Among the applications of capillary pressure, the role of positive pressure is of great importance. This type of pumping can be achieved by the introduction of aqueous droplets with various sizes at the inlets and outlets of the pump. Smaller droplets at the inlet generate higher positive capillary pressure, driving the flow towards the larger droplet at the outlet (Wu et al., 2019; Lee et al., 2020). The speed and direction of flow within the channel can be controlled by adjusting droplet size. This method requires only a device to deliver small amounts of liquid, making it easily applicable in most laboratory settings. However, the challenge of pre-programming fluid flow times often necessitates user intervention to control the assay (Fichera et al., 2024; Najmi et al., 2022). It is worth noting that increasing the generated pressure can be a challenging issue, since it can cause irreversible damage to the platform. Furthermore, if the inlet and outlet droplet sizes are too similar, reverse flow could occur, limiting the reliability of the pumping system (Aghajanloo et al., 2024; Fichera et al., 2024; Najmi et al., 2022). To address these challenges, Berthier et al. developed an analytical model to optimize design parameters for surface tension-driven flows (Najmi et al., 2022). Their work identifies two distinct phases in the behavior of the input droplet: an initial stable flow phase followed by a transition to an unstable regime. The stable flow phase can be extended by periodically refilling the inlet droplet, enabling continuous flow in the microchannel, improving system reliability (Berthier and Beebe, 2007).

Microfluidic systems exploiting capillary pressure can be broadly categorized into three platforms: closed systems, semi-open systems, and open channels. Traditional closed systems consist of completely enclosed channels, whereas semi-open and open channels have open inlets and outlets with an air-liquid interface exposed to the environment. Compared to closed channels, open capillary systems offer several advantages, including fluid accessibility along the entire pathway, ease of fabrication, and ease of surface treatments. Additionally, these open-channel designs reduce the risk of bubble entrapment, which is a common issue in closed channels (Ye et al., 2018; Berry et al., 2019). However, issues such as fluid evaporation, potential fluid loss, and contamination hinder their broader use (Berthier et al., 2019).

To enhance the practicality of open capillary platforms, studies have focused on innovative designs. For example, a V-shaped cross-sectional geometries enhance spontaneous capillary action, making it suitable for handling high-viscosity fluids like blood (Berthier et al., 2015).

Porous capillary pumps (CP) can also play vital roles in lateral flow assays and immunochromatographic strip tests (Casals‐Terré et al., 2020). These systems are used to analyse a wide range of biological samples, including urine (Armbruster et al., 2022), whole blood, plasma, sweat, and so on (Koczula and Gallotta, 2016). In these devices, the sample is introduced at the inlet and moves through a porous material to complete the test, allowing for qualitative or semi-quantitative analyses without requiring any expensive equipment. These systems rely on the capillary action of fluids within fibrous materials such as paper (Casals‐Terré et al., 2020; Tian et al., 2018; Chen et al., 2012; Jarrett et al., 2020; Mehrdel et al., 2021; Yamada et al., 2017), polyester (Dai et al., 2019), wool, superhydrophobic textiles (Agustini et al., 2021; Lu et al., 2020), cotton thread, knots (Safavieh et al., 2015), and other substrates (Xu et al., 2020; Reches et al., 2010; Jarujamrus et al., 2020; Zhao et al., 2020; Fu and Downs, 2017).

In addition to CPs, various types of capillary valves based on porous materials have been studied to control fluid flow. The integration of these elements can lead to autonomous sequential fluid distribution within these systems (Chen et al., 2012). Porous capillary systems are easy to fabricate and use, cost-effective, and capable of transporting large fluid volumes, making them especially suitable for healthcare applications (Yamada et al., 2017). However, challenges such as reproducibility and flow rate control persist. The heterogeneous flow within selected porous membranes limits further miniaturization and fluid control. Variations in pore size and distribution within the porous structure restrict the resolution, robustness, and reliability of biochemical tests. In microfluidic devices, uneven pore distribution can lead to irregular flow rates, causing delays or variations in sample transport. For example, in saliva-based diagnostic devices, uneven pore structures can lead to inconsistent fluid dynamics and hinder sensitivity (Kumari et al., 2023).

Vinoth et al. developed a microfluidic platform integrated with a porous filtration membrane for monitoring salivary biomarkers (Vinoth et al., 2023). Their findings showed that the heterogeneous pore structure of the filtration membrane effectively prevented the bulkier proteins, mitigating biofouling issues. However, variations in pore size and distribution may still influence fluid dynamics, affecting the overall reliability and sensitivity of such devices (Vinoth et al., 2023). Furthermore, in lateral flow assays, like pregnancy tests, inconsistent pore sizes can hinder the consistent capillary flow of reagents, leading to variability in test results (Kumari et al., 2023; Yetisen et al., 2013). On the other hand, capillary systems based on microchannels minimize contamination risks, require smaller sample volumes, and provide improved flow rate control through structural manipulation. Consequently, flow in solid material-based capillary systems is more predictable and repeatable under controlled conditions.

Despite these advances, controlling the flow has remained the primary challenge in utilizing passive capillary-driven microfluidics. To address this challenge, a variety of materials have been employed in the fabrication of microfluidic chips, including glass, silicon, and polymers (Ren et al., 2013). Most devices are constructed from polymeric materials such as poly (methyl methacrylate) (PMMA) and, notably, polydimethylsiloxane (PDMS) due to their optical transparency and flexibility (Niculescu et al., 2021). In some systems, a combination of solid and porous materials is used to harness the benefits of both structures, enhancing functionality and control within capillary systems. This hybrid approach leverages the predictable flow characteristics of solid channels while utilizing the absorption and distribution properties of porous materials, allowing for more versatile and adaptable microfluidic designs (Kumari et al., 2023; Ren et al., 2013; Niculescu et al., 2021). Furthermore, capillary phenomena are often integrated with other passive and active pumping methods to drive fluid within microfluidic systems more effectively. For instance, Liu et al. combined capillary action with air absorption in degassed PDMS to wick, meter, and mix fluids in microfluidic devices (Liu and Li, 2018). This integration highlights the potential for innovative solutions by combining capillary effects with complementary technologies, expanding the versatility and application of microfluidic systems across various fields. Indeed, recent advancements in capillary-driven microfluidic devices have led to innovative applications across a wide range of biological assays. These devices offer unique advantages over traditional platforms, such as high sensitivity, rapid processing times, and minimal sample requirements, making them ideal options for POC diagnostics and other time-sensitive applications. Table 2 provides a summary of recent research in this area, highlighting the diverse design of capillary devices, various detection targets, sample types, assay turnaround times, and detection limits.

Capillary systems are suitable for single or multi-step immunoassays (Desaegher et al., 2025; Chen et al., 2020; Lin et al., 2020). In a recent study, a capillary flow ELISA assay platform was integrated with a smartphone analyser for data display, transfer, storage, and analysis (Ghosh et al., 2020). The study demonstrated that this platform, using chemiluminescence-based detection for the malaria biomarker PfHRP2, achieved a limit of detection (LOD) of 8 ng/mL, which is sensitive enough to detect active malarial infection. Furthermore, the platform can be adapted for other biomarkers, allowing the development of low-cost, portable, point-of-care diagnostic tools with full networking capability (Ghosh et al., 2020).

In another study, Gervais et al. developed a one-step immunoassay chip based on their previous works on capillary-driven platforms (Gervais and Delamarche, 2009). The aim of the study was to detect C-reactive protein (CRP) cardiac markers with concentrations below 1 ng/mL within 14 min by simply injecting a droplet of the sample. The device included a sample collector, a delay valve, a resistor, a reaction chamber, a CP, and vents, which were the main elements of a capillaric circuit (Gervais and Delamarche, 2010; G Navascues Liquid Surfaces, 1979). The results demonstrated that the device was capable of extending the dynamic range of the test and improving the distribution of the detection antibody through a Dean flow mixer. The signal intensity in the reaction chambers varied according to the flow rate, allowing for precise control of the signal with just one sample handling step. This strategy enhanced the effectiveness of the test and reduced the need for sample dilutions or preconcentrations, making the chip more practical for point-of-care diagnostic applications. In another work, they designed a multi-parametric chip for the manipulation of assay parameters using six parallel flow paths for the detection of a panel of biomarkers. They incorporated the platform for the one-step immunoassay of C-reactive protein detection in human serum. The results of the study showed that the microfluidic chip, through its varied flow rates and controlled incubation times, could significantly improve detection sensitivity. In particular, the longer incubation times resulted in a fourfold increase in detection signal for C-reactive protein. This flexibility allows for better optimization of immunoassay conditions and enhances the chip’s versatility for various surface fluorescence immunoassays.

Previous reviews have primarily focused on specific aspects of capillary systems, such as the design of individual components (e.g., capillary pumps and valves) or their integration into diagnostic devices. Nevertheless, a comprehensive understanding of the interplay between key design parameters, such as wettability, geometry, and surface tension, and the overall performance of capillary-driven microfluidic systems across diverse applications remains elusive. Furthermore, critical challenges including precise control of flow rates, enhancing reproducibility, and scaling these systems for broader biomedical and environmental applications persist. This review aims to address these shortcomings by providing a comprehensive analysis of both the fundamental principles and the latest innovations in capillary-driven flow, with a particular focus on addressing persistent challenges and exploring opportunities to optimize device performance.

On this basis, the following sections will examine the fundamental concepts underlying capillary-driven flow in microfluidic devices, including an analysis of the governing equations and key design parameters of primary capillary elements. In addition, the section will explore the various applications of these principles, which are instrumental in enhancing the performance and versatility of capillary microfluidic systems.

Surface tension arises from the cohesive forces that exist between the molecules in a liquid. Within the liquid, molecular attractions are isotropic, with equal and opposite forces acting in all directions. But at the surface, the absence of molecules above creates an imbalance. This imbalance causes molecules to be pulled inward, minimizing the surface area and surface free energy. Surface tension is quantified as force per unit length (N/m).

In microfluidics, surface tension plays a crucial role in determining the behaviour of fluids within microchannels. It is responsible for several key phenomena, including meniscus formation, capillary rise and the spontaneous filling of channels. These effects are fundamental to the control of liquid spreading, movement and stability in microfluidic systems. For example, in diagnostic applications, surface tension ensures precise fluid transport, enabling accurate reactions and measurements. An understanding of and ability to manipulate surface tension enables designers to optimise the performance of microfluidic devices, ensuring consistent fluid flow and robust operation across a variety of applications (G Navascues Liquid Surfaces, 1979).

The typical visual indicator of surface tension is the contact angle (Figures 2A, B), the higher the surface tension the greater the contact angle (Kim et al., 2021). One of the novel technologies that take advantage of the physical phenomenon of surface tension is SAW (surface acoustic wave), this technology provides a method of generating accurate and consistent single or multiple droplets by jetting. This is achieved by directing acoustic energy onto liquid surfaces using transducers, resulting in a technique known as SAW-ADE. By creating a pressure gradient that exceeds the Rayleigh-Taylor instability threshold, and effectively overcomes surface tension. This pressure gradient causes a liquid protrusion that transforms into a jet, which eventually disintegrates into one or more droplets as a result of the Plateau-Rayleigh instability (Figure 2C) (Ning et al., 2023).

Wettability refers to a liquid’s ability to adhere to a solid surface or another liquid substrate. It is a crucial concept in microfluidics because it determines how fluids interact with surfaces, influencing flow behavior and the efficiency of fluid transport in microchannels.

To quantify wettability, the spreading parameter S is often used, defined by Equation 1: S = γ SG − γ SL + γ LG (1) Wherein γ represents the surface tensions at the solid/gas (γ _SG ), solid/liquid (γ _SL ), and liquid/gas (γ _LG ) interfaces. This parameter is particularly important in devices where precise fluid control is essential, such as point-of-care diagnostic chips (Figure 2D).

In this context, the contact angle is a key parameter in determining wettability. It is the angle formed between the tangent to the surface of the liquid and the surface of the solid at the line of contact between liquid, gas and solid. The static contact angle (θ) is given by Young’s equation (Equation 2): γ LG × cos ⁡ θ E = γ SG − γ SL (2) Wherein θ _E is the equilibrium contact angle. This equation establishes the relationship between surface tensions at the solid-liquid-gas interfaces, providing insights into how a liquid interacts with a surface. This understanding is vital for predicting and optimizing fluid behavior in microchannels, particularly in applications where rapid and uniform fluid distribution is essential.

Surfaces are categorized based on their contact angle (Figure 2E). When the spreading parameter S > 0 and θ < 90°, the surface is considered highly wettable, or superhydrophilic. For S < 0 and θ < 90°, the surface is hydrophilic but exhibits reduced wettability. Conversely, when S < 0 and θ > 90°, the liquid forms distinct droplets, signifying a hydrophobic surface. Finally, when θ > 150° the surface is considered superhydrophobic. These classifications are crucial for tailoring surface properties to specific microfluidic applications. In Figure 2F, it can be seen how surface coating can affect to wettability. Authors developed fluorine-free superhydrophobic coatings on copper mesh by utilizing a combination of PDMS and a multi-walled carbon nanotube/zinc oxide (MWCNTs/ZnO) composite through dip-coating methods. The image on the center shows coloured water in contact with the uncoated surface (naturally hydrophobic), while the other two images show the effect of the coating on the coloured water and also on other liquids (clearly superhydrophobic).

Under dynamic conditions, contact angles become more complex. For example, as a droplet moves across a non-ideal surface, the advancing front (θ _a ) and receding front (θ _r ) often differ due to surface irregularities or variations in wettability. This phenomenon, known as contact angle hysteresis, can impact the consistency of fluid flow. If not properly accounted for, hysteresis can introduce errors in fluidic performance, potentially affecting the accuracy of diagnostic readings or other applications requiring precise control.

It is worth noting that a fluid’s contact angle increases with the increase in its flow rate. To formulize the capillary phenomena, Bracke et al. proposed an equation (Equations 3, 4) to calculate the contact angle with the use of the capillary number and the velocity of the fluid (Safavieh et al., 2015; Bracke et al., 1989): C a = U μ γ (3) cos ⁡ θ a − cos ⁡ θ d cos ⁡ θ a + 1 = 2 × C a 0.5 (4)

In this equation, U denotes the filling front velocity, which is the speed at which the liquid front progresses through the microchannel. θ _a is the static advancing contact angle, and θ _d is the dynamic advancing contact angle. This dynamic perspective is crucial for designing microfluidic platforms that enable controlled fluid delivery, particularly in healthcare and environmental monitoring.

Capillary pressure is a pivotal force in microfluidic devices, where the small dimensions of the channels amplify the effects of surface interactions. This phenomenon, first explored by Washburn (1921) in the 1920s, arises from the adhesive forces between liquid molecules and the capillary wall, combined with the cohesive forces among the liquid molecules themselves. These interactions create a concave meniscus at the liquid’s surface, which draws the fluid upward due to the negative (suction) capillary force. This upward movement is a key mechanism enabling fluid transport in microchannels.

Capillary rise of water in a vertical capillary tube occurs when the tube is immersed in a liquid, causing the meniscus to rise against gravity to a height (h) above the surrounding liquid surface. At equilibrium, this height is determined when the forces are balanced according to the Young-Laplace equation, which takes into account the effects of gravity and surface tension (Figure 2G). In the horizontal capillary pipette filled with water, the water will flow into the horizontal capillary until it is opposed by a counterforce, such as pressure on the back end of the pipette, to hold it in position or push it back (Figure 2G). With a tapered micropipette, the position of the meniscus is outwards, requiring a sequence of increasing applied pressure to force the liquid surface downwards along the taper (Figure 2G) (Needham et al., 2019).

The pressure created by these capillary phenomena is described by the Young-Laplace equation, which considers the surface tension of the liquid, the geometric features of the channels (such as width, height, and length), and the surface chemistry (wettability) of the solid wall. In rectangular channels, common in microfluidic devices made through photolithography or rapid prototyping methods, the Young-Laplace equation (Equation 5) is expressed as: P = − γ cos ⁡ θ t + cos ⁡ θ b h + cos ⁡ θ l + cos ⁡ θ r w (5) Wherein P is the capillary pressure, and γ is the surface tension of the liquid. The angles θ _t and θ _b represent the contact angles at the top and bottom surfaces of the channel, respectively, while θ _l and θ _r are the contact angles at the left and right sides of the channel. The height and width of the channel are denoted by ℎ and w, respectively.

In microfluidic systems with wettable surfaces, where the contact angle θ is less than 90°, a negative capillary pressure pulls the fluid into the channel. This mechanism is fundamental to the operation of CPs. For optimal performance of capillary circuits, a contact angle θ of less than 60° is preferred. However, the actual effective capillary pressure can be lower than the predicted value due to discontinuities or imperfections in the capillary paths, which can affect the fluid flow dynamics.

Understanding and predicting flow rates in microchannels is vital for designing capillary systems that operate reliably across a range of conditions. This flow rate can be described using the Navier-Stokes equation. A pressure-driven, steady-state, fully developed laminar flow in a circular channel (Hagen-Poiseuille flow), is given by Equation 6: Q = π a 4 8 η L Δ p = 1 η Δ P R F (6) Wherein Q is the volumetric flow rate, a is the radius of the channel, η is the dynamic viscosity of the fluid, L is the length of the channel, and Δp is the pressure difference across the length of the channel.

In typical microfluidic fabrication, channels are often rectangular. Therefore, for rectangular channels, the flow rate Q is described as (Equation 7): Q = 4 h 2 Δ p π 3 η L ∑ n , o d d ∞ 1 n 3 2 h n π w − 2 h n π tanh   n π w 2 h = 8 h 3 w Δ p π 4 η L ∑ n , o d d ∞ 1 n 4 − 2 h π w 1 n 5 tanh   n π w 2 h =   h 3 w Δ p 12 η L 1 − ∑ n , o d d ∞ 1 n 5 − 192 π 5 h w tanh   n π w 2 h (7)

In these expressions, ℎ is the channel height, w is the channel width, and the summation accounts for the contribution of multiple modes in the rectangular geometry.

The equation (Equation 8) can be approximated by: Q ≈ h 3 w Δ p 12 η L 1 − 192 π 5 h w ∑ n , o d d ∞ 1 n 5 = h 3 w Δ p 12 η L 1 − 192 π 5 31 32 ζ 5 h w = h 3 w Δ p 12 η L 1 − 0.630 h w (8)

When the ℎ/w ratio converges to zero, this approximation approaches the exact solution. For the case where ℎ = w, the deviation from the exact solution is about 13%, and for ℎ = w/2, it decreases to 2%, showing that the approximation aligns well with exact results (Bruus, 2008). The sensitivity of flow rates to channel dimensions, as highlighted by this approximation, underscores the need for precise fabrication techniques in microfluidic design, particularly for applications like biochemical assays where consistent flow rates are essential to ensure uniform reagent mixing and precise reaction times. Additionally, the equation offers valuable insights into the impact of design modifications, such as reducing h/w ratios, which can enhance flow stability. This improved control overflow dynamic is vital for optimizing assay conditions, ensuring reproducibility, and enhancing the overall accuracy of diagnostic systems.

It is worth noting that Zimmerman et al. have presented a first-order approximation for the flow rate as the following (Equation 9) (Zimmermann, 2009): Q = 1 12 η 1 + 5 h 6 w R h 2 h w L Δ P = 1 η Δ P R F (9)

Here, R _ℎ is the hydraulic radius, and ΔP is the pressure difference across the microchannel, approximated by the pressure at the liquid-air interface, as described by the Young-Laplace equation for capillary pressure in a rectangular channel.

On the other hand, Song et al. proposed an alternative equation (Equation 10) for the average fluid flow velocity within the capillary channel (Saha et al., 2009; Song et al., 2011): u a v g = A 1 − exp − B t 2 B l (10) Where A = 4 σ     1 + β   cos ⁡ θ ρ h , B = 12 μ γ ρ h 2 , and l = A B 2 exp − B t + A t B + l 0 2 − A B 2 0.5 . The parameter equals 1 − − 19 β ∑ i = 1 , 3 , 5 ∞ tan   i 2 π β i π 5 , and l ₀ is the initial distance moved by the liquid.

These two sets of equations are utilized in various studies. Although both yield approximately similar values for fluid parameters in capillary channels, some differences may exist. Mohammed et al. compared experimental results for fluid flow rates within capillary circuits with the equations proposed by Zimmerman and Song, demonstrating that Song’s equation is more accurate, particularly for larger channel ratios (Mohammed and Desmulliez, 2014a). In their calculations for pump filling, they treated the pump as a single channel with a length equal to the sum of the interconnected channels within its structure. They found that Zimmerman’s equations are better suited for smaller channel ratios. In summary, these equations appear to offer different approaches to model fluid flow in capillary channels. Zimmerman’s equation tends to be more appropriate for smaller channel ratios, while Song’s equation provides better accuracy for larger channels.

This comprehensive investigation underscores the importance of selecting appropriate theoretical models for different channel geometries and provides a useful comparison between theoretical predictions and experimental observations in microfluidic systems.

To further understand and optimize fluid flow in microfluidic systems, researchers often turn to electrical analogies. Considering electrical analogies for capillary circuits, the hydraulic resistance of a rectangular microchannel can be expressed as follows (Equation 11): R c = ∆ P Q = 82 w × h 3 12 × μ × L × 1 − 192 × h π 5 × w × tanh π × w 2 × h − 1 ≅ 12 η L h 3 w 1 − 0 , 63 h w − 1   P a . s . m − 3 (11) Wherein ΔP is the pressure difference, Q is the flow rate, η is the fluid viscosity, w and ℎ are the width and height of the channel, respectively, and L is the channel length.

The overall resistance of the channel is determined by its smallest cross-sectional area. Unlike the constant resistance in electronic circuits, flow resistance increases as the channel fills, while capillary pressure remains constant for a fixed cross-section. The CP’s contribution to the overall resistance is negligible due to the interconnected nature of its structure. Thus, a constant flow rate can be maintained once the flow reaches the CP. However, the flow resistance and consequently the flow rate can be adjusted by changing the dimensions of other circuit elements, such as chambers, vias, and valves. Programmable fluid flow within a microfluidic channel can be achieved by manipulating its geometric dimensions. This approach is advantageous for applications requiring specific timing or precise flow rates.

In recent years, several studies have expanded upon these foundational concepts by developing innovative designs and models to optimize capillary systems. For instance, Safavieh et al. explored novel capillary pump architectures aimed at improving fluid management within microchannels (Safavieh et al., 2015). Their work highlights the development of two CP designs that mitigate bubble entrapment and achieve consistent flow rates. These designs, known as the serpentine pump and the leading-edge pump, use variable gap sizes between posts to direct the flow and avoid air bubbles, which can disrupt the consistency of fluid metering. The gap between adjacent posts in two subsequent rows (W1) is larger than the gap between two columns (W2). The posts are oval-shaped to function as a temporary stop valve, preventing liquid from leaking into the gap between rows and to avoiding sharp edges that increase the bubble entrapment. The oval shapes of the posts reduce bubble entrapment by eliminating sharp edges. They suggest three strategies to increase the reliability of temporary stop valves between the posts. 1. Increasing the W1/W2 ratio, 2. Decreasing the hydrophilicity of the sealing over the pump structure, and 3. Increasing the capillary pressure in each row to wick the liquid more robustly.

The serpentine CP channels liquid along a winding path, while the leading-edge CP channels it along one edge, allowing the fluid to fill row-by-row. In serpentine CP, the capillary channels follow a serpentine or zigzag shape, which slows down the fluid flow and increases the contact area between the liquid and the channel walls. This design optimizes capillary action by increasing the surface area available for the liquid to “stick” to, thus enhancing the fluid’s movement through the channel.

In the second variant, leading-edge CP, the smaller gap distance of microarrays from one of the edges creates higher capillary pressure and guides the fluid parallel to that specific edge. From the leading edge, the parallel fluid flow moves between the rows and fills the entire pump immediately. Safavieh et al. studied 0–45° angled structures for both CPs and indicated that in 45-degree angled CPs, the liquid’s bypass and consequently air entrapment have the minimum susceptibility. They also demonstrated that a gradient in pressure and flow rate can be generated by adjusting the gap size between microposts, thus achieving precise control over the fluid dynamics within the CP. This deterministic guidance of the filling front enables robust management of flow rate and volume, making these designs particularly suitable for applications requiring high accuracy. Figure 3 demonstrates the serpentine and the leading-edge CP.

Similarly, Tamayol et al. investigated the pressure drop across ordered arrays of cylinders embedded within microchannels (Tamayol et al., 2013). Their research provided valuable insights into how geometrical parameters, such as cylinder diameter and spacing can impact flow resistance. Using both experimental and analytical methods, they developed models that predict flow resistance in the creeping flow regime. Their findings revealed that the pressure drop can be minimized by optimizing the diameter and spacing of micro-cylinders while desired surface-area-to-volume ratios were maintained. Tamayol’s work involved fabricating silicon/glass samples and conducting pressure drop measurements across various nitrogen flow rates. Their results indicated that the porous medium approach offers a broad applicability, while the variable cross-section technique excels in denser cylinder arrays. These findings underscore the importance of channel architecture in determining flow resistance and highlight the potential for optimizing the design of microchannel to achieve specific flow characteristics.

Another interesting approach is the integration of electrical analogies with advanced geometric designs, enabling significant advances in microfluidic systems. By understanding and manipulating flow resistance and capillary pressure, researchers can design programmable CPs with precise control over fluid dynamics. The innovative designs developed by Safavieh and Tamayol establish robust frameworks for enhancing microfluidic applications, paving the way for more efficient and reliable passive microfluidic systems (Safavieh et al., 2015; Tamayol et al., 2013).

A CP is often preceded by a meandered channel, which acts as a flow resistance element. Together, the pump and this resistance allow for precise flow rate control. Assuming constant capillary pressure within the pump, the flow rate becomes a function of the flow resistance. The CP itself is designed to have negligible flow resistance compared to other circuit elements—often two to three orders of magnitude smaller. This design allows the flow rate to be adjusted by modifying the upstream resistance without altering the pump’s structure.

In addition to resistance, the geometry and material properties of the pump structure play a critical role. Porous materials in CPs can be approximated as a collection of cylindrical capillaries. According to the Hagen–Poiseuille equation (Equation 13), the resistance (R) of each capillary is: R = 8 μ L π r 4 (13)

For a porous material, its resistivity ( ρ ) depends on the porosity ( m ), defined as the pore area divided by the total cross-sectional area. This relationship is expressed as Equation 14: ρ = 8 μ π r 4 m (14)

This framework allows for a detailed understanding of how capillary pressure, internal resistivity, and porosity influence the flow rate in porous materials.

An attractive force between the capillary wall and liquid generates capillary pressure ( P c a p ), which drives liquid movement. This pressure can be described by Jurin’s law (Equation 15), which relates the maximum height ( H ) a liquid can rise to the liquid’s density ( D ), gravitational acceleration ( g ), and the material’s wettability (Jurin, 1719): H = 2 γ ⁡ cos ⁡ θ D g (15)

CP capacity, defined as the maximum liquid volume intake, is determined by the pump’s geometry and material properties. For rigid porous materials, capacity is proportional to porosity, while for hydrogels, capacity is characterized by the swelling ratio. Hydrogels, which lack well-defined pore sizes, can achieve high volume efficiencies, sometimes exceeding 99.5%. However, swelling alters their resistivity and suction pressure, complicating flow control compared to rigid porous materials (Ogawa et al., 1993).

Capillary pumps serve diverse purposes in microfluidic circuits, including triggering reactions, pumping fluids, and metering precise fluid volumes. They also function as waste reservoirs in biological assays. The shape, geometric parameters, and wettability characteristics of CPs can be tailored to control the liquid filling front, ensuring consistent flow and minimizing bubble entrapment. For example, covering the CP can prevent contamination and evaporation, though evaporation-based methods may introduce challenges in flow regulation.

When designing CPs, it is important to balance pressure and flow rate depending on the connected device’s resistance. For systems with significant fluidic resistance, small pore sizes are advantageous due to their ability to generate higher pressures. Conversely, in systems with low resistance, larger pores optimize flow rate by reducing internal resistance. This trade-off resembles the maximum power transfer principle in electrical circuits.

Capillary pumps can be categorized into four main types (see Figure 4), each with unique features and applications:

Among the key designs for the internal structure of capillary pumps, the simplest involves a microchannel extended just enough to hold the required fluid volume. Another design involves parallel channels, known as tree line pumps, in which the fluid is drawn through a branching tree-like network of capillary channels (Figure 5A). The liquid enters through the main channel and then is guided through smaller branches. The advantage of such designs is their high surface area and enhanced capillary action due to the multiple, closely spaced smaller channels, which can improve the pumping efficiency. In these pumps, increasing resistance as the filling front progresses results in a decreased flow rate. Tree line pumps can provide regions with different capillary pressures, allowing for multiple flow rates. When one branch fills, it disconnects from the fluid flow, isolating its resistance from other branches (Zimmermann et al., 2007).

Microstructured CPs offer an effective design alternative. These microarrays create parallel interconnected flow paths, enhancing capillary flow with a high surface-to-volume ratio. This design results in a high and stable flow rate in the circuit, minimizing air entrapment and reducing flow resistance (Zimmermann et al., 2007). The structure of microarrays or meshes also decreases the likelihood of clogging and flow stoppages due to their interconnected fluid paths, making them robust designs for CPs. Moreover, they require a reduced total footprint for the pump. Zimmerman et al. investigated various microarray shapes within a CP (Zimmermann et al., 2007). The gaps between posts in a row act as pinning sites, serving as Delay Valves (DV) that impede the progression of the filling front. Rounded micro-post peripheries minimize fluid pinning, enhancing flow performance. The CP with the smallest characteristic dimension generates the largest flow rate in the CS. In their study, the “Symmetric Line” design with minimal dimensions exhibited the highest calculated flow rate, yet also showed the most significant discrepancy between theoretical and experimental results was due to fluid delays at pinning sites. Conversely, the interlocked characteristic of the hexagon design results in a more uniform flow rate because of fewer pinning sites. Based on experimental data, they proposed a correction factor for the flow resistance of the pump, using the formula (Equation 16): b = v 2 + 4 v 2 = v 5 (16) Wherein v is the lateral and forward distance between the arrays.

It should be noted that the ΔP calculated from theoretical equations might show a larger value than the actual one obtained from experiments. This discrepancy arises from the discontinuities in the pump-filling channel caused by gaps between the posts. To ensure sufficient capillary pressure for the circuit, the CP can be connected to the rest of the CS using a gradually expanding configuration. Bubble entrapment, known as corner flow, can occur when the liquid filling front moves faster along one side of the CP and reaches the outlet before completely filling it. This issue can cause inaccuracies in metering and deviations from expected behavior and timing, such as incubation times in bioassays. Additionally, entrapped air unpredictably increases channel resistance, becoming significant when the contact angle (θ) is below 30°. Therefore, the optimal working contact angle for capillary devices is between 30 and 60°. Zimmerman et al. used micro-post CPs with indented side walls to slow down fluid progression along the peripheral edges, minimizing air entrapment (Zimmermann et al., 2007) However, this strategy is not always effective, as the filling front may bypass some rows, entrapping bubbles within the channel. When pumps are connected in series, this problem becomes more pronounced and critical, especially in assays requiring different flow rates at various stages. Zimmerman et al. also designed an “arc-shaped” bridge at the end of each pump to minimize fluid bypass risk. This structure acts as a delay valve, reducing the likelihood of bubble entrapment but weakening the pump’s suction pressure. Increasing the bridge size guides fluid more predictably and aims to prevent incomplete pump filling. However, it can fail when weaker pumps are connected. These hierarchical bridges are more suitable when connecting a slow pump to a fast one. For highly hydrophilic surfaces, adding flow resistance after arc bridges can further prevent fluid bypass by slowing the filling front’s progression. However, adding these elements will increase the device complexity and uncertainty, potentially leading to CP failure.

As discussed in previous sections, delay valves manage the liquid front, reducing bubble entrapment and fluid bypass risks (Makhinia et al., 2023; Dos Ramos et al., 2016; Lai and Chung, 2018; Nie et al., 2020), especially in the connected CPs. This can be achieved by decreasing the capillary pressure in the intersection of two pumps. These valves slow down the meniscus progression in merging channels by increasing junctional channel size. A flow resistance at the end part minimizes the chance of fluid speeding in the channel and its bypass. In one study, authors designed a manufacturable DV, featuring several parallel comb-like protrusions and intermediate channels that facilitate precise control over timing (Li et al., 2017a).

Stop valves feature a restriction followed by an abrupt cross-sectional change, stopping fluid progression (Zimmermann et al., 2008; Barman et al., 2021; Azizian et al., 2023a; Papadimitriou et al., 2018; Glière and Delattre, 2006). A narrower width before the abrupt change and a larger angle deviation in the abrupted channel ensures a more reliable stop valve functionality. Ideally, the enlargement of the cross-section should occur in all directions, but it requires complicated fabrication processes. However, the stop valves with alterations in cross-section in the plane are not robust enough, with creeping possibilities from the bottom and above surfaces. To mitigate the creeping, the sealing can be made of less hydrophilic materials (e.g., PDMS) to prevent the valve’s break in the top surface. The hydrophobic nature of the cover can prevent the corner flow which breaks the valve. Increasing the aspect ratio of the valve stops the leakage from the bottom (i.e., depth/width>10), while it increases the complexities associated with the valve’s microfabrication process. In 2006, Glière and Delattre (2006), using a two-level deep reactive ion etching (DRIE) technology, generated a novel capillary stop valve design that extended the channel section in both width and depth. They determined that the liquid meniscus could be stopped at the valve opening in all situations. Additionally, the resultant burst pressure was high enough to enable accurate regulation and monitoring.

The following expression can be used to predict the behaviour of a stop valve, indicating that if the abrupt enlargement of the conduit curvature is set as β > α, when α = π 2 − θ , then for a two-dimensional valve, the pressure barrier can be defined as Equation 17: ∆ P = 2 γ l a w   cos ⁡ θ − α sin ⁡ α sin ⁡ β cos ⁡ β + sin ⁡ β sin ⁡ α α sin ⁡ α − cos ⁡ α (17) Wherein, w is the width of the meniscus in the valve, α defines the half of the angle inscribed in the meniscus when its shape is approximated as an arc, θ indicates the contact angle of the meniscus with channel walls, β demonstrates the angle between new and old direction of the microchannel wall, and γ_la shows the liquid-air interface surface tension.

The combination of two stop valves, in which the sample in one of the valves triggers the other one in flow junctions (intersection) is called a trigger valve (Safavieh et al., 2015). The delay in the simultaneous arrival of two liquids might lead to bubble entrapment which can cause the blockage of channels, thus controlling the flow timing is necessary. Logical functions can be implemented in the system using TV as well. A liquid reaching the junction is stopped in the entrance due to the geometry of the valve until the other channel is filled with a second liquid, then the leakage between two valves triggers the liquids to progress into a common channel. The geometrical design of the valve can determine its functionality. The timing can be programmed for independent processes or for more than two fluids using a trigger valve to synchronize their movement. An n-to-1 AND (logical function) gate can be designed with the same concept. In 2004, Melin et al. (2004) found that the angle at which two valves meet each other and the spacing between them affects performance. The combination of a trigger valve with a long channel could be turned into an embedded timer within the capillary system. While the fluid was filling the meandered channel, the other filling front was stopped at the trigger valve junction until both streams meet and move forward. Two streams could be triggered with suction in the common outlet channel as well.

Conventional TVs are susceptible to leakage along the bottom or the top wall of the channel, similar to the stop valves. Safavieh proposed the design of a two-level trigger valve to increase the robustness of conventional ones (Figure 9A). A perpendicular flow within the wider channel triggers the fluid and breaks the stop valve. Two-level TV experience expansion both laterally and vertically at the bottom, while the top surface of the valve is covered with hydrophobic PDMS, minimizing the chance of creeping. They compared the proposed two-level TV with conventional ones and claimed that two-level TV with an aspect ratio of 1.5 would maintain the fluid for up to 20 min with no leakage in several experiments while the conventional TV with an aspect ratio of 3 experienced leakages within 2 s (Safavieh and Juncker, 2013).

Combining trigger and stop valves, Li et al. presented a comb-like capillary time-valve to delay fluid in a predetermined manner for bioassays (Li et al., 2017a). The valve contained multiple comb-like protrusions. The filling front stopped temporarily at the edge of comb shape structures and started to progress along the edge until it reached the edge of the chip and triggered the fluid’s longitudinal flow, similar to the working principle of serpentine CP.

CRV and RBV work by reducing cross-sections to increase capillary pressure and preventing downstream drainage. CRVs hold the fluid permanently, while the RBVs act as release valves, and burst the flow at pressures higher than a predetermined threshold. Figure 9B shows a CRV integrated into the capillary system of Juncker et al., which prevented the access channel from drying out, thus inhibiting bubble formation or cross-contamination. The dimensions of constrained parts of CRV and RBV determines their strength to stop the fluid. If the pressure within the valve is higher than the pressure of CP, it would be a CRV, which will drive the fluid to the end. But if the pressure of CP is higher, the RBV will form, which will burst once the pressure in one point drops below the threshold of the valve. The hysteresis between the advancing and receding contact angles necessitates the dimension of CP to be much less than RBV, to ensure the valve’s drainage.

Hitzbleck et al. introduced capillary soft valve (CSV) (Figure 9C), which halts the fluid flow by making an abrupt enlargement in its structure like a stop valve (Hitzbleck et al., 2012). The difference is that this normally closed valve is actuated with regular external pressure on its top layer, which can be simply made of PDMS, a soft enough material for this application. Gently pressing on the cover will increase the capillary pressure and opens the valve, causing wicking of the fluid forward consequently.

In addition to the main capillary-based microfluidic elements introduced above, some studies have introduced ancillary elements or considerations to improve capillary system functionality. As an example, Li et al. presented a wavelike buffer at the inlet port of a capillary-driven device to eliminate the effect of initial velocity and pressure and reduce the probability of bubble generation. The buffer reduces the flow disturbance by changing the longitudinal flow into a transversal one, which renders the injected fluid flow steadily to perform a more accurate test (Li et al., 2017b).