16.4 Viscosity and Alignment in Nematics 343
velocity gradient does not alter the uniaxial character of the alignment nor affect the
magnitude of the alignment.
The equation for the alignment tensor which underlies a unified theory valid both
for the isotropic and the nematic phases of a liquid crystal, can also be derived within
the framework of irreversible thermodynamics, see the following section. The flow
alignment and also the viscous properties of nematics are treated by this approach.
Dynamic phenomena, such as a time dependent and even chaotic response of the
alignment to a stationary shear rate are discussed in Sect.17.3.
Some historical remarks: The application of a Fokker-Planck equation to the flow
birefringence in colloidal dispersions was initiated by A. Peterlin and H.A. Stuart in
1939 and reviewed 1943 [160]. They used the torque caused by the flow as derived
by Jeffery [161]. The inclusion of a torque associated with the alignment, which
allows the treatment of both the isotropic and nematic phases, was first presented
by the author [157]. An independent derivation was given later by Doi [158], who
considered the application to rod-like polymers, see also [162]. In the literature, both
the generalized Fokker-Planck equation and the resulting equation for the second
rank alignment tensor are referred to asDoi-theoryorDoi-Hess-theory, see e.g.
[163]. Different assumptions were made for the closure of the hierarchy equations.
A discussion of the dynamic equations and the underlying physics is also presented
in [164–167].
16.4.5 Unified Theory for Isotropic and Nematic Phases
The first unified theory for the flow alignment and the viscous properties of liquid
crystals in the isotropic and nematic phases [168], as well as the study of the influence
of a shear flow on the phase transition [169] was based on a generalized version of
irreversible thermodynamics, where the alignment tensor is treated as an additional
macroscopic variable, as in Sect.16.3.6. As before, the point of departure for a
treatment within the framework of irreversible thermodynamics is an expression for
the contribution of the alignment to the free energy or the free enthalpy. Now it is
assumed that this contribution is proportional to the Landau-de Gennes potentialΦ,
its time change is proportional to−Φμνdaμν/dt, whereΦμνis the derivative ofΦ
with respect toaμν. When the co-rotational time derivative of the alignment is used
as in (16.69), the resulting entropy production is similar to the expression (16.70),
just withaμν(
δaμν
δt )irrevreplaced byΦμν(
δaμν
δt )irrev. The ensuing constitutive relations
are similar to (16.133), nowaμν, in the first of these equations replaced byΦμν.As
a consequence, the inhomogeneous equation for the alignment tensor analogous to
(16.74) contains the nonlinear relaxation termτa−^1 Φμν, instead ofτa−^1 aμν.
Motivated by the first three terms of (16.124), the more general ansatz
daμν
dt− 2 εμλκωλaκν− 2 κ∇μvκaκν=(
δaμν
δt)
irrev