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Towards a constitutive equation for colloidal glasses

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Towards a constitutive equation for colloidal glasses

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- Towards a constitutive equation for colloidal glasses
- 1996/7: SGR Model (Sollich et al) for nonergodic materials
- Phenomenological trap model, no direct link to microstructure
- Regimes: Newtonian, PLF, Herschel Bulkley
- Full study of aging possible: Fielding et al, JoR 2000
- Tensorial versions e.g. for foams, Sollich & MEC JoR 2004

f

- Towards a constitutive equation for colloidal glasses
- Colloidal Glasses: SGR doesn’t work well
- No PLF regime observed: tm diverges at glass transition (not before)
- Dynamic yield stress jumps discontinuously

PLF

“X”

- Towards a constitutive equation for colloidal glasses
- Mode Coupling Theory:
- Established approximation route for the glass transition of colloids
- Folklore / aspiration: captures physics of caging
- Links dynamics to static structure / interactions

- MCT for shear thinning and yield of glasses
- steady state: M. Fuchs and MEC, PRL 89, 248304 (2002)
- Towards an MCT-based constitutive equation
- J. Brader, M. Fuchs, T. Voigtmann, MEC, in preparation (2006)
- Schematic MCT: ad-hoc shear thickening / jamming
- steady state: C. Holmes, MEC, M. Fuchs, P. Sollich, J. Rheol. 49, 237 (2005)

MODE COUPLING THEORY OF ARREST

MCT: a theory of the glass transition in bulk colloidal suspensions

= collective diffusion equation with Langevin noise on each particle

MODE COUPLING THEORY OF ARREST

MCT: a theory of the glass transition in bulk colloidal suspensions

= collective diffusion equation with Langevin noise on each particle

MCT calculates correlator by projecting down to two particle level

Bifurcation on varying S(q,0) c(r) (i.e. concentration / interactions)

fluid state, Y(q,∞) = 0 amorphous solid, Y(q,∞) > 0

- MODE COUPLING THEORY OF YIELDING
- M. Fuchs and MEC, PRL 89, 248304 (2002):
- Incorporate advection of density fluctuations by steady shear
- no hydrodynamic interactions, no velocity fluctuations
- several model variants (full, isotropised, schematic)

- MODE COUPLING THEORY OF YIELDING
- M. Fuchs and MEC, PRL 89, 248304 (2002):
- Incorporate advection of density fluctuations by steady shear
- no hydrodynamic interactions, no velocity fluctuations
- several model variants (full, isotropised, schematic)
- apply projection / MCT formalism to this equation of motion
- Related Approach: K. Miyazki & D. Reichman, PRE 66, 050501R (2002)

MODE COUPLING THEORY OF YIELDING

Petekidis,

Vlassopoulos

Pusey JPCM 04

MODE COUPLING THEORY OF YIELDING

Petekidis,

Vlassopoulos

Pusey JPCM 04

syc found from

(isotropised) MCT

Fuchs & Cates 03

glasses

liquids

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)
- As before, apply MCT/ projection methodology to:

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)
- As before, apply MCT/ projection methodology to:

Now:

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)
- This is a bit technical but here goes.....

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)
- This is a bit technical but here goes.....

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)
- This is a bit technical but here goes.....

survival prob

for strain

stress contribution

per unit strain

infinitesimal

step strains

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)
- This is a bit technical but here goes.....

survival prob

for strain

stress contribution

per unit strain

infinitesimal

step strains

advected wavevector

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

three-time memory function

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

three-time memory function

instantaneous decay rate

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

three-time memory function

instantaneous decay rate,

strain dependent:

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

three-time memory

two-time correlators

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

three-time memory

two-time correlators

three-time vertex functions

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

three-time memory

two-time correlators

three-time vertex functions

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)
- No hydrodyamic fluctuations, shear thinning only
- Numerically challenging equations due to multiple time integrations
- Results for strep strain only so far
- Schematic variants are more tractable e.g.:

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)
- No hydrodyamic fluctuations, shear thinning only
- Numerically challenging equations due to multiple time integrations
- Results for strep strain only so far
- Schematic variants are more tractable e.g.:

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)
- No hydrodyamic fluctuations, shear thinning only
- Numerically challenging equations due to multiple time integrations
- Results for strep strain only so far
- Schematic variants are more tractable e.g.:

N.B.: can add jamming, ad-hoc, to this

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

decay curves

after step strain:

schematic model

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

long time stress

asymptote

after step strain:

schematic model

- TIME-DEPENDENT RHEOLOGY VIA MCT
- J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

long time stress

asymptote

after step strain:

isotropised model

- Steady-state schematic model + ad-hoc jamming
- Schematic MCT model + empirical stress-dependent vertex
- strain destroys memory : m(t) decreases with shear rate
- stress promotes jamming: m(t) increases with stress S
- a = 0 approximates Fuchs/MEC calculations

- C Holmes, MEC, M Fuchs + P Sollich, J Rheol 49, 237 (2005)

a =

jammability

by stress

v = glassiness

ZOO OF STRESS vs STRAIN RATE CURVES

kBT

a3

s

x

≈

BISTABILITY OF DROPLETS/GRANULES

fracture

shear

stress

strain rate

kBT

a3

s

x

≈

BISTABILITY OF DROPLETS/GRANULES

shear

stress

strain rate

fluid droplet at S < kBT/a3

kBT

a3

s

x

≈

BISTABILITY OF DROPLETS/GRANULES

shear

stress

capillary force maintains

stress S : kBT/a3<< S << s/x

strain rate

fluid droplet at S < kBT/a3

BISTABILITY OF DROPLETS/GRANULES

experiments: Mark Haw

1mm PMMA, index-matched

hard spheres f = 0.61

BISTABILITY OF DROPLETS/GRANULES

experiments: Mark Haw

1mm PMMA, index-matched

hard spheres f = 0.61

The End

CAPILLARY VS BROWNIAN STRESS SCALES

- Complete wetting: colloid prefers solvent to air
- Energy scale for protrusion DE = s p a2 >> kBT
- Stress scale for capillary forces Scap = DE/ a3 >> kBT/a3 = Sbrownian
- Capillary forces can overwhelm Brownian motion
- Possible route to static, stress-induced arrest, i.e. jamming

BISTABILITY OF DROPLETS/GRANULES

- Fluid droplet, radius R:
- unjammed, undilated
- isotropic Laplace pressure
- P ≈ s/R
- no static shear stress

BISTABILITY OF DROPLETS/GRANULES

- Fluid droplet, radius R:
- unjammed, undilated
- isotropic Laplace pressure
- P ≈ s/R
- no static shear stress

- Solid granule:
- dilated, jammed
- Laplace pressure
- s/R >P > - s/a
- static shear stress S ≈ P

ZOO OF STRESS vs STRAIN RATE CURVES

a =

jammability

by stress

v = glassiness

a =

jammability

by stress

v = glassiness

ZOO OF STRESS vs STRAIN RATE CURVES

a =

jammability

by stress

v = glassiness

RAISE CONCENTRATION AT FIXED INTERACTIONS

a =

jammability

by stress

v = glassiness

RAISE CONCENTRATION AT FIXED INTERACTIONS

The End