# Propagation of Elastic Waves in a One-Dimensional High Aspect Ratio Nanoparticle Phononic Crystal
## 3 Results and Discussion
### Infiltration with a Liquid  
In this last section, we have filled the grooves between the nanoridges with a polymer liquid with the commercial name 'Cargille'. This polymer has a density of 1000 kg/m\({}^{3}\) and a longitudinal velocity of 1670 m/s. The contrast between epoxy and Cargille comes from their different physical natures, i.e., liquid and solid, while the densities are almost the same (see Table 1). The nanoridge structure that was considered for the calculations presents the following parameters: \(a\) = 630 nm, \(h\) = 2.7 um, and \(w\) = 250 nm. In the numerical FEM model, free stress boundary conditions have been applied at the surface of the liquid, on top of the unit cell. At the liquid/solid interface, the normal component of the displacement, as well as the normal stress, are continuous. We do not include capillary forces, which at the considered wavelengths should be significant at much lower frequencies than those (GHz) considered in this work. The validity of the model is also supported by our previous investigations of solid/liquid phononic crystal, in which we obtained a good agreement between the experimental and the theoretical band structure using the same numerical model [24; 28]. As the structure is mixed, i.e., contains simultaneously a liquid and a solid, we have to adapt the method of calculation to these two different media. First, the solid part follows the elastic wave equation:  
\[-\rho\omega^{2}\mathbf{u}=\boldsymbol{\
abla}\boldsymbol{\sigma} \tag{3}\]  
where \(\rho\) represents the density, \(\omega\) represents the angular frequency, \(\mathbf{u}\) represents the displacement field vector, and \(\sigma\) represents the stress tensor. Second, the liquid part of the structure is computed with the pressure equation of propagation:  
\[\Delta p=-\frac{1}{c^{2}}\omega^{2}p \tag{4}\]  
where \(p\) represents the acoustic pressure, and \(c\) represents the sound velocity in the liquid. At the solid/liquid interface, the boundary conditions are given by the continuity of the displacements and stress normal components. These conditions are given by the two equations:  
\[\left\{ \begin{array}{c}\mathbf{n}.\frac{1}{\rho}\
abla p=-\mathbf{n}.\omega ^{2}\mathbf{u}\\ \sigma.\mathbf{n}=p.\mathbf{n}\end{array} \right. \tag{5}\]  
where \(\mathbf{n}\) is the normal direction at the solid/liquid interface, \(\boldsymbol{\sigma}\) is the stress tensor, \(\mathbf{u}\) is the displacement field in the solid, and \(p\) is the pressure field in the liquid part. We then calculated the dispersion curves of the phononic crystal infiltrated with the liquid (Cargille) in the two directions of propagation: parallel and perpendicular to the nanoridges. For the direction parallel to the nanoridges, we applied a band-sorting calculation to select, among the huge number of modes, those that presented an 'in-plane' polarization \(P_{xy}\) that was close to 1 (Equation (1)). To do so, the components of the displacement field in the liquid part have been deducted from the following equations:  
\[\left\{ \begin{array}{c}u_{x}=\frac{1}{\rho\omega^{2}}\frac{\partial p}{ \partial x}\\ u_{y}=\frac{1}{\rho\omega^{2}}\frac{\partial p}{\partial y}\\ u_{z}=\frac{1}{\rho\omega^{2}}\frac{\partial p}{\partial z}\end{array} \right. \tag{6}\]  
The results of the calculations are presented in Figure 8a where only the modes that satisfied the condition \(P_{xy}(w,q)\geq 80\%\) were kept. These curves can be compared with Figure 3a for the structure without the infiltration. Let us remind that in the latter case, the modes have been attributed to fundamental Lamb-like modes of the nanoridges and their harmonics. Similarly, we have performed the calculation of a 1D infinite multilayer array alternately constituted of epoxy and Cargille. The corresponding dispersion curves are presented with blue color on the same figure.