Miscibility of sodium chloride and sodium dodecyl sulfate in the adsorbed film and aggregate

SDS (Nacalai Tesque’s guaranteed reagent) was purified by recrystallizing it triply from water and twice from ethanol. Its purity was checked by no minimum on the surface tension vs. concentration curve around the CMC. The purified SDS was kept under a dried atmosphere and reduced pressure in a desiccator to prevent its hydrolysis. Sodium chloride is the same sample as that used in the previous study [10]. It was stored under a dried atmosphere in a desiccator. Water, deionized and distilled, was further purified by distilling it twice from dilute alkaline solution of potassium permanganate.

Equilibrium surface tension was measured by the drop volume technique previously described [30], as a function of the total molality \(\hat m\) of NaCl and SDS and the mole fraction \(\hat X_2 \) of SDS in the mixture. Taking into account the dissociation of NaCl and SDS into ions in solutions, \(\hat m\) and \(\hat X_2 \) are defined by and where m ₁ is the molality of NaCl and m ₂ the one of SDS. The density of aqueous NaCl solutions was used for the calculation of surface tension because the concentration of SDS is very low [31, 32]. A glass tip at which a solution drop is formed was weakly coated with dimethyl polysiloxane (Siliconize L-25, Fuji Systems, Japan) to give reproducible values of surface tension for anionic surfactants [33]. Temperature was kept at 298.15 K within 0.01 K by immersing the measuring cell in a water thermostat. The errors inherent in the measurement were less than 0.05 mN m⁻¹.

The aqueous solutions of a NaCl–SDS mixture became turbid at large \(\hat m\) and small \(\hat X_2 \). Figure 1 shows the surface tension γ vs. total molality \(\hat m\) curves for two typical \(\hat X_2 \) at which turbidity takes place. The γ of transparent solutions decreases with increasing \(\hat m\); in sharp contrast, that of turbid solutions (solid circles) increases. Therefore, the lowest \(\hat m\) at which turbidity occurs was determined by a break on the γ vs. \(\hat m\) curve. Another break is observed on the curve of the higher \(\hat X_2 \) at \(\hat m\) below the lowest \(\hat m\), indicating the onset of micelle formation. Turbidity hence occurs in micellar solutions at higher \(\hat X_2 \) and in monomeric solutions at lower \(\hat X_2 \). The solutions at concentrations far above the lowest \(\hat m\) became turbid within 20 min during its standing at 298.15 K in the water thermostat after preparation. However, the solutions at concentrations around the lowest \(\hat m\) were allowed to stand in the thermostat for 12 h before measurement so that equilibrium is reached because the lowest \(\hat m\) did not change after the aging of 12 h.

Figure 2a shows the surface tension vs. total molality curves at constant \(\hat X_2 \) for higher \(\hat X_2 \), and Fig. 2b shows those for lower \(\hat X_2 \) including the results in Fig. 1. The γ vs. \(\hat m\) curves of single NaCl and SDS are in good agreement with those in previous studies [34‐36]. The γ vs. \(\hat m\) curve of mixture changes its shape regularly with the change in \(\hat X_2 \) and has one or two breaks indicating the onset of micelle formation and/or turbidity. The onset of turbidity, which is the formation of crystalline particle, is denoted by an arrow for each \(\hat X_2 \) in Fig. 2b. The CMC \(\hat C\) of the mixture and the lowest \(\hat m\) to bring about turbidity, which we call critical particle concentration \(\hat C^{\text{P}} \), were determined from the breaks in Fig. 2. The \(\hat C\) and \(\hat C^{\text{P}} \) are plotted as a function of \(\hat X_2 \) in Fig. 3. The magnification in Fig. 3 shows that the \(\hat C\) vs. \(\hat X_2 \) and \(\hat C^{\text{P}} \) vs. \(\hat X_2 \) curves meet at \(\hat X_2 = 0.00042\). Micelles and particles coexist in equilibrium at the intersection; we denote the \(\hat C\) and \(\hat X_2 \) at the intersection by \(\hat C^{{\text{eq}}} \) and \(\hat X_2^{{\text{eq}}} \), respectively. It is noticed that the \(\hat C^{\text{P}} \) increases with decreasing \(\hat X_2 \) below the \(\hat X_2^{{\text{eq}}} \); however, it is almost constant above the \(\hat X_2^{{\text{eq}}} \). We will later examine the coexistence of micelles and crystalline particles at the \(\hat C^{\text{P}} \) for \(\hat X_2 \) higher than the \(\hat X_2^{{\text{eq}}} \). The logarithm of CMC of SDS vs. logarithm of the counterion concentration obtained from the \(\hat C\) vs. \(\hat X_2 \) curve, known as the Corrin–Harkins plot [1], gave a straight line below approximately 0.18 mol kg⁻¹ of the NaCl concentration. However, the \(\hat C\) vs. \(\hat X_2 \) curve is more useful than the Corrin–Harkins plot because the former gives the composition of mixed micelle as shown later. The surface tension γ ^C at the \(\hat C\) and γ ^Pat the \(\hat C^P \), taken from Fig. 2, are plotted against \(\hat X_2 \) in Fig. 4. It is worth noting that γ ^C decreases as \(\hat X_2 \) decreases, in contrast, γ ^P increases in the range of \(\hat X_2 \) below the \(\hat X_2^{{\text{eq}}} \) and remains almost constant above the \(\hat X_2^{{\text{eq}}} \), as depicted in the magnification.

The mole fraction \(\hat X_2^H \) of SDS in the adsorbed film is expressed by where the surface density \(\Gamma ^{{\text{H}}}_{1} \) of NaCl is defined as that \(\Gamma _{{\text{Cl}}^ - }^{\text{H}} \) of chloride ion Cl⁻, and \(\Gamma _2^{\text{H}} \) of SDS as \(\Gamma _{{\text{DS}}^ - }^{\text{H}} \) of dodecyl sulfate ion DS⁻ [37]. Numerical values of \(\hat X_2^H \) were evaluated by applying (refer to Appendix) to the \(\hat m\) vs. \(\hat X_2 \) curves in Fig. 5. Here, the \(\hat m\) vs. \(\hat X_2 \) curves were obtained by taking the \(\hat m\) values at a given γ from the γ vs. \(\hat m\) curves in Fig. 2, and \(\gamma _{1 \pm ,\hat m} \) is the dependence of mean activity coefficient \(\gamma _{1 \pm } \) of NaCl on \(\hat m\) defined by

The literature value of \(\gamma _{1 \pm } \) was used for the evaluation of \(\hat X_2^H \) [38]. To draw the phase diagram of adsorption, the \(\hat X_2^H \) values are shown in the form of \(\hat m\) vs. \(\hat X_2^H \) curve together with the \(\hat m\) vs. \(\ifmmode\expandafter\hat\else\expandafter\^\fi{X}_{2} \) curve, in Fig. 5 [37]. \(\hat X_2^H \) is larger than unity at higher \(\hat X_2 \), which shows the negative adsorption of Cl⁻ by the definition of \(\hat X_2^H \) (Eq. 3). Therefore, we conclude that the mixed adsorbed film of SDS with NaCl at higher \(\hat X_2 \) is composed of SDS, and adsorbed DS⁻ ions expel Cl⁻ ions from the adsorbed film. Figure 5 further exhibits that the electrostatic repulsion between DS⁻ and Cl⁻ is enhanced at higher γ because DS⁻ is less densely packed in the adsorbed film at higher γ. Such repulsion of coions by surfactant ions is also observed for the adsorbed film of a mixture of NaCl and dodecylammonium chloride (DAC) [8].

It is noticed that \(\hat X_2^H \) is smaller than unity at very small \(\hat X_2 \). The \(\hat X_2^H \) smaller than unity indicates that (1) NaCl and SDS are miscible in the adsorbed film, and (2) \(\Gamma _1^{\text{H}} \) is positive due to enhanced adsorption of Na⁺ and highly weakened repulsion between DS⁻ and Cl⁻ in the adsorbed film. To show the miscibility in terms of surface density, we calculated the surface densities of individual ions by substituting the values of \(\hat X_2^H \) in Fig. 5 and the total surface density \(\hat \Gamma ^H \) into the following equations: and where the surface density \(\Gamma _{Na^ + }^H \) of sodium ion Na⁺ is given by from electroneutrality condition and \(\hat \Gamma ^H \) was evaluated by applying Eq. 26 in Appendix to the γ vs. \(\hat m\) curves in Fig. 1.

Figure 6 represents those surface densities as a function of \(\hat X_2 \) at 40 mN m⁻¹. The gradual increase in \(\Gamma _{{\text{DS}}^ - }^{\text{H}} \) with decreasing \(\hat X_2 \) in the range of \(\hat X_2 \) from 1 to approximately 0.002 is due to the electrostatic screening of the repulsion between the head groups of adsorbed DS⁻ ions by the presence of adsorbed Na⁺. On the other hand, the negative \(\Gamma _{{\text{Cl}}^ - }^{\text{H}} \), the desorption of Cl⁻, in the range of \(\hat X_2 \) from 1 to 0.04 shows the repulsion between DS⁻ and Cl⁻ in the adsorbed film. Furthermore, \(\Gamma _{{\text{DS}}^ - }^{\text{H}} = \Gamma _{{\text{Na}}^{\text{ + }} }^{\text{H}} \,{\text{and}}\,\Gamma _{{\text{Cl}}^ - }^{\text{H}} = 0\,{\text{at}}\,\hat X_2 = 0.04\), which shows adsorbed DS^- is neutralized by Na⁺. Taking into consideration that Na⁺ adsorbs at the surface of an adsorbed film of a nonionic surfactant caused by ion–dipole interaction between Na⁺ and the polar head group of surfactant molecule [9, 11, 12], \(\Gamma _{{\text{Na}}^{\text{ + }} }^{\text{H}} \) larger than \(\Gamma _{{\text{DS}}^ - }^{\text{H}} \) at \(\hat X_2 \) below 0.04 shows that Na⁺, after the neutralization, further adsorbs at the surface of the adsorbed film probably due to attractive interaction between Na⁺ and the sulfinyl groups in the head group of adsorbed DS⁻ [39]. Moreover, Fig. 6 shows that the excess adsorption of Na⁺ over the adsorption of DS⁻ is accompanied by the positive adsorption of Cl⁻ so as to satisfy the electroneutrality condition. Such positive adsorption of inorganic coions is also observed for the NaCl–dodecyldimethylammonium chloride system at large NaCl concentrations [7]. It is worth noticing that below \(\hat X_2 = 0.001\), \(\Gamma _{{\text{DS}}^ - }^{\text{H}} \)decreases with decreasing \(\hat X_2 \) whereas \(\Gamma _{{\text{Na}}^{\text{ + }} }^{\text{H}} \) and \(\Gamma _{{\text{Cl}}^ - }^{\text{H}} \) increase. The decrease in \(\Gamma _{{\text{DS}}^ - }^{\text{H}} \) is probably due to the penetration of Na⁺ among the head groups of adsorbed DS⁻ [40‐42] and/or the repulsion between the head groups of neutralized DS⁻ caused by the Na⁺ accumulated around the head groups of DS⁻ in the adsorbed film.

Miscibility of NaCl and SDS in the micelle can be examined by using the mole fraction \(\hat X_2^{\text{M}} \) of SDS in the micelle defined by where \(N_1^{\text{M}} \) and \(N_2^{\text{M}} \) are respectively the excess numbers of NaCl and SDS in one micelle defined with respect to the spherical dividing surface around the micelle so as to make the excess number of water molecules zero [43]. The \(\hat X_2^{\text{M}} \) was numerically evaluated by applying (refer to Appendix) to the \(\hat C\) vs. \(\hat X_2 \) curves in Fig. 3.

Since crystalline particles in solutions are coagels of the hydrated multilayers with a stacked bilayer structure in equilibrium with the surrounding solution [44, 45] and the particle concentration is extremely small at the \(\hat C^{\text{P}} \), particle formation can be treated in a manner similar to micelle formation using the excess number \(N_1^{\text{P}} \) of NaCl and \(N_2^{\text{P}} \) of SDS in one particle defined with respect to the dividing surface, which makes the corresponding excess number of water molecules zero. The mole fraction \(\hat X_2^{\text{P}} \) of SDS in the particle defined by was calculated by applying to the \(\hat C^{\text{P}} \) vs. \(\hat X_2 \) curve below \(\hat X_2^{{\text{eq}}} \) in Fig. 3. We depict the resulting \(\hat X_2^{\text{M}} \) and \(\hat X_2^{\text{P}} \) in the form of \(\hat C\) vs. \(\hat X_2^{\text{M}} \) and \(\hat C^{\text{P}} \) vs. \(\hat X_2^{\text{P}} \) curves, together with the \(\hat C\) vs. \(\hat X_2 \) and \(\hat C^{\text{P}} \) vs. \(\hat X_2 \) curves, in Fig. 7. Figure 7 can be called phase diagram of aggregate formation because it shows the relations between the composition of aggregates and solution at the critical aggregate concentration \(\hat C\) or \(\hat C^{\text{P}} \). The finding that \(\hat X_2^{\text{M}} \) is larger than unity shows the expulsion of Cl⁻ from the micelle by DS⁻. In contrast, \(\hat X_2^P \) is smaller than unity and decreases with decreasing \(\hat X_2 \). We therefore conclude that sodium ions highly adsorb at the bilayer surfaces in a particle, like at the surface of the adsorbed film at extremely small \(\hat X_2 \).

The miscibilities in the adsorbed film and aggregate can be compared with each other by use of the phase diagram for adsorbed film–aggregate equilibrium. We have for the adsorbed film–micelle equilibrium and for the adsorbed film–particle equilibrium (refer to Appendix), where \(\widehat{X}^{{{\text{H,C}}}}_{2} \) and \(\hat \Gamma ^{H,C} \) are the \(\hat X_2^H \) and \(\hat \Gamma ^H \) at the \(\hat C\), respectively, and \(\hat X_2^{{\text{H}},{\text{P}}} \) and \(\hat \Gamma ^{H,P} \) are the corresponding ones at the \(\widehat{C}^{{\text{P}}} \).

The \(\hat X_2^{{\text{H}},{\text{C}}} \) was calculated by applying Eq. 14 to the γ ^C vs. \(\hat X_2 \) curve in Fig. 4 and substituting the \(\hat X_2^{\text{M}} \) values in Fig. 7 and the \(\hat \Gamma ^{H,C} \) values into Eq. 14. In a manner similar to the evaluation of \(\hat X_2^{{\text{H}},{\text{C}}} \), we obtained the numerical values of \(\hat X_2^{{\text{H,P}}} \) from the \(\gamma ^{{\text{P}}} \) vs. \(\hat X_2 \) curve. Figure 8 shows the \(\hat X_2^{{\text{H,C}}} \) and \(\hat X_2^{{\text{H,P}}} \) together with the \(\hat X_2^M \) and \(\hat X_2^{\text{P}} \) in the form of the surface tension vs. composition curves at the \(\hat C\) and \(\hat C^{\text{P}} \). It is noticed that \( \hat X_2^{\text{M}} > \hat X_2^{{\text{H,C}}} \) and \( \hat X_2^{\text{P}} < \hat X_2^{{\text{H,P}}} \). The former inequality stems from the fact that the ionic head groups of DS⁻ ions are less densely packed in the micelle than in the adsorbed film due to the difference in geometry between the adsorbed film and micelle. The latter one shows that the miscibility of NaCl and SDS in the adsorbed films of bilayers in a crystalline particle is larger than the corresponding one in the adsorbed film at water/air interface. Taking into consideration that the curvature effect on miscibility is negligible for the adsorbed films of the bilayers because the crystalline particles are much larger than the micelles, we can attribute the \(\hat X_2^{\text{P}} \) smaller than \(\hat X_2^{{\text{H,P}}} \) to larger adsorption of Na⁺ at the bilayer surfaces compared with the corresponding one at the bulk surface. The above view is substantiated by the fact evidenced on the foam films stabilized with SDS and with decyl methyl sulfoxide that the adsorption of Na⁺ at the surfaces of a Newton black film, a model of bilayer, is larger than the corresponding one of a thick film [17, 19‐22].

The decrease in γ of micellar solutions with increasing \(\hat m\) in Figs. 1 and 2 can be explained by comparing the γ ^C vs. \(\hat X_2^{\text{M}} \) curve in Fig. 8 with the γ ^C vs. \(\hat X_2 \) curve in Fig. 4. The corresponding increase in γ of turbid solutions with increasing \(\hat m\) at a given \(\hat X_2 \) can be explained by comparing the γ ^P vs. \(\hat X_2^{\text{P}} \) curve with the γ ^P vs. \(\hat X_2 \) curve because the two curves form a phase diagram of particle formation: A transparent solution at a given \(\hat X_2 \) becomes turbid, and particles appear at the \(\hat C^{\text{P}} \) and further increase in the total molality at the \(\hat X_2 \) causes an increase in the amount of particles and a decrease in the mole fraction of SDS in the solution surrounding the particles because the \(\hat X_2^{\text{P}} \) is larger than the \(\hat X_2 \) and the \(\gamma ^{{\text{P}}} \) increases with decreasing \(\hat X_2 \).

Let us consider the micelle-particle equilibrium in solutions in the range of \(\hat X_2 \geqslant \hat X_2^{{\text{eq}}} \) and \(\hat m \geqslant \hat C^{{\text{eq}}} \) shown in the magnification in Fig. 3. Mass balance relations are given by for the total solutes and for SDS, where \(\hat m^M \) and \(\hat m^P \)are the \(\hat m\) ascribed to the micelle and particle, respectively, and \(\hat X_2^{{\text{M,eq}}} \) and \(\hat X_2^{{\text{P,eq}}} \) are the \(\hat X_2^{\text{M}} \) and \(\hat X_2^{\text{P}} \) at the \(\hat X_2^{eq} \) (see Fig. 7). As \(\hat m\) increases at a given \(\hat X_2 \) above the \(\hat X_2^{eq} \) (see the magnification in Fig. 3), a micellar solution becomes turbid at the \(\widehat{C}^{{\text{P}}} \) where micelles are in equilibrium with an infinitesimal amount of crystalline particles, that is, \(\hat m^P \approx 0\). Eliminating the \(\hat m^M \) in Eqs. 16 and 17 and solving the resulting equation for \(\hat m\) at \(\hat m^P \approx 0\), we have the relation between \(\hat m\) and \(\hat X_2 \) for the micellar solutions at \(\widehat{C}^{{\text{P}}} \):

Here \(\hat m^I \) is defined as \(\hat m\) at which micelles coexist with an infinitesimal amount of particles at equilibrium. Equation 18 expresses the \(\hat m\) vs. \(\hat X_2 \) curve, one of the two micelle particle coexisting curves.

Further increase in \(\hat m\) at the \(\hat X_2 \) causes the disappearance of micelles in the turbid solution at \(\hat m^{II} \), the \(\hat m\) at which particles are in equilibrium with an infinitesimal amount of micelles, then we have \(\hat m^M = 0\). Eliminating the \(\hat m^{\text{P}} \) in Eqs. 16 and 17 and solving the resulting equation for \(\hat m\) at \(\hat m^M = 0\) give for the other coexisting curve \(\hat m^{II} \) vs. \(\hat X_2 \).

For the NaCl–SDS mixture, we obtain \(\hat C^{{\text{eq}}} = 1,465\,{\text{mmol}}\,{\text{kg}}^{ - 1} \) and \(X_2^{{\text{eq}}} = 0.00042\) from Fig. 3 and \(X_2^{{\text{M,eq}}} = 0.999\) and \(\hat X_2^{{\text{P,eq}}} = 0.731\) from Fig. 7. Substitution of those values into Eqs. 18 and 19 leads to \(\hat m^I \approx \hat m^{II} \approx \hat C^{eq} \) for \(\hat X_2 \) a little larger than the \(\hat X_2^{{\text{eq}}} \) because the \(\hat X_2^{{\text{eq}}} \) is much smaller than the \(\hat X_2^{{\text{M,eq}}} \) and \(\hat X_2^{{\text{P,eq}}} \). The experimental evidence shown in Fig. 3 that \(\hat C^{\text{P}} \) is almost \(\hat C^{{\text{eq}}} \) in the range of \(\hat X_2 \) above the \(\hat X_2^{{\text{eq}}} \) is consistent with the requirement \(\hat m^I \approx \hat m^{II} \approx \hat C^{{\text{eq}}} \) from Eqs. 18 and 19. Since micelle formation can be regarded as an appearance of macroscopic phases [46, 47], the number of degrees of freedom for the solutions of the present mixture in which micelles and particles coexist at equilibrium can be approximated to be two from the Gibbs phase rule. The experimental evidence at 298.15 K and atmospheric pressure in Fig. 4 that γ ^P is almost constant at \(\hat X_2 \) above the \(\hat X_2^{eq} \) satisfies the requirement of phase rule.

In a future study, we will apply the present thermodynamic approach to a mixture of calcium chloride (CaCl₂) and SDS in which strong electrostatic attraction is expected between calcium ion and dodecyl sulfate ion in the adsorbed film and micelle, and the salting out will occur at around the bulk molar ratio CaCl₂/SDS = 1/2.