Polydisperse solution of randomly branched homopolymers, inversion symmetry and critical and theta states

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Abstract
We discuss the phase behavior of a model of a binary mixture of randomly branched homopolymers in a solution. The monomer–solvent interaction is determined by a Boltzmann weight w. The theory has been presented recently and is obtained by approximating the underlying lattice by a Bethe lattice of the same coordination number q. Of special interest is the class of randomly branched polymers with inversion symmetry (see the text). This class includes linear polymers. The phase diagram for the special class of polymers is very simple. There is a line C of critical points in the dilute limit on which branched polymers become a critical object in a good solvent. This is an extension of the result due to de Gennes for linear chains in an athermal solution to the above class of branched polymers in any good solvent. The line C meets with another critical line C ′ for phase separation in a poor solvent. We identify the theta point as a tricritical point as first suggested by de Gennes for linear chains only. The theta point appears only in the limit of infinite polymers such that the second virial coefficient A 2 vanishes. We calculate various exponents and identify the order parameter. We point out a subtle difference between the theta state and the random walk state. However, the radius of gyration exponent does have its mean-field value of 1/2 in the theta state but only in d⩾3. There does not exist a tricritical point for randomly branched polymers without inversion symmetry.