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Towards understanding elastocapillarity: comparing wetting of soft and rigid plates
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 155105 (http://iopscience.iop.org/0953-8984/26/15/155105) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 155105 (10pp)
doi:10.1088/0953-8984/26/15/155105
Towards understanding elastocapillarity: comparing wetting of soft and rigid plates Xiang-Ying Ji1 and Xi-Qiao Feng1,2 Institute of Biomechanics and Medical Engineering, AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China 2 Center for Nano and Micro Mechanics, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China 1
E-mail: [email protected] Received 9 December 2013, revised 8 February 2014 Accepted for publication 11 February 2014 Published 27 March 2014 Abstract
Elastocapillarity plays a significant role in the buoyancy and water repellency of soft objects floating on water. In this paper, we analyze the wetting behavior of an elastic and circular plate pressing a liquid surface. The geometry and stability of axisymmetric infinite liquid menisci are investigated, and their qualitative difference from two-dimensional planar menisci is revealed. By comparing the wetting processes of rigid and elastic circular plates under pressing, we show that flexibility benefits both the maximal depth and buoyancy a plate can reach. The results are helpful not only for understanding the living behavior of some aquatic creatures but also for the design of biomimetic soft microrobotics. Keywords: wetting, deformation, buoyancy, water meniscus, analytical method (Some figures may appear in colour only in the online journal)
1. Introduction
question also facilitates the design of biomimetic aqueous robotics and some other advanced devices and systems used on water or other liquids. Elastocapillarity refers to those phenomena in which capillarity is significantly coupled with elasticity. For example, flexible hairs can become adhered into a bundle when they are wetted [6–11], and a thin sheet can fold spontaneously when a drop of water is placed onto it [12–15]. Both theoretical analysis and experimental observations have found that when a hydrophobic flexible fiber is pressed obliquely on a water surface, the elastic deformation induced by surface tension and hydrostatic pressure affects its water-repellent ability [16–18]. Vella et al examined the equilibrium of a floating raft composed of strips by assuming that the entire edge of the plate contacts the water surface [5]. Burton and Bush considered the condition of two rigid plates jointed by a torsion spring and extended the analysis to the case of continuous deformable bodies [19]. Their results acknowledged that flexibility benefits the floating of objects. In other words, due to the effect of elastocapillarity, a soft object can achieve a greater buoyancy than a rigid one. These findings help understand the structure– property–function relation of water strider legs. However, these previous works were based on a two-dimensional planar
Many biological organisms are highly water repellent. For example, such aquatic creatures as water striders and water spiders walk effortlessly on water, and the leaves of lotus and other hydrophytes reject the staining of water. The outstanding hydrophobicity of these biological materials is commonly attributed to their wax coatings and hierarchical surface structures at micro- and nano-scales. Take water striders as an example. Their long legs have a large water contact angle up to 150°, and a single leg pressed on a water surface can achieve a buoyancy of about 15 times the body weight of the entire insect [1, 2]. For such objects as water strider legs, with a hydrophobic surface and a characteristic size smaller than or comparable to the capillary length of water, surface tension plays an important role in the buoyancy when they move on water. Considerable effort has been directed toward understanding the mechanisms of buoyancy and water-repellent ability [3–5]. In most previous studies, the floating object was assumed to be rigid. However, those floating biological organisms are usually highly flexible. An intriguing issue thus arises: how do these insects utilize the flexibility in their wetting behavior? The answer to this 0953-8984/14/155105+10$33.00
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J. Phys.: Condens. Matter 26 (2014) 155105
hypothesis. To gain an insight into the elastocapillarity of soft objects, finer and three-dimensional models are desired. There are many other vivid instances in nature showing the significance of elastocapillarity on the survival of living beings. Corollas of Nymphoides flowers, which usually float on water surfaces, close when tethered downwards into water by their stems [20]. Such phenomena are quite common for flowers and leaves of aquatic plants [21, 22]. In the present paper, this wetting behavior is analyzed by considering an elastic circular plate pressing a water surface. The hydrostatic pressure acting on the lower surface and the surface tension along the boundary tend to deform the plate. Such a thin plate model of elastocapillarity is also applicable to artificial water appliances, such as rafts, buoys and robots [23, 24]. For a circular plate pressed on a liquid surface, both the plate and the meniscus undergo axisymmetric deformations. It is known that for a planar, two-dimensional liquid surface, the Young–Laplace equation controlling the meniscus morphology depends only on its single curvature, while for an axisymmetric meniscus, two principal curvatures are involved in its Young–Laplace equation. For this reason, the geometry and stability of an axisymmetric water meniscus are distinctly different from that of a two-dimensional one. In addition, the Young–Laplace pressure across a curved meniscus depends on the hydrostatic pressure, which increases with depth. As a result, it is hard to analytically solve the deformation of a soft circular plate strongly coupled with the as-yet-to-be-determined meniscus shape. Furthermore, in the case of a very large water surface, the meniscus approaches an asymptote at infinity, making the theoretical analysis more difficult. Recently, Su et al examined the axisymmetric meniscus induced by a cylinder vertically pressing a water surface [25]. They determined the meniscus profile by the perturbation method. Phan et al investigated the equilibrium of a water droplet staying on an oil surface and calculated the axisymmetric oil surface shape numerically [26]. Chan et al studied the dewetting process on a cylindrical fiber [27]. In these computations, only the outer region of meniscus approaching the horizontal surface at infinity, along which the curvature variation is small, was taken into account. However, a thin plate pressing a water surface could create a deep puddle, in which the slope along the liquid meniscus changes significantly, and the meniscus profile is beyond the scope of the aforementioned studies. Therefore, it is imperative to understand the properties of axisymmetric liquid menisci for studying the wetting behavior of floating circular plates. This paper is outlined as follows. In section 2, a framework is constructed to solve the geometry of an infinite and axisymmetric liquid meniscus in a cylindrical coordinate system. In section 3, the buoyancy–depth relation of a rigid circular plate pressing downwards is investigated to show the difference between axisymmetric and planar wetting problems. In section 4, the effect of elastocapillarity is further taken into account, and the strongly non-linear problem of a soft circular plate coupled with an infinite liquid meniscus is computed. Finally, the main conclusions drawn from this study are summarized.
Figure 1. (a) Schematic diagram of an axisymmetric infinite liquid meniscus. At points A and B, the slope angle φ is smaller and larger than 90 ° respectively. (b) Contact boundary condition between the plate and the meniscus.
2. Solution of axisymmetric infinite liquid menisci In this section, we will develop a method to analyze the geometry of infinite and axisymmetric infinite liquid menisci. Figure 1(a) shows an axisymmetric liquid meniscus with the effect of gravity. Refer to the coordinate system ( r , z ), where the vertically downward z-axis is the axisymmetric axis of the problem. The meniscus has two prominent geometric features: its two principal curvatures are of opposite signs and it approaches an asymptote at infinity, i.e. z → 0 as r →∞. The meniscus curve makes an angle φ measured from the negative direction of r-axis. The curve has an inflection point, at which the tangent is vertical, i.e. φ = 90°. Normalize the length parameters of z and r as z = κz , r = κr (1) −1 where κ = γ / ρg denotes the capillary length of the liquid, γ the surface tension, ρ the mass density and g the gravity. In terms of the dimensionless z and r , the Young–Laplace equation of axisymmetric menisci can be expressed as − ⎛ dr ⎞ ⎡ ⎛ dz ⎞2 ⎤ 2 ⎡ 1 d2z 1 dz ⎤ − sgn ⎜ ⎟ ⎢ 1+⎜ ⎟ ⎥ ⎢ + ⎥ = z. ⎝ dz ⎠ ⎣ ⎝ dr ⎠ ⎦ ⎣ 1+(dz / dr )2 dr 2 r dr ⎦ (2) 1
For completeness, the derivation of this equation is given in the appendix. Equation (2) is hard to solve analytically. Su et al have adopted a perturbation method to obtain an approximate solution in a limited range of small z values [25], but the full solution for a meniscus shape illustrated in figure 1(a) is still unavailable because of the strong non-linearity and liquid–structure coupling. Therefore, we here establish a numerical scheme to compute the shape of an axisymmetric infinite meniscus. 2
X-Y Ji and X-Q Feng
J. Phys.: Condens. Matter 26 (2014) 155105
Figure 3. φ1 versus H curves of plates with different radii R.
Figure 2. Inner regions of the axisymmetric menisci under a few representative values of r0, where the starting point O* of calculation is marked by a circle at the upper end of each curve.
where Δs = κΔs is a given infinitesimal increment of arc length, i = 0, 1, … , n with n being an integer satisfying φn ≤ 180 °. For some representative values of r0, the meniscus shapes calculated from the above methods are shown in figure 2. It is observed that the axisymmetric liquid menisci differ greatly from those under the planar, two-dimensional condition. The larger the rotation radius r0 of a meniscus, the deeper it can reach. When r0 is very large, the axisymmetric meniscus shape in the (r, z) coordinate system is close to that in the planar problem.
First, for the outer region of the meniscus far from the axisymmetric axis, the value of z is very small and satisfies dz / dr ≪1 and dz / dr 1, φ1max can approach 180°, demonstrating again that the axisymmetric problem with a plate radius larger than the capillary length can be well approximated by the planar solution.
(i) when R ⩽ 0.22, the pressing process is identical to the condition of θ = 180°, (ii) when 0.22 2.25, both the maximal depth and buoyancy are smaller than those of θ = 180°. Figure 6(b) also shows that a plate with θ 152°, if the plate radius is comparable to or smaller than the capillary length, the 5
X-Y Ji and X-Q Feng
J. Phys.: Condens. Matter 26 (2014) 155105
Figure 7. Variations of (a) the maximal depth Hmax and (b) the corresponding apparent contact angle φ1max with respect to the plate radius R under different intrinsic contact angles θ. Variations of (c) the maximal buoyancy Fmax and (d) the corresponding apparent contact angle φ1 F with respect to R under different values of θ.
maximal depth and buoyancy are independent of the intrinsic contact angle θ and are determined by the stability of the axisymmetric meniscus. For a few representative intrinsic contact angles, figures 7(a) and (b) plot the maximal depth Hmax and the corresponding apparent contact angle φ1 max as functions of the normalized plate radius R , respectively. When R is larger than a certain value, the penetration of the liquid surface is dictated only by the contact condition between the plate boundary and the meniscus. In this case, the maximal apparent contact angle φ1 max will be a constant and equal to the intrinsic contact angle, as shown by the horizontal lines in figure 7(b). One can also see that when θ = 90°, the maximal depth is independent of the plate radius. For plates with different intrinsic contact angles, the maximal buoyancies Fmax and the corresponding apparent contact angles φ1 F are shown in figures 7(c) and (d), respectively. The horizontal lines in figure 7(d) mean that the maximal buoyancy Fmax is totally determined by the liquid surface penetration. It is also found that for a superhydrophobic plate with a large contact angle (e.g. θ = 150°), the maximal buoyancy is independent of the liquid surface penetration and equals that for θ = 180°.
as the vertical component of displacement at point r with respect to the plate center (r = 0). For simplicity, a small deformation theory of linearly elastic plates is employed. The differential equation of the deflection curve of a circular thin plate is p d4w 2 d3w 1 d2w 1 dw + − + = , dr 4 r dr 3 r 2 dr 2 r 3 dr D
(10) where p ( r ) is the hydrostatic pressure acting on the lower surface of the plate. It is noticed that the hydrostatic pressure distribution p ( r ) varies with increasing depth H. D is the bending rigidity of the plate and defined as D=
Eδ 3 , 12(1 − ν 2 )
(11) where E, ν and δ are the Young’s modulus, Poisson’s ratio and thickness of the plate, respectively. The hydrostatic pressure p ( r ) is proportional to the depth and expressed as p = ρg ( H − w ), (12) where H denotes the depth of plate at r = 0. Introduce the following parameters normalized by the plate radius R:
4. Wetting of soft circular plates
4.1 Deformation equation of a circular plate
Now we consider a soft circular plate pressing a water surface, as shown in figure 8. Its deflection, w ( r ), is defined
∼ = w / R, w
r∼ = r / R, H͠ = H / R, ∼3 Eδ D . D͠ = = 12ρgR(1 − ν 2 ) ρgR 4
Then equation (10) is recast as 6
(13)
X-Y Ji and X-Q Feng
J. Phys.: Condens. Matter 26 (2014) 155105
where φ1 is the tangent angle of the meniscus at the plate boundary (figure 8). For a rigid plate φ1 equals the apparent contact angle, while for an elastic plate under deformation the apparent contact angle is written as φ1 − (dw / dr ) r =R. Letting γ∼=γ / D͠ ρgR 2=γR 2 / D, equation (20) becomes ∼ v dw ∼⎞ ⎛ d2w ⎜ ∼2 + ∼ ∼ ⎟ ⎝ dr r dr ⎠
∼ 1 dw ∼⎞ d ⎛ d2w + ⎜ ⎟ dr∼ ⎝ dr∼2 r∼ dr∼ ⎠
Figure 8. Schematic diagram of the deformation of an elastic thin plate pressing a liquid surface.
∼ 2 d3w ∼ 1 d2w ∼ 1 dw ∼ H͠ − w ∼ d4w + − + = . dr∼ 4 r∼ dr∼3 r∼2 dr∼2 r∼3 dr∼ D͠
⎛ ∼ ∼4 ∼ = H͠ + 1 C Bei ⎛⎜ r ⎞⎟ + C G 2,0 ⎜ − r w 1 0 2 0,4 ⎜ 256D͠ ⎝ D͠ 1/4 ⎠ 4 ⎝ ⎛ ⎛ r∼ ⎞ r∼ 4 2,0 ⎜ + C3Ber0 ⎜ 1/4 ⎟ + C4G0,4 − ⎜ 256D͠ ⎝ D͠ ⎠ ⎝
—— 1 , 1 ,0,0 2 2
—— 1 , 1 ,0,0 2 2
⎞ ⎟, ⎟ ⎠
(15)
Berv ( z ) = R[ Jv ( z e3π i/4 ) ] , Bei v ( z ) = I[ Jv ( z e3π i/4 ) ] , (17) with R and I designating the real part and the imaginary part of Jv respectively. G is the Meijer G function:
≡
1 2π i
∫
γL
) Π jm= 1Γ ( bj − s ) Π jn= 1Γ ( 1 − aj + s ) Π
Γ ( 1 − bj + s )
Γ ( aj − s ) Π
p j=n+1
q j=m+1
x s ds ,
(18)
where Γ is the Gamma function. To obtain the complete buoyancy–depth curve of the soft circular plate, we assume a displacement-controlled loading condition. The depth of the plate in a very small circular region of 0 ≤ r ≤ a is specified as H. Thus, we have the following two boundary conditions of displacement w: ∼ ∼ ∼ = (dw ∼ / dr∼ ) ∼ ∼ = 0, (19) w r =a r =a ∼ where a = a / R≪1. Along the boundary of the circular plate, the bending moment is zero and the shear force equals the vertical component of surface tension γ . Therefore, the boundary conditions at r = R are ⎛ d2w v dw ⎞ MR = ⎜ 2 + ⎟ ⎝ dr r dr ⎠
∫
r=R
= γ sin φ1, r=R
a
F = 2πRγ sin φ1 + ρgH ⋅ 2πr dr + 0 Its dimensionless form is
= 0,
d ⎛ d2w 1 dw ⎞ QR = − D ⎜ 2 + ⎟ dr ⎝ dr r dr ⎠
=− γ∼sin φ1 .
(21)
In section 4.1, we have shown how to calculate the deformation of an elastic plate with given H and φ1, but its coupling with the meniscus has not been considered. For a soft plate pressing a liquid surface, however, H and φ1 are interdependent. For a given depth H , the value of φ1 should be determined by considering the morphology of the liquid surface, i.e. the meniscus. This problem is highly non-linear because both the hydrostatic pressure and surface tension acting on the plate are coupled with its elastic deformation. As above, we normalize the length parameters by the plate radius R in the following analysis. For a specified value of r0, denoted as r0 − j, the meniscus curve normalized by the capillary length, Sj, can be calculated following the procedure ∼ described in section 2. Let Sj denote the dimensionless meniscus curve normalized by R, which can be obtained via coordinate conversion: r z r∼ = = r D͠ γ∼ , z∼ = = z D͠ γ∼ . (22) κR κR ∼ ∼ Let Cj label the position on Sj satisfying r∼ = 1, where z∼ = z∼1′_ j ∼ and φ1 = φ1_ j. For a given depth H͠ at the plate center, the elastic deformation can be calculated following the method in section 4.1 and the corresponding depth of the plate ∼ can be solved boundary is labeled as z∼1″_ j . A value of φ 1_ j ∼ ∼ ∼ satisfies all conditions from the condition of z1″_n = z1′_n. φ 1_ j of the meniscus geometry in section 2 and the plate deformation in section 4.1 and, therefore, is the actual value of φ1 for the specified plate depth H͠ . The corresponding meniscus morphology and the elastocapillary deformation of the plate are the actual solution. Thus, the actual buoyancy of a deformed plate can be derived by
(16) Jv ( z e3π i/4 ) =Berv ( z ) + iBei v ( z ) , where Jv is the ν-order Bessel function of the first kind, i.e.
(
r∼= 1
4.2 Calculation method
⎞ ⎟ ⎟ ⎠
where C1, C2, C3 and C4 are constants as yet to be determined. Functions Ber and Bei are defined by
, ap Gpm, q, n x ba11,,… … , bq
= 0,
From the four boundary conditions in equations (19) and (20), the parameters C1, C2, C3 and C4 in equation (15) can be determined. Thus, provided that the depth and contact condition of the plate are specified, its elastic deformation can be solved from the above theory.
(14)
Its general solution is
r∼= 1
(20) 7
∼ F=
∫a
R
ρg ( H − w ) ⋅ 2πr dr . (23)
∼2 ∼ F ∼D͠ sin φ + a h + = γ 1 2πρgR3 2
∫a∼ ( h∼ − w∼) r∼ dr∼. 1
(24)
X-Y Ji and X-Q Feng
J. Phys.: Condens. Matter 26 (2014) 155105
can be seen that the descending plate will experience a distinct deformation. For a given rigidity, the larger the plate, the more remarkable its deformation. On the plate boundary, the apparent contact angle is φ1 − (dw / dr ) r =R. Due to the large deformation and the considerable value of a sloping angle (dw / dr ) r =R, the apparent contact angle of the soft plate is significantly smaller than that of a rigid plate at the same depth. Therefore, the apparent contact angle is unlikely to reach the intrinsic value, provided that the material is hydrophobic. In this situation, the liquid penetration by the plate boundary is controlled mainly by the instability of the axisymmetric meniscus, rather than the solid–liquid contact condition along the plate boundary. Therefore, it is rational to disregard the effect of the intrinsic contact angle value in the elastocapillary problem under study. 4.4 Effects of plate size and rigidity
Now we analyze the effects of the radius and bending rigidity of a plate on its wetting behavior. First, the effect of rigidity is examined by fixing the plate radius. Figure 10(a) gives the buoyancy force–displacement curves for four representative values of the bending rigidity D, where we set R=1 mm. It is seen that a plate with a lower bending rigidity can reach a larger buoyancy and a larger depth. The deformed shapes of the plate at its maximal pressing depth are shown in figure 10(b) for the three cases of D =10−8 N m, 10−7 N m and 10−6 N m, respectively. Then we investigate the size effect of the plate on its elastocapillarity by keeping the bending rigidity D = 10−7 N m. Figure 10(c) shows the force–displacement curves of soft plates with different radii. The corresponding results of rigid plates with the same radii are also provided for comparison. Figure 10(d) shows the plate deflections at the maximal depth. It can also be observed from figure 10(c) that the flexibility of the plate enlarges both its maximal buoyancy and depth. Such a tendency is remarkable for larger and softer plates, but the contribution of elastic deformation to the buoyancy is insignificant for smaller and stiffer plates.
Figure 9. Deformations of plates with bending rigidity D =10−7 N m
and different radii during the descending process: (a) R = 0.1 mm, (b) R = 1 mm, (c) R = 2 mm.
5. Conclusions
∼ The two normalized buoyancies, F and F , are related by ∼ ∼ 3/2 (25) F = F R3κ 3 = F / ( D͠ γ∼ ) .
In this paper, the wetting behavior of a circular soft plate pressing downwards on a liquid surface has been investigated. The results show that axisymmetric wetting problems have some qualitative differences from planar, two-dimensional problems, since the geometry of an axisymmetric meniscus involves two principal curvatures while a two-dimensional meniscus involves only one. The wetting behavior of soft plates exhibits a distinct size effect. For a plate with radius smaller than the capillary length, the deviation of the axisymmetric wetting problem from the planar one is more remarkable. Another conclusion is that the flexibility of a plate benefits its water-repellency behavior. A plate with lower bending rigidity can achieve a larger buoyancy and a larger penetration depth. These results deepen our understanding of axisymmetric wetting and elastocapillary phenomena, and the design of biomimetic soft robotics.
4.3 Elastocapillarity of soft plates
For illustration, we take the following parameters of pure water for the liquid: surface tension γ = 0.072 N m−1, ρg=104 kg m−2 s−2, and the capillary length κ −1 = 2.7 mm. For the soft plate, we set Young’s modulus as E = 1 GPa and the bending rigidity D = 10−7 N m, which corresponds to a plate thickness of about 10 µm. The elastocapillary deformation of a plate in the descending process is calculated by using the method in section 4.2. Figures 9(a)–(c) show the results of plates with a few representative radii of R = 0.1 mm, 1.0 mm and 2.0 mm, respectively. In the calculation, we set the parameter a = 0.01R. It 8
X-Y Ji and X-Q Feng
J. Phys.: Condens. Matter 26 (2014) 155105
Figure 10. (a) Buoyancy–depth relations and (b) maximal deformations of plates with R =1 mm and different bending rigidities during the
descending process. (c) Buoyancy–depth relations and (d) maximal deformations of plates with D =10−7 N m and different radii.
Finally, it is pointed out that for a soft plate pressing against a liquid surface, both the plate and the meniscus may become unstable with increasing depth. The instabilities involved in this problem and their influences on the wetting behavior deserve further research.
Normalize all length parameters by the capillary length and define
Acknowledgments
dφ sin φ − = κ 2z. ds r
s = κs ,
z = κz ,
(A3)
r = κr .
(A4)
Then the dimensionless form of equation (A3) is
Support from the National Natural Science Foundation of China (Grant No 31270989) and the 973 Program of MOST (2013CB933033 and 2012CB934101) is acknowledged.
dφ sin φ − = z. ds r
(A5)
Use the geometric relations Appendix. Derivation of Young–Laplace equation for axisymmetric menisci The Young–Laplace equation of a meniscus reads
⎛1 1⎞ γ ⎜ + ⎟ = Δp = ρgz, ⎝ R1 R2 ⎠
(A1)
where Δp is the Young–Laplace pressure difference across the liquid surface and equals the hydrostatic pressure at depth z, and R1 and R2 are the two principal curvatures of the meniscus. In the axisymmetric case, one has
ds = R1dφ , r = − R2sin φ ,
dz dr = − cot φ , = − tan φ , dr dz ⎛ dr 1 π⎞ , = − cos φ = sgn ⎜ φ − ⎟ ⎝ ds 2 ⎠ 1+(dz / dr )2 ⎛ dz π⎞ = sin φ = sgn ⎜ φ − ⎟ ⎝ ds 2⎠
dz / dr 1+(dz / dr )2
.
(A6)
Thus, the Young–Laplace equation for axisymmetric menisci in equation (A5) can be rewritten in terms of z and r as ⎛ dr ⎞ ⎡ ⎛ dz ⎞2 ⎤ 2 ⎡ 1 d2z 1 dz ⎤ − sgn ⎜ ⎟ ⎢ 1+⎜ ⎟ ⎥ ⎢ + ⎥ = z. ⎝ dz ⎠ ⎣ ⎝ dr ⎠ ⎦ ⎣ 1+(dz / dr )2 dr 2 r dr ⎦ (A7) −1
(A2)
where s denotes the arc length. Substituting equation (A2) into (A1) and using the definition of capillary length leads to 9
X-Y Ji and X-Q Feng
J. Phys.: Condens. Matter 26 (2014) 155105
[13] Hure J, Roman B and Bico J 2011 Phys. Rev. Lett. 106 174301 [14] Jamin T, Py C and Falcon E 2011 Phys. Rev. Lett. 107 204503 [15] Liu J L and Feng X Q 2012 Acta Mech. Sin. 28 928 [16] Park K J and Kim H Y 2008 J. Fluid Mech. 610 381 [17] Vella D 2008 Langmuir 24 8701 [18] Ji X Y, Wang J W and Feng X Q 2012 Phys. Rev. E 85 021607 [19] Burton L J and Bush J W M 2012 Phys. Fluids 24 101701 [20] Roman B and Bico J 2010 J. Phys.: Condens. Matter 22 493101 [21] Jung S, Reis P M, James J, Clanet C and Bush J W M 2009 Phys. Fluids 21 091110 [22] Reis P M, Hure J, Jung S, Bush J W M and Clanet C 2010 Soft Matter 6 5705 [23] Jiang C G, Xin S C and Wu C W 2011 AIP Adv. 1 032148 [24] Yong J, Yang Q, Chen F, Zhang D, Du G, Si J, Yun F and Hou X 2014 J. Micromech. Microeng. 24 035006 [25] Su Y W, He S J, Ji B H, Huang Y and Hwang K C 2011 Appl. Phys. Lett. 99 263704 [26] Phan C M, Allen B, Peters L B, Le T N and Tade M O 2012 Langmuir 28 4609 [27] Chan T S, Gueudré T and Snoeijer J H 2011 Phys. Fluids 23 112103 [28] Ji X Y and Feng X Q 2013 Langmuir 29 6562
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Towards understanding elastocapillarity: comparing wetting of soft and rigid plates
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 155105 (http://iopscience.iop.org/0953-8984/26/15/155105) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 155105 (10pp)
doi:10.1088/0953-8984/26/15/155105
Towards understanding elastocapillarity: comparing wetting of soft and rigid plates Xiang-Ying Ji1 and Xi-Qiao Feng1,2 Institute of Biomechanics and Medical Engineering, AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China 2 Center for Nano and Micro Mechanics, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China 1
E-mail: [email protected] Received 9 December 2013, revised 8 February 2014 Accepted for publication 11 February 2014 Published 27 March 2014 Abstract
Elastocapillarity plays a significant role in the buoyancy and water repellency of soft objects floating on water. In this paper, we analyze the wetting behavior of an elastic and circular plate pressing a liquid surface. The geometry and stability of axisymmetric infinite liquid menisci are investigated, and their qualitative difference from two-dimensional planar menisci is revealed. By comparing the wetting processes of rigid and elastic circular plates under pressing, we show that flexibility benefits both the maximal depth and buoyancy a plate can reach. The results are helpful not only for understanding the living behavior of some aquatic creatures but also for the design of biomimetic soft microrobotics. Keywords: wetting, deformation, buoyancy, water meniscus, analytical method (Some figures may appear in colour only in the online journal)
1. Introduction
question also facilitates the design of biomimetic aqueous robotics and some other advanced devices and systems used on water or other liquids. Elastocapillarity refers to those phenomena in which capillarity is significantly coupled with elasticity. For example, flexible hairs can become adhered into a bundle when they are wetted [6–11], and a thin sheet can fold spontaneously when a drop of water is placed onto it [12–15]. Both theoretical analysis and experimental observations have found that when a hydrophobic flexible fiber is pressed obliquely on a water surface, the elastic deformation induced by surface tension and hydrostatic pressure affects its water-repellent ability [16–18]. Vella et al examined the equilibrium of a floating raft composed of strips by assuming that the entire edge of the plate contacts the water surface [5]. Burton and Bush considered the condition of two rigid plates jointed by a torsion spring and extended the analysis to the case of continuous deformable bodies [19]. Their results acknowledged that flexibility benefits the floating of objects. In other words, due to the effect of elastocapillarity, a soft object can achieve a greater buoyancy than a rigid one. These findings help understand the structure– property–function relation of water strider legs. However, these previous works were based on a two-dimensional planar
Many biological organisms are highly water repellent. For example, such aquatic creatures as water striders and water spiders walk effortlessly on water, and the leaves of lotus and other hydrophytes reject the staining of water. The outstanding hydrophobicity of these biological materials is commonly attributed to their wax coatings and hierarchical surface structures at micro- and nano-scales. Take water striders as an example. Their long legs have a large water contact angle up to 150°, and a single leg pressed on a water surface can achieve a buoyancy of about 15 times the body weight of the entire insect [1, 2]. For such objects as water strider legs, with a hydrophobic surface and a characteristic size smaller than or comparable to the capillary length of water, surface tension plays an important role in the buoyancy when they move on water. Considerable effort has been directed toward understanding the mechanisms of buoyancy and water-repellent ability [3–5]. In most previous studies, the floating object was assumed to be rigid. However, those floating biological organisms are usually highly flexible. An intriguing issue thus arises: how do these insects utilize the flexibility in their wetting behavior? The answer to this 0953-8984/14/155105+10$33.00
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J. Phys.: Condens. Matter 26 (2014) 155105
hypothesis. To gain an insight into the elastocapillarity of soft objects, finer and three-dimensional models are desired. There are many other vivid instances in nature showing the significance of elastocapillarity on the survival of living beings. Corollas of Nymphoides flowers, which usually float on water surfaces, close when tethered downwards into water by their stems [20]. Such phenomena are quite common for flowers and leaves of aquatic plants [21, 22]. In the present paper, this wetting behavior is analyzed by considering an elastic circular plate pressing a water surface. The hydrostatic pressure acting on the lower surface and the surface tension along the boundary tend to deform the plate. Such a thin plate model of elastocapillarity is also applicable to artificial water appliances, such as rafts, buoys and robots [23, 24]. For a circular plate pressed on a liquid surface, both the plate and the meniscus undergo axisymmetric deformations. It is known that for a planar, two-dimensional liquid surface, the Young–Laplace equation controlling the meniscus morphology depends only on its single curvature, while for an axisymmetric meniscus, two principal curvatures are involved in its Young–Laplace equation. For this reason, the geometry and stability of an axisymmetric water meniscus are distinctly different from that of a two-dimensional one. In addition, the Young–Laplace pressure across a curved meniscus depends on the hydrostatic pressure, which increases with depth. As a result, it is hard to analytically solve the deformation of a soft circular plate strongly coupled with the as-yet-to-be-determined meniscus shape. Furthermore, in the case of a very large water surface, the meniscus approaches an asymptote at infinity, making the theoretical analysis more difficult. Recently, Su et al examined the axisymmetric meniscus induced by a cylinder vertically pressing a water surface [25]. They determined the meniscus profile by the perturbation method. Phan et al investigated the equilibrium of a water droplet staying on an oil surface and calculated the axisymmetric oil surface shape numerically [26]. Chan et al studied the dewetting process on a cylindrical fiber [27]. In these computations, only the outer region of meniscus approaching the horizontal surface at infinity, along which the curvature variation is small, was taken into account. However, a thin plate pressing a water surface could create a deep puddle, in which the slope along the liquid meniscus changes significantly, and the meniscus profile is beyond the scope of the aforementioned studies. Therefore, it is imperative to understand the properties of axisymmetric liquid menisci for studying the wetting behavior of floating circular plates. This paper is outlined as follows. In section 2, a framework is constructed to solve the geometry of an infinite and axisymmetric liquid meniscus in a cylindrical coordinate system. In section 3, the buoyancy–depth relation of a rigid circular plate pressing downwards is investigated to show the difference between axisymmetric and planar wetting problems. In section 4, the effect of elastocapillarity is further taken into account, and the strongly non-linear problem of a soft circular plate coupled with an infinite liquid meniscus is computed. Finally, the main conclusions drawn from this study are summarized.
Figure 1. (a) Schematic diagram of an axisymmetric infinite liquid meniscus. At points A and B, the slope angle φ is smaller and larger than 90 ° respectively. (b) Contact boundary condition between the plate and the meniscus.
2. Solution of axisymmetric infinite liquid menisci In this section, we will develop a method to analyze the geometry of infinite and axisymmetric infinite liquid menisci. Figure 1(a) shows an axisymmetric liquid meniscus with the effect of gravity. Refer to the coordinate system ( r , z ), where the vertically downward z-axis is the axisymmetric axis of the problem. The meniscus has two prominent geometric features: its two principal curvatures are of opposite signs and it approaches an asymptote at infinity, i.e. z → 0 as r →∞. The meniscus curve makes an angle φ measured from the negative direction of r-axis. The curve has an inflection point, at which the tangent is vertical, i.e. φ = 90°. Normalize the length parameters of z and r as z = κz , r = κr (1) −1 where κ = γ / ρg denotes the capillary length of the liquid, γ the surface tension, ρ the mass density and g the gravity. In terms of the dimensionless z and r , the Young–Laplace equation of axisymmetric menisci can be expressed as − ⎛ dr ⎞ ⎡ ⎛ dz ⎞2 ⎤ 2 ⎡ 1 d2z 1 dz ⎤ − sgn ⎜ ⎟ ⎢ 1+⎜ ⎟ ⎥ ⎢ + ⎥ = z. ⎝ dz ⎠ ⎣ ⎝ dr ⎠ ⎦ ⎣ 1+(dz / dr )2 dr 2 r dr ⎦ (2) 1
For completeness, the derivation of this equation is given in the appendix. Equation (2) is hard to solve analytically. Su et al have adopted a perturbation method to obtain an approximate solution in a limited range of small z values [25], but the full solution for a meniscus shape illustrated in figure 1(a) is still unavailable because of the strong non-linearity and liquid–structure coupling. Therefore, we here establish a numerical scheme to compute the shape of an axisymmetric infinite meniscus. 2
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J. Phys.: Condens. Matter 26 (2014) 155105
Figure 3. φ1 versus H curves of plates with different radii R.
Figure 2. Inner regions of the axisymmetric menisci under a few representative values of r0, where the starting point O* of calculation is marked by a circle at the upper end of each curve.
where Δs = κΔs is a given infinitesimal increment of arc length, i = 0, 1, … , n with n being an integer satisfying φn ≤ 180 °. For some representative values of r0, the meniscus shapes calculated from the above methods are shown in figure 2. It is observed that the axisymmetric liquid menisci differ greatly from those under the planar, two-dimensional condition. The larger the rotation radius r0 of a meniscus, the deeper it can reach. When r0 is very large, the axisymmetric meniscus shape in the (r, z) coordinate system is close to that in the planar problem.
First, for the outer region of the meniscus far from the axisymmetric axis, the value of z is very small and satisfies dz / dr ≪1 and dz / dr 1, φ1max can approach 180°, demonstrating again that the axisymmetric problem with a plate radius larger than the capillary length can be well approximated by the planar solution.
(i) when R ⩽ 0.22, the pressing process is identical to the condition of θ = 180°, (ii) when 0.22 2.25, both the maximal depth and buoyancy are smaller than those of θ = 180°. Figure 6(b) also shows that a plate with θ 152°, if the plate radius is comparable to or smaller than the capillary length, the 5
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J. Phys.: Condens. Matter 26 (2014) 155105
Figure 7. Variations of (a) the maximal depth Hmax and (b) the corresponding apparent contact angle φ1max with respect to the plate radius R under different intrinsic contact angles θ. Variations of (c) the maximal buoyancy Fmax and (d) the corresponding apparent contact angle φ1 F with respect to R under different values of θ.
maximal depth and buoyancy are independent of the intrinsic contact angle θ and are determined by the stability of the axisymmetric meniscus. For a few representative intrinsic contact angles, figures 7(a) and (b) plot the maximal depth Hmax and the corresponding apparent contact angle φ1 max as functions of the normalized plate radius R , respectively. When R is larger than a certain value, the penetration of the liquid surface is dictated only by the contact condition between the plate boundary and the meniscus. In this case, the maximal apparent contact angle φ1 max will be a constant and equal to the intrinsic contact angle, as shown by the horizontal lines in figure 7(b). One can also see that when θ = 90°, the maximal depth is independent of the plate radius. For plates with different intrinsic contact angles, the maximal buoyancies Fmax and the corresponding apparent contact angles φ1 F are shown in figures 7(c) and (d), respectively. The horizontal lines in figure 7(d) mean that the maximal buoyancy Fmax is totally determined by the liquid surface penetration. It is also found that for a superhydrophobic plate with a large contact angle (e.g. θ = 150°), the maximal buoyancy is independent of the liquid surface penetration and equals that for θ = 180°.
as the vertical component of displacement at point r with respect to the plate center (r = 0). For simplicity, a small deformation theory of linearly elastic plates is employed. The differential equation of the deflection curve of a circular thin plate is p d4w 2 d3w 1 d2w 1 dw + − + = , dr 4 r dr 3 r 2 dr 2 r 3 dr D
(10) where p ( r ) is the hydrostatic pressure acting on the lower surface of the plate. It is noticed that the hydrostatic pressure distribution p ( r ) varies with increasing depth H. D is the bending rigidity of the plate and defined as D=
Eδ 3 , 12(1 − ν 2 )
(11) where E, ν and δ are the Young’s modulus, Poisson’s ratio and thickness of the plate, respectively. The hydrostatic pressure p ( r ) is proportional to the depth and expressed as p = ρg ( H − w ), (12) where H denotes the depth of plate at r = 0. Introduce the following parameters normalized by the plate radius R:
4. Wetting of soft circular plates
4.1 Deformation equation of a circular plate
Now we consider a soft circular plate pressing a water surface, as shown in figure 8. Its deflection, w ( r ), is defined
∼ = w / R, w
r∼ = r / R, H͠ = H / R, ∼3 Eδ D . D͠ = = 12ρgR(1 − ν 2 ) ρgR 4
Then equation (10) is recast as 6
(13)
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J. Phys.: Condens. Matter 26 (2014) 155105
where φ1 is the tangent angle of the meniscus at the plate boundary (figure 8). For a rigid plate φ1 equals the apparent contact angle, while for an elastic plate under deformation the apparent contact angle is written as φ1 − (dw / dr ) r =R. Letting γ∼=γ / D͠ ρgR 2=γR 2 / D, equation (20) becomes ∼ v dw ∼⎞ ⎛ d2w ⎜ ∼2 + ∼ ∼ ⎟ ⎝ dr r dr ⎠
∼ 1 dw ∼⎞ d ⎛ d2w + ⎜ ⎟ dr∼ ⎝ dr∼2 r∼ dr∼ ⎠
Figure 8. Schematic diagram of the deformation of an elastic thin plate pressing a liquid surface.
∼ 2 d3w ∼ 1 d2w ∼ 1 dw ∼ H͠ − w ∼ d4w + − + = . dr∼ 4 r∼ dr∼3 r∼2 dr∼2 r∼3 dr∼ D͠
⎛ ∼ ∼4 ∼ = H͠ + 1 C Bei ⎛⎜ r ⎞⎟ + C G 2,0 ⎜ − r w 1 0 2 0,4 ⎜ 256D͠ ⎝ D͠ 1/4 ⎠ 4 ⎝ ⎛ ⎛ r∼ ⎞ r∼ 4 2,0 ⎜ + C3Ber0 ⎜ 1/4 ⎟ + C4G0,4 − ⎜ 256D͠ ⎝ D͠ ⎠ ⎝
—— 1 , 1 ,0,0 2 2
—— 1 , 1 ,0,0 2 2
⎞ ⎟, ⎟ ⎠
(15)
Berv ( z ) = R[ Jv ( z e3π i/4 ) ] , Bei v ( z ) = I[ Jv ( z e3π i/4 ) ] , (17) with R and I designating the real part and the imaginary part of Jv respectively. G is the Meijer G function:
≡
1 2π i
∫
γL
) Π jm= 1Γ ( bj − s ) Π jn= 1Γ ( 1 − aj + s ) Π
Γ ( 1 − bj + s )
Γ ( aj − s ) Π
p j=n+1
q j=m+1
x s ds ,
(18)
where Γ is the Gamma function. To obtain the complete buoyancy–depth curve of the soft circular plate, we assume a displacement-controlled loading condition. The depth of the plate in a very small circular region of 0 ≤ r ≤ a is specified as H. Thus, we have the following two boundary conditions of displacement w: ∼ ∼ ∼ = (dw ∼ / dr∼ ) ∼ ∼ = 0, (19) w r =a r =a ∼ where a = a / R≪1. Along the boundary of the circular plate, the bending moment is zero and the shear force equals the vertical component of surface tension γ . Therefore, the boundary conditions at r = R are ⎛ d2w v dw ⎞ MR = ⎜ 2 + ⎟ ⎝ dr r dr ⎠
∫
r=R
= γ sin φ1, r=R
a
F = 2πRγ sin φ1 + ρgH ⋅ 2πr dr + 0 Its dimensionless form is
= 0,
d ⎛ d2w 1 dw ⎞ QR = − D ⎜ 2 + ⎟ dr ⎝ dr r dr ⎠
=− γ∼sin φ1 .
(21)
In section 4.1, we have shown how to calculate the deformation of an elastic plate with given H and φ1, but its coupling with the meniscus has not been considered. For a soft plate pressing a liquid surface, however, H and φ1 are interdependent. For a given depth H , the value of φ1 should be determined by considering the morphology of the liquid surface, i.e. the meniscus. This problem is highly non-linear because both the hydrostatic pressure and surface tension acting on the plate are coupled with its elastic deformation. As above, we normalize the length parameters by the plate radius R in the following analysis. For a specified value of r0, denoted as r0 − j, the meniscus curve normalized by the capillary length, Sj, can be calculated following the procedure ∼ described in section 2. Let Sj denote the dimensionless meniscus curve normalized by R, which can be obtained via coordinate conversion: r z r∼ = = r D͠ γ∼ , z∼ = = z D͠ γ∼ . (22) κR κR ∼ ∼ Let Cj label the position on Sj satisfying r∼ = 1, where z∼ = z∼1′_ j ∼ and φ1 = φ1_ j. For a given depth H͠ at the plate center, the elastic deformation can be calculated following the method in section 4.1 and the corresponding depth of the plate ∼ can be solved boundary is labeled as z∼1″_ j . A value of φ 1_ j ∼ ∼ ∼ satisfies all conditions from the condition of z1″_n = z1′_n. φ 1_ j of the meniscus geometry in section 2 and the plate deformation in section 4.1 and, therefore, is the actual value of φ1 for the specified plate depth H͠ . The corresponding meniscus morphology and the elastocapillary deformation of the plate are the actual solution. Thus, the actual buoyancy of a deformed plate can be derived by
(16) Jv ( z e3π i/4 ) =Berv ( z ) + iBei v ( z ) , where Jv is the ν-order Bessel function of the first kind, i.e.
(
r∼= 1
4.2 Calculation method
⎞ ⎟ ⎟ ⎠
where C1, C2, C3 and C4 are constants as yet to be determined. Functions Ber and Bei are defined by
, ap Gpm, q, n x ba11,,… … , bq
= 0,
From the four boundary conditions in equations (19) and (20), the parameters C1, C2, C3 and C4 in equation (15) can be determined. Thus, provided that the depth and contact condition of the plate are specified, its elastic deformation can be solved from the above theory.
(14)
Its general solution is
r∼= 1
(20) 7
∼ F=
∫a
R
ρg ( H − w ) ⋅ 2πr dr . (23)
∼2 ∼ F ∼D͠ sin φ + a h + = γ 1 2πρgR3 2
∫a∼ ( h∼ − w∼) r∼ dr∼. 1
(24)
X-Y Ji and X-Q Feng
J. Phys.: Condens. Matter 26 (2014) 155105
can be seen that the descending plate will experience a distinct deformation. For a given rigidity, the larger the plate, the more remarkable its deformation. On the plate boundary, the apparent contact angle is φ1 − (dw / dr ) r =R. Due to the large deformation and the considerable value of a sloping angle (dw / dr ) r =R, the apparent contact angle of the soft plate is significantly smaller than that of a rigid plate at the same depth. Therefore, the apparent contact angle is unlikely to reach the intrinsic value, provided that the material is hydrophobic. In this situation, the liquid penetration by the plate boundary is controlled mainly by the instability of the axisymmetric meniscus, rather than the solid–liquid contact condition along the plate boundary. Therefore, it is rational to disregard the effect of the intrinsic contact angle value in the elastocapillary problem under study. 4.4 Effects of plate size and rigidity
Now we analyze the effects of the radius and bending rigidity of a plate on its wetting behavior. First, the effect of rigidity is examined by fixing the plate radius. Figure 10(a) gives the buoyancy force–displacement curves for four representative values of the bending rigidity D, where we set R=1 mm. It is seen that a plate with a lower bending rigidity can reach a larger buoyancy and a larger depth. The deformed shapes of the plate at its maximal pressing depth are shown in figure 10(b) for the three cases of D =10−8 N m, 10−7 N m and 10−6 N m, respectively. Then we investigate the size effect of the plate on its elastocapillarity by keeping the bending rigidity D = 10−7 N m. Figure 10(c) shows the force–displacement curves of soft plates with different radii. The corresponding results of rigid plates with the same radii are also provided for comparison. Figure 10(d) shows the plate deflections at the maximal depth. It can also be observed from figure 10(c) that the flexibility of the plate enlarges both its maximal buoyancy and depth. Such a tendency is remarkable for larger and softer plates, but the contribution of elastic deformation to the buoyancy is insignificant for smaller and stiffer plates.
Figure 9. Deformations of plates with bending rigidity D =10−7 N m
and different radii during the descending process: (a) R = 0.1 mm, (b) R = 1 mm, (c) R = 2 mm.
5. Conclusions
∼ The two normalized buoyancies, F and F , are related by ∼ ∼ 3/2 (25) F = F R3κ 3 = F / ( D͠ γ∼ ) .
In this paper, the wetting behavior of a circular soft plate pressing downwards on a liquid surface has been investigated. The results show that axisymmetric wetting problems have some qualitative differences from planar, two-dimensional problems, since the geometry of an axisymmetric meniscus involves two principal curvatures while a two-dimensional meniscus involves only one. The wetting behavior of soft plates exhibits a distinct size effect. For a plate with radius smaller than the capillary length, the deviation of the axisymmetric wetting problem from the planar one is more remarkable. Another conclusion is that the flexibility of a plate benefits its water-repellency behavior. A plate with lower bending rigidity can achieve a larger buoyancy and a larger penetration depth. These results deepen our understanding of axisymmetric wetting and elastocapillary phenomena, and the design of biomimetic soft robotics.
4.3 Elastocapillarity of soft plates
For illustration, we take the following parameters of pure water for the liquid: surface tension γ = 0.072 N m−1, ρg=104 kg m−2 s−2, and the capillary length κ −1 = 2.7 mm. For the soft plate, we set Young’s modulus as E = 1 GPa and the bending rigidity D = 10−7 N m, which corresponds to a plate thickness of about 10 µm. The elastocapillary deformation of a plate in the descending process is calculated by using the method in section 4.2. Figures 9(a)–(c) show the results of plates with a few representative radii of R = 0.1 mm, 1.0 mm and 2.0 mm, respectively. In the calculation, we set the parameter a = 0.01R. It 8
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J. Phys.: Condens. Matter 26 (2014) 155105
Figure 10. (a) Buoyancy–depth relations and (b) maximal deformations of plates with R =1 mm and different bending rigidities during the
descending process. (c) Buoyancy–depth relations and (d) maximal deformations of plates with D =10−7 N m and different radii.
Finally, it is pointed out that for a soft plate pressing against a liquid surface, both the plate and the meniscus may become unstable with increasing depth. The instabilities involved in this problem and their influences on the wetting behavior deserve further research.
Normalize all length parameters by the capillary length and define
Acknowledgments
dφ sin φ − = κ 2z. ds r
s = κs ,
z = κz ,
(A3)
r = κr .
(A4)
Then the dimensionless form of equation (A3) is
Support from the National Natural Science Foundation of China (Grant No 31270989) and the 973 Program of MOST (2013CB933033 and 2012CB934101) is acknowledged.
dφ sin φ − = z. ds r
(A5)
Use the geometric relations Appendix. Derivation of Young–Laplace equation for axisymmetric menisci The Young–Laplace equation of a meniscus reads
⎛1 1⎞ γ ⎜ + ⎟ = Δp = ρgz, ⎝ R1 R2 ⎠
(A1)
where Δp is the Young–Laplace pressure difference across the liquid surface and equals the hydrostatic pressure at depth z, and R1 and R2 are the two principal curvatures of the meniscus. In the axisymmetric case, one has
ds = R1dφ , r = − R2sin φ ,
dz dr = − cot φ , = − tan φ , dr dz ⎛ dr 1 π⎞ , = − cos φ = sgn ⎜ φ − ⎟ ⎝ ds 2 ⎠ 1+(dz / dr )2 ⎛ dz π⎞ = sin φ = sgn ⎜ φ − ⎟ ⎝ ds 2⎠
dz / dr 1+(dz / dr )2
.
(A6)
Thus, the Young–Laplace equation for axisymmetric menisci in equation (A5) can be rewritten in terms of z and r as ⎛ dr ⎞ ⎡ ⎛ dz ⎞2 ⎤ 2 ⎡ 1 d2z 1 dz ⎤ − sgn ⎜ ⎟ ⎢ 1+⎜ ⎟ ⎥ ⎢ + ⎥ = z. ⎝ dz ⎠ ⎣ ⎝ dr ⎠ ⎦ ⎣ 1+(dz / dr )2 dr 2 r dr ⎦ (A7) −1
(A2)
where s denotes the arc length. Substituting equation (A2) into (A1) and using the definition of capillary length leads to 9
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J. Phys.: Condens. Matter 26 (2014) 155105
[13] Hure J, Roman B and Bico J 2011 Phys. Rev. Lett. 106 174301 [14] Jamin T, Py C and Falcon E 2011 Phys. Rev. Lett. 107 204503 [15] Liu J L and Feng X Q 2012 Acta Mech. Sin. 28 928 [16] Park K J and Kim H Y 2008 J. Fluid Mech. 610 381 [17] Vella D 2008 Langmuir 24 8701 [18] Ji X Y, Wang J W and Feng X Q 2012 Phys. Rev. E 85 021607 [19] Burton L J and Bush J W M 2012 Phys. Fluids 24 101701 [20] Roman B and Bico J 2010 J. Phys.: Condens. Matter 22 493101 [21] Jung S, Reis P M, James J, Clanet C and Bush J W M 2009 Phys. Fluids 21 091110 [22] Reis P M, Hure J, Jung S, Bush J W M and Clanet C 2010 Soft Matter 6 5705 [23] Jiang C G, Xin S C and Wu C W 2011 AIP Adv. 1 032148 [24] Yong J, Yang Q, Chen F, Zhang D, Du G, Si J, Yun F and Hou X 2014 J. Micromech. Microeng. 24 035006 [25] Su Y W, He S J, Ji B H, Huang Y and Hwang K C 2011 Appl. Phys. Lett. 99 263704 [26] Phan C M, Allen B, Peters L B, Le T N and Tade M O 2012 Langmuir 28 4609 [27] Chan T S, Gueudré T and Snoeijer J H 2011 Phys. Fluids 23 112103 [28] Ji X Y and Feng X Q 2013 Langmuir 29 6562
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