The Large Amplitude Oscillatory Strain Response of Aqueous Foam: Strain Localization and Full Stress Fourier Spectrum By F. Rouyer, S. Cohen-Addad, R. Höhler, P. Sollich, and S.M. Fielding The European Physical Journal E, 27 (2008) 309-321 Jason Rich McKinley Group Summer Reading Club Yielding, Yield Stresses, and Viscoelastoplasticity July 20, 2009
Foams A concentrated dispersion of a gas in a liquid (or solid) φ 0.95 is fairly typical Examples: Froth from a beer or soda Meringue Hair-styling Mousse Shaving cream Generally unstable, but can be metastable on certain time scales Foam coarsening and liquid drainage Foams stabilized with surfactants Gas volume fraction, surface tension and bubble size govern most properties Larson, Ronald G. Structure and Rheology of Complex Fluids. Oxford University Press, 1999 2
Foam Rheology Rheological studies restricted in time because of stability Typical properties: Highly elastic at low stress/strain (G ~ 5G ) Yield stress/strain ( rapidly with φ) Crossover to G > G above yield strain Wall slip and flow localization (esp. for lubricants like shaving cream) In roughened geometries, flow localization in middle of gap Dimensionless parameter: viscosity Capillary Number = Ca = η a γ σ = s surface tension Tells relative importance of viscous dissipation to elasticity arising from surface tension For typical parameters, viscous dissipation negligible unless γ > 10 s Larson, Ronald G. Structure and Rheology of Complex Fluids. Oxford University Press, 1999 4 1 3
Foam Rheology Models Viscoelastoplastic mechanical model OR does not explain the physical mechanism of foam rheology on the bubble or mesoscopic scale Soft Glassy Rheology (SGR) Model Many soft materials (foams, emulsions, pastes, concentrated suspensions, colloidal glasses) exhibit similar rheology Share features of structural disorder and metastability* *P. Sollich, F. Lequeux, P. Hebraud, M. E. Cates. Phys. Rev. Lett. 78 (1997) 2020. System evolves into energy wells that cannot be overcome thermally disorder and metastability 4
SGR Model - Basics Ω Consider a mesoscopic region A such that: We can define a local strain l that is ~ constant in the region It can be characterized with an average elastic constant, k Bubbles deform elastically Bubbles rearrange and relax l y = yield strain kl = local shear stress before yield max elastic energy Structural disorder is modeled by assuming a stochastic distribution of E (or equivalently, l y ) among the mesoscopic regions E 0 = energy scale (e.g. ~10 4 k B T or 0.5kγ 2 under shear) Note: time-independent After yielding, l is re-zeroed to the position of last yield 5
SGR Model Effective Noise Temperature This leads to the following probability P that a given mesoscopic element has yield energy E and local strain l Elastic deformations Macroscopic shear stress: Effective noise temperature, x 1/τ 0 = attempt frequency Y(t) = total yielding rate x = effective noise temperature Characterizes the interactions between mesoscopic regions that result in activated yield events through remote structural rearrangements x < 1 Yielding events Relaxation to new equilibrium positions (l reset to l = 0) glass phase, yield stress, continuous aging A constitutive equation x = 1 glass transition 1 < x power-law fluid behavior 6
SGR Model Rheological Predictions Linear viscoelasticity: Only strictly possible if x > 1 (no time-dependence or aging) Note that for 1 < x < 2, G /G = const For x 1, average over a few oscillation cycles Steady shear G x = 1 Increasing age G linear plot 7
SGR Model Much More Some references for further study (since I can only scratch the surface here): J. P. Bouchard J. Phys. I (France) 2 (1992) 1705. [An early attempt] P. Sollich, F. Lequeux, P. Hebraud, M. E. Cates. Phys. Rev. Lett. 78 (1997) 2020. [Basics of model] P. Sollich. Phys. Rev. E 58 (1999) 738 [More detail] * S. M. Fielding, P. Sollich, M. E. Cates J. Rheol. 44 (2000) 323. [Lots of detail] M. E. Cates, P. Sollich, J. Rheol. 48 (2004) 193. [extension to tensorial model] A detailed discussion of time translational invariance is found in * Necessary for an exact description of the rheology of aging materials Note that even up to now, physical interpretations of x are speculative at best The activation energy available for kicks that cause yielding These kicks both cause and arise from rearrangements (yielding) So kick energy scale is same as that from rearrangements, so x ~ 1 8
The current paper What did they do? LAOS on foams with increasing strain amplitude Allows transition from solid to liquid (i.e. yielding) to be probed Exploring Lissajous-(Bowditch) plots and full Fourier spectrum Exploring departure from linearity Flow localization study: Is there strain/flow throughout the gap? Ω Comparison to models: Mechanical model and SGR model 9
Samples and Experiments Gillette Shaving Cream AOK (aqueous foam of polymer surfactant) bubble diameters = 28 μm or 36 μm (depending on coarsening time) bubble diameters = 50 μm Experiments done in cylindrical Couette cell with grooved walls to prevent wall slip 0.2 mm Controlled strain rheometer: strain sweep, Γ 0 = 10-3 to 3 ω = 1 Hz or 0.3 Hz Measuring shear stress (torque) M(t) = torque Σ(t) = macroscopic shear stress H = cylinder height Flow localization measurement 10
Results: Lissajous-Bowditch curve Gillette foam, ω = 1 Hz Compare to ideal responses viscoelastic elastoplastic Σ(t) SGR Prediction (x = 1.1) ideal elastoplastic response Γ(t) 11
Results: Flow Localization Flow localization sets in above strain amplitude = Γ 0 = 0.6 ± 0.1 for Gillette sample, ω = 1 Hz, d = 28 μm Expected strain at middle of gap Flow localization is NOT directly from stress heterogeneity Neither the mechanical models nor SGR predict this But only sets in well-above the yield strain (Γ y = 0.15 ± 0.01) So comparison to models should be fine for all but highest strains 12
Results: Strain Dependent Moduli Predominantly linear elastic at small Γ 0 Crossover between G and G at large Γ 0 No distinctive features in moduli at flow localization Flow localization and yielding are separate phenomena Inflection in q upon flow localization 13
Results: Comparison to Models -1 Lack of sharp increase suggests a distribution of yield strains Consistent with mechanical model (solid lines) for Γ 0 >> Γ y But discrepancy for Γ 0 < Γ y 14
Results: Comparison to Models -1 SGR (dashed and dotted lines): from fitting, x = 1.07 for Gillette foam and 1.05 for AOK Fits G very well for all Γ 0 / Γ y Fits G well until Γ 0 / Γ y 2 Fits q well for AOK, but too steep for Gillette foam Possible reason for discrepancies: flow localization But they don t really know 15
Microstructure Connection and Conclusions Detailed microstructural insight, even with models, is elusive Princen* related shear modulus and yield stress to foam liquid fraction, surface tension, and bubble size for ideal 2-D foams. Allows prediction of some model parameters The authors suggest that perhaps x is somehow related to coarsening rate Conclusions Flow localization is distinct from wall slip and stress heterogeneity, and sets in at strain amplitudes well above Γ y Full nonlinear viscoelastic spectrum found Fits to both a mechanical model and SGR model are not perfect, but do fit well in certain strain regimes SGR Model also captures some features of nonlinear harmonics 16 *Princen HM, (1983). J Colloid lnteriface Sci 91:160, or see Larson, Ronald G. Structure and Rheology of Complex Fluids. Oxford University Press, 1999
Thank you for your attention! 17