text: Fischer and Jandera studied the effect of changing the concentration of methanol on CMC values for three commonly used surfactants.  Two cationic, hexadecyltrimethylammonium bromide (CTAB), and N-(a-carbethoxypentadecyl) trimethylammonium bromide (Septonex),  and one anionic surfactant, sodium dodecyl sulphate (SDS) were chosen for the experiment.  Generally speaking, the CMC increased as the concentration of methanol increased.  It was then concluded that the distribution of the surfactant between the bulk mobile phase and the micellar phase shifts toward the bulk as the methanol concentration increases.  For CTAB, the rise in CMC is greatest from 0&ndash;10% methanol, and is nearly constant from 10&ndash;20%.  Above 20% methanol, the micelles disaggregate and do not exist.  For SDS, the CMC values remain unaffected below 10% methanol, but begin to increase as the methanol concentration is further increased.  Disaggregation occurs above 30% methanol.  Finally, for Septonex, only a slight increase in CMC is observed up to 20%, with disaggregation occurring above 25%.
  As has been asserted, the mobile phase in MLC consists of micelles in an aqueous solvent, usually with a small amount of organic modifier added to complete the mobile phase.  A typical reverse phase alkyl-bonded stationary phase is used.  The first discussion of the thermodynamics involved in the retention mechanism was published by Armstrong and Nome in 1981. In MLC, there are three partition coefficients which must be taken into account.  The solute will partition between the water and the stationary phase (KSW), the water and the micelles (KMW), and the micelles and the stationary phase (KSM).
  Armstrong and Nome derived an equation describing the partition coefficients in terms of the retention factor, formally capacity factor, k¢.  In HPLC, the capacity factor represents the molar ratio of the solute in the stationary phase to the mobile phase.  The capacity factor is easily measure based on retention times of the compound and any unretained compound.  The equation rewritten by Guermouche et al. is presented here:
  :1/k¢ = [n • (KMW-1)/(f • KSW)] • CM +1/(f • KSW)
  Where:
  *k¢ is the capacity factor of the solute
  *KSW is the partition coefficient of the solute between the stationary phase and the water
  *KMW is the partition coefficient of the solute between the micelles and the water
  *f is the phase volume ratio (stationary phase volume/mobile phase volume)
  *n is the molar volume of the surfactant
  *CM is the concentration of the micelle in the mobile phase (total surfactant concentration - critical micelle concentration)
  A plot of 1/k¢ verses CM gives a straight line in which KSW can be calculated from the intercept and KMW can be obtained from the ratio of the slope to the intercept.  Finally, KSM can be obtained from the ratio of the other two partition coefficients:
  :KSM = KSW/ KMW
  As can be observed from Figure 1, KMW is independent of any effects from the stationary phase, assuming the same micellar mobile phase.
  The validity of the retention mechanism proposed by Armstrong and Nome has been successfully, and repeated confirmed experimentally.  However, some variations and alternate theories have also been proposed.  Jandera and Fischer developed equations to describe the dependence of retention behavior on the change in micellar concentrations.  They found that the retention of most compounds tested decreased with increasing concentrations of micelles.  From this, it can be surmised that the compounds associate with the micelles as they spend less time associated with the stationary phase.
  Foley proposed a similar retentive model to that of Armstrong and Nome which was a general model for secondary chemical equilibria in liquid chromatography.  While this model was developed in a previous reference, and could be used for any secondary chemical equilibria such as acid-base equilibria, and ion-pairing, Foley further refined the model for MLC.  When an equilibrant (X), in this case surfactant, is added to the mobile phase, a secondary equilibria is created in which an analyte will exist as free analyte (A), and complexed with the equilibrant (AX).  The two forms will be retained by the stationary phase to different extents, thus allowing the retention to be varied by adjusting the concentration of equilibrant (micelles).
  The resulting equation solved for capacity factor in terms of partition coefficients is much the same as that of Armstrong and Nome:
  :1/k¢ = (KSM/k¢S) • [M] + 1/k¢S
  Where:
  *k¢ is the capacity factor of the complexed solute and the free solute
  *k¢S is the capacity factor of the free solute
  *KSM is the partition coefficient of the solute between the stationary phase and the micelle
  *[M] may be either the concentration of surfactant or the concentration of micelle
  Foley used the above equation to determine the solute-micelle association constants and free solute retention factors for a variety of solutes with different surfactants and stationary phases.  From this data, it is possible to predict the type and optimum surfactant concentrations needed for a given solute or solutes.
  Foley has not been the only researcher interested in determining the solute-micelle association constants.  A review article by Marina and Garcia with 53 references discusses the usefulness of obtaining solute-micelle association constants. The association constants for two solutes can be used to help understand the retention mechanism.  The separation factor of two solutes, a, can be expressed as KSM1/KSM2.  If the experimental a coincides with the ratio of the two solute-micelle partition coefficients, it can be assumed that their retention occurs through a direct transfer from the micellar phase to the stationary phase.  In addition, calculation of a would allow for prediction of separation selectivity before the analysis is performed, provided the two coefficients are known.
  The desire to predict retention behavior and selectivity has led to the development of several mathematical models. Changes in pH, surfactant concentration, and concentration of organic modifier play a significant role in determining the chromatographic separation.  Often one or more of these parameters need to be optimized to achieve the desired separation, yet the optimum parameters must take all three variables into account simultaneously.  The review by Garcia-Alvarez-Coque et al. mentioned several successful models for varying scenarios, a few of which will be mentioned here.  The classic models by Armstrong and Nome and Foley are used to describe the general cases.  Foley's model applies to many cases and has been experimentally verified for ionic, neutral, polar and nonpolar solutes; anionic, cationic, and non-ionic surfactants, and C8, C¬18, and cyano stationary phases.  The model begins to deviate for highly and lowly retained solutes.  Highly retained solutes may become irreversibly bound to the stationary phase, where lowly retained solutes may elute in the column void volume.
  Other models proposed by Arunyanart and Cline-Love and Rodgers and Khaledi describe the effect of pH on the retention of weak acids and bases.  These authors derived equations relating pH and micellar concentration to retention.  As the pH varies, sigmoidal behavior is observed for the retention of acidic and basic species.  This model has been shown to accurately predict retention behavior.  Still other models predict behavior in hybrid micellar systems using equations or modeling behavior based on controlled experimentation.  Additionally, models accounting for the simultaneous effect of pH, micelle and organic concentration have been suggested.  These models allow for further enhancement of the optimization of the separation of weak acids and bases.
  One research group, Rukhadze, et al. derived a first order linear relationship describing the influence of micelle and organic concentration, and pH on the selectivity and resolution of seven barbiturates.  The researchers discovered that a second order mathematical equation would more precisely fit the data.  The derivations and experimental details are beyond the scope of this discussion.  The model was successful in predicting the experimental conditions necessary to achieve a separation for compounds which are traditionally difficult to resolve.
  Jandera, Fischer, and Effenberger approached the modeling problem in yet another way. The model used was based on lipophilicity and polarity indices of solutes.  The lipophilicity index relates a given solute to a hypothetical number of carbon atoms in an alkyl chain.  It is based and depends on a given calibration series determined experimentally.  The lipophilicity index should be independent of the stationary phase and organic modifier concentration.  The polarity index is a measure of the polarity of the solute-solvent interactions.  It depends strongly on the organic solvent, and somewhat on the polar groups present in the stationary phase.  23 compounds were analyzed with varying mobile phases and compared to the lipophilicity and polarity indices.  The results showed that the model could be applied to MLC, but better predictive behavior was found with concentrations of surfactant below the CMC, sub-micellar.
  A final type of model based on molecular properties of a solute is a branch of quantitative structure-activity relationships (QSAR).  QSAR studies attempt to correlate biological activity of drugs, or a class of drugs, with structures.  The normally accepted means of uptake for a drug, or its metabolite, is through partitioning into lipid bilayers.  The descriptor most often used in QSAR to determine the hydrophobicity of a compound is the octanol-water partition coefficient, log P. MLC provides an attractive and practical alternative to QSAR.  When micelles are added to a mobile phase, many similarities exist between the micellar mobile phase/stationary phase and the biological membrane/water interface.  In MLC, the stationary phase become modified by the adsorption of surfactant monomers which are structurally similar to the membranous hydrocarbon chains in the biological model.  Additionally, the hydrophilic/hydrophobic interactions of the micelles are similar to that in the polar regions of a membrane.  Thus, the development of quantitative structure-retention relationships (QRAR) has become widespread.
  Escuder-Gilabert et al. tested three different QRAR retention models on ionic compounds.  Several classes of compounds were tested including catecholamines, local anesthetics, diuretics, and amino acids.  The best model relating log K and log P was found to be one in which the total molar charge of a compound at a given pH is included as a variable.  This model proved to give fairly accurate predictions of log P, R > 0.9. Other studies have been performed which develop predictive QRAR models for tricyclic antidepressants and barbiturates.