Normal stresses at the gelation transition

A simple Rouse-type model, generalised to incorporate the effects of chemical crosslinks, is used to obtain a theoretical prediction for the critical behaviour of the normal-stress coefficients $\Psi_{1}$ and $\Psi_{2}$ at the gelation transition. While the exact calculation shows $\Psi_{2}\equiv 0$, a typical result for these types of models, an additional scaling ansatz is used to demonstrate that $\Psi_{1}$ diverges with a critical exponent $\ell = k+z$. Here, $k$ denotes the critical exponent of the shear viscosity and $z$ the exponent governing the divergence of the time scale in the Kohlrausch decay of the shear-stress relaxation function. For crosslinks distributed according to mean-field percolation, this scaling relation yields $\ell =3$, in a accordance with an exact expression for the first normal-stress coefficient based on a replica calculation. Alternatively, using three-dimensional percolation for the crosslink ensemble we find the value $\ell \approx 4.9$. Results on time-dependent normal-stress response are also presented.