Elastic response of wire frame glasses. I. Two dimensional model

We study the elastic response of concentrated suspensions of rigid wire frame particles to a step strain. These particles are constructed from infinitely thin, rigid rods of length L. We specifically compare straight rod-like particles to bent and branched wire frames. In dense suspensions, the wire frames are frozen in a disordered state by the topological entanglements between their arms. We present a simple, geometric method to find the scaling of the elastic stress with concentration in these glassy systems. We apply this method to a simple 2D model system where a test particle is placed on a plane and constrained by a random distribution of points with number density ν. Two striking differences between wire frame and rod suspensions are found: (1) The linear elasticity per particle for wire frames is very large, scaling like ν²L⁴, whereas for rods, it is much smaller and independent of concentration. (2) Rods always shear thin but wire frames shear harden for concentrations less than ∼K/k_BTL⁴, where K is the bending modulus of the particles. The deformation of wire frames is found to be important even for small strains, with the proportion of deformed particles at a particular strain, γ, being given by (νL²)²γ². Our results agree well with simple numerical calculations for the 2D system.