Macromolecules, Vol. 36, No. 4, 2003
Association of Adhesive Spheres 1339
ν/n from 0 at Cp to unity at about 3Cp can be understood
in terms of an increasing effective functionality of the
micelles, but for a quantitative description more realistic
computer simulations are required.
restructuration of the micelles. We will present a more
detailed study of the transition to a hard gel elsewhere.
The results presented here show similarities with
those obtained for suspensions of stabilized oil droplets
in the presence of difunctionalized PEO.37 For this
system phase separation is induced above of a critical
number of difunctionalized PEO chains per droplet.
They also observed slanted tie lines because the system
can gain entropy if the droplets in the dense phase
contain more difunctionalized PEO than those in the
dilute phase. The binodal should nevertheless be the
same as for adhesive spheres unless there is a distribu-
tion in the number of difunctionalized chains per droplet
already in the homogeneous phase. In the homogeneous
phase they also observed the formation of a transient
gel at a percolation concentration that increased with
decreasing PEO concentration.
The terminal relaxation time increases with increas-
ing PEO concentration. Close to Cp so-called super
bridges3 are formed, i.e., elastic network chains that
contain more than one PEO chain. The escape of any
chain end in the super bridge relaxes the stress which
decreases τr. The fraction and the length of super
bridges decrease with increasing concentration so that
τr increases with C. Figure 9 shows that τr is approxi-
mately independent of f at a given value of C/Cp and
thus a given value of G0/(nRT). This means that τr is
mainly determined by the fraction of bridges that is
elastically active.
At a fixed polymer concentration the viscosity de-
creases with decreasing f. However, at a fixed concen-
tration of difunctional chains the viscosity actually
increases with decreasing f. The increase of the volume
fraction of polymeric micelles compensates for the
reduction of the functionality of each micelle. For the
same reason an increase of the viscosity is often
observed if small surfactants are added to associative
polymers.1,33,34 Of course, in the latter case one also
modifies the structure of the micelles; notably excluded-
volume interactions between the micelles are reduced.
If one continues to add small surfactants, the viscosity
reaches a maximum and decreases with further addi-
tion. The reason is that at high concentrations of
surfactants the reduction of the functionality dominates,
and finally the micelles cease to percolate. For the
present system this situation cannot be reached even
for f ) 0.2 because addition of a large fraction of
monofunctionalized PEO leads to strong excluded-
volume interactions between the micelles and the
formation of a so-called hard gel.
Con clu sion
Hydrophobically end-capped PEO forms polymeric
micelles in aqueous solution with a number of hydro-
phobic groups per micelles that is independent of the
ratio of mono- and difunctionalized chains if the HLB
is kept the same. Increasing the temperature reduces
exclude-volume interaction between the micelles, but
their molar mass and hydrodynamic radius change very
little.
The micelles phase separate above a critical temper-
ature that decreases with increasing fraction of difunc-
tional chains. The system can be modeled as a collection
of adhesive spheres with an adhesion parameter that
is proportional to the number of difunctionalized chains
per micelle and inversely proportional to the thermo-
dynamic volume of the micelles. This model describes
semiquantitatively the phase diagrams and the concen-
tration dependence of the osmotic compressibility.
The mechanical properties can be interpreted in terms
of the formation of a transient network of bridging
micelles. The terminal relaxation time is determined by
the escape time of an end group from the multiplet
which is independent of the fraction of difunctionalized
chains. The elastic modulus has a very weak tempera-
ture dependence, and both the terminal relaxation time
and the viscosity have an Arrhenius temperature de-
pendence.
At higher concentrations the micelles closely pack,
and at a concentration Chg > Cp the system jams, which
is characterized by the arrest of flow. Chg is strongly
temperature dependent, but it is almost independent
of f. J amming of the micelles is induced by excluded-
volume interactions and is therefore controlled by the
thermodynamic volume of micelles, which itself is
almost independent of f. In fact, we have observed this
transition also for Brij700, which cannot bridge at all.
A temperature-dependent transition to a nonflowing gel
phase has been observed for diblock copolymers13 and
star polymers.25 In some cases the hard gel was pre-
ceded by a so-called soft gel which flows upon tube
inversal. The soft gel was attributed to temperature-
dependent attractive interactions between the mi-
celles.35,36 Lobry et al.36 determined the phase diagram
and the percolation threshold of a triblock copolymer
with a hydrophobic middle block (Pluronic). They in-
terpreted their results in terms of adhesive spheres in
a manner similar to our interpretation. However, the
attractive interaction between Pluronics is not well
understood. The terminal relaxation time of the soft gel
is very long for that system.
At a higher concentration the system shows a transi-
tion to a nonflowing gel that is caused by jamming of
closely packed micelles. This transition is strongly
temperature dependent but almost independent of the
fraction of difunctionalized chains.
Refer en ces a n d Notes
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For the present system the origin of the attractive
interactions is clear and can be quantified. In one case
we have determined the frequency dependence of G′′
down to very low frequencies (see Figure 10a). For this
system we observe the relaxation of the hard gel well
separated from that of the bridges. The relaxation of
the hard gel might be due to hopping or complete
(6) Abrahamse´n-Alami, S.; Alami, E.; Franc¸ois, J . J . Colloid
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