is calculated from the total surfactant concentration using a
mass balance equation and the equilibrium constant, Kmic, for
the process defined in eqn. (4), where n is the micellar
aggregation number.
The micellar aggregation number, n, is temperature depend-
ent and decreases from 121 to 32 between 45 °C and 25 °C,
according to static light scattering measurements, and pre-
3
sumably decreases further as the temperature is further
decreased. A temperature dependence of n is not included in the
present data treatment because the value of n itself has only a
small effect on the calculated curves shown in Fig. 1 and it
would be futile to include another fitting parameter to account
for its temperature dependence. Notwithstanding this, careful
examination of the curves shows that the cooperativity in
micelle formation at larger n should be reflected in a sharper
break at the c.m.t., whereas the best fit integer values of n are
only 3 or 4 at the concentrations of EPE used (Table 1).
However, like the changes in ln kobs in Fig. 1, changes in dye
solubilisation used to determine c.m.t. values do not show the
abrupt break expected of a highly cooperative aggregation
EPE = 1/n EPE
n
(4)
The dependence of Kmic on temperature is given by eqn. (5)
where DH0mic and
(5)
0
DS mic are the enthalpy and entropy of micellisation. The curves
E
0
in Fig. 1 are calculated from the best-fit values of K mic, DH mic
0
and DS mic (Table 1) obtained using non-linear regression of the
2
4
23
23
23
data at 8.5 3 10 mol dm and 2.55 3 10 mol dm EPE
at n values of 1, 5, and 15, respectively. (The value of n = 1 in
eqn. (4) corresponds to unimolecular micelle formation where
the poly(propylene oxide) converts from a loose hydrated
random coil, where the EPE molecule has little surfactant
property, to a more compressed, folded conformation.) It is
notable that the choice of the micellar aggregation number, n,
has only a small effect on the shape of the calculated curve and
the best-fit parameters shown in Table 1. This will be discussed
further in a later paragraph.
2
process with large n. Neither does the temperature dependence
of the molar excess heat capacity obtained from differential
scanning calorimetry. A recent paper on the latter shows that the
size of the cooperative unit of E28P E28 is about 2, much lower
48
than the value around 37 for n from static light scattering at 40
6
°C. It is therefore safe to assert that the low value of n obtained
in this study is not an artefact of the kinetic method of the
present work, but an actual property of the polymer solution.
This may possibly reflect a low n value at the onset of
micellisation, just after the c.m.t., and is related to the
polydispersity of the polymer, as discussed in Ref. 6.
The present data treatment assumes that DH0mic and DS0mic
are constant with temperature and this is in agreement with the
E
reported linearity of plots of inverse c.m.t. versus the logarithm
Table 1 shows that the best-fit values of K mic are almost
2
E
identical for the different chosen values of n, and, moreover, are
independent of the concentration of EPE within experimental
of EPE concentration. The treatment also assumes that K mic is
constant with temperature and this is acceptable because, in
fact, the dependence (results not shown) of kobs on the total EPE
concentration at temperatures above the c.m.t., where it
essentially equals [S]mic and eqn. (3) applies, yields best-fit
0
0
error. Best-fit values of DH mic and DS mic show a systematic
drop with increasing n, but this is not particularly large and
0
diminishes with increasing n. The best-fit values of DH mic and
E
DS
0
mic
at different concentrations of EPE are in perfectly
values of K mic that vary by no more than 15% between 30 °C
and 45 °C. Hence the concentration of ester remaining in the
bulk aqueous phase and still able to react with peroxide remains
constant within about 15% over this temperature range. This
explains why the slopes of the Arrhenius plots in Fig. 1 at
temperatures above the transition region are virtually identical
to the slope in the absence of polymer.
satisfactory agreement. The seemingly ‘too small to be true’
0
0
standard deviations in DH mic and DS mic translate into
reasonable experimental errors in Kmic after application of eqn.
0
0
(5). The values of DH mic and DS mic for the best-fit integer
values of n = 3 and 4 shown in Table 1 at the two different
concentrations of EPE, respectively, are in excellent agreement
5
21
with each other and close to the values, 2.96 3 10 J mol and
3
21 21
1
.09 3 10 J mol
K
of Hatton and co-workers obtained
Table 1 Best-fit values ± their standard deviations of the micellar
association constant of p-nitrophenyl trimethyl acetate and the enthalpy and
2
from the concentration dependence of the c.m.t.
61
entropy of micellization of the triblock copolymer, E27P E27, for various
micellar aggregation numbers at two polymer concentrations.
Notes and references
1 B. Chu and Z. Zhou in Non-ionic surfactants: polyoxyalkene block
copolymers, ed. V. M. Nace, Marcel Dekker Inc., New York, 1996, pp.
67–143.
2 P Alexandridis, J. F. Holzwarth and T. A. Hatton, Macromolecules, 1994,
27, 2414–2425.
3 G. Wanka, H. Hoffmann and W. Ulbricht, Macromolecules, 1994, 27,
4145–4159.
4 F. M. Menger and C. E. Portnoy, J. Am. Chem. Soc., 1967, 89,
4698–4703; A. K. Yatsimirski, K. Martinek and I. V. Berezin,
Tetrahedron, 1971, 27, 2855–2868; L. S. Romsted, C. A. Bunton and J.
Yao, Curr. Opin. Colloid Interface Sci., 1997, 2, 622–628.
5 D. M. Davies, N. D. Gillitt and P. M. Paradis, J. Chem. Soc., Perkin
Trans. 2, 1996, 659–666; D. M. Davies and N. D. Gillitt, J. Chem. Soc.,
Dalton Trans., 1997, 2819–2823; D. M. Davies and S. J. Foggo, J. Chem.
Soc., Perkin Trans. 2, 1998, 247–251; D. M. Davies, S . J. Foggo and P.
M. Paradis, J. Chem. Soc., Perkin Trans. 2, 1998, 1597–1602.
6 S. Hvidt, C. Trandum and W. Batsberg, J. Colloid Interface Sci., 2002,
KEmic/103
dm mol
DS mic/103
0
DH0mic/105
[
1
E
27
P
61
E
27]/
3
02 mol dm
23
3
21
J mol K21
21
21
n
J mol
0
2
0
2
.85
.55
.85
.55
1
1
3
3
4.18 ± 0.17
3.93 ± 0.31
—
2.34 ± 0.15
2.11 ± 0.17
—
6.94 ± 0.04
6.23 ± 0.05
—
a
3.91 ± 0.17
1.13 ± 0.04
3.24 ± 0.12
3.25 ±
0
2
0
.85
.55
.85
4a
4
5
4.18 ± 0.05
—
4.18 ± 0.06
1.14 ± 0.003
—
1.07 ± 0.001
0.008
—
3.02 ± 0.04
2
.74 ±
0.004
.41 ±
0.002
.99 ±
0.003
2
0
.55
.85
.55
5
3.80 ± 0.15
4.14 ± 0.11
3.76 ± 0.25
0.97 ± 0.002
0.87 ± 0.006
0.73 ± 0.013
2
15
15
1
2
a
Best-fit integer value of n at this concentration of polymer.
2
50, 243–250.
CHEM. COMMUN., 2003, 224–225
225