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

2

PaintsChocolate

Drilling muds

Ceramics

Cements

Shear thickening of model systems

3

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

5

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

6

[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 ! < !∗

7

frictionless particles

Ft → 0

FnRelative Viscosity

Volume Fraction

μ→0%& =

%%(

%&~ *+,- − * /0

Viscosity divergence at high stress ! > !∗

8

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 !(#)

9

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

2

Theory for shear thickening

10

[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

11

Contact Force Network

Strain

Stress !Average !

μ = 0.5#$ − # = 0. 005

Coordination number ! and viscosity divergence

12

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

13

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

14

!" #

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

15

Δ" = "$ − "

Δ& = &$ − &

1

0.36

Not true for all ', and Δ&Wyart and Cates, PRL 20131 3 ("$(') − ") ∼ (&$(') − &)

Normal contact force probability distribution ! "

16

" = $%⟨$%⟩

!(")

[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 #

17

" = %&⟨%&⟩

!(")

[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 #

18

" = %&⟨%&⟩

!(")

[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

19

! = #$⟨#$⟩

'(!)

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

20

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

21

Dense suspension team @ Edinburgh

Mr. Yang CuiRheology of non-Brownian suspensions composed of frictional and frictionless particles

Dr. J.P. Morrissey