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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 - PowerPoint PPT Presentation
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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
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