Florence Rouyer, Sylvie Cohen-Addad, Reinhard Höhler

Aqueous foams have complex rheological properties. They show elastic behavior when subjected to stresses below a yield stress and they flow as non-Newtonian fluids for stresses beyond this characteristic value. The yielding behavior may depend on the flow history of the sample and the flow geometry of the experiment, possibly involving strain localization effects. These phenomena as well as the difficulty to characterize fully the relevant structural and physicochemical properties of a foam may explain the discrepancy between experimentally obtained values of the yield stress that can be found in the literature.

Influence of flow geometry

Large oscillatory strains in a cylindrical Couette cell Inclined Plane

y = (1- )g sin h = (1- )g sin hf

hf

z

h

45° 45°

hf

Gillette 81%

(Coussot et al, 1996)

z

y

xz(1- )g sin (zmax-z)

y

y

gh

x(Di Federico, 1999)

Inclined Plane Oscillatory strain

0

50

100

0.7 0.8 0.9 1

y

Gas volume fraction

(f =1Hz)

f21mm

25mm 5 mm

0.0015

0.002

0.0025

0 0.5 1 1.5

Localization

r + rr

s

r  s /r – s/r

r  1/r2

s(r)

/sm

ax

1/r2

(mm

-2)

r (mm) (r)

= .5 = .7

= .5 = .7

0

0.5

1

21 22 23 24 25

= .9

• Strain localization is observed around = 0.6 0.1, well beyond yielding.

• Non-monotonous strain variation is a precursor of shear-banding.

• Localization is not located near the moving wall.

• Similar to observations in emulsions (Mason et al., 96).

•Yield stress data do not depend on the geometry

• Inclined plane set-up is not well adapted to the study of dry foam.

Effects of controlled stress or strain and frequency

Comparison with the literature

• Plotting y /G'0 allows to avoid uncertainties due to T and R determination.

•.With this scaling, our results agree with previous ones (St Jalmes, Durian 1999, Mason et al. 1995 & 1996).

• Low frequency oscillatory measurements of y/G'0 give values different from that of steady shear

measurements. Possible effect of localization in steady shear ?

0.1

1

0.01 0.1 1 10

G'/G'0 and G"/G'0

gamma*G'0/sigmaY

0.05

0.15

0.25

0.1 1 10

gammaY

frequence

G'/ G'0

G'' / G'0

y Frequency (Hz)

Controlled stress strain

y

y / G'0

• Controlled stress or strain experiments give similar results.

• y y / G'0

zmax

0

Elasto-plastic model

Gillette 92%, 0.3 Hz

Yie

ld S

trai

n

Nor

mal

ized

She

ar M

odul

us

s : curvilinear coordinate

smax

Shear-banding

1

10

100

1 10 100y

G'

G''

G'0

G''0

Stress amplitude (Pa)

Gillette 92%

Shea

r M

odul

us (

Pa)

-0.25

0

0.25

0 1 2 3 4

time (s)

-60

0

60

0 1 2 3 4

(Pa)

time (s)

Stre

ss

S

trai

n

Time (s)

-100

0

100

0 1 2 3 4

(Pa)

time (s)

-0.8

0

0.8

0 1 2 3 4

time (s)

Time (s)

Small amplitude Large amplitude

0

0.2

0.4

0.8 0.9 1

y

/G' 0

expe

rim

ents

Sim

u-

lati

ons

To ConcludeYielding measurement do not depend on geometry and controlled parameter (stress or strain), in the quasi-static regime .Shear modulus behavior of non-dissipative and dry foam is well described by the elasto-plastic model, but this model does not capture all the anharmonicity of the response.In low frequency oscillatory experiments, where localization effects can be excluded, yield strain is equal to yield stress divided by elastic shear modulus measured at small amplitude. The yield strain seems to be well defined for foams and emulsions, and varies as (c)/ .

Strain observation of the free sample surface in a Couette cell

()()()()()()()()()000000000ReRe*,*,',",2*,ititTittetGetGGiGGtedtTωωωγγσγγωσγωγωγωγωσγ−⎡⎤=⎣⎦⎡⎤=+Δ⎣⎦=+=∫

()()()20yy0' /' where 0.64ccccTTGRRGφφφσφφσφφφφ⇓ − −∝ ∝−≈∝1444444442444444443

Comparison with the elasto-plastic model *

* Weaire & Hutzler, 1999

0.1

1

0.01 0.1 1 10

G'/G'0 and G"/G'0

gamma*G'/sigmaY

Foam1 92%

gillette 92.5%

Foam2 93%

Influence of physico-chemical properties

0.1

1

0.01 0.1 1 10

G'/G'0 and G"/G'0

gamma*G'0/sigmaY

Foam1 92%

Foam1 97%

y

G0 y

2 4 6 8 10 12

-1

-0.5

0.5

1

2 4 6 8 10 12

-1

-0.5

0.5

1

y

time

time

(t)

(t)0.1 0.2 0.5 1 2 5 10

0.05

0.1

0.2

0.5

1

2

0.1 1 10

y

G'/ G0

G'' / G0

1

0.1

y= y /G0

Stress Residual

• The complex shear modulus is well described by the elasto-plastic model, all the more if the foam is dry and weakly dissipative.

----- SGR (Sollich 98)___ Elasto-plastic

f

gillette 30 min 1 Hz gillette 30 min .3 Hz 0.92 gillette 60 min 1 Hz foam1 1 Hz foam1 .3 Hz 0.97

r

0.1

1

0.01 0.1 1 10

G'/G'0 et G"/G'0

gamma/gammaYy

Foam 1:

Solution :AOK, PEO, LOH

Gas : N2 + perfluorohexan

produced in a porous tube

Bubble Radius: R = 23 µm

Surface tension : = 27 mN/m

Foam characteristics

Foam 2:

Solution :TTAB, glycerol, LOH

Gas : N2 + perfluorohexan

produced by whipping

Bubble Radius: R = 18 µm

Surface tension : T= 37 mN/m

Nor

mal

ized

She

ar M

odul

usG

'' / G

' 0

&

G

'/ G

' 0G

'' / G

' 0

&

G

'/ G

' 0

G''

/ G' 0

&

G'/

G' 0

Localization onset Localization onset

Gillette:

Commercial shaving cream

Bubble Radius: R = 10 µm

Surface tension : = 30 mN/m0.1

1

0.1 1

G'/G'0 and G"/G'0

sigma/sigmaY

Foam1 92%

gillette 92.5%

Foam2 93%

()()()()()0022Re*,ittGettresidualtωσγγωσσσ⎡⎤=+Δ⎣⎦Δ=∫∫

G''

/ G' 0

&

G'/

G' 0

yy

0.1

1

0.1 1

'G'/G'0 and G"/G'0

sigma / sigmaY

Foam1 92%

Foam1 97%

y

G''

/ G' 0

&

G'/

G' 0

0.001

0.01

0.1

1

0.01 0.1 1 10

residual

gamma/gammaY

resi

dual

y

• Elasto-plastic residual overestimates experimental values.

• SGR model residual agrees with experimental values, but experimental values of G' and G'' are not well described by this model.

emulsions

Princen, Kiss (1986 &1989)

Mason et al. (1995 & 1996) *

Mason et al. (1995 & 1996)

* * * oscillatory measurements

foams

St Jalmes, Durian (1999) *

Khan (1988)

Gillette *

Foam1 *

Gopal, Durian (1999)

Kraynik et al. (2000)

Durian (1995)

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

Is the yield stress of aqueous foam a well defined quantity ?Is the yield stress of aqueous foam a well defined quantity ?