Discontinuous Shear Thickening

Transcription

1 Discontinuous Shear Thickening dynamic jamming transition Ryohei Seto, Romain Mari, Jeffrey F. Morris, Morton M. Denn Levich Institute, City College of New York

2 First experimental data Williamson and Hecker (1931) Some Properties of Dispersions of the Quicksand Type 39.8% 40.0% 40.8% 42.8% 43.8% 44.4%

3 A review article Barnes (1989) Shear-Thickening (Dilatancy) in Suspensions of Nonaggregating Solid Particles dispersed in Newtonian liquids 57% 45%

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5 Hydrodynamics-based models Stokes flow + Hard spheres + Brownian forces Stokes equation microstructure g(r) Boundary conditions given by moving particles Stokesian Dynamics Durlofsky, Brady, and Bossis (1987)

6 Shear induced structures (hydroclusters?)? Wagner and Brady (2009)

7 Recent data R. Egres (Wagner s group) nm silica particle suspension 52% η/pa s 51% 50% 48% 44% 40% 36% 30% 20% γ/s 1

8 Modeling Stokesian Dynamics This work Hydrodynamic n-spheres approx. + 2-spheres exact (singularity) 2-spheres approx. (singularity cutoff) Shear-rate dependence Brownian force vs. flow ( Péclet number) repulsion vs. flow Particle interaction (no contact) friction (contact model)

9 Does the lubrication singularity prevent contacts? U U 2 U 1 However, finite forces can cause exponentially small gaps. F lub

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11 Two-sphere solution F 1 F 2 T 1 T 2 (hydrodynamic lubrication) A 11 A B B U U (r 1 ) = η A 21 A B B U 2 U (r 2 ) 0 B 11 B 12 C 11 C 12 Ω 1 Ω B 21 B 22 C 21 C 22 Ω 2 Ω A (X ) 1 A (X ) 1 h A (Y ) ( 1 log ) Regularization h + δ A (Y ) log ( 1 ) h h + δ B log ( 1 ) δ = 10 3 B log ( 1 ) h h + δ C log ( 1 ) C log ( 1 ) h h + δ

12 Hard-spheres + Newtonian fluid is Newtonian Hydrodynamic interactions follow linear relations. No essential shear-rate dependence Hard-sphere has no specific force scale. Trajectories are the same at low shear rates at high shear rates

13 Hard-spheres + Newtonian fluid + other force is non-newtonian A short-range repulsive force F R (h) Caexp ( κh ) n at low shear rate at high shear rate Shear-rate dependence

14 Frictional contact (Contact model in granular physics) h = 0 h < 0 F C,norm = k n ( h) n T C,norm = 0 ξ = 0 ξ F C,tang = k t ξ T C,tang = a n F C,tang Friction law F C,tang <µf C,norm

15 Modeling Stokesian Dynamics This work Hydrodynamic n-spheres approx. + 2-spheres exact (singularity) 2-spheres approx. (singularity cutoff) Shear-rate dependence Brownian force vs. flow ( Péclet number) repulsion vs. flow Particle interaction (no contact) friction (contact model)

16 Equation of motion Inertia zero limit (overdamped) X (t + t) = X (t) + U t force and torque balance equations ( ) F H T H F H + F C + F R = 0 T H + T C = 0 hydrodynamic lubrication (pairwise interactions) ( A B = η 0 B C )( U U ) Ω Ω + η 0 E

17 Simulation conditions Bidisperse system to avoid strong ordering Periodic boundary condition never migration Zero-inertia simulation never Bagnold s law relative viscosity ηr strain γ

18 Simulation result: Viscosity vs. Shear rate : relative viscosity : dimensionless shear rate 10 2 Ù 10 1 φ = Ù Ù Ù Ú Ú Ú Ú Ï ÚÙ Ï ÚÙ Ù Ú Ú Ï ÏÏ Ï 0.55 Ú 0.54 Ï Ï Ï

19 Comparison to experimental data experiment simulation 10 3 η/pa s 52% 51% 50% 48% 44% 40% 36% 30% 20% % Ù Ù Ù Ú Ú Ú Ù Ú ÚÏ Ú Ï Ú Ï Ú Ï Ù Ù 56% 55% 54% 52% Ï Ï Ï 50% 48% γ/s γ/s 1

20 Comparison to experimental data η/pa s Parameter to adjust? 10 3 Silica particles 57% 52% Ú Ï Ï Ï Ï Ú Ú Ï Ú Ú Ï ÏÏ Ï 56% 51% Ú Ú Ú 54% 50% 52% 48% 48% 44% a = 225 nm MW = 200 polyethylene glycol η 0 = Pa s Γ γ = 6πη 0a 2 F R (0) = F R (0) 0.1 nn γ/s 1 Double layer force F EDL (h = 0) 2πaε 0 ε r κψ 2 0 If 1/ = 5 nm and r = 20 ψ 0 45 mv

21 The same onset stress? ηr φ Ù Ï ÏÚ ÏÚ ÏÏ Ú Ú Ú Ú Ù Ù Ù Ï Ù Ù Ù Ú Ú Ú Ï Ï Ï σ η r Γ min σ φ

22 Frictional is essential! F C,tang <µf C,norm 10 2 µ = µ = 0.1 Ï Ï Ï Ï Ï Ï Ï Ï µ =

23 Frictional is essential! µ = 0 µ = 1

24 µ = 1 Ï Ï Ï Ï Ï Ï Ï n = 80 n = 1000 n =

25 Shear jammed state Bi et. al (Nature) ı ı ı ı Û Û Û Ûı Ûı ı Û Ì Ì Ì Û ÌÌ Û Ì Ì Û Ì Ì Rigid particles + Dense + Friction unflowable state (Jamming phase)

26 Phase diagram continuous shear thickening shear jammed state Γ viscosity minimum discontinuous shear thickening φ

27 Some insights for this discontinuity ı ı ı ı Û Û Û Û Ì Ûı Ì Ûı ı Û Û Ì ÌÌ Ì Ì Û Ì Ì

28 Mechanism of discontinuity Contact network γ φ

29 A kinetic model to understand the discontinuity

30 static friction dynamic friction

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32 Kinetic model (idea) Overcoming the repulsion to get into contact growth coalescence rotate breakup

33 Summary Our model can reproduce both continuous and discontinuous shear thickening. Frictional particles are jammed under shear. Growth of contact network (No hydrocluster, no Bagnold law, no migration, and less order-disorder transition)