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

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Towards a constitutive equation for colloidal glasses

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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

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

  • 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)


Towards a constitutive equation for colloidal glasses

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


Towards a constitutive equation for colloidal glasses

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


Towards a constitutive equation for colloidal glasses

  • 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)


Towards a constitutive equation for colloidal glasses

  • 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)


Towards a constitutive equation for colloidal glasses

MODE COUPLING THEORY OF YIELDING

Petekidis,

Vlassopoulos

Pusey JPCM 04


Towards a constitutive equation for colloidal glasses

MODE COUPLING THEORY OF YIELDING

Petekidis,

Vlassopoulos

Pusey JPCM 04

syc found from

(isotropised) MCT

Fuchs & Cates 03

glasses

liquids


Towards a constitutive equation for colloidal glasses

  • TIME-DEPENDENT RHEOLOGY VIA MCT

  • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

  • As before, apply MCT/ projection methodology to:


Towards a constitutive equation for colloidal glasses

  • TIME-DEPENDENT RHEOLOGY VIA MCT

  • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

  • As before, apply MCT/ projection methodology to:

Now:


Towards a constitutive equation for colloidal glasses

  • TIME-DEPENDENT RHEOLOGY VIA MCT

  • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

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


Towards a constitutive equation for colloidal glasses

  • TIME-DEPENDENT RHEOLOGY VIA MCT

  • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

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


Towards a constitutive equation for colloidal glasses

  • 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


Towards a constitutive equation for colloidal glasses

  • 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


Towards a constitutive equation for colloidal glasses

  • TIME-DEPENDENT RHEOLOGY VIA MCT

  • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)


Towards a constitutive equation for colloidal glasses

  • TIME-DEPENDENT RHEOLOGY VIA MCT

  • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

three-time memory function


Towards a constitutive equation for colloidal glasses

  • TIME-DEPENDENT RHEOLOGY VIA MCT

  • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

three-time memory function

instantaneous decay rate


Towards a constitutive equation for colloidal glasses

  • 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:


Towards a constitutive equation for colloidal glasses

  • TIME-DEPENDENT RHEOLOGY VIA MCT

  • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)


Towards a constitutive equation for colloidal glasses

  • TIME-DEPENDENT RHEOLOGY VIA MCT

  • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

three-time memory

two-time correlators


Towards a constitutive equation for colloidal glasses

  • 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


Towards a constitutive equation for colloidal glasses

  • 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


Towards a constitutive equation for colloidal glasses

  • 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.:


Towards a constitutive equation for colloidal glasses

  • 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.:


Towards a constitutive equation for colloidal glasses

  • 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


Towards a constitutive equation for colloidal glasses

  • TIME-DEPENDENT RHEOLOGY VIA MCT

  • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006)

decay curves

after step strain:

schematic model


Towards a constitutive equation for colloidal glasses

  • 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


Towards a constitutive equation for colloidal glasses

  • 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


Towards a constitutive equation for colloidal glasses

  • 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)


Towards a constitutive equation for colloidal glasses

a =

jammability

by stress

v = glassiness

ZOO OF STRESS vs STRAIN RATE CURVES


Towards a constitutive equation for colloidal glasses

kBT

a3

s

x

BISTABILITY OF DROPLETS/GRANULES

fracture

shear

stress

strain rate


Towards a constitutive equation for colloidal glasses

kBT

a3

s

x

BISTABILITY OF DROPLETS/GRANULES

shear

stress

strain rate

fluid droplet at S < kBT/a3


Towards a constitutive equation for colloidal glasses

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


Towards a constitutive equation for colloidal glasses

BISTABILITY OF DROPLETS/GRANULES

experiments: Mark Haw

1mm PMMA, index-matched

hard spheres f = 0.61


Towards a constitutive equation for colloidal glasses

BISTABILITY OF DROPLETS/GRANULES

experiments: Mark Haw

1mm PMMA, index-matched

hard spheres f = 0.61


Towards a constitutive equation for colloidal glasses

The End


Towards a constitutive equation for colloidal glasses

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


Towards a constitutive equation for colloidal glasses

BISTABILITY OF DROPLETS/GRANULES

  • Fluid droplet, radius R:

  • unjammed, undilated

  • isotropic Laplace pressure

  • P ≈ s/R

  • no static shear stress


Towards a constitutive equation for colloidal glasses

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


Towards a constitutive equation for colloidal glasses

ZOO OF STRESS vs STRAIN RATE CURVES

a =

jammability

by stress

v = glassiness


Towards a constitutive equation for colloidal glasses

a =

jammability

by stress

v = glassiness

ZOO OF STRESS vs STRAIN RATE CURVES


Towards a constitutive equation for colloidal glasses

a =

jammability

by stress

v = glassiness

RAISE CONCENTRATION AT FIXED INTERACTIONS


Towards a constitutive equation for colloidal glasses

a =

jammability

by stress

v = glassiness

RAISE CONCENTRATION AT FIXED INTERACTIONS


Towards a constitutive equation for colloidal glasses

The End


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