Scaling theory of star polymers and general polymer networks in bulk and semi-infinite good solvents

A scaling theory of general polymer networks in bulk and semi-infinite good solvents is derived by using the equivalence between the generating function for the total number of configurations and the multi-spin correlation function of the classical n-component Heisenberg model in the limit n → 0. For general polymer networks with fixed topology G composed of f linear chains with the same length l, the total number of configurations behaves for large f as Ng(l, l, ..., l) ∼ lγg- 1 μlf. The exponent γg is expressed exactly in terms of the corresponding exponents y ( f ) and γ s(f) of f-arm (free and center-adsorbed) star polymers. When g of the f arms of the star polymer are attached to the surface at their end points, the exponent γg γ11 ... 1 (f) is given in terms of that for f-arm star polymers y ( f ) and the exponents of linear chains as γ11 ... 1 (f) = γ (f) + v + g [γ11 - γ 1]. Also the exponent γg for comb polymers (with g 3-functional units) is reduced to a linear combination of the exponent of 3-arm star polymers, y (3 ), and the configuration number exponent of linear chains, γ, as γcomb (g ) = y + g [y (3) — y ]. The star-polymer exponents γ (f), γs (f) and γ11 ... 1 (f) are evaluated by means of mean-field theory, ε expansion and some general considerations. Our results for y ( f ) and γs(f) are $$ $$ where A ( f ) and B (f) are regular functions of f and are of O(ε 2). A ( f ) is found to be ε 2/64 + O(ε 3). Our first formula for γ ( f) is, however, inconsistent with Duplantier's exact result for the two-dimensional case, while our scaling relations for γg in terms of γ(f), γs( f) are consistent with his earlier results. The end-to-end distribution function pg (r) of a polymer network with general topology G is found to have a scaling form pg (r) ∼ r -dg( rL-ν) with the exponent ν of linear chains. Here L denotes the total length of chains. Thus the mean square end-to-end distance g behaves like L2ν . Scaling ideas are also applied to study the case where linear chains with different lengths construct the polymer networks. Relations to the direct method are also pointed out.