Solutions to Exercise 4-Amphiphilic Polymers,
KJM5530.
1) Rheology and the longest time of relaxation:
G'()  G
 * 2
;G''()  G
1   *2 2
 *2 
1   *2 2
Aqueous solutions of diblock (DB) and triblock (TB) and mixtures.
104
G'
103
G''
102
101
100
TB, 3 wt %
10-1
10-2
* = 0.003 s
10-3
10-4
10-2
10-1
100
101
104
103
102
101
TB, 5 wt %
100
10-1
10-2
* = 0.004 s
10-3
-4
10
10-2
10-1
100
101
104
103
102
101
100
TB : DB = 1 : 2
10-1
-2
5 wt %
10
-3
10
* = 0.002 s
10-4
-2
-1
0
1
G', G'' (Pa)
a)
102
G', G'' (Pa)
b)
102
G', G'' (Pa)
c)
10
10
10
10
102
Frequency (s-1)
We can see that the frequency dependencies of the dynamic
moduli can be described by a single Maxwell-element. We can
see that the time of relaxation is short.
1
Dynamic light scattering:
Aqueous solutions of diblock (DB) and triblock (TB) and mixtures.
100
g1(t)
1.0
0.8
10-1
TB
o
 = 90
0.00
0.01

t (s )
0.6
g1(t)
0.02

0.1 wt %
0.5 wt %
1.0 wt %
5.0 wt %
0.4
0.2
g1 (t) = Af exp[-t/f ] + As exp[-(t/s e )]
0.0
g1 (t) = Af exp[-t/f ] + As exp[-(t/s e )] + Av s exp[-(t/v s e )]
10-6
10-5
10-4
10-3
10-2
10-1
100
101
t (s)
In this case a more complex picture with much longer
relaxation times emerges. At all TB concentrations,
except the two highest concentrations, the relaxation
process can be described by a single exponential (short
times), followed by a stretched exponential (long times).
For the two highest concentrations a third mode appears.
The long-time tail of the correlation function is shifted
toward longer times as the concentration increases.
2
2.)
The DB polymer chain has one hydrophobic end,
while the TB polymer is hydrophobically modified
at both ends.
The micelles show
incipient overlap, but
no connectivity.
The bridges provide the
connectivity
of
the
system.
In the case of DB, a weak network consisting of
interpenetrating micelles is developed, while TB forms a
strong network, where bridges provide the connectivity.
3
The thermodynamic properties are different:
150
Cloud Point (oC)
2
100
PEG (M=8000)/water (S. Saeki et al. Polymer 17, 685 (1976))
Diblock (DB)/water
Triblock (TB)/water
50
1
0
0.1
1
10
Concentration (wt %)
3.) Findings from NMR, zero-shear viscosity, and
dynamic light scattering suggest -like behavior.
Aqueous solutions of diblock (DB) and triblock (TB) and mixtures.
a)
10-11
Ds (m2s-1)
10-12
D s = D 0exp(- c)
DB
TB : DB = 1 : 50
TB : DB = 1 : 10
TB : DB = 1 : 2
TB
10-13
10-14
10-1
100
Concentration (wt %)
101
b)
10-11
10-13
10-14
D0 (m2s-1)
Ds (m2s-1)
10-12
2x10-11
10-11
0.00
10-15
0.1
mTB
1
mTB + mDB
10-16
0.00
NMR:
3x10-11
0.1
mTB
0.1 wt %
1.0 wt %
3.0 wt %
5.0 wt %
1
mTB + mDB
i) The concentration dependence of the self-diffusion coefficient
is described by Ds ( c)  D 0 exp(   c ) , where D0 is the
4
diffusion coefficient at infinite dilution and  is a scaling
factor that depends on polymer molecular weight
(   M 1/ 4 ) and solvent quality. We observe =0.5 for DB
and =1 for TB (suggesting -like behavior).
Zero-shear viscosity:
103
102
101
105
104 TB
103
5
102  ~ c
101
100
10-1 100 101
Concentration (wt %)
100
Viscosity (mPas)
104
DB
TB : DB = 1 : 10
TB : DB = 1 : 2
TB
a)
Viscosity (mPas)
Viscosity (mPas)
104
10-1
b)
103
100
Concentration (wt %)
101
0.1 wt %
1.0 wt %
3.0 wt %
5.0 wt %
102
101
100
0.00
0.1
mTB
1
mTB + mDB
i) We may note (see the inset plot) that the concentration
dependence of the zero-shear viscosity for the TB polymer is
described by c5 (suggesting a -like behavior). Actually, for
entangled solutions of polystyrene at -conditions, the same
value of the power law exponent has been reported. This
indicates that the bridges are very effective in forming an
entangled network.
5
Dynamic light scattering: (Cooperative diffusion)
Aqueous solutions of diblock (DB) and triblock (TB) and mixtures.
-10
10
Dm (m2s-1)
a)
DB
TB : DB = 1 : 50
TB : DB = 1 : 10
TB : DB = 1 : 2
TB
10-11
0.04
-10
10
0.1
1
Concentration (wt %)
6
Dm (m2s-1)
b)
10-11
0.1 wt %
1.0 wt %
3.0 wt %
5.0 wt %
0.00
0.1
mTB
1
mTB + mDB
The concentration dependence of the cooperative
diffusion coefficient for TB is reminiscent of that
for a flexible high molecular weight polymer at
theta solvent conditions; i.e., in the semidilute
regime Dcc1.
6
The q-dependence of the slow mode:
6
5
Result from fitting to a third mode (vs)

~q
s, vs
4
3
2
s
DB
TB: DB= 1: 50
TB: DB= 1: 10
TB: DB= 1: 2
TB
1
0
-1
0.04
0.1
1
6
Concentration (wt %)
The slow mode becomes q-independent at high
concentrations.
This
suggests
enhanced
viscoelasticity. Connectivity established: Strong
viscoelastic response and long debridging time.
This behavior is typical for semidilute solutions of
flexible high molecular weight polymers at solvent conditions.
7
4.) At low and moderate concentrations of TB, the
decay of the correlation function can initially be
described by a single exponential, followed at
longer times by a stretched exponential. At high
concentrations a third mode appears, which also
can be described by a stretched exponential. This
mode can probably be attributed to the dynamics of
large clusters.
Aqueous solutions of diblock (DB) and triblock (TB) and mixtures.
100
g1(t)
1.0
0.8
10-1
TB
o
 = 90
0.00
0.01
0.6
g1(t)
0.02
t (s)
0.1 wt %
0.5 wt %
1.0 wt %
5.0 wt %
0.4
0.2
g1 (t) = Af exp[-t/f ] + As exp[-(t/s e )]
0.0
g1 (t) = Af exp[-t/f ] + As exp[-(t/s e )] + Av s exp[-(t/v s e )]
10-6
10-5
10-4
10-3
t (s)
8
10-2
10-1
100
101
5.)
Characteristic data for the HM-P+ and HM-Ppolymers.
HM-P+
100 000
positive
2
HM-P200 000
negative
99
Molecular weight
Charge
Concentration of charges in a 1 wt %
aqueous solution (mm)
Mean contour length between charges (Å) 100
Hydrophobic modification degree
5.4
(mol %)
Mean contour length between
100
hydrophobic tails (Å)
1
2.5
3
84
2
y = hydrophobically modified cationic polyelectrolyte
x = hydrophobically modified anionic polyelectrolyte
Charge neutralisation
y = hydrophobically modified cationic polyelectrolyte
x = anionic polyelectrolyte
0.0
0.2
0.4
0.6
mx
mx + my
Thuresson et al. Langmuir 1996, 12, 530.
9
0.8
1.0
i) We should note that in a mixture of two oppositely
charged polyelectrolytes, one unmodified and one
hydrophobically modified, an extended two-phase
domain is observed over the composition range.
ii) When both polyelectrolytes are hydrophobically
modified, a strong reduction of the two-phase region
occurs.
iii) In this case, the attractive hydrophobic interactions
give rise to the formation of mixed aggregates and this
will lead to a net charge of the macromolecular part of
the concentrated phase.
iv) This process leads to an entropy loss in the
counterion distribution on phase separation and to
counteract the increase in free energy of the system, the
concentrated phase swells, and the tendency of phase
separation is reduced or completely eliminated,
depending on the charge stoichiometry.
10
6.)
r=0 (HM-P+); r=1 (HM-P-)
This result suggests stronger interactions as the
two-phase region is approached.
Frequency =0.6 s-1
Dynamic viscosity (Pas)
101
100
2
10-1
10-2
10-2
0
10-1
100
r
103
s (s), * (s)
* = 1/freq. (G' = G'')
a)
102
101
2
100
s
*
10-1
1.0
0
10-2
10-1
100
b)

0.8
0.6
2
0.4
0.2
0
10-2
10-1
100
r
11
The results show that the complex viscosity, the
longest time of relaxation, and the slow relaxation
time all increase as the two-phase regime is
approached. The value of the stretched exponent 
decreases as phase separation is approached. All
these features suggest enhanced entanglement
couplings as the two-phase region is approached.
7.) The q-dependence of the slow mode:
A strong q-dependence of the slow mode (strong
couplings effects) is predicted by the coupling theory
through the parameter . The calculated values from the
theory of Ngai (2/) (see eq.11) is in good agreement
with the experimental ones.
12
105
-1
-1 -1 -1 -1
f (s ), s (s )
103
-1
-1
10
2.5 ± 0.3
s ~ q
2.5 ± 0.5
f ~ q
2
101
-1
f , r = 1
100
-1
s,r =1
-1
f , r = 0.015
10-1
-1
-1
s , r = 0.015
5.2 ± 0.3
s ~ q
-2
10
8x106
7
107
3x107
q (m-1)
b)
s
2/
6
s, 2/
a)
2.1 ± 0.1
f ~ q
104
-1
s ~
s
q
5
2
4
3
2
0
10-2
10-1
100
r
These results indicate strong coupling effects in the
vicinity of the two-phase regime. This observation
is consistent with the conjecture of strong
intermolecular interactions.
13
These results suggest that the network structure
becomes more “open” (heterogeneous network) in
the vicinity of the two phase region, i.e., the fractal
dimension decreases.
101
a)
S(q)
100
r=0
r = 0.00492
r = 0.0103
r = 0.0150
r = 0.0205
r=1
10-1
10-2
10-3
8x106
df
101
q (m-1)
107
3x107
b)
S(q) ~ q-df
2
100
10-1
0.0
10-2
10-1
100
r
The picture that emerges from these results is that the polymer
network undergoes a structural reorganization from a
homogeneous structure in the solutions of the pure
polyelectrolytes to an heterogeneous network containing
bundles of polycation and polyanion chains as the two-phase
region is approached.
14
8.) Aggregation kinetics:
Reaction-limited
cluster-cluster
aggregation
(RLCA)
The
sticking
probability
(or
collision
efficiency) is very low-the clusters need to collide
many times before they stick. Dense aggregates are
formed.
Fractal morphology: The structure of the clusters
in RLCA is characterized by its fractal dimension
df MRdf
df = 2.1
Reaction kinetics: Exponential kinetics: Rexp(bt)
15
Diffusion-limited
cluster-cluster
aggregation
(DLCA)
The sticking probability is equal to onecollision between particles always results in
irreversible sticking.
Produce aggregates of open structures. df  1.8
Reaction kinetics: Power law growth: R  t1/ d f
9.)
400
1.0 x 10-3 wt % PS-latex
a)
0.7 M NaCl
0.15 M NaCl
200
400
Rh (nm)
Rh (nm)
300
100
70
0
100
90
80
70
101
102
time (min)
9
50 100 150 200
time (min)
103
Intensity (arbitrary units)
2x10
109
b)
1.0 x 10-3 wt % PS-latex
0.7 M NaCl
start time
11 min
1 h 42 min
3 h 22 min
df = 2.2 ± 0.1
df = 2.0 ± 0.1
df = 2.2 ± 0.3
4x108
1.5x107
2x107
q (m-1)
3x107
16
No aggregation at low salinity.
The salt-induced aggregation exhibits three stages:
i) An initial slow aggregation (Rht0.04) process.
ii) A fast increase of the average cluster size.
iii) The third stage is characterized by a slower
increase in cluster size. (RLCA).
a) During the very early times of aggregation the
growth process is slow, probably due to the fact
the dominating clusters are of small size.
b) At longer times, a steep transition zone with a
rapid growth of the clusters is observed, followed
by a slower aggregation process at later times.
This regime is frequently addressed in the study
of fractal morphology and kinetics of colloids
undergoing aggregation.
c) Since the cluster size growth can be described by
an exponential at later times, and the fractal
dimension, determined from ILS, is close to 2.1
(see Fig.b), the results indicate RLCA behavior.
17
10.) Salt-induced aggregation in the presence of
polymer .
In this case the aggregation process is significantly
slowed down (Rh t0.04) and the behavior is independent of
polymer concentration (in the dilute range) and hydrophobicity.
1.0 x 10-9 wt %
Rh (nm)
103
a)
EHEC
1.0 x 10-1 wt %
5.0 x 10-2 wt %
5.0 x 10-3 wt %
1.0 x 10-4 wt %
 = 0.038 ± 0.003
 = 0.039 ± 0.003
 = 0.041 ± 0.003
102
 = 0.038 ± 0.003
101
1.0 x 10-1 wt %
5.0 x 10-2 wt %
5.0 x 10-3 wt %
1.0 x 10-4 wt %
103
b)
HM-EHEC
Rh (nm)
103
102
 = 0.041 ± 0.002
 = 0.038 ± 0.003
 = 0.038 ± 0.003
102
 = 0.038 ± 0.003
101
102
time (min)
18
103
i) In the salt-induced aggregation process of bare
PSL particles, duplex and triplets are formed in
the beginning, but after a while multiplets are
formed and the average size of the clusters
increases strongly.
ii) In the case of the polymer-coated particles, the
conjecture is that the formation of large clusters
is suppressed due to weaker attractive
interactions, steric hindrance and geometrical
reasons.
19
11.)
Illustration
of
the
variation
of
the
hydrodynamic thickness  h  R h  R h ,0 with the
level of surfactant addition for the systems
EHEC/SDS/PSL and HM-EHEC/SDS/PSL.
180
Hydrodynamic thickness (nm)
160
EHEC/SDS/PSL
HM-EHEC/SDS/PSL
140
25 °C
120
100
80
60
40
20
0
0
10
20
30
40
50
Concentration of SDS (mmolal)
i) In the absence of surfactant, the hydrodynamic
thickness is larger for the hydrophobically modified
analogue than for EHEC.
ii) A monotonous decrease of h for HM-EHEC, while
h for EHEC passes through a maximum (close to
cac) and then decreases.
iii) The conjecture is that the difference in behavior is
related to the difference in cac for EHEC/SDS and
HM-EHEC/SDS.
20
12.) The variation of the hydrodynamic radius for
the
system
EHEC/SDS/PSL
temperatures.
240
EHEC/SDS/PSL
SDS/PSL
200
15 °C
160
Hydrodynamic radius (nm)
120
80
0
10
20
30
40
50
240
EHEC/SDS/PSL
SDS/PSL
200
25 °C
160
120
80
0
10
20
30
40
50
240
EHEC/SDS/PSL
SDS/PSL
200
30 °C
160
120
80
0
10
20
30
40
50
Concentration of SDS (mmolal)
21
at
various
The variation of the hydrodynamic radius for the
system
HM-EHEC/SDS/PSL
temperatures.
300
HM-EHEC/SDS/PSL
SDS/PSL
250
15 °C
200
150
Hydrodynamic radius (nm)
100
0
10
20
30
40
50
300
HM-EHEC/SDS/PSL
SDS/PSL
250
200
25 °C
150
100
0
10
20
30
40
50
300
HM-EHEC/SDS/PSL
SDS/PSL
250
200
30 °C
150
100
0
10
20
30
40
Concentration of SDS (mmolal)
22
50
at
various
The variation of the hydrodynamic radius for the
systems
EHEC/CTAB/PSL
and
HM-
EHEC/CTAB/PSL.
Hydrodynamic radius (nm)
240
a)
EHEC/CTAB/PSL
CTAB/PSL
200
160
25 °C
120
80
Hydrodynamic radius (nm)
0.0
1
10
Concentration of CTAB (mmolal)
300
b)
HM-EHEC/CTAB/PSL
CTAB/PSL
250
200
25 °C
150
100
0.0
1
10
Concentration of CTAB (mmolal)
The differences between the systems of unmodified
and hydrophobically modified EHEC in the
presence of SDS as compared to CTAB can be
summarized in the following way:
23
a) The general behavior of Rh is similar for both
surfactants.
b) For EHEC, the interaction peak is located at
about 3 mmolal and 0.3 mmolal in the presence of
SDS and CTAB, respectively. These values are
close to their respective cac values. Almost
complete desorption is observed at their respective
cmc values (8 mmolal for SDS and 1 mmolal for
CTAB).
c) The interaction peak is more pronounced in the
presence of SDS. This is probably due to the fact
that the polymer-surfactant interaction is stronger
in the presence of the anionic surfactant.
24
d) For HM-EHEC, a progressive decrease of Rh
with
increasing
surfactant
concentration
is
observed. Almost complete desorption is found at
their respective cmc values.
e) The fact that the polymer-surfactant complexes
are negatively or positively charged does not seem
to be important.
25