#### 4.1. Span-80 Adsorption Layers at Paraffin–Oil/Water Interface

Figure 3,

Figure 4, and

Figure 5 show the experimental results for modulus

$|\epsilon *\left(i\hspace{0.17em}\omega \right)\hspace{0.17em}|$ and phase

$\phi =a\mathrm{tan}\left({\epsilon}_{i}/{\epsilon}_{r}\right)$ of the complex viscoelasticity measured for Span-80 adsorbed at paraffin–oil/water interface at an amplitude of 20% and different temperatures. The results at other amplitudes (5% and 10%) are similar, but the relative experimental errors are the smallest for the amplitude of 20%. The experimental points in

Figure 3,

Figure 4, and

Figure 5 are the averages of three different runs.

The experiments were planned for different concentrations of Span-80 in paraffin-oil created via a series of injections of small surfactant amounts into the matrix liquid. However, as it was reported before in [

20], a leakage has occurred from the injection syringe during the facility upload and installation on the ISS. As a consequence, the value of real Span-80 concentration in the majority of experiments was approximately within the range c = 2.0 × 10

^{−5}–3.0 × 10

^{−5}mol/dm

^{3}, as it has been estimated from the measured mean level of γ(t)-value (γ

_{0} = 23–26 mN/m) [

20]. Therefore, the viscoelasticity modulus for all injections was approximately the same, as it is seen from

Figure 3,

Figure 4, and

Figure 5. The exception are the results for the injection no. 0 (nominally pure paraffin-oil) at temperature 40 °C, shown in

Figure 5. These results were obtained under conditions of incomplete mixing of surfactant after the leakage, while the concentration in the vicinity of the drop interface remained smaller and the interfacial tension remained higher—within the range γ

_{0} = 31–33 mN/m (for more details see [

20]). This particular experiment was the first in the time-line. All subsequent experiments were performed after stirring actions resulting in a more homogeneous surfactant distribution.

All experimental data shown in

Figure 3,

Figure 4, and

Figure 5 demonstrate similar trends: the modulus is increasing, whereas the phase shift is decreasing with increasing frequency. These experimental data can be fitted by using the simplest (single-element) FMM model, as given by Equations (15)–(17), as it is seen from

Figure 3,

Figure 4, and

Figure 5. The model allows fitting both the modulus and the phase shift simultaneously, with the same parameter sets shown in the figure panels. The parameters of the model vary not in wide limits—α is about 0.19 and 0.2, τ is about 3.8 and 4.9 s, and Δγ

_{n}(0) is about 64 and 68 mN/m for most of the experiments. Only for the incipient experiment (injection no. 0 at temperature 40 °C) the parameters are slightly different—α = 0.21, τ = 3.5, and Δγ

_{n}(0) = 48 mN/m.

There are only obvious deviations of the model from the experimental data at the higher frequencies (mainly for the frequency 1 Hz). Most probably, such deviations can appear due to the increasing contribution of hydrodynamic effects, which can lead to a change of the system behavior (e.g., due to deviations of the drop shape from ideal spherical shape or more complicated relaxation mechanism).

The three parameters of the FMM model affect the shape of the modulus and phase shift vs. frequency dependencies in a different way. In particular, the phase shift does not depend on the multiplier before the frequency dependent part (“Δγ

_{n}(0)”) in Equations (15) and (16). From the remaining two parameters, only the parameter “α” influences the slope of the phase shift vs. frequency dependence, whereas the second one (“τ”) is responsible for the shift of the whole curve along the frequency axis and does not influence the slope. Therefore, it is very easy to obtain these two parameters using only the phase shift vs. frequency dependence. On the other hand, these two parameters should be the same for the modulus vs. frequency dependence. Thus, we can easily find the remaining third parameter (“Δγ

_{n}(0)”) by fitting this dependence. If the quality of fitting is good, this can be a criterion that the model is applicable for the considered system. The uncertainty in the fitting parameters can arise only from large experimental errors. For example, the slope of the phase shift vs. frequency dependence can vary within certain limits because the phase shift obtained in different experimental runs is slightly different, as it is seen in

Figure 3,

Figure 4, and

Figure 5.

The Lucassen–van den Tempel model does not fit these experimental data properly. It allows to fit with a good accuracy the modulus only, but it does not allow to fit the phase shift (these results are not shown here). For this model, the slope of the phase shift vs. frequency dependencies appears approximately two times larger than the experimental one, because the Lucassen–van den Tempel model (α = 0.5) predicts the small-frequency limit of the phase shift equal to 45°, whereas the FMM model with α = 0.2 gives a limit of about 20°, which is much more consistent with the experimental data.

#### 4.2. C_{13}DMPO/TTAB Adsorption Layers at Water/Hexane Interface

The second set of experiments was performed with mixed C

_{13}DMPO/TTAB adsorption layers at water/hexane interface. The surfactants were injected by increasing small amounts into the aqueous matrix phase. The injected non-ionic C

_{13}DMPO has penetrated into the n-hexane drop, according to the partitioning equilibrium (its hexane-to-water distribution coefficient is about 30, [

21,

26]). The ionic TTAB does not dissolve in n-hexane and remained in the aqueous phase. Thus, if the interface is expanded or compressed, the equilibrium is restored due to TTAB adsorption/desorption from the aqueous phase and C

_{13}DMPO adsorption/desorption from both liquid phases. Two examples of the modulus and phase shift of the complex viscoelasticity vs. frequency dependencies for this system are shown in

Figure 6 and

Figure 7.

It is seen from these two examples that the single-element fractional Maxwell model allows fit to only the modulus and does not allow fit to the phase shift for the system considered here. This system is more complex as compared to that considered in the previous section, because it consists of two surfactants instead of one, while one of them is dissolved in both contacting liquids. Comparing the results in

Figure 6 and

Figure 7with those in

Figure 3,

Figure 4, and

Figure 5 one can see that for mixed C

_{13}DMPO/TTAB adsorption layers at the water/hexane interface the phase shift is increasing with the frequency, while for Span-80 adsorbed at paraffin–oil/water interface it was decreasing. The single-element fractional Maxwell model predicts only a decreasing phase shift for all possible parameter sets. That is why this model does not fit the phase shift vs. frequency dependencies in

Figure 6 and

Figure 7. The classic Maxwell model and the Lucassen–van den Tempel model also predict only a decreasing phase shift as they are only particular cases of the fractional Maxwell model.

The analysis, presented in [

21], has shown that there are two possible reasons for an increasing phase shift vs. frequency dependencies for mixed C

_{13}DMPO/TTAB adsorption layers at water/hexane interface. Such behavior can be explained by either the presence of a second surfactant in the system, or the effect of curvature of the drop interface (or a superposition of both effects). Thus, for many systems the rheological behavior of an adsorption layer similar to that in

Figure 6 and

Figure 7 is quite expected. For such systems, the single-element fractional Maxwell model is not sufficient and more complicated models should be involved for their proper description.