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
達方電子課程講義
08/20/2011
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. 流變儀基本操作說明
Principal References: “Dynamics of Polymeric Liquids: Volume 1 Fluid Mechanics”
by R.B. Bird et al., 2nd Ed., Wiley-Interscience (1987);
N. J. Wagner, J. F. Brady, Physical today, October 2009
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
黏度η為定值
非牛頓流體
- 高分子溶液、膠體等
Small molecule
Macromolecule
V
●
V
Newton’s law of viscosity
V
 yx    V
Y
Deformable
黏度不為定值
(尤其在快速流場下)
I. 流 變 現 象 與 無 因 次 群 分 析
 非牛頓流體的三大特徵
 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
Time-dependent effects (搖變性)
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.
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 
Elastic force
  tflow
Viscous force
or We = 
(Re  103 for all cases)
-
描述非牛頓流體行為之程度
 : 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

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; vy  0; vz  0
Shear rate
Streamlines for elongational flow (b=0)
Elongation
rate

x
2

vy   y
2
vz   z
vx  
 The Stress Tensor
y
x
z
Shear Flow
Elongational Flow
Total stress
tensor*
Stress tensor
 yx
 p   xx
  p       yx
p   yy



 0
0



0 
p   zz 
0
 p   xx
  p      0



 0

0
p   yy
0
Hydrostatic pressure forces
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
H
(4) Symmetry in  ,    0
FIG. Concentric cylinder viscometer
Shear rate  :
WR
 1 1
R2  R1
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 R1H )  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  :

Shear - rate dependent
viscosity  ( ) :
W0
0
(homogeneous)
W0 : Angular velocity of cone
 0 : Cone angle
R: Radius of circular plate
FIG. 1.3-4. Cone-and-plate geometry
3T0
 ( ) 
2 R3W0
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  
H. Munstedt, J. Rheol. 24, 847-867 (1980)
導電銀漿流變性質的鑑定
1. Steady-state Viscosity
2. First normal stress difference
3. Linear viscoelasticity
4. Creep recovery
The Viscosity Curves of Steady Shear Flow
1e+6
1e+6
1e+5
1e+5
1e+4
1e+4
Viscosity ( Pa s )
Viscosity ( Pa s )
A
1e+3
1e+2
1e+1
1e+3
1e+2
1e+1
PP 25(TEK)
1e+0
0.0001
0.001
0.01
PP 25(TEK)
0.1
1
10
100
1e+0
0.0001
1000
0.001
0.01
Shear Rate ( 1/s )
1
10
100
1000
10
100
1000
435-2
1e+6
1e+6
1e+5
1e+5
1e+4
1e+4
Viscosity ( Pa s )
Viscosity ( Pa s )
0.1
Shear Rate ( 1/s )
C
1e+3
1e+2
1e+1
1e+3
1e+2
1e+1
PP 25(TEK)
PP 25(TEK)
1e+0
0.0001
B
0.001
0.01
0.1
1
Shear Rate ( 1/s )
10
100
1000
1e+0
0.0001
0.001
0.01
0.1
1
Shear Rate ( 1/s )
The Stress Curves of Steady Shear Flow
A
10000
B
10000
PP 25(TEK)
PP 25(TEK)
Stress ( Pa )
1000
Stress ( Pa )
1000
100
10
0.0001
100
0.001
0.01
0.1
1
10
100
10
0.0001
1000
0.001
0.01
Shear Rate ( 1/s )
0.1
1
10
100
1000
10
100
1000
Shear Rate ( 1/s )
C
435-2
10000
10000
PP 25(TEK)
PP 25(TEK)
Stress ( Pa )
Stress ( Pa )
1000
1000
100
100
0.0001
0.001
0.01
0.1
1
Shear Rate ( 1/s )
10
100
1000
10
0.0001
0.001
0.01
0.1
1
Shear Rate ( 1/s )
A
A
The 1st Normal Stress Curves of Steady Shear Flow
6000
3000
6000
PP
25
PP
CP 25(TEK)
25-4
4000
2000
1000
Normal Stress ( Pa )
Normal
Normal Stress
Stress (( Pa
Pa ))
4000
2000
0
0
-2000
-1000
-4000
-2000
-6000
-3000
-8000
-4000
0.001
0.0001
PP
PP25(TEK)
25
2000
0
-2000
-4000
-6000
0.01
0.001
0.010.1
0.1
1
1
10 10
100
100
-8000
0.0001
1000
1000
0.001
0.01
Shear Rate
Shear
Rate (( 1/s
1/s ))
10
100
1000
1000
PP 25
PP 25(TEK)
800
PP 25
25(TEK)
PP
600
2000
Normal Stress ( Pa )
Normal Stress ( Pa )
1
435-2
6000
0
-2000
-4000
400
200
0
-200
-6000
-8000
0.0001
0.1
Shear Rate ( 1/s )
C
4000
B
-400
0.001
0.01
0.1
1
Shear Rate ( 1/s )
10
100
1000
-600
0.0001
0.001
0.01
0.1
1
Shear Rate ( 1/s )
10
100
1000
A
B
PP 25(TEK)
PP 25(TEK)
1e+6
10000
1e+5
10000
1e+3
1e+4
1000

G' ; G'' ( Pa )
1000
Complex Viscosity  Pa s )
1e+4
G' ; G'' ( Pa )

Complex Viscosity  Pa s )
1e+5
1e+3
100
1e+2
1e+2
1e+1
0.01
100
0.1
1
10
1e+1
0.01
100
10
0.1
Angular Frequency ( 1/s )
10
100
Angular Frequency ( 1/s )
C
435-2
10000
1e+5
1e+3
100
Complex Viscosity 
1e+2
Storage Modulus G'
1e+4
1000
G' ; G'' ( Pa )

G' ; G'' ( Pa )
1000
10000

1e+4
Complex Viscosity  Pa s )
1e+5
Complex Viscosity  Pa s )
1
1e+3
100
1e+2
Loss Modulus G''
1e+1
0.01
10
0.1
1
Angular Frequency ( 1/s )
10
100
1e+1
0.01
10
0.1
1
Angular Frequency ( 1/s )
10
100
A
0.0006
Strain ,  ( 1 )
0.0005
0.0004
0.0003
0.0002
0.0001
0.0000
0
200
400
600
Time , t ( s )
800
1000
III. 流 變 儀 基 本 操 作 說 明
 流變儀初始化
 Torque校正(空氣校正)
 夾具選擇
 系統選擇
 Steady Shear Flow 量測
 G’ & G’’ 量測
流 變 儀 初 使 化
初始化
在初始化之前，需注意meas. System和meas. Cell是否為實驗的夾具和系統，
如果要更改，更改部分請詳見3. 夾具的選擇、4. 系統的選擇。
 Torque 校 正
Configuration
Torque 校正，即對空氣校正，每次實驗前必做
溫控系統
選擇設定
Torque校正
選擇所需夾具，開始校正
完成後，按OK即可
夾 具 的 選 擇
Edit Measuring Systems
選擇所需夾具
可更變所選夾具之
Css、Csr 值
1.夾具之Css、Csr值，可參照原廠設定，或是經過自行校正後所得之值
2.完成後，按OK即可。
系 統 之 選 擇
Meas. Cell
系統選擇於此畫面作更改即可
 Steady Shear Flow 量 測
Template Manager
於起始畫面，開啟新檔
Group
Type of
Workbook
先選擇所要量測之類型，再選擇量測種類
首先，選擇Group中的Standard Templates for Rotational Tests，再選
擇所屬之Flow & Viscosity Curve /Yield Fluid，即完成。
1.Setup Rheometer
2.Edit Time Setting
3.Edit Meas. Profile
依所標示順序做設定
1.Setup Rheometer
1.
2.
3.
1.再次確認夾具、系統是否正確，亦可於此更改
2.CC系統無須Zero Gap，CP、PP 在每次實驗前皆須要Zero Gap
Zerp Gap完成後即可放上欲量測樣品，調整轉子高度即可。
3.溫控設定
2.Edit Time Setting
Data數量
Data時間間隔
量測時間
於此設定所要量測之時間間隔、長度，Data數量
3.Edit Meas. Profile
Profire
將量測方式更改為Const. Shear Rate
Set Variable
Unit
Initial
於此可更改量測之變數、大小
Profire
將量測方式更改為Const. Shear Rate
1.
2.
1.上述動作完成後，即可開始量測。
2.編輯檔案名稱，開始量測。
 Example
Constant Shear Rate下，Time對Shear Rate圖。
於此，可點擊座標，更改、新增座標(變數、單位)
Axis
Variable
Group
Unit
於此可編輯作標之變數、單位
 G’ & G’’ 量 測
此與先前介紹一樣，首先，選擇Group中的Standard Templates for
Oscillatory Tests，再選擇所屬之Frequency Sweep /FS，即可。
1.Setup Rheometer
1.
2.
3.
此處與Steady Shear Flow一樣
2.Edit Time Setting
Meas. Points
設定量測Data數量
3.Edit Meas. Profile
1.
Unit
2.
Amplitude
1.變更Amplitude單位
2.設定Amplitude大小
3.設定Frequency範圍
3.
Frequency
1.
2.
完成後，即可開始量測並且編輯檔案名稱
 Example
同Steady Shear Flow，此處可編輯座標變數、單位