The xylem-inspired channel model comprises a straight channel embedded with a porous mesh structure (Figure 1B). This channel model is an open circular channel fabricated using a Tygon tube with an outer diameter of d = 0.7 mm as a channel mold (Mannino et al., 2015). The tube was tightly fixed with tension to form a straight line at the middle height of a Petri dish. Poly-dimethylsiloxane (PDMS) was partially cured using a 10:1 mixture of Sylgard-184 (Dow Corning) in the dish for 80 min at 50°C. The Tygon tube was then pulled out from PDMS. A piece of nylon mesh (Nylon mesh 100, APEC Industrial, Ltd.) was inserted between two excised cross sections of the circular PDMS channel. The fiber diameter and average opening width of the porous mesh were 100 and 150 μm, respectively. The contact angles of various kinds of Nylon were measured in the range from 74° to 94° (Fort, 1964). The wettability of Nylon reflects hydrophobic property of lignin that constructs various kinds of supporting tissues of xylem vessels (McCully et al., 2014). The excised cross-sectional channel surfaces were firmly adhered onto the mesh, and the channel model was completely cured for more than 2 h at 50°C. Based on the diameter of xylem vessels (Olson and Rosell, 2013; Rosell et al., 2017), the diameter of the xylem-inspired channel was scaled up by a factor of 10⁰ to 10².

The water flow in the channel was supplied using a syringe pump (neMESYS 290N, Cetoni GmbH) with varying flow velocity v from 0.2 to 0.8 mm/min. Rhodamine-B solution (Acros Organics) was diluted in distilled water to a concentration of 50 mg/l and used as working fluid for observing the movement of the air–water meniscus. The ratio of inertia force to viscous force, prescribed by the Reynolds number Re = ρvd/μ, was adjusted to the order of 10⁻¹ in consideration of sap flow in xylem vessels (Jensen et al., 2016). Herein, ρ and μ denote the density and viscosity of the working fluid. Capillary number Ca = μv/γ, the ratio of viscous force to surface tension, was additionally modulated to be comparable with that of real sap flow in the range of 10⁻⁵ (Jensen et al., 2016). Here, γ is the surface tension of water. The syringe contained 0.05 ml of air to regulate the internal pressure gradually.

During water-filling in the channel model, the movement of each air–water meniscus was sequentially captured by a high-speed camera (pco 1200 hs, PCO) at a frame rate of 10 fps. The position of the meniscus was determined by reaching the maximum location in the pixel intensity gradient. The temporal variation of the moving velocity of the meniscus was analyzed using ImageJ software (Rueden et al., 2017). The pressure in front of the mesh structure was measured with a pressure transducer (PX409-015GUSB, Omega engineering). The penetration process of the air–water meniscus through the mesh was visualized by synchrotron X-ray microscopy at the 6C Biomedical Imaging Beamline at Pohang Accelerator Laboratory (PAL, Pohang, Korea). The X-ray beam energy was modulated to 24 keV, and the sample-to-detector distance was fixed at 50 mm. X-ray images were consecutively captured at a frame rate of 10 fps by using a Zyla camera (Andor Zyla, Ireland) with a 10 × objective lens. The field of view of the images was 1.7 × 1.4 mm in size, and the corresponding spatial resolution was 0.65 μm per pixel. The xylem-inspired channel was mounted on a translation stage for vertical and horizontal positioning. The flow rate was remotely controlled to 4.62 μl/min by using a syringe pump (PHD 2000 and PHD Ultra, Harvard Apparatus). The contact line speed and contact angle of the air–water meniscus in contact with the mesh structure were estimated from the captured consecutive X-ray images.

The water-filling process in the bioinspired channel embedding a porous membrane structure was investigated by matching its characteristic numbers, Re and Ca, to real sap flow in xylem vessels of vascular plants (Figure 2A). We monitored the movement of the water plug which appeared as a dark region in the time-sequential images. It moved from the left side to the right side of the channel. The In the initial 20 s (−20 < t < 0 s), the water plug moved at a constant speed, as shown by the images obtained every 10 s. The spacing between vertical dotted lines was constant. At t = 0 s, the meniscus reached the porous mesh embedded at the middle of the channel. We defined the time at which the meniscus touched the mesh as t₀. The water meniscus stopped at the mesh for 16 s and then restarted toward the right side. The flow was accelerated immediately after passing through the membrane from t = 16 s to t = 19 s. The time of meniscus acceleration was defined as t_cr, and the time duration of meniscus stoppage was expressed as Δt_s. This flow passing through the mesh structure exhibited a stop-and-acceleration pattern similar to the water-refilling phenomenon at perforation plates in the embolized xylem vessels (Jeje and Zimmermann, 1979; Lee and Kim, 2008; Kim and Lee, 2010; Lee et al., 2013; Hwang et al., 2016).

Positional variations of the meniscus during the water-filling process in the xylem-inspired channel were analyzed with varying initial flow velocity v₀ (Figure 2B). The positional information of the meniscus and the time are expressed as dimensionless parameters L = l/l₀ and T = t/t^*, where l is the length of the water column inside the channel, l_o is the channel length, and t^* is the characteristic time l₀/v₀, respectively. The normalized positional variation of water, when v₀ is 0.2 mm/s, clearly exhibit the stop-and-acceleration flow pattern at the porous mesh. The stop-and-acceleration flow pattern can be classified into three stages on the basis of the characteristic times t₀ and t_cr. In the first stage (stage I; during t < t₀), the meniscus approached the mesh located at 0.5l₀ with a constant speed v₀. This result indicates water flow in a simple straight channel without any porous structure. In the next stage (stage II; t₀ ≤ t ≤ t_cr), the meniscus stopped at the mesh during Δt_s, after the meniscus came in contact with the mesh (t = t₀). As illustrated in Figure 2C, the stoppage time Δt_s was inversely proportional to v₀. The solid line is the linear fitting line with a slope of 0.282, and the corresponding R-squared value is 0.997. In stage III (t > t_cr), the meniscus completely passed the mesh and accelerated toward the right side. The maximum instantaneous velocity v_max was nearly invariant regardless of v₀ (Figure 2D). The accelerated water meniscus recovered the original velocity v₀ at the end of stage III. The flow pattern returned to stage I, where water-filling was negligibly affected by the presence of a porous mesh.

The temporal variations in pressure P at the front side of the mesh were also measured during the water-filling process in the xylem-inspired channel. The three different stages were also clearly discerned in the graph of normalized P vs. t (Figure 3). The normalized pressure values were obtained by dividing P by the P_avg which is defined as the average of maximum critical pressures of each flow rate cases, and t by t^*, respectively. As the meniscus approached the mesh at t < t₀, the pressure was remained close to the atmospheric pressure (stage I). When the meniscus reached the mesh at t = t₀, the pressure began to increase linearly, and the increasing rate was proportional to v₀ in stage II (t₀ ≤ t ≤ t_cr). After reaching maximum pressures ranged from 0.75 to 1.25 P_avg, the maximum pressure was released to the value near the atmospheric pressure. At this point, the meniscus completely passed through the mesh (stage III; t > t_cr). The pressure was returned close to the atmospheric pressure, which corresponded to stage I.

The dynamic behavior of the water meniscus through the mesh structure was visualized by X-ray imaging technique (Figure 4, Supplementary Video S1). Phase-contrast X-ray imaging technique enables the clear visualization of the interfaces between different phases inside the opaque porous structures. The method also allowed the analysis of the variations in the contact line and meniscus curvature (Figure 4, Supplementary Video S1) and contact angles of the menisci (Supplementary Information S1). The captured consecutive X-ray images show the dynamic motion of the water plug from the left side to the right side. Each arrow indicates the vector of flow velocity v. Before the water plug reached the porous mesh (stage I; t < t₀), the water meniscus moved with a constant velocity v₀ toward the mesh located at the middle of each image. After the meniscus reached contact with the mesh at t = t₀(= 0 s), the meniscus split into several smaller menisci to penetrate the mesh pores. When the water plug was anchored to the mesh (stage II; t₀ ≤ t ≤ t_cr), the menisci penetrated the mesh fibers with a lower velocity v compared with the initial velocity v₀. Meanwhile, the curvatures of the convex-shaped menisci were increased until the end of stage II (t = t_cr = 4.5 s). Immediately before the start of acceleration (t = t_cr), the contact angle between the water meniscus and the mesh fiber surface was measured to be 109.8° on average (Supplementary Information S1). Beyond the critical time t_cr, the menisci burst through the porous mesh. The menisci then instantaneously merged into a meniscus, which accelerated beyond v₀ toward the right side of the field of view for a short interval of 0.5 s (stage III; t > t_cr).

The penetration process of the water meniscus through the mesh pores in stage II is simplified in Figure 5A. To represent the pressure increase caused by meniscus advancement, we defined an angle α between the horizontal line and the tangent line of the meniscus curve at the mesh fiber. The shapes of the meniscus at t = t₀ and t = t_cr are indicated by a black dotted line and a solid line, respectively. The angle α increases, as the contact line of the meniscus curve moves upward along the mesh fiber surface in stage II and the pressure difference across the interface increases according to the formula expressed as follows:

where θ_ad is the contact angle of the water meniscus advancing along the mesh fiber surface. The distance between the centers of two mesh fibers, and the radius of the mesh fiber are denoted as 2R and a, respectively. The average moving velocity of the contact line was measured to be 15 μm/s from the captured consecutive X-ray images. Since Ca is in the order of 10⁻⁸, the effect of contact angle variation was negligible (Berthier, 2008). Thus, the advancing contact angle θ_ad was assumed to be constant as the measured value of 109.8 ± 7.3° (Supplementary Information S1). In the above-mentioned equation, the effective radius of the mesh pore R_eff = ξR−ζacosα was introduced to compensate for the shape effects of the actual mesh. The shape factor ξ reflects the rectangular shape and pore size irregularity which related to the pore size R and ζ modifies the woven structure of the mesh and squashing of mesh fiber that affect a. According to Equation (1), the pressure difference ΔP_L varies with α as shown in Figure 5D. It increases up to the maximum value of ΔP_{L, cr} as α increases to the critical angle α_cr at t = t_cr (Figure 5B). When s is maximized as ΔP_{L, cr}, the derivative of Equation (1) with respect to α becomes zero at t = t_cr as follows:

According to Equations (1) and (3), theoretical value of ΔP_{L, cr} and α_cr were thoroughly matched with the experimental result, 346.7 Pa and 35.7°, respectively, when ξ is about 5.4 and ζ is 8.1. The experimental value of α_cr was measured on the basis of the captured X-ray images. Detailed derivation of Equations (1–3) is provided in the Supplementary Information S2.

In the proposed model, the critical Laplace pressure ΔP_{L, cr}, at which the water begins to pass through the porous mesh does not depend on v₀. This result supports that the maximum pressure remained within a certain range, regardless of the initial flow velocity (Figure 3). Because the ΔP_{L, cr} is not a function of v₀, the case of water-refilling at a faster v₀ reaches the critical state ΔP_{L, cr}, and α_cr faster. This patterns implies that the pressure increasing rate is proportional to v₀, and Δt_s is inversely proportional to v₀ (Figure 2C). Moreover, under the proposed model, the critical Laplace pressure ΔP_{L, cr} increases with decreasing pore size and wettability of the porous mesh.

In stage III, the menisci that formed at each pore merged into a meniscus, which was then accelerated forward (Figure 5C). The flow acceleration at this stage can be mathematically described by balancing the inertial, capillary, viscous pressures, and the pressure drop caused by the hydraulic resistance of the mesh structure (R_m). At the instant of flow acceleration, the position of the water meniscus l(t) is expressed as

where r is the radius of the circular channel. When a water meniscus passes through the porous mesh, the pressure in front of the mesh increases up to the maximum Laplace pressure ΔP_{L, cr}. As the pressure increases to the maximum value of ΔP_{L, cr}, flow acceleration is driven by ΔP_{L, cr} and the pressure is instantaneously released to the atmospheric pressure. By contrast, the viscous force and pressure drop caused by hydraulic resistance of the mesh suppress the flow acceleration as resisting force. Through the dimensionless parameters L = l/l₀, T = t/t^*, and β = l_o/r, Equation (4) is transformed into

where ΔPm=πr2voRm denotes the pressure drop caused by R_m. The values of Re for the given initial flow velocities are in the range of 0.07–0.28. The average ΔP_{L, cr} was experimentally obtained to be 346.7 Pa, β and πr2Rm are 57.1 and 4335.5 Pa s/m, respectively.

At the initial phase of stage III (t = t_cr), the water meniscus burst from the mesh pores. Flow was accelerated by the accumulated maximum pressure difference ΔP_{L, cr} formed at the meniscus. Then, the flow decelerated to the initial velocity v₀ because of viscous dissipation and the hydraulic resistance R_m. Since the magnitudes of the terms representing ΔP_{L, cr}, the viscous force and the hydraulic resistance of the mesh exceed 10³, the other term is negligible. In addition, the derivative dL/dT is assumed to be V_max = v_max/v₀ at the initial stage of the flow acceleration. In this regard, Equation (5) can be simplified as

where L₀ is the dimensionless length of the water plug during the acceleration. From Equation (6), the relationship of Vmax~v0-1 is obtained. This relation supports that v_max is constant, regardless of v_o (Figure 2D). More detailed derivation of Equations (4–6) is provided in the Supplementary Information S3.

Water-refilling flows that pass through perforation plates in embolized xylem vessels have been reported to exhibit the stop-and-acceleration pattern. However, the hydraulic functions of perforation plates have not been thoroughly examined with the aid of theoretical support. The present experimental and theoretical investigations are thus helpful for understanding the hydraulic roles in terms of water-refilling and pressure regulation inside embolized xylem vessels. The refilling water plug begins to advance into pores of the mesh structure and appeared to cease moving at the mesh. However, in our real-time microscopic view, the menisci formed between the mesh fibers advanced continuously and immediately accelerated when the contact angle reached the critical value. The geometrical features of the mesh structure determined the critical pressure operating as a resisting force against water-filling flow. The pressure in the compartment filled with water was gradually increased up to the threshold value. The critical pressure difference also served as a driving force to instantaneously accelerate the water-filling flow.

These results imply that perforation plates in xylem vessels may function as hydraulic valves that raise the hydrostatic pressure inside the vessel compartment during water-refilling in embolized xylem vessels (Lee and Kim, 2008; Hwang et al., 2014). In this process, the refilling vessel could be pressurized when the meniscus passes the perforation plates. The pressure increase at the porous structures may help dissolve gas bubbles into the water inside the embolized xylem vessels (Tyree and Yang, 1992; Schenk et al., 2016). Scholars conjectured that the porous structures in xylem vessels briefly hinder axial water-refilling and then simultaneously increase the local pressure to enable radial water-refilling flow (Hwang et al., 2016).

Through the xylem-inspired channel composed of a channel embedded with a porous structure, the characteristics of water-refilling flow around a porous mesh were systematically examined. Our experimental and theoretical analyses were limited to a regularly shaped porous structure with large-scale pores and the pressure inside the xylem-inspired channel has a large difference with the pressure inside xylem vessels. However, this study can still contribute to the understanding of the hydraulic functions of perforation plates for embolism repair. The proposed xylem-inspired channel model can be also applied to validate many hypotheses on embolism repair and plant hydrodynamics.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.