Vincent Labiausse, Reinhard Höhler, Sylvie Cohen-Addad
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Transcript of Vincent Labiausse, Reinhard Höhler, Sylvie Cohen-Addad
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