334 16 Constitutive Relations
The three Miesowicz coefficients do not involve the viscosity coefficientη ̃ 3 .This
is different forη 45 corresponding to the case wherenpoints along the bisector in the
xy-plane. The resultingHelfrich viscosity coefficientη 12 = 4 η 45 − 2 (η 1 −η 2 ),cf.
(16.53), is given by
η 12 = 2 η ̃ 3. (16.106)
In the theoretical approach of Ericksen and Leslie, the local anisotropy of a nematic
liquid crystal is characterized by the directorn, which depends on the timetand
may also depend on the positionr. A frequently used ansatz for the pressure tensor
of a nematic is
−pνμ=α 1 nνnμnκnλ∇λvκ +α 2 nνNμ+α 3 Nνnμ (16.107)+α 4 ∇μvν+α 5 nνnλ∇λvμ+α 6 ∇νvλnλnμ,where
Nμ=∂nμ
∂t−εμλνωλnν (16.108)is the co-rotational time derivative of the director. For a constant director field,
(16.107) is equivalent to (16.101) when one has∇λvλ=0 andζ=0 applies. The
Leslie viscosity coefficientsαi,i= 1 , 2 , ..,6 are related to the viscosities introduced
in (16.101), by
η=1
2
α 4 +1
6
(α 5 +α 6 ), η ̃ 1 =1
2
(α 5 +α 6 ), η ̃ 2 =1
2
(α 2 +α 3 ), η ̃ 3 =1
2
α 1 ,
(16.109)γ 1 =α 3 −α 2 ,γ 2 =α 6 −α 5. (16.110)The symmetry (16.102) corresponds to theOnsager-Parodi relation[145]
α 2 +α 3 =α 6 −α 5. (16.111)The Miesowicz viscosities of nematic liquid crystals composed of prolate particles
obey the inequalitiesη 2 >η 3 >η 1 and consequentlyγ 2 =η 1 −η 2 is negative. For
nematics composed of discotic particles, the order of the Miesowicz coefficients is
reversed andγ 2 is positive. An affine transformation model considered next gives an
instructive insight into the anisotropy of the viscosity.
Theconnectionbetweenthefrictionpressuretensorinvolvingthevariousviscosity
coefficients of nematics and the alignment tensor is elucidated in Sects.16.4.4and
16.4.5. Alternative approaches are, e.g. found in [149, 150].
The rotational viscosity coefficientγ 1 has been measured for many nematics. The
full set of viscosity coefficients, viz.γ 1 ,γ 2 andη 1 ,η 2 ,η 3 ,η 12 has been determined
for a few liquid crystals only, e.g. cf. [151]. For model systems, the anisotropy of the