Critical temperature of infinitely long chains from Wertheim's perturbation theory

Wertheim's theory is used to determine the critical properties of chains fo rmed by m tangent spheres interacting through the pair potential u(r). It i s shown that within Wertheim's theory the critical temperature and compress ibility factor reach a finite non-zero value for infinitely long chains, wh ereas the critical density and pressure vanish as m(1.5). Analysing the zer o density limit of Wertheim's equation or state for chains it is found that the critical temperature of the infinitely long chain can be obtained by s olving a simple equation which involves the second virial coefficient of th e reference monomer fluid and the second virial coefficient between a monom er and a dimer. According to Wertheim's theory, the critical temperature of an infinitely long chain (i.e. the Theta temperature) corresponds to the t emperature where the second virial coefficient of the monomer is equal to 2 /3 of the second virial coefficient between a monomer and dimer. This is a simple and useful result. By computing the second virial coefficient of the monomer and that between a monomer and a dimer, we have determined the The ta temperature that follows from Wertheim's theory for several kinds of cha ins. In particular, we have evaluated Theta for chains made up of monomer u nits interacting through the Lennard-Jones potential, the square well poten tial and the Yukawa potential. For the square well potential, the Theta tem perature that follows from Wertheim's theory is given by a simple analytica l expression. It is found that the ratio of Theta to the Boyle and critical temperatures of the monomer decreases with the range of the potential.