A free volume theory on the chain length dependence of the diffusivity of linear polymers

A free volume theory was developed to account for the crossover of the chain length dependence of the center-of-mass self-diffusion coefficient of linear polymers from unentangled to entangled regimes. Similar to the original free volume theory of Cohen and Turnbull, this theory requires information about the free volume overlapping factor (α), critical free volume per bead (v), and mean free volume per bead (〈v_f,i〉). However, one additional parameter is needed and it is the critical fraction of beads having free volume greater than or equal to αv termed as φ⁺. Here, α and v can be readily determined from the intermolecular and intramolecular radial distribution functions (i.e., g(r) and g^intra(r)) obtained from molecular dynamics (MD) simulation and they were found to be 0.5 and 0.0257 nm³, respectively, for polyethylene melts and it is not dependent on N. 〈v_f,i〉 was determined using the generic van der Waals (GvdW) equation of state and it had a value of 0.01 nm³ and the volume available to each bead can also be determined by Voronoi tessellation (VT) on the corresponding MD simulation trajectories. VT yielded exact probability of finding a certain amount of free volume and the free volume distribution was found in the form of the gamma distribution that is consistent with the positron annihilation lifetime spectroscopy observation. Finally, φ⁺ was calculated using the experimentally measured activation energies for diffusion per polyethylene molecule with different chain lengths and was found to be approximately 0.22 that was in line with what was found from MD simulations.