Study of the premicellar state in aqueous solutions of sodium dodecyl sulfate by nuclear magnetic resonance diffusion

Self-diffusion coefficients of sodium dodecyl sulfate (SDS) were measured in aqueous solutions in the premicellar range of the SDS concentrations 7 – 34.7 mM and temperatures 30 – 90 (cid:1) C. Average effective hydrodynamic radii and aggregation numbers of SDS in the premicellar region were determined. At C < CMC at all temperatures, the SDS solution is the solution of monomers. At C > CMC, the increase of temperature leads to decrease in the effective hydrodynamic radii and the average aggregation numbers. At C > > CMC, it is impossible to reach the monomeric state by increasing the temperature.


| INTRODUCTION
Surfactants in aqueous solutions in the concentration range near the critical micelle concentration (CMC) are present in the premicellar state, [1,2] a transition state between the monomeric molecular solution of the surfactant and the micellar solution. At a concentration C < < CMC, the surfactant solution is a true molecular solution of surfactant monomers that is not yet capable of aggregating due to its low concentration. [3] At a concentration of C > > CMC, the surfactant solution is a solution of aggregates, micelles whose shapes change from spherical to ellipsoidal, spherocylindrical and filamentous (when the surfactant concentration is increased), until they form liquid crystal structures. [4] The premicellar region of surfactant solutions was investigated by various methods: fluorescence quenching, [5,6] nuclear magnetic resonance (NMR) spectroscopy [7,8] and NMR diffusion [9][10][11] and ESR. [12] Computer simulation methods were used to study the properties of the premicelles. [13,14] Thermodynamic analysis [15] shows that the formation of micelles also occurs at C < CMC. Premicelles are used in micellar catalysis [16][17][18] and in the processes of adsorption and creation of nanoscale structures. [19] In the premicellar state, the solution contains both monomers and dimers, as well as trimers, tetramers formed by surfactant molecules or ions. An increase in the number of n-mers as well as the degree of aggregation can be expected with an increase of the surfactant concentration in the solution up to the formation of spherical micelles. On the other hand, with increasing temperature, a decrease in the number of n-mers in the solution can be expected due to an increase in the destructive effect of thermal motion.
The aim of this work was to study the premicellar state in surfactant solutions in the concentration range below and above the CMC, depending on temperature. An anionic surfactant sodium dodecyl sulfate (SDS), CH 3 (CH 2 ) 11 OSO 3 Na, which has a wide range of applications in industry, pharmaceuticals and cosmetics, was chosen as the object of the study. We studied the translational mobility of kinetic units formed by SDS molecules in aqueous solutions, depending on concentration and temperature, by NMR diffusometry. The self-diffusion coefficients measured by NMR were used to calculate the average hydrodynamic radii of the kinetic units of SDS. In addition, we calculated the average aggregation numbers of n-mers in premicellar solutions of SDS.

| Materials
SDS and deuterated water (D 2 O, degree of substitution 99.9%) were purchased from Sigma-Aldrich. All NMR samples were prepared in 5-mm NMR tubes.

| Diffusion measurements
Diffusion coefficients were measured with an NMR spectrometer "Bruker Avance" ( 1 Н, 400 MHz) using pulse sequence of stimulated spin echo. [20] The amplitude of the stimulated spin echo signal is determined by the expression: where Т 1 and Т 2 are spin-lattice and spin-spin relaxation times, respectively; τ and τ 1 are time intervals in the pulse sequence; γ is the gyromagnetic ratio for protons; g and δ are the amplitude and duration of the magnetic field gradient pulses, respectively, D is the coefficient of self-diffusion, and t d = (Δ À δ//3) is the diffusion time, where Δ = (τ + τ 1 ). Measurements were carried out at δ = 1 ms, Δ = 49 ms, the number of scans NS = 16, the repetition time RT = 5 s; the amplitude of the pulse gradient g was chosen, taking into account the values of the self-diffusion coefficient, g max = 2.2 T/m. The processing of diffusion decays and the determination SDC were carried out using the free software Bruker TopSpin 3.5. Self-diffusion coefficients were measured as a function of temperature in the range from 30 to 70 C at SDS concentrations of 7, 11.5, 17.3, 26 and 34.7 mM. This range of concentrations is close to the critical concentration of SDS micelle formation; at 30 C, the CMC is 8.5 mM. [21,22] As the temperature rises, the degree of counterion binding decreases, the aggregation numbers decrease and the CMC increases, remaining within the 8-10 mM range. [22] Self-diffusion coefficients were determined from decays of the intensities of the proton lines of SDS methylene groups. The diffusion decays, the dependences of ln (A/A 0 ) on γ 2 δ 2 g 2 t d , had a linear form at all temperatures and throughout the studied concentration range. Figure 1 shows diffusion decays of spin echo signals of protons of SDS methylene groups at temperatures 30, 40, 50, 60 and 70 C and concentrations 7 and 34.7 mM, which correspond to the minimum and maximum studied concentrations of SDS.
The linear nature of the diffusion decays indicates a rapid (on the NMR scale) exchange of surfactant molecules between their possible states in solution: monomers, dimers, n-mers and micelles. Indeed, the lifetime of surfactants in micelles is $10 À6 s, [23] which is significantly less than the diffusion time t d $ 10 À3 s in the NMR experiment. Kinetic units formed by surfactant molecules or ions in solution have different translational mobility and, accordingly, have different diffusion coefficients. The resulting self-diffusion coefficient, D, measured by NMR, can be represented as a weighted average for all possible states: where D i and p i are diffusion coefficients and corresponding fractions of SDS in these states, and Σp i = 1. In the micellar range of concentrations and temperatures studied, which is usually limited by two states of surfactants (monomers and micelles), the concentration of monomers in solution is equal to the CMC, in this case [24] D ¼ where D mon and D mic are diffusion coefficients of the monomers and micelles of the surfactant, respectively. С is the total concentration of the surfactant in the solution.
On the basis of the experimentally measured diffusion coefficients, using the Stokes-Einstein relation (4), the effective hydrodynamic radii of diffusing particles were calculated. In this work, aqueous (D 2 O) solutions of SDS were studied in the concentration range from 7 to 34.7 mM. Such solutions can be considered infinitely dilute, because even at the maximum concentration there are about 1400 D 2 O molecules per one SDS molecule. The condition of infinite dilution made it possible to apply the Stokes-Einstein relation [25] to calculate the    [26] : where k is the Boltzmann constant, T is the absolute temperature and η is the dynamic viscosity.

| RESULTS AND DISCUSSION
The results of measurements of SDS in aqueous (D 2 O) solutions in the temperature range from 30 to 70 C are presented in Table 1.
These SDS values characterize the translational mobility of all kinetic SDS units and can be decomposed into separate components using relations (2) and (3), which characterize the mobility of the monomers, dimers, n-mers and micelles, provided that their relative concentrations in the solution are known. In this work, we limited ourselves to considering the average characteristics of SDS aggregates. The average effective radii of the aggregates and then the average aggregation numbers were calculated from the experimental values of SDS.
The average values of the effective hydrodynamic radii, R H , of the kinetic units of SDS depending on concentration and temperature are shown in Figures 2 and 3. The calculations used the values of the dynamic viscosity coefficients of heavy water. [27] How well the condition of infinite dilution is satisfied and how justified is the use of the solvent viscosity instead of the solution viscosity in calculations can be estimated by calculating the values of the hydrodynamic radii of water molecules R w from the values of SDC from Table 1. Calculations using relation (4) with 'stick' boundary conditions [28] showed that the value of R w does not depend on either temperature or SDS concentration and is equal to ≈1.3 nm, thereby confirming the correctness of the above assumptions.
As can be seen from Figure 2, the radii of the kinetic units of SDS at all temperatures asymptotically decrease with a decrease in the surfactant concentration in the solution to $0.35 nm. Obviously, this value corresponds to the effective hydrodynamic radius of SDS monomers. It follows from Figure 3 that with increasing temperature at SDS concentrations close to the CMC (C = 7, 11.5 and 17.3 mM), the radii of kinetic SDS units also approach 0.35 nm when extrapolated to a temperature of $90 C. Note that when the concentration of SDS is less than the CMC, the radius of the kinetic units of SDS remains F I G U R E 2 Concentration dependences of the average effective hydrodynamic radii of kinetic units of SDS in aqueous solutions (D 2 O) at temperatures: 1-30, 2-40, 3-50, 4-60, 5-70 С equal to 0.35 nm and is almost independent of temperature. It can be concluded that the effective hydrodynamic radius of SDS monomers is $0.35 nm. Calculation using the method of atomic increments [25] gave a value of 0.4 nm for the spherical approximation of the shape of SDS molecules.
Let us compare the average effective hydrodynamic radii of kinetic units of SDS (Figures 2 and 3) with the minimum radius of a spherical SDS micelle. The minimum radius of a spherical micelle is equal to the length of an extended aliphatic surfactant chain; for SDS micelles, this value is $2.5 nm, [29] which is confirmed by measuring the sizes and shapes of SDS micelles by quasi-elastic light scattering [29] and smallangle neutron scattering (SANS). [30,31] Consequently, because our studies are in the premicellar region of the surfactant state in solution, the radii of the kinetic units of SDS are less than those of spherical micelles, and the presence of monomers, dimers, trimers and SDS micelles should be expected in the solution. The average effective hydrodynamic radius of kinetic units remains less than the minimum radius of a spherical micelle.
The dependences of the diffusion coefficient of a surfactant in Arrhenius coordinates, ln(D) = f (1/T), have a linear form and can be characterized by the activation energy of the diffusion process: The presence in the solution of various types of n-mers and a change in their relative composition with a change in the surfactant concentration in the solution are manifested in the fact that the diffusion activation energy changes with the surfactant concentration ( Figure 4). The increase in E D with an increase in the concentration of SDS in solution is explained by an increase in the size of the n-mers, and reaching a plateau means the end of the premicellar concentration range. Thus, the premicellar region is bound from below by C ≈ 7 mM and from above by C ≈ 25-30 mM. Note that the E D of water molecules remains constant, does not depend on the presence of a surfactant in the solution and is equal to 19.6 kJ/mol. In the paper, [32] the value of ΔE D for pure heavy water is given as equal to 18.9 kJ/mol, according to measurements of the diffusion coefficient by the labeled atoms method.
A change in the average effective hydrodynamic radii of the kinetic units of SDS in solutions with a change in concentration or temperature means that the ratio of monomers, dimers, trimers and n-mers in solution changes. With an increase in the surfactant concentration, the degree of aggregation increases from n = 2, which corresponds to dimers, to N agr for spherical micelles. The aggregation numbers in spherical SDS micelles change with temperature from n = 68 (at 20 C) to n = 40 (at 40 C). [21] According to the results of dynamic light scattering, n = 60 (at 25 С) [29] ; according to SANS data, n = 65 [30] ; according to surface tension data, n = 68 [33] ; and according to geometric estimates, n = 56. [34] Let us determine the average aggregation numbers of kinetic units depending on the surfactant concentration in the solution by comparing the volumes of these units and monomers: In the approximation of the spherical shape of particles, one can relate the volumes to the effective hydrodynamic radii of the kinetic units R and monomers R mon and transform the expression (6) to the form The calculation results are shown in Figures 5 and 6.

| CONCLUSIONS
Based on the results of measurements of SDS in solution, we determined the boundaries of the premicellar state and calculated the average effective hydrodynamic radii and aggregation numbers of kinetic SDS units in the premicellar region. It was shown that the aggregation properties of SDS solutions in the premicellar region are determined by both concentration and temperature. At C = 7 mM, that is, at C < CMC, the SDS solution is a solution of monomers at all temperatures. The effective hydrodynamic radius of the SDS kinetic units is 0.35 nm, which corresponds to monomers. With an increase in the concentration of SDS, the sizes of the aggregates and the average values of the aggregation numbers increase. At C = CMC and temperatures from 30 to 60 C, the average aggregation numbers already correspond to dimers and trimers. With a further increase in concentration, the sizes of the aggregates and the aggregation number reach values typical for aqueous solutions of SDS micelles.
With an increase in temperature at all concentrations, we observe a decrease in the effective hydrodynamic radii and average aggregation numbers. At C = 7 mM at all temperatures, the aggregation numbers are equal to one-only surfactant monomers are present in the F I G U R E 6 SDS aggregation numbers in aqueous (D 2 O) solutions depending on concentration at different temperatures: 1-30, 2-40, 3-50, 4-60, 5-70 С solution. At concentrations of C = 11.5 and 17.3 mM, an increase in temperature allows the monomeric state of SDS to form in solution. At higher concentrations, it is impossible to reach the monomeric state by increasing the temperature.