Transcript Shear-induced normal stress differences in aqueous foams

Shear-induced normal stress differences in aqueous foams
Vincent
Labiausse, Reinhard Höhler, Sylvie Cohen-Addad
Visco-elastic behaviour of aqueous foams
Elastic normal stresses differences N1 and N2
Introduction
2
Princen’s law *:
G  G ' i G ''
G '   (  C )

d
10
solid
10
N1 = 11 - 22
N2 = 22 - 33
G'
2
G ''
11
1
33
3
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 ?
22
12
3
liquid
Complex shear modulus:
*
Shear modulus (Pa)
 Definition
 Stationary flow
Weissenberg effect:
1
10
0.001
0.01
0.1
1
Strain am plitude 
 Elastic regime
Poynting effect:
0
N1  12 
G 2
Valid for any elastic isotropic material
* Princen, Kiss 1986; Mason, Bibette, Weitz 1995; Saint-Jalmes, Durian 1999
Do foams, which are visco-elastic and plastic, obey the Poynting law ?
The first normal stress difference induced by oscillatory shear
Strain
Normal stress
• Measuring N1 in aqueous foams is difficult because of uncontrolled trapped
stresses superpose to applied stress : there are no data in the literature.
T
2
N  2 ,  0    N1 (t ) e2 i  t dt
T0
*
1
Tim e
• Effect of trapped stresses:
• A constitutive law of Mooney-Rivlin type, rigorously developed starting
from the physical ideas of the model of Doi and Otha:
With
trapped stresses
N /G
0.0 2

1
G

B6B
7

left Cauchy –Green tensor: B  F F
T
P
0.0 1
1
 Foam is described as an ensemble of
independent films.
 Initially, the films are randomly oriented.
 The deformation of the material is affine
(no rearrangements).
-0.0 1
-0.1
-0.05
0
0.05
1.4
 G  ,  0 
2
0
*
1.2
Without
trapped stresses
0
2 N1*  2  ,  0 
For elastic material,
Poynting law: P = 1
0.1
P
Examples:
1   2


 
shear 
 0
Effect of randomly oriented
trapped stresses on P:
0.0 3
1
Strain
 0

1 0
0 1 
0.8
Visco-elastic generalisation for
a nonlinear Maxwell liquid,
if t >>1: P = 1
0.6
0
* Doi and Ohta 1991
0.2
0.4
0.6
0.8
1
N ormalised strain energy
Höhler, Cohen-Addad, Labiausse, J.Rheol. 2004
Sample characteristics
Results and discussion
Foaming solution: Sodium a-olefine Sulfonate + PEO + Dodecanol
10
10
 = 97%
AOK-N2
Stability:
AOK-N2-C6F14
• No coalescence
• Negligible drainage
1
1
P
Controlled variation of the parameters:
P
• Mean bubble diameter <d>
·
• Coarsening rate <d>
Cone angle
Dry foams  = 97%
Foam types
0.1
0.01
·
<d>
(µm)
<d>
(µm/min)
AOK- N2 - C6F14
Gas: nitrogen + perfluorohexan
47
0.4
AOK- N2
Gas: nitrogen
156
4.6
10°
15°
0.1
Strain amplitude 
0.1
0.01
1
0
0.1
Strain am plitude 
Coarsening rate
1
0
10
Good agreement with the generalised Poynting law (0  0.1)
Significant deviations at low amplitudes (0 < 0.1) with the 10° cone
(trapped stresses stronger than with 15°)
Development of a new rheometer optimised for
measuring N1
Coarsening releases part of the stresses trapped due to the strain history.
=> more isotropic structure
 Cone and plate geometry:
Conclusions
• We
propose a non-linear viscoelastic constitutive model predicting the first normal stress
difference N1, based on a physical description of foams.
Stress heterogeneity
for  = 15°,   7%
2 FZ
N1 
 R2
R = 6 cm D. Hautemayou
• 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.
 Normal stress sensitivity (with equal surface  1dm²)
Commercial Bohlin rheometer (CVOR150):  0.1 Pa
Our optimised rheometer:  0.001 Pa
This work was presented at the 5th European Conference in foam, emulsions and applications, Champs-sur-Marne,
France, July 2004.