Micelle size distribution

Hence the sizes of spherical micelles are distributed around a most probable aggregation number M, which depends only on molecular details of the surfactants in this simplest approximation. Indeed, micelle size distributions at concentrations beyond the CMC have shown a marked peak at a given aggregation number in many simulations [37,111,112,117,119,138,144,154,157]. 

FIG. 10 Micelle size distribution for H2T2 surfactants within the Larson model. The dashed lines show fits to the expected form for spherical micelles (main peak) and cylindrical micelles (tail). Inset shows the tail of the distribution on a semi-logarithmic plot to demonstrate the exponential decay predicted for the cylindrical micelles. (From Nelson et al. [120].)... 

The micelle formation is not restricted to solvents for polystyrene but also occurs in very unpolar solvents, where the fluorinated block is expected to dissolve. Comparing the data, we have to consider that the micelle structure is inverted in these cases, i.e., the unpolar polystyrene chain in the core and the very unpolar fluorinated block forming the corona. The micelle size distribution is in the range we regard as typical for block copolymer micelles in the superstrong segregation limit.2,5,6 The size and polydispersity of some of these micelles, measured by DLS, are summarized in Table 10.3. 

Even with the used detail, both the MAM and the PSM models need some improvements for more realistic representation of available data. They could be refined to account for micelle size distribution (the MAM) and nonideal mixing with the solvent, once data on x, Cj, C2, N(x), and / (x) become available. 

In most cases when the micelle size distribution has been analyzed quantitatively the multiple equilibrium model has been used. This model can be formulated either, in analogy with Eq. (3.2), as a number of equilibria... 

Consider a micellar solution at equilibrium that is subject to a sudden temperature change (T-jump). At the new temperature the equilibrium aggregate size distribution will be somewhat different and a redistribution of micellar sizes will occur. Aniansson and Wall now made the important observation that when scheme (5.1) represents the kinetic elementary step, and when there is a strong minimum in the micelle size distribution as in Fig. 2.23(a) the redistribution of micelle sizes is a two-step process. In the first and faster step relaxation occurs to a quasi-equilibrium state which is formed under the constraint that the total number of micelles remains constant. Thus the fast process involves reactions in scheme (5.1) for aggregates of sizes close to the maximum in the distribution. This process is characterized by an exponential relaxation with a time constant Tj equal to... 

Here k is the off rate constant for the micelle at the mean aggregation number n262- and a is the standard deviation of the assumed gaussian micelle size distribution. a is the relative amount of micellized monomer... 

Table 5.1. Rate constants and micelle size distribution for a series of sodium alkyl sulfates at 25 °C. (Adapted from Ref.1O4))...

The structure of the AOT micellar system, as well as the state of water entrapped inside swollen micelles, have been characterized using different techniques, such as photon correlation spectroscopy (25), positron annihilation (26), NMR (27, 28), fluorescence (29-32) and more recently small angle neutron scattering (33). The existence of reversed micelles has been demonstrated in the domain of concentrations explored by protein extraction experiments. Their size (proportional to the molar ratio of water to surfactant known as wo), shape and aggregation number have been determined. Furthermore, the micelle size distribution is believed to be relatively monodisperse. 

Since the slow relaxation process is critically dependent on the micelle size distribution, kinetic measurements can be used to determine the standard deviation of the distribution. As can be seen in Table 19.7, the distribution is relatively narrow for surfactants with longer alkyl chains. 

Table 19.7. The CMC, the mean aggregation number N, and the standard deviation a of the micelle size distribution for a series of sodium alkyl sulfates, as determined from kinetic measurements. (From E. A. G. Aniansson et al, J. Phys. Chem., 80 (1976) 905)...

TABLE II Rate Constants and Micelle Size Distribution for a Series of Sodium Alkyl Sulfates at 25°C ... 

The micelle size distribution is determined by a balance between the energy gain AT endcap at each combination and the corresponding reduction in entropy of mixing [5.26]. 

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. 

Stop-flow experiments have been performed by Tuzar and Kratochvil [7] and more recently by Kositza et al. [128]. In analogy to low molar weight surfactants, it could be shown that two relaxation processes have to be considered for block copolymer micellar systems the first in the time scale of tens of microseconds, associated to unimer exchange between micelle and bulk solution, and the second, in the millisecond range, attributed to the rearrangement of the micelle size distribution. Major differences were observed between A-B diblock and A-B-A triblock copolymers, which could be explained by the fact that the escape of a unimer, which has to disentangle from the micellar core, might be much easier in a diblock than in a triblock structure. 

This, together with relation (8.9.9), determines the distribution of micellar sizes. In order to determine the micelle size distribution, one must first know all AG (A ), then solve Eq. (8.9.9) for Pm, i.e., the implicit equation... 

Aqueous micellar solutions provide some interesting theoretical problems, such as the prediction of the micelle size distribution, the most stable shapes of the micelles and the elucidation of their behavior at the CMC. From the practical point of view, the most important aspect of micellar solutions is their capability to solubilize solutes that are very sparingly soluble in pure water. This phenomenon occurs naturally in biological systems and has been exploited in many applications, as for example in the pharmaceutical and detergent industries. 

Returning to the micelle size distribution, we can estimate how this distribution changes upon the addition of solute a. Suppose again that the two-component system of surfactant and water behaves as an associated ideal dilute solution for which (8.9.11) and (8.9.13) are valid. By adding the solute a we can define the new species A(k, n), i.e., an aggregate with n monomers and k solute molecules a. Since the strong correlations between a and An are within the species A k, n), the assumption of associate ideality can be retained for all the species A k, n). Hence the analogue of (8.9.11) is now... 

On the other hand, 5G A k, n)) in (8.9.32) is almost the same as SG(A ) in (8.9.11). The reason is that the coupling work of M and A to the solvent are nearly independent of the extent of occupation of the micelles by a. If the micelles are very large compared with the solute a, and if the interaction of a with the micelle is weak, then all the quantities on the rhs of (8.9.32) are approximately independent of k hence the micelle size distribution is almost unaffected by a. This is indeed observed for simple nonpolar solutes. For larger or polar solutes, the effect of the solutes on the distribution could be significant. 

In kinetic measurements of micellization processes one generally obtains two relaxation times from which rate constants, the width of the micelle size distribution and information on the rarest intermediate single micelles can be deduced [1, 2]. From such measurements it is also possible, as we shall see, to calculate the mean lifetime of a micelle. Such a quantity is of course of interest in itself but further interest arises in connection with the role of the micelle as host for one or more solubilized or adsorbed molecules. 

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. 

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