Interaction Parameters for the Formation of Mixed Micelles and Partitioning of Solutes in Them: A Review

Abstract

When two or more surfactants are mixed, the critical micelle concentration and solubilization capability are changed, and a careful selection of the combination promotes the micelle formation and enhances the solubilizing capability. Thus, understanding the mechanism behind the phenomena is essential for controlling the physical properties of the mixed micelle. The interaction parameters β and B that describe the formation of mixed micelles and their partitioning of solutes, respectively, were proposed by Treiner four decades ago. In this work, data on the formation and partitioning in binary surfactant systems were collected. Although the data on the parameters β and B for polar solutes and theoretical development are still insufficient, the directions of research to acquire an in-depth understanding of the formation and partitioning of the mixed micelle are proposed.

1. Introduction

Spherical micelles and vesicles can be applied to drug delivery, cancer therapy, and the pseudostationary phase for separation [1,2,3,4]. When two or more surfactants are mixed, the critical micelle concentration (CMC) and solubilization capability are changed, and a careful selection of the combination promotes the micelle formation and enhances the solubilizing capability [5,6,7,8,9]. Thus, understanding the mechanism behind the phenomena is essential for controlling the physical properties of the mixed micelle.

The micelle formation of the mixed micelle can be described by the regular solution theory [5,6], more appropriately called the simple mixture model, established by Rubingh and coworkers [10]. The theory uses the interaction parameter β for a binary mixture of surfactants. A negative value β means synergistic interaction, while a positive value indicates an antagonistic interaction. The CMC of the binary mixed micelle at an arbitrary composition can be appropriately described using the CMC of the pure surfactant and β [11].

An equation can also describe the partition coefficients of the mixed micelle based on the regular solution theory. O’Connell and Prausnitz originally derived the equation to calculate the solubility of inert gases in a binary solvent [12,13], and Treiner and coworkers applied the theory to the mixed micelle system with minor modifications [14,15,16]. The equations have been used for various mixed micelle systems [5,6,15,16,17,18,19,20,21,22,23,24].

Treiner and coworkers carefully categorized binary surfactants and solute systems and concluded a linear relation between the interaction parameters β and B when an appropriate combination of binary surfactants (nine examples) and polar solute was used [16,21]. The exceptions are nonpolar solutes, nonionic surfactants with long oxyethylene chains, and amphiphilic compounds such as yellow OB that undergo aggregation [21]. It has been four decades since Treiner reported the relation and many binary mixed micelle systems partitioning various solutes. In addition, there has been progress in modeling mixed micelle [10,25,26,27,28] and mixed solvent solubility [29,30]. These progresses may enforce the claim made by Treiner or help to develop a more general model that explains the relation between β and B.

In this work, data on the formation and partitioning in binary surfactant systems were collected. The summarized data set may reveal the relationship between the two parameters.

2. Theories for the Formation and Partitioning of the Mixed Micelle

According to the regular solution theory, the CMC of the binary mixed micelle of surfactants 1 and 2 can be estimated using an interaction parameter (β) and the CMC of pure surfactants [5,6]. The parameter β is related to net (pairwise) interactions among surfactants 1 and 2 in the binary mixed micelle as β = N(W₁₁ + W₂₂ − 2W₁₂)/RT, where N is Avogadro’s number, W₁₁, W₂₂, and W₁₂ are the pairwise interaction energies between surfactants 1 and 2 in the micelles, R is the gas constant, and T is the temperature. If β > 0, then a repulsive (antagonistic) interaction is taking place between two surfactants, while β < 0 indicates an attractive (synergistic) interaction between the surfactants. The value of β = 0 indicates the ideal mixture of surfactants. Experimentally, only the mole fraction of total surfactant 1 (a₁) and surfactant 2 (a₂) are known, where a₁ + a₂ = 1 (see Figure 1).

The regular solution theory relates the mole fraction of surfactant 1 (x₁) in the binary mixed micelle (Figure 1), the CMC of the pure surfactant 1 (C₁) and binary mixed micelle (C₁₂), and the mole fraction of surfactant 1 in total mixed surfactants (a₁) as follows.

$$\beta =\frac{\mathrm{l}\mathrm{n}[a_1{C}_{12}/(x_1{C}_1\left)\right]}{{(1-x_1)}^2}$$

(1)

Since x₁ is the mole fraction of surfactant in the binary mixed micelle 1, x₂ = 1 − x₁ holds. If C₁, the CMC of the pure surfactant 2 (C₂), C₁₂, and a₁ are known, the unknown variable x₁ can be calculated by iteratively solving the following equation for x₁.

$${x_1}^2\ln \left[\frac{a_1{C}_{12}}{x_1{C}_1}\right]={(1-x_1)}^2\ln \left[\frac{a_2{C}_{12}}{(1-x_1)C_2}\right]$$

(2)

Experimentally, the x₁ value is determined using ultrafiltration [31,32,33,34], small angle neutron scattering [35,36], or a combination of ion-selective electrodes and ultraviolet-visible absorption spectroscopy (UV–Vis) [37]. The theoretical x₁ value obtained using the regular solution theory qualitatively agrees with the experiment [33,35]. According to the theory, x₁ (= 1 − x₂) at the CMC can be estimated by the following equation:

$$x_1=\frac{a_1{C}_2{f}_2{x}_2}{{a_{2}C}_1{f}_1}$$

(3)

$$f_1=\mathrm{e}\mathrm{x}\mathrm{p}\left[\beta {\left(1-x_1\right)}^2\right]$$

(4)

$$f_2=\mathrm{e}\mathrm{x}\mathrm{p}\left[\beta {x_1}^2\right]$$

(5)

where f₁ and f₂ are activity coefficients of the surfactants 1 and 2, respectively.

The values of x₁ and x₂ (= 1 − x₁) above the CMC can be calculated using the following equation:

$$x_1=\frac{a_1{C}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}{C_1{f}_1-C_2{f}_2+a_2{C}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}/x_2}$$

(6)

where C_total is the total surfactant concentration. The concentration of surfactants 1 and 2 in the mixed micelles (C_1,M and C_2,M, respectively) with the CMC of C₁₂ can be expressed as follows:

$$C_{1,\mathrm{M}}=x_1(C_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-C_{12})$$

(7)

$$C_{2,\mathrm{M}}=x_2(C_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-C_{12})$$

(8)

By defining synergism as “micelle formation at total mixed surfactant concentrations in the solution phase lower than the CMCs of both surfactants in the mixture”, Hua and Rosen derived the second condition for synergism (|ln(C₁/C₂)| < |β|) in addition to the first condition of β > 0 [38]. Likewise, they derived the second condition of antagonism as |ln(C₁/C₂)| < β in addition to the first criterion (β > 0).

O’Connell and Prausnitz derived the regular solution theory for Henry’s coefficient of the gas in the mixed solvent [12,13]. Treiner and coworkers applied the theory to the partition coefficients of the mixed micelle for solutes with slight modification as follows [14,15,20,21]:

$${\mathrm{l}\mathrm{n}K}_{12}=x_1\mathrm{l}\mathrm{n}{K}_1+\left(1-x_1\right)\mathrm{l}\mathrm{n}{K}_2+B{x}_1\left(1-x_1\right)$$

(9)

where K₁₂, K₁, and K₂ are partition coefficients of a binary mixed micelle of surfactant composed of 1 and 2, pure micelle composed of surfactant 1, and that of surfactant 2, respectively. The value B is the interaction parameter and should be the same as β; however, they point out that assuming β = B is quantitatively not entirely satisfactory, even in the case of nonpolar gases in simple liquid mixtures, because of the omission of the contribution of the solute in the interaction parameter [15]. Therefore, they determined B separately from β, including all surfactant/surfactant and solute/surfactant interaction terms. Using the experimental data of the partition coefficient and CMC value, Treiner and colleagues proposed the following empirical equation to relate the values of β and B [16]:

$$B=0.194+0.343\beta$$

(10)

However, the meaning of B remained ambiguous. Recently, the author and coworkers proposed a relation [28].

$$B=\beta -{\beta }_{123}$$

(11)

β₁₂₃ is an additional interaction parameter involving the two surfactants (surfactant 1 and 2) and one solute (regarding the solute as the third component, surfactant 3, of the mixed micelle). The parameter can be determined by measuring the CMC of the ternary system (two surfactants and a solute), the method of which is presented in the reference [28]. The information gives the change in the interaction parameter specific to the ternary system. Still, the evaluation of parameter β₁₂₃ requires evaluating the CMC in a ternary system and determining partitioning constants, treating one of the components as the solute. Such an example is limited so far [28], and further investigation is required to verify and find a relation between β₁₂₃ and β.

3. Results

Figure 2 summarizes the solute molecules used for determining the partition coefficients of the binary mixed micelles. Most of the solutes have a chromophore such as an aromatic ring. This could be because UV–Vis measurement is employed to evaluate the partition coefficient. In the case of UV–Vis measurement, the solute is the chromophore used in the spectroscopy assays. Since the relationship between B and β depends on the polarity of the solutes, the values of the octanol–water partition coefficient (logP) are estimated using ChemDraw 21.0.0.28.

Figure 3, Figure 4, Figure 5 and Figure 6 show the chemical structures of surfactants used for studying the formation and partitioning of mixed micelles from the surfactants reported by researchers. The anionic surfactants include perfluoroalkyl surfactant LFOS, which shows an antagonistic interaction with surfactants bearing alkyl groups, resulting in a positive β value. For the Gemini surfactants [39] and bile salts [40], reviews on the interaction parameter β have recently been reported. Some compounds were excluded due to their undefined structure [41].

Table 1. The interaction parameters of mixed micelles composed of anionic/anionic surfactants.

		Measurement of β				Measurement of B
Sys. ID	Surfactants	β	T [°C]	Method	Ref.	Solute	B	T [°C]	Method	Ref.
AA01	LFOS/LDS	2.2	23	NMR	[19]	1-pentanol	–	25	GC	[20]
AA02	LFOS/LDS	2.2	23	ST	[46]	1-hexanol	–	–	GC	[46]
AA03	LFOS/LDS	2.2	23	ST	[46]	2,2,3,3,4,4,4-heptafluorobutanol	–	–	GC	[46]
AA04	SPFO/SDeS	1.8	23	NMR	[19]	1-pentanol	–	25	GC	[14]
AA05	SPFO/SDS	2.1	23	NMR	[19]	1-pentanol	–	25	GC	[14]
AA06	SD2MeAla/SDLAla	−0.090	25	ST	[27]	norphenylephrine	0.20	25	MEKC	[27]
AA07	SD3MeAla/SDLAla	−0.090	25	ST	[27]	homatropine	0.16	25	MEKC	[27]
AA08	SD4MeAla/SDLAla	−0.090	25	ST	[27]	bupivacaine	0.21	25	MEKC	[27]

,

Table 2. The interaction parameters of mixed micelles composed of cationic/cationic surfactants.

		Measurement of β				Measurement of B
Sys. ID	Surfactants	β	T [°C]	Method	Ref.	Solute	B	T [°C]	Method	Ref.
CC01	CG1/CPC	−0.8	25	ST	[47]	anthracene	12.25	25	UV–Vis	[47]
CC02	CG1/CPC	−0.8	25	ST	[47]	pyrene	16.28	25	UV–Vis	[47]
CC03	CG4/CG2	−0.08	25	ST	[24]	naphthalene	−4.71	25	UV–Vis	[24]
CC04	CG4/CG2	−0.08	25	ST	[24]	naphthalene	2.58	25	UV–Vis	[24]
CC05	CG4/CG2	−0.08	25	ST	[24]	naphthalene	−2.13	25	UV–Vis	[24]
CC06	CG4/CG3	−0.14	25	ST	[24]	phenanthrene	−5.92	25	UV–Vis	[24]
CC07	CG4/CG3	−0.14	25	ST	[24]	phenanthrene	0.25	25	UV–Vis	[24]
CC08	CG4/CG3	−0.14	25	ST	[24]	phenanthrene	−4.01	25	UV–Vis	[24]
CC09	CG4/CG5	−0.29	25	ST	[24]	pyrene	−3.67	25	UV–Vis	[24]
CC10	CG4/CG5	−0.29	25	ST	[24]	pyrene	−8.22	25	UV–Vis	[24]
CC11	CG4/CG5	−0.29	25	ST	[24]	pyrene	10.5	25	UV–Vis	[24]
CC12	CG4/CG5	−0.38	25	ST	[48]	naphthalene	–	25	UV–Vis	[48]
CC13	CG4/CG5	−0.38	25	ST	[48]	pyrene	–	25	UV–Vis	[48]
CC14	CG4/CPC	−0.32	25	ST	[47]	phenanthrene	−1.06	25	UV–Vis	[49]
CC15	CG4/DTAB	−0.22	25	ST	[47]	phenanthrene	−17.6	25	UV–Vis	[49]
CC16	CG6/CTAB	−8.91	30	ST	[50]	naphthalene	1.04	30	UV–Vis	[50]
CC17	CG6/CTAB	−8.91	30	ST	[50]	anthracene	0.45	30	UV–Vis	[50]
CC18	CG6/CTAB	−8.91	30	ST	[50]	pyrene	2.73	30	UV–Vis	[50]
CC19	CG7/CTAB	−7.2	30	ST	[50]	naphthalene	1.26	30	UV–Vis	[50]
CC20	CG7/CTAB	−7.2	30	ST	[50]	anthracene	1.92	30	UV–Vis	[50]
CC21	CG7/CTAB	−7.2	30	ST	[50]	pyrene	1.39	30	UV–Vis	[50]
CC22	CG1/CTAB	−6.76	30	ST	[50]	naphthalene	4.17	30	UV–Vis	[50]
CC23	CG1/CTAB	−6.76	30	ST	[50]	anthracene	9.73	30	UV–Vis	[50]
CC24	CG1/CTAB	−6.76	30	ST	[50]	pyrene	11.21	30	UV–Vis	[50]
CC25	CTAB/HBzC	−0.47	25	ST	[51]	naphthalene	−0.03	25	UV–Vis	[51]
CC26	CTAB/HBzC	−0.47	25	ST	[51]	anthracene	0.01	25	UV–Vis	[51]
CC27	CTAB/HBzC	−0.47	25	ST	[51]	pyrene	−0.1	25	UV–Vis	[51]
CC28	DTAB/DEB	−0.14	25	ST	[51]	naphthalene	0.04	25	UV–Vis	[51]
CC29	DTAB/DEB	−0.14	25	ST	[51]	anthracene	−0.25	25	UV–Vis	[51]
CC30	DTAB/DEB	−0.14	25	ST	[51]	pyrene	−0.2	25	UV–Vis	[51]
CC31	TTAC/TBzC	0.92	–	CD	[21]	butobarbital	−0.60	25	UV–Vis	[21]

,

Table 3. The interaction parameters of mixed micelles composed of nonionic/nonionic surfactants.

		Measurement of β				Measurement of B
Sys. ID	Surfactants	β	T [°C]	Method	Ref.	Solute	B	T [°C]	Method	Ref.
NN01	Brij30/Tween20	−0.59	25	ST	[52]	nifedipine	–	25	UV–Vis	[52]
NN02	Brij30/Tween20	−0.59	25	ST	[52]	carbamazepine	–	25	UV–Vis	[52]
NN03	Brij35/Tween20	−2.67	25	ST	[52]	nifedipine	–	25	UV–Vis	[52]
NN04	Brij35/Tween20	−2.67	25	ST	[52]	carbamazepine	–	25	UV–Vis	[52]
NN05	Brij56/Tween40	−1.36	25	ST	[52]	nifedipine	–	25	UV–Vis	[52]
NN06	Brij56/Tween40	−1.36	25	ST	[52]	carbamazepine	–	25	UV–Vis	[52]
NN07	Brij58/Brij35	−1.23	25	ST	[53]	naproxen	9.39	25	UV–Vis	[53]
NN08	Brij58/Tween40	−2.67	25	ST	[52]	nifedipine	–	25	UV–Vis	[52]
NN09	Brij58/Tween40	−2.67	25	ST	[52]	carbamazepine	–	25	UV–Vis	[52]

,

Table 4. The interaction parameters of mixed micelles composed of nonionic/anionic surfactants.

		Measurement of β				Measurement of B
Sys. ID	Surfactants	β	T [°C]	Method	Ref.	Solute	B	T [°C]	Method	Ref.
NA01	Brij30/SC	−8.77	25	ST	[52]	nifedipine	–	25	UV–Vis	[52]
NA02	Brij30/SC	−8.77	25	ST	[52]	carbamazepine	–	25	UV–Vis	[52]
NA03	Brij35/SC	−7.97	25	ST	[52]	nifedipine	–	25	UV–Vis	[52]
NA04	Brij35/SC	−7.97	25	ST	[52]	carbamazepine	–	25	UV–Vis	[52]
NA05	Brij35/SDS	−2.5	25	ST	[18]	1-pentanol	–	25	GC	[20]
NA06	Brij35/SDS	−2.5	25	ST	[18]	heptabarbital	−2.1	25	UV–Vis	[21]
NA07	Brij35/SDS	−2.5	25	ST	[18]	reposal	−2.3	–	UV–Vis	[15]
NA08	Brij35/SDS	−2.5	25	ST	[18]	pentobarbital	−1.4	–	UV–Vis	[15]
NA09	Brij35/SDS	−2.5	25	ST	[18]	butobarbital	−1.7	–	UV–Vis	[15]
NA10	Brij35/SDS	−2.5	25	ST	[18]	barbital	−1.1	–	UV–Vis	[15]
NA11	Brij35/SDS	−2.5	25	ST	[18]	reposal (0.2 M NaCl)	−1.7	–	UV–Vis	[15]
NA12	Brij35/SDS	−2.71	25	ST	[53]	naproxen	25.8	25	UV–Vis	[53]
NA13	Brij35/SDS	−2.71	25	ST	[53]	propranolol	–	25	MEUF	[54]
NA14	Brij35/SDS	−2.71	25	ST	[53]	lidocaine	–	25	MEUF	[54]
NA15	Brij35/SDS	−2.71	25	ST	[53]	ephedrine	–	25	MEUF	[54]
NA16	Brij35/SDS	−2.71	25	ST	[53]	syringic acid	–	25	MEUF	[54]
NA17	Brij35/SDS	−2.71	25	ST	[53]	acetone	–	25	MEUF	[54]
NA18	Brij35/SDS	−2.71	25	ST	[53]	toluene	–	25	MEUF	[54]
NA19	Brij35/SDS	−2.71	25	ST	[53]	phenol	–	25	MEUF	[54]
NA20	Brij35/SDS	−2.71	25	ST	[53]	triphenylamine	7.09	25	UV–Vis	[23]
NA21	Brij35/SDS	−2.71	25	ST	[53]	triphenylphosphine	30.1	25	UV–Vis	[23]
NA22	Brij35/SDS	−2.71	25	ST	[53]	triphenyl phosphine oxide	9.69	25	UV–Vis	[23]
NA23	Brij35/SDS	−2.6	25	ST	[55]	pyrene	−1.34	25	UV–Vis	[55]
NA24	Brij35/SDS	−2.71	25	ST	[53]	naphthalene	–	25	UV–Vis	[56]
NA25	Brij35/SDS	−2.71	25	ST	[53]	anthracene	–	25	UV–Vis	[56]
NA26	Brij35/SDS	−2.71	25	ST	[53]	phenanthrene	–	25	UV–Vis	[56]
NA27	Brij35/SDS	−2.71	25	ST	[53]	pyrene	–	25	UV–Vis	[56]
NA28	Brij35/SDS	−2.71	25	ST	[53]	linalool	–	25	UV–Vis	[56]
NA29	Brij35/SDS	−2.71	25	ST	[53]	benzyl acetate	–	25	UV–Vis	[56]
NA30	Brij35/SDS	−2.71	25	ST	[53]	α-ionone	–	25	UV–Vis	[56]
NA31	Brij35/SDS	−2.71	25	ST	[53]	α-hexylcinnamaldehyde	–	25	UV–Vis	[56]
NA32	Brij35/SLE1S	2.01	–	FL	[57]	pyrene	1.33	–	UV–Vis	[57]
NA33	Brij35/SLE2S	2.01	–	FL	[57]	pyrene	0.68	–	UV–Vis	[57]
NA34	Brij35/SLE3S	2.01	–	FL	[57]	pyrene	0.40	–	UV–Vis	[57]
NA35	Brij56/SC	−10.42	25	ST	[52]	nifedipine	–	25	UV–Vis	[52]
NA36	Brij56/SC	−10.42	25	ST	[52]	carbamazepine	–	25	UV–Vis	[52]
NA37	Brij58/SC	−9.11	25	ST	[52]	nifedipine	–	25	UV–Vis	[52]
NA38	Brij58/SC	−9.11	25	ST	[52]	carbamazepine	–	25	UV–Vis	[52]
NA39	Brij58/SDS	−5.07	25	ST	[53]	naproxen	20.6	25	UV–Vis	[53]
NA40	Brij58/SDS	−2.1	25	ST	[55,58]	pyrene	−28.3	25	UV–Vis	[55]
NA41	Brij58/SDS	−6.18	30	ST	[55,58]	limonene	−0.02	30	UV–Vis	[59]
NA42	Brij58/SDS	−6.18	30	ST	[55,58]	eugenol	−0.40	30	UV–Vis	[59]
NA43	Tween20/SC	−8.08	25	ST	[52]	nifedipine	–	25	UV–Vis	[52]
NA44	Tween20/SC	−8.08	25	ST	[52]	carbamazepine	–	25	UV–Vis	[52]
NA45	Tween40/SC	−8.9	25	ST	[52]	nifedipine	–	25	UV–Vis	[52]
NA46	Tween40/SC	−8.9	25	ST	[52]	carbamazepine	–	25	UV–Vis	[52]
NA47	Tween80/SDDC	−4.2	30	ST	[60]	naphthalene	–	30	UV–Vis	[60]
NA48	Tween80/SDDC	−4.2	30	ST	[60]	phenanthrene	–	30	UV–Vis	[60]
NA49	Tween80/SDDC	−4.2	30	ST	[60]	pyrene	–	30	UV–Vis	[60]
NA50	TX100/SDS	−1.5	25	ST	[55]	pyrene	−2.81	25	UV–Vis	[55]
NA51	TX100/SDS	−1.5	25	ST	[55]	naphthalene	–	25	UV–Vis	[56]
NA52	TX100/SDS	−1.5	25	ST	[55]	acenaphthylene	–	25	UV–Vis	[56]
NA53	TX100/SDS	−1.5	25	ST	[55]	anthracene	–	25	UV–Vis	[56]
NA54	TX100/SDS	−1.5	25	ST	[55]	phenanthrene	–	25	UV–Vis	[56]
NA55	TX100/SDS	−1.5	25	ST	[55]	pyrene	–	25	UV–Vis	[56]
NA56	TX100/SDS	−4.02	30	FL	[61]	2-[3-(N-methyl- N-phenylamino)- 2-propenylidene] indanone	–	30	UV–Vis	[61]
NA57	TX405/SDS	−3.5	25	ST	[55]	pyrene	1.54	25	UV–Vis	[55]

,

Table 5. The interaction parameters of mixed micelles composed of nonionic/cationic surfactants.

		Measurement of β				Measurement of B
Sys. ID	Surfactants	β	T [°C]	Method	Ref.	Solute	B	T [°C]	Method	Ref.
NC01	Brij30/DEB	−3.66	25	ST	[51]	naphthalene	3.15	25	UV–Vis	[51]
NC02	Brij30/DEB	−3.66	25	ST	[51]	anthracene	6.96	25	UV–Vis	[51]
NC03	Brij30/DEB	−3.66	25	ST	[51]	pyrene	5.03	25	UV–Vis	[51]
NC04	Brij30/DTAB	−3.53	25	ST	[51]	naphthalene	2.34	25	UV–Vis	[51]
NC05	Brij30/DTAB	−3.53	25	ST	[51]	anthracene	4.11	25	UV–Vis	[51]
NC06	Brij30/DTAB	−3.53	25	ST	[51]	pyrene	4.58	25	UV–Vis	[51]
NC07	Brij35/CG4	−0.38	25	ST	[47]	phenanthrene	1.8	25	UV–Vis	[49]
NC08	Brij35/CG4	−1.27	25	ST	[48]	naphthalene	–	25	UV–Vis	[48]
NC09	Brij35/CG4	−1.27	25	ST	[48]	pyrene	–	25	UV–Vis	[48]
NC10	Brij35/CG5	−1.42	25	ST	[48]	naphthalene	–	25	UV–Vis	[48]
NC11	Brij35/CG5	−1.42	25	ST	[48]	pyrene	–	25	UV–Vis	[48]
NC12	Brij35/CTAB	−1.03	25	ST	[53]	naproxen	18.9	25	UV–Vis	[53]
NC13	Brij35/CTAB	−1.03	25	ST	[53]	lidocaine	–	25	MLC	[54]
NC14	Brij35/CTAB	−1.03	25	ST	[53]	coumarin	–	25	MLC	[54]
NC15	Brij35/CTAB	−1.03	25	ST	[53]	isovanillin	–	25	MLC	[54]
NC16	Brij35/CTAB	−1.03	25	ST	[53]	phenol	–	25	MLC	[54]
NC17	Brij35/CTAB	−1.03	25	ST	[53]	syringic acid	–	25	MLC	[54]
NC18	Brij35/CTAB	−1.03	25	ST	[53]	ferulic acid	–	25	MLC	[54]
NC19	Brij35/CTAB	−1.03	25	ST	[53]	vanillic acid	–	25	MLC	[54]
NC20	Brij35/DTAB	−3.98	25	ST	[53]	naproxen	21.4	25	UV–Vis	[53]
NC21	Brij35/DTAB	−3.98	25	ST	[53]	triphenylamine	11	25	UV–Vis	[23]
NC22	Brij35/DTAB	−3.98	25	ST	[53]	triphenylphosphine	22.9	25	UV–Vis	[23]
NC23	Brij35/DTAB	−3.98	25	ST	[53]	triphenylphosphine oxide	1.87	25	UV–Vis	[23]
NC24	Brij35/TTAB	−3.8	25	FL	[22]	reposal	−0.3	–	UV–Vis	[15]
NC25	Brij35/TTAB	−3.8	25	FL	[22]	pentobarbital	−0.4	–	UV–Vis	[15]
NC26	Brij35/TTAB	−3.8	25	FL	[22]	butobarbital	−0.4	–	UV–Vis	[15]
NC27	Brij35/TTAB	−3.8	25	FL	[22]	barbital	−0.3	–	UV–Vis	[15]
NC28	Brij35/TTAB	−2.17	25	ST	[53]	naproxen	25.6	25	UV–Vis	[53]
NC29	Brij56/CG1	−2.55	25	ST	[47]	anthracene	3.33	25	UV–Vis	[47]
NC30	Brij56/CG1	−2.55	25	ST	[47]	pyrene	4.06	25	UV–Vis	[47]
NC31	Brij58/CG6	−3.24	30	ST	[50]	naphthalene	2.97	30	UV–Vis	[50]
NC32	Brij58/CG6	−3.24	30	ST	[50]	anthracene	2.65	30	UV–Vis	[50]
NC33	Brij58/CG6	−3.24	30	ST	[50]	pyrene	2.57	30	UV–Vis	[50]
NC34	Brij58/CG7	−4.16	30	ST	[50]	naphthalene	1.05	30	UV–Vis	[50]
NC35	Brij58/CG7	−4.16	30	ST	[50]	anthracene	1.51	30	UV–Vis	[50]
NC36	Brij58/CG7	−4.16	30	ST	[50]	pyrene	0.2	30	UV–Vis	[50]
NC37	Brij58/CG1	−0.75	30	ST	[50]	naphthalene	1.9	30	UV–Vis	[50]
NC38	Brij58/CG1	−0.75	30	ST	[50]	anthracene	5.69	30	UV–Vis	[50]
NC39	Brij58/CG1	−0.75	30	ST	[50]	pyrene	5.31	30	UV–Vis	[50]
NC40	Brij58/CTAB	−2.99	25	ST	[53]	naproxen	22	25	UV–Vis	[53]
NC41	Brij58/CTAB	−3.05	25	ST	[51]	naphthalene	2.86	25	UV–Vis	[51]
NC42	Brij58/CTAB	−3.05	25	ST	[51]	anthracene	4.56	25	UV–Vis	[51]
NC43	Brij58/CTAB	−3.05	25	ST	[51]	pyrene	5.96	25	UV–Vis	[51]
NC44	Brij58/CTAB	−2.99	25	ST	[53]	triphenylamine	6.44	25	UV–Vis	[23]
NC45	Brij58/CTAB	−2.99	25	ST	[53]	triphenylphosphine	23.6	25	UV–Vis	[23]
NC46	Brij58/CTAB	−2.99	25	ST	[53]	triphenylphosphine oxide	9.62	25	UV–Vis	[23]
NC47	Brij58/DTAB	−5.25	25	ST	[53]	naproxen	34.3	25	UV–Vis	[53]
NC48	Brij58/HBzC	−2.98	25	ST	[51]	naphthalene	2.46	25	UV–Vis	[51]
NC49	Brij58/HBzC	−2.98	25	ST	[51]	anthracene	4.57	25	UV–Vis	[51]
NC50	Brij58/HBzC	−2.98	25	ST	[51]	pyrene	5.26	25	UV–Vis	[51]
NC51	Brij58/TTAB	−4.61	25	ST	[53]	naproxen	22.4	25	UV–Vis	[53]
NC52	TX100/CG4	−0.94	25	ST	[47]	phenanthrene	0.76	25	UV–Vis	[49]

and

Table 6. The interaction parameters of mixed micelles composed of anionic/cationic surfactants.

		Measurement of β				Measurement of B
Sys. ID	Surfactants	β	T [°C]	Method	Ref.	Solute	B	T [°C]	Method	Ref.
AC01	AOT/CG1	−3.33	25	ST	[47]	anthracene	6.25	25	UV–Vis	[47]
AC02	AOT/CG1	−3.33	25	ST	[47]	pyrene	7.69	25	UV–Vis	[47]
AC03	SC/DMIMB	−9.03	25	FL	[62]	phenothiazine	–	–	UV–Vis	[62]
AC04	SDC/DCP	−3.69	25	ST	[63]	ethenzamide	–	37	HPLC	[63]
AC05	SDC/DMIMB	−9.95	25	FL	[62]	phenothiazine	–	–	UV–Vis	[62]
AC06	SDeS/DeTAB	−13.2	23	ST	[6]	butobarbital	−4.5	25	UV–Vis	[21]
AC07	SDeS/DeTAB	−13.2	23	ST	[6]	1-pentanol	–	25	GC	[20]
AC08	SDS/DTAC	−8.09	25	CD	[64]	heptabarbital	−8.8	25	UV–Vis	[21]
AC09	TADPS/CPC	−18.3	25	ST	[65]	toluene	–	25	UV–Vis	[66]
AC10	TADPS/CTAB	−15.7	25	ST	[65]	1-octanol	−0.61	25	UV–Vis	[66]
AC11	TADPS/CTAB	−15.7	25	ST	[65]	n-heptane	–	25	UV–Vis	[66]
AC12	TADPS/CTAB	−15.7	25	ST	[65]	toluene	–	25	UV–Vis	[66]

are the previously reported mixed micelle systems. Some studies of partitioning solutes using binary micelles did not provide information on the interaction parameter B. Such cases may require a further analysis based on their data or reproducing the experiment may be beneficial to note. In the literature, various methods were employed to analyze the micellar formation and partitioning of the solutes: surface tension (ST); fluorescence (FL); nuclear magnetic resonance (NMR); conductance (CD); gas chromatography (GC); UV–Vis; micellar enhanced ultrafiltration (MEUF); micellar liquid chromatography (MLC); high-performance liquid chromatography (HPLC); and micellar electrokinetic chromatography (MEKC). The most popular method to determine is ST, while the most popular method to determine B is UV–Vis measurement. MEKC was recently introduced and related to the regular solution theory to analyze the mixed micelle [27]. The method can determine not only the partition coefficients [42,43] but also the kinetics of the solute permeation of the micelles [44,45]. Various methods were used to analyze the interaction parameters, but comparative studies of the different methods to determine the interaction parameter for the partitioning of solutes using the same surfactants and solute combination have not been reported to the best of the author’s knowledge. The systems of the binary micelles are categorized with the types of composing surfactants: anionic/anionic (AA); cationic/cationic (CC); nonionic/nonionic (NN); nonionic/anionic (NA); nonionic/cationic (NC); and anionic/cationic (AC) surfactants. In general, the interaction between the two surfactants is mainly due to electrostatic forces, and the strength of the attractive electrostatic interaction parameter β decreases in the order of AN > NA ≈ NC > AA ≈ CC ≈ NN surfactants [11,27].

4. Discussion

The empirical equation that relates the interaction parameters β and B is reported by Treiner (Equation (10)). However, Treiner and coworkers claim that it is necessary to distinguish the nonpolar solutes from the polar solutes to apply the equation [21]. The nonpolar solutes tend to show positive B values even if β is negative, which often contradicts the prediction by Equation (10). They explain the discrepancy by the location of the interacting nonpolar solutes in the mixed micelle. In the mixed micelle, the nonpolar solute should be in the hydrocarbon core, and the enthalpy of the mixing should be close to zero following ideal mixing β ~ 0, which gives a positive value for Equation (10). The other reason for the different behavior of nonpolar solvents from polar solvents is the change in the aggregation number. Some nonpolar solutes increase the aggregation number and induce a structural change from spherical to cylindrical micelles. The analysis is sensitive to the change, resulting in an increase in micellar solubilization, because the Laplace pressure is smaller for a cylinder than a sphere.

In this work, the values of logP are used just for the qualitative evaluation of the polarity of the solutes. More accurate and/or sophisticated parameters, such as experimentally determined logP and solubility parameters, could help analyze the relationship between β and B, which may be performed in the future work. The purpose of introducing the logP in this work is to distinguish polar and nonpolar solutes. Treiner and coworkers report that the borderline that may not follow Equation (10) is benzene (logP = 2.03) and 1-octanol (logP = 2.64). To safely apply Equation (10), logP = 2.03 was chosen as the border at the start. However, it was found that the logP value for anthracene was lower than benzene. Most of the B for anthracene is positive while β is negative (CC01, CC17, CC20, CC23, CC26, NC02, C05, C29, C32, C35, C38, C42, C49, and AC01), although there is an example with negative β and negative B (CC29). Therefore, the anthracene was regarded as at the borderline; thus, the borderline between polar and nonpolar was determined by the logP of anthracene (1.96). If logP ≥ 1.96, the solute is regarded as nonpolar, while if 1.96 > logP, the solute is regarded as polar. Many reports are complete for nonpolar solutes, and the determination of B for polar solutes is limited. Although the work and linear relation between β and B reported by Treiner and coworkers [16,21] is widely recognized [7,10,15,24,27,28,47,49,57,59,61,62,67], their model solutes 1-pentanol and barbital and their derivatives have seldom been reported by other researchers. The few reports on the polar solutes could be because the most popular measurement method is the UV–Vis measurement, and 1-pentanol without chromophore cannot be used. The barbital derivatives are the other solutes, but their structures are uncommon. The remaining polar systems were reported by Treiner and coworkers (CC31, NC24–27, NA06–11, AC06, and AC08) [15,21] and the author (AA06 and AA08) [27]. Figure 6 illustrates the distribution of β and B for nonpolar (red) and polar (blue) solutes. For the polar solutes, the β vs. B relation shows a linear relation, as pointed out by Treiner. The two examples AA06 and AA08 [27] were added to Treiner’s systems. Although the methods of determining the parameter B are GC, UV–Vis, and MEKC, the linearity is maintained, confirming the equation’s generality. However, it should be said that the amount of the data is insufficient, and the validity of Equation (10) remains questioned. Since the solute is polar, it is expected that there is no strong interaction between the surfactant and solute. However, even for the polar solute, β = B does not hold, which could be because the effect of the polar solute is not negligible. Yet, the empirical relation Equation (10) implies that there is some relationship behind the two interaction parameters. More data are accumulated for the nonpolar system. The authors reported many nonpolar systems with various kinds of solutes, but most of them show a negative β and positive B, as pointed out by Treiner and coworkers.

Further study will be necessary to enrich the understanding of formation and partitioning in mixed micelle systems. The author proposes directions in this field to acquire an in-depth understanding of the formation and partitioning of the mixed micelle. First, for nonpolar solutes, more examples are necessary to test Equation (10). The reports on the formation and partitioning of polar solutes using mixed micelles are rare, and the validity of the equation remains questioned. A reanalysis of reported systems with polar solutes may increase the amount of data. However, the effort will not give information for nonpolar solutes, and the applicability of the equation remains limited. Therefore, uncovering parameters governing the relationship between β and B should be clarified. Second, the accumulation of analyzed examples of surfactant/surfactant/solute systems is necessary to experimentally verify Equation (11) reported by the author [28] based on “the quasi-simple ternary mixture” model. The equation gives a clear relation between β and B. Still, the determination of β₁₂₃ requires the analysis of the ternary (two surfactants and a solute) system regarding the solute as the third component of the micellar phase. The example for the LDS (surfactant)/LFOS (surfactant)/n-hexanol (solute) ternary system was reported [28]. The resulting values of β and β₁₂₃ were 1.7 and −0.11, and B was calculated as 1.8, giving a slightly better fit for the prediction of the partition coefficient, which is encouraging. Regarding the “solute” as a third surfactant is uncommon, but such treatment may give helpful information to uncover the relation between β and B. Third, finding a new approach to relate β and B other than Equations (10) and (11) could be possible, taking similar models developed for cosolvent solubility by Joubyan [29,30].

5. Conclusions

In this work, information on the formation and partitioning of solutes using various mixed micelle systems is tabulated. Researchers report different cationic, anionic, and nonionic surfactants, and their combinations. The logP of anthracene was used to distinguish polar and nonpolar solutes. It was found that the information of parameter B is limited, and the Equation (10) proposed by Treiner cannot be verified. Three research directions for the formation and partitioning in mixed micelle systems are offered to acquire an in-depth understanding of the phenomenon. First, obtaining more data on the partitioning of polar solutes will be necessary. Second, an analysis of the ternary (two surfactants and a solute) system regarding the solute as the third component of the micellar phase will increase the information of β₁₂₃, which will give the information for surfactant–solute interaction. Third, finding a new theoretical model to relate β and B other than Equations (10) and (11) is important, where the cosolvent solubility by Joubyan could provide a hint.