collapsed insoluble block during the expulsion process. The parameter α(36π) was
fitted to be approximately equal to 3.3 instead of 4.8, as expected for spherical
globules. This disagreement was thought to be due to deviations from the spherical
conformation or to interfacial effects resulting from modifications of the surface
energy by the other block. The best value for τ 0 was fitted to 2.4 Â 10
À7 s, which
is comparable to a typical elemental time expected for polymer dynamics. In summary, the work of Choi et al. and, subsequently, the novel interpretation of the data of
Lund et al. strongly corroborate that the pseudo-logarithmic time decay in the
equilibrium kinetics is a consequence of core block polydispersity, which can be
rationalized by a double exponential dependence of the exchange rate on chain length.
In principle, both experiments essentially confirm the validity of the theory of
Halperin and Alexander. However, there remain several contradictions concerning
the exact mechanism, in particular the adopted chain conformation during the
activated step of the exchange process. This still needs to be delineated in future
experiments. These details of the mechanism can be more conveniently studied using
a monodisperse system. Such a system will be discussed in the next section.
4.4 n-Alkyl-PEO Polymeric Micelles
The chain exchange kinetics of n-alkyl-PEO (C n H 2nÀ1 , where n ¼ 18, 24, or 30)
polymeric micelles in water was studied by Zinn et al. [103]. Structurally, the n-alkylPEO polymers can be considered as hybrids between amphiphilic block polymers
(e.g., PEP-PEO or PEE-PEO) and nonionic C n (EO) m surfactants. With respect to the
exchange kinetics, these materials were taken as model system because the polydisperse hydrocarbon block is replaced by a relatively short but truly monodisperse (M w /
M n ¼ 1)) aliphatic chain. Accordingly, if the above considerations were true, the
relaxation kinetics was expected to follow a single exponential decay. Moreover,
variation of n should directly reflect the dependence on chain length and thus the
effect of polydispersity on the time decay of R(t). In fact, the kinetic measurements
reveal a strong dependence on the alkyl chain length, as depicted in Fig. 26, where the
neutron detector count rates are plotted versus time after mixing the two differently
labeled (H/D) micellar species at room temperature (22
C). The figure shows that
within five orders of magnitude in time up to 1,000 s, the count rate of the C 30 H 61 -
PE05 micelles (squares in Fig. 26) stays constant revealing no chain exchange. For the
C 18 H 37 -PEO5 micelles (triangles Fig. 26) on the other hand, full equilibration
(depicted by the solid line in Fig. 26) was already obtained after a few milliseconds.
Apparently, the relaxation process is too fast to be resolved by TR-SANS. Only for the
C 24 H 49 -PEO5 system (circles Fig. 26) could the full process of chain exchange be
measured, indicated by the continuous decay of the count rate from initial to final state
in a time frame of 100 s. Figure 27 shows the corresponding relaxation function on a
logarithmic time scale. The linear decay in this representation perfectly revealed the
theoretically expected single exponential decay: R(t) ¼ exp(Àt/τ 0 ) with a characteristic time τ 0 ¼ 44 s. It should be mentioned that evaluation of the data using either the
model-dependent, f exc , or model-independent method, R(t), (for details see Sect. 3.2.2)
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R. Lund et al.
fitted to be approximately equal to 3.3 instead of 4.8, as expected for spherical
globules. This disagreement was thought to be due to deviations from the spherical
conformation or to interfacial effects resulting from modifications of the surface
energy by the other block. The best value for τ 0 was fitted to 2.4 Â 10
À7 s, which
is comparable to a typical elemental time expected for polymer dynamics. In summary, the work of Choi et al. and, subsequently, the novel interpretation of the data of
Lund et al. strongly corroborate that the pseudo-logarithmic time decay in the
equilibrium kinetics is a consequence of core block polydispersity, which can be
rationalized by a double exponential dependence of the exchange rate on chain length.
In principle, both experiments essentially confirm the validity of the theory of
Halperin and Alexander. However, there remain several contradictions concerning
the exact mechanism, in particular the adopted chain conformation during the
activated step of the exchange process. This still needs to be delineated in future
experiments. These details of the mechanism can be more conveniently studied using
a monodisperse system. Such a system will be discussed in the next section.
4.4 n-Alkyl-PEO Polymeric Micelles
The chain exchange kinetics of n-alkyl-PEO (C n H 2nÀ1 , where n ¼ 18, 24, or 30)
polymeric micelles in water was studied by Zinn et al. [103]. Structurally, the n-alkylPEO polymers can be considered as hybrids between amphiphilic block polymers
(e.g., PEP-PEO or PEE-PEO) and nonionic C n (EO) m surfactants. With respect to the
exchange kinetics, these materials were taken as model system because the polydisperse hydrocarbon block is replaced by a relatively short but truly monodisperse (M w /
M n ¼ 1)) aliphatic chain. Accordingly, if the above considerations were true, the
relaxation kinetics was expected to follow a single exponential decay. Moreover,
variation of n should directly reflect the dependence on chain length and thus the
effect of polydispersity on the time decay of R(t). In fact, the kinetic measurements
reveal a strong dependence on the alkyl chain length, as depicted in Fig. 26, where the
neutron detector count rates are plotted versus time after mixing the two differently
labeled (H/D) micellar species at room temperature (22
C). The figure shows that
within five orders of magnitude in time up to 1,000 s, the count rate of the C 30 H 61 -
PE05 micelles (squares in Fig. 26) stays constant revealing no chain exchange. For the
C 18 H 37 -PEO5 micelles (triangles Fig. 26) on the other hand, full equilibration
(depicted by the solid line in Fig. 26) was already obtained after a few milliseconds.
Apparently, the relaxation process is too fast to be resolved by TR-SANS. Only for the
C 24 H 49 -PEO5 system (circles Fig. 26) could the full process of chain exchange be
measured, indicated by the continuous decay of the count rate from initial to final state
in a time frame of 100 s. Figure 27 shows the corresponding relaxation function on a
logarithmic time scale. The linear decay in this representation perfectly revealed the
theoretically expected single exponential decay: R(t) ¼ exp(Àt/τ 0 ) with a characteristic time τ 0 ¼ 44 s. It should be mentioned that evaluation of the data using either the
model-dependent, f exc , or model-independent method, R(t), (for details see Sect. 3.2.2)
126
R. Lund et al.