Relaxation Time for the Exchange Process

The following assumptions were made by Aniansson and Wall  

In Equation 3.9 N is the average micelle aggregation number, that is, the average number of surfactants making up a micelle, a quantity that is experimentally accessible by a number of techniques, a is the reduced surfactant concentration given by 

To a first approximation, the free surfactant concentration [Ai] can be taken as the cmc. Thus, l/Xj should increase linearly with C provided that all other quantities in Equation 3.9 are independent of C. From such plots, one can obtain k and provided that N is known. Notice that Tjj = N/k- is the average residence time of a given surfactant in the micelle whereas l/k is the residence time of any smTactant in the micelles. The variance characterizes the micelle polydis-persity. Moreover, since [AJ i=cmc, k+ can be obtained from 

Most of the assumptions, particularly (3) and (4) under which Equation 3.9 was derived, were later relaxed. Thus Equation 3.9 was shown to remain valid if k+ and k vary linearly with s. Numerical calculations showed that very large deviations from a linear dependence of the two rate constants on s are required in order that the fast process be characterized by more than one experimentally accessible relaxation time. ° An expression of in the case of not very small perturbations was also derived. 

Criticism concerning the validity of assumption (2) was shown to be unfounded. Nevertheless, it is noteworthy that Equation 3.9 strictly applies only to nonionic surfactants and dilute micellar solutions. 

---

Fig. 5. Cetyltrimethylammonium bromide 2.2 x 10" at 24 . Variation of the relaxation time for the exchange process upon addition of alcohol (A) methanol ( ) ethanol (x) propanol (d) butanol (+) pentanol and (0) hexanol. Note the similarity between these curves and those relative to the change of a with 1 values were measured by...

Figure 4.6 Variations of the relaxation time for the exchange process with the copolymer concentration at the temperature of 21.1, 21.4 and 40.0°C for F127, P123, and F88, respectively, where most of the copolymer is in the micellar state. Reproduced from Reference 32 with permission of Elsevier.

V-V exchange between two hydrogen halides has been observed by Chen, Stephenson, and Moore [218] utilizing laser-excited vibrational fluorescence. The vibration-vibration transfer rate for HC1-HI collisions was determined by measurement of the exponential decay of HC1 Av = 1 fluorescence following excitation by an HC1 laser pulse. The relaxation time for the process... 

Fig. 4K is discussed later. Suffice it to note here that the relaxation time for the activation-controlled process is inversely proportional to the exchange current density while x, the diffusional relaxation time, is independent of i. Thus, a large value of x /x indicates that the electrode reaction is slow, and vice versa. 

This section recalls the main aspects of the derivation of the expressions of the relaxation times for the surfactant exchange process and for the micelle formation/breakdown as done by Aniansson and WalP in 1974 and 1975, and the main extensions of this theory by Kahlweit et al. and HalP in the years that followed. 

In all instances where it could be measured, the relaxation time for the surfactant exchange process was found to obey Equation 3.9, that is, llx increased linearly with the... 

Molecular Motions and Dynamic Structures. Molecular motions are of quite general occurrence in the solid state for molecules of high symmetry (22,23). If the motion does not introduce disorder into the crystal lattice (as, for example, the in-plane reorientation of benzene which occurs by 60° jumps between equivalent sites) it is not detected by diffraction measurements which will find a seemingly static lattice. Such molecular motions may be detected by wide-line proton NMR spectroscopy and quantified by relaxation-time measurements which yield activation barriers for the reorientation process. In addition, in some cases, the molecular reorientation may be coupled with a chemical exchange process as, for example, in the case of many fluxional organometallic molecules. ... 

The measurement of the exchange time xm may provide useful kinetic information on the system. Kinetic parameters for the dissociation process may be obtained by performing relaxation measurements as a function of temperature. If it is assumed that the dissociation of the ligand from the paramagnetic site is a first order kinetic process, the dissociation rate constant r 1 is given by the Eyring relationship... 

Unpaired electronic density can be delocalized onto the various nuclei of the complex via through-bond scalar hyperfine interactions involving occupied orbitals containing s-character (direct interaction or polarization according to the Fermi mechanism, Wertz and Bolton (1986)). Random electron relaxation thus produces a flip-flop mechanism which affects the nuclear spin and increases nuclear relaxation processes (Bertini and Luchinat, 1996). Since these interactions are isotropic, they do not depend on molecular tumbling and re is the only relevant correlation time for non-exchanging semi-rigid complexes. Moreover, only electronic spin can be delocalized via hyperfine interactions (no orbital contribution) and the contact re-... 

---