Onsager virial expansion

In his paper7) Onsager has considered the liquid-crystalline transition in the system of rigid rods using two main assumptions a) the interaction of rods was assumed to be due to the pure steric repulsion (no attraction) b) the virial expansion method was used (for the details of the Onsager method see Sect. 2.4). Thus, the Onsager results... 

The virial expansion has enjoyed greater appeal, especially as applied to lyotropic systems. Onsager s classic theory rests on analysis of the second virial coefficient for very long rodlike particles. It was the first to show that a solution of hard, asymmetric particles such as long rods should separate into two phases above a threshold concentration that depends on the axial ratio of the particles. One of these phases should be anisotropic (nematic), the other completely isotropic. The former is predicted to be somewhat more concentrated than the latter, but it is the alignment (albeit imperfect) of the solute molecules that is the predominent distinction. The phase separation is a consequence of shape asymmetry alone intervention of intermolecular attractive forces is not required. 

Priest (1973) and Straley (1973), in terms of the classical virial expansion, the Onsager theory (referred to in Section 2.1) and the curvature moduli theory, derived the elastic constants of rigid liquid crystalline polymers. The free energy varies according to the change of the excluded volume of the rods due to the deformation. The numerical calculation of elastic constants (Lee, 1987) are shown in Table 6.2. 

Onsager treated the LC state by a virial expansion method. He deduced the relationship between the volume fraction of rods, the rod length, and the rod diameter for both ordered and isotropic phases [16]. 

The Onsager theory, or second virial expansion, is very successful in predicting the qualitative behavior of the isotropic-nematic phase transition of hard rods. H owever, it is an exact theory in the limit ofI/D—>oo. Straley has estimated that, for L/D < 20, the contribution of the third virial coefficient to the free energy in the nematic phase should be at least comparable to that of the second virial coefficient [22]. For more concentrated solutions, an alternative approach such as the Flory lattice model [3, 5] is required. The full phase diagram of rod-like colloidal systems has been obtained in computer simulations [23, 24]. The effects of polydispersity of rods [19, 25] and charged rods [10] are also important in the phase transitions. The comparison between Onsager theory and experimental results has been summarized by Vroege and Lekkerkerker [26]. 

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