Results and discussion

1. s T Shear Three types of 2D model systems • Periodic boundary conditions • µ2 = second central moment of the distribution of bubble coordination numbers, a measure of disorder • At the end of sample preparation, shear and normal stresses are relaxed Voronoi Relaxed Minimize energy using the Surface Evolver 50 bubbles m2 =1.5 Voronoi Sheared Relaxed 50 bubbles m2 =0.88 Voronoi Coarsened Relaxed Coarsening 50 bubbles m2 =1.12 100 bubbles D t The Rheology of Coarsening 2d Dry Foams: A Numerical Simulation Study Sébastien Vincent-Bonnieu, Reinhard Höhler, Sylvie Cohen-Addad Recent experiments have shown that 3D coarsening aqueous foams respond to a small constant shear stress by a creep deformation that linearly increases with time [1]. Moreover, in situ light scattering studies have shown that this phenomenon scales linearly with the rate of coarsening induced intermittent local bubble rearrangements. These findings are in qualitative agreement with 2D numerical foam simulations by Kermode et al [2] which demonstrate that coarsening leads to a viscoelastic relaxation. However, this pioneering work only has proved the existence of this effect, and a systematic quantitative analysis such as the one presented here has not been published to our knowledge. [1] Cohen-Addad S., Höhler R., Khidas Y. Interplay between interfacial rheology, bubble rearrangements and the slow macroscopic rheological response of aqueous foam. Abstract to EUFOAM 2004. [2] Weaire D., Hutzler S.: The Physics of Foams, Oxford University Press, New York 1999. Introduction Creepin 3D foam: Experimental results* Coarsening provokes topological changes Experiment: Strain response e(t) to an imposed stress. stress s 0 Origin of coarsening: This coarsening induces rearrangements : Yield stress Differences in Laplace pressure drive gas transfer between bubbles. Time T1 event Steady creep T2 event T2 & T1 events or Light scattering data suggest that linear viscoelastic creep is the consequence of coarsening induced bubble rearrangements*. Aim of this 2D simulation study: Clarify the creep mechanism on the bubble scale. Can these rearrangements explain macroscopic creep ? * Cohen-Addad, Höhler, Khidas, Phys. Rev. Lett. in press Samples and simulation technique Results and discussion The strain-jump scales linearly with stress on the average. Yield stress stress s 0 Voronoi Relaxed Average over 10 samples s = 0.15 • Strain-jumps de correspond to T1 Shear strain de Strain-jump 1 sample Average over 10 samples Strain evolution between jumps time s = 0.12 Strain evolution between successive rearrangements does not contribute to macroscopic creep. The onset of creep depends on the initial foam structure. Voronoi Relaxed Full agreement with experimental data (Cohen-Addad et al, abstract to EUFOAM 2004) Conclusions Numerical simulation creep using the surface evolver Mesoscopic model of the steady creep • Find structure of minimal energy • Make T1 or T2 rearrangements if necessary • Transfer gas between bubbles during time Dt • We have studied creep in 2D disordered coarsening dry foams using the Surface Evolver software. This work shows that: • The mechanism of steady creep on the bubble scale is strain relaxation via coarsening induced T1 rearrangements. • Using the simulation results, we have validated a schematic mesoscopic model of creep based on continuum mechanics. • Our findings are in good agreement with recent experimental results. • Hypothesis: upon a rearrangement in an area fraction f , foam locally and temporarily looses its elasticity. • Under constant stress s, this leads to an increase of macroscopic strain de, as if the macroscopic shear modulus G were reduced: Adjust strain to maintain an imposed stress Gas diffusion and numerical convergence occur simultaneously. The diffusion constant must be chosen small enough to ensure quasistatic conditions. • Since we observe in the simulations that most rearrangements • involve 4 bubbles, one would expect f 4 / 50  0.08. • This is in rough agreement with the simulation data: For our simulations diffusion constant = 0.001