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STAND: Surface Tension for Aggregation Number Determination Pablo F. Garrido,† Pilar Brocos,† Alfredo Amigo,† Luis García-Río,‡ Jesús Gracia-Fadrique,§ and Á ngel Piñeiro*,†

Langmuir 2016.32:3917-3925. Downloaded from pubs.acs.org by QUEEN MARY UNIV OF LONDON on 08/28/18. For personal use only.

†

Departamento de Física Aplicada, Facultade de Física, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain ‡ Departamento de Química Física, Centro Singular de Investigación en Química Biolóxica e Materiais Moleculares (CIQUS), Facultade de Química, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain § Departamento de Fisicoquímica, Facultad de Química, Universidad Nacional Autónoma de México, Ciudad Universitaria, 04510 México D.F., Mexico S Supporting Information *

ABSTRACT: Taking advantage of the extremely high dependence of surface tension on the concentration of amphiphilic molecules in aqueous solution, a new model based on the double equilibrium between free and aggregated molecules in the liquid phase and between free molecules in the liquid phase and those adsorbed at the air/liquid interface is presented and validated using literature data and fluorescence measurements. A key point of the model is the use of both the Langmuir isotherm and the Gibbs adsorption equation in terms of free molecules instead of the nominal concentration of the solute. The application of the model should be limited to non ionic compounds since it does not consider the presence of counterions. It requires several coupled nonlinear fittings for which we developed a software that is publicly available in our server as a web application. Using this tool, it is straightforward to get the average aggregation number of an amphiphile, the micellization free energy, the adsorption constant, the maximum surface excess (and so the minimum area per molecule), the distribution of solute in the liquid phase between free and aggregate species, and the surface coverage in only a couple of seconds, just by uploading a text file with surface tension vs concentration data and the corresponding uncertainties.

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INTRODUCTION

The surface tension of a solution at the liquid/air interface (σ) is defined as the free energy required to enlarge the surface per unit area. σ depends directly on the concentration and arrangement of molecules at the interface, but the relative and absolute amounts of these adsorbed molecules depend on their concentration in the bulk phase. Thus, this property is extremely sensitive to any concentration change in the liquid, even for involved multicomponent systems where the different chemical species interact and compete for space at the interface with each other and with the possible supramolecular complexes they can form. For many substances, solutions at almost negligible concentrations in the liquid phase may lead to the saturation of the interface. From the experimental point of view, it is really simple to get a reasonable estimation of the surface tension for a given solution just by counting the number of drops falling from a capillary syringe when a given volume is injected at a constant rate. Commercial instruments based on a variety of methods (pendant drop, bubble pressure, Wilhelmy plate, du Noüy ring, capillary rise, drop volume, among other)

Molecular aggregation as well as adsorption to interfaces between media of different polarity are necessary for many natural phenomena and industrial applications including the compartmentalization of living media into cells or liposomes, the rational design of new materials based on two or threedimensional supramolecular structures, and the encapsulation of dirty particles by cleaning products.1 All these phenomena rely on organized molecular self-assembly. There is a clear connection between liquid phase molecular aggregation and interfacial adsorption processes, in such a way that they are very difficult to separate from each other: usually, molecules with high affinity to interfaces between media of different polarity also have a clear ability to self-organize in the liquid phase. Given the importance of both processes, there is a large number of experimental (absorption/emission spectroscopy at different frequencies, particle or light scattering, electrical, magnetic, chemical, optical or mechanical properties, calorimetric measurements, and a wide range of microscopies) and computational techniques to address them.2 Yet, no experimental method is able to simultaneously assess both phenomena. © 2016 American Chemical Society

Received: February 6, 2016 Revised: April 2, 2016 Published: April 5, 2016 3917

DOI: 10.1021/acs.langmuir.6b00477 Langmuir 2016, 32, 3917−3925

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are available to quantify the surface tension of a liquid in a highly precise way.3 Some of these methods even allow getting the value of σ as a function of time under nonequilibrium regimes, including mechanical perturbations of the interface from which it is possible to obtain interfacial viscoelastic properties.4 Surface tension measurements are often employed to directly obtain the main features of surfactant molecules: the critical micelle concentration (cmc) and the σ value when the surface is saturated. Several attempts to obtain more information from surface tension measurements have been proposed, including the determination of equilibrium constants for supramolecular complex formation, 5 the surface excess6,7 and activity coefficients.8,9 Surface tension measurements are typically employed to characterize surfactant molecules. The average aggregation number (N) together with the cmc, are probably the most important parameters to be considered in the development of surfactant-based commercial products, as well as in fundamental applications involving these kinds of molecules. The cmc has been quantitatively defined in different ways, but it is generally understood as the free surfactant concentration beyond which all the added molecules become part of nanoaggregate structures called micelles. The combination of both numbers allows determining the concentration of micelles for a given nominal surfactant concentration and its ability to encapsulate a certain cosolute in aqueous solution. Other thermodynamic parameters such as the Gibbs energy, enthalpy or entropy corresponding to the micelle formation process are also key in the characterization of these molecules since they allow the introduction of rational chemical modifications in their structure to modify their stability, diffusion rate, and other physicochemical properties. The most accessible experiments aimed to determine micelle aggregation numbers require using a fluorescent probe and a quencher,10 although other methods including light scattering and small angle neutron scattering are also occasionally employed.11,12 The reliability of the results is often dependent on the physicochemical properties of the employed molecules. Additionally, these techniques present some limitations such as the temperature at which the experiment can be carried out and the accessible concentration ranges of the target surfactant. In the present paper we will introduce a model that allows getting the average aggregation number in the liquid phase, the micellization free energy, the adsorption constant, the maximum surface excess (and so the minimum area per molecule), the distribution of molecules into micelles and free monomers in the bulk phase, and the surface coverage, just from surface tension versus concentration measurements. We have named this model STAND as an acronym of “Surface Tension for Aggregation Number Determination” since the aggregation number is probably the most significant property provided by the proposed method. We will also show how the model can be easily applied in just a few seconds by using a homemade computational application that is publicly available from our web server. A variety of surfactant molecules, including alkyl glucopyranosides and alkyl maltopyranosides at different temperatures, together with two nonylphenol derivatives are employed to validate the method by comparing the results with complementary information obtained from fluorescence experiments also performed in this work and with other data available in the literature for the same systems.

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MATERIALS AND EXPERIMENTAL METHODS

Materials. Octyl-β-D-glucopyranoside (C8G1), decyl-β-Dglucopyranoside (C10G1), decyl-β-D-maltopyranoside (C10G2), dodecyl-β-D-maltopyranoside (C12G2), and tetradecyl-β-Dmaltopyranoside (C14G2) were purchased from Anatrace, kept in a dark place at 278 K, and used as received. Nonylphenol polyethylene-7 glycol ether (Tergitol NP7) and nonylphenol polyethylene-10 glycol ether (Tergitol NP10) were purchased from Sigma and used with no further purification. The analysis of nonyl phenol ethoxylates molecular weight distribution was performed in the Mass Spectrometry and Proteomics Core Facility at the University of Santiago de Compostela using a matrix-assisted laser desorption/ionization time-of-flight mass spectrometer (MALDI-TOF MS) with α-cyano-4-hydroxycinnamic acid as the matrix. Mass spectra were obtained using a MALDI-TOF Autoflex mass spectrometer (Bruker Daltonix).The weight-average molecular weights (Mw) for NP7 and NP10 are 576.07 and 671.35 g·mol−1, respectively. Water content was determined with a C20 coulometric Karl Fischer titrator from Mettler Toledo. Ultrapure water (Elix 3 purification system, Millipore Corp.) was used in the sample preparation. Solutions were carefully prepared by mass (balance model AT250, Mettler, Switzerland) with the procedure of diluting aliquots from a mother solution, in whose preparation the water content of the surfactants was considered. The relative uncertainty in the concentration (c) of the final solutions was less than 0.2% and 0.01% for the lowest and highest c values, respectively. Fluorescence. Steady-state fluorescence experiments were recorded using a Cary Eclipse instrument. Pyrene of the highest available purity was purchased from Aldrich and was used without further purification. The fluorescence spectrum of pyrene was measured with an excitation wavelength of 334 nm. All emission spectra measured were corrected for emission monochromator response and were background-subtracted using appropriate blanks. Slits and rate of acquisition were chosen for a convenient signal-to-noise ratio. The micelle aggregation number of surfactants can be determined by static luminescence quenching according to the Turro-Yekta method.13 Pyrene was selected as the fluorescent probe and cetylpyridinium chloride (CPC) was assigned to be its quencher. N can be calculated on the basis of the following equation: ⎛I ⎞ [Q] ln⎜ 0 ⎟ = N ⎝I⎠ [S] − cmc

(1)

where I0 and I are the fluorescent intensities of pyrene in the absence and presence of CPC; and [Q] and [S] are the concentration of CPC and surfactant (slightly above the cmc), respectively. The cmc values required to use eq 1 were obtained from surface tension measurements, unless specified in the Results section. This equation relies on the assumptions that the pyrene and CPC molecules are completely solubilized in the micelle phase and that their distributions obey Poisson statistics. As an example, fluorescence results for C12G2 are shown in Figure S1 of the Supporting Information (SI). Surface Tension. Equilibrium surface tensions (σ) were measured with a Lauda drop volume tensiometer (TVT 2 model, Germany) using the standard mode or the quasi-static mode depending on the surfactant characteristics. Capillaries with inner radius of 1.345 mm and 1.71 mm, and a 2.5 mL syringe were used. Both the equipment and the procedure were 3918


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Langmuir described in detail in the past.14,15 Depending on the working concentration and temperature, the surface tensions were determined with an uncertainty ranging from ±0.01 to ±0.05 mN·m−1 for the measurements with the standard mode and from ±0.1 to ±1 mN·m−1 for the measurements performed with the quasi-static mode, both at the 95% confidence level. The temperature of the external bath to which the measurement cell is connected was controlled within ±0.1 K.

Δμ0 = −(RT /N ) ln([M]) + (RT /N ) ln([SF]N ) ≈ RT ln([SF]) ≈ RT ln([ST]) = RT ln(cmc)

The latter simplification will NOT be considered in our model proposal. Note that the cmc in eq 6 is expressed in terms of molarity. It is quite common however to use the same expression with the cmc in molar fraction. It is easy to show that the term RT ln(1000/18) ≈ 4RT (where 1000 and 18 are the mass of 1 L of water and its molecular weight, respectively) should be subtracted from eq 6 to obtain the Δμ0 with the cmc expressed in molar fraction. Equilibrium of Free Surfactant Molecules between the Bulk Liquid Phase and the Air/Liquid Interface. Equations 2−4 account for the equilibrium of surfactant molecules between free monomers and micelles in the liquid phase, but surface tension at the air/liquid interface is only sensitive to the molecules that are present at the boundary between both phases. Thus, a connection between the surfactant concentration in the liquid bulk and at the interface is required to complete the model. The Langmuir isotherm will be employed for this aim:

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THEORY Description of the STAND Model. The equations employed to analyze surface tension as a function of concentration measurements obtaining a detailed simultaneous characterization of surfactant adsorption and micellization processes are presented in this section. The Monodisperse Mass Action Approach for Free Surfactant-Micelle Equilibrium in the Bulk Phase. Typically, two different types of models are applied to the analysis of micelle formation processes observed by different experimental techniques, namely, those based on the pseudophase separation and on the mass action approaches.16,17 The former type of models is normally easier to apply since it assumes the micelles as a different phase in equilibrium with monomers, disregarding the number of surfactant molecules in each nanoaggregate. In contrast, models from the second family may explicitly consider the size of the micelles approaching it from the number of surfactant molecules forming them. These models usually assume that all micelles are identical. Since the surface tension of a given surfactant hardly changes once the interface is saturated, this property is expected to be sensitive only to the micelles formed at concentrations around the cmc. Their size distribution in this region is expected to be narrow enough to justify the monodispersity assumption. Thus, the method proposed herein is based on the monodisperse mass action approach, which assumes the equilibrium between the N surfactant monomers (S) and the corresponding micelles (M): NS ↔ S N ≡ M

θ=

dπ = θ ΓmaxRT

(2)

(3)

d[SF] [SF]

(8)

π = σw − σ = ΓmaxRT ln(1 + β[SF])

(4)

(9)

The combination of eqs 3−5 with eq 9 allows simultaneously describing the micellization and adsorption processes of surfactant molecules from a single surface tension isotherm (the surface tension as a function of the surfactant nominal concentration). The output of the model would be the aggregation number of the micelles that are formed around the cmc (N), the micellization free energy (Δμ0, the opposite of the molar energy required to extract a surfactant molecule from a micelle), the adsorption constant (β), the maximum surface excess (Γmax) from which it is straightforward to extract the minimum area per molecule at the interface (Amin), the

where the loss of concentration due to adsorbed molecules, which would depend on the geometry and composition of the solution container, is considered to be negligible. The free energy for the micelle formation process per mole of surfactant molecule is given by Δμ0 = −(RT /N ) ln(K )

(7)

where π is the surface pressure, i.e., the difference between the surface tension of the solvent and that of the solution at any concentration (= σw − σ). It is important to mention that both the adsorption isotherm and the Gibbs adsorption equation are here expressed in terms of the f ree surfactant concentration. This is key for the success of our proposal. The nominal (total) surfactant concentration is typically used in these equations introducing a serious artifact since the free surfactant monomers are simultaneously in equilibrium with the micelles in the liquid phase and with the adsorbed molecules at the air/liquid interface. Thus, the use of the nominal surfactant concentration in the isotherm and in the Gibbs equation is only a reasonable approach at very low concentrations, where the concentration of micelles is negligible and all the surfactant molecules are free. Combining eqs 7 and 8 and integrating:

where [SF] is the molar concentration of free surfactant and [M] is the molar concentration of micelles. Thus, the total concentration of surfactant in a given solution can be expressed as [ST] = [SF] + N[M] = [SF] + NK[SF]N

β[SF] 1 + β[SF]

where θ, which can be expressed as the ratio between the surface excess and its maximum value (Γ/Γmax), is the fraction of occupied sites at the interface (dimensionless) and β is the adsorption constant (in M−1 units). The Gibbs adsorption equation connects the surface tension with θ and with the free surfactant concentration:

Note that the presence of ions forming part of micelles is not explicitly considered in the previous reaction and so, the application of this model would be less rigorous for ionic molecules. The overall equilibrium constant corresponding to eq 2 is given by

K = [M]/[SF]N

(6)

(5)

Equation 5 is often simplified at the cmc by assuming that, in that region, [M] ≪ [ST] and [ST] ≈ [SF]. Under these assumptions and for large N, by substituting eq 3 in eq 5: 3919


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Langmuir surface coverage, and the distribution of surfactant molecules between the solution and the micelles. It is worth noting that (i) the proposed model does not make use of the cmc at all since it is based on a continuous equilibrium throughout the whole concentration range; (ii) it does NOT make use of the approach introduced by eq 6; and (iii) it employs the Langmuir isotherm and the Gibbs adsorption equations in terms of the free concentration instead of the total concentration of surfactant molecules. Fitting Procedure. The application of the proposed model requires a nonlinear fitting of each experimental surface tension isotherm to the previous equations in order to get useful information. The simulated annealing and the Levenberg− Marquardt algorithms were sequentially employed in order to reach the global minimum for each system.18 Each surface tension value was normally distributed around the corresponding experimental concentration in order to account for the errors in the concentrations. The uncertainty of such concentration was employed as the width of the distribution at half-height. The uncertainties of the surface tension values were employed to perform a weighted fitting using the following objective function: s = (n − ν)−1∑ [(σi Exp − σi Teo)/s(σi)]2

Figure 1. Surface tension experimental measurements20 at four temperatures (see legend) for n-octyl-β-D-glucopyranoside (structure represented in the inset), together with the fittings obtained from the application of the STAND model (solid lines).

(10)

where n is the number of experimental points in the isotherm, ν is the number of fitting parameters, σiExp and σiTeo are the experimental and theoretical (given by eq 9) surface tension values, and s(σi) is the uncertainty of σiExp. All data were analyzed with a homemade software that is publicly available in our group Web site (http://smmb.usc.es/stand/stand.php). A more user friendly version of this software will be available soon in the platform AFFINImeter (https://www.affinimeter.com/).

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RESULTS The maximum surface excess (Γmax) and the cmc can be estimated simultaneously from a surface tension isotherm by integrating the Gibbs adsorption equation and assuming that Γ is a constant (= Γmax) in the surface saturation region:19,20 π = πmax − ΓmaxRT ln(cmc) + ΓmaxRT ln[ST]

Figure 2. Surface tension experimental measurements at four temperatures (see legend) for n-decyl-β-D-glucopyranoside (structure represented in the inset), together with the fittings obtained from the application of the STAND model (solid lines).

(11)

As usually seen in the literature for the analysis of surface tension isotherms, this method considers the total surfactant concentration in the Gibbs adsorption equation instead of the free surfactant concentration. Additionally, the obtained results depend on the concentration region selected to fit the data.21−23 The big advantage of this approach is that it only requires a linear fitting of a few σ-concentration points. By using the eq 6, the cmc obtained from eq 11 could be employed to estimate the micellization free energy. The STAND model introduced in the present work does not make use of any of the assumptions described in the previous paragraph, and it provides much more information. All the experimental surface tension data reported in this work were analyzed by using our homemade software (see Theory section). The fittings were excellent in all cases, as shown in Figures 1−4. Table 1 shows the parameters obtained from the application of the STAND model to the surface tension data shown in Figures 1−4 together with the minimum area per molecule and the micellization free energy estimated from eqs 11 and 6, respectively.

Figure 3. Surface tension experimental measurements at 298 K for ndecyl-β-D-maltopyranoside (structure represented in the inset), ndodecyl-β-D-maltopyranoside, and n-tetradecyl-β-D-maltopyranoside (see legend), together with the fittings obtained from the application of the STAND model (solid lines).

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2386 ± 14 2181 ± 10 1887.5 ± 8.8 1547 ± 10 20 753 ± 94 19 712 ± 86 17 886 ± 88 14 710 ± 70 26 447 ± 45 137 900 ± 580 202 000 ± 24 000 2 420 000 ± 160 000 6 760 000 ± 560 000 106 ± 20 83.2 ± 5.8 68.2 ± 3.6 75.7 ± 4.5 116 ± 15 140 ± 19 500 ± 260 180 ± 28 93.5 ± 5.0 25.09 ± 0.26 26.2 ± 3.1 29.8 ± 5.8 21.07 ± 0.59 C10G2 C12G2 C14G2 NP7 NP10

C10G1

283.15 298.15 310.15 323.15 283.15 298.15 310.15 323.15 298.15 298.15 298.15 298.15 298.15 C8G1

a The area per molecule and the micellization free energy obtained from eqs 11 and 6, respectively, are also included for comparison. The reference state for Δμ0 is in both cases the hypothetical state of unit mole fraction extrapolated along the Henry’s law line.

−17.57 ± 0.26 −19.09 ± 0.17 −20.07 ± 0.25 −20.98 ± 0.14 −23.54 ± 0.15 −25.26 ± 0.20 −26.35 ± 0.21 −27.32 ± 0.32 −25.058 ± 0.036 −30.8 ± 3.2 −37 ± 16 −34.1 ± 1.0 −34.9 ± 3.9 41.74 ± 0.55 42.50 ± 0.34 43.69 ± 0.48 44.22 ± 0.28 37.59 ± 0.16 38.30 ± 0.20 39.37 ± 0.21 41.04 ± 0.32 51.513 ± 0.050 47.4 ± 3.4 39 ± 12 53.1 ± 1.1 51.2 ± 3.9 0.034 0.016 0.0036 0.017 0.031 0.028 0.034 0.027 0.0015 0.011 0.15 0.21 0.058 ± ± ± ± ± ± ± ± ± ± ± ± ± −17.841 −18.834 −19.7807 −20.689 −23.295 −25.056 −26.296 −27.160 −24.7949 −29.998 −36.80 −33.11 −33.386 0.068 0.054 0.060 0.086 0.047 0.051 0.060 0.061 0.025 0.056 0.98 0.73 0.98 ± ± ± ± ± ± ± ± ± ± ± ± ± 40.372 40.143 40.461 40.997 35.032 35.373 36.172 37.133 48.585 37.506 12.76 47.85 59.24

Amin (Å )

Γmax (μmol/m ) β (L/mol) T (K) surfactant

N

4.1132 ± 0.0069 4.1367 ± 0.0056 4.1041 ± 0.0062 4.0505 ± 0.0085 4.7402 ± 0.0064 4.6945 ± 0.0068 4.5907 ± 0.0076 4.4719 ± 0.0073 3.4168 ± 0.0017 4.4275 ± 0.0066 13.0 ± 1.0 3.471 ± 0.053 2.803 ± 0.046

Δμ0 (kJ/mol) Amin (Å )

Δμ (kJ/mol)

2 0 2

In most cases, the results obtained from both methods are very similar, although the values coming from the STAND model are typically slightly lower (in absolute value) for both Amin and Δμ0. The differences observed for Δμ0 are likely due to the assumptions required to get the eq 6, and so the values obtained in that way are expected to be less reliable than those obtained from the STAND model. Differences in the same direction for Δμ0more negative for the less accurate expressionhave already been observed in the literature.24 For the minimum area per molecule, the values are, in general, quite similar, but the differences are significantly large for C12G2 and even larger for C14G2. This could be due to the significant loss of material in the liquid phase for these systems due to the adsorption of molecules to the interface (note that the concentration of adsorbed material was neglected in eq 4). It was reported that, for surfactants having 14 or more carbon atoms in their hydrophobic part, it is not a good approach to match the total and bulk equilibrium concentrations in the dilution zone.25,26 This means that the total surfactant concentration cannot be used to analyze the surface tension isotherms of these systems. It is worth noting that the analysis was much more userfriendly and less time-consuming using the STAND model than using eqs 11 and 6, since the application of the STAND model does not require selection of a specific concentration region for the fitting, it is fully automatized in our web tool, and it provides much more information than the second method. The most outstanding information provided by the STAND model is the aggregation number of the micelle. As we have already explained, it is not easy to get this parameter from alternative experiments and it is key for many surfactant-based industrial and fundamental applications. A list of aggregation number values obtained from different experimental methods in the literature10,27−63 as well as from the STAND model applied to surface tension measurements in this work, and from fluorescence measurements also performed in the present work are included in Table S1. For ease of reading, data at each temperature are ordered in this table by increasing N. It is seen that, in general, the aggregation number values automatically obtained from the STAND model compare well with those obtained from several independent methods. Note

2

Figure 4. Surface tension experimental measurements at 298 K for nonylphenol polyethylene-7 glycol ether (structure represented in the inset) and nonylphenol polyethylene-10 glycol ether (see legend), together with the fittings obtained from the application of the STAND model (solid lines).

STAND model

Table 1. Parameters Obtained from the Application of the STAND Model to the Surface Tension Data Shown in Figures 1−4a

results from eqs 11 and 6

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between the liquid and the surface phases. The surface coverage, as a measurement of the surfactant concentration at the interface, is also plotted in Figure 5. It is clearly seen that the concentration at the interface increases much faster than the surfactant concentration in the liquid phase. Additionally, it is seen that the surface becomes saturated just before the formation of the former micelles. This quantitative description provided by the STAND model matches with the expected behavior of a surfactant solution where the double equilibrium between free surfactant, micelles and adsorbed molecules is present. It is worth mentioning that these results could only be obtained by using the concentration of free surfactant in both the Langmuir isotherm and the Gibbs adsorption equation, in contrast to what is usually done. Given the large amount of information provided by the STAND model, the possible coupling between parameters has also been considered. Additionally, the uncertainty values of the fitted parameters are estimated by the web application developed to apply this model. Figure S6 (see SI) shows that the shape of the global minimum of the objective function is symmetric, and so the standard deviation obtained from the covariance matrix during the minimization is suitable as an estimation of the parameter uncertainties. The projections of the sampling values on two-dimensional spaces formed by combining different pairs of fitting parameters are shown in Figure 6. It is clearly seen that there is a strong correlation between Γm and β and another independent correlation between N and Δμ0. However, these correlations are restricted to the region of the space bounded by the uncertainties. This can be easily tested by fixing any of the parameters with values out of the confidence region and fitting the remaining parameters. When the fixed parameter is within the confidence region, the coupled parameter modifies its value and the quality of the fitting is good. However, if the fixed value is modified out of the confidence region, the quality of the fitting gets significantly worse. It is also worth noting that the uncertainty of the Δμ0 values is relatively small, and the values approached by eq 11 are out of the confidence region of this parameter. This means that the latter values cannot be employed in the STAND model as fixed values to reduce the number of fitting parameters; however they could be employed as a seed or initial value to find the final result.

that the comparison should be done at similar concentrations since the micelle shape, and thus the aggregation number, are expected to depend on the relative amount of surfactant in the solution. This is why the ratio between the total surfactant concentration employed to determine the aggregation number and the corresponding cmc ([ST]/cmc]) are specified in Table S1. When this number is far from unity, the comparison is not reliable. This is also the reason why we decided to determine the aggregation number from fluorescence measurements around the cmc region. In general, the results coming from fluorescence are in good agreement with those obtained from the STAND model. It is worth noting, however, that the uncertainty in the data coming from fluorescence is typically high due to the term ([S] − cmc) that appears in the denominator of eq 1. When the surfactant concentration approaches the cmc, that term is very small, and the resulting equation diverges. Additionally, the fluorescence method is only useful to provide the aggregation number of the micelles, while the STAND method applied to the surface tension data provides much more information in return for an equivalent experimental effort. As mentioned above, the distribution of surfactant molecules into micelles and free molecules in the solution, as well as the surface coverage, are also directly obtained from the STAND model. Figure 5 shows these data for the system C10G1 (similar

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Figure 5. Concentration of free surfactant (dotted line), surfactant molecules in micelles (dashed line) and surface coverage (solid line) as a function of the total concentration of n-decyl-β-D-glucopyranoside at 298 K. Data obtained from the application of the STAND model.

CONCLUSIONS AND PERSPECTIVES A new model able to simultaneously provide the following parameters from surface tension isotherms is presented in this work: (i) the average aggregation number of an amphiphile, (ii) the molar free energy of the aggregation process, (iii) the adsorption constant to the liquid/air interface, (iv) the maximum surface excess (and so the minimum area per molecule), (v) the distribution of solute in the liquid phase between free and aggregate species, and (vi) the surface coverage. The obtained aggregation number has been validated by comparing the results for a number of surfactant systems at different temperatures with literature data and with values obtained from the analysis of fluorescence measurements in the vicinity of the cmc. The model is based on the combination of the Langmuir isotherm with the Gibbs adsorption equation in terms of free molecules, and it considers the monomer-micelle equilibrium in the bulk phase. No counterions are considered, and so it is justified to be applied only for non ionic compounds. Although the Langmuir isotherm does not take into account the interactions between the adsorbed com-

results for the rest of the systems are shown in Figures S2−S5 of the SI). The results obtained for C14G2 (Figure S4) are unrealistic since the maximum surface coverage observed is about ∼0.7 and it should be much closer to 1, as observed for the rest of the systems. As mentioned above, such difference could arise from ignoring the amount of surfactant adsorbed to the liquid/air interface in eq 4. As expected, Figure 5 shows that the concentration of free surfactant molecules increases linearly up to a critical value where the micelles appear. After that point, the concentration of free surfactants is practically constant and its value could be taken as the cmc. It is important to recall that no discontinuity was introduced in the STAND model to consider the cmc as a fitting parameter that separates two different regimes in the isotherm. The sharp onset of micelle formation in the concentration scale arises naturally as a result of the equilibrium 3922


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Figure 6. Projections of normalized fitting parameters pairs obtained for each evaluation of the objective function(s). The blue−red color gradient represents the value of s normalized by its minimum value (smin). The value of the parameters corresponding to smin is employed to normalize them. Results obtained from the data of C8G1 at 298 K illustrating the correlation between N and Δμ0, and between Γmax and β, restricted to a relatively narrow range of values.

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pounds, the use of more involved isotherms was not considered convenient since they would require additional fitting parameters with the consequent increase in the uncertainties. Additionally, a web software designed to apply this model in a straightforward way is presented. As a result, it is now possible to get the information described above from a surface tension isotherm in just a few seconds. For the correct application of the model, it is important that all the experimental measurements are in thermodynamic equilibrium. The model could be corrected to explicitly consider the concentration change due to the loss of adsorbed material. This refinement is not trivial because the concentration change would depend on the area of the interface and also on the geometry and material of the sample container, which could also be able to adsorb solute molecules. However, this correction does not seem to be highly significant for most of the systems tested at the moment. Although the model has already been validated, including the confidence regions of each parameter as well as their coupling, and explicitly considering both the uncertainties in the surface tension measurements and also in the concentrations, the availability of a public web application will favor the possibility of a collaborative massive validation by many researchers from different laboratories. The model is expected to be extended to consider ionic compounds and multicomponent systems.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful for funding from the Spanish Ministry of Economy and Competitiveness MINECO (MAT2011-25501, MAT2015-71826-P and CTQ2014- 55208-P), from the Xunta de Galicia (AGRUP2015/11 and GRC2007/085) and from DGAPA PAPIIT-IN114015.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b00477. Table S1 and Figures S1−S6. (PDF) 3923


Article

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