Structural Changes of Aggregated Filler Particles in Elongated Rubbers through Two-Dimensional Pattern Reverse Monte Carlo Modeling

To investigate the three-dimensional (3D) structure of nanoparticle (NP) aggregates in rubber tire treads with a large system size, a particle-mesh-based two-dimensional (2D) pattern reverse Monte Carlo (PM-2DpRMC) method was applied to a series of 2D scattering patterns (2DSPs) of NPs during stretching and unloading. To characterize the structures of aggregated NPs, a high-connectivity cluster analysis based on the number of NPs within a specific cutoff distance was introduced. Similar to the reverse Monte Carlo (RMC) analysis of a one-dimensional scattering spectrum, PM-2DpRMC analysis for the case before stretching clarified that the characteristic cluster sizes of silica NPs in end-modified styrene–butadiene rubber (M-SBR) treads were smaller than those in nonmodified SBR (n-SBR) treads. Changes in the 3D configurations of NPs during stretching were modeled from a series of 2DSPs of NPs by repeating the stepwise deformation of the box of the periodic boundary condition, and this method was termed the “on-the-fly PM-2DpRMC” method. The high-connectivity cluster analysis elucidated that the NPs in the M-SBR treads aggregated more at higher elongation ratios (ε); however, the characteristic sizes of the NP aggregates in the n-SBR treads were slightly small at higher ε values. At a high ε of >140%, although the number of clusters in the M-SBR treads became close to those in the n-SBR treads, the difference between the characteristic sizes of NP aggregates was retained. Furthermore, the high-connectivity cluster analysis was mathematically more sensitive for the characterization of NP morphologies than a previously reported analysis based on the Voronoi cell volume. The high-connectivity cluster analysis helps clarify the justification of the on-the-fly PM-2DpRMC method and its numerical comparison with the PM-2DpRMC analysis starting from random NP configurations. Moreover, the results of this study elucidate the importance of a lower scattering wavenumber, q, limit in the modeling of NP aggregation.