Onsager critical concentration

However, as discussed below critical concentrations for cellulose, in a variety of solvents, and based on optical observations under crossed polars are much lower than predicted using eauation 1 and kw = 2 q. Como et al. (4 point out one has to consider the possibili that the lattice model does not accuratelv predict the values of V2 and that V2 values using the Onsager (28) and Isihara (30) theories are about half that predicted by equation 1. 

Modern-day interest in LCPs had its origin with the molecular theories of Onsager (5) and Flory (6). They predicted that rod-like molecules would spontaneously order above a critical concentration that depended on the aspect ratio of the molecule. These theories were later expanded to include other effects such as polydispersity (2) and partial rigidity (g). 

The Onsager and Flory theories are both statistical theories on rigid rod liquid crystalline polymers, but the former is a virial approximation while the latter is a lattice model. The first is more applicable to dilute solutions while the latter works especially well for high concentrations and a highly ordered phase. These theories with experiments, especially critical volume fractions 4>i and critical order parameter Sc at nematic-isotropic transition are made below. 

For lyotropic liquid crystals, the relationships between the chain length (molecular weight) and the formability of liquid crystals have been formulated. According to Onsager and Flory (Chapter 2), for any solutions of rigid rods there is a critical concentration v (the volume fraction of rods in the solution) determined by the length-to-diameter ratio (L/d) of the rods... 

The excluded volume theories of Onsager and Flory show that depending on the axial ratio of the rod-like particles, there is a critical concentration above which a nematic phase is formed. This concentration does not depend on the temperature of the system as these theories are essentially athermal, i.e. the part of the free energy leading to the anisotropy is an entropy term. 

There are several different theoretical approaches to the problem. The Landau molecular field theory was applied by de Gennes to liquid-crystal phase transitions. (89) The Maier-Saupe theory focuses attention on the role of intermolecular attractive forces.(90) Onsager s classical theory is based on the analysis of the second virial coefficient of very long rodlike particles.(91) This theory was the first to show that a solution of rigid, asymmetric molecules should separate into two phases above a critical concentration that depends on the axial ratio of the solute. One of these phases is isotropic, the other anisotropic. The phase separation is, according to this theory, solely a consequence of shape asymmetry. There is no need to involve the intervention of intermolecular attractive forces. Lattice methods are also well suited for treating solutions, and phase behavior, of asymmetric shaped molecules.(80,92,93)... 

For neutral rods the hard-core interaction leads to orientational correlations at higher concentrations. As predicted by Onsager and Flory [36,37], at a critical concentration a solution of rods phase separates into an isotropic and an anisotropic nematic phase. The latter is characterized by a long-range orientational and a liquid-like positional order. Charged rods may be simply viewed as rods with a larger effective diameter of the order of k" but the orientation of the rods is complicated by the fact that electrostatic interactions tend to disorient the rods because the electrostatic repulsion is strongest for a parallel array and weakest for a perpendicular orientation. 

The formation of lyotropic LCPs may be construed as the occurrence of a transition from an isotropic phase to an anisotropic phase. As early as 1949, Onsager (1949) predicted, on the basis of calculation of free energy, the formation of an anisotropic phase at a critical concentration of a solution when the aspect ratio of the molecule is sufflciently large. Several years later, using the concept of lattice in mixtures of polymer and solvent, Flory (1956) also developed a theory to predict, in terms of the aspect ratio of polymer molecule, the critical concentration at which the formation of an anisotropic phase is possible. This subject has been summarized in several review articles (Flory 1984 Grosberg and Khokhlov 1981 Papkov 1984). 

As the Kuhn segments of different chains are statistieally independent, the situation studied is equivalent to the well-known model of rigid rods. When the exeluded volume is considered within the framework of Onsager model [49], the system is eonsidered to possess critical concentration C r, where the system obtains orientational ordering. The value of this critical concentration is determined by the length and diameter of the rods ... 

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