Systematic review of centrifugal valving based on digital twin modeling towards highly integrated lab-on-a-disc systems

Abstract

Current, application-driven trends towards larger-scale 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 “Lab-on-a-Disc” (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 larger-scale 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, de-risks and lowers costs along the critical advancement from idea, layout, fluidic testing, bioanalytical validation, and scale-up to commercial mass manufacture.

Introduction

Since their inception in the early 1990s, an important design goal of microfluidic Lab-on-a-Chip, 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 law⁶ 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, high-performance computers, smartphones, and gadgets, as well as economic incentives, as the cost of material and (typically pattern-transfer based) manufacturing processes strongly scales with the surface area of the chip⁷. “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, microfluidics-enabled (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 surface-to-volume ratios towards miniaturization.

Over the recent decades, numerous “Lab-on-a-Chip” 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 time-to-result, by imposing highly controlled conditions under strict laminarity at low Reynolds numbers, or by scale-matching with bio-entities such as cells. On the other hand, miniaturization resides at the backbone of sample-to-answer automation and parallelization, e.g., as a crucial product requirement for deployment of bioassay panels at the point-of-use/point-of-care (PoC), and patient self-testing at home.

Lab-on-a-Chip 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 multi-branched channel architecture can usually not be properly washed to assure full regeneration of fluidic functionality, and also to avoid cross-contamination or carry-over of biosamples and reagents. Hence, in most cases, the chip is devised as single-use. The cost of material, equipment, process development, and machine time of this disposable, which is normally mass-produced by tool-based 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 Lab-on-a-Chip 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 mid-1990s^{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 “Lab-on-a-Disc” (LoaD) systems, biochemical assay kits are ported on the rotationally controlled scheme by dissecting the often conventional, possibly volume-reduced 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 batch-wise, rather than a continuous-flow 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 multiplexing⁶⁶. Most of these “passive” flow control schemes root in the interplay of the centrifugal pressure exerted on a rotor-based 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 lifting²⁰, lowering¹⁵ or accelerating⁶⁷ the spin rate across critical frequency thresholds.

Yet, at least as stand-alone, 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 PoC-compatible instrument; they also lack to provide a physical vapor barrier, hence making them unsuitable for longer-term 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 rotor-based power unit⁶⁸, e.g., for mechanical⁶⁹, laser-^{70,71,72,73,74,75} or heat-induced perforation of a film⁷⁶, melting of a wax plug^77,78,79, or magnetic and pressure-induced deflection^80,81,82, either during rotation or at rest. Also, passive, solvent-selective barriers have been explored^44,68,83, which only transmit flow upon distinct physico-chemical stimuli.

More recently, centrifugo-pneumatic (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 surface-tension stabilized liquid “piston” within a microchannel across the crest point of their outlet siphon. This type of “Lab-on-a-Disc” (LoaD) platform uses a gas-impermeable, dissolvable film (DF), which is initially protected by a neighboring gas pocket. Once a geometry-dependent 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 CP-DF scheme, which can be solely controlled by the system-innate 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 real-time 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 batch-wise 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 system-level.

The next section covers the basic mechanisms underlying common, rotationally controlled valving technologies; we distinguish between high- and low-pass 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 dissolvable-film (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 mini-discs, 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 “stop-and-go” 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

$$f_\omega = \varrho \cdot {{{{{\mathcal{R}}}}}} \cdot \omega ^2$$

(1)

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 non-inertial 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 v⁴⁹; 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 time-varying 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 − r₀ and mean radial position $\bar r\left( \omega \right) = 0.5 \cdot \left( {r + r_0} \right)$ between its confining upstream and downstream menisci r₀ and r, respectively.

Expressed in cylindrical coordinates with the radial position r and the (potentially disjunct) local cross section A(r), the integral

$$U_0 = \mathop {\int}\limits_{{{\Lambda }}\left( \omega \right)} {{\mathrm{d}}V} = \mathop {\int}\limits_{\mathop{{\check{r}}}\limits\left( \omega \right)}^{\hat r\left( \omega \right)} {A\left( r \right)} {\mathrm{d}}r = {{{{{\mathrm{const}}}}}}.$$

(2)

corresponds to the total liquid volume U₀ contained in the segment. The conservation of U₀ 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 cross-sectional function A(r) is preserved, i.e., dU₀/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 r-direction 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

$$p_\omega = {\it{\rho }} \cdot \bar r\Delta r \cdot \omega ^2$$

(3)

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 − r₀ of the liquid segment Λ(ω). The product

$$\bar r{{\Delta }}r = \frac{1}{2}\left[ {r + r_0} \right] \cdot \left[ {r - r_0} \right] = \frac{1}{2}\left[ {r^2 - r_0^2} \right]$$

(4)

in p_ω (3) can also be expressed by the front and rear radial positions of the menisci r and r₀, 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

$$p_V = p_0 \cdot \frac{{V_0}}{V}$$

(5)

of a gas volume that is compressed from an initial volume V₀ at p₀ to V < V₀ (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

$$p_{{\Theta }} = \frac{{4\sigma }}{D}\cos {{\Theta }}$$

(6)

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 semi-quantitative scale. During flow at a volumetric rate Q through a channel with round cross section A = π ⋅ (D/2)², a pressure drop

$$p_{{{{{{Q}}}}}} = \frac{{64}}{\pi } \cdot \frac{{\eta \cdot L \cdot Q}}{D}$$

(7)

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

$$p_m = \frac{{\varrho \cdot U \cdot R \cdot {\mathrm{d}}\omega {{{{{\mathrm{/}}}}}}{\mathrm{d}}t}}{A}$$

(8)

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 rotor-based pressure sources and pumps^37,66,100, by thermo-pneumatic 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 batch-mode 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

$$\underbrace {\varrho \cdot \bar r{{\Delta }}r \cdot \omega ^2}_{p_\omega } + p_ \to = p_ \leftarrow$$

(9)

between p_ω ∝ ω² (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

$$\omega = \sqrt {\frac{{p_ \leftarrow - p_ \to }}{{\varrho \cdot \bar r{{\Delta }}r}}}$$

(10)

can be attributed to a given Λ(ω) of a coherent liquid volume U₀ 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 batch-mode-processing, 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 high-pass and low-pass 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 mixing³², incubation of dissolved biomolecules with surface-immobilized capture probes, resuspension of dry-stored reagents, or to support mechanical cell lysis through fixed-geometry 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 ∝ ω² (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 high-pass and low-pass valves, respectively. In some valving schemes presented later, the rotational actuation may not be achieved immediately after crossing Ω; proper (high-pass) valving is only assured once a (slightly) elevated actuation frequency Ω* > Ω is reached.

Alternatively, other, non-centrifugal 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 p₀, i.e., p_V ↦ p₀, and normally p₀ ≈ 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)

$${{{{\Delta \Omega }}}}\left( {\left\{ {\gamma _k,{{\Delta }}\gamma _k} \right\}} \right) \approx \sqrt {\mathop {\sum}\limits_k {\left( {\frac{{\partial {{\Omega }}}}{{\partial \gamma _k}}{{\Delta }}\gamma _k} \right)} ^2}$$

(11)

can be calculated (and then systematically be optimized) by Gaussian error propagation, or through Monte-Carlo 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 U₀, 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 low-pass 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 system-level 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 ω⁹³. 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 non-overlapping 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 ω-range 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 0.9999³ ≈ 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 system-level robustness to 0.95³ ≈ 86%. Note that for the sake of simplicity, these back-of-the-envelope 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 high-pass and ω > max {Ω_i + M · ΔΩ_i} for low-pass valves. The order of release by venting simply relates to the sequence of the removal of the seals.

For rotationally actuated, simultaneous release of high-pass 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 tolerance-related bands {Ω_i ± M ⋅ ΔΩ_i} and $\{ {{\Omega }}_i^ \ast \pm M \cdot {{{{\Delta \Omega }}}}_i^ \ast \}$ do not overlap for all {i}.

Fig. 1: Fluidic multiplexing of rotationally actuated high-pass valving on the LoaD platform.

a Robust concurrent actuation of valves {i, j} of a step i sharing the critical spin rate Ω_i at a time T_i requires lifting of the spin rate ω across the interval of width 2 ⋅ M ⋅ max{ΔΩ_i,j}, which takes a time span ΔT_i. b In serial actuation of steps {i}, the frequency intervals {Ω_i ± M ⋅ ΔΩ_i} around the actuation frequencies {Ω_i} need to be separated in order to assure the correct order of release at times {T_i}. a Simultaneous actuation. b Sequential actuation.

Basic centrifugal flow control schemes

Sacrificial barriers

Apparently, straight-forward implementation of normally closed valves are removable materials for intermittently blocking liquids and gases. Various types of such sacrificial-barrier valves have been developed^104,105,106. However, most of them require external actuation by an instrument-based 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, ice¹⁰⁸, or (laser) irradiation⁷⁴. 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 centrifugo-pneumatic valving.

Centrifugo-capillary 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 well-defined (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).

Fig. 2: Centrifugo-capillary burst valve (not to scale).

In this hydrophobic constriction, a centrifugal pressure head $p_\omega \propto \bar r{{\Delta }}r$ (3) runs against the capillary counterpressure p_← = p_Θ (6) calculated by the contact angles Θ > 90° and Θ_rear ≈ 90°, and their channel diameters D ≪ D_rear for their front and rear menisci, respectively. The valve stays closed until a critical spin rate Ω_Θ (12) is surpassed. (Left) Radial alignment. (Right) Isoradial alignment where radial “squeezing” of the liquid within the isoradial outlet may become an issue.

In the hydrostatic approximation (9), a threshold frequency for a hydrophobic constriction

$${{\Omega }}_{{\Theta }} = \sqrt {\frac{{4\sigma }}{{\varrho \cdot \bar r \cdot {{\Delta }}r}}\left( {\frac{{\cos {{\Theta }}_{{{{{{\mathrm{rear}}}}}}}}}{{D_{{{{{{\mathrm{rear}}}}}}}}} - \frac{{\cos {{\Theta }}}}{D}} \right)} \quad \approx \quad \sqrt { - \frac{{4\sigma }}{{\varrho \cdot \bar r \cdot {{\Delta }}r}}\frac{{\cos {{\Theta }}}}{D}}$$

(12)

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 centrifugo-pneumatic 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 so-called “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 longer-term 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, well-localized coatings. Otherwise, retention rates Ω might shift, or flow might still creep, instead of being cleanly halted, as required for proper batch-mode processing.

Centrifugo-pneumatic 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 V₀ to V < V₀ 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 p₀ + δp₀ = p₀ ⋅ (1 + χ) at ω ≈ 0; the small offset δp₀ with δp₀/p₀ = χ ≪ 1 of the gas pressure p₀ 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.

Fig. 3: Centrifugo-pneumatic valving (not to scale).

a At rest (ω ≈ 0), the liquid stops at r = R in front of the radial outlet of length Z and a sufficiently narrow cross section A, which is followed by a compression chamber. At this point, the gas volume V_C + A · Z is at ambient pressure p₀, plus a small contribution δp₀ with χ = δp₀/p₀ ≪ 1 linked to (dynamic) filling effects. b At a critical spin rate Ω_V, the meniscus protrudes to r = R + Z at the transition to the pneumatic chamber. Now the pneumatic counterpressure has increased to p_V = p₀⋅(1 + χ) ⋅ (V_C + A ⋅ Z)/V_C, and a droplet of volume V_drop ≈ (4/3)π(D/2)³ located at r = r_drop ≈ R + Z emerges from the orifice. Centrifugo-pneumatic valving is triggered once the weight force $F_m = V_{{{{{{\mathrm{drop}}}}}}} \cdot f_\omega$ (1) of the droplet exceeds the counteracting surface tension force F_σ = πD_min⋅σ at its minimum diameter D_min = D/κ with κ > 1. While the exact dynamics of its detachment are unclear and hard to quantify, it is assumed that the hydrodynamic agitation caused by the detaching droplet disrupts the integrity of the liquid plug, thus causing successive release of the entire liquid into the compression chamber, while gas gradually escapes in the reverse direction to atmosphere. c CP valving with an isoradial outlet pinned at r = R. A minimum liquid volume U₀ ≥ U_iso is required for generating Δr > 0, and thus p_ω > p_V(Ω_V) for opening. Towards high field strength f_ω (1), the shape of the front meniscus progressively distorts. a Filling at rest. b State at critical spin rate ${{\Omega }}_V$. c Isoradial configuration.

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 = p₀⋅(1 + χ)⋅(V_C + A⋅Z)/V_C (5). The cross section A needs to be sufficiently small so that the surface tension sustains “piston-like” characteristics of the liquid plug. Under these conditions, we set

$$p_ \leftarrow = p_V = p_0 \cdot \left( {1 + \chi } \right) \cdot \left( {1 + \frac{{A \cdot Z}}{{V_{{{{{\mathrm{C}}}}}}}}} \right)$$

(13)

and p_→ = p₀ to obtain a critical spin rate (10)

$${{\Omega }}_V = \sqrt {\frac{{p_0}}{{\varrho \cdot \bar r \cdot {{\Delta }}r}} \cdot \left[ {\left( {1 + \chi } \right) \cdot \left( {1 + \frac{{A \cdot Z}}{{V_{{{{{\mathrm{C}}}}}}}}} \right) - 1} \right]} \approx \sqrt {\frac{{p_0}}{{\varrho \cdot \bar r \cdot {{\Delta }}r}} \cdot \frac{{A \cdot Z}}{{V_{{{{{\mathrm{C}}}}}}}}}$$

(14)

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 δp₀ ≈ 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 centrifugo-pneumatic 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

$${{\Omega }}_{{{{{{\mathrm{drop}}}}}}} \approx \sqrt {\frac{{\sigma \cdot \pi D_{{{{{{\mathrm{min}}}}}}}}}{{\varrho \cdot \left( {4/3} \right)\pi \left( {D/2} \right)^3 \cdot r_{{{{{{\mathrm{drop}}}}}}}}}} \approx \frac{1}{D}\sqrt {\frac{{6\sigma }}{{\varepsilon \cdot \varrho \cdot \left( {R + Z} \right)}}}$$

(15)

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 p₀ 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 p₀, 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(p₀).

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 tolerance-forgiving 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).

Fig. 4: Variation in the atmospheric pressure p₀.

a Barometric formula quantifies the decrease in atmospheric pressure from sea level to roughly the limit of habitable space on earth at roughly 4000 m altitude. b Compensation factor of the spin rate ω for altitude adjusted (standard) pressure p₀. For the given example, the critical spin rate would need to be lowered by about 20% from Ω/2π = 25 Hz at sea level to about 20 Hz in high altitude. In general, any known local pressure p₀, either caused by altitude or weather, can be flexibly accommodated by adjusting the spin rate ω according to $\sqrt {p_0{{{{{\mathrm{/}}}}}}p_{{{{{{\mathrm{std}}}}}}}}$ (23). c Shift of the retention rate Ω (23) with the (dimensionless) coefficient χ (14) representing potential dynamic effects entailing deviations of the effective pressure in the gas volume enclosed in the compression chamber from ambient p₀ at the point when it is pneumatically isolated from the liquid in the isoradial channel. a Changes of atmospheric pressure p₀ with altitude with respect to p_std. b Adjustment factor for the retention rate ${{\Omega }}$ (23) to compensate altitude. c Impact of dynamic filling effects characterized by the coefficient $\chi$ on the retention rate ${{\Omega }}$ (23).

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 centrifugo-pneumatic valves essentially proceeds at frequencies

$${{\Omega }}_{{{{{{\mathrm{cpv}}}}}}} = {{{{{\mathrm{min}}}}}}\left\{ {{{\Omega }}_V,{{\Omega }}_{{{{{{\mathrm{drop}}}}}}}} \right\}$$

(16)

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 step-wise 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 centrifugo-pneumatic flow control schemes mainly suitable for final valving steps into a dead-ended cavity, e.g., for aliquoting of liquid sample or reagents into detection chambers¹⁰⁹. Moreover, centrifugo-pneumatic valving requires powerful spindle motors, aerodynamic optimization, and mechanically well-balanced rotors, and may raise concerns about lab safety.

Venting

Opening the compression chamber to atmosphere, i.e., V_C ↦ ∞ leading to p_V ↦ p₀ (5), constitutes an alternative actuation mechanism for these CP-valves. 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 high-speed rotation (Fig. 5b). Especially in the context of “event-triggered” 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).

Fig. 5: Centrifugo-pneumatic valving by venting.

a Below the retention rate ω ≤ Ω_V (14), the liquid is kept outside the pneumatic chamber, which is closed by gas-impermeable or dissolvable film (DF) membranes. b Upon its dissolution, the pneumatic counter pressure p_V converges to the atmospheric pressure p₀, thus releasing the liquid at any ω > 0. c For this high-pass valve, liquid enters the compression chamber above a retention frequency Ω_cpv (16). After a sufficient volume U_DF = β · V_C with 0 < β < 1 has entered the compression chamber of (dead) volume V_C, the liquid wets and thus opens the DF to trigger flow at any ω > 0 through an outlet, which is, e.g., located in a lower layer connected through a vertical via concealed underneath the DF. a Retention phase. b Venting through membrane. c Venting through dissolvable film.

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 A₀, 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 < A₀. 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.

Fig. 6: Centrifugally controlled siphon valving with distributions Λ for retention (left) and release (right) (linearized display, not to scale).

The layout Γ features an inlet reservoir of cross section A₀ and a bottom at R, connected to an isoradial outlet of length L_iso, radial height h_iso and volume U_iso. The following siphon channel starts with an inbound segment of radial length Z and cross section A₁ between R and a crest point R_crest = R − Z, an inner isoradial channel of axial length L_crest, radial height h_crest and volume U_crest, and an outlet channel of cross section A₂ and volume U_out along the interval R_crest < r < R_cham leading to a collection chamber at R_cham > R. The ambient pressure p₀ in the vicinity of p_std applies to all vented chambers. The meniscus positions confining the liquid U₀ in the inlet, the inbound and outbound sections are r₀(ω), r₁ = r(ω = Ω) and r₂ = r(Ω^*), respectively. A net centrifugal pressure p_ω ∝ Δr⋅ω² (3) with Δr(ω) = r_i(ω) − r₀(ω) and i ∈ {1, 2} plus an axially directed pressure difference p_z = p_→ − p_← (assumed here as constant along the axial direction), which is composed of forward and backwards contributions p_→ and p_←, shape the liquid distribution Λ(ω). A first critical retention frequency Ω is set with p_ω(Ω) + p_z(Ω) = 0 at R_crest < r₁ < R in the inbound segment. Ω is usually chosen so that r₁(Ω) settles sufficiently below R_crest to account for M · ΔΩ (11) linked to tolerances {Δγ_i}. A second critical position r₂ is found in the outlet segment at a second spin rate Ω* at p_ω(Ω*) + p_z(Ω*) = 0, possibly after adding a liquid volume U_Δ to U₀ (a). Valving is practically possible if the calculated Ω and Ω* reside within the frequency envelope between ω_min and ω_max, suitable U₀ > U_iso and Γ, so the radial positions r₀, r₁ and r₂ are located within their allowed radial intervals R_min < r₀ < R, R_crest < r₁ < R and R_crest < r₂ < R_cham. a Priming by volume addition. For $p_ \to = p_ \leftarrow = p_0$, and thus $p_z = 0$, and ${{\Delta }}r = 0$ for all spin rates $\omega \, > \, 0$. The front meniscus resides at $r = r_1 \, > \, R_{{{{\mathrm{crest}}}}}$ in the inbound segment of cross section $A_1$ for all liquid volumes $U_0$ within $0 \, < \, U_0 - U_{{{{\mathrm{iso}}}}} \, < \, \left( {A_0 + A_1} \right) \cdot Z$. Then, an amount $U_{{\Delta }} \, > \, 0$ with $U_0 + U_{{\Delta }} = (A_0 + A_1) \cdot Z + U_{{{{\mathrm{iso}}}}} + U_{{{{\mathrm{crest}}}}} + A_2 \cdot (r_2 - R_{{{{\mathrm{crest}}}}})$ needs to be added to shift the meniscus across the crest channel of volume $U_{{{{\mathrm{crest}}}}}$ to $r_2 \, > \, R_{{{{\mathrm{crest}}}}}$ in the outbound channel of cross section $A_2$. The conservation of liquid volume $U_0$ between ${{\Omega }}$ and ${{\Omega }}^ \ast$ demands $U_0 = A_0 \cdot \left[ {R - r_0\left( {{\Omega }} \right)} \right] + U_{{{{\mathrm{iso}}}}} + A_1 \cdot \left[ {R - r_1\left( {{\Omega }} \right)} \right] = A_0 \cdot \left[ {R - r_0\left( {{{\Omega }}^ \ast } \right)} \right] + U_{{{{\mathrm{iso}}}}} + A_1 \cdot Z + U_{{{{\mathrm{crest}}}}} + A_2 \cdot \left[ {R_{{{{\mathrm{crest}}}}} - r_2\left( {{{\Omega }}^ \ast } \right)} \right] - U_{{\Delta }}$. After a critical position $R_{{{{\mathrm{crest}}}}} \, < \, r_2 = r_0 \, < \, R_{{{{\mathrm{cham}}}}}$ is reached, ${{\Delta }}r \, > \, 0$ holds for all $\omega \, > \, 0$ without further volume addition $U_{{\Delta }}$, so centrifugally driven forward pumping into the vented receiving chamber situated at $R_{{{{\mathrm{cham}}}}} \, > \, R$ kicks in. b Priming of low-pass siphon valves occurs for $p_z \, > \, 0$. (Left) At the retention rate $\omega = {{\Omega }}$, the front meniscus resides in the inbound section of cross section $A_1$ at $r_1$ with $R_{{{{\mathrm{crest}}}}} \, < \, r_1 \, < \, R$, so that ${{\Delta }}r \, < \, 0$ and $p_\omega = p_z$. (Right) At a second hydrostatic equilibrium defining release at ${{\Omega }}^ \ast \, < \, {{\Omega }}$ $\left( {{{{\mathrm{and}}}}\;U_{{\Delta }} = 0} \right)$, the meniscus has passed the crest channel of volume $U_{{{{\mathrm{crest}}}}}$ to arrive at a second position $r_2$ in the radial outlet of cross section $A_2$, so that for $\omega \, < \, {{\Omega }}^ \ast$, $p_\omega$ aligns parallel to the axial direction and $p_z$, hence constructively pumping the liquid into the (vented) outer recess at $R_{{{{\mathrm{cham}}}}} \, > \, R$. c Priming of high-pass siphon valves with $p_z \, < \, 0$. (Left) During retention $\left( {\omega \, < \, {{\Omega }}} \right)$ of the meniscus at $r_1$ in the inbound section of cross section $A_1$ with $R_{{{{\mathrm{crest}}}}} \, < \, r_0 \, < \, r_1 \, < \, R$, a liquid level difference ${{\Delta }}r \, > \, 0$ is needed to compensate the counterpressure $p_z$. (Right) For $\omega = {{\Omega }}^ \ast$ $\left( {{{{\mathrm{and}}}}\;U_{{\Delta }} = 0} \right)$, a critical point $R_{{{{\mathrm{cham}}}}} \, > \, r_2 \, > \, R_{{{{\mathrm{crest}}}}}$ is reached in the radially outbound channel of cross section $A_2$. For any $\omega \, > \, {{\Omega }} \ast$, the centrifugal pressure $p_\omega$ (3) exceeds $p_z$ to transfer the liquid into the (vented) outlet at $R_{{{{\mathrm{cham}}}}}$. Note that for pneumatically controlled principles, i.e., $p_ \leftarrow = p_V$, the receiving chamber needs to be sealed.

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 = r₁ with R_crest < r₁ < R for a loaded liquid volume U₀ > U_iso. The retention rate Ω is usually set so that the meniscus r₁ 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 r₁ 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 < r₂ = r(Ω*) < R_cham is situated in the outbound channel beyond which any further increase in Δr = r₂(ω) − r₀(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 r₁ = r(Ω) in the inbound and r₂ = r(Ω*) in the outbound segments.

Priming

In volume addition mode, priming is triggered by topping up U₀ 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 r₀, i.e., r > r₀.

For low-pass 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 high-pass 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 A₀ ⋅ [R − r₀(Ω)] + U_iso + A₁ ⋅ [R − r₁(Ω)] = A₀ ⋅ [R − r₀(Ω*)] + U_iso + A₁ ⋅ Z + U_crest + A₂ ⋅ [r₂ − 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 < r₀ < R, R_crest < r₁ < R, and R_crest < r₂ < 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) multi-step 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 p₀ + δp₀, with δp₀ = χ ⋅ p₀ and χ ≪ 1.

Fig. 7

Pneumatic siphon priming with p_z = p_→ = p_V and a vented outlet, i.e., p_← = p₀ (not to scale). a Liquid is loaded at ${{t}_{0}}$ and $\omega \approx 0$ to the inlet so the air in the side chamber of dead volume $V_{{{{\mathrm{side}}}}}$ is cut off from atmosphere at $p_0$ at an initial pressure $p_0 + \delta p_0 = p_0 \cdot \left( {1 + \chi } \right)$ with $\chi \ll 1$. b At the highest spin rate $\omega = {{\Omega }}_{{{{\mathrm{load}}}}}$, the gas pocket is centrifugally compressed to ${V}\left( {{{\Omega }}_{{{{\mathrm{load}}}}}} \right) \,<\, {V}_{{{{\mathrm{side}}}}}$ while assuring ${r}_1(\omega ) = {r}_2(\omega ) \,>\, {R}_{{{{\mathrm{crest}}}}}$. c At $\omega = {{\Omega }}_{{{{\mathrm{pps}}}}} \, < \, {{\Omega }}_{{{{\mathrm{load}}}}}$, the local pressure to $p_{{{{\mathrm{side}}}}} = p_0 \cdot \left( {1 + \chi } \right) \cdot V_{{{{\mathrm{side}}}}}/V\left( {{{\Omega }}_{{{{\mathrm{pps}}}}}} \right)$ sets the common liquid levels ${r} = {r}_0 = {R}_{{{{\mathrm{crest}}}}}$ in the inlet and inbound segment of cross section ${{A}_{1}}$, i.e., ${{\Delta }}r = 0$. d Once $\omega \,<\, {{\Omega }}_{{{{\mathrm{pps}}}}}^ \ast \,<\, {{\Omega }}_{{{{\mathrm{pps}}}}}$ has been adequately lowered, the front meniscus has passed the crest channel at ${R}_{{{{\mathrm{crest}}}}}$ of cross section $A_{{{{\mathrm{crest}}}}}$ to reach the second equilibrium position at $r_2$ in the outbound channel of cross section $A_2$, i.e., ${{r}_{2} = {r}\left( {\omega = {{\Omega }}_{{{{\mathrm{pps}}}}}^ \ast } \right)}$ For sufficiently small ${{A}_{1}}$, $A_{{{{\mathrm{crest}}}}}$ and $A_2$, liquid is centrifugally “pulleyed” from the inlet through the siphon channel into the outlet chamber commencing at $R_{{{{\mathrm{cham}}}}}$.

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 r₀(ω) = r₁(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 r₁(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 A₁, A_crest, and A₂ 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

$$p_{{{{{{\mathrm{side}}}}}}}\left( \omega \right) = \varrho \cdot \bar r_{{{{{{\mathrm{side}}}}}}}{{\Delta }}r_{{{{{{\mathrm{side}}}}}}} \cdot \omega ^2 + p_0 = p_0 \cdot \left( {1 + \chi } \right) \cdot \frac{{V_{{{{{{\mathrm{side}}}}}}}}}{{V\left( \omega \right)}}$$

(17)

with the mean value and difference $\bar r_{{{{{{\mathrm{side}}}}}}}$ and Δr_side deriving from the liquid levels r₀ 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 U₀ have to be configured so

$$\begin{array}{l}V\left( {{{\Omega }}_{{{{{{\mathrm{pps}}}}}}}^ \ast } \right) - V\left( {{{\Omega }}_{{{{{{\mathrm{pps}}}}}}}} \right) \ge \left[ {r_1\left( {{{\Omega }}_{{{{{{\mathrm{pps}}}}}}}} \right) - R_{{{{{{\mathrm{crest}}}}}}}} \right] \cdot \left( {A_0 + A_1} \right) + U_{{{{{{\mathrm{crest}}}}}}} + \left[ {r_2\left( {{{\Omega }}_{{{{{{\mathrm{pps}}}}}}}^ \ast } \right) - R_{{{{{{\mathrm{crest}}}}}}}} \right] \cdot A_2\end{array}$$

(18)

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 < r₁(Ω_cps) < R in the inbound segment of cross section A₁ with a negative offset Δr < 0, i.e., r₁ < r₀. This distribution Λ(Ω_cps) relates to a retention rate

$${{\Omega }}_{{{{{{\mathrm{cps}}}}}}} = \sqrt {\frac{{4\sigma \cos {{\Theta }}}}{{\varrho \cdot \bar r{{\Delta }}r \cdot D}}}$$

(19)

(neglecting the small capillary pressure at the meniscus in the large inlet reservoir for A₁/A₀ ≪ 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 r₂ establishes at $\omega = {{\Omega }}_{{{{{{\mathrm{cps}}}}}}}^ \ast$ with the meniscus at r₂ > R_crest in the outlet segment of cross section A₂. Any further progression r > r₂ 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 low-pass valve, capillary-action 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 capillary-driven priming at ω = 0, the time

$$T_{{\Theta }} = \frac{{4\eta \cdot l^2}}{{D \cdot \sigma \cos {{\Theta }}}}$$

(20)

for covering the axial distance l ≈ L + Z + L_crest + (R_cham − R_crest) scales with l², the viscosity of the liquid η, and inversely with its surface tension σ, cos Θ > 0, and the cross-sectional diameter of the (round) channel D.

Lost volume

Transfer by centrifugal siphoning (Fig. 6) is usually accompanied by a loss U_loss < U₀ + U_Δ of the original liquid volume U₀, 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/U₀ ≪ 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 A₁, A_crest, and A₂. 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.

Fig. 8: Residual volume in centrifugal siphoning.

After the transfer, part of the liquid U₀ is left in the siphon structure. This U_loss < U₀ sensitively hinges on the dynamics of flow, and the shape of critical parts of Γ, for instance, on the cross sections A₀, A₁, A₂, A_iso, and A_crest of the radial and isoradial segments; U_loss would ideally be 0, or at least reproducible, i.e., ΔU_loss ≈ 0. a Liquid $U_{{{{\mathrm{loss}}}}} \approx U_{{{{\mathrm{iso}}}}}$ residing in the siphon structure ${{\Gamma }}$ after “pulley” type siphoning failed to empty the isoradial channel of volume $U_{{{{\mathrm{iso}}}}}$. This volume $U_{{{{\mathrm{loss}}}}}$ detaches from the liquid as air is sucked into the liquid plug. $U_{{{{\mathrm{loss}}}}}$ and its spread ${{\Delta }}U_{{{{\mathrm{loss}}}}}$ may, for instance, be minimized through reduction of $A_1 \cdot Z$. b Liquid volume $U_{{{{\mathrm{loss}}}}} = U_{{{{\mathrm{iso}}}}} + \left( {A_0 + A_1} \right) \cdot Z$ remaining in ${{\Gamma }}$ with a purely overflow driven siphoning, which is favored by larger cross sections $A_2$ of the radially outbound channel, for which the liquid just “drizzels” into the outer chamber out after passing the crest point.

Centrifugo-pneumatic dissolvable-film siphon valving

The geometry Γ in Fig. 9 constitutes a hybrid of centrifugo-pneumatic (CP) valves (Fig. 3), sacrificial dissolvable-film (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.

Fig. 9: Operational principle of the siphon-shaped CP-DF siphon valve (linearized display, not to scale).

The structure Γ features a constant depth d. a Loading of isoradial segment with $U_{{{{\mathrm{iso}}}}}$ at $\omega = 0$ so a gas volume $V_{{{\mathrm{C}}}} + A \cdot Z$ is enslosed at a pressure $p_0 + \delta p = p_0 \cdot \left( {1 + \chi } \right)$ with respect to the atmosphere at $p_0$ and a small $\chi$ representing potential dynamic effects during priming. b Topping up liquid volume to $U_0$ so that the front meniscus settles at $r = R - Z$ when spinning at the retention rate $\omega = \Omega$. c At the release frequency ${{\Omega }}^ \ast \,>\, {{\Omega }}$, a sufficient liquid volume $U_{{{{\mathrm{DF}}}}} = \beta \cdot V_{{{{\mathrm{DF}}}}}$ corresponding to a fraction $0 \, < \, \beta \, < \, 1$ of the dead volume $V_{{{{\mathrm{DF}}}}}$ of the outer chamber has overflown to perforate the DF. d After the DF membrane has opened at $\omega \,>\, {{\Omega }}^ \ast$, or by venting $V_{{{\mathrm{C}}}} \, \mapsto \, \infty$, and absence of pulley effects, the liquid underneath the crest point $r \ge R - Z$ and $\alpha \cdot V_{{{{\mathrm{DF}}}}}$ with $0 \, < \, \alpha \, < \, \beta$ remains in ${{\Gamma }}$.

During retention of this high-pass 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 break-off 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 U₀ with smaller and more reproducible pressure offset δp₀ (and thus ${{\Delta }}\chi \mapsto 0$) than for the basic CP-DF valve (Fig. 3). The gas-tight DF initially isolating the final pneumatic chamber allows for rotationally controlled opening without external actuators, as well as venting mode, while also removing the end-point character of the receiving chamber familiar from basic CP valves (Fig. 3).

By virtue of these manifold synergistical benefits, we consider CP-DF 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 closed-form 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 CP-DF siphon valving, a somewhat hypothetical, multi-step 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 p₀ + δp₀ = p₀ ⋅ (1 + χ) with 0 ≤ χ ≪ 1, and the ambient pressure p₀ applying to the inlet, which is open to atmosphere, often at p ≈ p_std.

The total liquid volume is then topped up to U₀ = U_iso + A₀ ⋅ [R − r₀(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 = r₁(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

$$p_V\left( {R,{{\Gamma }},U_{{{{{{\mathrm{DF}}}}}}},p_0,\chi ,r} \right) = p_ \leftarrow = p_0 \cdot \left( {1 + \chi } \right) \cdot \left( {\frac{{V_{{{{{\mathrm{C}}}}}} + A \cdot Z}}{{V_{{{{{\mathrm{C}}}}}} + A \cdot \left[ {Z - \left( {R - r} \right)} \right] - U_{{{{{{\mathrm{DF}}}}}}}}}} \right)$$

(21)

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

$$r_0\left[ {R,{{\Gamma }},U_0,U_{{{{{{\mathrm{DF}}}}}}},r\left( \omega \right)} \right] = R - \frac{{U_0 - U_{{{{{{\mathrm{iso}}}}}}} - A \cdot \left[ {R - r\left( \omega \right)} \right] - U_{{{{{{\mathrm{DF}}}}}}}}}{{A_0}}$$

(22)

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_→ = p₀ = const., the hydrostatic equilibrium for the CP-DF 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, U₀, U_DF, p₀, χ, ω) can be found (in principle) for a given geometry of the CP-DF 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 U₀. 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.

Fig. 10: Meniscus position z = R − r as a function of the spin rate ν = ω/2π.

With growing ω, the meniscus rises in the radially inbound channel until it arrives at R_crest = R − Z when $\omega = {{{{\Omega /}}}}2\pi \approx 22\,{{{{{\mathrm{Hz}}}}}}$.

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

$${{\Omega }}\left( {R,{{\Gamma }},U_0,\chi } \right) = \sqrt {\frac{{p_0 \cdot \left[ {\left( {1 + \chi } \right) \cdot \frac{{V_{{{{{\mathrm{C}}}}}} + A \cdot Z}}{{V_{{{{{\mathrm{C}}}}}}}} - 1} \right]}}{{\varrho \cdot \bar r{{\Delta }}r}}} \approx \sqrt {\frac{{p_0}}{{\varrho \cdot \bar r{{\Delta }}r}} \cdot \frac{{A \cdot Z}}{{V_{{{{{\mathrm{C}}}}}}}}}$$

(23)

for the critical retention rate Ω of the CP-DF 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 gas-filled compression chamber (Fig. 11a). Ω also increases by extending the length of the radially inbound segment Z (Fig. 11b). As r₀ is linear in R and U₀ (22), the radial product $\bar r{{\Delta }}r$ is a square function in r₀, so Ω decreases with U₀ (Fig. 11c) and R (Fig. 11d), roughly following 1/U₀ and 1/R, respectively.

Fig. 11: Critical spin frequency Ω/2π as a function of individual experimental input parameters (for χ ≈ 0).

According to (23), Ω depends on (a) the compression volume V_C, roughly merging to ${{\Omega }} \propto 1{{{\mathrm{/}}}}\sqrt {V_{{{\mathrm{C}}}}}$. As for a given U₀, Δr rapidly shrinks with growing Z, while less affecting $\bar r$, Ω increases steeply towards large Z (b). For r₁(Ω) = R − Z, increasing the volume U₀ enlarges Δr faster than $\bar r$ decreases, so the retention rate Ω reduces with U₀ (c) and the radial position R (d). a Volume of compression chamber V_C. b Length of radially inbound sector Z. c Liquid volume U₀ loaded to Γ. d Radial position R of Γ.

Tuning of the critical spin rate

As r₀(Ω) (22) is linear in U₀/A₀, also the radial product $\bar r{{\Delta }}r$ (4), and thus Ω (23), remain unaltered for U₀/A₀ = const. Hence, an LUO requiring retention of a different liquid volume U₀ preserves the same critical spin rate Ω (23) as long as the cross section of the inlet A₀ 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 gas-filled 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 CP-DF 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 “event-triggering” ⁸⁸.

This transfer of U_DF = β ⋅ V_DF into the recess reduces the original liquid and gas volumes U₀ and V_C + A ⋅ Z, respectively, by U_DF, and typically U_DF ≪ U₀, while the forward meniscus remains pinned to r = R − Z. We calculate the release rate

$${{\Omega }}^ \ast = \sqrt {\frac{1}{{\varrho \cdot \bar r\left( {U_0 - \beta \cdot V_{{{{{{\mathrm{DF}}}}}}}} \right){{\Delta }}r\left( {U_0 - \beta \cdot V_{{{{{{\mathrm{DF}}}}}}}} \right)}} \cdot p_0 \cdot \left[ {\left( {1 + \chi } \right) \cdot \frac{{V_{{{{{\mathrm{C}}}}}} + A \cdot Z}}{{V_{{{{{\mathrm{C}}}}}} - \beta \cdot V_{{{{{{\mathrm{DF}}}}}}}}} - 1} \right]}$$

(24)

from (23) by considering the cutback of the loaded volume U₀ 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 CP-DF siphon valves through Γ, for instance, by the dead volume of the DF chamber outside r ≥ R_DF. For common CP-DF siphon valves, β ⋅ V_DF/V_C ≪ 1, so that the 0 < Ω_step/Ω ≪ 1.

The opening mechanism of the CP-DF siphon valve (Fig. 9) imposes the general volume condition A₀ ⋅ [R − Z − r₀(Ω)] ≥ U_DF for both, actuation by rotation or venting, to assure Δr > 0, and thus a non-vanishing 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 CP-DF 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 CP-DF 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 system-level 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

$$U_{{{{{{\mathrm{loss}}}}}}} = \left( {A_0 + A} \right) \cdot Z + U_{{{{{{\mathrm{iso}}}}}}} + \alpha \cdot V_{{{{{{\mathrm{DF}}}}}}}$$

(25)

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 A₀ 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 “cut-off”, e.g., by placing a sharp-edged “liquid knife” within a low dead-volume 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, system-level reliability for rapid and cost-efficient scale-up 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 high-pass valve, maximum field strengths $f_\omega \left( {{{{\hat{\mathrm \Omega }}}}} \right) \propto {{{\hat{\mathrm \Omega }}}}^2$ (1) for capillary burst (12)

$$f_{{\Theta }} \approx \varrho \cdot \bar r \cdot {{\Omega }}_{{\Theta }}^2 \approx \frac{{4\sigma }}{{{{\Delta }}r}}\frac{{\left( { - \cos {{\Theta }}} \right)}}{D}$$

(26)

and basic CP valves (16)

$$f_{{{{{{\mathrm{CP}}}}}}} \approx \varrho \cdot \bar r \cdot {{\Omega }}^2 \approx \frac{{p_0}}{{{{\Delta }}r}} \cdot \frac{{A \cdot Z}}{{V_{{{{{\mathrm{C}}}}}}}}$$

(27)

as well as for the CP-DF siphoning structure (23) of retention rate ${{{\hat{\mathrm \Omega }}}}$ cannot be exceeded during processing of an upstream LUO. For the low-pass 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

$$f_{{{{{{\mathrm{cps}}}}}}} \approx \varrho \cdot \bar r \cdot {{\Omega }}_{{{{{{\mathrm{cps}}}}}}}^2 \approx \frac{{4\sigma \cos {{\Theta }}}}{{{{\Delta }}r \cdot D}}$$

(28)

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 r₀ 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 − r₀ 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 r₀ and r. Note, however, that maximization of ${{\Omega }} \propto 1{{{{{\mathrm{/}}}}}}\sqrt {{{\Delta }}r}$ (23) hits a limit as the volume A₀ · Δr has to be sufficient to effectuate complete filling of the channel section extending between the position r₁ during retention to r₂ for still being able to trigger liquid release.

Bandwidth

The metric

$$\overline {{{{{\Delta \Omega }}}}} = \frac{{{{{{\Delta \Omega }}}}}}{{\omega _{{{{{{\mathrm{max}}}}}}} - \omega _{{{{{{\mathrm{min}}}}}}}}}$$

(29)

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 tolerance-forgiving 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 r₂ beyond the crest point at R_crest, or even further to deliver a minimum liquid volume U_DF to the outer chamber for ushering CP-DF 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

$$\overline {{{{{\Delta \Omega }}}}^ \ast } = \frac{{({{\Omega }}^ \ast + M \cdot {{{{\Delta \Omega }}}}^ \ast ) - \left( {{{\Omega }} - M \cdot {{{{\Delta \Omega }}}}} \right)}}{{\omega _{{{{{{\mathrm{max}}}}}}} - \omega _{{{{{{\mathrm{min}}}}}}}}} \approx \frac{{2 \cdot M \cdot {{{{\Delta \Omega }}}}}}{{\omega _{{{{{{\mathrm{max}}}}}}} - \omega _{{{{{{\mathrm{min}}}}}}}}}$$

(30)

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 · U₀ with 0 ≤ ζ < 1 of the originally loaded volume U₀ 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 CP-DF valving. In some bioassays, a systematic loss can be factored in by loading more liquid volume U₀ to the inlet. Still, accommodating U_loss (25) tends to increase the footprint of liquid handling structures decisively enters the output volume U₀ − 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

$$\bar U_{{{{{{\mathrm{loss}}}}}}} = \frac{{U_{{{{{{\mathrm{loss}}}}}}}}}{{U_0}}$$

(31)

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 ΔU₀ and ΔU_loss impact the precision of the inlet volume U₀ 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

$$\overline {{{\Delta }}U} _{{{{{{\mathrm{loss}}}}}}} = \frac{{{{\Delta }}U_{{{{{{\mathrm{loss}}}}}}}}}{{U_{{{{{{\mathrm{loss}}}}}}}}}$$

(32)

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 U₀.

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 source¹⁰⁰, 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) liquids¹¹⁵ or centrifugo-pneumatic siphoning⁶⁷. 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 multi-step 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

$$\overline {{{\Delta }}R} = \frac{{{{\Delta }}R_{{\Gamma }}}}{{R_{{{{{{\mathrm{max}}}}}}} - R_{{{{{{\mathrm{min}}}}}}}}}$$

(33)

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”)

$$\bar A = \frac{1}{{R_{{{{{{\mathrm{max}}}}}}} - R_{{{{{{\mathrm{min}}}}}}}}} \cdot \mathop {\int}\limits_{R_{{{{{{\mathrm{min}}}}}}}}^{R_{{{{{{\mathrm{max}}}}}}}} {\frac{{W\left( r \right)}}{{2\pi \cdot r}}} {\mathrm{d}}r$$

(34)

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 CP-DF 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 U₀ 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

$$T_{{{{{\mathrm{Q}}}}}} \approx \frac{{U_0}}{Q} = \frac{{8\pi \eta }}{{A^2}} \cdot \frac{{U_0 \cdot l}}{{\varrho \cdot \bar r{{\Delta }}r \cdot {{\Omega }}^2}}$$

(35)

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 r₀ 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, U₀) and Ω*, and their associated bandwidths ΔΩ and ΔΩ* of rotationally controlled valves through the shape and location R of Γ and U₀, 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 manufacturability⁹². For CP-DF siphon valves (Fig. 9), the far-ranging configurability of the retention rate Ω through the volume of the permanently gas-filled 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.

Table 1 Overview of rotationally controlled valving schemes according to common criteria represented by metrics.

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 U₀. 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 U₀.

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 tolerance-forgiving valve designs displaying predictable functionality along scale-up from prototyping for demonstrating proof-of-concept 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 sample-to-answer automation of multi-step/multi-reagent bioassay panels, the design-for-manufacture (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 Lab-on-a-Chip platforms and applications.

Outlook

Several extensions of the rudimentary digital twin approach are proposed, e.g., inclusion of previously introduced flow control by event-triggering, 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 anti-counterfeit features⁹⁴, 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 larger-scale integrated, fluidically functional designs with predictable reliability may enable foundry models that are commonplace in mature industries such as microelectronics and micro-electro-mechanical 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 multi-step, multi-reagent and multi-analyte bioassay panels.

Follow-up work is already planned on computer-aided, possibly automated optimization of integration density, robustness, and manufacturability. As an open platform concept¹¹⁹, the work is meant to encourage honing of design, modeling, simulation, and experimental verification, for instance, within a blockchain-incentivized participatory research model involving crowdsourcing by means of hackathons, citizen science, and fab/maker labs^{120,121,122,123}. Such community-based organization of research, which are already well-established 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.