Monte Carlo simulation of homopolymer chains. I. Second virial coefficient

The second virial coefficient, $A2,$ is evaluated between pairs of short chain molecules by direct simulations using a parallel tempering Monte Carlo method where the centers of mass of the two molecules are coupled by a harmonic spring. Three off-lattice polymer models are considered, one with rigid bonds and two with flexible bonds, represented by the finitely extensible nonlinear elastic potential with different stiffness. All the models considered account for excluded volume interactions via the Lennard-Jones potential. In order to obtain the second virial coefficient we calculate the effective intermolecular interaction between the two polymer chains. As expected this intermolecular interaction is found to be strongly dependent upon chain length and temperature. For all three models the θ temperature $(θn),$ defined as the temperature at which the second virial coefficient vanishes for chains of finite length, varies as $θn−θ∞∝n−1/2,$ where $n$ is the number of bonds in the polymer chains and $θ∞$ is the θ point for an infinitely long chain. Introducing flexibility into the model has two effects upon $θn;$ the θ temperature is reduced with increasing flexibility, and the $n$ dependence of $θn$ is suppressed. For a particular choice of spring constant an $n$-independent θ temperature is found. We also compare our results with those obtained from experimental studies of polystyrene in decalin and cyclohexane, and for poly(methyl methacrylate) in a water and tert-butyl alcohol mixture, and show that all the data can be collapsed onto a single universal curve without any adjustable parameters. We are thus able to relate both $A2$ and the excluded volume parameter $v,$ to the chain interaction parameter $z,$ in a way relating not only the data for different molecular weights and temperatures, but also for different polymers in different solvents.