*2.3. Methods*

The Langmuir monolayers were studied using a Langmuir trough KSV Nima model KN2002 (Biolin Scientific, Espoo, Finland), equipped with two Delrin barriers allowing for symmetric compression/expansion of the free liquid surface. The total surface area of the Teflon trough is 243 cm2. The surface tension, γ, was measured using a force balance fitted with a paper Wilhelmy plate (Whatman CHR1 chromatography paper, effective perimeter 20.6 mm, supplied by Sigma Aldrich, St. Louis, MO, USA), ensuring a zero contact angle. The surface pressure, Π, is obtained as the difference between the surface tension of the pure water/vapor interface γ*<sup>w</sup>* and γ, i.e., Π = γ*<sup>w</sup>* − γ.

The quasi-equilibrium isotherms for the monolayers were obtained measuring the surface pressure as the interfacial area available for the monolayer, *A*, is reduced at a fixed compression velocity of 2 cm2/min, which is equivalent to a compression rate (Δ*A*/*A*0)/Δ*t* of about 10−<sup>5</sup> s<sup>−</sup>1, with Δ*A*/*A*<sup>0</sup> being the amplitude of the deformation, represented as the ratio between the change of area Δ*A* (amplitude of deformation) and the reference interfacial area *A*<sup>0</sup> (generally the area in which the compression is started), and Δ*t* the time needed for the deformation. This compression rate allows for avoiding an undesired non-equilibrium effects during the determination of the isotherms [44].

The Langmuir trough also enables the study of the effects of the incorporation of nanoparticles on the dilational rheology of the DPPC monolayers using the oscillatory barrier method. A detailed description of the foundations of this method can be found elsewhere [45–47]. The oscillatory barrier method allows for evaluating the modulus of the complex dilational viscoelasticity *E* = Δγ/(Δ*A*/*A*), i.e., the variation, of the surface tension γ as a result of a harmonic change at a controlled frequency ν (in a range of frequencies from 10−<sup>3</sup> to 0.15 Hz) of the interfacial area which is written as follows:

$$A(t) = A^0 + \Delta A \sin(2\pi\nu t). \tag{1}$$

The harmonic change of the interfacial area (strain) results in a stress response ΔΠ = <sup>Π</sup><sup>0</sup> <sup>−</sup> <sup>Π</sup>(*t*), which is defined as the change of surface pressure between the reference state Π<sup>0</sup> and the instantaneous value of the surface pressure Π(*t*). When the deformation presents a small amplitude, i.e., deformation within the linear regime, the stress response also follows a sinusoidal profile with the same frequency than the deformation:

$$
\Pi(t) = \Delta \Pi \sin(2\pi \nu t + \phi),
\tag{2}
$$

where φ is a phase shift accounting for a possible delay of stress response (surface pressure change) in relation to the strain (area deformation). Considering the above-mentioned linear response, the stress can be considered proportional to the deformation *u*(*t*) = Δ*A*/*A*<sup>0</sup> (elastic term) and to the rate of deformation d*u*(*t*)/d*t* (viscous term), which allows one to write the stress as:

$$
\Pi(t) = \varepsilon u(t) + \eta(\text{d}u(t)/\text{d}t),
\tag{3}
$$

with ε and η being the dilational elasticity and viscosity, respectively. Considering the definition given by Equation (3) and assuming a generic harmonic perturbation, it is possible to obtain a definition for the complex dilational viscoelasticity:

$$
\varepsilon^\* = \varepsilon + 2\pi\nu\eta\mathbf{i} \tag{4}
$$

where i = (−1)1/2. The analysis of the curves corresponding to the strain and stress in terms of Equations (1) and (2) provide information about their amplitudes and the phase shift, enabling for the calculation of the dilational viscoelasticity. In the reported experiments, the amplitude of the dilational deformation *u(t)* = 0.01 was adopted, which allows the response to remain within the linear regime. It is worth noting that the conditions considered in our work for the evaluation of the mechanical response of lipid layers, and how particles' incorporation impact such response, are far from those corresponding to the characteristic values of the dynamics processes involved in biologically relevant systems, e.g., respiratory cycle, where higher values for the frequency of the deformation and its amplitude are expected. In the particular case of the respiratory cycle, the frequency and the deformation amplitude assume values around 0.3 Hz and 0.30–0.40, respectively [48]. However, the evaluation of the dilational response within the linear regime provides helpful information for analyzing the impact of incorporation of particles on the relaxation mechanisms leading to equilibration of the lipid layers, which serve as a preliminary assay towards the understanding of more complex dynamics situations than those appearing in biologically relevant systems.

A Brewster Angle Microscope Mulstiskop from Optrel (Sinzing, Germany) fitted with a He-Ne laser (λ = 614 nm) and coupled to the Langmuir through was used to obtain information about of the lateral organization of lipids and particles at the interface on the basis of Brewster Angle Microscopy (BAM) images of the interfacial textures.

## **3. Results**
