Vincent Labiausse, Reinhard Höhler, Sylvie Cohen-Addad

101

102

103

0.001 0.01 0.1 1

Strain amplitude ε0

Shear modulus (Pa)

G'

G''

Visco-elastic behaviour of aqueous foams

'()CGdγφφφ∝−

* Princen, Kiss 1986; Mason, Bibette, Weitz 1995; Saint-Jalmes, Durian 1999

*'''GGiG=+

Princen’s law *:

Since foams can undergo large elastic strains, their behaviour must present significant non-linear effects,

like for instance rubber. How can we study these effects which have been predicted but never

measured ?

Complex shear modulus: solid

liqu

id

plas

tic N1 = 11 - 22

N2 = 22 - 33

Elastic normal stresses differences N1 and N2

Stationary flowWeissenberg effect:

1

3

2 22

12

11

33

Definition

Elastic regimePoynting effect:

Valid for any elastic isotropic material

2112NGεε=;

Foam is described as an ensemble of independent films. Initially, the films are randomly oriented. The deformation of the material is affine (no rearrangements).

()167GBBσ−=−

left Cauchy –Green tensor:

21010001TBFFεεε⎛⎞+⎜⎟==⎜⎟⎜⎟⎝⎠

shear

Höhler, Cohen-Addad, Labiausse, J.Rheol. 2004* Doi and Ohta 1991

• Measuring N1 in aqueous foams is difficult because of uncontrolled trapped stresses superpose to applied stress : there are no data in the literature.

• A constitutive law of Mooney-Rivlin type, rigorously developed starting from the physical ideas of the model of Doi and Otha:

Do foams, which are visco-elastic and plastic, obey the Poynting law ?

-0.01

0

0.01

0.02

0.03

-0.1 -0.05 0 0.05 0.1

N1

/ G

Strain

Examples:

Without trapped stresses

With trapped stresses

• Effect of trapped stresses:

The first normal stress difference induced by oscillatory shear

()*2101022,()TitNNtedtTωωε=∫

()()10200**22,,NPGωεεωε≡

Effect of randomly oriented trapped stresses on P:

For elastic material,Poynting law: P = 1

Visco-elastic generalisation for a nonlinear Maxwell liquid,if >>1: P = 1

Time

StrainNormal stress

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

P

Normalised strain energy

Development of a new rheometer optimised for measuring N1

Cone and plate geometry:

β

Fz R

122ZFNRπ=

Stress heterogeneityfor β = 15°, 7%

Normal stress sensitivity (with equal surface 1dm²)Commercial Bohlin rheometer (CVOR150): 0.1 PaOur optimised rheometer: 0.001 Pa

R = 6 cm D. Hautemayou

Sample characteristics

• No coalescence• Negligible drainage

Stability:

• Mean bubble diameter <d>• Coarsening rate

Controlled variation of the parameters:

Foam types <d> (µm)

(µm/min)

AOK- N2 - C6F14

Gas: nitrogen + perfluorohexan

47 0.4

AOK- N2

Gas: nitrogen

156 4.6

·<d>Dry foams = 97%

Foaming solution: Sodium -olefine Sulfonate + PEO + Dodecanol

·<d>

0.1

1

10

0.01 0.1 1Strain amplitude ε

0

P

Results and discussion

Good agreement with the generalised Poynting law (ε0 0.1)

= 97%

10°

15°Cone angle

Significant deviations at low amplitudes (ε0 < 0.1) with the 10° cone (trapped stresses stronger than with 15°)

AOK-N2

Coarsening rate10

0.1

1

10

0.01 0.1 1Strain amplitude ε

0

P

Coarsening releases part of the stresses trapped due to the strain history. => more isotropic structure

AOK-N2-C6F14

Conclusions

• We propose a non-linear viscoelastic constitutive model predicting the first normal stress difference N1, based on a physical description of foams.

• We have carried out the first experimental study of N1 for aqueous foams.

• When the effects of trapped stresses are minimised, our results agree with the model.

Introduction

This work was presented at the 5th European Conference in foam, emulsions and applications, Champs-sur-Marne, France, July 2004.

Shear-induced normal stress differences in aqueous foamsShear-induced normal stress differences in aqueous foams