1. Synchotron image of chocolate mousse by Graner and Cloetens (Grenoble) Modelling the Response of Aqueous Foams (and emulsions?) Structured Fluids and Transitions, DCF10 Simon Cox

2. Ingredients: - Elasticity - Plasticity - Viscosity/Dissipation Recipes: - Microstructure calculations - Continuum modelling Shear start-up Parameters: Bubble size, liquid fraction, surface tension, disorder, …

3. Quasi-static 3D shear start-up Kraynik & Reinelt Surface Evolver

4. Disordered but not disorganized In equilibrium, soap films minimize their free energy = surface area. Implications: Each film has constant mean curvature; Three (and only three) films meet, at 120°, in a Plateau border; Plateau borders always meet symmetrically in fours. Liquid fraction Φ l measures size of PBs. Bubbles go from polyhedra (dry) to spheres (wet).

5. Static Shear Modulus, G γ = surface tension R = sauter mean radius (see Princen, Derjaguin, Stamenovic, …) G decreases slightly with increasing disorder. μ 2 (n) = second moment of distribution of the number of sides of each bubble. In the dry limit (Φ l → 0): G decreases strongly with increasing liquid fraction.

6. The Elastic Limit is reached when neighbour- switching topological (T 1 ) changes occur (plasticity): 3D 2D (Hele-Shaw) Instability occurs “pre-emptively”: the triangular face becomes unstable before it reaches zero area

7. In the dry limit (Φ l → 0) the yield stress decreases with bubble size: and decreases strongly with increasing liquid fraction. Above its yield stress, a foam behaves as a shear thinning fluid, suggesting a Herschel Bulkley model. The exponent is generally between 0.5 and 1. The precise dynamics are affected by: Drag on bounding surfaces Viscous flow in PBs Surfactant transport Film stretching and creation …

8. Liquid fraction Shear modulus Yield stress Hutzler, Weaire and Bolton 2D simulations (PLAT)

9. Bubble-scale models Surface Evolver (Brakke): quasi-static, dry (pressure) Potts model (e.g. Glazier, Graner): quasi-static, dry (many bubbles) Lattice Gas (e.g. Sun, Hutzler): wet Vertex Model (Kawasaki, Cantat): dry, dissipation at vertices Bubble Model (Durian): wet, viscous dissipation Viscous Froth Model: 2D dry, external dissipation (uses Surface Evolver)

10. Improved continuum models Phenomenological model (continuum counterpart of viscous froth model) of Weaire et al. for 2D flow: - balance gradient of shear stress with external viscous drag: - assume Bingham-like form for τ: where γ y is yield strain. Predicts velocity profiles for simple shearing flow:

11. Need local (tensorial) measures of strain & plasticity Strain: Graner et al. use the link l joining neighbouring bubble centres to define a texture tensor: and strain Plasticity: During a (2D) T 1 event, a link is destroyed and another created; this defines a rearrangement tensor: Allows quantitative comparison between simulation and experiment, and links T 1 s (local) with yielding (global).

12. Outlook Main thrust of current research: Including dissipative terms in continuum and microstructure models. Relating local description of foam state to macroscopic response. Application to Transient velocity profiles during shearing Slow oscillatory shear 2D Surface Evolver simulation

13. Outlook Main thrust of current research: Including dissipative terms in continuum and microstructure models. Relating local description of foam state to macroscopic response. Application to Transient velocity profiles during shearing (localisation) Microfluidic foam devices Include boundaries, particles 2D Viscous Froth Model Slow (v=0.8) Fast (v=1.2)

14. Outlook 2D Viscous Froth Model Slow (v=0.8) Fast (v=1.2) Main thrust of current research: Including dissipative terms in continuum and microstructure models. Relating local description of foam state to macroscopic response. Application to Transient velocity profiles during shearing (localisation) Microfluidic foam devices Include boundaries, particles