11.7 Scalar Product of Three Irreducible Tensors 197
=(u 1 ,u 2 ,r)=∑
′∑
1∑
2φ( 1 , 2 ,)(r)P(^1 ,^2 ,)(u 1 ,u 2 ,̂r). (11.62)The prime in the summation overindicates that only the “allowed” values are used,
as indicated in (11.60), viz.= 1 + 2 , 1 + 2 − 2 ,...,| 1 − 2 |.
The scalar functionsφ( 1 , 2 ,)(r)characterize the radial dependence of the differ-
ent angular contributions. The functionφ( 0 , 0 , 0 )is thespherical symmetric part,also
referred to asisotropic partof the potential. The dipole-dipole, dipole-quadrupole
and quadrupole-quadrupole interactions considered in Sect.10.4.5are examples for
the cases( 1 , 2 ,)equal to( 1 , 1 , 2 ),( 1 , 2 , 3 )and( 2 , 2 , 4 ). The anisotropic part
of the interaction betweenJanus spheres, these are spherical particles with two dif-
ferent ‘faces’, can be modeled, cf. [35], with three terms proportional tou 1 ·u 2 ,to
u 1 ·r−u 2 ·rand tou 1 ·ru 2 ·r−^13 u 1 ·u 2. These correspond to( 1 , 2 ,)equal
to( 1 , 1 , 0 ),( 1 , 0 , 1 )−( 0 , 1 , 1 )and( 1 , 1 , 2 ). A non-spherical interaction poten-
tial for nematic liquid crystals, introduced in [36], involves the anisotropic terms
u 1 u 1 : u 2 u 2 and(u 1 u 1 +u 2 u 2 ): rr, corresponding to( 1 , 2 ,)equal to
( 2 , 2 , 0 )and( 2 , 0 , 2 )+( 0 , 2 , 2 ). This potential is simpler than the commonly used
Gay-Berne potential, cf. [37].