Colloidal depletion gel elasticity arises from the packing of locally glassy clusters

In colloidal gels, attractive interactions among suspended particles drive a thermodynamic instability that promotes aggregation, arresting in a space spanning network structure possessing elasticity and a yield stress. As the attractive strength between particles increases, the gel elastic modulus increases. Surprisingly, this change cannot be accounted for by the immediate increase in bond stiffness between particles and clusters derived from the depletion interaction energy. Instead, the quantitative agreement between integrated experimental, computational, and graph theoretic approaches are used to understand the arrested state and the origins of the gel elastic response. The micro-structural source of elasticity is identified by the l-balanced graph partition of the gels into minimally interconnected clusters that act as rigid, load bearing units. The number density of cluster-cluster connections grows with increasing attraction, and explains the emergence of elasticity in the network through the classic Cauchy-Born theory. Clusters are amorphous and iso-static; the internal cluster particle density maps onto the known attractive glass line of sticky colloids at low attraction strengths and extends this line to higher strengths and lower particle volume fractions.