Radius of spherical micelle

Using the definition of the aggregation number (Equation 5.3), we can derive the following relationship between the surfactant volume and area with the radius of spherical micelles ... 

Bending elasticity moduli can be measured, or at least estimated experimentally. The spontaneous curvature is close to the reciprocal radius of spherical micelles in equilibrium with solubilizate, and therefore is rather straightforward to measure. The bending modulus is more difficult to measure experimentally.Both k and k are usually presented in units of thermal energy, ksT. Typical values of the bending modulus vary from 0.1 ( flexible monolayers) to 10 k T ( rigid monolayers). There is still very little data concerning the experimental value of the saddle splay... 

The different location of polar and amphiphilic molecules within water-containing reversed micelles is depicted in Figure 6. Polar solutes, by increasing the micellar core matter of spherical micelles, induce an increase in the micellar radius, while amphiphilic molecules, being preferentially solubihzed in the water/surfactant interface and consequently increasing the interfacial surface, lead to a decrease in the miceUar radius [49,136,137], These effects can easily be embodied in Eqs. (3) and (4), aUowing a quantitative evaluation of the mean micellar radius and number density of reversed miceUes in the presence of polar and amphiphilic solubilizates. Moreover it must be pointed out that, as a function of the specific distribution law of the solubihzate molecules and on a time scale shorter than that of the material exchange process, the system appears polydisperse and composed of empty and differently occupied reversed miceUes [136],... 

Size analyses of (using light scatter) some spherical micelles of SDS have indeed shown that the radius of the micelle is almost the same as the length of the SDS molecule. However, if the solute interferes with the outer polar part of the micelle, then the micelle system may change in such a way that the CMC and other properties change. This is observed in the case of the addition of dodecanol to SDS solutions. However, very small additions of solutes show very little effect on CMC. The data in Figure 3.18 show the change in the solubility of naphthalene in SDS aqueous solutions. 

Important parameters that control the size of micelles are the degree of polymerization of the polymer blocks, NA and NB, and the Flory-Huggins interaction parameter %. The micellar structure is characterized by the core radius Rc, the overall radius Rm, and the distance b between adjacent blocks at the core/shell-interface as shown in Fig. 1. b is often called grafting distance for comparisons to polymer brush models, b2 is the area per chain which compares to the area per head group in case of surfactant micelles. In the case of spherical micelles, the core radius Rc and the area per chain b2 are directly related to the number of polymers per micelles, i.e., the aggregation number Z=4nR2clb2. 

The data from the SANS study were fitted to a poly-disperse spherical core-plus-shell (or corona) model assuming no intermicellar interactions. The model Sharma and Bhatia used essentially fits three parameters, namely the radius of the micelle core, / i, the radius of the micelle shell, R2 and the micelle aggregation number. To fit the SANS data to the model, the authors assumed that the micellar cores were comprised only of hydrophobe (PPO) and D2O, while the shell was assumed to comprise only of hydrophile (PEO) and D2O, with no intermixing of the PPO and PEO chains. 

The radius of the spherical micelle cannot exceed a critical length k, roughly equal to but less than the fully extended length of the hydrocarbon chain. Therefore, once v/aok > 1/3, spherical micelles will not be able to form unless a >ao. This yields a critical condition for the formation of spherical micelles... 

Figure 4.11 Schematic representation of spherical reverse micellar structure formed in discontinuous cubic (b) phase. The radius of the micelle is r, the radius of the hydrophilic...

The micelle diameter can be calculated by measuring the micelle diffusion coefficient using the technique of dynamic light scattering (DLS). If one assumes that aU micelles are spherical in shape, the radius of a micelle in solution may be calculated by using the Stokes-Einstein relation ... 

The fully stretched length of the tail is 1.5415 nm. The radius of the micelle is R = 0.4258// nm, but it is required that i 7 or 3.621 or TV 58 at 25°C, which is less than the reported value of 62 and the similar value foimd in Example 4.1. Hence relatively short cylindrical micelles are expected for SDS. For cetyl pyridium chloride, = 16, 4 = 2.68 nm, and N 249. Its aggregation number is 138, and such micelles are correctly predicted to be spherical. For sodium octyl sulfate, N < 21.5. Actual values lie in the range of 24 to 31 and experimental measurements indicate that it forms spherical micelles and not cylindrical ones, showing that the simple geometric criterion employed is not always adequate. 

Under these considerations, the analysis of the energetics of size and shape of the micelles becomes of interest. The spherical shape would be the most stable structure if the monomers aggregate with a minimum of other constraints needed to satisfy the forces as described under Chap. 2.3, because this gives the minimum surface area of contact between the micelle and the solvent. On the other hand, if large constraints exist, other possible shapes, e.g. ellipsoids, cylinders or bilayers would be present [1,4]. It is obvious that micelles as formed by non-linear surfactants, e.g. bile salts etc., can not be analyzed by these theories, because steric hinderance gives rise to rather small aggregation numbers [1,3,4, 12,32,33,34,35,36,37,38,39,40]. In the case of spherical micelles of linear alkyl chain surfactants, with aggregation numberm, the radius, R, and total volume, V, and micellar surface area, A, we have ... 

Figure 15.5. Normalized (with respect to the self-diffusion of spherical micelles) self-diffusion coefficients of prolate-and oblate-shaped micelles as a function of the axial ratio r (keeping the short axis constant and equal to the radius of the sphere). Plots obtained according to equations (15.5) and (15.6) and by using the constant-area-to-enclosed-volume constraint...

On the assumption of spherical micelles, their radius could be obtained from the long spacing using ... 

A decrease of water surface tension is observed up to a certain concentration, called the critical micelle concentration (CMC). Above the CMC, surfactants create new liquid-like often near spherical stmctures, called micelles (Figure 5.2). Micelles are small semi-spherical agglomerates, all about the same size, where (in ordinary micelles) the non-polar tails are inside and the polar heads are outside. The radius of the micelles is of the order of the length of the surfactant molecules a few nanometres. 

As in the general case the radius of the micelle (R) is equal to or less than the critical length of the surfactant, then the condition for a spherical micelle is that CPP has to be equal or less than 1. The value of CPP = V3 is obtained exactly when R = Ic-... 

As the kinetics simply scale with the diffusion coefficient the choice of this parameter is unimportant. Hence, all reaction/survival yields are presented in the form D t, which factors out the effect of the diffusion coefficient. The choice of the encounter radius used to model reactions inside the micelle (reported alongside each result) is based on the data given in Ref. [10]. The radius of the micelle (reported alongside each result) was chosen based on the work by Bruce and co-workers [11]. For a spherical micelle the maximum radius R must be the same as the maximum extended length of the hydrophobic chain Ic. Hence, for a micelle to form a spherical structure the following condition must be satisfied... 

FIG. 4 Onion model of spherical water-containing reversed micelles. Solvent molecules are not represented. A, surfactant alkyl chain domain B, head group plus hydration water domain C, hulk water domain. (For water-containing AOT-reversed micelles, the approximate thickness of layer A is 1.5 nm, of layer B is 0.4 nm, whereas the radius of C is given hy the equation r = 0.17R nm.)... 

With the development of new instrumental techniques, much new information on the size and shape of aqueous micelles has become available. The inceptive description of the micelle as a spherical agglomerate of 20-100 monomers, 12-30 in radius (JJ, with a liquid hydrocarbon interior, has been considerably refined in recent years by spectroscopic (e.g. nmr, fluorescence decay, quasielastic light-scattering), hydrodynamic (e.g. viscometry, centrifugation) and classical light-scattering and osmometry studies. From these investigations have developed plausible descriptions of the thermodynamic and kinetic states of micellar micro-environments, as well as an appreciation of the plurality of micelle size and shape. 

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