Abstract
Current, applicationdriven trends towards largerscale integration (LSI) of microfluidic systems for comprehensive assay automation and multiplexing pose significant technological and economical challenges to developers. By virtue of their intrinsic capability for powerful sample preparation, centrifugal systems have attracted significant interest in academia and business since the early 1990s. This review models common, rotationally controlled valving schemes at the heart of such “LabonaDisc” (LoaD) platforms to predict critical spin rates and reliability of flow control which mainly depend on geometries, location and liquid volumes to be processed, and their experimental tolerances. In absence of largerscale manufacturing facilities during product development, the method presented here facilitates efficient simulation tools for virtual prototyping and characterization and algorithmic design optimization according to key performance metrics. This virtual in silico approach thus significantly accelerates, derisks and lowers costs along the critical advancement from idea, layout, fluidic testing, bioanalytical validation, and scaleup to commercial mass manufacture.
Introduction
Since their inception in the early 1990s, an important design goal of microfluidic LabonaChip, or micro Total Analysis Systems (µTAS)^{1,2,3,4,5}, has been to “cram more components onto integrated circuits”, and thus provide more functionality on a given piece of (chip) real estate. This objective is somewhat on the analogy of Moore’s law^{6} that has been guiding the miniaturization of microelectronics since the 1960s. Shrinking structural dimensions is reasoned by technical aspects, e.g., functional integration for enabling modern, highperformance computers, smartphones, and gadgets, as well as economic incentives, as the cost of material and (typically patterntransfer based) manufacturing processes strongly scales with the surface area of the chip^{7}. “Price per functional unit”, and thus the packing density, may hence be deemed a paramount driver of technology development.
While the general wish lists for cost and capabilities are quite alike, microfluidicsenabled (bio)analytical technologies can often not be downsized towards the nanoscale; this is, for instance, to still guarantee the presence of a minimum number of analyte molecules or particles in the (bio)sample for assuring sufficient statistics, for meeting limits of detection, for avoiding drastic changes in dominant fluidic effects, such adverse surface interactions, and evaporation, along increasing surfacetovolume ratios towards miniaturization.
Over the recent decades, numerous “LabonaChip” platforms have been developed, many of them conceived for decentralized biochemical testing^{8,9,10,11,12,13}. On the one hand, these microfluidic systems may enhance the analytical performance, e.g., through expediting the completion of transport processes, such driving diffusive mixing and heat exchange for short timetoresult, by imposing highly controlled conditions under strict laminarity at low Reynolds numbers, or by scalematching with bioentities such as cells. On the other hand, miniaturization resides at the backbone of sampletoanswer automation and parallelization, e.g., as a crucial product requirement for deployment of bioassay panels at the pointofuse/pointofcare (PoC), and patient selftesting at home.
LabonaChip systems frequently feature a modular setup where a microfluidic chip is inserted into a compact, rugged, and potentially portable instrument equipped with a control unit, sensors, actuators, and a pumping mechanism to process the liquid sample and reagents. The underlying, typically multibranched channel architecture can usually not be properly washed to assure full regeneration of fluidic functionality, and also to avoid crosscontamination or carryover of biosamples and reagents. Hence, in most cases, the chip is devised as singleuse. The cost of material, equipment, process development, and machine time of this disposable, which is normally massproduced by toolbased polymer replication schemes, such as injection molding, increases with the volume of bulk material and the surface area; in addition, the price tag on postprocessing, e.g., coatings, barrier materials, and reagents, as well as assembly steps, e.g., alignment of inserts and a lid, might be considerable, and may thus be commercially prohibitive for larger disc real estate.
Amongst various LabonaChip technologies addressing comprehensive process integration of bioanalytical protocols, we investigate here liquid handling by centrifugal microfluidics that has been successfully advanced in industry and academia since the mid1990s^{14,15,16,17,18,19,20,21,22,23,24,25,26,27,28} for various use cases, mostly in the context of biomedical in vitro diagnostics (IVD) for deployment at the PoC. Other applications comprise liquid handling automation for the life sciences, e.g., concentration/purification and amplification of DNA/RNA from a range of biosamples and matrices, process analytical techniques, and cell line development for biopharma, as well as monitoring the environment, infrastructure, industrial processes, and agrifood.
In most such “LabonaDisc” (LoaD) systems, biochemical assay kits are ported on the rotationally controlled scheme by dissecting the often conventional, possibly volumereduced protocol into a sequence of Laboratory Unit Operation (LUOs) such as metering/aliquoting^{29,30,31}, mixing^{32,33,34,35}, incubation, purification/concentration/extraction^{36,37}, homogenization^{38,39}, particle filtering^{40,41,42,43,44,45}, and droplet generation^{46,47,48}. These LUOs are overwhelmingly processed in a batchwise, rather than a continuousflow fashion, by transiently sealing their fluidic outlet with a normally closed valve, thus intermittently stopping the flow while continuing rotation within certain boundaries, e.g., for vigorous agitation of the liquid sample. These centrifugal LUOs and their linked downstream detection techniques have been comprehensively reviewed elsewhere^{24,25,26,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65}.
The various, rotationally controlled centrifugal platforms analyzed in this work are predominantly distinguished by their valving mechanisms, which critically determine their capability for functional multiplexing^{66}. Most of these “passive” flow control schemes root in the interplay of the centrifugal pressure exerted on a rotorbased liquid volume with a counteracting effect. Initial concepts were mainly based on interfacial tension to create burst valves or siphons primed by capillary action, which open by lifting^{20}, lowering^{15} or accelerating^{67} the spin rate across critical frequency thresholds.
Yet, at least as standalone, such capillary valving mechanisms tend to be hard to fine control and to stabilize over the lifetime of the chip, ranging from production, packaging, storage, and transport to eventual handling by the user and processing on a PoCcompatible instrument; they also lack to provide a physical vapor barrier, hence making them unsuitable for longerterm onboard storage of liquid reagents as an important feature for many PoC scenarios.
To this end, several, normally closed valving schemes employing sacrificial barriers for retention of liquids and their vapor were introduced. In most of their implementations, the barrier is opened by an instrument or rotorbased power unit^{68}, e.g., for mechanical^{69}, laser^{70,71,72,73,74,75} or heatinduced perforation of a film^{76}, melting of a wax plug^{77,78,79}, or magnetic and pressureinduced deflection^{80,81,82}, either during rotation or at rest. Also, passive, solventselective barriers have been explored^{44,68,83}, which only transmit flow upon distinct physicochemical stimuli.
More recently, centrifugopneumatic (CP) siphon valves were developed^{30,67,84,85,86} where the air is entrapped and centrifugally compressed by the incoming liquid during filling in a side chamber. Upon lowering the spin rate ω, the expansion of the pressurized volume pushes a surfacetension stabilized liquid “piston” within a microchannel across the crest point of their outlet siphon. This type of “LabonaDisc” (LoaD) platform uses a gasimpermeable, dissolvable film (DF), which is initially protected by a neighboring gas pocket. Once a geometrydependent critical spin rate is surpassed, the forward meniscus wets the DF to, at the same time, vent the compression chamber and open a downstream outlet. Based on this conceptually simple CPDF scheme, which can be solely controlled by the systeminnate spindle motor, the integration of LoaD systems has been substantially elevated^{83,87,88,89}.
This work will significantly support systematic layouts by providing a “digital twin”^{90,91}, i.e., a virtual representation that serves as the realtime virtual, in silico counterpart of a physical object or process, for optimizing fluidic performance, robustness, packing density, and manufacturability of rotationally controlled valving schemes for LoaD platforms^{92,93,94,95}. The first section surveys the fundamentals of centrifugal fields, continuity of mass and pressures contributing to hydrostatic equilibria at the core of valving liquid samples and reagents during batchwise processing of their upstream LUOs. We then outline the concepts of critical spin rates and their associated bandwidths as quantitative, key performance indicators for systematically assessing the impact of experimental and geometrical tolerances on operational reliability at component and systemlevel.
The next section covers the basic mechanisms underlying common, rotationally controlled valving technologies; we distinguish between high and lowpass actuation, depending on whether they release their liquid upon increase or reduction of the spin rate, respectively. In addition to sacrificial barriers, capillary and pneumatic principles, various techniques for priming and thus opening siphon valves are surveyed. After pointing out their numerous synergistical benefits, we designate a full section on siphon valves that run against the pneumatic counter pressure into an outlet chamber that is initially sealed by a dissolvablefilm (DF) membrane. Next, important performance metrics are defined, which guide the algorithmic design optimization^{92,93} towards fluidic LSI at high operational robustness before concluding with a comparison of passive valving techniques for LoaD platforms.
Note that, for convenience, the term “disc” will be used in general for designating the microfluidic, typically disposable device attached to the spindle motor. This alludes to the original idea to derive LoaD systems from common optical data storage technologies like CD or DVD. Yet, centrifugal microfluidic liquid handling does not depend on the outer shape of the usually polymeric “disc”, and many other formats, like minidiscs, segments, microscope slides, foils, or tubes, have been attached to the rotor in the meantime.
Flow control
This paper focusses on rotationally controlled valving at the pivot of enhancing functional integration and reliability of centrifugal LoaD systems operating in “stopandgo” batch mode between subsequent LUOs. We first look into the underlying general hydrostatic model before demonstrating its implementation for common centrifugal valving schemes.
Centrifugal field
Under rotation at an angular frequency ω = 2π · ν, a particle of mass m experiences a centrifugal force \(F_\omega = m \cdot {{{{{\mathcal{R}}}}}} \cdot \omega ^2\) with its center of mass located at the radial position \({{{{{\mathcal{R}}}}}}\). Within continuum mechanics underlying the modeling of fluidic systems, we consider the centrifugal force density
which applies to a fluid distribution Λ of density \(\varrho\). Note that in a suspension, \(\varrho\) designates the difference of densities between the (bio)particle and its surrounding medium.
Other pseudo forces (densities) arising in the noninertial frame of reference, but of less relevance to this review, are the Euler force (density) \(\left {f_{{{{{\mathrm{E}}}}}}} \right = \varrho \cdot {\mathcal{R}} \cdot {\mathrm{d}}\omega {{{{{\mathrm{/}}}}}}{\mathrm{d}}t\) pointing against the (vector of) the angular acceleration dω/dt, and the Coriolis force (density) \(f_v = 2\varrho \cdot \omega \cdot v\) acting on fluids moving at a (local) velocity v^{49}; for common centrifugal systems, f_{v} aligns in the plane of the disc, and perpendicular to the flow, with its direction opposite to the sense of rotation^{35,96,97}.
Liquid distribution
More generally, we describe microfluidic systems by (contiguous) liquid segments of constant density \(\varrho\), each containing a volume U_{0,i} which assume distributions {Λ_{i}(t)} within a given structure Γ at a timevarying spin speed ω(t). In the (quasi) static approximations assumed in this work, i.e., very slow changes dω/dt ≈ 0, we substitute the dependency on the time t by ω. Furthermore, for the sake of clarity, we look at each volume distribution Λ_{i}(ω) individually, for which we apply the notation Λ(ω). In response to a centrifugal field f_{ω} (1), Λ(ω) assumes a radial extension Δr(t) = r − r_{0} and mean radial position \(\bar r\left( \omega \right) = 0.5 \cdot \left( {r + r_0} \right)\) between its confining upstream and downstream menisci r_{0} and r, respectively.
Expressed in cylindrical coordinates with the radial position r and the (potentially disjunct) local cross section A(r), the integral
corresponds to the total liquid volume U_{0} contained in the segment. The conservation of U_{0} requires that the volume between their inner and outermost radial confinements \(\mathop{{\check{r}}}\limits\left( \omega \right)\) and \(\hat r\left( \omega \right)\), respectively, within the fixed structure Γ of crosssectional function A(r) is preserved, i.e., dU_{0}/dω = 0. While Eq. (2) captures the general case of a randomly shaped liquid distribution Λ, we will later introduce simplified geometries, essentially composed of rectangular cuboids, for which the integral along the radial rdirection over Λ can be replaced by an analytical expression.
Pressure contributions
Static pressures
Fluids shape according to the pressure distribution they are exposed to at a given location and time. The rotationally induced pressure head
derives from (1), and scales with the mean radial position \(\bar r = 0.5 \cdot \left( {r_0 + r} \right)\) and the radial extension Δr = r − r_{0} of the liquid segment Λ(ω). The product
in p_{ω} (3) can also be expressed by the front and rear radial positions of the menisci r and r_{0}, respectively. For typical values \(\varrho = 10^3\,{{{{{\mathrm{kg}}}}}}\,{{{{{\mathrm{m}}}}}}^{  3}\), \(\bar r = 3\,{{{{{\mathrm{cm}}}}}}\), and Δr = 1 cm, spin frequencies ν = ω/2π = 10 and 50 Hz roughly yield 12 and 300 hPa, respectively. So even for the faster rotational speeds ω, p_{ω} (3) only reaches about 1/3 of the standard atmospheric pressure p_{std} = 1013.25 hPa.
The pneumatic pressure
of a gas volume that is compressed from an initial volume V_{0} at p_{0} to V < V_{0} (law of Boyle–Mariotte) is of particular importance for this paper. By sufficient reduction of the final volume V, p_{V} (5) can, at least theoretically, assume randomly high values.
Also relevant to the small feature sizes in centrifugal microfluidics is the capillary pressure
as expressed for a liquid of surface tension σ in a channel (of round cross section) with a diameter D and the contact angle Θ between the liquid and the solid surface. For typical of values, e.g., σ = 72.8 mN m^{−1}, Θ = 120° and a channel diameter D = 100 μm, the counterpressure p_{Θ} (6) can only sustain centrifugal pressure heads p_{ω} (3) in the range of ν = ω/2π ≈ 12 Hz.
Dynamic effects
In this work, we primarily look at the hydrostatic approximation dω/dt ≈ 0 when dynamic pressure contributions are neglected. Yet, we briefly cover such effects on a semiquantitative scale. During flow at a volumetric rate Q through a channel with round cross section A = π ⋅ (D/2)^{2}, a pressure drop
is experienced by a liquid of viscosity η across its axial extension L (law of Hagen–Poiseuille). For accelerating a liquid segment of volume U traveling at a speed v through a channel of cross section A at a rate dv/dt = R ⋅ dω/dt with Q = A · v, a counterpressure
is to be provided by a valve to stay closed. The rotationally induced local acceleration dω/dt = τ_{spindle}/I_{disc} is limited by the (maximum) torque τ_{spindle} of the motor, and the moment of inertia of the disc (and its rotor) I_{disc}. For a solid disc of mass m_{disc}, (homogenous) density \(\varrho _{{{{{{\mathrm{disc}}}}}}} = {{{{{\mathrm{const}}}}}}.\) and radius R_{disc}, we obtain \(I_{{{{{{\mathrm{disc}}}}}}} \approx 0.5 \cdot m_{{{{{{\mathrm{disc}}}}}}} \cdot R_{{{{{{\mathrm{disc}}}}}}}^2\); however, strictly speaking, a LoaD cartridge exhibits cavities (with \(\varrho _{{{{{{\mathrm{disc}}}}}}} \approx 0\)) that are partially filled with a liquid distribution Λ = Λ(t) with a density \(\varrho \ne \varrho _{{{{{{\mathrm{disc}}}}}}}\) and (a center of mass) moving radially outbound over time t.
Active flow control
Also externally powered and pneumatic controllers have been employed in centrifugal LoaD platforms^{66,75,76,80,98,99,100,101}. The additional pressure p_{ext}(t) has, for instance, been generated by external or rotorbased pressure sources and pumps^{37,66,100}, by thermopneumatic actuation (law of Gay–Lussac), i.e., p_{T}(T) ∝ T(t) (with the absolute temperature T)^{31}, and by chemical reactions entailed by the expansion of gas volumes V(t), i.e., p(t) ∝ V(t)^{101}. These techniques may readily be accounted for by including p_{ext}(t) in the digital twin model. However, such active techniques tend to compromise the conceptual simplicity of the LoaD platform; rotationally controlled valving is thus the main focus of this paper.
Hydrostatic equilibrium
For the batchmode processing considered in the majority of centrifugal LoaD systems, flow is intermittently stopped by normally closed valves, i.e., the term p_{Q} ∝ Q (7) can be neglected. The spatial distribution of the liquid Λ(ω) is determined by the hydrostatic pressure equilibrium
between p_{ω} ∝ ω^{2} (3), and further contributions p_{→} and p_{←} driving the liquid segment along or against the axial direction of the channel, respectively.
To trigger valving, the equilibrium distribution Λ resulting from (9) is modulated through at least one flexibly controllable pressure constituent p_{ω}, p_{→}, or p_{←}. If the pressures p_{→} and p_{←} in the hydrostatic equilibrium (9) do not (explicitly) depend on ω, a spin rate
can be attributed to a given Λ(ω) of a coherent liquid volume U_{0} within a structure Γ as a function of the radial product \(\bar r{{\Delta }}r\) (4). A critical frequency ω = Ω is defined for Λ(Ω) representing the ω−boundary for retention of liquid, which is linked to a position of the front meniscus r = r(Ω). Note that for spin protocols ω(t) displaying steep ramps dω/dt ≠ 0, the inertial term p_{m} (8) will have to be incorporated in p_{←} or p_{→} or, depending on whether the disc is accelerated (dω/dt > 0) or slowed down (dω/dt < 0), respectively.
Laboratory unit operations
In batchmodeprocessing, valves need to close the outlet of an upstream LUO between the points in time of loading T_{load} and release T_{open}, while agitating sample or reagents by a frequency protocol ω_{LUO}(T_{load} < t < T_{open}). Most LUOs, such as plasma separation from whole blood, run fastest and most efficiently at high centrifugal field strengths \(f_\omega \propto {\mathcal{R}} \cdot \omega ^2\) (1) which, for a given layout Γ and its radial location ℛ, are established at high rates spin rates ω. Liquid retention is thus delineated by a threshold frequency Ω, and a resulting boundary for the field strength f_{ω}(ω = Ω) from (1), for which the conditions \({{{{{\mathrm{max}}}}}}\left[ {\omega _{{{{{{\mathrm{LUO}}}}}}}\left( {t} \right)} \right] \,<\, {\hat \Omega}\) or \({\mathrm{min}}\left[ {\omega _{\mathrm{LUO}}\left( t \right)} \right] \,>\, {{{\check{\Omega}}}}\) need to be met for highpass and lowpass valving, respectively.
Likewise, resilience of the valve to angular acceleration ramps \({{{{{\mathcal{R}}}}}} \cdot {\mathrm{d}}\omega {{{{{\mathrm{/}}}}}}{\mathrm{d}}t\) is important to agitate chaotic advection, as it is, for instance, required for liquid–liquid mixing^{32}, incubation of dissolved biomolecules with surfaceimmobilized capture probes, resuspension of drystored reagents, or to support mechanical cell lysis through fixedgeometry obstacles or suspended (possibly magnetic) beads^{102,103}.
Resulting, inertially induced pressure heads related to p_{m} (8) need to be factored into the calculation of the retention rates Ω. Also note that for supplying a given moment of inertia I_{rotor} of the rotor, such rotational acceleration dω/dt ≠ 0 requires sufficient torque τ_{spindle} delivered by the spindle motor.
Actuation
For common rotational actuation by the spin rate ω through p_{ω} ∝ ω^{2} (3), the liquid segment is retained upstream of the valve until a certain frequency threshold \(\omega = {{\Omega }} \in \left\{ {{{\hat{\mathrm \Omega }}}},{{{\check{\Omega}}}}\right\}\) is crossed, either surpassed (\(\omega \, > \, {{{\hat{\mathrm \Omega }}}}\)) or undershot (\(\omega \, < \, {{{\check{\Omega}}}}\)) for highpass and lowpass valves, respectively. In some valving schemes presented later, the rotational actuation may not be achieved immediately after crossing Ω; proper (highpass) valving is only assured once a (slightly) elevated actuation frequency Ω* > Ω is reached.
Alternatively, other, noncentrifugal pressure contributions to the equilibrium (9) may be modulated to prompt valving. Of particular interest for this work will be the venting of the compression chamber to level the pneumatic p_{V} (5) and atmospheric pressures p_{0}, i.e., p_{V} ↦ p_{0}, and normally p_{0} ≈ p_{std}. Note also that in absence inbound pressure gradients, e.g., created by capillary pressure p_{Θ} (6) or active sources p(t), the center of gravity \(\bar r\) (4) of the liquid distribution Λ may only move radially outwards due to the unidirectional nature of the centrifugal field f_{ω} (1) in the aftermath of valving.
Reliability
Tolerances and bandwidth
Due to statistical deviations {Δγ_{k}} in its input parameters {γ_{k}}, the experimentally observed retention frequency Ω (and Ω*) extends over an interval of standard deviation ΔΩ({γ_{k}, Δγ_{k}}). In the digital twin concept presented here, the spread of the critical spin rate Ω (10)
can be calculated (and then systematically be optimized) by Gaussian error propagation, or through MonteCarlo methods mimicking a large number of (virtual) test runs.
Using (11), we can directly relate the standard deviations ΔΩ in the critical spin rates Ω (10) to (the partial derivatives of) the fundamental experimental parameters {γ_{k}} and their precision for the pipetting or metering U_{0}, or for radial, vertical and lateral dimensions R, d, and w, resulting cross sections \({{\Delta }}A = \sqrt {\left( {w \cdot {{\Delta }}d} \right)^2 + \left( {d \cdot {{\Delta }}w} \right)^2}\) and (dead) volumes \({{\Delta }}V = \sqrt {\left( {wh \cdot {{\Delta }}d} \right)^2 + \left( {dh \cdot {{\Delta }}w} \right)^2 + \left( {dw \cdot {{\Delta }}h} \right)^2}\), delineating the valve structure Γ.
To avoid premature opening at \(\omega \, < \, {{{\hat{\mathrm \Omega }}}}\) (or \(\omega \, > \, \mathop{\Omega }\limits^{\smile}\) or in lowpass valving), the spin rate ω should be spaced by M · ΔΩ on either side of the nominal threshold value Ω, where M relates to the desired level of confidence; the aggregate rate of operational robustness P_{M} is mathematically evaluated by \({{{{{\mathrm{erf}}}}}}\big[M{{{{{\mathrm{/}}}}}}\sqrt 2 \big]\), with “erf” representing the error function; so, for M ∈ {1, 2, 3, 4, …}, valving reliability can be gauged at P_{M} ≈ {68%, 95%, 99.7%, 99.99%, …}. Hence, in the spirit of Six Sigma, these probabilities imply that, above M ≈ 6, the reliability of this (single) valving step is situated in the range of 1 to 10 defects per million opportunities (DPMO), for M ≥ 7, DPMOs are practically absent. The systemlevel reliability for N (independently operating) valves is calculated by (P_{M})^{N}, e.g., \(P_M^N \approx 77\%\) for M = 2 and N = 5.
Limited frequency space for multiplexing
The maximum degree of multiplexing is confined by the practically allowed range of spin rates ω^{93}. At its lower end, the rotationally induced pressure head p_{ω} (3) still has to dominate capillary effects to keep the liquids at bay, which tends to require ω ≥ ω_{min} ≈ 2π ⋅ 10 Hz. On its upper end, motor power and concerns of lab safety may impose ω ≤ ω_{max} ≈ 2π ⋅ 60 Hz. Independent rotational actuation of concurrently loaded valves {i} requires nonoverlapping bands {Ω_{i} ± M · ΔΩ_{i}} (assuming Ω* ≈ Ω); the finite extent of the practical range ω_{max} − ω_{min} thus restricts the highest number of rotationally triggered sequential valving steps to N as calculated from \(\omega _{{{{{{\mathrm{max}}}}}}}  \omega _{{{{{{\mathrm{min}}}}}}} \ge 2 \cdot M \cdot \mathop {\sum}\nolimits_{i = 1}^N {{{{{\Delta \Omega }}}}_i}\). Consequently, the available frequency envelope ω_{min} < ω < ω_{max} for fluidic multiplexing is best exploited by minimizing ΔΩ_{i}, and to stagger the bands {Ω_{i} ± M · ΔΩ_{i}} as closely as possible while avoiding overlap.
So, for example, a practically allowable ωrange within ω_{min} = 2π ⋅ 10 Hz ≤ ω ≤ ω_{max} = 2π ⋅ 60 Hz and a mean ΔΩ_{i}/2π = 1 Hz, and a 99.99% reliability expressed by M = 4 at component level would imply an (average) bandwidth of 2 ⋅ M ⋅ ΔΩ_{i}/2π = 2 ⋅ 4 ⋅ 1 Hz = 8 Hz, and thus provide proper operation of 50 Hz/8 Hz ≈ 6 concurrently loaded and serially triggered valving steps i; the reliability at system level would amount to 0.9999^{3} ≈ 99.97%. For M = 2, the width of the required frequency bands halves to provide space for of 50 Hz/4 Hz ≈ 12 frequency bands, at the expense of a drop in systemlevel robustness to 0.95^{3} ≈ 86%. Note that for the sake of simplicity, these backoftheenvelope calculations were based on fixed ΔΩ_{i}({γ_{k}, Δγ_{k}}), while these standard deviations actually tend to broaden towards higher spin rates ω.
Multiplexing
The digital twin approach will support the design of LoaD structures implementing multiplexed liquid handling protocols. Key flow control capabilities are the simultaneous and sequential release of several liquid volumes {U_{i,j}} loaded to rotational valving structures {Γ_{i,j}} located at radial positions {R_{i,j}}. During their concurrent retention, the common spin rate needs to follow ω < min {Ω_{i} − M · ΔΩ_{i}} for highpass and ω > max {Ω_{i} + M · ΔΩ_{i}} for lowpass valves. The order of release by venting simply relates to the sequence of the removal of the seals.
For rotationally actuated, simultaneous release of highpass valves {i, j} in the same step i at time T_{i} (Fig. 1a), the spin rate ω(t) needs to cross a zone min {Ω_{i,j} − M · ΔΩ_{i,j}} < ω < max {\({{\Omega }}_{i,j}^ \ast\) + M · Δ\({{\Omega }}_{i,j}^ \ast\)} centered at the (ideally identical) nominal critical rates Ω_{i} = {Ω_{i,j}} and \({{\Omega }}_i^ \ast = \{ {{\Omega }}_{i,j}^ \ast \}\) within an interval ΔT_{i}. For sequential actuation of valves {i} at times {T_{i}} (Fig. 1b), the critical spin rates {Ω_{i}} with Ω_{i−1} < Ω_{i} and T_{i−1} < T_{i}, must be spaced so that (the outer boundaries of) their tolerancerelated bands {Ω_{i} ± M ⋅ ΔΩ_{i}} and \(\{ {{\Omega }}_i^ \ast \pm M \cdot {{{{\Delta \Omega }}}}_i^ \ast \}\) do not overlap for all {i}.
Basic centrifugal flow control schemes
Sacrificial barriers
Apparently, straightforward implementation of normally closed valves are removable materials for intermittently blocking liquids and gases. Various types of such sacrificialbarrier valves have been developed^{104,105,106}. However, most of them require external actuation by an instrumentbased module. Examples are wax plugs^{71,72,73,78} and barrier films that are disrupted by knife cutters (xurography)^{69}, pressure sources, heat^{71,77,107}, ice^{108}, or (laser) irradiation^{74}. Such flow barriers may be trivially included in the pressure equilibrium (9) by a counter pressure jumping to infinity when the liquid arrives at the sacrificial material.
In rotationally controlled, sometimes also referred to as “passive” LoaD systems that are mainly considered here, a sealing membrane opens once the rotationally induced pressure head \(p_\omega \left( {R_{{{{{{\mathrm{seal}}}}}}}} \right) \propto {{\Omega }}_{{{{{{\mathrm{seal}}}}}}}^2 \, > \, p_{{{{{{\mathrm{seal}}}}}}}\) (3) applying at the location of the seal R_{seal} exceeds a minimum threshold p_{seal}. Yet, the typically large magnitude and sensitivity of the release frequency Ω_{seal} on manufacturing tolerances {Δγ_{k}} tends to result in large spreads ΔΩ_{seal}.
More recently, dissolvable films (DFs) that selectively disintegrate or become permeable upon contact with a specific solvent, e.g., of aqueous or organic nature, have been utilized for flow control^{44,68,87}. It has been shown for a wider range of assays that the dissolved molecules do not interfere with bioanalytical protocols or detection, or, even if, could be effectively removed from the flow path into a side chamber under the prevalent laminar flow conditions. To provide timing of their upstream LUOs according to the programmable spin protocol ω(t), DF valves have been combined with centrifugopneumatic valving.
Centrifugocapillary burst valves
Hydrophobic constrictions, and also hydrophilic expansions with sharply defined edges, have been frequently used in centrifugal microfluidic system to stop the flow at a welldefined (axial) position r = R along a channel^{20,49,50,54}. For a liquid segment driven by the centrifugal pressure p_{ω} (3) down a channel, such barriers exert a net counterpressure p_{←} composed of the capillary pressures p_{Θ} (6) of its radially outbound, front and rear menisci p_{Θ,front} and p_{Θ,rear}, respectively (Fig. 2).
In the hydrostatic approximation (9), a threshold frequency for a hydrophobic constriction
is obtained from inserting p_{←} = p_{Θ}(D, Θ) and p_{→} = p_{Θ}(D_{rear}, Θ_{rear}) ≈ 0 (6) into (10), which needs to be exceeded for the liquid volume to progress to r > R. Note that with Θ > 90° for a hydrophobic coating, cos Θ < 0. Often, hydrophobic barriers are designed with D/D_{rear} ≪ 1 and/or Θ_{rear} ≈ 90°, so that the contribution from the rear meniscus becomes negligible. Assuming a density \(\varrho = 1000\,{{{{{\mathrm{kg}}}}}}\,{{{{{\mathrm{m}}}}}}^{  3}\) and surface tension σ = 75 mN m^{−1} of water, its contact angle Θ ≈ 120° with Teflon, a mean radial position \(\bar r = 3\,{{{{{\mathrm{cm}}}}}}\) and radial extension Δr = 1 cm, and a constriction diameter D = 100 μm, we obtain threshold spin rates in the region of Ω_{Θ}/2π ≈ 10 Hz. Note that at such low spin speeds ω < Ω_{Θ}, detachment of a droplet, as outlined later in the context of the centrifugopneumatic valve in (15), is not expected as typically Ω_{Θ} ≪ Ω_{drop} (Fig. 2).
Hydrophilic expansions with Θ < 90° also produce a capillary stop. However, their retention frequencies Ω tend to be much smaller, and they sensitively depend on the exact shape, surface tension σ and contact angle Θ at the solid–liquid–gas interface. Similar geometrical features are thus often used for transient pinning of the meniscus, or, as socalled “phase guides” for shaping the front of creeping flows, e.g., during capillary priming of microfluidic chips.
Moreover, note that both types of capillary valves do not to curb evaporation, which leads to volume loss and exposure of the connected fluidic network to humidity; these valves are thus unsuitable for use in longerterm liquid storage. Also, capillary barriers often involve significant manufacturing and assembly challenges, as all four walls, with one of them usually represented by a flat lid, need to display homogeneous, welllocalized coatings. Otherwise, retention rates Ω might shift, or flow might still creep, instead of being cleanly halted, as required for proper batchmode processing.
Centrifugopneumatic burst valves
Pneumatic retention
For rotational flow control, the centrifugal pumping by p_{ω} (3) can be opposed by a pneumatic pressure p_{←} = p_{V} (5) arising from the compression of a gas volume from V_{0} to V < V_{0} enclosed at the downstream end of the structure Γ. As outlined in Fig. 3a, this counter pressure p_{V} may differ from its initial value p_{0} + δp_{0} = p_{0} ⋅ (1 + χ) at ω ≈ 0; the small offset δp_{0} with δp_{0}/p_{0} = χ ≪ 1 of the gas pressure p_{0} at the volume V_{C} + A · Z represents a departure from the hydrostatic approximation attributed to dynamic effects during filling. It may be explained by air that is drawn with the flow of liquid into the compression chamber, and either needs to be quantified empirically, or by advanced simulation.
At ω = Ω_{V} (Fig. 3b), the original gas volume is reduced by A · Z to V_{C}, thus increasing its pressure to p_{V} = p_{0}⋅(1 + χ)⋅(V_{C} + A⋅Z)/V_{C} (5). The cross section A needs to be sufficiently small so that the surface tension sustains “pistonlike” characteristics of the liquid plug. Under these conditions, we set
and p_{→} = p_{0} to obtain a critical spin rate (10)
to position the front meniscus at r = R + Z. For typical values, \(\bar r \approx R = 3\,{{{{{\mathrm{cm}}}}}}\), Δr = 1 cm, a volume ratio A · Z/V_{C} ≈ 1/10 and δp_{0} ≈ 0, this estimate provides a release threshold in the region of Ω_{V}/2π ≈ 22 Hz. An isoradial variant of the valve (Fig. 3c) tends to display a tilted meniscus surface, thus compromising the validity of the formula for Ω_{V} (14) towards large ω.
Droplet release
To effectuate basic centrifugopneumatic valving, a droplet of volume V_{drop}≈(4/3)π(D/2)^{3} ≪ V_{C} located at the radial position r_{drop} ≈ R + Z is pulled by the centrifugal force \(F_m = V_{{{{{{\mathrm{drop}}}}}}} \cdot f_\omega = V_{{{{{{\mathrm{drop}}}}}}} \cdot \varrho \cdot r_{{{{{{\mathrm{drop}}}}}}} \cdot \omega ^2\) (1) out of the orifice to the compression chamber. While the exact mechanism is somewhat obscure, we consider a simplified model akin to goniometric measurement of surface tension; detachment of the hanging droplet is suppressed until its surface tension force F_{σ} = σ ⋅ πD_{min} applying at its minimum cross section of diameter D_{min} = D/ε with ε > 1 cannot support its weight force \(F_m \approx \varrho \cdot V_{{{{{{\mathrm{drop}}}}}}} \cdot r_{{{{{{\mathrm{drop}}}}}}} \cdot \nu ^2\) anymore. This model leads to a critical spin rate
for droplet release with D_{drop} ≈ D.
Inserting typical values D ≈ 200 μm, σ ≈ 75 mN m^{−1}, ε ≈ 1.5, \(\varrho \approx 1000\,{{{{{\mathrm{kg}}}}}}\,{{{{{\mathrm{m}}}}}}^{  3}\) and R ≈ 3 cm in (15), we obtain Ω_{drop}/2π ≈ 80 Hz. This very coarse “back of the envelope” calculation reveals that the threshold spin rate for droplet release Ω_{drop} (15) sensitively depends on the diameter of the outlet D.
Compensation of ambient pressure
The main systematic error in the threshold spin rate Ω_{V} (14) is introduced by its dependence on the actual ambient (atmospheric) pressure p_{0} from its nominal (standard) value p_{std} = 1013.25 hPa at sea level, which remains rather constant at a given geolocation, and over the course of a bioassay, typically minutes to an hour. By timely local measurement of p_{0}, e.g., by a commodity pressure sensor mounted to the instrument, the spin protocol ω(t) can be flexibly adjusted by the factor \(\sqrt {{{\Delta }}p_0{{{{{\mathrm{/}}}}}}p_{{{{{{\mathrm{std}}}}}}}}\) to compensate the dependency Ω = Ω(p_{0}).
Figure 4a shows the reduction of the atmospheric pressure with altitude up to the highest human settlements by about 30% (left), and the required compensation of the spin rate ω to assure proper retention of liquid volumes by about 3%, 6%, 9%, and 12% at 500 m, 1000 m, 1500 m, and 2000 m, respectively (Fig. 4b). Note that a toleranceforgiving design would then make sure that the (lower) centrifugal field f_{ω} (1) would still be sufficient to carry out the upstream LUO, possibly by also extending the length of its correlated time interval T_{open} − T_{load} in the spin protocol ω(t).
Similar considerations can be applied for the compensation of χ ≠ 0 (14). As portrayed in Fig. 4c, the valve geometry should either be tuned to widely suppress such dynamic effects, i.e., χ ≈ 0, or to at least stabilize χ, i.e., Δχ ≈ 0; a finite, but constant χ can thus be accounted for by an adjusted spin rate protocol ω(t), as already described above for compensating deviations of the local ambient from standard atmospheric pressure p_{std}.
Rotational actuation
Followingly, droplet release triggering the opening of centrifugopneumatic valves essentially proceeds at frequencies
and may be associated with rather large uncertainties ΔΩ_{cpv} caused by effects that are hard to quantify by the simple (hydrostatic) modeling presented here.
It is surmised that the detachment of a (first) hanging drop above Ω_{cpv} (16) severely disrupts the surface of the liquid plug, so that a certain portion of the compressed air can escape through the narrow outlet, and thus gradually vent the compression chamber. This partial pressure release has a bigger impact on the pneumatic counter pressure p_{V} (5) than the loss of liquid volume to the chamber on the radial product \(\bar r{{\Delta }}r\) in p_{ω} (3). Consequently, more liquid will protrude into the compression chamber to progressively complete the transfer. Such stepwise liquid transfer has indeed been experimentally observed (qualitatively) in the region ω ≈ Ω_{cpv}. It was accompanied by a large spread ΔΩ_{cpv}, which may reflect the sensitivity of Ω_{drop} (15) to is experimental input parameters.
Their comparatively high burst frequencies Ω_{cpv} in (16), combined with their large spread ΔΩ_{cpv}, make such basic centrifugopneumatic flow control schemes mainly suitable for final valving steps into a deadended cavity, e.g., for aliquoting of liquid sample or reagents into detection chambers^{109}. Moreover, centrifugopneumatic valving requires powerful spindle motors, aerodynamic optimization, and mechanically wellbalanced rotors, and may raise concerns about lab safety.
Venting
Opening the compression chamber to atmosphere, i.e., V_{C} ↦ ∞ leading to p_{V} ↦ p_{0} (5), constitutes an alternative actuation mechanism for these CPvalves. While this principle would allow high retention frequencies Ω_{V} (14), and thus vigorous agitation for its upstream LUO (Fig. 5a), it turned out to be challenging to provide a conceptually simple mechanism for perforating the pneumatic chamber during highspeed rotation (Fig. 5b). Especially in the context of “eventtriggered” valving concepts^{88,89,110}, venting of compression chambers, which are initially by sealed dissolvable film (DF) membranes, has been implemented through arrival of a sufficient volume of ancillary liquid at strategic locations on the disc (Fig. 5c).
Centrifugal siphon valving
Layout and liquid distribution
For understanding the core principle underlying centrifugal siphoning, Fig. 6 displays a basic design Γ with an inner reservoir of cross section A_{0}, and a bottom at R which is connected by an isoradial segment of volume U_{iso} of radial length L_{iso} and height h_{iso} to a siphon channel of (constant) cross section A < A_{0}. Its isoradial section at the crest point R_{crest} = R − Z has a volume capacity U_{crest}, axial length L_{crest} and radial height h_{crest}. The radially directed outlet channel of volume U_{out} extends between R_{crest} and the final receiving chamber starting at R_{cham} > R. All parts of the siphon structure up to the crest point are, for the sake of simplicity, chosen to have the same depth d, typically 1 mm, and a small fraction of that beyond that point.
We consider adjustment of the liquid distribution Λ(ω) between (hydrostatic) equilibria p_{ω}(ω) + p_{z}(ω) = 0, with p_{ω} from (3), and an axially directed pressure head p_{z} = p_{→} − p_{←}, with forward and reverse contributions p_{→} and p_{←}, respectively, in response to a (slowly) changing spin rate ω = ω(t). A first equilibrium distribution Λ(Ω) can be found in the inbound segment at r = r_{1} with R_{crest} < r_{1} < R for a loaded liquid volume U_{0} > U_{iso}. The retention rate Ω is usually set so that the meniscus r_{1} stays well below R_{crest} to factor in a safety margin M · ΔΩ (11) resulting from tolerances {Δγ_{i}} in the input parameters {γ_{i}}, and the targeted level of reliability denoted by M. Optionally, the meniscus in the inbound segment may be “pinned” to a fixed target position r_{1} by a low capillary barrier, or by a local widening of the channel cross section (which would only slightly change the following calculations).
A second critical point R_{crest} < r_{2} = r(Ω*) < R_{cham} is situated in the outbound channel beyond which any further increase in Δr = r_{2}(ω) − r_{0}(ω), e.g., induced by topping up a liquid volume U_{Δ} or modulating ω, leads to a growth in Δr, and hence the pumping force p_{ω} (3). Different types of siphon valves can be categorized by their priming mechanism to assure p_{ω}(ω) + p_{z}(ω) > 0 for migrating between r_{1} = r(Ω) in the inbound and r_{2} = r(Ω*) in the outbound segments.
Priming
In volume addition mode, priming is triggered by topping up U_{0} with U_{Δ} > 0. Figure 6a shows the simplest case for p_{z} = 0, so pumping initiates at any spin rate ω > 0 once the outlet channel is reached to assure Δr > 0, so that the liquid level r in the radially outbound channel has fallen below the inner meniscus in the inlet reservoir r_{0}, i.e., r > r_{0}.
For lowpass siphon valving (Fig. 6b), p_{z} > 0 and U_{Δ} = 0, a threshold Ω* < Ω can be determined to guarantee pumping for ω < Ω*. Conversely, according to the basic highpass siphoning concept illustrated in Fig. 6c, liquid is released when p_{ω} + p_{z} > 0 along the entire path of the front meniscus to the end of the outlet at R_{cham}, and during release into the chamber.
Note that, for a given design Γ, the liquid distributions Λ(ω) need to obey the continuity of volume (2) as expressed by A_{0} ⋅ [R − r_{0}(Ω)] + U_{iso} + A_{1} ⋅ [R − r_{1}(Ω)] = A_{0} ⋅ [R − r_{0}(Ω*)] + U_{iso} + A_{1} ⋅ Z + U_{crest} + A_{2} ⋅ [r_{2} − R_{crest}] − U_{Δ} (Fig. 6). Moreover, for valid solutions Λ(Ω) and Λ(Ω*), the menisci at r_{i} with i ∈ {0, 1, 2} need be situated within the corridors R_{min} < r_{0} < R, R_{crest} < r_{1} < R, and R_{crest} < r_{2} < R_{cham}, while ω_{min} ≤ ω ≤ ω_{max} needs to hold for both critical spin rates ω = Ω and ω = Ω*.
Pneumatic priming
The same principle used for generating the counterpressure p_{←} in the basic pneumatic valving mode (Fig. 3) can also be sourced for priming the siphon valve^{84,111}, i.e., p_{→} = p_{V} (5). To this end, a side chamber of dead volume V_{side} is laterally connected to the inlet reservoir (Fig. 7). In a (somewhat idealized) multistep procedure, a first liquid volume U_{iso} is loaded at small ω ≈ 0 (Fig. 7a). At this stage, a gas volume V_{side} of the same size as the side chamber is disconnected from the main valving structure by the incoming liquid, which experiences a pressure p_{0} + δp_{0}, with δp_{0} = χ ⋅ p_{0} and χ ≪ 1.
In the next stage (Fig. 7b), the spin rate ω is (steeply) increased to Ω_{load} for shrinking the enclosed gas volume to V(Ω_{load}) < V_{side} while r_{0}(ω) = r_{1}(ω) = r(ω) > R_{crest}. Then (Fig. 7c), a retention rate Ω_{pps} ≪ Ω_{load} is set so that the enclosed gas expands to V(ω = Ω_{pps}) expands, while r_{1}(Ω_{pps}) stays well below R_{crest} to allow for tolerances ΔΩ (11), thus still preventing overflow. At \(\omega = {{\Omega }}_{{{{{{\mathrm{pps}}}}}}}^ \ast \,<\, {{\Omega }}_{{{{{{\mathrm{pps}}}}}}}\) (Fig. 7d), the liquid level arrives above the crest channel, i.e., \(r_2\left( {{{\Omega }}_{{{{{{\mathrm{pps}}}}}}}^ \ast } \right) \le R_{{{{{{\mathrm{crest}}}}}}}\). Mainly depending on the cross sections A_{1}, A_{crest}, and A_{2} of the inbound, crest, and outlet sections, respectively, liquid is either transferred into the outer chamber at R_{cham} by overflow, or liquid pulley mechanisms.
In more detail, the gas pressure in the side chamber amounts to
with the mean value and difference \(\bar r_{{{{{{\mathrm{side}}}}}}}\) and Δr_{side} deriving from the liquid levels r_{0} and r_{side} in the inlet and the side chamber, respectively (Fig. 7). For pneumatic siphon priming to unfold, i.e., to reach Δr > 0, the geometry Γ and liquid volume U_{0} have to be configured so
holds for the gas volume displaced from the side chamber into the main structure while reducing the spin rate ω from Ω_{pps} to \({{\Omega }}_{{{{{{\mathrm{pps}}}}}}}^ \ast\).
Capillary priming
For priming by capillary pressure p_{Θ} (6), the outlet displays a hydrophilic coating to provide a (constant) contact angle \(0 \, < \, {{\Theta }} \, \ll \, 90^\circ\) at all interfacial surfaces, and hence p_{z} = p_{→} = p_{Θ} > 0. In such a siphon valve (Fig. 6b), the meniscus stops at a first equilibrium position R_{crest} < r_{1}(Ω_{cps}) < R in the inbound segment of cross section A_{1} with a negative offset Δr < 0, i.e., r_{1} < r_{0}. This distribution Λ(Ω_{cps}) relates to a retention rate
(neglecting the small capillary pressure at the meniscus in the large inlet reservoir for A_{1}/A_{0} ≪ 1), which results in Ω_{cps}/2π ≈ 10 Hz and 16 Hz for water under typical conditions, and Θ = 70° and 0°, respectively; any spin frequency ω > Ω_{cps} will retain the liquid.
The second equilibrium position r_{2} establishes at \(\omega = {{\Omega }}_{{{{{{\mathrm{cps}}}}}}}^ \ast\) with the meniscus at r_{2} > R_{crest} in the outlet segment of cross section A_{2}. Any further progression r > r_{2} of the meniscus for \(\omega \,<\, {{\Omega }}_{{{{{{\mathrm{cps}}}}}}}^ \ast\) will then grow Δr to set p_{ω} + p_{Θ} > 0, and thus trigger continuous siphoning. As for the other mechanisms for siphon priming, the choice of the critical rates Ω and Ω* needs to consider their standard deviations ΔΩ and ΔΩ* (11) induced by experimental tolerances {Δγ_{i}}, and the required reliability quantified by the factor M.
As a lowpass valve, capillaryaction primed siphons are particularly suitable for LUOs requiring strong centrifugal fields f_{ω} (1). The spread ΔΩ_{cps} of the threshold frequency Ω_{cps} (12), which might be related to poor definition of the diameter D contact angle Θ, is normally of minor practical relevance, as long as Θ stays well below 90°.
In purely capillarydriven priming at ω = 0, the time
for covering the axial distance l ≈ L + Z + L_{crest} + (R_{cham} − R_{crest}) scales with l^{2}, the viscosity of the liquid η, and inversely with its surface tension σ, cos Θ > 0, and the crosssectional diameter of the (round) channel D.
Lost volume
Transfer by centrifugal siphoning (Fig. 6) is usually accompanied by a loss U_{loss} < U_{0} + U_{Δ} of the original liquid volume U_{0}, plus U_{Δ} for the case of priming by volume addition (Fig. 8). In “pulley”type of siphoning, the separation of this residual volume U_{loss} occurs when air is drawn into the filled outlet channel during forward pumping, so the initially coherent liquid plug tears apart (Fig. 8a). This residual volume ideally vanishes U_{loss}/U_{0} ≪ 1, or exhibits a small spread ΔU_{loss}/U_{loss} ≪ 1; however, in practice, U_{loss} and ΔU_{loss} sensitively depend on the hydrodynamic processes and the shape of Γ, and tend to decrease with the cross sections A_{1}, A_{crest}, and A_{2}. Overflow driven liquid transfer running without a pulley mechanism (Fig. 8b) tends to reduce the spread ΔU_{loss}, while producing larger absolute losses U_{loss}.
Centrifugopneumatic dissolvablefilm siphon valving
The geometry Γ in Fig. 9 constitutes a hybrid of centrifugopneumatic (CP) valves (Fig. 3), sacrificial dissolvablefilm (DF) barriers (Fig. 5), and centrifugal siphoning (Fig. 6). Its transition between the two hydrostatic equilibrium distributions Λ_{i∈{1,2}} results from a centrifugally induced pumping pressure p_{ω} (3) running against a pneumatic back pressure p_{z} = p_{←} = p_{V} (9) from the (initially) sealed receiving chamber. This configuration thus eliminates the need for priming by interim addition V_{Δ} (Fig. 6a), hard to manufacture and define circumferential hydrophilic coating Θ < 90° of the narrow outlet channel (Fig. 6b), and difficult to control pneumatic charging of a side chamber (Fig. 7). Liquid transfer merely relies on volume overflow through channel segments exhibiting sufficiently large cross sections A.
During retention of this highpass siphon valve ω < Ω, the meniscus stabilizes in the radially inbound section of the siphon channel, thus effectively dampening inertial overshoot propelled by inertia p_{m} (8) at finite flow rates Q > 0, suppressing premature droplet breakoff of CP valves (Fig. 3), and radial squeezing of the meniscus for alternative layouts with isoradially directed outlets (Fig. 2, right and Fig. 3, right). Even without direct experimental data, the scheme provides better overall management of loading U_{0} with smaller and more reproducible pressure offset δp_{0} (and thus \({{\Delta }}\chi \mapsto 0\)) than for the basic CPDF valve (Fig. 3). The gastight DF initially isolating the final pneumatic chamber allows for rotationally controlled opening without external actuators, as well as venting mode, while also removing the endpoint character of the receiving chamber familiar from basic CP valves (Fig. 3).
By virtue of these manifold synergistical benefits, we consider CPDF siphon valves as a key enabler for microfluidic large(r)scale integration (LSI) at high operational reliability, and thus designate a separate section for them. For sake of clarity, we use a simplified geometry Γ (Fig. 9) to represent the valving structures and the resulting, quasi static liquid distributions Λ that lend themselves to a description by closedform analytical formulas, rather than the previous integrals as, e.g., occurring in (2). The basic concept has been outlined and experimentally validated in a series of prior publications^{87,88,112}.
Functional principle
Loading
To best illustrate the basic principle of the CPDF siphon valving, a somewhat hypothetical, multistep loading procedure is portrayed in Fig. 9. At rest ω ≈ 0, a liquid volume U_{iso} completely fills the isoradial section of radial position R, length L, and height h. This way, a pneumatically isolated gas pocket occupies a volume V_{C} + A · Z. The product A · Z represents the volume of the inbound siphon segment of cross section A and length Z, while V_{C} is mainly composed of the volumes V_{C,0} of the large chamber at its inner end, the segmented internal channel V_{int}, and the final, shallow recess chamber volume V_{DF} positioned at R_{cham}, i.e., typically V_{int} + V_{DF} ≪ V_{C,0} < V_{C}. The pressure in this gas pocket corresponds to p_{0} + δp_{0} = p_{0} ⋅ (1 + χ) with 0 ≤ χ ≪ 1, and the ambient pressure p_{0} applying to the inlet, which is open to atmosphere, often at p ≈ p_{std}.
The total liquid volume is then topped up to U_{0} = U_{iso} + A_{0} ⋅ [R − r_{0}(Ω)] + A ⋅ [R−Z], so that, at the retention rate ω = Ω, the liquid distribution Λ(Ω) places its front meniscus in the inbound segment at r = r_{1}(Ω) = R_{crest} = R−Z (Fig. 9b). For ω < Ω, r stays in the interval R − Z < r(ω) < R.
Pneumatic pressure
Due to the compression of the enclosed gas volume by A ⋅ (R − Z), the resultant increase in the pneumatic counterpressure
can hence be expressed by r, with the liquid volume in the DF chamber U_{DF} = 0 vanishing during retention at ω ≤ Ω.
Meniscus position
Considering that the position of the rear meniscus in the inlet reservoir
is linear in r, the radial product \(\bar r{{\Delta }}r\) in (4), and thus also the driving pressure \(p_\omega \propto \bar r{{\Delta }}r\) (3), are square functions in r. With p_{→} = p_{0} = const., the hydrostatic equilibrium for the CPDF siphon valves \(p_\omega + p_0 = p_V\) (9) can be written as a cubic function in r. Given the algebraic nature of the equation, any advanced symbolic or generic numerical solver can readily produce the results shown.
Consequently, algebraic solutions r = r(R, Γ, U_{0}, U_{DF}, p_{0}, χ, ω) can be found (in principle) for a given geometry of the CPDF siphon valve Γ, which are parametrized by common experimental parameters, such as the spin rate ω, the radial position R of Γ, its compression volume V_{C} and the loaded liquid volume U_{0}. Figure 10 displays the rise of the meniscus z = R − r in the inbound segment of the siphon channel until the crest point R_{crest} = R − Z is reached at the critical frequency ω = Ω ≈ 22 Hz.
Liquid retention
Critical spin rate
When the front meniscus of Λ assumes r = R_{crest} at the upper end of the inbound segment (Fig. 9a), inserting p_{V} (21) into (10) provides
for the critical retention rate Ω of the CPDF siphon valve.
Figure 11 examines the dependence of the critical spin rate Ω on key experimental parameters. The retention rate Ω is highly configurable, reducing with growing volume V_{C,0} of the permanently gasfilled compression chamber (Fig. 11a). Ω also increases by extending the length of the radially inbound segment Z (Fig. 11b). As r_{0} is linear in R and U_{0} (22), the radial product \(\bar r{{\Delta }}r\) is a square function in r_{0}, so Ω decreases with U_{0} (Fig. 11c) and R (Fig. 11d), roughly following 1/U_{0} and 1/R, respectively.
Tuning of the critical spin rate
As r_{0}(Ω) (22) is linear in U_{0}/A_{0}, also the radial product \(\bar r{{\Delta }}r\) (4), and thus Ω (23), remain unaltered for U_{0}/A_{0} = const. Hence, an LUO requiring retention of a different liquid volume U_{0} preserves the same critical spin rate Ω (23) as long as the cross section of the inlet A_{0} is adjusted by the same factor (Γ might also feature a partitioned inlet reservoir in which compartments are flexibly connected by individually configurable barriers).
Figure 11 also reveals that the retention rate Ω (23) may be tuned in the range 10 Hz < Ω/2π < 70 Hz. Considering the effort to optimize manufacturing processes to a specific design, it is usually wise to leave the essential, liquid carrying parts of Γ unaltered when adjusting the critical spin rate Ω (23) to the requirements of the assay protocol. Therefore, tuning of Ω (23) is preferentially implemented by the rather large volume of the main compression chamber V_{C,0}, while preserving the other sectors of Γ. In situations when the radial position R needs to be moved, e.g., through spatial requirements, the relation (23) provides a recipe for compensating the shift in R by adjusting V_{C,0}.
Note that the permanently gasfilled sections only contribute with their total (dead) volume V_{C} to Ω (23), but they can be partitioned, distributed and located anywhere, as long as being in unfettered pneumatic communication with each other. For instance, the compression volume V_{C} might be constituted by a smaller “attachment” to the inner end of the radially inbound section which is connected through a channel of tiny cross section to a larger chamber placed where space would still be available in a multiplexed (disc) layout (see also the advanced geometry displayed in Fig. A1 of Appendix A3).
Liquid release
Modes
Up to now, the considerations have primarily focused on the barrier function of CPDF valving for 0 < R − r < Z by keeping ω < Ω. The opening condition is captured by the overflow of a minimum volume U_{DF} = β ⋅ V_{DF} with 0 < β < 1 to sufficiently wet and disintegrate the DF, hence venting the outer chamber of total volume V_{DF}; Fig. 9c represents the example of β = 0.5 for a central location of the DF in a recess of round cross section. After opening the DF at ω > Ω* > Ω (Fig. 9, bottom, left), or by perforation of a seal (Fig. 9c), the pneumatic compartment is vented, i.e., \(V_{{{{{\mathrm{C}}}}}} \, \mapsto \, \infty\) and \(p_V \, \mapsto \, p_0\), and, consequently, any spin rate ω > 0 will propel further liquid transfer. On the analogy of Fig. 5, an additional seal, or the DF, might be opened by an external actuator^{69,113}, or by a preceding liquid handling step, e.g., through “eventtriggering” ^{88}.
This transfer of U_{DF} = β ⋅ V_{DF} into the recess reduces the original liquid and gas volumes U_{0} and V_{C} + A ⋅ Z, respectively, by U_{DF}, and typically U_{DF} ≪ U_{0}, while the forward meniscus remains pinned to r = R − Z. We calculate the release rate
from (23) by considering the cutback of the loaded volume U_{0} upstream of the crest point and the compression volume by U_{DF} = β ⋅ V_{DF} in \(\bar r{{\Delta }}r\) (4) and V = V_{C} − βV_{DF} in p_{V} (5), respectively. These volume reductions lead to a defined increment of the spin rate Ω_{step} = Ω* − Ω, which grows with U_{DF}. The gap Ω_{step} can thus be tuned for CPDF siphon valves through Γ, for instance, by the dead volume of the DF chamber outside r ≥ R_{DF}. For common CPDF siphon valves, β ⋅ V_{DF}/V_{C} ≪ 1, so that the 0 < Ω_{step}/Ω ≪ 1.
The opening mechanism of the CPDF siphon valve (Fig. 9) imposes the general volume condition A_{0} ⋅ [R − Z − r_{0}(Ω)] ≥ U_{DF} for both, actuation by rotation or venting, to assure Δr > 0, and thus a nonvanishing centrifugal field p_{ω} ∝ Δr (3), to drive liquid transfer through the outlet for any ω > 0 subsequent to the removal of the DF or seal of the compression chamber. Note that strictly speaking, ω < Ω describes “clean” retention without overflow into the DF chamber while, in principle, ω < Ω* would be sufficient, as long as {Δγ_{k}} = 0.
Reliability
For consistent CPDF siphon valving, the forward meniscus needs to stay at r(ω = Ω − M ⋅ ΔΩ) > R − Z during retention; for reliable rotational actuation, ω ≥ Ω* + M ⋅ ΔΩ*. Beyond the factors impacting the standard deviation ΔΩ, the uncertainty ΔΩ_{step} is thus mainly determined by the definition of the volume fraction β · V_{DF}. For typical experimental conditions Ω_{step}/Ω ≪ 1 and ΔΩ ≈ ΔΩ*, so robust rotational actuation comes down to ω ≥ Ω* + M ⋅ ΔΩ* ≈ Ω + M ⋅ ΔΩ; hence, a “forbidden” frequency band of approximate width 2 · M · ΔΩ around ω = Ω must be crossed for reliable switching the CPDF siphon valve (see also Fig. 1).
Residual volume
As already investigated in the context of basic siphon valving (Fig. 8), the accuracy and precision of the transferred liquid volume directly enters the mixing ratios underpinning bioanalytical quantitation, and also the Ω (23) and Ω* (24) for subsequent valving steps, and, consequently, critically impacts systemlevel reliability of microfluidic LSI.
Neglecting inertial and interfacial effects, and assuming purely Δr > 0 driven overflow across the crest channel, and a fraction α · V_{DF} with 0 < α < β < 1 remaining in the recess for the DF, the volume
constitutes an (approximate) upper boundary of liquid “swallowed” after the transfer (Fig. 9d), with α ≈ β in common application cases. U_{loss} (25) displays a direct contribution of U_{iso}, and increases linearly with Z as well as the cross sections A_{0} and A. Note that, especially for the here assumed, sufficiently large cross section A, “pulley”type siphoning is largely suppressed, therefore optimizing volume precision by minimizing ΔU_{loss}; such metering might be further improved via the proper definition of a liquid “cutoff”, e.g., by placing a sharpedged “liquid knife” within a low deadvolume section.
Rotational valving schemes
The objective of the digital twin concept presented here is to advise the choice and layout of rotationally controlled valving techniques at the pivot of LoaD systems featuring high functional integration density with “in silico” predictable, systemlevel reliability for rapid and costefficient scaleup of manufacture from prototyping (for initial fluidic testing) to pilot series (for initial bioanalytical testing) and commercial mass fabrication. This section proposes a repertoire of quantitative metrics which guide the selection of the type and layout of rotationally controlled valving for a given scenario. Note that the model underlying the digital twin presented here contains various simplifications, so experimental verification is still needed.
Performance metrics
Critical frequencies and field strengths
For a given highpass valve, maximum field strengths \(f_\omega \left( {{{{\hat{\mathrm \Omega }}}}} \right) \propto {{{\hat{\mathrm \Omega }}}}^2\) (1) for capillary burst (12)
and basic CP valves (16)
as well as for the CPDF siphoning structure (23) of retention rate \({{{\hat{\mathrm \Omega }}}}\) cannot be exceeded during processing of an upstream LUO. For the lowpass mechanisms, there is, per definition, only a critical rate \({{\check{\Omega}}}\) for valve opening at \(\omega \, < \, {{\check{\Omega}}}\). In case of the capillary primed siphoning (Fig. 6b), there is a minimum field strength
which will have to be calculated numerically for the pneumatic priming mechanism (Fig. 7). In most LUOs, such a minimum field strength f is of minor practical relevance. Note that for particle separation by f_{ω} (1), \(\varrho\) represents the density differential to the suspending medium.
For the CP valves (with χ = 0), we find, by revisiting at (16) and (23), that the difference p_{←} − p_{→} becomes \(p_0 \cdot A \cdot Z{{{{{\mathrm{/}}}}}}V_{{{{{\mathrm{C}}}}}}\), so \({{{\hat{\mathrm \Omega }}}} \propto \sqrt {A \cdot Z{{{{{\mathrm{/}}}}}}V_{{{{{\mathrm{C}}}}}}}\). Mathematically, its scaling with \(1{{{{{\mathrm{/}}}}}}\sqrt {V_{{{{{\mathrm{C}}}}}}}\) allows raising the retention rate \({{{\hat{\mathrm \Omega }}}}\) to any required value by simply downsizing the compression volume V_{C}. The same holds for capillary burst valves with p_{←} = p_{Θ} ∝ σ ⋅ cos Θ/D (6) with vanishing p_{→} ≈ 0 when shrinking the diameter of the constriction D. However, in practice, reducing V_{C} and D is limited by the minimum feature sizes of the manufacturing technologies, and growing spread ΔΩ (11), and also the product σ · cos Θ has upper limits for capillary valves.
Apart from its linearity in \(\sqrt {p_ \leftarrow  p_ \to }\), we also observe that the retention rate \({{{\hat{\mathrm \Omega }}}} \propto 1{{{{{\mathrm{/}}}}}}\sqrt {\bar r{{\Delta }}r}\) (10) can be increased by minimizing the geometrical product \(\bar r{{\Delta }}r\) representing the radial coordinates of the liquid distribution Λ within the valving structure Γ. This dependence unravels a clear advantage of siphoning strategies where the Δr and \(\bar r\) only refer to the radial distribution of Λ between the menisci r_{0} and r in the inlet and inbound section, respectively, while the outer volumes extending between r and R (plus U_{iso}) do not enter \({{{\hat{\mathrm \Omega }}}}\) (10), and can thus be sized “randomly”.
Hence, in contrast to the basic capillary (Fig. 2) or CP (Fig. 3) modes where Δr = R + Z − r_{0} and \(\bar r = 0.5 \cdot \left( {R + Z + r_0} \right)\) must hold during retention, siphon valving can be geared for high retention rates \({{{\hat{\mathrm \Omega }}}}\) (10) by “hiding” the bulk liquid volume outside r, while minimizing the radial extension Δr or the mean position \(\bar r\) of the inner liquid distribution Λ in the radial interval between r_{0} and r. Note, however, that maximization of \({{\Omega }} \propto 1{{{{{\mathrm{/}}}}}}\sqrt {{{\Delta }}r}\) (23) hits a limit as the volume A_{0} · Δr has to be sufficient to effectuate complete filling of the channel section extending between the position r_{1} during retention to r_{2} for still being able to trigger liquid release.
Bandwidth
The metric
reflects the statistical spread of the critical frequency Ω to variations in the experimental input parameters with respect to the practically available spin rate corridor between ω_{min} and ω_{max}. Minimization of \(\overline {{{{{\Delta \Omega }}}}}\) (29) can thus guide the development of toleranceforgiving designs.
For rotationally actuated siphon valving (Fig. 6), the retention and release frequencies Ω and release Ω* can be modified separately. Both spin rates need to be suitably spaced to account for their individual spreads ΔΩ and ΔΩ*; in addition, the differential in the spin rate ω needs to allow lifting the meniscus past the second (unstable) equilibrium distribution Λ to r_{2} beyond the crest point at R_{crest}, or even further to deliver a minimum liquid volume U_{DF} to the outer chamber for ushering CPDF siphon valving (Fig. 9). This requires reserving a band \({{\Omega }}  M \cdot {{{{\Delta \Omega }}}} \,<\, \omega \,<\, {{\Omega }}^ \ast + M \cdot {{{{\Delta \Omega }}}}^ \ast\) for the spin rate ω. Towards LSI, it is thus favorable to minimize the metric
in order to be able to “squeeze” as many fluidic operations as possible into the available frequency range.
Volume loss
Upon completion of valving, a part U_{loss} = ζ · U_{0} with 0 ≤ ζ < 1 of the originally loaded volume U_{0} might remain in the structure Γ (Fig. 8). While ζ ↦ 0 for the basic valve setups implementing radially directed flow (Figs. 2 and 3), the emptying of the siphon structures (Fig. 6) runs against the centrifugal pressure head p_{ω} (3) in the inbound section once gas has compromised the integrity of the liquid plug to create a segment characterized by Δr < 0.
As sketched in Fig. 9, such volume loss U_{loss} may be approximated by (25) for CPDF valving. In some bioassays, a systematic loss can be factored in by loading more liquid volume U_{0} to the inlet. Still, accommodating U_{loss} (25) tends to increase the footprint of liquid handling structures decisively enters the output volume U_{0} − U_{loss}, and thus the frequencies Ω (23) and Ω* (24) of subsequent valving steps in LSI. A metric guiding design optimization might thus be
which obviously vanishes when minimizing U_{loss} (31).
Volume precision
In the same way as the absolute amount of liquid determines the critical frequencies Ω (23) and Ω* (24) of subsequent flow control operations and concentrations in assays, their statistical spreads ΔU_{0} and ΔU_{loss} impact the precision of the inlet volume U_{0} in the next step. While, again, systematic losses may be factored into the valve design Γ and spin protocol ω(t), stochastic fluctuations may even interrupt liquid handling sequences as the minimum amount of liquid needed to reach Ω* (24) may not be available in the inlets of a subset of valves. We define the dimensionless ratio
as the metric to be minimized, i.e., \(\overline {{{\Delta }}U} _{{{{{{\mathrm{loss}}}}}}}\, \mapsto \, 0\), for enhancing the reliability of multiplexed valving. Alternatively, U_{loss} may also be referenced in (32) to U_{0}.
Radial extension
Radial space is precious on centrifugal LoaD systems. To illustrate this, we consider that the centrifugal field f_{ω} (1) is unidirectional, i.e., it cannot (directly) pump liquids towards the center of rotation; such centripetal pumping would require the provision of power, e.g., connection of a pressure source^{100}, chemical reaction^{105,110}, imbibition^{89,114}, or potential energy in the centrifugal field, e.g., through the simultaneous displacement of a centrally stored (ancillary) liquids^{115} or centrifugopneumatic siphoning^{67}. Such methods, while technically feasible and successfully demonstrated, would somewhat compromise the conceptual simplicity of the LoaD paradigm.
In purely rotationally controlled LoaD systems considered in this work, the LUOs of (serial) assay protocols are therefore typically aligned in a radially outbound sequence arranged in the order of their execution. This also implies that the reservoirs taking up the sample and reagents to be processed may need to be located centrally. In multistep assay protocols, the radial confinement of the disc between R_{min}, e.g., given by the size of an inner hole to clamp the disc to the spindle (R_{min} = 75 mm for optical data storage media), plus some space for bonding to a lid, and the largest radius R_{max} (in the range of 55 mm for a CD format) at which structures can still be placed, limits the number of LUOs that can be automated. A design goal may therefore be to radially compress each structure of extension ΔR_{Γ} of the LUO and its downstream control valve. \({{\Delta }}R_{{\Gamma }} = \hat r  \mathop{{\check{r}}}\limits\) will often correspond to the difference between the minimum radial position of the inner meniscus \(\big[\mathop{{\check{r}}}\limits = {{{{{\mathrm{min}}}}}}[r_0\left( \omega \right)]\) over the course of valving ω(t), and the radially outer edge \(\hat r\) of the final receiving chamber. The metric
might thus be chosen to guide optimization of radial space for a rotationally valved LUO.
Real estate
The total area available on the round LoaD device \(A_0 = {\int}_{R_{{{{{{\mathrm{min}}}}}}}}^{R_{{{{{{\mathrm{max}}}}}}}} {2\pi \cdot rdr} = \pi \left( {R_{{{{{{\mathrm{max}}}}}}}^2  R_{{{{{{\mathrm{min}}}}}}}^2} \right)\) is shared between LUOs and their intermittent valves. Therefore, any space savings through the clever design of Γ will enhance the potential for multiplexing. Furthermore, the unidirectional nature of liquid transport implies that the reservoirs taking up the sample and reagents to be processed may need to be located near the axis of rotation.
Overall, these boundary conditions, which are intrinsic to LoaD systems, make central real estate more scarce and thus precious, which we reflect by the metric (“price tag”)
where W(r) represents the total azimuthal width of the valve structure Γ at a radial location r, e.g., the length of the isoradial channel L in the simplified geometry of the CPDF siphon valve (Fig. 9). Note that for finite thickness of the fluidic substrate, typically on the order of 1.2 mm for optical storage media derived formats, the area of sectors containing the liquid volume U_{0} loaded to the valve cannot be arbitrarily reduced.
Valving time
The interval between prompting the opening of a valve and the completion of the liquid transfer through its structure Γ to the subsequent stage involves different processes, which depend on the selected valving mechanism. For the core modes of hydrophobic barriers (Fig. 2) and CP valving (Fig. 3), a transfer time
is obtained for a centrifugally driven flow propelled by a pressure differential \(p = p_\omega = \varrho \cdot \bar r{{\Delta }}r \cdot {{\Omega }}^2\) (3) of a liquid of density \(\varrho\) and viscosity η through the radial outlet of length l and cross section A.
The approximation (35) neglects start up and exit effects when the channel is only partially filled, and assumes constant \(\bar r{{\Delta }}r\) to deliver a stable pumping pressure p. However, the product \(\bar r\left( t \right){{\Delta }}r\left( t \right)\) changes over the course of liquid transfer. As previously outlined, for siphon valving, \(\bar r\) and Δr are calculated from r_{0} and r, l refers to the aggregate axial length of the siphon and outlet channels, and Ω needs to be replaced by the release frequency Ω* for rotational actuation modes.
By assuming typical values U = 10 μl, A = (100 μm)^{2}, l = 1 cm, a mean radial position \(\bar r = 3\,{{{{{\mathrm{cm}}}}}}\), Δr = 1 cm, Ω = 2π ⋅ 25 Hz, and a density \(\varrho = 1000\,{{{{{\mathrm{kg}}}}}}\,{{{{{\mathrm{m}}}}}}^{  3}\)and viscosity η = 1 mPa s roughly corresponding to water, we arrive at an order of magnitude for T_{Q} ≈ 3.4 s (35) for the basic radial valve configurations. When extending l by a factor of 5 and reducing Δr by the same factor to account for siphoning, T_{Q} (35) increases by a factor of 25 to about 1.5 min.
For release mechanisms implementing DFs, the dissolution time T_{DF} of the membrane adds to T_{Q} (35). T_{DF} can be set by the formulation and thickness of the film, and may further require a minimum pressure p_{DF} on the film located at R_{DF} during wetting. Values for T_{DF} can range from seconds to minutes, and may display large standard deviation ΔT_{DF}. Note that various “timing” modules have been developed for LoaD systems, e.g., to delay or synchronize liquid handling time spans required for assay biokinetics^{89,111}.
Configurability
The previous deliberations and formulas allowing to maintain or tune the retention, burst and release rates Ω = Ω(R, Γ, U_{0}) and Ω*, and their associated bandwidths ΔΩ and ΔΩ* of rotationally controlled valves through the shape and location R of Γ and U_{0}, play an important role for assay automation and parallelization. This digital twin will enable in silico tools offering high predictive power for configuring designs Γ that are optimized for functional integration, reliability and manufacturability^{92}. For CPDF siphon valves (Fig. 9), the farranging configurability of the retention rate Ω through the volume of the permanently gasfilled compression chamber V_{C,0}, which can be located “anywhere” on the disc, and through reduction of Δr by “hiding” liquid volume on the outer part of the structure Γ, provide major benefits (Fig. 11).
Configurability might also be vital regarding other, collective aspects of LSI, such as mechanical balancing of the disc featuring cavities with moving liquid distribution Λ(t), minimizing its moment of inertia \(I_m = 0.5\pi \cdot \varrho _{{{{{{\mathrm{disc}}}}}}} \cdot R_0^4\), increasing heat transfer, optimization of mold flow for its mass replication, and interfacing that is compatible with standard liquid handling robotics and workflows.
Comparison
Table 1 compiles select metrics with their typical scaling behavior and value ranges for the previously outlined valving schemes. Note that, due to the plethora of parameters, their wide value ranges, and refined designs, absolute assessments cannot be made; yet the digital twin concept presented here will help choosing and optimizing the valving concept for a given LoaD application.
Summary and outlook
Summary
We have surveyed basic, rotationally controlled valving techniques and modeled their critical spin rates Ω and other performance metrics as a function of their radial positions R, geometries Γ and loaded liquid volumes U_{0}. The underlying digital twin approach allows to efficiently select, configure and optimize the valve towards typical design objectives, such as retention at high field strength during the processing of a Laboratory Unit Operation (LUO) in the inlet reservoir, or to accommodate different reagent volumes U_{0}.
The modeling presented here specifically correlates retention rates Ω and their standard deviations ΔΩ with experimental input parameters displaying statistical spreads resulting from pipetting, material properties, ambient conditions, and, in particular, the lateral and vertical precision of the manufacturing technique. As a major benefit, this digital twin allows to engineer toleranceforgiving valve designs displaying predictable functionality along scaleup from prototyping for demonstrating proofofconcept to pilot series and, eventually, mass manufacture and extended bioanalytical testing. Experimental validation should be implemented once a production technology becomes available that can supply a sufficiently large, and thus statistically relevant number of LoaD devices for thorough characterization on the path to regulatory compliance.
Towards large(r)scale integration (LSI) of fluidic function underpinning comprehensive sampletoanswer automation of multistep/multireagent bioassay panels, the designformanufacture (DfM) capability of the digital twin thus allows maximizing the packing density in real and frequency space while assuring reliability at the system level.
The high predictive power of the in silico approach can thus substantially curb the risk, cost, and time for iterative performance optimization towards high technology readiness levels (TRLs), and thus efficiently supports systematic Failure Mode & Effects Analysis (FMEA), and advancement towards commercialization. The general formalism developed for functional and spatial optimization may well be adopted for other LabonaChip platforms and applications.
Outlook
Several extensions of the rudimentary digital twin approach are proposed, e.g., inclusion of previously introduced flow control by eventtriggering, rotational pulsing, and delay modules, or further increase of real estate by vertical stacking of multiple fluidic layers. Valving performance can be improved by the sophistication of layouts^{92,93}, e.g., with refined shapes, rounded contours, and anticounterfeit features^{94}, and migration from the hydrostatic model to computational fluidic dynamic (CFD) simulation. An advanced design tool could also include the bioassay kinetics. Moreover, virtual prototyping could be extended by including the simulation of the manufacturing processes of the layouts themselves. This would be particularly suitable for more complex methods like mold flow for injection molding or 3D printing.
Regarding the bigger picture, the ability to create largerscale integrated, fluidically functional designs with predictable reliability may enable foundry models that are commonplace in mature industries such as microelectronics and microelectromechanical systems (MEMS)^{7}. These efforts might be supported by existing initiatives aiming at standardization of interfaces, manufacture, and testing^{116,117,118}. As valving assumes a similar role on centrifugal LoaD platforms as transistors for the emergence of integrated circuits (ICs) in (digital) electronics, the community is well equipped with the presented digital twin approach to develop large(r)scale integrated “bioCPUs” Centrifugal Processing Units for implementing multistep, multireagent and multianalyte bioassay panels.
Followup work is already planned on computeraided, possibly automated optimization of integration density, robustness, and manufacturability. As an open platform concept^{119}, the work is meant to encourage honing of design, modeling, simulation, and experimental verification, for instance, within a blockchainincentivized participatory research model involving crowdsourcing by means of hackathons, citizen science, and fab/maker labs^{120,121,122,123}. Such communitybased organization of research, which are already wellestablished in the thriving field of blockchain, are particularly attractive for centrifugal microfluidic technologies as key intellectual property (IP), which was mainly filed throughout the 1990s and early 2000s, has now entered the public domain.
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Appendices
Appendix
Default geometry
The structure Γ, loaded liquid volumes U_{0} and radial positions R can be varied across a multidimensional parameter space, e.g., to tune retention rates Ω, or other key performance indicators. Table 2 gives of overview or generic values which can be used to initiate optimization.
Consistency checks
Not all in silico designed structures Γ turn out to provide proper valving with given volumes U_{0} and critical spin rates Ω. One reason is manufacturability, the other fluidic function. Table 2 already introduces some rudimentary sanity checks as necessities, but not guaranteeing sufficiency for proper valving. As for other parts of this work, we exclude analysis of biochemical function, which may, for instance, be related to surface adsorption of (bio)molecules in high surfacetovolume ratio channels, denaturing of biomolecules due to chemical agents leaching from bulk materials, and assembly processes, viability of cells or reaction kinetics.
Fluidic function
A common design goal is to set a certain retention and release rates Ω and Ω* that are consistent with the valving sequence in frequency space for a liquid volume U_{0} prescribed by the assay protocol. At least when discarding manufacturing restrictions, the basic radial layouts Γ displayed in Figs. 2 and 3 may be configured to any given critical frequencies Ω, as long as the inlet reservoir can accommodate U_{0}, i.e., U_{0} < A_{0} · (R − R_{min}) for the hydrophobic barrier. Furthermore, the cross section of the outlet A needs to be sufficiently small to that surface tension maintains the integrity of the liquid plug.
However, proper functioning of the siphontype valves (Fig. 6) requires more complex design rules, such as fundamental correlations between radial positions, linear dimensions and volumes are already included for (CPDF) siphon valves. For example, to allow centrifugally driven outflow, the key radial positions of Γ need to be staggered according to R_{crest} < R < R_{out} for transitioning between the hydrostatic equilibria at r_{1} = r(ω) and r_{2} = r(Ω*) in the inbound and outbound segments at ω = Ω and Ω*, respectively. So R_{crest} < r_{1} < R and R_{crest} < r_{2} < R_{out} needs to hold for the two targetted equilibrium positions of the front menisci. These conditions imply ranges 0 < U_{0} − A_{0} ⋅ r_{0}(R, Γ, U_{0}, Ω) − U_{iso} ≤ A ⋅ (R − R_{crest}) and 0 < U_{0} − A_{0} ⋅ r_{0}(R, Γ, U_{0}, Ω*) − U_{iso} – A ⋅ Z − U_{crest} ≤ A ⋅ (R_{out} − R_{crest}) for the loaded liquid volume U_{0}. In addition, the priming pressure \(p_{{{{{{\mathrm{prime}}}}}}} = p_\omega + p_ \to  p_ \leftarrow\) needs to stay positive along the entire, ωcontrolled changeover of highpass (Ω < ω < Ω*) and lowpass (Ω* < ω < Ω) siphon valves between the two equilibrium positions r_{1} = r(R, Γ, U_{0}, Ω) to r_{2} = r(R, Γ, U_{0}, Ω*) of their front meniscus.
Manufacturability
The choice of the valving technology must also comply with manufacturing restrictions^{124,125} of each scheme availed of during scaleup from prototyping to pilot series production and mass fabrication. While early centrifugal microfluidic platforms were often based on capillary pumping and valving, the local definition of contact angles Θ on all walls including the lid and its stabilization over time under different ambient conditions during storage, transport, and deployment at the end user proves to be challenging. This work, therefore, emphasized valving schemes that would not require a coating step during manufacture.
For each manufacturing technique, there are also technical and economical limitations regarding shapes, aspect ratios, geometrical feature sizes, and their tolerances. For instance, minimum (lateral) dimensions of precision milling (of channels) are imposed by practicable tool diameters, in many cases about 200 μm, but rarely smaller than 100 μm; the tool radius furthermore determines the minimum curvature of (lateral) corners. As milling is a common way to prototype polymer LoaD substrates, and also for patterning replication tools, these restrictions apply to positive and negative, i.e., toolbased structures of the original design. Note also that while milling offers a powerful structuring in a wide range of materials, machine times, tooling cost (and wear) and process development of subsequent replication can go rampant when increasing demands on specifications such as surface quality (on floor and side walls), optical finish, wobble, tool wear other deviations from “native” 2.5 to 3dimensional geometries.
Especially for common polymer mass replication schemes like injection molding, a minimum wall thickness δW between all cavities needs to be enforced. For instance, in the CPDF siphoning valve in Fig. 9, amongst the distances to be monitored are Z > δW and L = w_{0} − w > δW. Smooth demolding sets upper limits on aspect ratios, and commonly necessitates the inclusion of draft angles, i.e., wall inclinations of the order of 5°–15°. More complex criteria may need to be accounted for, like an even distribution of the hydrodynamic resistance in the tool to avoid shadowing and inhomogeneous solidification during mold flow for the typically central, (compression)injection of the hot melt. Such collective mechanisms may induce adverse effects, such as wobble and eccentricity of the disc, ridges to compromise pressuretight bonding, or optical artifacts possibly interfering with detection and customer expectation.
In addition to such “designformanufacture” (DfM) considerations of each scheme individually, it also needs to be factored in that the technology along manufacturing scaleup might involve significantly diverging capabilities in (technoeconomically) achievable tolerances or shapes; therefore, the design restrictions introduced by the least capable scheme will have to be accounted for to assure seamless scaleup from prototyping to commercial production.
In particular, tool making and optimization of mold flow are decisive cost drivers for microfluidic systems; layouts for new applications, i.e., centrifugally automated assay protocols, should ideally be derived, as much as possible, from designs that have already been previously validated, while only varying parameters that are assumed to be less critical on behalf of fluidics, biology, and manufacturing.
Advanced design
In Fig. 12, we finestructure the original geometry for CPDF siphoning of Fig. 9; each compartment may have its specific depth, width, and height to allow a wider space for multiparameter optimization according to the performance metrics (26–35). Note that permanently gasfilled parts of the compression volume V_{C} may be moved to “any” location available on the disc as long as it is connected by a pneumatic conduit to the DF chamber.
Further variations might include inclination angles with respect to the radial and azimuthal directions, rounded shapes, branched structures to prevent blockage of air flow by residual liquid, liquid knifes for accurate metering of dispensed liquid volumes, lowthreshold capillary stops for transient pinning of the meniscus, and draft angles for proper demolding.
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Ducrée, J. Systematic review of centrifugal valving based on digital twin modeling towards highly integrated labonadisc systems. Microsyst Nanoeng 7, 104 (2021). https://doi.org/10.1038/s41378021003173
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DOI: https://doi.org/10.1038/s41378021003173
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