Aggregate size distributions in an aqueous solution containing either charged or neutral surfactants are investigated using Raman multivariate curve resolution (Raman-MCR) spectroscopy and analyzed with the aid of a multi-aggregation chemical potential surface (MCPS) modeling strategy. Total least squares decompositions of the concentration-dependent Raman-MCR spectra are used to quantify the free and micelle surfactant populations, and the surfactant’s C–H stretch frequency is used as a measure of its average aggregation state. MCPS predictions relate the experimental measurements to the underlying surfactant aggregate size distribution by fitting either the component concentrations or the average C–H frequency to MCPS predictions, and thus determine the critical micelle concentration (CMC) and estimate the corresponding micelle size and polydispersity. The Raman-MCR spectra of aqueous 1,2-hexanediol, sodium octanoate, and sodium dodecyl sulfate, measured both below and above CMC, provide critical tests of the assumed functional form of the MCPS and the presence of low-order premicellar aggregates. Our results indicate that the low-order aggregate population gradually emerges as the CMC is approached and then remains nearly concentration-independent after the appearance of micelles.

## I. INTRODUCTION

The biological, industrial, and environmental importance of micelles and their intrinsically interesting behavior have motivated a large body of prior experimental^{1–6} and theoretical^{7–19} studies, establishing that micelles are liquid-like droplets of nanometer dimension whose core hydration, surface structure, and salt sensitivity remain the subjects of research.^{5,6,18,19} A characteristic signature of micelle formation is their relatively abrupt appearance at the corresponding critical micelle concentration (CMC). Thus, although micelle formation is a single-phase equilibrium structural transformation, it has some similarities to a macroscopic phase transition, such as the formation of water droplets when the atmosphere reaches 100% humidity.^{2,20} Open questions remain regarding the concentration dependence of the low-order and high-order aggregate size distributions, both below and above CMC. Here, we address these questions using an analysis strategy that combines probe-free Raman multivariate curve resolution (Raman-MCR) spectroscopic measurements and a multi-aggregation chemical potential surface (MCPS) modeling strategy, applied to micelles formed from both neutral (1,2-hexanediol, 12HD) and ionic (sodium octanoate, C8OONa, and sodium dodecyl sulfate, SDS) surfactants. The MCPS predictions are constrained by the experimental concentration dependence of the Raman-MCR spectral components assigned to the free- and micelle-bound surfactants and the surfactant C–H frequency shift used to track the average surfactant aggregation state. Our results provide a sensitive probe-free measure of CMC and a critical test of premicellar low-order aggregation, as well as estimates of the size and polydispersity of the subsequently formed micellar aggregates.

The vast number of prior experimental studies of micelle-forming aqueous surfactant solutions are exemplified by a definitive early compilation of CMC measurements for 720 different surfactants obtained using 71 different experimental techniques^{1} and a more recent compilation that includes experimental CMC and aggregation numbers (but not polydispersities).^{21} Although some optical spectroscopic methods have used probe chromophores to detect the formation of micelles,^{3,4,22} a number of experimental methods are probe-free, including surface tension, conductivity, light scattering, refractive index, and x-ray diffraction.^{1,3} Prior probe-free vibrational spectroscopic studies of micelle-forming systems^{2} include an early Raman study of aqueous sodium carboxylates (including C8OONa), in which concentration-dependent changes in the C–C and C–H vibrational sub-bands were used to infer the gradual unfolding of surfactant chains with increasing concentration below CMC.^{23} The authors of the latter study were not convinced that their observations were consistent with premicellar aggregation, regarding which earlier non-spectroscopic studies had reached conflicting conclusions.^{24,25} More recently, Raman multivariate curve resolution (Raman-MCR) has been used to probe the micelle structure^{5} and the influence of salts on CMC.^{6} Here, we demonstrate the utility of combining Raman-MCR and MCPS to critically test aggregate size distribution predictions, thus providing additional evidence of the formation of premicellar aggregates and a new probe-free means of quantifying the CMC and the size and polydispersity of the corresponding micelles.

Prior theoretical descriptions of micelle formation include the pioneering work of Debye,^{7} Tanford,^{8–10} Israelachvili *et al.*,^{11,12} and others,^{13,14} as well as early^{15,16} and recent^{17–19} computer simulations (and references therein). The present MCPS theoretical framework has much in common with that previously described by Tanford and Israelaschvili. However, rather than being based on assumptions regarding the surfactant structure and micelle shape, here we use experimental Raman-MCR spectra to critically test MCPS predictions and constrain the functional form of the MCPS that self-consistently predicts the formation of both low-order (premicellar) and high-order (micellar) aggregates.

## II. MULTI-AGGREGATION CHEMICAL POTENTIAL SURFACE (MCPS) PREDICTIONS

The following MCPS-based theoretical description of micelle formation relates the surfactant excess chemical potential to the resulting equilibrium aggregate size distribution, as obtained from the corresponding equilibrium constants and free energies associated with the formation of aggregates of variable sizes. More specifically, the MCPS describes the dependence of the free energy of a surfactant molecule on the size (*n*) of the aggregate in which it is contained. Although the MCPS is not the same as the potential of mean force (PMF) plotted as a function of a one-dimensional inter- or intra-molecular separation coordinate, it is closely related to the PMF associated with a hypothetical process in which the surfactant is sequentially dissolved in a series of fully equilibrated aggregates of increasing *n*.

The equilibrium constant for an aggregation reaction of the form *nM ⇌ A* may be expressed as follows:

where *β* = 1/RT, *R* = *N*_{A}*k*_{B} is the molar Boltzmann constant, and *T* is the absolute temperature. The second equality in Eq. (1) introduces a monomer-based notation for the concentration *C*_{n} of surfactants that are in aggregates of size *n*. In other words, the concentration of aggregates of size *n* is $[A]=1nCn$, and thus, [*M*] = *C*_{1} is the concentration of free (non-aggregated) monomers. The associated aggregation free energy Δ*G*° may also be expressed using the monomer-based notation for the surfactant chemical potentials,

where Δ*G*°_{A} and Δ*G*°_{M} are the formation Gibbs energies (in a 1 M aqueous solution) of aggregates of size *n* and free monomers, respectively, *μ*°_{n} is the chemical potential of a monomer contained in an aggregate of size *n*, and Δ*μ*°_{n} is the MCPS. In other words, *μ*°_{n} is equivalent to the Ben-Naim pseudo-chemical potential of a monomer in an aggregate of size *n* at 1 M concentration.^{26,27}

The MCPS, Δ*μ*°_{n}, dictates the corresponding equilibrium aggregate size distribution, *C*_{n}, since, at equilibrium, the chemical potentials of the free and bound monomers must be equal to each other. In other words, at the equilibrium, *μ*_{1} = *μ*_{n}, where *μ*_{1} = *μ*_{1}° + *RT* ln *C*_{1} and *μ*_{n} = *μ*°_{n} + *RT* ln *C*_{n} (assuming that the solution is sufficiently dilute that one may neglect interactions between free monomers and between aggregates). Since the same is true for aggregates of any size, this relation implies that the chemical potentials of the surfactants in aggregates of any size must all be equal to each other at the equilibrium, in keeping with Eq. (1).

The above equilibrium condition [Eqs. (1) and (2)] yields the following expression for the equilibrium concentration of monomers in aggregates of size *n*:

We further generalize the definition of CMC by defining the *n*-dependent *critical aggregation concentration**C*_{A} as that free-monomer concentration at which *C*_{1} = *C*_{n}. This definition, combined with Eq. (3), yields the following expression for *C*_{A} as a function of *n*:

Note that if the micelle aggregation number *n*^{*} is large (*n*^{*} ≫ 1), then at *n* = *n*^{*}, both $nn$ and *n*/(*n* − 1) become approximately equal to 1, and thus, $CA*\u2248e+\beta \Delta \mu \xb0n*$. The latter approximate equality, or equivalently $\Delta \mu \xb0n*\u2248RT\u2061ln\u2061CA*$ or $\Delta G\xb0\u2248nRT\u2061ln\u2061CA*$, is often taken as defining the relationship between the critical micelle concentration and either $\Delta \mu \xb0n*$ or Δ*G*°.^{12,28} In other words, $CA*$ is essentially equivalent to previous definitions of CMC. Although the precise experimental value of CMC for a given surfactant solution is slightly dependent on the experimental method used to detect the onset of micelle formation, the resulting CMC value is necessarily close to the free monomer concentration after micelles have begun to form and, thus, is necessarily approximately equivalent to $CA*$. The above more general (*n*-dependent) definition of *C*_{A} [Eq. (4)] is preferable, as it is applicable to aggregates of any size and assures that *C*_{A} is always equivalent to the value of *C*_{1} at which *C*_{1} = *C*_{n} (or, equivalently, [*M*] = *n*[*A*]).

Equations (3) and (4) may be combined to obtain the following relation between the equilibrium concentrations of aggregated *C*_{n} and free *C*_{1} monomers:

Note that Eq. (5) implies that *C*_{n} = *C*_{1} when *C*_{A} = *C*_{1}.

Equations (3)–(5) are quite general, as they may be applied to situations in which aggregates of various sizes are simultaneously in equilibrium with each other, as dictated by the underlying MCPS, Δ*μ*°_{n}. In other words, if we know how Δ*μ*°_{n} depends on *n*, then we can use Eq. (3) to predict the resulting aggregate size distribution, *C*_{n}. Conversely, Δ*μ*°_{n} may be determined from experimental (or simulation) measurements of the aggregate size distribution *C*_{n}.

The formation of micelles from surfactants typically produces aggregates with a relatively narrow size distribution, centered around some particular aggregate size *n*^{*}, with a polydispersity of $\sigma \u2248n*$,^{9,12} although the values of *n*^{*}, and particularly *σ*, are more difficult to experimentally determine than the critical micelle concentration. This behavior implies that Δ*μ*°_{n} has a minimum value near *n*^{*} and, thus, that a chemical potential gradient drives the system toward the formation of aggregates near this special size, with a polydispersity *σ* related to the curvature of the chemical potential surface near *n* = *n*^{*}.

The simplest functional form that is consistent with the above behavior is a quadratic function of Δ*n* = *n* − *n*^{*},

The value of the constant, *a*, in Eq. (6) may be determined using Eq. (4), combined with the fact that Δ*n* = 0 when *n* = *n*^{*},

Note that the absence of an additional linearly Δ*n*-dependent term in Eqs. (6) and (7) is required in order to assure that the MCPS minimum occurs at *n* = *n*^{*}. In other words, any additional linearly Δ*n*-dependent term in Eqs. (6) and (7) could be removed by re-defining Δ*n* such that the MCPS minimum occurs at Δ*n* = 0.

If it is further assumed that *μ*°_{n} is a linear function of *n* at small *n*, as would be expected if the binary surfactant contact free energy is approximately independent of *n* (for free monomers and low-order aggregates), then that implies that Δ*μ*°_{n} must approach 0 as *n* approaches 1 and, thus, that *σ* is equal to the following function of *n*^{*} and $CA*$:

Although the *σ* values obtained using Eq. (8) are clearly not simply equal to $n*$, they are not too far from $n*$ for short-chain surfactants, such as C8OONa and 12HD (with $CA*>0.1$ M), but significantly smaller than $n*$ for larger surfactants, such as C12OONa and SDS (with $CA*<0.1$ M), in disagreement with some previous experimental and theoretical estimates.^{9,12,29} However, given the difficulty of experimentally determining *σ*, it is not clear whether such discrepancies can be considered to provide critical tests of the above predictions (as further discussed in Secs. IV and V below).

Alternatively, rather than assuming that *β*Δ*μ*°_{n} is a quadratic function of *n*, we may consider how the MCPS predictions would change if we assumed that *C*_{n} is a perfect Gaussian function when the total surfactant concentration is that at which $C1=CA*$, leading to the following Gaussian MCPS prediction:

Note that the only difference between the above quadratic and Gaussian MCPS functions is that *n*^{*} is replaced by *n* everywhere (except in Δ*n* = *n* − *n*^{*}). However, despite the similar appearance of the two expressions, the functional forms of Δ*μ*°_{n} are quite different, as are the resulting *C*_{n} predictions, particularly at small *n* (as further illustrated and discussed below).

The aggregate size distribution predictions pertaining to the above two models may be obtained by inserting *β*Δ*μ*°_{n} into Eq. (3). Thus, the quadratic potential model yields the following predicted aggregate size distribution:

Note that when $C1=CA*$ and *n* ≈ *n*^{*}, the above expression reduces approximately to a Gaussian distribution $Cn\u2248CA*\u2061exp[\u2212\Delta n2/2\sigma 2]$. On the other hand, if we assume that the distribution is exactly Gaussian when $C1=CA*$, then Eqs. (3) and (9) yield the following predicted aggregate size distribution:

Note that this distribution is, once again, obtainable from the quadratic prediction upon replacing *n*^{*} by *n* everywhere (except in Δ*n*).

Both of the above aggregation models have three input parameters: *n*^{*}, $CA*$, and *σ* [which may be related to each other by Eq. (8)]. The results predict aggregate concentrations as a function of both the aggregate size, *n*, and total monomer concentration, *C*_{T} (or free monomer concentration, *C*_{1}). Although both the models lead to similar high-order aggregate size distributions, the Gaussian model strongly suppresses the formation of low-order aggregates. Thus, self-consistency of the predictions with experimental measurements may be used to determine whether low-order premicellar aggregates are experimentally detectable.

The above expressions for Δ*μ*°_{n} and *C*_{n} pertain to all values of *n* > 1. When *n* = 1, then Δ*μ*_{1}° = *μ*_{1}° − *μ*_{1}° = 0 and *C*_{n} = *C*_{1}. Thus, the free monomer concentration *C*_{1} is an independent variable whose value is determined by the total monomer concentration $CT=C1+\u22112\u221eCn$, and thus, *C*_{T} may be obtained from *C*_{1} and *C*_{n} [predicted using either Eq. (7) or Eq. (9)]. Alternatively, given *C*_{T} and *C*_{n}, one may determine *C*_{1} by numerically solving $C1=CT\u2212\u22112\u221eCn$ to obtain the value of *C*_{1} that is consistent with the desired (experimental) value of *C*_{T}.

Figure 1 compares the quadratic and Gaussian MCPS predictions obtained using experimental values of $CA*$ and *n*^{*} for aqueous sodium dodecanoate (C12OONa), with *σ* determined using Eq. (8).^{5} The predictions of the quadratic and Gaussian MCPS models are clearly quite similar, except with regard to the low-order aggregate distribution (at small *n*), as most clearly seen in the inset of Fig. 1(a), and to a lesser extent in the difference between the predicted free (*C*_{1}) and micelle-bound (*C*_{T} − *C*_{1}) surfactant concentrations shown in Fig. 1(b), particularly when $CT\u2264CA*$. Note that for both the quadratic and Gaussian predictions, $CA*$ is equal to the value of *C*_{1} at which free- and micelle-bound surfactant concentrations are equal to each other, corresponding to the crossing point of the two curves in Fig. 1(b).

For systems containing a significant population of low-order aggregates (both below and above $CA*$), it can further be useful to distinguish the low-order and high-order aggregate population by including the low-order aggregates in the effective free monomer concentration. This can be accomplished by introducing a fourth parameter *n*_{L} equal to the maximum value *n* that is included in the low-order aggregate population. More specifically, the following expression is used to predict the effective free monomer concentration, *C*_{f}, which includes the influence of partially hydrated low-order aggregates, and thus, the corresponding effective concentration of micelle-bound aggregates is *C*_{m} = *C*_{T} − *C*_{f}:

Note that the coefficient in parentheses is equal to 1 when *n* = 1 and decreases linearly toward zero as *n* approaches *n*_{L} + 1. Thus, this coefficient effectively treats low-order aggregates as partially free, in proportion to the fractional aggregation state of the surfactant. Thus, *n*_{L} may be viewed as an effective maximum surfactant coordination number, above which the aggregates are treated as belonging to the micelle population. The value of *n*_{L} can either be fixed to some reasonable coordination number, such as *n*_{L} = 10, or can be treated as an adjustable parameter whose value is obtained by fitting experimental data. The chosen value of *n*_{L} only has a significant influence on *C*_{f} for systems in which there is a significant low-order aggregate population, as it will invariably be the case that *C*_{f} ≈ *C*_{1} when the low-order aggregate population is negligible.

The results shown in Fig. 1 are obtained assuming that *n*_{L} = 1, which is equivalent to treating all aggregates with *n* > 1 as part of the micelle population. Quite similar C12OONa results are obtained when assuming 1 ≤ *n*_{L} ≤ 10, all of which predicts that *C*_{f} ≈ *C*_{1}. In other words, these MCPS predictions imply that C12OONa has a relatively small population of low-order aggregates (both below and above $CA*$).

For systems with a more significant population of low-order aggregates, predictions obtained with *n*_{L} = 1 differ significantly from those obtained with *n*_{L} > 1, and *n*_{L} > 1 is required in order to obtain good agreement with the experimental Raman-MCR results. However, any value of *n*_{L} that produces good experimental agreement leads to approximately the same effective $CA*$ value, corresponding to the value of *C*_{f} at which *C*_{f} = *C*_{m} (as further illustrated and discussed below).

In this work, we have used three different means of connecting the above predictions with experimental Raman-MCR measurements. One method involves obtaining *C*_{f} and *C*_{m} by decomposing experimental spectra into free and micelle populations. The other method relies on the experimental average C–H Raman-band frequency of the surfactant as a measure of the average surfactant aggregation state (as a function of surfactant concentrations). A third method involves a hybrid strategy that uses the MCPS to fit, extrapolate, and reconstruct experimental spectra (as described in the Appendix). The close agreement between the results obtained using all three analysis strategies confirms the self-consistency of the experimental and MCPS predictions, providing strong evidence of the presence of premicellar aggregates in some of the surfactant solutions.

## III. RAMAN-MCR SPECTRAL DECOMPOSITION

Aqueous stock solutions of 12HD (1,2-hexanediol, Sigma-Aldrich, $\u226598%$), C8OONa (sodium octanoate, Sigma-Aldrich, $\u226598%$), and SDS (sodium dodecyl sulfate, MP Biomedicals, $\u226599%$) were prepared using ultrapure filtered water (Milli-Q, 18.2 MΩ cm) by weighing either solute into 25 ml volumetric flasks. Solutions of lower concentrations were prepared in 2 ml volumetric flasks by diluting appropriate volumes of the stock solutions.

Raman measurements of the surfactant solutions and pure water were performed in 2 ml cylindrical glass vials that were equilibrated to 20.0 °C in a temperature-controlled cell holder (Quantum Northwest). These measurements were made using a home-built Raman system, as previously described.^{6} Briefly, the Raman instrument includes an Ar-ion 514.5 nm excitation laser (with ∼20 mW of power at the sample cell), a 300 mm spectrograph (Acton Research, Inc.), a grating with either 1200 grooves per mm (for 12HD and C8OONa) or 300 grooves per mm (for SDS), and a thermoelectrically cooled CCD camera (Princeton Instruments, Inc.). Each Raman spectrum was obtained using 5 min of signal averaging.

Two rounds of self-modeling curve resolution (SMCR) were used to decompose the measured Raman spectra.^{6,30,31} In the first round of SMCR, two pairs of pure water and surfactant solution spectra were decomposed to generate minimum-area surfactant-correlated (SC) spectra, containing the surfactant C–H stretch bands and an O–H stretch band arising from water molecules that are perturbed by the surfactant. Subsequently, a second SMCR decomposition was performed to decompose the first-round SMCR SC spectra into free- and micelle-bound spectral components. For this second spectral decomposition, the monomer spectrum was constrained to the measured SC spectrum of a dilute surfactant (below CMC). The micelle-bound spectrum is identified as that member of the second-round SMCR rotational-ambiguity family of spectra that produced nearly concentration-independent free monomer concentrations at high total surfactant concentrations. More specifically, trial micelle-bound spectra are obtained from linear combinations of the free-surfactant SC spectrum and the second-round SMCR minimum-area micelle-correlated spectrum. Essentially identical free- and micelle-bound spectra can also be determined by extrapolating the measured first-round SC spectra to the low- and high-concentration limits (as described in the Appendix).

A total least squares (TLS) regression was used to decompose all the first-round SC spectra into a linear combination of free- and micelle-bound components to obtain *C*_{f} and *C*_{m}.^{6,32,33} These TLS fits were performed either using only the C–H stretch band or both the C–H and O–H stretch bands. The self-consistency of the spectral decomposition results is further verified as described in the Appendix.

The average C–H frequencies were obtained using $\u27e8\omega \u27e9=\u222b\omega 1\omega 2\omega I(\omega )d\omega $, where *ω* is the C–H vibrational frequency and *I*(*ω*) is the C–H band shape normalized to unit area between *ω*_{1} and *ω*_{2}. For 12HD, *ω*_{1} = 2836 cm^{−1} and *ω*_{2} = 2976 cm^{−1}, for C8OONa, *ω*_{1} = 2831 cm^{−1} and *ω*_{2} = 2986 cm^{−1}, and for SDS, *ω*_{1} = 2820 cm^{−1} and *ω*_{2} = 2991 cm^{−1}. The resulting average C–H frequencies were plotted as a function of total surfactant concentration, *C*_{T}, and fit to MCPS predictions to obtain *n*^{*}, *σ*, *n*_{L}, and the effective critical surfactant concentration $CA*$, as well as the average frequencies of the free- and micelle-bound surfactant, *ω*_{f} and *ω*_{m}, as further described below. The measured ⟨*ω*⟩ values were related to MCPS predictions using the following expression:

where *C*_{T}, *ω*_{f}, and *ω*_{m} are the experimental total surfactant concentration, limiting ⟨*ω*⟩ C–H frequencies, and *C*_{f}/*C*_{T} is the effective fraction of free surfactants obtained using Eq. (12).

## IV. RESULTS

Figure 2 illustrates how the measured Raman spectra of aqueous solutions, containing the neutral 12HD surfactant, are sequentially decomposed using two rounds of SMCR to quantify the concentrations of the free, *C*_{f}, and micelle-bound, *C*_{m}, components as a function of the total 12HD concentration, *C*_{T}. Figure 2(a) shows the C–H and O–H stretch bands in the raw Raman spectra of aqueous 12HD solutions of various concentrations. Figure 2(b) shows the 12HD SC spectra (normalized to the same C–H band area) obtained from the first-round SMCR decomposition of pairs of spectra obtained from pure water and 12HD solutions of different concentrations. The decrease in the SC O–H band area with increasing *C*_{T} reveals the aggregation-induced surfactant dehydration.

The dotted-red and dotted-purple curves in Fig. 2(c) are the free- and micelle-bound SC spectra, respectively. These are equivalent to the first-round SC spectrum at *C*_{T} = 0.2M and the micelle-bound SC spectrum that produced a nearly constant *C*_{f} when $CT>CA*$, as shown in Fig. 2(d). The points in (d) are the TLS *C*_{f} and *C*_{m} concentrations, and the curves are the MCPS predictions obtained by fitting the experimental points. The self-consistency of these predictions is further demonstrated by comparison with independent MCPS fits to experimental ⟨*ω*⟩ values (as described below) and using the alternative TLS- and MCPS-based decomposition strategy (as described in the Appendix).

Figure 3 shows the results obtained by using the MCPS to fit experimental average surfactant C–H frequencies ⟨*ω*⟩ in solutions containing either neutral 12HD, in Figs. 3(a)–3(c), or ionic C8OONa surfactants, in Figs. 3(d)–3(f). Figures 3(a) and 3(d) show ⟨*ω*⟩, obtained from the corresponding first-round SMCR SC spectra, plotted as a function of *C*_{T}. The concentration-dependent shift in ⟨*ω*⟩ is assumed to correlate approximately linearly with the average volume fraction of surfactants in the first solvation shell of each surfactant. Thus, the observed non-linear concentration dependence of ⟨*ω*⟩ indicates that a relatively abrupt increase in aggregation occurs at the micelle formation concentration near 0.7 M for 12HD and near 0.4 M for C8OONa. Moreover, the approximately linear concentration dependence of ⟨*ω*⟩ at lower concentrations implies that there is some premicellar interaction between the surfactants.

The curves in Figs. 3(a) and 3(d) are fits to the experimental ⟨*ω*⟩ values using either the quadratic MCPS [solid curves, obtained using Eqs. (8), (10), and (12)] or the Gaussian MCPS [dashed curves, obtained using Eqs. (11) and (12)]. Note that the Gaussian model, which strongly suppresses premicellar aggregation, predicts that ⟨*ω*⟩ should be concentration-independent below $CA*$, as evidenced by the flat top of the dashed curves in Figs. 3(a) and 3(d). Thus, the markedly better fits of ⟨*ω*⟩ to the quadratic as opposed to the Gaussian model confirm that there is significant premicellar aggregation occurring in both these surfactant solutions.

The curves in Figs. 3(b) and 3(e) are obtained from MCPS fits to the ⟨*ω*⟩ results in Figs. 3(a) and 3(d), while the points were obtained from the TLS spectral decompositions (as described above and illustrated in Fig. 2). The good agreement between the curves and points in Figs. 3(b) and 3(e), thus, confirms the remarkable self-consistency of the quadratic MCPS predictions with the measured C–H shifts and spectral decompositions, all of which clearly identify $CA*$ as the value of *C*_{f} when the *C*_{f} and *C*_{m} curves cross. Note that these $CA*$ values are close (but not identical) to those obtained by linearly extrapolating *C*_{m} to zero, as previously described.^{6} The present effective $CA*$ values are preferable as they pertain to the more precise definition of $CA*$ expressed in Eq. (4).

Figures 3(c) and 3(f) show the MCPS, *β*Δ*μ*°_{n} (green curves), and aggregation size distributions, *C*_{n} (purple points). The corresponding $CA*$ predictions pertain to the value of *C*_{f} at which *C*_{f} = *C*_{m} (or $CT\u22482CA*$). Note that both the quadratic and Gaussian fits yield similar $CA*$ values, but the quadratic MCPS predictions contain significantly more low-order aggregates (with *n* ≤ 10).

The quadratic fits shown in Fig. 3 were obtained by numerically optimizing the values of the parameters $CA*$, *n*^{*}, and *n*_{L} and the two limiting frequencies *ω*_{f} and *ω*_{m}. The effective $CA*$ (as well as *ω*_{f} and *ω*_{m}) values are quite robust, but the MCPS predictions are less sensitive to the precise values of *n*^{*} and *n*_{L}, although optimal fits are obtained using the values given in Fig. 3. More specifically, the effective $CA*$ values obtained by fitting either *C*_{f} or ⟨*ω*⟩ are self-consistent to better than 10%, while similar MCPS predictions can typically be obtained when varying *n*^{*} and *n*_{L} by 30% or more.

The Gaussian MCPS fits shown in Fig. 3 were obtained using the same *n*^{*}, *σ*, and *n*_{L} as the quadratic MCPS fits, with new best-fit values of $CA*$, *ω*_{f}, and *ω*_{m}, resulting in an effective $CA*$ (obtained from *C*_{f} and *C*_{m} points) that is essentially the same as that obtained from the quadratic MCPS fits. However, the Gaussian fits were less stable than the quadratic fits when *n*^{*}, *σ*, or *n*_{L} were also allowed to vary. Moreover, the Gaussian MCPS predictions are fundamentally inconsistent with our experimental observation that ⟨*ω*⟩ is not constant below $CA*$ and are also unphysical in predicting a large positive value of Δ*μ*_{2}°, implying large positive surfactant dimer contact free energy.

The different minimum values of *β*Δ*μ*°_{n} (near *n* = *n*^{*}) obtained from the quadratic and Gaussian MCPS fits are a consequence of the absence of low-order aggregates in the Gaussian MCPS predictions. In other words, although both the quadratic and Gaussian fits use the same value of *n*_{L} > 1, the Gaussian predictions imply that the best fit $CA*$ is equal to the value of *C*_{1} at which *C*_{1} = *C*_{m}, while the quadratic predictions yield a different value $CA*$, when defined in the same way. However, both the quadratic and Gaussian fits yield essentially identical effective $CA*$ values, equal to the value of *C*_{f} at which *C*_{f} = *C*_{m}.

The above Raman-MCR-based strategy is most readily applicable to the surfactant with relatively high critical concentrations of $CA*>0.1$ M ($>$ 100 mM) because of the difficulty of obtaining high-quality SC spectra at lower concentrations. However, we have found that it is possible to extend this method to surfactants with $CA*$ values down to ∼10 mM (or, more generally, ∼10 mg/ml). The feasibility of doing so is illustrated in Fig. 4, which contains Raman-MCR measurements and MCPS predictions for aqueous SDS (which has previously been reported to have a CMC of 8.1 ± 0.1 mM, *n*^{*} = 69 ± 5, and *σ* ∼ 13).^{12,29} At the lowest concentrations (below ∼0.02 M), it was not possible to accurately determine the shape of the broad hydration shell O–H band (peaked near 3500 cm^{−1}), and thus, the TLS fits were performed only on the C–H band.

The points in Fig. 4(b) are the experimental surfactant average C–H frequencies ⟨*ω*⟩ (obtained from the first-round SMCR SC spectra), and the green curve is the quadratic MCPS prediction obtained from a best fit to the points [with *σ* from Eq. (8) and *n*_{L} = 1]. The excellent agreement between the experimental and MCPS predictions and the constant value of ⟨*ω*⟩ below $CA*$ confirm that there is no detectable premicellar aggregation in SDS. The error bars on the MCPS parameters reflect the range of values over which very similar MCPS fits are obtained. The value of $CA*$ is in excellent agreement with that previously reported, while the *n*^{*} value is more uncertain but agrees roughly with the reported values of ∼60–80,^{12,21,29} and equally good agreement with MCPS predictions could be obtained assuming the latter *n*^{*} values. The fact that the micelle polydispersity *σ* ∼ 3 ± 1 is significantly smaller than a previously reported experimental value of 13^{29} is further discussed below.

## V. DISCUSSION

The present results provide strong evidence that the shape of the MCPS is approximately describable as a quadratic function of *n* and is inconsistent with functional forms that strongly suppress the formation of low-order aggregates, such as the Gaussian MCPS. We have further assumed that the MCPS is approximately concentration-independent, as it must be if the surfactant concentration is sufficiently low that one may safely neglect the influence of interactions between free monomers and aggregates. However, the true shape of the MCPS is undoubtedly not precisely quadratic and is expected to become concentration-dependent at high concentrations. For example, some surfactants have been found to produce micelles whose size increases with increasing concentration or to produce supramolecular lattices of different structures at different surfactant concentrations.^{10,12} Although concentration-dependent micelle size distributions may arise from a concentration-independent MCPS with a broad minimum,^{12} they may also result from an MCPS that is concentration-dependent or has a more complicated functional form, perhaps with multiple minima. The present results also do not address the temperature dependence of the MCPS, which is likely to be significant and non-linear, given that the critical micelle concentrations are often non-monotonic functions of temperature (as is the case for the solubility of oily molecules).^{10,12}

Any physically realistic MCPS is likely to have a near zero-intercept, in the sense that Δ*μ*°_{n} smoothly approaches zero as *n* approaches 1. The reason for this is that any abrupt jump in Δ*μ*°_{n} between *n* = 1 and *n* = 2 (which differs significantly from that between *n* = 2 and *n* = 3) would imply that the contact free energy between a pair of surfactants is strongly *n*-dependent. Note that the Gaussian MCPS has such unphysical behavior, as it implies that Δ*μ*_{2}° − Δ*μ*_{1}° is large and positive, while Δ*μ*_{3}° − Δ*μ*_{2}° is negative. It is this unphysical behavior that is responsible for the strong suppression of low-order aggregates by the Gaussian MCPS.

Our results further imply that the concentration of low-order aggregates is expected to decrease with decreasing critical micelle concentration because the probability of forming low-order aggregates necessarily decreases with decreasing concentration (as is the case for any dimerization reaction). This is the case despite that the contact free energy between a pair of surfactants is expected to become more favorable (more negative) with increasing chain length and decreasing CMC. For example, our results imply that the contact free energies of 12HD and C8OONa are both ∼−0.1 kJ/mol, while those for the longer-chain surfactants C12OONa and SDS are ∼−0.3 and −0.4 kJ/mol, respectively. More generally, the quadratic MCPS predictions imply that the contact energy should become more favorable with decreasing $CA*$ at constant *n*^{*} and less favorable with increasing *n*^{*} at constant $CA*$.

The robustness of the above results is further demonstrated using the alternative spectral decomposition strategy described in the Appendix. These results imply that it is only possible to reliably determine $CA*$ if high-quality Raman-MCR SC spectra are obtainable down to $CT\u2248CA*$. Higher concentration spectra can be used to confirm that the solutions are composed of a mixture of two species, but cannot unambiguously determine either the free-surfactant SC spectrum or $CA*$. However, if $CA*$ has been independently determined, then the strategy described in the Appendix may be capable of reconstructing the free-surfactant SC spectrum and, thus, extrapolating experimentally derived *C*_{f} and *C*_{m} component concentrations below the experimentally accessible concentration range.

The value of *σ* = 46 ± 10 that we obtained for 12HD using Eq. (8) is in reasonably good agreement with previously reported values *σ* = 40 ± 10 and *σ* = 35 ± 4 obtained from light scattering and vapor pressure measurements, respectively (as reported in the Note Added in Proof of Ref. 34). Our value of *σ* = 3 ± 1 for SDS is close to *σ* ∼ 4 obtained from molecular dynamics simulations,^{17} but is significantly smaller than *σ* ∼ 13 inferred from kinetic relaxation measurements.^{29} Note that Eq. (8) is only expected to hold if the MCPS is very nearly a quadratic function all the way down to *n* = 1. If the true *σ* of SDS is, in fact, significantly larger than 3, then that would imply that the true MCPS of SDS deviates significantly from a quadratic *n*-dependence and has a smaller curvature near *n* = *n*^{*}.

## SUPPLEMENTARY MATERIAL

See the supplementary material for additional concentration-dependent Raman-MCR spectra and C–H intensities.

## ACKNOWLEDGMENTS

This work was supported by a grant from the National Science Foundation (Grant No. CHE-1763581).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available within this article and its supplementary material.

### APPENDIX: HYBRID ANALYSIS STRATEGY

Here, we describe an alternative hybrid experimental-MCPS analysis strategy that may be used to obtain self-consistent $CA*$ and *n*^{*} values from Raman-MCR SC spectra measured over a range of surfactant concentrations. This hybrid strategy differs from that used to obtain the results shown in Figs. 1–4 in that the TLS fits to the experimental SC spectra are performed and extrapolated with the aid of the MCPS, rather than using purely experimentally determined ⟨*ω*⟩, or *C*_{f} and *C*_{m}, followed by MCPS fits.

The essential element of this hybrid strategy is that, rather than fitting ⟨*ω*⟩ using Eq. (13), a similar Eq. (A1) is used to fit the TLS coefficients, *S*_{L} (or *S*_{H} = 1 − *S*_{L}), obtained by decomposing each measured SC spectrum into a linear combination of a particular pair of low- and high-concentration SC spectra, SC_{L} and SC_{H}, respectively. In other words, each SC spectrum may be expressed as SC = *S*_{L}SC_{L} + *S*_{H}SC_{H}, where SC is a concentration-dependent spectrum and *S*_{L} and *S*_{H} are concentration-dependent coefficients,

The parameters $SL0$ and $SL\u221e$ are the *C*_{T} → 0 and *C*_{T} → *∞* limiting values of *S*_{L}, respectively, obtained from fits to MCPS predictions. In other words, the MCPS predictions provide a functional form for *S*_{L} that is used to extrapolate *S*_{L} to its limiting values. As a result, the TLS-reconstructed free-surfactant SC spectrum is SC_{F} = $SL0$SC_{L} + $(1\u2212SL0)$SC_{H}, and the corresponding reconstructed micelle-bound SC spectrum is SC_{M} = $SL\u221e$SC_{L} + $(1\u2212SL\u221e)$SC_{H}.

The points in Fig. 5 are the TLS coefficients obtained using two different pairs of experimental reference SC_{F} and SC_{M} spectra from (a) C8OONa and (b) SDS solutions. The insets show the resulting free monomer fractions *C*_{f}/*C*_{T} obtained by re-scaling the *S*_{L} values to a range from 0 to 1, using the limiting TLS coefficients $SL0$ and $SL\u221e$. The essentially perfect agreement between the *C*_{f}/*C*_{T} points obtained from the two different TLS decompositions represents a significant additional validation of our assumption that all the measured SC spectra can be accurately represented as a linear combination of free- and micelle-bound component spectra.

The curves through the points in Fig. 5 are the MCPS predictions obtained by fitting the points to obtain optimal values of the $CA*$, *n*^{*}, $SL0$, and $SL\u221e$ (where the latter two values depend on the chosen SC_{L} and SC_{H} reference spectra). The value of *σ* is obtained from Eq. (8), and the value of *n*_{L} is treated as an additional adjustable parameter when fitting the C8OONa spectra. For the SDS spectra, *n*_{L} is set to 1, but essentially identical fits are obtained for 1 ≤ *n*_{L} ≤ 10 because, in this case, the MCPS predictions imply that there are so few low-order aggregates that only *C*_{1} has an appreciable concentration when $CT<CA*$, and *C*_{1} remains nearly constant above $CA*$.

The effective $CA*$ values listed in the caption of Fig. 5 were obtained as described above (in Secs. III and IV). The fits to MCPS predictions are relatively insensitive to *n*^{*}, whose optimal value depends primarily on the experimental TLS coefficients obtained near $CA*$ over a concentration range of $0.5CA*\u2264CT\u22642CA*$. In performing TLS fits, it is often convenient to begin by setting *n*^{*} to some reasonable value (such as 50, 75, or 100) when fitting $CA*,SL0$, and $SL\u221e$ and then use the later best-fit values as initial guesses when performing an additional fit to simultaneously optimize *n*^{*}, $SL0$, and $SL\u221e$, whose values are given in the caption of Fig. 5, with error bars reflecting the range of variations of each parameter over which nearly equally good fits to the experimental points can be obtained.