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Cellular fluidics

Abstract

The natural world provides many examples of multiphase transport and reaction processes that have been optimized by evolution. These phenomena take place at multiple length and time scales and typically include gas–liquid–solid interfaces and capillary phenomena in porous media1,2. Many biological and living systems have evolved to optimize fluidic transport. However, living things are exceptionally complex and very difficult to replicate3,4,5, and human-made microfluidic devices (which are typically planar and enclosed) are highly limited for multiphase process engineering6,7,8. Here we introduce the concept of cellular fluidics: a platform of unit-cell-based, three-dimensional structures—enabled by emerging 3D printing methods9,10—for the deterministic control of multiphase flow, transport and reaction processes. We show that flow in these structures can be ‘programmed’ through architected design of cell type, size and relative density. We demonstrate gas–liquid transport processes such as transpiration and absorption, using evaporative cooling and CO2 capture as examples. We design and demonstrate preferential liquid and gas transport pathways in three-dimensional cellular fluidic devices with capillary-driven and actively pumped liquid flow, and present examples of selective metallization of pre-programmed patterns. Our results show that the design and fabrication of architected cellular materials, coupled with analytical and numerical predictions of steady-state and dynamic behaviour of multiphase interfaces, provide deterministic control of fluidic transport in three dimensions. Cellular fluidics may transform the design space for spatial and temporal control of multiphase transport and reaction processes.

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Fig. 1: Cellular fluidics: a nature-inspired, unit-cell-based platform.
Fig. 2: Architected design for predictable capillary rise.
Fig. 3: Numerical simulations and high-speed videography reveal dynamics.
Fig. 4: Transpiration in cellular fluidics.
Fig. 5: Gas absorption in cellular fluidics.
Fig. 6: Preferential fluid pathways for continuous flow and selective patterning.

Data availability

Source data for calculated porosities and interfacial areas of different unit cells are included with the paper. Reasonable requests for additional data that support the findings of this study will be honoured upon securing release approval from the Lawrence Livermore National Laboratory and the US Department of Energy. Source data are provided with this paper.

Code availability

Reasonable requests for the code that support the findings of this study will be honoured upon securing release approval from the Lawrence Livermore National Laboratory and the US Department of Energy.

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Acknowledgements

We are grateful to the late J. J. Vericella for his contributions to the CO2 capture technologies that enabled this work. We thank R. Goldsberry and J. Long for support with figure illustrations and artwork; D. Macknelly for providing guidance on lattice generation; C. Ye and J. Mancini for assistance with high-speed video experiments; and J. Knipe, S. Pang, X. Xia, V. Beck and C. Spadaccini for discussions. This work was performed under the auspices of the US Department of Energy by the Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344 and LDRD 19-SI-005 with information management number LLNL-JRNL-811671.

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Authors

Contributions

N.A.D. designed structures, devices and experiments, performed 3D printing, capillary rise tests and analytical calculations, recorded images and videos, and wrote the manuscript. E.J.F. designed and fabricated devices, performed capillary rise, transpiration, and active gas and liquid flow experiments, recorded images and videos, carried out thermal and Schlieren imaging and analysis, and colorimetric analysis of CO2 absorption. H.B.G. fabricated structures, performed capillary rise, active flow, transpiration and CO2 absorption experiments and analysis, performed selective metallization, recorded images and videos. J.R.D. designed structures, carried out numerical simulations and analysis, performed high-speed videography experiments and analysis. M.C.R. performed CO2 absorption experiments and analysis. B.D.M. designed and assembled the Schlieren imaging set-up and participated in data collection. J.T.D. fabricated structures and carried out electroless plating. S.E.B. proposed and supervised the project. E.B.D. conceived the idea, proposed and supervised the project, and wrote the manuscript. All authors participated in discussions of the research and revisions of the manuscript.

Corresponding author

Correspondence to Eric B. Duoss.

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Competing interests

A patent application has been filed by the Lawrence Livermore National Security LLC on the basis of the cellular fluidics concept described here (US Patent Application Serial No. 16/549,543), on which N.A.D., S.E.B., V. Beck, S. Chandrasekaran, J.R.D., E.B.D., J. Feaster, J. Knipe, J. Mancini, J. Oakdale, F. Qian and M. Worsley are listed as inventors.

Additional information

Peer review information Nature thanks Sung Kang, Abraham D. Stroock and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Effects of cellular architecture on fluid transport.

a, Unit cell types used in this study. b, Porosity as a function of relative strut diameter D* can be used to determine the liquid-holding capacity of a given cell type. c, The cell design can be optimized for a desired liquid–solid and gas–liquid interfacial area by choosing an optimal D*. d, Illustration of the repeating structural motif in BCC arrays. Owing to the symmetry of the BCC unit cell, the structure defined by the green control volume (0, 0, 0) is also repeated in the magenta control volume, offset by a half-cell distance (L/2, L/2, L/2). e, Geometric considerations for a simplified analysis of capillary rise in BCC structures. Assuming the force equilibrium always occurs at the central node where Fadh is at a minimum (Fig. 2a), the contact perimeter can be approximated by a circle circumscribing the elliptical cross-section. The capillary height is explicitly solved for from the force equilibrium at the node. For an array of cells, the contact perimeter in the control volume can be approximated by the dashed lines, resulting in a greater contact perimeter and higher capillary rise.

Source data

Extended Data Fig. 2 Numerical simulations of surface-tension-driven and actively pumped liquid flow.

a, Simulations of capillary rise in an array of BCC cells. Similar trends in the spatiotemporal evolution of the gas–liquid and liquid–solid interfaces are observed as for a single column of cells (Fig. 3c). The liquid fronts at the outer corners accelerate and decelerate in an alternating manner. The overall velocity decreases over time. See Supplementary Video 1. b, Simulations of flow through a channel of simple cubic structures of different strut diameters were carried out on the outlined domain of 1.5 mm cells using quarter symmetry. The gas–liquid interface evolution and pressure map along the central cross-section plane are shown at time steps of 2, 10 and 20 ms. The white circles represent the cross-sections of the cell struts. For a cell with 500-μm struts, the gas–liquid interface initially advances, then recedes. c, Flow rate variation over time for cubic cells with different strut diameters. The negative magnitude of the flow rate (black line) corresponds to the receding interface in a cell with 500 μm. d, Average pressure variation over time. In cells with 500-μm struts, the average pressure becomes positive, indicating that the interface will not propagate in the desired direction and suggesting a potential leak scenario. In cells with 600-μm struts, the pressure hovers around zero, indicating that this strut thickness can be considered as the minimum critical value, and that slightly thicker struts are needed to ensure flow in the desired direction. e, The maximum pressure at the interface across the first cell (0 to L) corresponds to the swelling of the liquid in the cell before it propagates into the subsequent cell. Exceeding the Laplace pressure of the advancing interface indicates a leak can occur (Supplementary Fig. 6).

Extended Data Fig. 3 Gas–liquid transport under different environmental conditions.

a, Illustration of water evaporation for cellular fluidics with and without a liquid reservoir and filter paper (left). Evaporative mass loss of the test structures at different relative humidity (middle). When supplied with a liquid reservoir, the evaporation rate of cellular fluidics is comparable to filter paper (right). b, Flow cell used for measurements of humidification and pressure drop for gas flow across wetted devices (left). Comparable levels of humidification are achieved between cellular fluidic devices and filter paper (middle); Data are mean ± s.d.; n = 6. However, significant pressure drops were observed across filter paper, whereas no appreciable pressure drop could be detected for the cellular fluidics. Gas-driven liquid ejection was observed for both the filter paper (11–15 ml min−1) and a cellular fluidic device (>25 ml min−1) with no gas pores (right). Incorporating open (non-liquid-filled) gas pores allowed robust operation with no liquid ejection at all tested flow rates. c, Flow cell used for measurements of pressure drop of liquid flow across a uniform isotruss lattice (left). The data agree well with calculations from the Ergun equation, implying that pressure drop due to inertial effects is significant at higher flow rates (middle). Pressure drops calculated for lattices of different cell types at a liquid velocity of 10 cm s−1 (right). The circle symbol in the right plot corresponds to the circle symbol data point in the middle plot.

Extended Data Fig. 4 Effect of unit cell coordination number on CO2 absorption rate.

a, Images (top) of the hierarchical lattice filled with a solution of Na2CO3 and a pH-sensitive colour indicator that changes from blue to yellow upon CO2 saturation. The rate of absorption in the lattice is significantly faster than a liquid pool of the same sorbent volume. b, Hierarchical structure 750 s after exposure to CO2, with analysed regions of interest outlined. Scale bar, 5 mm. c, Colorimetric map depicting the reaction rate during the first 500 s. The extent of CO2 absorption is estimated by the colour change from blue to yellow (the sum of green and red channels minus the blue channel for each pixel). Higher reaction rates occur at cells with a lower coordination number—that is, more gas–liquid interfaces (blue arrows)—compared to cells with a higher coordination number (black arrows). The enlarged images of the outlined regions show high rates of reaction occurring at interfaces in the centre of each unit cell. Although it is expected that the reaction will occur more rapidly at struts and corners, where the liquid layer is the thinnest and has the highest surface-area-to-volume ratio, the thin liquid layer results in high reflectivity and low levels of pH indicator, which limits the measurement accuracy in these regions. The cell at the intersection with a higher coordination number shows a slower reaction rate than adjacent cells. d, The extent of reaction progression for cells selected in b. Cells with a coordination number 5 (black) show lower levels of reaction than cells with coordination number 2 (blue and pink).

Extended Data Fig. 5 Actively driven flow in channels of simple cubic cells with uniform and graded density structure.

a, In the uniform density structure, the pressure drop across the channel causes the liquid to halt its propagation in the desired direction and break the surface tension at a gas–liquid interface facing an undesirable direction, which results in a leak. b, In the graded density structure, the retaining capillary force along the desired flow direction better matches pressure drop across the channel, which results in uniform and leak-free flow through the entire structure. Scale bars, 5 mm.

Extended Data Fig. 6 Capillary rise in Nx × Ny × Nz arrays of isotruss and simple cubic cells.

a, For isotruss cells, similar to BCC cells (Fig. 2), capillary rise of Nx × Ny × Nz arrays is higher compared to a single column of the same relative density. b, The same effect is not observed in arrays of simple cubic cells, where there is no strut structure that is internal to the cubic cell. Scale bars, 5 mm.

Extended Data Fig. 7 Transpiration experiments comparing cellular fluidic structures with filter paper.

Transpiration was observed by the water level drop in the reservoir and the concentration of green dye at the tip of the structures. a, A cellular fluidic structure and filter paper with an equivalent surface area exhibit similar evaporation rates, plotted as the decrease in liquid level in the reservoir over time. b, Filter paper was cut to match the shape of the cellular fluidic structure, but the filter paper collapsed under its own weight after 90 min of use. Conversely, the cellular fluidic component is a mechanically robust structure that exhibits no sign of collapse or deformation. c, Transpiration from the stochastic tree structure and filter paper holding equivalent volumes of liquid. Scale bars, 5 mm.

Extended Data Table 1 Dimensionless parameters derived for cellular fluidics
Extended Data Table 2 Parameter values of cellular fluidic structures

Supplementary information

Supplementary Information

This file contains Supplementary Methods (analytical calculations) and Supplementary Figures (unmodified images from Figs. 2d, 2f and 6c, evaporation front analysis from Fig. 4c, simulated velocity from Fig. 3c and thermal imaging data from Fig. 4b without smoothing, microscopy of micro-roughness effects, illustration of filter paper testing cell).

Peer Review File

Supplementary Video 1

Capillary rise – simulations and experiments. Simulation (left) and high-speed video (right) of single cell filling and capillary rise in a 1×1×Nz cell column shown in Fig. 3. High-speed video of capillary rise in a column. Simulation of capillary rise in a Nx×Ny×Nz array (Extended Data Fig. 2a). High-speed video of capillary rise in an array.

Supplementary Video 2

Multiphase processes in cellular fluidics. Video of transpiration of green-dyed ethanol through a stochastic tree structure. Infrared videos corresponding to data in Figs. 4b and 4c. Schlieren imaging used to calculate evaporative flux shown in Fig. 4d. Temporal color change of the hierarchical structure is used to estimate CO2 absorption in Extended Data Fig. 4.

Supplementary Video 3

Active flow. Videos of active flow through 1D and 2D cellular fluidic structures, and demonstration of programmed fluidic and selectively metallized pathways in 3D.

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Dudukovic, N.A., Fong, E.J., Gemeda, H.B. et al. Cellular fluidics. Nature 595, 58–65 (2021). https://doi.org/10.1038/s41586-021-03603-2

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