Complex Fluids & Molecular Rheology Lab., Department of Chemical Complex Fluids & Molecular Rheology Lab., Department of Chemical EngineeringEngineering

流 變 學 之 簡 介 與 應 用流 變 學 之 簡 介 與 應 用An Introduction to Rheology and Its ApplicationsAn Introduction to Rheology and Its Applications

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)

Mini-symposia organized in the 「 2004 世界流變會議」

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

1. 計算流變2. 流體的不穩性3. 泡沫、乳液、界面活性劑4. 食品、生物材料5. 材料加工6. 微結構模擬7. 奈料科技、微流體

8. 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

Scope of Rheology

Rheology is the science of fluids. More specifically, the study of Non-Newtonian Fluids

流體

為何需要流變學家 ? Macromolecules are easily deformable Chain interactions are complicated Processings typically involve flows Try to make Rheology not an issue

什 麼 是 流 變什 麼 是 流 變 ((Rheology)Rheology)??

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

yx Y

VV

YNewton’s law of viscosity

V

黏度 η 為定值

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

二次流與不穩定現象

特徵時間與無因次群分析

非牛頓黏度 (Non-Newtonian Viscosity) - Shear Thinning

非 牛 頓 流 體 的 特 徵非 牛 頓 流 體 的 特 徵

p

牛頓流體(甘油加水 )

非牛頓流體(高分子溶液 )

Flow curve for non-Newtonian Fluids

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

Concentric Cylinders

牛 頓 流 體 的 不 穩 定 性牛 頓 流 體 的 不 穩 定 性 : : 慣 性 效 應慣 性 效 應

;T a 1.3 94 Re 4

Laminar Secondary Turbulent

Onset of Secondary

Flow

Turbulent

Taylor vortices

Ta (or Re) plays the central role!

(Ta) 41.3 Centrifugal force

Taylor number Viscous force Ta 141; Re 322

Ta 387; Re 868 Ta 1,715; Re 3,960

非 牛 頓 流 體 的 不 穏 定 性非 牛 頓 流 體 的 不 穏 定 性 : : 黏 彈 性 效 應黏 彈 性 效 應

收縮流道

De 0 0.2 1 3 8

牛頓流體(葡萄糖漿 )

非牛頓流體(0.057% 聚丙烯醯胺 /葡萄糖 溶液 )

flowDe or We = t Elastic forceViscous force

:

Re for all cases)31( 0

- 描述非牛頓流體行為之程度流體的特徵或 “鬆弛” 時間流動系統的特徵時間tflow : : 剪切速率

“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

微觀的角度

流變的性質主要決定於

流 變 性 質 的 微 觀 流 變 性 質 的 微 觀 (( 分 子分 子 ) ) 成 因成 因

● ●

流體組成性質

流場因素

flowDe t

Dilute/EntangledPolydispersityFlexibilityLinear/BranchedChain interactions

Flow strengthFlow kinematics

Competition between relaxation & deformation rates

Deformable

V

Small moleculeMacromolecule

典型製程之流場強度範圍

-1 ( ) s

High-speed coating

Injection molding

Lubrication

Sedimentation

Rolling

Pipe flow

Extrusion

Spraying

Chewing

710510310110110310510

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).]

Secondary flow

Primary Flow

Secondary Flow

Non-Newtonian Fluids

Secondary Flows and Instabilities

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]

Primary Flow

Secondary Flow

Newtonian Fluids

Melt instability

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)]

Sharkskin Melt fracture

Taylor-Couette flow for dilute solutions

Flow visualization of the elastic Taylor-Couetteinstability in Boger fluids.[http://www.cchem.berkeley.edu/sjmgrp/]

Taylor vortex

R1R2

[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).]

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

Two standard types of flows, shear and shearfree, are frequently used to characterize polymeric liquids

典 型 均 勻 流 場典 型 均 勻 流 場

Steady simple shear flow

xv y

; 0; 0x zy yxv y v v

Streamlines for elongational flow (b=0)

2

2

x

y

z

v x

v y

v z

(a) Shear (b) Shearfree

Shear rate

Elongationrate

The Stress Tensor

x

y

z

0

0

0 0

xx yx

yx yy

zz

p

p p

p

0 0

0 0

0 0

xx

yy

zz

p

p p

p

Shear Flow Elongational Flow

yx

xx yy

yy zz

Shear Stress:

First Normal Stress Difference:

Second Normal Stress Difference:

zz xx Tensile Stress:

Total stresstensor*

Hydrostatic pressure forces

Stress tensor

流 變 儀 夾 具 與 流 場 特 性(a) Shear

(b) Elongation

Cone-and-Plate

Concentric Cylinder Parallel Plates

Capillary

Moving Clamps

Pressure Flow:

Drag Flows:

0 0b Uniaxial Elongation ( , ) :

適 用 流 場 強 度 與 濃 度 範 圍

-1γ (s )

Homogeneousdeformation:*

Nonhomogeneousdeformation: Parallel

Plates

(a) Shear

(b) Elongation

Capillary

3 2 1 0 1 2 3 4 510 10 10 10 10 10 10 10 10

Cone-and-Plate

Concentric Cylinder

Concentrated Regime Dilute Regime

-1 (s )

For Melts & High-Viscosity Solutions

Moving clamps

*Stress and strain are independent of position throughout the sample

Concentric Cylinder

1 1 0

0

r zv RW v v

Assumptions:

(1) Steady, laminar, isothermal flow

(2) only and

(3) Negligible gravity and end effects

(4) Symmetry in ,

FIG. Concentric cylinder viscometer

1 1

2 1

W R

R R

21

( )2 R

T

H

1 2, :

:

R R

H

Radii of inner and outer cylinders

Height of cylinders

:Shear rate ( ) : Shear-rate dependent viscosity

1

:

:

W

T

Angular velocity of inner cyl

Torque on inner cyli

inder

nder

1R

2R

H

1W

基 礎 黏 度 量 測

(homogeneous)

11 1(2 )r r R

T

T R H R

where the torque acting on the

surface of the inner cylinder is:

Cone-and-Plate Instrument

FIG. 1.3-4. Cone-and-plate geometry

03

0

3( )

2 W

T

R

1 2 2

2( )

F

R

0

0

W

:

:

T

F Force required to keep tip of cone

in contact with c

Torque on plate

ircular plate

:Shear rate

0

( , ) 0

0.1 rad ( 6 )

rv r v v

(1) Steady, laminar, isothermal flow

(2) only;

(3)

(4) Negligible body forces

(5) Spherical liquid boundary

Assumptions:

(From p.205 of ref 3)

(homogeneous)

1( ) : The first normal stress

difference coefficient ( ) : Shear-rate dependent

viscosity

0

0

:

:

:

W

R

Angular velocity of cone

Cone angle

Radius of circular plate

2 2

20 0

Rr drT d

Uniaxial Elongational Flow

max max max 0ln ( )t L L :Hencky strain

0

max

:

:

L

L

Initial sample length

Maximum smaple length

( ) ( )zz rr F t A t ( ):

( ):

F t

A t

Total force per unit area exerted by the load cell

Instantaneous corss- sectional area of the sample

Device used to generate uniaxial elongational flows by separating Clamped ends of the sample

:The Normal Stress Difference

00

0 0

( )

( ) tzz rr

F tA e

: The Transient Elongational Viscosity

0

0

:

: A

Elongation rate

Initial cross- sectional area of the sample

z

r

典 型 剪 切 流 量 測 模 式

I. 穩 態 剪 切 流

( )yx yx 2

1

22

( )

( )

xx yy yx

yy zz yx

Exp a: Steady Shear Flow

Non-Newtonian viscosity η of a low-density polyethylene at several Different temperatures

The shear-rate dependent viscosity ηis defined as:

The first and second normal stress coefficients are defined as follows:

0

s

s

] lim[c c

srel

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 of dilute polystyrene Solutions, With various solvents, as a function of reduced shear rate β

Intrinsic Viscosity:

Relative Viscosity:

s

:

:

Solution viscosity

Solvent viscosity

: c Mass concentration

II. 小振幅反覆式剪切流 : 黏性與彈性檢定Exp b: Small-Amplitude Oscillatory Shear Flow

Oscillatory shear strain, shear rate, shear stress, and first normal stress difference in small-amplitude oscillatory shear flow

0( ) sinyx t t Shear strain:

0( ) cosyx t t Shear rate:

The oscillates with frequency ,

but is not in phase with eith shear s

shear s

traier the

o

n

shea

tre

r

ss

r rate

0( ) sin( )yx A t Shear Stress:

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 solidcurves are calculated from the generalized Maxwell model, Eqs. 5.2-13 through 15

0 0( ) sin co( ) syx GG t t

It is customary to rewrite the above equations to display the in-phase and out-of-phase parts of the shear stress

Storage modulus

Loss modulus

III. III. 拉 伸 流 黏 度 量 測 與 特 徵拉 伸 流 黏 度 量 測 與 特 徵

Shearfree Flow Material Functions

( )zz xx

0 0b For Uniaxial Elongational Flow ( , ) :

Elongation viscosity and viscosity

for a polystyrene melt as functions of elongation

rate and shear rate, respectively

0Zero-elongation-rate

elongational viscosity

0Zero- shear-rate

viscosity

:

:

Elongational viscosity

Elongation rate

Elongational Stress Growth Function

+

Time dependence of the elongational

stress growth viscosity for four polystyrene melts

The number average and weight averagemolecular weights of the samples:

0(0, )t t

The abrupt upturn, or " ,"

occurs at a roughly constant value of

Hencky st

strain hard

r

ening

ain

Monodisperse, but with atail in high M.W. (GPC results)

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

Master curves for the viscosity and first normal Stress coefficient as functions of shear rate fora low-density polyethylene melt

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.

According to the Reptation Theory:

Newtonian Power law

Zero-shearviscosity, 0

critical/1 time,Relaxation

0(0)

0 dNG 0

(0)N0 d

(0)N, where the "plateau modulus" is temperature insensitiveG G

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.

TT

TTc

TTc

TTcaT

001

002

001log

:equation Ferry)-Landel-(Williams WLF

WLF 溫度重整因子 :

J. D. Ferry, Viscoelastic Properties of Polymers, 3rd ed., Wiley: New York (1980).

TT

TTc

TTc

TTcaT

001

002

001log

:equation Ferry)-Landel-(Williams WLF

WLF temperature shift parameters

II. 分子量的效應(Molecular Weight Dependences)

Molecular weight,

Mw

Zero-shear viscosity,

0

Relaxation time,

Diffusivity, DG

< Mc ~ Mw ~ Mw2 ~ 1/Mw

> Mc ~ Mw3.4 ~ Mw

3 ~ 1/Mw2

For linear polymer melts

Mc (=2Me): critical molecular weightMe: 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 1.6x105 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)

Linear Polymer Star Polymer Pom-Pom Polymer

IV. 高分子結構的影響 (Molecular Architecture)

polybutadiene Polyisoprene Polyisoprene

S. C. Shie, C. T. Wu, C. C. Hua, Macromolecules 36, 2141-2148 (2003)

C. C. Hua, H. Y. Kuo, J Polym Sci Part B: Polym Phys 38, 248-261 (2006)

V. 溶劑品質及其對高分子溶液的影響 (Effects of Solvent Quality for Polymer Solutions)

T. Kotaka et al., J. Chem. Phys. 45, 2770-2773 (1966).

22

solvent

solutionr 1

: viscosityRelative

ckc

1

: viscositySpecific

solvent

solventsolutionsp

r

sp

0

Intrinsic viscosity:

cc

An example of viscosity versus concentration plots for polystyrene (Mw=7.14106 g/mol) in benzene at30 C. White circles: plot of sp / c vs. c; black circles: plot of (lnr)/c vs. c. (1) Zimm-Crothers viscometer(3.710-3 ~7.610-2 dyn/cm2); (2)Ubbelohde viscometer (8.67 dyn/cm2); (3)Ubbelohde viscometer (12.2 dyn/cm2).

[cf. p109]

Polystyrene, Mw = 7.14x106 g/mol

Weissenberg Number

0.0001 0.001 0.01 0.1 1 10

[]

(ml/

g)

100

1000

benzene(30 oC)1-chlorobutane(38 oC)trans-decalin(23.8 oC)

Polystyrene, Mw = 7.14x106 g/mol

Weissenberg number

0.0001 0.001 0.01 0.1 1 10

[]

/ [ ]

0 0.6

0.8

1.0

1.2

benzene(30 oC)1-chlorobutane(38 oC)trans-decalin(23.8 oC)

Superposition of Intrinsic Viscosity Data on Various Solvent Systems:

T. Kotaka et al., J. Chem. Phys. 45, 2770-2773 (1966).

Magnitude of intrinsic viscosity-temperature & SolventFlow curve

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.

Essential Scaling Laws:

Root mean square

end-to-end distanceSolvent condition

Temperature

T

Index

<R2>end-to-end 1/2

Good T > 3/5 T = 1/2

Bad T < 1/3

Scaling law of polymer size and molecular weight (<R2>end-to-end 1/2 ~ Mw

).

Phase Separation by Temperature-Induced Solvent Quality Changes:

The (temperature, weight fraction) phase diagram for the polystyrene-cyclohexane system for samplesof Indicated molecular weight.

S. Saeki et al, Macromolecules 6, 246-250(1973).

TU: upper critical solution temperatureTL: lower critical solution temperature

Coil-Globule Transition due to Changes in Solvent Quality:

H. Yang et al., Polymer 44, 7175-7180 (2003).

Poly(N-isopropylacrylamide) in water

coil globule

X. Wang et al., Macromolecules 31, 2972-2976 (1998).

Mw = 4.45x105 g/mol, c = 6.65x10-4 g/mlMw = 1.00x107 g/mol, c = 2.50x10-5 g/ml

coil

globule

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

The Maxwell model(for melts or concentrated solutions)

I. 線性黏彈性分析 (Linear Viscoelasticity)

10 by / and by replace

solidHookean afor stressshear

fluidNewtonian afor stressshear

GμtG

τ

y

uGτ

yxyx

yx

xyx

yxyx

Relaxation modulus, G(t):The nature of fluid

The nature of flow

''' dttet

t

t tt )(})/{()(

:form integral theb.

:form aldifferenti thea.

1/)(10

0

γ τ

γτ

τ

1

Other Transformation Relationships

yx

yx

t

yxyx

dssG

dtttGt

0

0)(

)()(

''

s = t-t’

η0 is zero-shear viscosity

0

cos)()()(

dsssGG

'"

0

sin)()()(

dsssGG

"'

η’ is dynamic viscosity

0

2

00

])([

)(

dssG

dsssGJ

e

Je0 is steady- state compliance

(eq2)

N

k

tk

keGtG1

/)(

G0

t (s)

G(t

) (P

a)

A spectral decomposition of five-constant model combined with eq2.

G1

G2

G3

G4

G5

G(t

) (P

a)

(eq1) )/exp(

)(lim)(

0

0

tG

ttG yx

The single

exponential mode, eq1, with relaxation time λ=0.1 s and G0=105 Pa.

The single mode dose not fit typical data well. A logical improvement on this model is to try several relaxation times , shown as eq2.

C. H. Macosko, Rheology Principles, Measurements, and Applications, Wiley-VCH: New York (1994).

(eq1)

k k

kkGG

22

22

1

'

(eq2)

k k

kkGG

221

"

G’(

Pa)

ω(s-1)

ω(s-1)

G”(

Pa)

k 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

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.

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.

Relaxation times and moduli for LDPE at 150 ℃

線性黏彈性實驗數據轉換法則 (Transformation between linear viscoelastic data)

])/()/[()()( 22 "' GG

The Cox-Merz rule

G’,

G”

(pa)

ω (1/s)

η, η

* (

Pa

．

s)

)/1(, sDynamic moduli measured in small-amplitude oscillatory experiments for a monodisperse solution PS2Ma:measurements were conducted at 25℃

Comparison between steady-shear and complex viscosities for a monodisperse solution, PS2Ma:measurements were conducted at 25℃

Sample Mw (10-6

g /mol)PDI Solvent Wt % Zeq η0

(Pa ． s)Je

0 (Pa-1) τrep (s) T (k)

PS2Ma 2.0 1.09 DEP 7 6.5 1.73×104 3.66×10-4 6.33 298

η +

(P

a

．

s)

t (s)

Transient viscosity growth for a monodisperse solution,PS2Ma, following startup of steady shearing at various shear rates; measurements were conducted at 25 ℃

][

)(

0

220

0

0

*0

ωGJ e /'

0eJrep 0

Y-H Wen, H-C Lin, C-H Li, C-C. Hua, Poymer 45, 8551-8559 (2004)

Material properties for PS solutions

Flow geometry : Cone and plate (25 mm diameter, cone angle 2 °)

Laun’s rule

aGGGΨ ])/(1)[/(2)( 221 "''

Sample Mw (10-6

g /mol)PDI Solvent Wt % Zeq η0

(Pa ． s)Je

0 (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

)/1( s

Ψ1 (

Pa.

s2 )

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

Ψ1 (

Pa.

s2 )

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 ℃

Material properties for PS solutions

Flow geometry : Cone and plate (25 mm diameter, cone angle 2 °)

Y-H Wen, H-C Lin, C-H Li, C-C. Hua, Poymer 45, 8551-8559 (2004)

a was original given as 0.7

基礎流變參數的取得 (Retrieval of Fundamental Material Constants from Linear Viscoelastic Data)

0 0N e 2

d

12/G cRT M

0N

20

0,1

0

20

20

5

6

2 G

GJe

e ~ M0

d ~ M3

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 = 8.9x103 r/mol (L9) to Mw = 5.8x105 g/mol (L18).

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford Science: New York (1986), pp 229-230.

00 N dG

00

( , ) yxG t

Relaxation Modulus:*

0 00lim ( , ) ( )G t G t

For small shear strains

0

0 0t t

0

The shear strain can be induced by applying a

large, constant shear rate for a short time

interval , so that

1

0

0

( , )

( , )

G t

G t

The relaxation modulus (open symbols) and normal stress

relaxation function (solid symbols) for a low-density polyethylene melt

In this limit, the shear stress is linear in strain

1

0

0

( , )1

( , )

G t

G t

The Lodge-Meissner Rule:

Stress Relaxation after a Sudden Shearing Displacement (Step-Strain Stress Relaxation)

60 w

0

( , ) 20 % 1.8 10

( , )

G t M

G t

The stress relaxation modulus for polystyrene ( ) in Aroclor.

Part (a) shows how varies with shear strain. In (b) the data are superposed by vertical

to sh

shif

ow t

ting

he s 0( , )G t imilarity in at large times regardless of the imposed shear stain

G’

and

G”

(Pa)

Experiment (Set 3)τ e = 2.0 ×10-4 s; Zeq = 42

ω (1/s)

Determination of model parameters

Sample Solution Zeq τe (s) τR (s) τd,0 (s) τi (s) (Pa)

Set 3 PS/DEP 42 2.0 × 10-4 0.71 89.4 0.157 5.0 × 103

Essential model parameters and time constants

melte,

W

solne,

Weq M

M

M

MZ

Zeq : number of entanglements Φ : polymer volume fraction Me,melt = 13,300 for PS, α = 1.3

II. 非線性黏彈性分析 (Nonlinear Viscoelasticity)

Flow geometry : Cone and plate (25 mm diameter and cone angle 4 °)

Y. H. Wen, C. C. Hua, J Polym Sci Part B: Polym Phys 44, 1199-1211 (2006)

:)0(NG

Stretch relaxation of a 1-D Rouse chain

F(t)Q(t)GλtG yx

yx )(4

15),( )0(2

N

Tube model formulation for single –step strain flows

:)0(NG Plateau modulus

M

iii tgGtGtF

1

)0( )/exp(/)()( N

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.

eq

222 )( ltltλ )( λ (t) : Primitive chain length normalized by its equilibrium value

A14)(eq 8

odd

R

R

eq

eq

:

222

1

2

22

22

)/exp(1

2

)/2exp(1

1

1

)0(

)(

p

N

p

tpp

tpNltl

ltl

F(t) : the time-dependent tube survival probability describing the linear stress relaxation.

Qyx(γ) : the yx component of the orientation tensor

Set 3

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

Properties of relaxation modes utilized to fit linear stress relaxation data

eq

222 )0( ltl

t (s)

G(t

, γ)

(Pa)

Theory /data comparison for nonlinear stress relaxation

N

F(t)QGtλtG yx

yx )(4

15),( )0(2 )(

Baumgaertel, Schausberger, and Winter (BSW) model

Linear viscoelastic measurements for elongational flow properties

)/1(][)(

)/exp()(

)(

021

0

hHHH

tH

tG

genn

d

PS390K:closed symbols

G’

G”

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

(kPa) 257

η0 (MPa s) 755

)0(NG

e

n

n

HG

e

01)0( N

)1/()1/(

)(

122

111

00

gn

en nHnH

ssG

ge

d

Slope:ng

Slope:ne

H1 describes the rubbery behavior at low and intermediate ω, while H2 describes the glassy behavior at large ω

A. Bach, K. Almdal, H. K. Rasmussen, O. Hassager, Macromolecules 36, 5174-5179 (2003)

h(x) is the Heaviside step function

PS melt properties at 130 ℃

PS390K

A. Bach, K. Almdal, H. K. Rasmussen, O. Hassager, Macromolecules 36, 5174-5179 (2003)

11.0 s PS390K

3 η0

10003.0 s