A surfactant concentration model for the systematic determination of the critical micellar concentration and the transition width

molecules Review A Surfactant Concentration Model for the Systematic Determination of the Critical Micellar Concentration and the Transition Width Wajih Al-Soufi *,† and Mercedes Novo †   Citation: Al-Soufi, W.; Novo, M. A Surfactant Concentration Model for the Systematic Determination of the Critical Micellar Concentration and the Transition Width. Molecules 2021, 26, 5339. https://doi.org/10.3390/ molecules26175339 Academic Editor: Katarzyna Szymczyk Received: 30 July 2021 Accepted: 11 August 2021 Published: 2 September 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Department of Physical Chemistry, Faculty of Science, University of Santiago de Compostela, E-27002 Lugo, Spain; [email protected] *Correspondence: [email protected] † Both authors contributed equally to this work. Abstract: The critical micellar concentration (cmc) is a fundamental property of surfactant solutions. Many proposed methods for the definition and determination of the cmc from property-concentration plots yield values, which depend on the studied property, on the specific technique used for its analysis and in many cases on the subjective choice of the chosen type of plot and concentration interval. In this focus review, we revise the application of a surfactant concentration model we proposed earlier that defines the cmc directly based on the surfactant concentration. Known equations for the concentration-dependence of different surfactant properties can then be combined with this concentration model and fitted to experimental data. This modular concept makes it possible to determine the cmc and the transition width in a systematic and unambiguous way. We revise its use in the literature in different contexts: the determination of the cmc of surfactants and their mixtures from different properties (electrical conductivity, NMR chemical shift, self-diffusion, surface tension, UV-Vis absorption, fluorescence intensity and fluorescence correlation). We also revise the dependence of the width of the transition region on composition, detailed studies of the properties of fluorescent probes and the aggregation of non-surfactant systems, namely amyloid peptides. Keywords: surfactants; micelles; critical micelle concentration; concentration model; data analysis 1. Introduction The critical micellar concentration (cmc) is a fundamental property of a surfactant solution. At surfactant concentrations near and above the cmc the surfactant molecules start to aggregate and form micelles. The formation of micelles affects many properties of a surfactant solution introducing usually a marked change at the cmc in plots of a property versus the surfactant concentration. Any property of the surfactant solution affected by the formation of micelles can in principle be used to determine the cmc: electrical conductivity of ionic surfactants, surface tension, NMR chemical shift, UV-Vis absorption, fluorescence spectroscopy of a probe molecule, ion-selective electrodes, isothermal titration calorimetry, densimetry, ultrasound spectroscopy, translational diffusion determined by light scattering or fluorescence correlation spectroscopy and many more. Many methods for the definition and determination of the cmc from property-concentration plots have been proposed over the last 70 years [ 1 – 9 ] and are still published [ 10 – 12 ]. Most of these methods determine the cmc from the intersection of two straight lines drawn or fitted to the two branches below and above the cmc, assuming that the property depends linearly on the concentrations of free surfactant and micelles. In the case of a sigmoidal propertyconcentration curve, such as those obtained from absorption or fluorescence of a probe, the cmc is arbitrarily defined as one of the characteristic points of the curve. Other methods propose to determine the cmc from the maxima or zero-crossings of the nth-order derivative of the measured data or of some arbitrary curve fitted to it. These methods yield experimental values of the Molecules 2021,26, 5339. https://doi.org/10.3390/molecules26175339 https://www.mdpi.com/journal/molecules Molecules 2021,26, 5339 2 of 22 cmc, which depend on the studied property, on the specific technique used for its analysis and in many cases on the subjective choice of the investigator on the type of plot and the concentration interval used. The methods proposed so far define the cmc indirectly as the point of significant change in the measured property, for example a break in the surface tension, in the electrical conductivity of the solution or in the fluorescence emission of a probe molecule. However, the critical micellar concentration is a feature of the surfactant aggregation, and should therefore be defined and analysed directly based on the surfactant concentration. This distinction is relevant for several reasons. Firstly, because a “critical” change in the measured property does not necessarily coincide exactly with the concentration of the critical change at the molecular level. This may seem counterintuitive, but is a direct consequence of the nonlinear dependence of some properties on the concentration. We will revise this effect in more detail in this contribution. Secondly, a cmc-definition based on the surfactant concentration itself avoids the ambiguity in its definition based on different properties and procedures. The big variety of empiric procedures for the determination of the cmc from experimental data is probably also due to the complex physicochemical models and numeric simulations of the micellization process [13–23]. Although increasingly sophisticated and precise, these models and simulations are not aimed towards the analysis of experimental data and therefore do not offer fitting functions with the cmc as an adjustable parameter that could be fitted to experimental data. Of course, it would be desirable to derive such a function from theoretical principles. Nearly ten years ago we proposed a surfactant concentration model, which we derived from established empiric features of surfactant solutions [ 9 ]. This model allows one to calculate the concentrations of monomeric and micellized surfactants in solution for a given total surfactant concentration. It depends on only two adjustable parameters, the cmc and the width of the transition region around the cmc. We then combined this concentration model with the known equations for the concentration dependence of several properties of surfactant solutions, specifically with those of electrical conductivity, NMR chemical shift, self-diffusion, surface tension, UV-Vis absorption, steady-state and time resolved fluorescence intensity and fluorescence correlation [ 9 , 24 , 25 ]. The resulting set of equations for the concentration dependence of the properties proved to fit excellently the experimental data and gave consistent values among the properties of the cmc and of the transition width. The separation between the concentration model and the experimental property has many advantages. The cmc is now directly a feature of the surfactant concentration and is common to all derived properties. Its value can be unambiguously and precisely determined from fits to experimental data, even by simultaneous (global) fits of data from different techniques. Another advantage is that, once we prove that the concentration model is sufficiently precise for a given surfactant system, we can test the validity and the limitations of the equations describing the concentration dependence of the solution property. The objective definition of the cmc as part of the concentration model and its determination from data regression also avoids errors inherent to subjective graphical procedures. Finally, the model we proposed also introduces naturally the transition region between purely monomeric and dominantly micellized surfactant. It is known that micelles are already formed at concentrations below the cmc and there is some controversy whether these pre-micelles should be considered as proper species [23,26,27]. In this focus review, we summarize shortly the surfactant concentration model and its application to different properties. Then we revise its use in the literature in different contexts: the determination of the cmc of surfactants and their mixtures, as basis of theoretical models or new derived properties, the discussion of the width of the transition region, detailed studies of the properties of fluorescent probes and the aggregation of non-surfactant systems, namely peptides. Molecules 2021,26, 5339 3 of 22 This work is divided into the following sections: (1) Introduction; (2) The Surfactant Concentration Model; (3) Application to Different Techniques; (4) Influence of Nonlinear Property-Concentration Relationships on the cmc Value; (5) Analysis with Sparse Data Points; (6) Conclusions. Examples of the application of the concentration model to several surfactant systems and properties is presented of referenced: conductivity data of the anionic surfactant SDS (Sections 3.1.1 and 5), of mixtures of two cationic surfactants, tetradecyltrimethylammonium bromide (TTAB), and methylammonium dodecylethylmethacrylate bromide (Section 3.1.2), of two antidepressant surface-active drugs, imipramine (IMP) and clomipramine (ClMP) hydrochloride (Section 3.1.3). Surface tension of a small molecule hydrogelator with surfactant-like properties, Fmoc-phenylalanine (Section 3.2), and of the mixture of the anionic surfactant, dodecylbenzene sulphonic acid (DBSA) and poly (ethylene glycol) (PEG) (Section 3.2). NMR-Chemical shift of two cationic (CTAB, TTAB), one anionic (SDS), and a nonionic surfactant (Triton X-100) (Section 3.3). Fluorescence of the probes perylene, pyrene, Coumarine 153 (C153) and 2-anilinonaphthalene-6-sulfonic acid (ANS) in Triton X-100 (Section 3.4). The spectral ratios I/III of pyrene in TX100 and SDS (Section 3.4.1). UV-Vis absorbance of Triton X-100 (Section 3.5). Fluorescence Correlation Spectroscopy (FCS) of Rhodamine 123, Coumarine 152 and BODIPY-labelled cholesterol in the presence of Triton X-100 (Section 3.6) and of Amyloidβ (1–42) peptides (Section 3.7). Conductivity and density measurements of SDS and sodium lauryl ether sulfate (SLES) (Section 3.8). The dependence of the ZnS nanoparticle radius on the CTAB concentration (Section 4). 2. The Surfactant Concentration Model We derived the model in detail before [ 9 ]. Here we will summarize its main properties. The starting point for the derivation of the concentration model is the definition of the cmc itself. Already in 1955 Phillips proposed to define the cmc as “the point corresponding to the maximum change in gradient in an ideal property-concentration relationship” [ 1 ]. This so-called “Phillips-condition” has been extensively used in the literature with more than 650 citations so far. It is an empiric condition, not derived from a more fundamental theory of micellization. As discussed by Rusanov, several other analogue conditions could also be used [28]. The Phillips condition refers originally to the change in the “property-concentration relationship”, whereas we applied the condition directly to the surfactant concentration. We define the cmc as the total surfactant concentration [S] 0 at which the change in the gradient of the monomer concentration [S 1 ] with respect to [S] 0 is maximal, that is where the third derivative of [S1] is zero: d3[S1] d[S]0[S]0=cmc =0 (1) In the same way, we transfer the concepts developed by García-Mateos et al. [ 3 ] for conductivity data directly to the surfactant concentration. They applied a probabilistic point of view to the pseudophase separation model and interpret the distribution of counterions or of monomers in the aqueous and micellar phases with the aid of a binomial distribution model between micellized or free states. The degree of micellization represents then the distribution function, which becomes a continuous normal distribution when the number of micellized surfactants is high enough (central limit theorem). After some substitutions and normalisation, they show that the second derivative of the electrical conductivity with respect to the surfactant concentration is a Gaussian function centred at the cmc with some width σ . They fit conductivity data with a double numerical integration of this Gaussian function. Molecules 2021,26, 5339 4 of 22 We apply this concept to the surfactant concentration. We describe the second derivative of [S 1 ] with respect to [S] 0 by a Gaussian function centred at the cmc, with amplitude A and width σas given in Equation (2) (Figure 1c): d2[S1 d[S]2 0 =−A √2πσe−([S]0−cmc)2 2σ2(2) Figure 1. Surfactant concentration model. ( a ) Concentration of monomeric surfactant [S 1 ] and of micellized surfactant [S m ] as function of the relative total surfactant concentration s 0 = [S] 0 /cmc (Equations (5)–(8)) for three values of the relative transition width r(solid: r= 0.1, dot: r= 0.3, dash: r= 0.5). ( b ) First derivative [S 1 ] 0 = d[S 1 ]/d[S] 0 (Equation (9)) and (differential) degree of micellization γm = [S m ] 0 = d[S m ]/d[S] 0 (Equation (10)) ( c ) Second derivative [S 1 ]” = d 2 [S 1 ]/d[S] 02 (Equation (2)). The vertical thick line indicates the cmc. (Adapted with permission from [9]). The standard deviation σ of the Gaussian (half width at e−1/2 ) is a measure of the width of the transition region around the cmc. In this region, cmc ±σ , the concentration of surfactant monomers still increases with [S] 0 , but first micelles are already formed. The smaller σ , the sharper is the transition between the two linear regions below and above the cmc. In order to facilitate the comparison of transition widths σ of surfactants with different cmc values, we define the relative transition width r: r=σ/cmc (3) Molecules 2021,26, 5339 5 of 22 Ideally, below the cmc, the concentration of monomeric surfactant [S1] is equal to the total surfactant concentration [S]0and stabilizes at the value of the cmc itself: [S1] = [S]0, cmc, [S]0<< cmc [S]0>> cmc (4) The concentration of micellized surfactant, [S m ], is determined from the difference between the total concentration and that of monomeric surfactants: [Sm]=[S]0−[S1](5) The concentration of micelles, [M], depends on the mean aggregation number n: [M]=[Sm]/n(6) After double algebraic integration of the Gaussian of Equation (2) and proper normalization using Equations (1)–(6) we obtain the central equation of the concentration model, the monomer concentration [S 1 ] as a function of the total surfactant concentration [S] 0 (Figure 1a) [9]: [S1] = cmc −A 2 q2 πσe−([S]0−cmc)2 2σ2+ ([S]0−cmc)erf[S]0−cmc √2σ−1! =cmc"1−A 2 q2 πre−(s0−1)2 2r2+ (s0−1)erfs0−1 √2r−1!# (7) with the relative total surfactant concentration s0=[S]0/cmc and the amplitude A(For r< 0.4 the amplitude A≈1): A=2 1+q2 πre−1 2r2+erf1 √2r(8) This model assumes a constant aggregation number n, also at the onset of micelle formation at the cmc. This is in line with simulations [ 29 ] and thermodynamic models [ 20 ] and proved to describe well the behaviour of several surfactants [9]. The surfactant concentration model, Equations (5)–(8), can be plugged into equations that describe derived properties obtained with different techniques. The resulting equations for the property-concentration relationships can be easily implemented as fitting functions in analysis software. Functions for OriginPro (OriginLab Corporation, Northampton, MA, USA) and other software packages can be downloaded from the webpage of the authors. The first derivative [S 1 ] 0 of the monomer concentration of Equation (7) is given by the error function erf(x), which is a sigmoid type function centred at x=0 (in this case at the cmc) with limiting values erf(0)=0, erf(−∞) = −1, and erf(∞) = 1 (Figure 1b). d[S1] d[S]0 =A 21−erf[S]0−cmc √2σ (9) [S1]0is directly related to the differential degree of micellization γ: γ=d[Sm] d[S]0 =1−d[S1] d[S]0 (10) Molecules 2021,26, 5339 6 of 22 The second derivative [S 1 ] 00 is of course the Gaussian function of Equation (2) (Figure 1c). The value of the (fractional) degree of micellization αm = [S m ]/[S] 0 (or molar fraction of micellized surfactant Xm=αm) at the cmc is given approximately by: αm(cmc) = [Sm] [S]0cmc ≈r √2π≈0.4 ×r(11) For a typical value of r= 0.1 only 4% of the surfactant molecules form micelles at the cmc. 3. Application to Different Techniques 3.1. Electrical Conductivity of Ionic Surfactant Solutions Data of the conductivity (specific conductance) κ of a solution of an ionic surfactant of concentration [S]0can be fitted with the linear model [9,30] κ=a[S1] + b[Sm] + κs =a[S1] + b([S]0−[S1]) + κs(12) The parameters aand bhave units of molar conductivity and are the slopes of the limiting straight lines κ1=a[S]0+κs and κ2=a·cmc +b·([S]0−cmc) + κs observed at low and high concentrations of surfactant, respectively [ 9 ]. κs is the residual conductivity of the solvent without surfactant. The concentration [S 1 ] is calculated from [S] 0 using Equation (7) of the concentration model. The molar conductivity (equivalent conductance) Λmis given by: Λm= (κ−κs)/[S]0=a[S1]/[S]0+b[Sm]/[S]0(13) This model reproduces conductivity data of SDS, DTAB, CTAB and LAS very well, and extracts precise cmc values even in cases with very small change in the slope below and above the cmc [9]. 3.1.1. Comparison to the Carpena Model The method proposed by Carpena et al. [ 6 ] pursued the same aim as our concentration model, to eliminate the imprecision and subjectivity inherent to the determination of the cmc by graphical methods. Their method is specific for conductivity data and clearly inspired the more general approach we proposed ten years later. The authors observed that experimental data of the differential conductivity Λdi f f =dκ/d[S]0 are well fitted by a Boltzmann type sigmoidal centred at the cmc with the transition width ∆[S]0: Λdi f f Carpena =dκ/d[S]0=a−b 1+e([S]0−cmc)/∆[S]0 +b(14) Here we use the same nomenclature as in Equation (13). Integration of this sigmoidal and proper normalisation leads to an expression of the conductivity as a function of [S]0: κCarpena =κs+a[S]0+∆[S]0(b−a)ln 1+e([S]0−cmc)/∆[S]0 1+e−cmc/∆[S]0!(15) This expression can be fitted to conductivity data in order to obtain the cmc and the width of the transition ∆ [S] 0 . The authors and others applied this method successfully to a variety of ionic surfactant systems. How does this method compare to the concentration model (Equations (7) and (12)) we proposed? (i) Both methods are based on empirical models. The concentration model relies on the Phillips condition for the cmc and on the description of the degree of micellization as Molecules 2021,26, 5339 7 of 22 a continuous normal distribution function, derived from a probabilistic description of the distribution of free and micellized surfactants. The Carpena method starts with the Boltzmann function for the description of the differential conductivity. Other conditions or sigmoidal functions would lead to other models with similar results. However, the concentration model is derived from more general assumptions that make it applicable to any property-concentration relationship. (ii) The value of the transition width ∆ [S] 0 of the Carpena method, has no direct relation to that of the transition width σ =r · cmc of the concentration model, which stems from the width of the Gaussian function of Equation (2). Both widths define the slopes of the corresponding sigmoidal functions at the inflection point [S]0=cmc: d2κ d[S]2 0 (cmc)Carpena =b−a 4∆[S]0 d2κ d[S]2 0 (cmc)≈b−a √2πσ,(r<0.4, A≈1)(16) The ratio σ/∆[S]0≈4/√2π= 1.60 yields sigmoids with the same slope at the cmc. The best fit between both curves is obtained with a ratio σ/∆[S]0 = 1.75. In practice, fits of conductivity data with both methods yield transition width with ratios σ/∆[S]0 between 1.5 and 1.8. (iii) The values of the cmc determined by the two methods are in practice identical. The sigmoidal Boltzmann function of Equation (14) is not identical to the sigmoidal error function of Equation (9), but very similar. The fit of the SDS conductivity data of our original paper (Figure 2in [ 9 ]) with Equation (15) yields cmc Carpena = 8.096 ± 0.005 mM, identical to cmc = 8.099 ± 0.005 mM obtained with Equation (12). The same is true for the slopes aand b. The reduced χ2 values are very similar with χ2 = 0.436 and χ2 = 0.427 for Equations (12) and (15), respectively. The transition width is ∆ [S] 0 = 0.519 mM. Compared to the transition width from the concentration model σ =r · cmc = 0.112 × 8.099 mM = 0.907 mM, the ratio σ / ∆ [S] 0 = 1.75 falls within the expected interval indicated above. 3.1.2. Example of the Use for Surfactant Mixtures Pereyra et al. [ 31 ] studied mixtures of two cationic surfactants, a conventional one, tetradecyltrimethylammonium bromide (TTAB), and a polymerizable surfactant (PS), methylammonium dodecylethylmethacrylate bromide. They analyzed conductivity data of the mixtures using three methods, the excess specific conductivity method, the Carpena method discussed above [6], and the concentration model (Equations (7) and (12)). The cmc values they obtain from fits with our surfactant model (Equation (12)) are virtually identical to those from fits with the Carpena method for all mixtures. The excess specific conductivity method depends on a graphical determination of the intersection of two straight lines. It yields cmc values, which coincide well with those from the fits at low mole fraction of PS, but not at higher ones. This is probably due to the limitations of the graphical method. At high PS content, the cmc lies at the upper end of the studied surfactant concentration interval. This, together with the increase of the transition width, makes the graphical determination of the limiting line very unreliable. The fit of the full curve with one of the models, our model or Carpena’s, is much less affected by this problem. The authors also analyse the variation of the transition width with composition. The relative transition width rof the TTAB-PS mixtures increases with increasing mole fraction of PS, from r= 0.11 for pure TTAB up to r= 0.33 for pure PS (Figure A2). Interestingly, the variation of rwith αPS is well fitted with Clint’s model derived for the cmc of surfactant mixtures (Figure A2). The ratio σ / ∆ [S] 0 = 1.58–1.73 of the transition width determined from the two methods is not constant but falls within the expected interval (see above). Molecules 2021,26, 5339 8 of 22 The plots of the Gaussian functions (Equation (2)) for the different mixtures the authors show seem to be a very good way to illustrate the change in the transition width. These Gaussians can be easily calculated with the functions we provide. The authors also compare the concentrations of free and bound counterions, determined in ion-selective-electrode (ISE) experiments, with those calculated with the concentration model Equations (7) and (12). They find good agreement at low PS content but deviations at PS rich mixtures. It will be definitely worth to analyse these results in more detail and to extend the concentration model to ISE data. 3.1.3. Global Fit of Conductivity Data. Application to Weakly Surface-Active Drugs One of the advantages of the concentration model (7) and (12) over graphical or procedural methods is the possibility to perform global analysis, sharing some of the parameters of the model among the fits to all curves of a series, for example among conductivity titration curves measured at different temperatures. Global fits of a model to series of data reduces the number of fitting parameters and thus the effect of parameter correlation [31–36]. Global fitting helps to extract the cmc and the transition width even for surfactants presenting only a very small change in the slopes of their conductivity-concentration plots and wide transition intervals. In order to illustrate global fitting we use electrical conductivity data published by López Fontán et al. of aqueous solutions of two antidepressant surface-active drugs, imipramine (IMP) and clomipramine (ClMP) hydrochloride. The data set consists of titrations at 19 concentrations and 31 temperatures between 283 K and 313 K [ 37 ]. Figure 2shows the data for ClMP (diamonds) and Figure A1 that of IMP. The determination of the cmc is difficult due to the small change in the slopes and the very wide transition interval, especially in the case of IMP. The authors estimated the cmc values from the maxima of the second order derivatives of the conductivity data with a local polynomial regression method (LPRM) based on nonparametric regression [ 8 ]. The authors do not indicate uncertainties. Their cmc values (red triangles in Figure 3) show discontinuities within each series and, importantly, opposite temperature dependencies. In the case of IMP the temperature dependence is slightly convex with an overall decrease in the cmc with increasing temperature, whereas for ClMP a steady increase is obtained. Individual fits of the concentration model (Equation (12)) to each of the curves are excellent at all temperatures for both drugs. However, at high temperatures and especially in the case of IMP the uncertainty of the determination of cmc and rfrom these individual fits is huge, leading to unreliable results. We observed, though, that the values of the relative transition width rdo not significantly change within their errors with temperature, and that global fits of Equation (12) with a shared value of ryield very similar residual patterns and practically the same value of the reduced χ2 as the individual fits. This justifies the application of global fits, assuming a temperature independent transition width r. Figure 2shows the results of the global fit of Equation (12) to the whole dataset of ClMP and Figure A1 to that of IMP. The fit was performed with a commercial software (Origin Pro). The black and grey diamonds are the experimental data values. The thin grey lines are the fit curves. The lower panels in the figures indicate the residual errors. The temperature dependence of the cmc values (blue circles in Figure 3) shows now no discontinuities and is very similar for both drugs. The cmc values are relatively high (24 ± 1 × 10 −3 mol kg −1 for ClMP and 38 ± 2 × 10 −3 mol kg −1 for IMP, both at 298 K) and increase about 10 × 10 −3 mol kg −1 in the studied 30 K temperature interval. The fit of a second order polynomial gives nearly identical values for both drugs, except for a constant offset (See Figure 2and Table A1 in the Appendix A). The values of the slopes aand bobtained from the global fit are plotted as insets in Figure 2together with the ratio b/a= 0.41–0.48. The increase in temperature causes a decrease of abut an increase b, resulting in a strong rise of the ratio b/a. Molecules 2021,26, 5339 9 of 22 The relative transition widths of the two drugs do not depend on the temperature and are very wide (ClMP: r= 0.53 ± 0.04, IMP: 0.94 ± 0.08). This is in line with the approximate rule that the transition width increases with increasing cmc, both in absolute terms as well as relative to the cmc [2]. Figure 2. Conductivity of clomipramine hydrochloride (ClMP) in pure water as function of molal concentration and temperature. Black and grey diamonds: experimental data [ 37 ]. Grey and black lines: global fit of Equations (7) and (12) with r= 0.53 ± 0.04, the parameters indicated in the insets, and the cmc shown in Figure 3. Vertical grey lines: range of cmc values obtained from the fit. Right Inset: slopes aand bfrom the fit in units of mS cm −1 kg mol −1 . Left Inset: slope ratio b/a. Lower panel: residuals of the fit. Figure 3. Values of the cmc of IMP and ClMP at different temperatures obtained by different analysis methods from the data of Figures 2and A1. Blue circles: from global fits of Equation (12) with shared r. Red triangles: from LPRM [ 37 ]. The continuous line represents a second order polynomial fit through the values from the global fit (Table A1). Molecules 2021,26, 5339 16 of 22 values of apparent molar volumes of monomeric and micellized surfactants using the concentration model, Equations (5)–(8). Then they determined the values of the micelle-water partition coefficient, K MW , for each pair of the perfume-surfactant system by fitting UV spectra of the studied fragrance materials obtained at all concentrations of surfactant to the relation between the absorbance of the solution and the concentration of the micellized surfactant as determined by Equations (5)–(8). The authors obtained excellent fits and could extract valuable information from relatively small changes in the experimental data. 4. Influence of Nonlinear Property-Concentration Relationships on the cmc Value The concentration model uses the Phillips condition, Equation (1), to define the cmc as the point of maximum change in the gradient of the concentration of monomeric surfactant. As pointed out before, other conditions could have been used. However, once defined, the position of the cmc should be independent of the experimental property, the technique or the representation used to determine its value. The cmc refers to the surfactant and not to a dependent property. As long as the property φ or its representation depends linearly on the surfactant concentration (Equation (26)), the cmc defined by Equation (1) also corresponds to the point of maximum change in gradient of the property φ or its representation. In fact, Phillips originally defined the condition specifically for linear properties (Footnote in [1]). φ=A[S1] + B[Sm](26) Many properties are linear functions of the surfactant concentration, such as conductivity, Equation (12), NMR-chemical shift, Equation (18), or UV-Vis absorbance, Equation (22). These properties yield in general linear property-concentration curves with a break in the slope at the cmc. However, others depend nonlinearly on the concentration: molar conductivity, Equation (13), surface tension, Equation (17), fluorescence intensity of a probe molecule, Equation (20), pyrene band ratio, Equation (21), absorbance ratio, Equation (23), the mean diffusion coefficient of a dye in micellar exchange, Equation (24), and calorimetric data, such as isothermal titration calorimetry (ITC). These properties generally lead to more or less sigmoidal property-concentration curves. For these nonlinear properties, the correspondence between the point of maximum gradient of the concentration and that of the property is lost. Even worse, the position of these curves relative to the cmc depends on intrinsic parameters such as the width of the transition region, a binding equilibrium constant (fluorimetry), the adsorption equilibrium constant (ST), etc. The example of Figure 4 illustrates the important influence these parameters have on the property-concentration plot (in this case the binding equilibrium constant in fluorimetry). It is principally an impossible task to define or to determine a coherent cmc value from these nonlinear properties as long as the intrinsic parameters are not taken into account. This applies to procedures analysing derivatives of first, second, third or even higher orders of the experimental data, but also to graphical methods looking for intersections of straight lines [ 10 – 12 ]. As said before, the critical micellization concentration, cmc, is a property of the surfactant concentration itself, not of other derived experimental properties. Properties that depend linearly on the surfactant concentration are well behaved and directly reflect the changes in the concentrations of monomeric and micellized surfactants. Here most methods yield correct and coherent cmc values, apart from impurities and other experimental errors. However, those properties that depend nonlinearly on the surfactant concentration distort the surfactant concentration dependence and require a detailed analysis in order to relate their characteristics with the cmc. A general method for the determination of the cmc from linear and nonlinear properties should be based on a common definition of the cmc directly related to the surfactant concentration itself and then derive the experimental properties from this concentration. The concentration model we propose is not the only possible and can still be refined and adapted, but it has so far proven to reproduce excellently several properties, to yield Molecules 2021,26, 5339 17 of 22 coherent cmc values and to contribute to a better understanding of how properties depend on the surfactant concentrations, the cmc and the transition width [23,24,64]. The concentration model also allows one to deduce new property-concentration relationships and to validate them experimentally. For example, in order to clarify the nucleation role of the cationic surfactant CTAB, Dvorsky et al. derived an expression for the mathematical dependence of the ZnS nanoparticle radius on the CTAB concentration based on Equation (7) that reproduced the experimental data within a relative uncertainty as low as 0.34% [65]. 5. Analysis with Sparse Data Points The concentration model can also be used in those cases where the transition region is not well defined due to the lack of data points. The determination of the cmc with the concentration model is robust and mostly independent of the value of the transition width. The uncertainty in the cmc value depends more on the overall quality of the data points than on the presence or not of values near the cmc. In the Figure 7we compare the fit of a dataset of 332 points of the conductivity of SDS with that of the same data eliminating all but 10 points. The results of the fits are given in Table 2. The fit to all points gives precise values of the cmc and the relative transition width already presented before [9]. Of course, the transition width is very poorly defined in the fit to the reduced data set. However, fits with fixed values of r= 0.1 or r= 0.001 lead within error limits to the same value of the cmc as with the full data set. Fixing the value of rto a small value effectively determines the cmc at the intersection of the two straight lines determined by the slopes aand b. Figure 7. Conductivity of SDS in aqueous solution at 298 K. Full data set: the small circles are the 332 data points. The red solid line behind the circles indicates the fit of model Equation (12) to this data. Reduces data set: the big squares are 10 data points taken from the full data set and shifted upwards for better visibility. Orange continuous line and grey dashed line are fits of Equation (12) with fixed values r= 0.1 and 0.001, respectively. Dashed grey lines: concentrations [S 1 ] and [S m ] from Equations (5) and (7). Dotted grey line: second derivative of [S 1 ] (Equation (2)). Vertical grey line: cmc as given by the fit. Lower panel: residuals of the fit to the full data set. (Adapted with permission from [9]). Molecules 2021,26, 5339 18 of 22 Table 2. Results of the fit with the concentration model Equations (7) and (12) to the data of SDS in aqueous solution at 25 ◦C shown in Figure 7. Data cmc/mM ra(1) b(1) all points 8.099 ±0.005 0.112 ±0.001 66.74 ±0.03 26.43 ±0.01 10 points 8.12 ±0.02 0.08 ±0.03 66.6 ±0.1 26.37 ±0.08 10 points 8.10 ±0.02 0.1 fixed 66.75 ±0.09 26.35 ±0.08 10 points 8.12 ±0.02 0.001 fixed 66.60 ±0.09 26.37 ±0.07 (1) Unit: µS cm−1mM−1. 6. Conclusions We have shown that the combination of a surfactant concentration model with known equations for the concentration-dependence of different surfactant properties makes it possible to determine the cmc and the transition width of surfactant solutions and their mixtures in a systematic and unambiguous way. We revised its application to a variety of properties, such as electrical conductivity, NMR chemical shift, self-diffusion, surface tension, UV-Vis absorption, fluorescence intensity and fluorescence correlation. The model equations we propose yield for different properties coherent cmc values for a given type of surfactant. It also opens the way to develop new models for property-concentration relationships or to detect and overcome the limitations of existing ones. The concentration model allows one to extract cmc values from nonlinear property-concentration relationships, such as ST or fluorescent probes, which are consistent with those determined from linear properties and with the proper definition of the cmc based on the surfactant concentration. The model yields also an objective measure of the width of the transition interval around the cmc. Its dependence on composition or temperature or other parameters can yield valuable information not exploited so far. The determination of the cmc from nonlinear fits, even global ones, yields not only objective and consistent values, but also information about their statistical uncertainty of great relevance for the correct interpretation of observed variations. Author Contributions: Both authors have contributed equally to the manuscript. Both authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Xunta de Galicia, grant number ED431B 2019/18. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in Figures 2,3and A1 are available in [ 37 ] and that of Figure A2 in the SI of [ 31 ]. Fitting functions of the concentration model are available on the web page of the authors. Conflicts of Interest: The authors declare no conflict of interest. Molecules 2021,26, 5339 19 of 22 Appendix A Figure A1. Conductivity of imipramine hydrochloride (IMP) in pure water as function of molal concentration and temperature. Black and grey diamonds: experimental data [ 37 ]. Grey and black lines: global fit of Equations (7) and (12) with r= 0.94 ± 0.08, the parameters indicated in the insets, and the cmc shown in Figure 3. Vertical grey lines: range of cmc values obtained from the fit. Right Inset: slopes aand bfrom the fit in units of mS cm −1 kg mol −1 . Left Inset: slope ratio b/a. Lower panel: residuals of the fit. Table A1. Polynomial fit cmc =a+b·T+c·T2 of the cmc values of IMP and ClMP hydrochloride at different temperatures obtained from global fits of the data of Figures 2and A1. cmc/mol kg−1a/mol kg−1b/mol kg−1K−1c/mol kg−1K−2 ClMP 0.30 ±0.04 (−2.2 ±0.2) ×10−3(4.3 ±0.4) ×10−6 IMP 0.29 ±0.03 (−2.1 ±0.2) ×10−3(4.2 ±0.4) ×10−6 Figure A2. Relative transition width ras function of the mole ratio αPS of mixtures of TTAB and PS reported by Pereyra et al. [ 31 ]. 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