Tensors for Physics

16.3 Viscosity and Non-equilibrium Alignment Phenomena 327

The pertaining complex shear modulusG(ω)is given by

G(ω)=−iωτMη(ω)=G′(ω)−iG′′(ω), (16.83)

G′(ω)=G

(ωτM)^2

1 +(ωτM)^2

, G′′(ω)=G

ωτM

1 +(ωτM)^2

.

For low frequencies whereωτM1, the real partη′of the viscosity approaches the
viscosityη, its imaginary partη′′,aswellasG′andG′′vanish. For high frequencies
whereωτM 1, the real partG′ofG(ω)approaches the “high frequency shear
modulus”G, whereasG′′,aswellasη′andη′′become zero.
Maxwell derived the ‘Maxwell model’ equation from the Boltzmann equation for

the velocity distribution function of a gas. In that case,pμνis the kinetic contribution

pkinμν to the symmetric traceless friction pressure tensor, see the first equation of

(16.59) withτM−^1 =νpandνpa=0. Here the Maxwell relaxation time is determined
by a Boltzmann collision integral and one hasG=p 0 =nkBT.
Multiplication of the kinetic equation (12.120) for the pair correlation function
byrνFμ, and use of (12.107) yields a Maxwell model equation for the potential
contribution of the pressure tensor. In this case,τMis equal to the relaxation time
τintroduced in (12.120) and the high frequency shear modulusGis given by the
Born-Green expression (16.35).
The Maxwell model equation can also be derived within the framework of irre-
versible thermodynamics. Taking into account that the entropy density contains a

contribution proportional toG−^1 pμν pμν, whereGis the high frequency shear
modulus. The ensuing entropy production associated with the second rank tensors
is proportional toG−^1 pμνdpμν/dt.TheExtended Irreversible Thermodynam-
ics, cf. [132, 133], takes into account additional contributions to the entropy and
consequently to the entropy production, which then contains time derivatives of
‘non-conserved’ quantities, like of the friction pressure considered here, or of the
heat flux. An expression for the extended non-equilibrium entropy, valid for gases, is
derived in Exercise12.3. More general schemes for the treatment of non-equilibrium
phenomena are presented in [134, 135].
In molecular fluids and colloidal dispersions containing non-spherical particles,
the visco-elastic behavior is associated with the dynamics of the alignment as
described by (16.74), cf. [136, 137]. The constitutive equation (16.133) for the fric-
tion pressure tensor is equivalent to

pνμ=− 2 ηiso∇νvμ+ pνμ

align

, pνμ

align

=

√

2

ρ

m

kBT

τap

τa

aμν. (16.84)

Here pνμ

align
is the part of the pressure tensor associated with the alignment. The
viscosity coefficientηiso, corresponding to a situation, where the alignment vanishes,
is smaller than the Newtonian viscosityη=ηNewpertaining to the case where the
time derivative of the alignment and effects nonlinear in the shear rate vanish. These