  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.
