Morphology of Ionic Micelles as Studied by Numerical Solution of the Poisson Equation.

The existing concepts of the ionic micelle structure were specified. It was noted that the composition of dispersed phase particles in a liquid dispersion medium should necessarily include adsorbed counterions rigidly bound to these particles. By numerical solution of the Poisson equation for the two most often used approximations, the Poisson-Boltzmann (PB) model and the Jellium-approximation (JA), the electric potential decay from the Stern potential of dispersed phase particles was defined. A new methodological approach to analyze the reaction of micelle potential decay based on small variability of the CMC value was proposed. It made possible to determine the dimension parameter, which in the presence of weak thermal effects approximately corresponds to the micelle hydrodynamic radius, and to calculate the electrokinetic potential of micelles. The results of theoretical calculations were compared with our previous experimental data on the thickness of the SDS micelle hydrophilic layer obtained by SAXS. A good agreement between the calculated and measured values was obtained, and it was noted that for low concentrations the experimental values are more correctly described by the PB model, but for concentrations greater than 100 mM the JA model is more preferable. It was found that the slipping plane is located near the outer Stern plane and is separated from it only by a few molecular layers of water. The influence stronger than the thermal one can shift the slipping plane closer to the micelle core. Accordingly, the smallest hydrodynamic micelle size is determined by the outer Stern plane. The results of our work allowed us to conclude that the micelle is not something soft and watery, but according to its specified structure, it is a more solid-like particle than was previously assumed. The proposed approach can be extended to investigate other effects of a physicochemical nature, in particular, those observed with the addition of an external electrolyte or nanoparticles.

Introduction

Surfactants in molecular form and organized assemblies are of great fundamental, technological, and commercial interest stimulating further insight into their physical–chemical properties and enormous prospective applications, including pharmacology, agrochemistry, household chemistry, cosmetics, biotechnology, and nanomedicine.[1−5] In addition, surfactants are of considerable importance in the oil industry, for example for enhanced oil recovery, oil transportation and processing.[6−9] Many novel technologies are difficult without nanoparticle stabilization in water or oil. In particular, surfactants are needed for obtaining carbon nanotube dispersions.[10−13] The development of surfactant-based technologies and applications demands reliable information about the physicochemical characteristics of surfactants in various structural states.

The present study is performed in view of the critical analysis of the existing picture on micelle morphology and size in water solution of ionic surfactants. The motivation of this work is based on the variety of structural definitions[14,15] and the contradictory experimental data on micelle structure (morphology) known from the literature.[16−32] Unfortunately, the published measured and calculated values for the micelle basic sizes, for example for the most representative ionic surfactant sodium dodecyl sulfate (SDS), differ among themselves significantly. So, for the radius of the SDS micelle hydrocarbon core, the values vary from 1.50 to 1.93 nm.[16−24] The size of the micelle central part (named aggregate of surfactant ions, micellar particles, solid phase particles) was estimated as 2–2.5 nm,[25−31] and its hydrodynamic radii were evaluated as 3–3.2 nm (at 10 mM)[29] and as 3.5 nm (at 20 mM).[32] With all this going on, the micelle size determination has been often undertaken at various concentrations and using different methods that are sensitive to different physical peculiarities of micelles. Also, the discrepancies in micelle size are due to subjective reasons because the used models reflect ideas of different authors about the structure of a micelle and its parts. The situation gets even more complicated in the case of ionic surfactants when the counterions play an additional role in micelle construction and determine its real physical size.[33−36]

In an aqueous medium at low concentrations the ionic surfactants, which generally belong to the 1–1 electrolyte family, dissociate completely, and exist as a solution of surfactant ions and their counterions. With the increase of the surfactant content to the critical micelle concentration (CMC) the micellar transition is observed, when surfactant molecules begin to self-assemble and form micelles. The CMC value is usually determined by the abrupt change in different physical–chemical properties of surfactant solution.[37] The charged spherical micellar surface adsorbs (condensates) a portion of counterions, the value of which is determined considering many factors, the main being the molecular geometry of surfactant molecules and solution conditions (surfactant concentration, temperature, pH, and the presence of background electrolyte).[33,38−41]

In various experiments, micelles manifest themselves as charged particles of the dispersed phase, each composed of a core, formed by surfactant ions with a part of counterions rigidly bound to its surface.[34] Every micelle is surrounded by a diffuse layer containing the remaining counterions. The part of the diffuse layer, limited by the slipping plane (also named shear surface), moves with the particle during its thermal diffusive motion. This slipping plane determines the micelle hydrodynamic radius rζ, which is often considered as the micelle size. The electric potential of the slipping plane, called the electrokinetic or ζ-potential of micelle, is an experimentally determined quantity. The goal of the present work is the theoretical and computational study of interconnection between the micelle spherical morphology and its hydrodynamic parameters (hydrodynamic radius and electrokinetic potential).

The development of ideas about the micelle structure and itemization of naming for separate inner parts of micelles (hydrocarbon core, aggregate, micelle core, micellar particles, micelle), as well as the tendency to call the dispersed phase as a micelle, led to confusion in many concepts, names, and scales. Since there is still no complete uniformity even in determining the radius of a micellar particle, we tried to specify the existing concepts on micelle morphology and structure. One of the main aims of this work was to assess the relationship between morphological and hydrodynamic parameters of the micelle. Since there are no equations connecting these parameters, such estimation can be performed only numerically. The obtained results made it possible to estimate the approximate number of molecular layers of bound water (about 5), which separate a solid micellar particle from its hydrodynamic surface. That is, in fact, a micelle is not something soft and watery, as previously thought, but according to its properties it is a much more solid-like particle.

Theoretical Basis

Classic Conception of the Micelle Structure

In this section, we briefly consider the details of micelle morphology. The classic conception of ionic micelles suggests that at CMC the Nagg surfactant molecules (usually 50–100) cluster into roughly spherical aggregates having the radius Rag. The inner part of the aggregate, micelle hydrocarbon core, with radius Rhc is formed by hydrophobic hydrocarbon chains of surfactant ions (Figure ). The head groups of surfactants form a charged micelle shell facing an aqueous medium and containing a certain number of water molecules.[39−41] The charge of aggregate qag = ±eNagg determines a significant potential Ψ(Rag) of its surface. A part of counterions is adsorbed on the outer surface of this shell, forming the Helmholtz adsorption layer. The adsorption of counterions leads to partial compensation of micelle aggregate charge ±eNagg up to the value of micellar particle charge qmic = ±αeNagg and to a significant lowering of the electric potential of the Helmholtz layer to the value Ψ(Rmic). Here, α is the degree of micelle ionization and e is the elementary electric charge. The Helmholtz counterion monolayer is separated from the next diffuse layer by a hypothetical boundary known as the Stern plane.[40,42] The shell, confined by the Stern plane, contains surfactant head groups, adsorbed counterions, and water molecules as well. The Helmholtz layer is often called the Helmholtz–Stern layer, or sometimes the Stern layer. The aggregate of surfactant ions with the adsorbed counterions forms the micelle particle of radius Rmic (Figure ), which represents a solid-like micelle core.[43] The electric charge ±αeNagg, distributed on the surface of the micellar particle, defines its Stern electric potential Ψ(Rmic). In fact, the size of Rmic is determined by the dimension of surfactant ions, presented at every instant in micelle in the form of contact ion pairs with adsorbed counterions.

Figure 1

Main characteristic radial dimensions of a micelle: Rhc is the radius of the micelle hydrocarbon core, Rag is the radius of the surfactant ion aggregate, Rmic is the radius of the micellar particle (dispersed phase particle or micelle core), rζ is the hydrodynamic radius of the micelle. Dehydrated counterions are indicated by white circles; hydrated counterions are shown by white circles with a blue rim.

It should be emphasized that the composition of the solid-like particle of the dispersed phase in the liquid dispersion medium should include the adsorbed counterions rigidly bound to this particle (adsorbed counterions are also a part of the dispersed phase). Therefore, the notions of a micelle particle and a micelle core should be identical. The remaining counterions are distributed in a bulk liquid phase, comprising the so-called diffuse part of the double layer or ionic micelle atmosphere. On the whole, the ionic micelles are electrically neutral. Below we used exactly the same concept for our mathematical calculations of the electric potential decay around the micellar particle (micelle core).

In addition, the modified models of micelle morphology are known, for instance, the Stern–Graham model,[44] associated with the separation of the adsorption layer into two parts: (1) the Helmholtz ionic monolayer with rigidly adsorbed dehydrated counterions is the closest to the micelle core and (2) the successive layer with hydrated counterions occupying several interatomic distances. In this case, the layer with hydrated counterions is called the Stern layer, and its boundaries are called the inner and outer Stern planes. The thickness of the Stern layer (outer Stern plane) is determined by the size of the solvent-separated ion pair (linking the surfactant ion and the counterion).[34,45] The shell of the micelle hydrocarbon core, including the Helmholtz–Stern layer, represents the concentrated mixture of hydrocarbons, electrolyte, and water having unique structural and chemical properties, which promotes various chemical processes.[46−48] Immediately after the Helmholtz–Stern layer the diffuse part of the ionic double layer is located containing all of the remaining counterions. This ionic atmosphere of the micellar particle is formed due to the thermal motion of counterions in an attractive electrostatic field of a charged micelle surface, which tends to equalize counterion concentration within the solution bulk.[39,49] The first few water layers, located immediately after the Helmholtz layer, are bound rather rigidly to the micelle core having increased local viscosity. Because of the hindered orientation mobility of bound water molecules its local permittivity is sufficiently small.[50]

Under the influence of some physical forces (e.g., electric, magnetic, acoustic in nature) or under the ordinary thermal Brownian motion the micelle double ionic layer gets disturbed. The interface, which separates bound and bulk water, is called the slipping (shear) plane. Its radius determines another micellar characteristic dimension, called the micelle hydrodynamic radius rζ, which is often considered the micelle size (Figure ). The electric potential at the slipping plane is called the electrokinetic or ζ-potential.

The location of the slipping plane depends on magnitude of the applied external disturbance. Therefore, the micelle slipping plane can slightly alter its position in different experiments even at equal surfactant concentrations and temperatures, or we can assume that micelles have a slipping layer of a certain thickness instead of the slipping plane. This slipping layer should be located outside the Helmholtz–Stern layer right after the layer of bound water with high ordering and increased viscosity. It may be suggested that in the presence of strong intermicelle interactions in concentrated systems, the slipping planes may drift closely to the outer Stern surface. The weaker action will hold the slipping plane away from the micelle center. The lower limit of ζ-potential corresponds to the weakest influence on the position of the slipping plane, corresponding only by the Brownian thermal motion. The hydrodynamic radius of micelles rζ has the maximum value in the presence of only thermal impact. In the absence of external disturbances, the ambiguity in determining the hydrodynamic radius and ζ-potential of micelles disappears.

The value of ζ-potential is determined by electric charges bounded by the slipping plane and therefore indicates the degree of electrostatic repulsion of micelles. It is known[51] that a ζ-potential greater than 30 mV (positive or negative) indicates the stability of colloid dispersions. Any additional influence (for example, by ultrasound) leads to the displacement of the slipping plane closer to the micelle core, resulting in a decrease of the hydrodynamic radius rζ, an increase of the ζ-potential, and in the advancement of dispersion stability as a whole.

Problem Formulation

A spherical micelle, as a structural arrangement is characterized by sizes and potentials associated with them: radius of the micellar particle Rmic = a (for simplicity, in the following we will use designation a for the radius of a micellar particle or micelle core) and its surface potential Ψ(Rmic) = Ψ(a), as well as the hydrodynamic radius of micelle rζ and its electrokinetic or ζ-potential.[39,40] Since there is no theoretical relation between these parameters, we have tried to eliminate, in the present work, the existing inconsistencies. In our calculation model, we are proceeding from the fact that the slipping (shear) surface divides the solution around a micelle into two parts, differing in viscosity. The potential decay curves calculated by us numerically indicate the existence of two regions associated with an uneven decrease in the bulk density of counterions with an increase in distance from the micellar particle when very fast potential decay turns into a very slow one. So, the enlarged counterion concentration near micelle can be associated with the increased solution viscosity, and the breakpoint of decay can be correlated with the hydrodynamic radius of micelles. However, it is impossible to specify this distance only by the shape of numerically calculated decays. Therefore, we have developed a new approach that allowed us to determine this boundary distance.

The novelty of our study is the proposed methodological approach to analyze the reaction of micelle potential decay on small variability of the CMC value. Since different experimental techniques demonstrate the abrupt change in various physical–chemical properties of surfactant solution upon micellization[37] in some concentration range near the average CMC value, we assumed that the proposed approach would take into account small fluctuations of the micelle microenvironment, inevitable upon micellar transition. As a result, we determined that at certain distances from the micelle core there is the region with redistribution of counterion density (concentration) with a well-marked maximum. We tried to relate the determined position of this maximum rmax to the micelle hydrodynamic radius rζ as well as to the Debye length. The validity of the developed approach was successfully confirmed by the obtained agreement between calculated positions of maxima and our early experimental SAXS results on the thickness of the hydrophilic layer of SDS micelles.[32]

In the theoretical computations of parameters for the diffuse part of the micelle double ionic layer the finite size of ions is ignored. The self-consistent field method is used in our calculations. The independent motion of ions in this field is allowed. In the framework of the applied method the electrostatic potential Ψ(), created near the spherical micellar core, and the electric field strength E() = –dΨ()/dr at the point with radius vector from the center of the micelle are associated with the spatial distribution of charge density around the micelle by means of the Poisson equation[40]where ρ is an average charge density at the point of the diffuse layer, where potential is calculated, ε is the dielectric permittivity of the continuous phase (water), and ε0 is the electric constant (SI system is used). The Poisson equation is supplemented with the boundary conditions, according to which the potential changes from the value Ψ(a) on the surface of the micelle core to zero at infinity. The main problem in solving eq is the determination of the spatial charge density distribution ρ and its dependence on potential Ψ.

The results can be generalized over the case of a 1–1 electrolyte using the approach described in the paper.[52] When solving the Poisson equation, one usually operates with the dimensionless potential Φ = ±e·Ψ(r)/(k·T) = Ψ(r)/Ψ0, where e is the elementary charge and k is the Boltzmann constant. For temperature T = 298 K, the combination of constants is equal (k·T)/e = Ψ0 = 25.7 mV. The choice of the sign on the right-hand side of this relation is defined by the condition that the dimensionless potential Φ must be essentially positive. Since the surface potential of micellar core Ψ(a) is usually equal to dozens of millivolts the value of the dimensionless potential Φ(a) = Φ0 corresponds to several units. It means that the ratio Φ0 ≪ 1 used for certain approximations[40] does not satisfy the case of surfactant micelles. In particular, in the case of sodium dodecyl sulfate (SDS) the known values on the micellar core surface potential are very inconsistent (ranging from −60 to −140 mV).[20,27,30,53] It is known that for Ψ(a) = −82 mV the calculations give Φ0 = 3.19.[53]

There are various approaches to simplify the Poisson equation, associated with the description of ion distribution in solution using various models.[54−57] Usually to describe the disordered concentrated solutions, the concept of ionic strength I is introducedwhich highlights not only the ions concentration in solution but also the magnitude of the ion–ion electrostatic (Coulomb) interactions. Here, C is the concentration of the type i ions in solution and Z is their charge number. In the premicellar state, the ionic strength is equal to the surfactant concentration, i.e., I = C. For the case of the micellar solution, the concept of ionic strength becomes ambiguous, depending on the choice of the model describing charge distribution in solution. Two of the most often used models are the Poisson–Boltzmann (PB) and the Jellium-Approximation (JA) ones. The PB model assumes that a spatial distribution of all particles, including micelles, is described by the Boltzmann distribution. In the PB model, micelles are identified with multiple charged ions with a strong Coulomb repulsion.[58] Such a picture of micelle solutions leads to the interrelation

In the JA model, it is assumed that the uniformly distributed micelles, located in solution near the equilibrium positions, are surrounded by the ionic atmosphere, which obeys the Boltzmann distribution. For these models, the simplified form of the Poisson equation (but still requiring numerical solution) is given below in the next section. In the JA model, micelles are the isolated objects with zero total charge due to the ordered arrangement of counterions around micelles.[20,59] In this case, the ionic strength of surfactant micellar solution I is determined according to another relation

The distribution of micelles over the bulk is determined not by their Coulomb repulsion but according to the statistical nature of solution disorganization, which is responsible for the thermal motion of particles. In the paper,[60] four methods of the ionic strength calculation were considered. It was noted that SDS micelles make significant contributions to solution conductivity but not to the effective ionic strength. Also, it was shown that for the ionic strength calculation it is more correct to use relation (.

Dimensionless Micelle Potential in Linear Approximation

It is known that for the case of a small electrostatic potential of particle surface, i.e., for Φ0 ≪ 1, the Poisson equation for both models in linear approximation reduces to the form[40]

This equation can be solved not only numerically but also analytically. For this case the Debye length λ = (ε·ε0kT/2e2I)1/2, determined by the ionic strength I, is introduced to describe the properties of the system. The numerical value of parameter λ depends on the concentration of ions in solution, namely, on the ionic strength of solution. In the case of flat charged surface, the parameter λ has a simple physical meaning specifying the distance at which the potential decreases e = 2.72 times. Despite the fact that for spherical particles this interpretation ceases to be valid, the Debye length λ, which is related to the potential decay rate is compared with the thickness of the ionic atmosphere around a charged spherical particle. This approach is not very correct for micelles since the approximation Φ0≪1 is not satisfied here. However, the simplicity of calculating the parameter λ and the possibility to write down the analytical expression for potential decay allow us to use eq to compare the obtained micelle parameters for two models. A comparison of Debye lengths performed for two models in linear approximation showed that λPB < λJA, since at equal surfactant concentrations the ionic strength value in the PB model is much larger compared with the JA one. This means that in the PB model potential decreases faster and the thickness of ionic atmosphere around micelle is less than that for the JA model. The solution of eq for r > a represents the following function[61]For linear approximation the shape of potential decay Φ(r/a) as the function of relative distance for r > a has a shape similar to the curves shown in Figure (obtained for nonlinear approximation). They are characterized by a very sharp decrease in the range of r values close to a and by a much slower decay at a large distance. The range, where noticeable changes in potential are detected, is several (3–4) times greater than λ. Therefore, in many studies, the parameter λ is compared not with the actual thickness of the ionic atmosphere but with a distance to the micellar slipping plane. However, this approach also has certain drawbacks. In particular, the calculations show that at this distance the numerical values of Φ are too small to correspond to ζ-potential because of the exponential potential decay. For real micelles, when the condition Φ ≪ 1 is not satisfied, the shape of potential decay is changed, the introduction of the constant λ becomes incorrect, and relation ( for calculation of the ionic atmosphere thickness λ can no longer be applied. Thus, for the case of highly charged particles the equations exist, which allows numerical calculation of the dimensionless potential decay at various surfactant concentrations. However, there is no criterion, which allows correct estimation of the ionic atmosphere thickness and its alteration in the presence of impacts of any physical or chemical nature.

Figure 2

Decays of dimensionless potential of SDS micelles calculated within the PB (left) and JA (right) models for nonlinear approximation as a function of r/a for eight surfactant concentrations indicated in order, increasing from right to left.

Dimensionless Micelle Potential in Nonlinear Approximation

For numerical calculations within the PB and JA models the transformed Poisson equation[57] was used. For the case of the 1–1 valence ionic surfactant in the presence of the external 1–1 electrolyte with concentration CE the equations for both PB and JA models can be written in the common form In the PB model, there is an additional term APB = 1 – exp[−ZΦ], where Z = αNagg is the charge number of the micelle core. In the JA model, this term is absent. For both models, the equations require a numerical solution.

It should be noted that all calculations were carried out within the selected models, which require the experimentally defined data on the size of micelle core a, the potential of its surface Ψ(a), the number of surfactant aggregation in micelle Nagg, the degree of micelle ionization α, and the critical micelle concentration CCMC. The following parameters for SDS micelles were used in our calculations: Nagg = 64, a = 2.3 nm, CCMC = 8 mM, and Ψ(a) = −82 mV. We have taken the value α = 0.25, although the known values at T = 298 K lie in the interval 0.25–0.29.[41,60,62,63] The choice of SDS was specified by its prevalence in scientific studies and our experience to work with SDS.[12,13,32,33]

To numerically solve the resulting equations the bvpsuite1.1 software package was used.[64] The results of calculations coincide with the results of the paper[65] and with those in the paper.[66] This software package was used to calculate the dependence of dimensionless micellar potential on the relative distance from the micelle core surface and to compare the obtained results for nonlinear approximation within the PB and the JA models. For both models in the absence of an external electrolyte (hereinafter CE = 0), the decrease in dimensionless potential of SDS micelle was calculated for eight surfactant concentrations (16, 24, 32, 64, 128, 256, 512, and 1024 mM) at T = 298 K as a function of relative distance r/a. The obtained results, shown in Figure , Tables and 2 depict the main parameters for both PB and JA models, calculated in the framework of both linear and nonlinear approximations.

Table 1

Ionic Strength I, Relative Debye Length Λ, and Value rmax/a at Different Surfactant Concentrations C for the PB Modela

				linear approximation		nonlinear approximation
C (mM)	D (nm)	D/2a	IPB	ΛPB	rmax/a	rmax/a	ΛPB
16	23.7	5.15	25	1.83	1.54	1.56	1.85
24	18.8	4.09	42	1.64	1.44	1.50	1.75
32	16.4	3.57	59	1.54	1.39	1.47	1.69
64	12.4	2.69	127	1.37	1.29	1.40	1.56
128	9.61	2.09	263	1.26	1.21	1.33	1.44
256	7.55	1.64	535	1.18	1.16	1.27	1.34
512	5.96	1.30	1079	1.13	1.12	1.22	1.27
1024	4.72	1.03	2167	1.09	1.08	1.18	1.21

D and D/2a are explained in the text.

Table 2

Ionic Strength I, Relative Debye Length Λ, and Value rmax/a at Different Surfactant Concentrations C for the JA Modela

				linear approximation		nonlinear approximation
C (mM)	D (nm)	D/2a	IJA	ΛJA	rmax/a	rmax/a	ΛJA
16	23.7	5.15	9	2.39	1.78	1.69	2.17
24	18.8	4.09	10	2.32	1.75	1.67	2.12
32	16.4	3.57	11	2.26	1.72	1.64	2.05
64	12.4	2.69	15	2.08	1.65	1.56	1.87
128	9.61	2.09	23	1.87	1.56	1.47	1.69
256	7.55	1.64	39	1.67	1.46	1.37	1.51
512	5.96	1.30	71	1.50	1.37	1.29	1.37
1024	4.72	1.03	135	1.36	1.28	1.20	1.24

D and D/2a are explained in the text.

We note that the shape of potential decays for linear and nonlinear approximations differ little for both models. The differences are much more essential between the models. First of all, it is associated with different values of ionic strength at equal surfactant concentrations (forth column of Tables and 2). Values of the relative Debye length Λ = (a + λ)/a, calculated in linear approximation for different surfactant concentrations in both models, are given for subsequent comparison.

The tables also show the distance between centers of micelles D at various concentrations of SDS, calculated by formulaand the radius of spheres per micelle, D/2a, in relative units (NA is Avogadro’s number).

For nonlinear approximation, a slower decrease in potential curves is observed for the JA model. A comparison of numerical data, obtained for dimensionless potential, indicates the right shift of the central part of curves for the PB model and the left shift for the JA model. They correspond to the increase in the effective thickness of the ionic atmosphere for the PB model and its decrease for the JA model for highly charged spherical particles indicating a convergence of results obtained within different models for nonlinear approximation. We also note the nonzero values of potential at the boundaries between micelles, starting from the concentration of C = 64 mM, which can lead to the appearance of a self-consistent field and the ordered arrangement of micelles in solution.

Results and Discussion

Calculation of Micelle Boundaries

For the case of highly charged particles, it is possible to use eq , which allows making numerical calculations of dimensionless potential decay at various surfactant concentrations. The shape of potential decay curves allow us to suggest the presence of an uneven decrease in the bulk density of counterions with the increase of the distance from the micellar particle. As is clearly seen in Figure (more clearly demonstrated for high surfactant concentrations), a very fast potential decay turns sharply into a very slow one. Since the enlarged counterion concentration near micelle can be associated with the increased solution viscosity, the breakpoint of these curves can be compared with the hydrodynamic radius of micelles. However, it is difficult to determine this distance only from Figure . The situation is aggravated by the fact that the curves were calculated numerically and their analytical function is unknown.

The approach proposed by us, which takes into account small variations in CMC, is free from this drawback. It allows one to determine the position of a surface separating the total diffuse layer from the region with the increased concentration of counterions, which may be a consequence of increased viscosity of solvent surrounding micelle. Therefore, there is every reason to consider this surface as the slipping (shear) surface and to correlate the obtained distance with the micelle hydrodynamic radius.

So, to estimate micelle hydrodynamic radius, the following estimation procedure was carried out. To model the small fluctuations of the micelle microenvironment we solved eq for two close CMC values, namely 8.3 and 8.0 mM. For every of two models a couple of sets (CCMC = 8.3 and 8.0 mM) for dimensionless micelle potential with the same values of main parameters, corresponding to SDS micelles, was calculated. The analysis of difference curves ΔΦ = Φ8–Φ8.3 of the dimensionless potentials Φ8 and Φ8.3, calculated for two CMC values, making it possible to determine the domain of the diffuse layer, in which alterations of the counterion concentration take place. The location of this domain, namely the distance rmax, corresponding to the maximum amplitude of difference curves, indicates a breakpoint of potential decay, i.e., it can be correlated with the radius of the micelle slipping plane rζ (Figure ).

Figure 3

Difference of dimensionless electrostatic potential ΔΦ calculated for two close CMC values 8.3 and 8.0 mM as a function of r/a within the PB (left) and JA (right) models in order of increasing surfactant concentration from top to bottom.

For both the PB and JA models difference curves have a clearly visible maximum. The analysis of obtained results shows that the calculated positions rmax/a of the ΔΦ maxima are close to the relative Debye length Λ (see Tables and 2). For the nonlinear approximation, when the use of the Debye length Λ is incorrect, the position rmax/a of the ΔΦ maxima can be chosen as a parameter related to the micelle hydrodynamic radius. The numerical estimations of such regions of active counterion redistribution, caused by fluctuations in the micellar microenvironment, can be carried out for any arbitrary shape of potential decay.

To find the correlation between parameters Λ and rmax/a for linear approximation, the analysis of extremes for difference curves was performed analytically. The functional dependencies Φ8 and Φ8.3, determined by eq have small discrepancies because of the difference in parameter λ depending on I, and thereby, on the CMC. The resulting difference function ΔΦ = Φ8–Φ8.3 was analyzed for extreme as a function of relative distance r/a. For this purpose, we found the r/a values at which the derivative of the difference function reaches zero. It was found that for linear approximation the relative Debye length Λ = (a + λ)/a and position of the difference function ΔΦ maximum, namely the relative distance rmax/a, is related to each other by a quadratic equation, which allows us to write the ratio

The generalization of relation ( made it possible to calculate parameter Λ in a nonlinear approximation.

Comparison of the Calculated Data with the Experiment

The rmax/a concentration dependencies, calculated in linear and nonlinear approximations, are shown in Figure . The lower solid curve corresponds to the PB model and the upper one to the JA model. For nonlinear approximation, namely for real micelles, the positions rmax/a are shown by triangles (PB model) and squares (JA model). The analysis of data, presented in Figure , indicates that the points corresponding to nonlinear approximation are shifted toward larger distances in comparison with linear case in the PB model and toward shorter distances in the JA model. Similar alterations were observed when we compared the decay of dimensionless potential Φ as a function of relative distance in both cases under consideration. In addition, the values rmax/a for difference curves in both models differ much less for nonlinear approximation than for the linear one. It is important that the results obtained for highly charged particles within both models are in an agreement and their difference does not exceed 10%.

Figure 4

Calculated rmax/a values for linear (solid curves) and nonlinear approximations (triangles - PB model, squares - JA model) and values (a+Tshell)/a (asterisks) calculated from the SAXS experiment on hydrophilic layer thickness Tshell for SDS micelles.[32]

In addition to theoretically calculated values of rmax/a, one can see in Figure the experimental data on the hydrophilic layer thickness for SDS micelles (asterisks), obtained by us previously using small-angle X-ray scattering (SAXS).[32] These data, obtained in the form of concentration dependence of micelle hydrophilic layer thickness Tshell can be considered as a definite estimation of micelle electric double layer thickness. For comparison, we recalculated our previous experimental data to a relative form (a + Tshell)/a for the same radius of the micelle core as was taken before, namely a = 2.3 nm. One can see that the results of theoretical calculations are in good agreement with our present experimental data (Figure ). It is also remarkable that at low surfactant concentrations the experimental values are described more correctly by the PB model, while at concentrations more than 100 mM, when the intermicellar interactions cannot be neglected, the JA model proves to be preferable. However, with concentration increase micelles are distributed more evenly and described better by the JA model.

Estimation of Micelle Boundaries

To clarify the physical–chemical meaning of parameter rmax/a, the values of dimensionless potential Φ(rmax/a) were estimated. It turns out that Φ(rmax/a) values are almost unchanged in the range of studied surfactant concentrations with small discrepancies only at large concentrations. The values of Φ(Λ) did not exhibit this trend. Moreover, our calculations show that in the PB model the micelle electrostatic potential at distance r is about 2.5 times less than its maximum value Ψ(a) = −82 V at the micelle core surface and about 3 times less for the JA model. For both models Ψ(a) corresponds to −30 mV, i.e., to the threshold of dispersion stability.[51] Moreover, ζ-potential of SDS micelles in the absence of salts turned out to be −27 mV.[67] On the contrary, the obtained Φ(Λ) values are too small to correspond to the ζ–potential. Both these facts allowed us to suggest that the calculated value of r can be taken as the estimate of micelle hydrodynamic radius or shear surface radius rζ (micelle size) and that the potential on this surface can be compared with the electrokinetic or ζ-potential.

Thus, even in linear approximation for the estimation of the micelle hydrodynamic radius, it is more appropriate to use the r value calculated with the help of eq . This equation can be extended to the case of highly charged spherical particles. The data presented in Tables and 2 show that the discrepancy between the Debye length for linear and nonlinear variants does not exceed 12%. This means that an approximate estimation of micelle hydrodynamic radius is possible without a numerical solution of differential eq . In addition, eq in the form ofcan be applied to find micelle hydrodynamic radius for any surfactant if data on the micelle core radius and the magnitude of ionic strength of solution at a certain temperature are available. It should also be noted that the dependence of sufficiently close values such as a + λ and rmax on temperature and ionic strength of solution has a fundamentally different character as follows from eq (.

The data presented in Tables and 2 allow the determination of micelle characteristic size r, which is close to or identical to the micelle hydrodynamic radius in the presence of weak thermal effects. For greater correctness of calculations, one should restrict himself to surfactant concentrations that do not greatly exceed the CMC, since with the increase in concentration, surfactant micelles rather quickly lose their spherical shape and alter the size. At low concentrations, the PB model is preferable. In this case, the rmax/a values are close to 1.5, giving a hydrodynamic radius of about 3.5 nm for SDS micelles. Also, interesting results were obtained when estimating the number of molecular layers of water, located between the micellar particle surface (inner Stern plane) and shear surface with radius rmax (slipping plane). Since the thickness of this layer is approximately rmax– a = 1.2 nm it cannot contain more than 5 molecular layers of water with its molecular size of about 0.27–0.3 nm. Of course, in the Stern layer water is strongly ordered and forms the highly packed pseudophase[68] but the value of 5 layers cannot be greatly exceeded.

The obtained results confirm the hypothesis that the micelle hydrodynamic slipping plane is located near the outer Stern plane and separated from it by only two or three molecular layers of water. The influence of effects, which are stronger than the thermal ones, can shift the slipping plane closer to the micelle core. Therefore, it can be assumed that the smallest micelle hydrodynamic size is determined by the outer Stern plane, namely, it is limited by the distance at which the quantum-mechanical interactions are manifested, linking surfactant ions and counterions to the solvent-separated ion pairs.

The proposed approach can be extended to study other physicochemical effects in micellar solutions of surfactants. In particular, the suggested approach can be applied to study the influence of external electrolytes or nanoparticles on micellar systems, which will affect the characteristic properties of systems, including the CMC values.[12,13,69]

Conclusions

In this work, the existing concepts of the structure of ionic surfactant micelles were refined. By numerically solving the Poisson equation for the two most frequently used approximations, the Poisson–Boltzmann (PB) and the Jellium-approximation (JA) models, the electrostatic potential decay caused by a micelle particle was determined. All calculations were made under the assumption that the composition of a micellar particle of a solid-like dispersed phase in a liquid dispersion medium includes adsorbed counterions rigidly bound to this particle. The shape of decay curves indicates the presence of an uneven decrease in the bulk density of counterions with the increase of distance from the micellar particle. The inflection point of the potential decay curve can be correlated with the position of the micelle hydrodynamic radius. For its determination, the difference curves corresponding to small alterations in the CMC value were considered. Since different experimental techniques give only average CMC values, we assumed that the proposed approach will take into account small fluctuations in the micelle microenvironment, which are inevitable during the micellar transition. Difference curves show that at certain distances from the micelle core there is a region of the counterion density (concentration) redistribution with a well-defined maximum. The position of this maximum is correlated with the boundary separating the region with high solution viscosity and increased concentration of counterions near the micelle from the total diffuse layer in the solution bulk. The obtained distance was recognized as the micelle hydrodynamic radius.

So, we have developed a new methodological approach that allows one to determine the parameter approximately corresponding to the micelle hydrodynamic radius and to calculate the micelle electrokinetic potential. The results of theoretical calculations were compared with our previous experimental data on the thickness of the hydrophilic layer of SDS micelles obtained in SAXS experiments. Good agreement between the calculated and measured data was obtained. It was shown that at low surfactant concentrations, the experimental values are more correctly described by the PB model and at concentrations above 100 mM, when the interactions between micelles cannot be neglected, the JA model is preferable.

The number of molecular layers of water located between the surface of the micellar particle (inner Stern plane) and micelle slipping plane was estimated as not more than five. This confirms the hypothesis that the micelle hydrodynamic slipping plane is located near the outer Stern plane and is separated from it by only a few molecular layers of water. Because of different impacts, stronger than the thermal one, and a shift in the slipping plane closer to the micelle core, it can be assumed that the smallest micelle hydrodynamic size will be determined by the outer Stern plane. The results of our work allow us to conclude that in fact, a micelle is not something soft and watery, but from its structural properties it is found to be much more a solid-like particle than previously assumed.

Various options for estimating the ionic strength of ionic surfactants micellar solutions, the correlation of calculated and experimental data was discussed. It was noted that the suggested approach can be extended to other effects in surfactant micellar solutions, in particular, to study the influence of external electrolytes or nanoparticles.