Force distribution and contact network analysis of sheared ... · •Justified...
Transcript of Force distribution and contact network analysis of sheared ... · •Justified...
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
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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
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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
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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