Visual Programming Language

A Theoretical and Experimental Investigation
on Short-Time Stretch Relaxation of Entangled
Polymer Solutions
Y. H. Wen (溫玉合) and C. C. Hua (華繼中)
Department of Chemical Engineering, National Chung Cheng University
Introduction

Historical Sketch



de Gennes (1971)
Doi and Edwards (1986)
The tube (or reptation) model and its derivatives have proven
very successful, especially in describing the linear viscoelasticity
of entangled polymer liquids
Linear
relaxation

A Story about Nonlinear Viscoelasticity
Reptation time
Rouse time
 R  2.4 s
 d  545 s
Chain Retraction
(nonlinear relaxation)
Chain Retraction
(nonlinear relaxation)
Orientation Relaxation
(linear relaxation)
 d /  R  3Zeq
 PS/DEP solution (Mw=5.5x106 g/mol; PI=1.01; Zeq=77)

Application of the Rouse chain in two distinct cases
Rouse chain
Dilute solution


Concentrated solution
Only terminal chain retraction is captured for the case of concentrated systems
Objectives of the current investigation:
The Rouse Model Prediction
on
Short-time Chain Retraction
Nonlinear Stress
Relaxation Data in
Single-step Strain Flows
 For time scales <  R
 The impact of polymer entanglement ?
Formulation of Stress Relaxation in Single-Step
Strain Flows

Nonlinear stress relaxation modulus:
G(t ,  ) 
 yx 15 (0) 2
 GN  (t ) Qyx ( )  (t ) 
4

 yx
 (t ) : tube survival probability

 (t ) : dimensionless chain stretching GN(0)
: shear stress
: strain
: plateau modulus
Q yx ( ) : D-E universal function
 The Rouse model
(w/o IA assumption)
 2 (t ) 1
1 N
  exp(tp 2 /  R )
2

 (t  0 ) 1 N p1

where N  Z eq and  (t  0 )  E  u
N
Z eq
eq
: No. of Rouse modes
: No. of entanglements per chain at equilibrium
Experimentally Determined Model Parameters
PS/TCP solution
105
104
G 
G' and G" (Pa)

103
G
102
101
G'
G"
100
e= 1.3 x 10-3 s; Zeq = 19
10-1
10-3
10-2
10-1
100
101
102
 (rad/s)
 Number of entanglements per chain at equilibrium, Z eq
Zeq 
Mw 1

M e,melt
2
 The Rouse time,  R ( Z eq  e )
φ: volume fraction of polymer
Theory/Data Comparisons
PS/DEP solution
  0.3
102
  0.4
103
G(t,) (Pa)
G(t,) (Pa)
103
PS/DEP solution
 9
101
 9
102
101
Exp. (Zeq=42)
100
Exp. (Zeq=118)
Rouse model
10-1
Rouse model
100
100
101
t (s)
Parameters: Zeq  42;  R  0.35 s
102
10-1
100
101
102
103
t (s)
Parameters: Zeq  118;  R  2.10 s

Self-consistently Renormalized Rouse Modes
(a) t = 0 (at equilibrium)
Stretching
(b) t = 0+
 A different number of
 entanglements per chain
 N is a dynamic variable
A Renormalized Rouse model:
Retraction
(c) t <
 2 (t ) 1
1 N
2

exp(

t
p
/ R )

2

 (t  0 ) 1 N p1
where N  Z (t ) 
 (t )
 (t  0 )
Zeq
PS/DEP solution
  0.3
102
  0.4
103
G(t,) (Pa)
G(t,) (Pa)
103
PS/DEP solution
 9
101
 9
102
101
100
10-1
Exp. (Zeq=42)
Exp. (Zeq=118)
Rouse model
Renormalized Rouse model
Rouse model
Renormalized Rouse model
100
101
t (s)
Parameters: Zeq  42;  R  0.35 s
100
102
10-1
100
101
102
103
t (s)
Parameters: Zeq  118;  R  2.1 s

Theory/Data Comparisons for Various Polymer Species
PS/TCP solution
PαMS/PCB solution
103
104
  0.5
  0.3
102
G(t,) (Pa)
G(t,) (Pa)
103
  10
102
Exp. (Zeq=19)
Rouse model
Renormalized Rouse model
101
 7
101
Exp. (Zeq=45)
Rouse model
Renormalized Rouse model
100
100
10-1
100
10-1
101
100
101
102
t (s)
t (s)
1,4-PB/FO solution
PMMA/PCB solution
103
  0.3
G(t,) (Pa)
G(t,) (Pa)
  0.5
105
  3.33
Exp. (Zeq=26)
104
 5
102
Exp. (Zeq=17)
Rouse model
Renormalized Rouse model
101
Rouse model
Renormalized Rouse model
10-1
100
t (s)
101
0.1
1
10
t (s)
100
Conclusions





The instantaneous entanglement property has a significant
impact on short-time chain retraction of entangled polymer
solutions.
Self-consistent mode renormalization leads to better agreement
with experimental data.
Remaining discrepancies might result from
(a) Inaccuracy of short-time relaxation data and/or
(b) Tube pressure associated with a deformed polymer network
Acknowledgements

National Science Council
(93-2116-E-194-001)

Excellency Project of the Ministry of Education of ROC
(91-E-FA04-2-4A)

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