# Interfacial Dilational Viscoelasticity of Adsorption Layers at the Hydrocarbon/Water Interface: The Fractional Maxwell Model

^{1}

^{2}

^{3}

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^{*}

## Abstract

**:**

_{13}DMPO/TTAB mixed surfactants/aqueous-solution matrix. The fractional constitutive single-FMM is demonstrated to embrace the standard Maxwell model (MM) and the Lucassen–van-den-Tempel model (L–vdT), as particular cases. The single-FMM adequately fits the Span-80/paraffin-oil observed results, correctly predicting the frequency dependence of the complex viscoelastic modulus and the inherent phase-shift angle. In contrast, the single-FMM appears as a scarcely adequate tool to fit the observed behavior of the mixed-adsorption surfactants for the C

_{13}DMPO/TTAB/aqueous solution matrix (despite the single-FMM satisfactorily comparing to the phenomenology of the sole complex viscoelastic modulus). Further speculations are envisaged in order to devise combined FMM as rational guidance to interpret the properties and the interfacial structure of complex mixed surfactant adsorption systems.

## 1. Introduction

## 2. Materials and Method

#### 2.1. Materials

^{®}, (CAS 110-54-3), (c) tridecyl dimethyl phosphine oxide (C

_{13}DMPO), (CAS 186953-53-7) and (d) tetradecyl trimethyl ammonium bromide (TTAB), (CAS 1119-97-7).

#### 2.2. Apparatus

_{13}DMPO/TTAB mixed solution matrix. Typical video frames of the drops inside the liquid matrices were, for example, visualized in Figure 4 of [20] and in Figure 1 of [21].

#### 2.3. Experimental Procedure

_{13}DMPO/TTAB adsorbed at water/hexane interface), shows that the mean-level values of interfacial tension, γ, as a function of frequency, is constant for all amplitudes (but random disturbances inherent in the intermittent piezo push-pull action), albeit γ attains a somewhat smaller level for the final 20%-amplitude oscillation sequence.

## 3. Brief Outline of the Applied Model

#### 3.1. The Interfacial Dilational Viscoelastic Modulus

_{0}, Δγ and ΔA are the oscillation amplitudes of the interfacial tension and the interfacial area, respectively, ϕ the phase shift between perturbation and response, A

_{0}is the mean interfacial area of the sinusoidal cycle, ε

_{r}and ε

_{i}are the real and imaginary parts, respectively, of the complex viscoelasticity ε*(iω).

_{n}(t) = Δγ(t)/(ΔA/A

_{0}).

_{n}(t) relaxation function vanishes at zero value in a time interval at least one order of magnitude shorter than the time-interval of the non-equilibrium condition evolution, at the start of the interfacial layer excitation.

#### 3.2. Maxwell Model

_{n}(0) is the interfacial tension value at time t = 0 s for Δγ

_{n}(t) = Δγ(t)/(ΔA/A

_{0}).

#### 3.3. Fractional Maxwell Model

_{1}, τ

_{1}, and α, and a spring, E

_{2}, connected in a series. The constitutive equation for this model is given by [11,15]:

_{2}and ${\tau}^{\alpha}\hspace{0.17em}=\hspace{0.17em}\left({E}_{1}/{E}_{2}\right)\times \text{\hspace{0.17em}}{\tau}_{1}{}^{\alpha}$

_{n}(0), τ and α, the determined modulus of the complex function ε*(iω) for a single-element fractional Maxwell model reads:

#### 3.4. Lucassen–van den Tempel Model

_{D}is the characteristic frequency of diffusional relaxation.

_{D}= (2τ)

^{-1}and ε

_{0}= Δγ

_{n}(0), which corresponds to a purely diffusional relaxation. In contrast, the case of α = 1 (i.e., the standard Maxwell model) corresponds to a purely kinetic controlled adsorption, as can be concluded from Figure 8.2 in [25]. Note, however, in the case of a mixed adsorption kinetics (diffusion complicated by an adsorption barrier) the single-element fractional Maxwell model cannot be applied as this case requires two different relaxation times for a proper description.

## 4. Results and Discussion

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

^{−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.

_{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.

_{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.

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

_{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.

_{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.

_{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.

## 5. Conclusions

_{13}DMPO and TTAB surfactants cannot be described by a single-element FMM. For this case, obviously, rheological schemes with more elements are required.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Dynamic interfacial properties of mixed C

_{13}DMPO/TTAB adsorption layers at the n-hexane/aqueous solution interface as a function of frequency, at temperature T = 15 °C, at a relative interfacial-area oscillation-amplitudes of A = 5%, 10%, and 20%, for a mixed aqueous solution at concentration C

_{C13DMPO}= 2.0 × 10

^{−7}mol/dm

^{3}and concentration C

_{TTAB}= 4.5 × 10

^{−5}mol/dm

^{3}. Right axis: mean-level of the interfacial tension oscillations. Left axis: modulus of the interfacial dilational viscoelasticity. Red open up-triangles A = 5%, blue solid circles A = 10%, violet open down-triangles A = 20%.

**Figure 2.**Cole-Cole plot of the complex viscoelasticity described by the fractional Maxwell model with different parameter α.

**Figure 3.**Modulus and phase shift of the complex viscoelasticity measured for the successive injections no. 0, 1, and 2 at an amplitude of 20% and temperature 20 °C for Span-80 adsorbed at paraffin–oil/water interface (inj. 0, blue squares, injection 1, green squares, and injection 2, magenta squares). The lines are the results of calculations by using the fractional Maxwell model (FMM) model, Equations (15)–(17), with the parameter set α = 0.2, τ = 3.9 s, and Δγ

_{n}(0) = 68 mN/m.

**Figure 4.**Modulus and phase shift of the complex viscoelasticity measured for the successive injections no. 0 and 1 at an amplitude of 20% and temperature 30 °C for Span-80 adsorbed at paraffin–oil/water interface (injection 0, blue squares, injection 1, green squares). The lines are the results of calculations by using the FMM model, Equations (15)–(17), with the parameter set α = 0.2, τ = 3.8 s, and Δγ

_{n}(0) = 67 mN/m.

**Figure 5.**Modulus and phase of the complex viscoelasticity measured for the successive injections no. 0 and 1 at an amplitude of 20% and temperature 40 °C for Span-80 adsorbed at paraffin–oil/water interface (injection 0, blue squares, and injection 1, green squares). The blue and green lines are the results of calculations by using the FMM model, Equations (15)–(17), with the parameter sets α = 0.21, τ = 3.5 s, and Δγ

_{n}(0) = 48 mN/m and α = 0.19, τ = 4.9 s, and Δγ

_{n}(0) = 64 mN/m, respectively.

**Figure 6.**Modulus and phase shift of the complex viscoelasticity measured for mixed C

_{13}DMPO/TTAB adsorption layers at water/hexane interface at an amplitude of 10% and temperature 15 °C (the surfactant concentrations in the aqueous phase are C

_{C13DMPO}= 2.0 × 10

^{−7}mol/dm

^{3}and C

_{TTAB}= 4.5 × 10

^{−5}mol/dm

^{3}). The lines are the results of calculations by using the FMM model, Equations (15)–(17), with the parameter sets α = 0.38, τ = 1.7 s, and Δγ

_{n}(0) = 29 mN/m.

**Figure 7.**Modulus and phase shift of the complex viscoelasticity measured for mixed C

_{13}DMPO/TTAB adsorption layers at water/hexane interface at an amplitude of 10% and temperature T = 20 °C (the surfactant concentrations in the aqueous phase are C

_{C13DMPO}= 2.2 × 10

^{−5}mol/dm

^{3}and C

_{TTAB}= 4.5 × 10

^{−5}mol/dm

^{3}). The lines are the results of calculations by using the FMM model, Equations (15)–(17), with the parameter sets α = 0.2, τ = 15 s, and Δγ

_{n}(0) = 16 mN/m.

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**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. Interfacial Dilational Viscoelasticity of Adsorption Layers at the Hydrocarbon/Water Interface: The Fractional Maxwell Model. *Colloids Interfaces* **2019**, *3*, 66.
https://doi.org/10.3390/colloids3040066

**AMA Style**

Loglio G, Kovalchuk VI, Bykov AG, Ferrari M, Krägel J, Liggieri L, Miller R, Noskov BA, Pandolfini P, Ravera F,
et al. Interfacial Dilational Viscoelasticity of Adsorption Layers at the Hydrocarbon/Water Interface: The Fractional Maxwell Model. *Colloids and Interfaces*. 2019; 3(4):66.
https://doi.org/10.3390/colloids3040066

**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. 2019. "Interfacial Dilational Viscoelasticity of Adsorption Layers at the Hydrocarbon/Water Interface: The Fractional Maxwell Model" *Colloids and Interfaces* 3, no. 4: 66.
https://doi.org/10.3390/colloids3040066