CM4655 Morrison Lecture 5 2013
CM4655 Polymer Rheology Lab
Fitting Linear-Viscoelastic Spectra
to G’, G” data
Prof. Faith A. Morrison
Michigan Technological University
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© Faith A. Morrison, Michigan Tech U.
Single-Mode Maxwell Model

 0 
G t
  0 
  0   ( t t  ) / 
 (t ) dt 
e

 
t
 (t )    


g 0

relaxation time parameter
modulus parameter
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© Faith A. Morrison, Michigan Tech U.
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CM4655 Morrison Lecture 5 2013
Maxwell’s model combines viscous and
elastic responses
Spring (elastic) and
dashpot (viscous) in series:
initial state
no force
Dtotal
final state
force, f, resists
displacement
Displacements are
additive:
f
Dtotal  Dspring  Ddashpot
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© Faith A. Morrison, Michigan Tech U.
Predictions of the (single-mode) Maxwell Model

 0 
G t
Step shear
strain
  0 
G (t ) 
  0   ( t t  ) / 
 (t ) dt 
e




t
 (t )    
 0 t / 
e

G1  G 2  0
Does predict a reasonable
relaxation function in step
strain (but no normal stresses;
no nonlinearities).
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© Faith A. Morrison, Michigan Tech U.
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CM4655 Morrison Lecture 5 2013
Step Shear Strain Material Functions
Kinematics:
t0
0


 (t)  lim  0  t  
 0
t 
0
  constant   0
(t)x2 


v 0 
 0 

123
Material Functions:
G(t,  0) 
 21(t,  0)
First normal-stress
relaxation modulus
G1 
Second normalstress relaxation
modulus
G2 
0
Relaxation
modulus
 11   22
 02
  22   33
 02
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© Faith A. Morrison, Michigan Tech U.
Step-Shear-Strain Material Function G(t) for Maxwell Model
1.2
 G t 
1.0
o
0.8
1
0.6
 G t 
0.4
o
0.2
0.1
0.0
0
2
4
6
t

8
10
0.01
0.001
0.01
0.1
1
t
10

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© Faith A. Morrison, Michigan Tech U.
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CM4655 Morrison Lecture 5 2013
Comparison to experimental data
10,000
G(t), Pa
1,000
100
10
1
10
100
time, s
1,000
10,000
Figure 8.4, p. 274 data from Einaga et al.,
PS 20% soln in chlorinated diphenyl
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© Faith A. Morrison, Michigan Tech U.
We can improve this fit by adjusting the Maxwell
model to allow multiple relaxation modes
  k   ( t  t  ) / k
e
 (t ) dt 

  k 
t
 ( k )    
N
 (t )   ( k )
k 1
Generalized
Maxwell
Model
 N  k  ( t  t  ) / k 
     e
 (t ) dt 


  k 1 k
t
2N parameters (can fit anything)
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© Faith A. Morrison, Michigan Tech U.
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CM4655 Morrison Lecture 5 2013
Generalized Maxwell model combines
Maxwell-elements in parallel
5 element Maxwell model
is equivalent to this
physical system:
 5  k  ( t  t  ) / k 
     e
 (t ) dt 


  k 1 k
t
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© Faith A. Morrison, Michigan Tech U.
Predictions of the Generalized Maxwell Model
 N  k  ( t  t  ) / k 
     e
 (t ) dt 


  k 1 k
t

G (t )   k e t / 
k 1 k
N
Step shear
strain
G1  G 2  0
k
This function can fit any
observed data; note that the
GMM does not predict shear
normal stresses.
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© Faith A. Morrison, Michigan Tech U.
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CM4655 Morrison Lecture 5 2013
Fitting G(t) to Generalized Maxwell Model
10,000
4
 gk et 
k
G(t), Pa
k 1
k k
gk
1
2
3
4
500
900
1300
1600
450
100
30
5.0
1,000
k=1
100
k=4
Figure 8.4, p. 274 data
from Einaga et al., PS
20% soln in chlorinated
diphenyl
k=3
k=2
10
1
10
100
1000
10000
time, t
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© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear Material Functions
Kinematics:
(t)x2 


v 0 
 0 

123
(t)  0 cost

0  0

Material Functions:
 21(t,  0)
0
G( ) 
0
cos 
0
Storage modulus
 G sin t  G cos t
( is the phase
difference
between stress
and strain)
G( ) 
0
sin 
0
Loss modulus
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© Faith A. Morrison, Michigan Tech U.
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CM4655 Morrison Lecture 5 2013
Predictions of the Generalized Maxwell Model (GMM)

 k  ( t t  ) / 
e

  k 1 k
t
3
    
k
Small-amplitude
oscillatory shear

 (t ) dt 

 k k  2
2
k 1 1  (k  )
N
k
G( )  
1

(k  ) 2
k 1
N
G( )  
GMM
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© Faith A. Morrison, Michigan Tech U.
Predictions of (single-mode) Maxwell Model in SAOS
G( ) 
g112 2
11 2

1  (1 ) 2 1  (1 ) 2
G( ) 
g11
1

2
1  (1 )
1  (1 ) 2
10
1
0.1


 G" 1 


 1 
0.01


 G ' 1 


 1 
0.001
-4
10
0.0001
-5
10
0.00001
0.001
0.01
0.1

1
10
100
1000
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© Faith A. Morrison, Michigan Tech U.
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CM4655 Morrison Lecture 5 2013
Predictions of (multi-mode) Maxwell Model in SAOS
 k k  2
2
k 1 1  (k  )
N
 k
G( )  
1

(k  ) 2
k 1
N
1.E+07
1.E+06
G( )  
G' (Pa)
G'' (Pa)
1.E+05
k k (s)
1.E+04
kgk(kPa)
k(s)
1
2
3
4
5
1.E+03
1.E+02
1.E+01
1.E+00
gk(kPa)
2.3E-3
16
3.0E-4 140
3.0E-5
90
3.0E-6 400
3.0E-7 4000
Figure 8.8, p. 284
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07 data from
Vinogradov, PS melt
aT, rad/s
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© Faith A. Morrison, Michigan Tech U.
Predictions of (multi-mode) Maxwell Model in SAOS
 k k  2
2
k 1 1  (k  )
N
 k
G( )  
2
k 1 1  (k  )
N
1.E+06
1.E+05
G( )  
G' (Pa)
G" Pa
moduli
1.E+04
1.E+03
k
50
5
1
0.5
0.2
0.1
0.05
0.02
0.02
0.01
0.0007
1.E+02
1.E+01
1.E+00
1.E-01
1.E-04
1.E-02
1.E+00
1.E+02
aTrad/s
gk
4.42E+02
3.62E+03
4.47E+03
3.28E+03
9.36E+03
1.03E+04
1.71E+03
1.18E+02
2.55E+04
3.33E+04
1.83E+05
1.E+04
1.E+06
Figure 8.10, p. 286
data from Laun,
PE melt
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© Faith A. Morrison, Michigan Tech U.
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CM4655 Morrison Lecture 5 2013
We can use Excel Solver to
solve for the parameters of the
GMM model.
Demo file:
www.chem.mtu.edu/~fmorriso/cm4650/Demo_fitting_LVE_spectrum_new.xls
Reference: Faith A. Morrison, Understanding Rheology,
Oxford, 2001, pp281-285
See also Rheology Bulletin, January 2007, volume
76(2), article by J. M. Dealy for discussion of
significance of relaxation spectra.
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© Faith A. Morrison, Michigan Tech U.
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