Capillary Wicking on Heliamphora minor-Mimicking Mesoscopic Trichomes Array

Abstract

Liquid spontaneously spreads on rough lyophilic surfaces, and this is driven by capillarity and defined as capillary wicking. Extensive studies on microtextured surfaces have been applied to microfluidics and their corresponding manufacturing. However, the imbibition at mesoscale roughness has seldom been studied due to lacking fabrication techniques. Inspired by the South American pitcher plant Heliamphora minor, which wicks water on its pubescent inside wall for lubrication and drainage, we implemented 3D printing to fabricate a mimetic mesoscopic trichomes array and investigated the high-flux capillary wicking process. Unlike a uniformly thick climbing film on a microtextured surface, the interval filling of millimeter-long and submillimeter-pitched trichomes creates a film of non-uniform thickness. Different from the viscous dissipation that dominated the spreading on microtextured surfaces, we unveiled an inertia-dominated transition regime with mesoscopic wicking dynamics and constructed a scaling law such that the height grows to 2/3 the power of time for various conditions. Finally, we examined the mass transportation inside the non-uniformly thick film, mimicking a plant nutrition supply method, and realized an open system siphon in the film, with the flux saturation condition experimentally determined. This work explores capillary wicking in mesoscopic structures and has potential applications in the design of low-cost high-flux open fluidics.

1. Introduction

Liquid touching superlyophilic rough surfaces spontaneously spreads into the texture, driven by capillarity; this is called capillary wicking [1,2,3,4,5,6,7] and is of significant interest in a broad range of applications such as microfluidics [8,9,10,11], the design of self-cleaning surfaces [12], and lab-on-a-chip technology [13,14,15]. Extensive research on the wetting of microtextured surfaces [1,2,3,4,5,6,10,16,17,18,19,20,21,22,23] demonstrates that liquid behavior on surfaces with microscopic structures is influenced by the combined effects of the surface wetting properties, structure shapes, and liquid properties. In detail, surface energy serves as primary driving force, while viscous dissipation hinders the spreading or imbibition of liquid [24,25], and the surface roughness, defined as the ratio between the actual and projected surface areas, enhances the release of surface energy [1,2,26]. The scaling law known as the Lucas–Washburn equation [27] states that the macroscopic wicking distance increases to 1/2 the power of time due to the balance between surface tension and viscosity. The initial stage is dominated by inertia [28], which follows a power law of 1. Furthermore, a microscopic view of a fringe film extending into the roughness interval was addressed recently, in which the meniscus grew to 1/3 the power when the front was far from the microscale structure [4,29]. However, microtextured surfaces cannot achieve high-flux wicking due to the limitation posed by their structure size and film thickness. To utilize spontaneous capillary wicking in practical applications, it is essential to explore and understand the dynamics of wetting the modified surfaces of large-scale structures.

Inspired by the plant strategy of modifying the surface with millimeter-long trichomes to manipulate their interaction with water for survival, we systematically investigated the capillary wicking on Heliamphora minor, a representative species of South American pitcher plants, and revealed that a lubrication layer spontaneously forms on the pubescent wall to capture insects as well as assist in draining excess water from the pitcher [30,31]. With the speed and resolution improvement of additive manufacturing, the rapid preparation of mesoscale structures is convenient, allowing for the fabrication of a mimetic millimeter-long trichomes array for further study of the wicking dynamics under various experimental conditions. We discovered that the wicking on mesoscopic trichomes forms a non-uniformly thick film which enables high flux flow and explained the profile of this film with respect to the concept of the catenary line. Moreover, we demonstrated that the wicking dynamics on mesoscopic structures differs from that on microscopic structures, with the distance growing to 2/3 the power of time. Additionally, we used the mimetic trichomes array to construct an open siphon and examined the change of flux with height. Through a simulation using salt and sugar, we demonstrated the mass transportation in a thick wicking film, displaying an active method utilized by plants for nutrient supply. Overall, this work provides innovative insight into the design of mesoscopic structures that achieve spontaneous capillary wicking, and advances our understanding of this phenomenon for practical applications requiring high flux.

2. Materials and Methods

2.1. Biological Characterization and the Fabrication of Biomimetic Samples

Heliamphora minor, whose lid is smaller than the pitcher (Figure 1a, images taken by Nikon D750, Toyko, Japan), was cultivated in a greenhouse with the environment maintained at a temperature of 23–28 °C, and a humidity of 90–95%. The sectional view from the middle (Figure 1b) presents the pitcher’s inside wall morphology, with a constriction that splits the wall into two zones; the upper is covered with millimeter-long trichomes and called the pubescent zone, and the lower is the smooth zone. A stereomicroscope (Figure 1c, DSX 1000, Olympus, Toyko, Japan) and scanning electron microscopy at 10 kV (Figure 1d, SEM, SU8020, Hitachi, Toyko, Japan) were utilized for detailed characterization of trichomes on the inside wall (Figure 1e for statistical analysis by Origin 2024). The trichomes are downward-pointing and densely arranged, with a base diameter, d, of 49.3 ± 18.1 μm, and a mutual pitch, p = 176.1 ± 36.9 μm. The trichomes’ length, L, ranges from 153.4 μm to 922.2 μm, with an average of 547.1 μm. In general, the trichomes’ distribution is in mesoscale, which is ubiquitous in nature [32].

Contact angle measurements of the pubescent zone (Figure 1f–h, DCAT 21, DataPhysics, Filderstadt, Germany) indicate that the wettability is superhydrophilic and water spontaneously wicks into the trichomes as soon as it touches the surface. The lightning-fast wicking process was observed and recorded using a high-speed camera at 1000 fps (Figure 1i,j, FASTCAM Mini UX100, Photron, Toyko, Japan); water injected by a syringe from a distribution pump (TS-1A, Longer Pump, Baoding, China) at a rate of 200 μL s⁻¹ spread radially on the pubescent wall and filled the interval at the same time. Therefore, the film thickness increases as the height becomes higher. Finally, the film covers the trichomes and serves as the lubrication layer. Mimicking the pubescent zone, a polymerization-based 3D printer (Mini 8K, Phrozen Tech., Taiwan, China) using an ultraviolet liquid crystal display to initiate the photo reaction of the monomer methyl methacrylate—TiO₂–SiO₂ mixture was used (Figure 1k), and the layer resolution and pixel resolution were 20 μm and 18 μm, respectively. After preparation, the mimetic samples were ultrasonically washed in commercial cleaning solutions (Go Print Co., Ningbo, China) for 5 min, dried by N₂ flow and post-cured under UV light for 3 min. Oxygen plasma treatment (DT-03, OPS Plasma Technology Co., Shenzhen, China) at an RF power of 300 W for 10 min was performed to adjust surface superhydrophilicity. SEM images of the printed mimetic samples sputtered with a thin layer of platinum (EM ACE, Leica, Wetzlar, Germany) at 10 kV were obtained (Figure 1l). The particles of TiO₂ and SiO₂ increase surface roughness and enable chemical modification (Figure 1m for energy-dispersive spectrometer).

2.2. Capillary Wicking, High-Flux Siphon Application, and Mass Transportation

The fabricated biomimetic trichomes array was inserted into a dish filled with blue-dyed water (Figure 2a as a schematic), and the dish diameter was 10 cm, which was large enough to avoid the wicking process disturbing the surface. A colored 4K resolution high-speed camera at 1000 fps (Wave, Freefly, WA, USA) was used to observe and record the wicking dynamics from the side, front, and oblique view. The water fringe front extending on substrates of various trichome geometries was tracked by Tracker software (6.0.10 version), starting from the moment of contact with the water level. Additionally, we utilized micro-computed tomography (Micro-CT, Skyscan 1272, Bruker, German) to reconstruct the wicking film profile and investigate the formed curvature gradient. Furthermore, mimicking the biological function of assisting drainage from pitcher, we modified an n-shaped arc surface of mesoscale trichomes, which was also prepared by 3D printing, and put the higher side of the model in contact with water tank, measuring the mass of water flowing out using an analytical balance (104E, Mettler Toledo, Switzerland). The flux was reported in the form of the measured mass divided by time. Finally, we dissolved CuSO₄, FeCl₃, and glucose, respectively, into water and repeated the wicking process. After long enough time for evaporation, the solute crystallization from the solution process was recorded using a Nikon D750, which reveals the mass transportation inside the wicking film.

3. Results and Discussion

3.1. Capillary Wicking Height and Film Thickness

Once water touched the mesoscopic trichomes array, wicking between the trichomes was observed on the hydrophilic and superhydrophilic substrates (Figure S1) whose contact angles were smaller than the hydrophilic–hydrophobic boundary of 65° [33]. Conversely, water did not climb on the hydrophobic substrate. The front of the advancing meniscus exhibits a parabolic shape (Figure 2b) until it is fully developed at approximately 50 ms. Then, water climbing on the substrate and filling the intervals of the trichomes occur at the same time, which results in a film of non-uniform thickness and only wets the surface of the trichomes at higher heights (Figure 2c).

Similar to the capillary rise in a tube with a small radius (Figure S2), where the equilibrium rising height decreases as the diameter increases, known as Jurin’s law [34], the equilibrium wicking height, H, decreases as the pitch increases (Figure 2d). Notably, the mesoscopic structures of the discontinuous periodical biomimetic trichomes differ from the continuous tube wall. On the former substrate, the meniscus must extend from the first row to reach the second row (Figure 2e, inset) to ensure a continued wicking process. The pitch limitation, p_m, of our fabricated material was determined to be 2200 µm, above which no obvious capillary wicking and interval filling were observed. This limitation is attributed to an insufficient driven pressure in the first interval, while the maximum pitch of two closet trichomes satisfies the condition where the pressure at the advancing meniscus in the first interval, P_M, is balanced by the hydrostatic pressure, P_h, such that

P_M = P_air − σL/p_m² = P_h = P_air − ρgp_m

(1)

where σ is liquid–air surface tension, ρ is the density of the liquid, and g is the acceleration of gravity. The maximum pitch obtained from Equation (1) is p_m = (Ll_c)^1/3, where l_c is (σ/ρg)^1/2 and called the capillary length. For the L = 2 mm used in our experiments, Equation (1) predicted a pitch limitation of 2.4 mm which is consistent with our results.

In addition, the film thickness in single interval changes with time, not monotonically increasing but oscillating around the final thickness. This oscillation starts after the water imbibes into the interval and lasts for O(0.1 s), which is particularly evident in intervals at lower heights, indicating that previously filled intervals act as a source to supply the meniscus extending to the next row and repeating the filling process.

3.2. Microscopic Capillary Wicking Dynamics

An enlarged observation of the capillary wicking process (Figure 3a) shows that the advancing meniscus front (red dotted line) extends over the substrate at the same velocity, but water first wets only the bottom of trichomes. Driven by capillary forces, F_c, along the perimeter of trichomes (Figure 3b), water gradually fills the interval and eventually covers almost the whole surface. The competition between the extension of the advancing meniscus on the substrate and the water spreading on the surface of trichomes contributes to the oscillation of film thickness in single interval that delays the spreading process.

Racing to the nth interval from the bottom to the top, the capillary forces, F_c = σπd_n, where π is the circular constant and d_n is the contact diameter of the trichomes in the nth row, decrease as the water spreads towards the tip of the trichomes due to their smaller d_n. Additionally, the film thickness in the nth interval, e_n, is non-uniform and decreases as the row number, n, increases (Figure 2c). Consequently, the capillary force along the perimeter of the trichomes varies with n and deforms the liquid–air interface in each interval to different extent during the filling process. Because e_n is obviously influenced by the trichome length, L, as longer trichomes support a thicker film, the ratio between e_n and L, denoted as α_n, is utilized as a dimensionless quantity to describe the profile of the film. In the case of trichome lengths of 1 mm, 2 mm, and 3 mm, the variation of α_n with n shows a similar tendency, suggesting a self-similarity of the film profile.

To investigate the relationship between the thickness and the meniscus profile, we employ the catenary line concept for detailed analysis and description (Figure 3d). As shown in the left schematic, we consider an arbitrary arc of a small length, PQ, in the nth interval, and the difference in the z direction between points P and Q is denoted as ∆z. The force balances between the surface tension force, T(z), and gravity, G(z), in the horizontal and vertical directions are

T(z) sin(θ(z)) = T(z + ∆z) sin(θ(z + ∆z)),

(2)

T(z + ∆z) cos(θ(z + ∆z)) − T(z) cos(θ(z)) = G(z) = ρgS_n[φ(z + ∆z) − φ(z)]

(3)

where φ(z) is the path of the profile from point B to P, θ(z) is the tangential angle with respect to the z axis, and S_n~e_np is the cross-sectional area perpendicular to the z axis. Considering an infinitely small step as ∆z approaches 0, and dividing Equations (2) and (3) by ∆z, their differential forms are given as

d(Tsinθ)/dz = 0,

(4)

d(Tcosθ)/dz = ρgS_ndφ/dz

(5)

Equation (4) implies that Tsinθ = F, where F is a constant comparable to the surface tension force along the contact perimeter of the trichomes, which scales as σd_n. Substituting T into Equation (5) as well as the path, φ, by the profile function, f(z), obtains an equation describing the f(z):

−f″/(f′)² = K_n (1 + (f′)²)^1/2

(6)

where f′ and f″ denote the first and second derivatives of the profile, f, and K_n = ρgS_n/F is a constant related to the film thickness, e_n. The boundary conditions are rigorously adjusted by the curvature at the contact points A and B to satisfy the Laplace pressure balance of the hydrostatic pressure.

The general solution of Equation (6) describes the meniscus profile in the nth interval (see Appendix A for the mathematical derivation). To visualize the profile, we plot normalized z coordinates, z* = z/p, against normalized y coordinates, y* = y/L, for different film thickness ratios α (Figure 3e). As the film thickness in the interval decreases, the meniscus becomes flatter and the curvature of the meniscus, κ, induced by the deformation of the profile, f, is

κ = −K_n/(K_nz + C)²

(7)

where C is a negative constant determined by the boundary condition of the profile. As indicated by Equation (7), the curvature-determined Laplace pressure increases from the bottom to the top of each interval (z increases from 0 to p). In other words, a pressure gradient forms inside the film at the nth interval, which propels the meniscus front to extend on substrate to reach the (n + 1)th row of trichomes. As the meniscus advances, the capillary force along the perimeter of the trichomes drives the liquid spreading along the trichomes to fill the (n + 1)th interval and finally renews the pressure gradient, with the coefficient K_n modified to K_n+1.

The repeated behavior of substrate climbing and interval filling results in different meniscus profiles in each interval along the wicking direction (Figure 3f). Cross-sectional views perpendicular to the z axis and in the y–z plane prove that this pressure gradient; that the pressure at the bottom is large, and small at the top. In a single interval (Figure 3g), the release of the trichomes’ surface energy accelerates the flow, as water reaches the base of the trichomes while viscous dissipation slows down the flow, which is consistent with previous work on the wetting of rough surfaces decorated with sparsely arranged microscopic structures [4]. In general, the capillary force acting on the perimeter of a trichome leads to a catenary line-shaped meniscus profile, which induces a pressure gradient, propelling the repeated climbing–filling process that eventually forms a non-uniformly thick film of height H.

3.3. Macroscopic Capillary Wicking Dynamics

Different from the capillary rise in a closed tube [34] of a large diameter that enables a higher flux (larger amount of water raised) but results in a slower speed and lower height, capillary wicking in a mesoscopic trichomes array with different lengths but a fixed pitch and diameter exhibits a similar height change over time, as shown in Figure 4a, which implies that the interval filling process and substrate climbing process are independent. In other words, a faster wicking speed, higher wicking height, and thicker wicking film allow for the achievement of a higher flux at the same time.

In addition, capillary rise is usually classified into three regimes [35,36] (Figure S2): the initial stage is dominated by inertia following a scaling law, H~t; the final stage of thin-film wetting is influenced by viscosity, which follows Tanner’s law [37,38,39], H~t^1/10; and the regime between the two stages is called the inertia–viscosity transition, which is described by the Lucas–Washburn equation as H~t^1/2. However, mesoscopic structures enhance surface energy release and reduce the velocity gradient, which determine the viscous drag. The Reynolds number, Re = ρue_n/μ, is O(10²), indicating that inertia dominates over viscosity, where u is the wicking velocity and μ is the viscosity of the liquid. The Bond number, Bo = ρgHe_n/σ, is O(10⁻²), implying that the influence of gravity is negligible. Therefore, the governing equation for the mesoscopic wicking dynamics of a single interval column takes the following form:

d(mu)/dt = udm/dt + mdu/dt = F = σ*p

(8)

where σ* is the equivalent surface tension considering the surface roughness [1,4] and m is the mass of the wicking liquid, which can be expressed as kρHpe_n, where k is a geometric factor and u = H′ is the first derivative of H. Considering that the thickness relates to the wicking height as e_n~βH (Figure 3c), m is expressed as k′ρH²p, where k′ = kβ and a different scaling law can be obtained from Equation (8) (see Appendix A for detail):

$$H~{\left(\frac{3{\sigma }^{*}}{2{\mathrm{k}}^{\prime }\rho }t\right)}^{2/3}$$

(9)

which describes the transition regime dynamics of mesoscopic wicking and is in accord with the experimental results.

Furthermore, we confirm the general applicability of the new scaling law regarding the transition regime of capillary wicking on a mesoscopic trichomes array. The height change over time for different liquids wetting our fabricated materials is reported in Figure 4c, which reveals a similar trend, but the low surface tension of viscous liquids such as silicone oil exhibits a slower climbing rate compared to high-surface-tension liquids with a low viscosity, such as water. Dimensionless height, H/l_c, with respect to normalized time, σt/μ, was used to assess deviations from the expected tendency of different liquids (Figure 4d). The results indicate that all liquids follow the scaling law at the beginning, but deviate from their expected tendencies in the late stage because the viscosity-negligible condition that Re is large does not hold for thin films of viscous liquids such as silicone oil and ethylene glycol in our experiments.

3.4. High-Flux Open Siphon Applications and Mass Transportation

Subsequently, we curved the straight plate into “n” shape (Figure 5a), so that the film on the trichomes array acts as a water path connecting two levels and facilitating water transfer from the higher level to the lower level, functioning as an open siphon device [40,41,42]. Compared to the “n” shape device without a trichomes array modification (Figure 5b), water spontaneously wicks between the trichomes once it touches the modified region and follows the trichomes’ path, flowing into the other side of the device. As water accumulates in the outlet side, the meniscus rises to touch the modified region, enabling continuous water transfer until the meniscus levels on the inlet side and outlet side are equal. Similar to a traditional siphon in a closed tube, the height difference between the inlet and outlet, h_out − h_in, where h_in is the height difference from the inlet tank’s bottom to irs ridge top and h_out is the height difference from the outlet tank’s bottom to its ridge top, influences the siphon flux [43] (Figure 5c).

However, as reported in previous work, the flux of an open siphon exhibits a maximum flat stage [42], such that increases in the two-sided height difference have little influence on the flux. Instead, the maximum flux is influenced by the inlet height difference, h_in, which determines the driving pressure difference across the free surface. The flux of our trichomes array achieved an open siphon, which is also strongly influenced by the h_in, which the maximum flux reduces to half when the h_in increases from 1.00 cm to 1.25 cm. Except for the maximum flux, flux under small two-sided height differences is enhanced by capillarity if the h_in is small, which is attributed to a larger Laplace pressure difference across the meniscus for a small h_in. Siphon applications that require high flux such as irrigation systems and drainage systems can be enhanced by adding a trichomes array with an optimized height difference on their surface.

The application of water wicking on a trichomes array extends beyond liquid transfer, which can be utilized for mass transportation as well. Various solutions carrying mineral salts or nutrients, such as glucose, wick on the biomimetic trichomes array and form a non-uniformly thick film (Figure 5d). The thinner film at higher levels evaporates first, causing solute crystallization to cover the surface. Over time, as the film gets thinner, solutes from the resources are transported to the surface at high levels (Figure 5e). In this way, a pitcher plant like H. minor can carry nutrients such as sugar and digestive liquid from the glandular organ at the bottom of pitcher to the upper pubescent zone, which attracts insects after crystallization. This process requires further investigation of the biological subject [31].

4. Conclusions

Inspired by the capillary wicking on the pubescent inside wall of Heliamphora minor, we fabricated a biomimetic array of millimeter-long and sub-millimeter-pitched trichomes through LCD-based 3D printing, with a high resolution of up to 18 μm, to investigate mesoscopic wicking dynamics. Our results demonstrate that the water wicking on a trichomes array consists of two independent processes: the filling intervals of the trichomes and the meniscus advancing on the substrate. The filling of the trichomes’ intervals is driven by capillary forces along the perimeter of the trichomes, resulting in the formation of a film with non-uniform thickness, and the film profile at each interval is described by a catenary line. Additionally, we unveiled that the transition regime of wicking on a mesoscopic-trichomes-modified surface is dominated by inertia. The dynamics follows a scaling law of H~t^2/3, which is distinct from the H~t^1/2 observed in traditional capillary rising or wicking on surfaces modified with microscopic structures. Because the thickness and wicking dynamics are independent, high flux and fast speed can be achieved at the same time, which can improve the flow efficiency and has potential in applications such as high-flux siphons. Finally, we examined the mass transportation inside the film and observed thin-film evaporation, resulting in solute crystallization at higher levels first, which indicates a method utilized by the plant to transport nutrients in the digestive fluid from the pitcher to the upper pubescent zone, assisting in the attraction of insects. This work, exploring the capillary wicking phenomena on mesoscopic structures, advances our understanding of high-flux wetting behaviors and improves the design of efficient liquid transfer devices.