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On-board reagent storage and release by solvent-selective, rotationally opened membranes: a digital twin approach

DOI:10.1007/s10404-022-02519-1
Authors:
Jens Ducrée at Dublin City University
Jens Ducrée
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Abstract and Figures

Decentralized bioanalytical testing in resource-poor settings ranks among the most common applications of microfluidic systems. The high operational autonomy in such point-of-care/point-of-use scenarios requires long-term onboard storage of liquid reagents, which also need to be safely contained during transport and handling, and then reliably released just prior to their introduction to an assay protocol. Over the recent decades, centrifugal microfluidic technologies have demonstrated the capability of integrated, automated and parallelized sample preparation and detection of bioanalytical protocols. This paper presents a novel technique for onboard storage of liquid reagents which can be issued by a rotational stimulus of the system-innate spindle motor, while still aligning with the conceptual simplicity of such “Lab-on-a-Disc” (LoaD) systems. In this work, this highly configurable reagent storage technology is captured by a digital twin, which permits complex performance analysis and algorithmic design optimization according to objectives as expressed by target metrics.
Basic storage and release mechanism (linearized, 2-dimensional display of cylindrical coordinates, dimensions not to scale). The two-pronged structure, referred to as Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document}, is divided into upper and lower radial sections of cross sections A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document} and A′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{\prime }$$\end{document}, and a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a$$\end{document} and a′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a^{\prime }$$\end{document}, and heights of H,H′,h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H,H^{\prime } ,h$$\end{document} and h′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{\prime }$$\end{document}, respectively. An isoradial channel at an inner radial position R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R$$\end{document} possesses an axial length L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{L}$$\end{document}, a radial extension H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}$$\end{document} and cross section A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{A}$$\end{document}; a section of radial height h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\fancyscript{h}$$\end{document}, cross section a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\fancyscript{a}$$\end{document} and length is centered at to provide a further degree of freedom. (Left) In a first step at rest (ω=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega =0$$\end{document}), the aqueous reagent and an immiscible ancillary liquid of volumes U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U$$\end{document} and U′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U^{\prime }$$\end{document}, densities ϱ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varrho$$\end{document} and ϱ′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varrho^{\prime }$$\end{document}, and viscosities η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta$$\end{document} and η′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta^{\prime }$$\end{document}, are loaded to the left and right reservoirs with initial meniscus positions r0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{0}$$\end{document} and r0′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{0}^{\prime }$$\end{document} of their liquid distributions Λω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda }\left( \omega \right)$$\end{document} and Λ′ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda^{\prime } \left( \omega \right)$$\end{document}, respectively; their phase interface z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z$$\end{document} is targeted to be located within the center of the -segment of axial length and centered at in the isoradial channel, or at least . These compartments are then isolated at pneumatic pressures pV(ω=0)=pV′(ω=0)=p0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{V}(\omega =0)={p}_{V}^{\prime}(\omega =0)={p}_{0}$$\end{document} (9) from the ambient pressure at p0≈pstd=1013.25hPa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{0}\approx {p}_{\mathrm{std}}=1013.25 \, \mathrm{hPa}$$\end{document}. (Right) Before its (default) on-site usage, both seals are removed from the disc. During rotation at (theoretically any) ω>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega >0$$\end{document}, the initial difference Δpωϱ,ϱ′,U,U′,R,Γ,ω,z=pωϱ,U,R,Γ,ω,z-pωϱ′,U′,R,Γ,ω,z>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {p}_{\omega }\left(\varrho ,{\varrho }^{\prime},U,{U}^{\prime},R,\Gamma ,\omega ,z\right)={p}_{\omega }\left(\varrho ,U,R,\Gamma ,\omega ,z\right)-{p}_{\omega }\left({\varrho }^{\prime},{U}^{\prime},R,\Gamma ,\omega ,z\right)>0$$\end{document} resulting from the pressure heads pω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{\omega }$$\end{document} (4) drives the liquid distributions Λω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda }\left( \omega \right)$$\end{document} and Λ′ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda^{\prime } \left( \omega \right)$$\end{document} towards hydrostatic equilibrium Δpωϱ,ϱ′,U,U′,R,Γ,ω,z=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {p}_{\omega }\left(\varrho ,{\varrho }^{\prime},U,{U}^{\prime},R,\Gamma ,\omega ,z\right)=0$$\end{document} (4), and consequently to new inner meniscus positions r>r0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>{r}_{0}$$\end{document} and r′<r0′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r}^{\prime}<{r}_{0}^{\prime}$$\end{document}. For triggering the reagent release through an outlet, e.g., located in a lower disc layer connected by a vertical via, the liquid–liquid interface at z=z(ϱ,ϱ′,U,U′,R,Γ,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=z(\varrho ,{\varrho }^{\prime},U,{U}^{\prime},R,\Gamma ,\omega )$$\end{document} in hydrostatic equilibrium needs to cover the DF at , which features an axial extension , that was so far protected by the ancillary liquid, i.e., . Note that the geometry Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document} is deliberately composed of cuboid segments to facilitate calculations. The default values for the parameters indicated are compiled in Table 1. Evidently, by suitably adjusting the dissolution characteristics of the DF, also non-aqueous liquids may be stored and released with the same mechanism
Basic storage and release mechanism (linearized, 2-dimensional display of cylindrical coordinates, dimensions not to scale). The two-pronged structure, referred to as Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document}, is divided into upper and lower radial sections of cross sections A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document} and A′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{\prime }$$\end{document}, and a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a$$\end{document} and a′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a^{\prime }$$\end{document}, and heights of H,H′,h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H,H^{\prime } ,h$$\end{document} and h′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{\prime }$$\end{document}, respectively. An isoradial channel at an inner radial position R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R$$\end{document} possesses an axial length L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{L}$$\end{document}, a radial extension H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}$$\end{document} and cross section A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{A}$$\end{document}; a section of radial height h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\fancyscript{h}$$\end{document}, cross section a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\fancyscript{a}$$\end{document} and length is centered at to provide a further degree of freedom. (Left) In a first step at rest (ω=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega =0$$\end{document}), the aqueous reagent and an immiscible ancillary liquid of volumes U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U$$\end{document} and U′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U^{\prime }$$\end{document}, densities ϱ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varrho$$\end{document} and ϱ′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varrho^{\prime }$$\end{document}, and viscosities η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta$$\end{document} and η′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta^{\prime }$$\end{document}, are loaded to the left and right reservoirs with initial meniscus positions r0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{0}$$\end{document} and r0′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{0}^{\prime }$$\end{document} of their liquid distributions Λω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda }\left( \omega \right)$$\end{document} and Λ′ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda^{\prime } \left( \omega \right)$$\end{document}, respectively; their phase interface z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z$$\end{document} is targeted to be located within the center of the -segment of axial length and centered at in the isoradial channel, or at least . These compartments are then isolated at pneumatic pressures pV(ω=0)=pV′(ω=0)=p0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{V}(\omega =0)={p}_{V}^{\prime}(\omega =0)={p}_{0}$$\end{document} (9) from the ambient pressure at p0≈pstd=1013.25hPa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{0}\approx {p}_{\mathrm{std}}=1013.25 \, \mathrm{hPa}$$\end{document}. (Right) Before its (default) on-site usage, both seals are removed from the disc. During rotation at (theoretically any) ω>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega >0$$\end{document}, the initial difference Δpωϱ,ϱ′,U,U′,R,Γ,ω,z=pωϱ,U,R,Γ,ω,z-pωϱ′,U′,R,Γ,ω,z>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {p}_{\omega }\left(\varrho ,{\varrho }^{\prime},U,{U}^{\prime},R,\Gamma ,\omega ,z\right)={p}_{\omega }\left(\varrho ,U,R,\Gamma ,\omega ,z\right)-{p}_{\omega }\left({\varrho }^{\prime},{U}^{\prime},R,\Gamma ,\omega ,z\right)>0$$\end{document} resulting from the pressure heads pω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{\omega }$$\end{document} (4) drives the liquid distributions Λω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda }\left( \omega \right)$$\end{document} and Λ′ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda^{\prime } \left( \omega \right)$$\end{document} towards hydrostatic equilibrium Δpωϱ,ϱ′,U,U′,R,Γ,ω,z=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {p}_{\omega }\left(\varrho ,{\varrho }^{\prime},U,{U}^{\prime},R,\Gamma ,\omega ,z\right)=0$$\end{document} (4), and consequently to new inner meniscus positions r>r0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>{r}_{0}$$\end{document} and r′<r0′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r}^{\prime}<{r}_{0}^{\prime}$$\end{document}. For triggering the reagent release through an outlet, e.g., located in a lower disc layer connected by a vertical via, the liquid–liquid interface at z=z(ϱ,ϱ′,U,U′,R,Γ,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=z(\varrho ,{\varrho }^{\prime},U,{U}^{\prime},R,\Gamma ,\omega )$$\end{document} in hydrostatic equilibrium needs to cover the DF at , which features an axial extension , that was so far protected by the ancillary liquid, i.e., . Note that the geometry Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document} is deliberately composed of cuboid segments to facilitate calculations. The default values for the parameters indicated are compiled in Table 1. Evidently, by suitably adjusting the dissolution characteristics of the DF, also non-aqueous liquids may be stored and released with the same mechanism
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Net pneumatic (counter-)pressure ΔpV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta p}_{V}$$\end{document} (10) as a function of (left) the axial coordinate z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z$$\end{document} with the equilibrium ΔpV=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {p}_{V}=0$$\end{document} at , and (right) scaling the initial volumes V0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{0}$$\end{document} and V0′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{0}^{\prime}$$\end{document} by applying a factor ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi$$\end{document} to their heights while pinning the meniscus position at the end of the -section . Reducing the initial gas volumes V0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{0}$$\end{document} and V0′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{0}^{\prime}$$\end{document} hence stabilizes the actual meniscus position z~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{z }$$\end{document} in the vicinity of during transport. Default values (Table 1) are used in this example
Net pneumatic (counter-)pressure ΔpV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta p}_{V}$$\end{document} (10) as a function of (left) the axial coordinate z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z$$\end{document} with the equilibrium ΔpV=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {p}_{V}=0$$\end{document} at , and (right) scaling the initial volumes V0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{0}$$\end{document} and V0′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{0}^{\prime}$$\end{document} by applying a factor ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi$$\end{document} to their heights while pinning the meniscus position at the end of the -section . Reducing the initial gas volumes V0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{0}$$\end{document} and V0′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{0}^{\prime}$$\end{document} hence stabilizes the actual meniscus position z~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{z }$$\end{document} in the vicinity of during transport. Default values (Table 1) are used in this example
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Resilience quantified by τβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{\beta }$$\end{document} (14) of the liquid distributions Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda$$\end{document} and Λ′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda }^{\prime}$$\end{document} contained in the structure Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document} as a function of (left) the length of the isoradial segmented centered at , and (right) its cross section aq=a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a}_{q}=\fancyscript{a}$$\end{document}. As inferred from its definition (14), the resilience τβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{\beta }$$\end{document} steeply increases towards high flow resistance R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}$$\end{document} (12), i.e., with lq/aq2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${l}_{q}/{\fancyscript{a}}_{q}^{2}$$\end{document}. The gridlines mark the values when inserting default parameters (Table 1)
Resilience quantified by τβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{\beta }$$\end{document} (14) of the liquid distributions Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda$$\end{document} and Λ′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Lambda }^{\prime}$$\end{document} contained in the structure Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document} as a function of (left) the length of the isoradial segmented centered at , and (right) its cross section aq=a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a}_{q}=\fancyscript{a}$$\end{document}. As inferred from its definition (14), the resilience τβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{\beta }$$\end{document} steeply increases towards high flow resistance R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}$$\end{document} (12), i.e., with lq/aq2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${l}_{q}/{\fancyscript{a}}_{q}^{2}$$\end{document}. The gridlines mark the values when inserting default parameters (Table 1)
… 
Algorithmically optimized layouts Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document} obtained with given design metrics for stabilizing the liquid–liquid interface at during transport. (Left) Pneumatic ΔpV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {p}_{V}$$\end{document} (10): Both ports closed, (Center), Resilience τβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{\beta }$$\end{document} (14): Both ports open, and (Right) Combination of stabilization of pneumatics and resilience ΔpV·τβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {p}_{V}\cdot {\tau }_{\beta }$$\end{document} with both ports closed, which finds the right balance between partially contradictory design guidelines
Algorithmically optimized layouts Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma$$\end{document} obtained with given design metrics for stabilizing the liquid–liquid interface at during transport. (Left) Pneumatic ΔpV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {p}_{V}$$\end{document} (10): Both ports closed, (Center), Resilience τβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{\beta }$$\end{document} (14): Both ports open, and (Right) Combination of stabilization of pneumatics and resilience ΔpV·τβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {p}_{V}\cdot {\tau }_{\beta }$$\end{document} with both ports closed, which finds the right balance between partially contradictory design guidelines
… 
Burst frequency Ω/2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega /2\pi$$\end{document} required to establish when both inlet ports remain sealed during rotation when reducing the ancillary volume U′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U^{\prime }$$\end{document} by a factor of χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi$$\end{document}. Towards χ↦1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi \mapsto 1$$\end{document}, Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document} reaches values that are way beyond the capabilities pf typical LoaD instruments, thus effectively preventing release. Yet, spin rates below the practical upper limit ω/2π≈100Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega /2\pi \approx 100\, \mathrm{Hz}$$\end{document} can be achieved below χ≈0.55\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi \approx 0.55$$\end{document}. Default parameters (Table 1) are employed, except that the cross sections A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{A}$$\end{document} and a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\fancyscript{a}$$\end{document} of the isoradial channel have been reduced by a factor of 3 to still assure its complete filling
+2
Burst frequency Ω/2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega /2\pi$$\end{document} required to establish when both inlet ports remain sealed during rotation when reducing the ancillary volume U′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U^{\prime }$$\end{document} by a factor of χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi$$\end{document}. Towards χ↦1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi \mapsto 1$$\end{document}, Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document} reaches values that are way beyond the capabilities pf typical LoaD instruments, thus effectively preventing release. Yet, spin rates below the practical upper limit ω/2π≈100Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega /2\pi \approx 100\, \mathrm{Hz}$$\end{document} can be achieved below χ≈0.55\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi \approx 0.55$$\end{document}. Default parameters (Table 1) are employed, except that the cross sections A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{A}$$\end{document} and a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\fancyscript{a}$$\end{document} of the isoradial channel have been reduced by a factor of 3 to still assure its complete filling
… 
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Microfluidics and Nanofluidics (2022) 26:39
https://doi.org/10.1007/s10404-022-02519-1
RESEARCH PAPER
On‑board reagent storage andrelease bysolvent‑selective,
rotationally opened membranes: adigital twin approach
JensDucrée1
Received: 8 October 2021 / Accepted: 21 December 2021 / Published online: 26 April 2022
© The Author(s) 2022
Abstract
Decentralized bioanalytical testing in resource-poor settings ranks among the most common applications of microfluidic
systems. The high operational autonomy in such point-of-care/point-of-use scenarios requires long-term onboard storage of
liquid reagents, which also need to be safely contained during transport and handling, and then reliably released just prior to
their introduction to an assay protocol. Over the recent decades, centrifugal microfluidic technologies have demonstrated the
capability of integrated, automated and parallelized sample preparation and detection of bioanalytical protocols. This paper
presents a novel technique for onboard storage of liquid reagents which can be issued by a rotational stimulus of the system-
innate spindle motor, while still aligning with the conceptual simplicity of such “Lab-on-a-Disc” (LoaD) systems. In this
work, this highly configurable reagent storage technology is captured by a digital twin, which permits complex performance
analysis and algorithmic design optimization according to objectives as expressed by target metrics.
Keywords Reagent storage· Centrifugal microfluidics· Lab-on-a-Disc· Rotational actuation· Process integration· Digital
twin
1 Introduction
The automation of bioanalytical assay panels has been a para-
mount objective of microfluidic technologies since their debut
in the early 1990s (Manz etal. 1990; Auroux etal. 2002;
Reyes etal. 2002; Whitesides 2006; Janasek etal. 2006). In
the meantime, these Lab-on-a-Chip devices have pervaded
manifold application spaces, primarily in biomedical point-of-
care and global diagnostics, liquid handling automation for the
life sciences, process analytical techniques and cell line devel-
opment for biopharma, as well as monitoring the environment,
infrastructure, industrial processes and agrifood (Gijs etal.
2010; Nge etal. 2013; Liu etal. 2014; Mauk etal. 2017; Yuan
and Oleschuk 2018; Olanrewaju etal. 2018). Compliance with
the relevant workflows, infrastructure, operator skill and com-
petitive cost of ownership/per test are vital for their deploy-
ment in locations outside sophisticated medical infrastructure.
Various miniaturized liquid handling platforms have been
introduced, which may be distinguished by their pumping
scheme; among them are pressure sources, capillary force,
electrokinetics, electrowetting on dielectric, bulk and surface-
acoustic waves. Throughout the last about 30years, centrifu-
gal microfluidic platforms have been at the center of com-
mercial and academic endeavors. Driven by a rugged spindle
motor, these low-complexity “Lab-on-a-Disc” (LoaD) sys-
tems (Schembri etal. 1992, 1995; Abaxis (Piccolo Express)
2021; Andersson etal. 2007; Inganas etal. 2005; Gyros
Protein Technologies 2021; Madou and Kellogg 1998; Shea
2003; Smith etal. 2016a; Kong etal. 2016a; Maguire etal.
2018; Gorkin etal. 2010a; Burger etal. 2016; Aeinehvand
etal. 2018; Sciuto etal. 2020; Ducrée 2021a; Ramachandraiah
etal. 2013; Thompson etal. 2016a; Krauss etal. 2019; Abi-
Samra etal. 2011a; Thompson etal. 2016b; Watts etal. 2007;
Kim etal. 2013; Moschou etal. 2006) excel through their
high-performance centrifugal sample preparation, and their
modular setup featuring an instrument ("player”) spinning a
microfluidic disc carrying the sample and reagents.
Mostly operating in batch-mode, biosamples are pre-
conditioned through a series of centrifugally implemented
Laboratory Unit Operations (LUOs), such as metering (Mark
etal. 2008; Keller etal. 2015), mixing (Grumann etal. 2005;
Ducrée etal. 2006a, b; Burger etal. 2020), incubation, puri-
fication/concentration/extraction (Strohmeier etal. 2015a;
* Jens Ducrée
jens.ducree@dcu.ie
1 School ofPhysical Sciences, Dublin City University,
Glasnevin, Dublin9, Ireland
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Microfluidics and Nanofluidics (2022) 26:39
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39 Page 2 of 17
Brassard etal. 2019), homogenization (Karle etal. 2009;
Kido etal. 2007), particle filtering (Haeberle etal. 2006;
Steigert etal. 2007; Kinahan etal. 2016a; Dimov etal.
2014; Gaughran etal. 2016; Zehnle etal. 2017), and droplet
gene ration (Haeberle etal. 2007; Schuler etal. 2015, 2016),
while transiently held back in each step by a downstream
valve. Note that some assay steps may also be realized by
transferring functionalized magnetic particles between rea-
gent chambers (Czilwik etal. 2015; Grumann etal. 2004).
Akin to the pick-up heads familiar with digital data stor-
age technologies like CD, DVD or Blu-ray, most read-out
schemes for LoaD systems are based on optical detection
(Maguire etal. 2018; Gorkin etal. 2010a; Burger etal. 2016;
Ducrée etal. 2007; Lutz etal. 2011; Tang etal. 2016; Duffy
etal. 1999; Azimi-Boulali etal. 2020; Strohmeier etal. 2015b;
Kong etal. 2016b; Aeinehvand etal. 2017, 2019; Hess etal.
2019; Nguyen etal. 2019; Rombach etal. 2020; Homann etal.
2021; Madadelahi etal. 2020; Miyazaki etal. 2020; Brennan
etal. 2017; Delgado etal. 2016). While the radially directed,
outwards pointing centrifugal field is independent of the outer
contours of the microfluidic chip, a disc shape complies with
the rotational symmetry and supports mechanical balance, lay-
out, and mold flow in common mass manufacturing schemes
such as (compression-)injection molding; yet, deviations from
the common 12-cm diameter and 1.2-mm thick CD format are
quite common, e.g., smaller “mini-discs”, disc segments, tubes
(Kloke etal. 2014; Mark etal. 2009a; Haeberle etal. 2008), or
rectangular, e.g., microscope slides (Morais etal. 2006). Fur-
thermore, the alignment of the inlet ports, outlets and detection
chambers may also be important for seamless interfacing with
standard well-plate formats, liquid handling robotics, and asso-
ciated equipment like readers. For simplicity, we refer to all
these LoaD variants as “discs” in the following.
Liquid volumes concurrently residing on a rotor experience
the same spin rate, and are thus simultaneously driven by the
rotationally induced centrifugal field, which further depends on
their individual radial locations. Hence, and other than for most
conventional Lab-on-a-Chip systems, valving represents a key
ingredient for automating sequential liquid handling protocols
of the LoaD. In principle, the disc could be halted for valve
opening, for instance, by a manual or instrument-based, e.g.,
mechanical, thermal or radiation-based actuator (Mishra etal.
2017; Kinahan etal. 2016b; SpinX Technologies 2021; Abi-
Samra etal. 2011b; Kong etal. 2015; Al-Faqheri etal. 2013;
García-Cordero etal. 2009, 2010; Torres Delgado etal. 2018).
However, it is often preferrable to keep the disc-based liquid
volumes at bay through, at least moderate, centrifugation; such
suppression of flow during spinning involves co-rotating power
sources, e.g., pneumatic pumps (Clime etal. 2015, 2019) or
electro-thermal or radiative units for melting sacrificial barriers
film (Torres Delgado etal. 2018; Kinahan etal. 2018; Delgado
etal. 2018).
However, this work focuses on rotationally controlled val-
ving concepts, which have been chosen by many researchers by
virtue of to their smooth alignment with the low overall com-
plexity of the LoaD platform (Ducrée 2021b). In these passive
valves, the centrifugal pressure head propelling liquid segments
towards the perimeter of the disc is combined with other pres-
sure contributions that are independent of external power. In
their high-pass variants, the centrifugal driving force is, for
instance, opposed by a capillary barrier, while low-pass siphon
valves typically feature hydrophilic coatings in their inbound
sections, or pneumatic effects, so that valving is ushered in by
reducing the spin rate below a critical threshold. In addition,
various centrifugo- or thermo-pneumatic flow control mecha-
nisms (Godino etal. 2013; Schwemmer etal. 2015a, 2015b;
Zhao etal. 2015; Zehnle etal. 2015; Henderson etal. 2021;
Gorkin etal. 2012; Kinahan etal. 2014, 2015; Mishra etal.
2015) have been engineered for creating forward or reverse
pressure differentials.
To provide full-fledged sample-to-answer automation for
conforming with the needs of point-of-care applications, the
disc has to be pre-loaded with liquid or dry reagents; this way
the user only needs to introduce the sample while being relieved
from dealing with the logistics and loading of reagents from
separate stocks. In addition to the capabilities of many tech-
niques conceived for temporarily retaining liquid volumes
while carrying out LUOs along with assay protocols, valves
for longer-term storage also need to impede evaporation, diffu-
sion into the bulk material, or exposure to ambient contaminants
and humidity over shelf lives of many months, sometimes even
years, and possibly even harsh ambient conditions and rough
handling during transport.
In addition to (dry) storage within matrices (Zhang etal.
2016; Hin etal. 2018; Eker etal. 2014; Rombach etal. 2014;
Tijero etal. 2015), liquid buffers and reagents have been
retained by physical barriers (Deng and Jiang 2019), like flex-
ible membranes (Baier etal. 2009; Margell etal. 2008; Czurratis
etal. 2015; Kazemzadeh etal. 2019), blisters/pouches (Smith
etal. 2016b), ampoules (Krauss etal. 2019; Hoffmann etal.
2010), cartridges (Kloke etal. 2014; Li etal. 2020), or wax
plugs (Abi-Samra etal. 2011b; Kong etal. 2015; Al-Faqheri
etal. 2013; Wang etal. 2019); similar to the previously dis-
cussed active valves, these containers may be opened by stimuli
such as mechanical piercing, illumination by a high-power laser,
or by local heating. Further performance criteria of suitable rea-
gent storage valves are their compatibility with manufacture
and assembly, the suppression of possible leaching of chemicals
from solids during extended periods of phase contact. Last but
not least, the fluidic performance needs to be assured in terms
of the reliability of the release mechanism, and the recovery
ratio, accuracy, and precision of the delivered liquid volume.
Dovetailing the centrifugal microfluidics covered in
this work, also directly rotationally induced opening is, at
least theoretically, possible. Yet, for existing mechanisms,
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Microfluidics and Nanofluidics (2022) 26:39
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Page 3 of 17 39
it needs to be considered that, e.g., for limited motor power
and safety, there is an upper limit for practically achiev-
able spin rates, and the reservoir might have to be placed
centrally, which would severely restrict obtainable pressure
heads to fractions of common atmospheric pressures. Thus,
the valve controlling the reagent reservoir has to yield at
rather small pressure heads. Reliability is additionally
impaired as the mechanical strength of a required desig-
nated mechanical weak or yield point is hard to define, thus
smearing out the associated spin frequency threshold for
triggering liquid release. Critical operational robustness
hence demands setting a high rotational speed for opening,
which tends to counteract the options for fluidic multiplex-
ing of concurrently loaded liquids (Ducrée 2021a, c).
This paper focuses on a novel type of rotationally actu-
ated valve; during storage of an aqueous reagent, a (water)
dissolvable film (DF) presents a diffusion barrier which
is initially protected by an oleophilic, ancillary liquid
having a certain, specific density (Gaughran etal. 2016;
Mishra etal. 2015, 2017, 2020; Ducrée 2021d; Lu etal.
2020). Upon spinning, a centrifugo-hydrostatic equilib-
rium is reached, in which the interface between the two
immiscible liquids contacts and thus dissolves the DF. In
an idealized model, the reagent will be issued at any finite
spin rate situated within its practical limits imposed by
motor power, operational robustness and safety.
In addition to the general prerequisites for on-board
storage of liquid (and potentially also for protecting dry
/ lyophilized) reagents, such valves need to meet further
specifications. During logistics and manual handling,
acceleration due to shaking and terrestrial gravity act
on the liquid volumes, while the meniscus between the
immiscible fluids needs to stay near its default rest posi-
tion to prevent premature opening. Moreover, manufactur-
ing and dispensing precision, evaporation rates, and struc-
tural fidelity affect the release mechanism, and potentially
enclosed or emerging gas bubbles need to be tolerated.
This work first elaborates the operational principles
for rotationally controlled on-board storage and release
valving enabled by an immiscible ancillary liquid. In the
initial section, strategic features of the basic layout are
introduced, and first motivated in a mostly qualitative
manner. Next, design objectives are quantified in terms of
key performance indicators. The individual or collective
optimization of these “KPIs” is implemented based on a
“digital twin” (Digital Twin 2021; Marr 2017; Grieves
etal. 2017), which constitutes a virtual representation of
a real-world physical system derived on the underlying
functional model. The digital twin method allows virtual-
izing the optimization of the novel reagent storage tech-
nology, thus substantially reducing cost and time scales
for product development.
2 Working principle
Figure1 illustrates the fundamental mechanism underpinning
storage and release of liquid reagents described in this work. In
the portrayed, exemplary fluidic structure, which is referred in
the following to as
Γ
, two reservoirs of upper and lower cross
sections
A
and
a
, and heights
H
and
h
on the left, and
A′
and
a′
, and
H′
and
on the right, hold the aqueous reagent and the
immiscible ancillary liquid of densities
𝜚
and
𝜚′
, respectively.
These containers are interconnected by an isoradial channel
placed at an (inner) radial position
R
of axial length
L
, radial
height
H
, and of cross section
A
, which accommodates a
water-dissolvable film (DF) of axial extension
𝛿Z
at the (mean)
position on the
z
-axis.
For the duration of storage and transport (
𝜔=0
), the phase
interface between the immiscible liquids must reside within a
(coaxial) segment of a cross section
𝒶
of axial length , at
a target position , or at least . For
activation and release in hydrostatic equilibrium at an angular
spin rate
𝜔=2𝜋⋅𝜈>0
, the meniscus needs to move beyond
the location of the DF at .
The conservation of liquid volumes is generally expressed
as
for general geometries
Γ
by the integral over the function
A(𝓇)
representing the dependence of the (compartmental-
ized) cross section
A
on the radial coordinate
𝓇
between the
inner and outer radial confinements of the liquid distribu-
tions
Λ
or
Λ�
; this liquid volume
U
(1) spans between its
inner and other radial confinements
̌r
and
̂r
, respectively.
To simplify the math without compromising the outcomes
of this work, we consider a structure
Γ
composed of cuboidal
segments in Fig.1, with
̌r=r0
or
r
(and
r′
0
and
r′
for the ancil-
lary liquid), and
̂r=R+𝒽
(for both arms). Instead of having
to (numerically) solve the possibly complex integral
U(̌r,̂r)
in (1), the liquid volumes can then be expressed (assuming
R−h−H≤r≤R−h
) by the algebraic formulas
for the aqueous reagent and (assuming
R−h�−H�≤r�≤R−h�
)
for the ancillary liquid. Equations(2) and (3) link the loaded
liquid volumes
U
and
U′
to the radial position
R
of
Γ
, their
structural parameters
A,A′
,a,a′
,h,h′
,
and
L
as
represented by
Γ
, their liquid levels
r
and
r′
, and their phase
interface at
0<z<L
; the liquid levels at rest can therefore
be expressed as
r=r(U,R,Γ)
in (2) and
r�(
U
�
,R,Γ
)
in (3).
(1)
U
(̌r,̂r)=
∫
̂r
̌r
A(𝓇)d
𝓇
(2)
U=A⋅(R−h−r)+a⋅h+A⋅z
⇒
r=r(U,R,Γ,z)
(3)
U�
=A
�
⋅
(
R−h
�
−r
�)
+a
�
⋅h
�
+A⋅(L−z)⇒r
�(
U
�
,R,Γ,z
)
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Microfluidics and Nanofluidics (2022) 26:39
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39 Page 4 of 17
2.1 Loading, storage andtransport
Following a suitable, well-reproducible, possibly closed-
loop controlled experimental loading procedure at
𝜔=0
,
the two liquids are introduced at ambient pressure
p0
(typi-
cally
p0≈pstd
with the standard atmospheric pressure
pstd =1013.25 hPa
) to place the phase interface at
z
within
the center of the isoradial segment at . For a starting posi-
tion of the initial interface after properly loading
U
and
U′
at
𝜔=0
, the conservation of mass (1) trivially yields
the initial filling levels
r0=r0(U,Γ,Z
)
and
r�
0
=r
�
0(
U
�
,Γ,Z
)
.
To suppress evaporation and contamination during subse-
quent storage, transport and handling, each reservoir is then
isolated from ambient by a membrane exhibiting good bar-
rier properties.
2.2 Liquid release
In response to spinning at a finite spin speed
𝜔>0
, the cen-
trifugal pressure heads
are induced to move and reshape the liquid distributions
Λ
and
Λ�
, which are then confined by their (inner) menisci
at the radially inner
r,r′
and common outer positions
R
, respectively. (Note that
H∕R≪1
and
𝒽∕R≪1
are
assumed throughout.) The products
rΔr
and
r�Δr�
in (4)
are composed of the mean radial positions
r=0.5 ⋅(R+r)
and
r�
=0.5 ⋅
(
R+r
�)
, and the liquid level differences
Δr=R−r
and
Δr�=R−r�
.
(4)
p𝜔
=𝜚⋅rΔr⋅𝜔
2
and p
�
𝜔
=𝜚
�
⋅r
�
Δr
�
⋅𝜔
2
Fig. 1 Basic storage and release mechanism (linearized, 2-dimen-
sional display of cylindrical coordinates, dimensions not to scale).
The two-pronged structure, referred to as
Γ
, is divided into upper
and lower radial sections of cross sections
A
and
A′
, and
a
and
a′
,
and heights of
H
,
H′
,
h
and
h′
, respectively. An isoradial channel at
an inner radial position
R
possesses an axial length
L
, a radial exten-
sion
H
and cross section
A
; a section of radial height
𝒽
, cross sec-
tion
𝒶
and length is centered at to provide a further
degree of freedom. (Left) In a first step at rest (
𝜔=0
), the aque-
ous reagent and an immiscible ancillary liquid of volumes
U
and
U′
, densities
𝜚
and
𝜚′
, and viscosities
𝜂
and
𝜂′
, are loaded to the left
and right reservoirs with initial meniscus positions
r0
and
r′
0
of their
liquid distributions
Λ(𝜔)
and
Λ�(𝜔)
, respectively; their phase inter-
face
z
is targeted to be located within the center of the -segment
of axial length and centered at in the isoradial channel,
or at least . These compartments are then iso-
lated at pneumatic pressures
pV
(𝜔=0)=p
�
V
(𝜔=0)=p
0
(9) from
the ambient pressure at
p0≈pstd =1013.25 hPa
. (Right) Before
its (default) on-site usage, both seals are removed from the disc.
During rotation at (theoretically any)
𝜔>0
, the initial difference
Δ
p
𝜔(
𝜚,𝜚
�
,U,U
�
,R,Γ,𝜔,z
)
=p
𝜔
(𝜚,U,R,Γ,𝜔,z)−p
𝜔(
𝜚
�
,U
�
,R,Γ,𝜔,z
)
>
0
resulting from the pressure heads
p𝜔
(4) drives the liquid dis-
tributions
Λ(𝜔)
and
Λ�(𝜔)
towards hydrostatic equilibrium
Δ
p
𝜔(
𝜚,𝜚
�
,U,U
�
,R,Γ,𝜔,z
)
=
0
(4), and consequently to new
inner meniscus positions
r>r0
and
r′
<r
′
0
. For triggering the
reagent release through an outlet, e.g., located in a lower disc
layer connected by a vertical via, the liquid–liquid interface at
z=z(𝜚
,
𝜚�
,
U
,
U�
,
R
,
Γ
,
𝜔)
in hydrostatic equilibrium needs to cover
the DF at , which features an axial extension , that was so
far protected by the ancillary liquid, i.e., . Note that the geome-
try
Γ
is deliberately composed of cuboid segments to facilitate calcu-
lations. The default values for the parameters indicated are compiled
in Table1. Evidently, by suitably adjusting the dissolution character-
istics of the DF, also non-aqueous liquids may be stored and released
with the same mechanism
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Microfluidics and Nanofluidics (2022) 26:39
1 3
Page 5 of 17 39
In default actuation mode, the pneumatic seals are
removed from the disc prior to launching the centrifugal
assay protocol
𝜔(t)
. While rotating at sufficiently high
𝜔
, so
that
p𝜔∝𝜔2
(4) overcomes unfavorable stiction and capil-
lary effects occurring in real-world systems, a hydrostatic
equilibrium
establishes which can be rewritten
and therefore only relates the two liquid levels
r
=r
(
𝜚,𝜚
�
,U,U
�
,R,Γ
)
and
r�
=r
�(
𝜚,𝜚
�
,U,U
�
,R,Γ
)
to the
radial position
R
of
Γ
, the loaded liquid volumes
U
and
U′
,
and their densities
𝜚
and
𝜚′
, but not to
𝜔
. The new, centrifu-
gally stabilized position of the phase interface is defined by
Δp𝜔(𝜚
,
𝜚�
,
U
,
U�
,
R
,
Γ
,
z)=0
(5), and hence directly obtained
from
r
and
Γ
, i.e.,
z=z(𝜚,𝜚�
,U,U�
,R,Γ)
.
Consequently, the centrifugally triggered release through
the DF at comes down to , which predicates on
Δp𝜔(z)>0
(5) during the transition of the meniscus at
z
all along the way from the position during storage at to
the DF at . This condition implies that a minimum
reagent volume
(5)
Δ
p
𝜔
=p
𝜔
−p
�
𝜔
(6)
𝜚
⋅(R+r)⋅(R−r)=𝜚
�
⋅
(
R+r
�)
⋅
(
R−r
�)
depends on the partial derivatives
𝜕z∕𝜕𝛾k
with
𝛾k∈{𝜚
,
𝜚�
,
U
,
U�
,
R
,
Γ}
, and the standard deviations
Δ𝛾k∈ {Δ𝜚,Δ𝜚�
,ΔU,ΔU�
,ΔR,ΔΓ}
, evaluated at the critical
positions and , respectively. Note that strictly speaking,
Eq.(8) only holds for small deviations
{Δ𝛾k
}. Alter natively,
as used for robustness analysis further below, Monte-Carlo
methods (see Fig.7 below) can be employed to compute
Δz
at the two target positions and .
According to this functional model, operational robust-
ness during storage, transport and rotation caused by sta-
tistical variations
{Δ𝛾k}
is tightly linked to squeezing the
interval of the actual interface positions
z±0.5 ⋅M⋅Δz
(8) within the narrow segment after loading at rest, i.e.,
, and for reliable release at
𝜔>0
, . The factor
M
quan-
tifies the targeted degree of operational robustness, with
68%, 95%, 99.7%, 99.99%,…
for
M∈{1,2,3,4,…}.
3 Design characterization andoptimization
The digital twin (Ducrée 2021b; Digital Twin 2021; Marr
2017) developed here allows to configure the free experi-
mental parameters
{𝛾k}
for achieving key performance goals,
while staying commensurate with design-for-manufacture and
scale-up of fabrication (Ducrée 2019). Such design optimiza-
tion is facilitated by the multi-segmented structure
Γ
(Fig.1)
featuring cross sections
A,a,A′
,a′
,A
and
𝒶
with respect to
the axial direction (and their respective axial heights / lengths
and
L
). The core motivation of this draft lay-
out is now briefly outlined on a qualitative, heuristic manner;
note that due to the huge variety of possible application cases
in the multi-dimensional parameter space
{𝜚,𝜚�
,U,U�
,R,Γ}
,
only computational optimization towards well-defined target
metrics is likely to lead to good-quality results.
The liquid volumes
U
and
U′
are chosen to settle the
menisci
r
and
r′
in the inner (wider) region of their reservoirs
with cross sections
A
and
A′
, so that effects of volume devia-
tions, whether related to systematic loss by evaporation or
dispenser precision
ΔU
and
ΔU�
, on the initial liquid levels
r0
, and
r′
0
, and thus the centrifugal equilibrium (5) determin-
ing
z
at
𝜔>0
, are largely mitigated. For given volumes
U
and
U′
, the lower radial segments of the reservoirs display
smaller cross sections
a
and
a′
; for instance,
a<A
ampli-
fies
Δr=r�−r
, and thus the net pressure
p𝜔∝Δr
(4) for
pumping the minimum volume fraction
UΔ
(7) of
U
to reach
, as required for prompting disintegration of the DF.
(7)
(for ) needs to be displaced for valve
opening, and the shift of the left liquid level from
r0
to
r>r0
,
while the meniscus of the ancillary liquid moves radially
inbound from
r′
0
to
r′
<
r′
0
. Given that all meniscus positions
r
0,r,r
′
0
and
r′
remain in their respective inner compartments
of cross sections
A
and
A′
during this reconfiguration of
the liquid segments towards centrifugo-hydrostatic equilib-
rium from to , we obtain and
.
2.3 Reliability
In practical applications, tolerances, mainly in the geometri-
cal dimensions, as quantified by the standard deviations here
collectively referred to as
ΔΓ
, and in the liquid volumes
ΔU
and
ΔU�
after pipetting, impact the spread
Δz
of the interface
z
from their target positions and at
𝜔=0
and
𝜔>0
,
respectively. Using (5), (2) and (3) for the meniscus position
z=z(𝜚
,
𝜚�
,
U
,
U�
,
R
,
Γ)
, its standard deviation
(8)
Δ
z

𝜕z
𝜕𝛾k
,
Δ𝛾k
=
k
𝜕z(𝜚,𝜚�,U,U�,R,Γ)
𝜕𝛾k
⋅Δ𝛾k
2
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Microfluidics and Nanofluidics (2022) 26:39
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39 Page 6 of 17
The additional segment centered at in the isoradial
channel featuring a cross section
𝒶
over an axial exten-
sion has been introduced for supporting the definition
of the liquid–liquid interface, and to suppress the shift of
the meniscus by a high flow resistance scaling with
during storage and transport. Evidently, it is critical that the
actual position of the meniscus at
̃z
, when factoring in exper-
imental tolerances in
{𝜚,𝜚�
,U,U�
,R,Γ,}
, remains between
the edges, i.e., (while avoiding enclo-
sure of gas between the liquid phases). A large extension
is desirable for improving the tolerance to discrepancies in
the loading procedure of the two liquids.
3.1 Loading
The reservoirs are filled at the factory (while
𝜔=0
) with
the two immiscible liquids, in the main exemplary case
considered here with water and FC-72 (3M™ Fluorinert™
Electronic Liquid FC-72), of target volumes
U
and
U′
and
densities
𝜚
and
𝜚′
through their designated inlet ports. The
loading procedure should be highly reproducible, and ideally
be monitored until the actual meniscus location
̃z
matches
, or at least settles sufficiently central in their designated
channel section around , i.e., . For the
low contact angles that are often observed between an oil-
based ancillary liquid and the polymer surface, it is advis-
able to first load the aqueous phase or to coat the wall with
a suitable agent.
3.2 Transport
When taken out of its storage, a LoaD may experience vari-
ous accelerations
𝛽
, e.g., repetitively during manual han-
dling, walking or in a moving vehicle, or punctually and
more forcefully when its full weight hits solid ground, e.g.,
after falling from a height. The directions of the resulting,
usually unintended forces tend to be oriented randomly but
may possess components parallel to the designated radial
axis during spinning. The unknown number, magnitude,
orientation and duration of such arbitrary inertial effects
make it impossible to exactly quantify their impact on the
deviation of the actual meniscus position
̃z
from its target
value. As successive accelerations might point in opposite
directions, and thus somewhat neutralize their effect on
the meniscus position, it is mostly likely that a single hard
impact aligned in
𝓇
-direction will compromise the liquid dis-
tributions
Λ
and
Λ�
. We consider here two main mechanisms
for pinning the phase interface during transport.
3.2.1 Pneumatic stabilization
Air-tight membranes seal the inlet ports of the reservoirs
after loading the reagent and ancillary liquid to prevent
evaporation. In addition, these fluid barriers also pneumati-
cally stabilize
Λ
and
Λ�
by virtue of Boyle’s law
stating that a change of an originally confined gas volume
V0
at
p0
to
V
alters the pressure to
pV≠p0
(9). So, for instance,
according to the conservation of liquid volume
dU∕dt =0
(1), transport-related disruption may induce a shift of the
liquid level
r
on the (left) aqueous side towards the center
of rotation, which ensues a peripheral displacement of the
liquid level
r′
of the ancillary fluid (right) towards
R
(Fig.1).
The resultant changes in the gas volumes
V
and
V′
lead
to an increase in
pV
and a reduction of p
′
V
, thus seeking to
restore . The driving pressure (difference)
should consequently be maximized through adjusting the
z
-dependent, displaced liquid volume
UΔ(z)
in (7), and the
initial gas volumes
V0
and
V′
0
underneath the seal for steady-
ing the interface position near during transport. Fig-
ure2(left) illustrates the dependency of the effective counter
pressure
ΔpV
(10) in response to (left) a shift
z
from the
default Figure2(right) reveals that the restor-
ing pressure
ΔpV<0
(10), evaluated at the downstream
boundary of the -segment , approaches
0 towards scaling the initially enclosed gas volumes
V0
and
V′
0
by the (same) factor
𝜉
.
3.2.2 Geometrical factors
It would be disastrous from an operational point of view if
the full volume in (7) was displaced so that the aque-
ous reagent already reaches to open the DF during
storage, transport and handling. This fatal event would hap-
pen in case the liquid distributions
Λ
and
Λ�
experienced an
acceleration
𝛽
with a strong component parallel to the radial
𝓇
-direction of the centrifugal field (Fig.1) for a sufficiently
long duty cycle
𝜏
. Note that even though not discussed here
for the sake of clarity, the interface may also leave the des-
ignated isoradial region, i.e., , under the
impact of
𝛽
to disrupt the phase interface.
In response to an acceleration
𝛽
with a major component
parallel to the radial
𝓇
-axis, the liquid distributions
Λ
and
Λ�
seek hydrostatic equilibrium (5). The resulting flow is driven
by the pressure differential
(9)
p
V=p0⋅
V
0
V
(10)
Δ
pV(z)=p�
V(z)−pV(z)=p0⋅
[
V
�
0
V�(z)−V0
V(z)
]
=p0⋅
[
V�
0
V�
0
−U
Δ
(z)−V0
V
0
+U
Δ
(z)
]
(11)
p𝛽
≈𝛽⋅
[
𝜚⋅Δr−𝜚
�
⋅Δr
�]
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Microfluidics and Nanofluidics (2022) 26:39
1 3
Page 7 of 17 39
(initially) applying along the axial
z
-axis towards , and
throttled by the aggregate hydrodynamic resistance
of the structural segments indexed by
q
, possessing a cross
section
Aq
, and filled over an axial length
lq
with the reagent
and ancillary liquid with viscosities
𝜂
or
𝜂′
, respectively.
The numerical coefficients
cq
amount to
8𝜋
for round cross
sections. This law of Hagen-Poiseuille delivers a volume
flow rate
which is governed by
p𝛽
(11) and {
Rq}
(12). As the outlet
opens for
̇
UV
⋅
𝜏≥UΔ(z)
in (7), the duty cycle
𝜏=UΔ∕̇
UV
ought to be maximized to best suppress operationally
disastrous premature release of reagent during transport.
However, since
𝛽
is unknown, we define a resilience (evalu-
ated for )
expressed in units of
s3
⋅m−1
, to be maximized by adjusting
to the structure
Γ
(Fig.1) for mitigating unfavorable effects
owing to transport conditions.
This design goal of maximizing
𝜏𝛽
(14) translates into
enlarging the dead volumes (7) of the isoradial chan-
nel extending between on the left, and on the right,
which turns out to be mostly determined via the part having
the (larger) cross section
A
. Furthermore, the flow rate
̇
UV
(13) should stay small; this means that the (initial) liquid
level difference
r�−r
should be minimized, e.g., by
(12)
R
=
∑
q
Rq=
∑
q
cq⋅
𝜂
q
⋅l
q
A2
q
(13)
̇
U
V=dU
dt =p𝛽
R≈𝛽⋅
𝜚⋅Δr−𝜚
�
⋅Δr
�
q
cq⋅𝜂q⋅lq∕A2
q
adjusting the cross sections or the reservoirs
A,a,A′
and
a′
,
while the dominant flow resistance
R
(12), as imposed by
the isoradial segment of length and cross section
Aq=𝒶
, and scaling with the geometrical ratio ,
should be large. As the properties of the reagent are usually
prescribed by the assay, an ancillary liquid possessing a high
viscosity
𝜂′
would also be beneficial to increase
𝜏𝛽
(14). Fig-
ure3 shows the dependency of the resilience
𝜏𝛽
(14) on (left)
the length , and (right) the cross section
𝒶
of the -seg-
ment, which accounts for the biggest impact on the flow
resistance
R
(12) through .
3.2.3 Design optimized fortransport
By maximizing the quantities
ΔpV
(10) and
𝜏𝛽
(14) within
the practical ranges of their input parameters
{𝛾k}
and their
tolerances {
Δ𝛾k
}, the parametrized structure
Γ
can be algo-
rithmically optimized to accomplish these design goals.
Figure4 shows (left) the layouts for the highest restoring
pressure
ΔpV
(10), (center) the resilience
𝜏𝛽
(14), and (right)
their product
ΔpV⋅𝜏𝛽
.
While the reasoning behind these designs is rather com-
plicated, we point out some key characteristics of the opti-
mization in Fig.4. The left, pneumatically stabilized design
shows a minimized gas volume above the reagent to raise the
pV
(9) at the initial position (Fig.1), thus preventing
advancement of the phase interface during storage towards
the DF. To still enable rotational actuation within the allowed
frequency envelope, the gas volume above the ancillary liquid
is kept noticeably larger. The resilience
𝜏𝛽
(14) in
the central layout displaying open reservoirs is improved with
Fig. 2 Net pneumatic (counter-)pressure
ΔpV
(10) as a function of
(left) the axial coordinate
z
with the equilibrium
ΔpV=0
at ,
and (right) scaling the initial volumes
V0
and
V′
0
by applying a factor
𝜉
to their heights while pinning the meniscus position at the end of
the -section . Reducing the initial gas volumes
V0
and
V′
0
hence stabilizes the actual meniscus position
̃z
in the vicinity
of during transport. Default values (Table1) are used in this exam-
ple
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Microfluidics and Nanofluidics (2022) 26:39
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39 Page 8 of 17
the high flow resistance
R
(12), as imposed by the narrow
section in the horizontal channel. The compromise design on
the right-hand side of Fig.4 features a tiny outer part of the
radial channel on the reagent side to maintain a large
Δr
, and
thus
Δp𝜔
(4), so the counter pressure required for opening the
DF at (Fig.1) can still be provided.
4 Actuation
4.1 Pneumatic modes
In the case of the default scenario portrayed in Fig.1, the
inlets of the reservoirs for the aqueous reagent and the ancil-
lary liquid are vented during rotation
(𝜔>0
), which can
be interpreted as
V0
↦
∞
and
V�
0
↦
∞
in (10), and thus
ΔpV
↦
0
(10). So, as long as the centrifugally induced
pressure head remains positive, i.e.,
Δp𝜔>0
(5) for ,
the meniscus will reach , and thus wet and open the
DF at the outlet. However, for centrifugo-pneumatic actua-
tion, a critical spin rate
Fig. 3 Resilience quantified by
𝜏𝛽
(14) of the liquid distributions
Λ
and
Λ�
contained in the structure
Γ
as a function of (left) the length
of the isoradial segmented centered at , and (right) its cross
section
aq
=𝒶
. As inferred from its definition (14), the resilience
𝜏𝛽
steeply increases towards high flow resistance
R
(12), i.e., with
lq
∕𝒶
2
q
. The gridlines mark the values when inserting default param-
eters (Table1)
Fig. 4 Algorithmically optimized layouts
Γ
obtained with given
design metrics for stabilizing the liquid–liquid interface at dur-
ing transport. (Left) Pneumatic
ΔpV
(10): Both ports closed, (Center),
Resilience
𝜏𝛽
(14): Both ports open, and (Right) Combination of
stabilization of pneumatics and resilience
ΔpV⋅𝜏𝛽
with both ports
closed, which finds the right balance between partially contradictory
design guidelines
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Microfluidics and Nanofluidics (2022) 26:39
1 3
Page 9 of 17 39
first needs to be surpassed, i.e.,
𝜔>Ω
, before trigging rea-
gent release. Here, the ambient pressure
p0
applies to open,
and
pV
and
p′
V
(9) to sealed ports. Such a finite
Ω>0
(15)
proves especially advantageous for multiplexed flow control
(Ducrée 2021b, c).
As the default experimental parameters
𝜚
,
𝜚′
,
U
,
U′
,
R
and
Γ
(Table1) are geared to result in
Δp𝜔=0
(5) at ,
and thus
Ω↦∞
, centrifugo-pneumatic actuation requires
adjusting a subset of these variables. In general, the digi-
tal twin allows to calculate the changes required within
the rather intricate correlation between the parameters
𝜚
,
𝜚�
,
U
,
U�
,
R
,
Γ
and
Ω
in order realize certain design targets,
e.g., a release threshold
Ω
(15) that is commensurate with
a rotationally automated, multiplexed assay protocol. The
example portrayed in Fig.5 shows the scaling of the critical
spin rate
Ω∕2𝜋
(15) with the ancillary volume
𝜒⋅
U′
. Spindle
speeds
Ω∕2𝜋
(15) within the experimentally feasible range
smaller than
100 Hz
only emerge below
𝜒
≈
0.55
. It follows
that, to make sure that the isoradial channel is always filled
with the ancillary liquid, the cross sections
A
and
𝒶
are both
reduced by a factor of 3 with respect to the values in Table1.
Note that Eq.(10) also discloses
ΔpV∝p0
. This scaling
with the atmospheric pressure
p0
only affects the overall
magnitude of
ΔpV
, but not the direction and ratio of the
pneumatic pressures. We therefore refer to previous work on
centrifugo-pneumatic valving where the impact and possible
compensation of variation in the atmospheric pressure by
weather, and particularly local altitude, have been examined
in more detail (Ducrée 2021a, b, c, d).
(15)
Ω=√
p�
V−pV
𝜚r
Δ
r
−
𝜚�r
�
Δ
r�
The critical spin rate
Ω
(15), furthermore, deter-
mines the maximum (density) of the centrifugal field
f
𝜔
=𝜚
part
⋅R
LUO
⋅Ω
2
that can be sustained by the reagent
valve, e.g., while an LUO, for instance, plasma extraction, is
simultaneously processed to separate blood cells of (relative)
density
𝜚part
at
𝜔<Ω±M⋅ΔΩ
.
4.2 Systematic volume losses
Due to evaporation or absorption at rates
̇
U
and
̇
U′
, liquid vol-
umes
U
and
U′
may appreciably decline during storage over
time periods
T
, typically lasting months up to a few years, by
𝛿U=̇
U⋅T
and
𝛿U�=̇
U�⋅T
. Even for open inlet ports dur-
ing rotation, the reduced volumes
U−𝛿U
and
U�−𝛿U�
may
lead to
(5), and thus cause valve malfunction by failure to dissolve
the DF at . (In addition, these volume losses
𝛿U
and
𝛿U′
may also affect the outcome of quantitative bioassays.)
The main influence of the lost liquid volumes
𝛿U
and
𝛿U′
on the density-weighted radial products
rΔr
and
r
�Δr�
in
the centrifugal equilibrium (6) manifests through
Δr=R−r
and
Δr�=R−r�
via
r(U
,
𝛿U
,
A)=(U−̇
U
⋅
T)∕A
and
r
�
(
U�
,𝛿U�
,A�
)
=(U�−
̇
U�⋅T)∕A
�
. Enlarged cross sections
A
and
A′
thus reduce the effect of evaporation or other losses
𝛿U
and
𝛿U′
of the reagent and ancillary phases at
̇
U
and
̇
U′
.
Figure6 shows that the prerequisite
Δp𝜔>0
for is
assured for nearly 2.5years at exemplary annual evaporation
losses of 5% and 1% for the reagent and the ancillary liquid,
respectively. This condition of a positive centrifugal pres-
sure differential
Δp𝜔
(10) also entails that sufficient liquid
volumes
U
and
U′
remain available along all locations of the
interface , to eventually displace
UΔ
(7) from
Fig. 5 Burst frequency
Ω∕2𝜋
required to establish when both
inlet ports remain sealed during rotation when reducing the ancil-
lary volume
U′
by a factor of
𝜒
. Towards
𝜒
↦
1
,
Ω
reaches values
that are way beyond the capabilities pf typical LoaD instruments,
thus effectively preventing release. Yet, spin rates below the practical
upper limit
𝜔∕2𝜋≈100 Hz
can be achieved below
𝜒≈0.55
. Default
parameters (Table1) are employed, except that the cross sections
A
and
𝒶
of the isoradial channel have been reduced by a factor of 3 to
still assure its complete filling
Fig. 6 Systematic reductions
𝛿U=̇
U⋅T
and
𝛿U
�
=̇
U
�
⋅T
of the
originally loaded liquid volumes
U
and
U′
at rates
̇
U
and
̇
U′
, thus
changing the centrifugal pressure balance p
𝜔∕
𝜔
2
(6) over a time
period
T
. In this example, annual loss rates of
̇
U
and
̇
U′
of 5% and
1% are assumed, and
̇
U>̇
U′
is compensated by loading 10% more
reagent volume
U
. The curve uncovers that the opening condition
Δ
p
𝜔∕
𝜔
2
>
0
at is assured beyond a typical minimum storage
period of 24months
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Microfluidics and Nanofluidics (2022) 26:39
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39 Page 10 of 17
the reagent side through the isoradial segment towards the
ancillary reservoir.
If evaporation through the DF membrane turns out to
be unacceptably high, its formulation ought to be altered,
a coating applied, or its thickness enlarged. Note also that
leaks in the assembly or permeation through the bulk mate-
rial should be investigated. Moreover, even if the reagent
valve still properly opens, the forwarded liquid volume
might still be insufficient to actuate subsequent valves, e.g.,
by failing to generate enough centrifugal pressure head
p𝜔
(4) for overcoming a critical frequency threshold, such as
Ω
(15).
4.3 Statistical tolerances
Consistent reagent release hinges upon
in hydrostatic equilibrium (5) at
𝜔>0.
The standard devia-
tion
Δz
(8) is affected by the unavoidable spreads {
Δ𝛾k
} of
the experimental input parameters {
𝛾k
}, mainly within the
geometrical dimensions
ΔΓ
and
ΔU
delineating the structure
Γ
, and the loaded liquid volumes
U
and
U′
, respectively.
The histogram in Fig.7 displays the distribution of
̃z
with
a mean
z=19.97 mm
close to and a standard
deviation
Δz=4.02 mm
(8) as acquired from a Monte-Carlo
simulation with 1000 runs using the default values and real-
istic tolerances
Δd
and
Δw
in vertical and lateral machining,
and pipetting the liquid volumes
ΔU
and
ΔU�
, as listed in
Table1. This digital-twin based Monte-Carlo method may
be regarded as a virtualized manufacture and testing, which
is very useful in view of the common paucity of physical
devices throughout prototyping.
4.4 Gas enclosure
During priming or storage, bubbles may emerge from or
between the two liquids, e.g., driven by the vapor pressure
of the ancillary phase. For developing a semi-quantitative
understanding of their influence on valving, we consider the
case of a gas volume
Vg,0
entrapped, after loading, at ambient
pressure
p0
in the center of the isoradial -segment. Upon
reaching hydrostatic equilibrium during spinning at
𝜔>0
,
its original volume
Vg,0
is compressed to
while now residing near . To still open the DF in pres-
ence of the entrapped gas, the liquid volumes
U
and
U′
need
to be sized so that the meniscus at
z
was to shift by a further
(16)
V
g(𝜔)=Vg,0 ⋅
p
0
p
0
+p
𝜔
=Vg,0 ⋅
p
0
p
0
+𝜚⋅rΔr⋅𝜔
2
0.5
⋅V
g(
𝜔
)∕A
(16) in the isoradial
z
-axis, compared to the
absence of the bubble. (Alternatively, the position of the
DF at can be appropriately adjusted to provide bubble
tolerance.)
Assuming symmetrical displacement into each lateral res-
ervoir (which is, strictly speaking, only the case for
𝜚=𝜚�
),
the liquid levels
r
and
r′
in hydrostatic equilibrium (5) are
lifted by about
0.5 ⋅Vg(𝜔)∕A
and
0.5
⋅
Vg(𝜔)∕A�
towards
the center of rotation, respectively. The impact of such an
entrapped gas bubble is assessed here in a back-of-the-enve-
lope calculation; for this, we assume
V0=1μl
,
p0=pstd
,
𝜈=𝜔∕2𝜋=25 Hz
and default values for the other param-
eters (Table1), to arrive at
V≈0.9Vg,0
and
𝛿r≈𝛿r�≈1mm
.
5 Renements
To illustrate the very concept and potential of a digital twin
for optimizing the long-term storage and release mecha-
nism towards strategic design goals, we introduced a basic
structure
Γ
(Fig.1); this way, engineering objectives could
be quantified and expressed by algebraic formulas without
obfuscating the underlying mechanisms; moreover, these
equations can be solved on reasonable time scales by com-
monly available computing resources.
Pegging the forward meniscus of the first introduced liq-
uid by a minor capillary barrier, occasionally referred to as
Fig. 7 Monte-Carlo simulation of the distribution of actual meniscus
position
̃z
when targeting at centrifugal equilibrium
Δp𝜔=0
found for the default parameters
𝜚,𝜚′
,U,U′
,R
and
Γ
while
factoring in their respective tolerances
ΔU,ΔU�
and
ΔΓ
(Table1).
After 1000 (time consuming) runs, the histogram features a mean
position
z=19.99 mm
with a standard deviation
Δz=0.387 mm
.
The vertical lines indicate (magenta) the default center and limits of
the DF at and , respectively, and (red shades)
with
M= {1,2,3}
with the standard deviation
Δz
(8) of the
̃z
-distribution
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Microfluidics and Nanofluidics (2022) 26:39
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Page 11 of 17 39
phase guide, at , is amongst many possible improve-
ments. Trapping of an interstitial bubble after filling the sec-
ond liquid may be prevented by an (initially) gas-permeable
membrane, located near , or a local outlet to be sealed
after priming has been completed. Stiction of the progress-
ing meniscus, e.g., caused by capillary pinning owing to
manufacturing-related artefacts or dust, may be overcome by
choosing a sufficiently elevated spin rate
𝜔≫0
for reaching
hydrostatic equilibrium (5) at .
In our own, naturally limited set of similar assay imple-
mentations (Lu etal. 2020; Mishra etal. 2020), we did not
observe any adverse effect of the ancillary liquid on the bio-
analytical performance. Yet, a general, blanket guarantee
cannot be issued a priori. To avoid possible interference with
the assay protocol, the immiscible ancillary liquid can be
cleanly removed under prevalent laminar flow conditions
through an additional, radially outer side pocket. Through
centrifugal stratification, this reservoir can be designed to
retain the entire volume of the higher-density ancillary liq-
uid, while the released liquid reagent overflows towards the
next stage. A corresponding structure has been proposed
in the context of high-quality centrifugal separation of the
plasma phase (Haeberle etal. 2006) (Czugala etal. 2013;
Yao etal. 2021) or bands (Kinahan etal. 2016a) from whole
blood.
Specific configurations of the reagent storage technique
have been tested for premature opening upon possibly
adverse conditions during storage, manual handling and
transport. The valve proved to be stable, the main impact
was evaporation, which was therefore included in the digital
twin simulation (Fig.6) (Lu etal. 2020; Mishra etal. 2020).
It is assumed that a malicious, brute force approach would
be required for inducing valve opening by deformation of
the disc.
The permanently gas-filled parts of reservoirs may be
placed at distal locations, as long as they are still pneumati-
cally connected through conduits, e.g., to make efficient use
of precious disc real estate required for multiplexed assay
panels. The pneumatic seals may also be removed through
a secondary mechanism, e.g., akin to venting procedures
implemented for centrifugo-pneumatic valves based on
mechanical (Kinahan etal. 2016b), laser- (García-Cordero
etal. 2010; Mishra etal. 2018) or pneumatic (Keller etal.
2015; Hess etal. 2019; Kinahan etal. 2016c, 2018; Godino
etal. 2012, 2013; Schwemmer etal. 2015a; Zhao etal. 2015;
Zehnle etal. 2015; Ducrée 2021c; Mishra etal. 2018; Gorkin
etal. 2010b, 2011; Mark etal. 2008, 2009b) principles.
6 Summary andoutlook
6.1 Summary
A novel technology has been introduced for Lab-on-a-Disc
systems, which offers a physical evaporation barrier and sta-
bilization of liquid distributions during long-term storage,
transport, and handling. Initially protected by an immiscible
ancillary liquid, the aqueous reagent contacts a dissolvable
film by centrifugal displacement. The convoluted interde-
pendencies governing the operational principle over its mul-
tiparameter space have been modelled to characterize system
robustness and behavior in silico. The resulting digital twin
further enables computational design optimization towards
given performance objectives within practically achievable
regimes of the experimental input parameters and their toler-
ances. In addition, systematic volume losses, e.g., through
evaporation during storage, artefacts such as enclosed gas
bubbles, and statistical deviations in the governing parame-
ters can be factored in, thus permitting algorithmic enhance-
ment the operational robustness of the valving mechanism.
6.2 Outlook
Evidently, the work presented only represents a blueprint
for setting up digital twins to efficiently characterize and
improve other functional elements of (centrifugal) micro-
fluidic Lab-on-a-Disc systems. Its simplified representation
of the valving structure by cuboidal elements can be sig-
nificantly refined to optimize flow, e.g., by curved contours
of the compartments, their inclination with respect to the
radial orientation, and fins to guide the relocation of liquids
and gases. Similar to the entrapped bubble, further parasitic
effects observed during experimental testing, and additional
elements of the layout can be included in the digital twin.
Such enhancements will require a more complex computa-
tional fluid dynamic (CFD) simulation, which should also
include inertia of the liquid and elastic components, like the
sealing membranes, which may bend or even yield under
pressure.
It is well known that tests with real systems will show
effects that are not included in the digital-twin modelling.
So, it is well expected that experimental validation will
remain a substantial tool for arriving at a product. Never-
theless, in particular during the early stage of development
where manufacturing and testing are mostly manual, only
smaller numbers of fluidic chips are available, which usually
precludes collecting sufficient statistics for proper device
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Microfluidics and Nanofluidics (2022) 26:39
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39 Page 12 of 17
performance and reliability analysis. Extensive experimental
validation of a wide spectrum of use cases, and sophistica-
tion of the rudimentary digital twin model, by the scientific
community is highly encouraged.
The digital twin presented in this work can then mark-
edly expedite the design iteration of microfluidic systems by
providing virtual prototyping and testing. Such a tool hence
empowers failure mode and effects analysis (FMEA) regard-
ing unavoidable tolerances, and in silico optimization of the
layout for given design targets. Adding similar programs for
simulating manufacturing and biochemical processes would
also be desirable to combine with the digital twin for fluidics
presented here.
On the bigger picture, the digital twin approach can boost
microfluidic industries by standardization (Heeren 2012;
Stavis 2012; Reyes etal. 2021), interpreted in a way that
validated boundary conditions issued by foundries can be
incorporated in computational design software to guaran-
tee manufacturability, reliability and targetted performance
within given cost limits. Moreover, the digital twin model-
ling published here, in tandem with simultaneously evolv-
ing, exponential technologies such as artificial intelligence
(AI) and digital manufacture in an increasingly virtualized
”metaverse” (Metaverse 2021; Meta 2021), lends itself for
open platform concepts which can leverage crowdsourcing
of brains, hands, infrastructure and equipment, e.g., coor-
dinated by the rapidly emerging, decentralized blockchain
technology (Ducrée etal. 2020a, b, 2021; Ducrée 2020,
2021e).
Appendix
The default parameters of the basic structure
Γ
in Fig.1
are listed in Table1. The tolerances in vertical and lateral
dimensions are
Δd=30 𝜇m
and
Δw=20 𝜇m
, respectively.
The minimum wall thickness between fluidic cavities is
set to
1mm
to account for scale-up of production by com-
mon injection molding schemes, and to provide adequate
bonding surface. The properties of water and “FC-72”
(3M™ Fluorinert™ Electronic Liquid FC-72) represent-
ing an aqueous reagent and an immiscible ancillary liquid
(at 25℃) are used. The volume of the ancillary fluid
U′
is
chosen to settle at
𝜔>0
(for both reservoirs open).
For the definition of liquid volumes,
ΔU=ΔU�=100 nl
are assumed, and a minimum gap
𝛿H
between the fill-
ing level and the seal is implemented to facilitate proper
experimental loading. Fluctuations in
p0
with respect to
the standard atmospheric pressure
pstd
are limited to the
range of typical weather conditions, i.e., about 4%; the
impact of the local altitude on
p0
is more pronounced
when operating the LoaD in mountainous regions. The
effect of variances in and
R
on the
meniscus position
z
is assumed to be negligible.
Table 1 Default dimensions
and boundary conditions for
experimental parameters of
the basic valving structure
Γ
(Fig.1)
Isoradial channel
R
=
30 mm
L
=
30 mm
H
=
3mm
Isoradial -Segment
𝒽=2mm
DF Region
Cross sections (depth
×
width)
A=1mm × 10 mm
A
�=1mm × 10 mm
A=1mm × 3mm
a=1mm × 5mm
a�=1mm × 4.5 mm
𝒶=1mm × 2mm
Depths
D=d=1mm
D�=d�=1mm
D=𝒹=1mm
Reservoir heights
H=15 mm
H�=10 mm
h=5mm
h�=2.5 mm
Minimum dimensions Vertical
≥300 μm
Lateral
≥200 μm
Wall Thickness
≥1mm
Structurable area:
Rmin =7.5 mm
Rmax
=
55 mm
Geom. tolerances Vertical: 30
μm
Lateral: 20
μm
Liquid volumes
U=160 μl
U�≈60.68 μl
ΔU=ΔU�=100 nl
Minimum filling gap
𝛿H=𝛿H�=1mm
Liquid densities
𝜚=
997
kg
⋅
m−3
𝜚�=
1680
kg
⋅
m−3
Liquid viscosities
𝜂=1.0016 mPa ⋅s
𝜂�
=
0.64
mPa ⋅
s
Ambient pressure
p0=pstd =1013.25 hPa
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Microfluidics and Nanofluidics (2022) 26:39
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Page 13 of 17 39
Funding Open Access funding provided by the IReL Consortium.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article's Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article's Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
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