Crossover between entropic and interfacial elasticity and osmotic pressure in uniform disordered emulsions

Abstract

We develop a simple predictive model of the osmotic pressure Π and linear shear elastic modulus G^′_p of uniform disordered emulsions that includes energetic contributions from entropy and interfacial deformation. This model yields a smooth crossover between an entropically dominated G^′_p ∼ k_BT/a³ for droplet volume fractions ϕ below a jamming threshold for spheres, ϕ_c, and an interfacially dominated G^′_p ∼ σ/a for ϕ above ϕ_c, where a and σ are the undeformed radius and interfacial tension, respectively, of a droplet and T is the temperature. We show that this model reduces to the known ϕ-dependent jamming behavior G^′_p(ϕ) ∼ (σ/a)ϕ(ϕ − ϕ_c) as T → 0 for ϕ > ϕ_c of disordered uniform emulsions, and it also produces the known divergence for disordered hard spheres G^′_p(ϕ) ∼ (k_BT/a³)ϕ/(ϕ_c − ϕ) for ϕ < ϕ_c when σ → ∞. We compare predictions of this model to data for disordered uniform microscale emulsion droplets, corrected for electrostatic repulsions. The smooth crossover captures the observed trends in G^′_p and Π below ϕ_c better than existing analytic models of disordered emulsions, which do not make predictions below ϕ_c. Moreover, the model predicts that entropic contributions to the shear modulus can become more significant for nanoemulsions as compared to microscale emulsions.