Polymer network of fixed topology in a high-molecular weight solvent

Abstract

We consider a polymer network G with fixed topology made of long polymer chains of common polymerization degree N. The network is assumed to be dissolved in a melt of short linear polymer chains of polymerization degree P < N. Probe N-chains and mobile P-chains are assumed to be of the same chemical nature. Let Φ P be any physical property (partition function, gyration radius, …) associated with the network. The purpose is to investigate quantitatively the behavior of Φ p in the long-chain limit (N → ∞, at fixed P). Assume that the same physical quantity behaves for a low-molecular weight solvent (P=1) as: Φ 1 N X+σ, where x is the critical exponent when G is ideal, and the additional one, σ, which is universal but it depends on the topology of G, characterizes the swelling of N-chains. We show that the asymptotic behavior of Φ P for a high-molecular weight solvent reads: Φ P ∼ P −2 σ/(4− d) N X+ σ , provided that N > P 2/(4−d) (d =2, 3 is the space dimensionality and 4 is the critical dimension of the system). However, at dimension 3, when N < P 2, the network becomes ideal. Thus, a crossover between ideal and real networks occurs along the curve of equation N ∼ P 2 in the (P, N)-space. In dimension 2, we find that the network is always swollen by mobile chains, as long as N > P, which is the condition defining the system in the beginning. The main conclusion is that, in the presence of a high-molecular weight solvent, the asymptotic behavior is renormalized multiplicatively by a P ζ-factor, whose exponent ζ is related to the swelling exponent σ relative to a solvent of small molecules through: ζ = −2σ/(4−d). Finally, this unified description can be extended to polymer networks near interacting surfaces.