Tensors for Physics

15.2 Isotropic↔Nematic Phase Transition 279

whereP 2 (x)=^32 (x^2 −^13 )is the second Legendre polynomial. Frequently, the
quantityS 2 is denoted bySand referred to asMaier-Saupe order parameter.
Theoretical approaches to thephase transition isotropic↔nematicare discussed
next.

15.2.2 Landau-de Gennes Theory

TheLandau-de Gennes theoryfor the phase transition isotropic↔nematic is based
on finding the minimum of a free energyFaassociated with the alignment. This free
energy is written as
Fa=NkBTΦ, (15.10)

whereNis the number of particles,kBis the Boltzmann constant,Tis the tempera-
ture andΦis a dimensionless thermodynamic potential function which depends on
the scalar invariantsI 2 ,I 3 of the alignment tensor. In the Landau de Gennes theory
the ansatz

Φ=ΦLdG≡

1

2

AI 2 −

1

3

BI 3 +

1

4

CI 22 , A=A 0

(

1 −

T∗

T

)

, A 0 ,B,C> 0 ,

(15.11)

or explicitly,

ΦLdG=

1

2

Aaμνaνμ−

1

3

B

√

6 aμνaνκaκμ+

1

4

C(aμνaνμ)^2 , (15.12)

is made. The phenomenological coefficientsA 0 ,B,C>0areassumedtobepracti-
cally constant in the vicinity of the phase transition, andT∗is a pseudo-critical tem-
perature, which is somewhat below the isotropic-nematic transition temperatureTni.
The equilibrium value of the alignment is inferred from the minimum of the free
energy, which in turn, is obtained by putting the first derivative of the potentialΦ
with respect to the alignment tensor equal to zero. To compute the derivative∂Φ/∂a,
replaceainΦ(a)bya+δawhereδais a small distortion. Then find the factor in
the term ofδΦ=Φ(a+δa)−Φ(a)which is linear inδa. For the present case,
use of

δΦ=

(

∂Φ

∂I 2

∂I 2

∂aμν

+

∂Φ

∂I 3

∂I 3

∂aμν

)

δaμν,

with

∂I 2

∂aμν

δaμν= 2 aμνδaμν,

∂I 3

∂aμν

δaμν= 3

√

6 aνκaκμδaμν= 3

√

6 aνκaκμδaμν,