1!

1

Rheology, Diffusion and Plastic Correlations in Jammed Suspensions

Arka Prabha Roy

PhD Thesis Proposal Civil and Environmental Engineering

Carnegie Mellon University

Committee: Craig E. Maloney (advisor)

Jacobo Bielak Michael Widom

Alan J.H. McGaughey

March 25, 2014

2!Soft Jammed Materials

Particles can be •  Solid (colloidal particles) •  Fluid (oil droplets) •  Gas (air bubbles)

My work •  Athermal •  Deformable particle •  Jammed

� > �RCP

Technological applications •  Device Fabrication •  Coating Industry •  Food/ personal care

3!

Special property •  At low T and small load,

it behaves like an elastic solid

•  Above the yield stress, it flows like a viscous fluid

Yield stress behavior

[1]&Jyo*&R.&Seth&and&Robert&T.&Bonecazze,&The&University&of&Texas&at&Aus*n&[2]&Sudeep&DuEa&and&Daniel&Blair,&Georgetown&University&[3]&M.&E.&Mobius,&G.&Katgert&and&M.&Van&Hecke&&Lab,&Leiden&University&

Microgel suspension Oil-water emulsionFoam

4!Yield Stress development above Jamming transition (I)

Durian Group

Viscous : Yield stress :

� > �J

� < �J ⌘ ⇠ ��1/2

� = �Y +A��

Herschel Bulkley law

Nordstrom&et&al.&[PRL&105,&175701&(2010)]&

•  Dense aqueous colloidal suspension of hydrogel particles (NIPA)

•  10% polydispersity with average diameter 1micron

Solid like Viscous Fluid

4&

5!Yield Stress development above Jamming transition (II)

•  Simulations using a micro-mechanical model pressurelubrication~ pressureelastic

•  Experiments with hydrogel (pNIPA) particles and viscous oil-water emulsions

� = �Y +A��

Herschel Bulkley law

Seth&et&al.&[Nature&Materials&10,&838&(2011)]&

� = 1/2

6!Research question

How rheology of soft jammed suspensions is related to the diffusivity and spatial structure of the particle rearrangements?

Vertical displacement

� = 10�7

7!

~Fij =@V (rij)

@~rij

�

r

V (r) / �2

Particle Model

Overdamped dynamics: 50:50 bi-disperse mixture in simple shear

�t = 10�2⌧D

⌧D = b�20/✏

Timescale:

~FDi = b. ~�vi

~FEi =

X

j

~Fij

~ri = ~vi

~

�vi = ~vi � yi�x

~�vi =~FEi /b

7&b = 1D.J.&Durian&[PRL&75,&4780&(1995)]&

8!Rheology

•  Slow rate : yield stress behavior •  Fast rate : non-Newtonian rheology

� = �Y +A��Herschel Bulkley law

10!7

10!6

10!5

10!4

10!3

10!2

10!3

10!2

!

"

� / �1/3

Flow

Stress(�)

Shear rate (�)

�Yield

[1]&Ethan&PraE&and&Michael&Dennin&[PRE&67,&051402,&2003]&[2]&Mobius&et&al.&[Europhysics&LeEer&90,&44003&(2010)]&

Foam rheology experiment

[1]

[2]� ⇡ 0.36± 0.05

� = 1/3

9!Stress vs strain (low shear rate)

1.45 1.5 1.552.5

3

3.5

4

4.5

5

5.5x 10−3

!

"

Strain(�)

Stress(�)

Elastic loading

Plastic drop

At steady state •  Majority of time : loading

elastically •  Minority of time : plastic

dissipation

� = 10�7

10!Energy dissipated during the stress drop

1.45 1.5 1.550

20

40

60

80

100

!

!

Strain(�)

� = 10�7

1.45 1.5 1.552.5

3

3.5

4

4.5

5

5.5x 10−3

!

"

Elastic loading

Plastic drop

Stress(�)

Strain(�)

• � is energy dissipated per unit strain

• Like instantaneous decorrelation rate

• Energy change in a�ne deformation : �

• Input power

• Dissipation rate

Ono&et&al.&[PRE&67,&061503&(2003)]&

11!Overlapping plastic events for increasing rate

•  With increasing rate, motion becomes uniform, burstiness disappears

1.2 1.25 1.30

40

80

120

Strain(�)

�

1.2 1.25 1.30

0.002

0.004

0.006

0.008

0.01

Stress(�)

Strain(�)

8⇥ 10�5

8⇥ 10�3

1⇥ 10�7

10!7

10!6

10!5

10!4

10!3

10!2

10!3

10!2

!

"

slope : 1/3

Flow

stress

Shear rate (�)

12!Movies : Intermittent energy dissipation near yield stress

Slow : � = 10

�7

102

104

106

108

1010

10!5

10!3

10!1

!/!

P(!

)

1 ! 10"7

2 ! 10"7

4 ! 10"7

8 ! 10"7

P(�) plotted against �/�

10!4

10!2

100

102

104

10!5

10!3

10!1

!

P(!

)

1 ! 10"7

2 ! 10"7

4 ! 10"7

8 ! 10"7

Probability distribution of �

• Peak at low �

• Power law distribution

• Large � cuto↵

Quasistatic scaling : � / b · �v2/� / b�

13!Movies : Energy dissipation in flow

Fast : � = 8⇥ 10�3

� distributions are :

• power law at small �

• Gaussian at large �

10−2 100 10210−6

10−4

10−2

100

!

P(!

)

FastSlow

Probability distribution of � for di↵erent rates

14!

Fast rate

Slow rate

Plastic Displacements near yield stress and flow

•  Slow rate : transient slip-line like features, span the simulation cell.

•  Fast rate : correlated behavior goes

away.

Transverse displacement (�y) calculated over strain (�� = 2.5%)

10!7

10!6

10!5

10!4

10!3

10!2

10!3

10!2

!

"

slope : 1/3Stress(�)

Shear rate (�)

15!

(a) (b)Vertical&displacement&Horizontal&displacement&

C.E.&Maloney&and&M.&Robbins&[J.&Phys.:&Condens.&MaEer&20,&244128&(2008)]&

Characteristic strain in quasistatic limit

•  Displacement discontinuity:

•  Characteristic strain relieved in each event:

��TSL ⇠ a/L

a

Typical response in glass (Lennard Jones)

16!

•  Single particle displacement appears to be fickian and follows flat distribution:

•  Elementary slip lines have:

•  No. of events:

•  Mean Squared Displacement:

&&&

�y 2 [�a/2, a/2]

h�y2iTSL = a2/12

Ab-initio estimate : quasistatic diffusion

16&

�a/2 +a/2

P (�y)

�y

=

✓aL

12

◆��

•  Effective QS Diffusion constant

D = lim��!1

⌧�y2

2��

�=

aL

24

C.E.&Maloney&and&M.&Robbins&[J.&Phys.:&Condens.&MaEer&20,&244128&(2008)]&

NTSL = ��/(a/L)

h�y2i = NTSL · h�y2iTSL

17!Displacement statistics

10−2 100 10210−2

10−1

100

!!

!!y2"/!!

!"

� 2 [1⇥ 10�05, 8⇥ 10�03]

Di↵usive

Increasing Rate

10−2 100

10−1

100

!!

"

↵ = 3h�y2i2/h�y4iNon-Gaussian parameter

• Di↵usive/ Fickian : < �y2 >⇠ ��

• Gaussian : ↵ ⇡ 1

•  Slow rate :

•  Fast rate : Gaussian for all ��

Gaussian & Fickian above �� ⇠ 1

18!Long time diffusivity

D = lim��!1

⌧�y2

2��

�Effective Diffusion Coefficient :

L = 40

10−7 10−6 10−5 10−4 10−3 10−2

10−1

100

!

D

D

D / ��1/3

Shear rate (�)

DQS ⇡ 0.84• Displacement discontinuity

a ⇠ 0.5

• Strain needed for a slip

��TSL ⇠ 12L

19!System size dependence

10−7 10−5 10−3

10−1

100

!

D

L = 40L = 80

Di↵usion

D / ��1/3

10−1 100 101

10−3

10−2

L! 1/3

D/L

L = 40L = 80

slope: -1

•  Flow stress is size independent •  QS effective diffusion coefficient

grows linearly with size, but overall has same dependence on rate

10−7 10−5 10−310−3

10−2

!

"

L = 40L = 80

� / �1/3

Rheology

Flow

stress(�)

Shear rate (�)

20!Argument for diffusion-rheology coupling

•  Deformation occurs from uncorrelated slip lines of length

•  Linear elasticity :

•  Time to form a slip line

•  Diffusive approximation:

•  Assume:

•  Then,

D / ⇠ ln(L/⇠)

finite rates : ⇠ << L

� � �Y / µ⌧ �

⇠2 ⇠ ⌧

⇠

⌧

� � �Y / µ⇠2� / µD2�

/ µ�1/3

A.&Lemaitre&and&C.&Caroli&[PRL&103,&065501&(2009)]&

21!Spatial structure of vertical displacement field

x

y

102 ! Cuy

−40 0 40−40

0

40

0

0.2

0.4

0.6

0.8

1

1.2

102 ⇥ Cuy

Slow : � = 10

�7

Cuy (~R) = huy(~r + ~R)uy(~r)i~r

Real space displacement-displacement correlation function,

•  Strong correlations along Y correspond to the vertically extended features in the displacement image.

22!Correlation in vertical displacement field

x

y

102 ! Cuy

−40 0 40−40

0

40

0

0.2

0.4

0.6

0.8

1

1.2

102 ⇥ Cuy

10 20 30 40−0.4

−0.2

−0

0.2

0.4

0.6

0.8

1

x

Cuy(x

)/C

uy(x

=1)

! = 10!7

! = 10!6

! = 10!5

! = 10!4

! = 10!3

Cuy along x

Decreasing rate

10−8 10−6 10−4 10−2100

101

102

!

" L

Strain rate (�)

slope : �1/3

Correlationlength(⇠)

•  Characteristic length varies with strain rate. •  At very slow rate the correlation saturates at system size. •  Agrees with the argument that effective diffusion

coefficient goes with the length-scale.

⇠ / ��1/3

A.&Lemaitre&and&C.&Caroli&[PRL&103,&065501&(2009)]&

23!Power spectrum of vertical displacement

10−2 10−1100

101

102

103

104

105

kx/2!

Suy/Suy(kx=

!)

" = 10!7

" = 10!6

" = 10!5

" = 10!4

" = 10!3

Along kx

10−2 10−1100

101

102

103

104

105

ky/2!

Suy/Suy(ky=

!)

" = 10!7

" = 10!6

" = 10!5

" = 10!4

" = 10!3

Along ky

Transverse :

• peaks are observed for high �

• quasi-static : / k�2.5x

Longitudinal :

• relatively insensitive to �

• quasi-static : / k�1.5y

kx/2!ky/2!

log10(Suy)

−1 0 1−1

0

1

−2

−1

0

1

2

3

� = 10�7

log10(Suy )

Suy (~k) =h|uy(~k)|2i

Np

•  Power in displacement:

24!Slip lines visible in velocity field

Slow : � = 10

�7

Active

Quiescent

Typical

24&

Slow rate

• Slip-line like features are

visible when active

Fast rate

• Short length correlations observed

at each instant - surprising

Fast : � = 8⇥ 10�3

25!Velocity correlations

x

y

1010 ! Cvy

−40 0 40−40

0

40

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.41010 ⇥ Cvy

Slow : � = 10

�7

10 20 30 40−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

Cvy(x

)/C

vy(x

=1)

! = 10!7

! = 10!6

! = 10!5

! = 10!4

! = 10!3

Correlations in vertical velocities along x

⇠ / ��1/3

Correlation length

Cvy (~R) = hvy(~r + ~R)vy(~r)i~r•  Real space velocity correlation

26!Yielding mechanism

•  Local rearrangement (yields) •  Redistributes stress according to

Eshelby •  Strain field:

✏xy

=

a2�✏0⇡

cos 4✓

r2

Force dipole

Eshelby solution [Picard et al., 2004]

Stress response due to shear transformation

Picard&et&al.&[The&European&Physical&Journal&15,&371&(2004)]&

27!Strain correlation

x

y

103 ! C!

−40 0 40−40

0

40

−0.05

0

0.05

0.1103 ⇥ C✏

Slow : � = 10

�7•  Symmetrized strain

•  Real space correlation

✏ =1

2

✓@u

y

@x

+@u

x

@y

◆

C✏(~R) = h✏(~r + ~R)✏(~r)i~r

•  Quadrupolar symmetry, similar to Eshelby

•  Corrrelation decays with rate,

•  Sharp cutoff for fast rates

C✏ /1

r

100 101 10210−3

10−2

10−1

100

101

x

C!(x)/C

!(x=

1)

" = 10!7

" = 10!6

" = 10!5

" = 10!4

" = 10!3

Strain correlations along x for different rates

C✏ / r�1

28!Eshelby response in strain-rate field

Strain per unit time : ✏ = 12

�@v

x

@y

+ @v

y

@x

�

C✏(~R) = h✏(~r + ~R)✏(~r)i~r

x

y

1012 ! C !

!40 0 40!40

0

40

!1

!0.5

0

0.5

11012 ⇥ C✏

� = 10�7

x

y

107 ! C !

!40 0 40!40

0

40

!0.5

!0.25

0

0.25

0.5107 ⇥ C✏

� = 10�3

100

101

102

10!4

10!3

10!2

10!1

100

101

x

C!(x

)/C

!(x

=1)

" = 10!7

" = 10!6

" = 10!5

" = 10!4

" = 10!3

C✏ / r�2

Strain-rate correlation for different rate

29!Summary

•  Bursty, intermittent dissipation at low rate

•  Smooth, uniform motion at fast rate

•  Yield stress behavior with no-Newtonian rheology

•  Effective diffusion decreases with increasing rate

•  Correlation length saturates to system size at QS regime, decays with rate in a similar power law,

•  Strain rate correlations show Eshelby response, •  Strain correlations show similar quadrupolar

symmetry but the response is much long ranged than Eshelby,

D / ��1/3

⇠ / ��1/3

� � �Y / �1/3

10−8 10−6 10−4 10−2100

101

102

CorrLength(⇠ L

)

slope : �1/3

DisplacementVelocity

Shear rate (�)

10−7 10−6 10−5 10−4 10−3 10−2

10−1

100

!

D

D

D / ��1/3

Shear rate (�)

10!7

10!6

10!5

10!4

10!3

10!2

10!3

10!2

!

"

� / �1/3

Flow

Stress(�)

Shear rate (�)

C✏ /1

r2

C✏ /1

r

30!Proposed and future work

Exhaustive study of Pair Drag Model

•  Diffusion •  Rheology •  Correlation

Role of inertia on the spatial structures and correlations

10−5 10010−4

10−3

10−2

10−1

!"D

#xy

0.06250.25141664

! !1/2

Rheology for Pair Drag model

� � �Y / �1/2

~FDi = b

X

j

(~vi � ~vj)

~vi ~vj

Drag force :

Experiments with microgel suspension shows,

� � �Y / �1/2

31!Acknowledgements

•  Kamran Karimi and Akanksha Garg •  Members of NTPL (Prof. Alan McGaughey)

Thank You&