Aggregation numbers

Aggregation numbers for many surfactants have been found to fall in the range of 50-100 molecules, although that can vary significantly according to structure and conditions. Some typical aggregation numbers for various surfactant types are given in Table 15.3. 

Because the size and dispersity of micelles are sensitive to many internal (hydrophobic structure, head group type) and external (temperature, pressure, pH, electrolyte content) factors, it is sometimes difficult to place too much significance on reported values of m. However, some generalizations can be made that are usually found to be true are as follows  

In aqueous solutions, it is generally observed that the longer the hydro-phobic chain for an homologous series of surfactants, the larger will be the aggregation number. 

A similar increase is seen when there is a decrease in the hydrophiUcity of the head group—for example, a higher degree of ion binding in an ionic material or a shorter polyoxyethylene chain in nonionics. 

Factors that result in a reduction in the hydrophilicity of the head group such as high electrolyte concentrations will also cause an apparent increase in aggregation number. 

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Other properties of association colloids that have been studied include calorimetric measurements of the heat of micelle formation (about 6 kcal/mol for a nonionic species, see Ref. 188) and the effect of high pressure (which decreases the aggregation number [189], but may raise the CMC [190]). Fast relaxation methods (rapid flow mixing, pressure-jump, temperature-jump) tend to reveal two relaxation times t and f2, the interpretation of which has been subject to much disagreement—see Ref. 191. A fast process of fi - 1 msec may represent the rate of addition to or dissociation from a micelle of individual monomer units, and a slow process of ti < 100 msec may represent the rate of total dissociation of a micelle (192 see also Refs. 193-195). 

Micellization is a second-order or continuous type phase transition. Therefore, one observes continuous changes over the course of micelle fonnation. Many experimental teclmiques are particularly well suited for examining properties of micelles and micellar solutions. Important micellar properties include micelle size and aggregation number, self-diffusion coefficient, molecular packing of surfactant in the micelle, extent of surfactant ionization and counterion binding affinity, micelle collision rates, and many others. 

Other solubilization and partitioning phenomena are important, both within the context of microemulsions and in the absence of added immiscible solvent. In regular micellar solutions, micelles promote the solubility of many compounds otherwise insoluble in water. The amount of chemical component solubilized in a micellar solution will, typically, be much smaller than can be accommodated in microemulsion fonnation, such as when only a few molecules per micelle are solubilized. Such limited solubilization is nevertheless quite useful. The incoriDoration of minor quantities of pyrene and related optical probes into micelles are a key to the use of fluorescence depolarization in quantifying micellar aggregation numbers and micellar microviscosities [48]. Micellar solubilization makes it possible to measure acid-base or electrochemical properties of compounds otherwise insoluble in aqueous solution. Micellar solubilization facilitates micellar catalysis (see section C2.3.10) and emulsion polymerization (see section C2.3.12). On the other hand, there are untoward effects of micellar solubilization in practical applications of surfactants. Wlren one has a multiphase... 

The reasons for self-assembly and the mechanisms necessary conditions for the aggregation into micelles, mono- or bilayers, structure of aggregates, distribution of aggregation numbers, etc. 

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]. 

At small N, correction terms come into play, which account for the ends of the cylinders. In particular, the aggregation number of cylindrical micelles in this simple picture must always be larger than M, the most probable aggregation number of a spherical micelle. Putting everything together, the expected size distribution has a peak at M which corresponds to spherical micelles, and an exponential tail at large N which is due to the contribution of cylindrical micelles. 

FIG. 11 Eigenvalues of the radius of gyration tensor (dots largest, squares middle triangles smallest) of micelles vs aggregation number N in an oif-lattiee model of H2T2 surfaetants. The mieelle size distribution for this partieular system has a peak at 28. (From Viduna et al. [144].)... 

Figure 20 shows the plot of the surface tension vs. the logarithm of the concentration (or-lg c-isotherms) of sodium alkanesulfonates C,0-C15 at 45°C. In accordance with the general behavior of surfactants, the interfacial activity increases with growing chain length. The critical micelle concentration (cM) is shifted to lower concentration values. The typical surface tension at cM is between 38 and 33 mN/m. The ammonium alkanesulfonates show similar behavior, though their solubility is much better. The impact of the counterions is twofold First, a more polarizable counterion lowers the cM value (Fig. 21), while the aggregation number of the micelles rises. Second, polarizable and hydrophobic counterions, such as n-propyl- or isopropylammonium and n-butylammonium ions, enhance the interfacial activity as well (Fig. 22). Hydrophilic counterions such as 2-hydroxyethylammonium have the opposite effect. Table 14 summarizes some data for the dodecane 1-sulfonates.

The following equations were derived from experimental data [74] to calculate aggregation numbers of sodium alkane 1-sulfonates ... 

The aggregation numbers of alcohol sulfates can be as low as 20 for sodium octyl sulfate in water solution but usually range from 60 to 150 for higher alcohol sulfates. 

Aggregation numbers increase as the length of the hydrophobic chain increases. The presence of electrolytes also causes an increase in the aggregation... 

The logarithm of the micellar molecular weight (M) and consequently the aggregation number of sodium dodecyl sulfate at 25°C in aqueous sodium chloride solutions is linearly related to the logarithm of the CMC plus the concentration of salt (Cs), both expressed in molar units, through two equations [116]. Below 0.45 M NaCl micelles are spherical or globular, and Eq. (18) applies ... 

The aggregation numbers of sodium octyl, decyl, dodecyl, and tetradecyl sulfates at temperatures between 20°C and 60°C have been studied by Vass by small-angle neutron scattering [130]. Table 22 shows the aggregation numbers of some alcohol and alcohol ether sulfates. 

Alkyl chain Cation Medium Aggregation number Ref. 

Molecular micellar weights can be calculated from the aggregation number. In the case of sodium dodecyl sulfate it is around 15,000 (aggregation number = 52) and it increases to 30,000 (aggregation number = 104) in diluted NaCl solutions. 

Van Paassen [57] describes the CMC of some polyether carboxylates with different fatty chains and EO degrees (Fig. 2). In an extensive study, Binana-Limbele et al. [59] investigated the micellar properties of the alkylpolyether carboxylates of the general formula CnH + OCF CH OCI COONa with n = 8, x = 5, and n = 12 and x = 5,1, and 9, by means of electrical conductivity (CMC, apparent micellar ionization degree) and time-resolved fluorescence probing (micelle aggregation number A7) as a function of temperature and surfactant concentration (Table 1). 

It has been found that the CMC values are higher and the micelle aggregation numbers smaller than those of the corresponding nonionic surfactants. The CMC increases with increasing EO chain, which is, according to the authors, opposite to the results for sodium alkyl ether sulfate. 

From the apparent ionization degree it was concluded that the EO chain probably behaves as part of the headgroup. As with Aalbers [49], a low surface charge of the sodium alkyl ether carboxylate micelles was mentioned. The micelle aggregation number N increases with the C chain much more than for the corresponding nonionic surfactants. In the case of C8 there was no influence of temperature. A small decrease was found with increasing EO, but much smaller than in the case of nonionics. 

Based on studies on the influence of NaCl and pentanol on N for n = 12, x = 7 and n = 12, x = 9 it has been found that NaCl increases and pentanol decreases the micelle aggregation number N. Qualitatively this is similar to the ionic surfactants however, changes are smaller in the case of the classical ionic surfactants. 

Highly monodisperse reversed micelles are formed by sodium bis(2-ethylhexyl) sul-fosuccinate (AOT) dissolved in hydrocarbons that are in equilibrium with monomers whose concentration (cmc) is 4 X 10 M, have a mean aggregation number of about 23, a radius of 15 A, exchange monomers with the bulk in a time scale of 10 s, and dissolve completely in a time scale of 10 s [1,2,4,14], Other very interesting surfactants able to form reversed micelles in a variety of apolar solvents have been derived from this salt by simple replacing the sodium counterion with many other cations [15,16],... 

From electrophoretic data for SLS micelles under various ionic conditions (22 ), values of 80 for the aggregation number and 23 for the effective charge of the kinetic micelles can be used. This gives the following formula for the eluant total ionic strength. 

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