Shear Localization/Banding

Michael Dennin

UC Irvine

What’s in this talk?

• Why study shear banding?

• Summary of experimental results.

• Brief comments on theory/modelling.

Shear Banding/Localization

• Two or more “distinct” flow regimes

• Flow regimes distinguished by different rates of strain

• Average property – “steady state”

Three of many experiments

Coussot, et al., PRL 88, 218301 (2002) Bocquet,et al., PRE 65, 011307 (2001).

Debregeas,et al., PRL 87 (2001)

GRANULAR

2D FOAM

SUSPENSION

All of these examples are in Couette geometries

General Issues

• Inhomogeneous applied stress.

• Interesting flow curves (stress as a function of rate of strain).

• Discontinuities in the rate of strain.

• Changes in the microscopic structure of the material.

• Impact of boundaries (2D issue mainly)

• Path in “parameter” space.

Foam issues

• Composition of “fluid walls” including stabilizers.

• Sample preparation.

• Pre-shear conditions.

• Dimensionality.

• “wall drag”.

• Flow induced structural changes.

Question: When is shear banding the coexistence of

two distinct states?

Jamming Phase Diagram

Liu and Nagel, Nature v 396, 1998

The “J-point”

Flow + jammed state

WARNING

• Equilibrium systems minimize a free energy – coexistence occurs at unique and well defined points.

• Nonequilibrium systems do not necessarily obey a minimization principle – coexistence of states can be more complicated.

Equilibrium case

Example: Thermal Convection

State of the system depends on the path in parameter space!

Kolodner, et al., PRL 60, 1723 (1988)

Summary of Experiments

Wall drag

Confined Bubbles

Debregeas,et al., PRL 87 (2001)

Couette Geometry: two plates

No top

4.5 5.0 5.5 6.0 6.54.5x10-4

5.0x10-4

5.5x10-4

6.0x10-4

4 6 8 100

2x10-4

4x10-4

6x10-4

v(r)

/r (

s-1)

radial position (cm)

radial position (cm)

v(r

)/r

(s-1

)

J. Lauridsen, G. Chanan, M. Dennin, PRL, V 93, 018303 (2004).

Parallel Shear

Direct Comparison

-10 -5 0 5 10

-1.0

-0.5

0.0

0.5

1.0

Bubble

velo

city

/Belt

velo

city

y/<d>

shear rate=0.0014 s-1

shear rate=0.0028 s-1

shear rate=0.014 s-1

-10 -5 0 5 10

-0.4

-0.2

0.0

0.2

0.4

Bubble

velo

city

/Belt

velo

city

y/<d>

shear rate=0.0014 s-1

shear rate=0.0028 s-1

shear rate=0.014 s-1

System without a top System with a top

Wang, Krishan, Dennin, PRE V. 73, 031401 (2006).

Dispersity/Boundaries

Katgert, Phys. Rev. Lett. 101, 058301 (2008

bidisperse monodisperse

More Couette

Outer and inner shear bands.

Krishan and Dennin, PRE 78, 051504 (2008).

Discontinuities – is it all about attractions?

Review paper: Dennin, J. Physics: Cond. Matter 20, 283103 (2008).

Bubble Raft

4.5 5.0 5.5 6.0 6.5 7.00.0

0.2

0.4

0.6

0.8

1.0

v(

r)/(r

)

radial position (cm)

Yield stress fluid

Power law fluid

J. Lauridsen, G. Chanan, M. Dennin, PRL, V 93, 018303 2004).

Effective Viscosity: stress/(strain rate)

-3 -2 -1 01

2

3

4

log

(vis

cosi

ty)

log (strain rate)

1/3 1/3 1/3(0.8 mN/m)( / ) (1.8 mNs /m)( / ) y a d dt d dt

3D Case

Rodts et al, Europhys. Lett. 69, 636 (2005)

Interesting aside …

0.00 0.05 0.10 0.15 0.200.00

0.02

0.04

0.06

0.08

0.10

Cri

tica

l Ra

te o

f Str

ain

(s-1

)

(s-1)

“discrete”

“continuum”

Gilbreth, et al., Phys. Rev. E 74, 051406 (2006).

Rodts et al, Europhys. Lett. 69, 636 (2005)

Discontinuous vs Continuous

G. Ovarlez, K. Krishan, R. Höhler, S. Cohen-Addad, in preparation

Leiden Results

• See later talks for pictures

• Couette flow in bubble raft – continuous shear band.

BASE+MAc, 20 wt % Glycerol

z, mm

0.0 0.5 1.0 1.5 2.0 2.5 3.0

V, m

m/s

0.0

0.1

0.2

0.3

0.4

0.5

10.13 s

cr

Parallel shear (thanks to Denkov)

Experimental results

Lessons from other systems

• Unstable flow curves

• Impact of system interactions – attractive/repulsive

• Impact of structural changes (and connection to unstable flow curves)

• Changes in density resulting in changes in other properties

Theories/models

• 2D: Extra drag terms

• Other systems: nonlinear flow curves/unstable regions => structural changes.

• Stress focusing from T1 events

What next?

Careful study of attractions in foams …

Other issues …

Critical Strains/ Time evolution

Rouyer, et al. Phys. Rev. E 67, 021405 (2003)

Below critical strain: linear

Above critical strain: nonlinear

Time dependence of critical radius

Gilbreth, et al., Phys. Rev. E 74, 051406 (2006).

4.5 5.0 5.5 6.0 6.54.5x10-4

5.0x10-4

5.5x10-4

6.0x10-4

4 6 8 100

2x10-4

4x10-4

6x10-4

v(r)

/r (

s-1)

radial position (cm)

radial position (cm)

v(r

)/r

(s-1

)

Value of critical radius depends on averaging time.Wang, et al.Phys. Rev. Lett. 98, 220602 (2007)

Path in Phase space

40 50 60 70 80 900

5

10

15

20

25

s/r

r(mm)

regular quench quench w/delay adiabatic decrease

All four curves are for the same rotation rate in a Couette geometry.All four curves take a different “path in phase space”.

Return to Coexistence Idea

Summary

• What nonequilibrium transitions occur in driven foams?

• Are shear bands the coexistence of different nonequilibrium states?

• What are the “microscopic” mechanisms for shear banding => attractive interactions in foam?