## 3 Design Optimization
### Performance Metrics & Design Criteria
#### 3.3.11 General Design Guidelines  
Fitting a structure \(\Gamma\) into a given, still free \"slot\" in both, real and frequency space therefore represents a primary design goal directing the layout of multiplexed LoaD systems. Without a universal design recipe owing to the plethora of possible application cases, a few important, partially counteracting trends can still be pointed out. The retention rate \(\Omega\) (6) of a CP-DF siphon valve is chiefly affiliated with its radial position \(R\) and geometry \(\Gamma\), the liquid volume \(U_{0}\) of density \(\varrho\), and the ambient pressure \(p_{0}\). Large \(\Omega\) are achieved toward high compression ratios \(V_{C}/V\) between the initial and final volumes of the entrapped gas and small radial products \(\overline{r}\Delta r\). Consequently, high field strengths \(f_{\omega}\propto R\cdot\Omega^{2}\) (8) at narrow \(\Delta\Omega\) (Figure 2b) are best established at outer radial positions \(R\). The quotient \(V_{C}/V\) is directly related to the statically defined structure \(\Gamma\). Moreover, the product \(\overline{r}\Delta r\) immediately derives from the meniscus positions \(r_{0}\) and \(r\) in the inlet reservoir and inbound segment, respectively. In the general case, \(r_{0}\) and \(r\) confining the liquid distribution \(\Lambda\) emerge in response to the centrifugal, pneumatic and ambient pressures \(p_{\omega}\) (1), \(p_{V}\) (2), and \(p_{0}\). Yet, for \(\omega=\Omega\), \(r=R_{\text{crest}}=R-Z\) (Figure 1a), so \(r_{0}\) becomes a function of \(\Gamma\) and \(U_{0}\), and overall, \(\Omega=\Omega(R,\ \Gamma,U_{0},\varrho,p_{0})\). The free experimental parameters to set \(\Omega\) are typically \(R,\Gamma\) and \(U_{0}\), while \(\varrho\) is prescribed by the assay and \(p_{0}\) by the local altitude, temperature and weather. Furthermore, any tolerances \(\Delta R\), \(\Delta\Gamma\), \(\Delta U_{0}\), \(\Delta q\) and \(\Delta p_{0}\) in their corresponding input parameters \(R\), \(\Gamma\), \(U_{0}\), \(e\) and \(p_{0}\), and possibly \(\chi\
eq 0\) (16), echoing the filling dynamics, lead to a spread \(\Delta\Omega\) in the retention rate \(\Omega\). In most application setups, the actual ambient pressure \(p_{0}\
eq p_{\text{std}}\) may be measured and remains reasonably constant over the course of a bioassay. Hence, statistical fluctuations \(\Delta p_{0}\) can be neglected, i.e., \(\Delta p_{0}/p_{0}\ll 1\), which can also be assumed for the density, i.e., \(\Delta\varrho/\varrho\ll 1\). Irreproducible filling procedures (Figure 1) can lead to statistical deviations \(\Delta\chi\approx 0\), which may have a significant impact on \(\Omega\) (16) if they exceed the permille range (Figure 8b). Measures such as slow, quasi-static loading of the liquid, an optimized shape of the inlet reservoir and the tiny isoradial channel imposing a significant hydrodynamic resistance to induce more reproducible laminar flow are thus advisable. If these considerations are considered, the standard deviation \(\Delta\Omega\) mainly represents a function of the pipetting precision \(\Delta U_{0}\) and the dimensional manufacturing tolerances \(\{\Delta\gamma_{k}\}\) of \(\Gamma\); importantly, these variations are either well known from manuals or the literature, or can rather easily be determined for a particular material and its processing technique from comparatively simple, general-purpose test structures to generally inform \(\Delta\Omega\) for a wide range of designs \(\Gamma\). This way, the (costly) manufacture of a large, statistically sufficient number of specific LoaD devices can be bypassed. Restricting the band width \(\Delta\Omega\) turns out to be particularly difficult towards large \(\Omega\). Figure 2 advises that \(\Delta\Omega\) is reduced toward rising compression volumes \(V_{\text{C}}\) and outer radial positions \(R\). This is reasoned by the decreasing (relative) influence of manufacturing tolerances \(\{\Delta\gamma_{k}\}\) defined on an absolute scale toward growing structural dimensions of \(\Gamma\).