Mean end - to - end distance of branched polymers

We use Monte Carlo methods to calculate the mean end-to-end distance of randomly branched polymer molecules. Molecular realisations are brown on the square and simple cubic lattices, and consist of N bifunctional monomers and N, polyfunctional branching units of functionality f. We treat three cases in which the monomer branches are either random walks, self-avoiding walks (saw), or SAW'S except that different branches may join at their end points to form closed loops. For the first case, we find that the mean end-to-end distance L varies with N as L for fixed branching probability p , consistent with previous theoretical predictions. For the latter two cases we find that for fixed p . L N u , where Y = 0.57 i 0.06 on the square lattice and 0.45 *0.06 on the simple cubic lattice. Although the number of closed loops formed per realisation is quite small, it does appear possible that loop formation may play a dominant role in the N+oo limit. An important class of polymer configurations is formed when N bifunctional monomers and Nf polyfunctional branching units of functionality f connect in a random way to form a branched structure (see figure 1). The theoretical treatment of such randomly branched polymer configurations was initiated by Zimm and Stockmayer (1949), who calculated the radius of gyration when the number of polyfunctional units is a small fixed number. Subsequently, de Gennes (1968) considered a more general problem in which the number of branching units is variable. Using diagrammatic techniques, de Gennes calculated the probability distribution for the branched structure given the probability distribution function for the linear chain. These two analyses were restricted only to the case in which the monomer branches formed random walks. (This is often called the 'Gaussian' branched polymer.) This approximation is used to describe the theta temperature, where the Van der Waals attraction of neighbouring molecules is balanced by the excluded volume constraint. Under this restriction, de Gennes showed that the mean end-to-end distance L varies as t L N "G(Nf)-w ( l a ) where vsc=; is the exponent for the single-chain random walk, and (U = a in all dimensions d. Thus for a fixed number of branching units single-chain behaviour occurs. If N, is not fixed, but is determined by a random probability, then (N , ) N and hence equation ( l a ) can be rewritten as