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Understanding Rigidity in Colloidal Gels

Glasses and gels are the two paradigmatic examples of dynamically arrested, disordered solids. In contrast to glasses, gels in attractive colloids can form at relatively low densities [1, 2]. Quoting from [1]: “Gels are amorphous materials composed of a liquid phase and an interconnected network like solid phase, which can be formed at very low solid volume fractions. They are widespread, both in nature and in industry – from the pharmaceutical industry (agar gel) to the construction sector (cement hydrate or aluminosilicate gels in cement) – or in everyday life: the bread gets its rubbery texture from the formation of an elastic network of gluten protein, and the formation of protein networks is vital for the successful production of cheese. All soft matter, in fact, from proteins to colloids and polymers easily self-assemble into such weakly elastic solids. They generally present a heterogeneous structure which is the result of the coupling between the kinetics of aggregation, phase separation or demixing, and arrest of the microscopic dynamics”. Unlike crystalline solids, where elasticity emerges as a consequence of broken symmetry, the origin of rigidity in amorphous solids is far from clear. Rigidity of gels, which occur at low densities is even harder to understand than the rigidity of glasses, which occur at high densities. Low-density gels are characterized by sparse networks of colloidal particles, which are sustained through attractive interactions. This geometric percolation is certainly necessary for gelation but it cannot be a sufficient condition for the emergence of mechanical rigidity. For covalently bonded networks, such as polymer gels, the idea that not all parts of a connected network are necessarily rigid is well established [3]. In contrast, discussion of colloidal gels have largely focussed on geometric percolation, with the assumption that all persistent clusters are rigid. The relation between mechanical rigidity and the connectivity of the gel network has received less attention. The distinctive feature of a solid is its resistance to shear. The challenge faced in defining and computing the shear modulus of an amorphous solid was laid out succinctly in a seminal paper on rubber elasticity by Deam and Edwards [4]. They presented an elegant statistical mechanics approach that relied on the concept of conserved topologies to compute differences in free energies between sheared and unsheared states. An alternative to this approach, that