“The Large Amplitude Oscillatory Strain Response of Aqueous Foam ... · “The Large Amplitude...

“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

Jason Rich, McKinley Group Summer Reading Club, 7/20/09

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Larson, Ronald G. Structure and Rheology of Complex Fluids. Oxford University Press, 1999

Foams• A concentrated dispersion of a gas in a liquid (or solid)

– 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 scalesFoam “coarsening” and liquid drainage

Foams stabilized with surfactants

Gas volume fraction, surface tension and bubble size govern most properties

0.95φ ≈


3Larson, Ronald G. Structure and Rheology of Complex Fluids. Oxford University Press, 1999

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 )– Crossover to G″ > G′ above yield strain– Wall slip and flow localization

– In roughened geometries, flow localization in middle of gap• Dimensionless parameter:

– Tells relative importance of viscous dissipation to elasticity arising from surface tension

– For typical parameters, viscous dissipation negligible unless

φ

(esp. for lubricants like shaving cream)

viscosityCapillary Number Casurface tensionsaη γ σ= = =

4 110 sγ −>


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Foam Rheology Models• Viscoelastoplastic mechanical model

• 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*

OR

“…does not explain the physical mechanism of foam rheology on the bubble or mesoscopic scale”

*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


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SGR Model - Basics

• Structural disorder is modeled by assuming a stochastic distribution of E (or equivalently, ly) among the mesoscopic regions

• After yielding, l is re-zeroed to the position of last yield

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

ly = yield strainkl = local shear stress before yieldmax elastic energy

E0 = energy scale (e.g. ~104kBT or 0.5kγ2 under shear)

Note: time-independent


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• This leads to the following probability P that a given mesoscopicelement has yield energy E and local strain l

• Macroscopic shear stress:

1/τ0 = attempt frequencyY(t) = total yielding ratex = effective noise temperature

SGR Model – Effective Noise Temperature

• “Effective noise temperature”, x– Characterizes the interactions between mesoscopic regions that result

in “activated” yield events through remote structural rearrangements

Elastic deformations Yielding events

Relaxation to new equilibrium positions

(l reset to l = 0)A constitutive

equation

x < 1 glass phase, yield stress, continuous aging

x = 1 glass transition

1 < x power-law fluid behavior


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Increasing ageG′

G″

x = 1

SGR Model – Rheological Predictions

Note that for 1 < x < 2, G′/G″ = const

linear plot

• Linear viscoelasticity:– Only strictly possible if x > 1 (no time-dependence or aging)

– For x ≤ 1, average over a few oscillation cycles• Steady shear


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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


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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?

• Comparison to models:– Mechanical model and SGR model

Ω


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Samples and Experiments

• Experiments done in cylindrical Couette cell with grooved walls to prevent wall slip

• Controlled strain rheometer: strain sweep, Γ0 = 10-3 to 3– ω = 1 Hz or 0.3 Hz

• Measuring shear stress (torque)

• Flow localization measurement

Gillette Shaving CreamAOK

(aqueous foam of polymer surfactant)

bubble diameters = 28 μm or 36 μm(depending on coarsening time) bubble diameters = 50 μm

0.2 mm

M(t) = torqueΣ(t) = macroscopic shear stressH = cylinder height


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Results: Lissajous-Bowditch curve• Compare to ideal responses

viscoelastic elastoplastic

Σ(t)

Γ(t)

• ideal elastoplasticresponse

Gillette foam, ω = 1 Hz

SGR Prediction (x = 1.1)


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Results: Flow Localization• Flow localization sets in above strain amplitude = Γ0 = 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 (Γy = 0.15 ± 0.01)

– So comparison to models should be fine for all but highest strains


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Results: Strain Dependent Moduli

• Predominantly linear elastic at small Γ0

• Crossover between G′ and G″ at large Γ0

• No distinctive features in moduli at flow localization– Flow localization and yielding are

separate phenomena– Inflection in q upon flow localization


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Results: Comparison to Models

• Consistent with mechanical model (solid lines) for Γ0 >> Γy

• But discrepancy for Γ0 < Γy

-1

Lack of sharp increase

suggests a distribution of yield strains


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Results: Comparison to Models

• SGR (dashed and dotted lines): from fitting, x = 1.07 for Gillette foam and 1.05 for AOK

• Fits G″ very well for all Γ0 / Γy

• Fits G′ well until Γ0 / Γy ≈ 2• Fits q well for AOK, but too steep for Gillette foam• Possible reason for discrepancies: flow

localization– But they don’t really know

-1


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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 Γy

– 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

*Princen HM, (1983). J Colloid lnteriface Sci 91:160, or see Larson, Ronald G. Structure and Rheology of Complex Fluids. Oxford University Press, 1999


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