Micro and Macro in the Dynamics of Dilute Polymer Solutions

breakfastclerkMechanics

Jul 18, 2012 (4 years and 11 months ago)

324 views

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

Coarse-grained Models

Department of Chemical Engineering
Oldroyd-B 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):
Department of Chemical Engineering
The Non-dimensional Oldroyd-B Model

Viscosity Ratio

=
Weissenberg
Number

Department of Chemical Engineering
The High Weissenberg Number Problem



Computations with the Oldroyd-B 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)
,
DEVSS-G

(1995)
,
AVSS

(1996)
,
DAVSS

(1999)
,
DEVSS-TG

(2002)



Viscoelastic flow computations are not yet safe and routine!
Department of Chemical Engineering
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

Coarse-grained 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
.

Department of Chemical Engineering
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
Department of Chemical Engineering
Elements of a Molecular Theory


The polymer molecule can be replaced with
a coarse-grained model


There is a vast separation of time-scales
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
Department of Chemical Engineering
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

Department of Chemical Engineering
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

Department of Chemical Engineering
Conformation Tensor

Department of Chemical Engineering
Hookean Dumbbell and Oldroyd-B are the same!



Polymer stress is given by:


The conformation tensor is given by:
Equivalent to the Oldroyd-B model with

Department of Chemical Engineering
Shear Flows



Shear stress
:



Shear rate
:



Viscosity
:
Velocity,
V
y
x
z
Surface area,
A
Force,
F

h
Department of Chemical Engineering
Shear Thinning
Zero-shear-
rate
viscosity,
η
0

Power
law
regime


The Hookean Dumbbell model fails to predict shear thinning!
Department of Chemical Engineering
Extensional Flows
1
Λ

x
y
z
Extensional viscosity
Department of Chemical Engineering
Hookean Dumbbell Model in Extensional Flows



Polymer molecules unravel
close to a critical


Stress becomes unbounded
at


The polymer molecules
undergo a
coil-stretch
transition!

Department of Chemical Engineering
Finite Size Effects


The use of linear springs leads to the prediction of
an unbounded stretch


Bead-rod model


Most realistic


Difficult to simulate

N
k

Kuhn steps
Department of Chemical Engineering
Nonlinear Springs


The solution lies in using
non-linear springs that
account for the finite
contour length of the chain


The flexibility of the polymer
leads to different non-linear
force laws:



F
initely
E
xtensible
N
onlinear
E
lastic
(FENE) force law


Worm-like-chain (WLC) force law

Q/Q
max
F
Department of Chemical Engineering
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
Department of Chemical Engineering
Non-Linear Phenomenon need BDS



Non-linear microscopic effects cannot be accounted for
at the macroscopic scale without closure approximations


Example of a closed form equation is the FENE-P model




Brownian dynamics simulations lead to exact predictions
of models with non-linear physics on the microscopic scale
Department of Chemical Engineering
Filament Stretching Rheometer
10
0
10
1
10
2
10
3
0
1
2
3
4
5
6
7
Trouton
Ratio
Hencky
Strain
SM-1
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
Department of Chemical Engineering
Extensional Viscosity: FENE chains and Experiment



Need to adjust parameters to obtain a good fit to data

JOR, 44, 291-322, 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

Coarse-grained 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
Oldroyd-B 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 FENE-CR fluid around a sphere
and a cylinder
Department of Chemical Engineering


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 FENE-CR model


results in
loss of convergence

Flow of a FENE-CR model around a cylinder
Department of Chemical Engineering
The Continued Use of the Oldroyd-B Model



The notion of self-correction

(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
ultra-dilute solution
, the governing
equations for the Oldroyd-B model can be solved for
arbitrarily large values of
Department of Chemical Engineering
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
polynomial-based 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 Oldroyd-B!

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

Coarse-grained 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


Cross-over from
θ
to good solvent behaviour
Department of Chemical Engineering
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
Department of Chemical Engineering
Consequences of HI


Diffusivity ~ without HI


Diffusivity ~ with HI
M
1
M
1


HI ensures drag is conformation dependent
Department of Chemical Engineering
Coil-Stretch hysteresis


Two disparate states can
exist at the same strain rate
A cross-slot cell

Wi

Department of Chemical Engineering
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 strain-rates, but
different quench strains.
— Constant strain rate
— Constant stress
Sridhar, Nguyen, Prabhakar, Prakash, PRL
98,
167801 (2007)
Department of Chemical Engineering
Glassy dynamics and hysteresis


Experimental observations can be reproduced by
Brownian dynamics simulations
Wi
0
= 10
Wi
= 0.5
Department of Chemical Engineering
Successive Fine Graining



SFG
: A procedure to systematically increase
the number of beads in the chain


Bead-rod predictions obtained in the limit

N



N
k
(the number of
Kuhn steps
)

Department of Chemical Engineering
SFG & Experimental Observations


Benchmark data from
uniaxial extensional flow
experiments, carried out in
a filament stretching
rheometer, by Sridhar and
co-workers


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, 163-182.

Department of Chemical Engineering
Closure Approximations


HI can be incorporated in a constitutive
equation:


Zimm theory


Consistent Averaging


Gaussian Approximation
Department of Chemical Engineering
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)
Department of Chemical Engineering
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 “linear-viscoelastic”
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

Coarse-grained Models

Department of Chemical Engineering
Flow Around a Confined Cylinder


Geometry and Boundary Conditions

Department of Chemical Engineering
Region of Interest: Wake of the Cylinder

Region of Interest
(wake of the cylinder)
Department of Chemical Engineering
Symmetry Line Simplifications


The Conformation Tensor Equation is a
linear ODE
along
the Symmetry Line — can be integrated numerically

is the non-dimensional

component of conformation tensor
non-dimensionalized using


At steady state, this can be re-arranged to:
Department of Chemical Engineering
Nature of the Maxima


At the maxima along the symmetry line, denoted by
,


As a result,


If , then,
Department of Chemical Engineering
Downstream in Dilute Solutions
Coil-stretch 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 Oldroyd-B 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

Micro-Macro
Simulations

Velocity and
Stress fields

Microscopic
Physics

Conservation Laws

Coarse-grained Models

Department of Chemical Engineering
Viscoelastic Free Surface Flows
Slot-Coating 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)
Micro-Macro 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
Micro-Macro scheme is
stable and appears free
of the High Weissenberg
Number Problem
Bajaj, Bhat, Prakash, Pasquali, J. Non-Newtonian Fluid Mech. 140, 87, 2006
Department of Chemical Engineering
BCF with Non-Linear Dumbbells


HI and FENE have a significant influence


Need to extend to bead-spring chains
Bajaj, Bhat, Prakash, Pasquali, J. Non-Newtonian 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
non-linear microscopic
phenomenon
to obtain accurate predictions at the
macroscopic scale and to overcome
computational difficulties


The future lies in
closure approximations
and
Multi-scale simulations