Crossover to Tricriticality in Polymer Solutions

An example of competition between mesoscales occurs in a polymer solution near the critical point of phase separation. In this system, two mesoscales can be tuned independently the correlation length of the critical fluctuations by the proximity to the critical point of demixing and the radius of gyration Rg by the degree of polymerization n.  

In the mean-field approximation the field-dependent potential (osmotie pressure) can be presented in the form of a truncated Landau expansion in powers of the order parameter 

The dimensionless osmotic compressibility is related to a dimensionless Gibbs energy of mixing (per monomer), AG, by a Legendre transformation n = ( 0) - AG. This yields the expression 

Equation (7.74) is equivalent to the virial expansion of the osmotic pressure near the theta point given by eq 7.32 with 

The critical demixing point can be found from the stability conditions 

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