第五讲 高分子溶液动力学和流变学R

算有显著影响。例如n值为0.33时，若不进行非牛顿性修正，粘度值将高估50％。
Linear Viscoelasticity
Linear viscoelasticity (Boltzmann叠加原理) (Boltzmann Superpositional Principle)
T G(t t ') (t ')dt '
G' (b J [Pa]
)
)
The Non-Linearity Monitor
5 4 3 2 1 0 1 2 3 4 5
0
TIME [seconds]
FUNDAMENTAL THIRD HARMONIC RESULTANT
6.28
sin 6.28 0.5 sin 18.84 sin 6.28 + 0.5 sin 18.84
• Rheology - a Bridge between Micro-Macro.
Event
Time Scale
Fast Slow
Monitoring Linearity of Samples
105 102
10
1
10
4
10
0
10 HDPE Filled Polymer
-1
103 0 10
Material Functions in Steady Shear Flow
Function Symbol & Definition SI Unit C.G.S.Unit
Shear Strain Shear Stress Shear Rate Viscosity First Normal Stress Function Second Normal Stress Function First Normal Stress Coefficient Second Normal Stress Coefficient Zero Shear Viscosity
R 2 R 0
令


w
R
r dr
2 0
R

r 2 dv z
0
R
r
dr
w
R
w
d
其中
w为靠壁应力。代入上式：
R3 2 V 3 ( )d w 0
令牛顿流体靠壁剪切速率
4V 4 3 3 R w
按Leibnitz规则：
Deformation History
Transient Dynamic Steady
Shear
Stretch Bi-Axial
Deformation Form
30 Material functions describe the complex fluids from different sides

Stress

G
Stress Modulus shear strain
Viscosity shear rate
Complex Fluid
Soft matter
(viscoelastic, partial memory, non-linear, complexity, softness)
拉伸形变 Tensile Deformation
A: Area = W * t
1
x
F: Force V: Velocity
z
3
Stress: T12 = = F/A Strain: F12= = x/y = x/H Strain Rate: 2D12= d/dt = V/H
.
L
F: Force V: Velocity Stress: T11= = F/A Strain: F11 = = L/L Strain Rate: D11 = d/dt
因此是测粘流。可写出：
Tzz T Tzr 0
不可压缩流体的运动方程为：
Tzr Trr 0
0 0 T
( ( ) ) g P T t
在柱坐标中分解，略去零项，则有：
P 1 z分量： (rTzr ) z r r T P 1 (rTrr ) r分量： r r r r 1 P 0 分量： r
另一种推导方法： to consider the force balance on the cylinder 2
r P 2 rL
2
r P P r 2 rL 2L
以下求切速率。
总流量
V

2 rv z ( r ) dr v z r
2 0

R
0
2 rv z ( r ) dr
• In the non-linear regime, a material’s response is no longer sinusoidal. • 三阶谐频振幅与基频的 比例 The Non-linearity Monitor is a normalized ratio of the third harmonic to the fundamental.
，，
管流一维流动速度分布：
v1 vZ (r )
0
v2 vr 0
vZ 0 z
v3 v 0
对不可压缩流体，连续性方程为： ，则有
写出应变率张量
0 1 v D Z 2 r 0
vZ r 0 0
0 0 0
t
T (t t' ) (t' )dt'

t
Dynamic Oscillation
G’= Go cosδ, G”= Go sinδ G*= G’ + i G” Stress relaxation Creep
动态频率扫描是最常用的流变测量之一
Polypropylene at 180°C
RHEOLOGY
Mechanics
连续介质力学 Continuum mechanics( stress, strain, strain rate) Constitutive equation Measurement
Rheometry
物料函数 Material function
Structural
minimicronanomolecular
r
ConePlate
平行板


r
Parallel Plate
圆管
z
z
r
Pipe
狭缝流
R

Slit
Z
Y
X
Deformation Forms
Monoaxial
extension (elongation)
Planar
Uni-axial
Bi-axial
Non-uniaxial
Shear
Deformation History
sp
C
)C 0
0

Rate Dependence (Steady Testing)
Material Functions in Steady Shear Flow （PDMS in silicone oil）
N1 - 第一法向应力差
Viscosity Function of Non-Newtonian Fluids


w
0
2 ( ) d ( w ) N


B ( x ) I d B( x) B A I ( x , y ) dy dy I ( x , B ) I ( x , A) A ( x ) x dx A ( x ) x x
（3.17）式对
w 微分得到：
Non-Newtonian fluid mechanics Mechanics
Engineering
Physics & Chemistry material sci. & technology
简单流动
Three Major Directions
y
2
x
剪切形变 Shear Deformation
A: Area

Same Viscosity here,but Different Curves!
Shear Rate
要导出流变量的测量公式，也就是将应力、切变速率用直接测定的边界量表示 出来是一个流体力学边值问题。这就要就设定的边界条件和初始条件求解连续性 方程和运动方程的联立偏微分方程组。我们在此仅举一例说明这一方法，读者可 举一反三推导其他公式或查阅文献。 例 一般非牛顿流体在毛细管流中相应特性的推导。
则靠壁处切速率为：
4V 3n 1 w ( ) 3 4n R

3n 1 )( w ) N ( 4n
注意上表达式中未对
w w ( )


作具体假定，因此它是对一般非牛顿流体的切速率的表达式，它等于牛顿流体值
( w ) N 乘一个因子。这一非牛顿性修正也称Rabinowitz修正。n值对粘度的计
1 N 1
2
2 N2
0 ( )
s

2
dyn·s2cm -2
P P
0
Specific Viscosity Specific Increment Intrinsic Viscosity
r
s
－
－
η sp ＝η r－1
－
m 3kg-1
－ －
(
105 104
Commanded Strain or Stress