shear strain

0

0.2

0.4

0.6

Results and discussionSamples and simulation technique

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.

Creep in 3D foam: Experimental results*

0

0.004

0.008

0 100 200 300 400

J(t) =

ε(t)/

σ (Pa

-1 )

( )Time s

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.

Experiment: Strain response ε(t) to an imposed stress.

σ 0

Steady creep

Yield stress

Time

* Cohen-Addad, Höhler, Khidas, Phys. Rev. Lett. in press

stress

Can these rearrangements explain macroscopic creep ?

Origin of coarsening:

Differences in Laplace pressure drive gas transfer between bubbles.

T2 & T1 events

T2 event

T1 event

This coarsening induces rearrangements :

or

Coarsening provokes topological changes

Numerical simulation creep using the surface evolver

0.045

0.05

0.055

0.06

0 0.002 0.004 0.006

Creep recovery

Diffusion constant (A.U.)

Quasistatic Regime

For our simulations diffusion constant = 0.001

• Find structure of minimal energy• Make T1 or T2 rearrangements if necessary• Transfer gas between bubbles during time t

Adjust strain to maintain an imposed stress

t

σT

Gas diffusion and numerical convergence occur simultaneously.The diffusion constant must be chosen small enough to ensurequasistatic conditions.

Strain-jumps correspond to T1

time

Sh

ear

stra

in

Yield stress

0

0.1

0.2

0.3

0.4

0 1000 2000 3000

Averages over 10 samples :

shear strain

time

Voronoi RelaxedVoronoi Sheared RelaxedVoronoi Coarsened Relaxed

The onset of creep depends on

the initial foam structure.

-0.04

-0.02

0

0.02

0.04

-0.4 -0.2 0 0.2 0.4

<δε>

stress

()rate of T1 eventsεδε×∝&

δεσ∝

Strain evolution between jumps

Strain-jump

Strain evolution between successive rearrangements does not contribute to macroscopic creep.

Full agreement with experimental data (Cohen-Addad et al, abstract to EUFOAM 2004)

The strain-jump scales linearly with stresson the average.

Average over 10 samples

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.

Conclusions

Mesoscopic model of the steady creep

•Hypothesis: upon a rearrangement in an area fraction foam locally and temporarily looses its elasticity.

•Under constant stress , this leads to an increase of macroscopic strain , as if the macroscopic shear modulus G were reduced:

GGδφ≅−

Gσε= GGGσδδε=− Gσδεφ=

•Since we observe in the simulations that most rearrangementsinvolve 4 bubbles, one would expect 4 / 50 0.08.

•This is in rough agreement with the simulation data: 0.1Gδεφσ==

0.5G= 0.05φ=

• 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

100 bubbles

Three types of 2D model systems

Voronoi Coarsened

Relaxed

Minimize energy using the Surface

Evolver

Coarsening

Shear Voronoi Sheared Relaxed

50 bubbles =0.88

50 bubbles =1.12

Voronoi Relaxed

50 bubbles =1.5

Introduction

Voronoi Relaxed

0

0.12

0.15

Voronoi Relaxed

σ

stre

ss

1 sampleAverage over 10 samples

The Rheology of Coarsening 2d Dry Foams: A Numerical Simulation Study

The Rheology of Coarsening 2d Dry Foams: A Numerical Simulation Study