Tensors for Physics

15.2 Isotropic↔Nematic Phase Transition 283

15.2.3 Maier-Saupe Mean Field Theory.

Although Maier and Saupe [80] followed a different line of reasoning, the essence
of their theory for the isotropic-nematic phase transition is based on a mean field
approach. More specifically, it is assumed that a molecule feels an orienting internal
field caused by the orientation of its neighbors which, in turn, is proportional to the
second rank alignment tensor. By analogy with (12.30), the Hamilton function for
the orientational interaction is determined by

H=H(MS)∼−aμνuμuμ, −H(MS)/kBT=βMSaμνφμν,βMS=T∗/T.

(15.24)

The equilibrium distribution function is proportional to exp[βMSaμνφμν]. Thus
aμν=〈φμν〉evaluated with this distribution leads to a nonlinear equation for the
alignment tensor, from which the phase transition behavior can be inferred.
For a uniaxial alignment, and this is the case treated by Maier and Saupe, one has

aμν=

√

3 / 2 nμnν withaμνaμν=a^2 anda=

√

5 〈P 2 (n·n)〉. Then the Hamilton
function reduces to

H(MS)=−

√

5 kBT∗aP 2 (n·u)=− 5 kBT∗SP 2 (n·u), S=〈P 2 (n·u)〉.(15.25)

The self-consistency relation determining the equilibrium values of the order para-
meter is

a=J(a), J(a)=Z(a)−^1

√

5

∫ 1

0

P 2 (x)exp[(T∗/T)

√

5 aP 2 (x)]dx,

Z(a)=

∫ 1

0

exp[(T∗/T)

√

5 aP 2 (x)]dx. (15.26)

In terms ofa ̃ =(T∗/T)aandF(a ̃) = J((T/T∗)a ̃), the relation (15.26)is
equivalent to

T

T∗

a ̃=F(a ̃),

where nowF(a ̃)is a function which does not depend onT. The intersection of
the straight lines, cf. Fig.15.4, with the curve yields the self-consistent value for
a ̃=(T∗/T)a.
The equilibrium value for the order parameteracan be plotted as function ofT/T∗
by a parametric plot ofF(x)/xviaF(x), withxinstead ofa ̃, in the appropriate range,
see Fig.15.5. The pertaining Gibbs free energy has a minimum with the value 0 at
T=Tni= 1. 099 T∗≈ 1. 1 T∗. The order parameter, at the coexistence temperature
isani = 0. 98 ≈1, corresponding to the value 0.44 for the Maier-Saupe order
parameterSat the phase transition temperatureTni. The right end of the curve in