*3.3. Interactions of Particles with DPPC at the Interface*

Additional insights on the impact of the particles incorporation into DPPC monolayers can be obtained using the concepts of the thermodynamics of interfacial mixtures [63,64]. This approach helps to understand how the interactions between DPPC and particles modify the behavior of the mixed monolayer in relation to what happens in those cases without DPPC–particles interactions, i.e., ideal mixture conditions. The interfacial area of an ideally mixed monolayer at a fixed value of the surface pressure is defined as follows:

$$A^{\rm id} = \propto\_{\rm DFPC} A\_{\rm DFPC} + \propto\_{\rm P} A\_{\rm P} \tag{6}$$

where *A*DPPC and *A*<sup>p</sup> correspond to the areas per mass unit of DPPC and particles at the considered values of surface pressure for a monolayer of the pure compounds, respectively, and *x*DPPC and *x*<sup>p</sup> are referred to the weight fractions of DPPC and particles at the interface in the mixed monolayer, respectively. The differences associated with the mixing process at a fixed value of the surface pressure can be evaluated in terms of the excess area *A*<sup>E</sup> defined as follows:

$$A^{\to} = A\_{12} - A^{\text{id}} \tag{7}$$

where *A*<sup>12</sup> is referred to the area per mass unit corresponding to the mixed monolayer at a fixed value of the surface pressure. The *A*<sup>E</sup> provides information related to the mutual miscibility between the compounds forming the monolayer, which is governed by the cohesion forces existing between them. Figure 7 shows the dependences of the *A<sup>E</sup>* on the weight fraction of particles at the interface for different values of the surface pressure.

**Figure 7.** Dependences of the *A*<sup>E</sup> 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 dependences of *A*<sup>E</sup> on the surface pressure and the values of *x*<sup>p</sup> are common to DPPC monolayers upon the incorporation of SiO2 and CB particles. The results provide evidence of an enhanced interaction between the particles and the DPPC molecules at the interface with the increase of the packing of the film, i.e., *A*<sup>E</sup> decreases with the increase of the compression degree of the monolayers. This may be explained considering the existence of a forced cohesion of lipids and particles at the interface as a result of the reduction of the available interfacial area. On the other side, the increase of the weight fraction of particles at the interface leads to a decrease of *A*E, which may be ascribed to the enhancing of the DPPC–particles cohesion interaction. However, it is worth mentioning that the average cohesion between the DPPC molecules is reduced with respect to what happens in monolayers of pure DPPC at the water/vapor interface. The experimental results show that the interactions between DPPC and particles are repulsive (*A*<sup>E</sup> values > 0) until *x*<sup>p</sup> has reached high values when the excess area becomes negative. The repulsive interactions between particles and DPPC may be explained assuming the existence of a hindered lateral packing of the lipid due to the particles' incorporation. The enhanced miscibility appearing at the highest values of *x*p is explained considering the emergence of many-bodies interactions favoring the lateral packing of the monolayer [65].

The differences of the *A*<sup>E</sup> values found between monolayers after the incorporation of SiO2 and CB particles may be again ascribed to the different organization of the particles within the DPPC monolayers as a function of their chemical nature. Therefore, the lowest values of *A*<sup>E</sup> found upon the incorporation of SiO2 are explained by the better distribution of these particles within the lipid layer, and the possible role of the electrostatic interactions between the silanol groups on the surface of the particles and the ammonium terminal group of the DPPC.
