Department of Chemical Engineering
Micro and Macro in the
Dynamics of Dilute Polymer
Solutions
Ravi Prakash Jagadeeshan
Department of Chemical Engineering
Complex Fluid Mechanics
Kinetic
Theory
(Microscopic)
Continuum
Mechanics
(Macroscopic)
Constitutive
Equations
Experimental
Validation
Homogeneous
Flows
Complex
Flows
Closed form
equations
Velocity and
Stress fields
Simulations
(BDS etc)
Simulations
(FEM etc)
Rheological Properties in
Shear and Extensional Flows
Stress
Calculator
Phenomenology
Microscopic
Physics
Conservation Laws
Coarsegrained Models
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OldroydB Model for Polymer Solutions
In polymer solutions, solvent contribution to stress
is significant. Total stress tensor :
Solvent contribution to stress:
Polymer contribution to stress (UCM Model):
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The Nondimensional OldroydB Model
Viscosity Ratio
=
Weissenberg
Number
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The High Weissenberg Number Problem
Computations with the OldroydB model
break down
at
Weissenberg number
Accompanied by
large stresses and stress gradients
in
narrow regions of the flow domain
Driving force
for the development of
Numerical techniques

EEME
(1989),
EVSS
(1990),
DEVSS
(1995)
,
DEVSSG
(1995)
,
AVSS
(1996)
,
DAVSS
(1999)
,
DEVSSTG
(2002)
Viscoelastic flow computations are not yet safe and routine!
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HWNP in Benchmark Flows
•
Most computations of benchmark flow
break down
at
Flow around a cylinder
Lid driven cavity
4:1 Contraction flow
Department of Chemical Engineering
Complex Fluid Mechanics
Kinetic
Theory
(Microscopic)
Continuum
Mechanics
(Macroscopic)
Constitutive
Equations
Experimental
Validation
Homogeneous
Flows
Complex
Flows
Closed form
equations
Velocity and
Stress fields
Simulations
(BDS etc)
Simulations
(FEM etc)
Rheological Properties in
Shear and Extensional Flows
Stress
Calculator
Phenomenology
Microscopic
Physics
Conservation Laws
Coarsegrained Models
Department of Chemical Engineering
The Molecular Approach
Microstructure cannot be ignored for
complex
fluids.
Aim of
molecular rheology
is to predict properties
from a molecular point of view.
Chemical structure of
different polymer molecules is
not important because of the
existence of
universal behavior
.
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Origin of Universal Behavior
The large scale features of a polymer
molecule, such as:
its stretchability
its orientability
and the large number of degrees of freedom
are responsible for its universal behavior
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Elements of a Molecular Theory
The polymer molecule can be replaced with
a coarsegrained model
There is a vast separation of timescales
between the motion of the polymer and the
solvent molecules
The solvent molecules are replaced by the
influence they have on the polymer
molecules. They exert:
a
Brownian
force
a
drag
force
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Equation of motion
HYDRODYNAMIC
FORCE
BROWNIAN
FORCE
SPRING
FORCE
NO
INERTIA
The Hookean Dumbbell Model
F
b
F
s
F
d
F
d
F
s
Q
F
b
Dumbbell
model
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The Stress Tensor
The stress tensor describes the
forces transmitted across an
arbitrary plane in the fluid
The polymer molecules contribute
to the stress
by carrying momentum across the
surface
by straddling the plane
n
Stress is given by Kramers expression:
where is the stress tensor
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Conformation Tensor
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Hookean Dumbbell and OldroydB are the same!
Polymer stress is given by:
The conformation tensor is given by:
Equivalent to the OldroydB model with
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Shear Flows
Shear stress
:
Shear rate
:
Viscosity
:
Velocity,
V
y
x
z
Surface area,
A
Force,
F
h
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Shear Thinning
Zeroshear
rate
viscosity,
η
0
Power
law
regime
The Hookean Dumbbell model fails to predict shear thinning!
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Extensional Flows
1
Λ
x
y
z
Extensional viscosity
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Hookean Dumbbell Model in Extensional Flows
Polymer molecules unravel
close to a critical
Stress becomes unbounded
at
The polymer molecules
undergo a
coilstretch
transition!
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Finite Size Effects
The use of linear springs leads to the prediction of
an unbounded stretch
Beadrod model
•
Most realistic
•
Difficult to simulate
N
k
Kuhn steps
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Nonlinear Springs
The solution lies in using
nonlinear springs that
account for the finite
contour length of the chain
The flexibility of the polymer
leads to different nonlinear
force laws:
F
initely
E
xtensible
N
onlinear
E
lastic
(FENE) force law
Wormlikechain (WLC) force law
Q/Q
max
F
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FENE Models in Extensional Flows
b
:
dimensionless maximum stretchable length of the molecules
Bounded stresses and
extensional viscosity
The
Hookean Dumbbell
model is recovered as
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NonLinear Phenomenon need BDS
Nonlinear microscopic effects cannot be accounted for
at the macroscopic scale without closure approximations
Example of a closed form equation is the FENEP model
Brownian dynamics simulations lead to exact predictions
of models with nonlinear physics on the microscopic scale
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Filament Stretching Rheometer
10
0
10
1
10
2
10
3
0
1
2
3
4
5
6
7
Trouton
Ratio
Hencky
Strain
SM1
Fluid
Tr
=
3
(a)
(b)
(c)
(d)
Toronto
De
= 12
Monash
De
= 14
M.I.T.
De
= 17
Strain
Extensional Viscosity
Extensional viscosity first measured accurately at Monash University
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Extensional Viscosity: FENE chains and Experiment
Need to adjust parameters to obtain a good fit to data
JOR, 44, 291322, 2000
Department of Chemical Engineering
Complex Fluid Mechanics
Kinetic
Theory
(Microscopic)
Continuum
Mechanics
(Macroscopic)
Constitutive
Equations
Experimental
Validation
Homogeneous
Flows
Complex
Flows
Closed form
equations
Velocity and
Stress fields
Simulations
(BDS etc)
Simulations
(FEM etc)
Rheological Properties in
Shear and Extensional Flows
Stress
Calculator
Phenomenology
Microscopic
Physics
Conservation Laws
Coarsegrained Models
Department of Chemical Engineering
Physics in the Constitutive Equation
Rallison and Hinch (1988) argued that the HWNP has
a physical origin in the
unrealistic behaviour
of the
OldroydB model
Rallison and Hinch suggest the use of a finitely
extensible spring
Chillcott & Rallison (1988) model a dilute solution as a
suspension of
finitely extensible
dumbbells
Examine the flow of a FENECR fluid around a sphere
and a cylinder
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CR found that at high
,
polymers are most
highly
stretched
close to the rear
stagnation point
in the flow
around a cylinder.
Stagnation
Points
Region of
High Stretch
Solutions exist
for all examined for the FENECR model
results in
loss of convergence
Flow of a FENECR model around a cylinder
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The Continued Use of the OldroydB Model
The notion of selfcorrection
(Any real flow will adapt in order to avoid infinite stresses)
Solutions exist for
arbitrarily large values of , it’s just
that current numerical techniques are unable to resolve
the high stress and stress boundary layers
Wapperom and Renardy (2005) used a
Lagrangian
technique
to solve viscoelastic flow past a cylinder and
showed that for an
ultradilute solution
, the governing
equations for the OldroydB model can be solved for
arbitrarily large values of
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The Log Conformation Tensor Model
Stress is
exponential
in regions of high deformation
rates or stagnation points
Numerical instability
caused by failure to balance
exponential growth with convection
Inappropriateness of
polynomialbased approximations
to represent the stress
Resolution is to change variable to the
matrix logarithm
of the
conformation tensor
Fattal & Kupferman (2004)
Still no sign of mesh convergence for OldroydB!
Department of Chemical Engineering
Complex Fluid Mechanics
Kinetic
Theory
(Microscopic)
Continuum
Mechanics
(Macroscopic)
Constitutive
Equations
Experimental
Validation
Homogeneous
Flows
Complex
Flows
Closed form
equations
Velocity and
Stress fields
Simulations
(BDS etc)
Simulations
(FEM etc)
Rheological Properties in
Shear and Extensional Flows
Stress
Calculator
Phenomenology
Microscopic
Physics
Conservation Laws
Coarsegrained Models
Department of Chemical Engineering
Physics at the Mesoscale
Two important long range interactions between
different parts of the polymer chain must be
taken into account
–
Hydrodynamic Interactions
For an accurate prediction of
dynamic
properties:
Rheological behaviour
Excluded Volume Interactions
For an accurate prediction of
static
properties:
Scaling with molecular weight
Crossover from
θ
to good solvent behaviour
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Hydrodynamic Interactions
The motion of one bead
disturbs the solvent velocity
field near another bead
The presence of
hydrodynamic interactions
couples the motion of one
bead to the motion of all the
other beads
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Consequences of HI
Diffusivity ~ without HI
Diffusivity ~ with HI
M
1
M
1
HI ensures drag is conformation dependent
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CoilStretch hysteresis
Two disparate states can
exist at the same strain rate
A crossslot cell
Wi
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Stress hysteresis
Multiple values of stress at identical strain rates!
The strains at quench are
indicated next to the triangle
symbols.
Triangles of the same color
have nearly the same post
quench strainrates, but
different quench strains.
— Constant strain rate
— Constant stress
Sridhar, Nguyen, Prabhakar, Prakash, PRL
98,
167801 (2007)
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Glassy dynamics and hysteresis
Experimental observations can be reproduced by
Brownian dynamics simulations
Wi
0
= 10
Wi
= 0.5
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Successive Fine Graining
SFG
: A procedure to systematically increase
the number of beads in the chain
Beadrod predictions obtained in the limit
N
→
N
k
(the number of
Kuhn steps
)
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SFG & Experimental Observations
Benchmark data from
uniaxial extensional flow
experiments, carried out in
a filament stretching
rheometer, by Sridhar and
coworkers
Results are for 2 million
molecular weight
polystyrene
Parameter free
predictions
0
2
4
6
8
1
0
1
0
5
1
0
6
1
0
7
1
0
8
1
0
9
!
"
p
!
M
o
d
e
l
E
x
p
t
.
(
a
)
W
i
=
9
.
6
0
2
4
6
8
1
0
1
0
5
1
0
6
1
0
7
1
0
8
1
0
9
!
"
p
!
M
o
d
e
l
E
x
p
t
.
(
b
)
W
i
=
1
8
0
2
4
6
8
1
0
1
0
5
1
0
6
1
0
7
1
0
8
1
0
9
!
"
p
!
M
o
d
e
l
E
x
p
t
.
(
c
)
W
i
=
4
7
0
2
4
6
8
1
0
1
0
5
1
0
6
1
0
7
1
0
8
1
0
9
!
"
p
!
M
o
d
e
l
E
x
p
t
.
(
d
)
W
i
=
8
5
Prabhakar, R., Prakash, J. R. and Sridhar, T. (2004)
J. Rheol.,
116, 163182.
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Closure Approximations
HI can be incorporated in a constitutive
equation:
Zimm theory
Consistent Averaging
Gaussian Approximation
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Hysteresis prediction
The Gaussian approximation assumes the distribution to be
Gaussian
Accounts for
fluctuations
in hydrodynamic interactions
Prabhakar, Prakash and Sridhar, JOR, 506, 925, 2006
Department of Chemical Engineering
CaBER
Measurements of filament radius can be
used to determine relaxation times
Basis for the development of the
Ca
pillary
B
reakup
E
xtensional
R
heometer
CaBER
Newtonian
Filament
Viscoelastic
Filament
Anna & McKinley, J. Rheol., 45, 115 (2001)
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Concentration dependent
λ
Clasen, Verani, Plog, McKinley, Kulicke, Proc. XIV Int. Congr. On Rheology, 2004, Korea
Recent experiments by
Clasen et al. show a
puzzling dependence of
relaxation time on
concentration even when
c << c
*
!
What is the source of
this concentration
dependence?
Department of Chemical Engineering
Comparison With Experiment
Inclusion of HI though the two
fold normal approximation
Agreement over a wide range
of concentrations
All curve fits are obtained with
the same “linearviscoelastic”
relaxation time
Prabhakar, Prakash and Sridhar, JOR, 506, 925, 2006
Department of Chemical Engineering
Complex Fluid Mechanics
Kinetic
Theory
(Microscopic)
Continuum
Mechanics
(Macroscopic)
Constitutive
Equations
Experimental
Validation
Homogeneous
Flows
Complex
Flows
Closed form
equations
Velocity and
Stress fields
Simulations
(BDS etc)
Simulations
(FEM etc)
Rheological Properties in
Shear and Extensional Flows
Stress
Calculator
Phenomenology
Microscopic
Physics
Conservation Laws
Coarsegrained Models
Department of Chemical Engineering
Flow Around a Confined Cylinder
Geometry and Boundary Conditions
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Region of Interest: Wake of the Cylinder
Region of Interest
(wake of the cylinder)
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Symmetry Line Simplifications
The Conformation Tensor Equation is a
linear ODE
along
the Symmetry Line — can be integrated numerically
is the nondimensional
component of conformation tensor
nondimensionalized using
At steady state, this can be rearranged to:
Department of Chemical Engineering
Nature of the Maxima
At the maxima along the symmetry line, denoted by
,
As a result,
If , then,
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Downstream in Dilute Solutions
Coilstretch transition is responsible for the HWNP!
Computations beyond a threshold
Wi
are not
possible using FEM as simulations break down
as
Department of Chemical Engineering
Upper bound on
Wi
?
Is Wi
≈
0.7 the maximum computable
Wi
for OldroydB fluids?
and
Bajaj, Pasquali, Prakash, To appear in J. Rheol., 2007
Department of Chemical Engineering
Complex Fluid Mechanics
Kinetic
Theory
(Microscopic)
Continuum
Mechanics
(Macroscopic)
Experimental
Validation
MicroMacro
Simulations
Velocity and
Stress fields
Microscopic
Physics
Conservation Laws
Coarsegrained Models
Department of Chemical Engineering
Viscoelastic Free Surface Flows
SlotCoating Flow
The image part with relationship ID rId4 was not found in the ﬁle.
The image part with
relationship ID rId5 was not
found in the ﬁle.
WEB
AIR
OUTFLOW
COATING BEAD
DIE
The image part with
relationship ID rId6 was not
found in the ﬁle.
W
a
l
l
M
o
v
i
n
g
S
u
b
s
t
r
a
t
e
F
i
n
i
t
e
E
l
e
m
e
n
t
M
e
s
h
E
n
s
e
m
b
l
e
o
f
P
o
l
y
m
e
r
M
o
l
e
c
u
l
e
s
I
n
f
l
o
w
O
u
t
f
l
o
w
Finite Elements/Brownian
Configuration Fields Method
Collaboration with Matteo
Pasquali (Rice)
MicroMacro Simulations
Department of Chemical Engineering
Stress, Velocity & Conformations
Re = 0.0, Ca=0.1, Q=0.3,
β
=0.75
Stress
Velocity
STRONGLY
EXTENDED
RELAXED
Conformations
MicroMacro scheme is
stable and appears free
of the High Weissenberg
Number Problem
Bajaj, Bhat, Prakash, Pasquali, J. NonNewtonian Fluid Mech. 140, 87, 2006
Department of Chemical Engineering
BCF with NonLinear Dumbbells
HI and FENE have a significant influence
Need to extend to beadspring chains
Bajaj, Bhat, Prakash, Pasquali, J. NonNewtonian Fluid Mech. 140, 87, 2006
Department of Chemical Engineering
Acknowledgents
Students & Postdocs:
Dr Satheesh Kumar
Dr Prabhakar Ranganathan
Dr P Sunthar
Dr Mohit Bajaj
Funding Sources:
ARC Discovery
Monash University
VPAC Expertise program
Department of Chemical Engineering
Conclusions
With kinetic theory, nearly
quantitative
agreement
with experimental results can be obtained for both
macroscopic
and
microscopic
quantities
It is essential to include
nonlinear microscopic
phenomenon
to obtain accurate predictions at the
macroscopic scale and to overcome
computational difficulties
The future lies in
closure approximations
and
Multiscale simulations
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