Micelle size, calculation

A Inverse micelle size calculated from the dielectric property... 

From the micelle size, the surface occupied by the hydrophilic part of one polyester molecule can be calculated, and thus the surface occupied by one carboxy group. This calculation is done assuming the density of the polyester in the micelle to be similar to bulk polyester and all the carboxy groups located on the micelle surface (conformation A and B given in Figure 2). 

To calculate micelle size and diffusion coefficient, the viscosity and refractive index of the continuous phase must be known (equations 2 to 4). It was assumed that the fluid viscosity and refractive index were equal to those of the pure fluid (xenon or alkane) at the same temperature and pressure. We believe this approximation is valid since most of the dissolved AOT is associated with the micelles, thus the monomeric AOT concentration in the continuous phase is very small. The density of supercritical ethane at various pressures was obtained from interpolated values (2B.). Refractive indices were calculated from density values for ethane, propane and pentane using a semi-empirical Lorentz-Lorenz type relationship (25.) Viscosities of propane and ethane were calculated from the fluid density via an empirical relationship (30). Supercritical xenon densities were interpolated from tabulated values (21.) The Lorentz-Lorenz function (22) was used to calculate the xenon refractive indices. Viscosities of supercritical xenon (22)r liquid pentane, heptane, decane (21) r hexane and octane (22.) were obtained from previously determined values. 

A, B, C, D depend on the extracted protein and are functions of AGo, oc, e, A i, n, pi, Na, z. Their numerical value have been calculated from experimental data on solubilization of ribonuclease and concanavalin A in AOT/isooctane with a good correlation to the model equation. The great interest of this model is that all the assumptions necessary for its elaboration make it very simple, and at the same time, a promising tool of quantification of protein solubilization thermodynamics, even if some further refinements are still needed. It can be noted that there are more parameters than can be adjusted from experimental data. As a consequence, the model can provide no value for n, related to the micelle size, which could have permitted an interesting comparison with that predicted by Caselli et al. s model. 

The problem of calculating the distribution of micelle sizes reduces to that of establishing the dependence of ACe(i) on i. Since there is good evidence that the equilibrium mixture at and above the c.m.c. contains only a low concentration of species other than those of a size close to the mean micelle size, we must seek a form of AG°(i) which leads to this result. Some early theories assumed that the standard free energy is a linear function of i. This would mean Lhal AG (i, i + 1), which is equal to AG"(i + 1) — AGe(i), is constant. However, wc saw in Chapter 9 that, if this is so, above a certain concentration aggregates will grow to macroscopic size, contrary to the limiting sizes associated with miccllisation. Other theories have set AG (i) equal to zero for all values of i except lor a particular value of i at the c.m.c. This implies that miccllisation occurs by the. simultaneous association of i monomer molecules, which is physically unrealistic. 

Furthermore, the most probable micelle size T is calculated from the maximum of the distribution function by taking logarithms of both sides of Eq. 10, rearranging, and differentiating ... 

At low surfactant concentration, is a monotonically decreasing function of s. Above the cnK the distribution function exhibits a maximum corresponding to the most probable micelle size J. Further information, including expressions for calculating 1 for rod- and disclike micelles and for transitions between the various micellar shapes, may be taken from Rusanov s article. 

Calculations of micelle sizes and aggregation numbers have been made from static and time-resolved quenching experiments in which deviations from the behavior in homogeneous solutions due to compartmentalization of the reactants in micelles (cage effect) were quantitatively exploited (cf. Section IV). It is assumed that the number of quenchers per micelle follows Poisson statistics, and it must be guaranteed that the time which donors and acceptors spend in a micelle is long compared to fluorescence lifetimes of donors and that quenching by the acceptors is the main deactivation process of the donors. 

The apparent hydrodynamic diameters of the droplets (or the correlation length), as calculated using the Stokes-Einstein equation for a number of different systems, are given in Table 2. These early findings showed that the micelle sizes measured in near-critical and supercritical solutions were similar to those found for conventional water-in-oil microemulsions in liquid alkane. At lower fluid densities, DLS probes the combined effect of the collective diffusion coefficient of the micelle cluster and that of the individual micelles. 

For those systems near a phase transition, the apparent hydrodynamic diameter of the droplets (or the correlation length), as calculated using the Stokes-Einstein equation, appears to decrease as pressure increases [2,4,39]. For example, the apparent hydrodynamic diameter of a microemulsion droplet (for [surfactant] = 150 mM and 5) in supercritical xenon [2] decreases from 6.5 to 4.5 nm as pressure is increased from 350 to 550 bar (10 bar = 1 MPa). This effect is due to the change in the extent of micelle clustering rather than an actual change in the micelle size. 

At R = 10 the acoustic method gave a slightly larger diameter than expected. This could be as a result of the constrained state of the bound water in the swollen reverse micelles. The water under these conditions may exhibit different thermal properties from those of the bulk water used in ttie particle size calculations. Also, at the low R values R< 10 or < 2.4% water), the attenuation spectrum is not very large as compared to the background heptane signal. 

It is not difficult to carry out the statistical calculations for these two degrees of freedom (size and flexibility), since they are largely independent of one another. The energy AE endcap does not depend on micelle length, nor on the various bending configurations the micelle may explore. Statistical analysis shows that mean micelle size (L) varies in a continuous way [5.14], but very rapidly with AE endcap- In fact,... 

The surface area, average particle size, marber of polymeri -zation sites and number of soap micelles were calculated for each time interval. 

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. 

Increasing temperature decreases the solubilization capacity of non-ionics even though the micelles grow in size [160], but if the molar ratio of steroid to surfactant (and not micelle) is calculated this value increases for steroids in polysorbate 40 and tetradecylammonium bromide (TDABr) [165] (see Table 6.21). This topic is further discussed in Section 5.3.3. 

Above the CMC of each surfactant, linear enhancements in HCB solubility were observed, similar to trends reported for HOCs in micellar solutions (7,5,75). The corresponding WSR values for Tween 60, Tween 80 and Triton X-100, calculated using Equation 1, were 0.59 g/kg, 0.63 g/kg and 0.35 g/kg, respectively. The lower HCB solubilization capacity of TritonX-100 is consistent with solubility correlations developed by PenneU et al. (5) for a range of surfactants and HOCs. This behavior is attributed to the greater alkyl chain length, and hence larger micelle size of Tween 60 and Tween 80 relative to that of Triton X-100. 

Coexistence of micelles of different size at certain surfactant chemical potential is always expected. The size distribution can be derived from the SCF calculations. The central quantity is the excess Helmholtz energy of the micelle, F [5,18]. At fixed values of the chemical potentials ( t ), T and p, the excess Helmholtz energy has a minimum at the most likely micelle size. Very close to this minimum, and in first approximation, the excess Helmholtz energy is found to be a quadratic function of the micellar size ... 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

In order to test further the applicability of 1-pyrene carboxaldehyde as a fluorescent probe, we applied Keh and Valeur s method (4) to determine average micellar sizes of sulfonate A and B micelles. This method is based on the assumption that the motion of a probe molecule is coupled to that of the micelle, and that the micellar hydrodynamic volumes are the same in two apolar solvents of different viscosities. For our purposes, time averaged anisotropies of these systems were measured in two n-alkanes hexane and nonane. The fluorescence lifetime of 1-pyrene carboxaldehyde with the two sulfonates in both these solvents was found to be approximately 5 ns. The micellar sizes (diameter) calculated for sulfonates A and B were 53 5A and 82 lOA, respectively. Since these micelles possesed solid polar cores, they were probably more tightly bound than typical inverted micelles such as those of aerosol OT. Hence, it was expected that the probe molecules would not perturb the micelles to an extent which would substantially affect the micellar sizes measured. 

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