Transcript Polymer Rheology
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
, 2 nd 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
, Wiley VCH (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 analytical tools , and phenomenology, general concepts 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 Non-Newtonian Fluids fluids . More specifically, the study of
Y
流體 牛頓流體 水、有機小分子溶劑等 非牛頓流體 高分子溶液、膠體等
V
黏度不為定值 (尤其在快速流場下)
Newton’s law of viscosity
yx
V
V Y
黏度
η
為定值 Small molecule
Macromolecule
V
● Deformable
I.1
Shear Thinning
/
Thickening
Dilatants (Shear thickening)
Newtonian Fluids Pseudoplastics (Shear thinning)
0 pleatau
Dilatants (Shear thickening)
Newtonian Fluids 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
, 2 nd 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 = 10 6 g/mol) in 100 cm 3 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, Wiley Interscience (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,
t
flow : 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 min at 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 Non-Newtonian Fluids Secondary Flow Primary Flow 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 ]
I.5
Secondary Flows and Instabilities
(cont.)
Secondary flow Newtonian Fluids Primary Flow Secondary Flow Newtonian fluid (N): water-glycerin Non-Newtonian Fluids Non-Newtonian fluid (P): 100 ppm polyacrylamide in water-glycerin Primary Flow Secondary Flow 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 R 1 R
[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).]
2
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 the De, Wi numbers 機械量測 本質方程式 flow pattern 模流分析 Traditional route Modern (predictive) route 基本流變性質
G N
0 microrheology microscopy/spectroscopy birefringence/dichroism light/ neutron scatterings particle tracking 光學量測 molecular orientation / alignment particle size distribution/ diffusivity micro/mesoscopic structures 分子動力理論 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
” by R. B. Bird et al. For more details, you are referred to the textbooks and references cited therein.
(Vol. 1)
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 characterize polymeric liquids and shearfree , are frequently used to (a) Shear (b) Shearfree
v x
y
FIG. 3.1-1.
Steady simple shear flow
v x
y x y
;
v y
0;
v z
0 Shear rate
FIG. 3.1-2.
Streamlines for elongational flow (
b
=0)
v x v y v z
Elongation rate
x
2
y
2
z
The Stress Tensor
y x z
Shear Flow
Total
stress tensor*
p
Stress tensor
p
yx
xx p
yx
yy
0 0 Hydrostatic pressure forces 0 0
p
zz
Shear Stress :
yx
First Normal Stress Difference :
xx
Second Normal Stress Difference :
yy yy
zz
Elongational Flow
p
p
xx
0 0 0
p
yy
0 0 0
p
zz
Tensile Stress :
zz
xx
*See § 2.2
Classification of Flow Geometries
(a) Shear Pressure Flow: Capillary Drag Flows: Concentric Cylinder (b) Elongation Uniaxial Elongation (
b
0 , 0 ): Cone-and Plate Moving Clamps Parallel Plates
Typical Shear/Elongation Rate Range & Concentration Regimes for Each Geometries
(a) Shear
Concentrated Regime Dilute Regime
Homogeneous deformation:* Nonhomogeneous deformation: Cone-and Plate Concentric Cylinder (b) Elongation Parallel Plates Capillary 1 2 3 4 5 Moving clamps
For Melts & High-Viscosity Solutions
-1
(s )
*Stress and strain are independent of position throughout the sample
Viscometric Functions & Assumptions Example:
Concentric Cylinder
Assumptions :
(1) Steady, laminar, isothermal flow (2)
v
1 only and
v r
v z
0 (3) Negligible gravity and end effects 0 (From p.188 of ref 3)
R
1
W
1
R
2
H
Shear rate :
R W R
1 1 2
R
1 (homogeneous)
FIG.
Concentric cylinder viscometer Shear -rate dependent viscosity where the torque acting on the 2
T R
1 2
H
T
r
1 (2
R H
1 )
R
1 , 2 : Radii of inner and outer cylinders
H
: Height of cylinders
W
1 : Angular velocity of inner cyl inder
T
: Torque on inner cyli nder
Flow Instability in a Concentric Cylinder Viscometer for a Newtonian Liquid Taylor number (Ta) Centrifugal force Viscous force 41.3
Laminar
Secondary
Turbulent T
Secondary
a
Onset
Flow
4 1.3
;
of
Re 9 4 Ta 141; Re 322 Ta (or Re) plays the central role!
Taylor vortices Ta 387; Re 868 Ta
Turbulent
1, 715; Re 3, 960
Rod Climbing
is not a subtle effect, as demonstrated on the cover by Ph.D. student Sylvana Garcia-Rodrigues from Columbia. Ms. Garcia-Rodrigues is studying rheology in the Mechanical Engineering Department at U. of Wisconsin-Madison, USA The Apparatus shown was created by UWMadison Professors Emeriti John L. Schrag and Arthur S. Lodge. The fluid shown is a 2% aqueous polyacrylamide solution Carlos Arango Sabogal ( 2006 ) , and the rotational speed is nominally 0.5 Hz. Photo by
Example 2.3-1:
Interpretation of Free Surface Shapes in the Rod-Climbing Experiment (N)
I. Phenomenological Interpretation:
(P) (F) : For the Newtonian fluids the surface near the rod is slightly depressed and acts as a sensitive manometer for the smaller pressure near the rod generated by centrifugal force (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 and alignment of the polymer molecules along the streamlines. The thermal motions make the polymer molecules act as small “rubber bands” wanting to snap back
‧
z P
r
(N) :1
r
: 2
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
d d
ln
r p
) 2
21
d
(
22
33
11 21
22 )
v
1 2 (N) : For Ne wton ia n fluids the first and second normal stress differences, 11 22 and 22 33 , are both z ero in shear flow, and eq. 2.3-5 shows that the normal pressure exerted on the l ubric a ted li d
inc reases
with the radius (P) : For polymer ic fluids we have already indicated that 11 with a numerical value much larger than that of 22 22 is pratically always negative 33 . We see that the normal stresses may caus e th e total normal pressure to
decrease
in the radial direction.
Examples 1.3-4 & 10.2-1:
Cone-and-Plate Instrument
Assumptions :
(1) Steady, laminar, isothermal flow (2) only;
v r
v
0 (3) 0 0.1 rad ( 6 ) (4) Negligible body forces (5) Spherical liquid boundary (From p.205 of ref 3)
FIG. 1.3-4.
Cone-and-plate geometry Shear rate
W
0
0 : (homogeneous)
W
0 : Angular velocity of cone 0 : Cone angle
R
: Radius of circular plate Shear -rate dependent viscosity The first normal stress difference coefficient 1 3
T
2 2
0 1
R
3
W
0
R T
: Torque on plate
F
:
T
2
R r
2 0 0 Force required to keep tip of cone
F
in contact with c ircular plate 2 2
dr d
Example:
p.530
Uniaxial Elongational Flow Hencky strain :
max
t
max ln (
L
max
L
0 )
L
0 : Initial sample length
L
max : Maximum smaple length The Normal Stress Difference :
zz
rr
( ): Total force per unit area exerted by the load cell Instantaneous corss - sectional area of the sample
z
The Transient Elongational Viscosity :
r
(
zz
0
rr
)
A e
0
0 0
t
0 : Elongation rate
A
0 : Initial cross - sectional area of the sample
FIG. 10.3-1a.
Device used to generate uniaxial elongational flows by separating 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:
Disk Instrument o o Measurement of the Viscometric Functions in the Parallel-
Problem 1B.5: Problem 1D.2:
Parallel-Disk Viscometer Viscous Heating in Oscillatory Flow
2.2. Basic Vector/Tensor Manipulations
Vector Operations (Gibbs Notation)
Vector:
u
i u i
i
u
1 1
u
2 2
u
3 3
z x y
1 3 2 3 1 The Kronecker delta
ij
1 Dot product: Let
ij ij
i
, if , if
i
i u i
i
j u j
j
ij
(
u v i j
)(
i
j
) (
u v i i
)
ii
i
j j
i u v i i
3 2
u
2
The permutation symbol
ijk
ijk ijk ijk
, if
ijk
123 231 , or 312 , if
ijk
213 0 if any two indices are alike Cross product:
ijk i u i
i
(
u v i j
)(
ijk k j
)
u
Using
i
j
j
j
k
3 1
ijk k
ij
(
u v i j
)(
i
j
)
1 2
u
1
v
1
u
2
v
2
3
u
3
v
3
Tensor Operations
Tensor:
i j
i j ij
1 1 11
2 1 21
1 2 12
2 2 22
1 3 13
2 3 23
3 1 31
3 2 32
3 3 33 2 2 ( 21 , 22 , 23 ) 1 ( 11 , 12 , 13 ) Stresses acting on plane 1 1 3 3 ( 31 , 32 , 33 )
FIG.
The stress tensor The
total
momentum flux tensor for an incompressible fluid is: Normal stresses
p
21
31 11
p
12
32 22
13
p
23 33
p
Hydrostatic pressure forces Stress tensor or Momentum flux tensor
Example:
The total momentum flux through a surface of orientat ion
n
is:
ijk
i j
ij n k
i j ij
(
i j
k
)
k n k
k
Let
k
j
ij
ij n j i
Some Definitions & Frequently Used Operations:
Transpose : † Gradient : :
i
ij
i j ij
i
x i
† Cartesian coordinate Laplacian: :
i
2
x i
2 Cartesian coordinate Unit tensor : = :
ij
i j ij
1 0
ij
i j ji
0 1 0 0 0 0 1
ij
i j
v j
x i
ik
ijkl
(
k
x i ik v ij k
)(
i jkl l
ijk
(
ij v jl
)(
i l
) )
ijk
ijk
ij
ij ji k
v j
x i
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.
Different temperatures The shear-rate dependent is defined as: Non-Newtonian viscosity viscosity η η of a low-density polyethylene at several The first and second normal stress coefficients are defined as follows:
yx
yx
xx
yy
1 2
yx
2
yy zz
2
yx
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 Intrinsic Viscosity: [ ]
c
lim 0
c
s s
c
: Mass concentration
FIG. 3.3-4.
Intrinsic viscosity of polystyrene Solutions, With various solvents, as a function of reduced shear rate β Relative Viscosity: rel
s : Solution viscosity s : Solvent viscosity
II.
Unsteady Shear Flow Material Functions Exp b:
Small-Amplitude Oscillatory Shear Flow The shear s tre ss but is not in phase with eith er the shear s trai n o r shea r rate Shear Stress:
yx
A
0 sin( ) Shear rate:
yx
0 cos
t
Shear strain:
yx
0 sin
t
FIG. 3.4-2.
and first normal stress difference in small-amplitude oscillatory shear flow Oscillatory shear strain, shear rate, shear stress,
It is customary to rewrite the above equations to display the in-phase out-of-phase parts of the shear stress and
yx
G
0 Storage modulus sin
t
G
0 co s
t
Loss modulus
FIG. 3.4-3.
Storage and loss moduli, G’ and G”, as functions of frequency temperature of
T 0
=423 K ω at a reference 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 + linear viscoelastic envelope, goes through a maximum, and then approaches the steady - state value.
FIG. 3.4 -7.
Shear stress growth function + 0 ) data for a low - density polyethylene melt. The solid curve is calculated from Eq. 5.3- 25 with the spectrum in Table 5.3- 2
FIG. 3.4 - 8.
Shear stress growth function 0 ) ( for two polymer solutions (a and b) and an aluminum 0 ) soap solution (c)
FIG. 3.4 - 9.
First normal stress growth function 1 for the low - density polyethylene melt 0 )
Exp e:
Stress Relaxation after a Sudden Shearing Displacement (Step-Strain Stress Relaxation) Relaxation Modulus:*
0 )
yx
0 0 0 0 For small shear strains lim 0 0
0 ) In this limit, the shear stress is linear in strain The Lodge-Meissner Rule:
G
1
0 ) 0 ) 1
FIG. 3.4 -15.
The relaxation modulus
G t
relaxation function
G
1 0 ) (open symbols) and normal stress 0 ) (solid symbols) for a low - density polyethylene melt *
Example 5.3-2:
Stress Relaxation after a Sudden Shearing Displacement
FIG. 3.4 -16.
The stress relaxation modulus
G t
Part (a) shows how 0 ) 20 %
M
w 6 ) in Aroclor.
0 ) varies with shear strain. In (b) the data are superposed by vertical shif ting to show the similarity in
G t
0 ) at large times regardless of the imposed shear stain
2.4. Material Functions in Elongational Flows
Shearfree Flow Material Functions
Zero - elongation-rate elongational viscosity 0 For Uniaxial Elongational Flow (
b
0 , 0 ):
zz
xx
: Elongational viscosity : Elongation rate Zero - shear -rate viscosity 0
FIG. 3.5-1.
for a polystyrene melt as functions of elongation rate and shear rate, respectively
Elongational Stress Growth Function The number average and weight average molecular weights of the samples:
FIG. 3.5-2.
Monodisperse, but with a tail in high M.W. (GPC results) Time dependence of the elongational The abrupt upturn, or " strain hard ening ," occurs at a roughly constant value of Hencky st r ain 0
t
Chapter III Optical Characterizations
Topics in Each Section
§
3-1 Introduction to Rheo-Optics Method 3-1.1.
3-1.2.
Introduction & Review of Optical Phenomena Characteristic Dimension & Optical Range §
3-2 Typical Experimental Set-ups 3-2.1
3-2.2
Flow Dichroism and Birefringence Measurements [for Case Study 1-2 ] Combined Rheo-Optcial Measurements (including Rheo-SALS) [for Case Study 3] §
3-3 Information Retrieval in Individual Measurements Case Study 1: Case Study 2: Case Study 3:
Flow Dichroism and Birefringence of Polymers Dynamics of Multicomponent Polymer Melts 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) Force (related to the stress) (b) 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 Polymer Contribution Spatial Resolution Molecular Labeling Sensitivity Time Scale
Molecular orientation and shape Dominates the measured signal Possible Possible Good precision Shorter Dissipation and/or storage of energy Relatively small for dilute solutions Impossible Impossible Longer
I.
When incident electromagnetic radiation interacts with matter, three broad classes of phenomena are of interest: • • Transmission of Light 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 II.
• • Scattering Radiation 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 III.
• • 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 I
0 exp(
l
)
I
: Intensity of the transmitted light
I
0 : Intensity of the incident light : Turbidity
l
: Path length
I
smaller
I
0 The loss of intensity is du e to absorp tion and/or scatt ering of the light independen t of the polarisation state of the l ight
Dichriosm For dichroic materials, light is attenuated differently with different s polarization state depen dent
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 anisotropic scattered light depends on the polarisation direction due to the nature of the microstructure under shear flow
Birefringence A material is called birefringence, when the refra ctive inde x is polarisation state dependent Resulting in a pha se differe nce between light with different polarisation stat es after having pa ssed the birefringence material
FIG .
Linear polarised light is generally transmitted as elliptically a birefringent material polarized light through 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:
S
0
M
11 (90 )
S
1 Polarizer
M
9 (
m
PEM
S
2
M
S
S
3
M
11
S
4 Analyzer Light source Sample Detector
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
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) ] DAQ card oscilloscope Lock in amplifier PEM controller Polarizer PEM Head 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 I dc , I ω , and I 2ω are determined with lock in amplifier.
The Measured Intensity
I
for a Sample with Coaxial Birefringence and Dichroism oriented at an Angle θ is:
S
4
I
0 2
m
m
I
: Intensity of the transmitted light
I
0 : Intensity of the incident light on the PEM : Orientation angle of the sample : Retardance of the sample m : Retardance of the PEM given by m
A
sin
t I
I
0
2
m
[Frattini, P. L. and G. G. Fuller, J. Rheol. 28 , 61-70 (1984) ]
m
Fourier series expansion of cos δ
m
the intensity at the detector, and sin δ
m
, and
I
I dc
I
sin
t
I
2
cos 2
t
........
where I dc = I 0 /2 I ω =2cos2θsinδJ 1 (A) I dc I 2ω =2cos2θsin 2θ(1-cosδ)J 2 (A) I dc
The birefringence Δn of the sample is defined from the retardance by
2
d
[Frattini, P. L. and G. G. Fuller, J. Rheol. 28 , 61-70 (1984) ]
Solid State Birefringence Measurement
Analyer( 45 ° ) PEM PMT Quarter wave(sample 1/4 λ ) Polarizer(+45 ° ) [Hinds Instruments, Inc., PEM-100 technical note] laser
A=0.3125 λ
I I
0
1 2 1 cos
A
sin(
λ/4 0.3125
λ
t
)
50kHz
1.0
I/I 0 0.5
0.0
0.0000
0.0001
0.0002
0.0003
0.0004
time(s) Fig. Waveforms obtained from calculation 0.0005
Fig. Waveforms obtained from oscilloscope 0.15
I/I 0 0.10
0.05
0.00
0 2e-5 6e-5 4e-5 time(s) Fig. Waveforms obtained from DAQ card 8e-5
A= 0.5 λ
I I
0
1 2 1 cos
λ/4
A
sin(
t
)
0.5λ 50kHz
1.0
I/I 0 0.5
0.0
0.0000
0.0001
0.0002
0.0003
0.0004
time(s) Fig. Waveforms obtained from calculation 0.0005
Fig. Waveforms obtained from oscilloscope 0.10
I/I 0 0.05
0.00
0 2e-5 6e-5 4e-5 time(s) Fig. Waveforms obtained from DAQ card 8e-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
1
Small-Angle Light Scattering (SALS)
A monochromatic beam of laser light impinges on a sample and is scattered into a 2D detector 2 3 4 Hammouda, B.,
Probing Nanoscale Structures: The SANS Toolbox
(unpublished book)
Brief Introduction to SALS
Description
or light Particle 2 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 Phase difference : 2 2 (QP OR) (
S 0
) Geometry of the path length difference 2 , where
s
Conventionally, the quantitiy
q
( is defined as the scattering vec tor 2
s
0 4
n
sin ) 2 By the law of cosines
Angular Range
θ
=1 o to 10 o
Probed Length Scale Application
~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
Double-slit Fraunhofer pattern
Small structural length
Schematic Drawing of SALS Apparatus
1 Monochromatic Light Polarier 2 Collimation Objective lens Lens Mirror Pinhole
Spatial filter & Beam expander
Iris Iris 3 Scattering 1 to 10 Sample Iris Mirror Analyzer 4 Detection CCD Lens 1 Lens 2
Beam expansion
Spatial filter Beam expansion
Spatial filter
SALS Apparatus
1 Monochromatic Light
Polarizer 1,2 Laser
2 Collimation 3 Scattering 4 Detection
Sample stage Lens 1,2 Laser pointer CCD Mirror Spatial filter Mirror Iris 1,2,3
Mirror
Calibration: Diffraction Pattern of a Pinhole
Polarier Using a 50 μm pinhole as a sample, whose diffraction pattern is known (airy func.) Objective lens Lens Analyzer Pinhole Iris
Spatial filter & Beam expander 1
Iris
Pinhole
Lens 1 Iris Mirror Lens 2
Measured diffraction pattern Airy function 0.1
CCD
0.01
0.001
0.0001
0 2 4 k a sin
6 8 10
Calibration: Scattering of a PS Colloidal Dispersion
Polarier Objective lens Lens Mirror Pinhole
Spatial filter & Beam expander
Iris 100-nm-diameter PS colloidal dispersion Iris
Sample 10 6
Lens 1 Iris Mirror Analyzer Lens 2 CCD The Rayleigh-Gans-Debye theory predicts that the scattering profile of the measured sample is of no angular dependence , as was confirmed experimentally
10 5 10 4 10 3 100 200 300 Pixel 400 500
Versatile Optical Rheometry
Iris
PEM
Iris Lens Objective lens Polarier
Flow-LS
(large-angle detection)
CCD
Rheo-SALS
Lens Couette cell Analyzer
Rheology
Pinhole
Spatial filter & Beam expander
Screen with aperture (from PEM) 1f 2f Photodiode
Rheo-Birefringence
Lock-in amplifiers
Rheo-SALS (under construction)
12.67cm
10cm 37.5cm
18.75cm
Flow cell Rheometer Lens 1 f=10 cm Lens 2 f=12.5 cm 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 Temperature Controller 10 o C to 70 o C
30 to 150
Polarizer 1 Circulating water Sample cell Photomultiplier tube Polarizer 2
30 to 150
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
Kishbaugh and McHugh studied monodisperse polystyrenes dissolved in decalin. In most cases, solutions were in the dilute to semidilute transition region, i .e.,
c c
1 . In the high shear rate range where the
reversible shear thickening
occ urred (i.e., 500 s -1 10, 000 s -1 ) Note that only data for the case of
M
w =1.54 x 10 6 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
exhibited by a lower molecular weight sample (
M
w drop concentration are similar to those =4.3 x 10 plateau 5 ). Namely, the dichroism rises to a plateau, while viscosity undergoes a monotonic with shear rate to an eventual Newtonian • At
higher
region of 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
shear thickening
in which the viscosity continuously rises, while the dichrois m 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 dichroism , which may take place as indicated in the
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 : Onset of the shear - enhanced concentration fluctuations a : Onset of the anomalies in the rheological and scattering behaviors
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 c : Onset of the shear - enhanced concentration fluctuations Change of the sign Due to the stringlike structures oriented parallel to the flow dir.
Chains weakly orient along the flow dir.
Chains in the strings with their end-to-end vectors parallel to the flow dir.
a : Onset of the anomalies in the rheological and scattering behaviors
Continued
Comparisons with Mechanical Characterizations: 6.0
wt% PS/DOP solution (
c c
30 )
M
w 6 ;
M
w
M
n 1.06
Mechanical
FIG
n
), and dichroism (
n
) of the solution Notice that the behavior of the shear viscosity is also classified into three regimes
References
(1) Baaijens, J. P. W.,
Evaluation of Constitutive Equations for Polymer Melts and Solutions in Complex Flows
, Eindhoven University of Technology, Department of Mechanical Engineering, Eindh oven, The Netherlands (1994).
(2) Collyer, A. A. and L. A. Utracki,
Polymer Rheology and Processing
, Elsevier Science Publishers Ltd, London (1990).
(3) Fuller, G . G.,
Optical Rheometry of Complex Fluids
, Oxford University Press, New York (1995).
(4) Kishbaugh, A. J. and A. J. MuHugh, "A Rheo - Optical Study of Shear - Thickening and Structure Formation in Polymer Solutions. Part I. Experim ental ,"
Rheo Acta
32
, 9 - 24 (1993).
( 5) Kornfield, J. A., G. G. Fuller, and D. S. Pearson, "Infrared Dichroism Measurements of Molecular Relaxation in Binary Blend Melt Rhe ology,"
Macromolecules
22
, 1334 -1345 (1989).
(6) Kume, T., T. Hashimoto, T. Takahashi, and G. G. Fuller, "Rheo - Optical Studies of Shear - Induced Structures in Semidilute Polystyrene Solutions,"
Macromolecules
30
, 7232 - 7236 (1997) .
(7) Lenstra, T. A. J., Colloids near phase transition lines under shear, Ph.D. thesis, University of Utrecht, Netherlands (2001). (http: //igitur - archive.library.uu.nl/dissertations/1952394/ inhou d.htm) (8) Macosko, C. W.,
Rheology : Principles, Measurements, and Applications
, Wiley - VCH, New York (1994).
(9)
Rheo - Physics of Multiphase Polymer Systems : C Rheo - Optical Techniques
, Technomic Publ. Co., Lancaster, PA (1994).
haracterization by
(10) Tapadia, P., S. Ravindranath, and S.- Q. Wang, "Banding in Entangled Polymer Fluids in Oscillatory S hear ing,"
Phys Rev Lett
96
, 196001 (2006).
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
Polystyrene, Mw = 7.14x10
6 g/mol Polystyrene, Mw = 7.14x10
6 g/mol 1.2
1000 1.0
100 0.0001
benzene(30 o C) 1-chlorobutane(38 o C)
trans
-decalin(23.8 o C) 0.001
0.01
0.1
Weissenberg Number 1 Magnitude of intrinsic viscosity -temperature & Solvent Flow curve 10 0.8
benzene(30 o C) 1-chlorobutane(38 o C)
trans
-decalin(23.8 o C) 0.6
0.0001
0.001
0.01
0.1
Weissenberg number 1 10 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
(
Root mean square end-to-end distance
IV.1 Effects of Solvent Quality
P MS in cyclohexane 1.0
0.9
0.8
0.7
0.6
0.5
0.4
15 poor 20 25 -temperature 30 T ( o C) 35 40 45 good Sample Poly( methylstyrene) Mw g/mol 1.14
× 10 6 Mw / Mn (SEC) 1.11
50 Advantages of P MS: 1.
High plasticized speed 2.
3.
4.
5.
Good temperature tolerance Contamination resistance Compatibility with other additives 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).
T U : upper critical solution temperature T L : lower critical solution temperature
IV.1 Effects of Solvent Quality
Mw = 1.00x10
7 Poly(N-isopropylacrylamide) in water g/mol, c = 2.50x10
-5 g/ml Mw = 4.45x10
5 g/mol, c = 6.65x10
-4 g/ml coil globule coil globule X. Wang et al., Macromolecules
31
, 2972-2976 (1998).
x PNIPAM/water, heating cooling PNIPAM/SDS/water, cooling H. Yang et al., Polymer
44
, 7175-7180 (2003).
IV.1 Effects of Solvent Quality on intrinsic viscosity
(i) The Rouse model:
N A M
s
N
2
b
2
36
N
A
:
Avogadro constant
M
:
Molecular weight
(ii) The Zimm model for Θ solvent: 0 .
425
N A M
(
N b
) 3
N
:
number of segments per polymer
b
:
effective bond length
:
friction constant
(iii) The Zimm model for good solvent:
N A M N
3
b
3
η
s
:
solvent viscosity
ν is equal to the α We write the molecular weight dependence of [η] as
M
1 0 3 .
5 1 0 .
8 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, M w < M c Zero-shear viscosity, 0 Relaxation time, Diffusivity, D G ~ M w ~ M w 2 ~ 1/M w > M c ~ M w 3.4
~ M w 3 ~ 1/M w 2 M c (=2M e ): critical molecular weight M e : 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 viscosity
r
solution solvent 1 :
c
k
2
c
2 Specific
sp viscosity :
solution
solvent solvent
r
1 Intrinsic viscosity
sp
c
c
0 : [cf. p109] An example of viscosity versus concentration plots for polystyrene (Mw=7.14
10 6 30 C. White circles: plot of sp g/mol) in benzene at /
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/cm 2 ); (2)Ubbelohde viscometer (8.67 dyn/cm 2 ); (3)Ubbelohde viscometer (12.2 dyn/cm 2 ).
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 M w dependence and concentration effect , the zero-shear viscosity can be estimated by [H] : Hydrophobe content in the monomer feed
N
H : Number of hydrophobe per micelle 0
C
4 ~ 5
M
w
3 .
4 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 polybutadiene Star Polymer Polyisoprene Pom-Pom Polymer Polyisoprene
IV.3 Retrieval of Fundamental Material Constants
Zero-shear viscosity, 0 Newtonian Power law 0 Relaxation time, 1 / critical Fig 3.3-1 (p 105) in the textbook
IV.3 Retrieval of Fundamental Material Constants
J e
0
G
2 0 2 0 1 , 0 2 0 2 6 5
G
N
G
N
cRT
/
M
e 12 0 2 d 0
G
N d e d ~ M 0 ~ M 3 Theoretical results of (a) G(t) and (b) G’( ) for polymer melts.
Storage modulus vs. frequency for narrow distribution polystyrene melts. Molecular weight ranges from Mw = 8.9x10
3 r/mol (L9) to Mw = 5.8x10
5 g/mol (L18).
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 time dependent material response.
WLF (Williams log
a T
c
2 0
c
1 0
T T
Landel
T
0
T
0 Ferry) equation
c
1 0
T
T
T
T
0 :
IV.4 Time-Temperature Superposition
WLF temperature shift parameters WLF (Williams log
a T
c
2 0
c
1 0
T T
Landel
T
0
T
0 Ferry) equation
c
1 0
T
T
T
T
0 : 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 2.4x10
6 g/mol, concentration from 0.255 to 0.55 g/cm 3 , and temperature from 303 to 333 K.
to Fig 3.6-5 (p 146) in the textbook.
The ability to measure viscoelasticity of low viscosity fluids without TTS data shifting
The relaxation times for low-viscosity fluids are usually quite short fall in the time domain of milliseconds or below.
and 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) torsion resonator(TR) provides a way to characterize the dynamic properties of these low-viscosity fluids. or
The ability to measure viscoelasticity of low viscosity 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 low viscosity 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.
2.
3.
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 , 781 795(2010) 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) 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.
Incompressible Newtonian fluids:
τ
-
γ
f
temperature, pressure, composition
Incompressible generalized Newtonian flu ids:
τ
f
scalar invariants of
γ
,....
V.1
Models for Generalized Newtonian Fluids
The Carreau-Yasuda model 0 1
a
n
1 /
a
: viscosity : relaxation , 0 : zero shear visc time,
n
osity, : infinite : power law exponent,
a
shear visc : dimensionl osity, : shear rate ess parameter 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
V.1
Models for Generalized Newtonian Fluids
The Eyring model The Eyring equation was the first 0 arcsinh expression obtained by a molecular 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 time dependent 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) a. the differential form: 1
t
0 b. the integral form:
t
0 / 1
e
The nature of flow / 1 shear stress for a Newtonian fluid
yx
yx
shear stress for a Hookean solid
τ yx
G
u
y x
τ
replace
yx
G
by 0
t yx
and
μ
/
yx G
by 1
Relaxation modulus, G(t): The nature of fluid
V.2 Constitutive Equations for Generalized Linear Viscoelasticity
The Jeffreys model (for dilution solutions) a. the differential form: 1
t
0 2
t
b. the integral form:
t
0 1 1 2 1
e
/ 1 2 0 2 1
t
t
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
0
γ
2
γ
Convected time derivative
γ
τ γ
n
1
D Dt D
γ
T Dt
τ
T
γ
τ
τ
γ
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 τ
s
τ
s
τ
p
s
γ
τ
p
1
τ
p
1
p
τ
p
τ
p
p
γ
The model contains four parameters: a relaxation time, 1 ; the zero-shear rate 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
M
t
t
W
I
1
I
1 ,
I
2
W
I
1
I
, 2
I
2
d t
The factorized Rivlin-Sawyers model:
τ
t
M
t
t
1
I
1 ,
I
2 2
I
1 ,
I
2
M W
t I
1 ,
I t
2 : time or
i
-
I
1 , dependent
I
2 : strain factor dependent
d t
factor
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 Polymer properties Governing equations (balance equations of mass, momentum and energy) Power-law constitutive equation Finite element method Stretching section 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 1 Roller 2 Roller 3
CAEFF (Center for Advanced Engineering Fibers and Films) software
Polymer properties Density Glass transition temperature Surface tension Melt shear modulus Maximum crystallization rate Maximum rate temperature Crystallization half width temperature Avrami index Maximum percent crystallinity 0.85 (Kg/m 3 ) 253 (K) 35 (dyn/cm) 9x10 8 (Pa) 0.55 (1/s) 65 (K) 60 (K) 3 70 (%)
V.4 Simulations of Complex Flow Processing
Model properties Orientation hardening parameter Frictional coefficient Room temperature Initial tensile stress Initial percent crystallinity Crystallization rate Amorphous shear stress Poisson ratio Rubber elasticity activation energy Pre-exponential shear strain rate Activation volume Activation parameter Mass flow rate 9 0.6
298 (K) 5x10 5 (Pa) 45 (%) 0.1
8.5x10
6 (Pa) 4.3x10
5 (Pa) 1.5x10
5 (Pa) 10800 2.3x10
7 4.7x10
-29 3.65
1.3x10
-6 (Kg/s) Heat capacity parameters C
s
1 C
s
2 C
s
3 C
l
1 C
l
2 C
l
3 Hf 0.25 (cal/g/ C) 7.0x10
-4 (cal/g/ C 2 ) 0 0.32 (cal/g/ C) 5.7x10
-4 (cal/g/ C 2 ) 0 30 (cal/g) Roller parameters Temperature Radius Roller 1 Velocity of Roller 2 Roller 3 35 ( C) 8 (cm) 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
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 .
Right video : two layers containing shear thickening fluids and Nylon.
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 T
B 6
a
The mean square of the particle’s displacement is
x
2
Dt
Accordingly, the diffusivity sets the characteristic time scale for the particle’s Brownian motion.
t particle
a
2
D
A dimensionless number known as Péclet number , Pe
Pe
a
2
D
a
3
k
B
T
VI.2 Onset of shear thickening : Péclet number
Low shear rate ( Pe <<1 , ) 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 Pe>>1 The flow-induced density fluctuations are known as hydroclusters which lead to an increase in viscosity.
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.
The formation of hydroclusters is reversible , so reducing the shear rate returns the suspensions to a stable fluid At (Pe>>1) regime, the strong hydrodynamic coupling between particles leads to the formation of hydroclusters (red particles) which cause an increase in viscosity.
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 .
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)