Heterogeneity Effects in Highly Cross-Linked Polymer Networks

Despite their level of refinement, micro-mechanical, stretch-based and invariant-based models, still fail to capture and describe all aspects of the mechanical properties of polymer networks for which they were developed. This is for an important part caused by the way the microscopic inhomogeneities are treated. The Elastic Network Model (ENM) approach of reintroducing the spatial resolution by considering the network at the level of its topological constraints, is able to predict the macroscopic properties of polymer networks up to the point of failure. We here demonstrate the ability of ENM to highlight the effects of topology and structure on the mechanical properties of polymer networks for which the heterogeneity is characterised by spatial and topological order parameters. We quantify the macro- and microscopic effects on forces and stress caused by introducing and increasing the heterogeneity of the network. We find that significant differences in the mechanical responses arise between networks with a similar topology but different spatial structure at the time of the reticulation, whereas the dispersion of the cross-link valency has a negligible impact.


Consider all connections between the 4 slots and all monomers exchanges
Final loop is reach ?

New network configuration
No Write outputs, process done Yes P (n): probability that a network chain contains n monomers P (r|n): probability that a chain of length n has a end-to-end distance r n chain : number of monomers per chains A.6 B.6 C.6 D.6 E.6 Figure S7. Histogram of the chain orientations at λ = 1 (after mechanical relaxation) for systems with mean node connectivity equal to 6. θ is the tilt angle of the chains with respect to axis x. For systems C.6 to E.6, the distribution of chain orientation is almost isotropic.  Figure S8. Correlations between the centrality and the local node density for systems with mean connectivity 6. The error bar is the standard error of the mean. For C.6 (regular structure, random topology), no clear correlation. For D.6 (random structure and topology) the centrality of the nodes is on average positively correlated with the local node density.  Figure S9. Correlations between the centrality and the local node density for systems with mean connectivity 8. The error bar is the standard error of the mean. The same remarks hold as for connectivity 6. For C.8 (regular structure, random topology), no clear correlation. For D.8 (random structure and topology) the centrality of the nodes is on average positively correlated with the local node density.

Estimation of the maximal extension for regular lattices of polydisperse chains
We make the following approximations or assumptions for estimating the maximal extension in our periodic boxes: The maximal extension is the length of the shortest path (in number of monomers) • connecting any node to its first image in the direction of extension. This means that we do not consider that the possibility of direct self-connection beyond the first periodic image could be more restrictive.
The shortest path is one of the percolating paths having the smallest number of • chains. This means that we do not consider the possibility that the shortest path would have more chains but fewer monomers in total. The probability p(l) for a path to have a length l is independent of the probability • of length of the other paths. l will also be considered as a continuous variable.
Let us call n the smallest number of chains required to percolate across the box in the direction of extension, and N the number of such percolating paths having n chains. l 0 will be the average length of the chains and σ their standard deviation.
The total length of a path of n chains is a sum of n normal variates. Its distribution is 1 p(l) = 2 n π σ 2 −1/2 exp − 1 2 n l − n l 0 σ 2 (1) The probability for a path to have a length no shorter than L is with erfc the complementary error function. The probability that, among the N shortest path candidates, none is shorter than L is (assuming independent probabilities) So finally, the probability that the shortest path has length L is The most probable value of l min is such that dp(l min =L) dL = 0. So we have to solve This equation is solved numerically. The size and geometry of System B.6 is such that n = 12 and N = 144, l 0 = 27 (Kuhn units) and σ = 5. We find l min ≈ 281, which is about 13 % shorter than in System A.6 (l min = 324).
In this case, l min ≈ 333, which is about 37 % smaller than in System A.8 (l min = 528).