Elasticity of bidisperse attractive particle systems

Bidisperse particle systems are common in both natural and engineered materials, and it is known to influence packing, flow, and stability. However, their direct effect on elastic properties, particularly in systems with attractive interactions, remains poorly understood. Gaining insight into this relationship is important for designing soft particle-based materials with desired mechanical response. In this work, we study how particle size ratio and composition affect the shear modulus of attractive particle systems. Using coarse-grained molecular simulations, we analyze systems composed of two particle sizes at fixed total packing fraction and find that the shear modulus increases systematically with bidispersity. To explain this behavior, we develop two asymptotic models following limiting cases: one where a percolated network of large particles is stiffened by small particles, and another where a small-particle network is modified by embedded large particles. Both models yield closed-form expressions that capture the qualitative trends observed in simulations, including the dependence of shear modulus on size ratio and relative volume fraction. Our results demonstrate that bidispersity can enhance elastic stiffness through microstructural effects, independently of overall density, offering a simple strategy to design particle-based materials with tunable mechanical properties.