15.5 Energetic Coupling of Order Parameter Tensors 295
or the molecular alignment and the anisotropy of the pair-correlation function, in the
first coordination shell. The dimensionless free energyΦ=Φ(a,b)is written as
Φ=Φa+Φb+Φab,Φab=c 1 aμνbμν+√
6 c 2 aμκaκνbμν, (15.51)whereΦa=Φa(a)andΦb=Φb(b)are e.g. expressions of Landau-de Gennes
type with coefficientsAa,Ba,CaandAb,Bb,CbandΦab=Φab(a,b), with the
coefficientsc 1 ,c 2 characterizes the coupling between the two tensors. A term
∼aμκbκνbμνis possible, but disregarded here for simplicity. The derivatives of
the potential with respect to the tensors are
∂Φ
∂aμν=Φμνa +c 1 bμν+c 22√
6 aμκbκν,∂Φ
∂bμν=Φμνb +c 1 aμν+c 2√
6 aμκaκν. (15.52)In thermal equilibrium, both expressions are zero. Then it is possible to determine
bas function ofafrom the second equation of (15.52) and to insert it into the first
equation of (15.52). This yields a derivative of a Landau-de Gennes potential, cf.
(15.13), with renormalized coefficientsA,B,C.
For the special caseΦb=^12 AbbμνbμνwithAb=1 due to an appropriate choice
of the normalization ofb, the result is
A=Aa−c^21 , B=Ba+ 3 c 1 c 2 , C=Aa− 2 c^22. (15.53)The derivation is deferred to the following exercise.
The renormalized coefficientsAandCare smaller thanAaandCa, irrespective
of the sign ofc 1 andc 2. The coefficientBis larger or smaller thanBadepending on
whether the coupling coefficientsc 1 andc 2 have equal or opposite sign.
The treatment of relaxation processes and other non-equilibrium phenomena of
the kind presented in Chap. 17 is based on differential equations which contain the
derivatives (15.52) of the relevant potential function, e.g. see [96].
15.3 Exercise: Renormalization of Landau-de Gennes Coefficients
Consider the special case whereΦb=^12 Abbμνbμν, for simplicity putAb=1.
Determinebμνfrom∂∂Φbμν =0 with the help of the second equation of (15.52).
Insert this expression into the first equation of (15.52) to obtain a derivative of a
Landau-de Gennes potential with coefficientsA,B,Cwhich differ from the original
coefficientsAa,Ba,Cadue to the coupling between the tensors.
Hint: use relation (5.51)fora,viz.aμκaκλaλν =^12 aμνaλκaλκ.