流變學之簡介與應用
An Introduction to Rheology and Its Applications
Complex Fluids & Molecular Rheology Lab., Department of Chemical
Engineering
課程大綱
I. 流變現象與無因次群分析
II. 基礎量測系統與功能
III. 影響流變行為的主要因素
IV. 實驗分析原理與技術
Principal References: “Dynamics of Polymeric Liquids: Volume 1 Fluid Mechanics”
by R.B. Bird et al., 2nd Ed., Wiley-Interscience (1987)
Scope of Rheology

Mini-symposia organized in the 「2004 世界流變會議」
1.
2.
3.
4.
5.
6.
7.

計算流變
流體的不穩性
泡沫、乳液、界面活性劑
食品、生物材料
材料加工
微結構模擬
奈料科技、微流體
非牛頓流體力學
9. 融熔高分子
10. 高分子溶液
11. 流變量測、實驗方法
12. 固體、複合物
13. 懸浮物、膠體
14. 應用流變、一般論文
8.
A rheologist should be familiar with the following subjects





輸送現象
統計力學
高分子物理
膠體科學
分子動態理論
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
黏度η為定值
非牛頓流體
- 高分子溶液、膠體等
黏度不為定值
(尤其在快速流場下)
 為何需要流變學家?




Newton’s law of viscosity
V
 yx    V
Y
Macromolecules are easily deformable
Chain interactions are complicated
Processings typically involve flows
Try to make Rheology not an issue
I. 流 變 現 象 與 無 因 次 群 分 析
 非牛頓流體的三大特徵
 二次流與不穩定現象
 特徵時間與無因次群分析
非牛頓流體的特徵
 非牛頓黏度 (Non-Newtonian Viscosity)
- Shear Thinning
p

Flow curve for non-Newtonian
Fluids
牛頓流體
(甘油加水)
非牛頓流體
(高分子溶液)
 正向力差的效應 (Normal Stress Differences)
- Rod-Climbing
牛頓流體 (水)
非牛頓流體 (稀薄高分子溶液)
 記憶效應 (Memory effects)
- Elastic Recoil
-
Open Syphon Flow
牛 頓 流 體 的 不 穩 定 性: 慣 性 效 應
Concentric Cylinders
Taylor number (Ta) 
Centrifugal force
 41.3
Viscous force
Onset of
Secondary
Flow
Ta  41.3; Re  94
Ta  141; Re  322
Ta (or Re) plays the central role!
Laminar  Secondary  Turbulent
Taylor
vortices
Turbulent
Ta  387; Re  868 Ta  1, 715; Re  3,960
非 牛 頓 流 體 的 不 穏 定 性: 黏 彈 性 效 應
“The mountains flowed before the Lord”
[From Deborah’s Song, Biblical Book of Judges, verse 5:5],
quoted by Markus Reiner at the Fourth International Congress
on Rheology in 1963
De 
(Re  103 for all cases)
Elastic force
  tflow or We =  - 描述非牛頓流體行為之程度
Viscous force
 : 流體的特徵或 “鬆弛” 時間
tflow : 流動系統的特徵時間
 : 剪切速率
De  0
0.2

牛頓流體
(葡萄糖漿)
3
1
收
縮
流
道
非牛頓流體
(0.057% 聚丙烯醯胺/葡萄糖 溶液)
8
流 變 性 質 的 微 觀 (分 子) 成 因
 微觀的角度
V
●
●
Small molecule
Deformable
Macromolecule
 流變的性質主要決定於











流體組成性質







流場因素





Dilute/Entangled
Polydispersity
Flexibility
Linear/Branched
Chain interactions
Flow strength
Flow kinematics
 De   tflow
Competition between
relaxation & deformation rates

典型製程之流場強度範圍
Lubrication
High-speed coating
Rolling
Spraying
Injection molding
Pipe flow
Chewing
Extrusion
Sedimentation
105
103
101
101
 (s-1 )
103
105
107
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).]
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
 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)]
 Taylor-Couette flow for dilute solutions
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/]
II. 基 礎 量 測 系 統 與 功 能
 剪切流與非剪切流
 流變儀夾具選擇與應用
 基礎流變量測模式與功能
典型均勻流場
 Two standard types of flows, shear and shearfree, are frequently used to
characterize polymeric liquids
(b) Shearfree
(a) Shear
vx   y
Steady simple shear flow
vx   yx y; v y  0; vz  0
Shear rate
Streamlines for elongational flow (b=0)
Elongation
rate
vx  
vy  

2

2
vz   z
x
y
 The Stress Tensor
y
x
z
Shear Flow
Elongational Flow
Total stress
tensor*
Stress tensor
 yx
 p   xx
  p       yx
p   yy



 0
0

Hydrostatic pressure forces


0 
p   zz 
0
 p   xx
  p      0



 0

0
p   yy
0
Shear Stress :  yx
First Normal Stress Difference :  xx   yy
Second Normal Stress Difference :  yy   zz
Tensile Stress :  zz   xx


0 
p   zz 
0
流 變 儀 夾 具 與 流 場 特 性
(a) Shear
Pressure Flow:
Capillary
Drag Flows:
Concentric Cylinder
(b) Elongation
Cone-andPlate
Uniaxial Elongation (b  0,   0):
Moving
Parallel
Plates
適 用 流 場 強 度 與 濃 度 範 圍
(a) Shear
Concentrated Regime







Homogeneous
deformation:*
Cone-andPlate
Nonhomogeneous
deformation:
(b) Elongation
Dilute Regime
103
Concentric Cylinder
Parallel
Plates
Capillary
102 101 100
Moving clamps
101
102
103
104
γ (s-1 )
105
 (s-1 )
For Melts & High-Viscosity Solutions
*Stress and strain are independent of position throughout the sample
基 礎 黏 度 量 測
Concentric Cylinder
W1
R1
Assumptions :
(1) Steady, laminar, isothermal flow
(2) v  R1W1 only and vr  vz  0
R2
(3) Negligible gravity and end effects
(4) Symmetry in  ,    0
FIG. Concentric cylinder viscometer
Shear rate  :
WR
 1 1
R2  R1
H
Shear -rate dependent viscosity  ( ) :
(homogeneous)
T
 ( ) 
2 R12 H 
where the torque acting on the
surface of the inner cylinder T is :
T   r

r  R1
(2 R1 H )  R1

R1 , R2 : Radii of inner and outer cylinders
W1: Angular velocity of inner cylinder
H : Height of cylinders
T : Torque on inner cylinder
Cone-and-Plate Instrument
Assumptions :
(1) Steady, laminar, isothermal flow
(2) v (r , ) only; vr  v  0
(3)  0  0.1 rad (  6 )
(4) Negligible body forces
(5) Spherical liquid boundary
(From p.205 of ref 3)
Shear rate  :

W0
0
FIG. 1.3-4. Cone-and-plate geometry
Shear - rate dependent
viscosity  ( ) :
 ( ) 
(homogeneous)
W0 : Angular velocity of cone
 0 : Cone angle
R: Radius of circular plate
3T 0
2 R 3W0
The first normal stress
difference coefficient 1 ( ) :
2F
1 ( ) 
 R 2 2

2
 
R
r  
T : Torque on plate T  0
0
F : Force required to keep tip of cone
in contact with circular plate
2
  2
drd

Uniaxial Elongational Flow
Hencky strain :
 max   tmax  ln ( Lmax L0 )
L0 : Initial sample length
Lmax : Maximum smaple length
The Normal Stress Difference :
 zz   rr   F (t ) A(t )
F (t ): Total force per unit area exerted by the load cell
A(t ): Instantaneous corss - sectional area of the sample
z
r
The Transient Elongational Viscosity   :
 
( zz   rr )
0

F (t )
A0 e 0t
0
 0 : Elongation rate
A0 : Initial cross - sectional area of the sample
Device used to generate uniaxial
elongational flows by separating
Clamped ends of the sample
典型剪切流量測模式
I.
穩態剪切流
Exp a: Steady Shear Flow
Non-Newtonian viscosity η of a low-density polyethylene at several
Different temperatures
The first and second normal stress
The shear-rate dependent viscosity η coefficients are defined as follows:
is defined as:
2
 yx   ( ) yx
 xx   yy  1 ( ) yx
 yy   zz   2 ( ) yx2
Relative Viscosity:
 rel 

s
: Solution viscosity
s : Solvent viscosity
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:
  s 
[ ]  lim 

c 0
 cs 
c: Mass concentration
Intrinsic viscosity of dilute polystyrene
Solutions, With various solvents, as a function
of reduced shear rate β
II.
小振幅反覆式剪切流: 黏性與彈性檢定
Exp b: Small-Amplitude Oscillatory Shear Flow
The shear stress oscillates with frequency  ,
but is not in phase with either the shear strain
or shear rate
Shear Stress : yx   A() 0 sin(t   )
Shear rate:  yx (t )   0 cos t
Shear strain:  yx (t )   0 sin t
Oscillatory shear strain, shear rate, shear stress, and first normal stress difference in
small-amplitude oscillatory shear flow
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() 0 sin t  G() 0 cos t
Loss modulus
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
III. 拉 伸 流 黏 度 量 測 與 特 徵
 Shearfree Flow Material Functions
For Uniaxial Elongational Flow (b  0,   0):
Zero - elongation - rate
elongational viscosity 0
 zz   xx   ( )
 : Elongational viscosity
 : Elongation rate
Elongation viscosity  and viscosity
 for a polystyrene melt as functions of elongation
rate and shear rate, respectively
Zero - shear - rate
viscosity 0
Elongational Stress Growth Function  
The abrupt upturn, or "strain hardening,"
occurs at a roughly constant value of
Hencky strain  (0, t )   0t
The number average and weight average
molecular weights of the samples:
Monodisperse, but with a
tail in high M.W. (GPC results)
Time dependence of the elongational
stress growth viscosity  + for four polystyrene melts
III. 影 響 流 變 行 為 的 主 要 因 素
 時間－溫度疊合原理
 分子量及其分佈的效應
 高分子結構的影響
 溶劑品質及其效應
I. 時間－溫度 疊合原理 (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
According to the Reptation Theory:
0  GN(0)  d , where GN(0) the "plateau modulus" is temperature insensitive
Newtonian
Power law
Zero-shear
viscosity, 0
0  G0N(0)  d
Relaxation time,   1 / critical
     0 
WLF 溫度重整因子:
 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 - Landel - Ferry) equation :
 c10 T  T0   c10 T  T0 
log aT  0

c2  T  T0
T  T
WLF temperature shift parameters
WLF (Williams - Landel - Ferry) equation :
 c10 T  T0   c10 T  T0 
log aT  0

c2  T  T0
T  T
J. D. Ferry, Viscoelastic Properties of Polymers, 3rd ed., Wiley: New York (1980).
II. 分子量的效應
(Molecular Weight Dependences)
For linear polymer melts
Molecular
weight,
Mw
Zero-shear
Relaxation Diffusivity,
viscosity,
time, 
DG
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).]
A “Time-Temperature-Molecular Weight-Concentration”
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.
III. 分子量分佈的影響
H. Munstedt, J. Rheol. 24, 847-867 (1980)
IV. 高分子結構的影響 (Molecular Architecture)
Linear Polymer
Star Polymer
Pom-Pom Polymer
polybutadiene
Polyisoprene
Polyisoprene
C. C. Hua, H. Y. Kuo, J Polym Sci Part B: Polym Phys 38,
248-261 (2006)
S. C. Shie, C. T. Wu, C. C. Hua,
Macromolecules 36, 2141-2148 (2003)
V. 溶劑品質及其對高分子溶液的影響
(Effects of Solvent Quality for Polymer Solutions)
Relative viscosity :
solution
2
r 
 1   c  k   c 2  
solvent
Specific viscosity :
solution  solvent
sp 
 r 1
solvent
Intrinsic viscosity:
 sp 
    
 c  c 0
[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).
Superposition of Intrinsic Viscosity Data on Various Solvent
Systems:
Polystyrene, Mw = 7.14x106 g/mol
Polystyrene, Mw = 7.14x106 g/mol
1.2
1000
[] / []0
[] (ml/g)
1.0
benzene(30 oC)
1-chlorobutane(38 oC)
trans-decalin(23.8 oC)
100
0.0001
0.001
0.01
0.8
o
benzene(30 C)
o
1-chlorobutane(38 C)
trans-decalin(23.8 oC)
0.6
0.1
1
Weissenberg Number
Magnitude of intrinsic viscosity
-temperature & Solvent
Flow curve
10
0.0001
0.001
0.01
0.1
1
10
Weissenberg number
T. Kotaka et al., J. Chem. Phys. 45, 2770-2773 (1966).
Essential Scaling Laws:
 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
T>
T=
T<
Index

3/5
1/2
1/3
Phase Separation by Temperature-Induced Solvent Quality
Changes:
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
Coil-Globule Transition due to Changes in 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).
PNIPAM/water, heating
cooling
PNIPAM/SDS/water, cooling
H. Yang et al., Polymer 44, 7175-7180 (2003).
IV. 實 驗 分 析 原 理 與 技 術
 線性黏彈性與轉換關係
 非線性應力鬆弛與分析
I. 線性黏彈性分析 (Linear Viscoelasticity)
 The Maxwell model
(for melts or concentrated solutions)
a. the differenti al form :
τ
τ  1
  0 γ
t
b. the integral form :
t
τ (t )    {( 0 / 1 )e

 ( t t' ) / 1
shear stress for a Newtonian fluid
 yx    yx
shear stress for a Hookean solid
τ yx  G
The nature of flow
} γ (t' )dt'
Relaxation modulus, G(t):
The nature of fluid

τ yx 
  yx
u x
y
   yx
G t
replace  by  0 and μ / G by 1
Other Transformation Relationships
t
 yx (t )    G (t  t' ) yx dt'



   G ( s )ds  yx s = t-t’
 0

η0 is zero-shear viscosity
 0 yx
G" ( )

 ' ( )   G ( s ) cos s ds


G' ( )
 " ( )   G ( s ) sin s ds
0

η’ is dynamic viscosity
0
J
0
e


[

0

0
sG ( s)ds
G ( s)ds ]2
Je0 is steady- state compliance
 yx (t )
G (t )  lim
 0

 G0 exp( t /  ) (eq1)
G(t) (Pa)
G0
The single mode dose not fit typical
data well. A logical improvement on
this model is to try several relaxation
times , shown as eq2.
The single
exponential mode,
eq1, with relaxation
time λ=0.1 s and
G0=105 Pa.
G1
N
G (t )   Gk e
k 1
 t / k
(eq2)
G(t) (Pa)
G2
G3
G4
G5
C. H. Macosko, Rheology Principles, Measurements, and Applications,
Wiley-VCH: New York (1994).
t (s)
A spectral
decomposition of
five-constant model
combined with eq2.
G"   Gk
k
G”(Pa)
 2 2k
G'   Gk
(eq1)
2 2
1



k
k
k
(eq2)
2 2
1   k
Relaxation times and moduli
for LDPE at 150℃
λk (s) Gk (Pa)
1
103
1.00
2
102
1.80×102
3
10
1.89×103
4
100
9.80×103
5
10-1
2.67×104
6
10-2
5.86×104
7
10-3
9.48×104
8
10-4
1.29×105
G’(Pa)
k
ω(s-1)
ω(s-1)
Dynamic shear moduli for LDPE at 423 K. Data
were collected at different temperatures and
shifted according to time-temperature
superposition. The solid curves are calculated
from G(t) using eq1-2.
Spectral decomposition of the storage
and loss moduli for LDPE at 423 K. The
moduli are calculated by eq1-2 with the
Gk and λk given in left table.
線性黏彈性實驗數據轉換法則
(Transformation between linear viscoelastic data)
Flow geometry : Cone and plate
(25 mm diameter, cone angle 2 °)
The Cox-Merz rule
 0   * ( )  0
 ()   ( )    [(G' /  )  (G" /  ) ]  

2
2
J e0   0 2 [G' /ω 2
 rep  0 J e0
 0
]
Material properties for PS solutions
Sample
Mw (10-6 g
PDI
Solvent
/mol)
Wt %
Zeq
η0 (Pa．s) Je0 (Pa-1)
τrep (s)
T (k)
PS2Ma
7
6.5
1.73×104
6.33
298
G’,G” (pa)
3.66×10-4
η + (Pa s)
DEP
η, η* (Pa s)
1.09
．
2.0
．
t (s)
ω (1/s)
Dynamic moduli measured in small-amplitude
oscillatory experiments for a monodisperse solution
PS2Ma:measurements were conducted at 25℃
,(1/ s)
Comparison between steady-shear and
complex viscosities for a monodisperse
solution, PS2Ma:measurements were
conducted at 25℃
Transient viscosity growth for a
monodisperse solution,PS2Ma, following
startup of steady shearing at various shear
rates; measurements were conducted at 25
℃
Y-H Wen, H-C Lin, C-H Li, C-C. Hua, Poymer 45, 8551-8559 (2004)
Laun’s rule
Ψ 1 ( )  2(G' /  2 )[1  (G' / G" ) 2 ]a
 
Material properties for PS solutions
a was original
given as 0.7
Flow geometry : Cone and plate
(25 mm diameter, cone angle 2 °)
Mw (10-6 g
/mol)
PDI
Solvent
Wt %
Zeq
η0 (Pa．s)
Je0 (Pa-1)
τrep (s)
T (k)
PS2Mb
2.0
1.09
DEP
20
32.2
1.17×105
6.0×10-4
70.2
313
Ψ1 (Pa.s2)
Ψ1 (Pa.s2)
Sample
 (1 / s)
Transient behavior of first normal stress difference
coefficient for a monodisperse solution, PS2Mb,
following startup of steady shearing at various
shear rates; measurements were conducted at 40 ℃
 (1 / s)
Comparison between experimentally measured first normal
stress difference stress difference coefficient(points) and
predictions (lines) based on Laun’s rule for a monodisperse
solutions, PS2Mb; measurements were conducted at 40 ℃
Y-H Wen, H-C Lin, C-H Li, C-C. Hua, Poymer 45, 8551-8559 (2004)
基礎流變參數的取得 (Retrieval of Fundamental Material Constants
from Linear Viscoelastic Data)

 G 
6
J e0   2 2 
 1,20 
0 
  0  0 20 5GN
GN 0  cRT / M e 
120
 2 d
0  GN0 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.
Stress Relaxation after a Sudden Shearing Displacement (Step-Strain Stress Relaxation)
Relaxation Modulus:*
 yx
G (t ,  0 )  
0
The shear strain  0 can be induced by applying a
large, constant shear rate  0 for a short time
interval t , so that  0 t   0
For small shear strains
lim G(t ,  0 )  G(t )
 0 0
In this limit, the shear stress is linear in strain
The Lodge-Meissner Rule:
G (t ,  0 )
1
G1 (t ,  0 )
The relaxation modulus G(t ,  0 ) (open symbols) and normal stress
relaxation function G1 (t ,  0 ) (solid symbols) for a low - density polyethylene melt
The stress relaxation modulus G(t ,  0 ) for 20 % polystyrene ( M w  1.8 106 ) in Aroclor.
Part (a) shows how G(t ,  0 ) varies with shear strain. In (b) the data are superposed by vertical shifting
to show the similarity in G(t ,  0 ) at large times regardless of the imposed shear stain
II. 非線性黏彈性分析 (Nonlinear Viscoelasticity)
Determination of model parameters
Flow geometry : Cone and plate (25
mm diameter and cone angle 4 °)
Essential model parameters and time constants
Sample
Solution
Zeq
τe (s)
τR (s)
τd,0 (s)
τi (s)
GN( 0) :
(Pa)
Set 3
PS/DEP
42
2.0 × 10-4
0.71
89.4
0.157
5.0 × 103
G’ and G” (Pa)
Z eq 
MW
MW 


M e,soln M e,melt
Zeq : number of entanglements
Φ : polymer volume fraction
Me,melt = 13,300 for PS, α = 1.3
Experiment (Set 3)
τ e = 2.0 ×10-4 s; Zeq = 42
ω (1/s)
Y. H. Wen, C. C. Hua, J Polym Sci Part B: Polym Phys 44, 1199-1211 (2006)
Tube model formulation for single –
step strain flows
 yx 15 2
G(t ,  ) 
 λ (t)GN( 0)Qyx ( ) F(t) 

4
F (t )  G (t ) / GN( 0 )
Set 3
F(t) : the time-dependent tube
  g i exp( t / i ) survival probability describing
i 1
the linear stress relaxation.
M
GN( 0) : Plateau modulus
λ(t )  l (t )
2
2
l
2
eq
λ (t) : Primitive chain length
normalized by its equilibrium value
Qyx(γ) : the yx component of the
orientation tensor
Stretch relaxation of a 1-D Rouse chain
l (t )  l
2
2
eq
l 2 (t  0  )  l 2
eq


 1  1 N
2

  exp( 2 p t /  R )
  1  N p 1


2 
8
2
exp(

p
t
/

)

R  (eq A14)
  1  p:odd p 2 2

 2  l 2 (t  0 )
l2
eq
Properties of relaxation
modes utilized to fit linear
stress relaxation data
gi
λi
0.1989
0.133
0.1579
0.275
0.0859
0.571
0.1820
1.19
0.0816
2.46
0.1377
5.11
0.0841
10.6
0.0643
22.0
0.0029
45.8
0.0055
95.0
The first term in eqA14 may be plausibly
described as arising from the contribution
of local segmental-length fluctuation.
The second term in eqA14 represents the
contribution from the fluctuations of entire
chain length.
Theory /data comparison for
nonlinear stress relaxation
G(t, γ) (Pa)
 yx 15 2
G(t ,  ) 
 λ (t )GN( 0)Qyx ( ) F(t) 

4
t (s)
Linear viscoelastic measurements for elongational
flow properties
G’
Slope:ne
Baumgaertel, Schausberger, and Winter
(BSW) model
G”
G (t )  
Slope:ng
H ( )

exp( t /  ) d

n
H ( )  [ H1n  H 2  ]h(1   / 0 )
0
PS390K:closed
symbols
g
e
h(x) is the Heaviside step function
PS melt
properties
at 130 ℃
H1 describes the rubbery behavior at
low and intermediate ω, while H2
describes the glassy behavior at large ω
Mw (g/mol)
3.9×105
Mw/Mn
1.06
λ0 (s)
2.1×104
ne
0.16
ng
0.7
H1λ0ne
(Pa)
4.17×104
H2λ0-ng (Pa)
20
GN( 0 ) (kPa)
257
η0 (MPa s)
755
(0)
N
G
H
 1 0
ne
ne

 0   G ( s ) ds 
0
H11
ne 1
1 n g
/( ne  1)  H 2 2
/(1  ng )
A. Bach, K. Almdal, H. K. Rasmussen, O. Hassager, Macromolecules 36,
5174-5179 (2003)
3 η0
PS390K
  0.1 s 1
PS390K
  0.0003 s 1
A. Bach, K. Almdal, H. K. Rasmussen, O. Hassager, Macromolecules 36,
5174-5179 (2003)