# Dynamic Properties of Mixed Cationic/Nonionic Adsorbed Layers at the N-Hexane/Water Interface: Capillary Pressure Experiments Under Low Gravity Conditions

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{13}DMPO at the water/hexane interface. While the non-ionic surfactant C

_{13}DMPO is soluble in both bulk phases, water and hexane, the cationic surfactant TTAB is only soluble in the aqueous phase. The interfacial layer is thus formed by TTAB molecules adsorbing from the aqueous phase while the C

_{13}DMPO molecules adsorb from the aqueous phase, and transfer partially into the hexane phase until both the equilibrium of adsorption and the distribution between the two adjacent liquid phases is established. The experimental constrains as well as all possible influencing parameters, such as interfacial and bulk phase compressibility, interfacial curvature, calibration of pressure and absolute geometry size, are discussed in detail. The experimental results in terms of the dilational interfacial viscoelasticity of the mixed adsorption layers in a wide range of oscillation frequencies show that the existing theoretical background had to be extended in order to consider the effect of transfer of the non-ionic surfactant across the interface, and the curvature of the water/hexane interface. A good qualitative agreement between theory and experiment was obtained, however, for a quantitative comparison, additional accurate information on the adsorption isotherms and diffusion coefficients of the two studied surfactants in water and hexane, alone and in a mixed system, are required.

## 1. Introduction

_{13}DMPO), which are well studied at the interface between their aqueous solutions and air and various oil phases [2,3,20,21,22] and references therein. Mixtures of these two surfactants have not been investigated yet, and also the number of adsorption studies for other surfactant mixtures at liquid interfaces are rather limited [23,24,25,26].

_{13}DMPO in water have been investigated at the water/hexane interface at different concentrations and compositions. A complication for the present mixed surfactant system is the solubility of the non-ionic C

_{13}DMPO in both bulk phases, water and hexane [20,27], while TTAB is only soluble in the aqueous phase. Thus, after formation of the interface, TTAB molecules only adsorb until equilibrium has been established while C

_{13}DMPO molecules adsorb from the aqueous phase, and transfer partially until not only the equilibrium of adsorption but also of the distribution in the two adjacent liquid phases is reached.

## 2. Experimental Technique and Procedure

#### 2.1. Materials

^{®}(CAS 110-54-3) was purchased from Sigma-Aldrich (Merck) and used as received.

_{13}DMPO, (CAS 186953-53-7) was purchased from Gamma-Service, Berlin, Germany, with high purity and used as received.

#### 2.2. Measurement Cell

_{13}DMPO.

^{3}and of the matrix-cell is about V = 68.2 cm

^{3}, gradually increasing after each injection of surfactant solution, up to a maximum of 70.0 cm

^{3}. During the instrument storage and transport the two chambers are separated by a closed main valve (positioned in the safe state).

_{13}DMPO (concentration in the syringe#1 aqueous solution, c = 0.002 mol/dm

^{3}) and of TTAB (concentration in the syringe#2 aqueous solution, c = 0.45 mol/dm

^{3}), in a pre-established sequence, triggered by a built-in time line.

#### 2.3. Measurement Sequence

_{13}DMPO-surfactant (see Table 1). Then, the nitrogen bubbles were purged out from the capillary and subsequently the gas phase became dissolved in the aqueous matrix, as often visually ascertained by inspection of the telemetered images, up to the TL termination.

_{13}DMPO concentration value is affected by a 0.3% uncertainty, in connection with the estimated 0.3% inaccurate knowledge of the aqueous-matrix volume. A more important uncertainty can be caused in the C

_{13}DMPO concentration due to its dissolution in hexane, as it is discussed in detail below.

#### 2.4. Data Acquisition, Transmission and Pre-Processing

- (a)
- noise filtering,
- (b)
- data misalignment check inside the telemetered digital-unit packets and, in case required, synchronization of pressure data with optical data,
- (c)
- selection of oscillation sequences, for each concentration sample, at each amplitude and at each temperature,
- (d)
- in flight check of calibration parameters and, in case required, calibration adjustment.

#### 2.4.1. Optical Calibration In-Flight Check

#### 2.4.2. Adjustment of Pressure Sensor Calibration

## 3. Theory

#### 3.1. Dilational Viscoelasticity of Mixed Adsorption Layers

_{0}, with γ

_{0}and A

_{0}being the equilibrium interfacial tension and area, i the complex unit, ω = 2πf the angular frequency, and f the frequency of oscillations.

_{1}= Г

_{1}(c

_{1}, c

_{2}) and Г

_{2}= Г

_{2}(c

_{1}, c

_{2}) by the surfactants concentrations c

_{1}and c

_{2}, D

_{1}and D

_{2}are the diffusion coefficients, and $\mathrm{B}=1+\sqrt{\frac{\mathrm{i}\mathsf{\omega}}{{\mathrm{D}}_{1}}}{\mathrm{a}}_{11}+\sqrt{\frac{\mathrm{i}\mathsf{\omega}}{{\mathrm{D}}_{2}}}{\mathrm{a}}_{22}+\frac{\mathrm{i}\mathsf{\omega}}{\sqrt{{\mathrm{D}}_{1}{\mathrm{D}}_{2}}}\left({\mathrm{a}}_{11}{\mathrm{a}}_{22}-{\mathrm{a}}_{12}{\mathrm{a}}_{21}\right)$. The expression Equation (4) transforms to the Lucassen and van den Tempel equation, Equation (3), in the case when one of the two surfactant concentrations, c

_{1}or c

_{2}, turns to zero.

_{1}and D

_{2}by their effective counterparts

_{j}. Thus, to calculate the surface viscoelasticity modulus we have to know these parameters.

#### 3.2. Equilibrium Equation of State and Adsorption Isotherms of the Mixed Adsorption Layer

_{01}and E

_{02}, and four coefficients a

_{i,j}. These 6 parameters should be determined from the surface equation of state $\gamma =\gamma ({\mathsf{\Gamma}}_{1},{\mathsf{\Gamma}}_{2},\mathrm{T})$ and the adsorption isotherms ${\mathsf{\Gamma}}_{1}={\mathsf{\Gamma}}_{1}\left({\mathrm{c}}_{1}^{\mathsf{\alpha}},{\mathrm{c}}_{2}^{\mathsf{\alpha}}\right)$ and ${\mathsf{\Gamma}}_{2}={\mathsf{\Gamma}}_{2}\left({\mathrm{c}}_{1}^{\mathsf{\alpha}},{\mathrm{c}}_{2}^{\mathsf{\alpha}}\right)$ (or ${\mathsf{\Gamma}}_{1}={\mathsf{\Gamma}}_{1}\left({\mathrm{c}}_{1}^{\mathsf{\beta}},{\mathrm{c}}_{2}^{\mathsf{\beta}}\right)$ and ${\mathsf{\Gamma}}_{2}={\mathsf{\Gamma}}_{2}\left({\mathrm{c}}_{1}^{\mathsf{\beta}},{\mathrm{c}}_{2}^{\mathsf{\beta}}\right)$). Thus, to calculate the surface viscoelasticity modulus by using Equation (4) we have to know the equilibrium equation of state and adsorption isotherms of the mixed adsorption layers [34,35]. Unfortunately, for the mixed adsorption layers of TTAB and C

_{13}DMPO at hexane/water interface considered here, such equations are presently unknown. Therefore, for the analysis of the experimental data we will use a simplified approach assuming that the individual adsorption layers of TTAB and C

_{13}DMPO can be characterized by the Langmuir model with adsorptions and interfacial tension given by the equations

_{0}is the interfacial tension of the pure hexane/water interface (which is about 51.1 mN/m [20,38]), ${\mathsf{\Gamma}}_{\mathrm{j}\infty}$ are the limiting adsorptions, and ${\mathrm{b}}_{\mathrm{j}}^{\mathsf{\alpha}}$ are the equilibrium adsorption constants. Note, Equation (7) defines the adsorptions with respect to the concentrations in the phase α, but they can be also defined with respect to the concentrations in the phase β, as these concentrations are in equilibrium according to Equation (5). The parameters ${\mathsf{\Gamma}}_{\mathrm{j}\infty}$ and ${\mathrm{b}}_{\mathrm{j}}^{\mathsf{\alpha}}$ for the individual adsorption layers of TTAB and C

_{13}DMPO at the hexane/water interface can be found in literature (e.g., in [20,38]).

_{1∞}and Г

_{2∞}. In a general case, when Г

_{1∞}and Г

_{2∞}are different, these equations are inconsistent with thermodynamic requirements [33].

#### 3.3. Dilational Viscoelasticity of Curved Interfaces

_{0}is the equilibrium drop radius, and c

_{α}is the surfactant concentration in the external liquid phase. With account for Equation (5) this expression can be easily transformed to the case when the adsorption is defined with respect to the concentration in the internal phase β, $\mathsf{\Gamma}=\mathsf{\Gamma}\left({\mathrm{c}}_{\mathsf{\beta}}\right)$. In the limiting cases of a very small (K << 1) or very large (K >> 1) distribution coefficient Equation (14) is reduced to the respective expressions by Joos for the surfactant diffusion and adsorption from only one side of the drop interface. For large drop radii and large oscillation frequencies, when $\left|{\mathrm{n}}_{\mathsf{\alpha}}{\mathrm{r}}_{0}\right|>>1$ and $\left|{\mathrm{n}}_{\mathsf{\beta}}{\mathrm{r}}_{0}\right|>>1$, Equation (14) takes the form of the Lucassen and van den Tempel equation, Equation (3), with the effective diffusion coefficient ${\mathrm{D}}_{\mathrm{e}\mathrm{f}}={\left(\sqrt{{\mathrm{D}}_{\mathsf{\alpha}}}+\mathrm{K}\sqrt{{\mathrm{D}}_{\mathsf{\beta}}}\right)}^{2}$. This corresponds to the case of a locally flat interface with adsorption of one surfactant from the both contacting liquids.

## 4. Results and Discussion

#### 4.1. Experimental Results—Low Frequency Range

_{0}= 0.270 ± 0.004 mm, slightly larger than the inner capillary radius (0.25 mm).

_{13}DMPO and TTAB concentrations. These dependencies will be used here for the analysis of the dynamics of the mixed interfacial layers formed at the water/hexane interface under different experimental conditions.

_{13}DMPO concentration at a fixed TTAB concentration the viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ initially increases (for the concentrations between 4.0 × 10

^{−7}and 8.0 × 10

^{−6}mol/dm

^{3}) and then decreases (for concentrations higher than 8.0 × 10

^{−6}mol/dm

^{3}). Figure 4 shows a similar behavior of the viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ with respect to the TTAB concentration at a fixed C

_{13}DMPO concentration: the modulus initially increases (for the concentrations between 4.5 × 10

^{−5}and 2.2 × 10

^{−4}mol/dm

^{3}) and then decreases (for concentrations higher than 2.2 × 10

^{−4}mol/dm

^{3}). Such effect of concentration is typical also for individual surfactants solutions, as will be discussed further below.

^{−3}and 4.5 × 10

^{−3}mol/dm

^{3}in the presence of 2.2 × 10

^{−5}mol/dm

^{3}C

_{13}DMPO (injections 6-4 and 6-5) the viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ becomes very low, of the order of few mN/m only. It should be noted that the concentration of 2.2 × 10

^{−3}mol/dm

^{3}is close to and the concentration 4.5 × 10

^{−3}mol/dm

^{3}is above the CMC for aqueous TTAB solutions, which is of about 3.5 × 10

^{−3}mol/dm

^{3}[40]. Such small modulus values are typical for micellar surfactant solutions, where the exchange of monomers with the micelles supports to keep the monomer concentration at the interface almost constant (if the oscillation frequency is not very high), and, as consequence, the interfacial tension practically does not change [41]. At very small modulus values, the determination of the phase shift may become problematic because the relative errors in determining the real and imaginary parts of the modulus strongly increase. This can even cause some small negative phase shift values obtained for the injection 6-5. Therefore, these results are not analysed here.

#### 4.2. Comparison with Model Calculations

#### 4.2.1. Parameter Sets of the Model

_{13}DMPO presented in literature. The parameters of the Langmuir isotherm for adsorption layers of C

_{13}DMPO at the hexane/water interface are presented in [20]: ${\mathsf{\Gamma}}_{1\infty}=$2.1 × 10

^{−6}mol/m

^{2}and ${\mathrm{b}}_{1}^{\mathsf{\alpha}}=$1.25 × 10

^{7}dm

^{3}/mol (for definiteness, phase α is water here). The diffusion coefficient of C

_{13}DMPO in water was obtained in [20,27] to be about ${\mathrm{D}}_{1}^{\mathsf{\alpha}}$ = (0.9–2.2) × 10

^{−9}m

^{2}/s. These values are several times higher than the typical values for surfactants in aqueous solutions, which can be probably explained by a certain contribution of a convective transfer. Such convective mixing is less probable under microgravity conditions, therefore, the correct diffusion coefficient of C

_{13}DMPO in water should be probably smaller. According to the results in [27], the diffusion coefficient of C

_{13}DMPO in hexane is smaller than in water. One could expect the opposite, as the viscosity of hexane is about three times lower than that of water. A strong solvent-surfactant interaction could probably explain this fact, according to the authors. In a recent study by Fainerman et al. [42] the diffusion coefficient of C

_{13}DMPO in hexane was found to be ${\mathrm{D}}_{1}^{\mathsf{\beta}}$ = (3.2–4.5) × 10

^{−9}m

^{2}/s for a water drop in hexane, and ${\mathrm{D}}_{1}^{\mathsf{\beta}}$ = (2–3) × 10

^{−9}m

^{2}/s for a hexane drop in water. The diffusion coefficient of C

_{13}DMPO in water was found to be ${\mathrm{D}}_{1}^{\mathsf{\alpha}}$ = (1–2) × 10

^{−9}m

^{2}/s, i.e., practically the same as in [27]. The distribution coefficient for C

_{13}DMPO between hexane and water was obtained to be of about K

_{1}= 34 in [27] and of about K

_{1}= 30 in [42], which is quite close to each other. With these values for the diffusion coefficients in hexane and water and the distribution coefficient, the effective diffusion coefficient of C

_{13}DMPO, ${\mathrm{D}}_{1}^{\mathrm{e}\mathrm{f}}={\left(\sqrt{{\mathrm{D}}_{1}^{\mathsf{\alpha}}}+{\mathrm{K}}_{1}\sqrt{{\mathrm{D}}_{1}^{\mathsf{\beta}}}\right)}^{2}$, will be of the order of 10

^{−6}m

^{2}/s. Such large value is explained by the high solubility of C

_{13}DMPO in hexane.

^{−6}mol/m

^{2}and ${\mathrm{b}}_{2}^{\mathsf{\alpha}}=$9.4 × 10

^{5}dm

^{3}/mol. It is seen that the difference between ${\mathsf{\Gamma}}_{1\infty}$ and ${\mathsf{\Gamma}}_{2\infty}$ is small, therefore the approximate Equations (9)–(13) can be applied. It is seen that the solubility of TTAB in water is much higher than the solubility of C

_{13}DMPO. At the same time the solubility of TTAB in hexane is practically negligible, which is typical for ionic surfactants. Therefore, the effective diffusion coefficient of TTAB is determined mainly by its diffusion coefficient in the aqueous phase, ${\mathrm{D}}_{2}^{\mathrm{e}\mathrm{f}}={\left(\sqrt{{\mathrm{D}}_{2}^{\mathsf{\alpha}}}+{\mathrm{K}}_{2}\sqrt{{\mathrm{D}}_{2}^{\mathsf{\beta}}}\right)}^{2}\approx {\mathrm{D}}_{2}^{\mathsf{\alpha}}$, which is about 1 × 10

^{−10}mol/m

^{2}, according to [38].

#### 4.2.2. Viscoelasticity Modeling for Mixed and Individual Solutions

_{13}DMPO concentrations and a fixed TTAB concentration and are presented in Figure 5. The frequency range used for the calculations was much broader than in the experiments. It is seen that similarly to the experimental data the calculated viscoelasticity modulus increases with the frequency in the whole frequency range. In contrast, the phase shift versus frequency dependence is not monotonous, it also demonstrates an increase with the frequency, but only in a limited interval of intermediate frequencies. This is a typical behaviour for surfactants mixtures [34,35]. Hence, one of the possible explanations of the increasing phase shift with frequency is the presence of the second surfactant in the solution. It can be assumed that for the system considered here, that the applied frequencies related to the low frequency range approximately coincide with this limited frequency interval where both the viscoelasticity modulus and the phase shift are increasing. Another possible explanation of the increasing phase shift could be related to the curvature effect of the interface, as will be considered below.

_{13}DMPO and TTAB according to the Lucassen and van den Tempel model, Equation (3), by using the same adsorption parameter sets. The results are presented in Figure 6. The viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ for individual solutions demonstrates a qualitatively similar behaviour as for the mixed solutions presented in Figure 5, i.e., the modulus increases with the frequency. In contrast, the phase shift does not have a frequency interval, where it increases with the frequency. For diffusion-limited adsorption of a single surfactant at a flat interface the phase shift should decrease continuously with increasing frequency from 45° at very small frequencies to zero at very high frequencies, as it follows from the Lucassen and van den Tempel model. Thus, the increasing phase shift with the frequency observed in our experiments can be an indication of the significance of the fact that the solutions studied here are surfactants mixtures.

_{D1}and ω

_{D2}, which characterize the rates of diffusional relaxation of the two surfactants in the mixture. These characteristic frequencies are determined by the effective diffusion coefficients ${\mathrm{D}}_{1}^{\mathrm{e}\mathrm{f}}$ and ${\mathrm{D}}_{2}^{\mathrm{e}\mathrm{f}}$, which depend on the diffusion rate in both liquid phases. The effective diffusion coefficients depend also on the surfactants’ distribution coefficients K

_{j}. The diffusion in the phase, where the surfactant is more soluble, is more significant for the surface dilational viscoelastic modulus. The surface dilational viscoelastic modulus of a mixture depends also on some equilibrium parameters: E

_{0j}, Г

_{0j}and the ratios a

_{12}/a

_{22}and a

_{21}/a

_{11}.

_{D1}and ω

_{D2}, are shown on the curve for the concentration ${\mathrm{c}}_{1}^{\mathsf{\alpha}}=$ 8.0 × 10

^{−6}mol/dm

^{3}in Figure 5. In this particular case, the difference between these two frequencies is large, their ratio is larger than two orders of magnitude. The limited frequency interval, where the phase shift increases with frequency, is located between these two frequencies. If we would decrease the effective diffusion coefficient ${\mathrm{D}}_{1}^{\mathrm{e}\mathrm{f}}$ for C

_{13}DMPO by two or three orders of magnitude, then the difference between the frequencies ω

_{D1}and ω

_{D2}will be much smaller, and a frequency interval with increasing phase shift would not be observed. This fact demonstrates that the apparently large value of the effective diffusion coefficient ${\mathrm{D}}_{1}^{\mathrm{e}\mathrm{f}}$ for C

_{13}DMPO discussed above is consistent with the observed system behavior. Note, the real diffusion coefficients of C

_{13}DMPO in the two liquids are by three orders of magnitude smaller than ${\mathrm{D}}_{1}^{\mathrm{e}\mathrm{f}}$.

_{D2}increases and the difference between the frequencies ω

_{D1}and ω

_{D2}becomes much smaller, i.e., the frequency interval with increasing phase shift becomes much shorter or disappears (for smaller C

_{13}DMPO concentrations). Thus, the smaller equilibrium constant ${\mathrm{b}}_{2}^{\mathsf{\alpha}}$ obtained in Ref. [38] is more consistent with the experimental results presented above.

#### 4.2.3. Effect of the Isotherm Parameters

_{13}DMPO can be characterized by the Langmuir model for which all necessary parameters are available. A detailed comparison of the experimental data with the results of the model calculations shows that the use of the Langmuir model is probably sufficient for a qualitative interpretation of the observed dynamic system behavior, however it is not sufficient for a more precise quantitative analysis. In particular, one has to pay attention to very high viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ values for the individual and mixed adsorption layers, which exceed 100 mN/m for higher frequencies, as it is seen in Figure 5 and Figure 6. It was shown previously [44] and references therein] that such large viscoelasticity modulus values can be a consequence of neglecting the finite size of the adsorbing molecules in the Langmuir or Frumkin models. At high surface coverages one can improve formally the agreement of the models with experimental data by assuming that the molar areas can depend linearly on the surface pressure as

_{0j}are the molar areas of the surfactants at zero surface pressure, ε

_{j}are coefficients of the linear dependence, and ∏ = γ

_{0}− γ is the surface pressure (γ

_{0}and γ are the surface tensions of the interface in absence and in presence of surfactants, respectively). Sometimes, an alternative dependence is used [35,38]

^{−7}mol/dm

^{3}and ${\mathrm{c}}_{2}^{\mathsf{\alpha}}=$ 4.5 × 10

^{−5}mol/dm

^{3}we obtained surface coverages of θ

_{1}= 0.055 and θ

_{2}= 0.924, respectively, with the total surface coverage θ

_{1}+ θ

_{2}= 0.978. Note, the adsorption constants ${\mathrm{b}}_{\mathrm{j}}^{\mathsf{\alpha}}$ for oil/water interfaces are usually much larger than for the air/water interface, which for the studied concentrations results in large ${\mathrm{b}}_{\mathrm{j}}^{\mathsf{\alpha}}{\mathrm{c}}_{\mathrm{j}}^{\mathsf{\alpha}}$ values (much larger than 1) and large θ

_{j}values. Hence, even for the smallest concentrations the total surface coverage is close to unity, i.e., the studied adsorption layers are in a compressed state. For such compressed layers the effect of variation of the molar areas could be significant. Thus, the very high viscoelasticity modulus values seen in Figure 5 and Figure 6 could be a consequence of the neglected variation of the molar areas. Also, the characteristic frequencies, ω

_{D1}and ω

_{D2}, could be incorrect for the same reason. Therefore, the calculations performed here based on the Langmuir model are not sufficient for a more detailed quantitative comparison with the experimental data.

_{13}DMPO. Thus, a more general model, which takes into account the surfactants interaction, would be more adequate for the considered system.

^{−5}mol/dm

^{3}). However, our results discussed above (Figure 4) show that there is obviously a maximum of the viscoelasticity modulus at the TTAB concentration of about 2.25 × 10

^{−4}mol/dm

^{3}, i.e., the maximum is shifted to a higher TTAB concentration in the presence of C

_{13}DMPO. Also, the height of the maximum for mixed solutions is much smaller than in absence of C

_{13}DMPO. These differences could be an indication of the importance of interactions between the surfactants in the studied system.

_{13}DMPO.

_{13}DMPO concentration, because of its dissolution in hexane. The volume of the hexane drop is much smaller than the volume of the aqueous matrix-cell, therefore, the amount of C

_{13}DMPO dissolved in the drop can be neglected. However, the drop is connected to the hexane reservoir, and a part of C

_{13}DMPO can diffuse to this reservoir though the capillary connecting it with the drop. This diffusion process is very slow, but the total time of the experiment is large, which may lead to a gradual decrease of the C

_{13}DMPO concentration in the matrix-cell. Nevertheless, this concentration decrease should not be large because the measurements for each subsequent concentration are performed shortly after the injection of a new portion of C

_{13}DMPO during the time, which is not sufficient for the diffusion of C

_{13}DMPO between the drop and the hexane reservoir. Moreover, the injected amounts are continuously increasing, therefore, the loss of small surfactant amounts from the previous injections should be insignificant.

#### 4.2.4. Effect of Curvature of the Drop Interface

_{0}= 0.27 mm and diffusion coefficient D = 10

^{−9}m

^{2}/s, one obtains $\mathrm{D}/2\mathsf{\pi}{\mathrm{r}}_{0}^{2}$ = 2.2 × 10

^{−3}s

^{−1}, which is not too far from the lower frequencies in this study. Therefore, the possibility of curvature effect should be also considered. In Figure 7, one can see the viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ and phase shift φ(ω) of C

_{13}DMPO adsorption layers calculated according to Equation (14) for different drop radii with D

_{α}= D

_{β}= 10

^{−9}m

^{2}/s, c

_{α}= 8.0 × 10

^{−7}mol/dm

^{3}and the Langmuir isotherm parameters as presented above. At higher frequencies the curves for $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ and φ(ω) for different drop radii approach those for the infinite radius. The larger is the drop radius, the closer is the respective curve to that for the infinite radius (flat surface). However, the $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ dependencies for the finite radii have a plateau and the respective φ(ω) dependencies are not monotonous at smaller frequencies, in contrast to the case of infinite radius. It is important that there is a frequency range where the phase shift φ(ω) increases with frequency.

_{0}, and the interface behaves like locally flat. Accordingly, the phase shift decreases with increasing frequency in this case and the viscoelasticity modulus gradually increases up to its high-frequency limit (see Equation (3)). In contrast, if the oscillation frequency is very small and, additionally, the surfactant solubility in the external phase is also small or negligible (K >> 1), then the diffusion layer thickness ${\mathsf{\delta}}_{\mathrm{D}}=\sqrt{\mathrm{D}/\mathsf{\omega}}$ is much larger than the drop radius r

_{0}, and the adsorption equilibrium between the interface and the drop volume establishes during a time much smaller than one oscillation period. In this case, the interface and the surfactant concentration within the drop oscillate with almost the same phase and the phase shift between the area oscillations and the interfacial tension oscillations becomes very small. Thus, the phase shift should decrease with the frequency decrease left from the local maximum. Also under these conditions, the amplitudes of the interfacial tension oscillations practically do not depend on the rate of interfacial area expansion or compression. This explains the appearance of a plateau on the viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ versus frequency dependencies at intermediate frequencies. The level of this plateau is determined by the surfactant depletion in the drop due to its adsorption at the interface—the smaller is the drop the higher is the depletion [45].

#### 4.3. Experimental Results and Analysis—High Frequency Range

_{A,B}and V

_{A,B}are the effective bulk elasticities and volumes of chamber A and chamber B (n-hexane reservoir), respectively, δV

_{Act}is the volume variation produced by the piezo-actuator, and R

_{C}is the complex hydrodynamic resistance of the capillary with the attached meniscus and adjacent liquid. In the approximation of a quasi-stationary flow this resistance is given by the equation [28,46]

_{1}and G

_{2}are the coefficients describing the viscous and inertia contributions due to the flow in the bulk of liquids, $\frac{\mathrm{d}\mathrm{r}}{\mathrm{d}{\mathrm{V}}_{\mathrm{m}}}$ and $\frac{\mathrm{d}ln\mathrm{A}}{\mathrm{d}{\mathrm{V}}_{\mathrm{m}}}$ are the geometric parameters—the derivatives of the drop radius r and the relative surface area lnA on the meniscus volume V

_{m}, and γ

_{eq}and r

_{0}are the equilibrium interfacial tension and drop radius.

_{B}, is negligible, and the pressure difference, δP

_{AB}, is practically the same as the pressure variation in chamber A, δP

_{A}(see Equation (18)). However, for frequencies above 10 Hz the pressure variation in chamber B becomes also significant (due to resonance effects), and one has to take it into account.

_{C}is determined not only by the hydrodynamic effects (viscosity and inertia) in the bulk of liquids but also by the properties of the drop interface—the equilibrium interfacial tension γ

_{eq}and the interfacial viscoelasticity modulus E. This means that the oscillating flow through the capillary connecting the chambers A and B depends on the equilibrium (γ

_{eq}) as well as on the dynamic (E = E(iω)) properties of the interfacial layer formed between the two contacting liquids. Accordingly, the measured pressure variations in the chambers A and B should also depend on these interfacial characteristics. However, the interfacial characteristics strongly depend on the composition of the interfacial layer, which is determined by the surfactants’ concentrations in the bulk solutions. As a result, the measured pressure signals strongly depend on the solutions’ composition.

_{13}DMPO and TTAB with different concentrations. As it is seen, these characteristics strongly depend on the surfactants’ concentrations. The relative amplitude of the pressure oscillations decreases with C

_{13}DMPO concentration and the minimum and maximum in the frequency characteristic gradually disappear (Figure 8). At the same time, the phase shift depends only slightly on the C

_{13}DMPO concentration. With increasing TTAB concentration the maximum in the amplitude-frequency characteristic becomes more pronounced and shifts to smaller frequencies (Figure 9). The phase shift decreases by its absolute value (i.e., increases with respect to −π) for the two highest TTAB concentrations at small frequencies.

_{eq}decreases with increasing surfactant concentrations what should lead to an increase of the plateau level. However, the effect of the interfacial viscoelasticity E(iω) is opposite in phase and is stronger in this concentration range and, therefore, the general trend is a decrease of the plateau level with increasing surfactant concentrations. The plateau level gradually rises up with increasing frequency which reflects an increase of the viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ and an increase of the hydrodynamic contribution.

_{2}ω

^{2}by the elastic contribution due to the bulk elasticity and the contribution of the interface in the denominator of Equation (18) [28,46]. The height of this maximum is determined by the dissipative processes in the system. With increasing C

_{13}DMPO concentration, the contribution of the dilational interfacial viscosity ${\mathsf{\eta}}_{\mathrm{S}}\left(\mathsf{\omega}\right)={\mathsf{\omega}}^{-1}\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|sin\left(\mathsf{\varphi}\left(\mathsf{\omega}\right)\right)$ becomes larger, therefore, the resonance maximum decreases (see Figure 8). However, with a further increase in the TTAB concentration both real and imaginary parts of the interfacial viscoelasticity E(iω) become very small and, as a result, the resonance maximum moves toward smaller frequencies and slightly increases again (Figure 9). These results show that the viscoelastic characteristics of the interface can have a strong effect on the dynamic properties of the system as a whole.

## 5. Conclusions

_{13}DMPO and TTAB surfactants at the interface between water and hexane at different surfactants’ concentrations were studied in a wide frequency range. The measured interfacial dilational viscoelasticity as a function of frequency and of temperature reveal that the water/hexane interface in contact with mixed C

_{13}DMPO/TTAB solutions exhibits a viscoelastic behavior and that the adsorption from both contacting liquids is important. The phase shift versus frequency dependences measured at different surfactants’ concentrations are rising up, in contrast to the decreasing curves usually obtained for single surfactant adsorption layers at flat interfaces. The possible explanations of the increasing phase shifts with the frequency of oscillations are: (i) the effect of mixture, when two surfactants in the mixture are characterized by different characteristic relaxation frequencies, and (ii) the effect of curvature of the interface, when the radius of curvature is comparable with the diffusion layer thickness. To quantify the relative contribution of each of these two effects, a more rigorous information about the adsorption isotherms is required.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Mixed non-ionic/cationic surfactant components of the aqueous matrix solution (C

_{13}DMPO + TTAB) and of the adjoining hydrocarbon dispersed phase (pure n-hexane drop).

**Figure 2.**Fitting of differential pressure vs. inverse radius, r, for injection 6-5 at T = 15 °C for a growing drop experiment, with filtered reference pressure and adjusted calibration parameters (linear fit, slope = 4.55 mNm

^{−1}, offset = 0.00 Pa).

**Figure 3.**Viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ and phase shift φ(ω) as functions of the frequency for C

_{13}DMPO concentrations of 4.0 × 10

^{−7}, 8.0 × 10

^{−7}, 4.0 × 10

^{−6}, 8.0 × 10

^{−6}and 2.2 × 10

^{−5}mol/dm

^{3}(injections 2-1 to 6-1) at a fixed TTAB concentration of 4.5 × 10

^{−5}mol/dm

^{3}(T = 20 °C; ampl. 10%); the lines are guides for the eye.

**Figure 4.**Viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ and phase shift φ(ω) as functions of frequency for TTAB concentrations of 4.5 × 10

^{−5}, 2.2 × 10

^{−4}, 4.5 × 10

^{−4}and 2.2 × 10

^{−3}mol/dm

^{3}(injections 6-1 to 6-4) at a fixed C

_{13}DMPO concentration of 2.2 × 10

^{−5}mol/dm

^{3}(T = 20 °C; ampl. 10%); the lines are guides for the eye.

**Figure 5.**Viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ and phase shift φ(ω) of a mixed adsorption layer as functions of frequency calculated according to Equations (4), (6) and (11)–(13) for ${\mathrm{c}}_{1}^{\mathsf{\alpha}}=$ 8.0 × 10

^{−7}, 4.0 × 10

^{−6}and 8.0 × 10

^{−6}mol/dm

^{3}at fixed ${\mathrm{c}}_{2}^{\mathsf{\alpha}}=$ 4.5 × 10

^{−5}mol/dm

^{3}.

**Figure 6.**Viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ and phase shift φ(ω) of individual adsorption layers as functions of frequency calculated according to Equation (3) for ${\mathrm{c}}_{1}^{\mathsf{\alpha}}=$ 8.0 × 10

^{−7}, 4.0 × 10

^{−6}and 8.0 × 10

^{−6}mol/dm

^{3}at ${\mathrm{c}}_{2}^{\mathsf{\alpha}}=$ 0 (full lines) and for ${\mathrm{c}}_{2}^{\mathsf{\alpha}}=$ 4.5 × 10

^{−5}mol/dm

^{3}at ${\mathrm{c}}_{1}^{\mathsf{\alpha}}=$ 0 (dashed line).

**Figure 7.**Viscoelasticity modulus $\left|\mathrm{E}\left(\mathrm{i}\mathsf{\omega}\right)\right|$ and phase shift φ(ω) of C

_{13}DMPO adsorption layers at the drop interfaces with radii r

_{0}= 0.1, 0.27, 0.5 mm and infinite as functions of frequency, calculated according to Equation (14) for D

_{α}= D

_{β}= 10

^{−9}m

^{2}/s and c

_{α}= 8.0 × 10

^{−7}mol/dm

^{3}with the Langmuir isotherm parameters given in the text.

**Figure 8.**Amplitude- and phase-shift-frequency characteristics for the pressure in chamber A (aqueous matrix-cell) measured for mixed solutions of C

_{13}DMPO with concentrations of 4.0 × 10

^{−7}, 8.0 × 10

^{−7}, 4.0 × 10

^{−6}, 8.0 × 10

^{−6}and 2.2 × 10

^{−5}mol/dm

^{3}(injections 2-1 to 6-1) and TTAB with a fixed concentration of 4.5 × 10

^{−5}mol/dm

^{3}(T = 20 °C; ampl. 10%); the lines are guides for the eye.

**Figure 9.**Amplitude- and phase-shift-frequency characteristics for the pressure in chamber A (aqueous matrix-cell) measured for mixed solutions of TTAB with concentrations of 4.5 × 10

^{−5}, 2.2 × 10

^{−4}, 4.5 × 10

^{−4}, 2.2 × 10

^{−3}and 4.5 × 10

^{−3}mol/dm

^{3}(injections 6-1 to 6-5) and C

_{13}DMPO with a fixed concentration of 2.2 × 10

^{−5}mol/dm

^{3}(T = 20 °C; ampl. 10%); the lines are guides for the eye.

Date DOY 2014 | Injection n # | Injection C_{13}DMPO Syringe #1 (mm^{3}) | Concentration C_{13}DMPO (mol/dm^{3}), in Matrix Cell | Injection TTAB Syringe #2 (mm^{3}) | Concentration TTAB (mol/dm^{3}), in Matrix Cell | TTAB/C_{13}DMPOConcentration Ratio |
---|---|---|---|---|---|---|

183–184 | 0-0 | - | 0 (*) | - | 0 (*) | - |

184 | 1-0 | 7 | 2.0 × 10^{−7} (*) | - | 0 (*) | - |

184–188 | 1-1 | 2.0 × 10^{−7} | 7 | 4.5 × 10^{−5} | 225.00 | |

188–190 | 2-1 | 7 | 4.0 × 10^{−7} | - | 4.5 × 10^{−5} | 112.50 |

191–194 | 3-1 | 14 | 8.0 × 10^{−7} | - | 4.5 × 10^{−5} | 56.25 |

195–200 | 4-1 | 112 | 4.0 × 10^{−6} | - | 4.5 × 10^{−5} | 11.25 |

200–202 | 5-1 | 140 | 8.0 × 10^{−6} | - | 4.5 × 10^{−5} | 5.625 |

202–206 | 6-1 | 507.5 | 2.2 × 10^{−5} | - | 4.5 × 10^{−5} | 2.00 |

206–209 | 6-2 | - | 2.2 × 10^{−5} | 28 | 2.2 × 10^{−4} | 10.00 |

209–210 | 6-3 | - | 2.2 × 10^{−5} | 35 | 4.5 × 10^{−4} | 20.00 |

211–216 | 6-4 | - | 2.2 × 10^{−5} | 280 | 2.2 × 10^{−3} | 100.00 |

216–219 | 6-5 | - | 2.2 × 10^{−5} | 350 | 4.5 × 10^{−3} | 200.00 |

Total injection | 787.5 | 700 |

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## Share and Cite

**MDPI and ACS Style**

Loglio, G.; Kovalchuk, V.I.; Bykov, A.G.; Ferrari, M.; Krägel, J.; Liggieri, L.; Miller, R.; Noskov, B.A.; Pandolfini, P.; Ravera, F.;
et al. Dynamic Properties of Mixed Cationic/Nonionic Adsorbed Layers at the N-Hexane/Water Interface: Capillary Pressure Experiments Under Low Gravity Conditions. *Colloids Interfaces* **2018**, *2*, 53.
https://doi.org/10.3390/colloids2040053

**AMA Style**

Loglio G, Kovalchuk VI, Bykov AG, Ferrari M, Krägel J, Liggieri L, Miller R, Noskov BA, Pandolfini P, Ravera F,
et al. Dynamic Properties of Mixed Cationic/Nonionic Adsorbed Layers at the N-Hexane/Water Interface: Capillary Pressure Experiments Under Low Gravity Conditions. *Colloids and Interfaces*. 2018; 2(4):53.
https://doi.org/10.3390/colloids2040053

**Chicago/Turabian Style**

Loglio, Giuseppe, Volodymyr I. Kovalchuk, Alexey G. Bykov, Michele Ferrari, Jürgen Krägel, Libero Liggieri, Reinhard Miller, Boris A. Noskov, Piero Pandolfini, Francesca Ravera,
and et al. 2018. "Dynamic Properties of Mixed Cationic/Nonionic Adsorbed Layers at the N-Hexane/Water Interface: Capillary Pressure Experiments Under Low Gravity Conditions" *Colloids and Interfaces* 2, no. 4: 53.
https://doi.org/10.3390/colloids2040053