Filler network model of filled rubber materials to estimate system size dependence of two-dimensional small-angle scattering patterns

We proposed a filler network toy (FN-toy) model in order to approximately forecast changes in two-dimensional scattering patterns (2DSPs) of nanoparticles (NPs) in crosslinked polymer networks in ultrasmall-angle X-ray scattering (USAXS) experiments under uniaxial elongation. It enables us to estimate the system size dependence of the 2DSP of the NPs. In the FN-toy model, we considered NPs connected by harmonic springs with excluded-volume interactions among the NPs. In this study, we used the NP configurations estimated by reverse Monte Carlo (RMC) analysis for USAXS data observed in SPring-8 experiments on filler-filled styrene butadiene rubber (SBR). In the FN-toy model, we set a bond between every pair of NPs whose distance is less than Cd, where d is the diameter of an NP and C is a parameter that characterizes network properties. We determined the optimal value of C by comparison with 2DSPs of the NPs at 200% elongation for end-modified and unmodified SBR. These 2DSPs are obtained from the results of a largescale coarse-grained molecular dynamics (CGMD) simulation with 8,192 NPs and 160 million Lennard-Jones (LJ) particles in previous works. For the end-modified SBR, the fitted value is C = 1.367 and for the unmodified SBR, C = 1.258. The difference in C can be regarded as originating from the difference in polymer-NP interactions. We found that the harmonic potential used in the current FN-toy model is not sufficient to reproduce stress-strain curves and local structures of NPs obtained in the previous CGMD simulations, although the FN-toy model can reproduce the 2DSPs. Using the FN-toy model with the fitted value of C, we calculated the 2DSPs of 65,536 and 524,288 NPs, whose initial positions were estimated by RMC analysis for the same USAXS data. It was found that CGMD simulations with 10 billion LJ particles and 524,288 NPs can provide a high-resolution 2DSP that is comparable to the 2DSP observed in USAXS experiments.