Article pubs.acs.org/Langmuir
Bubbles and Drops on Curved Surfaces Majid Soleimani,† Reghan J. Hill,*,† and Theo G. M. van de Ven‡ †
Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 0C5, Canada Department of Chemistry, McGill University, Montreal, Quebec H3A 2K6, Canada
‡
S Supporting Information *
ABSTRACT: Surface curvature affects the shape, stability, and apparent contact angle of sessile and pendant drops. Here, we develop an approximate analytical solution for non-axisymmetric perturbations to small spherical drops on a flat substrate, assuming a fixed contact angle and fixed drop volume. The analytical model is validated using numerical solutions of the Laplace equation from the Surface Evolver software. We investigate the effects of surface curvature on drop shape, pressure, and surface energy, ascertaining the energy-gradient force that drives lateral drop migration. By balancing this force with the viscous resistance/drag force, in the lubrication approximation, we predict velocities of the order of 0.1 mm s−1 for 1 mm diameter drops of water with a 30° contact angle on a substrate with a curvature gradient of 0.01 mm−2, achieved, for example, on a harmonic surface with a wavelength of 4 cm and an amplitude of 4 mm.
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INTRODUCTION Bubbles and drops on curved surfaces are ubiquitous in technological applications and in nature. Their shapes are controlled by wettability, which depends on the chemical and physical properties of the substrate and its morphology. Because of its numerous applications,1 wettability analysis by means of drop morphology has been the subject of many studies.2 For example, numerical and experimental methods have been described for contact-angle measurement of heterogeneous curved surfaces by extending the axisymmetric approximation for drop shape, because traditional contact-angle measurements are restricted to flat, ideal, horizontal surfaces. Iliev and Pesheva3 proposed a numerical method for specifying the local contact angle of a drop when the contact line is known for a given drop volume and capillary length. Guilizzoni4 experimentally explored the shapes and contact angles of drops on curved surfaces using image processing, spline fitting, and numerical integration, extracting the drop contour in a number of cross sections. This procedure requires only a side view and can be used for fluids with unknown surface tension, without needing to measure the drop volume. However, recent investigations by Extrand and Moon5 showed that substrate curvature does not affect the intrinsic contact angle. They deposited small liquid drops at the apex of polymer spheres to analyze the wetting of structured surfaces with spherical or cylindrical features, concluding that the intrinsic contact angles are the same as on flat, horizontal surfaces. Another influence of substrate curvature is to impart a topological change in the interface configuration. Eral et al.6 experimentally and analytically studied the equilibrium morphology of a drop on a sphere as a function of the contact angle and drop volume. They used electrowetting to precisely control the wetting parameters, which allowed all equilibrium © 2013 American Chemical Society
morphologies to be accessed. This showed that the partially engulfing morphology is energetically more favorable than the spherically symmetric, completely engulfing morphology, as proven earlier by Smith and van de Ven.7 Other examples of bubbles and drops on curved surfaces occur in elastocapillarity,8−11 fog harvesting,12 electrowetting,13,14 and electrodeposition15 processes. For a liquid−gas interface to be at equilibrium, there are two hydrostatic conditions that must be satisfied simultaneously.16 The first is the Laplace condition, which requires that the pressure jump across a liquid−gas interface be equal to the product of interfacial tension γ and mean curvature 2H: ⎛1 1 ⎞ p0 − p = 2Hγ = γ ⎜ + ⎟ R2 ⎠ ⎝ R1
(1)
where R1 and R2 are the principal radii of curvature and p and p0 are the liquid and gas pressures, respectively. The second is the Dupré−Young condition at the contact line: γ cos α = γsg − γsl (2) where γsl and γsg are the solid−liquid and solid−gas interfacial tensions, respectively, and α is the equilibrium contact angle. Equation 2 gives16 n⃗ i ·n⃗ s = cos α
(3)
where n⃗i and n⃗s are the unit normal vectors on the interface and substrate, respectively. The equilibrium shape is obtained by solving eq 1 with eq 3 and a prescribed volume and substrate shape. Received: August 9, 2013 Revised: October 3, 2013 Published: October 4, 2013 14168
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Perturbation methods are useful in many capillary problems.17−21 A notable analysis closely related to this study was undertaken by Myshkis et al.16 They derived perturbation equations for a general interface by perturbing all the relevant parameters. However, because the final set of differential equations is very complicated, they demonstrated existence conditions and a general solution form, not an exact solution. Although many studies have addressed drop- or bubbleshape analysis on curved surfaces, no analytical solution has been derived for non-axisymmetric curved surfaces that furnishes their profiles, internal pressure, and interfacial energy. Here, we propose a general solution for the equilibrium drop shape on a curved surface in the absence of gravity. One motivation is to understand the elastocapillary-driven wood-fiber deformation during drying. As wood fibers dry, bubbles or drops form in the lumen or between entangled fibers.22 The capillary pressure may play a key role in fiber deformation, and it substantially affects fiber strength and conformation.23,24 Here, we assess the effect of surface curvature on the drop shape, internal pressure, and surface energy. The results might be used to identify conditions that change an open lumen to a flattened structure upon drying. We consider a spherical drop on a flat surface, which we perturb to compute the resulting drop shape. The new interface and contact line are obtained by solving eq 1 while satisfying eq 3 at a constant volume. The solution has a singularity that we remove by restructuring the solution and matching the new form to the original one at the boundary of the problematic neighborhood. This solution is then applied to a drop inside or outside a circular tube and verified numerically using Surface Evolver.25 The resulting interface shape, drop pressure, and energy match the numerical computations well for all contact angles, even for large values of the perturbation parameter. Because the energy changes with substrate curvature, drops are expected to migrate to the lowest-energy positions. This possibility is studied theoretically and verified with Surface Evolver for a drop on a harmonic surface. We estimate the lateral velocity by balancing the lateral energy-gradient force with the viscous resistance force, in the lubrication approximation.
Figure 1. Perturbation (dashed curves) of an initially spherical drop on an initially flat substrate (solid curves).
and the volume enclosed by the interface and the support is obtained by integrating eqs 4 and 5, giving V=
z 0 = −cos α + εh(r , θ )
(8)
where ε is the perturbation parameter and θ is the azimuthal coordinate, and ∞
h(r , θ ) = A 0(r ) +
∑ [A n(r) cos(nθ) + Bn(r) sin(nθ)] n=1
(9)
Note that h(r,θ) is continuous at r = 0, and An,r(r), Bn,r(r), nAn(r), and nBn(r) are O(1) values with the subscripts r and θ denoting partial derivatives. Because of the substrate perturbation, the contact line deviates from a circle. We approximate the perturbed contact line by adding a perturbation to eq 6 r = sin α + εf (θ ), z = − cos α + εh(r , θ )
THEORY Consider a spherical fluid−fluid interface on a planar supporting substrate with a three-phase contact angle α, as shown in Figure 1. The origin O is at the sphere center, and the distances are scaled with sphere radius R0. In cylindrical coordinates (r,θ,z), the interface is z(r ) = ± 1 − r
(7)
Now we perturb the substrate by adding a disturbance to eq 5, giving
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2
⎞ 2π ⎛⎜ 3 1 − cos3 α − sin 2 α cos α⎟ ⎠ 3 ⎝ 2
(10)
where f(θ) has the same form as eq 9. Similarly, the interface deviates from a sphere. If α ≤ π/2, then z(r , θ ) = − 1 − r 2 + εg (r , θ )
(11)
where ∞
g (r , θ ) = ξ0(r ) +
(4)
∑ [ξn(r) cos(nθ) + ηn(r) sin(nθ)] n=1
and the substrate is z 0 = −cos α
(5)
For a flat surface, the contact line is a circle r = sin α , z = − cos α
(6)
(12)
⎡ 1 ∂ (rzr )⎢1 r ∂r ⎣
+
2⎤
⎡
Here, we must specify the functions ξn(r) and ηn(r) using the known functions An(r) and Bn(r) and by solving eq 1 with appropriate boundary conditions. From eq 1, the mean curvature in cylindrical coordinates when z = z(r,θ) is26 2⎤
( zr ) ⎥⎦ + zr ⎢⎣zr 2 − 2 zr zrθ + ( zr ) ⎥⎦ + zr θ
r
⎡ 2 ⎢⎣1 + zr +
θ
3/2 zθ 2 ⎤ ⎥ r ⎦
θ
θθ 2
(1 + zr 2) = 2H
( )
(13)
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Because the drop pressure is a constant that depends only on the mean interfacial curvature, the perturbed curvature is
(r 4 − r 2)ξ0, rr + (4r 3 − r )ξ0, r =
k H=1+ε (14) 2 Substituting eqs 11 and 14 into eq 13 and retaining only the terms up to order ε give
kr 2 1 − r2
(r 4 − r 2)ξn , rr + (4r 3 − r )ξn , r + n2ξn = 0 (r 4 − r 2)ηn , rr + (4r 3 − r )ηn , r + n2ηn = 0
(16)
kr 2
(r 4 − r 2)grr + (4r 3 − r )gr − gθθ =
1 − r2
(15)
which must be solved with a constant volume, specified substrate shape, and the Dupré−Young condition. To satisfy eq 3, the perturbed substrate and interface normal vectors
Now, because {sin(nθ),cos(nθ)} is a complete orthogonal system, from eqs 12 and 15 ⎡ ⎤ h (r , θ ) n⃗ s = ⎢εhr (r , θ ), ε θ , −1⎥ + O(ε 2) ⎣ ⎦ r
⎡ ⎤ 1 − r2 n⃗ i = ⎢r + εgr (r , θ )(1 − r 2)3/2 , ε gθ (r , θ ), − 1 − r 2 + εrgr (r , θ )(1 − r 2)⎥ + O(ε 2) r ⎢⎣ ⎥⎦
where hr(r,θ) and hθ(r,θ) are assumed to be O(1), which furnish additional conditions on An(r) and Bn(r). Satisfying eq 3 at the perturbed contact line, which is given in eq 10, and using eq 17 give f (θ ) = cos αhr (sin α , θ) − cos3 αgr (sin α , θ)
z[sin α + εf (θ ), θ ] = z 0[sin α + εf (θ), θ ]
(17) (19)
Substituting eqs 8 and 11 into eq 19 and using eq 18 give −sin α cos2 αgr (sin α , θ) + g (sin α , θ) = h(sin α , θ)
(18)
− sin αhr (sin α , θ )
Another boundary condition comes from the specified substrate shape. At the contact line, the interface and the substrate intersect, requiring
(20)
Then, substituting eqs 9 and 12 into eq 20 furnishes the boundary conditions for eq 16:
−sin α cos2 αξn , r(sin α) + ξn(sin α) = A n(sin α) − sin αA n , r (sin α) −sin α cos2 αηn , r (sin α) + ηn(sin α) = Bn(sin α) − sin αBn , r (sin α)
(21)
Using eq 21 and demanding a finite g(0,θ), the solution of eq 16 is available in the Supporting Information, giving
g (r , θ ) = C 0 +
k 2 1 − r2
⎧ ⎪ (n +
∞
+
∑⎨
1 − r 2 )[2(1 −
⎪
n=2
⎩
rn 1 − r2
where C0 = A 0(sin α) − sin αA 0, r (sin α) −
Dn =
k cos α 2
[A n(sin α) − sin αA n , r (sin α)] sin α (1 − n2)[2(1 − cos α) − sin 2 α]n /2
2π
sin α + εf (θ )
2π
sin α + εf (θ )
V=
∫0 ∫0
−
∫0 ∫0
[Bn(sin α) − sin αBn , r (sin α)] sin n α (1 − n2)[2(1 − cos α) − sin 2 α]n /2
⎫ ⎪ [Cn cos(nθ ) + Dn sin(nθ )]⎬ ⎪ ⎭
(22)
The only remaining unknown is the perturbation to the mean curvature (or pressure), which requires specifying k. With a constant volume
n
Cn =
1 − r 2 ) − r 2]n /2
⎡ ⎤ 2 ⎣ − 1 − r + εg (r , θ )⎦r d r d θ [−cos(α) + εh(r , θ )]r dr dθ (24)
Equations 7 and 24 give
(23)
k=
In eq 22, the first harmonic is not included because the equilibrium condition precludes satisfying the Laplace and Dupré−Young conditions simultaneously. The first mode has previously been excluded in the contact-line perturbation17 and contact-angle perturbations.27
{
−2
∫0
sin α
A 0(r )r dr + sin 2 α[A 0(sin α)
}
− sin αA 0, r (sin α)]
⎛ ⎞ 1 ⎜cos α − 1 + sin 2 α cos α⎟ ⎝ ⎠ 2 (25)
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The solution is now complete for α ≤ π/2, but as r → 1, eq 22 becomes singular. Therefore, to remove this singularity, we bring the singular part under the square root, giving z = − 1 − r 2 + εγ1(r , θ) + ε 2γ2(r , θ) + εγ3(r , θ)
as region 2 in Figure 2. Almost the same technique was used by Chesters,21 but he did not keep the second-order term under the square root. This may not be correct, because at the maximal value of r the imaginary part of z is O(ε) and cannot be neglected. After matching eqs 22 and 26 at r = 1 − |ε| (see the Supporting Information), we find
(26)
where γ1−γ3 are found by matching eqs 22 and 26. This new form is then used in the interval [1 − |ε|,rmax], which is illustrated F (r , θ ) =
k + 2
∞
∑ {nr −n[Cn cos(nθ) + Dn sin(nθ)]} n=2
γ1(r , θ ) = −2F +
F2 F2 sgn(ε), γ2(r , θ ) = 2 4
∞
γ3(r , θ ) = C0 +
∑ {r −n(2 − r 2)n/2 − 1(2 − r 2 − n2)[Cn cos(nθ) + Dn sin(nθ)]} (27)
n=2
If sin α > 1 − |ε|, then the contact line is in the interval [1 − |ε|,rmax]. Setting α = π/2 − |ε|β gives 1 sin α = 1 − ε 2β 2 and cos α = εβ (28) 2 so from eq 3 we find the contact line equation in this interval r=1+ε
γ1(1, θ ) 2
− γ2(1, θ ) +
−
For contact angles α > π/2, there is little change in the equations upon passing the z = 0 plane, because the mean curvature and the right-hand side of eq 4 change sign there. Accordingly, there are two solutions that must match at rmax. We apply the same procedure described above to α > π/2, furnishing drop shapes for all contact angles. Depending on the contact angle, we divide the problem into four separate cases (Figure 2), because the drop-shape equation varies near r = 1, and the contact-line equation should be adapted correspondingly. When α > π/2, for regions 1 and 4
1 2⎧ ε ⎨[β sgn(ε) − hr (1, θ )]2 2 ⎩ ⎪
⎪
γ1(1, θ )2 ⎫ ⎬ 4 ⎭ ⎪
⎪
g (r , θ ) = C 0 ±
(29)
k 2 1 − r2
∞
±
⎧ ⎪ (n ±
∑⎨
1 − r 2 )[2(1 ∓
⎪
n=2
⎩
1 − r 2 ) − r 2]n /2
rn 1 − r2
where the top sign (in the pair ± or ∓) identifies region 1 and the bottom sign identifies region 4 (k, C0, Cn, and Dn are defined in eqs 23 and 25). For regions 2 and 3
⎫ ⎪ [Cn cos(nθ ) + Dn sin(nθ )]⎬ ⎪ ⎭
we have 2π
z = ∓ 1 − r 2 + εγ1(r , θ ) + ε 2γ2(r , θ ) + εγ3(r , θ )
Alg =
∫0 ∫0
A sl =
∫0 ∫0
sin(α) + εf (θ )
(31) 2π
where the top sign identifies region 2 and the bottom sign identifies region 3 with the same γ1−γ3 as in eq 27. The maximal value of r for cases 3 and 4 is rmax = 1 + ε
γ1(1, θ ) 2
⎡ γ (1, θ ) γ (1, θ )2 ⎤ ⎥ + ε 2⎢ 2 − 1 ⎢⎣ 2 ⎥⎦ 8
ϕ(r , θ , z) = z + cos α − εh(r , θ ) = 0
|∇ϕ| r dr dθ |∇ϕ·ez|
(37)
+ sin αA 0, r (sin α)] + O(ε 2)
(38)
⎡ k ⎤ A sl = π sin 2 α + 2πε⎢− sin α + cos αA 0, r (sin α)⎥ + O(ε 2) ⎣ 2 ⎦ (39)
so the total energy (to within an arbitrary constant) from eq 33 is
(33)
where subscripts sg, sl, and lg denote solid−gas, solid−liquid, and liquid−gas interfaces, respectively. With 1 − r 2 − εg (r , θ ) = 0
(36)
Alg = 2π (1 − cos α) + 2πε[−k(1 − cos α) (32)
E =γAlg + γsgA sg + γslA sl
S(r , θ , z ) = z ±
sin(α) + εf (θ )
|∇S| r dr dθ |∇S ·ez|
which become
Energy. In the absence of gravity, the total energy is the surface energy
=γAlg − γA sl cos α + constant
(30)
⎛ ⎞ E 1 = 2π ⎜1 − cos α − cos α sin 2 α⎟ 2 ⎝ ⎠ 2 γR 0 sin α ⎡ ⎤ A 0(r )r dr + sin 2 αA 0(sin α)⎥ + 2πε⎢ −2 ⎣ ⎦ 0
(34)
∫
(35) 14171
(40)
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Figure 2. Configurations for various contact angles. Small circles identify singularities at r = 1.
Application to Circular Tubes. As an example, we consider the perturbation of a spherical drop on a plane that is deformed into a cylinder, as shown in Figure 1. In Cartesian coordinates, the cylinder surface is described by 2
2
2
(z ± R + R 0 cos α) + x = R
⎡ k z( r , θ ) = − 1 − r 2 + ε ⎢C 0 + ⎢⎣ 2 1 − r2 ⎤ ⎛ ⎞ 2 2 + C2⎜⎜ − 2 − 1⎟⎟ cos(2θ )⎥ ⎥⎦ r ⎝ r2 1 − r2 ⎠
(41)
• region 2
where R is the cylinder radius. Scaling with drop radius R0 ⎛ ⎞2 ⎛ 1 ⎞2 1 ⎜z − + cos α⎟ + x 2 = ⎜ ⎟ ⎝ ⎠ ⎝ε⎠ ε
z = − 1 − r 2 + εγ1(r , θ ) + ε 2γ2(r , θ ) + εγ3(r , θ ) (42)
{ 14 r [1 + cos(2θ)]}
(45)
• region 3
where, in this example, ε = R0/R with |ε| ≪ 1. Note that ε < 0 (>0) implies that the drop is inside (outside) the tube. Now we write eq 42 explicitly in z, linearize it in ε, and transform to cylindrical coordinates, giving z 0 = −cos α + ε
(44)
z=
1 − r 2 + εγ1(r , θ ) + ε 2γ2(r , θ ) + εγ3(r , θ ) (46)
• region 4
2
z( r , θ ) =
(43)
Comparing eqs 8 and 43 furnishes A0(r) and A2(r), and from eqs 22, 26, 30, and 31, we find (see Figure 2)
⎡ k 1 − r 2 + ε ⎢C 0 − ⎢⎣ 2 1 − r2
⎤ ⎛ ⎞ 2 2 − C2⎜⎜ + 2 + 1⎟⎟ cos(2θ )⎥ ⎥⎦ r ⎝ r2 1 − r2 ⎠
• region 1 14172
(47)
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where C0 = −
1 k sin 2 α − cos α 4 2
C2 =
sin 4 α 24(1 − cos α) − 12 sin 2 α
k=
−3 sin 4 α 4(2 cos α − 2 + sin 2 α cos α)
(48)
and γ1(r , θ) = − k − γ2(r , θ) =
⎤2 4C2 2C2 1⎡1 θ + + θ cos(2 ) cos(2 ) ⎢ ⎥⎦ sgn(ε) 2⎣2 r2 r2
⎤2 2C2 1⎡k ⎢⎣ + 2 cos(2θ)⎥⎦ 4 2 r
γ3(r , θ) = C0 −
⎛2 ⎞ + 1⎟C2 cos(2θ) ⎝ r2 ⎠
⎜
(49)
Numerical Solution. We verified the foregoing analytical approximation using Surface Evolver, which is for studying equilibrium surfaces shaped by surface tension. From a userdefined initial guess of the shape, the program advances the surface to lower-energy states by displacing mesh vertices. The user can progressively refine the mesh to achieve a desired accuracy. Details of the algorithm and the program implementation are available in the user manual.28 For a drop inside or outside a circular tube, Surface Evolver computes the equilibrium interface with a fixed volume (eq 7) and zero density. For example, from a pyramid as the initial shape, with one of the faces restricted to the tube wall, computations converged to nine significant digits in the total energy using 20480 facets. An analysis demonstrating first-order convergence of the energy with respect to the number of facets is available as Supporting Information. Moreover, we demonstrate, using the pressure perturbation, that the errors of the analytical approximation are O(ε2). Drop shapes for two representative examples are shown in Figure 3. Quantitative comparisons with the analytical theory are examined below.
Figure 4. Drop profiles with an α of 60° and ε values of −0.1 (blue) and −0.3 (red) (top). Drop profiles with α values of 50° (blue), 90° (red), and 120° (green) and an ε of −0.1 (middle). Scaled surfacedisplacement perturbation profile with an α of 50° and ε values of −0.1 (blue), −0.3 (red), and −0.5 (green) (bottom). Symbols are from Surface Evolver, and lines are the perturbation approximations.
value for the perturbation parameter, and we do not expect to obtain the same correspondence when an increasing ε. When ε = −0.1, the transition is smooth, and inward bending of the surface stretches the drop normal to the tube axis, also compressing it along the tube axis. Drop profiles in the tube cross section with a constant surface curvature and several contact angles are shown in Figure 4 (middle). As in the top panel, where ε = −0.1 for both hydrophobic and hydrophilic surfaces, the transitions between regions are smooth. Displacement profiles scaled with ε, i.e., g(r,θ), from eq 11 are shown in Figure 4 (bottom). For small values of ε, the numerical solution corresponds well with the perturbation solution, deviating gradually with an increasing |ε|. However, even when ε = −0.5, the difference is small compared to g(r,θ). Next, we compare the numerical and perturbation approximations of the pressure and energy. As explained in Theory, the pressure difference across the interface changes with substrate curvature, and this change is equal to k(α)ε, with k(α) given by eq 25. Note that k depends on only the axisymmetric part of perturbation A0(r), because the other terms are periodic in θ and vanish upon being integrated with respect to θ from
Figure 3. Drop shapes from Surface Evolver with an α of 90° and ε values of −0.5 (left) and 0.5 (right).
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RESULTS Drop profiles in the tube cross section are shown in Figure 4 (top) for ε values of −0.1 and −0.3 with an α of 60°. When ε = −0.3, there is a nonsmooth transition between regions 1 and 2 in Figure 2, and the perturbation solution deviates from the numerics. Because of the larger errors in the contact-line calculation, the drop profile and the substrate do not match well at the contact line. Note that an |ε| of 0.3 is a large absolute 14173
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Figure 5. Scaled pressure perturbation (k from eq 48) vs contact angle (left). Symbols are from Surface Evolver with ε values of −0.1 (blue) and 0.5 (red), and the line is the perturbation approximation. Scaled internal drop pressure vs aspect ratio (right). Symbols are from Surface Evolver, and lines are the perturbation approximations with α values of 30° (red), 90° (blue), and 150° (green).
Figure 6. Scaled energy perturbation vs contact angle (left). Symbols are from Surface Evolver with ε values of −0.1 (blue) and 0.5 (red), and the line is the perturbation approximation. Scaled energy vs aspect ratio (right). Symbols are from Surface Evolver, and lines are the perturbation approximations with α values of 30° (red), 90° (blue), and 150° (green).
capillary pressure and an accompanying force that impacts fiber collapse. In Theory, we showed that the surface energy changes with the substrate curvature when the volume and the contact angle are fixed. This change is expressed as a function of contact angle in eq 40, retaining only the first-order terms in ε. The energy change depends only on the axisymmetric part of the perturbation. Figure 6 (left panel) shows how the energy changes with α with constant substrate curvatures (ε) of −0.1 and 0.5. As in the left panel of Figure 5, the largest difference between the Surface Evolver and perturbation results occurs when α = 90°. The energy change increases with contact angle for hydrophilic surfaces and decreases for hydrophobic surfaces, with the maximum at α ≈ 90°. Figure 6 (right panel) shows surface energy versus ε with three different contact angles. Here, the energy changes that accompany the changing substrate curvature are small compared to the initial energy. The total surface energy increases with curvature when the drop is outside the tube and decreases when it is inside the tube.
0 to 2π when the drop volume is calculated. Thus, the nonaxisymmetric part of the perturbation does not change the internal pressure. Figure 5 (right panel) compares k(α)ε to Surface Evolver computations with three contact angles. Note that the internal pressure increases with substrate curvature when the drop is outside the tube and decreases when it is inside the tube. Accordingly, because the interface curvature and internal pressure are related by eq 1, the drop curvature increases when the drop is outside the tube and decreases when it is inside the tube. When ε = 0, the drop is spherical and the three lines in Figure 5 intersect. While the slopes of the lines in the right panel of Figure 5 equal k and are independent of ε, they vary with the contact angle. Figure 5 (left panel) compares the perturbation approximation of k with Surface Evolver computations with curvatures (ε) of −0.1 and 0.5. Here, the theory agrees well with the numerics for all contact angles when ε < 0.3 but deviates with an increasing ε, with a maximal deviation occurring when α ≈ 90°. Here, Surface Evolver computations exhibited the slowest convergence, possibly because of the coincidence of the contact line with the singularity in the drop profile at the extremum (r = 1); interestingly, the pressure perturbation becomes nonlinear in ε when α ≈ 90° when ε > 0.2. Note that k decreases with an increasing α, indicating that the perturbed pressure decreases with an increasing contact angle. For very hydrophobic surfaces (α > 140°), curvature does not significantly influence the pressure. For hydrophilic surfaces, which are characteristic of wood fibers, however, the pressure perturbation can be a significant fraction of the unperturbed pressure. Thus, for example, when wood fibers dry, the air bubbles that grow inside the fiber lumen from micrometer dimension holes in the cell wall (pit holes) may produce a
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AUTONOMOUS DROPLET MOTION The foregoing energy change manifests as an energy-gradient force that drives lateral drop migration. To quantify this force, we consider a drop on a harmonic surface z(r , θ ) = A cos(κx)
(50)
where A, κ = 2π/λ, and λ are the amplitude, wavenumber, and wavelength, respectively. Figure 7 shows a drop on such a surface where a local coordinate system O′x′z′ is positioned at the drop center, and the osculating circle identifies the surface curvature. If the surface is locally approximated by a circle, this problem is analogous to the drop on the circular tube 14174
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the highest surface curvature, thus minimizing its surface energy. The force is plotted as a function of x in Figure 9 (left panel). While the force vanishes at x = 0, this is not a position of stable equilibrium. The surface valleys are the energetically favored (stable) positions for drops and bubbles. Note that, because there is no viscosity and a no-slip boundary condition in the Surface Evolver computation, there is no time scale. Nevertheless, because the force perturbs the equilibrium three-phase contact line, the apparent contact angle is the dynamic contact angle, and the drop moves. If we neglect energy dissipation in the precursor film of the moving contact line and in the external fluid, then the tangential component of this force (Ft) is balanced by a viscous resistance −ζηR0U, where ζ is a dimensionless friction coefficient and η is the fluid viscosity. The resulting drop velocity is
Figure 7. Sinusoidal surface showing the osculating circle (dashed line) and its parabolic approximation (dashed−dotted line).
considered above. Thus, with a parabolic approximation of the circle 1 z′(r , θ ) = εx′2 (51) 2 and eq 40 furnishes the drop energy. Because the curvature is independent of the coordinate system, we can express ε as a function of x, for example ε=−
Aκ 2 cos(κx) 1 = R {1 + [Aκ sin(κx)]2 }3/2
U=
dE π dε = − sin 4 α dx 4 dx
(54)
Note that three sources of dissipation are present with a moving contact line:30 bulk hydrodynamic dissipation, viscous dissipation in the precursor film, and dissipation at the real contact line at the end of the precursor film. Although de Gennes30 proved that only the dissipation at the real contact line can be neglected for very small contact angles, it has been popular to neglect the precursor film dissipation.17,31−33 The spherical drop-shape assumption during drop motion implies that the advancing and receding contact angles are equal to the thermodynamic contact angle, deviating from the theory of de Gennes.30 Thus, contact-angle hysteresis is not considered. Here, we estimate ζ using a lubrication approximation, which restricts the solution to small contact angles where the drop height is small with respect to its radius. Daniel et al.33 showed that this approximation is valid when α < 45°. The shear stress at the substrate−drop interface is
(52)
which furnishes a lateral force Fx = −
γR 0Ft γ dε 1 π = − sin 4 α ζηR 0 4 ζη dx {1 + [Aκ sin(κx)]2 }1/2
(53)
This result is validated by Surface Evolver in Figure 8, where a drop is seen to migrate (from right to left) to a position where
τ=η
Figure 8. Drop motion induced by surface curvature on a sinusoidal surface calculated according to Surface Evolver (time advancing from left to right). The drop migrates from right to left, toward the position with the lowest surface energy.
3U h(r , θ )
(55)
where h(r,θ) is the local drop height. Neglecting the effects of gravity and motion on drop shape, we assume the drop surface is spherical. To avoid the contribution of a nonintegrable singularity in τ at the drop periphery, the integration domain must be truncated near the contact line. Following de Gennes,30 the cutoff distance is furnished by consideration of van der Waals interactions, giving
the substrate has the largest negative (inward) curvature. This curvature-driven drop motion seems to explain the experiments of Renvoisé et al.29 in which a liquid bridge in a tapered tube was observed to self-propel itself to the end with
Figure 9. Scaled force (Fx = −dE/dx) and energy (E) (left) and scaled migration velocity (U) (right) vs scaled position for a 1 μL drop on a surface for which A/R0 = 5, λ/R0 = 40, and α = 30°. 14175
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Figure 10. Dimensionless friction coefficient (lubrication approximation) vs contact angle with contact radius R0 sin α values of 10−3 (blue), 10−4 (red), 10−5 (green), and 10−4 (yellow) (left). Maximal scaled velocity on a wrinkled surface vs contact angle with Γ (=κ2AR0 sin α) values of 0.001 (blue), 0.01 (red), and 0.1 (green) (right).
δ=
AH 6πγα 4
the drop profile. The solution has a singularity at the extremum (r = 1) when the contact angle exceeds 90°, but this can be removed by reconstructing the solution and using the new form in a small neighborhood near the extremum. Calculating the internal pressure and the surface energy of the perturbed drop, we showed that, at a constant volume, the pressure and the energy change with the axisymmetric part of the perturbation. The solution was then used to compute the shape of a drop inside or outside a circular tube, and these analytical results were compared to numerical solutions from Surface Evolver. The drop profiles and their perturbations correspond well with the numerical computations for all contact angles, although they depart with O(ε2) errors with an increase in perturbation parameter ε. We compared the calculated pressure and energy with the numerical results, showing that the internal pressure and surface energy increase with substrate curvature when the drop is outside the tube and decrease when it is inside the tube. Comparing the pressure and the energy at a constant substrate curvature, but different contact angles, revealed that the pressure change decreases with the contact angle, while the surface energy change attains a maximum when the contact angle is 90°. Extending the example to a drop on a harmonic surface, we showed that the most energetically favored (stable) positions for drops are the surface valleys. Indeed, drops are expected to migrate to these positions, driven by a force that is proportional to the surface-curvature gradient. We estimated the drop velocity by balancing the energy-gradient force with the viscous resistance obtained from a lubrication approximation and found that this velocity increases with surface-curvature gradient, contact angle, and drop radius.
(56)
where AH is the Hamaker constant and δ is the cutoff distance. With an AH of ≈10−20 J and a γ of ≈0.05 J m−2, we find δ≈
10−10 m rad−2 α2
(57)
Next, integrating the shear stress over the truncated contact area of a spherical drop gives ⎧ ⎫ ⎡ ⎤ δ tan α ζ = 6π ⎨ − cos α ln⎢ ⎥ + (1 − cos α)⎬ ⎣ R 0(1 − cos α) ⎦ ⎩ ⎭ ⎪
⎪
⎪
⎪
(58)
34
which is the same as that derived by Subramanian et al. when we replace sphere radius R0 in eq 54 with contact radius R0 sin α. As shown in Figure 10 (left panel), plotting ζ at a fixed contact radius (R0 sin α = constant) demonstrates the effect of the contact angle on the constant contact area. Note that ζ decreases with drop size and contact angle and diverges with vanishing contact angle. From ζ, we can estimate the drop velocity. For example, as shown in Figure 9 (right panel) for a 1 μL drop with an η of 0.001 Pa s and a γ of 0.07 N m−1 on a wrinkled surface at α = 30°, A/R0 = 5, and λ/R0 = 40, the velocity is ≈0.2 mm s−1. Dimensional analysis shows that the dimensionless drop velocity Uη/γ depends on three independent dimensionless parameters: α, κR0 sin α, and A/(R0 sin α). Note that Uη/γ is the capillary number, which is the ratio of the characteristic viscous shear stresses to the capillary stresses. As shown in Figure 10 (right panel), this scaled velocity is small, indicating (as we required for the lubrication analysis) that the drop migrates with negligible deformation. Changes in κR0 sin α and A/(R0 sin α) represent changes in the substrate-curvature gradient; thus, we set Γ = κ2AR0 sin α as a convenient alternate dimensionless parameter. Accordingly, an increase (decrease) in Γ indicates an increase (decrease) in substrate-curvature gradient dε/dx or drop radius R0. Therefore, from Figure 10, the drop velocity increases monotonically with contact angle, drop size, and substrate-curvature gradient.
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ASSOCIATED CONTENT
S Supporting Information *
Details of the analytical solution of eq 16 in the main text and an analysis of the numerical and perturbation theory errors. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
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*E-mail:
[email protected].
SUMMARY A non-axisymmetric perturbation analysis was performed for a drop or a bubble supported on a flat substrate in the absence of gravity. For a spherical cap on a planar substrate, a standard perturbation method was modified by adding a non-axisymmetric perturbation to the substrate and calculating the perturbation to
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We gratefully acknowledge financial support for this work from the NSERC Innovative Green Wood Fibre Network. 14176
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