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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

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Towards a constitutive equation for colloidal glasses

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  1. 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

  2. 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”

  3. 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)

  4. 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

  5. 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

  6. 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)

  7. 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)

  8. MODE COUPLING THEORY OF YIELDING Petekidis, Vlassopoulos Pusey JPCM 04

  9. MODE COUPLING THEORY OF YIELDING Petekidis, Vlassopoulos Pusey JPCM 04 syc found from (isotropised) MCT Fuchs & Cates 03 glasses liquids

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

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

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

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

  14. 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

  15. 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

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

  17. TIME-DEPENDENT RHEOLOGY VIA MCT • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006) three-time memory function

  18. TIME-DEPENDENT RHEOLOGY VIA MCT • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006) three-time memory function instantaneous decay rate

  19. 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:

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

  21. TIME-DEPENDENT RHEOLOGY VIA MCT • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006) three-time memory two-time correlators

  22. 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

  23. 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

  24. 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.:

  25. 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.:

  26. 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

  27. TIME-DEPENDENT RHEOLOGY VIA MCT • J. Brader, M. Fuchs, T. Voigtmann and MEC, in preparation (2006) decay curves after step strain: schematic model

  28. 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

  29. 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

  30. 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)

  31. a = jammability by stress v = glassiness ZOO OF STRESS vs STRAIN RATE CURVES

  32. kBT a3 s x ≈ BISTABILITY OF DROPLETS/GRANULES fracture shear stress strain rate

  33. kBT a3 s x ≈ BISTABILITY OF DROPLETS/GRANULES shear stress strain rate fluid droplet at S < kBT/a3

  34. 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

  35. BISTABILITY OF DROPLETS/GRANULES experiments: Mark Haw 1mm PMMA, index-matched hard spheres f = 0.61

  36. BISTABILITY OF DROPLETS/GRANULES experiments: Mark Haw 1mm PMMA, index-matched hard spheres f = 0.61

  37. The End

  38. 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

  39. BISTABILITY OF DROPLETS/GRANULES • Fluid droplet, radius R: • unjammed, undilated • isotropic Laplace pressure • P ≈ s/R • no static shear stress

  40. 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

  41. ZOO OF STRESS vs STRAIN RATE CURVES a = jammability by stress v = glassiness

  42. a = jammability by stress v = glassiness ZOO OF STRESS vs STRAIN RATE CURVES

  43. a = jammability by stress v = glassiness RAISE CONCENTRATION AT FIXED INTERACTIONS

  44. a = jammability by stress v = glassiness RAISE CONCENTRATION AT FIXED INTERACTIONS

  45. The End

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