Equation-Oriented Mixed Micellization Modeling of a Subregular

Article pubs.acs.org/IECR

Equation-Oriented Mixed Micellization Modeling of a Subregular Ternary Surfactant System with Potential Medical Applications Romina B. Pereyra,† Erica P. Schulz,*,† Guillermo A. Durand,‡ José L. Rodriguez,† Rosanna M. Minardi,† Hernán A. Ritacco,§ and Pablo C. Schulz† †

Departamento de Química, INQUISUR (Universidad Nacional del Sur - CONICET), 8000 Bahía Blanca, Argentina Departamento de Ingeniería Química, PLAPIQUI (Universidad Nacional del Sur - CONICET), 8000 Bahía Blanca, Argentina § Departamento de Física, IFISUR (Universidad Nacional del Sur - CONICET), 8000 Bahía Blanca, Argentina ‡

S Supporting Information *

ABSTRACT: The aqueous tricomponent surfactant mixture dodecyltrimethylammonium bromide (DTAB), sodium 10-undecenoate (SUD), and sodium dodecanoate (SDD) has been studied over the complete triangular diagram. This system is highly nonideal and has a coacervate domain. The mixtures do not precipitate in any proportion. There is a wide region of the phase diagram with positive deviations of the critical micelle concentration (CMC). The thermodynamic analysis has been done with the recently developed equation-oriented mixed micellization model (EOMMM) since the widely employed multicomponent regular solution theory (MRST) failed to represent this subregular ternary surfactant system. The advantages of the EOMMM over the MRST are discussed. The biocide properties of DTAB and SUD against microorganisms, the high CMCs that ensure high concentrations of biologically active monomers, and the possibility of having a system that remains liquid in any proportion make the present system attractive for the design of bactericide and antifungal preparations for the medicine, food, and cosmetic industries.

1. INTRODUCTION At a certain concentration, namely, the critical micelle concentration (CMC), surfactants self-assemble in aqueous solutions to form aggregates called micelles. At the onset of micelle formation, there are striking changes in the behavior of the solution properties, such as the surface tension and light scattering. Surfactant mixtures are commonly used instead of pure amphiphiles because the mixtures often have enhanced properties compared with the sum of the properties of the pure components (synergism).1 The determination of the composition of the mixed micelles is a major problem since its value is fixed by the partition equilibria of the species between the aggregate and the surrounding medium. Only the composition of the total micellar solution is accessible to the experimenter a priori. Therefore, the composition of the mixed micelles has to be experimentally determined or calculated on the basis of a thermodynamic model parametrized with physicochemical properties, mainly the CMC.2 There are two types of thermodynamic methods for dealing with micelle formation in surfactant solutions: (i) phenomenological models such as the pseudophase separation model and the mass-action model and (ii) molecular thermodynamic methods (see Phenomenological Models and Molecular Thermodynamic Methods in the Supporting Information (SI) and references therein). These two types of methods are generally considered in the literature as parallel treatments for micelle formation. © XXXX American Chemical Society

The phenomenological methods mainly aim to estimate the mixed micelle composition, while the molecular thermodynamic theories are generally used to predict CMCs and other properties. Regular solution theory (RST or Rubingh’s method) for binary surfactant mixtures and its extension to multicomponent systems, multicomponent regular solution theory (MRST) (see the sections about RST and MRST methods in the SI, and references therein) are widely used phenomenological models that consider the micelles as a pseudo-microphase (pseudophase separation model). They treat the nonidealities via the regular solution approximation and are based on symmetric Margules formulations.3,4 The application of these models is quite simple, but they have been extensively criticized.5,6 The equation-oriented mixed micellization model (EOMMM) is a new approach based on equation-oriented optimization and Margules asymmetric formulations7 (contemplating both symmetric and asymmetric thermodynamic behaviors since the symmetric formulations are a particular case of the asymmetric ones) that is not restricted to the number of components and guarantees the applicability of the Gibbs−Duhem relation8 (for details, see the Received: Revised: Accepted: Published: A

June 21, 2017 September 1, 2017 September 7, 2017 September 7, 2017 DOI: 10.1021/acs.iecr.7b02549 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

When mixing in a multicomponent system is nonideal, phase separation may occur, giving rise to binodes separating coexisting phases. Within each binode is contained the spinodal surface, which separates unstable and metastable regions. In binary systems, when the second derivative of the Gibbs energy with respect to composition is negative (∂2G/∂x2 < 0), a spinodal decomposition occurs, and the system decomposes into two phases, one solute-rich and the other solute-depleted (e.g., in alloys). Thermodynamic criteria for spinodal decomposition in ternary systems have been developed considering that when ΔG < 0 the system is unstable and decomposes spontaneously by an infinitesimal composition fluctuation.12 Therefore, it is evident at this point that higher-order Taylor series or more complex multicomponent equations are required to represent the Gibbs free energy of ternary systems with decomposition. There has been a proliferation of equations for activity coefficients, but none has been proposed that can represent all type of solutions.13 Effort has been devoted to develop a set of thermodynamically consistent equations in which two or more different binary equations could be mixed in the multicomponent computations.14 The subregular or asymmetric mixing model has been used extensively in geology and metallurgy. Although in such fields phases containing four or more components are common, the extension of the subregular model to these systems is not intuitively obvious, and most attempts in the literature are in error. There are two common approaches in the development of multicomponent excess free energy models.15 The “projected multicomponent models” consist in moving from the binaries to the multicomponent space by combining the binaries according to empirical schemes to generate expressions for the multicomponent excess free energy. The “power series multicomponent models” are based on an appropriate multicomponent polynomial function to represent the excess Gibbs free energy of mixing that is truncated after the number of terms demanded by the nature of the data. Truncation after the third-degree terms (in which the combined power of the product of mole fractions equals 3) leads to a formulation in which the binaries have subregular behavior. In order to compare the two approaches (MRST and EOMMM), in the present work we have studied a ternary system that must be, prima facie, highly nonideal. The system is an aqueous mixture of sodium dodecanoate (SDD), sodium 10-undecenoate (SUD), and dodecyltrimethylammonium bromide (DTAB). This system has a potential practical use in health applications as an antiseptic soap and has a coacervate region in its triangular phase diagram, i.e., a region in which two immiscible liquid phases having the same components but different concentrations coexist. We have experimentally determined the CMC and the degree of ionization of the micelles as functions of the global mixture composition. The thermodynamic behavior of this system has been analyzed with both procedures: the MRST and the EOMMM employing a fourth-order Margules equation.16,17 In a previous work, we studied the aqueous binary system DTAB−SUD. Both components have biocide properties: DTAB is bactericide and SUD is a bactericide and fungicide (see The Biological Effects of Surfactants in the SI). This system does not precipitate in any proportion and shows a coacervate domain.18 However, for practical applications as an antiseptic, for instance for surgeons’ hand soap, it is convenient to dilute these components with a cheaper surfactant. Thus, we are now studying the DTAB−SUD−SDD mixture. Since the biological activity of

section about the EOMMM in the SI). Equation-oriented optimization simultaneously solves a system of equations in order to find the minimum/maximum of an objective function subject to a set of constraints. The EOMMM finds the Margules parameters (see Generalized Margules Formulations in the SI) and the micelle compositions that globally minimize the total free energy of micellization. It has been recognized as a main drawback that the original RST, and thus the MRST, deal with ionic surfactants as nondissociated components. However, the EOMMM contemplates the dissociation of ionic surfactants through the r parameter and proper expressions for the activities of each component in the micelles. Thus, the EOMMM can be employed for nonionic or ionic surfactants with or without the presence of a supporting electrolyte. The RST and MRST employ symmetric Margules formulations, as shown in Table 1 of Mukhopadhyay et al.7 Thus, the RST and MRST consider that the excess free energy Gexc can be approximated by a second-order Taylor series, that is to say, it considers symmetric (or strictly regular) solutions. The excess free energy for a binary asymmetric or subregular solution is Gexc = xixj(xjWij + xiWji), where xi is the mole fraction of component i in mixed micelles. However, as the RST employs Margules formulations for symmetric regular solutions, Wij = Wji = βRT, where β is the intramicellar interaction parameter of the RST. When xi → 0, ln γi → ln γ∞ i = Wij/RT = βij, so the interaction parameter βij can be interpreted as the energy of introducing a molecule of i into a pure-j micelle. Thus, in symmetric systems the excess free energy when the mole fraction of j in micelles (xj) approaches zero (infinite dilution) is equal to the excess free energy when xi approaches zero, and the limiting activity coefficients are equal. Assuming symmetry in systems that are evidently asymmetric can lead to mistaken conclusions through erroneous thermodynamic properties and may also mask useful information. Interesting conclusions can be derived from the results of the EOMMM considering asymmetric Margules formulations.9 Considering symmetric Margules formulations in the MRST implies that the ternary interactions can be completely described by binary interaction parameters (without ternary interaction parameters). The authors of the MRST thought this would enhance the value of the model as a predictive tool since the multicomponent result would be obtained without the introduction of new adjustable parameters beyond those established for the binary case.10 We will demonstrate in the present work that this cannot be taken for granted, at least in the system here addressed. The subregular Margules model for arbitrary numbers of components and expressions for higher derivatives of the free energy for use in binode and spinode location have been derived by Helffrich and Wood.11 It has been pointed out that ternary mixing parameters exist independent of their component binaries. Binary interaction coefficients only account for the forces arising between pairs of molecules. A ternary interaction parameter represents new interactions involving triplets of different molecular species and is unrelated to pairwise behavior. Thus, it is impossible to estimate ternary interaction coefficients having knowledge of only the mixing properties of the binaries. Ternary interaction parameters are in general negligible, but this says only that pairwise interactions dominate, not that triplet interactions cannot arise or that they are necessarily insignificant when they do arise. Besides, it has been concluded that quaternary or higher-order coefficients are not required for multicomponent subregular solutions. B

DOI: 10.1021/acs.iecr.7b02549 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article


and the micelles agglomerate because of the disappearance of the electrostatic energy barrier. Rodlike micelles agglomerate in bundles in the DTAB-rich side of the phase diagram, while spherical or globular micelles agglomerate in clusters in the SUDrich side. Because of the curvature imposed by the inclusion of the terminal double bond of SUD in the micelles’ surface and the hydration barrier, the system does not precipitate but gives the micelle agglomerates.22 The coacervate domain in the three-component system is shown in Figure 1 in a triangular phase diagram. It is evident that

these substances is given by the monomeric (nonmicellized) species, it is important to determine the CMCs of these mixtures as well as other properties. Since the bicomponent systems DTAB−SUD18 and SUD−SDD19 are both nonideal, the DTAB− SUD−SDD mixture has been presumed to be highly nonideal.

2. MATERIALS AND METHODS 2.1. Materials. Sodium dodecanoate (CH3(CH2)10COO−Na+) was obtained by neutralization of dodecanoic acid (≥99%), whereas sodium 10-undecenoate (CH2CH−(CH2)8COO−Na+; 98%) and dodecyltrimethylammonium bromide (CH3(CH2)11N+(CH3)3Br−; ∼99%) were purchased from Sigma-Aldrich (Argentina). Solutions were prepared with triply distilled water. Concentrated solutions of the pure amphiphiles were mixed in the appropriate proportions to obtain the different surfactant mixtures with different mole fractions of surfactant i without considering the water (αi, i.e., ∑αi = 1). 2.2. Methods. The CMC was determined by means of specific conductivity (κ) measurements using an Antares II conductimeter from Instrumentalia and an immersion cell. The conductivity was measured after successive additions of measured volumes of concentrated solution of surfactant to water. The device was calibrated as usual with KCl solutions. The critical micelle concentrations of the pure components (CMCi) and binary (CMCij) and ternary (CMCijk) mixtures were determined conductimetrically at 25 °C in a thermostatized bath. The CMC was determined from conductivity data employing the excess specific conductivity (Δκ = κ − κextrapolated, where κextrapolated is the specific conductivity extrapolated from the preCMC straight line at very low concentrations). Δκ was plotted against the total concentration, as proposed by Miura and Kodama,20 to magnify the conductivity changes at the CMC. This is very convenient when the difference in the slopes above and below the CMC is small since the error in the determination of the change of slopes is minimized, as illustrated in Figure S1 in the SI. The micelle degree of ionization, i.e., the fraction of the charge of the micellized surfactant ions that is not neutralized by the counterions in the Stern layer, can be computed with the Evans equation21 (see Evans Equation for the Micelle Ionization Degree in the SI) using the specific conductivity versus concentration data. The boundaries of the two-phase region (coacervate domain) in the tricomponent system were established by visual observation of the concentrations at which the coacervate disappears as SDD is added. Different compositions of SUD−DTAB mixtures within the bicomponent coacervate domain were titrated with a SDD solution until the coacervate disappeared. In order to complete the boundaries between the micellar and coacervate domains, some micellar solutions having the three components were titrated with different SDD−SUD mixtures until the coacervate appeared.

Figure 1. Triangular diagram for the three-component system DTAB− SUD−SDD. The region of existence of the coacervate is shown in green.

the inclusion of SDD in a certain proportion causes the dissolution of the coacervate, probably by increasing the surface potential of aggregates and thus their mutual repulsion, as can be deduced from the conditions for the formation of the coacervate in the bicomponent system.22 3.2. The CMC. 3.2.1. The Binary Systems. The nonideal bicomponent systems SUD−DTAB18 and SUD−SDD19 were previously studied, and we have now determined the CMC for the DTAB−SDD system. Tables SI, SII, and SIII in theSI report the experimental and ideal CMCs. The critical micelle concentrations of the pure components are CMCSUD = 0.120 mol·dm−3,18 CMCSDD = 0.021 mol·dm−3,19 and CMCDTAB = 0.015 mol·dm−3.18 In order to apply the original MRST, the binary interaction parameters have to be determined. As usually happens when applying the original RST, different values for the intramicellar interaction parameter were obtained for each mixture composition and differ in the extreme compositions, as can be seen in Figures S2−S4. However, the RST gives reasonable results only when the intramicellar interaction parameters do not differ very much, as happens in several binary systems where the micelle compositions computed with this procedure agree with those experimentally determined,23−26 while in others this does not occur.27,28 Hoffmann and Pössnecker29 showed that the error considerably increases when one component dominates in the mixed micelle. Thus, to minimize this error, an average β was obtained by discarding the extreme compositions: for αSDD between 0.2 and 0.8 we obtained the average value of βDTAB,SDD = −14.5 ± 1.1 in kBT units. In our previous works we obtained βSDD,SUD= −7.4 ± 1.119 and βDTAB,SUD = −5.5 ± 1.118 in kBT units. 3.2.2. The CMC in the Tricomponent System. The CMCs for about 80 mixtures were determined by conductimetry

3. RESULTS 3.1. Phase Diagram: The Coacervate Domain. Let us discuss first the binary system DTAB−SUD. This system shows a coacervate region between αSUD = 0.43 and 0.6218 where two phases are present, i.e., these phases are aqueous solutions of both surfactants with different concentrations. The upper solution is a diluted solution of mixed micelles, while the lower solution is a concentrated solution of agglomerated micelles. The coacervate appears when the micelle ζ potential goes to zero C


Article


Figure 2. Surface plots of the critical micelle concentrations as functions of the overall mixture composition: (a) experimental; (b) ideal; (c) obtained with the MRST; (d) obtained with the EOMMM.

and are represented on the triangular diagram in Figure 2a. Figure 2b corresponds to the ideal CMC according to Clint’s relation (see the SI), and Figure 2c,d corresponds to CMCs calculated with MRST and EOMMM, respectively. The same results are reported as heat maps in Figure S5. The surfactant mixtures within the coacervate domain were not considered in the model. The EOMMM as presented in Schulz and Durand8 was modified in order to improve the virtues of the equationoriented approach in fitting the CMCs of a considerable number of mixtures. First, the EOMMM was solved to global optimality as in Schulz and Durand,8 but the minimization of the sum of total free energies as the objective function was replaced with the minimization of the sum of quadratic errors in the calculated critical micelle concentration, CMCijk. Then a second optimization was performed on the results, this time minimizing the total free energy and allowing the already determined CMCijk to deviate by at most 5% of its value. Figure S6 shows the absolute relative error of the CMCijk calculated with the EOMMM and MRST. Although the scale in the figures is until 1, in the case of the MRST some values are higher, up to 3.5. Each mixture has an average 90% absolute relative error when calculated with the MRST and 30% with the EOMMM. Moreover, the MRST fails to represent the CMC behavior close to the coacervate domain, while a quite good representation is attained with a fourth-order Margules formulation in the EOMMM.16,17 In order to study the tricomponent system in detail, we considered the mixtures with compositions on the bisectors of the three angles of the triangular phase diagram, i.e., when αSDD = αSUD (with αDTAB as the independent variable), αSDD = αDTAB (with αSUD as the independent variable), and αSUD = αDTAB (with αSDD as the independent variable), as shown in Figure 3. Some points of the plots of the CMC along the bisectors

Figure 3. Critical micelle concentrations for the tricomponent system SUD−SDD−DTAB along the three bisectors of the angles of the triangular phase diagram. CMCijk experimental, ● ; ideal,  ; computed by the MRST, ▲; computed with the EOMMM, □; coacervate formation, ■.

αSDD = αDTAB and αSUD = αDTAB are within the coacervate region and correspond to the minimal concentration at which the coacervate appears. Although the behavior is in general nonideal, the mixtures along the bisector αSDD = αSUD approach ideality. 3.3. The Mixed Micelle Ionization Degree. The micelle degree of ionization, i.e., the fraction of the charge of the micellized surfactant ions that is not neutralized by the counterions in the Stern layer, was calculated with the Evans equation (see the SI). Figure 4 shows the micelle ionization degree (α) for all of the mixtures in the triangular phase diagram. The α values in the tricomponent system are higher than those in the bicomponent ones (Figures S7 and S8). This is caused by the partial D


Article


Figure 4. Micelle ionization degree as a function of the mixture composition.

Figure 5. Mixed micelle mole fractions of the components (xDTA+, xUD− and xDD−) calculated with the EOMMM as a function of the total mixture mole fractions (αDTAB, αSUD, and αSDD).

complicated because of the existence of two different counterions (e.g., Cl− and Br−) and steric effects (such as those caused by the presence of a benzyl group or bulky, highly hydrated nonionic headgroups) that are superimposed with the electrostatic interactions caused by charged headgroups or by the intercalation of nonionic headgroups among ionic ones. This latter effect seems to be influenced by the sign of the charge in the ionic headgroup.22,31 Because the MRST model failed in describing the behavior of multicomponent systems,36 the thermodynamic analysis will be based on the results of the EOMMM model. The EOMMM equations here used for the ternary system correspond to a fourth-order Margules equation with Bij = Bji:

mutual neutralization of the oppositely charged headgroups and also by the inclusion of the double bond of the undecenoate tail in the Stern layer, which increases the surface of the micelles and thus releases counterions because of the reduction of the surface potential.30

4. DISCUSSION 4.1. EOMMM Equations. There are some works on mixtures of three dissimilar surfactants: sodium dodecylbenzenesulfonate, tetradecyltrimethylammonium bromide (C14TAB), and polyoxyethyleneoctylphenols;31 hexadecyltrimethylammonium bromide (C16TAB), hexadecylbenzyldimethylammonium chloride (C16BzDAC), and polyoxyethylene(17)-cethyl ether (Brij 58);32 hexadecylpyridinium chloride (C16 PyC), Tween-40, and Brij 56;33 combinations of C16TAB, C16BzDAC, dodecyltrimethylammonium bromide (C12TAB), dodecylethyldimethylammonium bromide (C12EDAB), Brij-58, and Brij −30;34 and C16PyC, C16TAB, and Brij-56.35 The nonidealities detected in these systems are mainly attributed to the interactions among the dissimilar polar head groups. In some cases, the interpretation is

Gexc = x1x 2(A 21x1 + A12 x 2) + x12x 2 2B12 + x1x3(A31x1 + A13x3) + x12x32B13 + x 2x3(A32 x 2 + A 23x3) + x 2 2x32B23 + x1x 2x3(A 21 + A13 + A32 + J1x1 + J2 x 2 + J3x3) (1) E


Article


Figure 6. Total free energy of micellization of the mixed surfactant system as a function of the composition of the overall surfactant mixture (left) and the mixed micelle mole fractions (right). The void region in the left diagram corresponds to the coacervate domain.

Figure 7. Ideal free energy of mixed micellization as a function of the overall surfactant mixture composition (left) and the mixed micelle composition (right). The void region in the left diagram corresponds to the coacervate domain.

and J3 = 226607.377, where the subscript 1 stands for SDD, 2 for SUD, and 3 for DTAB. 4.2. Mixed Micelle Mole Fractions and Total and Excess Free Energies of Micellization. Figure 5 shows the mixed micelle mole fractions of the components obtained from the EOMMM as a function of the total mole fractions of the components in the mixtures. The maximum mole fraction of each component in the mixed micelles is near the side representing its mole fraction in the triangular diagram, for instance xDD− is maximum along the αSDD side. However, there are some regions in which secondary maxima appear. Figure S9 shows these values along the bisectors of the three angles of the phase diagram. The plots show interruptions in the compositions falling in the coacervate domain in the three bisectors, where the mixed micelles are predominantly composed of the soap ions (UD− and DD−). The total free energy of micellization can also be obtained from the model and is plotted in Figures 6 and S10. The more thermodynamically favored regions are those corresponding to

RT ln(γ1) = x 2 2[A12 + 2x1(A 21 − A12 + B12 ) − 3B12 x12] + x32[A13 + 2x1(A31 − A13 + B13) − 3B13x12] + x 2x3[A 21 + A13 − A32 + 2x1(A31 − A13) + 2x3(A32 − A 23) − 3B23x 2x3 + J1x1(2 − 3x1) + J2 x 2(1 − 3x1) + J3x3(1 − 3x1)]

(2)

The expressions for the activity coefficients of the other components can be obtained by rotating the subscripts as 1 → 2 → 3 → 1 because this rotation leaves the expression for Gexc unchanged (see Generalized Margules Formulations in the SI). The Margules parameters obtained were A12 = −82989.6297, A13 = −37607.0508, A21 = −72712.3972, A23 = −76860.9245, A31 = −67111.1709, A32 = 32747.7379, B12 = 239930.148, B13 = 186514.41, B23 = 121622.652, J1 = 439862.453, J2 = −200876.527, F


Article


Figure 8. Excess free energy of mixed micellization as a function of the overall surfactant composition (left) and the mixed micelle composition (right). The void region in the left diagram corresponds to the coacervate domain.

many double bonds in the Stern layer decreases the surface potential, thus releasing Na+ ions to the micelle Gouy−Chapman layer. The effect is similar to that produced by the inclusion of a nonionic surfactant in the mixed micelle.19 The catanionic systems SUD−DTAB and SDD−DTAB show high α values. Since α reflects the fraction of counterions that are not in the micelle Stern layer, this means that more counterions are released from the micelle surface because the micelle charged surface has a low electrical potential. This is the case because part of the charge of one component is neutralized by the headgroups of the other component. The α values in the tricomponent system are higher than in the bicomponent ones (see Figures 4 and S8). This is caused by the partial mutual neutralization of the oppositely charged headgroups and also by the inclusion of the double bond of the undecenoate tail in the Stern layer, which increases the surface of micelles and thus releases counterions because of the reduction of the surface potential.

cationic−anionic mixtures, while the anionic−anionic ones are less favored. In many systems, such as alloys, spinodal decomposition occurs when the second derivative of the Gibbs energy with respect to composition is negative, i.e., a spontaneous decomposition occurs into two phases, one solute-rich and the other solute-depleted. Jiang et al.12 discussed a thermodynamic criterion of spinodal decomposition in ternary systems. Thus, the condition for decomposition of a relative maximum in free energy between two minima (closed convex region) is similar to what is observed around the coacervate region. The plot of the total free energy of mixed micellization as a function of the micelle composition (Figure 6 right) shows that the more stable micelles are those richer in DTA+ and UD− ions, while the micelles richer in DD− are the less favored. The attractive interaction between the cationic and anionic headgroups is energetically favorable. Besides, the inclusion of the terminal −CHCH2 groups of the undecenoate chain between the charged headgroups10 favors the aggregation because they reduce their mutual repulsion. As shown in Figure 7, the ideal free energy of mixed micellization follows the same tendency as the total free energy, but the regions of minimal free energy are larger. From the energy values, it can be seen that the actual mixed micelles are more stable than the ideal ones, as reflected in the negative values of the excess free energy of mixed micellization in Figure 8. The maximum values of the excess free energy are around the coacervate domain. Thus, inside the coacervate region the excess free energy is positive, leading to the separation into two phases. The plot of the excess free energy as a function of the mixed micelle composition (Figure 8 right) shows the complexity of the interactions in the tricomponent system, reinforcing the opinion that these systems cannot be adequately studied with a model that does not consider thermodynamic asymmetry and ternary interaction parameters. 4.3. The Mixed Micelle Ionization Degree. The degrees of micelle ionization (α) of the bicomponent systems are shown in Figures S7 and S8. The SDD−SUD binary system shows an almost invariant α value, which seems reasonable because the two polar headgroups are equal. However, when SUD clearly predominates (αSUD > 0.8), α is augmented because the inclusion of

5. POTENTIAL USE FOR MEDICAL APPLICATIONS As mentioned in the introduction, both SUD and DTAB have biological effects as described in detail in The Biological Effects of Surfactants in the SI. In general, only the monomeric or nonmicellized surfactant species are biologically active. Micelles act as a reservoir of monomers to restore the monomers that are adsorbed in the cells’ membranes. In the tricomponent system, there is a large region in which the experimental CMC is higher than the ideal one (Figure 9), which indicates that a high amount of unmicellized surfactant molecules are available for biological activity. Figure 9 shows the region in the three-component mixtures in which the experimental CMC is higher than the ideal one. As can be seen, the region is close to the center of the diagram, i.e., a significant part of the system may be formed by the cheaper common soap, SDD, maintaining a relatively high concentration level of the active surfactants, SUD and DTAB. The biological activity of surfactants is generally enhanced when they are mixed with other substances and surfactants, when their hydrophobicity is enough to be included in the cell membrane bilayer, and when a high CMC ensures a high concentration G


Article


one and thus the amount of unmicellized surfactant molecules available for biological activity is higher.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b02549. Phenomenological models, molecular thermodynamic methods, regular solution theory (RST), the multicomponent regular solution theory (MRST), the equation-oriented mixed micellization model (EOMMM), biological effects of surfactants, and the Evans equation for the micelle ionization degree (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Erica P. Schulz: 0000-0003-4732-6406

Figure 9. Experimental minus ideal CMCs as a function of the total compositions. The region of the triangular diagram in which the experimental CMC is higher than the ideal one is highlighted in green. The void region in the diagrams corresponds to the coacervate domain.

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS R.B.P. received a fellowship of the Science Research Council of Buenos Aires Province (CIC). E.P.S. is an adjunct, G.A.D. is an assistant, and H.R. is an independent researcher at CONICET. This work was supported by Grants PGI-UNS 24/Q073 and PGI-UNS 24/F067 of Universidad Nacional del Sur and PICT 2013 (D) Nro 2070 of Agencia Nacional de Promoción ́ Cientifica y Tecnológica (ANPCyT) and PIP-GI 2014 Nro 11220130100668CO (CONICET).

of biologically active monomers. All of these conditions are fulfilled by at most a significant part of the compositions in the ternary phase diagram presented, making this system an attractive model for the rational formulation of other soluble systems with different biologically active surfactants. On the basis of the considerations detailed in The Biological Effects of Surfactants in the SI, the system here studied may be a basis to design bactericide preparations in the medicine, food, and cosmetic industries.

■

REFERENCES

(1) Phenomena in Mixed Surfactant Systems; Scamehorn, J. F., Ed.; ACS Symposium Series, Vol. 311; American Chemical Society: Washington, DC, 1986. (2) Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series, Vol. 501; American Chemical Society: Washington, DC, 1992. (3) Rubingh, D. N. Mixed Micelle Solutions. In Solution Chemistry of Surfactants, Vol. 1; Mittal, K. L., Ed.; Springer: New York, 1979; pp 337− 354. (4) Holland, P. M.; Rubingh, D. N. Nonideal multicomponent mixed micelle model. J. Phys. Chem. 1983, 87, 1984−1990. (5) Letellier, P.; Mayaffre, A.; Turmine, M. Thermodynamics of Mixed Micelles: Determination of the Aggregate Composition. J. Colloid Interface Sci. 2008, 327, 186−190. (6) Letellier, P.; Mayaffre, A.; Turmine, M. Thoughts on the ideal behavior of mixed micelles and the approppiate application of regular solution theory (RST). J. Colloid Interface Sci. 2011, 354, 248−255. (7) Mukhopadhyay, B.; Basu, S.; Holdaway, M. J. A discussion of Margules-type formulations for multicomponent solutions with a generalized approach. Geochim. Cosmochim. Acta 1993, 57, 277−283. (8) Schulz, E. P.; Durand, G. A. Equation oriented mixed micellization modeling based on asymmetric Margules-type formulations. Comput. Chem. Eng. 2016, 87, 145−153. (9) Pereyra, R. B.; Schulz, E. P.; Durand, G. A.; Ritacco, H.; Schulz, P. C. Thermodynamic analysis of an asymmetric system: Aqueous sodium dehydrocholate-hexadecyltrimethylammonium bromide mixed micelles. Colloids Surf., A 2016, 509, 675−683. (10) Holland, P. M. Nonideal mixed micellar solutions. Adv. Colloid Interface Sci. 1986, 26, 111−129. (11) Helffrich, G.; Wood, B. Subregular model for multicomponent solutions. Am. Mineral. 1989, 74, 1016−1022.

6. CONCLUSIONS The SUD-SDD-DTAB mixed system is highly nonideal and has a coacervate domain in its triangular phase diagram. The binary systems have been analyzed under the light of the EOMMM. The MRST, which assumes symmetric thermodynamic behavior and neglects ternary interactions, failed to model this system with complex interactions among the three components. Therefore, the recently developed EOMMM approach has been employed, and a fourth-order Margules equation was required to give a proper representation, especially in the surroundings of the coacervate domain. Besides, the original EOMMM was modified in order to improve the virtues of the equation-oriented approach in fitting the CMCs of a considerable number of mixtures. The present work shows the versatility of the EOMMM and its advantages over the generally used MRST. It can be concluded from the ionization degree determinations that as a result of the inclusion of the terminal double bonds in the SUD tail into the mixed micelle surface and the partial mutual neutralization of charges by the inclusion of cationic and anionic headgroups in the Stern layer, the majority of the counterions (Br− and Na+) are released from the micelles to the Gouy− Chapman diffuse layer. Since both SUD and DTAB have antimicrobial and antifungal activities, the ternary surfactant system studied here has a high potential for use in food and pharmaceutical applications such as bioactive soap. A region of the triangular phase diagram has been identified where the experimental CMC is higher than the ideal H


Article


(34) Dar, A. A.; Rather, G. M.; Das, A. R. Mixed micelle formation and solubilization behavior toward polycyclic aromatic hydrocarbons of binary and ternary cationic-nonionic surfactant mixtures. J. Phys. Chem. B 2007, 111, 3122−3132. (35) Chakraborty, T.; Ghosh, S.; Moulik, S. P. Micellization and related behavior of binary and ternary surfactant mixtures in aqueous medium: cethylpyridinium chloride (CPC), acetyl trimethyl ammonium bromide (CTAB) and Polyoxyethylene (10) cetyl ether (Brij-58) derived system. J. Phys. Chem. B 2005, 109, 14813−14820. (36) Schulz, E. P.; Rodriguez, J. L.; Minardi, R. M.; Miraglia, D. B.; Schulz, P. C. On the applicability of the regular solution theory to multicomponent systems. J. Surfactants Deterg. 2013, 16 (5), 795−803.

(12) Jiang, B.; Zhang, M.; Wei, Q.; Xu, Z. Thermodynamic criterion of spinodal decomposition in ternary systems. Acta Metall. Sin. (Engl. Lett.), Ser. B 1991, 4 (2), 75−81. (13) Peng, D- Y. Extending the Van Laar Model to Multicomponent Systems. Open Thermodyn. J. 2010, 4, 129−140. (14) Chien, H. H. Y.; Null, H. R. Generalized Multicomponent Equation for Activity Coefficient Calculation. AIChE J. 1972, 18 (6), 1177−1183. (15) Cheng, W.; Ganguly, J. Some aspects of multicomponent excess free energy models with subregular binaries. Geochim. Cosmochim. Acta 1994, 58 (18), 3763−3767. (16) Gokcen, N. A.; Reddy, R. G. Thermodynamics, 2nd ed.; Plenum Press: New York, 1996; p 183. (17) Hala, E.; Pick, J.; Fried, V.; Vilim, O. Vapor−Liquid Equilibrium, 2nd ed.; Pergamon Press: New York, 1967. (18) Sierra, M. B.; Morini, M. A.; Schulz, P. C. The catanionic system sodium undecenoate-dodecyltrimethylammonium bromide at low concentration. Colloid Polym. Sci. 2004, 282 (6), 633−641. (19) Sierra, M. B.; Morini, M. A.; Schulz, P. C.; Junquera, E.; Aicart, E. Effect of double bonds in the formation of sodium dodecanoate and sodium 10-undecenoate mixed micelles in water. J. Phys. Chem. B 2007, 111, 11692−11699. (20) Miura, M.; Kodama, M. The Second CMC of the Aqueous Solution of Sodium Dodecyl Sulfate. I. Conductivity. Bull. Chem. Soc. Jpn. 1972, 45, 428−431. (21) Evans, H. C. Alkyl sulphates. Part I. Critical micelle concentrations of the sodium salts. J. Chem. Soc. 1956, 0, 579−586. (22) Sierra, M. B.; Messina, P. V.; Morini, M. A.; Ruso, J. M.; Prieto, G.; Schulz, P. C.; Sarmiento, F. The nature of the coacervate formed in the aqueous dodecyltrimethylammonium bromide-sodium 10-undecenoate mixtures. Colloids Surf., A 2006, 277, 75−82. (23) Makayssi, A.; Lemordant, D.; Treiner, C. Structural change and micellar composition in aqueous solutions of binary cationic surfactant mixtures as deduced from cross-flow ultrafiltration experiments. Langmuir 1993, 9, 2808−2813. (24) Moya, S. E.; Schulz, P. C. The Aggregation of the Sodium Dodecylsulfate - n-Octanol -Water System at Low Concentration. Colloid Polym. Sci. 1999, 277 (8), 735−742. (25) Peyre, V. Determination of activities of mixed micelles involving neutral surfactants. Langmuir 2002, 18, 1014−1023. (26) Rodríguez, J. L.; Minardi, R. M.; Schulz, E. P.; Pieroni, O. I.; Schulz, P. C. The Composition of Dodecyltrimethylammonium Bromide-Hyamine Mixed Micelles in Water. J. Surfactants Deterg. 2012, 15, 147−155. (27) Hines, J. D.; Thomas, R. R.; Garrett, P. R.; Rennie, G. K.; Penfold, J. Investigation of mixing in binary surfactant solutions by surface tension and neutron reflection: strongly interacting anionic/zwitterionic mixtures. J. Phys. Chem. B 1998, 102, 8834−8846. (28) Eads, C. D.; Robosky, L. C. NMR Studies of binary surfactant mixture thermodynamics: molecular size model for asymmetric activity coefficients. Langmuir 1999, 15, 2661−2668. (29) Hoffmann, H.; Pössnecker, G. The mixing behavior of surfactants. Langmuir 1994, 10, 381−389. (30) Schulz, P. C.; Rodríguez, J. L.; Minardi, R. M.; Sierra, M. B.; Morini, M. A. Are the mixtures of homologous surfactants ideal? J. Colloid Interface Sci. 2006, 303, 264−271. (31) Graciaa, A.; Ben Ghoulam, M.; Marion, G.; Lachaise, J. Critical concentrations and compositions of mixed micelles of sodium dodecylbenzenesulfonate, tetradecyltrimethylammonium bromide and polyoxyethyleneoctylphenols. J. Phys. Chem. 1989, 93, 4167−4173. (32) Dar, A.; Rather, G. M.; Ghosh, S.; Das, A. R. Micellisation and interfacial behavior of binary and ternary mixtures of model cationic and nonionic surfactants in aqueous NaCl medium. J. Colloid Interface Sci. 2008, 322, 572−581. (33) Ghosh, S. Surface chemical and micellar properties of binary and ternary surfactant mixtures (cetyl Pyridinium Chloride, Tween-40 and Brij-56) in an aqueous medium. J. Colloid Interface Sci. 2001, 244, 128− 138. I


Equation-Oriented Mixed Micellization Modeling of a Subregular

Recommend Documents