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LIQUID CRYSTALS with COLLOIDS - DETAILS from Oshri and Priel

  • Generalized Gay Berne model:
    different types of molecules with distinct energetic anisotropies, i.e, different ratio of well-depth between end to end and side by side configurations: for each particle, in addition to the difference in the size and shape of the particles. Our expressions are based on a molecular dynamics study http://mw.concord.org/modeler/articles/beyondlj.pdf and we thank the NSF funded Concord Consortium for advice.
    The potential is given by (summing over all pairs of molecules):

    where σ is the intermolecular separation, which has now a similar but different form, depending on

    We notice that exchanging i<--> j is equivalent to <-->1/, and as expected the orientation dependent molecular separation σ is symmetric under exchange i <-->j. di denotes the breadth of molecule i and li denotes the length of molecule i. For identical particles li=lj, di=dj: we can see that = 1
  • The scale parameter σ0 now is determined as the contact parameter between the pair of molecules, and is different for each pair:
    The shape anisotropy parameter χ is now defined by: And we can see that for identical particles this definition degenerates to the familiar definition of χ in the classical Gay-Berne model (where all particles are identical). Since σee = 2l, σss = 2d, κ = l/d. Hence, also the definition of σ(ui,uj,rij) converges to the original definition (recalling that in this case =1) in the classical GB model.
    Another special case is when one molecule of the pair is spherical, i.e d = l, so that χ=0 and = 0 or infinity. In this case the separation σ depends only on the angle between the nonspheric molecule and the vector beween the molecules (and as expected - the energy doesn't depend on the orientation of the spherical molecule). In this limit σ approaches:

    The well depth of the interaction potential is defined: and
    with and ,
    where ε0 is now different for each molecule. For identical particles, again, we can see that the definition degenerates to the familiar definion of ε(ui,uj,rij) in the classical Gay Berne model.
    In the most degenerate case when the particles are spherical and identical it can be seen that the equations reduce back to the Lennard Jones potential.