The Entropy of Mixing in Self-Assembly and the Role of Surface Tension in Modeling the Critical Micelle Concentration

Abstract

A theory for the micelle formation of nonionic head-tail amphiphiles (detergents) in aqueous solutions is derived based on the traditional molecular thermodynamic modeling approach and a variant of the Flory–Huggins theory that goes beyond lattice models. The theory is used to analyze experimental values for the critical micelle concentration of n-alkyl-ß-D-maltosides within a mass action model. To correlate those parts of the micellization free energy, which depend on the transfer of hydrophobic molecule parts into the aqueous phase, with molecular surfaces, known data for the solubility of alkanes in water are reanalyzed. The correct surface tension to be used in connection with the solvent-excluded surface of the alky tail is ~30 mN/m. This value is smaller than the measured surface tension of a macroscopic alkane–water interface, because the transfer free energy contains a contribution from the incorporation of the alkane or alkyl chain into water, representing the change in free volume in the aqueous phase. The Flory–Huggins theory works well, if one takes into account the difference in liberation free energy between micelles and monomers, which can be described in terms of the aggregation number as well as the thermal de Broglie wavelength and the free volume of the detergent monomer.

1. Introduction

The self-assembly in water of amphiphilic molecules (detergents, surfactants)—in particular those with a head-tail architecture—is a vital process whose understanding bears implications for many research fields, such as detergency [1], biochemistry [2,3,4,5], and biophysics [6,7,8,9]. In their seminal paper on the theory of self-assembly of hydrocarbon amphiphiles, Israelachvili et al. [10] quoted the opening paragraph of a review by Parsegian [11], “Despite enormous progress in understanding the genetics and biochemistry of molecular synthesis we still have only primitive ideas of how linearly synthesized molecules form the multimolecular aggregates that are cellular structures. We assume that the physical forces acting between aggregates of molecules and between individual molecules should explain many of their associative properties; but available physical methods have been inadequate for measuring or computing these forces in solids and liquids.” Israelachvili et al. [10] continue by discussing the burden of linking physical concepts like force and free energy to the observable properties of condensed matter and conclude their own opening paragraph by stating, “To be convincing, and to have any hope whatever of reducing to some semblance of order the vast complexity of those intricate multimolecular structures that are the subject of biology, any successful theories of self-assembly must have as a minimal requirement extreme simplicity to make them accessible to the biologist who has enough concerns of his own not to be dragged into the subtleties of modern physics” (italics by the present author). Some 50 years later, we face similar problems. It seems rather that in the area of life science, the percentage of scientific studies requiring simplicity to avoid the subtleties of modern physics (e.g., statistical physics [12]) has increased.

The focus of our own research in this context [13,14,15,16] is on understanding the self-assembly of mild, nonionic detergents such as n-alkyl-ß-D-maltosides, where the hydrophilic head is a sugar moiety (maltose) and the hydrophobic tail an alkyl chain. These detergents are used to isolate membrane protein complexes from the native membrane [17] and to keep them solubilized in aqueous solution as detergent–protein complexes (PDCs) [8,18,19] for the purpose of crystallizing them for structural characterization [6,17,19,20]. An important quantity characterizing detergents is the critical micelle concentration (CMC), which is the minimal total detergent concentration required for the formation of globular detergent aggregates, termed micelles [21]. Building on the traditional molecular thermodynamic modeling (TMT) approach [10,22,23,24,25,26,27,28,29,30], we searched for a formula that links the CMC to a sum of contributions to the free energy change associated with this self-assembly. In the spirit of the simplicity alluded to above, the TMT approach seeks to explain each part of the sum in terms of molecular properties and comprehensible physical models. Our hope is that such a model, which in the first instance describes only the detergent, can ultimately be incorporated into a model for a more complex system, containing, e.g., membrane proteins, vesicles, etc., and serve as a prototype for modeling the formation of other types of aggregates such as the detergent belt surrounding a membrane protein in a PDC in aqueous solution [8,13]. It should be noted that we call our modeling approach traditional to distinguish it from the computer simulation-molecular thermodynamic (CS-MT) approach, in which the micellization free energy is decomposed in a different way and the modeling is supported by classical molecular dynamics (MD) simulations. We refer the reader to the work by Blankschtein and colleagues [28,29,30] for more information about this method. In the present paper, we focus on the TMT approach.

In a biochemical context, the problem arises that the self-assembly of the detergent takes place in an aqueous medium that is not pure water but contains a number of ingredients such as buffer and salt. Since the majority of models built within the TMT approach refer to pure water, the question arises of how to incorporate the “salt effects” in a sufficiently simple way into the model. An example of an answer to this question is the work by Carale et al. [27]. As shown below, in Section 2.2, there are two contributions to the micellization free energy, termed $g_{\mathrm{t}\mathrm{r}}$ and $g_{\mathrm{i}\mathrm{n}\mathrm{t}}$, which are related to the contact of hydrophobic molecular surfaces with water. The first term, $g_{\mathrm{t}\mathrm{r}}$, refers to the transfer of the alkyl tail of the detergent monomer into the micelle interior and is modeled on the basis of transfer free energies of alkanes into water. According to Carale et al. [27], salt effects may be modeled by using transfer free energies of alkanes into aqueous salt solutions instead. The second term, $g_{\mathrm{i}\mathrm{n}\mathrm{t}}$, refers to the formation of an interface between the hydrophobic interior of the micelle and the surrounding water (partially shielded by the detergent head groups), and it is modeled by considering the interfacial area of the micellar core and an interfacial tension believed to be identical with the measurable interfacial tension of a macroscopic alkane–water interface [22]. The solution to the problem of salt effects, proposed by Carale et al. [27] in this context, is to use the measured interfacial tensions of alkane–aqueous electrolyte interfaces [31].

Motivated by the idea that the cosolute effects on micelle formation might be incorporated into the theory by simply changing the surface tension, we recast $g_{\mathrm{t}\mathrm{r}}$ in terms of the molecular surface of the alkyl tail and proposed to lump it together with $g_{\mathrm{i}\mathrm{n}\mathrm{t}}$ [14]. Since data on the interfacial tension of alkanes and buffer solutions are generally not available, Bothe et al. [15] devised a fitting procedure, in which experimental CMC data of a series of detergents with the same head group, but alkyl tails of different length, were combined with the molecular properties of the detergents to yield the remaining contributions to the micellization free energy. The surface tension $\sigma$, describing the change in interaction between hydrophobic molecular surfaces of the detergent and the aqueous phase upon micelle formation, was then obtained from the slope value of a linear regression. The resulting value of $\sigma \approx$ 30 mN/m [15] was significantly smaller than the macroscopic value of $\sigma \approx$ 50 mN/m [14,22,31]. This discrepancy cannot be traced back to the buffer ingredients. Rather, it is related to an old problem whereby surface tensions associated with molecular surfaces (“microscopic” surface tensions) are found to be smaller than their measurable “macroscopic” counterparts [32,33].

Some 30 years ago, in their 1991 paper in Science, Sharp et al. [34] suggested that the discrepancy between “microscopic” and “macroscopic” surface tensions is due to the neglect of molecular volume differences between solute and solvent. It is at this stage that the entropy of mixing comes into play. In the ideal-mixing models usually employed, the entropy of mixing for a simple solution consisting of the solvent (1) and one solute (2) has the form $\mathsf{\Delta }S=-k_{\mathrm{B}}\left(N_1\ln X_1+N_2\ln X_2\right)$, where $N_i$ and $X_i$ are, respectively, the particle number and mole fraction of species $i$ ($i=\mathrm{1,2}$). The simplest way to take into account the volume differences between molecular species is to use the theory developed by Flory and Huggins for polymer solutions based on lattice models [12,35,36,37]. According to this theory, one has to replace the mole fractions $X_i$ in the above equation for the entropy of mixing by volume fractions $Y_i$ (see below, Equation (10)). As we shall discuss in detail in the present paper (and as has been known since 1947 [38]), the same result can be obtained under the appropriate approximations, without invoking lattice models. Accordingly, Bothe et al. [15] speculated that their finding of a small surface tension value could be an artifact of the used ideal-mixing model.

The goal of the present paper is to clarify this issue by developing a Flory–Huggins type of theory for describing the CMC in combination with the TMT approach and—related to the latter—reanalyzing the solubility of alkanes in water in terms of surface tensions. As it turns out, the “microscopic” surface tensions are correct, which bears implications not only for modeling micellization, but also for understanding hydrophobic solvation in aqueous media of a complex composition in general. A key aspect is that Sharp et al. [34] used a modified version of the Flory–Huggins theory, which has been the subject of debate [39]. Here, we attempt an interpretation of this theory and try to explain why it does not work in modeling micelle formation. In this respect, it was helpful to connect the theory to the concept of the pseudo-chemical potential (PCP) introduced by Ben-Naim [40,41]. This connection revealed a deeper problem in the modeling of micellization that is related to the treatment of kinetic energy in statistical mechanics.

The present paper must deal with some of the subtleties of physics. To make these more accessible, the paper contains a lot of explanations of theoretical concepts and partly has the character of a review. We hope that this is to the benefit of the reader who wants to understand the theory.

2. Theory

2.1. Free Energy of the Micellar Solution and the Entropy of Mixing

2.1.1. Definition of Free Energy and a Model for the Entropy of Mixing

To start with, we follow traditions and decompose the Gibbs free energy $G$ of the micellar solution as follows:

$$G=G_{\mathrm{f}}+G_{\mathrm{m}\mathrm{i}\mathrm{x}}+G_{\mathrm{i}\mathrm{n}\mathrm{t}},$$

(1)

where $G_{\mathrm{f}}$ is often referred to as the free energy of formation and contains the reference chemical potentials, ${\mu }_i^{*},$ and the particle numbers, $N_i,$ of all chemical species in the solution:

$$G_{\mathrm{f}}=N_W{\mu }_W^{*}+\sum _{\nu \ge 1}{N}_{\nu }\nu {\mu }_{\nu }^{*}+\sum _{\alpha }{N}_{\alpha }{\mu }_{\alpha }^{*}.$$

(2)

Here, the indices $\mathrm{W}$, $\nu$, and $\alpha$ stand, respectively, for the solvent water, detergent micelles with aggregation number $\nu$ (including monomers, for which $\nu =1$), and cosolutes (e.g., salt, buffer). We use the term “reference chemical potential” and the notation with an asterisk to distinguish ${\mu }_i^{*}$ from other standard chemical potentials to be discussed below, in Section 2.1.3. In the context of Equation (2), ${\mu }_{\mathrm{W}}^{*}$ represents a Raoult’s law standard state (i.e., pure water) [42,43], while for all the solutes, ${\mu }_i^{*}$ represents a Henry´s law standard state (i.e., infinite dilution in pure water) [42,43]. It is easy to see that the total number of detergent molecules in the solution is $N_{\mathrm{d}\mathrm{e}\mathrm{t}}=\sum _{\nu }\nu N_{\nu }$, if one takes into account that $N_{\nu }$ is the number of micelles that contain $\nu$ detergent molecules. Accordingly, we have to distinguish two different ways of balancing the particle numbers as follows:

$$N_{\mathrm{t}\mathrm{o}\mathrm{t}}=N_W+N_{\mathrm{d}\mathrm{e}\mathrm{t}}+\sum _{\alpha }{N}_{\alpha },$$

(3)

$$\overline{N}=N_W+\sum _{\nu }{N}_{\nu }+\sum _{\alpha }{N}_{\alpha }.$$

(4)

In Equation (3), all individual molecules are counted, whereas in Equation (4), a micelle counts as one particle. Mole fractions will be defined as $X_i=N_i/N_{\mathrm{t}\mathrm{o}\mathrm{t}}$ for all species except the detergent, for which we set $X_{\nu }=\nu N_{\nu }/N_{\mathrm{t}\mathrm{o}\mathrm{t}}$.

To keep the theoretical treatment as simple as possible, we consider only diluted detergent solutions (i.e., with detergent concentrations around the CMC), so that no interactions between detergent molecules, except micelle formation, need to be taken into account. Note that this excludes micelle–micelle interactions and thus puts limits to the applicability of the model. In contrast, we will consider interactions (other than aggregation) between detergent and cosolutes, as well as between cosolutes, by using a mean-field model of the Bragg–Williams type adapted to the Flory–Huggins theory for the free energy of interaction [12]:

$$G_{\mathrm{i}\mathrm{n}\mathrm{t}}={\beta }^{-1}\left[\sum _{\alpha ,\nu }{J}_{\alpha \nu }{Y}_{\alpha }{Y}_{\nu }+\sum _{\alpha ,\alpha \prime }{J}_{\alpha \alpha \prime }{Y}_{\alpha }{Y}_{\alpha \prime }\right].$$

(5)

Here, $J_{\alpha \nu }$ and $J_{\alpha \alpha \prime }$ are coupling constants (also called exchange or mixing parameters), $Y_i$ is the volume fraction of species $i$, and $\beta ={\left(k_{\mathrm{B}}T\right)}^{-1}$ ($k_{\mathrm{B}}$ and $T$ are, respectively, the Boltzmann constant and the absolute temperature.) The factor 1/2, used to avoid double counting of interactions in the second sum in Equation (5), is absorbed into the constant $J_{\alpha \alpha \prime }$. Note that solute–solvent interactions are covered by the reference chemical potentials, ${\mu }_i^{*}$, of the solutes.

The free energy of mixing, $G_{\mathrm{m}\mathrm{i}\mathrm{x}}$, is related to the entropy of mixing, $S_{\mathrm{m}\mathrm{i}\mathrm{x}},$ by $G_{\mathrm{m}\mathrm{i}\mathrm{x}}=-T{S}_{\mathrm{m}\mathrm{i}\mathrm{x}}$. Motivated by the work of Hildebrand [38], we will use a model of $S_{\mathrm{m}\mathrm{i}\mathrm{x}}$ that takes into account molecular volume effects. To illustrate the basic idea, let us consider a gas with a van der Waals (vdW) type equation of state, where only the excluded volume parameter $b$ matters, but the attractive interaction parameter $a$ is set to zero. Then, the entropy change for an isothermal expansion of the gas from a volume $V_1$ to a volume $V_2$ is given as follows:

$$\mathsf{\Delta }S=k_{\mathrm{B}}\ln \left({\displaystyle \frac{V_2-b}{V_1-b}}\right).$$

(6)

Apparently, it is the difference $V-b$ that determines the entropy change. For further interpretation and to avoid confusion, we have to discuss the meaning of the parameter $b$. It is sometimes interpreted as the excluded volume (i.e., the volume around the center of mass (COM) of one molecule, into which the COM of another molecule cannot enter), which is different from the actual volume of a molecule (Figure 1). However, as pointed out recently by Melnyk et al. [44], “the physical meaning of this parameter, termed covolume, is not immediately clear”. Therefore, in many practical applications, $b$ is merely an adjustable parameter. In the treatment by Hildebrand [38], it is the actual volume of a molecule (or one mole of molecules), so that the difference $V-b$ is the free volume (i.e., the empty space not occupied by molecules, cf. Figure 1).

Since we are talking here about the entropy related to the translational motion of molecules, it would make sense to relate entropy to accessible volume, which is the volume that can be reached by the COMs of the molecules. However, the accessible volume cannot be determined easily, as illustrated in Figure 1. Therefore, we follow Hildebrand and use the free volume for the further development of the theory. This strategy introduces a systematic error, however (see below and the discussion in Section 4). In the following, we will denote by $b_i$ the actual volume of one molecule of species $i$, so that the free volume in a gas with $N_i$such molecules is $V-N_i{b}_i$.

In a liquid, the molecules are densely packed, so it is useful to consider a molecule as moving in a cage instantaneously formed by neighboring molecules (Figure 2). At first glance, this seems to exclude diffusion, since the molecule cannot leave the cage. However, if we take an average of all the possible configurations of molecules in the liquid—as it is done in statistical thermodynamics to describe equilibrium states—all arrangements of molecules that may be produced by the diffusion processes are actually taken into account. Thus, the cage picture suffices if we are not interested in diffusion kinetics. Then, we can define different types of volumes associated with an individual molecule in the liquid, as illustrated in Figure 2. We consider the molecules to interact via a hard-core potential, i.e., they may collide elastically like billiard balls and do not interact otherwise. This part of the intermolecular interaction is the most important one in a theory that aims to describe effects due to molecular volume. For simplicity, the molecules are described as spheres in Figure 2, but the theory to be developed is not restricted to this special case. Furthermore, the drawings in Figure 2 are projections into the plane of the figure, so that spheres become circles and surfaces bounding a volume become contours surrounding an area.

We first look at a simple liquid, in which the central and neighboring molecules are of the same type $i$ and, hence, have the same size. Imagine that we freeze the neighbors and allow only the central molecule to move as in Figure 2a. Then, the latter can sample the whole cage within the green circumference and so occupies the orange-colored area on average. To obtain the occupied volume, ${\mathrm{v}}_i,$ of molecular species, $i$, in the pure liquid, we have to unfreeze the neighbors and average the orange-colored area over all possible arrangements of neighbors. The result of this procedure is the volume per molecule of the liquid. It is related to the molar volume, $v_i$, which can be determined from experimental data, at least for pure liquids (see below), by $v_i=N_{\mathrm{A}}{\mathrm{v}}_i,$ with $N_{\mathrm{A}}$ being Avogadro´s number. Due to the thermal motion, the packing of molecules in the liquid is not the densest possible, so that ${\mathrm{v}}_i>b_i$ and ${\mathrm{v}}_i$ depends on pressure, $P$, and temperature, $T$.

The accessible volume, ${\tilde{\omega }}_i,$ per molecule in the liquid is the volume that is sampled by the COM of the central molecule during its motion (area within the blue circumference in Figure 2a) and averaged over all neighbor configurations. Clearly, ${\mathrm{v}}_i>{\tilde{\omega }}_i$. In addition, it is suggested by Figure 2a, albeit not proven, that it is reasonable to assume that ${\mathrm{v}}_i$ is approximately proportional to ${\tilde{\omega }}_i$. The situation in the liquid is different from that in a gas (cf. Figure 1), since in the former, the molecules are so densely packed that their excluded volumes always overlap. This overlap is of a certain order of magnitude and probably depends less on the actual configuration of the molecules than in a gas (meaning that the spread of overlap values is probably smaller in the liquid). We could try to build our theory upon this proportionality between ${\tilde{\omega }}_i$ and ${\mathrm{v}}_i$. However, we shall not pursue this idea in the present paper. Here, we want to explore the possibilities and limits of the Flory–Huggins theory [12,35,36,37] and the related ideas by Hildebrand [38]. So, we proceed—like Hildebrand—with the concept of the free volume, ${\omega }_i$, which is illustrated in Figure 2b. The yellow-colored area corresponds to the difference between the instantaneous cavity volume and the molecular volume. If we average this volume difference over all possible configurations of the central molecule and the neighbors, we arrive at the free volume per molecule, which can be expressed as ${\omega }_i={\mathrm{v}}_i-b_i$.

With this preparation in mind, we now look at a binary liquid mixture. To obtain the entropy of mixing, we consider the transfer of $N_1$ molecules of type 1 and $N_2$ molecules of type 2 with molecular volumes $b_1$ and $b_2$, respectively, from the pure state, where they occupy the respective volumes, ${\mathrm{v}}_1$ and ${\mathrm{v}}_2$, to a solution with total particle number $N_1+N_2$. Since the molecules under consideration are, in general, polyatomic, they have internal degrees of freedom (DOFs) due to vibrations, rotations, and librations. An important simplification suggested by Hildebrand [38] is to assume that the energy in these internal DOFs is not significantly different in the pure state and in the solution, so that the contribution of the internal DOFs to the entropy change can be neglected. Under these conditions, the entropy of mixing is obtained by considering for each component the expansion (or possibly compression) from its free volume, $N_i\left({\mathrm{v}}_i-b_i\right),$ in the pure state to its free volume, $V-N_1{b}_1-N_2{b}_2$, in the mixture, where $V$ is the actual volume of the solution, as follows:

$${\displaystyle \frac{\mathsf{\Delta }S}{k_{\mathrm{B}}}}=N_1\ln \left\{{\displaystyle \frac{V-N_1{b}_1-N_2{b}_2}{N_1({\mathrm{v}}_1-b_1)}}\right\}+N_2\ln \left\{{\displaystyle \frac{V-N_1{b}_1-N_2{b}_2}{N_2({\mathrm{v}}_2-b_2)}}\right\}.$$

(7)

This equation can also be derived from statistical mechanics (Appendix A), albeit with some roughness in the spatial integration, which is related to the difference between accessible and free volume (to be discussed in Section 4).

At this stage, it is still possible that due to interactions with molecules of different types, the volume occupied by one molecule in the mixture is different from the volume it occupies in the pure phase, i.e., there is an excess volume (cp. Figure 2a,c). An important further simplification is the assumption that the solution is additive, i.e., excess volumes are neglected, and the volume of the mixture can be written simply as $V=N_1{\mathrm{v}}_1+N_2{\mathrm{v}}_2$. Then, the entropy of mixing becomes

$${\displaystyle \frac{\mathsf{\Delta }S}{k_{\mathrm{B}}}}=N_1\ln \left\{{\displaystyle \frac{N_1{\omega }_1+N_2{\omega }_2}{N_1{\omega }_1}}\right\}+N_2\ln \left\{{\displaystyle \frac{N_1{\omega }_1+N_2{\omega }_2}{N_2{\omega }_2}}\right\}.$$

(8)

We would like to link the free volumes to more handy quantities, like the mole fraction $X_i$ and the volume fraction $Y_i$. To this end, we make another simplifying assumption discussed by Hildebrand [38]; we assume that ${\omega }_i$ is proportional to $b_i$, with a species-independent proportionality constant $\varsigma$, i.e., ${\omega }_i=\varsigma b_i$. Then, ${\mathrm{v}}_i=\left(\varsigma +1\right)b_i$ is also proportional to $b_i$ and

$$\begin{gathered} {\displaystyle \frac{N_1{\omega }_1}{N_1{\omega }_1+N_2{\omega }_2}}={\displaystyle \frac{N_1\varsigma b_1}{N_1\varsigma b_1+N_2\varsigma b_2}}={\displaystyle \frac{N_1\left(\varsigma +1\right)b_1}{N_1\left(\varsigma +1\right)b_1+N_2\left(\varsigma +1\right)b_2}} \\ ={\displaystyle \frac{N_1{\mathrm{v}}_1}{N_1{\mathrm{v}}_1+N_2{\mathrm{v}}_2}}={\displaystyle \frac{N_1{\mathrm{v}}_1}{V}}=Y_1, \end{gathered}$$

(9)

so that

$$\mathsf{\Delta }S=-k_{\mathrm{B}}\left(N_1\ln Y_1+N_2\ln Y_2\right).$$

(10)

Thus, we can express the entropy of mixing in a very simple way in terms of particle numbers and volume fractions. This result has been obtained by Flory and Huggins based on a lattice model [35,36,37]. In Appendix B, we show a variant of such a lattice model along the lines of Hill´s treatment of the Flory–Huggins theory [12] and adapted to the problem of micelle formation. In the following, we will call any theory of solutions that is based upon expressing the entropy of mixing in terms of volume fractions as in Equation (10), and that is derived as above based upon Hildebrand´s considerations [38], a Flory–Huggins (FH) theory. Thus, the term “FH” is used here in a more general meaning than is usually the case in the literature. Our FH theory is not limited to polymer solutions, nor is it built upon lattice models.

Turning back to our micellar solution, we now have for the free energy of mixing the following equation:

$$G_{\mathrm{m}\mathrm{i}\mathrm{x}}={\beta }^{-1}\left(N_{\mathrm{W}}\ln Y_{\mathrm{W}}+\sum _{\nu }{N}_{\nu }\ln Y_{\nu }+\sum _{\alpha }{N}_{\alpha }\ln Y_{\alpha }\right).$$

(11)

2.1.2. The Micellar Size Distribution

We apply the method of minimizing the Gibbs free energy to derive the micellar size distribution. To this end, we minimize $G$ in Equation (1) with respect to a variation in the numbers $N_{\nu }$ under the constraint that the number $N_{\mathrm{d}\mathrm{e}\mathrm{t}}=\sum _{\nu }\nu N_{\nu }$, defined above, is constant. The result is as follows [15]:

$$\mathsf{\lambda }={\displaystyle \frac{1}{\nu }}{\left({\displaystyle \frac{\partial G}{\partial N_{\nu }}}\right)}_{T,P,N_{\mathrm{W}},\left\{N_{\alpha }\right\},\left\{N_{\nu \prime }\right\}},$$

(12)

where $\lambda$ is to be identified with the chemical potential, ${\mu }_1,$ of detergent monomers in the solution. The evaluation of the partial derivative in Equation (12) is detailed in the Supplementary Text S1. Comparing the case of general $\nu$ with the case $\nu =1$, we obtain from Equation (12) the following:

$$\begin{array}{cc}-\beta \nu \left({\mu }_{\nu }^{*}-{\mu }_1^{*}\right)=\hfill & \ln {\displaystyle \frac{X_{\nu }}{\nu X_1^{\nu }}}+\ln {\displaystyle \frac{c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_{\nu }}{{\left(c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_1\right)}^{\nu }}}+1-\nu \hfill \\ & +{\displaystyle \frac{1}{V}}{\displaystyle \sum _{\alpha }}{Y}_{\alpha }\left({\mathrm{v}}_{\nu }{J}_{\alpha \nu }-\nu {\mathrm{v}}_1{J}_{\alpha 1}\right)-{\displaystyle \frac{\overline{N}+L}{V}}\left({\mathrm{v}}_{\nu }-\nu {\mathrm{v}}_1\right).\hfill \end{array}$$

(13)

Here, we introduced the total molarity $c_{\mathrm{t}\mathrm{o}\mathrm{t}}=N_{\mathrm{t}\mathrm{o}\mathrm{t}}/\left(V{N}_{\mathrm{A}}\right)$ of the solution. The quantity $L$ contains detergent–cosolute as well as cosolute–cosolute interactions and is defined in Equation (S7). The first term on the right-hand side (rhs) of Equation (13) can be interpreted as the logarithm of the equilibrium constant $K_{\nu }=X_{\nu }/\left(\nu X_1^{\nu }\right)$ for the formation of a micelle ${\mathrm{D}}_{\nu }$ from $\nu$ monomers $\mathrm{D},$ according to the following:

$$\mathsf{\nu }\mathrm{D}\rightleftharpoons {\mathrm{D}}_{\nu }.$$

(14)

Note that $X_{\nu }/\nu$ is the mole fraction of micelles of size $\nu$. The use of $K_{\nu }$ implies a mass action law, but since we allow for various numbers $\nu$, we are working with a multi-equilibrium model in the present stage of development.

The second term on the rhs of Equation (13) originates from the fact that we describe the free energy of mixing in terms of volume fractions rather than mole fractions, where the latter approach would represent an ideal-mixing model. It is possible to formulate the mass action law in terms of volume fractions with the following equilibrium constant:

$${\mathcal{K}}_{\nu }={\displaystyle \frac{Y_{\nu }}{Y_1^{\nu }}}={\displaystyle \frac{N_{\nu }{\mathrm{v}}_{\nu }{V}^{\nu }}{V{\left(N_1{\mathrm{v}}_1\right)}^{\nu }}}={\displaystyle \frac{c_{\mathrm{t}\mathrm{o}\mathrm{t}}{N}_{\mathrm{A}}{N}_{\nu }{\mathrm{v}}_{\nu }{\left(N_{\mathrm{t}\mathrm{o}\mathrm{t}}\right)}^{\nu }}{N_{\mathrm{t}\mathrm{o}\mathrm{t}}{\left(c_{\mathrm{t}\mathrm{o}\mathrm{t}}{N}_{\mathrm{A}}{N}_1{\mathrm{v}}_1\right)}^{\nu }}}={\displaystyle \frac{X_{\nu }{c}_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_{\nu }}{\nu X_1^{\nu }{\left(c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_1\right)}^{\nu }}}=K_{\nu }{\left(c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_1\right)}^{\left(1-\nu \right)},$$

(15)

so that the first and second term on the rhs of Equation (13) together can be interpreted as the logarithm of ${\mathcal{K}}_{\nu }$. Another variant often employed is to write the mass action law in terms of molarities, yielding yet another equilibrium constant:

$${\mathbb{K}}_{\nu }={\displaystyle \frac{\left[{\mathrm{D}}_{\nu }\right]}{{\left[\mathrm{D}\right]}^{\nu }}}={\displaystyle \frac{N_{\nu }/\left(N_{\mathrm{A}}V\right)}{{\left(N_1/\left(N_{\mathrm{A}}V\right)\right)}^{\nu }}}={\displaystyle \frac{N_{\mathrm{t}\mathrm{o}\mathrm{t}}{X}_{\nu }/\nu }{{\left(N_{\mathrm{t}\mathrm{o}\mathrm{t}}{X}_1\right)}^{\nu }}}{\displaystyle \frac{{\left(N_{\mathrm{A}}V\right)}^{\nu }}{N_{\mathrm{A}}V}}=K_{\nu }{c}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\left(1-\nu \right)}.$$

(16)

It has recently been suggested that equilibrium constants should always be formulated with dimensionless quantities [45] to avoid problems with units such as the factor $c_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\left(1-\nu \right)}$ in Equation (16). However, as is obvious from Equation (15), not all equilibrium constants formulated with dimensionless quantities are equivalent.

The term $1-\nu$ also originates from the use of volume fractions in $G_{\mathrm{m}\mathrm{i}\mathrm{x}}$ (more about this in Section 2.1.3), while the fourth term on the rhs of Equation (13) is the difference between micelle–cosolute and monomer–cosolute interactions weighted by the occupied volume ${\mathrm{v}}_{\nu }$. The last term follows from parts of $G_{\mathrm{m}\mathrm{i}\mathrm{x}}$ and $G_{\mathrm{i}\mathrm{n}\mathrm{t}}$ and contains the difference, ${\mathrm{v}}_{\nu }-\nu {\mathrm{v}}_1$, between the volumes occupied by a micelle and $\nu$ monomers. This difference merits a comment; the concept of an additive solution implies that the volume occupied by a species in solution is the same as that it occupies in a hypothetical pure solute phase. Since micelles and monomers are treated as different species, it is still possible that ${\mathrm{v}}_{\nu }\ne \nu {\mathrm{v}}_1$. However, to arrive at the FH theory, we had to make the additional assumption that the volume occupied by a molecule is proportional to its molecular volume, where the proportionality constant $\left(\varsigma +1\right)$ is the same for all species (see above). Thus,

$${\mathrm{v}}_{\nu }=\left(\varsigma +1\right)b_{\nu }=\left(\varsigma +1\right)\nu b_1=\nu {\mathrm{v}}_1.$$

(17)

We then conclude that ${\mathrm{v}}_{\nu }-\nu {\mathrm{v}}_1=0$ in FH theory. Clearly, this conclusion necessarily follows from a lattice model (cf. Appendix B), but it also derives rather directly from Hildebrand´s considerations [38].

Taking into account Equation (17), Equation (13) simplifies to

$$-\nu g_{\nu }^{*}=\ln {\displaystyle \frac{X_{\nu }}{\nu X_1^{\nu }}}+\ln {\displaystyle \frac{c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_{\nu }}{{\left(c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_1\right)}^{\nu }}}+1-\nu +\sum _{\alpha }{Y}_{\alpha }{j}_{\alpha \nu },$$

(18)

where we have introduced the abbreviations $g_{\nu }^{*}=\beta \left({\mu }_{\nu }^{*}-{\mu }_1^{*}\right)$ and

$$j_{\alpha \nu }={\displaystyle \frac{\nu {\mathrm{v}}_1}{V}}\left(J_{\alpha \nu }-J_{\alpha 1}\right).$$

(19)

Equation (18) can be rearranged to give the micellar size distribution:

$$X_{\nu }=X_1^{\nu }{\left(c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_1\right)}^{\left(\nu -1\right)}\exp \left\{-\left(\nu g_{\nu }^{*}+1-\nu +\sum _{\alpha }{Y}_{\alpha }{j}_{\alpha \nu }\right)\right\},$$

(20)

where we assumed $v_{\nu }=\nu v_1$, based on Equation (17), and $v_i=N_{\mathrm{A}}{\mathrm{v}}_i$. This distribution function differs from the one obtained with an ideal-mixing model by the pre-exponential factor ${\left(c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_1\right)}^{\left(\nu -1\right)}$ (which is actually ${\nu \left(c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_1\right)}^{\left(\nu -1\right)}$, canceling out the factor $\nu$ in the ideal-mixing model) and the term $1-\nu$ in the exponent. The last term in the exponent would also occur in an ideal-mixing model but would be formulated in terms of $X_{\alpha }$ rather than $Y_{\alpha }$.

2.1.3. Reference States and Standard States

In our model, the chemical potential of a detergent molecule in a micelle is as follows (cf. Equation (12) and Supplementary Material S1):

$$\begin{array}{cc}{\mu }_{\nu }\hfill & ={\displaystyle \frac{1}{\nu }}{\left({\displaystyle \frac{\partial G}{\partial N_{\nu }}}\right)}_{T,P,N_{\mathrm{W}},\left\{N_{\alpha }\right\},\left\{N_{\nu \prime }\right\}}\hfill \\ & ={\mu }_{\nu }^{\mathrm{*}}+{\left(\beta \nu \right)}^{-1}\left[\ln \left({\displaystyle \frac{X_{\nu }}{\nu }}\right)+\ln \left(c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_{\nu }\right)+1-{\displaystyle \frac{\overline{N}{\mathrm{v}}_{\nu }}{V}}+{\displaystyle \frac{{\mathrm{v}}_{\nu }}{V}}\left(\sum _{\alpha }{J}_{\alpha \nu }{Y}_{\alpha }-L\right)\right].\hfill \end{array}$$

(21)

If we have no cosolutes, this reduces to

$${\mu }_{\nu }={\mu }_{\nu }^{\mathrm{*}}+{\left(\beta \nu \right)}^{-1}\left[\ln \left({\displaystyle \frac{X_{\nu }}{\nu }}\right)+\ln \left(c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_{\nu }\right)+1-{\displaystyle \frac{\overline{N}{\mathrm{v}}_{\nu }}{V}}\right].$$

(22)

Here, ${\mu }_{\nu }^{\mathrm{*}}$ refers to a reference state of a micelle at infinite dilution in water. Thus, ${\mu }_{\nu }^{\mathrm{*}}$ contains detergent–detergent interactions within the micelle as well as interactions of the micelle with water. (For $\nu =1$, it contains only the interaction of the detergent monomer with water). This Henry´s law standard state, in which the properties of an infinitely diluted system are extrapolated to $X_{\nu }=1$ [42,43], is somewhat fictitious. However, Phillips had the insightful idea to envisage such a state as a hypothetical pure phase of micelle hydrates [46]. Since we do not allow for micelle–micelle interactions, the hypothetical pure phase, albeit condensed, consists of non-interacting micelle hydrates. Such a reference state is useful in computations, as it allows us to model ${\mu }_{\nu }^{\mathrm{*}}$ (or ${\mu }_{\nu }^{\mathrm{*}}-{\mu }_1^{\mathrm{*}}$, which is needed in Equation (20)) on the basis of a single micelle in water. Besides the internal DOFs of the detergent molecules, detergent–detergent interactions within the micelle and interactions of the micelle with its hydration shell (possibly including water molecules penetrating the micelle) belong to the internal DOFs of the micelle hydrate. These internal DOFs are assumed to remain unaffected by putting the micelle into the solution.

A physically appealing view has been provided by Ben-Naim [40,41], who introduced the PCP (Appendix C). It is worthwhile to consider this concept, as it sheds light on the mysterious term $1-\overline{N}{\mathrm{v}}_{\nu }/V$ in Equation (22) (and other aspects; see below). The easiest way to envisage the chemical potential, $\mu$, is to view it as the free energy difference between an $\left(N+1\right)$- and an $N$-particle system, as shown in Equation (A13). The PCP $\eta$ (as defined in Appendix C) is then obtained by adding the $\left(N+1\right)$th particle at a fixed position (which is arbitrary and denoted by the position vector ${\mathbf{R}}_0$ in Figure 3; Equation (A14)). The only free energy change that can be caused in this way by a non-interacting particle with no internal DOFs is due to a decrease in the free volume. This is the essential proposition made by Equation (A20), where the pseudo-chemical potential is defined by a partial differentiation. The difference $\mu -\eta$ is referred to as the liberation free energy and has three independent contributions due to removing the constraint of a fixed position from the added particle, originating from the particle´s kinetic energy, its expansion into the free volume, and its indistinguishability from the other particles of the same type (Appendix C).

If we assume that the solution is additive and the volume occupied by a molecule is proportional to its actual volume (cf. Section 2.1.1), we see that the term $1-\overline{N}{\mathrm{v}}_{\nu }/V$ corresponds to the difference $\mathsf{\Delta }{\eta }_{\nu }={\eta }_{\nu }-{\eta }_{\nu }^{*}$ of the micelle´s PCP between the solution and the pure hydrate phase (times $\beta$; Appendix C). Note that $\mathsf{\Delta }{\eta }_{\nu }$ does not contain contributions from the internal DOFs of the micelle hydrate. Then, Equation (22) becomes

$${\mu }_{\nu }={\mu }_{\nu }^{\mathrm{*}}+{\nu }^{-1}\left[{\beta }^{-1}\ln Y_{\nu }+\mathsf{\Delta }{\eta }_{\nu }\right].$$

(23)

We are now able to explain the term $1-\nu$ occurring in the micellar size distribution in Equation (20). It arises from the difference

$$\beta \left(\mathsf{\Delta }{\eta }_{\nu }-\nu \mathsf{\Delta }{\eta }_1\right)=1-{\displaystyle \frac{\overline{N}{\mathrm{v}}_{\nu }}{V}}-\nu \left(1-{\displaystyle \frac{\overline{N}{\mathrm{v}}_1}{V}}\right)=1-{\displaystyle \frac{\overline{N}{\mathsf{\nu }\mathrm{v}}_1}{V}}-\nu \left(1-{\displaystyle \frac{\overline{N}{\mathrm{v}}_1}{V}}\right)=1-\nu ,$$

(24)

which has its origins in the finite molecular volumes and vanishes in ideal-mixing models that neglect these volumes.

So far, we have defined the PCP ${\eta }_i$ only for featureless non-interacting particles. If we include particle–particle interactions and internal DOFs, we arrive at Equation (A28) for the PCP, which is now termed ${\tilde{\mu }}_i$, in accordance with Ben-Naim [40] (Appendix D). The decomposition of the Gibbs free energy of the solution according to Equation (1) and the use of FH theory for $G_{\mathrm{m}\mathrm{i}\mathrm{x}}$ implies that we split any solute–solute interaction into a hard-core part, which contributes to $G_{\mathrm{m}\mathrm{i}\mathrm{x}}$ and yields the form of ${\eta }_i$ found for non-interacting particles with finite volume (Supplementary Material S3) as well as a remaining part described approximately by $G_{\mathrm{i}\mathrm{n}\mathrm{t}}$. The solute–water interactions can be considered as being part of ${\tilde{\mu }}_i$ according to Equation (A28), but we do not specify here the form of the potential energy, $B_i,$ of this interaction.

The next question is how is ${\tilde{\mu }}_i$ related to ${\mu }_i^{\mathrm{*}}$? By definition, ${\tilde{\mu }}_i$ does not contain the liberation free energy. We have to find out, where in $G$ (Equation (1)) the three contributions to the liberation free energy (see above and in Appendix C) occur. Certainly, they are not contained in $G_{\mathrm{i}\mathrm{n}\mathrm{t}}$. In $G_{\mathrm{m}\mathrm{i}\mathrm{x}}$, we subtract the hard-core part of the free energy of the pure solutes from that of the mixture. This merely accounts for the change in the free-volume contribution to the liberation free energy due to mixing. Hence, the pure-substance free-volume contribution and the other contributions to the liberation free energy are contained in $G_{\mathrm{f}}$. So, we have the following:

$${\mu }_i^{\mathrm{*}}={\tilde{\mu }}_i^{\mathrm{*}}+{\beta }^{-1}\ln \left\{{\displaystyle \frac{N_i{\mathsf{\Lambda }}_i^3}{N_i{\omega }_i}}\right\},$$

(25)

where ${\mathsf{\Lambda }}_i$ is the thermal de Broglie wavelength of particles of type $i$ (defined below) and the asterisk in ${\tilde{\mu }}_i^{\mathrm{*}}$ reminds us that it refers to the limit of infinite dilution in pure water. In Equation (25), the three contributions to liberation are ${\mathsf{\Lambda }}_i^3$ (kinetic energy), $N_i$ (indistinguishability), and $N_i{\omega }_i$ (free volume). This choice of the reference chemical potential is ultimately dictated by the use of $G_{\mathrm{m}\mathrm{i}\mathrm{x}}$ in Equation (1), which is based on the free energy difference between the solution and the pure components (Appendix A). Note that the term $\mathsf{\Delta }{\eta }_{\nu }$ in Equation (23) serves to remove the pure-substance free-volume contribution to the PCP and add the appropriate free-volume contribution of the mixture. However, in the micellization free energy, which is the difference between micelles and monomers, the mixture contributions cancel out according to Equation (24), which is a consequence of the approximations necessary to arrive at the FH theory (specifically, Equation (17)).

For micelles, Equation (25) has to be modified slightly because we defined the reference chemical potential by $\nu {\mu }_{\nu }^{\mathrm{*}}$ (cf. Equation (2)), so that

$$\nu {\mu }_{\nu }^{\mathrm{*}}={\tilde{\mu }}_{\nu }^{\mathrm{*}}+{\beta }^{-1}\ln \left\{{\displaystyle \frac{{\mathsf{\Lambda }}_{\nu }^3}{{\omega }_{\nu }}}\right\}.$$

(26)

Here, ${\tilde{\mu }}_{\nu }^{\mathrm{*}}$, ${\mathsf{\Lambda }}_{\nu }$, and ${\omega }_{\nu }$ refer to a whole micelle, whereas ${\mu }_{\nu }^{\mathrm{*}}$ is the reference chemical potential of one detergent molecule in the micelle. Equation (26) is illustrated in Figure 3a, if one replaces ${\mu }_{\nu }^{\mathrm{*}}$ and ${\eta }_{\nu }^{\mathrm{*}}$ by $\nu {\mu }_{\nu }^{\mathrm{*}}$ and ${\tilde{\mu }}_{\nu }^{\mathrm{*}}$, respectively.

In the presence of cosolutes, we may define the standard chemical potential ${\mu }_{\nu }^0$ by

$${\mu }_{\nu }^0={\mu }_{\nu }^{\mathrm{*}}+{\displaystyle \frac{{\mathrm{v}}_{\nu }}{\beta \nu V}}\left(\sum _{\alpha }{J}_{\alpha \nu }{Y}_{\alpha }-L\right)={\mu }_{\nu }^{\mathrm{*}}+{\displaystyle \frac{{\mathrm{v}}_1}{\beta V}}\left(\sum _{\alpha }{J}_{\alpha \nu }{Y}_{\alpha }-L\right).$$

(27)

It is then possible to write Equation (21) in the simple form

$${\mu }_{\nu }={\mu }_{\nu }^0+{\nu }^{-1}\left[{\beta }^{-1}\ln Y_{\nu }+\mathsf{\Delta }{\eta }_{\nu }\right].$$

(28)

The standard state described by ${\mu }_{\nu }^0$ corresponds to an infinite dilution in the aqueous medium containing all cosolutes $\alpha$. With the abbreviation $g_{\nu }^0=\beta \left({\mu }_{\nu }^0-{\mu }_1^0\right)$, we can write the micellar size distribution as follows:

$$Y_{\nu }=Y_1^{\nu }{e}^{\nu -1}{e}^{-\nu g_{\nu }^0}.$$

(29)

The advantage of using this standard state is that the effects of the cosolutes on the micellization equilibria in Equation (14) can be described by a shift in the equilibrium constants ${\mathcal{K}}_{\nu }$. Let ${\mathcal{K}}_{\nu }^{*}$ be the equilibrium constant in the absence of cosolutes (i.e., in pure water). Then,

$${\displaystyle \frac{{\mathcal{K}}_{\nu }}{{\mathcal{K}}_{\nu }^{*}}}=e^{-\nu \left(g_{\nu }^0-g_{\nu }^{*}\right)}=\exp \left\{-\sum _{\alpha }{Y}_{\alpha }{j}_{\alpha \nu }\right\}.$$

(30)

It is possible to consider the detergent–cosolute interactions as part of the PCP (cf. Appendix D), which we denote as ${\tilde{\mu }}_{\nu }^0$ to distinguish it from ${\tilde{\mu }}_{\nu }^{\mathrm{*}}$. Then, ${\mu }_{\nu }^0$ can be related to ${\tilde{\mu }}_{\nu }^0$ via

$$\nu {\mu }_{\nu }^0={\tilde{\mu }}_{\nu }^0+{\beta }^{-1}\ln \left\{{\displaystyle \frac{{\mathsf{\Lambda }}_{\nu }^3}{{\omega }_{\nu }}}\right\}$$

(31)

in analogy to Equation (26). Next, we compute the difference

$$\nu {\mu }_{\nu }^0-\nu {\mu }_1^0={\tilde{\mu }}_{\nu }^0-\nu {\tilde{\mu }}_1^0+{\beta }^{-1}\ln \left\{{\displaystyle \frac{{\mathsf{\Lambda }}_{\nu }^3{\omega }_1^{\nu }}{{\mathsf{\Lambda }}_1^{3\nu }{\omega }_{\nu }}}\right\},$$

(32)

which is required for the micellization free energy. By definition, the thermal de Broglie wavelength is given by ${\mathsf{\Lambda }}_i=h/\sqrt{2\pi m_i{k}_{\mathrm{B}}T}$, where $m_i$ is the mass of particle $i$. Neglecting the penetration of water or cosolutes into the micelle, we have $m_{\nu }=\nu m_1$ and thus ${\mathsf{\Lambda }}_{\nu }={\mathsf{\Lambda }}_1/\sqrt{\nu }$ (at constant $T$). With ${\mathsf{\omega }}_{\nu }=\varsigma b_{\nu }=\varsigma \nu b_1=\nu {\mathsf{\omega }}_1$, we obtain

$$\nu g_{\nu }^0=\beta \left({\tilde{\mu }}_{\nu }^0-\nu {\tilde{\mu }}_1^0\right)-{\displaystyle \frac{5}{2}}\ln \nu -\left(\nu -1\right)\ln \left\{{\displaystyle \frac{{\mathsf{\Lambda }}_1^3}{{\omega }_1}}\right\}.$$

(33)

Equation (33) states that the micellization free energy can be written as the PCP difference plus the terms describing the liberation free energy difference between the micelle and $\nu$ monomers. Here, the problem arises that the change in liberation free energy is not accounted for in the traditional modeling of the micellization free energy (see below).

2.2. Mass Action Model and Molecular Thermodynamic Modeling

2.2.1. Mass Action Model and Definition of the Critical Micelle Concentration

When we talk about a mass action model, we mean that we consider only one type of micelle with a fixed aggregation number $m$. This implies that we neglect the polydispersity of the micelles, and $\nu \in \left\{1,m\right\}$. At this stage, it is useful to introduce the abbreviations $x=y+z$, $y=X_1$, $z=X_m$, and $\mathbb{c}=c_{\mathrm{t}\mathrm{o}\mathrm{t}}{v}_{1}e$, where $e$ is Euler´s number. Then, the mass balance of detergent reads as follows:

$$x=y+y^m{\mathbb{c}}^{\left(m-1\right)}{e}^{-m{g}_m^0}.$$

(34)

Here, the factor ${\mathbb{c}}^{\left(m-1\right)}$ (actually $m{\mathbb{c}}^{\left(m-1\right)}$ with $m$ canceling) comprises all corrections to an ideal-mixing model that are due to the consideration of finite molecular volumes (except those contained in $g_m^0$, originating from $G_{\mathrm{i}\mathrm{n}\mathrm{t}}$).

According to Bothe et al. [15], the CMC can be defined by

$${\left({\displaystyle \frac{d^{3}y}{d{x}^3}}\right)}_{x=X_{\mathrm{C}\mathrm{M}\mathrm{C}}}=0,$$

(35)

where $X_{\mathrm{C}\mathrm{M}\mathrm{C}}$ is the total detergent mole fraction at the CMC, while the CMC in molarity units is given by $\mathrm{C}\mathrm{M}\mathrm{C}=c_{\mathrm{t}\mathrm{o}\mathrm{t}}{X}_{\mathrm{C}\mathrm{M}\mathrm{C}}$. If $X_{\mathrm{C}\mathrm{M}\mathrm{C}}$ is small (i.e., CMC < 30 mM) and $x$ remains in the vicinity of $X_{\mathrm{C}\mathrm{M}\mathrm{C}}$ (i.e., $c_{\mathrm{t}\mathrm{o}\mathrm{t}}{X}_{\mathrm{C}\mathrm{M}\mathrm{C}}$ < 50 mM), then $c_{\mathrm{t}\mathrm{o}\mathrm{t}}$ (which is in the order of 55 M) will not change much and can be considered as constant. Then, $\mathbb{c}$ is a constant, too. The exponent in Equation (34) must not depend on $x$, as the standard micellization free energy refers to a standard state at infinite dilution. This is strictly true for $g_m^{*}$. However, $g_m^0$ depends on $Y_{\alpha }$, based on the definition of ${\mu }_{\nu }^0$ in Equation (27). One may wonder whether this connection leads to an indirect dependence of $g_m^0$ on $x$. Since the volume fractions sum up to unity, they are not all independent. If we choose the volume fraction of water as the dependent variable, all solute volume fractions are independent of each other and $Y_{\alpha }$ does not depend directly on $x$. It may still depend indirectly on $x$ via $c_{\mathrm{t}\mathrm{o}\mathrm{t}}$, but with constant $c_{\mathrm{t}\mathrm{o}\mathrm{t}}$, $Y_{\alpha }$ is a constant too. So, the exponent in Equation (34) does neither depend on $x$ nor on $y$. Related difficulties discussed recently [16] are the artifacts of the ideal-mixing model and the used form of $G_{\mathrm{i}\mathrm{n}\mathrm{t}}$.

Under the condition that the exponent is a constant as regards derivatives with respect to $x$ and $y$, it has been shown by Bothe et al. [15], that Equation (35) is equivalent to the condition

$${3\left({\displaystyle \frac{d^{2}z}{d{y}^2}}\right)}^2=\left(1+{\displaystyle \frac{dz}{dy}}\right){\displaystyle \frac{d^{3}z}{d{y}^3}}.$$

(36)

From this condition, the ratio $y/x=\left(2m^2-m\right)/\left(2m^2-2\right)$ at the CMC follows (which is the same as found by Bothe et al. for the ideal-mixing model [15]). The relationship between $g_m^0$ and $X_{\mathrm{C}\mathrm{M}\mathrm{C}}$ is given by

$$g_m^0={\displaystyle \frac{\left(m-1\right)}{m}}\ln \left\{\mathbb{c}{X}_{\mathrm{C}\mathrm{M}\mathrm{C}}\right\}+{\tau }_m$$

(37)

with

$${\tau }_m={\displaystyle \frac{1}{m}}\ln \left\{{\displaystyle \frac{{\left(2m^2-m\right)}^m}{\left(m-2\right){\left(2m^2-2\right)}^{\left(m-1\right)}}}\right\}.$$

(38)

Here, ${\tau }_m$ is different from the corresponding term obtained by Bothe et al. [15], which is ${\tau \prime }_m={\tau }_m+m^{-1}\ln m$. The various factors and terms depending on $m$ are plotted in Figure S3. For $m>50$, the prefactor $(m-1)/m$ is close to unity and can be neglected, while ${\tau }_m<0.085$—albeit clearly smaller than ${\tau \prime }_m$—is not necessarily negligible. We then obtain

$$Y_{\mathrm{C}\mathrm{M}\mathrm{C}}=v_1\mathrm{C}\mathrm{M}\mathrm{C}=e^{g_m^0-\left({\tau }_m+1\right)},$$

(39)

where $Y_{\mathrm{C}\mathrm{M}\mathrm{C}}$ is the total volume fraction of detergent at the CMC. Alternatively, one may compute $g_m^0$ from the experimentally determined CMC values according to

$$g_m^0=\ln \left\{v_1\mathrm{C}\mathrm{M}\mathrm{C}\right\}+{\tau }_m+1.$$

(40)

2.2.2. Thermodynamic Cycle Approach and Molecular Thermodynamic Modeling

In the following, we will denote the PCP part of $g_m^{*}$ or $g_m^0$, which we wish to model, as $g_{\mathrm{m}\mathrm{i}\mathrm{c}}$. Thus, $g_{\mathrm{m}\mathrm{i}\mathrm{c}}$ corresponds to a standard free energy difference per detergent molecule in the micelle, divided by the thermal energy that is devoid from any contribution from the liberation free energy, related to the COM motion of micelles and monomers. The notion of $g_{\mathrm{m}\mathrm{i}\mathrm{c}}$ as a PCP difference is a new aspect of the present paper. In molecular thermodynamic modeling [14,22,25,26], $g_{\mathrm{m}\mathrm{i}\mathrm{c}}$ is decomposed on the basis of a thermodynamic cycle (Figure 4), in which the components, in our notation, are also to be understood as the standard free energy difference per detergent molecule divided by the thermal energy. Some of these components may seem like contributions from liberation, but these refer to internal DOFs of the micelles and thus do not belong to the liberation free energy separated off in Equation (33). We will consider only nonionic detergents. The first step in the cycle (A → B) involves the separation of the head groups from the alkyl tails requiring the free energy $g_{\mathrm{s}\mathrm{e}\mathrm{p}}$ and yielding alkane chains in aqueous solution next to head group molecules. Since $g_{\mathrm{s}\mathrm{e}\mathrm{p}}$ is supposed to be recovered in step C → D and thus cancels out, we need not model it. Step B → C is the condensation of the singly dispersed alkane chains into an alkane droplet becoming the hydrophobic core of the micelle. This process contributes three free energy components. The first is $g_{\mathrm{t}\mathrm{r}}$, which corresponds to the transfer free energy of alkanes from a diluted solution in water into a pure alkane phase. This contribution is traditionally modeled on the basis of experimentally determined transfer free energies, which will be the topic of Section 2.3.

When the alkane chains form a droplet, a significant portion of their molecular surfaces become shielded from the aqueous environment. The use of $g_{\mathrm{t}\mathrm{r}}$ implies assuming the loss of all alkane–water contacts. However, in contrast to a pure alkane phase, an alkane droplet in water still has a certain surface area $A$ (per detergent molecule in the micelle = per alkane molecule in the droplet) that is in contact with water. Therefore, the free energy $g_{\mathrm{i}\mathrm{n}\mathrm{t}1}=\beta {\sigma }_{\mathrm{h}\mathrm{w}}A$ is considered, where ${\sigma }_{\mathrm{h}\mathrm{w}}$ is the interfacial tension between a pure hydrocarbon phase (h) and water (w) in the traditional modeling approach [22].

The packing of the alkane chains in the droplet is different from that in a pure alkane phase. Furthermore, one end of each alkane chain has to be close to the droplet´s surface to be prepared for the reattachment of the head group (which represents the different packing of alkyl chains in the micelle interior, compared to alkane chains in a liquid hydrocarbon). This free energy contribution, $g_{\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}}$, can be modeled based on lattice models of the micelle interior [47,48]. We will use

$$g_{\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}}={\displaystyle \frac{3{\pi }^2{R}_{\mathrm{s}}^2}{80\mathcal{N}{\mathcal{L}}^2}},$$

(41)

which is applicable to spherical or nearly spherical micelles [22]. In Equation (41), $\mathcal{N}$ is the number of segments representing the alkyl chain in the lattice model with length $\mathcal{L}=4.9 \AA$, following the lattice definition used by Dill and Flory [48], where $\mathcal{N}=\left(n+1\right)/3.6$ with 3.6 being the number of methylene groups per lattice site and $n$ is the number of carbon atoms in the alkyl chain. $R_{\mathrm{s}}$ is the radius of the spherical micelle interior. For oblate spheroidal micelles, $R_{\mathrm{s}}={\left(a{b}^2\right)}^{1/3}$, where $a$ and $b$ are the minor and major radii, respectively, of the spheroidal micelle [14].

In the final step C → D, the head groups are reattached. This makes three contributions to the free energy. The first is that $g_{\mathrm{s}\mathrm{e}\mathrm{p}}$ is recovered and drops out of the free energy balance. The second is that the attached head groups shield a certain portion, $A_0,$ of the alkane droplet´s surface from the aqueous phase, yielding $g_{\mathrm{i}\mathrm{n}\mathrm{t}2}=-\beta {\sigma }_{\mathrm{h}\mathrm{w}}{A}_0$. Note that $A_0$ is the shielded surface per detergent molecule and thus corresponds to the effective area covered by one head group. Both interfacial contributions can be lumped together as follows:

$$g_{\mathrm{i}\mathrm{n}\mathrm{t}}=g_{\mathrm{i}\mathrm{n}\mathrm{t}1}+g_{\mathrm{i}\mathrm{n}\mathrm{t}2}=\beta {\sigma }_{\mathrm{h}\mathrm{w}}\left(A-A_0\right).$$

(42)

Finally, the head groups attached to the micelle interact with each other. For non-ionic detergents, it is common to model this interaction by a hard-core repulsion akin to a two-dimensional vdW gas and similar in spirit to our treatment of the entropy of mixing. As discussed by Bothe et al. [15] based on earlier work [22,23], this free energy contribution can be written as follows:

$$g_{\mathrm{s}\mathrm{t}}=-\ln \left(1-{\displaystyle \frac{A_{\mathrm{p}}}{A}}\right).$$

(43)

Note that the effective cross-sectional area, $A_{\mathrm{p}},$ of the polar head group is usually assumed to be different from the effective area, $A_0$, covered by the head group, whereas the surface area, $A$, is the same in Equation (43) as in Equation (42).

Taking everything together, we have the following decomposition:

$$g_{\mathrm{m}\mathrm{i}\mathrm{c}}=g_{\mathrm{t}\mathrm{r}}+g_{\mathrm{i}\mathrm{n}\mathrm{t}}+g_{\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}}+g_{\mathrm{s}\mathrm{t}}.$$

(44)

2.2.3. Surface-Based Description of the Transfer Term

It is well known that the transfer free energies of alkanes into water show a nice linear correlation with the alkane´s molecular surface $S$ (see Section 3.1). Accordingly, it has been proposed [14] to recast the transfer term $g_{\mathrm{t}\mathrm{r}}$ in the form

$$g_{\mathrm{t}\mathrm{r}}=-\beta {\sigma }_{\mathrm{h}\mathrm{w}}S,$$

(45)

where $S$ is now to be understood as the molecular surface of the alkyl tail of the detergent, and ${\sigma }_{\mathrm{h}\mathrm{w}}$ is the same as in Equation (42). A problem occurs here however, as there are several ways of defining the molecular surface [49].

Lee and Richards [50] defined the solvent-accessible surface (SAS)—with the corresponding SAS area referred to as SASA—by rolling a sphere of radius $r_{\mathrm{p}}$ (called the probe) over the vdWsurface of a molecule (Figure 5). The latter can be regarded as the net surface of a molecule represented by a set of overlapping spheres $M$, each having the vdW radius of the respective atom. According to Sanner et al. [49], the SAS “can also be perceived as the topological boundary of a set of spheres $M\prime$, obtained by increasing the radius of each sphere by the value $r_{\mathrm{p}}$“ (inflated vdW surface). Richards [51] defined the molecular surface, consisting of patches lying either on the atoms (contact surface) or on the probe (reentrant surface). Sanner et al. [49] prefer to define it “as the topological boundary of the union of all possible probes having no intersection with $M$”. Since this surface literally defines whether a point in space lies inside or outside the molecule, it is called the solvent-excluded surface (SES)—with the corresponding area referred to as SESA—based on the work of Greer and Bush [52]. Since Connolly [53] provided equations to compute the analytical description of that surface, it is also referred to as the Connolly surface. With reference to Section 2.1.1, we may consider the volume enclosed by the SAS as the excluded volume of the molecule, as the center of the probe is not able to enter this volume. On the other hand, the volume enclosed by the SES can be taken as the actual volume $b_i$ of a molecule of type $i$. Please, do not confuse the solvent-excluded surface with the excluded volume; the latter is enclosed by the SAS and not the SES.

To model $g_{\mathrm{t}\mathrm{r}}$, we computed the SASA and SESA for a series of alkanes in the extended conformation with the number of carbon atoms $n$ ranging from 2 to 12 (

Table 1. SASA and SESA values (Ų) for alkanes with $n$ carbon atoms computed with the MSMS program [49] (for details, see Supplementary Material S4).

$\mathit{n}$	SASA	SESA
2	197.708	80.386
3	230.336	100.662
4	262.777	120.903
5	295.248	141.159
6	327.515	161.338
7	359.942	181.592
8	397.300	204.457
9	425.002	222.204
10	457.362	242.415
11	492.184	263.975
12	522.711	283.188

; Supplementary Material S4). Both areas depend linearly on $n$, with slopes of 32.6 ± 0.2 Ų for the SASA and 20.34 ± 0.08 Ų for the SESA (Figure 6). These slopes provide the group contributions $S_2^{\mathrm{S}\mathrm{A}\mathrm{S}\mathrm{A}}$ and $S_2^{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}$ of a methylene group to the SASA and SESA, respectively, of the alkyl tail (where the lower index indicates the number of hydrogen atoms in the methylene group). The group contributions of the final methyl group (with three hydrogen atoms) can be obtained from half the value of ethane ($n=2$) and are found to be $S_3^{\mathrm{S}\mathrm{A}\mathrm{S}\mathrm{A}}$ = 98.9 ± 0.2 Ų and $S_3^{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}$ = 40.19 ± 0.08 Ų, where we assumed the same errors as for the methylene group contributions. We can then write the transfer term as follows:

$$g_{\mathrm{t}\mathrm{r}}=-\beta {\sigma }_{\mathrm{M}}\left[S_3^{\mathrm{M}}+S_2^{\mathrm{M}}\left(n-1\right)\right].$$

(46)

Here, the index $\mathrm{M}$ stands for either SASA or SESA. Note that the appropriate value for the interfacial tension ${\sigma }_{\mathrm{M}}$ depends on the chosen surface (more about this in Section 3.1).

It has been proposed to merge $g_{\mathrm{t}\mathrm{r}}$ and $g_{\mathrm{i}\mathrm{n}\mathrm{t}}$ into one term, $g_{\mathrm{t}\mathrm{r}}^{*}$, that describes the free energy contribution due to a net change in the area of hydrophobic molecular surfaces exposed to water [14,15]. Because of the definition of A in Equation (42) as the molecular surface of the micelle interior (see Section 2.2.2), it is reasonable to use here the SESA rather than the SASA. Defining the effective area difference in hydrophobic molecular surfaces as

$$A_{\mathrm{e}\mathrm{f}\mathrm{f}}=A-A_0-\left[S_3^{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}+S_2^{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}\left(n-1\right)\right],$$

(47)

we obtain

$$g_{\mathrm{t}\mathrm{r}}^{*}=\beta {\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}^{*}{A}_{\mathrm{e}\mathrm{f}\mathrm{f}}$$

(48)

with the appropriate interfacial tension ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}^{*}$ describing the interaction with water based on the SESA. Then, we have $g_m^{*}=g_{\mathrm{t}\mathrm{r}}^{*}+g_{\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}}+g_{\mathrm{s}\mathrm{t}}+g_{\mathrm{l}\mathrm{i}\mathrm{b}}$ with

$$g_{\mathrm{l}\mathrm{i}\mathrm{b}}=-{\displaystyle \frac{5}{2m}}\ln m-\ln \left\{{\displaystyle \frac{{\mathsf{\Lambda }}_1^3}{{\omega }_1}}\right\},$$

(49)

where we assumed $\left(m-1\right)/m \approx 1$ (cf. Equation (33)). It remains an open question, whether ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}^{*}$ can be identified with the interfacial tension, ${\sigma }_{\mathrm{h}\mathrm{w}},$ of a macroscopic alkane–water interface, as traditionally used in molecular thermodynamic modeling [22].

Another problem is that in many applications, cosolutes are present that may modify the solubility of hydrophobic groups in the aqueous phase and hence affect the value of the effective surface tension to be used in Equation (48). If we assume that the cosolutes do not change $g_{\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}}$ nor $g_{\mathrm{s}\mathrm{t}}$ and have no influence on the aggregation number $m$, we can model the cosolute effect by introducing an effective surface tension ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}^0$, so that

$$g_{\mathrm{t}\mathrm{r}}^0=\beta {\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}^0{A}_{\mathrm{e}\mathrm{f}\mathrm{f}}$$

(50)

and $g_m^0=g_{\mathrm{t}\mathrm{r}}^0+g_{\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}}+g_{\mathrm{s}\mathrm{t}}+g_{\mathrm{l}\mathrm{i}\mathrm{b}}$. The relationship between ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}^0$ and ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}^{*}$ can be inferred from Equation (30) as

$${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}^0={\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}^{*}+{\displaystyle \frac{\beta }{m{A}_{\mathrm{e}\mathrm{f}\mathrm{f}}}}\sum _{\alpha }{Y}_{\alpha }{j}_{\alpha m}.$$

(51)

In the present paper, we do not specify the interaction constants $j_{\alpha m}$. Equation (51) merely serves to provide a molecular interpretation of the difference between ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}^0$ and ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}^{*}$.

2.3. Alkane Partitioning and Transfer Free Energy

2.3.1. Alkane Solubility in Water and the Transfer Free Energy as PCP Difference

At the heart of the traditional molecular thermodynamics modeling approach is the use of alkane solubility data to infer $g_{\mathrm{t}\mathrm{r}}$ in Equation (44) [22]. To understand alkane solubility in water, we consider a liquid n-alkane phase in contact with liquid water at room temperature (298.15 K) and neglect the penetration of water molecules into the alkane phase. At equilibrium, we have to equate the chemical potential of the alkane (index H for hydrocarbon) in the pure alkane phase (index h)

$${\mu }_{\mathrm{H}}^{\mathrm{h}}={\tilde{\mu }}_{\mathrm{H}}^{\mathrm{h}}+{\beta }^{-1}\ln \left\{{\displaystyle \frac{{\mathsf{\Lambda }}_{\mathrm{H}}^3}{{\omega }_{\mathrm{H}}}}\right\}$$

(52)

with the chemical potential of the same alkane in a purely aqueous environment (index aq)

$${\mu }_{\mathrm{H}}^{\mathrm{a}\mathrm{q}}={\tilde{\mu }}_{\mathrm{H}}^{\mathrm{a}\mathrm{q}}+{\beta }^{-1}\ln \left\{{\displaystyle \frac{N_{\mathrm{H}}{\mathsf{\Lambda }}_{\mathrm{H}}^3}{{V-N}_{\mathrm{W}}{b}_{\mathrm{W}}{-N}_{\mathrm{H}}{b}_{\mathrm{H}}}}\right\}$$

(53)

where $N_{\mathrm{H}}$ and $N_{\mathrm{W}}$ are the numbers of alkane and water molecules, respectively, in the aqueous phase, and $V$ is the volume of this phase. The PCPs are given by

$${\tilde{\mu }}_{\mathrm{H}}^{\mathrm{h}}=-{\beta }^{-1}\ln q_{\mathrm{H}}^{\mathrm{h}}-{\beta }^{-1}\ln {〈e^{-\beta B_{\mathrm{H}}^{\mathrm{h}}}〉}_{\mathrm{h}}$$

(54)

$${\tilde{\mu }}_H^{\mathrm{a}\mathrm{q}}=-{\beta }^{-1}\ln q_H^{\mathrm{a}\mathrm{q}}-{\beta }^{-1}\mathit{ln}{〈e^{-\beta B_H^{\mathrm{a}\mathrm{q}}}〉}_{\mathrm{a}\mathrm{q}}$$

(55)

Note that both the internal partition functions $q_{\mathrm{H}}^{\mathrm{h}}$ and $q_H^{\mathrm{a}\mathrm{q}}$ of the alkane and the interaction energies $B_{\mathrm{H}}^{\mathrm{h}}$ and $B_H^{\mathrm{a}\mathrm{q}}$ of an alkane molecule added at a fixed position with the surrounding medium differ, in general, in the two phases h and aq. The averages ${〈\dots 〉}_{\mathrm{h}}$ and ${〈\dots 〉}_{\mathrm{a}\mathrm{q}}$ are over all configurations of the particles in the respective phase at constant $T$ and $P$, except for the newly added alkane particle (cf. Appendix D). From ${\mu }_{\mathrm{H}}^{\mathrm{h}}={\mu }_{\mathrm{H}}^{\mathrm{a}\mathrm{q}}$, it follows that

$${\tilde{\mu }}_{\mathrm{H}}^{\mathrm{a}\mathrm{q}}-{\tilde{\mu }}_{\mathrm{H}}^{\mathrm{h}}={\beta }^{-1}\ln \left\{{\displaystyle \frac{{V-N}_{\mathrm{W}}{b}_{\mathrm{W}}{-N}_{\mathrm{H}}{b}_{\mathrm{H}}}{N_{\mathrm{H}}{\omega }_{\mathrm{H}}}}\right\} .$$

(56)

With the assumption that the aqueous phase is additive, and the free volume of the alkane molecule is the same in both phases and proportional to its molecular volume, we arrive at the following:

$$\beta \left({\tilde{\mu }}_{\mathrm{H}}^{\mathrm{a}\mathrm{q}}-{\tilde{\mu }}_{\mathrm{H}}^{\mathrm{h}}\right)=-\ln Y_{\mathrm{H}},$$

(57)

where $Y_{\mathrm{H}}$ is the volume fraction of alkane in the aqueous phase at saturation.

A way to arrive at an ideal mixing model is to assume that the ratio $\varphi =v_{\mathrm{H}}/v_{\mathrm{W}}$ of molar volumes of alkane and water equals unity. As shown in Section 2.3.2, the volume fraction $Y_{\mathrm{H}}$ then goes over into the mole fraction $X_{\mathrm{H}}$, and we obtain

$$\beta \left({\tilde{\mu }}_{\mathrm{H}}^{\mathrm{a}\mathrm{q}}-{\tilde{\mu }}_{\mathrm{H}}^{\mathrm{h}}\right)=-\ln X_{\mathrm{H}}.$$

(58)

To understand how the present treatment of alkane solubility is related to the work of Sharp et al. [34], it is useful to look at the problem from a slightly different angle. This is shown in Supplementary Material S5.

2.3.2. Linking Alkane Solubility Mass Ratios to Volume and Mole Fractions

The available solubility data of liquid alkanes are given as the mass ratio $\xi =m_{\mathrm{H}}/m_{\mathrm{W}}$ of alkane to water in the saturated aqueous phase [54]. To exploit Equations (57) and (58), we have to express $Y_{\mathrm{H}}$ and $X_{\mathrm{H}}$ as functions of $\xi$. To this end, we introduce the ratio of molar volumes $\varphi =v_{\mathrm{H}}/v_{\mathrm{W}}$ and the ratio of molar masses $\zeta =M_{\mathrm{H}}/M_{\mathrm{W}}$. While $M_{\mathrm{W}}$ = 18.015 g/mole, $M_{\mathrm{H}}$ depends on the number of carbon atoms $n$ in the alkane, according to

$$M_{\mathrm{H}}=2M_3+\left(n-2\right)M_2$$

(59)

with $M_2$ = 14.027 g/mole and $2M_3$ = 30.07 g/mole. The molar volumes can be calculated as the ratio of the molar mass to the mass density $\rho$. For water, ${\rho }_{\mathrm{W}}$ = 0.9973 (see [16] and references therein), yielding $v_{\mathrm{W}}$ = 18.064 cm³/mole. The densities of liquid alkanes at 298.15 K are given in

Table 2. Data of linear alkanes (H) with $n$ carbon atoms relevant to link solubility data to volume fractions $Y_{\mathrm{H}}$ and mole fractions $X_{\mathrm{H}}$ in water (W).

$\mathit{n}$	${\mathit{\rho }}_{\mathbf{H}}$ in g/cm3 a	${\mathit{M}}_{\mathbf{H}}$ in g/mole b	${\mathit{v}}_{\mathbf{H}}$ in cm3/mole c	$\mathit{\varphi }=\frac{{\mathit{v}}_{\mathbf{H}}}{{\mathit{v}}_{\mathbf{W}}}$d	$\mathit{\zeta }=\frac{{\mathit{M}}_{\mathbf{H}}}{{\mathit{M}}_{\mathbf{W}}}$e	$\mathit{\xi }·{10}^6$ f	$\mathsf{{\rm Y}}·{10}^6$	$\mathsf{\chi }·{10}^6$	${\mathit{Y}}_{\mathbf{H}}·{10}^6$	${\mathit{X}}_{\mathbf{H}}·{10}^6$
5	0.62124	72.151	116.140	6.429	4.005	38.5 ± 2.0	61.802	9.613	61.798	9.613
6	0.65507	86.178	131.555	7.283	7.283	9.5 ± 1.3	14.462	1.304	14.462	1.304
7	0.67965	100.205	147.436	8.162	8.162	2.93 ± 0.20	4.300	0.359	4.300	0.359
8	0.69842	144.232	163.558	9.054	9.054	0.66 ± 0.942	0.942	0.073	0.942	0.073

^a Mass density of pure liquid alkane at 298.15 K; data from [55]. ^b Molar mass of alkane. ^c Molar volume of liquid alkane at 298.15 K. ^d Molar volume ratio. ^e Molar mass ratio. ^f Experimentally determined mass ratio in saturated aqueous solution at 298.15 K; data from [54].

and were taken from Aucejo et al. [55]. As expected, the resulting molar volumes at 298.15 K are slightly larger than those at 293.15 K, reported by McAuliffe [54]. Since we have to restrict the analysis to liquid alkanes, for which solubility data are available [54], we consider here only alkanes from n-pentane to n-octane.

With the introduced quantities, we can express the volume fraction as

$$Y_{\mathrm{H}}={\displaystyle \frac{\mathsf{{\rm Y}}}{1+\mathsf{{\rm Y}}}} , \mathsf{{\rm Y}}={\displaystyle \frac{\xi \varphi }{\zeta }}$$

(60)

and the mole fraction as

$$X_{\mathrm{H}}={\displaystyle \frac{\mathsf{\chi }}{1+\mathsf{\chi }}} , \mathsf{\chi }={\displaystyle \frac{\xi }{\zeta }}$$

(61)

A comparison of Equations (60) and (61) shows that $X_{\mathrm{H}}$ is obtained from $Y_{\mathrm{H}}$ by setting $\varphi =1$. Due to the low solubility of the alkanes in water, it holds that practically $Y_{\mathrm{H}}=\mathsf{{\rm Y}}$ and $X_{\mathrm{H}}=\chi$, as can be seen from Table 2.

3. Results

3.1. Surface-Based Analysis of Alkane Solubility

To obtain $g_{\mathrm{t}\mathrm{r}}$ in terms of molecular surfaces, we have to correlate $\beta \left({\tilde{\mu }}_{\mathrm{H}}^{\mathrm{a}\mathrm{q}}-{\tilde{\mu }}_{\mathrm{H}}^{\mathrm{h}}\right)$, either in the form $-\ln X_{\mathrm{H}}$ (ideal mixing model) or in the form $-\ln Y_{\mathrm{H}}$ (FH theory), with the molecular surface areas, which can be the SASA or the SESA (Figure 7). The value of ${\sigma }_{\mathrm{M}}$ obtained from linear regression for the ideal mixing model, based on the SASA (

Table 3. Values for the surface tension ${\sigma }_{\mathrm{M}}$ (in mN/m) obtained from correlating $\beta \left({\tilde{\mu }}_{\mathrm{H}}^{\mathrm{a}\mathrm{q}}-{\tilde{\mu }}_{\mathrm{H}}^{\mathrm{h}}\right)$ representing the solubility of alkanes from n-pentane to n-octane in water ($T$ = 298.15 K) with the SASA or the SESA of the alkane molecules. If $s$ is the slope of the corresponding line in Figure 7, then ${\sigma }_{\mathrm{M}}=s/\beta$. The errors originate from linear regression.

$\mathit{\beta }\left({\tilde{\mathit{\mu }}}_{\mathbf{H}}^{\mathbf{a}\mathbf{q}}-{\tilde{\mathit{\mu }}}_{\mathbf{H}}^{\mathbf{h}}\right)$	SASA	SESA
$-\ln X_{\mathrm{H}}$	19.8 ± 1.6	31.9 ± 2.4
$-\ln Y_{\mathrm{H}}$	17.1 ± 1.2	27.4 ± 1.7

), is in good agreement with literature data [34,39,56]. Note that our value of 19.8 mN/m corresponds to 28 cal/(mole Ų), which has to be compared to 29 cal/(mole Ų), reported by De Young and Dill [56] and to 31 cal/(mole Ų), found by Sharp et al. [34]. Our value is systematically smaller, since we used slightly larger values of the vdW radii to compute the SASA. Another possible source of discrepancy is the use of a smaller data set for the alkane solubilities employed in the present work. We used only values for liquid n-alkanes, reported directly by McAuliffe [54], to be consistent with our theory and to avoid any possible bias from a processing of the data by others. Altogether, the agreement is very satisfactory.

Our value of ${\sigma }_{\mathrm{M}}$, obtained for the FH theory based on the SASA (Table 3), corresponds to 25 cal/(mole Ų) and is consistently slightly smaller than the value of 27 cal/(mole Ų), determined by De Young and Dill [56]. Thus, we confirm that the use of volume fractions instead of mole fractions causes only a small decrease in ${\sigma }_{\mathrm{S}\mathrm{A}\mathrm{S}\mathrm{A}}$. The values of ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}$ are consistently larger by a factor of 1.6 for both models, which is due to the SESA being correspondingly smaller than the SASA (cf. Table 1, Figure 5).

In the above analysis, we interpret the PCP difference ${\tilde{\mu }}_{\mathrm{H}}^{\mathrm{a}\mathrm{q}}-{\tilde{\mu }}_{\mathrm{H}}^{\mathrm{h}}$ as the transfer free energy. More precisely, the PCP difference corresponds to the free energy change due to taking out an alkane molecule from a fixed position in the alkane phase and putting it at a fixed position into the aqueous phase. Thus, it is devoid of any contributions from liberation. The PCP difference encompasses all changes that the alkane molecule experiences when going from one phase to the other, which includes changes in the internal partition function (represented by $q_{\mathrm{H}}^{\mathrm{h}}$ and $q_{\mathrm{H}}^{\mathrm{a}\mathrm{q}}$) and changes in the alkane–environment interaction (represented by $B_{\mathrm{H}}^{\mathrm{h}}$ and $B_{\mathrm{H}}^{\mathrm{a}\mathrm{q}}$; cp. Equations (54) and (55)). This free energy difference is also referred to as contact free energy. In the words of Chan and Dill [39], it is “the free energy of swapping environments of the solute”. It is precisely this contact free energy that we have to correlate with molecular surfaces if we want the corresponding surface tension, ${\sigma }_{\mathrm{M}},$ to be a meaningful descriptor of the alkane–water contact and a measure of hydrophobic solvation. Note that the contact free energy is temperature-dependent and has an entropic contribution.

Sharp et al. [34] advocated for the use of a different equation (see Equation (S43)), which we re-derived in Supplementary Material S5 and which De Young and Dill [56] refer to as “FH corrected”. Note that some authors call this “corrected” version the “Flory–Huggins theory”. This is not the case in the present paper. As explained in Section 2.1.1, we refer to the use of a volume-fraction-based entropy of mixing according to Equation (10) as the FH theory. When we apply the “corrected” version (see Figure S1), we obtain ${\sigma }_{\mathrm{S}\mathrm{A}\mathrm{S}\mathrm{A}}$ = 28.1 mN/m (Table S1), which corresponds to 40 cal/(mole Ų) and has to be compared to 43 cal/(mole Ų), reported by De Young and Dill [56], and to 47 cal/(mole Ų), found by Sharp et al. [34]. Again, the agreement is satisfactory given the different vdW radii underlying the SASA and the smaller set of solubility data used in the present work. Thus, we confirm that employing the “FH corrected volume fraction” yields larger surface tension values.

Tuñón et al. [57] combined the “FH corrected” theory with the SESA and obtained ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}$ = 67 cal/(mole Ų). This value is fairly close to 50 mN/m (72 cal/(mole Ų)) for the macroscopic interfacial tension, ${\sigma }_{\mathrm{h}\mathrm{w}}$, of an alkane–water interface [14,22]. We basically confirm this result with our value of 44.8 mN/m (64.6 cal/(mole Ų); Table S1). These results seem to suggest using the “FH corrected” theory in combination with the SESA and the macroscopic interfacial tension. However, as we shall see, this procedure does not work in the modeling of micelle formation.

3.2. Application to Alkyl Maltosides

In the following, we will analyze the same experimental CMC data for n-alkyl-ß-D-maltosides that have been investigated recently by Bothe et al. [15], based on an ideal mixing model. These data were obtained with a fluorescence technique [2,14,58] that has been shown [15] to be compatible with the definition of the CMC by Equation (35). The two data sets from [14,15] and [2,58] will be referred to as “Bothe et al.” and “Alpes et al.“, respectively.

To analyze the experimental data, we recast Equation (40) in the form

$$g_m^0=\ln \left\{{\displaystyle \frac{\mathrm{C}\mathrm{M}\mathrm{C}}{c_{\mathrm{t}\mathrm{o}\mathrm{t}}}}\right\}+\ln \left\{v_1{c}_{\mathrm{t}\mathrm{o}\mathrm{t}}\right\}+{\tau }_m+1,$$

(62)

where $\mathrm{C}\mathrm{M}\mathrm{C}/c_{\mathrm{t}\mathrm{o}\mathrm{t}}=X_{\mathrm{C}\mathrm{M}\mathrm{C}}$. Then, we define the quantities

$$\mathsf{\gamma }=-\ln \left\{{\displaystyle \frac{{\mathsf{\Lambda }}_1^3}{{\omega }_1}}\right\},$$

(63)

$$\mathsf{\Gamma }=\ln \left\{X_{\mathrm{C}\mathrm{M}\mathrm{C}}\right\}+\ln \left\{v_1{c}_{\mathrm{t}\mathrm{o}\mathrm{t}}\right\}+{\tau }_m+1+{\displaystyle \frac{5}{2m}}\ln m-\left(g_{\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}}+g_{\mathrm{s}\mathrm{t}}\right)-\gamma ,$$

(64)

and

$$\mathsf{\Phi }=\beta A_{\mathrm{e}\mathrm{f}\mathrm{f}},$$

(65)

so that $\mathsf{\Gamma }=\mathsf{\sigma }\mathsf{\Phi }$, where $\sigma$ is the sought surface tension. The latter can be obtained from a linear regression of $\mathsf{\Gamma }$ versus $\mathsf{\Phi }$ for a set of detergents with the same head group, but different numbers, $n$, of carbon atoms in the alkyl tail, provided that we capture the $n$-dependence of the various terms in $\mathsf{\Gamma }$ correctly. As in [14], we do not aim at a prediction of aggregation numbers, but use values from the literature. These values, their sources, as well as the limitations of this approach have been discussed by Bothe et al. [15]. Likewise, we take values for $g_{\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}}$ and $g_{\mathrm{s}\mathrm{t}}$ from this earlier work, whereas values of $A_{\mathrm{e}\mathrm{f}\mathrm{f}}$ are computed from the group contributions determined in Section 2.2.3, together with previously reported values for $A$ and $A_0$. All the necessary data and their sources are compiled in Tables S2 and S3. The determination of the molar volumes, $v_1,$ of detergent monomers is described in Supplementary Material S6.

The problem is that we do not know $\gamma$. The results of Bothe et al. [15] show that when using the model parameters $A$, $g_{\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}}$, and $g_{\mathrm{s}\mathrm{t}}$ from the ideal-mixing model (see Table S2), $\mathsf{\Gamma }\prime =\ln \left\{X_{\mathrm{C}\mathrm{M}\mathrm{C}}\right\}+{\tau \prime }_m-\left(g_{\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}}+g_{\mathrm{s}\mathrm{t}}\right)$ correlates linearly with $\mathsf{\Phi }$ with zero intercept. Thus, the data suggest that $\Sigma -\gamma =0$ with

$$\Sigma =\ln \left\{v_1{c}_{\mathrm{t}\mathrm{o}\mathrm{t}}\right\}+1+{\displaystyle \frac{3}{2m}}\ln m.$$

(66)

Indeed, this turns out to be approximately correct.

In Figure 8, we show the correlation of

$$\mathsf{\Gamma }=\ln \left\{X_{\mathrm{C}\mathrm{M}\mathrm{C}}\right\}+\ln \left\{v_1{c}_{\mathrm{t}\mathrm{o}\mathrm{t}}\right\}+{\tau }_m+1+{\displaystyle \frac{5}{2m}}\ln m-\left(g_{\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}}+g_{\mathrm{s}\mathrm{t}}\right)$$

(67)

(i.e., leaving $\gamma$ aside) with $\mathsf{\Phi }$. The result for both experimental data sets is a linear correlation with a slope close to the expected surface tension (cp. The slope values in

Table 4. Values for the surface tension ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}$ (in mN/m, slope) obtained from correlating experimental CMC values with the SESA difference due to micelle formation. The errors originate from linear regression.

Source of Experimental CMC Data	Slope/mN m−1	Intercept	Slope with Intercept Fixed to 3.7/mN m−1
[14,15]	29.8 ± 0.8	4.0 ± 0.5	29.1 ± 0.1
[2,58]	27.5 ± 0.9	3.4 ± 0.5	28.1 ± 0.2

with 27.4 mN/m from Table 3) and an intercept in the order of 4 (Figure 8a). The difference in intercepts is not significant based on the errors from linear regression, so that equally satisfactory fits can be obtained by fixing the intercept to the same value for both data sets (Figure 8b). The basic result is that we confirm that it is the “microscopic” surface tension that has to be used in the modeling of micelle formation.

For the detergents studied, the quantity, $\Sigma ,$ defined in Equation (66) varies slightly from 4.1 to 4.2 when going from $n=8$ to $n=12$. If $\Sigma \approx \gamma$, the weak $n$-dependence of $\Sigma$ can explain why the correlation can be described well with a constant intercept. Taking $\gamma$ between 3.0 and 4.2, we can estimate from Equation (63) that the ratio ${\omega }_1/{\mathsf{\Lambda }}_1^3$ of the free volume of the detergent monomer to the third power of its thermal de Broglie wavelength is between 20 and 67.

The experimental CMC data refer to micelle formation in aqueous solutions containing salt and buffer (specifically, [14,15]: 100 mM piperazine-1,4-bis-(2-ethanesulfonic acid), pH = 7.0, 5 mM CaCl₂, $c_{\mathrm{t}\mathrm{o}\mathrm{t}}$ = 54.65 M; [2,58]: 150 mM KCl, $c_{\mathrm{t}\mathrm{o}\mathrm{t}}$ = 55.32 M). It is possible that the values for ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}$ found from fitting the CMC data are larger than those found from the alkane solubility data, because the surface tension is increased by the cosolutes. However, the differences are within the error margins, so we cannot draw any viable conclusions about the cosolute effects. Irrespectively, to predict the CMC of the detergents in pure water ($c_{\mathrm{t}\mathrm{o}\mathrm{t}}$ = 55.56 M), we used the value ${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}$ = 27.4 mN/m, suggested by the alkane solubility data, and an intercept correction of 3.2, assuming that there is a correlated effect of the cosolutes (or their absence) on slope and intercept. In addition, there might be an effect due to the slight temperature differences, which were taken into account in the computation of $\mathsf{\Phi }$. It turns out that experimental CMC data from the literature can well be reproduced in this way (Table S4).

4. Discussion

In the present work, we sought to solve a problem in the modeling of the self-assembly of amphiphiles into micelles that is related to the change in the contact free energy of hydrophobic molecular surfaces in contact with the aqueous phase. Following traditional approaches, we modeled the contact free energy in terms of molecular surfaces and an associated surface tension. In agreement with earlier findings [32,33,39,56], we found the appropriate surface tension value (${\sigma }_{\mathrm{S}\mathrm{E}\mathrm{S}\mathrm{A}}$ = 27–30 mN/m, Table 3 and Table 4) to be significantly smaller than the value typically associated with macroscopic alkane–water interfaces (50–53 mN/m [14,22,31]), which cannot be explained with the effects of cosolutes [31]. Apart from the problem of using different surface definitions (SASA vs. SESA), there has been the proposal in the literature that the discrepancy in surface tension values is due to the neglect of molecular volume differences between solvent and solute [34]. Since a proposed solution to this issue [34] is based on a variant of the Flory–Huggins theory, we approached the subject by deriving a Flory–Huggins type of theory for the micellization problem. It should be noted that the use of such a theory in the context of micellar solutions is not new. In a study of entropy models related to micelle formation and phase separation in surfactant solutions—which to our knowledge is the only one of this sort available in the literature—Nagarajan [24] also tested a Flory–Huggins model. In fact, his Equation (25) for the micellar size distribution is essentially the same as our Equation (29). However, Nagarajan did not say anything about the derivation of this equation. Therefore, we had to start from scratch. The advantage of this line of action was that we learned about the implications of the Flory–Huggins theory. In particular, we did not derive it from a lattice model, but based on approximations suggested earlier by Hildebrand [38].

The starting point of Hildebrand´s approach is Equation (7), which formulates the entropy of mixing in terms of changes in the free volume for each molecular species, including solute and solvent. In the derivation of this equation from classical statistical mechanics, we had to assume that the integration over the spatial coordinates in the partition function yields the free volume (Appendix A). This is actually wrong given our definition of the free volume. To obtain Equation (A3) from Equation (A2), we had to integrate over the empty space not occupied by molecules. The integration variables ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ are the COM positions of the molecules. Consequently, the integration over ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ in Equation (A2) actually yields the accessible volume. The free volume is obtained by reinterpreting the integration variables as describing points in space rather than COM coordinates, or by neglecting the volumes $b_i$ during the integration over ${\mathbf{q}}_i$. The latter variant introduces a contradiction, however. We encounter, here, an old problem in condensed matter physics. As discussed by Barrat and Hansen [59] for the case of spherical molecules, it is “important to realize that, while the excluded volume associated with each individual sphere is $v_{\mathrm{e}\mathrm{x}}$, the total volume, $V_{\mathrm{e}\mathrm{x}}$, from which the center of a test sphere is excluded is less then $N\times v_{\mathrm{e}\mathrm{x}}$, since the exclusion spheres of neighboring particles can overlap” (cf. Figure 1). This problem makes the determination of the accessible volume practically impossible, if not determined numerically. In contrast, the free volume in Hildebrand´s theory—as it is perceived by Flory—“is considered to represent the difference between the actual volume of the liquid (or the amorphous polymer) and the minimum volume which it would occupy if its molecules were packed firmly in contact with each other. Incompressible molecules with rigid dimensions are implied in this definition of a free volume. The unrealistic nature of this implication undermines precise determination, or even an exact definition, of the free volume. “The concept has proved useful nevertheless” (see footnote on p. 506 of [37]; italics by the present author). In view of these difficulties, we rigorously defined the free volume by the space not occupied by molecules. Then, in order to base Equation (7) on statistical mechanics, we have to reinterpret the integration variables in Equation (A2), in order to recover Hildebrand´s formula.

Barrat and Hansen [59] pointed out that each particle in a condensed liquid is trapped, on average, in a cage of neighboring molecules. It is then useful to ask for the volume that the COM of a molecule can sample while it rattles in its solvent cage. In a hard-core description, where we neglect the subtleties of intermolecular interactions, the SES of each molecule and its neighbors define the ultimate limits of the molecular motion in the cage. Note that the green circumference in Figure 2a can be considered as the SES of the inner cavity wall. The volume of the cage—on average—is in fact ${\mathrm{v}}_i$, which is related to the molar volume by $v_i=N_{\mathrm{A}}{\mathrm{v}}_i$ and can in principle, at least for pure substances, be determined experimentally (cf. the molar volumes of alkanes in Table 2). This is an important aspect in practical applications, since it allows linking the theoretical concepts to measurable quantities. If we imagine the molecule moving in its cage, it is reasonable to assume that the volume sampled by the COM of the molecule (i.e., the accessible volume ${\tilde{\omega }}_i$; area within the blue circumference in Figure 2a) is at least approximately proportional to ${\mathrm{v}}_i$. However, in Hildebrand´s approach, it is assumed that the average difference between the volume of the cage and the volume of the molecule itself (i.e., the free volume ${\omega }_i={\mathrm{v}}_i-b_i$; cf. yellow-colored area in Figure 2b) is proportional to $b_i$. As a consequence, ${\mathrm{v}}_i$ is also proportional to $b_i$. The next step necessary to arrive at the FH theory is to assume that ${\omega }_i$ is proportional to $b_i$, with the same proportionality constant for all molecule types. This assumption entails a corresponding species-independent proportionality between ${\mathrm{v}}_i$ and $b_i$. Only with this tactic are we able to convert the ratio of free volumes between the pure substances and the mixture into the volume fraction according to Equation (9). What are the implications of this approximation? Imagine that we decrease all spheres in Figure 2b. The size ratio between cavity and central molecule is probably only weakly affected by this procedure, suggesting that the same proportionality constant, $\varsigma +1$ (cf. Section 2.1.1), can be applied to different pure substances to a reasonable approximation. A problem occurs however, if we place a solute molecule into a solvent. This situation is depicted in Figure 2c,d. Here, the solvent molecules forming the cage are smaller than the central solute molecule. As suggested by the figure, this change has an effect on the ratio between ${\mathrm{v}}_i$ and $b_i$. Ignoring this effect appears to be a gross oversimplification. One could now argue that when using the reference states introduced in Section 2.1.1, we are actually not comparing the mixture with the pure solute, but with a hypothetical phase of pure solute hydrate. For the solvent, we compare the mixture with the pure solvent, nonetheless. The ratio between ${\mathrm{v}}_i$ and $b_i$ can then be assumed to be similar in the mixture and the reference state for both solvent and solute. The problem remains however, since the solute molecules are surrounded by solvent molecules of a different size, whereas solvent molecules are mainly surrounded by other solvent molecules having the same size. So, the ratio between ${\mathrm{v}}_i$ and $b_i$ is different for solvent and solute and choosing the same proportionality constant, $\varsigma +1$, for both means that we neglect an important consequence of the volume difference between molecular species.

Can we do better? A comparison of Figure 2a with Figure 2c suggests that the ratio between the occupied volume, ${\mathrm{v}}_i$ (green), and the accessible volume, ${\tilde{\omega }}_i$ (blue), is much less affected by a size change in the neighboring molecules. Although the pictures are qualitative and do not provide proof, they deliver an idea; what if we replace ${\omega }_i$ by ${\tilde{\omega }}_i$ in Equation (8)? With such a replacement, we could solve a number of problems as follows: (i) We would describe the entropy of mixing in terms of a change in the volume accessible to the COMs of the molecules and thus relate it to the freedom in translational motion. (ii) We could derive the equation consistently from statistical mechanics, since the accessible volume is obtained by integration over the COM coordinates. (iii) The assumption that ${\mathrm{v}}_i$ is proportional to ${\tilde{\omega }}_i$ with the same proportionality constant for all molecular species is probably a better approximation than the same assumption for the proportionality between ${\mathrm{v}}_i$ and $b_i$. The full consequences of this idea remain to be explored.

Further implications of the FH theory become apparent, when we consider the concept of the PCP [40,41]. We see that the micellization free energy modeled in the TMT approach is actually a PCP difference and is not sufficient to model micelle formation. In the alkane partitioning problem, the kinetic energy contribution cancels out, according to Section 2.3.1, so that the PCP difference is related to the volume fraction in a simple way within the FH theory (see Equation (57)). In contrast, there is a contribution from kinetic energy and free volume changes in the self-assembly problem. Curiously, this contribution is found to be largely canceled by the terms necessary to convert mole fractions into volume fractions. This seems to indicate that the parameters of the TMT model taken over from ideal-mixing approaches work well only because of error compensation. On the other hand, assuming that these parameters are actually good enough and the cancelation is not merely due to error compensation, we estimated the ratio ${\omega }_1/{\mathsf{\Lambda }}_1^3$ of the free volume of the detergent monomer to the third power of its thermal de Broglie wavelength. The result is that the characteristic length scale for the rattling of the COM position of a detergent monomer in its solvent cage is between three and four times the thermal de Broglie wavelength. Given that the latter is on the order of 0.046 Å for the detergents studied here, this would imply a COM motion amplitude on the order of 0.14 to 0.18 Å. At present, we cannot say whether this is a realistic value. Future research should re-evaluate the parameters from TMT modeling, before any conclusions about the free volume can be drawn.

The correlation plots in Figure 8 indicate that those parts of the micellization free energy that depend on the contact of hydrophobic molecular surfaces with the aqueous environment can well be modeled with a surface tension that is similar to that obtained from the analysis of alkane solubility in water. The crucial point here, is that the transfer free energy involves the full PCP, including the part caused by the free volume $\eta$. This makes sense because, besides the hydrocarbon–water interaction, the change in the free volume in the aqueous phase matters in both the transfer of the alkane from water into the pure alkane phase (or vice versa) and the transfer of the detergent´s alkyl tail from water into the micelle interior. Accordingly, the same surface tension has to be used in both cases if the transfer free energy is related to the SESA of the respective hydrocarbon.

In contrast, Sharp et al. [34] for the SASA and Tuñón et al. [57] for the SESA, found higher surface tension values from a correlation of the quantity ${\overline{\mu }}_i$ in Equation (A29), which we confirmed (Figure S2, Table S1). Our analysis reveals that the “microscopic” and “macroscopic” surface tensions differ because the latter does not take into account the free volume contribution, $\eta$. This could potentially make sense because the free volume change due to the incorporation of an alkane molecule into the aqueous phase is not of relevance for the interfacial energy of a macroscopic alkane surface in contact with water. In other words, the molecular volume ratio of alkane and water is relevant for the solvation of the alkane in water, but not for the alkane molecules at the macroscopic interface.

The different surface tensions associated with the molecular surface of an alkane molecule in water, on the one hand, and a macroscopic alkane–water interface, on the other hand, can be rationalized by the behavior of the water molecules at these interfaces. The cross-section of an alkane chain has a diameter that is small enough so that water molecules can wrap around the chain and form a hydrogen bond network. Albeit different from bulk water, the network is strong enough to limit the free energy penalty of solvating the alkane. This situation corresponds to the case of a small hydrophobic solute [60,61,62]. In contrast, the macroscopic alkane–water interface corresponds to the limiting case of a large hydrophobic solute. Here, the hydrogen bond network is perturbed so severely that a larger free energy penalty results. Consequently, the surface tension, which is a free energy per unit area, is larger for the macroscopic than for the molecular surface. It remains an open question how this difference in the behavior of water molecules is related to the difference between ${\tilde{\mu }}_i$ and ${\overline{\mu }}_i$, discussed above.

Related to the problem of breaking hydrogen bonds, there is a possible systematic error in our TMT model. In the expression for $g_{\mathrm{t}\mathrm{r}}^{*}$ in Equation (48), the “microscopic” surface tension is combined with the effective area, $A_{\mathrm{e}\mathrm{f}\mathrm{f}}$, which according to Equation (47) consists of different types of surfaces. One is the molecular surface $S$ of the alkyl tail and the other the surface $A$ of the micellar core. It is well possible that the water molecules near these surfaces behave differently, so that different surface tensions have to be used for the two types of surfaces. In addition, the detergent head groups perturb the water molecules near the micellar core in a way that is difficult to quantify. The mere reduction in the area $A$ by the parameter $A_0$ is possibly insufficient to account for the effect of the head groups on the interfacial water. So, was the merging of $g_{\mathrm{t}\mathrm{r}}$ and $g_{\mathrm{i}\mathrm{n}\mathrm{t}}$ premature? This is one of the many questions that need to be answered in the future.

This seems to be bad news for our attempts to model cosolute effects in a simple way. Recall that our theoretical development was motivated by the practical requirement to infer model parameters of the theory from experimental data, so that the CMC can be understood on the basis of, e.g., surface tensions of buffer solutions. If the “microscopic” surface tension is relevant for the micellization free energy, what is then the relevance of measured macroscopic alkane–aqueous electrolyte interfacial tensions? And how do cosolutes affect any of the two surface tensions? What is clearly needed is a better understanding of the relationship between molecular and macroscopic surface tensions. A key aspect in this understanding is the behavior of water molecules near these surfaces and their interaction with the cosolutes.

Provided we solve the problems discussed above and obtain useful values for the surface tensions, what can we do with the model except for computing CMCs? One application lies in the field of membrane protein research. As discussed in earlier work [13,14,16], detergents are required to extract membrane proteins from the native membrane and to keep it in an aqueous solution afterwards by forming a detergent belt around the hydrophobic protein surfaces [8,9]. The formed PDCs must not aggregate, in order to allow for protein crystallization needed in X-ray crystallography [6,17,19], or for single-particle analysis in the framework of cryo-electron microscopy [7]. Both techniques are of utmost importance in biophysics, to gain information about the molecular structure of the proteins with implications for many areas such as bioengineering or drug design. The detergent belt in the PDC is formed in a way similar to a micelle [13,16]. There is a contribution to the free energy of belt formation due to the transfer of the detergent´s alkyl tail from the aqueous phase into the belt, as well as due to the interaction of the belt´s hydrophobic core with water molecules. Both contributions may be modeled by using the appropriate surface tension values in order to understand the influence of detergent concentration and cosolutes on the stability or aggregation behavior of the PDCs. Of particular interest are precipitating agents such as poly(ethylene glycol) (PEG) that are added to drive the PDC solution into the supersaturated regime and induce crystallization [63]. We are just at the beginning of understanding how PEG affects the detergent and its interplay with the protein [16]. Since PEG is a polymer, it is useful to consider micelle and belt formation in the context of the FH theory in order to enable an analysis of molecular volume effects.

The implications of the present results extend further, since the description in terms of surface tension can be applied to any process, in which the number of hydrophobic surfaces in contact with water changes. Accordingly, the SASA (rather than the SESA) is often used as a measure for the solubility of nonpolar molecules in water. In fact, alkanes are only a special case. A very important case in biophysics is the problem of protein folding, which is believed to be determined by the tendency of hydrophobic molecular parts to avoid contact with water [64]. The dominant role of the hydrophobic effect in protein folding is controversial, however [65]. Another example is the opening of a channel for the translocation of proteins through membranes, which is sealed by interactions between hydrophobic groups in its center [66,67]. In all these cases, the question arises of how cosolutes influence the equilibrium between the two states in the process, and whether it can be modeled by employing surface tension concepts.

5. Conclusions

In an attempt to achieve the “extreme simplicity” in the modeling of micelle formation claimed by Israelachvili et al. [10], we devised a theory, in which all contacts between hydrophobic molecule parts and water are modeled by the SESA and an appropriate surface tension [14,15]. This surface tension turns out to be smaller by a factor of about 3/5 than the interfacial tension of a macroscopic alkane–water interface. It has been suggested that this discrepancy is due to the neglect of volume differences between solute and solvent in the theory [34]. Therefore, we re-derived a formalism that combines the FH theory [35,36,37]—the simplest way to incorporate molecular volume differences—with the TMT approach [22,25]. This theory practically replaces the description of the solution in terms of mole fractions (ideal mixing) by one in terms of volume fractions, which can be seen from the expression for the entropy of mixing in Equation (10), in particular. The derivation was based on ideas published some time ago by Hildebrand [38], which shed light on the implications of the FH theory. Further insight was gained by considering the concept of the PCP introduced by Ben-Naim [40,41]. It turned out that the traditionally modeled micellization free energy corresponds to a PCP difference between micelles and monomers and has to be complemented by a term describing the liberation free energy difference. The neglect of the latter in earlier treatments might be the reason for the observed failure of the FH theory in predicting CMCs [24]. It remains to be clarified to what extent earlier models based on mole fractions depend on error compensation.

Another open problem is to properly link the FH theory to statistical mechanics beyond lattice models. Hildebrand´s approach uses the concept of free volume. While practical for discussions of molecular volume effects (e.g., the definition of the PCP for hard-core molecules in Equation (A20)), the concept suffers from inconsistencies in the evaluation of statistical integrals. We propose using the concept of the accessible volume per molecule in the liquid instead. The usefulness of this proposal remains to be explored.

The difference between “microscopic” and “macroscopic” surface tension is not due to the neglect of molecular volume ratios. In fact, incorporating molecular volume effects via the FH theory slightly decreases the value of the surface tension. The reason for the high surface tension found by Sharp et al. [34] for the SASA and by Tuñón et al. [57] for the SESA is the separation of the term $1-v_{\mathrm{H}}/v_{\mathrm{W}}$ (cf. Figure S1). Here, $v_{\mathrm{H}}/v_{\mathrm{W}}$ is the molar volume ratio of hydrocarbon and water. This term is part of the PCP difference, $\mathsf{\Delta }\tilde{\mu }$, in the transfer problem, which is the same for the transfer of the detergent´s alkyl tail into the micelle as for the transfer of an alkane molecule from water into a pure liquid alkane phase. It describes the free energy change due to incorporation of a voluminous alkane molecule into the aqueous phase (change in the free volume) and consequently depends on the ratio $v_{\mathrm{H}}/v_{\mathrm{W}}$. In contrast, the quantity relevant to describe the contact free energy at a macroscopic alkane–water interface, which could be $\mathsf{\Delta }\overline{\mu }=\mathsf{\Delta }\tilde{\mu }-{\beta }^{-1}\left(1-v_{\mathrm{H}}/v_{\mathrm{W}}\right)$ (cf. Equation (S43)), should not contain contributions from the incorporation of alkane into the aqueous phase. Curiously, the ratio of surface tensions, obtained from analyzing $\mathsf{\Delta }\overline{\mu }$ compared to $\mathsf{\Delta }\tilde{\mu }$ is 44.8/27.4 ≈ 1.6, is coincidentally the same as the SASA/SESA surface tension ratio. This, together with the fact the conversion factor from mN/m to cal/(mole Ų) is similar (~1.4), led to some confusion in the past [15].

Since cosolutes (salts, buffer) influence the solubility of hydrophobic substances in water, the idea was put forward that modified surface tension values could be used to model the influence of cosolutes on micelle formation in practical applications of the theory [15,27]. The problem here is that the “microscopic” surface tension has to be used, which is not generally available for buffered solutions, if not based on fits to solubility data that are likewise not generally available. There is hope however, that a better understanding of the relationship between surface tensions of molecular and macroscopic surfaces will help to link the modeling of micelle formation to quantities that can be measured in the laboratory. In this way, we could not only gain knowledge about the CMC and its changes under various conditions of practical significance, but also improve our understanding of other processes, such as membrane proton solubilization or protein folding, in which the contacts between hydrophobic surfaces and water are changed. By employing measured surface tensions in the modeling of these processes, we could—after some struggle with the subtleties of condensed matter physics—revert to simplicity.