Micellisation of Binary Mixtures of Surfactants Na-DeoxycholateâNa

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Micellisation of Binary Mixtures of Surfactants Na-Deoxycholate−NaDecyl Sulfate and Na-Hyodeoxycholate−Na-Decyl Sulfate in Water Solutions: Rational Development of the Thermodynamic Model for the Excess Gibbs Energy (GE) Mihalj Poša,* Ana Pilipović, Vesna Tepavčević, and Stoja Obradović Faculty of Medicine, Department of Pharmacy, University of Novi Sad, Hajduk Veljkova 3, 21000 Novi Sad, Serbia S Supporting Information *

ABSTRACT: Micellisation of two binary mixtures, sodium decyl sulfate (NaDS)−sodium deoxycholate (NaDC) and sodium decyl sulfate (NaDS)−sodium hyodeoxychlolate (NaHDC), is examined, in consideration with the different molar fractions of the monomers (x = 0.1−0.9). In bile acids, the OH groups attached to steroid skeletons are spatially ecraned, while the sulfate group of NaDS is not ecraned. Thus, NaDC and NaHDC in relation to NaDS have different coordination numbers in the micelle pseudophase. On the basis of stereochemical analysis, rational design shows that the excess Gibbs energy (GE) of formation of the binary mixed micelle throughout the whole composition range can be described with a two-parameter Margules function, while the GE function from the regular solution theories is applicable only in the narrow interval of the bile salts molar fractions (x = 0.5− 0.8). Coefficient of activity (hypothetical) of the infinitely diluted mixed micelle of NaDS in the presence of NaDC shows significant thermodynamic stabilization of decyl sulfate ions in comparison with the infinitely diluted hypothetical mixed micelle with NaHDC. This stabilization of decyl sulfate ions is probably due to the cooperative binding of NaDS by parallel OH groups of NaDC. mixture.5 In real mixed micelles, related to the Gibbs energy of mixing, there is the excess Gibbs energy (GE) in the formation of the ideal mixed micelle (ΔidGmM ij ), i.e., the Gibbs energy of = ΔidGmM ± GE.7,9 If the formation of real micelle is ΔreGmM ij ij E there is G < 0, the real mixed micelle is thermodynamically more stable than that of the ideal mixed micelle.6,7 Since micelle systems have a wide industrial application (pharmaceutical, food, oil, etc.),10−13 in terms of finding their ecological trail, it is important to find such a mixture of surfactants that forms micelles at the lowest possible total concentration. If at E some αi value GE < 0 and |ΔidGmM ij | ≪ |G |, it is possible that the cmcij has a lower value than even that from the cmc value of a more hydrophobic micelle building unit.7 The inclusion of additive into the association of an amphiphile will affect its physicochemical properties, such as the degree of ionization, reaction rates, and clouding or phase separation.14−18 Bile salts are steroid biosurfactants synthesized in the liver of vertebrates, and aside from their physiological, i.e., regulator, roles19−21 (in metabolism), they have an application in pharmaceutical formulations.22−25 Bile salts form relatively

1. INTRODUCTION In concentrations above the critical micelle concentration (cmci), surfactants in water solution form micelles, aggregates whose interior is hydrophobic, while the external boundary surface with water solution is hydrophilic.1,2 Binary mixtures of surfactants form mixed micelles at critical micelle concentration (cmcij), which represents the total concentration of the binary mixture of surfactants in a water solution. If p,T = const., the cmcij depends on the individual molar fractions of surfactants in the binary mixture (αi).3−5 Formation of ideal binary mixed micelle includes the same thermodynamic processes as the formation of monocomponent micelles out of its building units, that is, between the different micelle building units of the mixed micelle, there are the same intermolecular interactions as those between the building units of the monocomponent micelle.6,7 The driving force for the formation of ideal binary mixed micelles is of the entropic origin and represents the change in the Gibbs energy of mixing of monocomponent micelles as the pseudophase.6−8 For the binary mixture of surfactants that form ideal mixed micelle, the critical micelle concentration depends on the αi linearly, and it changes from the cmc value of the one building unit (cmci) to the cmc value of the second building unit (cmcj), i.e., the value of the cmcij is always lower than that of the cmc value of the less hydrophobic surfactant of the binary © XXXX American Chemical Society

Received: October 7, 2017 Accepted: January 29, 2018

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DOI: 10.1021/acs.jced.7b00880 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Figure 1. Examined surfactants.

Table 1. Provenance and Purity of Materials Used compound

source

CAS number

mass fraction purity

purification method

analysis method

sodium decyl sulfate (NaDS) sodium hyodeoxycholate (NaHDC) sodium deoxycholate (NaDC) pyrene NaCl

Sigma Sigma-Aldrich Calbiochem Aldrich Sigma-Aldrich

142-87-0 10421-49-5 302-95-4 129-00-0 7647-14-5

>0.980 >0.990 >0.980 >0.980 >0.990

none none none none none

none none none none

2. MATERIALS AND METHODS 2.1. Materials. Summary of provenance and purity of the studied materials is given in Table 1. 2.2. Spectrofluorimetric Measurements of cmc. Fluorescence measurements were carried out using an Agilent Cary Eclipse fluorescence spectrophotometer (Agilent Technologies, Santa Clara, California, USA). Pyrene was used as a fluorescence probe molecule. All solutions of surfactans were prepared using pyrene saturated water with NaCl (0.3 mol kg−1). For the preparation of a stock solution (15 mmol kg−1, for each examined system), the exact mass of a surfactant or binary mixture of surfactants in the defined molar ratio is dissolved in the corresponding amount of pyrene water with NaCl (0.3 mol kg−1).Working solutions were obtained from the stock solution by dilution (from 0.1 mmol kg−1 to 6 mmol kg−1 per steps of 0.1 mmol kg−1 for the binary mixtures of NaDC− NaDS. and from 0.1 mmol kg−1 to 2 mmol kg−1 per steps of 0.1 mmol kg−1 for the binary mixtures NaHDC−Na-DS). Fluorescence emission spectra of these solutions were recorded by employing an excitation wavelength of 334 nm. The intensities of the first (I1) and the third (I3) vibration bands of pyrene emission spectrum were measured at 373 and 384 nm, respectively (298 K). Critical micelle concentrations of pure surfactants, as well as cmc values of binary surfactant mixtures, were determined as intersections of two straight lines of different tangents in plane (I1/I3) − cT (Figure 2A and Tables S1−S21).29 2.3. Surface Tension Measurements. Surface tension (γ) measurements were carried out on a Krüss Easy Dyne tensiometer (Krüss GmbH,Hamburg, Germany) using the du Noüy ring method. To obtain the critical micelle concentration of the surfactants, surface tension was monitored as a function of each surfactant concentration (NaCl 0.3 mol kg−1 in water solution). Temperature was kept constant at 298 K (Figure 2B and Tables S22−S42). The tensiometers were calibrated using deionized, double distilled water, and surface tension values (70.9 ± 0.2 mN m−1) at 298 K agreed well with the literature (71−72 mN m−1).30

small micelles with several building units (2−16), and they have low capacity for the solubilization of a hydrophobic guest molecule.26,27,28 In bile canaliculus, bile salts form mixed micelles with phospholipids.19 For pharmaceutical formulations, especially those that need micelle solubilization, it is important to find the real mixed micelles of bile salts and certain surfactants, which would be thermodynamically more stable than ideal mixed micelles, i.e., those whose building units have synergistic interactions. The aim of this article is the examination of micellization (thermodynamic stabilization) of two binary mixtures of surfactants, sodium deoxycholate (bile salt)−sodium decyl sulfate and sodium hyodeoxycholate (bile salt) −sodium decyl sulfate (Figure 1). As the decyl sulfate possess a n-decane hydrophobic segment, it has good conformational flexibility in relation to a conformationally rigid steroid skeleton (whereby the length of the n-decane sequence corresponds to the length of the longitudinal axis of the cyclopentanoperhydrofenantren molecule graph). The n-decane chain can efficiently increase the hydrophobic domain of the mixed micelle when switching its conformation from all antiperiplanar (all-ap) to partially or whole (all) synclinal conformation (all-sc), i.e., it can efficiently adjust to the space that determines micelle−water interface. Since the examined micelle building units structurally differ, configurational entropy exists in the real mixed micelles, in comparison with that of the ideal binary mixed micelles, i.e., configuration entropy in this case presents a contribution to the excess entropy (SE). The formation of hydrogen bonds between the steroid OH groups of bile salts and the sulfate group of decyl sulfate is conditioned by the orientation of the decyl sulfate anion related to the other the steroid OH groups in micelle, i.e., whether they are cis-oriented (decyl sulfate anion and steroid OH group are on the same side of the steroid skeleton) or trans-oriented. Thus, the regular solution theory (RST) with SE = 0 approximation is not suitable for the examined systems, and the further goal is rational development of the novel thermodynamic model for the description of the excess Gibbs energy. B



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[RSTeqution, {x, x0}], where x0 responded to the starting point in the iterative procedure (x0 > 0 and x0 < 1), i.e., the solution of the RST equation was searched around the point x0. FindRoot iteratively applied the Newton’s procedure. If Newton’s method did not give the results, the software Mathematica was applied to the secant method. If FindRoot did not provide a solution near the originally set point x0, or the solution was not logically acceptable (x0> 1), than the point x0 was reset to a new value. Margules function of the second order was fitted using the software MATLAB 7.14 (The MathWorks).

3. RESULTS AND DISCUSSIONS Critical micelle concentrations of water solutions of the examined binary mixtures of surfactants, i.e., critical micelle concentrations of pure (monocomponent) surfactants, are determined in the presence of NaCl (0.3 mol kg−1). In this way, it can be taken that the ionic strength of a water solution of pure surfactants and of a water solution of the binary mixtures of surfactants is constant. Thus, in the thermodynamic expression, which describes the state of the binary mixed micelles, the coefficients of the activity for the monomer surfactants in water solutions are eliminated, i.e., if water phase electrostatic interactions between ionic surfactant monomers and micelle ions are not taken into account, but only the coefficients of activities of the surfactants in the micelle pseudophase are used.7,31,32 Critical micelle concentrations of the examined mixture of surfactants are determined using the spectrofluorimetric and tensiometric method. In the spectrofluorimetric method, a pyrene probe molecule is used. This method is invasive, since pyrene enters the hydrophobic interior of the micelle. Usually, in invasive methods, the probe molecules distort the micelle structure and consequently the detected critical micelle concentrations are lower than ones detected by non invasive methods (tensiometric method in this work). However, it is known that pyrene spends a relatively short time in one micelle, i.e., in a short interval of time it resides in many micelles. Therefore, it is taken that this probe molecule only slightly disturbs or does not disturb the structure of the micelle.33 This is confirmed by our measurements, according to which the critical micelle concentrations determined by the spectrofluorimetric method and by the tensiometric method do not statistically differ (Table 2).

Figure 2. (A) Dependence of the ratio of pyrene emission spectre of the first and third vibration bands (I1/I3) on the total concentration of the bile acid salt (ct) (cmc = critical micelle concentrations). (B) Dependence of the surface tension (γ) on the logarithm of the total concentration of the bile acid salt (log ct) (sodium deoxycholate (NaDC)−sodium decyl sulfate (NaDS); NaDC molar ratio in the binary surfactant mixture αB = 0.6; T = 295 K; p = 1.00 × 105 Pa).

2.4. Calculations. Molar ratio of the component i in the binary mixed micelle, according to the RST equation, 1 = md md 2 2 ((xmd i ) ln(aicmc/cmcixi ))/(1 − xi ) ln((1 − αi) cmc/cmcj (1 md 2−4 md − xi )) (xi = model dependent molar ratio of the component i in the binary mixed micelle, αi = molar ratio of the surfactant i in the binary mixture of surfactants, cmci and cmci = critical micelle concentrations of the components i and j, cmc = critical micelle concentration of the binary mixture of surfactants), was calculated using the software Mathematica 11 (Wolfram), by means of the function FindRoot

Table 2. Experimental Values of Critical Micelle Concentrations (cmc)a,b,c Na-deoxycholate−Na-decyl sulfate −1

Na-hyodeoxicholate−Na-decyl sulfate −1

αB

cmc (mmol kg ) SF

cmc (mmol kg ) TM

cmc (mmol kg−1) SF

cmc (mmol kg−1) TM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

7.581 4.025 3.684 3.344 3.213 3.103 3.051 2.988 2.968 3.183 2.989

7.605 4.201 3.835 3.391 3.201 3.150 3.140 3.051 3.095 3.255 3.00

7.581 1.423 1.350 1.293 1.321 1.351 1.514 1.493 1.695 1.650 6.730

7.605 1.485 1.391 1.302 1.337 1.366 1.524 1.501 1.706 1.659 6.741

αB = NaDC or NaHDC molar ratio in the binary surfactant mixture, SF = spectrofluorimetry, TM = tensiometry, water solution of NaCl: 0.3 mol kg−1, T = 295.15 K; p = 1.00 × 105 Pa. bStandard uncertainties, u, are u(T) = 0.1 K, u(p) = 0.002 MPa, u(α) = 0.01, and u(cmc) = 0.008 mmol kg−1 (level of confidence = 0.68). cIn each measuring unit of a molality (mol per kilogram of solvent), the mass of a solvent refers to the pure water. a

C



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In the literature, describing the excess Gibbs energy (GE) of the formation of the binary mixed micelles usually uses eq 1 from the regular solution theory (RST)2−5,7,31,32,34 GE = RTβijxixj

(1)

where x corresponds to the molar fraction of surfactant in the pseudo-micelle phase (binary mixed micelle), R and T are the universal gas constant and the thermodynamic temperature on which mixed micelles are formed. In eq 1, βij is the interaction parameter between the two structurally different micelle building units. When β ij < 0, real mixed micelle is thermodynamically more stable than the hypothetical ideal mixed micelle (GE < 0). When βij > 0, the binary mixed micelle is less stable than the ideal binary mixed micelle (GE > 0).7 The coefficient of the interaction, βij (z, T, ΔEij), is the function of the interaction energy (ΔEij) and the coordination number of the micelle surfactants (z).3,4 In the RST approximation, it is taken that both surfactants of the mixed micelle get the same coordination numbers. The RST function, GE (eq 1), is symmetrical, in relation to the composition of the mixed micelle xixj.3−5 For the binary mixtures of surfactants whose molecular structures significantly differ, experimentally determined GE functions are not symmetrical (on T = const.). This is due to the fact that z differs for two structurally different building units of the mixed micelle, and z is the function of the composition of the mixed micelle.35 3.1. Thermodynamic Model for Formation of Binary Mixed Micelle from Structurally Different Building Units. Since GE mainly originates from intermolecular interactions of polar groups of different molecules of surfactants on the interface of the mixed micelle (i.e., GE originates from intermolecular interactions and conformational states of the building units that are not present in monocomponent micelles),7,8 it is necessary to make the stereochemical analysis of the availability of polar groups for the examined surfactants. Sodium decylsulphate (NaDS) is a derivative of n-alkane, while NaDC and NaHDC are OH derivatives of 5β-cholanoic acid. The structure of the hydrophobic parts of the examined surfactants determines different coordination numbers (z) for the OH groups of bile acids and for the sulfate group of NaDS. In bile acids (salts), the steroid skeleton spatially hinders a part of the space around OH groups.26,36,37 In Figure 3, the molecular graph of DC is represented with 2D local polar coordinate systems, in which the center steroid OH groups are located. The position of the C12 OH (D) group is out of the C2 symmetric axis of the main molecular subgraph (SG), while the position of the C3 OH (D) group is on the C2 symmetric axis, thus the C12 OH group is spatially ecraned under the azimuth angle (ϕ) of 180°, while the C3 OH group is ecraned under the azimuth angle of 90° (i.e., it is less spatially ecraned than the C12 OH group). Steric analysis shows that the steroid skeleton C6 OH group of the hyodeoxycholic acid is ecraned to the same degree as the C12 OH group of DC. The decyl chain of DS at the interface of the mixed micelle, does not locally or spatially ecrane the sulfate group (Figure 4). Besides, the carbon skeleton of the steroid system of rings (Figure 3) takes part in the spatial ecraning of the OH groups of the bile salts sin-axial (sa) hydrogens. Figure 5 shows that the C12 OH group (of the surfactant NaDC) possess 3 sa hydrogens that sterically prevent formation of the hydrogen bond with the sulfate group of the NaDS surfactant, that is located on the opposite side of the steroid skeleton, in relation

Figure 3. In the molecular graph of the steroid skeleton of deoxycholic acid (DC), the C3 and C12 OH groups are spatially hindered (SH = spatial hindering, SG = molecular subgraph, ϕ = azimuth angle, C2 = axis of symmetry of the second order).

Figure 4. From the projection of the polar head of the decyl sulfate anion (DS) in the 2D plane of the micelle surface, it can be seen that the decyl chain does not ecrane the approach to the sulfate group (the sulfate group is the point below which the alkyle chain is placed in the hydrophobic interior of the micelle, ϕ = azimuth angle).

to the position of the C12 OH group (due to the flexibility of the decyle chain sulfate group without a sa hydrogen that could be folded through the steroid skeleton). C24 carboxylate function participates in the spatial ecraning of the C12 OH D



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Figure 5. The deoxycholic acid anion C12 OH group is spatially ecraned with sin-axial (sa) hydrogens, i.e., with C24 carboxylate function (NP), so that the formation of the hydrogen bond with the sulfate group depends on the relative orientation of the decyl sulfate anion related to the steroid core. The C3 OH group is not ecraned with sin-axial hydrogens, therefore, the formation of the hydrogen bond with the sulfate group is not conditioned with the relative orientation of the decyl sulfate anion related to the steroid skeleton (2D lattice shows a difference in the coordination numbers of the two OH groups of deoxycholic acid anion).

group, due to the relative vicinity of the C17 side chain. The C3 OH group of NaDC and NaHDC surfactants does not have sa hydrogens, so it is less spatially ecraned in the formation of the hydrogen bond with the sulfate group of the second building unit (Figure 5). Although the C6 α equatorial (αe) OH group of the hyodeoxycholic acid also does not have sa hydrogens, α axial hydrogens (αa) from the A ring of the steroid skeleton (i.e., sinclinal (sc) C5 methilene group) spatially prevent the sulfate group from the decyl sulfate to form a hypothetical hydrogen bond on the opposite side of the steroid skeleton in relation to C6 OH group (Figure 6). The above stereochemical analysis shows that in the 2D lattice of the micelle interface OH groups of the examined bile salts, i.e., sulfate groups, have different coordination numbers (Figure 6). If OH groups of the steroid system of the bile acids salts have the average value of the coordination number zB (average value of C12 and C3 OH groups for NaDC, i.e., for C6 and C3 OH groups for NaHDC) and the number of bile acid anion forms, N B , the number of interactions, ω ij (hypothetical hydrogen bonds), with the sulfate groups of the decyl sulfate (NaDS) in the micelle interface (xs are molar fractions of the NaDS surfactant in the binary mixed micelle) is

ωij = 2z BNBxS

Similar to eq 2, if the coordination number of the sulfate group of NaDS is zS, the number of interactions (ω ji) between N S anions of the decyl sulfate and anions of bile salt is (xB, molar fraction of bile salt in binary mixed micelle) ωji = 2zSNSx B

(3)

However, eqs 2 and 3 overestimate the number of interactions. Namely, with the increase in the molar fraction of micellar bile acid anions (xB) in the hydrophobic phase, their cooperative binding becomes apparent, i.e., two anions of bile acid mutually bind over the α side of the steroid skeleton building the hydrogen bonds between OH groups.36 This cooperative binding of bile salts in the mixed micelle, on the global level (in a system as a whole), prevents interaction of the steroid OH groups with the sulfate group of NaDS (conformational effect of the steroid skeleton exposes the spatial shielding on the local level, which is manifested with different values of the coordination numbers of two building units). The probability that the approach of sulfate group to the steroid OH group in the mixed micelle phase is not obstructed by formation of dimmer aggregates of bile acid anions (pB) is proportional to the molar fraction of NaDS in the binary mixed micelle pseudophase, i.e., pB ∝ xS (4)

(2) E



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i.e., if the latest expression is multiplied by 1/2 (in the order of interaction type A−B, i.e., B−A would not be counted twice): ω Σ = z BNBpB xS + zSNSpS x B

(6)

By multiplying both terms of eq 6 by N/N, N = NB + NS, the next equation is obtained: ω Σ = z BpB Nx BxS + zSpS NxSx B = Nx BxS(z BpB + zSpS ) (7)

If the interaction energy, Δeij, and N (Avogadro’s number) equal L, the molar excess internal energy of the formation of the binary mixed micelle is U E = LΔeijx BxS(z BpB + zSpS ) = x BxS(z BΔEijpB + zSΔEijpS )

(8)

By introducing the coefficient of the interaction and taking into account eqs 4 and 5, eq 9 is obtained U E = RTx BxS(βBxS + βSx B)

(9)

while: βB =

z zB ΔEij and βS = S ΔEij RT RT

The formation of the mixed micelles is examined using constant pressure. By implementing the approximation that the change in the volume during the formation of the mixed micelles out of the monocomponent micelles is zero, i.e., that there is equality UE = HE, the excess enthalpy for the formation of the binary mixed micelle is HE = RTx BxS(βBxS + βSx B) Figure 6. The Newman projection formula (NP) shows that the C6 OH group of the hyodeoxycholic acid anion (HDC) is spatially ecraned with α-axial αa hydrogen’s of the steroid skeleton and with a synclinal sc methylene group. The 2D lattice shows a difference in the coordinate numbers of the C6 OH group of the HDC and the sulfate group of the decyl sulfate anion (DS).

(10)

It is known that in the formation of the binary mixed micelles, the effect of the entropic−entalpic compensation (EEC) exists,38 expressed with the next equation:

H = TCS + b

(11)

,where TC is the temperature of compensation, while b is the intercept (energetic value).38 By applying the EEC to eqs 11 and 10, the excess entropy is

With the increase in the molar fraction of the NaDS anion in the mixed micelle, grows the probability for switching of the decyl chain from the all antiperiplanar (all-ap) conformation to the twisted (globular) conformation (that, in the micelle phase fills grooves between the decyl segments in all-ap conformation).33 The decyl chain in the globular conformation prevents the approach of the bile acid anion to the sulfate group of the all-ap decyl sulfate (the globular decyl segment is more voluminous than the all-ap decyl segment and it is placed beside the all-ap conformation, i.e., in the groove of the mixed micelle). The probability that the twisted conformation of the decyl chain on the global level (in the binary mixed micelle as the system) does not prevent the formation of the hypothetical hydrogen bonds between the sulfate and the steroid OH groups (pS) is proportional to the concentration of the bile acid anion: pS ∝ x B (5)

RTx BxS(βBxS + βSx B) = TcSE + b

SE = R

T b x BxS(βBxS + βSx B) − Tc Tc

(12)

Taking into account eqs 10 and 12, the excess Gibbs energy of the formation of the binary mixed micelle can be presented with the expression (with GE = HE − TSE): ⎛ T⎞ b GE = RTx BxS(βBxS + βSx B)⎜1 − ⎟ + Tc ⎠ Tc ⎝

(13)

i.e., by introducing modified coefficients of the interaction: βBκ =

Bearing in mind eqs 2 and 3, the total number of interactions between the different building units of the binary mixed micelle is

z ⎛1 zB ⎛ 1 1⎞ 1⎞ ⎜ − ⎟ΔEij and βSκ = S ⎜ − ⎟ΔEij R ⎝T Tc ⎠ R ⎝T Tc ⎠

with the expression: GE = RTx BxS(βBκ xS + βSκ x B) +

ω Σ = 2(z BNBpB xS + zSNSpS x B) F

b Tc

(14)



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b/Tc ≪ 1 usually exists,38 therefore making eq 14 for the value | GE| > 1 GE ≈ RTx BxS(βBκ xS + βSκ x B)

and f mM ≠ 1. In order to calculate the composition of the binary j mixed micelle, i.e., the values of the coefficients of relative activity, the RTS starts from the assumption that excess Gibbs energy (degree of deviation from the ideal state) of the formation of the binary mixed micelle behaves according to the Porter equation for the excess Gibbs energy of liquid mixtures (RT β mb ij = Aij). This means that the model predetermines, i.e., accepts, that the GE function is symmetrical and it is valid even in the case when the excess entropy is not zero:40,41

(15)

.Eq 15 corresponds to the two parameter Margules equation.39 3.2. Comparison of the RST Excess Gibbs Energy, GE, with the Thermodynamic Model, in which GE Is Described with the Two Parameter Margules Function. Generally, in the formation of the binary mixed micelle for each micelle building unit, the chemical potential from the micelle pseudo-phase is equalized to the chemical potential from the water phase, i.e., solution (equilibrium is observed on the critical micelle concentration):

GE = Aij xixj

(18)

From eq 1, i.e., eq 18 with the application of the term (∂ nGE/∂ ni)p,T,nj = RT ln f md = RT βmd (∂ nxixj/∂ni)p,T,nj, model i ij md dependent coefficients of activity (f i ) are derived:

μi0 + RT ln cmci + RT ln f imM ximM = μi0 + RT ln cmcijαi

ln f imd = βijmd (xjmd)2 , i.e., ln f jmd = βijmd (ximd)2

(16)

where cmci = critical micelle concentration of the surfactant i, mM f mM = coefficient of the relative activity, is the molar i , i.e., xi fraction of the building unit I in the micelle pseudo-phase, cmcij is the critical micelle concentration of the examined binary mixture of surfactants with αi molar fraction of the component i, and μ0i represents the standard chemical potential of the surfactant i (extrapolated value of the chemical potential from infinitely diluted aqueous solution to the concentration value of 1 moldm−3).39 If f mM = 1, i.e., f mM = 1, then the binary mixed i j micelle is ideal and the critical micelle concentration of the binary mixture of the surfactants behaves according to the Clint expression (derived from the eq 16 and from xmM + xmM = 1):5 i j αj α 1 = i + cmcij cmci cmcj (17)

(19)

By applying the equivalence of chemical potentials (eq 16) and eq 19 in the RST, the next expression is derived:2−8 1=

(1 −

(ximd)2 ln(αicmc /cmciximd) md 2 xi ) ln((1 − αi)cmc /cmcj(1

− ximd))

(20)

from which the model dependent composition of the binary micelles (xmd i ) is determined iteratively. By knowing the composition of the mixed micelle from eqs 16 and 19, the model dependent coefficient of interaction can be determined (β md ij ), i.e., the model dependent excess Gibbs energy (Tables 3 and 4). The function of the model dependent excess Gibbs energy (GEmd) (eq 18) from the molar fraction of any building unit of the binary mixed micelle, for both of the examined binary mixtures of surfactants, is not symmetric (Figure 8), which means that the RST model for the excess Gibbs energy (model for describing the intermolecular interactions in the examined binary mixed micelles) is not acceptable. For the application of the function (eq 15) (the twoparameter Margules function), it is necessary to determine the composition (molar fraction) of the binary mixed micelle from the critical micelle concentrations of the pure surfactants and the critical micelle concentration of the binary mixtures of the surfactants by some model independent method. In this article, the Rodenas procedure is used.42 The molar fraction of some of the mixed micelle building units is (model independent molar fraction xmi i ): αiαj ∂cmc + αi ximi = − cmc ∂αi (21)

In the formation of the ideal binary mixed micelle, the critical micelle concentration of the binary mixture of surfactants linearly changes from the higher critical micelle concentration of the one pure component to the lower critical micelle concentration of the other pure component (Figure 7). For

Taking into account the equality of chemical potentials of the surfactants in a micelle and in a water phase (eq 16) and using the model independent molar fractions of the surfactants in the mi micelle pseudo-phase (xmi i , xj , (eq 21)), the critical micelle concentrations of the binary mixture of the surfactants (cmcij), the critical micelle concentrations of the pure surfactants (cmci and cmcj), and with the application of the model independent of the excess Gibbs energy for the binary mixture, GEmi = RT mi mi mi E (xmi i ln f i + xj ln f j ), the next equation is derived for Gmi (Tables 3 and 4):

Figure 7. Dependence of critical micelle concentration (cmc) on the content of the binary mixture of surfactants (αB) (NaDS = sodium decyl sulfate, NaHDC = sodium hyodeoxycholate, NaDC = sodiun deoxycholate; cmcid = the ideal critical micelle concentration; T = 295 K; p = 1.00 × 105 Pa).

both of the examined binary mixtures of surfactants, the cmcij deviates from the ideal linear dependence, and this deviation is higher in the system sodium hyodeoxicholate−sodium decyl sulfate ( Figure 7). This means that in the formed mixed micelles, some interactions exists that are not present in the monocomponent micelles of the building units, i.e., f mM ≠1 i

⎛ αicmcij αjcmcij ⎞ E ⎟ Gmi = RT ⎜⎜ximi ln mi + xjmi ln mi xi cmci xj cmcj ⎟⎠ ⎝

(22)

The Gibbs energy (eq 22) is fitted with the two parameter Margules function (eq 15) (Figure 9 and Table 5). By using the G



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Table 3. Thermodynamic Parameters of Micellization for Binary Mixtures of Surfactants Sodium Deoxycholate (NaDC)− Sodium Dodecyl Sulfate (NaDS)a,b Rodenas34

RST αB

xmd B

βmd ij

f md B

f md S

GEmd (kJ/mol)

xmi B

f mi B

f mi S

GEmi (kJ/mol)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.356 0.441 0.511 0.573 0.632 0.69 0.755 0.824 0.957

−2.35 −1.87 −1.76 −1.57 −1.46 −1.33 −1.27 −1.19 0.47

0.377 0.557 0.656 0.750 0.820 0.882 0.926 0.963 1.000

0.741 0.694 0.631 0.595 0.556 0.525 0.483 0.445 1.544

−1.339 −1.143 −1.091 −0.957 −0.843 −0.703 −0.583 −0.427 0.047

0.192 0.335 0.439 0.516 0.577 0.631 0.689 0.763 0.863

0.700 0.734 0.763 0.831 0.899 0.969 1.014 1.040 1.109

0.590 0.583 0.550 0.526 0.483 0.436 0.381 0.331 0.307

−1.225 −1.143 −1.123 −1.005 −0.913 −0.805 −0.716 −0.572 −0.175

αB = NaDC molar ratio in the binary surfactant mixture, xB = molar ratio of the NaDC in the binary mixed micelle, βij = interaction coefficients, f B = activity coefficient of NaDC in the mixed micelle, f S = activity coefficient of NaDS in the mixed micelle, GE the excess Gibbs energy; md = model dependent, mi = model independent, T = 295 K, and p = 1.00 × 105 Pa. bStandard uncertainties, u, are u(T) = 0.1 K, u(p) = 0.002 MPa, u(α) = 0.01, u(x) = 0.002, u(β) = 0.03, u( f B) = 0.03, u(f SU) = 0.03, and u(GE) = 5.02 kJ mol−1 (level of confidence = 0.68). a

Table 4. Thermodynamic Parameters of Micellization for Binary Mixture of Surfactants Sodium Hyodeoxycholate (NaHDC)− Sodium Dodecyl Sulfate (NaDS)a,b Rodenas34

RST αB

xmd B

βmd ij

f md B

f md S

GEmd (kJ/mol)

xmi B

f mi B

f mi S

GEmi (kJ/mol)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.391 0.432 0.460 0.483 0.506 0.532 0.556 0.589 0.624

−8.06 −7.38 −7.12 −6.82 −6.65 −6.25 −6.49 −6.37 −7.33

0.053 0.092 0.125 0.162 0.198 0.253 0.279 0.341 0.353

0.280 0.251 0.221 0.202 0.180 0.170 0.133 0.108 0.057

−4.788 −4.494 −4.389 −4.223 −4.126 −3.860 −3.975 −3.824 −4.266

0.126 0.223 0.300 0.366 0.429 0.497 0.580 0.685 0.822

0.167 0.179 0.191 0.214 0.233 0.271 0.267 0.293 0.268

1.339 0.637 0.396 0.285 0.203 0.160 0.101 0.065 0.026

0.075 −1.818 −2.832 −3.368 −3.771 −3.888 −4.273 −4.208 −4.280

a αB = NaHDC molar ratio in the binary surfactant mixture, xB = molar ratio of the NaHDC in the binary mixed micelle, βij = interaction coefficients, f B = activity coefficient of NaHDC in the mixed micelle, f S = activity coefficient of NaDS in the mixed micelle, GE the excess Gibbs energy, md = model dependent, mi = model independent, T = 295 K, and p = 1.00 × 105 Pa. bStandard uncertainties, u, are u(T) = 0.1 K, u(p) = 0.002 MPa, u(α) = 0.01, u(x) = 0.002, u(β) = 0.03, u(f B) = 0.03, u(f SU) = 0.03, and u(GE) = 4.10 kJmol−1 (level of confidence = 0.68).

Figure 8. Dependence of the model dependent excess Gibbs energy (GE) on the molar fraction of bile salt (xB) in the binary mixed micelle. For acceptance of the model, GE should be symmetric in relation to xB = 0.5 (example: sodium deoxycholate (NaDC)−sodium decyl sulfate (NaDS); T = 295 K, p = 1.00 × 105 Pa).

Figure 9. Dependence of the model independent excess Gibbs energy (GE) on the molar fraction of the bile salt (xB) in the binary mixed micelle. The red line represents the two parameter Margules function (TP) obtained by fitting of the model independent GE (sodium deoxycholate (NaDC)−sodium dodecyl sulfate (NaDS); T = 295 K and p = 1.00 × 105 Pa), and the gray line represents the regular solution theory (RST) GE function.

partial differential, (∂ nGE/∂ni)p,T,nj = RT ln f i, on the GE function (eq 15), the next equations are obtained for the Margules-type activity coefficients of the building units in the mixed micelle: ln fB = xS(βSκ + 2(βBκ − βSκ )x B)

ln fS = x B2(βBκ + 2(βSκ − βBκ )xS)

(24)

By applying limit values on the above functions: ln f B∞ = lim ln fB = βSκ

(23)

xB→ 0

H

(25) DOI: 10.1021/acs.jced.7b00880 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


Article

infinitely diluted state, when there are only decyl sulfate anions around bile acid anions. For both of the examined mixtures of the surfactants for the parameter βκS = ln f ∞ B , such values are obtained that statistically do not differ from the used bile acid anion. This means that ecraning of the C12 OH group of the NaDC and the C6 OH group of the NaHDC is about the same degree (Figures 5 and 6), i.e., that both anions have the same average coordination numbers. Relatively small negative values of parameter βκS = ln f ∞ B (Table 5) show that bile acid anions in the mixed micelles are, to a small extent, stabilized (surrounded especially with NaDS surfactants), when compared to that of the state in the monocomponent micelles. The small stabilization effect of the bile acid anions suggests that even in monocomponent micelles, steroid OH groups form hydrogen bonds, which was confirmed by the molecule dynamic simulations.43 The parameter βκB = ln f ∞ S statistically significantly differs depending on the type of the bile salt in the

Table 5. Parameters of the Two Parameter Margules Function of the Excess Gibbs Energy (GE)a GE = RTxBxS(βκBxs + βκSxB) binary mixtures of surfactants sodium deoxycholate (NaDC)−sodium dodecyl sulfate (NaDS) sodium hyodeoxycholate (NaHDC)− sodium dodecyl sulfate (NaDS)

βκB

= ln f ∞ βκS = ln f ∞ S B

R2

−8355

−0.037

0.9562

−5.75

−0.032

0.9624

a κ βS

= ln f ∞ B = coefficients of activity of bile acid anions at an infinitely diluted state, βκB = ln f ∞ S = coefficient of activity of decyl sulfate anion in an infinitely diluted micelle pseudo-phase.

ln f S∞ = lim ln fS = βBκ xS → 0

(26)

parameters of the Margules function (eq 15) for the excess Gibbs energy get an additional meaning. The parameter βκS represents the activity coefficients of the bile acid anions at the

Figure 10. If a binary mixed micelle of the examined surfactants contains a bile acids anion, deoxycholate anion (DC) cooperative binding happens to the decyl sulfate anion (DS). In the binary mixed micelle with the hyodeoxycholate anion (HDC), due the hydrophobicity of the α, as well as of the β side of the C ring of the steroid skeleton, HDC could have two orientations in the micelle phase, and for both orientations, cooperative binding of the DS anion is not possible. I



Article

examined binary mixture of the surfactants (Table 5). βBκ = ln f∞ S represents a coefficient of the activity of the decyl sulfate anion in infinitely diluted micelle pseudophase, i.e., when each anion of NaDS is surrounded with bile acid anions, or NaDS anions are interlinked at a distance where they do not exhibit electrostatic repulsive interactions. The great absolute value of the parameter βκB = ln f ∞ S for the anion of NaDS in the presence of NaDC probably originates from the parallel orientation of the α-axial C12 OH group and α-pseudoaxial C3 OH group of the steroid skeleton. The anion of NaDC is by the β side and the C7 lateral side of the steroid skeleton in contact with the hydrophobic pseudo-phase of the mixed micelle, so the parallel OH groups of NaDC are switched toward the sulfate groups of NaDS. The distance between the parallel OH groups allows simultaneous formation of H-bonds for both OH groups with one anion of NaDS (cooperative binding) (Figure 10). Other anions of NaDS in the vicinity of the NaDC steroid skeleton are free, i.e., their configuration entropy does not change (lowers), which is related to the steroid skeleton of cholanoic acid (without steroid OH groups) since the center of the 2D lattice is where free anions of NaDS can take any cell, while for the anion of NaDS that has a hydrogen bond directly in front of the steroid OH groups of the NaDC, this results in negative excess enthalpy, due to H-bonds and negative configuration entropy.44 If an anion of NaDS is surrounded with anions of NaHDC, the parameter βκB = ln f ∞ S has a lower absolute value, which is related to the state when NaDC anions are present (Table 5). In the steroid skeleton of the NaHDC, the orientation of the α-pseudoaxial C3 OH group and the orientation of the α-equatorial C6 OH group form an angle of 30°, which stereochemicaly does not allow simultaneous formation of H-bonds between two steroid OH groups and the sulfate group of NaDS. Both OH groups of the steroid skeleton of NaHDC form H-bonds with different anions of NaDS. In this case the same number of H-bonds is formed as in the system with NaDC, but there is a greater reduction of the configuration entropy due to the binding (fixing) of the position of the two anions of NaDS (Figure 10). The Margules two parameter function (eq 15) describes the excess Gibbs energy in the whole range of the composition of the mixed micelle, while the RST function (eq 1) exactly describes the dependence of GE for the composition of the micelle pseudophase, but only in the range of the molar fraction of bile salts from 0.5 to 0.8 (Figure 9). Since the range of the composition of the mixed micelle from xB = 0.5 to xB = 0.8 is far from the system (mixed micelle) with the property xB → 0, the RST coefficient of the interaction (βmd ij ) testifies different intensities of the interactions between the building units of the mixed micelle and the parameters of the Margules function (eq 15). Higher thermodynamic stabilization of the NaHDC−NaDS mixed micelle than that of the NaDC−NaDS mixed micelle in the range of xB = 0.5 to xB = 0.8 is probably the consequence of the fact that the C6 OH group of NaHDC is in equatorial orientation and lies tangentially to the spheroid interface of the micelle, where the sulfate groups are located. NaDC possesses an axial C12 OH group normally oriented to the spheroid surface of the micelle, i.e., it is oriented toward the interior of the solution. The axial C12 OH group lies tangentially on the micelle surface if the steroid skeleton of NaDC is in contact with the hydrophobic phase of the micelle over the C7 lateral side of the steroid skeleton. However, with the composition of the mixed micelle from xB = 0.5 to xB = 0.8, there is a problem in the packing of the anion of the bile acids, i.e., the steroid

skeleton has fewer possibilities of spatial integration into the binary mixed micelle.

4. CONCLUSIONS Concerning a binary mixed micelle with the steroid surfactant as one building unit and the classic surfactant (hydrophobic conformationally flexible hydrogen carbon chain and polar head- rigid sphere) as the second building unit, it is necessary to use the two parameter function (eq 15) for describing the excess Gibbs energy. In an infinitely diluted mixed micelle, the decyl sulfate anion is, to a greater extent, more thermodinamically stabilized in the presence of an anion of the deoxycholic acid, than it is in the presence of the hyodeoxycholic acid anion. The orientation of the steroid OH groups of the deoxycholic acid allows for cooperative binding of the decyl sulfate by Hbonds.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.7b00880. Experimental values of the I1/I3 ratio of the pyrene emission spectrum as a function of the logarithm of the total concentration log ct of the surfactants binary mixture and the experimental values of surface tension, γ, as a function of the logarithm of the total concentration log ct of the surfactants binary mixture (PDF)

■

AUTHOR INFORMATION

Corresponding Author

* Tel: +381 6311 400 15; Fax: +381 21 422 760; E-mail: [email protected], [email protected]. ORCID

Mihalj Poša: 0000-0002-8044-2655 Funding

This study was supported by the Ministry of Science and Technological Development of the Republic of Serbia (Project No. 172021). Notes

The authors declare no competing financial interest.

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REFERENCES

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