16.4 Viscosity and Alignment in Nematics 339
Ω ̇λ=−νrΩλ+θ−^1 Tλsyst+θ−^1 Tλflct.Hereνr>0 is a rotational damping coefficient due to a frictional torque,θis the
relevant moment of inertia,TsystandTflctdenote the systematic and the fluctuating
torques acting on a particle. The systematic torque can be derived from a Hamiltonian
functionHaccording toTλsyst=−LλH(u),cf.(12.45). It is recalled that
Lλ=ελκτuκ∂
∂uτ.
In the following, the dimensionless functionH =−βH is used, whereβ =
(kBT)−^1 , andTis the temperature. Then one hasβTλsyst=LλH(u), and the
generalized Fokker-Planck equation pertaining to this Langevin equation reads
∂f(u)
∂t+ωλLλf(u)−ν 0 Lλ(Lλf(u)−f(u)LλH(u))= 0. (16.119)The relaxation frequencyν 0 is related to the rotational damping coefficientνrby
ν 0 =kBT
θνr. (16.120)
For spherical particles with radiusR, one hasνr ∼ R^3 , see the Exercise10.3,
provided that hydrodynamics applies. For non-spherical particlesνrdepends on the
shape of the particle, but an effective radiusReffcan be defined such thatνr∼Reff^3 ,
andν 0 ∼R−eff^3. For vanishing vorticity, the kinetic equation (16.119) is similar to
(12.46), where the systematic torque was assumed to be due to external orienting
fields. Here, however, the orienting torques are due to the viscous flow and an internal
field which, in turn, is caused by the alignment of the surrounding particles. Both
torques are of the formT
syst
λ ∼ελμκuμFκνuνsuch that
H=Fμνφμν,φμν=ζ 2 uμuν,ζ 2 =√
15
2
. (16.121)
The specific expression for the symmetric traceless tensorFμνis
Fμν=( 6 ν 0 )−^1 R∇μvν+T−^1 T∗aμν, aμν=〈φμν〉. (16.122)The first term in (16.122) involving theshape parameterRdescribes the orient-
ing effect of the velocity gradient. When hydrodynamics applies, this parameter is
given by
R=