Micelle volume fraction, calculation

Fig. 3.19 Critical micelle temperature versus the polymer concentration calculated for an aqueous solution of Pluronic P105 (PEO37PPO56PEO37) (Linse 1994b). Results from a polydisperse model (MJMn = 1.2) are shown as solid lines and for the monodisperse polymer as a dashed line. The curves for the polydisperse system are labelled in terms of the number of components representing the polydisperse polymer. Points with constant micellar volume fractions (criterion of the cmc) are represented by dotted curves, the volume fraction being indicated. Experimental data from Alexandridis et al. (1994a) are also included as filled squares.

If one considers solely the consecutive equilibria, the concentration of monomer can only increase with increasing total amphiphile concentration even above the CMC. (Apart from the trivial decrease in the monomer concentration calculated on the total volume which may arise when the micelles occupy a substantial volume fraction). However, if one realizes that micelles are not only composed of amphiphile, the result may be different. Thus counterion binding helps to stabilize the micelles and for ionic surfactants it can be predicted that the monomer activity may decrease with increasing surfactant concentration above the CMC. Good evidence for a decreasing monomer concentration above the CMC has been provided in the kinetic investigations of Aniansson et al.104), and recently Cutler et al.46) demonstrated, from amphiphile specific electrode studies, that the activity of dodecylsulfate ions decreases quite appreciably above the CMC for sodium dodecylsulfate solutions (Fig. 2.14). 

Fig. 15. Volume fraction distribution calculated, assuming homogeneous spheres of constant voluminosity, from electron micrographs of casein micelles from three cows (Holt et at., 1978a, and unpublished work by the same authors).

Calculations of the small-angle x-ray scattering expected from a disordered array of reverse micelles (whose dimensions can be accurately determined for this system since the interfacial area and volume fractions are well known) differ markedly from measured scattering spectra, except in the most water-rich microemulsion mixtures. Only at the highest water contents which form microemulsions alone, are conductivity and X-ray spectra consistent with water-filled reverse micelles embedded within an oil continuum. 

Another possible extension is to consider an excess oil phase which is a mixtnre of two or more species. Provided that mixing within the micelle can still be considered ideal and that activity coefficients for all species in the bulk oil mixture are known, an expression for for each solnte is readily obtained. Micelles formed from surfactant mixtures can be treated provided that micelle composition is known or can be calculated from theories of mixed micelles such as regular solution theory and that solubilization is low enough not to affect micelle shape or composition. Finally, nonideal mixing in the micelles can be included if some model for the nonideality is available as well as data for evaluating the relevant parameters. Perhaps the simplest scheme for incorporating nonideality with nonpolar solutes is to use volume fractions instead of mole fractions in the spirit of Flory-Huggins theory. 

Consider an ensemble of disk-like micelles composed of a single surfactant species. By analogy with the calculation of cylindrical micelles, find the probability distribution for finding a disk of a given size as a function of the surfactant volume fraction. Contrast the probability distribution for disks with that of cylindrical micelles (where a broad distribution of sizes exists) and comment on the reason for the difference. 

Here, is the hard-sphere volume fraction. We use (p (volume fraction of copolymer) instead of for the calculation of Do because the exact volume of micelles is unknown. The concentration of Sii4C3EO is = 2 vol% so that the calculated values. Do, differs from D by -5% and this difference is noted as an error bar. 

The swelling of micelles by an added homopolymer has also been studied, to a modest degree, by simulations, and one set of results is shown in Figure 17. It summarizes the outcomes of 11 different sets of extensive calculations done for different copolymers, homopolymers, and volume fractions. In these simulations, autocorrelation times for both the copolymers and the homopolymers were calculated and monitored. In some cases, the homopolymer was solubilized within the cores, and in others it formed a separate, large aggregate containing nearly all the homopolymers. This phase behavior for these 11 systems is summarized in Figure 17. It is consistent with the predictions of... 

FIGURE 4.36 Plot of disjoining pressure, II, vs. film thickness, h comparison of experimental data for a foam film from Ref. [461] (thin-film pressure balance) with the theoretical curve (the solid line) calculated by means of Equation 4.230. The film is formed from 200 mM aqueous solution of the nonionic surfactant Tween 20. The volume fraction of the micelles (cj) = 0.334) is determined from the film contact angle the micelle diameter id = 1.2 nm) is determined by dynamic light scattering. The points on the horizontal axis denote the respective values of h for the stratification steps measured by a thin-film pressure balance. 

To understand the causes at the basis of the remarkable differences observed in the NaAOTAV/bmimBF and the NaAOT/W/bmimBr phase diagrams, the bmim adsorption isotherms in both systems were compared. For a suitable description of the results, the effective volume fraction of the interface ( etE- Avolume fraction (3> q.j.) and the bmim+ volume fraction ( bmmi) respectively, reported in ordinate and in abscissa in Hgure 1.11.The volume fraction of the micellized bmim+ () was calculated under the assumption that the self-diffusion coefficient of the micellized bmim is much lower than that of free bmim+ in aqueous solution. When such a condition holds. Equation 1.3 simplifies in... 

In order to determine the aggregation number n and the volume Pin, we performed a contrast-variation experiment at constant dC(6)PC concentration (50 mM). The contrast, = ( m/Kn) — Ps> was changed by various D2O/H2O mixtures. Figure 4 presents the Guinier plots for dC(6)PC micelles in the presence of 2 M sorbitol with various contrasts. Equation (1) also indicates that and n can be determined from the position of the zero intensity and the slope of the plot of 7(0) as a function of contrasts, respectively. Figure 5 is a plot of the square root of 7(0) vs. the volume fraction of H20.7(0) vanishes in a solvent containing 16.1 vol% D2O. This corresponds to = 6.93 X 10 A . From this value and the slope of the straight line in Fig. 5, we found n = 32 and Fm = 640 A. In this calculation, the value of the micellar concentration estimated from the above measurement was also taken into consideration. 

As an illustration, let us consider experimental data [374] for micellar solutions of the ionic surfactant sodium nonylphenol polyoxyethylene-25 sulfate (SNP-25S) at concentration 3.35 X 10 M. The stepwise thinning of a film formed from such a micellar solution is shown in Fig. 25. Comparative experiments with 0.1 Af added NaCl and without any added NaCl have been performed. From the experimental number concentration of the micelles, n, their effective diameter and volume fraction, tp, have been calculated by means of Eqs. (223) and (226). 

This formula could be extended to ionic micelles or polymers as in the depletion case. For that purpose, the micellar (or pol)mier molecular) diameter d should be replaced by do = d + 2/k and the film thickness h,hy h- 2b. The pressure Po is calculated as explained above (see eqn. 10.18). Eqn. (10.23) is valid for low volume fractions of the small species (or effective volume fraction, when charge effects are present) not exceeding 0.1, and takes into account both oscillatory and depletion interactions. At higher concentrations one may use more elaborate expressions. ... 

The addition of smaller colloidal species (polymer molecules, micelles, microemulsions) to an emulsion may lead to destabilisation of the system due to depletion attraction or steric stabilisation if these species adsorb at the droplet interface. A model calculation, illustrating the depletion destabilisation, is presented in Figure 10.6. All the parameters are as in Figure 10.2a but in this case nonionic micelles are also present [see eqns. (10.17)-(10.19)]. The micellar diameter is chosen to be 10 nm and the volume fraction equals 0.1. 

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