functions, cf. Sect.17.1, are determined, the transport
coefficients then are obtained with the help of Green-Kubo relation, viz. as an integral
over the time. Viscosity coefficients were inferred from the dependence of the flow
resistance on the strength of orienting electric or magnetic fields, both in experiments
and in NEMD computer simulations [154]. Results from NEMD computations were
presented in [155] for the viscosity coefficientsγ 1 ,γ 2 andη 1 ,η 2 ,η 3 ,η 12 , as func-
tions of the density of the Gay-Berne fluid. For densities approaching the nematic↔
smectic A transition, the smallest of the viscosity coefficient, viz.η 1 increases and
it overtakesη 3 andη 2. The rod-like particles form disc-like clusters in anticipation
of the smectic layers.
16.4.2 Perfectly Oriented Ellipsoidal Particles
With the help of an volume conserving affine transformation, cf. Sect.5.7, the viscous
behavior of a fluid composed of perfectly oriented ellipsoidal particles can be mapped
onto that of a fluid of spherical particles [146, 147]. The affine transformation model
provides a good qualitative description for the anisotropy of the viscosity [148] of
nematics.
Within the framework of this model, the binary interaction potentialΦof the
non-spherical particles are expressed in terms of the interaction potentialsΦsphof a
reference fluid composed of spherical particles according to
Φ(r)=Φsph(rA), rμA=A^1 μν/^2 rν.The vectorrin real space is linked withrAvia an affine transformation, as given
by (5.5.3). This means, the equipotential surfaces are ellipsoids. In simple liquids,
the pressure is mainly determined by its potential contribution, see (16.29). For
simplicity, the kinetic contribution to the friction pressure tensor is disregarded here.
The spatial derivative is transformed according to
∇μA=A−μν^1 /^2 ∇ν.The transformation of the pressure tensorPνμ, between the affine and the real space is
PνμA =A^1 νλ/^2 PλκA−κμ^1 /^2.