Post on 06-Oct-2020
Force distribution and contact network analysis of sheared dense suspensions
Formative Formulation, University of CambridgeMarch 18, 2019
Ranga Radhakrishnan, J. R. Royer, W. C. K. Poon, and J. Sun
EP/N025318/1
Dense suspensions of particles
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PaintsChocolate
Drilling muds
Ceramics
Cements
Shear thickening of model systems
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Royer, Blair, Hudson PRL 2016
2 !m Silica spheres in
Glycerol+water
Universality of shear thickening in colloidal and non-Brownian model systems.
– Guy, Hermes, Poon PRL 2015
#∗
Mechanism of shear thickening
4
Repulsions prevent contact
Frictional contact
! < !∗ ! > !∗!∗ = &∗
'(Stress scale
[1] Fernandez et al., PRL 2013, [2] Seto, Mari, Morris, PRL 2013, J. Rheol. 2014, [3] Wyart and Cates, PRL 2014 [4] Guy, Hermes, Poon, PRL 2015, [5] Lin et al, PRL 2015, [6] Royer, Blair, Hudson, PRL 2016, [7] Clavaud et al PNAS 2017, [8] Comtet et al, Nat. Comm. 2017
Low Viscosity
High Viscosity
Repulsive force scale
Discrete Element Method Simulations
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LAMMPS (Sandia)
Ness and Sun, PRE 2015Non Brownian particles
Linear (Hooke) spring force
Coulomb criterionFt≤ μ Fnkn
ktμ
1.4 dd
Regularised lubrication Stokes drag
Hydrodynamic forces
h
Imposed shear rate "̇
Model of shear thickening
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[1] Mari, Seto, Denn, Morris, PRL 2013, J. Rheol. 2014 [2] Wyart and Cates, PRL 2014 [3] Ness CJ, and Sun J, Soft Matter 2016 [4] Clavaud et al PNAS 2017, [5] Comtet et al, Nat. Comm. 2017
Frictionlessparticle contact
Frictional contact
! < !∗ ! > !∗!∗ = &∗
'(Stress scaleCritical load
Critical load modelMax. Tangential Force
0
μ
Normal force
F*
Viscosity divergence at low stress ! < !∗
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frictionless particles
Ft → 0
FnRelative Viscosity
Volume Fraction
μ→0%& =
%%(
%&~ *+,- − * /0
Viscosity divergence at high stress ! > !∗
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frictionless particles
Fn > F*
frictional particles
Ft → 0 Ft≤ μ Fn
Fn
FtFnRelative Viscosity
Volume Fraction
μ=0
μ=0.5
&'()&*(,)
&./~ &* , − & 231
Fraction of frictional contacts !(#)
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Stress
φ=0.55φ=0.53φ=0.49
[1] Mari, Seto, Denn and Morris, PRL 2013[2] Wyart and Cates, PRL 2014
!(#)Frictionalcontact
exp(−#∗
# )
Frictionlesscontact
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Theory for shear thickening
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[1] Wyart and Cates, PRL 2014[2] Singh, Mari, Denn and Morris, J. Rheol 2018
Stress
Relative Viscosity
φ=0.55
0.53
0.49
!" # = % # !&+ 1 − % # !*+,
-.(#)~ !" # − ! 23
3
DEM simulations for a microscopic understanding
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Contact Force Network
Strain
Stress !Average !
μ = 0.5#$ − # = 0. 005
Coordination number ! and viscosity divergence
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3 22
1 ! = 2
[1] Lerner, During, and Wyart, PNAS 2012[2] Wyart and Cates, PRL 2013
Δ! = !% − !Soft Modes
'( ∼ Δ!*+
(!%(-) − !) ∼ (/%(-) − /)
0 = 2.1 (2D)0 = 2.7 (3D)
For - → 0
Assumed to be true for all -1 3
[1]
[2]
! = 10%&
Δ( = () − (
+)
+
Coordination number deficit Δ+ for ! → 0
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3 22
1 + = 2
! = 10%&
1+) − +
Δ(
! = 10%&
1 3 satisfied
Lerner, During, and Wyart, PNAS 2012Wyart and Cates, PRL 2013
Δ+
Critical contact number and volume fraction
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!" #
Isotropic compressed packing of spheres [3]
Sheared granular spheres [2]
Suspensions (Our results)
$" = !"!" + 2 ( [1]
[1] Edwards and Oakeshott, Physica A 1989, [2] Sun and Sunderesan J. Fluid Mech. 2011[3] Silbert, Soft Matter 2010
Critical volume fraction $" #
Critical contact number
Contact number deficit Δ" vs. distance to jamming
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Δ" = "$ − "
Δ& = &$ − &
1
0.36
Not true for all ', and Δ&Wyart and Cates, PRL 20131 3 ("$(') − ") ∼ (&$(') − &)
Normal contact force probability distribution ! "
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" = $%⟨$%⟩
!(")
[1,2]
Fit for * = 10-.
Packed amorphous spheres [2]
/ exp(−")
FtFn 45 − 4 ≈ 0.005
[1] Mueth, Jaegger and Nagel Phys. Rev. E 1998,[2] Blair, Muggenberg, Marshall, Jaegger and Nagel Phys. Rev. E 2001
! " for varying #
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" = %&⟨%&⟩
!(")
[1,2]
Fit for + = 10./
Packed amorphous spheres [2]
0 exp(−")
FtFn
[1] Mueth, Jaegger and Nagel Phys. Rev. E 1998,[2] Blair, Muggenberg, Marshall, Jaegger and Nagel Phys. Rev. E 2001
+ = 0.5
! " for varying stress #
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" = %&⟨%&⟩
!(")
[1,2]
Fit for + = 10./
Packed amorphous spheres [2]
0 exp(−")
FtFn
[1] Mueth, Jaegger and Nagel Phys. Rev. E 1998,[2] Blair, Muggenberg, Marshall, Jaegger and Nagel Phys. Rev. E 2001[3] Ness and Sun, Soft Matter 2016
+ = 0.5
Contact force distribution to frictional contacts
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! = #$⟨#$⟩
'(!)
Stress
φ=0.55φ=0.53φ=0.49*(+)
!∗ = #∗⟨#$ + ⟩
Guy, Hermes and Poon PRL 2015
* + ∼ ./∗
0' ! 1!
Exponential tail of '(!) gives rise to exponential *(+)2
Conclusions
• !" # , %"(#) and (()*) of suspensions are similar to granular packings, and sheared granular materials
• Justified + , ∼ exp(−,∗/ ,) in the Wyart-Cates model
• Contact number deficit, %"(#) –Z, does not explain viscosity divergence
and linear interpolation for the jamming volume fraction !3
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Future Work
• Network properties: 3 cycles, efficiency etc. to understand the
dynamics of shear thickening
• Use connections with granular materials to build better models
2
31
Thanks
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Dense suspension team @ Edinburgh
Mr. Yang CuiRheology of non-Brownian suspensions composed of frictional and frictionless particles
Dr. J.P. Morrissey