Dynamics of polyelectrolytes andliving polymers in shear flow
P. B. Sunil Kumar
Department of Physics, IIT Madras.

1
Tuesday 10 August 2010

Hydrophilic

Hydrophobic(a) Spherical Micelle, (b) Cylindrical Micelle, (c) Reverse Micelle, (d) Bilayer,(e) Hexagonal phase, (f) Vesicles

Cylindrical micelles also called living polymers
Can undergo scission/recombination under thermal fluctuations
Tuesday 10 August 20102

Micellar solutions are examples of living polymers

Oscillating solid sphere
Shear induced gelation

Jayaraman et al. PRE, 67, 65301 (2003)

Shear banding

Liu et al. PRL, 77, 2121 (1996)
Lerouge et al.
PRL, 81, 5457 (1998)

Rheocaos : R. Bandyopadhyay, G. Basappa, and A. K. Sood, Phys. Rev. Lett. 84, 2022 (2000).
Tuesday 10 August 2010

Rheology of wormlike micellar fluids from BD and MD simulations

S334

J T Paddinget al3] Cates M E and Candau S J 1990 J. Phys.: Condens. Matter
2 6869Rheology of wormlike micellar fluids from BD and MD simulations
4] Raghavan S R and Kaler E W 2001 Langmuir 17 300
Padding and Boek (2004): Atomistic MD
5] Boek E S, Jusufi A, Loewen H and Maitland G C 2002 J. Phys.:
Matter[3] Condens.
Cates M E andCandau14
S J 9413
1990 J. Phys.: Condens. Matter 2 6
simulations erucate than for EHAC
6] Boek E S, Padding J T, Anderson V, Tardy P, Crawshaw J and [4]Pearson
J RS A
2005
J. ENon-Newton.
FluidMech.
RaghavanR andKaler
W 2001 Langmuir
17 300surfactants
to compute the mechanical[5] Boek E S, Jusufi A, Loewen H and Maitland G C 2002 J. Phys.
126 39[6] Boek E S, Padding J T, Anderson V, Tardy P, Crawshaw J and P
atomistic
micellessuch as7] properties
Bautista F, deofSantosJ M, Puig
J E and ManeroO 1999 J. Non-Newton.
Fluid Mech. 80 93
126 39elasticmodulus
and8] persistence
Padding J T andlength,
Boek E S
2004 Europhys.
Lett.
66 756[7] Bautista F, de Santos J M, Puig J E and Manero O 1999 J. Non[8] Padding J T and Boek E S 2004 Europhys. Lett. 66 756
9] scission
Padding J Tenergy
and Boek E S 2004 Phys. Rev. E 70 031502
Figure 1. Snapshot of an MD simulation of an EHAC wormlike micelle in a 3% NaCl solutio[9] Padding J T and Boek E S 2004 Phys. Rev. E 70 031502
the surfactant molecules are shown: carbon (light blue), oxygen (red), nitrogen (dark blu0] Kröger M and Makhloufi R 1996 Phys. Rev. E 53Only
2531[10] Kröger M and Makhloufi R 1996 Phys. Rev. E 53 2531
and hydrogen (white).
KrögerM
2004
Phys.
Rep.
390453
Kröger M 2004
Phys.mode
Rep. 390 453
MD simulations of micelles consisting of coarse-grained(CG)surfactants[11]Oelschlaeger
C,WatonG,
BuhlerE, Candau S J and Cates M E
1] Oelschlaeger
Waton
Buhler E,Candau
S J andCatesM E 2002 Langmuir
18 3076
to optimizeC,theCGG,modelwithrespect
tothestructure
factorS(q)oftheatomistic
2. Mechanical properties fromdynamics[12] molecular
Cates M E andCandau simulations
S J 2001 Europhys.
Lett. 55 887
2] Cates M E and Candau S J 2001 Europhys.
Lett. 55 887
and Yu Jto
1995
Phys. Rev. Lett. 74 4329
micelle at large values of the wavevector q. The[13]CGO’Shaughnessy
model isBused
extrapolate
3] O’Shaughnessy B and Yu J 1995 Phys.
Rev.atomistic
Lett. 74MD
4329
Using
simulations,wecanfindrelationships
between[14] Cates M E 1987 Macromolecules 20 2296 the chemistry an
thestructure
factor
for
small
qin
order
toobtainreliablevaluesthecarry
micelleproperties of a wormlike
micelle.
To thisPfor
end,
weoutKsimulationsof Condea sma[15] Briels
W J, Mulder
and denOtter W
2004 J. Phys.:4] Cates M E 1987 Macromolecules 20mechanical2296[16]consisting
Boek
E S,ofden
Otter
Wnumber
K, Briels
J and Iakovlev
D 2004inPhi
bending
persistence
length. micelle,segmentof aJ.
wormlikea limitedofW
surfactant
molecules,th5] Briels
W J,rigidity
Mulder Pandand denOtter W
K 2004
Phys.:Condens.
Matter16
S3965[17]isden
Otter W inK, water
Shkulipa
S A and Briels
W J 2003concentratio
J. Chem. Ph
example EHAC [5]. The worm
immersedcontaining
the required6] Boek
ES,
den
Otter
WK,
BrielsWJand
Iakovlev
D2004
Phil.
Trans.
R.
Soc.
A362
1625
Figure 2. Structure factors Sα (q) of a tensionless EHAC
α =x, y Typically,in black andredMagid[18]
L J 1998 J.box
Phys.hasChem.
B 102 4064
of saltworm;(NaCl)ions.
theMD simulation
dimensionsof the order of 10 nm
respectively.
ThedashedlineshowstheS(q)fora
coarse-grained
worm
simulation[17],using
5 Chem. Phys.
7] den Otter W K, Shkulipa S A and Briels
WJ
2003
J.
119
2363
[19]
DoiMandEdwards
SF
1986
The
Theoryof Polymer
Dynamic
three dimension
and contains O(10 ) atoms. Periodic boundary conditions are applied inσ = 0.6 nm.
[20] Padding
J T segment
and Brielsof
WaJ wormlike
2001 J. Chem.
Phys. 1152846 thso 4064
effectively we are considering
an infinite
micelle.
Typically,
8] Magid L J 1998 J. Phys. Chem. B 102
Paddingand Briels timeW J 2002
J. Chem.
Phys. 117meaningf
925
simulations
require ofDynamics
the order of(Oxford:
10 ns ofJ T
simulation
to obtain
statistically9] Doi M and Edwards S F 1986 The Theoryof Polymer
Clarendon)[21] AllenM P andTildesley
J 1987
Simulationof Liquensemble averages. A simulation
snapshotis shownin D
figure1. ComputerThis system
contains
640] Padding J T and Briels W J 2001 J. Chem.
Phys.
1152846[22]PaddingJTandBriels
WJ
2003+−J. Chem. Phys. 118 10276
surfactant and 40 000 water molecules together with Na and Cl ions corresponding to a 3%[23] Anderson V J 2005 unpublished results
Padding J T and Briels W J 2002 J. Chem.
Phys.
117solution.
925 Coulomb(by weight)
NaClinteractions are treated using Ewald summations.
First, Simulation
we calculate ofradialdistributionfunctionsas ensemble averages over the partic1] Allen M P and Tildesley D J 1987 Computer
Liquids(Oxford:Clarendon)
Padding,
BoekandBriels((2005):amesoscopic
modelofwormlike
micelles,Figure 3. (a) Mesoscopic model of a wormlike micelle.
(b) Thedynamics of From
break-up
and functions, we find a radius of the worm r = 2.3 nm. The
coordinate
trajectories.
these2] Padding
JTandBriels
WJ
2003
J.
Chem.
Phys.
11810276
recombination of a wormlike
micelle.
representedby chains
whichwecanbreakrecombine
canbybesubjectedtodetermine
the andtensionlessstate. This isandachieveda seriesof MD simulations,where w3] Anderson V J 2005 unpublished results
compress/stretch the worm at constant volume [16]. We observe that the surface tension !

shear flow. For this model, where the smallest length-scale is the persistence

decreases
linearlytowith
adial distribution function. The persistence lengthis calculated
be ofthethelengthorderof the worm segment L z . The elastic modulus K L is thelength, the elastic modulus , calculated
scissionenergy and persistence length are taken
from23 nm. This value is in agreement with experimental
valuesfor wormlike micellar
4KV
as input
fromWetheatomisticand methods
CGMD.
e lengths
reportedparameterin the literature [18].
are currently
developing
L for=(L z − L z 0 )(1−!P
g Tuesday
the scission
energyrequired
tobreakaworm,
aswellas
theactivation
barrier
LLzz0
10 August 2010

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Particle Dynamics ( DPD) is a particle based “meso-fluid”
F#$$#&%(#7+"&%'(#)*+".23%0#)$"8F[F?
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MD with special thermostat
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Tuesday 10 August 2010

MDPD-­‐	  	  Basic	  model
N "dpd" particles interacting via conservative, dissipative and random pair wise interactions

Dissipative & random forces related
through fluctuation-dissipation theorem
Pair-wise random and dissipative forces
conserve momentum
Resulting in correct description ofhydrodynamics
Soft interactions allow for longer time steps,and therefore much longer times can beprobed via DPD as opposed to MD• P. J. Hoogerbrugge and J. M. V. A. Koelman, Europhys. Lett. 19, 155 (1992)
Tuesday 10 August 2010

DPD model for living polymers ( LP)For,
W – W, W – I
W – A, I – I, I – A
For, A-A

r1r5r39

Tuesday 10 August 2010

k3
The three-body bending potential
U3body = (1 − rjk /r3 )2 (1 − r̂ij .r̂jk )(2I1 – A’
The aij is set to 25 in thesimulation.
1 – A2 & A’1 – A2 – I2Equilibrium simulations are carried out in a cubic bwith periodic boundary conditions in all directions. At
we have few long micelles and as we increases the concetr
longerwhich eventually forms net= k3branched(1 − r̂ij · r̂micelles
Three body potentials getUijkjk ) for A1I1A1’ and A2I2A2’eter set I). Figures 3 diﬀerent structures formed for diﬀ
Forof
LPshear we observe a fluidof micelles. In the absence
function of concentration.At a threshold concentration, tFor solvent
scattering function indicates Zimm-like dynamics at larg
diﬀusive dynamics at small wave vectors. The kinetics ofbination
demonstrates
existence of mean field andL = 40clearly, Periodic
Boundary Condition• Simulation box size
regimesof the dynamics
of particles
192000 of living polymers at low and• density ρ = 3 , number= 1.0 , A-A = 2.0• Resting length A-Irespectively[5].
apply shear to system by keeping the high density mo= 200• Spring constant kWe-6 secsystems are subjected to s
τ, withτ oftheof 10z =40r
Whenthese• Time step = 0.01 andc . order
10 I)
undergoes a transition
to a lamellar(parameter set
S. Thakur et al Soft Matter 6, 489, 2010
tures(parameter set II). And we observe a shear-thinnig
Tuesday 10 August 2010

Fluid-gel transition
C -- % ratio of number of monomers to the total number of particles
Phase behavior was characterized by diffusivity of trimers•Fluid phase at low C•Gel phase at higher C•Transition point C = Cp = 3.5•Branched point ---- coordination number of particle is > 2
Tuesday 10 August 201011

Organization of polymers
C=2Fluid

C=3

Increasing C

C=4GelC=512

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Segment length distributionlΦ

Average polymer length
Monomer concentration

Branching reduces the average segment
lengthCluster size distribution
branching of polymers lead torandom percolation• Exponential distribution at low C <Cp• Power law distribution at C = Cp:
random percolation• Gel phase contains spanning clusters
and small segments• M. E. Cates and S. J. Candau, J. Phys 2,6869 (1990)
Tuesday 10 August 201013

Self-intermediate scattering function
N: total number of monomers

Relaxation time

Zimm dynamics at large q:
Diffusive at low q:
Tuesday 10 August 2010

S. Thakur et al

Soft Matter 6, 489, 201014

First recombination time τR = scission time (t2 )–recombination time (t1 )Two classes of recombination kinetic: Mean field (MF) and Diffusion controlled (DC)• Ben O’ Shaughnessy and Jane Yu, PRL, 74, 4329 (1995)

tdtdtstd15
Tuesday 10 August 2010

stress relaxation•We find a residual stress is due to the spanning clusters with stretched bonds• Early time oscillation is the result of bond potential within a trimer• G. Faivre and J. L. Gardissat,
Macromolecule, 19, 1988 (1986)

1010

1

1

G(t)

G(t)

• G(t) decays as:

0.1

0.1

0.01

0.01
0.1

1t

C=3.0
Tuesday 10 August 2010

10100

0.1

1t

10

C=4.010016

ShearV*zyx-V*• Density of wall = 20* Density of fluids• Periodic boundary in x & y directions
17• Results were verified using Lees-Edward method to rule out wall effect
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Shear induced Lamellar phase• First layer forms near theshear boundaries• All the polymers within arange of r5 is pulled towards
the boundary• Next layer forms exactly at adistance r5 from the first•

S. Thakur et al
Tuesday 10 August 2010

Soft Matter 6, 489, 201019

Shear induced structures

xzxyy

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• Lamellar phase in the system with C = 518

Shear induced structures

xzyxy
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• Columnar phase in the system with C = 5• Layer spacing is decided by r520

Stress and viscosity•Alignment of polymers leads to decrease in viscosity
21
Tuesday 10 August 2010

1000
100n(s)Cluster size distribution for a tetramer
building block.

2%
2.5%
3%
4%
5%

Percolation at 3% concentration

1010.10.01
110100s

1000

10000

1000

Percolation at 2.75% concentration

For larger segments the percolation is at
a lower concentration

Tuesday 10 August 2010

100n(s)Cluster size distribution for a pentamer
building block.

2%
2.5%
2.6%
2.75%
3%
4%1010.10.01

110100s1000
2010000

Diffusion coefficient as a function of concentration for different building
blocks

0.001

D0.0001

1e-05

TrimerTetramer
Pentamer
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1/C wt %Tuesday 10 August 2010121

Trimer

a1 = a2 = 0.1

a1 = a2 = 0.5

a1 = a2 = 0.1

a1 = a2 = 1.0

Tetramer

Pentamer

a1 = a2 = 1.0

a1 = a2 = 1.0

Tuesday 10 August 2010γ̇ = 1L$)#+)2%M$3-52%FA;;>H!"!#$%&'( )*+,*-'./-'01/2*23-4+*'#567187
9')*+,*-'/.'":8*;;*+8*'.<+=*='36'!"#$%&'()"$*&+(,'&-$.#)#&/0"$1,2'3&+(,'Sc of DPD fluid is around 1 , while that for a real fluid isabout 1000 !!
B/33/C+(/D$%C+)(/,5$%27'+E/,3%FBGBH

Higher Sc fluid ?

:+'%I$%*/J$2%I7%43/'6%+%2/**$)$'(%(#$)E-3(+(%/'%BGBKFigure 4: Temperature (KB T )as a function of dpd timesteps inδt!"#$%& '(()"'*+,-!"#$./01$)&$0,2+$)3"&2'24,5!"#$%&'()*$ +,-.%'/)01-%$('")2$-2$(34)")$5-63%-7'%")4($-7')%5-6%3&-'-8'9/$((-.)5"%)+:")30;'((3/5-)0-7%)04)7($-"3-'.<:5"-"#$-=)0$&'")4-2)5435)",-53-"#'"-"#$-*"))$*2,6*+3722,0839$)>Sc :,;<= 63%-'-()?:).@-)5-3+"')0$.A
5,BC-DC-E3/$F-Europhys. Lett. >? !GHHHA-GIJ@A0'37*'B,(')'3$2$) !
'BB"#&,2",280$,2+$,C7&*"&72A!"#$%&'($)*+,$%-*%./*$0%1 &'2/+'%&'3(/(4($%-*%"$,#'-5-67%8+2)+39%:#$''+/9%&'2/+9%;<1=>%?+'@%A;;>

AA

Figure 5: Sc as a function of diﬀerent Γ both for serial and parall
Tuesday 10 August 2010

Early stage

Sc=7, γ̇=1

steady state

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Summary of living polymer results•

A coarse grained model was developed for studying the dynamics of living
polymer solutions• The model, based on Dissipative Particle Dynamics, predicts a fluid to geltransition as the concentration of polymer is increased• The self intermediate scattering function, scission and recombination kinetics
and stress correlation function shows that the model reproduces expected results.
• We show the emergence of lamellar and columnar phase when the solution issubjected to simple shear• The gelation point ( as evident from the cluster size distribution) shifts to lower
monomer concentration as the size of the basic building block ( molecularweight) is increased.
• For larger molecular weight, higher shear stress was required to obtain thelayering transition

We now switch to polyelectrolytes
22Tuesday 10 August 2010