*3.4. Interfacial Dilational Rheology*

The above discussion was devoted to the study of the adsorption at interfaces with fixed surface areas. However, from a technological point of view, the understanding of the response of the interface against external mechanical perturbations is essential because this provides important insights into the relaxation processes involved in the equilibration of interfacial layers [25,48,62,63]. The dependences of the dilational viscoelastic moduli (ε represents the dilational elastic modulus and ε" the viscous modulus) on the surfactant concentration and the deformation frequency provide complementary information for the better understanding of the complexity of the mechanism involved in the equilibration of the interfaces, helping to give a more detailed picture of the physical processes governing the formation of adsorption layers from polymer-surfactant solutions [64]. It must be stressed that for both PDADMAC-SLMT and PDADMAC-SLES solutions, the values of ε" are negligible in relation to those of ε , with the ratio ε"/ε decreasing as the surfactant concentration increases. Hence, for the sake of simplicity only the behavior of ε will be discussed. Figure 5 shows the concentration dependences of the elastic modulus for PDADMAC-SLMT and PDADMAC-SLES layers.

The results indicate that the dependence of ε on the frequency is expected for the formation of layers at fluid interfaces, with ε increasing with the deformation frequency. Furthermore, the concentration dependence of ε is similar to that found for layers of surface active materials at fluid interfaces [46], with ε increasing with the surfactant concentration from the value corresponding to the clean interface, reaching a maximum and then dropping down again to quasi-null values for the highest surfactant concentrations. A careful examination of the values obtained for the elasticity modulus for each system indicate that PDADMAC-SLES layers present values that are more than twice those obtained for PDADMAC-SLMT solutions independent of the considered frequency. This is again indicative of the different features of the interfacial layers. For PDADMAC-SLES layers, the spreading of material along the interface leads to the formation of extended complexes that can build a cross-linked network, increasing the elastic modulus of the interfacial layers. This cross-linking process is not possible when the interfacial layer is formed by compact kinetically trapped aggregates, as in PDADMAC-SLMT layers, leading to lower values of the elastic modulus of the interface.

**Figure 5.** (**a**) Concentration dependences of the elastic modulus for PDADMAC-SLMT adsorption layers as were obtained from oscillatory barrier experiments performed at different frequencies. (**b**) Concentration dependences of the elastic modulus for PDADMAC-SLES adsorption layers, obtained from oscillatory barrier experiments performed at different frequencies. Note: ( and ) ν = 0.01 Hz; ( and -) ν = 0.05 Hz; (Δ and ) ν = 0.10 Hz. The lines are guides for the eyes. For the sake of clarity, only results corresponding to some of the explored frequencies (ν) are shown, with the other frequencies presenting similar dependences. The results correspond to PDADMAC-surfactant mixtures containing a fixed PDADMAC concentration of 0.5 wt.%, and left to age for one week prior to measurement.

The frequency dependences of the elasticity modulus can be described in terms of the rheological model proposed by Ravera et al. [64,65] (see Figure 6a for an example). According to this model the frequency dependence of the viscoelastic modulus accounts for the initial adsorption of the polymer-surfactant complexes at the water–vapor interface as a diffusion-controlled process that is coupled to a second step associated with the internal reorganization of the adsorbed layers. Thus, taking into account the aforementioned framework, it is possible to describe the complex viscoelastic modulus with the following expression:

$$\varepsilon^\* = \frac{1+\xi+i\xi}{1+2\xi,+2\xi^2} \left[ \varepsilon\_0 + (\varepsilon\_1 - \varepsilon\_0) \frac{1+i\lambda}{1+\lambda^2} \right] \tag{2}$$

where <sup>ξ</sup> <sup>=</sup> <sup>√</sup> ν*D*/ν, with ν<sup>D</sup> and ν being the characteristic frequency of the diffusion exchange and the frequency of deformation, respectively, and λ = ν1/ν, with ν<sup>1</sup> being the characteristic frequency of the extra relaxation process. Additionally, ε<sup>0</sup> and ε<sup>1</sup> represent the Gibbs elasticity and the high frequency limit elasticity within the frequency range considered, respectively. The validity of the discussed model, beyond confirming the complexity of the mechanisms involved in the equilibration of the interfacial layers formed by polyelectrolyte-surfactant solution, provides a description of the processes involved. It is expected that the equilibration of the interfacial occurs in the first stage by the diffusion-controlled adsorption of the kinetically trapped aggregates, and then such complexes undergo different reorganization processes depending on their nature. The existence of a two-step mechanism is in agreement with the picture proposed by Noskov et al. [45] for the equilibration of adsorption layers of PDADMAC-SDS at the water–vapor interface.

Figure 6b,c show the concentration dependences for the characteristic frequencies of the two dynamic processes appearing for the interfacial layers. As may be expected considering the different nature of the dynamic processes involved in the equilibration of the interfacial layer, ν1, which is the frequency corresponding to the interfacial relaxation process, presents higher values than those associated with the diffusional transport, νD, for both PDADMAC-SLMT and PDADMAC-SLES solutions. This behavior can be explained by assuming that the interfacial relaxation process, involving the reorganization of materials at the interface, occurs only when a certain degree of material is adsorbed at the interface.

The results show that both ν<sup>D</sup> and ν<sup>1</sup> increase in concentration for both studied systems. This increase can be explained in the case of ν<sup>D</sup> as a result of the enhanced surface activity of the kinetically trapped aggregates, as the surfactant concentration increases due to their higher hydrophobicity. Furthermore, the values of ν<sup>D</sup> are in a similar range for PDADMAC-SLMT and PDADMAC-SLES, which is in agreement with the similar origin of the process in both systems and the similarities of the complexes formed according to the above discussion. The slightly smaller values of ν<sup>D</sup> found for PDADMAC-SLMT than for PDADMAC-SLES may result from different sizes of the complexes formed in the solution. The dependence of ν<sup>1</sup> is assumed to be because of the increase of surfactant in solution leading to an increase of the surface excess of complexes at the interface, which facilitates their reorganization within the interface. The higher values of ν<sup>1</sup> for PDADMAC-SLMT solution than PDADMAC-SLES solutions, at almost one order of magnitude, are ascribable again to the differences in the structure of the interfacial layers. Thus, the diffusion of extended complexes within the interface can occur across longer distances within the interface than that of compact aggregates, and consequently this process involves longer time scales.

**Figure 6.** (**a**) Examples of frequency dependences of the elastic modulus for interfacial layers of PDADMAC-SLMT (-) and PDADMAC-SLES (-) and for solutions with surfactant concentration of 0.1 mM. Symbols represent the experimental data and the lines are the theoretical curves obtained from the analysis of the experimental results in term of the theoretical model described by Equation (2). (**b**) Concentration dependences of ν<sup>D</sup> for PDADMAC-SLMT (-) and PDADMAC-SLES (). (**c**) Concentration dependences of ν<sup>1</sup> for PDADMAC-SLMT (-) and PDADMAC-SLES (-). (**b**,**c**) Symbols represent the experimental data and the lines are guides for the eyes. The results correspond to PDADMAC-surfactant mixtures containing a fixed PDADMAC concentration of 0.5 wt.%, and left to age for one week prior to measurement.

#### **4. Conclusions**

The mechanisms involved in the equilibration of interfacial layers formed by adsorption of PDADMAC and two different anionic surfactants (SLMT and SLES) have been studied by surface tension (equilibrium and dynamics) and interfacial dilational rheology measurements. The combination of the interfacial characterization with studies on the association phenomena occurring in solution has evidenced that even the formation of kinetically trapped aggregates in the bulk occurs following similar patterns in both studied systems. These evolve following mechanisms depending of the specific chemical nature of the surfactant involved.

The equilibration of the interfacial layers formed by polyelectrolyte oppositely charged surfactants can be explained on the basis of a two-step mechanism. The first step is common to the different systems studied and is related with the diffusion-controlled incorporation of kinetically trapped aggregates at the water–vapor interfaces. Such aggregates can remain as compact aggregates at the interface, as in PDADMAC-SLMT solutions, or can undergo dissociation and spreading along the interface due to Marangoni flows, as in PDADMAC-SLES solutions. These different mechanisms result from differences in the hydrophobicity of the formed aggregates and the possibility to establish a cohesion interaction, such as a hydrogen bond, with the interface. On the basis of the obtained results, it can be concluded that there are no general laws governing the equilibration of the interfacial layers formed by the adsorption of polyelectrolyte-surfactant solutions at the fluid interface, with the process being primarily controlled by the specific nature of the chemical compounds involved and the interactions involved in the equilibration of the interface. This study contributes to the understanding of the fundamental basis describing the interfacial behavior of polyelectrolyte-surfactant solutions in conditions similar to that used in industrial application. Thus, the obtained result can help to exploit the interfacial behavior of these systems in technologically relevant conditions.

**Author Contributions:** Conceptualization, L.F.-P., A.A., and S.L.; Methodology, E.G., L.F.-P., A.A., and S.L.; Software, E.G.; Validation, E.G., F.O. and R.G.R.; Formal Analysis, E.G.; Investigation, E.G., L.F.P., A.A., S.L., F.O., and R.G.R.; Resources, R.G.R. and F.O.; Data Curation, E.G.; Writing—Original Draft Preparation, E.G.; Writing—Review and Editing, E.G., F.O. and R.G.R.; Visualization, E.G.; Supervision, E.G., F.O. and R.G.R.; Project Administration, R.G.R.; Funding Acquisition, R.G.R. and F.O.

**Funding:** This work was funded by MINECO under grant CTQ-2016-78895-R.

**Acknowledgments:** The CAI of spectroscopy from Universidad Complutense de Madrid is acknowledged for granting access to its facilities.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
