Virial coefficients asymptotic behavior

An open problem is the asymptotic behavior (with increasing n) of the virial coefficients of the hard sphere fluid. Since the only forces in this system are repulsive, it was conjectured that all the virial coefficients are necessarily positive. While this classical conjecture has neither been disproven nor verified the basis for it is untenable. Thus, the sixth virial coefficient of a fluid composed of three-dimensional hard cubes in which the forces are all repulsive is negative. ... 

We now present briefly more explicit calculations of the mutual virial coefficients obtained with the use of des Cloizeaux direct renormalization method for blends of linear flexible polymers in a common good solvent, a common 0-solvent and a selective solvent and for blends of rodlike polymers and flexible polymers in a 0-solvent (marginal behavior). These calculations enable one to find (universal) prefactors relating the mutual virial coefficient to the chain volume (in Eq. 7) in the asymptotic limit. Moreover they give the corrections to the scaling behavior which explicitly depend on the interactions between unlike monomers and are actually responsible for the phase separation of flexible polymer blends in a good solvent. 

We have seen in section II that in a good solvent A-S interactions play only a marginal role in dilute solutions and excluded volume interactions dominate. In the asymptotic limit of infinite molecular weight, polymers A and B are not distinguishable, i.e., the dimensionless mutual virial coefficient g B tends to the same universal asymptotic limit g as g and Sbb (36)). As a result, contrary to the case of a common 0-solvent or of a selective solvent, the phase separation arises from the corrections to the scaling behavior. Let us consider the symmetric caseiV = N and suppose that the solution is sufficiently dilute so that the virial expansion is valid. For such a case = 1/2 and Eq. (30) leads to a critical concentration... 

In a dilute solution in a common good solvent for both blocks, the interactions between. different copolymers may be studied using the same direct renormalization procedures as the interactions between two homopolymers A and B equivalent to the two blocks.As for blends, in the asymptotic limit of infinite molecular masses, the chemical difference between the two blocks is irrelevant and the dimensionless virial coefficient gc between block copolymers defined by Eq. (10) is equal to the same value g as for homopolymers. The interactions which may provoke the formation of mesophases are here again due to the corrections to the scaling behavior ... 

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