Theories of micelle formation

Two general approaches have been employed in attempting to describe the process of 

Equation (6.20) represents the formation of a cationic micelle from N surfactant ions D+ and (N-p) firmly held counterions X. Whenever the thermodynamics of a process is under consideration, it is important to define the standard states of the species. In this example, the standard states are such that the mole fractions of the ionic species are unity and the solution properties are those of the infinitely dilute solutions. The equilibrium constant may be written in the usual way 

---

Nagarajan R. Theory of micelle formation quantitative approach to predicting micellar properties from surfactant molecular structure. Surface Sci Ser 1997 70 1-81. 

Tanford, C. (1974),Theory of micelle formation in aqueous solutions,/. Phys. Chem., 78, 24. 

Critical micelle concentration ( >cmc is expected to decrease strongly with diminished diblock asymmetry rc as low rc values favor easier creation of highly curved micelle interfaces. Theory of micelle formation [231,260] also indicates that the overall copolymer degree of polymerization Nc, as well as the anchor -homopolymer interaction parameter %AP have to be considered to explain properly the onset of micelle segregation as observed by Shull et al. [260]. Using this theory, experimenters are able to choose systems where only individual copolymers segregate. 

It is usually observed that the critical micelle concentration for a surfactant is relatively sharp and characteristic. Although the detailed theory of micelle formation can become quite complex, the sharpness of the cmc can be explained conceptually in terms of the law of mass action. If Q denotes the total concentration of surfactant in solution, Q the fraction of surfactant present as free molecules, and Cm that in the aggregated state. Equation (15.3) may be written... 

M.D. Whitmore, J. Noolandi, Theory of micelle formation in block copolymer-homopolymer blends, Macromolecules 18 (1985) 657-665. 

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 section we consider the formation of micelles when AB diblock copolymers are placed in a solvent that is good for A but bad for B. Many of the theories specialize to the case when the solvent is the homopolymer A. First of all we review some basic theory of micelle formation. For a given number of copolymer chains in the solvent, there will exist a chemical equilibrium between isolated chains and all clusters of two or more chains. In dilute solution, where interactions between micelles and single copolymer chains may be neglected, one obtains the result... 

As an even more explicit example of this effect Figure 6 shows that EPM is able to reproduce fairly well the experimentally observed dependence of the particle number on surfactant concentration for a different monomer, namely methyl methacrylate (MMA). The polymerization was carried at 80°C at a fixed concentration of ammonium persulfate initiator (0.00635 mol dm 3). Because methyl methacrylate is much more water soluble than styrene, the drop off in particle number is not as steep around the critical micelle concentration (22.) In this instance the experimental data do show a leveling off of the particle number at high and low surfactant concentrations as expected from the theory of particle formation by coagulative nucleation of precursor particles formed by homogeneous nucleation, which has been incorporated into EPM. 

Scamehorn et. al. (20) also presented a simple, semi—empirical method based on ideal solution theory and the concept of reduced adsorption isotherms to predict the mixed adsorption isotherm and admicellar composition from the pure component isotherms. In this work, we present a more general theory, based only on ideal solution theory, and present detailed mixed system data for a binary mixed surfactant system (two members of a homologous series) and use it to test this model. The thermodynamics of admicelle formation is also compared to that of micelle formation for this same system. 

A simple mean field theory for micelle formation by a diblock copolymer in a homopolymeric solvent was developed by Leibler et al. (1983). This model enables the calculation of the size and number of chains in a micelle and its free energy of formation. The fraction of copolymer chains aggregating into micelles can also be obtained. A cmc was found for low copolymer contents even for weak incompatibilities between components. Leibler et al. (1983) emphasize that fora finite aggregation number p, the cmc is a region rather than a well-defined concentration and some arbitrariness is involved in its definition. 

Hurterr, P. N., J. M. H. M. Scheutjens, T. A. Hatton, and T. Alan. 1993. Molecular modeling of micelle formation and solubilization in block copolymer micelles. 1. Aself-consistent mean- eld lattice theory. Macromolecule26 5592-5601. 

A number of statistical thermodynamic theories for the domain formation in block and graft copolymers have been formulated on the basis of this idea. The pioneering work in this area was done by Meier (43). In his original work, however, he assumed that the boundary between the two phases is sharp. Leary and Williams (43,44) were the first to recognize that the interphase must be diffuse and has finite thickness. Kawai and co-workers (31) treated the problem from the point of view of micelle formation. As the solvent evaporates from a block copolymer solution, a critical micelle concentration is reached. At this point, the domains are formed and are assumed to undergo no further change with continued solvent evaporation. Minimum free energies for an AB-type block copolymer were computed this way. 

The last reported diblock copolymer family that formed tubular aggregates in block-selective solvents was poly(phenylquinoline)-fc/ock-polystyrene or PPQ-PS, where PPQ was a rigid-rod block [47]. Such tubes are not discussed further for the following reasons First, the tubes had diameters of several micrometers and were not nanotubes. Second, the formation mechanism and chain packing in such tubes were not well understood at all. While Halperin [48] has developed a scaling theory for micelle formation from rod-coil diblock copolymers with the rod block forming the core, the theory did not apply to the PPQ-PS system as the block-selective solvents used were good for the rod PPQ block rather than the coil PS blocL... 

Here, v is the hydrocarbon chain volume, a<> is the optimal head group surface area, and is the critical chain length. For packing parameters between and 1, amphiphiles should form flexible bilayers (vesicles), or planar bilayers as the packing parameter approaches 1. On the contrary, a packing paramter less than j should hinder the formation of bilayers in particular, the theory predicts micelle formation. 

The formation of ME systems can be explained with reference to the self-assembly theory of micelle and bilayer forming surfactant molecules where the volume of the surfactant is denoted v, its head group surface area a, and its... 

However, the proposal of narrow necks as proton connectors between inverted micelles has spurred debates and caused confusion. As there is no experimental evidence for the formation of such necks, their structure, ion content, and protonconducting ability have remained elusive. A modern version of the theory of neck formation, proposed by loselevich et al. (2004), addressed some of these issues. 

The molecular thermodynamic theory for micelle formation has heen worked out with increasing sophistication following the pioneering work of Israelachvili, Mitchell, and Ninham.26 The most comprehensive reports on micelle formation are those of Nagarajan and Ruckenstein and of Shiloah and Blankschtein. Many other theoretical approaches have been used in recent years to account for the formation of micelles and their properties thermod3mamics of small systems, the self-consistent field lattice model, the scaled particle theory, and Monte-Carlo and molecular dynamics MD simulations. MC and DC simulations are presently much in favor due to the increased availability of fast computers. A prediction common to all these theories is that micelles represent a thermodynamically stable state and that micellar solutions are single-phase systems. Several recent results of MD and (MC) simulations are in agreement with experimental results. ... 

---