LETTERS PUBLISHED ONLINE: 25 SEPTEMBER 2011 | DOI: 10.1038/NMAT3119 A micromechanical model to predict the flow of soft particle glasses Jyoti R. Seth 1 , Lavanya Mohan 1 , Clémentine Locatelli-Champagne 2 , Michel Cloitre 2 * and Roger T. Bonnecaze 1 Soft particle glasses form a broad family of materials made of deformable particles, as diverse as microgels 1 , emulsion droplets 2 , star polymers 3 , block copolymer micelles and proteins 4 , which are jammed at volume fractions where they are in contact and interact via soft elastic repulsions. Despite a great variety of particle elasticity, soft glasses have many generic features in common. They behave like weak elastic solids at rest but flow very much like liquids above the yield stress. This unique feature is exploited to process high-performance coatings, solid inks, ceramic pastes, textured food and personal care products. Much of the understanding of these materials at volume fractions relevant in applications is empirical, and a theory connecting macroscopic flow behaviour to microstructure and particle properties remains a formidable challenge. Here we propose a micromechanical three-dimensional model that quantitatively predicts the nonlinear rheology of soft particle glasses. The shear stress and the normal stress differences depend on both the dynamic pair distribution function and the solvent- mediated EHD interactions among the deformed particles. The predictions, which have no adjustable parameters, are successfully validated with experiments on concentrated emulsions and polyelectrolyte microgel pastes, highlighting the universality of the flow properties of soft glasses. These results provide a framework for designing new soft additives with a desired rheological response. Soft particle glasses share common features with hard sphere glasses such as non-ergodicity and caged dynamics. However, whereas hard sphere colloids experience only forces due to excluded volume, soft particles at high volume fraction are compressed against each other by bulk osmotic forces and form flat facets at contact, with the average deformation depending on particle elasticity and volume fraction. The solvent forming the continuous phase is localized in thin films separating the particles (Supplementary Fig. S1). We model a soft particle glass as a suspension of N non-Brownian, elastic spheres (Fig. 1a), which are dispersed at a volume fraction above the random close-packing of hard spheres (φ>φ c ≈ 0.64), in a solvent with viscosity η S . The particles are slightly polydisperse in size, with an average radius R, which prevents them from crystallizing under flow. We characterize the contact between two particles by the overlap distance h αβ and the relative deformation ε αβ = h αβ /R c , where R c is the contact radius (Fig. 1b). The suspension is subject to an imposed shear flow in the plane (x , y ) with shear rate ˙ γ , particle and fluid inertia being neglected. As two compressed particles move past one another, a flow of solvent develops inside the liquid films separating the facets. This generates a net positive pressure, causing a further elastic 1 Department of Chemical Engineering and Texas Materials Institute, The University of Texas at Austin, Austin, Texas 78712, USA, 2 Matière Molle et Chimie, UMR 7167 CNRS-ESPCI, Ecole Supérieure de Physique et Chimie Industrielles, 10 rue Vauquelin, 75005 Paris, France. *e-mail:michel.cloitre@espci.fr. y x n || n ⊥ u u , || r γ α αβ αβ β β u α a b Figure 1 | Structure and interactions of a model soft glass. a, Typical configuration of jammed elastic spheres at φ = 0.8; ˙ γ is the applied shear rate. b, Schematic showing pair-wise interactions between particles α and β with radii R α and R β centered at x α and x β and translating with velocities u α and u β . r αβ is the centre-to-centre distance. h αβ = R α + R β − r αβ is the overlap distance; the thickness of the lubricating film separating the facets is much smaller that the overlap distance. The deformation of the particles is characterized by ε αβ = h αβ /R c , where R c = R α R β /(R α + R β ) is the contact radius. u αβ,‖ is the component of the relative velocity parallel to the facets. The elastic force f e αβ and the EHD drag force f EHD αβ are parallel to the unit vectors normal (n ⊥ ) and parallel (n ‖ ) to the facets, respectively. deformation of the particles, which self-consistently maintains the lubricating films and makes particle motion possible. This mechanism shares strong similarities with the elastohydrodynamic (EHD) slippage of soft particles compressed against solid surfaces 5 . The interaction between α and β is composed of a central repulsive force f e αβ associated with the elastic contact between the two particles, coupled to an EHD drag force f EHD αβ , due to the motion of α relative to β . The elastic force f e αβ between soft particles such as elastomeric particles 6 , microgels 7,8 , and emulsion droplets 7 can be modelled using Hertzian-like potentials. The classical Hertz theory applies at rest and near equilibrium 6 , where ε αβ less than 0.1. When the glass flows at high shear-rates, ε αβ can be much larger and Hertz theory underestimates the contact force. We used a modified approximate expression 6 , which is valid up to ε αβ ≈ 0.6 : f e αβ = 4/3CE ∗ ε n αβ R 2 c n ⊥ , where E ∗ = E /2(1 −ν 2 ) is the contact modulus (E : Young modulus; ν = 0.5: Poisson’s ratio for incompressible spheres). The values of n and C vary with the degree of compression (Supplementary Fig. S2). It is interesting to note that the elastic energy associated with the elastic contact forces is generally much larger than kT (Methods), indicating that the origin of the dynamics resides in the elastic properties of the particles themselves. The EHD drag 838 NATURE MATERIALS | VOL 10 | NOVEMBER 2011 | www.nature.com/naturematerials © 2011 Macmillan Publishers Limited. All rights reserved