Towards a constitutive equation for colloidal glasses

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

Towards a constitutive equation for colloidal glasses

Colloidal Glasses: SGR doesn’t work well

• No PLF regime observed: m 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 glassessteady state: M. Fuchs and MEC, PRL 89, 248304 (2002)

• Towards an MCT-based constitutive equationJ. Brader, M. Fuchs, T. Voigtmann, MEC, in preparation (2006)

• Schematic MCT: ad-hoc shear thickening / jammingsteady 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, (q,∞) = 0 amorphous solid, (q,∞) > 0

MODE COUPLING THEORY OF YIELDINGM. 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 YIELDINGM. 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,VlassopoulosPusey JPCM 04

MODE COUPLING THEORY OF YIELDING

Petekidis,VlassopoulosPusey JPCM 04

yc found from

(isotropised) MCTFuchs & Cates 03

glasses

liquids

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

As before, apply MCT/ projection methodology to:

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

As before, apply MCT/ projection methodology to:

Now:

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

This is a bit technical but here goes.....

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

This is a bit technical but here goes.....

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

This is a bit technical but here goes.....

survival probfor strain

stress contributionper unit strain

infinitesimalstep strains

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

This is a bit technical but here goes.....

survival probfor strain

stress contributionper unit strain

infinitesimalstep strains

advected wavevector

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

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

three-time memory function

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

three-time memory function

instantaneous decay rate

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

three-time memory function

instantaneous decay rate,strain dependent:

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

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

three-time memory two-time correlators

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

three-time memory two-time correlators

three-time vertex functions

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

three-time memory two-time correlators

three-time vertex functions

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

decay curvesafter step strain:schematic model

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

long time stressasymptoteafter step strain:schematic model

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

long time stressasymptoteafter 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

• = 0 approximates Fuchs/MEC calculations

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

v = glassiness

= jammabilityby stress

ZOO OF STRESS vs STRAIN RATE CURVES

BISTABILITY OF DROPLETS/GRANULES

kBT a3

≈

shearstress

strain rate

fracture

BISTABILITY OF DROPLETS/GRANULES

kBT a3

≈

shearstress

strain rate

fluid droplet at kBT/a3

BISTABILITY OF DROPLETS/GRANULES

kBT a3

≈

shearstress

capillary force maintainsstress kBT/a3

fluid droplet at kBT/a3

strain rate

BISTABILITY OF DROPLETS/GRANULES

experiments: Mark Haw1m PMMA, index-matchedhard spheres = 0.61

BISTABILITY OF DROPLETS/GRANULES

experiments: Mark Haw1m PMMA, index-matchedhard spheres = 0.61

The End

• Complete wetting: colloid prefers solvent to air

• Energy scale for protrusion E = a2 >> kBT

• Stress scale for capillary forces cap = E/ a3 >> kBT/a3 = brownian

• Capillary forces can overwhelm Brownian motion

• Possible route to static, stress-induced arrest, i.e. jamming

CAPILLARY VS BROWNIAN STRESS SCALES

Fluid droplet, radius R:

• unjammed, undilated• isotropic Laplace pressure≈/R• no static shear stress

BISTABILITY OF DROPLETS/GRANULES

Fluid droplet, radius R:

• unjammed, undilated• isotropic Laplace pressure≈/R• no static shear stress

Solid granule:

• dilated, jammed• Laplace pressure /R /a• static shear stress ≈

BISTABILITY OF DROPLETS/GRANULES

v = glassiness

= jammabilityby stress

ZOO OF STRESS vs STRAIN RATE CURVES

v = glassiness

= jammabilityby stress

ZOO OF STRESS vs STRAIN RATE CURVES

v = glassiness

= jammabilityby stress

RAISE CONCENTRATION AT FIXED INTERACTIONS

v = glassiness

= jammabilityby stress

RAISE CONCENTRATION AT FIXED INTERACTIONS

The End