Abstract

Based upon the resistor analogy and using the ideal loop gas approximation (ILGA), it is shown that only pending loops reduce reduce the modulus of an otherwise perfect network made of monodisperse strands and junctions of identical functionality. Thus, the cycle rank of the network with pending structures removed (cyclic and branched) is sufficient to characterize modulus, if the resistor analogy can be employed. It is further shown that it is impossible to incorporate finite cycles into a polymer network such that individual network strands are at equilibrium conformations while maintaining simultaneously a force balance at the junctions. Therefore, the resistor analogy provides only an approximation for the phantom modulus of networks containing finite loops. Improved approaches to the phantom modulus can be constructed from considering a force balance at the junctions, which requires knowledge of the distribution of crosslink fluctuations in imperfect networks. Assuming loops with equilibrium conformations and a force balance at all loop junctions, a lower bound estimate for the phantom modulus G(ph )approximate to (xi  c(f)L(1))kT/V is obtained within the ILGA for endlinked model networks and in the limit of L1 << xi. Here, L, is the number of primary ("pending") loops, xi is the cycle rank of the network, k is the Boltzmann constant, V is the volume of the sample, and T is the absolute temperature. c(f) is a functionalitydependent coefficient, that is, approximate to 2.56 for junction functionality f = 3 and approximate to 3.06 for f = 4, while it converges quickly toward approximate to 4.2 in the limit of large f. Further corrections to the phantom modulus beyond finite loops are addressed briefly.
