*3.1. PDADMAC-Surfactants Assembly in Solution*

The equilibrium condition implies that the chemical potential of all the species in both the bulk and at the interfaces are the same. Therefore, any physical understanding of the latter implies knowledge of the behavior of the different species in the bulk. Figure 1a shows the surfactant-binding isotherms deduced from electromotive force (EMF) measurements. Comparing the curves of EMF obtained for surfactants and PDADMAC-surfactant solutions, it is possible to obtain the binding isotherms for the corresponding surfactant to PDADMAC chains following the approach described by Mezei and Meszaros [50]. The results point out a high degree of binding over the whole range of studied compositions, providing an additional confirmation of the high efficiency of PDADMAC in binding anionic surfactants. Campbell et al. [38] found for PDADMAC-SDS solutions binding degrees of surfactant to PDADMAC close to 0.3 in the vicinity of the isoelectric point (surfactant concentration around 0.2 mM). The extrapolation of such results in similar conditions to those considered in this work, i.e., polymer concentration 50-fold the one used by Campbell et al. [38,52], and assuming that the binding is not significantly modified either for the surfactant structure or for the differences in the ionic strength, takes the binding degree at charge neutralization to a value <1%. This is just the situation found here, where binding isotherms evidence that the amount of free surfactant in solution remains below 10%, even for the highest surfactant concentrations. The low concentration of free surfactant in solution allows us to assume hereinafter that the bulk has a negligible free-surfactant concentration.

Figure 1b shows the dependence of the optical density of the samples on the surfactant concentration for the solutions of PDADMAC and the two surfactants. Similar qualitative concentration dependences of the optical density were found for both polyelectrolyte-surfactant systems. It may safely be expected that all of the studied compositions for PDADMAC-surfactant solutions fall in an equilibrium one-phase region, showing optically transparent solutions. This comes from the fact that the number of surfactant molecules available in solution is not high enough to neutralize the charge of all the monomers in the polyelectrolyte chains, thus leading to the formation of undercompensated cationic complexes in solution. Indeed, considering the high polymer concentration, simple calculations suggest the existence of around 36 monomers for each surfactant molecule for a surfactant concentration of approximately 1 mM. Therefore, assuming the complete binding of surfactant molecules to the polymer chains, around 35 monomers remain positively charged in the complexes, supporting the

formation of transparent samples within the entire concentration range. However, contrary to what was expected for solutions with compositions far from the neutralization, the solutions formed by undercompensated complexes show an increase of the turbidity for the highest surfactant concentration. Therefore, for such concentrated solutions, the system should get close to the onset of where the two phase region occurs, even though no signature of charge neutralization was found from electrophoretic mobility measurements. This results from the mixing protocol used for solution preparation, which proceeds during the initial step by mixing a concentrated polymer solution with a concentrated surfactant solution. This precursor solution is them diluted up to the stated bulk composition. It may be expected that this methodology leads, due to the Marangoni stress created, to the formation of persistent kinetically-trapped aggregates that persist even upon dilution, leading to the appearance of a two-phase system far off the real neutralization point of the system [7,54]. These results contrast with those reported in other mixtures studied in the literature. In such systems, the increase on the optical density of the samples results from the formation of charge compensated complexes. The last is associated with the transition from a composition region, in which the charge of the complexes is governed by the excess of charged monomers to another region, in which the excess of bound surfactant to the polymer chain controls the charge of the formed complexes, i.e., a charge inversion transition [20,38,55]. The above results show that the production of kinetically-trapped aggregates during mixing can lead to turbid mixtures far from the real equilibrium phase separation [3]. Preliminary results have shown that the above discussed scenario changes significantly when the interaction of PDADMAC with betaine derived surfactants is considered. In such systems, even the polyelectrolyte-surfactant interactions occur through the negatively charged group in the terminal region of the polar head, the formation of kinetically-trapped aggregates is hindered, probably as a result of the electrostatic repulsion associated with the positively charged groups in the zwitterionic surfactant [56].

**Figure 1.** (**a**) Binding isotherms for surfactants on PDADMAC as a function of the initial concentration of surfactant in bulk. (**b**) Surfactant concentration dependences of the optical density of the solution, measured at 400 nm. Note: (-) = PDADMAC-SLMT; (-) = PDADMAC-SLES solutions. 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.

#### *3.2. Equilibrium Adsorption at the Water–Vapor Interface*

The evaluation of the surface pressure of solutions containing surface active compounds helps to understand the mechanisms involved in the equilibration of the water–vapor interface. Figure 2a shows the surface pressure dependences on the surfactant concentrations and on the PDADMAC concentration for the adsorption of the two surfactants and the polymer at the water–vapor interface (note that all solutions were prepared with the same pH and inert salt concentration as the polyelectrolyte-surfactant solutions). The results show that the surface activity of PDADMAC is negligible, at least up to concentrations that are 20-fold the one used in our work. This is in good agreement with the previous study by Noskov et al. [57] and with the negligible surface excess found for PDADMAC using neutron reflectrometry [38].

**Figure 2.** Results obtained using a drop profile analysis tensiometer: (**a**) Surface pressure dependence on surfactant concentration for the adsorption of pure SLES (-) and SLMT () at the water–vapor interface; cmc for both surfactants is marked. The inserted panel represents the surface pressure dependence on PDADMAC concentration for the adsorption of pure PDADMAC at the water–vapor interface. (**b**) Surface pressure dependence of SLMT concentration for pure SLMT () and PDADMAC–SLMT (-) solutions. (**c**) Surface pressure dependence of SLES concentration for pure SLES (-) and PDADMAC–SLES (-) solutions. The lines are guides for the eyes. The results for PDADMAC-surfactant mixtures correspond to mixtures containing a fixed PDADMAC concentration of 0.5 wt.%, and left to age for one week prior to measurement.

The adsorption behavior of SLMT and SLES is the expected for typical ionic surfactants. The Π increases with the bulk concentration up to the point that the surfactant concentration overcomes the threshold defined by the critical micellar concentration (cmc). Afterwards, Π remains constant with further increases of surfactant concentration. It is worth mentioning that the results obtained using different tensiometers (surface force tensiometer with Pt Wilhelmy as a probe plate and drop profile analysis tensiometer) agree within the combined error bars for the adsorption of both surfactants at the water–vapor interface. The surface pressure isotherms allow one to estimate the cmc of the pure surfactants, which showed values of around 10−<sup>2</sup> and 10−<sup>1</sup> mM for SLES and SLMT, respectively.

The comparison of the results obtained for the adsorption of pure surfactants at the water–vapor interface with those obtained for the adsorption of PDADMAC-surfactant solutions shows that for the lowest surfactant concentrations the surface pressure values are similar for pure surfactant and polyelectrolyte-surfactant solutions. This is the result of the low coverage of the interface (see Figure 2b,c). In such conditions, the surface excess is not high enough to produce any significant change in the surface free energy, and hence the Π values remain close to those of the bare water–vapor interface. The increase of the surfactant concentration leads to the increase of Π for both surfactant and polyelectrolyte-surfactant solutions. This increase starts for surfactant concentrations around one order of magnitude lower when polyelectrolyte-surfactant solutions are considered, which is a signature of the existence of a synergetic effect for the increase of the surface pressure as a result of the interaction in the solution of the polyelectrolyte and the surfactant. This is in agreement with previous results reported in the literature for several polyelectrolyte-surfactant systems [3,31,50,58]. The above-mentioned synergetic effects do not influence the adsorption behavior of solutions formed by PDADMAC and zwitterionic surfactants derived from the betaines, as was shown in preliminary results. This could be ascribed to the aforementioned differences in the aggregation process occurring in the bulk [56].

The study of the surface tension isotherms obtained for polymer-surfactant mixtures using different tensiometric techniques can help to understand the complexity of the interfacial behavior appearing when faced with these systems. Figure 3a,b shows that the surface tension isotherms obtained using different tensiometers reveal different features for PDADMAC-SLMT and PDADMAC-SLES solutions. PDADMAC-SLES solutions show similar surface pressure isotherms within the combined error bars, independent of the tensiometer used, and no evidences of the appearance of non-regular trends, either as surface tension peaks [38] or surface tension fluctuations [7], on the dependence of the surface pressure with the surfactant concentration were found. This contrasts with the results obtained for PDADMAC-SLMT solutions, in which the use of a surface force tensiometer with a Pt Wilhelmy plate as probe led to results that were significantly different to those obtained using the other tensiometers. The existence of such differences was previously reported in a study by Noskov et al. [31].

**Figure 3.** Surface pressure isotherms for solutions of PDADMAC with the two surfactants, obtained using different tensiometers. (**a**) Isotherms for PDADMAC-SLMT solutions. (**b**) Isotherms for PDADMAC-SLES solutions. Note: ( and ) Surface force tensiometer with Pt Wilhelmy plate as contact probe; ( and -) surface force tensiometer with paper Wilhelmy plate as contact probe; (Δ and ) drop profile analysis tensiometer. The lines are guides for the eyes. The results correspond to PDADMAC-surfactant mixtures containing a fixed PDADMAC concentration of 0.5 wt.% left to age for one week prior to measurement.

The differences found in the tensiometric behavior of PDADMAC-SLES and PDADMAC-SLMT solutions are correlated to differences in the equilibration mechanism of the interface. Assuming that the assembly of the polyelectrolyte-surfactant in solutions leads to the formation of kinetically trapped aggregates in both cases, this can evolve following different mechanisms upon adsorption at fluid interfaces. For PDADMAC-SLMT solutions, the appearance of surface tension fluctuations far from the phase separation region may be associated with the fact that upon adsorption at the water–vapor interface of the kinetically trapped aggregates can remain as isolated aggregates embedded at the interface. These do not dissociate spontaneously to form a kinetically trapped film at the interface. As a consequence, the trapped aggregates may adsorb onto the rough surface of the Pt Wilhelmy plate, changing its contact angle, which results in non-reliable surface tension values for the considered aggregates. This scenario is in agreement with the neutron reflectometry results obtained by Llamas et al. [50]. Their results showed a monotonic increase of the surface excess at the interface with the surfactant concentration, confirming that the surface tension fluctuations do not result from fluctuations of the interface composition. The behavior changes significantly when the adsorption of PDADMAC-SLES solutions is considered. In this case, the absence of surface tension fluctuation or significant differences in the results obtained using different tensiometers suggests the existence of dissociation and spreading of the kinetically trapped aggregates upon adsorption at the interface. Thus, the equilibration of the interface after the adsorption of the kinetically trapped aggregates occurs because of its dissociation, which is followed by the spreading of the complexes across the interface as a result of Marangoni flow associated with the lateral heterogeneity of the interface [38,42,50,59]. The differences in the adsorption mechanisms of PDADMAC-SLES and PDADMAC-SLMT complexes at the water–vapor interface may be explained on the bases of the molecular structures of the surfactant and the possibility to establish a cohesion interaction with the surrounding media. SLMT presents a hydrophobic tail formed by an alkyl chain, which tends to minimize the number of contact points with water, which favors the formed aggregates remaining as compact aggregates at the water–vapor interface upon adsorption. On the contrary, the presence of oxyethylene groups in SLES makes the dissociation and spreading of the complexes easier as a result of the possible formation of hydrogen bonds of the surfactant molecules with water. Surprisingly, studies on the adsorption of PDADMAC-SLES and PDADMAC-SLMT mixtures onto solid surfaces have evidenced a scenario compatible with that described for the adsorption at the fluid interfaces, where PDADMAC-SLES films present a topography reminiscent of the formation of extended complexes attached to the interface, whereas PDADMAC-SLMT films present a higher lateral heterogeneity [51,60]. Further confirmation of the discussed mechanisms may be obtained from the analysis of the adsorption kinetics at the water–vapor interface of the polyelectrolyte-surfactant solutions.
