Shear viscosity of a crosslinked polymer melt

TL;DR

This paper derives an exact relation between shear viscosity and resistor network resistance in a crosslinked polymer melt, revealing a logarithmic divergence at the vulcanization transition and establishing a scaling relation among critical exponents.

Contribution

It introduces a minimal mesoscopic model linking shear viscosity to resistor network resistances and proves a new scaling relation for critical exponents in crosslinked polymer melts.

Findings

Viscosity diverges logarithmically near the critical point.

Exact relation between viscosity and resistor network resistance is established.

Scaling relation $k=eta- ext{phi}$ is proven for realistic crosslink ensembles.

Abstract

We investigate the static shear viscosity on the sol side of the vulcanization transition within a minimal mesoscopic model for the Rouse-dynamics of a randomly crosslinked melt of phantom polymers. We derive an exact relation between the viscosity and the resistances measured in a corresponding random resistor network. This enables us to calculate the viscosity exactly for an ensemble of crosslinks without correlations. The viscosity diverges logarithmically as the critical point is approached. For a more realistic ensemble of crosslinks amenable to the scaling description of percolation, we prove the scaling relation $k=\phi-\beta$ between the critical exponent $k$ of the viscosity, the thermal exponent $\beta$ associated with the gel fraction and the crossover exponent $\phi$ of a random resistor network.