Phantom chain simulations for fracture of star polymer networks with various strand densities

Despite many attempts, the relation between fracture and structure of polymer networks is yet to be clarified. For this problem, a recent study for phantom chain simulations [Macromolecules, 56, 9359 (2023)] has demonstrated that the fracture characteristics obtained for polymer networks with various node functionalities and conversion ratios lie on master curves if they are plotted against cycle rank. In this study, we extended the simulation to the effect of prepolymer concentration on the relationships between cycle rank and fracture characteristics within the concentration range of 1<c/c^*<8, concerning the overlapping concentration c^*. We created networks from sols of star-branched phantom bead-spring chains via end-linking reaction between different chains through Brownian dynamics simulations with varying the number of branching arms f from 1 to 8, and the conversion ratio {\phi}_c from 0.6 to 0.95. For the resultant networks, cycle rank {\xi} was consistent with the mean-field theory. The networks were uniaxially stretched with energy minimization until break to obtain modulus G, strain at break {\epsilon}_b, stress at break {\sigma}_b, and work for fracture W_b. With the branch point density \u{psion}_br, G/\u{psion}_br, {\epsilon}_b, {\sigma}_b/\u{psion}_br, and W_b/\u{psion}_br of the data for various f and {\phi}_c draw master curves if plotted against {\xi}. The master curves depend on c; as c increases, all the mechanical characteristics monotonically increase. If we plot {\sigma}_b/\u{psion}_br and W_b/\u{psion}_br against G/\u{psion}_br, the data for various f and {\phi}_c lie on master curves but depending on c. Consequently, the fracture characteristics are not solely described by modulus for the examined energy-minimized phantom chain networks.