## 3 Design Optimization
### Performance Metrics & Design Criteria
#### 3.3.3 Retention Rate and Field Strength  
Liquids loaded to high-pass valves (Figure 1) are held back for \(\omega<\Omega\propto\left(\overline{r}\Delta r\right)^{-1/2}\) (5); considering that \(\overline{r}\) closely relates to the distance from the center of rotation \(r=R\), the structure \(\Gamma\) may be configured for a given liquid volume \(U_{0}\) (exceeding the dead volume of \(\Gamma\) outside \(R_{\text{crest}}\)) to minimize the difference in liquid levels \(\Delta r\) between the inlet reservoir and the radially inbound section; at least mathematically, \(\Omega\) (5) can then be tuned to any arbitrarily high value.  
For the specific case of CP-DF siphon valves (Figure 1) primarily investigated here, the scaling \(\Omega\propto\left(V_{\text{C}}/V-1\right)^{1/2}\) (6) reveals that \(\Omega\) also increases with the compression ratio \(V_{\text{C}}/V\) between the initial and the final volumes \(V_{\text{C}}\) and \(V\) of the enclosed gas pocket, respectively.  
However, especially minimizing the compression volume \(V_{\text{C}}\), or the liquid level difference \(\Delta r\) (not shown), entails a sharply increasing \(\Delta\Omega\) (Figure 2a). Hence, in most cases, maximizing the retention rate \(\Omega\) (5) needs to be carried out while confining \(\Delta\Omega\) (7) to reasonable territories in \(\omega\)-space.  
As a typical LUO requiring maximization of \(\Omega\) while limiting \(\Delta\Omega\) for compatibility with fluidic LSI, we look at the centrifugal sedimentation, which is governed by the field strength \(f_{\omega}=\Delta\varPhi\cdot\mathcal{R}\cdot\Omega^{2}\) (8) experienced by a particle of density differential \(\Delta\varPhi\) with respect to the suspending medium at the radial position \(\mathcal{R}\) and retention rate \(\Omega\). The metric  
\[\overline{f}_{\omega}=\frac{\mathcal{R}\cdot\Omega^{2}}{R_{\text{max}}\cdot \omega_{\text{max}}^{2}} \tag{10}\]  
Figure 3: Comparison between structures \(\Gamma\) in their (**a**) basic and (**b**) multi-segmented versions for typical tolerances \(\left\{\Delta\gamma_{k}\right\}\) (see Appendix A.1) at identical radial position \(R=3\) cm and release rate \(\Omega/2\pi=25\) Hz after optimization of \(\Delta\Omega\). Most notably, \(\Gamma\) in (**b**) displays a wider inner radial section of the reservoir and overall increased width of the inbound segment to “pin” the menisci at \(r_{0}\) and \(r=R_{\text{crest}}\), and thus \(\overline{r}\Delta r\), to counter variation in \(\Omega\) (5) and (6) via \(\overline{r}\Delta r\).  
may be used for optimizing the field strength \(f_{\omega}\) on a disc. In many cases, \(\mathcal{R}\approx R\).  
Figure 4 reveals how the multi-segmented structure \(\Gamma\) (Figure 1d) can be adjusted to minimize \(\Delta\Omega\)  while providing the same field strength \(f_{\omega}\) (8) at both (representative) radial locations \(R=3\) cm (a) and 5 cm (b). Such a design task may occur if the location of the particle sedimentation needs to be radially shifted, for instance, toward the center, to occur at an earlier stage of a serial liquid handling sequence, or to be placed where space is more ample in the outer region of the disc.