2019/08/14 by Michael Lang · 53 citations
Chemistry · Materials Science · Physics and Astronomy · #Advanced Polymer Synthesis and Characterization #Composite material #Elasticity (physics) #Materials science #Polymer #Polymer chemistry #Polymer composites and self-healing #Polymer crystallization and properties #Polymer science #cond-mat.soft
paper · pdf · doi:10.1021/acs.macromol.9b00996
published in Macromolecules 52(16), 6266-6273 (American Chemical Society)
openalex publication_date 2019/08/14 · arxiv created 2021/03/30 · arxiv updated 2021/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Based upon the resistor analogy and using the ideal loop gas approximation (ILGA), it is shown that only pending loops 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 cross-link 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 ≈ (ξ – c f L 1 ) kT / V is obtained within the ILGA for end-linked model networks and in the limit of L 1 ≪ ξ. Here, L 1 is the number of primary (“pending”) loops, ξ 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 functionality-dependent coefficient, that is, ≈2.56 for junction functionality f = 3 and ≈3.06 for f = 4, while it converges quickly toward ≈4.2 in the limit of large f . Further corrections to the phantom modulus beyond finite loops are addressed briefly.