Free Energy of Rigid Rod Solutions

We consider a dispersion of rigid rod-like molecules in a volume V. Let Q (r, ) be the number density of rods with an orientation u (or its solid angle Q) at a position r. The free energy F of the dispersion at the level of second virial approximation is given by  

The first three terms correspond to the translational entropy of rods and the first integral term shows the entropy change due to ordering. In a disordered phase, this term becomes zero because/(r, k) = 1. In an ordered phase, we have peaks in/(r, u) and then the first integral term remains with nonzero values. The last term shows the interaction between rods. 

Although, the third and higher virial coefficients are neglected in Eq. (2.5), it can be applied to describe basic properties of phase behavior in any particle system. The second virial approximation corresponds to the simplest version of density functional theories [16, 17] and is the theoretical instrument responsible for producing a qualitative rather than quantitative understanding. 

In the following, we assume that the system is spatially uniform but non-uniform for orientation. Using the orientational distribution function/( ), the density is given by g (r, u) = q/( ) and Eq. (2.5) can be expressed as  

The equilibrium distribution is determined by the condition that the free energy is a minimum for the distribution with the normalization condition  

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