Florence Rouyer, Sylvie Cohen-Addad, Reinhard Höhler

1. Florence Rouyer, Sylvie Cohen-Addad, Reinhard Höhler r + dr r ds g Small amplitude Large amplitude w = 2p f 21mm 25mm Stress Strain Time (s) Time (s) G'0 G''0 G' G'' Shear Modulus (Pa) Gillette 92% z hf Stress amplitude (Pa) a G0 sy G'/ G0 G'' / G0 1 0.1 0.1 1 10 g/gy Controlled stressstrain gy  sy / G'0  foams St Jalmes, Durian (1999) *  Khan (1988)  Gillette *  Foam1 *  Gopal, Durian (1999)  Kraynik et al. (2000)  Durian (1995) emulsions Princen, Kiss (1986 &1989)  Mason et al. (1995 & 1996) *  Mason et al. (1995 & 1996) * * * oscillatory measurements sy f /G'0 f F  gillette 30 min 1 Hz  gillette 30 min .3 Hz 0.92  gillette 60 min 1 Hz  foam1 1 Hz  foam1 .3 Hz 0.97 f Is the yield stress of aqueous foam a well defined quantity ? Aqueous foams have complex rheological properties. They show elastic behavior when subjected to stresses below a yield stress and they flow as non-Newtonian fluids for stresses beyond this characteristic value. The yielding behavior may depend on the flow history of the sample and the flow geometry of the experiment, possibly involving strainlocalization effects. These phenomena as well as the difficulty to characterize fully the relevant structural and physicochemical properties of a foam may explain the discrepancy between experimentally obtained values of the yield stress that can be found in the literature. Influence of flow geometry Localization Strain observation of the free sample surface in a Couette cell Large oscillatory strains in a cylindrical Couette cell Inclined Plane  g = .5  g = .7  g = .5  g = .7 Gillette 81% s(r)/smax 1/r2 (mm-2) hf Dh 5 mm smax g(r) r (mm) 45° 45° s : curvilinear coordinate g (r)ds /dr – s/r s (r) 1/r2 z g = .9 zmax Dh s < sy • Strain localization is observed around g = 0.6  0.1, well beyond yielding. • Non-monotonous strain variation is a precursor of shear-banding. • Localization is not located near the moving wall. • Similar to observations in emulsions (Mason et al., 96). 0 Shear-banding s > sy sxz= (1- f)rg sina (zmax-z) sy (Di Federico, 1999) a x (Coussot et al, 1996) Comparison with the elasto-plastic model * sy = (1- f)rg sinaDh = (1- f)rg sina hf g0 sy gy= sy /G0 g (t) s(t) * Weaire & Hutzler, 1999 time  Inclined Plane  Oscillatory strain (f =1Hz) • Yield stress data do not depend on the geometry • Inclined plane set-up is not well adapted to the study of dry foam. sy Normalized Shear Modulus time Gas volume fraction f Influence of physico-chemical properties Effects of controlled stress orstrainand frequency Foam characteristics Gillette 92%, 0.3 Hz Gillette: Commercial shaving cream Bubble Radius: R = 10 µm Surface tension : T = 30 mN/m G'/ G'0 G'' / G'0 G'' / G'0 & G'/ G'0 G'' / G'0 & G'/ G'0 Normalized Shear Modulus Foam 1: Solution :AOK, PEO, LOH Gas : N2 + perfluorohexan produced in a porous tube Bubble Radius: R = 23 µm Surface tension : T = 27 mN/m Yield Strain g/gy s/sy Elasto-plastic model g/gy Foam 2: Solution :TTAB, glycerol, LOH Gas : N2 + perfluorohexan produced by whipping Bubble Radius: R = 18 µm Surface tension : T= 37 mN/m Frequency (Hz) G'' / G'0 & G'/ G'0 G'' / G'0 & G'/ G'0 • Controlled stress or strain experiments give similar results. • gy  sy / G'0 Comparison with the literature s/sy g/gy experiments • The complex shear modulus is well described by the elasto-plastic model, all the more if the foam is dry and weakly dissipative. Stress Residual Simu- lations ----- SGR (Sollich 98) ___ Elasto-plastic Localization onset Localization onset • Plotting sy /G'0 allows to avoid uncertainties due to T and R determination. • .With this scaling, our results agree with previous ones (St Jalmes, Durian 1999, Mason et al. 1995 & 1996). • Low frequency oscillatory measurements of sy/G'0 give values different from that of steady shear measurements. Possible effect of localization in steady shear ? G'' / G'0 & G'/ G'0 r residual To Conclude Yielding measurement do not depend on geometry and controlled parameter (stress or strain), in the quasi-static regime . Shear modulus behavior of non-dissipative and dry foam is well described by the elasto-plastic model, but this model does not capture all the anharmonicity of the response. In low frequency oscillatory experiments, where localization effects can be excluded, yield strain is equal to yield stress divided by elastic shear modulus measured at small amplitude. The yield strain seems to be well defined for foams and emulsions, and varies as (f-fc)/ f. g/gy g/gy • Elasto-plastic residual overestimates experimental values. • SGR model residual agrees with experimental values, but experimental values of G' and G'' are not well described by this model. This work was presented at the 5th European Conference in foam, emulsions and applications, Champs-sur-Marne, France, July 2004.