Upload
arka-prabha-roy
View
69
Download
3
Tags:
Embed Size (px)
Citation preview
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