Application of computer simulation free-energy methods to compute the free energy of micellization as a function of micelle composition. 2. Implementation

In this paper, the implementation of the CS-FE/MT model introduced in article 1 is discussed, and computer simulations are performed to evaluate the feasibility of the new theoretical approach. As discussed in article 1, making predictions of surfactant/solubilizate aqueous solution behavior using the CS-FE/MT model requires evaluation of DeltaDeltaG for multiple surfactant-to-solubilizate or surfactant-to-cosurfactant transformations. The central goal of this article is to evaluate the quantitative accuracy of the alchemical computer simulation method used in the CS-FE/MT modeling approach to predict DeltaDeltaG for a single surfactant-to-solubilizate or for a single surfactant-to-cosurfactant transformation. A hybrid single/dual topology approach was used to morph the ionic surfactant sodium dodecyl sulfate (SDS) into the ionic solubilizate ibuprofen (IBU), and a dual topology approach was used to morph the nonionic surfactant octyl glucoside (OG) into the nonionic solubilizate p-aminobenzoate (PAB). In addition, a single topology approach was used to morph the nonionic surfactant n-decyl dimethylphosphine oxide (C10PO) into the nonionic cosurfactant n-decyl methyl sulfoxide (C10SO), the nonionic surfactant octylsulfinyl ethanol (C8SE) into the nonionic cosurfactant decylsulfinyl ethanol (C10SE), and the nonionic surfactant n-decyl methyl sulfoxide (C10SO) into the nonionic cosurfactant n-octyl methyl sulfoxide (C8SO). Each DeltaDeltaG value was computed by using thermodynamic integration to determine the difference in free energy associated with (i) transforming a surfactant molecule of type A into a cosurfactant/solubilizate molecule of type B in a micellar environment (referred to as DeltaG2) and (ii) transforming a surfactant molecule of type A into a cosurfactant/solubilizate molecule of type B in aqueous solution (referred to as DeltaG1). CS-FE/MT model predictions of DeltaDeltaG for each alchemical transformation were made at a number of simulation conditions, including (i) different equilibration times at each value of the coupling parameter lambda, (ii) different data-gathering times at each lambda value, and (iii) simulation at a different number of lambda values. For the three surfactant-to-cosurfactant transformations considered here, the DeltaDeltaG values predicted by the CS-FE/MT model were compared with DeltaDeltaG values predicted by an accurate molecular thermodynamic (MT) model developed by fitting to experimental CMC data. Even after performing lengthy equilibration and data gathering at each lambda value, physically unrealistic values of DeltaDeltaG were predicted by the CS-FE/MT model for the transformations of SDS into IBU and of OG into PAB. However, more physically realistic DeltaDeltaG values were predicted for the transformation of C10PO into C10SO, and reasonable free-energy predictions were obtained for the transformations of C8SE into C10SE and C10SO into C8SO. Each of the surfactant-to-cosurfactant transformations considered here involved less extensive structural changes than the surfactant-to-solubilizate transformations. As computer power increases and as improvements are made to alchemical free-energy methods, it may become possible to apply the CS-FE/MT model to make accurate predictions of the free-energy changes associated with forming multicomponent surfactant and solubilizate micelles in aqueous solution where the chemical structures of the surfactants, cosurfactants, and solubilizates differ significantly.     

Quantifying the hydrophobic effect. 2. A computer simulation-molecular-thermodynamic model for the micellization of nonionic surfactants in aqueous solution

In this article, the validity and accuracy of the CS-MT model is evaluated by using it to model the micellization behavior of seven nonionic surfactants in aqueous solution. Detailed information about the changes in hydration that occur upon the self-assembly of the surfactants into micelles was obtained through molecular dynamics simulation and subsequently used to compute the hydrophobic driving force for micelle formation. This information has also been used to test, for the first time, approximations made in traditional molecular-thermodynamic modeling. In the CS-MT model, two separate free-energy contributions to the hydrophobic driving force are computed. The first contribution, gdehydr, is the free-energy change associated with the dehydration of each surfactant group upon micelle formation. The second contribution, ghydr, is the change in the hydration free energy of each surfactant group upon micelle formation. To enable the straightforward estimation of gdehydr and ghydr in the case of nonionic surfactants, a number of simplifying approximations were made. Although the CS-MT model can be used to predict a variety of micellar solution properties including the micelle shape, size, and composition, the critical micelle concentration (CMC) was selected for prediction and comparison with experimental CMC data because it depends exponentially on the free energy of micelle formation, and as such, it provides a stringent quantitative test with which to evaluate the predictive accuracy of the CS-MT model. Reasonable agreement between the CMCs predicted by the CS-MT model and the experimental CMCs was obtained for octyl glucoside (OG), dodecyl maltoside (DM), octyl sulfinyl ethanol (OSE), decyl methyl sulfoxide (C10SO), decyl dimethyl phosphine oxide (C10PO), and decanoyl-n-methylglucamide (MEGA-10). For five of these surfactants, the CMCs predicted using the CS-MT model were closer to the experimental CMCs than the CMCs predicted using the traditional molecular-thermodynamic (MT) model. In addition, CMCs predicted for mixtures of C10PO and C10SO using the CS-MT model were significantly closer to the experimental CMCs than those predicted using the traditional MT model. For dodecyl octa(ethylene oxide) (C12E8), the CMC predicted by the CS-MT model was not in good agreement with the experimental CMC and with the CMC predicted by the traditional MT model, because the simplifying approximations made to estimate gdehydr and ghydr in this case were not sufficiently accurate. Consequently, we recommend that these simplifying approximations only be used for nonionic surfactants possessing relatively small, non-polymeric heads. For MEGA-10, which is the most structurally complex of the seven nonionic surfactants modeled, the CMC predicted by the CS-MT model (6.55 mM) was found to be in much closer agreement with the experimental CMC (5 mM) than the CMC predicted by the traditional MT model (43.3 mM). Our results suggest that, for complex, small-head nonionic surfactants where it is difficult to accurately quantify the hydrophobic driving force for micelle formation using the traditional MT modeling approach, the CS-MT model is capable of making reasonable predictions of aqueous micellization behavior.     

Molecular-thermodynamic theory of micellization of multicomponent surfactant mixtures: 1. Conventional (pH-Insensitive) surfactants

A molecular-thermodynamic (MT) theory was developed to model the micellization of mixtures containing an arbitrary number of conventional (pH-insensitive) surfactants. The theory was validated by comparing predicted and experimental cmc's of ternary surfactant mixtures, yielding results that were comparable to, and sometimes better than, the cmc's determined using regular solution theory. The theory was also used to model a commercial nonionic surfactant (Genapol UD-079), which was modeled as a mixture of 16 surfactant components. The predicted cmc agreed well with the experimental cmc, and the monomer concentration was predicted to increase significantly above the cmc. In addition, the monomer and the micelle compositions were predicted to vary significantly with surfactant concentration. These composition variations were rationalized in terms of competing steric and entropic effects and a micelle shape transition near the cmc. To understand the packing constraints imposed on ternary surfactant mixtures better, the maximum micelle radius was also examined theoretically. The MT theory presented here represents the first molecular-based theory of the micellization behavior of mixtures of three or more conventional surfactants. In article 2 of this series, the MT theory will be extended to model the micellization of mixtures of conventional and pH-sensitive surfactants.     

Micelle–monomer equilibria in solutions of ionic surfactants and in ionic–nonionic mixtures: A generalized phase separation model

On the basis of a detailed physicochemical model, a complete system of equations is formulated that describes the equilibrium between micelles and monomers in solutions of ionic surfactants and their mixtures with nonionic surfactants. The equations of the system express mass balances, chemical and mechanical equilibria. Each nonionic surfactant is characterized by a single thermodynamic parameter — its micellization constant. Each ionic surfactant is characterized by three parameters, including the Stern constant that quantifies the counterion binding. In the case of mixed micelles, each pair of surfactants is characterized with an interaction parameter, β, in terms of the regular solution theory. The comparison of the model with experimental data for surfactant binary mixtures shows that β is constant — independent of the micelle composition and electrolyte concentration. The solution of the system of equations gives the concentrations of all monomeric species, the micelle composition, ionization degree, surface potential and mean area per head group. Upon additional assumptions for the micelle shape, the mean aggregation number can be also estimated. The model gives quantitative theoretical interpretation of the dependence of the critical micellization concentration (CMC) of ionic surfactants on the ionic strength; of the CMC of mixed surfactant solutions, and of the electrolytic conductivity of micellar solutions. It turns out, that in the absence of added salt the conductivity is completely dominated by the contribution of the small ions: monomers and counterions. The theoretical predictions are in good agreement with experimental data.     

Molecular dynamics simulation and thermodynamic modeling of the self-assembly of the triterpenoids asiatic acid and madecassic acid in aqueous solution

The self-assembly behavior of the triterpenoids asiatic acid (AA) and madecassic acid (MA), both widely studied bioactive phytochemicals that are similar in structure to bile salts, were investigated in aqueous solution through atomistic-level molecular dynamics (MD) simulation. AA and MA molecules initially distributed randomly in solution were observed to aggregate into micelles during 75 ns of MD simulation. A "hydrophobic contact criterion" was developed to identify micellar aggregates from the computer simulation results. From the computer simulation data, the aggregation number of AA and MA micelles, the monomer concentration, the principal moments of the micelle radius of gyration tensor, the one-dimensional growth exhibited by AA and MA micelles as the aggregation number increases, the level of internal ordering within AA and MA micelles (quantified using two different orientational order parameters), the local environment of atoms within AA and MA in the micellar environment, and the total, hydrophilic, and hydrophobic solvent accessible surface areas of the AA and MA micelles were each evaluated. The MD simulations conducted provide insights into the self-assembly behavior of structurally complex, nontraditional surfactants in aqueous solution. Motivated by the high computational cost required to obtain an accurate estimate of the critical micelle concentrations (CMCs) of AA and MA from evaluation of the average monomer concentration present in the AA and MA simulation cells, a modified computer simulation/molecular-thermodynamic model (referred to as the MCS-MT model) was formulated to quantify the free-energy change associated with optimal AA and MA micelle formation in order to predict the CMCs of AA and MA. The predicted CMC of AA was found to be 59 microM, compared with the experimentally measured CMC of 17 microM, and the predicted CMC of MA was found to be 96 microM, compared with the experimentally measured CMC of 62 microM. The AA and MA CMCs predicted using the MCS-MT model are much more accurate than the CMCs inferred from the monomer concentrations of AA and MA present in the simulation cells after micelle self-assembly (2390 microM and 11,300 microM, respectively). The theoretical modeling results obtained for AA and MA indicate that, by combining computer simulation inputs with molecular-thermodynamic models of surfactant self-assembly, reasonably accurate estimates of surfactant CMCs can be obtained with a fraction of the computational expense that would be required by using computer simulations alone.     

Thermodynamic consideration of the mixed micelle of surfactants

In conformity with the conclusion obtained previously, the mixed micelle formation of surfactants was treated thermodynamically as the appearance of a macroscopic bulk phase with the aid of the excess thermodynamic quantities similar to those used for the adsorbed film. The composition of surfactant in the mixed micelle and the thermodynamic quantities of micelle formation were found to be evaluated by applying the thermodynamic equations derived. These equations were extended so as to be applicable to any kind of surfactant mixture. It was shown that the critical micelle concentration vs. composition of surfactant curves form a diagram analogous to the phase diagram of binary mixture. Applying the equation to the published data on typical surfactant systems, this thermodynamic approach was proved to be useful to clarify the miscibility of surfactants in the micellar state.     

Quantifying the hydrophobic effect. 3. A computer simulation-molecular-thermodynamic model for the micellization of ionic and zwitterionic surfactants in aqueous solution

In this article, the validity and accuracy of the CS-MT model introduced in article 1 for oil aggregates and in article 2 for nonionic surfactants is further evaluated by using it to model the micellization behavior of ionic and zwitterionic surfactants in aqueous solution. In the CS-MT model, two separate free-energy contributions to the hydrophobic driving force for micelle formation are computed using hydration data obtained from computer simulation: gdehydr, the free-energy change associated with dehydration, and ghydr, the change in the hydration free energy. To enable straightforward estimation of gdehydr and ghydr for ionic and zwitterionic surfactants, a number of simplifying approximations were made. Reasonable agreement between the CMCs predicted using the CS-MT model and the experimental CMCs was obtained for sodium dodecyl sulfate (SDS), dodecylphophocholine (DPC), cetyltrimethylammonium bromide (CTAB), two 3-hydroxy sulfonate surfactants (AOS-12 and AOS-16), and a homologous series of four DCNA bromide surfactants with a dimethylammonium head attached to a dodecyl alkyl tail and to an alkyl side chain of length CN, having the chemical formula C12H25CNH2N+1N(CH3)2Br, with N = 1 (DC1AB), 2 (DC2AB), 4 (DC4AB), and 6 (DC6AB). For six of these nine surfactants, the CMCs predicted using the CS-MT model are closer to the experimental CMCs than the CMCs predicted using the traditional molecular-thermodynamic (MT) model. For DC2AB, DC4AB, and DC6AB, which are the most structurally complex of the ionic surfactants modeled, the CMCs predicted using the CS-MT model are in remarkably good agreement with the experimental CMCs, and the CMCs predicted using the traditional MT model are quite inaccurate. Our results suggest that the CS-MT model accurately quantifies the hydrophobic driving force for micelle formation for ionic and zwitterionic surfactants in aqueous solution. For complex ionic and zwitterionic surfactants where it is difficult to accurately quantify the hydrophobic driving force for micelle formation using the traditional MT modeling approach, the CS-MT model represents a very promising alternative.     

Quantifying the hydrophobic effect. 1. A computer simulation-molecular-thermodynamic model for the self-assembly of hydrophobic and amphiphilic solutes in aqueous solution

Surfactant micellization and micellar solubilization in aqueous solution can be modeled using a molecular-thermodynamic (MT) theoretical approach; however, the implementation of MT theory requires an accurate identification of the portions of solutes (surfactants and solubilizates) that are hydrated and unhydrated in the micellar state. For simple solutes, such identification is comparatively straightforward using simple rules of thumb or group-contribution methods, but for more complex solutes, the hydration states in the micellar environment are unclear. Recently, a hybrid method was reported by these authors in which hydrated and unhydrated states are identified by atomistic simulation, with the resulting information being used to make MT predictions of micellization and micellar solubilization behavior. Although this hybrid method improves the accuracy of the MT approach for complex solutes with a minimum of computational expense, the limitation remains that individual atoms are modeled as being in only one of two states-head or tail-whereas in reality, there is a continuous spectrum of hydration states between these two limits. In the case of hydrophobic or amphiphilic solutes possessing more complex chemical structures, a new modeling approach is needed to (i) obtain quantitative information about changes in hydration that occur upon aggregate formation, (ii) quantify the hydrophobic driving force for self-assembly, and (iii) make predictions of micellization and micellar solubilization behavior. This article is the first in a series of articles introducing a new computer simulation-molecular thermodynamic (CS-MT) model that accomplishes objectives (i)-(iii) and enables prediction of micellization and micellar solubilization behaviors, which are infeasible to model directly using atomistic simulation. In this article (article 1 of the series), the CS-MT model is introduced and implemented to model simple oil aggregates of various shapes and sizes, and its predictions are compared to those of the traditional MT model. The CS-MT model is formulated to allow the prediction of the free-energy change associated with aggregate formation (gform) of solute aggregates of any shape and size by performing only two computer simulations-one of the solute in bulk water and the other of the solute in an aggregate of arbitrary shape and size. For the 15 oil systems modeled in this article, the average discrepancy between the predictions of the CS-MT model and those of the traditional MT model for gform is only 1.04%. In article 2, the CS-MT modeling approach is implemented to predict the micellization behavior of nonionic surfactants; in article 3, it is used to predict the micellization behavior of ionic and zwitterionic surfactants.     

Triton X-100/CTAB在乙二醇/水混合溶剂中的热力学性质和胶团化行为

利用表面张力法, 研究了非离子表面活性剂Triton X-100和离子表面活性剂十六烷基三甲基溴化铵(CTAB)混合体系在混合极性溶剂乙二醇/水(乙二醇的体积分数分别为5%、10%和20%)中的热力学性质和胶团化行为. 结果表明, 混合体系在乙二醇水溶液中存在协同效应, 临界胶束浓度随乙二醇含量的增加而增大. 利用Rubingh和Maeda模型计算了混合物中各组分在胶团相中的组成、相互作用参数以及自由能的贡献. 在实验研究的乙二醇浓度范围内, 发现该非离子/离子混合体系在离子组分摩尔分数约为0.3时, 协同效应最强.     

Use of the semi-equilibrium dialysis method in studying the thermodynamics of solubilization of organic compounds in surfactant micelles. Systemn-hexadecylpyridinium chloride-phenol-water

The semi-equilibrium dialysis method has been used to infer solubilization equilibrium constants or, alternatively, activity coefficients of solutes solubilized into micelles of aqueous surfactant solutions. Methods are described for inferring the concentrationa of monomers of the organic solute and of the surfactant on both sides of the dialysis membrane, under conditions where the organic solute is in equilibrium with both the high-concentration (retentate) and low-concentration (permeate) solutions. By using a form of the Gibbs-Duhem equation, activity coefficients of both phenol (the solubilizate) and n-hexadecylpyridinium chloride (the surfactant) are obtained for aqueous solutions at 25°C throughout a wide range of relative compositions of surfactant and solubilizate within the micelle. The apparent solubilization constant, K=[solubilized phenol]/([monomeric phenol][micellar surfactant]), is found to decrease significantly as the mole fraction of phenol in the micelle increases.     

Excess free energy,enthalpy and entropy of surfactant-surfactant mixed micelle formation

Enthalpies of dilution and osmotic coefficients of sodium decylsulfate (NaDeS)-dodecyldimethylamine oxide (DDAO) mixtures in water were determined at 298 and 310 K, respectively. From the enthalpies of dilution, the apparent and then the partial molar relative enthalpies of the surfactant mixtures were calculated. From the osmotic coefficients, calculated at 298 K, the non-ideal free energies were derived. The latter were combined with the partial molar relative enthalpies to obtain the non-ideal entropies. From the apparent molar properties, using a previously reported approach, the excess thermodynamic properties for the surfactant-surfactant mixed micelle formation in water were evaluated as functions of the mixture composition at some total micellized concentration. In the whole range of the mixture composition, the excess free energy is negative, indicating that the mixed micelle formation is favoured with respect to that of pure micelles. This process is governed by the enthalpy and/or the entropy, depending on the mixture composition. The effect of the alkyl chain length was also studied by comparing the present results to those of the sodium dodecylsulfate-DDAO mixture.     

Mixed micelle formation and solubilization behavior toward polycyclic aromatic hydrocarbons of binary and ternary cationic-nonionic surfactant mixtures

Water solubility enhancements of polycyclic aromatic hydrocarbons (PAHs), viz., naphthalene, anthracene and pyrene, by micellar solutions at 25 degrees C using two series of surfactants, each involving two cationic and one nonionic surfactant in their single as well as equimolar binary and ternary mixed states, were measured and compared. The first series was composed of three surfactants, benzylhexadecyldimethylammonium chloride (C16BzCl), hexadecyltrimethylammonium bromide (C16Br), and polyoxyethylene(20)mono-n-hexadecyl ether (Brij-58) with a 16-carbon (C16) hydrophobic chain; the second series consisted of dodecyltrimethylammonium bromide (C12Br), dodecylethyldimethylammonium bromide (C12EBr), and polyoxyethylene(4)mono-n-dodecyl ether (Brij-30) with a 12-carbon (C12) chain. Solubilization capacity has been quantified in terms of the molar solubilization ratio, the micelle-water partition coefficient, the first stepwise association constant between solubilizate monomer and vacant micelle, and the average number of solubilizate molecules per micelle, determined employing spectrophoto-, tensio-, and flourimetric techniques. Cationic surfactants exhibited lesser solubilization capacity than nonionics in each series of surfactants with higher efficiency in the C16 series compared to the C12 series. Increase in hydrophobicity of head groups of cationics by incorporation of ethyl or benzyl groups enhanced their solubilization capacity. The mixing effect of surfactants on mixed micelle formation and solubilization efficiency has been discussed in light of the regular solution approximation (RSA). Cationic-nonionic binary combinations showed better solubilization capacity than pure cationics, nonionics, or cationic-cationic mixtures, which, in general, showed increase with increased hydrophobicity of PAHs. Equimolar cationic-cationic-nonionic ternary surfactant systems showed lower solubilization efficiency than their binary cationic-nonionic counterparts but higher than cationic-cationic ones. In addition, use of RSA has been extended, with fair success, to predict partition coefficients of ternary surfactant systems using data of binary surfactants systems. Mixed surfactants may improve the performance of surfactant-enhanced remediation of soils and sediments by decreasing the applied surfactant level and thus remediation cost.     

Volumetric behavior and micelle formation of aqueous surfactant mixtures

A thermodynamic treatment of the volumetric behavior of surfactant mixtures in water have been developed on the basis of the thermodynamic treatment of mixed micelle by Motomura et al. Densities of aqueous solutions of mixtures of decyltrimethylammonium bromide (DeTAB) and dodecyltrimethylammonium bromide (DTAB) have been measured as a function of total molality at constant compositions. The apparent molar volumes of the mixtures have been derived from the density data and the mean partial molar volume of monomeric surfactant mixture V _t ^w , the molar volume of mixed micelle V^M/N _t ^M , the voluem of formation of mixed micelle _W ^M V, and the composition of surfactant in the mixed micelle have been evaluated. The V _t ^W , V^M/N _t ^M , and _W ^M V have been observed to depend on the composition. The linear dependence of V _t ^W and V^M/N _t ^M on the composition indicates that the mixing of DeTAB and DTAB is ideal both in the monomeric and micellar states. This has been confirmed further by the shape of the critical micelle concentration vs. composition curves.     

Effect of the Head Group of Surfactant on the Miscibility of Sodium Chloride and Surfactant in an Adsorbed Film and Micelle

The thermodynamic treatment of a surfactant mixture was applied to the mixture of sodium chloride, NaCl, with octyl methyl sulfoxide (OMS) and that with decyldimethylphosphine oxide (DePO). The surface tension of aqueous solutions of the mixtures was measured as a function of the total concentration and the composition of the mixtures at 298.15 K. The total surface densities of the mixtures and the composition of the adsorbed films and micelles were evaluated by applying thermodynamic equations to the expeimental results. It was found that the adsorbed film and micelle are almost composed of the surfactant and there is slight attractive interaction between the ions of NaCl and the head groups of OMS and DePO molecules in the adsorbed films and micelles. A difference in the miscibility of NaCl and surfactant was observed between the OMS and DePO systems and attributed to the difference in the hydration of the head group between OMS and DePO molecules. The comparison of these results with those of the mixtures of NaCl with tetraethylene glycol monooctyl ether (C(8)E(4)) and dodecylammonium chloride (DAC) indicated that the small difference in the miscibility in an adsorbed film and micelle among these nonionic surfactant systems arises from the difference in hydration and structure of the head groups and the large one between the nonionic surfactant and DAC systems results from electrostatic interactions between dodecylammonium and sodium ions. Copyright 2001 Academic Press.     

Conductometric study of the mixing behaviour of sodium benzoate and salicylate with an anionic surfactant sodium lauroylsarcosinate

Effects of two anionic hydrotropes – sodium benzoate (NaBz) and sodium salicylate (NaSal) – on the mixed-micelle formation with an amino-acid-based surfactant – sodium lauroylsarcosinate (SLS) – in water were investigated by the conductometric method. Specific conductivity was measured for SLS/NaBz/water and SLS/NaSal/water systems to determine the critical micelle concentration (cmc). Using the regular solution theory for non-ideal mixing, the pairwise interaction parameter, β₁₂, and micellar composition, χ, were estimated in the mixed micelle. The cmc values of the surfactant–hydrotropes mixtures were generally lower than those predicted from the ideal mixing theory. The β₁₂ values are generally negative for the two systems at all mole fractions with an average value of ?2.83 for the SLS/NaBz and ?3.31 for SLS/NaSal systems, respectively, indicative of a strong attractive interaction between the SLS/NaBz and SLS/NaSal mixed micelle. The calculated thermodynamic parameters of micellisation all indicated spontaneity in mixed-micelle formation for the systems studied.     

Markov chain model for the critical micelle concentration of surfactant mixtures

An extension of the Markov chain model (MC) for micellization is proposed, which allows the distribution of the surfactants between the monomer solution and the micelles in a mixed surfactant system to be predicted. The dependence of the critical micelle concentration (cmc) on the composition of the solution is investigated. The equilibrium thermodynamic relation between cmc and micelle composition is discussed. The case of ternary mixtures is analyzed, and theoretical triangular diagram is constructed according to MC. Available experimental data for binary and ternary mixtures agree well with the new MC theory. The dependence of MC parameters on the structure of the surfactants is discussed. Comparison of MC with the simple mixture (“regular solution”) model is presented. The parameters of the MC theory are related to the interaction parameter β _SM of the simple mixture model.