*3.4. E*ff*ect of Particles' Incorporation in the Dilational Response of DPPC Langmuir Monolayers*

The above discussion was focused on the impact of the particles' incorporation on the equilibrium properties of DPPC monolayers. However, biological systems are highly dynamic system, and hence the study of the effects associated with particles incorporation into DPPC monolayers in the response against mechanical deformations is a useful tool for a preliminary evaluation of the impact of the particles on the functionality of lipids layers. For this purpose, the influence of particles in the response of the DPPC to dilational deformations has been studied using the oscillatory barrier method at a fixed deformation amplitude within the linear response regime (1% of the initial area). These studies inform the modification of the relaxation mechanisms involved in the re-equilibration of the lipid layer as a result of the incorporation of particles [45,59]. For this purpose, the analysis of the frequency (ν) dependences of the viscoelastic modulus (|*E*|) for the pure DPPC monolayers and upon the incorporation of the particles is performed. Figure 8 shows, for the sake of an example, some of the frequency dependences obtained for the interfacial dilational viscoelasticity at different surface pressures for the mixed monolayers containing different weight fractions of particles.

Most of the viscoelastic modulus–deformation frequency curves obtained show the existence of inflexion points which are associated with the characteristic frequency of the reorganization of the molecules and particles within the interface The incorporation of particles, independently of their chemical nature, modifies the relaxation mechanism of the lipid monolayer, i.e., the characteristic relaxation frequencies, as is evidenced from the experimental curves. The characteristic relaxation frequencies can be estimated fitting the experimental data to the following theoretical expression which enables the description of interfacial relaxation occurring in insoluble adsorption layers [66]

$$|E| = \left[ (E\_1 \, ^2 + \lambda^2 E\_0 \, ^2)/(1 + \lambda^2) \right]^{1/2},\tag{8}$$

where λ = νR/ν, with ν<sup>R</sup> being the characteristic relaxation frequency, and *E*<sup>0</sup> and *E*<sup>1</sup> are the lower and upper limits of the elasticity within the considered frequency range. Note that, when insoluble monolayers are concerned, *E*<sup>0</sup> coincides with the quasi-static dilational elasticity obtained from the isotherm. The theoretical curves obtained using the model defined by Equation (8) are shown in

Figure 8 together with the experimental data, and the values of the characteristic frequencies ν<sup>R</sup> of the dilational response of mixed monolayers are shown in Figure 9.

**Figure 8.** Experimental (symbols) and calculated using Equation (8) (lines) dependences of the interfacial dilational viscoelasticity modulus on frequency. (**a**–**c**) dependences of |*E*| on ν for DPPC after the incorporation of different *x*<sup>p</sup> values at fixed surface pressure values (data for CB): (•, **—**) 0, (•,**—**) 0.10, (•,—) 0.33 and (•,—) 0.75. (**d**,**e**) dependences of |*E*| on ν for DPPC after the incorporation of particles with different chemical nature at fixed surface pressure values and *x*<sup>p</sup> = 0.10: (•, **—**) pure DPPC, (•,**—**) DPPC upon incorporation of SiO2 particles and (•,—) DPPC upon incorporation of CB particles. (**g**,**f**) Dependences of |*E*| on ν for DPPC after the incorporation of particles with different chemical nature at fixed surface pressure values and *x*<sup>p</sup> =0.75: (•, **—**) pure DPPC, (•,**—**) DPPC upon incorporation of SiO2 particles and (•,—) DPPC upon incorporation of CB particles. (Note that the value of the |*E*| for the lowest frequency value was assumed as the quasi-static dilational elasticity obtained from the isotherm, with the value of the frequency being the compression ratio which for the here experiments is about 10−<sup>5</sup> Hz).

**Figure 9.** ν*<sup>R</sup>* dependences, obtained using Equation (4), on the particle weight fraction, *xp*, for different values of the surface pressure Π for DPPC monolayers upon incorporation of CB (**a**) and SiO2 (**b**): (**—**•**—**) 3 mN/m, (**—**•**—**) 7.5 mN/m, (**—**•**—**) 20 mN/m and (—•—) 40 mN/m. Notice that the lines are guides for the eyes.

The analysis of the viscoelastic modulus-deformation frequency curves obtained for monolayers of pure DPPC shows the existence of an inflexion point, with the exception of that obtained for the lowest value of Π. The absence of such relaxation process for the lowest value of the surface pressure is explained assuming the low interfacial density, which allows the free reorganization of the DPPC molecules within the interface. This takes the relaxation process to time-scales below (higher frequencies) those tested by oscillatory barrier experiments.

The results show that the incorporation of particles modifies the relaxation mechanism of DPPC molecules at the water/vapor interface from the lowest values of surface pressure. SiO2 and CB particles' incorporation leads to the emergence of a relaxation process at Π values about 3 mN/m (LE phase for pure DPPC monolayers), with the time-scale for such relaxation process being faster when the incorporation of CB particles is concerned (0.001–0.01 Hz for CB particles incorporation vs. 0.0001–0.001 Hz for SiO2 ones). This results from the most important role of the steric hindrance associated with the incorporation of SiO2 particles, which makes it the lateral reorganization of the lipid molecules within the monolayer more difficult than when the incorporation of CB particles is considered. The origin of the emergence of a relaxation process may be explained considering the increase of the interfacial density associated with the presence of the particles, which reduces the time-scales involved in the reorganization of the molecules at the interface. The decrease of the time-scale involved in the reorganization process is stronger as the interfacial density of the particles increases. Therefore, it is possible to assume a slowing down of the velocity of this relaxation as a result of an increased role of the steric hindrance.

The approaching of the LE–LC coexistence phase provides evidence again of the differences on the effect of the incorporation of SiO2 and CB particles. The incorporation of CB particles does not modify significantly the relaxation mechanism, with a relaxation process presenting νR~10<sup>−</sup>3–10−<sup>4</sup> Hz appearing independently of the considered monolayer, i.e., for pure DPPC monolayers and upon the incorporation of CB particles. This is explained considering that, within the phase coexistence region, the nucleation of the LC phase associated with the disappearance of the LE one is found in both cases. Thus, the relaxation process should be ascribed to the exchange of the lipid molecules between the LC and LE phase. However, the introduction of SiO2 particles leads to a slowdown of the relaxation process for almost one order of magnitude [31]. This may be explained considering that the incorporation of SiO2 particles hinders partially the phase coexistence as result of the stronger steric hindrance associated with the particles incorporation which modifies the lateral reorganization of the lipid molecules at the interface.

Once the phase coexistence is overcome, the incorporation of CB does not lead to any significant change of the relaxation mechanism, with the relaxation frequency remaining in values about 10−<sup>4</sup> Hz, independently of the considered state and the *x*<sup>p</sup> value. This may be understood assuming that CB particles are simple obstacles that limit the average cohesion of the DPPC monolayers, shifting the phase behavior, without any significant change on the lipid lateral packing. However, the situation appears to be different when the incorporation of SiO2 particles is concerned, and the relaxation mechanism was found to be dependent on the *x*<sup>p</sup> value. The incorporation of SiO2 particles increases the value of ν<sup>R</sup> with the *x*<sup>p</sup> value up to reach a maximum value, and then a decrease of ν*<sup>R</sup>* with the increase of the *x*<sup>p</sup> was observed with the enhancing of the lateral packing of the monolayer. This is explained assuming the complexity of the interactions balance involved in DPPC monolayers upon the incorporation of SiO2 particles, which leads to the existence of coupled dynamics on the rheological response of the mixed monolayers against dilational deformations. It is worth mentioning that the increase of the interfacial density of the particles may induce a similar lateral packing of the monolayer for lower values of the surface pressure, and this may explain the characteristic features found for the *x*p dependence of the relaxation frequencies. Notice that, for the highest values of the lateral packing, the characteristic relaxation frequency appears to be similar for pure DPPC monolayers and upon incorporation of SiO2 that corresponds to the mixed layer appearing in larger time-scales. This subtle difference can be again ascribed to the role of the steric hindrance interactions, which makes it possible that the relaxation process may include complex rearrangements involving both the particles and the lipid molecules.

Figure 10 shows the dependences of *E*<sup>1</sup> on the *x*<sup>p</sup> value. The *E*<sup>1</sup> values obtained prove clearly an increase with the *x*p value when the incorporation of CB particles is considered; this may be understood considering that particles occupy partially the area available for the lipid organization, i.e., behave as obstacles, driving to a prior packing of the DPPC at the interface (packing occurs at higher values of the reduced area). About the dependences *E*1, the impact of SiO2 approximates the CB particle one. This aspect can be explained assuming the importance of the occupancy of the interfacial area by the incorporated particles in their impact, independently of their chemical nature.

**Figure 10.** *E*<sup>1</sup> dependences, obtained using Equation (4), on the particle weight fraction, *x*p, for different values of the surface pressure Π for DPPC monolayers upon incorporation of CB (**a**) and SiO2 (**b**): (**—**•**—**) 3 mN/m, (**—**•**—**) 7.5 mN/m, (**—**•**—**) 20 mN/m and (—•—) 40 mN/m. Notice that the lines are guides for the eyes.
