Capillary Flow in Open Microgrooves: Bifurcations and Networks

The Fastest Capillary Flow in Root-like Networks under Gravity

Capillary flow has garnered significant attention due to its unique dynamic characteristics that require no external force. Creating a quantitative analytical model to evaluate capillary flow behaviors in root-like networks is essential for enhancing fluid control properties in functional textiles. In this study, we explore the capillary dynamics within root-like networks under the influence of gravity and derive the most rapid capillary flow via structural optimization. The flow time in a capillary is dominated by the capillary pressure, viscous pressure loss, and gravity, each of which exhibits diverse sensitivities to the structures of root-like networks. We scrutinize various structural parameters to understand their impact on capillary flow in root-like networks. Subsequently, optimal structural parameters (namely, the mother tube diameter and diameter ratio) are identified to minimize capillary flow time. Moreover, we discovered that the correlation between flow time and distance for capillary flow in root-like networks does not obey the classical Lucas-Washburn equation. These results affirm that root-like networks can enhance capillary flow, providing critical insights for numerous capillary-flow-dependent engineering applications.

On the Dynamic Contact Angle of Capillary-Driven Microflows in Open Channels

The true value of the contact angle between a liquid and a solid is a thorny problem in capillary microfluidics. The Lucas-Washburn-Rideal (LWR) law assumes a constant contact angle during fluid penetration. However, recent experimental studies have shown lower liquid velocities than those predicted by the LWR equation, which are attributed to a velocity-dependent dynamic contact angle that is larger than its static value. Inspection of fluid penetration in closed channels has confirmed that a dynamic angle is needed in the LWR equation. In this work, the dynamic contact angle in an open-channel configuration is investigated using experimental data obtained with a range of liquids, aqueous and organic, and a PMMA substrate. We demonstrate that a dynamic contact angle must be used to explain the early stages of fluid penetration, i.e., at the start of the viscous regime, when flow velocities are sufficiently high. Moreover, the open-channel configuration, with its free surface, enhances the effect of the dynamic contact angle, making its inclusion even more important. We found that for the liquids in our study, the molecular-kinetic theory is the most accurate in predicting the effect of the dynamic contact angle on liquid penetration in open channels.

On the dynamic contact angle of capillary-driven microflows in open channels

The true value of the contact angle between a liquid and a solid is a thorny problem in capillary microfluidics. The Lucas-Washburn-Rideal (LWR) law assumes a constant contact angle during fluid penetration. However, recent experimental studies have shown lower liquid velocities than predicted by the LWR equation, which are attributed to a velocity-dependent dynamic contact angle that is larger than its static value. Inspection of fluid penetration in closed channels has confirmed that a dynamic angle is needed in the LWR equation. In this work, the dynamic contact angle in an open channel configuration is investigated using experimental data obtained with a range of liquids, aqueous and organic, and a PMMA substrate. We demonstrate that a dynamic contact angle must be used to explain the early stages of fluid penetration, i.e., at the start of the viscous regime, when flow velocities are sufficiently high. Moreover, the open channel configuration, with its free surface, enhances the effect of the dynamic contact angle, making its inclusion even more important. We found that for the liquids in our study, the molecular-kinetic theory (MKT) is the most accurate in predicting the effect of the dynamic contact angle on liquid penetration in open channels.

Capillary Filling in Open Rectangular Microchannels with a Spatially Varying Contact Angle

Capillary flow in microchannels is important for many technologies, such as microfluidic devices, heat exchangers, and fabrication of printed electronics. Due to a readily accessible interior, open rectangular microchannels are particularly attractive for these applications. Here, we develop modifications of the Lucas-Washburn model to explore how a spatially varying contact angle influences capillary flow in open rectangular microchannels. Four cases are considered: (i) different uniform contact angles on channel sidewalls and channel bottom, (ii) contact angles varying along the channel cross section, (iii) contact angle varying monotonically along the channel length, and (iv) contact angle varying periodically along the channel length. For case (i), it is found that the maximum filling velocity is more sensitive to changes in the wall contact angle. For case (ii), the contact angles can be averaged to transform the problem into that of case (i). For case (iii), the time evolution of the meniscus position no longer follows the simple square-root law at short times. Finally, for case (iv), the problem is well described by using a uniform contact angle that is a suitable average. These results provide insights into how to design contact-angle variations to control capillary filling and into the influence of naturally occurring contact-angle variations on capillary flow.

Nanomechanical Sensing for Mass Flow Control in Nanowire-Based Open Nanofluidic Systems

Open nanofluidic systems, where liquids flow along the outer surface of nanoscale structures, provide otherwise unfeasible capabilities for extremely miniaturized liquid handling applications. A critical step toward fully functional applications is to obtain quantitative mass flow control. We demonstrate the application of nanomechanical sensing for this purpose by integrating voltage-driven liquid flow along nanowire open channels with mass detection based on flexural resonators. This approach is validated by assembling the nanowires with microcantilever resonators, enabling high-precision control of larger flows, and by using the nanowires as resonators themselves, allowing extremely small liquid volume handling. Both implementations are demonstrated by characterizing voltage-driven flow of ionic liquids along the surface of the nanowires. We find a voltage range where mass flow rate follows a nonlinear monotonic increase, establishing a steady flow regime for which we show mass flow control at rates from below 1 ag/s to above 100 fg/s and precise liquid handling down to the zeptoliter scale. The observed behavior of mass flow rate is consistent with a voltage-induced transition from static wetting to dynamic spreading as the mechanism underlying liquid transport along the nanowires.

Enhanced capillary pumping using open-channel capillary trees with integrated paper pads

The search for efficient capillary pumping has led to two main directions for investigation: first, assembly of capillary channels to provide high capillary pressures, and second, imbibition in absorbing fibers or paper pads. In the case of open microfluidics (i.e., channels where the top boundary of the fluid is in contact with air instead of a solid wall), the coupling between capillary channels and paper pads unites the two approaches and provides enhanced capillary pumping. In this work, we investigate the coupling of capillary trees-networks of channels mimicking the branches of a tree-with paper pads placed at the extremities of the channels, mimicking the small capillary networks of leaves. It is shown that high velocities and flow rates (7 mm/s or 13.1 μl/s) for more than 30 s using 50% (v/v) isopropyl alcohol, which has a 3-fold increase in viscosity in comparison to water; 6.5 mm/s or 12.1 μl/s for more than 55 s with pentanol, which has a 3.75-fold increase in viscosity in comparison to water; and >3.5 mm/s or 6.5 μl/s for more than 150 s with nonanol, which has a 11-fold increase in viscosity in comparison to water, can be reached in the root channel, enabling higher sustained flow rates than that of capillary trees alone.

The growth of capillary networks by branching for maximum fluid access

Here we document the deterministic evolution of capillary networks that morph by connecting more and more branches to water sources. The network grows with the objective of extracting in steady state higher and higher liquid flow rates. Growth happens through the generation of tree-shaped structures and the geometrical configuration of the dendritic network evolves as the number of connected sources increases. We present a novel methodology to generate capillary architectures and show how the evolution of the network leads to pump higher volumetric flow rates by capillary suction. The results suggest that networks generated within a plane lead to higher flow rates than networks generated within a three-dimensional domain, for the same volume of fluid.

A machine learning-based framework to design capillary-driven networks

We present a novel approach for the design of capillary-driven microfluidic networks using a machine learning genetic algorithm (ML-GA). This strategy relies on a user-friendly 1D numerical tool specifically developed to generate the necessary data to train the ML-GA. This 1D model was validated using analytical results issued from a Y-shaped capillary network and experimental data. For a given microfluidic network, we defined the objective of the ML-GA to obtain the set of geometric parameters that produces the closest matching results against two prescribed curves of delivered volume against time. We performed more than 20 generations of 10 000 simulations to train the ML-GA and achieved the optimal solution of the inverse design problem. The optimisation took less than 6 hours, and the results were successfully validated using experimental data. This work establishes the utility of the presented method for the fast and reliable design of complex capillary-driven devices, enabling users to optimise their designs via an easy-to-use 1D numerical tool and machine learning technique.

Open-Channel Capillary Trees and Capillary Pumping

Velocity of capillary flow in closed or open channels decreases as the flow proceeds down the length of the channel, varying as the inverse of the square root of time or as the inverse of travel distance. In order to increase the flow rate-and extend the duration of the flow-capillary pumps have been designed by mimicking the pumping principle of paper or cotton fibers. These designs provide a larger volume available for the wicking of the liquids. In microsystems for biotechnology, different designs have been developed based on experimental observation. In the present paper, the mechanisms at the basis of capillary pumping are investigated using a theoretical model for the flow in an open-channel "capillary tree" (i.e., an ensemble of channels with bifurcations mimicking the shape of a tree). The model is checked against experiments. Rules for obtaining better designs of capillary pumps are proposed; specifically, we find (1) when using a capillary tree with identical channel cross-sectional areas throughout, it is possible to maintain nearly constant flow rates throughout the channel network, (2) flow rate can be increased at each branch point of a capillary tree by slightly decreasing the areas of the channel cross section and decreasing the channel lengths at each level of ramification within the tree, and (3) higher order branching (trifurcations vs bifurcations) amplify the flow rate effect. This work lays the foundation for increasing the flow rate in open microfluidic channels driven by capillary flow; we expect this to have broad impact across open microfluidics for biological and chemical applications such as cell culture, sample preparation, separations, and on-chip reactions.

Investigation of Capillary Filling Dynamics of Multicomponent Fluids in Straight and Periodically Constricted Microchannels

An extensive study of capillary flow of fluids with various viscosities in straight and periodically constricted microchannels with different surface wettability is presented. Capillary filling speed in hydrophilic, less hydrophilic, and hydrophobic microchannels were experimentally monitored and compared with the Washburn theoretical model. For all liquids, a linear relationship was found between the square of propagation distance and time, which is expected for Newtonian fluids. Experimental results indicated slower velocity compared to the theoretical prediction due to simplifications of the Washburn model. Capillary filling speed of fluids into long-fluororinated chain silane modified channels confirmed the expected lyophobic nature of the coating (i.e., not favorable for either hydrophilic or hydrophobic liquids). Presence of the precursor film ahead of the three-phase contact line in the microscopic level was demonstrated. White light and fluorescent images confirmed the presence of precursor film and capillary evaporation at the interface. Evaporation enhanced the deviation between experimental and theoretical results due to continuous wettability alteration of penetrating fluid.

Droplet Incubation and Splitting in Open Microfluidic Channels

Droplet-based microfluidics enables compartmentalization and controlled manipulation of small volumes. Open microfluidics provides increased accessibility, adaptability, and ease of manufacturing compared to closed microfluidic platforms. Here, we begin to build a toolbox for the emerging field of open channel droplet-based microfluidics, combining the ease of use associated with open microfluidic platforms with the benefits of compartmentalization afforded by droplet-based microfluidics. We develop fundamental microfluidic features to control droplets flowing in an immiscible carrier fluid within open microfluidic systems. Our systems use capillary flow to move droplets and carrier fluid through open channels and are easily fabricated through 3D printing, micromilling, or injection molding; further, droplet generation can be accomplished by simply pipetting an aqueous droplet into an empty open channel. We demonstrate on-chip incubation of multiple droplets within an open channel and subsequent transport (using an immiscible carrier phase) for downstream experimentation. We also present a method for tunable droplet splitting in open channels driven by capillary flow. Additional future applications of our toolbox for droplet manipulation in open channels include cell culture and analysis, on-chip microscale reactions, and reagent delivery.