Free energy micellization, single

The mixed cmc behavior of these (and many other) mixed surfactant systems can be adequately described by a nonideal mixed micelle model based on the psuedo-phase separation approach and a regular solution approximation with a single net interaction parameter B. However, the heats of micellar mixing measured by calorimetry show that the assumptions of the regular solution approximation do not hold for the systems investigated in this paper. This suggests that in these cases the net interaction parameter in the nonideal mixed micelle model should be interpreted as an excess free energy parameter. 

The simplest way in which a process occurs by itself is when it is under thermodynamic control. The folding of a protein, or the self-assembly of micelles at the critical micelle concentration (cmc) are examples of spontaneous processes the latter are characterized by a negative free-energy change, as the self-orgaiuzed product has a lower energy than the single components. ... 

Although there are some aspects of micellization that we have not taken into account in this analysis —the fact that n actually has a distribution of values rather than a single value, for example —the above discussion shows that CMC values expressed as mole fractions provide an experimentally accessible way to determine the free energy change accompanying the aggregation of surfactant molecules in water. For computational purposes, remember Equation (3.24), which states that x2 n2/n, for dilute solutions. This means that CMC values expressed in molarity units, [CMC], can be converted to mole fractions by dividing [CMC] by the molar concentration of the solvent, [solvent] x2 [CMC]/[solvent] for water, [solvent] = 55.5 mole liter... 

Analogous equations to (3.20-3.26) can be written for triblock copolymer micelles in a homopolymeric solvent (Balsara et at. 1991 ten Brinke and Hadziioannou 1987). However, in a BAB triblock copolymer where the solvent is selective for the A block, for a single micelle the A block must be looped. Then each chain enters the core twice, and eqn 3.21 must be multiplied by two, with a similar multiplier of the analogous term in eqn 3.22. An additional contribution must be added to the free energy of the corona due to looping. Balsara et at. (1991) estimated this to be... 

Block or graft copolymers in a selective solvent can form structures due to their amphiphilic nature. Above the critical micelle concentration (CMC), the free energy of the system is lower if the block copolymers associate into micelles rather than remain dispersed as single chains. Often the micelles are spherical, with a compact core of insoluble polymer chains surrounded by a corona of soluble chains (blocks) [56]. Addition of a solvent compatible with the insoluble blocks (chains) and immiscible with the continuous phase leads to the formation of swollen micelles or polymeric micro emulsion. The presence of insoluble polymer can be responsible for anomalous micelles. 

The interpretation of rate (or product) data in terms of probe location in a micelle is similar conceptually to the interpretation of rate (or product) data in terms of conformational shapes. In effect, the various positions of a probe in a micelle can be viewed as conformations of the probe micelle associate. In this regard, the Curtin-Hammett principle9a,b) states that if two or more conformations are in rapid equilibrium relative to reaction from either shape, and in which each form gives a characteristic product (or if only one form yields a product), the ratio of products (or the rate of formation of a single product) is independent of the energy difference between the conformers and depends only on the relative free energy levels of the transition statutes. 

A similar approach to mixed micelles was also proposed by Nagarajan [18, 56], where most of the contributions to the standard free energy difference between the single dispersed and the aggregated states were expressed as linear functions of individual contributions. For example, the transfer free energy contribution was presented as... 

Combining (84) and (74), one finds a closed expression for the free energy of a spherical micelle with a quenched PE corona as a function of a single structural... 

Fig. 4 Illustration of the chain expulsion process of a single chain from a star-like micelle with core radius and corona thickness D and a corresponding schematic free energy profile, F(y), along the reaction coordinate. In the calculatimis given in the text the reference state is chosen according to F(P + 1) = 0 so thatF = for the expulsion...

An essentially equivalent approach to that of small-systems thermodynamics has been formulated by Corkill and co-workers and applied to systems of nonionic surfactants [94,176]. As with the small-systems approach, this multiple-equilibrium model considers equilibria between all micellar species present in solution rather than a single micellar species, as was considered by the mass-action theory. The intrinsic properties of the individual micellar species are then removed from the relationships by a suitable averaging procedure. The standard free energy and enthalpy of micellization are given by equations of similar form to Equations 3.44 and 3.45 and are shown to approximate satisfactorily to the appropriate mass-action equations for systems in which the mean aggregation number exceeds 20. 

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