## Appendix B Derivation of the Model of the Leakage Rates  
Persson's Theory of Contact Mechanics--Derivation of the Average Separation Height \(u(p)\) of Two Contact Surfaces  
Consider the frictionless contact between two elastic solids with surface profiles \(h_{1}(x)\) and \(h_{2}(x)\). The elastic contact problem of these solids is equivalent to the contact between a rigid solid (substrate) with surface profile \(h(x)=h_{1}(x)+h_{2}(x)\) being in contact with an elastic solid (block) with a flat surface, whose Young's modulus \(E\) and Poisson ratio \(\
u\) are given by [35]:  
\[\frac{1-\
u^{2}}{E}=\frac{1-\
u_{1}^{2}}{E_{1}}+\frac{1-\
u_{2}^{2}}{E_{2}},\] (A21)  
where \(E_{1},E_{2}\) and \(\
u_{1},\
u_{2}\) are the Young's moduli and the Poisson ratios of the two solids, respectively. For the valves without Parylene coating, the effective Young's modulus for the steel-steel contact is of the same order of magnitude as the Young's modulus of stainless steel. For the valves with Parylene coating, the effective [PERSON]'s modulus is significantly decreased since the [PERSON]'s modulus of parylene is much smaller than the [PERSON]'s modulus of steel [26; 27]. The contact between the two surfaces depends on the pressure that squeezes them together. When the applied contact pressure p is increased, the average surface separation \(u\) is decreased, so the pressure p is again a function of \(u\): \(p=p(u)\). In order to setup an equation, we consider the energy balance \(U_{\text{el}}(u)\) stored in the deformed surface in the contact region and the mechanical work performed by the applied external pressure \(p_{contact}\)[30]:  
\[\int_{u}^{\infty}\!\!du^{\prime}A_{0}p\big{(}u^{\prime}\big{)}=U_{\text{el}}(u) \tag{100}\]  
or equivalently:  
\[p(u)=-\frac{1}{A_{0}}\frac{dU_{\text{el}}}{du}, \tag{101}\]  
where \(A_{0}\) is the nominal contact area. For small pressures, it can be assumed that the elastic energy is linearly proportional to the load, i.e., \(U_{\text{el}}(u)=u_{0}A_{0}p(u)\)[30]. The characteristic length \(u_{0}\) depends on the surface roughness but not on the contact pressure [30]. Thus, Equation (101) can be simplified to:  
\[p(u)=-u_{0}\frac{dp}{du} \tag{102}\]  
Or:  
\[p(u)=p_{c}e^{-u/u_{0}}, \tag{103}\]  
which is in good agreement with experimental data [30]. Many surfaces of interest are self-affine fractals for \(q_{0}<q<q_{1}\), where \(q_{0}\) and \(q_{1}\) are the lower and upper cutoff wave vectors. If a self-affine surface is magnified in a direction perpendicular to it, then the surface 'looks the same', i.e., the statistical properties of the surface are invariant under scale transformation. The surface roughness of a self-affine fractal can be described by a fractal geometry \(D_{f}\) with \(2<D_{f}<3\), whereby a value of 2 corresponds to a perfectly smooth surface and the limiting case of \(D_{f}=3\) is equivalent to a three-dimensional body. For most surfaces, the fractal dimension is smaller than 2.3 [36]. [PERSON] also derived expressions for \(p_{c}\) and \(u_{0}\) for a self-affine fractal surface [30]. The final simplified expressions for the contact pressure \(p(u)\) and the average surface separation \(u(p)\) are given by:  
\[p(u)=\beta eq_{0}h_{rms}E^{*}e^{-\alpha u/\gamma h_{rms}} \tag{104}\]  
And:  
\[u(p)\ =\gamma\alpha^{-1}h_{rms}\log\left(\beta eq_{0}h_{rms}\frac{E^{*}}{p} \right). \tag{105}\]  
In the equations above, \(q_{0}\) is the upper cutoff wavevector in the fractal analysis, \(h_{rms}\) is the root-mean-square value of the surface roughness, \(E^{*}=E/\left(1-\
u^{2}\right)\) and \(\epsilon=0.7493\)[30]. The constant \(\gamma<1\) (but of order unity) takes into account that the contact is not perfect, and the elastic energy stored in the contact region is less than the average elastic energy [30]. For simplification, it is set to \(\gamma=1\) in the following calculations. \(\alpha\) and \(\beta\) are factors in the context of the theory of contact mechanics and are functions of the fractal dimension and the ratio \(q_{1}/q_{0}\).