Simulation%20of%20complex%20fluids%20:%20a%20point%20of%20view%20of%20a%20physicist

1
Simulation of complex fluids a point of view of
a physicist
F. Lequeux Francois.lequeux_at_espci.fr

• 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

2
Pastes

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

3
Inertia
At the scale of the particles
h solvant viscosity 10-3 Pa.s a particle
radius 10-6 m r fluid density 103 kg/m3
Shear rate
At the scale of the flow
m paste apparent viscosity 103 Pa.s
Inertia effect are relevant only at very large
scale, never at the particles scale
4
Typical Particles interactions
Potentiel energy
Repulsive (requires chemistry)
Surface distance
Potentiel energy
Attractive (most of the situations)
Surface distance
5
Pastes simple classification
Non Brownian
Grains in solution, ceramic paste
Liquid Concrete
Repulsive
Attractive
Interaction strength
Clays suspensions, yoghurt
Tooth paste, paints, coating
Brownian
6
Rheology of pastes problems
Non Brownian
Sand in water, ceramic paste
Concentration gradient ! 2 constituents equations
Liquid Concrete
Repulsive
Attractive
Interaction strength
No Concentration gradient ! (at least at rest)
Clays suspensions, yoghurt
Tooth paste, paints, coating
Brownian
7
Ceramic pastes some problems
Ceramic Particles in a fluid
Risk of complete jamming ( if the solvent flows
faster than the particles )
Avoid concentration gradient in the flow ? use
visco-plastic suspending fluid
Avoid concentration gradient in the molded part ?
work near the maximum packing volume fraction
8
Rheology of pastes problems
Non Brownian
Sand in water, ceramic paste
Concentration gradient !
Shear dependant structure ? Strong Thixotropy
Liquid Concrete
Repulsive
Attractive
Interaction strength
No Concentration gradient !
Tooth paste, paints, coating
Clays suspensions, yoghurt
Brownian
9
Clays suspensions
Typical protocol for a reproducible experiment
Measurement
Measure at time t2
Stir at time t1
Mix at time t0
10
Rheology of pastes problems
Non Brownian
Sand in water, ceramic paste
Concentration gradient !

• Granular constitutive equations
• 2 components model
• (f.i. Pouliquen, IUSTI, Marseille)

Shear dependant structure
Liquid Concrete
Repulsive
Attractive
Interaction strength
No Concentration gradient !
Easiest situation
Clays suspensions, yoghurt
Brownian
Tooth paste, paints, coating
11
Repulsive paste
Dynamics is arrested at rest ? yield stress But
thermal motions are not negligible ? glass
behavior ( like glassy polymer)
12
First step toward  microscopics  plastic
events

• Starting point Flow occurs via local plastic
  rearrangements associated with a microscopic
  yield stress
• T1 events in foams (Princen)
• STZ (Argon, Spaepen, Falk-Langer, )
• Simulations of molecular systems
  (Maloney-Lemaitre)

Kabla Debrégeas, 2002
13
Second step from individual events to global
rheology
1- Localized plastic events relax the stress
(Princen)
2. leading to a global stress reorganization
? Collective Complex dynamics ? Mechanical Noise ?
14
(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
15
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, 010501 (2005)
16
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) p2934-2937
• - C.Derec, A. Ajdari, F. Lequeux Mechanics near
  a jamming transition  a minimalist model
  Faraday Discuss, (1999) 112 p 195-207
• Average the state by a scalar f the rate of
  plastic events

17
Repulsive paste
Poor mans model Maxwell fluid ( single
relaxation mode) C. Derec, A. Ajdari, F. Lequeux
Rheology and aging, a simple approach. Eur.
Phys. J. E 4, 355 361 (2001)
f is the rate of plastic jump f ?0 at rest (yield
stress)
Equation for f linear expansion
Slowing down after flow ? No other time scale
than f
18
How to measure f at rest
Evolution at rest
consequence
Use a small sollicitation i.e. in the linear
regime Step strain of Creep
19
See also C. Derec, A. Ajdari, G. Ducouret, F.
Lequeux  Aging and rheology of colloidal
concentrated suspensions Phys. Rev E 67, 061403
(2003)
Borrega, Cloitre, Monti, Leibler C.R. Physique
2000
And m is about 1
Experimentaly
20
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
21
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, 84-87 (3 July 2008) Spatial
cooperativity in soft glassy flows J. Goyon, A.
Colin, G. Ovarlez, A. Ajdari L. Bocquet
U
U
-U
-U
22
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

23
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