Capillary Rise in a Microchannel of Arbitrary Shape and Wettability

Nov 21, 2012 - Capillary rise in an asymmetric microchannel, in which both contact angle ... The higher height dominates in the short channel regime, ...
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Capillary Rise in a Microchannel of Arbitrary Shape and Wettability: Hysteresis Loop Zhengjia Wang,† Cheng-Chung Chang,‡ Siang-Jie Hong,† Yu-Jane Sheng,*,‡ and Heng-Kwong Tsao*,† †

Department of Chemical and Materials Engineering, National Central University, Jhongli, Taiwan 320, R.O.C. Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 106, R.O.C.

‡

ABSTRACT: Capillary rise in an asymmetric microchannel, in which both contact angle (wettability) and open angle (geometry) can vary with position, is investigated based on free-energy minimization. The integration of the Young−Laplace equation yields the general force balance between surface tension and gravity. The former is surface tension times the integration of cos θu along the contact line, where θu depicts the local difference between contact angle and open angle. The latter comes from the total volume right underneath the meniscus. For the same channel height, multiple solutions can be obtained from the force balance. However, the stable height of capillary rise must satisfy stability analysis. Several interesting cases have been studied, including short capillary, truncated cone, hyperboloid, and two different plates. As the tube length is smaller than Jurin’s height, the angle of contact will be tuned to fulfill the force balance. While only one stable state is seen for divergent channels, two stable states can be observed for convergent channels. Three regimes can be identified for the plot of the stable height of capillary rise against the channel height. The higher height dominates in the short channel regime, while the lower height prevails in the tall channel regime. However, both solutions are stable in the intermediate regime. Surface Evolver simulations and experiments are performed to validate our theoretical predictions. Our results offer some implications for water transport to the tops of tall trees. A small bore at the uppermost leaf connected to a larger xylem conduit corresponds to a convergent channel, and two stable heights are possible. The slow growth of the tree can be regarded as a gradual rise of the convergent channel. Consequently, the stable height of capillary rise to the top of a tall tree can always be achieved.

I. INTRODUCTION Capillarity involving wetting of solid surfaces by liquid is ubiquitous in various applications and phenomena such as microfluidics,1−7 oil recovery,8 wicking fabrics,9,10 and liquid transport in porous media,11 to name a few. When a long narrow tube is plunged into wetting liquid, some of the liquid rises up to a final height due to the balance between wetting and gravity. Capillary rise has many important consequences, including water retention in soil and water transport in plants. Liquid rises in cylindrical capillaries have been successfully described by the Jurin’s law (1718), and the height of capillary rise, the so-called Jurin’s height, is inversely proportional to the tube radius.2 Through introducing Young’s relation (1805), the height is related to the wettability of the channel wall in terms of the intrinsic contact angle, (θ*).12 It is related to interfacial tensions by cos θ* = (γsg − γsl)/γlg, where γsg, γsl, and γlg denote the interfacial tension of solid/gas, solid/liquid, and liquid/gas, respectively; however, Jurin’s law requires corrections of meniscus shape when the height variation of the meniscus is no longer small compared to the Jurin’s length. Liquid rise against gravity in capillaries with heightdependent cross sections has been investigated based on the Young−Laplace equation.13 Through the study of a cone-shape micropipet dipped in a liquid reservoir, the phase diagram of meniscus behavior in terms of the cone-opening angle, (α), against the intrinsic contact angle, (θ*), is obtained. At a fixed © 2012 American Chemical Society

open angle, the variation of the meniscus location with the contact angle shows a discontinuity across the critical contact angle, θ*c . That is, as the contact angle of a solvophilic cone is decreased, the meniscus height grows continuously until reaching θc* and it jumps to the top of the cone for all θ < θc*. However, as the open angle is less than a certain value, |α| < α*, the change of the meniscus height against the contact angle is always continuous. A similar discontinuous behavior associated with a jump down to the bottom of an inverted cone takes place for solvophobic walls. In addition, the multiple solutions are predicted for a capillary with periodic width modulations but not for a simple cone. The law of capillary rise can also be obtained from the minimization of the free energy, which consists of the gravitational potential energy and terms associated with interfacial energy.2,14,15 In general, the first derivative of the free energy leads to a result similar to that simplified from the Young−Laplace equation. For microchannels with uniform cross sections such as vertical cylindrical tubes, only solid− liquid interfacial energy needs to be accounted for because the meniscus does not change with the vertical position. Nonetheless, for capillary rise in a conduit of a nonuniform cross Received: September 9, 2012 Revised: November 21, 2012 Published: November 21, 2012 16917

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section, the term associated with the liquid−gas interfacial energy arises in the minimization of free energy. Where the tube narrows or widens, the liquid−gas area and the surface free energy of the meniscus are both reduced or elevated. Recently, capillary rise in a cone has been employed as an illustration of a first-order phase transition and a spinodal point.16 Starting with the assumption that the meniscus is spherical, the free energy is written down as a fourth degree polynomial in the height. Three solutions are obtained from free-energy minimization, including one completely filled cone and two solutions of partially filled cone. The stability analyses (the second derivative of the free energy) show that for solvophilic cones the solution of a filled cone is always stable, while the solution of a partially filled cone with a lower meniscus is only stable as the height of the cone apex exceeds the spinodal. In other words, when the position of the cone is above the spinodal point, the free energy has two minima, indicating the existence of multiple stable heights. The experiment of silicone oil in a PDMS cone has been conducted to demonstrate the presence of hysteresis. That is, there exist two stable heights of capillary rise for the same cone height, dependent on the cone descension or ascension. However, since the tip of the cone is truncated, the “filled cone” state is not realistic. Despite the fact that the liquid rise in capillaries with heightdependent cross sections have been investigated for a long period of time, a general expression of capillary rise in a conduit with arbitrary shape, like Jurin’s law for cylindrical capillaries, is not available. In addition, the multiple stable heights and hysteresis loop for convergent and divergent microchannels, such as a truncated cone or an inverted cone, are not thoroughly explored. The convergent channel is referred to as the conduit with its apex positioned away from the bulk liquid. A well-known example of capillary rise in a convergent channel is water transport from the xylem tube to the uppermost leaf in plants. Moreover, in this case, the rise of water reaches the top of the conduit which can be inclined and asymmetric, as shown in Figure 1. If a tall tree is modeled as a cylindrical capillary, water transport to the treetop implies that the channel height

always miraculously equals the Jurin’s height. What happens if the channel height is less than the Jurin’s height? In that case, does the Jurin’s law still apply? In this paper, the aforementioned problems are explored theoretically. Surface Evolver (SE) simulations and experiments are also performed to examine our theory. In Capillary Rise in a Microchannel of Arbitrary Shape and Wettability, on the basis of free-energy minimization, we shall derive a general force balance for the capillary rise in a microchannel with arbitrary shape and wettability. The experiments and SE simulations are briefly described in Experimental and Simulation Methods. A few interesting cases are studied in Short Capillary, Truncated Cone, Hyperboloid, and Two Different Plates. Finally, the implication for water transport to the top of a tall tree is offered in Implication for Water Transport to the Tops of Tall Trees.

II. CAPILLARY RISE IN A MICROCHANNEL OF ARBITRARY SHAPE AND WETTABILITY For a microchannel of arbitrary shape and wettability, the free energy of the system consists of the interfacial free energy and gravitational energy

∬A

F(h) =

+

γlg 1 + (∇z u)2 dA + p

∬A

(γsl − γsg) dA sl

∫ ∫ ∫V ρgz dV

(1)

l

The height of capillary rise, h, is defined as the minimum height of the meniscus, and the stable height can be acquired from the free-energy minimization. The first term on the right-hand side of eq 1 represents the liquid−gas contribution. zu denotes the meniscus profile, and it is defined with respect to the level of bulk liquid as shown in Figure 1. As the variation of meniscus height is small compared to h, one has zu(x,y) ≈ h. The domain, Ap, is the projection of the meniscus to the horizontal plane. The second term represents the solid−liquid contribution. The term, (γsl − γsg), which is equal to the term, −γlg cos θ*, can vary with position, and Asl denotes the area of the solid−liquid interface. Here, θ* denotes the intrinsic contact angle. The gravitational potential energy can be separated into two terms, including the contribution involving the meniscus profile and the remainder involving the channel boundary.

∫ ∫ ∫V

l

ρgz dV =

∬A

p

ρgz u2 dA + 2

∫ ∫ ∫V −V l

ρgz dV

eff

where the effective volume is Veff = ∫ ∫ Apzu dA. The meniscus profile zu(x,y) corresponding to the freeenergy minimum can be determined by the calculus of variations, δF/δzu = 0. As a result, the meniscus must satisfy the Euler−Lagrange equation15 ⎡ ∇zu ρgz u = γlg∇·⎢ ⎢ 1 + (∇z )2 ⎣ u

⎤ ⎥ ⎥ ⎦

on A p (2)

where ▽·p = [(∂px)/(∂x)] + [(∂py)/(∂y)]. Equation 2 is also a consequence of the Young−Laplace equation, which describes the pressure difference across the interface between two static fluids. In order to decide the meniscus shape, the angle of contact at which the meniscus meets the solid boundary is required. It is expressed by θ(Ωp), where Ωp is the projection of the triple contact line lying on the channel wall to the horizontal plane

Figure 1. Illustration of capillary rise in a microchannel. In the vicinity of the meniscus, the opening angle, (α), with respect to the channel wall and the contact angle, (θ), are shown. The general force balance between capillary force (γlg cos θu) and effective rising volume is indicated. 16918

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condition θ = θ* must be satisfied for the movable contact line. This phenomenon is attributed to the edge effect, which corresponds to the boundary minimum instead of the local minimum.17,18

and depicts the boundary of Ap. In terms of the meniscus profile, zu, the boundary condition becomes ∇z u·n p = cot θu ,

along Ω p

(3)

where np represents the exterior unit normal of Ωp. θu is the angle between the direction of gravity and the liquid surface (meniscus) near the contact line measured within the fluid, as illustrated in Figure 1. It is related to the contact angle, θ, and opening angle, α, by θu = θ − α. Here, α denotes the opening angle which can vary with the position and is the angle between the direction of gravity and the solid surface (wall) near the contact line. Note that α > 0 for the convergent channel and α < 0 for the divergent case. Under the axisymmetric condition, zu(r), the boundary condition is simplified to ∂zu/∂r = cot θu. By integrating eq 2 and using Green’s theorem and boundary condition, one obtains the force equation ρg

∬A

z u dA − γlg p

∮Ω

cos θu dl = 0 p

III. EXPERIMENTAL AND SIMULATION METHODS A. Materials and Experimental Methods. Two experiments are conducted: capillary rise in a short capillary and a truncated cone. The capillary tube with an inner diameter of 0.87 mm (Kimble Chase, Gerresheimer) is made of borosilicate glass. The plastic pipet tip (200 μL, yellow) is used as a truncated cone. It is made from polypropylene and purchased from Kwo-Yi Company. The inner diameters of the bottom and top orifices are 5.2 and 0.53 mm, respectively. They can be used to determine the opening angle of the cone, α is ∼2.6°. Both capillary tube and pipet tip are transparent, and the meniscus height can clearly be observed. Before conducting the experiments, the tube and pipet tip are rinsed with alcohol, cleaned by sonication with deionized water in an ultrasonic cleaning tank for 2 min, and then dried in a stream of nitrogen gas. The density and surface tension of isopropanol (99.5%, TEDIA COMPANY, INC.) purchased from ECHO Chemical Company Ltd. are 785 kg/m3 and 0.0212 N/m, respectively. The capillary length, lc = (γlg/ρg)1/2, is calculated to be lc = 1.66 mm. The capillary tube or plastic pipet tip is mounted with a linear stage (xyz) which allows objects to move vertically. The experiment is conducted by slowly rotating the knob to descend (ascend) the tube or pipet tip into (off) the liquid isopropanol contained in the serum glass vial. During the descending/ascending processes, the side view of the meniscus in a capillary tube or pipet tip is recorded by the lens of Optem 125C and an enlarged image of the height variation across the meniscus is obtained.19,20 Each image is captured after the knob rotation is stopped for two seconds to ensure that the meniscus is in local minimum. A high-resolution digital camera is also mounted and aimed perpendicular to the microchannel to acquire the height of the cone (H) and the meniscus’ bottom (h) simultaneously. It takes about five minutes to complete each descending/ascending experiment. All data are analyzed by MultiCam Easy 2007 (Shengtek). When a cylindrical capillary tube of radius, r, is used, there will be a degree of image distortion present that complicates the measurement problem.21 Since the diameter of the tube is smaller than the capillary length associated with isopropanol, the contact angle can be determined by the assumption of a spherical meniscus, θ = 90° − 2 tan−1(Δh/r), where Δh denotes the distance between the bottom of the meniscus to the contact line. B. Surface Evolver Simulation. Numerical simulations by public domain finite element SE package developed by Brakke are performed to examine our analytical results.22 The basic concept of SE is to minimize the energy of a surface subject to various forces and constraints. The minimization is achieved by evolving the surface down the energy gradient. SE models surfaces as unions of triangles with vertices that are iteratively moved from an initial tentative shape until a minimum energy configuration is obtained. It has been applied in the studies of various wetting phenomena.23,24 In our simulation, the system consisting of a microchannel immersed in a container is represented by 2 809 vertices, 8 088 edges, and 5 280 facets. Two constraints are used: the vertices of the contact line must lie on the surface of the channel, and the total liquid volume in the system is kept identical during iteration. Since the size of

(4)

Note that for a cylindrical tube, eq 4 can be obtained directly from dF/dh. This general expression of capillary rise indicates the balance between surface tension and gravity. The surface tension force is surface tension times the integration of cos θu along the contact line, where θu depicts the local difference between the angle of contact and open angle (see Figure 1). It can be regarded as the vertical component of the surface tension force. The effective gravitational force comes from the total volume right beneath the meniscus, as shown in Figure 1. Equation 4 can be directly applied to a typical example, a capillary with periodic width modulations, r(z) = r0 + rm sin(qz), where r0 and rm represent the mean capillary width and the modulation amplitude, and q is its wavenumber. Then the force balance is simply given by ρg(πr2h) ≅ 2πrγlg cos(θ* − α), where the open angle is α = π/2 − cot−1(dr/dz). Due to its nonlinearity, there may be multiple solutions for h. Although the general expression can be used to determine the equilibrium height of the capillary rise (h*), the stability of the solution has to be decided by the condition ∂ 2F |h * > 0, ∂h2

where

∂F |h * = 0 ∂h

(5)

To utilize eq 4, one has to know θu, which is related to the angle of contact, θ. The angle of contact can be obtained from the virtual displacement of the contact line, zcu = zu(Ωp), under the condition that the meniscus satisfies the Young−Laplace equation. The free-energy minimization corresponding to the small displacement, δzcu, indicates that the liquid−gas interfacial energy gain must be compensated by the solid−liquid interfacial energy loss. As a result, δF/δzcu = 0 gives the angle of contact, θ(Ωp) cos θ = cos θ*

This consequence indicates that the angle of contact is always equal to the intrinsic contact angle as the tube length (H) is greater than the height of capillary rise (h*) (i.e., H > h*). However, for a short capillary in which the theoretical height of capillary rise exceeds the tube length, one has δzcu = 0 at the top of the channel. As a result, the angle of contact must be altered to satisfy eqs 2 or 4 by keeping the height of capillary rise at h = H. In general, the angle of contact becomes θ > θ*, and it declines with increasing H. In other words, θ may exceed the intrinsic contact angle θ* for a fixed triple contact line, but the 16919

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Therefore, for a vertical capillary of radius r, the stable height is given by

the microchannel is much smaller than the dimension of the container, the loss of the level of the liquid in the container can be neglected. The calculation was done until the total freeenergy difference converged within the acceptable tolerance of 10−5, and generally it takes about 1.5 h of CPU time on a PC Intel Core i7−2600 3.4 GHz running Windows 7.

⎧ h ̅ *with θ = θ*, as H ≥ h ̅ * ⎪ h* = ⎨ ⎪ ⎩ H with θ = θ ̅(H ), as H < h ̅ *

Figure 2a illustrates the contact angle and height of capillary rise as functions of the height of the vertical tube, h*(H) and

IV. SHORT CAPILLARY, TRUNCATED CONE, HYPERBOLOID, AND TWO DIFFERENT PLATES The exact force balance and the second derivative of the free energy in a microchannel of arbitrary shape are obtained based on free-energy minimization. The former can be employed to find the possible heights of capillary rise, while the latter is used to determine their stability. Several cases will be considered to elucidate the application of the general expression of force balance, eq 4, including (1) short vertical tube, (2) circular and inverted cones, (3) vertical hyperboloid, and (4) two different plates, which are not necessarily parallel. A. Short Vertical Tube with its Height Less Than Jurin’s Height. For a long, cylindrical tube of radius r (α = 0), the general expression can be reduced to the well-known result ρgV = 2πrγlg cos θ* = 2πr(γsg − γsl)

(6)

where V denotes the liquid volume within the tube, and 2πr is the contour length of the triple contact line. Equation 6 is typically explained as the force balance between the upward pull of surface tension and the downward gravity of the liquid volume associated with the capillary rise. Since the liquid−gas area (meniscus) does not vary with its height, the increment of the gravitational potential energy is actually compensated by the gain of the solid−liquid interfacial energy at equilibrium (i.e., F(h) ≅ ρg(πr2h)(h/2) − 2πrh(γsg − γsl). Here, the height variation of the meniscus is neglected. Consequently, the upward driving force for a vertical tube originates from solid− liquid tension based on the free-energy analysis. When the capillary is long enough, the solution from eq 6 or dF/dh = 0 is given by

Figure 2. The height of capillary rise and contact angle as a function of the height of the vertical tube for r = 0.44 mm, θ* = 10°, and lc = 1.66 mm. (a) Comparison among the theory, the SE simulations, and the experiments are made. (b) Pictures of SE simulations and experiments are shown for various capillary heights.

h ̅ * = 2lc2 cos θ*/r

θ(H), respectively. It is shown that the numerical result from the SE and experiment is in good agreement with our theoretical result. Two regimes separated at H = h̅* are clearly identified. For a short tube with H < h̅*, θ decreases, but h* grows with increasing H. On the other hand, as H ≥ h̅*, h* and θ are constant and do not depend on the height of the tube, H. Some pictures of the SE and experiment have been demonstrated in Figure 2b. According to the analysis of the capillary rise in a vertical tube, the vertical component of the capillary force per unit length is expressed as f = γlg cos θ, where θ = θ* or θ̅. This expression has been generalized to any triple contact line and the capillary force (fc) acting on a triple line Ω is14

where lc = (γlg/ρg)1/2 denotes the capillary length, and h̅* is referred to as Jurin’s height. Since d2F/dh2 = ρgπr2 > 0, this solution is always stable for a long tube. For a short capillary, the height of capillary rise, H, is smaller than h̅*, determined from eq 6. However, the general force balance is still valid. According to eq 4 and ignoring the variation of the meniscus, we have ρg(πr2H) = 2πγlgr cos θ. Note that the height of capillary rise is fixed at H, but the contact angle is θ̅(H), instead of θ*. As a result, the angle of contact (θ̅) is determined as ⎛ 1 rH ⎞ θ ̅ = cos−1⎜ 2 ⎟ > θ* as H < h ̅ * ⎝ 2 lc ⎠

(7)

F=

Evidently, the angle of contact is always greater than the intrinsic contact angle, as the channel height is less than the Jurin’s height. The stability of this solution is decided by δ 2F = 2π

∫0

r

(8)

∫Ω (fc) dl = ∫Ω (γlg cos θ)n dl

(9)

where dl denotes the differential length of the contact line and n is the unit vector lying on the solid surface with its direction perpendicular to dl. However, the force balance based on the above expression, eq 9, often fails to explain the capillary rise of channels with varying cross sections, as will be discussed later. B. Circular Cone and Inverted Cone. As shown in Figure 3a, the truncated cone (right circular cone) is a convergent

⎡ ⎤ (δz u′)2 x⎢γlg + ρg (δz u)2 ⎥ dx > 0. 2 3/2 ⎢⎣ (1 + z u′ ) ⎥⎦

The solid−liquid contribution vanishes, and δ2F > 0 ensures the minimum of the free energy with θ = θ̅. 16920

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⎛1 r H⎞ h0 = H and θ ̅(H ) = α + cos−1⎜ 0 2 ⎟ ⎝ 2 lc ⎠

(12)

where H ≤ H+ = 2l2c cos θ*/r0. As α → 0, the cone becomes a vertical tube and both h− and h0 can be reduced to eq 8 for H ≥ h* and H < h*, respectively. According to the above results, h− exists for the cone height exceeding H−, while h0 exists only for the cone height shorter than H+. In the regime H− ≤ H ≤ H+, two stable heights of capillary rise can be seen. Certainly, the observed height depends on the initial condition. If the cone top is first beneath the liquid level and then is gradually pulled up, h0 is seen as increasing H, in Figure 4a. On the other hand, h− is observed as decreasing H if the empty cone is first placed in contact with the liquid level and then is gradually pushed down. The phase diagram of the stable height versus cone height at a given Figure 3. Schematics of capillary rise in (a) a truncated cone (normal and inverted), (b) a hyperboloid, and (c) two different plates. The geometrical features of the microchannels are shown as well.

channel with α > 0, while the inverted cone is a divergent one with α < 0. Consider the condition that the radius of the cone is always small compared to the capillary length. Thus, the meniscus can be reasonably approximated by a spherical cap. The detailed shape of the meniscus can be further neglected for the calculation of the derivative of free energy. It is justified when the height variation of the meniscus surface is negligible as compared to the total height. However, the simplification of the meniscus profile at the formulation of the free energy (F) (i.e., eq 1) will result in the failure to obtain the Young− Laplace equation from dF/dh = 0. It works only for α = 0 (vertical tube) because there is no change of the meniscus with the height. As a consequence, it is more accurate to make such an assumption at the level of dF/dh. According to eq 4, one has dF |h * = 0 = ρg (πru2h) − 2πruγlg cos θu dh

(10)

The radius of a circular cone is ru = r0 + (H − h) tan α, where H denotes the height of the cone above the liquid level, and r0 is radius at H. If θ = θ*, there are three solutions of h* h± =

H̅ (1 ± 2

1 − 8(lc/H̅ )2 cos θu/tan α ) and h0 = H̅ (11)

where the capillary length is lc = (γlg/ρg) and H̅ = H + r0/tan α denotes the distance from the cone apex to the liquid level. The stability of h* (h0 > h+ > h−) can be determined by d2F/ dh2|h*. It is found that h− is stable, while h+ is unstable. Nonetheless, the first solution h− exists only when H ≥ H− = lc(8 cosθu/ tan α)1/2 − r0/tan α. With the third solution, h0 comes from ru = 0 and d2F/dh2|h0 = 2πγlg cos θu tan α > 0 indicates it is numerically stable. However, it is not physical because of h0 > H. This result reveals that there exists a solution close to the top of the cone. This unrealistic solution is due to the assumption of θ = θ*. When the height of capillary rise reaches the top of the cone, the angle of contact must be altered to satisfy the force balance, eq 8. Consequently, the second stable solution is obtained due to the edge effect 1/2

Figure 4. The height of capillary rise in a truncated cone as a function of the cone height for r0 = 0.25 mm, α = 2.58°, θ° = 15°, and lc = 1.66 mm. (a) Comparison among the theory, the experiments, and the SE simulations are made. Lines, hollow symbols, and symbols with a cross inside denote results from eq 11, the SE, and the experiments, respectively. (b) Pictures of SE simulations and experiments are shown for various cone heights. (c) The variation of the angle of contact with the cone height. The comparison is between the theory and the SE simulations. 16921

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contact angle and an open angle [i.e., h*(H; α, θ*)] is shown in Figure 4a. In order to examine our theory for multiple stable states, both the SE and experiment have been performed. Consider a system of a plastic pipet cap in isopropanol, and the relevant parameters are α = 2.58°, r0 = 0.25 mm, lc = 1.66 mm, and θ* = 15°. The boundaries for the regime of two stable states are predicted to be H−/lc = 9.83, and H+/lc = 12.97. When H increases gradually, h0 jumps down to h− at H+/lc = 13.3 in the SE and 12.97 in the experiment. When H decreases gradually, h− jumps up to h0 at H−/lc = 9.96 in the SE and 10.02 in the experiment. Both the simulation and the experiment results agree quite well with our theoretical prediction. Some pictures of the SE and experiment have been illustrated in Figure 4b. In the regime of the two stable states, the contact angle also has two solutions as shown in Figure 4c, besides the existence of two heights of capillary rise. For an inverted cone, eq 11 is still valid but with α < 0. Moreover, −H̅ depicts the distance from the liquid level to the apex of the inverted cone as shown in Figure 3a. When θ = θ*, there are three solutions of h* again and one has h− > h0 > h+. According to d2F/dh2 at h*, it is found that h− > 0 and h+ < 0 are stable, while h0 is unstable. According to eq 11, one has h+ < H̅ , and thus h+ is unrealistic because it corresponds to the height below the apex. As H < h−, similar to a short capillary, the solution begins to follow eq 12 due to the edge effect. Unlike a circular cone, there exists only one stable solution for an inverted cone. That is ⎧ h− with θ = θ*, as H ≥ h− h* = ⎨ ⎩ H with θ = θ ̅(H ), as H < h−

The solution has to be obtained numerically. Nevertheless, the asymptotic analysis can be performed to give analytical expressions, which provide the qualitative behavior of h*(H). Two asymptotic limits can be obtained, |h − H|/b = ε ≪ 1 and | h − H|/b ≫ 1. The former case, (|h − H|/b = ε ≪ 1), corresponds to the meniscus in the vicinity of H, and the height of capillary rise to the order of ε is ⎛ 1 − 2lc2 cos θ*/aH ⎞ ⎟ h 0 ≃ H ⎜1 − 1 + 2lc2 sin θ*/b2 ⎠ ⎝

The solution h0 can be more accurate by expanding eq 14 to the order of ε2. Moreover, two solutions are obtained in this limit, h0 and h+. The latter case (|h − H|/b ≫ 1) depicts the meniscus far away from H, and the height is given by h± ≃

8blc2 cos[θ* − tan−1(a /b)] ⎞⎟ ⎟ aH2 ⎠

(16)

On the other hand, the solutions h0 and h+ disappear as the term inside the square root of h0 or h+ obtained from the order of ε2 becomes negative. Therefore, only one stable state (h−) exists when the neck height is greater than H+, which is determined from

(13)

⎛ 3b2 − a 2 ⎞ 4H+2 − 4lc2 cos θ*⎜ ⎟H+ ⎝ ab2 ⎠ 2 2⎤ ⎡ ⎛ 2 ⎛ 2lc2 sin θ* ⎞ ⎥ b2 ⎞⎛ lc cos θ* ⎞ 2 ⎢ ⎟ + b ⎜1 + ⎟ − 8⎜1 − 2 ⎟⎜ ⎢⎣ ⎝ b a ⎠⎝ b2 ⎠ ⎝ ⎠ ⎥⎦ =0

This consequence reveals that similar to the truncated cone, there exists three regimes separated by H− and H+ for the variation of the height of capillary rise with the neck height, h*(H). In the first regime, H < H−, only one stable state exists, h* = h0, and the meniscus is located near the neck height. In the second regime, H− ≤H < H+, there are two stable states, h0 and h−. In addition to the one near the neck, the other one is close to the liquid level. The solution h+ is unstable because of (d2F/dh2)h+ < 0. In the third regime, H > H+, there is one stable state, h−, which is near the liquid level. Consider a hyperboloid with a = 0.4 mm, b = 1.0 mm, and θ* = 70°. Then, the plot of the height of capillary rise against the neck height is shown in Figure 5. According to the asymptotic analysis, the boundaries separating the three regimes are estimated as H−/lc = 4.56, and H+/lc = 5.46. The numerical solution of eq 14 is depicted in Figure 5, and it is found that H−/lc = 4.39 and H+/lc = 5.52. The simulation results from the SE are also shown, and they agree quite well with the theory. The boundaries decided by SE are H−/lc = 4.35 and H+/lc = 5.74. The slight difference between our theory and the SE may be attributed to the neglect of the height variation of the meniscus. It is obvious that both truncated cone and hyperboloid have three regimes in h*(H) and exhibit the same qualitative behavior. From the viewpoint of geometry,

(z − H )2 r2 − =1 a2 b2

where the neck is located at H and its radius is a. For z < H, the channel is convergent and its shape is similar to a cone. The opening angle, α = tan−1{a(z − H)/b2(1 + [(z − H)2/b2)]1/2}, varies with z. The solution of the height in the hyperboloid can be obtained simply by considering the local minimum of the free energy, and the contact angle is always maintained at the intrinsic contact angle, θ*. According to eq 4, the height of capillary rise is determined by ⎡ (h − H )2 ⎤ (h − H )2 2 a 2 π h ⎢1 + − 2 a π l 1 + ⎥ c ⎦ ⎣ b2 b2

=0

1−

H − ≃ {8blc2 cos[θ* − tan−1(a/b)]/a}1/2

As α → 0, the inverted cone also becomes a vertical tube, and h− can be reduced to eq 8 for H ≥ h̅*. C. Circular Hyperboloid: Neck Effect. Owing to the edge effect (boundary minimum), two stable states exist for a truncated cone, and the contact angle is no longer equal to the intrinsic contact angle for the solution at the top of the cone. In order to avoid the edge effect, a microchannel with the hyperboloid shape is considered. As shown in Figure 3b, the hyperboloid is like a hourglass and can be described mathematically by

⎧ ⎡ ⎪ ⎢ a(h − H ) −1 × cos⎨θ* + tan ⎢ 2 ⎪ ⎢ b2 1 + (h −2H ) ⎣ b ⎩

⎛ H⎜ 1± 2 ⎜⎝

Evidently, one has h0 > h+ > h−. With both solutions, h± vanishes as the term inside the square root of eq 16 becomes negative. That is, only stable state (h0) is present when the neck height is less than H−, which is

⎪ ⎪

(15)

⎤⎫ ⎥⎪ ⎥⎬ ⎥⎪ ⎦⎭ (14) 16922

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where H represents the height of the two plates and D0 the separation between them at the top. Since eq 19 is a quadratic equation, there are two solutions, h±, in which θui* = θi* − αi is fixed at its intrinsic value. They are expressed as Db 2(tan α1 + tan α2)

h± =

× [1 ±

where Db = D0 + H(tan α1 + tan α2) is the separation between the two plates at the bottom. The stability of h± can be determined by the second derivative of F

Figure 5. The height of capillary rise as a function of the center of circular hyperboloid with a = 0.4 mm, b = 2 mm, θ* = 70°, and lc = 2.71 mm.

d2F = ρg[D b − 2(tan α1 + tan α2)h] dh2

both are convergent channels for z < H, but the radius of the hyperboloid grows gradually for z > H, while the radius of the truncated cone jumps to infinity at z = H, instantaneously. For the gradual radius change, the height of capillary rise exceeds H (h* > H), and the contact angle is kept at the intrinsic contact angle (θ = θ*). On the other hand, for the sudden change of radius, the height of capillary rise remains at the height of the channel (h* = H) but the contact angle exceeds the intrinsic value (θ > θ*). The comparison between truncated cone and hyperboloid reveals that the multiple stable heights always exist for convergent channels, and the presence of the edge is the asymptotic limit of the gradual change.25 Nonetheless, the edge effect leads to the variation of the contact angle as the contact line reaches the edge.17,18 D. Two Different Plates. Rather than the capillary rise in the short tube, cone, and hyperboloid, the effect of the asymmetry will be investigated in this section. Consider two different plates with the width, W, much larger compared to the gap. As illustrated in Figure 3c, the two plates are not necessarily vertical but rather have some opening angles, α1 and α2. In addition, the contact angles of both walls can be different (i.e., θ1* and θ2*). The free energy per unit width is then given by F = W

∫x

x2

1

⎛

∑ ⎜ρ g i=1

⎝

hi3tan αi h ⎞ − γlg cos θi i ⎟ 6 cos αi ⎠

h0 = H and cos θu̅ =

∫x

x2

1

(17)

⎡ (h 2 tan α1 + h22 tan α2) ⎤ ⎥ z u d x = ρ g ⎢V − 1 2 ⎦ ⎣ = γlg(cos θu1 + cos θu2)

D0H 2lc2

(22)

Note that the contact lines are pinned at the top of both plates, h1 = h2 = H. When the angle difference of one of the plates, say plate 1 with θ*u1 ≥ θ*u2, declines to its intrinsic value, a further increment of H leads to the decay of the contact line on the plate 1 (i.e., h1 < H with the contact angle θ*1 ). However, the other plate remains pinned at the top of plate 2, where h2 = H and θu2 ≥ θ*u2 (or θ2 ≥ θ*2 ). In other words, in the second regime, the plate with its contact line pinned at the top possesses the angle difference greater than its intrinsic value (i.e., max(θ*u1, θ*u2) > θu ≥ min(θ*u1, θ*u2), while the other plate reaching its intrinsic contact angle, θ*u , shows downward displacement of its contact line. Since |h2−h1| < lc, the height of capillary rise can still be represented by h0 ≃ H. Finally, when both angles of difference reach their intrinsic values, this solution is not stable anymore upon increasing H. The inset of Figure 6a illustrates the variation of the height with the channel height for two different plates with α1 = −5° and α2 = 10°. The wettabilities of the two plates are θ*1 = 50° and θ*2 = 60°. Three regimes separated by H− and H+ can be identified. Similar to the truncated cone and hyperboloid, only one stable height exists, h* = h0, for the first regime, H < H−, and one stable height, h* = h−, for the third regime, H > H+. In the second regime, H− ≤ H < H+, there are two stable states, h0 and h−. The boundary heights can be determined from eqs 20 and 22,

where hi represents the height of capillary rise at the position of the contact line, xi. By calculus of variations, δF = 0, the one-dimensional Young−Laplace equation is obtained. Subject to the boundary conditions, (dzu/dx)x1 = −cot θu1 and (dzu/dx)x2 = cot θu2, the integration of the Young−Laplace equation yields ρg

(21)

For convergent channels, one has (tan α1+ tan α2) > 0 and h+ > h−. According to eq 21, one has (d2F/dh2)h+ < 0 and (d2F/ dh2)h− > 0. Therefore, h− is stable. On the contrary, for divergent channels, (tan α1 + tan α2) < 0 and h− > h+. The stability analysis indicates that h− is a stable solution. For convergent channels, in addition to h− given in eq 20, there may exist another stable solution, h0, which satisfies eq 19. That is, the height of capillary rise reaches its maximum but the contact angle no longer fulfills Young’s equation. Depending on H, there are two regimes for θu. In the first regime in which H is smaller, the angle difference θu = θ − α is the same for both * , θu2 * ), although the two plates plates, θ̅ = θu1 = θu2 ≥ max(θu1 have different wettabilities and open angles. The height of capillary rise is

⎛ ρgz u2 ⎞ ⎜γlg 1 + (z u′)2 + ⎟ dx 2 ⎠ ⎝

2

+

* + cos θu2 * )(tan α1 + tan α2)/D b2 ] 1 − 4lc2(cos θu1 (20)

(18)

where θui = θi − αi. Equation 18 can be obtained from eq 4 as well. Neglecting the meniscus variation, one has h2 ≈ h1 = h. The height of capillary rise is then determined by h[D0 + (H − h)(tan α1 + tan α2)] ≅ lc2(cos θu1 + cos θu2)

H− =

(19) 16923

* + cos θu2 * )(tan α1 + tan α2) − D0 2lc (cos θu1 tan α1 + tan α2

(23)

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hyperboloid are also different from those of eq 9. That is, eq 9 is only valid for a straight capillary (α = 0) but not for microchannels with varying cross sections. Equation 4 shows the force balance between an upward pull by surface tension, γlg cos(θ − α) and the downward gravity of the liquid volume just below the meniscus, the effective rising volume. However, the effective rising volume is usually not equal to the real liquid volume. Moreover, the force balance is a necessary condition for determining the height of capillary rise. The stability of the height requires the second derivative of the free energy to be greater than zero.

V. IMPLICATION FOR WATER TRANSPORT TO THE TOPS OF TALL TREES Most plant physiologists accept the “cohesion-tension theory” as the explanation for the ascent of sap.26 In this qualitative theory, the move of water depends upon three important physical-chemical properties of water, which actually correspond to capillary rise (cohesion), cavitation (tension), and the hydrated wall (low contact angle), respectively. In this section, we focus only on the implication of the general force balance and loop hysteresis in the capillary rise of a tall tree. The height to which water in a tree rises is dependent on the size of the transport conduits. If one cuts down a tree and looks inside, the capillary dimensions of the relatively large conduits (the xylem tube) are on the order of 100 μm.27 As a result, the capillary rise is about 0.1 m. If capillary pressure alone were to explain the water rise to the treetop of a 100 m tall tree, such as the coastal redwoods of California, a capillary radius of about 100 nm is required. It was suggested that the relevant capillary dimension is the air−water interfaces in the cell walls of the uppermost leaves. The matrix of cellulose microfibrils is highly wettable, and the spacing among them yields effective pore diameters of about 10 nm. It has been pointed out that it is not necessary for the capillary to have a small bore throughout its length. Only the bore at the meniscus (i.e., in the uppermost leaf) is relevant.27 This consequence has been proved in our general expression of force balance, eq 4. Note that a microchannel containing corners or cusps on its cross section is not considered in the derivation of eq 4. Liquid filaments extend to infinity in the corners or cusps.28 Nonetheless, the elevation of the liquid column is still inversely proportional to the characteristic dimension of the cross section of the tube. For the solutions satisfying the force balance, nevertheless, there exists the issue of physical stability. A small bore at the uppermost leaf connected to a larger xylem conduit reveals the presence of a convergent microchannel. As a consequence, multiple stable heights are possible, as described in the aforementioned analyses. However, the final state depends on the initial condition. The liquid will rise to a stable height corresponding to the larger xylem conduit if the microchannel is initially empty. In other words, the liquid will not rise of its own accord to the stable height near the top of the convergent channel because it will not be able to traverse the larger conduit of the channel. This situation is, nonetheless, stable if the liquid is sucked up to the top and then the suction is removed. How does a tall tree acquire such large negative (suction) pressures from above? As demonstrated in our experiments, the gradual rise of an initially immersed cone is able to maintain the stability of the meniscus on the top of the truncated cone as long as the force balance is satisfied. Note that the contact angle in the vicinity of the small pore mouth can be tuned up to fulfill the force balance when the

Figure 6. (a) The angle of contact corresponding to the meniscus at the top as a function of the channel height for two different plates with D0 = 0.5 mm, α1 = −5°, α2 = −10°, θ1* = 50°, θ2* = 60°, and lc = 2.71 mm. The inset shows the height of capillary rise as a function of the channel height. (b) The height of capillary rise as a function of θ2* for two different plates with D0 = 0.5 mm, Db = 0.9lc, α1 = −5°, α2 = 10°, θ1* = 50°, and lc = 2.71 mm. The comparison between theory (lines) and SE simulations (symbols) are made, and the pictures from the SE are shown as well.

and H+ =

* + cos θu2 *) lc2(cos θu1 D0

(24)

They are H−/lc = 5.32 and H+/lc = 6.59, which are in good agreement with those of the SE, as shown in the inset of Figure 6a. The change of contact angles of the two plates are illustrated in Figure 6a. Similar to the height variation, three regimes can be seen. For H ≥ H+, one has θ1 = θ*1 (θu1 = θ*u1) and θ2 = θ*2 (θu2 = θ*u2). On the other hand, for H < H′− = 2l2c min(cosθu1 * ,cos θu2 * ), two angles of difference are the same, θu1 = θu2 > min(θu1 * , θu2 * ). In the intermediate regime H−′ ≤ H < H+, one has θ1 = θ*1 (θu1 = θ*u1) and θ2 = θ*2 (θu2 = θ*u2). Note that H′− is not necessarily equal to H−. In addition to the channel height, H, the intrinsic contact angle θi* also influences the existence of h− and h0 according to eq 20 and 24. As shown in Figure 6b, for a larger value of θ*2 and a fixed value of θ*1 , only lower meniscus h− exists. As θ*2 is decreased, H+ grows and thereby h0 appears for θ2* being lower than a critical value. That is, two stable heights occur for a smaller θ2*. Note that the variation of h− with θ*2 is still continuous, and there is no discontinuous rise to an upper meniscus, h0, that can happen only by ascending the channel. If one applies the concept of capillary force, eq 9, to this case, the resulting force balance would become ρgV = γlg(cos θ1 cos α1 + cos θ2 cos α2). This outcome is not consistent with our expression of force balance, eq 18. In fact, the force balances obtained from the general expression (eq 4) for cone and 16924

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microchannel is not high enough. The slow growth of the tree can be regarded as a gradual rise of the convergent channel. As long as water transport to the pores on the uppermost leaves is not interrupted throughout the entire course of the tree’s growth to 100 m, this stable height can be achieved without resorting to suction.

Article

AUTHOR INFORMATION

Corresponding Author

*Y.-J. S.: e-mail, [email protected]. H.-K. T.: e-mail, hktsao@ cc.ncu.edu.tw. Notes

The authors declare no competing financial interest.

■ ■

VI. CONCLUSION In this paper, capillary rise in an asymmetric microchannel, in which both contact angle (wettability) and open angle (geometry) can vary with position, is investigated based on the minimization of free energy (F). Surface Evolver simulations and experiments are performed to validate our theoretical analyses. The calculus of variation, δF = 0, yields the Young−Laplace equation, and its integration leads to the balance between surface tension and gravity. The surface tension force is surface tension times the integration of cos θu along the contact line, where θu depicts the local difference between the contact angle and open angle. The effective gravitational force comes from the total volume right beneath the meniscus. This consequence proves that only the bore at the meniscus is relevant. Although this general expression reduces to the well-known result for a vertical capillary, the effective gravitational force does not originate from the total liquid mass in a channel above the liquid level. For the same channel height, multiple solutions can be obtained from the force balance. However, the stable height of capillary rise has to satisfy the stability analysis δ2F > 0. Several interesting cases have been studied to elucidate our theory, including short capillary, truncated cone, hyperboloid, and two different plates. When the height of a vertical tube (H) is shorter than the height of the capillary rise in a long tube (h̅*), the force balance is still obeyed. The liquid rises to the top of the tube, but the apparent contact angle is tuned up because of the edge effect. As H is increased, the contact angle declines accordingly until it reaches the intrinsic contact angle (θ*). For capillary rise in the truncated cone, there are two stable heights: one stays on the top of the cone and the other solution is near the liquid level. Three regimes can be identified for the plot of the stable height (h*) against the cone height (H). The solution of higher height dominates in the short cone regime, while the solution of lower height prevails in the tall cone regime. However, both solutions are stable in the intermediate regime. The above theoretical predictions are confirmed by the SE simulations and experiments. For capillary rise in the hyperboloid and between two plates, observation of similar behavior reveals that the multiple stable heights and hysteresis loop always exist for convergent channels. The force balances derived from free-energy minimization for the aforementioned examples are distinctly different from those based on the force acting on the contact line, which is generalized from capillary rise in vertical tubes. Our results offer some implications for water transport to the tops of tall trees. A small bore at the uppermost leaf connected to a larger xylem conduit reveals the presence of a convergent microchannel, and two stable heights are possible. As demonstrated in our experiments, the gradual rise of an initially immersed cone is able to maintain the stability of the meniscus on the top of the truncated cone. The slow growth of the tree can be regarded as a gradual rise of the convergent channel. As a consequence, during the course of the tree’s growth to 100 m, the stable height of capillary rise to the treetop can always be achieved without resorting to suction.

ACKNOWLEDGMENTS This research work is supported by National Science Council of Taiwan. REFERENCES

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