Micelles translational entropy

We first assume that the aggregates formed by the copolymer chains have a bimodal distribution single chains and monodisperse spherical micelles. In this simplified model, the critical micelle concentration is a sharp transition if we neglect micelles translational entropy. We then discuss the micelle size distribution. 

Usually the discussion of the ODT of highly asymmetric block copolymers in the strong segregation limit starts from a body-centred cubic (bcc) array of the minority phase. Phase transitions were calculated using SOFT accounting for both the translational entropy of the micelles in a disordered micelle regime and the intermicelle free energy [129]. Results indicate that the ODT occurs between ordered bcc spheres and disordered micelles. 

Althogh, AG evaluated by Equation 9 takes into account the loss in translational entropy of counter ions upon micellar assoclatlon(3,4), it is doutfull that the term (m/n) RT ln[X], can Include all the effects of interionic interaction in micelle formation. 

In spite of the fact that the concentration of surfactants in the outer solution is assumed to be smaller than the critical micelle concentration, inside the network, micelles are supposed to be formed. The reason for this assumption is, first of all, intensive adsorption of surfactants on the network as a result of the ion exchange reaction. Moreover, in Refs. [38, 39], it was shown that critical concentration of micelles formation c c" within a polyelectrolyte network is much less than that in the solution of surfactant c° . Indeed, when a micelle is formed in solution immobilization of counter ions of surfactant molecules takes place, because these counter ions tend to neutralize the charge of micelles (see Fig. 13), whereas there is no immobilization of counter ions when the micelles are formed in the network the charge of micelles is neutralized by initially immobilized network charges which do not contribute to the translational entropy (Fig. 13). 

The total free energy of the micellar phase contains contributions from the translational entropy of the micelles, the entropy of mixing of homopolymers and copolymers and their interaction outside the micelles. It can be written as (Leibler et al. 1983)... 

The ability of the surfactant to increase its translational entropy lowers the surface tension. However, we note that this expression is only valid for insoluble surfactants at low concentrations, so the tendency for y to become negative at large a is just an indication that these approximations are breaking down. For soluble surfactants, one cannot consider fixed Ng. Rather, one has to equate the chemical potentials of the surfactants on the surface and in the bulk. For small concentrations, the surface tension is still reduced in a linear manner, but at large concentrations, the reduction in surface tension saturates due to the formation of micelles in the bulk this is discussed in Chapter 8. 

A frequently used simplifying approximation is based on neglecting the translational entropy of micelles (i.e., the second term in (1) is omitted). This approximation is justified as long as the aggregation number in an equilibrium micelle is large. Then, (2)-(4) reduce to ... 

A comparison of (10) and (14) indicates that because of the stability condition, (8), the exact CMC [defined by (10)] is smaller than the CMC obtained from the approximate analytical model, (14). Accounting for the translational entropy of micelles leads, therefore, to a lower value of the aggregation number, Peq(c) < po, and a lower CMC. 

The free energy, (1), complemented by the contribution due to counterions, cf. (15), can be minimized with respect to Cmic nd P- When the translational entropy of micelles is neglected, the optimal aggregation number, p = po, is still given by (11). The concentration of unimers that coexist with micelles, and thus the CMC, is, however, significantly larger than for neutral (uncharged) amphiphiles ... 

As discussed in Sect. 2.1, the origin of this increase in the CMC is the translational entropy penalty for the localization of counterions in the coronaupon the association of block copolymers into micelles. 

In the low salt limit, atCp > the coronal contribution to the free energy is dominated by the translational entropy of counterions entrapped inside the corona, int = k TabNAi riCp - 1). In this case, all results of the blob model are recovered both for osmotic starlike and crew-cut spherical micelles (59), (61), and (62). 

Under the so-called salt dominance conditions, the association of block copolymers into micelles does not lead to significant losses in the translational entropy of counterions (whose concentrations inside the corona and in the bulk of the solution are approximately equal). Therefore, within the accuracy of the main term, the CMC is controlled by the hydrophobicity of the block B ... 

Translational entropy of micelles Dead time of mixing Flory exponent Volume of component i Sample volume in cm ... 

In order to calculate the free energy of micellization accurately, one needs to take into account the enthalpic terms describing the interactions between the block copolymer and solvent, the free energy of the micelle (Fniiceiie), the mixing free energy (Fniix), as well as the translational entropy associated with micelles and free chains (5ni). 

The translational entropy associated with the micelles and the unaggregated block copolymer chains can be written using the Hory-Huggins theory as ... 

Here, the first term is the free energy of the micelles (Ff is the free energy per amphiphilic molecule in a micelle comprising/ molecules), the second term is the translational entropy of... 

Strayley extended the ideas of Onsager to chiral particles (not necessarily micelles) [43]. Onsager had shown that a disordered arrangement of aniso-metric particles costs translational entropy with increasing concentration of the particles. The increase of translational entropy can be avoided by the introduction of orientational order. Particles having a chiral shape consume less space by packing in a twisted manner predetermined by their shape, see Figure 14.18(a). 

In conclusion, solvation changes and loss of some of the translational entropy in forming a transition state are two important factors responsible for catalysis and rate enhancements observed with micellar systems. In this respect they resemble enzymes. Another formal similarity between enzymatic and micellar catalysis is the strong hydrophobic binding with the substrate. However, the fact that micelles are of limited rigidity results in poor specificity in catalysis and only moderate rate enhancements are obtained. 

The micellization is associated with a loss of translational entropy. This entropy favors the dissociated state and smaller aggregation numbers. The transfer free energy of the hydrophobic tails from water into the hydrophobic cores, -5kT, is the driving force for the micellization. The micellization process in-... 

---