## 2 Materials, Modeling and Characterization
### Modeling  
The first step is to calculate the transfer impedance \(Z\) of a circular micro-aperture of radius \(a\), assimilated to a short cylindrical tube whose inner axial particle velocity \(\overline{v}\), averaged over the tube cross-sectional area, is driven by the pressure difference \(\Delta p\), with harmonic time-dependence (\(\mathrm{e}^{\mathrm{i}\omega t}\)), exerted on either side of the tube, so that \(\Delta p=Z\,\overline{v}\). Following [PERSON]'s derivation of \(Z\) for micro-perforations filled with a viscous fluid [1], \(v\) satisfies the linearized equation of momentum conservation in cylindrical coordinates:  
\[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial v}{\partial r} \right)-\mathrm{j}\left(\frac{\mathrm{Sh}}{a}\right)^{2}\!\!v=\frac{1}{\mu} \frac{\Delta p}{t}, \tag{1}\]  
which is expressed in terms of the Shear number \(\mathrm{Sh}=a/\delta_{v}\) and the ratio between the aperture radius and the thickness \(\delta_{v}=\sqrt{\mu/\left(\omega\,\rho_{0}\right)}\) of the viscous boundary layer, with \(\mu\) as the air dynamic viscosity, \(\rho_{0}\) as the air density and \(\omega\) as the angular frequency. Depending on the values of the Knudsen number, \(\mathrm{Kn}=\widetilde{\Lambda}/a\), the ratio between the molecular mean-free path of air particles (\(\widetilde{\Lambda}=68\) nm under normal conditions of pressure and temperature) and the aperture radius \(a\), different boundary conditions are fulfilled over the aperture wall surface \(\Sigma\). If \(\mathrm{Kn}\leq 10^{-3}\), which is the case for MCP2, UMPPs and MPPs, continuum laws stay valid, and a no-slip boundary condition is satisfied. \[v_{\Sigma}=0, \tag{2}\]  
Figure 2: (**a**) Magnification (factor 10) of MCP2 revealing 60\({}^{\circ}\) staggered near-circular micro-channels with diameters of 200 \(\upmu\)m and hole pitches of 268 \(\upmu\)m; (**b**) Photograph of UMPP showing aligned distributions of circular holes with diameters of 0.3 mm and hole pitches of 3 mm; (**c**) Photograph of MPP showing aligned distributions of circular holes with diameters of 0.88 mm and hole pitches of 3 mm. which is fulfilled over the aperture walls (\(r=a\)). If \(10^{-3}\leq\text{Kn}\leq 10^{-1}\), which is the case for MCP1, a slip-flow regime holds due to tangential momentum exerted on the aperture walls [12]. Equation (2) is then replaced by the following boundary condition [13]:  
\[v_{\Sigma}=B_{v}\frac{\partial\varpi_{\Sigma}}{\partial r}, \tag{3}\]  
which is satisfied over the aperture walls, with \(B_{v}=\text{Kn}(2-\sigma_{v})/\sigma_{v}\) expressed in terms of \(\sigma_{v}\), the tangential momentum accommodation coefficient, which is equal to 0.9 for an air-glass interface [14]. Solving Equations (1) and (2) for \(v\), dividing by the perforation ratio and adding outer correction terms lead to the effective transfer impedance for an MPP [1]:  
\[Z_{\text{MPP}}=\frac{\mu\,\text{Sh}}{\sqrt{2}a\,\sigma}+\text{j}\omega\frac{ \rho_{0}\,t}{\sigma}\bigg{\{}\Big{[}1-F\Big{(}\text{Sh}\sqrt{-\text{j}}\Big{)} \Big{]}^{-1}+\frac{\delta\,a}{t}\bigg{\}}, \tag{4}\]  
which is expressed in terms of \(F(x)=2J_{1}(x)/[x\,J_{0}(x)]\) with the \(J_{1}\) and \(J_{0}\) Bessel functions of the first kind of orders 1 and 0, respectively. The inner term, \(\text{j}\omega\,\rho_{0}\,t\,\left[1-F\big{(}\text{Sh}\sqrt{-\text{j}}\big{)} \right]^{-1}\sigma^{-1}\), accounts for viscous dissipation and inertial effects inside the holes, whereas the outer terms, \(\mu\,\text{Sh}/\big{(}\sqrt{2}a\,\sigma\big{)}\) and \(\text{j}\omega\,\rho_{0}\delta\,a/\sigma\), respectively account for outer frictional losses and air motion inertial effects at the inlet/outlet of the holes, with \(\delta=16/(3\pi)\), the added-length correction factor [1]. An \"effective mass\" per unit area, \(\rho_{0}(t+\delta\,a)/\sigma\), is defined using Equation (4), which steadily increases for MCP2, MPPs and UMPPs, but that is negligible for MCP1.