Thermodynamics of micelle formation

Micelle formation can be regarded as a process in which a monomer is transferred from aqueous phase to micellar phase. The process involves the reversible aggregation of monomers of amphiphile to form a micelle of aggregation number m [1,10,14]  

In these derivations, ideal solution behaviour has been assumed, as otherwise activities will have to be used. Under these conditions the chemical potentials are given as [1,9,10,11,15]  

From these relations the standard free energy of mi cel 11zatlon. aG°, per monomer is found to be given as [101  

If the aggregation number, m, is assumed to be independent of temperature, then the enthalpy of micelle formation, aH, can be estimated by using the Clausius-Clapeyron relationship  

The entropy of micelle formation, aS° (= aH° obtained from equations [8] and [9]. 

The thermodynamic formulation of micellization is based on the assumption that micelles exist in equilibrium with micelle-forming surfactant monomers as expressed by Equation 1.4 for nonionic surfactants.  

Values of Fitting Parameters, CMCg, and 6 Calculated from Equation 1.2 for Different Surfactants in the Presence of /i-Bu4NBr and Urea 

Similar arguments can lead to Equation 1.8 for ionic univalent surfactants where P is the fraction of charges 

Based on the assumption used in the derivation of Equation 1.7 and Equation 1.8, Zana derived an equation. Equation 1.9, for the relationship between CMC of a surfactant in solution and its free energy of miceUization, AG°m. for a general type of ionic surfactant. In Equation 1.9, i and j represent respective number of charged groups of valency z and number of alkyl chains connected by some spacer 

Although Equation 1.7 and Equation 1.8 have very often been used for the calculation of free energy of miceUization of nonionic and ionic univalent surfactants, respectively, there is a so-called mathematical paradox in the derivation of Equation 1.7 and Equation 1.8. Eor instance, lim N 0, 0 which means 

The mass-action model should be verified before we discuss micelle thermodynamics. Recent progress in electrochemical techniques makes it possible to measure monomeric concentrations of surfactant ions and counterions, and determination of the micellization constant has become possible. The first equality of (4.24) has three parameters to be determined— K , n, and m, which are the most important factors for the mass-action model of micelle formation. For monodisperse micelles, the following equations result from (4.13) and (4.14), respectively  

The value of m/n is evaluated from a slope of the relation obtained by plotting log [S] against log [G] above the CMC, because log[M ] is negligibly small compared to log K . The value of m/n is also obtained from Eq. (4.37) if [S] and [G] are available for each surfactant concentration. Equation (4.38) is rewritten with (4.35) and (4.36) as 

The three micellization parameters can be evaluated from three bulk concentrations of S at different surfactant concentrations. 

If reference data on SDS are used to evaluate m/n (Fig. 4.8), the three micellization parameters are log K = 230, n = 64, and m = 46.7 in units of molar concentration. On the other hand, the bulk concentrations [S] and [G] can be evaluated using (4.39) at a given surfactant concentration (Fig. 4.9). The above numerical values yield excellent agreement between calculated and observed monomer concentrations at higher surfactant concentrations. However, lower parameter values are necessary to give good 

A linear relationship between log CMC and log[G] has been observed in a number of experiments, indicating that the approximation made above is correct. 

Two main approaches to the thermodynamic analysis of the micellization process have gained wide acceptance. In the phase separation approach the micelles are considered to form a separate phase at the CMC, whilst in the mass-action approach micelles and unassociated monomers are considered to be in association-dissociation equilibrium. In both of these treatments the micellization phenomenon is described in terms o.f the classical system of thermodynamics. Theories of micelle formation based on statistical mechanics have also been proposed [16Q-162] but will not be considered further. The application of the mass-action and phase-separation models to both ionic and non-ionic micellar systems will be briefly outlined and their limitations discussed. More recent developments in this field will be presented. 

In this approach the micelle, and in the case of ionic micelles this includes the counterions, is treated as a separate phase. It is clear from previous sections that 

To calculate the thermodynamic parameters for the micellization process we require to define the standard states. The hypothetical standard state for the surfactant in the aqueous phase is taken to be the solvated monomer at unit mole fraction with the properties of the infinitely dilute solution. For the surfactant in the micellar state, the micellar state itself is considered to be the standard state. 

If and //m are the chemical potentials per mole of the unassociated surfactant in the aqueous phase and associated surfactant in the micellar phase, respectively, then since these two phases are in equilibrium 

If it is assumed that the concentration of free monomers is low then the activity of surfactant monomer, as, may be replaced by the mole fraction of monomers Xs and Equation 3.11 becomes 

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The thermodynamics of micelle formation has been studied extensively. There is for example a mass action model (Wennestrdm and Lindman, 1979) that assumes that micelles can be described by an aggregate Mm with a single aggregation number m, so that the only descriptive equation is mMi Mm. A more complex form assumes the multiple equilibrium model, allowing aggregates of different sizes to be in equilibrium with each other (Tanford, 1978 Wennestrdm and Lindman, 1979 Israelachvili, 1992). 

There are two basic approaches to modeling the thermodynamics of micelle formation. The mass action model views the micelles as reversible complexes of the monomer that are aggregating and predicts the sharp change in tendency of incremental surfactant to form micelles instead of monomer at the CMC this sharp transition is a consequence of the relatively large number of molecules forming the aggregate. The mass action model predicts that micelles are present below the CMC but at very low concentrations. The ocher major model used to describe micelle formation is the pseudophase separation model, which views micelles as a separate thermodynamic phase in equilibrium with monomer. Because micelle formation is a second-order phase transition, micelles are not a true thermodynamic phase, and this model is an approximation. However, the assumption that there are no mi-celies present below the CMC, and that the surfactant activity becomes constant above the CMC. is close to reality. and the mathematical simplicity of the pseudophase... 

Tanford, C. 1974. Thermodynamics of micelle formation Prediction of micelle size and size distribution. Proceedings of the National Academy of Sciences. 71, 1811. 

The thermodynamics of micelle formation can be analyzed with varying degrees of complexity. The simplest is to consider that a phase separation into... 

Chaterjee, A. Moulik, S.P. Sanyal, S.K Mishra, B.K. Puri, P.M. Thermodynamics of Micelle Formation of Ionic Surfactants A Critical Assessment for Sodium Dodecyl Sulfate, Cetyl Pyridinium Chloride and Dioctyl Sulfosuccinate (Na Salt) by Microcalorimetric, Conductometric, and Tensiometric Measurements. /. Phys. Chem. 

The enthalpy of micellization of many surfactants in aqueous solution has been determined in the past, using mostly cell type and flow microcalorimeters [6-8]. These determinations were based on measurements of the excess heat associated with dilution of a surfactant from a concentration above the cmc to a concentration below the cmc, which results in demicellization of the preexisting micelles. One diffleulty with these determinations relates to the dependence of the heat evolution (AQ) on the initial and final concentrations, probably due to secondary self-aggregations of the surfactants at high concentrations and/or pre-micellar dimer formation at low surfactant concentrations [6,9], These difficulties are at least partially responsible for the lack of consistent data on the thermodynamics of micelle formation [6]. 

Hence, determination of the cmc and its dependence on temperature are theoretically sufficient for complete characterization of the thermodynamics of micelle formation For any given temperature, AGjJuc can be calculated directly from the (temperature-dependent) cmc, A/f c (which is also temperature-dependent) can... 

S Paula, W Sus, J Tuchtenhagen, A Blume. Thermodynamics of micelle formation as a function of temperature A high sensitivity titration calorimetry study. J Phys Chem 99 11742-11751 (1995). 

Now that the mass-action model has been supported by a number of observations, we move to the thermodynamics of micelle formation based on this model. As would be predicted from the above discussion, micelle formation can be well expressed by a single association constant, even though the process strictly involves multiple association equilibria. The error is less than 5%, for example, for micelles having an aggregation number more than 50. For nonionic surfactants, the standard free energy change AG° per mole of surfactant molecules follows directly from the equilibrium constant and is given from (4.21) and [S] = Q by... 

It thus seems that the basic physics of the process of micellization is well understood, but one can hardly expect the theories to be terribly quantitative. Some properties, such as the dimensions of the micelle, are not overly sensitive to the details of the approximation scheme, but other properties, such as cmc, the aggregation number and the thermodynamics of micelle formation are much more volatile in their behaviour. The theories presented all assumed a monodisperse micelle distribution, but in fact one can use the methods to calculate the full distribution from equations (42) and (43), and indeed the distribution does turn out to be narrowly peaked. One can also use the theories to estimate the relative stabilities of spherical micelles vis-a-vis non-spherical micelles, infinite cylinders and bilayers, and preliminary studies indeed indicate the possibility of infinite cylinders at copolymer concentrations less than the cmc. The possible formation of these and other structures should be more thoroughly investigated. 

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