Deformation behaviour of homogeneous and heterogeneous bimodal networks

Transcription

1 Deformation behaviour of homogeneous and heterogeneous bimodal networks Natasha Kamerlin 1,2 and Christer Elvingson 1, 1 Department of Chemistry - Ångström Laboratory, Uppsala University Box 523, S Uppsala, Sweden 2 Department of Mathematics, Uppsala University Box 480, S Uppsala, Sweden Supporting Information Constructing a homogeneous bimodal network Constructing a heterogeneous bimodal network Table S1. Figure S1. Figure S2. Figure S3. Figure S4. Parameters used when generating the heterogeneous networks. Probability distribution for the chain end-to-end distances for the short and long chains in a homogeneous and heterogeneous bimodal network with f s = 0.90, at equilibrium and at failure. Segmental chain orientation as a function of strain for networks with f s = 0.90, for various degrees of clustering. The fraction of ruptured short and long chains for homogeneous and heterogeneous networks with f s = 0.65 and f s = Stress-strain curves for networks with f s = 0.90, for various degrees of clustering. S1

2 Constructing a homogeneous bimodal network Tetravalent cross-linking nodes are inserted at random positions within a cylinder, subject to the constraint that no pair of nodes may be found within a minimum distance of one another. For each node, the indices of the four nearest unsaturated cross-linking nodes are found and stored. During this search, nodes within proximity start to become fully occupied, meaning that some of the connections towards the end of the procedure may end up being between nodes that are far apart. These long connections are eliminated through a process of reconnection.[1] Next, polymer chains are introduced by placing N s = 6 or N l = 30 beads along the straight lines connecting each pair of nodes. In this way, it is possible to avoid any trapped entanglements during network formation. The resulting bimodal network structures are also free from defects, such as loops or dangling chain ends. These starting structures are subsequently equilibrated, prior to imposing a strain. Constructing a heterogeneous bimodal network The method of constructing a bimodal network containing clusters of short chains is similar to the one described above for a homogeneous network. However, there are some important differences. Here, cross-linking nodes are first randomly distributed within a number of (non-overlapping) fictitious spheres that are contained in a cylindrical volume. Since the spheres represent the domains of short chains, we will refer to these nodes as shortchain nodes. Each sphere holds a fixed number of short-chain nodes, n, which has been set to n = 20, unless stated otherwise. Additional crosslinking nodes, which we will refer to as long-chain nodes, are distributed S2

3 in the volume excluding the spheres. Next, each short-chain node is connected to its four nearest unsaturated neighbours using a chain of length N s = 6. Note that a cross-linking node near the centre of a sphere will likely connect to other short-chain nodes within the same spherical domain, whereas a node closer to the border of the sphere might also connect to outside nodes that do not belong to the sphere. This outside node might be a long-chain node. If the volume fraction of fictitious spheres (φ spheres ) is large (i.e., if the average distance between short-chain nodes belonging to different clusters is small), the connection might, however, be to a shortchain node belonging to another cluster, meaning that two clusters form an agglomerate (and the average number of inter-cluster connections, ζ int, increases). The long-chain nodes are now used to interconnect the clusters through chains of length N l = 30, forming a closed network. Conversely, for a sufficiently small φ spheres, this approach could result in the generation of isolated clusters that are not connected to the larger network. The resulting network configurations were visualised to ensure that no isolated clusters are present. The number of clusters (C) in each cylindrical system is adjusted to give a fraction of short chains corresponding to f s =0.65, 0.80, and 0.90, where we have used a larger value of φ spheres at larger f s to promote the formation of agglomerates. Values of specific simulation parameters related to generating the network, as well as the resulting average number of connections between clusters, can be found in Table S1. S3

4 Table S1: Fraction of short chains (f s), number of clusters (C), number of nodes per cluster (n), ratio of the cylinder and sphere diameter (D/d), volume fraction of the spheres (φ spheres ), average number of inter-cluster connections ( ζ int), average number of connections per cluster to long-chain nodes ( ζ ext). f s C n D/d φ spheres ζint ζext S4

5 Homogeneous,short Homogeneous,long Heterogeneous,short Heterogeneous,long 0.2 P(R ee ) R ee Figure S1. Probability distribution for the chain end-to-end distances (scaled with the particle diameter) for the short and long chains in a homogeneous and heterogeneous bimodal network with f s = The solid lines show the equilibrium distribution and the dashed lines show the distribution for the deformed networks at failure. S5

6 n=0 n=20 n=30 n=40 n= P α Figure S2. Segmental chain orientation as a function of strain for homogeneous (n = 0) and heterogeneous networks with f s = 0.90, where the solid lines correspond to the short chains and the dashed lines to the long chains. In order to investigate the effect of the degree of heterogeneity, networks with larger clusters were generated by increasing the number of nodes n within each sphere and simultaneously reducing the volume fraction of the spheres, in order to suppress inter-cluster-connections. S6

7 Homogeneous,f s =0.90 Homogeneous,f s =0.65 Heterogeneous,f s =0.90 Heterogeneous,f s = γ α Figure S3. The fraction of ruptured chains for homogeneous and heterogeneous networks with f s = 0.65 and f s = 0.90, where the short chains correspond to the solid lines and the long chains to the dashed lines. S7

8 n=0 n=20 n=30 n=40 n=50 σ α Figure S4. Stress-strain curves for networks with f s = 0.90, with various degrees of inter-cluster connections (i.e., with the number of nodes per cluster n = 0 50, where n = 0 corresponds to the homogeneous network). References [1] N. Kamerlin and C. Elvingson. Tracer diffusion in a polymer gel: simulations of static and dynamic 3d networks using spherical boundary conditions. J. Phys. Condens. Matter, 28:475101, S8