Simulation of complex fluids : a point of view of a physicist Simulation are well working for polymer But for pastes … What are pastes ? Model for yield stress fluids : from microscopy to constitutive equations Yield stress and aging F. Lequeux

Pastes Particles in an incompressible solvant No inertia Thermal motion - or not Interactions ( attractive, repulsive)

Inertia  solvant viscosity Pa.s a: particle radius m  fluid density 10 3 kg/m 3 Shear rate At the scale of the particles At the scale of the flow  : paste apparent viscosity 10 3 Pa.s Inertia effect are relevant only at very large scale, never at the particles scale

Typical Particles interactions Potentiel energy Surface distance Repulsive (requires chemistry) Attractive (most of the situations)

Pastes simple classification Interaction strength Attractive Repulsive Non Brownian Brownian Grains in solution, ceramic paste Liquid Concrete Clays suspensions, yoghurt Tooth paste, paints, coating

Rheology of pastes : problems Interaction strength Attractive Repulsive Non Brownian Brownian Sand in water, ceramic paste Liquid Concrete Clays suspensions, yoghurt Tooth paste, paints, coating Concentration gradient ! 2 constituents equations No Concentration gradient ! (at least at rest)

Ceramic pastes : some problems Avoid concentration gradient in the molded part  work near the maximum packing volume fraction Avoid concentration gradient in the flow  use visco-plastic suspending fluid Risk of complete jamming ( if the solvent flows faster than the particles ) Ceramic Particles in a fluid

Rheology of pastes : problems Interaction strength Attractive Repulsive Non Brownian Brownian Liquid Concrete Clays suspensions, yoghurt Tooth paste, paints, coating Concentration gradient ! No Concentration gradient ! Shear dependant structure  Strong Thixotropy Sand in water, ceramic paste

Clays suspensions Typical protocol for a reproducible experiment : Mix at time t=0 Measurement Stir at time t 1 Measure at time t 2

Rheology of pastes : problems Interaction strength Attractive Repulsive Non Brownian Brownian Liquid Concrete Clays suspensions, yoghurt Tooth paste, paints, coating Concentration gradient ! No Concentration gradient ! Shear dependant structure Granular constitutive equations  2 components model  (f.i. Pouliquen, IUSTI, Marseille) Easiest situation Sand in water, ceramic paste

Repulsive paste Dynamics is arrested at rest  yield stress But thermal motions are not negligible  glass behavior ( like glassy polymer)

First step toward « microscopics » : plastic events Starting point : Flow occurs via local plastic rearrangements associated with a microscopic yield stress å T1 events in foams (Princen) å STZ (Argon, Spaepen, Falk-Langer, … ) å Simulations of molecular systems (Maloney-Lemaitre) Kabla & Debrégeas, 2002

1- Localized plastic events relax the stress… U -U U U 2. … leading to a global stress reorganization  Collective Complex dynamics ? Mechanical Noise ? Second step : from individual events to global rheology t (strain) YY G . (Princen)

U -U U U t (strain) YY G . (Princen) Ideally, work with domains each with a proper state (stress/strain relation) These domains are mechanically coupled They move (flow field) With eventually some thermal activation And some time scale ( time scale of a plastic rearrangement)  Very complex

Ideally, work with domains each with a proper state (stress/strain relation) These domaines are mechanically coupled They move (flow field) With eventually some thermal activation And some time scale ( time scale of a plastic rearrangement)  Very complex These models lead to self organized criticality for shear rate  0, and Temperature =0 (reminiscent to fracture, or earthquake like model) G. Picard, A. Ajdari, F. Lequeux, L. Bocquet, “Slow flows of yield stress fluids: Complex spatiotemporal behavior within a simple elastoplastic model” Phys. Rev E 71, (2005)  =  =0.005  =0.05  =2

Repulsive pastes Various type of approximations -field of stress distribution with approximate coupling -P. Sollich, F. Lequeux, P. Hébraud, M.E. Cates : « Rheology of soft glassy materials » Physical Review Letters 78 p 2020 ‑ 2023 (1997) -P. Hébraud, F. Lequeux : " A naive mode-coupling model for the pasty rheology of soft glassy materials » Phys. Rev. Lett. (1998) p C.Derec, A. Ajdari, F. Lequeux “Mechanics near a jamming transition : a minimalist model” Faraday Discuss, (1999) 112 p  Average the state by a scalar f :the rate of plastic events

Repulsive paste Poor man’s model : Maxwell fluid ( single relaxation mode) C. Derec, A. Ajdari, F. Lequeux “Rheology and aging, a simple approach”. Eur. Phys. J. E 4, (2001) f is the rate of plastic jump f  0 at rest (yield stress) Slowing down after flow  No other time scale than f Equation for f : linear expansion

How to measure f at rest Use a small sollicitation – i.e. in the linear regime Step strain of Creep consequence Evolution at rest

Creep experiments- in the linear regime - probe the spontaneous rearrangements : experimental protocol Thermal or mechanical rejuvenation (pre-shear !) Rheological Test (creep /step- strain/…) time Quench Or strain cessation Waiting time Borrega, Cloitre, Monti, Leibler C.R. Physique 2000 Experimentaly And  is about 1 See also : C. Derec, A. Ajdari, G. Ducouret, F. Lequeux : “Aging and rheology of colloidal concentrated suspensions“ Phys. Rev E 67, (2003)

Repulsive paste Yield stress  degeneracy Stress is not determined at rest ( f =0). It depends on the shear history Pasty systems are non-ergodic : arrested dynamics  Yield Stress  degenerated state at rest  This leads to technical difficulties in the modelisation

Repulsive paste : Beyond mean field approximation for the fludity Local coupling : f is not a local quantity but exhibit a range of propagation ( a few tens of particles). This has been recently observed experimentally. Nature 454, (3 July 2008) Spatial cooperativity in soft glassy flows J. Goyon, A. Colin, G. Ovarlez, A. Ajdari & L. Bocquet U -U U

Conclusion Polymer melt flow modelisation is well achieved ( for small shear rate at least) granular matter (dried) is nearly well understood ( at least good constitutive equations, based on physical arguments are able to reproduce experiments) Paste rheology understanding is poor because :  most of them are complex systems – not well characterized  even the simplest fluids (repulsive colloidal suspensions) exhibit complex physics Similar problems can be found for modeling plastic flow of solid polymer

Thanks to Theoreticians Physicists A. Ajdari (now in St Gobain) L. Bocquet (in Lyon, France) M. Cates, P. Sollich (UK) Experimentalists P. Hebraud (PhD) C. Derec (PhD) G. Picard (PhD) G. Ducouret PPMD/ESPCI Mathematicians C. LeBris E. Cances S. Boyaval I. Catto