Micellar size distributions

Figure 4. Micellar size distribution curves for a 10% w/w solution of poloxamer at 40 C exposed to irradiation doses of A, 0 B, 1.52 C, 2.53 D, 3.55 and E, 4.56 Mrad.

E. Ruckcnstcin, R. Nagarajan Critical micelle concentration. A transition point for micellar size distribution, JOURNAL OF PHYSICAL CHEMISTRY 79 (1975) 2622-2626. 

Critical Micelle Concentration. A Transition Point for Micellar Size Distribution... 

The equation assumed by Ben-Naim and Stillinger for the free energy of miceUization is used to demonstrate that, contrary to their assertion, the critical concentration Catt corresponding to a transition in the shape of the micellar size distribution has a value close to the conventional cmc. The ratio cmc fC is shown to decrease sharply from the large value of 20 obtained by Ben-Naim and Stillinger, to a value of 1.55, if one of the parameters (c) is assigned a physically more plausible value. This value accounts better for the cooperative self-association of the surfactant molecules. 

Figure 1. Qualitative micellar size distribution. The curves represent size distributions when the total surfactant concentration is less than Coo, equal to Cat, larger than C, and far above C, ...

Using a chemical potential with this n dependence [Eq. (12-20)], along with the law of mass action, Eq. (12-4), Israelachvili (1992) has shown that the micellar size distribution well above the critical micelle concentration (CMC) (where the CMC is roughly equal to exp(Eoo/kBT)) is approximated by... 

Polydispersity of simple bile salt micelles can only be assessed by modem QLS techniques employing the 2nd cumulant analysis of the time decay of the autocorrelation function [146,161]. These studies have shown, in the cases of the 4 taurine conjugates in 10 g/dl concentrations in both 0.15 M and 0.6 M NaCl, that the distribution in the polydispersity index (V) varies from 20% for small n values to 50% for large n values [6,146]. Others [112] have foimd much smaller V values (2-10%) for the unconjugated bile salts in 5% (w/v) solutions. Recently, the significance of QLS-derived polydispersities have been questioned on the basis of the rapid fluctuation in n of micellar assemblies hence V may not actually represent a micellar size distribution [167-169]. This argument is specious, since a micellar size distribution and fast fluctuations in aggregation number are identical quantities on the QLS time scale (jusec-msec) [94]. 

Note that the A-W theory was derived under the assumption that the micellar size distribution is Gaussian, independent of concentration, and can be taken to be essentially the equilibrium distribution. Thus, in this linear relaxation process, the micellar population is imagined to be shifted to a new mean aggregation value. 

Nagarajan, R. and Ruckenstein, E., Critical micelle concentration A transition point for micellar size distribution A statistical thermodynamical approach, J. Colloid Interface Sci., 60, 221, 1977. 

Inspection of Figs 5 and 6 shows that at high probe concentration, limiting values of D tend toward Di. Thus, a reasonable estimate of D for monodisperse systems can be obtained by measuring D at relatively large probe concentrations, for example, 2 to 5 mM. For polydisperse systems, additional equilibria must be considered. A three-state model was used to develop a version of Eq. (12) for two micellar size distributions, and applied to systems with coexisting globular and rodshaped micelles [4]. 

Although the mass action approach could account for a number of experimental results, such as the small change in properties around the c.m.c., it has not escaped criticism. For example, the assumption that surfactants exist in solution in only two forms, namely single ions and micelles of uniform size, is debatable. Analysis of various experimental results has shown that micelles have a size distribution that is narrow and concentration dependent. Thus, the assumption of a single aggregation number is an oversimplification and, in reaUty, there is a micellar size distribution. This can be analyzed using the multiple equilibrium model, which can be best formulated as a stepwise aggregation [2],... 

This is another version of (4.69). Now we need to specify an explidt expression of fi in order to calculate the micellar size distribution. 

The theoretical developments based on the effects of geometry on molecular aggregation have shown that physical characteristics of surfactants such as cmc, aggregate size and shape, and micellar size distribution (polydispersity) can be quantitatively described without relying on a detailed knowledge of the specific energetic components of the various molecular interactions. It is also useful in that it applies equally well to micelles, vesicles, and bilayer membranes the latter lie outside the normal models of association processes. For that reason, the geometric approach warrants a somewhat closer look. 

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