Universal scaling and characterization of gelation in associative polymer solutions

A Brownian dynamics simulation algorithm is used to describe the static behaviour of associative polymer solutions across a range of concentrations into the semidilute unentangled regime. Predictions for the fractions of stickers bound by intra-chain and inter-chain association, as a function of system parameters such as the number of stickers on a chain, the number of backbone monomers between stickers, the solvent quality, and monomer concentration are obtained. A systematic comparison between simulation results and scaling relations predicted by the mean-field theory of Dobrynin (Macromolecules, 37, 3881, 2004) is carried out. Simulation results validate the predictions of the mean-field theory across a wide range of parameter values in all but one of the scaling regimes. The value of the des Cloizeaux exponent, $\theta_2 = 1/3$, proposed by Dobrynin for sticky polymer solutions, is shown to lead to a collapse of simulation data for all the scaling relations considered here. Three different signatures for the characterisation of gelation are identified, with each leading to a different value of the concentration at the sol-gel transition. The Flory-Stockmayer expression relating the degree of inter-chain conversion at the sol-gel transition to the number of stickers on a chain is found not to be validated by simulations, while an alternative expression is satisfied by all three gelation signatures. Simulation results confirm the prediction of scaling theory for the gelation line that separates sol and gel phases, when the revised Flory-Stockmayer expression is used. Phase separation is found to occur with increasing concentration for systems in which the backbone monomers are under $\theta$-solvent conditions, and is shown to coincide with a breakdown in the predictions of scaling theory.