Solutions to Exercise 2-Amphiphilic Polymers,
KJM5530.
1) When measuring viscoelastic properties by
rheological methods, we distinguish between
transient measurements (relaxation) and oscillatory
measurements. In a transient measurement, a
momentary deformation is applied to the sample
and the response is measured as a function of time.
For instance the shear relaxation modulus G(t).
In an oscillatory shear experiment, a harmonic
oscillating deformation () of frequency  is
applied on the sample and the response () is
measured.
1
The deformation can be described by:
   0 sin(t )
(1)
We can separate the response in an elastic and a
viscous part:
 G  G'   G'  0 sin(t )
(2)
og
   '
d
 '  0 cos(t )
dt
(3)
The total response is:
   G     G'  0 sin(t )  '  0 cos(t ) (4)
From the definition G´´=´ we obtain:
  G'  0 sin(t )  G''  0 cos(t )
(5)
From trigonometric rules we have:
  0 sin(t   )  0 cos( )sin(t )  0 sin( )cos(t )
2
(6)
Identification gives:
G 
0
cos( )
0
G  
0
sin( )
0
tan( ) 
(7)
sin( ) G 

cos( ) G
A harmonic oscillating function can be expressed
by means of complex quantities:
 *   0 e it
i  1
(8)
For the elastic response:
*G  G'  *  G'  0e it
(9)
For the viscous response:
 *   *
d *
 '  0 ie it  i'  *
dt
3
(10)
By using the relation G´´=´, the complex
response function can be written as:
*  G'  *  i'  *  ( G' iG'') *
(11)
The complex dynamic modulus G* can be written
as:
G* 
*


 G' iG''  0 e i  0 (cos   sin  )
*
0
0
(12)
In an analogous way the complex viscosity * can
be written as:
*  ' i''
G * ( G'2  G''2 )1/ 2
* 

i

G''
' 

G'
'' 

4
(13)
All these parameters can be determined from
oscillatory shear measurements. The most frequent
used quantities are: the storage modulus G’, the
loss modulus G’’, the loss angle , the dynamic
viscosity ´, and the complex viscosity *.
2) The concept time of intersection * 
1
, or
2f *
the longest relaxation time corresponds to the
frequency of intersection f*, where G´ equals G´´.
Below we show results from an oscillatory shear
study on aqueous solutions of a hydrophobically
modified polyacrylamide in the presence of SDS
(Macromol. Chem. Phys. 199, 2385 (1998)).
5
3) Incipient gels can be described by the following
equation:
t
m(t) =S  (t  t') n (t')dt'
-
(- < t' < t)
6
m = The shear stress
(t) = the rate of deformation of the sample
S = the gel strength parameter (depends on the
crosslinking density and the molecular chain
flexibility)
n = the relaxation exponent
The linear viscoelastic behavior at the gel point is
described by this equation, and this equation
relates the instantaneous stress m(t) to the strain
history given by the rate of deformation.
Shear modulus: G(t) = S t-n
G(t) = shear relaxation modulus
n = relaxation exponent
Oscillatory shear: G'  G''   n
Phase angle : Independent of frequency
7
The most simple method to determine the gel point
is the “tilted test tube method”, where the gel point
is determined by tilting a test tube containing the
solution. In this approach, the gel point is
determined as the point where the system stops to
flow. The following rheological methods give are
more accurate determination of the gel point.
Method 1: A multifrequency plot of tan  versus
gelation parameter, e.g., temperature, polymer
concentration, or time.
8
-1
 = 0.15 s
2.5
a)
-1
 = 0.3 s
-1
 = 0.5 s
-1
 = 0.7 s
2.0
-1
 = 0.9 s
-1
tan 
 = 1.0 s
tg = 317 min
1.5
1.0
0.5
c(PVA) = 8%
c(GA) = 9 mM
260
280
300
320 340
Time (min)
360
380
400
Method 2: By plotting the “apparent” viscoelastic
exponents n´ and n´´, obtained from the frequency
dependence of G´ and G´´ at each condition
(temperature,
concentration,
or
time)
of
measurement, against the actual gelation variable
(temperature, concentration, or time) and observing
a crossover where n´=n´´=n.
9
4 % EHEC
8 mmolal SDS
1.0
G´~ n´
gel point
G´´~ n´´
1.5
n', n"
n', n"
1.0
1.0
tg = 404 min a)
tg = 317 min
0.8
0.8
b)
0.6
0.6
0.4 c(PVA) = 4%
0.4 c(PVA) = 8%
0.2 c(GA) = 13 mM
0.2 c(GA) = 9 mM
0.0
0.0
350
400
450
300 350 400 450
1.0
1.0
tg = 126 min d)
tg = 255 min
c) 0.8
0.8
0.6
0.6
0.4 c(PVA) = 10%
0.4 c(PVA) = 10%
0.2 c(GA) = 13 mM
0.2 c(GA) = 22 mM
0.0
0.0
50
100
150
1.0150 200 250 300 350
tg = 179 min
Time (min)
0.8
e)
0.6
0.4 c(PVA) = 12%
0.2 c(GA) = 9 mM
0.0
100
150
200
250
Time (min)
n'
n"
0.5
30
1.0
103
0.5
G', G" (Pa)
tan 
0.0
35
40
45
o
Temperature ( C)
n' = 0.40 ± 0.01
n" = 0.40 ± 0.01
-1
 = 0.09 s
2
-1
 = 0.1 s
10
1
10
-1
 = 0.3 s
G'
G"
T = 36 oC
-1
gel point
-1
 = 0.7 s
-1
 = 0.8 s
0
10
10
Frequency (s-1)
0.0
32
-1
 = 0.6 s
-1
 = 1.0 s
34
36
o
Temperature ( C)
10
38
4) This model provides an interpretation of the
change of the value of the relaxation exponent in
gelling systems of various nature. This model takes
into account that variations in the strand length
between crosslinking points may give rise to
changes of the excluded volume interactions.
Increasing strand length  enhanced excluded
volume effect.
Case a) The excluded-volume effect for a
polydisperse system is fully screened:
n = d(d + 2 - 2df)/2(d +2 - df)
Case b) Unscreened excluded-volume interactions:
n = d/(df + 2)
n varies from 1 to 3/5 as df varies from 1 to 3.
11
Incipient gel networks with high values of n have
low fractal dimensions and are said to be “open”,
and networks with low values of n have higher
fractal dimensions and are “tight”.
5) The most common phenomena observed in the
non-linear viscoelastic regime are shear-thinning
and shear-thickening. In these cases the viscosity
depends on the shear rate.
Associating polym. a)
b)
“Ordinary” polym.
12
6) We have observed that the following factors can
affect the value of the gel temperature.
i) The polymer concentration and amount of ionic
surfactant.
ii) The type of surfactant (SDS (anionic) or CTAB
(cationic))
iii) The level of added salt.
iv) The amount and distribution of the hydrophobic
microdomains. A random distribution promotes
gelation.
An illustration of how the gel point, the values of
the relaxation exponent and the fractal dimension
are affected by polymer concentration and SDS
concentration is given in the Figure below.
13
Gel Point (oC)
42
40
38
36
1 % EHEC
2 % EHEC
4 % EHEC
34
32
30
0.7
5
10
0.6
15
20
At the gel point
n
0.5
0.4
0.3
0.2
0.1
2.4
5
10
15
20
At the gel point
df
2.2
Model of Muthukumar:
2.0
n=d(d+2-2df)/2(d+2-df)
1.8
5
10
15
20
Concentration of SDS (mmolal)
i) The value of the gel point temperature increases
with increasing surfactant concentration (at a
given polymer conc.) and decreases with polymer
concentration at a given surfactant concentration.
These findings illustrate the delicate balance
between swelling and connectivity.
The network contains smaller "lumps" as the level
of surfactant conc. increases. The formation of
large "lumps" (high polymer concentration and low
surfactant concentration) gives rise to a more
heterogeneous network than those formed under
lower EHEC concentrations or higher levels of
surfactant addition. This should lead to a more
"open" network structure (i.e., the fractal
dimension df decreases).
14
The gel-network is stronger in EHEC/SDS system
than in EHEC/CTAB system. It is frequently
observed that the interactions are strongest in
systems with an anionic surfactant.
15
We can see that the slow relaxation time has higher
values for the EHEC/SDS system than for the
EHEC/CTAB system. The polymer concentration
is 1 wt %.
16
The gel strength parameter is higher, except at the
highest polymer concentration, for the EHEC/SDS
system.
120
0.45
100
0.40
0.35
60
n
S (Pa sn )
80
40
0.30
20
S
EHEC/CTAB (4 mmolal)
n
0
S
n EHEC/SDS (4 mmolal)
0
2
0.25
4
Concentration (wt%)
7) The Figures below illustrate how the maximum
of the dynamic viscosity changes with polymer and
surfactant
concentration
cSDS/cEHEC) and temperature.
17
(and
the
ratio
Kjøniksen et al. Macromolecules 1998, 31, 1852.
Dynamic viscosity (Pas)
102
101
100
10-1
10-2
10-3
0.5 wt % EHEC
2.0 wt % EHEC
1.0 wt % EHEC
4.0 wt % EHEC
10-4
0
2
Dynamic viscosity (Pas)
10
10
Concentration of SDS (mmolal)
100
101
100
10-1
10-2
10-3
0.5 wt % EHEC
2.0 wt % EHEC
1.0 wt % EHEC
4.0 wt % EHEC
10-4
0.00
0.1
cSDS/cEHEC
1
We can see that the maximum of the dynamic
viscosity is located at the same surfactant-polymer
ratio, independent of the total polymer
concentration. This suggests that the ratio plays a
crucial role for the optimum of the network
strength.
18
102
102
10 oC
Dynamic viscosity (Pas)
1
10
10
100
100
10-1
10-1
10-2
10-2
10-3
0.0
-1
10
0
10
102
-3EHEC
0.510
wt %
1 wt % EHEC
0.0
2 wt % EHEC
4 wt % EHEC
2
100
10-1
10-1
10-2
10-2
-1
100
40 oC
101
100
0.0
10-1
10
30 oC
101
10-3
20 oC
1
10-3
0
10
10
cSDS/cEHEC
0.0
10-1
100
cSDS/cEHEC
We can see that the maximum of the dynamic
viscosity is shifted toward lower values of the
surfactant-polymer ratio as the temperature
increases. This is probably related to the lower
critical aggregation concentration (cac; the onset
of surfactant binding to the polymer) observed at
elevated temperatures. We may say that the
hydrophobicity of the polymer increases with
temperature. At high surfactant concentrations the
network is disrupted.
19
8) Give a brief description of the model of Cabane
et al. for the temperature-induced gelation of
EHEC-surfactant systems.
This model emphasises the delicate interplay
between “lumps” (hydrophobic associations) and
swelling. The “lumps” provide the connectivity of
the gel-network and swelling is due to the repulsive
forces that are mediated through the adsorption of
an ionic surfactant to the polymer.
The
“lumps”
increase
with
increasing
temperature and decrease at higher levels of
surfactant
addition
(confirmed
experiments).
20
by
SANS