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 glasses steady state: M. Fuchs and MEC, PRL 89, (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,  (q,∞) = 0 amorphous solid,  (q,∞) > 0

MODE COUPLING THEORY OF ARREST The MCT equations (Goetze 1992): Vertex V(q,k) depends only on S(q,0) Glass transition: arrest on varying  or interactions Arrest predicted at  glass = 0.52 for hard spheres (rather too low)

MODE COUPLING THEORY OF YIELDING M. Fuchs and MEC, PRL 89, (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, (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, R (2002)

MODE COUPLING THEORY OF YIELDING Petekidis, Vlassopoulos Pusey JPCM 04

MODE COUPLING THEORY OF YIELDING Petekidis, Vlassopoulos Pusey JPCM 04  y c 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   = 0 approximates Fuchs/MEC calculations C Holmes, MEC, M Fuchs + P Sollich, J Rheol 49, 237 (2005)

v = glassiness  = jammability by stress ZOO OF STRESS vs STRAIN RATE CURVES

BISTABILITY OF DROPLETS/GRANULES k B T a 3 ≈  shear stress strain rate fracture

BISTABILITY OF DROPLETS/GRANULES k B T a 3 ≈  shear stress strain rate fluid droplet at  k B T/a 3

BISTABILITY OF DROPLETS/GRANULES k B T a 3 ≈  shear stress capillary force maintains stress  k B T/a 3  fluid droplet at  k B T/a 3 strain rate

BISTABILITY OF DROPLETS/GRANULES experiments: Mark Haw 1  m PMMA, index-matched hard spheres  = 0.61

BISTABILITY OF DROPLETS/GRANULES experiments: Mark Haw 1  m PMMA, index-matched hard spheres  = 0.61

The End

Complete wetting: colloid prefers solvent to air Energy scale for protrusion  E =  a 2 >> k B T Stress scale for capillary forces  cap =  E/ a 3 >> k B T/a 3 =  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  = jammability by stress ZOO OF STRESS vs STRAIN RATE CURVES

v = glassiness  = jammability by stress ZOO OF STRESS vs STRAIN RATE CURVES

v = glassiness  = jammability by stress RAISE CONCENTRATION AT FIXED INTERACTIONS

v = glassiness  = jammability by stress RAISE CONCENTRATION AT FIXED INTERACTIONS

The End