Micelle size distribution curve

The expression of T2 is much more complex than and its dependence on alcohol concentration cannot be simply assessed. However for a qualitative discussion, we recall that I/X2 is proportional to (cmc) " (8), where r is the number of amphiphilic ions constituting the associated species at the minimum of the micelle size distribution curve, and which may be considered as micelle nuclei (8). The value of r for normal detergents, in the absence of alcohol, is of about 10 (8,30). In previous studies rapid changes of have been explained in terms of changes of cmc and r (31). A similar explanation is likely to hold for the effect of addition of alcohol on T2, but the situation may somewhat be more complex as alcohol may also be incorporated to micelle nuclei. If we neglect this effect, take r = 10 and consider the effect of O.IM pentanol on the T2 of a O.IM TTAB solution, one is led to assume a decrease of r from 10 to 6 to explain the observed decrease of T2, as the cmc is decreased by a factor of 2 upon addition of O.IM pentanol. 

As pointed out in Subsection II.A.3, the values of Tg can be used to obtain information on the premicellar aggregates Ar at the minimum s = r of the micelle size distribution curve (see Figure 3.1), provided the measurements are performed on dilute solutions, where micelle formation/breakdown proceeds via stepwise reactions (4). The data were generally analyzed under the simplif3dng assumption that the concentration of the aggregates is much smaller than that of the other species in the narrow passage and kr k Equation 3.14 then reduces to... 

The length of the cosurfactant alkyl chain (carbon number nil) a very strong effect on the relaxation time T2 for tke micelle formation/breakdown. Indeed, the longer the cosurfactant (alcohol), the more it partitions in the micelles and the more it affects the micelle size distribution curve. Some representative resrdts are shown in Figure 3.14 for alcohol additions to CTAC. Similar results were reported for alcohol additions to tetradecyl and hexadecylpyridinium... 

I/X2. Similar results were reported for the SDS/PVP sys-tem.2 2 These variations are probably related to the decrease of the surfactant aggregation number upon increasing polymer concentration and the corresponding changes in the micelle size distribution curve when the surfactant aggregates on the polymer.The submicellar aggregates at the minimrun of the distribution curve may be stabihzed when forming on the polymer. 

There has been some preliminaiy chemical relaxation investigations of the kinetics of protein/surfactant systems.Both studies involved the bovine serumalbu-mine/SDS system and used the p-jmnp with conductivity detection. The introduction of the protein in the micellar solution of SDS was found to result in an increase of I/X2, a result similar to that obtained with regular pol5aners. This effect probably also reflects changes in the micelle size distribution curve. The surfactant aggregates bound to proteins are also smaller than in the absence of protein. 

Figure 4.12 Schematic representation of the shape of the micelle size distribution curve (w is the fraction of micelle of aggregation number P t is the time) during the first and second relaxation processes (see text). Reproduced from Reference 37 with permission of the American Chemical Society.

A consequence of the cooperativity is that, as schematized in Fig. 2.23, the size distribution curve has a deep minimum the appearance of a pronounced minimum in the size distribution curve is a good criterion for the formation of proper micelles. However, aggregation of amphiphilic compounds is of course not restricted to micelle formation and as indicated in Fig. 2.23 different types of size distribution curves... 

When the system has reached its quasi-equilibrium state a slower process, involving the relaxation to the true equilibrium, becomes measureable. This process involves a change in the number of micelles. The formation or dissolution of a micelle involves according to scheme (5.1) the appearence of aggregates of size at the minimum of the size distribution curve, and since these aggregates occur with low probability the process can be a very slow one. Aniansson and Wall showed that this process is also characterized by an exponential decay with a relaxation time r2,... 

A in particle size distribution curves. Colloidal gold, silver, platinum and platinized cadmium sulfide were generated in Aerosol-OT reversed micelles or in microemulsions by in situ photolysis of the appropriate ions (Figure 3.15). Under suitable conditions, each assembly contained approximately eight Au " ions, which firstly led to the formation of Aug clusters. ... 

The on rate constant, /c+, is diffusion-controlled and depends little on surfactant and micelle size (cf. Table 19.5). The off rate constant, on the other hand, is strongly dependent on alkyl chain length, micelle size, etc. Because of the co-operativity in micelle formation there is a very deep minimum in the size distribution curve. This leads to a two-step approach to equilibrium after a perturbation. In a fast step, quasiequilibrium is reached under the constraint of a constant total number of micelles. The redistribution of unimers between abundant micelles is a fast process. In order to reach a true equilibrium, the number of micelles must change. Because of the stepwise process, this also involves the very rare intermediate micelles. Therefore, this process is slow. 

As noted in particular in the analysis of kinetic data [10-15], there are aggregates over a wide range of aggregation numbers, from dimers and well beyond the most stable micelles. However, for surfactants with not too high a c.m.c., the size distribution curve has a very deep minimum, the least stable aggregates being present in concentrations many orders of magnitude below those of the most abundant micelles. For surfactants with predominantly spherical micelles, the polydispersity is low and there is then a particularly preferred micellar size. 

Size distribution curves have also been calculated by Tanford from a theoretical treatment of micelle formation [96, 97]. 

Some models of micelle growth take into account the fact that the surfactants located in the part connecting the cylindrical body of rodlike micelles to the endcaps are at a still higher chemical potential than those in the endcaps. This leads one to predict the existence of a second critical concentration, sometimes referred to as second cmc, above which micelles start growing. It also leads one to predict the coexistence of spherical and rodhke micelles, i.e., a bimodal distribution of micelle sizes. This is in contrast to the two-chemical potential approach that predicts a continuous growth of all micelles and a unimodal size distribution curve. 

The behavior of mixed micelles conforms to existing theories. A remarkable effect of alcohols on the relaxation associated with micelle formation/breakdown has been shown and attributed to the effect of alcohol on the size distribution curve. The possibility of a coexistence of two populations of micelles, one alcohol-rich and the other surfactant-rich, may explain the observed kinetic behavior. Small alcohol-rich micelles may play the role of carrier between surfactant-rich micelles. 

The critical concentration, Ccrit, is defined as the concentration at which there is a change in the size distribution of the surfactant aggregates from a monotonically decreasing curve (below Ccrit), to one that possesses a maximum and aminimum above the critical concentration. The transition in shape occurs at a size distribution that displays an horizontal inflexion point. The critical micelle concentration (CMC) has a value close to the critical value Ccrit [3.1 ] and [3.2],... 

Two well separated relaxation times can always be observed by relaxation measurements on micellar systems. (1) The best model that can account for the two processes has been proposed by Aniansson and Wall. (2) For the derivation of the relaxation expressions, the micellar distribution curve was divided into three different regions The monomers and oligomers, the nuclei in the distribution minimum and finally the proper micelles around the distribution maximum. If the equilibrium is suddenly perturbed, the reequilibration process between the proper micelles of different size can proceed rapidly. The number of the micelles is not changed by this process. Aniansson showed that the monomer relaxation leads exactly to a single relaxation time... 

Figure 3.15 Changes of the distribution curve of micelle size. From Lang et al. [224]. Aniansson and co-workers proposed the following expression for the fast process...

One of the systems studied was a polystyrene-b/ocfc-poly(ethylene/propylene) (37300 59700 Mn) copolymer in decane. Electron microscopy studies showed that the micelles formed by the block copolymer were spherical in shape and had a narrow size distribution. Since decane is a selectively bad solvent for polystyrene, the latter component formed the cores of the micelles. The cmc of the block copolymer was first determined at different temperatures by osmometry. Figure 13 shows a plot of n/cRT against c (where c is the concentration of the solution) for T=91A °C. The sigmoidal shape of the curve stems from the influence of concentration on the micelle/unassociated-chain... 

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