Polymer Rheology
(高分子流變學)
Instructor: Prof. Chi-Chung Hua
(華繼中 教授)
Complex Fluids & Molecular Rheology Laboratory,
National Chung Cheng University, Chia-Yi 621, Taiwan, R.O.C.
國立中正大學
複雜流體暨分子流變實驗室
Homepage: http://www.che.ccu.edu.tw/~rheology/

Textbook
R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids. Vol
I: Fluid Mechanics, 2nd edition, Wiley-Interscience (1987).

References
1. R. G. Larson, The Structure and Rheology of Complex Fluids, Oxford University
Press (1998).
2. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford Science:
New York (1986).
3. C. W. Macosko, Rheology-Principles, Measurements, and Applications, WileyVCH (1994).
4. G. G. Fuller, Optical Rheometry of Complex Fluids, Oxford University Press
(1995).
Scope and Goal

Rheology is a science that concerns the mechanical stresses
arising during processing of complex fluids, as well as the
microstructures that develop in responses to the external flow.

This course focuses on the phenomenology, general concepts,
analytical tools, and applications that are central to the interest
of researchers and engineers in related fields.

Rheology will not necessarily become your expertise after this
course; rather, you might find yourself indulged in a fantastic
world rich in the physics of a broad diversity of fluids.
Course Outline

Non-Newtonian Flows: Phenomenology (2 weeks)

Mechanical Characterizations: Measurements and Material Functions (3 weeks)

Optical Characterizations: Flow Birefringence/Dichroism and Light Scattering (3
weeks)

General Analyses: Scaling Laws, Time-Temperature Superposition, Solvent Quality,
and Fundamental Material Constants (3 weeks)

Constitutive Equations and Modeling of Complex Flow Processing (2 weeks)

Colloidal Rheology (1 week)

Student Presentations on Ongoing Researches and Future Perspectives (2 weeks)
Chapter Zero
Introduction of Rheology
Terminology

What is Rheology?
It normally refers to the flow and deformation of “non-classical materials”
or the so-called non-Newtonian Fluids.

What are “non-classical materials” ?
They include rubber, molten plastics, polymer solutions, slurries & pastes,
electrorheological fluids, blood, muscle, composites, soils, paints etc.
[Excerpt from the website of the Institute of Non-Newtonian Fluid Mechanics (INNFM),
http://innfm.swan.ac.uk/innfm_mms/index.html]
Rheological Properties—from Microscopy to
Macroscopy
Kinetic Theory
Fluid Mechanics
Rheological parameters acting as a “link” between monomer structure and final properties of a polymer.
[Reproduce from M. Gahleitner, “Melt rheology of polyolefins”, Prog. Polym. Sci., 26, 895 (2001).]
Rheological Circle
[Reproduced from C. Clasen and W. M. Kulicke, “Determination of viscoelastic and rheo-optical material
functions of water-soluble cellulose derivatives”, Prog. Polym. Sci., 26, 1839 (2001).]
Chapter I
Non-Newtonian Flows:
Phenomenology
“The mountains flowed before the Lord”
[From Deborah’s Song, Judges, 5:5]
Contents of Chapter I

Viscosity Thinning/Thickening (pp. 60-61)

Normal Stress Differences and Elasticity (pp. 62-69, 72-83)

Thixotropy

The Deborah/Weissenberg numbers (pp. 92-95)

Flow Regimes of Typical Processing

Secondary Flows and Instabilities (pp.69-72)

Length/Time scales & Probing Techniques
什 麼 是 流 變 (Rheology)?

Rheology is the science of fluids. More specifically, the study of
Non-Newtonian Fluids
Y

流體














牛頓流體
- 水、有機小分子溶劑等
V
 yx  
V
V
Y
黏度η為定值
非牛頓流體
- 高分子溶液、膠體等
Small molecule
Macromolecule
V
●
Newton’s law of viscosity
Deformable
黏度不為定值
(尤其在快速流場下)
I.1 Shear Thinning/Thickening
Dilatants
(Shear thickening)
Dilatants
(Shear thickening)
Newtonian Fluids
0 pleatau
Newtonian Fluids
Pseudoplastics
(Shear thinning)
Pseudoplastics
(Shear thinning)
(a) Shear stress vs shear rate and (b) log viscosity vs log shear rate for Dilatants, Newtonian fluids
and Pseudoplastics. For very high shear rates the pseudoplastic material reaches a second Newtonian
pleatau.
[Reproduced from G. M. Kavanagh and S. B. Ross-Murphy, “Rheological characterisation of polymer
gels”, Prog. Polym. Sci., 23, 533 (1998).]
I.1 Shear Thinning/Thickening (cont.)
Tube flow and “shear thinning”. In each part, the Newtonina
behavior is shown on the left (N); the behavior of a polymer
on the right (P). (a) A tiny sphere falls at the same rate through
each; (b) the polymer flows out faster than the Newtonian fluid.
[Reproduced from R. B. Bird, R. C. Armstrong and O. Hassager,
Dynamics of Polymeric Liquids. Vol I: Fluid Mechanics,
2nd edition, Wiley-Interscience (1987), p. 61.]
[Retrieved from the video of
Non-Newtonian Fluid Mechanics
(University of Wales Institute of
Non-Newtonian Fluid Mechanics,
2000)]
I.2 Normal Stress Difference and Elasticity

Rod-Climbing
Fixed cylinder with rotating rod. (N) The Newtonian
liquid, glycerin, shows a vortex; (P) the polymer solution,
polyacrylamide in glycerin, climbs the rod.
[Reproduced from R. B. Bird, R. C. Armstrong and O. Hassager,
Dynamics of Polymeric Liquids. Vol I: Fluid Mechanics,
2nd edition, Wiley-Interscience (1987), p. 63.]
[Retrieved from the video of
Non-Newtonian Fluid Mechanics
(University of Wales Institute of
Non-Newtonian Fluid Mechanics,
2000)]
I.2 Normal Stress Difference and Elasticity
(cont.)

Extrudate Swell (also called “die swell”)
Behavior of fluid issuing from orifices. A stream of Newtonian
fluid (N, silicone fluid) shows no diameter increase upon
emergence from the capillary tube; a solution of 2.44 g of
polymethylmethacrylate (Mn = 106 g/mol) in 100 cm3 of
dimethylphthalate (P) shows an increase by a factor in diameter
as it flows downward out of the tube.
[Reproduced from A. S. Lodge, Elastic Liquids,
Academic Press, New York (1964), p. 242.]
[Retrieved from the video of
Non-Newtonian Fluid Mechanics
(University of Wales Institute of
Non-Newtonian Fluid Mechanics,
2000)]
I.2 Normal Stress Difference and Elasticity
(cont.)

Tubeless Siphon
When the siphon tube is lifted out of the fluid,
the Newtonian liquid (N) stops flowing; the
macromolecular fluid (P) continues to be
siphoned.
[Reproduced from R. B. Bird, R. C. Armstrong
and O. Hassager, Dynamics of Polymeric Liquids.
Vol I: Fluid Mechanics, 2nd edition, WileyInterscience (1987), p. 74.]
[Retrieved from the video of
Non-Newtonian Fluid Mechanics
(University of Wales Institute of
Non-Newtonian Fluid Mechanics,
2000)]
I.2 Normal Stress Difference and Elasticity
(cont.)

Elastic Recoil
An aluminum soap solution, made of aluminum
dilaurate in decalin and m-cresol, is (a) poured from
a beaker and (b) cut in midstream. In (c), note that
the liquid above the cut springs back to the breaker
and only the fluid below the cut falls to the container.
[Reproduced from A. S. Lodge, Elastic Liquids,
Academic Press, New York (1964), p. 238.]
A solution of 2% carboxymethylcellulose
(CMC 70H) in water is made to flow under a
pressure gradient that is turned off just before
frame 5.
[Reprodeced from A. G. Fredrickson, Principles
and Applications of Rheology, © Prentice-Hall,
Englewood cliffs, NJ (1964), p. 120.]
I.3 Time-dependent effect_Thixotropy
Thixotropy behavior
Anti-thixotropy behavior
A decrease (thixotropy) and increase (anti-thixotropy) of the apparent viscosity
with time at a constant rate of shear, followed by a gradual recovery when the
motion is stopped
The distinction between a thixotropic fluid and a shear thinning fluid:
 A thixotropic fluid displays a decrease in viscosity over time at a constant
shear rate.
 A shear thinning fluid displays decreasing viscosity with increasing shear
rate.
I.3 The Deborah/Weissenberg Number

Dimensionless groups in Non-Newtonian fluid mechanics:
the Deborah number (De)
De   / t flow
: the characteristic time of the fluid, tflow: the characteristic time of the flow system
the Weissenberg number (We)
We  
: the characteristic strain rate in the flow

Dimensionless groups in Newtonian fluid mechanics:
the Reynolds number (Re)
Re  LV  / 
L: the characteristic length; V,  and  are the velocity, the density and the viscosity of fluid
I.3 The Deborah/Weissenberg Number (cont.)
Streak photograph showing the streamlines for the flow downward through an axisymmetric sudden
contraction with contraction ratio 7.675 to 1 as a function of De. (a) De = 0 for a Newtonian glucose syrup.
(b-e) De = 0.2, 1, 3 and 8 respectively for a 0.057 % polyacrylamide glucose solution.
[Reproduced from D. B. Boger and H. Nguyen, Polym. Eng. Sci., 18, 1038 (1978).]
I.4 Flow Regimes of Typical Processing
Typical viscosity curve of a polyolefin- PP homopolymer, melt flow rate (230 C/2.16 Kg) of 8 g/10 minat 230 C with indication of the shear rate regions of different conversion techniques.
[Reproduced from M. Gahleitner, “Melt rheology of polyolefins”, Prog. Polym. Sci., 26, 895 (2001).]
I.5 Secondary Flows and Instabilities

Secondary flow
Newtonian Fluids
Non-Newtonian Fluids

Secondary Flow
Primary Flow
Secondary flow around a rotating sphere in
a polyacrylamide solution.
[Reporduce from H. Giesekus in E. H. Lee, ed.,
Proceedings of the Fourth International Congress
on Rheology, Wiley-Interscience, New York (1965),
Part 1, pp. 249-266]

Secondary Flow
Primary Flow
I.5 Secondary Flows and Instabilities (cont.)

Secondary
flow
Newtonian Fluids
Non-Newtonian Fluids
Secondary Flow
Secondary Flow
Primary Flow
Primary Flow
Newtonian fluid (N):
water-glycerin
Non-Newtonian fluid (P):
100 ppm polyacrylamide
in water-glycerin
Steady streaming motion produced by a long cylinder oscillating normal to its axis. The cylinder is
viewed on end and the direction of oscillation is shown by the double arrow. The photographs do not
show streamlines but mean particles pathlines made visible by illuminating tiny Spheres with a
stroboscope synchronized with the cylinder frequency.
[Reproduced from C. T. Chang and W. R. Schowalter, Nature, 252, 686 (1974).]
I.5 Secondary Flows and Instabilities (cont.)

Melt instability
Sharkskin
Melt fracture
Photographs of LLDPE melt pass through a capillary tube
under various shear rates. The shear rates are 37, 112, 750
and 2250 s-1, respectively.
[Reproduced from R. H. Moynihan, “The Flow at Polymer
and Metal Interfaces”, Ph.D. Thesis, Department of Chemical
Engineering, Virginia Tech., Blackburg, VA, 1990.]
[Retrieved from the video of
Non-Newtonian Fluid Mechanics
(University of Wales Institute of
Non-Newtonian Fluid Mechanics,
2000)]
I.5 Secondary Flows and Instabilities (cont.)

Taylor-Couette flow
Taylor vortex
R1
R2
[S. J. Muller, E. S. G. Shaqfeh and R. G. Larson,
“Experimental studies of the onset of oscillatory
instability in viscoelastic Taylor-Couette flow”,
J. Non-Newtonian Fluid Mech., 46, 315 (1993).]
Flow visualization of the elastic Taylor-Couette
instability in Boger fluids.
[http://www.cchem.berkeley.edu/sjmgrp/]
I.6 Length/Time Scales & Probing Techniques
[Reproduced from G. M. Kavanagh and S. B. Ross-Murphy, “Rheological characterisation of polymer
gels”, Prog. Polym. Sci., 23, 533 (1998).]
流體加工性質
macrorheology
基本流變性質
the De, Wi numbers
機械量測
microrheology
microscopy/spectroscopy
birefringence/dichroism
light/ neutron scatterings
particle tracking
光學量測
G
0
N
molecular orientation / alignment
particle size distribution/ diffusivity
micro/mesoscopic structures
本質方程式
分子動力理論
flow pattern
模流分析
Traditional route
Modern (predictive) route
monomer mobility, elastic modulus etc.
量子、原子、多尺度計算
物質特性
(化學合成)
Chapter II
Mechanical Characterizations
*Most of the figures appearing in this file are taken from the textbook “Dynamics of Polymeric Liquids” (Vol. 1)
by R. B. Bird et al. For more details, you are referred to the textbooks and references cited therein.
Topics in Each Section
 §2-1 Rheometry
 Shear and Shearfree Flows
 Flow Geometries & Viscometric Functions
 §2-2 Basic Vector/Tensor Manipulations
 Vector Operation
 Tensor Operation
 §2-3 Material Functions in Simple Shear Flows
 Steady Flows
 Unsteady Flows
 §2-4 Material Functions in Elongational Flows
2.1. Rheometry
 Two standard kinds of flows, shear and shearfree, are frequently used to
characterize polymeric liquids
(b) Shearfree
(a) Shear
vx   y
FIG. 3.1-1. Steady simple shear flow
v x   yx y ;
v y  0;
vz  0
Elongation
rate
vx  
Shear rate
vy  
FIG. 3.1-2. Streamlines for elongational flow (b=0)

x
2

2
vz   z
y
 The Stress Tensor
y
x
z
Shear Flow
Total stress
tensor*
Elongational Flow
Stress tensor
 p   xx

  p  
 yx





0

 yx
0
p   yy
0
0
p   zz





 p   xx

  p  
0





0

0
0
p   yy
0
0
p   zz





Hydrostatic pressure forces
S h e a r S tre ss :  yx
First N o rm a l S tre ss D iffe re n ce :  xx   yy
S e co n d N o rm a l S tre ss D iffe re n ce :  yy   zz
Tensile Stress :  zz   xx
*See §2.2
 Classification of Flow Geometries
(a) Shear
Pressure Flow:
Capillary
Drag Flows:
Concentric Cylinder
(b) Elongation
Cone-andPlate
U n ia x ia l E lo n g a tio n ( b  0 ,   0 ) :
Moving
Parallel
Plates
 Typical Shear/Elongation Rate Range & Concentration
Regimes for Each Geometries
(a) Shear
Concentrated Regime








Homogeneous
deformation:*
Cone-andPlate
Nonhomogeneous
deformation:
10
(b) Elongation
Dilute Regime
Concentric Cylinder
Parallel
Plates
3
10
2
10
1
Capillary
10
0
10
1
10
2
10
3
10
4
-1
γ (s )
10
5
Moving
clamps
 (s )
-1
For Melts & High-Viscosity Solutions
*Stress and strain are independent of position throughout the sample
 Viscometric Functions & Assumptions
Example:
Concentric Cylinder
W1
R1
A ssu m p tio n s :
(1 ) S te a dy, la m in a r, iso th e rm a l flo w
(2 ) v  R1W 1 o n ly a n d v r  v z  0
R2
(3 ) N e gligible gra vity a n d e n d e ffe cts
H
(4 ) S ym m e try in  ,     0
(From p.188 of ref 3)
FIG. Concentric cylinder viscometer
S h e a r ra te  :
 
S h e a r - ra te d e p e n d e n t v isc o sity  (  ) :
W 1 R1
R 2  R1
 ( ) 
(homogeneous)
T
2 R1 H 
2
w h e re th e to rqu e a ctin g o n th e
su rfa ce o f th e in n e r cylin de r T is :
T   r

r  R1
(2  R1 H )  R1

R1 , R 2 : R a dii of in n e r a n d ou te r cylin de rs
W 1 : A n gu la r ve lo city o f in n e r cyl in de r
H : H e igh t of cylin de rs
T : To rqu e o n in n e r cyli n de r

Flow Instability in a Concentric Cylinder Viscometer for a Newtonian Liquid
Tay lo r n u m b e r (T a) 
C e n trifu g al fo rce
V isco u s fo rce
 41.3
Laminar  Secondary  Turbulent
Onset of
Secondary
Flow
T a  4 1.3; R e  9 4
T a  141; R e  322
Ta (or Re) plays the central role!
Taylor
vortices
Turbulent
T a  387; R e  868 T a  1, 715; R e  3, 960
Rod Climbing
is not a subtle effect, as demonstrated on th
cover by Ph.D. student Sylvana Garcia-Rodri
from Columbia. Ms. Garcia-Rodrigues is stud
rheology in the Mechanical Engineering
Department at U. of Wisconsin-Madison, US
The Apparatus shown was created by
UWMadison Professors Emeriti John L. Schra
and Arthur S. Lodge. The fluid shown is a 2%
aqueous polyacrylamide solution, and the
rotational speed is nominally 0.5 Hz. Photo b
Carlos Arango Sabogal (2006)
Example 2.3-1:
Interpretation of Free Surface Shapes in the Rod-Climbing Experiment
(N)
(P)
I. Phenomenological Interpretation:
(F): For the Newtonian fluids the surface near the rod is slightly depressed and acts a
sensitive manometer for the smaller pressure near the rod generated by centrifugal fo
(P): The Polymeric fluids exhibit an extra tension along the streamlines, that is, in the
direction. In terms of chemical structure, this extra tension arises from the stretching
alignment of the polymer molecules along the streamlines. The thermal motions make
the polymer molecules act as small “rubber bands” wanting to snap back
‧P
z

r
 :1
r :2
(N)
z :3
(P)
II. Use of Equations of Change to Analyze the Distribution of the Normal Pressure
The resultant formula derived in this example is:
d ( 33  p )
d ln r
 2 21
d
d  21
( 22   33 )  ( 11   22 )   v1
2
(2.3  5)
(N ) : Fo r N e w to n ia n flu id s th e first a n d se c o n d n o rm a l stre ss d iffe re n c e s,  1 1   2 2 a n d  2 2   3 3 , a re
b o th z e ro in sh e a r flo w , a n d e q . 2 .3 - 5 sh o w s th a t th e n o rm a l p re ssu re e x e rte d o n th e l u b ric a te d li d
in c reases w ith th e ra d iu s
(P) : Fo r po lym e r ic flu ids w e h a ve a lre a dy in dica te d th a t  11   22 is pra tica lly a lw a ys n e ga tive
w ith a n u m e rica l va lu e m u ch la rge r th a n th a t o f  22   33 . W e se e th a t th e n o rm a l stre sse s m a y
ca u s e th e to ta l n o rm a l pre ssu re to decrease in th e ra dia l dire ctio n .
Examples 1.3-4 & 10.2-1:
Cone-and-Plate Instrument
A s s u m p tio n s :
(1 ) S te a d y , la m in a r, iso th e rm a l flo w
(2 ) v ( r ,  ) o n ly ; v r  v  0
(3 )  0  0.1 rad (  6 )
(4 ) N e g lig ib le b o d y fo rc e s
(5 ) S p h e ric a l liq u id b o u n d a ry
(From p.205 of ref 3)
S h e a r ra te  :
 
S h e a r - ra te de pe n de n t
The first norm al stress
visco sity  (  ) :
difference coefficient  1 ( ) :
W0
0
FIG. 1.3-4. Cone-and-plate geometry
 ( ) 
(homogeneous)
3T  0
2 R W 0
3
2F
 1 ( ) 
T  
2

R 
2
R
r  
2
W 0 : A n gu la r ve lo city o f co n e
T : T o rqu e o n pla te
 0 : C o n e a n gle
F : Fo rce re qu ire d to ke e p tip o f co n e
R : R a diu s o f circu la r pla te
0
in co n ta ct w ith c ircu la r pla te
0
2
  2
dr d 

Example: p.530
Uniaxial Elongational Flow
H e n c k y stra in :
 max   t max  ln ( Lmax L0 )
L 0 : In itia l sa m p le le n g th
L m ax : M a x im u m sm a p le le n g th
The Norm al Stress Difference :
 zz   rr   F ( t ) A ( t )
F ( t ): To ta l fo rce pe r u n it a re a e xe rte d by th e lo a d ce ll
A ( t ): In sta n ta n e o u s co rss - se ctio n a l a re a o f th e sa m ple
The Transient Elongational Viscosity 

z
r
:
F (t )



( zz   rr )
0

A0 e
  0t
0
 0 : Elongation rate
A0 : Initial cross - sectional area of the sam p le
FIG. 10.3-1a. Device used to genera
uniaxial elongational flows by separati
Clamped ends of the sample
Exptl. data see §2.4
 Supplementary Examples
 Capillary:
o Example 10.2-3: Obtaining the Non-Newtonian Viscosity from the Capillary
 Concentric Cylinders
o Problem 10B.5: Viscous Heating in a Concentric Cylinder Viscometer
 Parallel Plates:
o Example 10.2-2: Measurement of the Viscometric Functions in the ParallelDisk Instrument
o Problem 1B.5: Parallel-Disk Viscometer
o Problem 1D.2: Viscous Heating in Oscillatory Flow
2.2. Basic Vector/Tensor Manipulations
 Vector Operations (Gibbs Notation)
Vector:
u
u
i
i
3
x  1

y  2
z  3

 u1 1  u 2  2  u 3 3
i
u
3
1
Th e K ron ecker delta  ij
1

  ij   1,


  ij  0 ,
Dot product:

u  v    u i i
 i
Le t j  i

i
(A .2  1, 2)
if i  j

 
    u j j  
  j

 ( u v )
i
if i  j
i
ii

uv
i
i
i
 (u v
i
ij
j
)( i   j )
2
2
Th e perm u tation sym bol  ijk
  ijk   1,

  ijk   1,

  ijk  0 ,
if ijk  123 , 231, o r 312
if ijk  321, 132 , o r 213
(A .2  3, 4, 5)
if a n y tw o in dice s a re a like
Cross product:

u  v    u i i
 i

 (u v
i
j

 
    u j j  
  j

 (u v
i
j
)( i   j )
ij
)(  ijk  k )
ijk
3
U sin g  i   j 

k 1
ijk
k
(A .2  15)
1
2
3
u  v  u1
u2
u3
v1
v2
v3
 Tensor Operations
2
Tensor:
 


i
i

 2  ( 21 ,  22 ,  23 )
 ij
j
 1  ( 11 ,  12 ,  13 )
j
Stresses acting on plane 1
 1 1 11   1 2 12   1 3 13
1
  2  1 21   2  2 22   2  3 23
  3 1 31   3 2 32   3 3 33
3
 3  ( 31 ,  32 ,  33 )
FIG. The stress tensor
The total momentum flux tensor for an incompressible fluid is:
Normal
stresses
 p   11

 
 21


 
31

 12
p   22
 32
 13


 23
 p 




p   33 
Hydrostatic pressure
forces
Stress tensor or
Momentum flux
tensor
Example:
T h e to ta l m o m e n tu m flu x th ro u g h a su rfa c e o f o rie n ta t io n n is :

n  

 i


 

  i j ij     n k  k 
j

  k
ij
n k ( i  j   k )
ijk
Let k  j


ij
ij
n j i
Some Definitions & Frequently Used Operations:
†


†
T ra n sp o se :  :    i  j ij  

 ij

G ra d ie n t :  :

i
Cartesian
coordinate

i

j
L a p la cia n :  :    
Cartesian
coordinate
i



2
 x
i
v 

2
i
1

U n it te n so r :  :   i  j  ij  0

=
ij
0

0
1
0
0

0

1 
j
 xi
ij
  
 xi
2

ij

i
v 
ji
v j
  ik
k
 xi
ik
 (
ij
v k )( i 
ij
v jl )( i  l )
l )
ijkl
v 


 (
ijk

v 
ijk
k
ijk
 v 

jkl


ij
ij

ji
v j
 xi
2.3. Material Functions in Simple Shear Flows
 Preface:
 A variety of experiments performed on a polymeric liquid will yield a host
of material functions that depend on shear rate, frequency, time, and so
on.
 The fluid behavior can be better understood by means of representative
rheological data.
 Descriptions of the nature and diversity of material responses to simple
shearing or shearfree flow are given.
FIG. 3.4-1. Various types of simple shear experiments used in rheology
I.
Steady Shear Flow Material Functions
Exp a: Steady Shear Flow
FIG. 3.3-1. Non-Newtonian viscosity η of a low-density polyethylene at several
Different temperatures
The shear-rate dependent viscosity η The first and second normal stress
is defined as:
coefficients are defined as follows:
 xx   yy    1 ( ) yx
2
 yx   (  )  yx
 yy   zz    2 ( ) yx
2
Relative Viscosity:  rel 

s
 : S olu tion viscosity
 s : S olven t viscosity
FIG. 3.3-2. Master curves for the viscosity and first
normal stress difference coefficient as functions of shear
rate for the low-density polyethylene melt shown in
previous figure
  s 
Intrinsic Viscosity: [ ]  lim 

c 0
 c s 
c : M a s s c o n c e n tra tio n
FIG. 3.3-4. Intrinsic viscosity of polystyrene
Solutions, With various solvents, as a function
of reduced shear rate β
II.
Unsteady Shear Flow Material Functions
Exp b: Small-Amplitude Oscillatory Shear Flow
T h e sh e a r s tre ss o sc illa te s w ith fre q u e n c y  ,
b u t is n o t in p h a se w ith e ith e r th e sh e a r s tra i n
o r sh e a r ra te
S h e a r S tre ss :  yx   A ( )  sin ( t   )
0
S h e a r ra te : 
S h e a r s tra in : 
FIG. 3.4-2. Oscillatory shear strain, shear rate, shear stress,
and first normal stress difference in small-amplitude oscillatory
shear flow
( t )   co s  t
0
yx
( t )   sin  t
0
yx
It is customary to rewrite the above equations to display the in-phase and
out-of-phase parts of the shear stress
Storage modulus
 yx   G  ( )  sin  t  G  ( )  co s  t
0
0
Loss modulus
FIG. 3.4-3. Storage and loss moduli, G’ and G”, as functions of frequency ω at a reference
temperature of T0=423 K for the low-density polyethylene melt shown in Fig. 3.3-1. The solid
curves are calculated from the generalized Maxwell model, Eqs. 5.2-13 through 15
Exp c: Stress Growth upon Inception of Steady Shear Flow
Transient Shear Stress:

 yx    0
Fo r la rg e sh e a r ra te s 
+
d e p a rts fro m th e
lin e a r visco e la stic e n ve lo p e , g o e s th ro u g h
a m a x im u m , a n d th e n a p p ro a ch e s th e
ste a d y - sta te va lu e .
+
FIG . 3 .4 - 7 . Shear stress grow th function  ( t ,  0 ) data for a low - density polyethylene m elt.
The solid curve is calculated from Eq. 5 .3 - 25 w ith the spectrum in Table 5.3 - 2

 F IG . 3 .4 - 8 . S h e a r stre ss gro w th fu n ctio n  ( t ,  0 )  (  0 )
fo r tw o po lym e r so lu tio n s (a a n d b) a n d a n a lu m in u m
so a p so lu tio n (c)

FIG . 3 .4 - 9 . First norm al stress grow th function  1 ( t ,  0 )
for the low - density polyethylene m elt
Exp e: Stress Relaxation after a Sudden Shearing Displacement (Step-Strain Stress
Relaxation)
Relaxation Modulus:*
G (t ,  0 )  
 yx
0
T h e sh e a r stra in  0 c a n b e in d u c e d b y a p p ly in g a
la rg e , c o n sta n t sh e a r ra te  0 fo r a sh o rt tim e
in te rv a l  t , so th a t  0  t   0
For small shear strains
lim G ( t ,  0 )  G ( t )
00
In th is lim it, th e s h e a r s tre s s is lin e a r in s tra in
The Lodge-Meissner Rule:
G (t ,  0 )
G  1 (t ,  0 )
1
F IG . 3 .4 - 1 5 . T h e re la x a tio n m o du lu s G ( t ,  0 ) (o pe n sym bo ls) a n d n o rm a l stre ss
re la x a tio n fu n ctio n G  1 ( t ,  0 ) (so lid sym bo ls) fo r a lo w - de n sity po lye th yle n e m e lt
*Example 5.3-2: Stress Relaxation after a Sudden Shearing Displacement
F IG . 3 .4 - 1 6 . T h e stre ss re la xa tio n m o du lu s G ( t ,  0 ) fo r 20 % po lystyre n e ( M w  1.8  10 ) in A ro clo r.
6
Pa rt (a ) sh o w s h o w G ( t ,  0 ) va rie s w ith sh e a r stra in . In (b) th e da ta a re su pe rpo se d by ve rtica l sh if tin g
to sh o w th e sim ila rity in G ( t ,  0 ) a t la rge tim e s re ga rdle ss o f th e im po se d sh e a r sta in
2.4. Material Functions in Elongational Flows
 Shearfree Flow Material Functions
Zero - elon gation - rate
Fo r U n ia x ia l E lo n g a tio n a l Flo w ( b  0 ,   0 ) :
elon gation al viscosity  0
 zz   xx   ( )
 : E lo n gatio n al visco sity
 : E lo n gatio n rate
Zero - shear - rate
viscosity  0
F IG . 3 .5 - 1 . E lo n ga tio n visco sity  a n d visco sity
 fo r a po lystyre n e m e lt a s fu n ctio n s o f e lo n ga tio n
ra te a n d sh e a r ra te , re spe ctive ly
Elongational Stress G row th Function 

T h e a b ru p t u p tu rn , o r " stra in h a rd e n in g , "
o c c u rs a t a ro u g h ly c o n sta n t v a lu e o f
H e n c k y st r a in  (0, t )   0 t
The number average and weight average
molecular weights of the samples:
Monodisperse, but with a
tail in high M.W. (GPC results)
FIG . 3 .5 - 2 . Tim e dependence of the elongational
stress grow th viscosity 
+
for four polystyrene m elts
Chapter III
Optical Characterizations
Topics in Each Section
 §3-1 Introduction to Rheo-Optics Method
3-1.1. Introduction & Review of Optical Phenomena
3-1.2. Characteristic Dimension & Optical Range
 §3-2 Typical Experimental Set-ups
3-2.1 Flow Dichroism and Birefringence Measurements
[for Case Study 12]
3-2.2 Combined Rheo-Optcial Measurements (including Rheo-SALS)
[for Case Study
3]
 §3-3 Information Retrieval in Individual Measurements
Case Study 1: Flow Dichroism and Birefringence of Polymers
Case Study 2: Dynamics of Multicomponent Polymer Melts
Case Study 3: Combined Rheo-Optcial Measurements
References
3-1. Introduction to Rheo-Optics Method
3-1.1 Introduction & Review of Optical Phenomena

A rheological measurement entails the measurement of:
(a)
(b)

Force (related to the stress)
Displacement (related to the strain)
In a rheo-optical experiment, both the force and optical properties of the
sample are measured
Table: A comparison of some important features in optical and mechanical measurements
Optical
Mechanical
Measured Quantity
Molecular orientation and shape
Dissipation and/or storage of energy
Polymer Contribution
Dominates the measured signal
Relatively small for dilute solutions
Spatial Resolution
Possible
Impossible
Molecular Labeling
Possible
Impossible
Sensitivity
Good precision
-
Time Scale
Shorter
Longer
When incident electromagnetic radiation interacts with matter, three broad
classes of phenomena are of interest:

I.
Transmission of Light
•
•
II.
The light can propagate through the material with no change in direction or
energy, but with a change in its state of polarization
Birefringence; “Dichroism”; Turbidity
Scattering Radiation
•
•
III.
The radiation can be scattered (change in direction) with either no change
in energy (elastic) or a measruable change in energy (inelastic)
Static Light, X-Ray, and Neutron Scattering; Dynamic Light Scattering
Absorption and Emission Spectroscopies
•
•
Energy can be absorbed with the possible subsequent emission of some or
all of the energy
Fluorescence; Phosphorescence
3-1.2 Characteristic Dimension & Optical Range
Typical levels of structures in polymeric systems are listed below
Structural length scales probed by various techniques
3-2. Typical Experimental Set-ups
3-2.1. Flow Dichroism and Birefringence of Polymers in Shear Flows

Basic Concepts:
Turbidity
The Lambert-Beer’s Law:
I : In ten sity o f th e tra n sm itted ligh t
I
I0
 exp(  l )
I 0 : In ten sity o f th e in ciden t ligh t
 : Tu rbidity
l : Pa th len gth
 W h e n lig h t p a sse s th ro u g h a m a te ria l th e in te n sity o f th e tra n sm itte d lig h t I w ill b e
sm a lle r th a n th e in te n sity I 0 o f th e in cid e n t lig h t b e a m
 T h e lo ss o f in te n sity is d u e to a b so rp tio n a n d /o r sca tt e rin g o f th e lig h t
 In m o st m a te ria ls  is in d e p e n d e n t o f th e p o la risa tio n sta te o f th e l ig h t
Dichriosm
 Fo r dich ro ic m a te ria ls, ligh t is a tte n u a te d diffe re n tly w ith diffe re n t
po la riza tio n sta te s, th a t is,  i s po la riza tio n sta te de pe n de n t
EX 1: Polariod Sun Glasses (A daily Eexample of dichrism resulting from absorption)
FIG. Representation of a Polaroid sheet. Light with a polarization direction parallel to
the aligned polymers is absorbed more strongly as compared to light with a polarization
direction perpendicular to the polymers
EX 2: Colloids under Shear Flow
The total amount of scattered light depends on the polarisation direction due to the
anisotropic nature of the microstructure under shear flow
Birefringence
 A m a te ria l is ca lle d bire frin ge n ce , w h e n th e re fra ctive in de x is po la risa tio n
sta te de pe n de n t
 R e su ltin g in a ph a se diffe re n ce be tw e e n ligh t w ith diffe re n t po la risa tio n sta t e s
a fte r h a vin g pa sse d th e bire frin ge n ce m a te ria l
FIG. Linear polarised light is generally transmitted as elliptically polarized light through
a birefringent material
More specifically, the polarisation direction of the light can be decomposed into
a component parallel to the direction where the refractive index is large and a
component parallel to the direction where the refractive index is small
After having traversed the sample, there is a relative phase shift of the two field
components and the sum of the two fields is generally elliptically polarised

Phase-Modulated Flow Birefringence Design:
M 9 ( m , 45  ) M S ( ,  )

M 11 (90 )
S0
Polarizer
Light source
S1
PEM
S2

M 11 (  4 5 )
S3
Analyzer
S4
Detector
Sample
FIG. Phase-Modulated Flow Birefringence schematic
• In transmission exps, one is normally concerned with the measurement of
light polarization
The Stokes Vector Entering the Detector is:



S 4  M 11 (  45 )  M S ( ,  )  M 9 ( m , 45 )  M 11 (90 )  S 0
The specific Mueller matrix components
(optical properties) of the sample can
be identified
[Frattini, P. L. and G. G. Fuller, J. Rheol. 28, 61-70 (1984) ]

Typical Arrangement for Flow Birefringence and Dichroism
Measurements:
FIG. Schematic of the experimental set-up for dichroism and birefringence measurements
A somewhat simpler set-up can be sued:
(1) Dichroism only: removal of R2, P2, and D2
(2) Turbidity only: removal of R1, R2, P2, and D2
BS’,BS: Beam splitter
D1-D3: Detectors
P1,P2: Polarizers
R1,R2: Rotating quarter wave platelets

Optical Train Mounted on a Rheometer
FIG Experimental apparatus for determination
of flow birefringence and flow dichroism
FIG A combination of rheomechanical and
rheo-optical measurements
http://www.chemie.uni-hamburg.de/tmc/kulicke/rheology/rheo2.htm
Flow Birefringence Setup
[Frattini, P. L. and G. G. Fuller, J. Rheol. 28, 61-70 (1984) ]
oscilloscope
DAQ
card
Polarizer
Lock in
amplifier
PEM
Head
PEM
controller
PMT
Analyzer
Reflecting
mirror
PEM：The PEM100
photoelastic modulator is
an instrument used for
modulating or varying
(at a fixed frequency) the
polarization of a beam of
light.(δ=A sinωt)
Lock in amplifier：
Experimentally Idc,
Iω, and I2ω are
determined with lock
in amplifier.
The Measured Intensity I for a Sample with Coaxial Birefringence
and Dichroism oriented at an Angle θ is:
I

 I   0 1  cos 2 sin  sin  m  cos 2 sin 2 (1  cos  ) cos  m  
   2

S4     

  

  

 


I : In ten sity o f th e tran sm itted lig h t
I 0 : In ten sity o f th e in cid en t lig h t o n th e P E M
 : O rien tatio n an g le o f th e sam p le
 : R etard an ce o f th e sam p le
 m : R etard an ce o f th e P E M g iven b y  m  A sin  t
I 
I0
2
1  cos 2 sin  sin  m  cos 2 sin 2 (1  cos  ) cos  m 
[Frattini, P. L. and G. G. Fuller, J. Rheol. 28, 61-70 (1984) ]
Fourier series expansion of cos δm and sin δm , and
the intensity at the detector,
I  I dc  I  sin  t  I 2  cos 2 t  ........
where Idc = I0 /2
Iω =2cos2θsinδJ1(A) Idc
I2ω =2cos2θsin 2θ(1-cosδ)J2(A) Idc
The birefringence Δn of the sample is defined from

the retardance by  n 
2 d
[Frattini, P. L. and G. G. Fuller, J. Rheol. 28, 61-70 (1984) ]
Solid State Birefringence Measurement
Analyer(45°)
PEM
PMT
laser
Quarter
wave(sample1/4λ)
Polarizer(+45
°)
[Hinds Instruments, Inc., PEM-100 technical note]
A=0.3125 λ
I

I0
1
2
1  cos    A sin(  t )  
λ/4 0.3125λ 50kHz
1 .0
I/I 0
0 .5
0 .0
0 .0 0 0 0
0 .0 0 0 1
0 .0 0 0 2
0 .0 0 0 3
0 .0 0 0 4
tim e (s )
Fig. Waveforms obtained from calculation
0 .0 0 0 5
Fig. Waveforms obtained from oscilloscope
0 .1 5
0 .1 0
I/I0
0 .0 5
0 .0 0
0
2 e -5
4 e -5
6 e -5
tim e (s )
Fig. Waveforms obtained from DAQ card
8 e -5
A= 0.5 λ
I
I0

1
2
1  cos    A sin(  t )  
λ/4
0.5λ
50kHz
1 .0
I/I 0
0 .5
0 .0
0 .0 0 0 0
0 .0 0 0 1
0 .0 0 0 2
0 .0 0 0 3
0 .0 0 0 4
tim e (s )
Fig. Waveforms obtained from calculation
0 .0 0 0 5
Fig. Waveforms obtained from oscilloscope
0 .1 0
I/I0
0 .0 5
0 .0 0
0
2 e -5
4 e -5
6 e -5
tim e (s )
Fig. Waveforms obtained from DAQ card
8 e -5
3-2.2. Combined Rheo-Optical Measurements

Optical Setup for Shear-Small-Angle-Light-Scattering (SALS)
Mesurements [Kume et al. (1997)]
FIG. Schematic diagram of the experimental setup for shear-light scattering: one-dimensional detector (photodiode
array), cone-and-plate type shear cell to generate Couette flow, and coordinate system used in this study
Small-Angle Light Scattering (SALS)
A monochromatic beam of laser light impinges on
a sample and is scattered into a 2D detector

1
2
3
4
Hammouda, B., Probing Nanoscale Structures: The SANS Toolbox (unpublished book)
Brief Introduction to SALS
Description
A monochromatic beam of laser light impinges
on a sample and is scattered into a 2D detector;
the interference of the scattered light is of
interest in scattering experiments
Particle 1

or light
Phase difference   :
 
Particle 2
2

(Q P  O R ) 
2

  2  s  r , w here s 
Geometry of the path length difference
Probed Length Scale
Application
S  S0

C onventionally, the quantitiy q (  2  s 
is defined as the scattering vec tor
Angular Range
(S 0  r  S  r )
4 n

sin

)
2
By the law of cosines
θ =1o to 10o
~500 nm to ~5 μ m
Larger systems, such as polymer solutions, gels,
colloids, micelles, etc, contain structures that
fall into the mesoscopic region (100 nm to 2 μ
m)
II. Why Large (Small) Structural Length Can be Probed at
Small (Large) Angles?
Focus on the “red” only (λ=633 nm)!
Large structural
length
slits are closer
Small structural
length
FIG. The diffraction pattern illustrated in Fig. (a) was captured by a 40x objective
imaging of the lower portion of the line grating in Fig 2(b), where the slits are closer together.
In Fig. (c), the objective is focused on the upper portion of the line grating, and more
spectra are captured by the objective.
Large structural
length
Small structural
length
Double-slit Fraunhofer pattern
Schematic Drawing of SALS Apparatus
1 Monochromatic Light
2 Collimation
Polarier
Objective
lens
3 Scattering
Lens
4 Detection
  1 to 10
Sample
Analyzer

Mirror
Pinhole
Iris
CCD
Iris
Spatial filter
& Beam expander
Iris
Lens 1
Spatial filter
Beam expansion
Spatial filter
Mirror
Lens 2
Beam expansion
SALS Apparatus
2 Collimation
3 Scattering
4 Detection
1 Monochromatic Light
Sample stage
Laser
Polarizer 1,2
Laser pointer
Lens 1,2
CCD
Mirror
Spatial filter
Mirror
Iris 1,2,3
Calibration: Diffraction Pattern of a Pinhole
Using a 50 μm pinhole as
a sample, whose diffraction
pattern is known (airy func.)
Polarier
Objective
lens
Lens
Analyzer

Pinhole
Iris
Iris Pinhole
Iris
Mirror
Lens 1
Spatial filter
& Beam expander
Lens 2
1
M ea su red d iffra ctio n
p a ttern
A iry fu n ctio n
0 .1
I(  ) / I(0 )
Mirror
CCD
0 .0 1
0 .0 0 1
0 .0 0 0 1
0
2
4
6
k a sin 
8
10
Calibration: Scattering of a PS Colloidal Dispersion
100-nm-diameter PS
colloidal dispersion
Polarier
Objective
lens
Lens
Analyzer

Mirror
Pinhole
Iris
CCD
Iris Sample
Mirror
Lens 1
Spatial filter
& Beam expander
10
S catterin g in ten sity (a.u .)
The Rayleigh-Gans-Debye theory predicts
that the scattering profile of the measured
sample is of no angular dependence, as was
confirmed experimentally
Iris
Lens 2
6
105
104
103
100
200
300
P ix el
400
500
Versatile Optical Rheometry
Lens
Iris PEM Iris
Objective
lens
Polarier
Pinhole
Spatial filter
& Beam expander
Flow-LS (large-angle detection)
Couette
cell

Rheology
CCD
Analyzer
Rheo-SALS

Lens
Screen with aperture
(from PEM)
1f
2f
Photodiode
Rheo-Birefringence
Lock-in amplifiers
Rheo-SALS
(under
construction)
Flow cell
12.67cm
10cm
Rheometer
Lens 1
f=10 cm
37.5cm
Lens 2
f=12.5 cm
18.75cm
Lens 3
f=6 cm
9 ± 0.15 cm
18 ± 0.62 cm
Butterfly Pattern (abnormal type, in this example)
Uniaxially
stretched
with “butterfly” scattering
patterns
Isointensity curves for the
uniaxially stretched sample
(calculated)
This anisotropy is the source of unusual
“butterfly” scattering patterns: density
fluctuation are the largest along the
stretching direction
Bastide et al., “Scattering by deformed swollen gels: butterfly isointensity patterns,” Macromolecules 23, 1821 (1990)
Complementary to SALS: Multi-Angle Dynamic/Static Light Scattering
  30 to 150
Temperature
Controller
10oC to 70oC
Polarizer 1
Sample cell
Polarizer 2
Photomultiplier tube
  30 to 150
Circulating water
Detection arm
3-3. Information Retrieval in Individual Measurements
CASE STUDY 1: Flow Dichroism and Birefringence of Polymers in Shear
Flows

A Rheo-Optical Study of Shearing Thickening and Structure Formation
in Polymer Solutions [Kishbaugh and Muhugh (1993); Figs.
Reproduced from Sondergaard and Lyngaae-Jorgensen (1995)]
FIG. Schematic of photoelastic modulation rheo-optical device. Optical elements in the alignment configuration
K ish ba u gh a n d M cH u gh stu die d m o n o dispe rs e po lystyre n e s disso lve d in de ca lin . In m o st ca se s,

so lu tio n s w e re in th e dilu te to se m idilu te tra n sitio n re gio n , i .e ., c c  1 . In th e h igh sh e a r ra te
ra n ge w h e re th e reversible shear thickening o cc u rre d (i.e ., 500 s
-1
   10, 000 s )
-1
Note that only data for the case
of Mw=1.54 x 106 is shown in
the following 3 pages
One-to-one correlation
between the onset of shear
thickening and the occurrence
of a maximum in the dichroism
Dichroism
Viscosity
• The viscosity and dichroism patterns for the
lowest concentration are similar to those
exhibited by a lower molecular weight sample
(Mw=4.3 x 105). Namely, the dichroism rises to
a plateau, while viscosity undergoes a monotonic
drop with shear rate to an eventual Newtonian
plateau
• At higher concentrations, a dramatic and
distinctive pattern emerges. One sees a
shaper rise in the dichroism to an eventual
maximum, while the viscosity simultaneously
drops to a minimum. This is followed by a
region of shear thickening in which the
viscosity continuously rises, while the
dichroism decreases and eventually turns
negative
• This figure shows that, in this range,
the orientation angle dropped to a
constant near-alignment with the
flow axis
• Throughout the entire flow curve,
the birefringence exhibits a steady
monotonic increase with shear rate
• These data offer strong evidence
that the overall orientation of the
chain segments is independent of
the structuring processes, which
may take place as indicated in the
dichroism
CASE STUDY 2: Dynamics of Multicomponent Polymer Melts

Infrared Dichroism Measurements of Molecular Relaxation in Binary Blend Melt
Rheology [Kornfield et al. (1989)]
1. Chains are identical in chemical composition, but differ in M.W.. Isotopic labeling
with
deuterium (D) can be used to distinguish one M.W. component from another
2. At 2,180 cm-1 the C-D bond absorbs but the C-H bond does not
3. The most interesting result is that the longest relaxation time of the the shorter
chains is a strongly increasing function of the volume fraction of longer chains. This
contrasts with the predictions of the basic reptation model
CASE STUDY 3: Combined Rheo-Optical Measurements

Rheo-Optical Studies of Shear-Induced Structures in Semidilute
Polystyrene Solutions [Kume et al. (1997)]
1. Shear-induced structure formation in semidilute solutions of high molecular
weight polystyrene was investigated using a wide range of rheo-optical techniques
2. The effects of shear on the semidilute polymer solutions could be classified into
some regimes w.r.t. shear rate
 c : O n se t o f th e sh e a r - e n h a n ce d
co n ce n tra tio n flu ctu a tio n s
 a : O n se t o f th e a n o m a lie s in th e
rh e o lo gica l a n d sca tte rin g be h a vio r s
FIG. A complete picture of the shear-induced phase separation and structure formation from a wide range of
techniques on the same polymer solutions
Continued
Homogeneous
solution
Strong butterfly-type
LS pattern
Streaklike
LS pattern
Oblate-ellipsoidal structures
Long stringlike
structures
Shear-microscopy
results
Change of the sign
Due to the stringlike structures oriented parallel to the flow dir.
Chains weakly orient
along the flow dir.
 c : O n se t o f th e sh e a r - e n h a n ce d
co n ce n tra tio n flu ctu a tio n s
Chains in the strings with their
end-to-end vectors parallel to the
flow dir.
 a : O n set o f th e an o m alies in th e
rh eo lo gical an d scatterin g beh avio r s
Continued
Comparisons with Mechanical Characterizations:

6.0 w t% P S /D O P so lu tio n ( c c  30 )
M w  3.84  10 ; M
6
Mechanical
w
M n  1.06
FIG Th e plots of sh e a r viscosity ( ), bire frin ge n ce (  n ), a n d dich roism (  n ) of th e solu tion
a s a fu n ction of sh e a r ra te (  )
Notice that the behavior of the shear viscosity is also classified into three regimes
References
(1 ) B a a ije n s, J. P . W ., E valuation of C onstitutive E quations for P olym er M elts and Solutions in C om plex F low s ,
E in d h o v e n U n iv e rsity o f T e ch n o lo g y , D e p a rtm e n t o f M e ch a n ica l E n g in e e rin g , E in d h o v e n , T h e N e th e rla n d s
(1 9 9 4 ).
(2 ) C o lly e r, A . A . a n d L . A . U tra ck i, P olym er R heology and P rocessing , E ls e v ie r S cie n ce P u b lish e rs L td , L o n d o n
(1 9 9 0 ).
(3 ) Fu lle r, G . G ., O ptical R heom etry of C om plex F luids , O x fo rd U n iv e rsity P re ss, N e w Y o rk (1 9 9 5 ).
(4 ) K ish b a u g h , A . J. a n d A . J. M u H u g h , "A R h e o - O p tica l S tu d y o f S h e a r - T h ick e n in g a n d S tru ctu re Fo rm a tio n
in P o ly m e r S o lu tio n s. P a rt I. E x p e r im e n ta l , " R heo A cta 3 2 , 9 - 2 4 (1 9 9 3 ).
( 5 ) K o rn fie ld , J. A ., G . G . Fu lle r, a n d D . S . P e a rso n , "In fra re d D ich ro ism M e a su re m e n ts o f M o le cu la r R e la x a tio n
in B in a ry B le n d M e lt R h e o lo g y , " M acrom olecules 2 2 , 1 3 3 4 - 1 3 4 5 (1 9 8 9 ).
(6 ) K u m e , T ., T . H a sh im o to , T . T a k a h a sh i, a n d G . G . Fu lle r, "R h e o - O p tica l S tu d ie s o f S h e a r - In d u ce d S tru ctu re s
in S e m id ilu te P o ly sty re n e S o lu tio n s , " M acrom olecules 3 0 , 7 2 3 2 - 7 2 3 6 (1 9 9 7 ) .
(7 ) L e n stra , T . A . J., C o llo id s n e a r p h a se tra n sitio n lin e s u n d e r sh e a r, P h .D . th e s is, U n iv e rsity o f U tre ch t,
N e th e rla n d s (2 0 0 1 ). (h ttp : //ig itu r - a rch iv e .lib ra ry .u u .n l/d isse rta tio n s/1 9 5 2 3 9 4 / in h o u d .h tm )
(8 ) M a co sk o , C . W ., R heology : P rinciples, M easurem ents, and A pplications , W ile y - V C H , N e w Y o rk ( 1 9 9 4 ).
(9 ) S o n d e rg a a rd , K . a n d J. L y n g a a e - Jo rg e n se n , R heo - P hysics of M ultiphase P olym er Syste m s : C haracterization by
R heo - O ptical T echniques , T e ch n o m ic P u b l. C o ., L a n ca ste r, P A (1 9 9 4 ).
(1 0 ) T a p a d ia , P ., S . R a v in d ra n a th , a n d S . - Q . W a n g , "B a n d in g in E n ta n g le d P o ly m e r Flu id s in O scilla to ry S h e a r in g , "
P hys R ev L ett 9 6 , 1 9 6 0 0 1 (2 0 0 6 ).
Chapter IV General Analyses:
Scaling Laws, Time-Temperature
Superposition, Solvent Quality, and
Fundamental Material Constants
Fig 3.3-1 (p 105) in the textbook
Content of Chapter IV

Effects of Solvent Quality (pp. 139-143)

Molecular-Weight Scaling Laws (pp.143-150)

Retrieval of Fundamental Material Constants

Time-Temperature Superposition: Application and Failure
(pp. 105-108, 139-143)

The ability to measure viscoelasticity of low viscosity fluids
without TTS data shifting

Case Study
IV.1 Effects of Solvent Quality
6
6
P o lystyren e, M w = 7 .1 4 x 1 0 g /m o l
P o lystyren e, M w = 7 .1 4 x 1 0 g /m o l
1 .2
1000
[ ] / [ ] 0
[ ] (m l/g )
1 .0
0 .8
o
b en zen e(3 0 C )
o
1 -ch lo ro b u tan e(3 8 C )
o
tra n s-d ecalin (2 3 .8 C )
o
benzene(30 C )
0 .6
o
1-chlorobutane(38 C )
o
trans-decalin(23.8 C )
100
0 .0 0 0 1
0 .0 0 1
0 .0 1
0 .1
1
W eissen b erg N u m b er
Magnitude of intrinsic viscosity
-temperature & Solvent
Flow curve
10
0 .0 0 0 1
0 .0 0 1
0 .0 1
0 .1
1
10
W eissen b erg n u m b er
Fig 3.3-4 (p 107) in the textbook, or
T. Kotaka et al., J. Chem. Phys. 45, 2770-2773 (1966).
IV.1 Effects of Solvent Quality


The solvent quality is an index describing the strength of
polymer-solvent interactions.
This interaction strength is a function of chemical species of
polymer & solvent molecules, temperature, and pressure.
Scaling law of polymer size and molecular weight (<R2>end-to-end 1/2 ~ Mw).
Root mean square
end-to-end distance
Solvent condition
Good
<R2>end-to-end 1/2

Bad
Temperature
T
Index

T>
T=
T<
3/5
1/2
1/3
IV.1 Effects of Solvent Quality
P  M S in cyclohexane
1 .0
poor
good
Sample
Mw
g/mol
Mw / Mn
(SEC)
Poly(methylstyrene)
1.14×106
1.11
[ ] (m l/g)
0 .9
0 .8
0 .7
0 .6
-temperature
0 .5
0 .4
15
20
25
30
35
o
T ( C)
40
45
50
Advantages of PMS:
1. High plasticized speed
2. Good temperature tolerance
3. Contamination resistance
4. Compatibility with other additives
5. Environment friendly
N. Hadjichristidis et al., Macromolecules 24, 6725-6729 (1991).
IV.1 Effects of Solvent Quality
The (temperature, weight fraction) phase diagram
for the polystyrene-cyclohexane system for samples
of Indicated molecular weight.
S. Saeki et al, Macromolecules 6, 246-250(1973).
TU: upper critical solution temperature
TL: lower critical solution temperature
IV.1 Effects of Solvent Quality
Poly(N-isopropylacrylamide) in water
Mw = 4.45x105 g/mol, c = 6.65x10-4 g/ml
Mw = 1.00x107 g/mol, c = 2.50x10-5 g/ml
coil
globule
coil
globule
x
X. Wang et al., Macromolecules 31, 2972-2976 (1998).
PN IPA M /w ater, heating
cooling
PN IPA M /SD S/w ater, cooling
H. Yang et al., Polymer 44, 7175-7180 (2003).
IV.1 Effects of Solvent Quality on intrinsic viscosity
(i) The Rouse model:
  
N
N b 
2
A
Ms
2
N A : Avogadro constant
36
M : Molecular
(ii) The Zimm model for Θ solvent:
   0 . 425
NA
(
N b)
N : number
3
M
(iii) The Zimm model for good solvent:
  
N
A
N
3
b
3
weight
of segments per polymer
b : effective bond length
 : friction constant
η s : solvent viscosity
ν is equal to the α
M
We write the molecular weight dependence of [η] as
  

1

  0 .5
 3  1  0 . 8

M

Rouse model (Θ solvent)
Zimm model (Θ solvent)
Zimm model (good solvent)
Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics. P113~114
IV.2.1 Molecular-Weight
Dependence
For linear polymer melts
Molecular
weight, Mw
Zero-shear
Relaxation Diffusivity,
viscosity,
DG
time, 
0
< Mc
~ Mw
~ Mw2
~ 1/Mw
> Mc
~ Mw3.4
~ Mw3
~ 1/Mw2
Mc (=2Me): critical molecular weight
Me: entangled molecular weight
Plot of constant + log 0 vs. constant + log M for nine different
polymers. The two constants are different for each of the polymers,
and the one appearing in the abscissa is proportional to
concentration, which is constant for a given undiluted polymer.
For each polymer the slopes of the left and right straight line
regions are 1.0 and 3.4, respectively.
[G. C. Berry and T. G. Fox, Adv. Polym. Sci. 5, 261-357 (1968).]
IV.2.2 Concentration Effect
Relative
r 
viscosity
 solution
 solvent
Specific
 sp 
:
 1   c  k   c  
viscosity
2
:
 solution   solvent
Intrinsic
 solvent
viscosity
  sp
   
 c



 c 0
2
 r 1
:
[cf. p109]
An example of viscosity versus concentration plots for polystyrene (Mw=7.14106 g/mol) in benzene at
30 C. White circles: plot of sp / c vs. c; black circles: plot of (lnr)/c vs. c. (1) Zimm-Crothers viscometer
(3.710-3 ~7.610-2 dyn/cm2); (2)Ubbelohde viscometer (8.67 dyn/cm2); (3)Ubbelohde viscometer
(12.2 dyn/cm2).
T. Kotaka et al., J. Chem. Phys. 45, 2770-2773 (1966).
IV.2.2 Concentration Effect for semi-dilute solution
PAM copolymer with
hydrophobic blocks
Pure polyacrylamides
• The viscosity of polymer solutions increases
steeply ( roughly in proportion to C 4~5 ) above
the overlap concentration.
• Combining the Mw dependence and
concentration effect , the zero-shear viscosity
can be estimated by
[H] : Hydrophobe content in the monomer feed
0  C
4~5
M
3 .4
NH : Number of hydrophobe per micelle
w
Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics.P157
Enrique J. R. et al., Macromolecules 32,8580- 8588 (1999)
IV.2.3 Impact of Molecular
Weight Distribution
H. Munstedt, J. Rheol. 24, 847-867 (1980)
IV.2.4 Molecular Architecture
Linear Polymer
Star Polymer
Pom-Pom Polymer
polybutadiene
Polyisoprene
Polyisoprene
IV.3 Retrieval of Fundamental Material Constants
Newtonian
Power law
Zero-shear
viscosity, 0
     0 
Relaxation

time,   1 /  critical
Fig 3.3-1 (p 105) in the textbook
IV.3 Retrieval of Fundamental Material Constants
 G
0
J e   2 2
  0
1 , 0

6



2
0 

5G N
   0 2 0
0 
G N  cRT / M e 
12  0
 d
2
0
0  GN  d
e ~ M0
d ~ M3
Storage modulus vs. frequency for narrow distribution
polystyrene melts. Molecular weight ranges from
Mw = 8.9x103 r/mol (L9) to Mw = 5.8x105 g/mol (L18).
Theoretical results of (a) G(t) and (b) G’()
for polymer melts.
M. Doi and S. F. Edwards, The Theory of Polymer
Dynamics, Oxford Science: New York (1986), pp 229-230.
IV.4 Time-Temperature Superposition

Time-temperature superposition holds for many polymer melts
and solutions, as long as there are no phase transitions or other
temperature-dependent structural changes in the liquid.

Time-temperature shifting is extremely useful in practical
applications, allowing one to make prediction of timedependent material response.
WLF (Williams
- Landel - Ferry) equation
 c1 T  T 0 
0
log a T 
c  T  T0
0
2
 c1 T  T 0 
0

T  T
:
IV.4 Time-Temperature Superposition
WLF temperature shift parameters
WLF (Williams
- Landel - Ferry) equation
 c1 T  T 0 
0
log a T 
c 2  T  T0
0
:
 c1 T  T 0 
0

T  T
J. D. Ferry, Viscoelastic Properties of Polymers, 3rd ed., Wiley: New York (1980).
IV.4 Time-Temperature Superposition
Non-Newtonian viscosity of a low-density
polyethylene melt at several different temperatures.
Master curves for the viscosity and first normal
Stress coefficient as functions of shear rate for
a low-density polyethylene melt
Fig 3.3-1 and 3.3-2 (pp105-106) in the textbook.
IV.4 Time-Temperature Superposition
A master curve of polystyrene-n-butyl benzene solutions. Molecular weights varied from 1.6x10 5 to
2.4x106 g/mol, concentration from 0.255 to 0.55 g/cm3, and temperature from 303 to 333 K.
Fig 3.6-5 (p 146) in the textbook.
The ability to measure viscoelasticity of lowviscosity fluids without TTS data shifting

The relaxation times for low-viscosity fluids are usually quite short and
fall in the time domain of milliseconds or below.

G’ and G” measurements must cover an enormously wide scale of
times or frequencies in order to capture the relaxation process of these
fluids.

Conventional rheometers are usually limited to frequencies
≦100 Hz due to inertial effects. This range of frequencies is insufficient
to reach the true high-frequency, limiting behavior of these fluids.

High-frequency rheometry such as piezoelastic axial vibrator (PAV) or
torsion resonator(TR) provides a way to characterize the dynamic
properties of these low-viscosity fluids.
The ability to measure viscoelasticity of lowviscosity fluids without TTS data shifting
PAV gives reliable mechanical
spectra for frequencies between 1
and 4000 Hz
The TR can be used only at given
(high) frequencies
The ability to measure viscoelasticity of lowviscosity fluids without TTS data shifting
FIG. 1. Fluid: DEP-10 wt% of monodisperse
PS Mw=210000. (■) η*, (●) G”, and (▲)
G’.
FIG. 2. Fluid: DEP-2.5 wt% of monodisperse
PS Mw=110000. (■) η*, (●) G”, and (▲)
G’.
Figure1 shows that the combination of mechanical rheometer and PAV
give a reasonable match in the overlapping region.
Figure 2 shows that the LVE data for both PAV and TR. Overlapping
data are not possible using these two rheometers. However, consistency
between the two data sets appears reasonable.
Reference
1. Vadillo, D.C., T.R. Tuladhar, A.C. Mulji, and M.R. Mackley, “The
rheological characterization of linear viscoelasticity for ink jet fluids using
piezo axial vibrator and torsion resonator theometers,” J. Rheol. 54, 781795(2010)
2. Crassous, J., R. Regisser, M. Ballauff, and N. Willenbacher,
“Characterization of he viscoelastic behavior of complex fluids using the
piezoelastic axial vibrator,” J. Rheol. 49, 851-863 (2005)
3. Fritz, G., W. Pechhold, N. Willenbacher, and N. J. Wagner,
“Characterization complex fluids with high frequency rheology using
torsional resonators at multiple frequencies,” J. Rheol. 47, 303-319 (2003)
Chapter V Constitutive Equations and
Modeling of Complex Flow Processing
Content of Chapter V

Models for Generalized Newtonian Fluids

Constitutive Equations for Generalized Linear Viscoelasticity

Objective Differential/Integral Constitutive Equations

Simulations of complex Flow Processing

Case Study
V.1 Models for Generalized Newtonian Fluids



In many industrial problems the most important feature of
polymeric liquids is that their viscosities decrease markedly as
the shear rate increases.
The generalized Newtonian model incorporates the idea of a
shear-rate-dependent viscosity into the Newton’s consitutive
equation.
The generalized Newtonian model cannot, however, describe
normal stress effects or time-dependent elastic effects.
Incom pressible N ew tonian fluids:
τ  -  γ ,   f  tem perature, pressure, com position 
Incom pressible generalized N ew tonian flu ids:
τ  - γ ,   f  scalar invariants of γ , .... 
V.1 Models for Generalized Newtonian Fluids

The Carreau-Yasuda model
 
0 

 1    
a

 n 1  / a
 : viscosity ,  0 : zero - shear visc osity,   : infinite
 : relaxation

time, n : power - law exponent,
- shear visc osity,  : shear rate
a : dimensionl
The power-law model
  m 
n 1
n < 1, shear-thinning (pesudoplastic) fluids
n = 1 and m = , Newtonian fluids
n > 1, shear-thickening (dilatant) fluids
ess parameter
V.1 Models for Generalized Newtonian Fluids

The Eyring model
 arcsinh   
   0 





T he E yring equation w as the first   


expre ssion obtained by a m olecular theory .
The Bingham model
  

   0   0 / 


 0 : yield stress ,  

0
0
τ : τ  / 2
Other empirical  functions in the generalized Newtonian fluid model (see
Table 4.5-1, p 228 in the textbook)
V.2 Constitutive Equations for Generalized Linear
Viscoelasticity

Goal: To introduce an equation that can describe some of the timedependent motions of fluids under a flow with very small displacement
gradients

Why do we concern the linear viscoelasticity (LVE) of fluids?
(1) To interrelate molecular structure with the linear mechanical
responses
(2) To proceed to the subject of nonlinear viscoelasticity

How to combine the idea of viscosity and elasticity into a single
constitutive equation that describes various rheological features?
A natural combination of the Newton’s law for Newtonian fluids & the
Hookean law for perfect elastic solids.
V.2 Constitutive Equations for Generalized Linear
Viscoelasticity

The Maxwell model
(for melts or concentrated solutions)
shear stress for a Newtonian
 yx     yx
a. the differential form :
  1

t
shear stress for a Hookean solid
  0 
The nature of flow
τ yx   G
b. the integral form :
t
 t   

 0 / 1  e
  t  t   / 1
fluid

  t   dt 

Relaxation modulus, G(t):
The nature of fluid

τ yx 
   yx
G
t
u x
y
    yx
replace  by  0 and μ / G by  1
V.2 Constitutive Equations for Generalized Linear
Viscoelasticity

The Jeffreys model
(for dilution solutions)
a. the differential form :
  1

 

  0     2

t
t 

b. the integral form :
  0
 t     
 
 1
t


 02
 2    t  t   / 1
2
  t  t      t   dt 
1 
e
1 
1


Relaxation modulus, G(t)
(contribution of both polymer and solvent)
V.3 Objective Differential/Integral Constitutive Equations

Quasi-linear model is obtained by reformulating the linear viscoelastic
model.
The convected Jeffreys model or Oldroyd’s fluid B
τ   1 τ 1    0 γ 1    2 γ  2  
Convected
time derivative
γ 1   γ
γ  n 1  
τ 1  


D
Dt
D
Dt


γ  n    v   γ  n   γ  n    v 

T

τ   v   τ  τ   v 
T
The convected Jeffreys model is derived from the kinetic theory for dilute
solutions of elastic Hookean dumbbell.
If 2 = 0, the model reduces to the convected Maxwell model.
V.3 Objective Differential/Integral Constitutive Equations

Nonlinear differential model
The Giesekus model:
τ  τs  τp
τ s   s γ
τ p  1 τ p 1   


1
p
τ
p
 τ p    p γ
The model contains four parameters: a relaxation time, 1; the zero-shearrate viscosities (s and p) of solvent and polymer; and the dimensionless
“mobility factor”, .
 is associated with anisotropic Brownian motion and/or anisotropic
hydrodynamic drag on the polymer molecules.
V.3 Objective Differential/Integral Constitutive Equations

Nonlinear integral models
The factorized K-BKZ model:
τ t  
 W I1 , I 2 
W I1 , I 2 
 

M
t

t



0 

I1
I 2


t
0  
The factorized Rivlin-Sawyers model:
τ t  
t

 M t  t   1  I 1 , I 2  0    2  I 1 , I 2 
0 
d t 

M t  t   : time - dependent
factor
W  I 1 , I 2  or  i  I 1 , I 2  : strain dependent
factor
 dt

V.3 Objective Differential/Integral Constitutive Equations

Advantages of nonlinear integral models:
(1) they include the general linear viscoelastic fluids
(2) they provide a framework of constitutive equations with molecular and
empirical origins
(3) it is possible to use these constitutive equations to interrelate various
material functions

Disadvantages of nonlinear integral models:
(1) the models generally predict too much recoil in elastic recoil experiments
(2) these models have been omitted for the cases of memory-strain coupling
V.4 Simulations of Complex Flow Processing


relaxation
section


Stretching
section
Polymer properties
Governing equations
(balance equations of mass,
momentum and energy)
Power-law constitutive
equation
Finite element method
A. Makradi et al, J. Appl. Polym. Sci. 100,
2259-2266 (2006).
V.4 Simulations of Complex Flow Processing
1D Post Draw model for
IPP Spinning
Roller 2
Roller 3
Polymer properties
Density
0.85 (Kg/m3)
Glass transition temperature
253 (K)
Surface tension
35 (dyn/cm)
Melt shear modulus
9x108 (Pa)
Maximum crystallization rate
0.55 (1/s)
Maximum rate temperature
65 (K)
Crystallization half width
temperature
60 (K)
Avrami index
3
Maximum percent crystallinity
70 (%)
Roller 1
CAEFF (Center for Advanced Engineering
Fibers and Films) software
V.4 Simulations of Complex Flow Processing
Model properties
Heat capacity parameters
Orientation hardening parameter
9
Cs1
0.25 (cal/g/C)
Frictional coefficient
0.6
Cs2
7.0x10-4 (cal/g/C2)
Room temperature
298 (K)
Cs3
0
Initial tensile stress
5x105 (Pa)
Cl1
0.32 (cal/g/C)
Initial percent crystallinity
45 (%)
Cl2
5.7x10-4 (cal/g/C2)
Crystallization rate
0.1
Cl3
0
Amorphous shear stress
8.5x106 (Pa)
Hf
30 (cal/g)
Poisson ratio
4.3x105 (Pa)
Rubber elasticity
1.5x105 (Pa)
Roller parameters
 activation energy
10800
Temperature
35 (C)
Pre-exponential shear strain rate
2.3x107
Radius
8 (cm)
Activation volume
4.7x10-29
Activation parameter
3.65
Mass flow rate
1.3x10-6
(Kg/s)
Roller 1
Velocity of Roller 2
Roller 3
80 (m/s, conter-clockwise)
80, 160 (m/s, clockwise)
160 (m/s, conter-clockwise)
V.4 Simulations of Complex Flow Processing
Velocity of Roller 2 = 160 m/s
Velocity of Roller 2 = 80 m/s
Chapter VI Shear Thickening in
Colloidal Dispersions
Content of chapter VI

Introduction to shear thickening fluids

Onset of shear thickening : the Péclet number

Lubrication hydrodynamics and hydroclusters

Controlling shear thickening fluids: to modify colloidal
surface
VI.1 Introduction to the shear thickening fluids

The unique material properties of increased energy dissipation
combined with increased elastic modulus make shear thickening
fluids ideal for damping and shock-absorption applications.

Example:
The different velocity at which a quarter –inch steel ball
required to penetrate various layers
For single layer of Kevlar is measured at about 100 m/s
For Kevlar formulated with polymeric colloids is about 150 m/s
For Kevlar formulated with silica colloids is about 250 m/s
VI.1 Introduction to shear thickening fluids
 Right video : two layers containing
shear thickening fluids and Nylon.
The popular interest in cornstarch and
water mixers known as “oobleck” is
due to their transition from fluid-like
to solid-like behavior when stressed.
 Left video : three layers containing
neat Nylon.
VI.1 Introduction to shear thickening fluids
Beyond the critical stress, the fluid’s
viscosity decreases (shear thinning).
At high shear stress, its viscosity
increases (shear thickening)
The viscosity of colloidal latex dispersions, as a
function of applied shear stress.
The actual nature of the shear thickening will depend on the parameters
of the suspended phase: phase volume, particle size (distribution),
particle shape, as well as those of the suspending phase (viscosity and
the details of the deformation.)
VI.2 Onset of shear thickening : the Péclet number
 Fluid drag on the particle leads to the Stokes-Einstein relationship:
D 
k BT
6 a
a : particle's hydrodynam ic radius
 Dt
 The mean square of the particle’s displacement is
Accordingly, the diffusivity sets the characteristic time scale for the
2
particle’s Brownian motion.
a
x
t particle 
2
D
 A dimensionless number known as Péclet number, Pe

Pe 
 a
D
2

a
3
k BT
VI.2 Onset of shear thickening : Péclet number

 Low shear rate ( Pe <<1 ,   t particle ) is close to equilibrium that
Brownian motion can largely restore equilibrium microstructure on
the time scale of slow shear flow.
 Pe ～1, shear thinning is evident around that regime.
 At high shear rates or stress (Pe >>1), deformation of colloidal
microstructure by the flows occurs faster than Brownian motion can
restore it. Accordingly, the High Pe triggers the onset of shear
thickening.
VI.3 Lubrication hydrodynamics and hydroclusters
Pe~1
Pe<<1
 The flow-induced density
fluctuations are known as
hydroclusters which lead to an
increase in viscosity.
 The formation of hydroclusters
is reversible, so reducing the
shear rate returns the
suspensions to a stable fluid
Pe>>1
 At (Pe<<1) regime, random collisions among particles
make them naturally resistant to flow.
 As the shear rate increase (Pe~1), particles become
organized in the flow, which lowers their viscosity.
 At (Pe>>1) regime, the strong hydrodynamic
coupling between particles leads to the formation of
hydroclusters (red particles) which cause an increase
in viscosity.
Normalized lubrication force
VI.3 Lubrication hydrodynamics and hydroclusters
The force required to drive two particles
together is lubrication force, as well as
the force is the same one required to
separate two particles.
Distance between particle surfaces
 In simple shear flow, particle trajectories are strongly coupled by the
hydrodynamic interaction if the particle are close together.
 When two particles approach each other, rising hydrodynamic pressure
between them squeezes fluid from the gap.
VI.3 Lubrication hydrodynamics and hydroclusters
Red region indicates the most probable particle position as nearest neighbors
 At low Pe number (0.1), the distribution of neighboring particles is isotropic.
 At Pe =0.1, shear distortion appears in neighbor distribution, such that
particles are convected together along the compression axes
 At high Pe regime, particles aggregate into closely connected clusters, which
manifest as yet greater anisotropy in the micro structure.
Particles are more closely packed and occupy a narrow region (red) ,
indicative of being trapped by the lubrication forces.
Viscosity (Pa*s)
VI.3 Lubrication hydrodynamics and hydroclusters
Pe>>1
Pe~1
■ :the viscosity of concentrated colloidal suspension
● : stochastic motion of particles component
▲ : hydrodynamic interaction component
Pe~1
Pe<<1
Pe>>1
Shear stress (Pa)
 The equilibrium microstructure is set by balance of stochastic and interparticle force,
including electrostatic and van der Waals force, but is not affected by hydrodynamic
interactions.
 The low shear (Pe<<1) viscosity has two components, one due to interparticle force,
and the other due to hydrodynamic interactions.
 At Pe~1 regime, the stochastic motion dominates the flow behavior
 At high shear rates (Pe>>1), hydrodynamic interactions between particles dominate over
stochastic ones.
VI.4 Controlling shear thickening fluids: to modify
colloidal surface
 The addition of a polymer “brush” grafted or absorbed onto the particles’
surface can prevent particles from getting close together.
 The figure shows that shear thickening is suppressed by imposing a purely
repulsive force field.
 With the right selection of grafted density, molecular weight, and solvent , the
onset of shear thickening moves out of the desired processing regime
References

N. J. Wagner, J. F. Brady, Physical today, October 2009

B. J. Maranzano, N. J. Wagner, J. Chem. Phys. 114, 10514 (2001)