Spontaneous Movement of a Droplet on a Conical Substrate: Theoretical Analysis of the Driving Force.

Experiments and simulations have shown that a droplet can move spontaneously and directionally on a conical substrate. The driving force originating from the gradient of curvatures is revealed as the self-propulsion mechanism. Theoretical analysis of the driving force is highly desirable; currently, most of them are based on a perturbative theory with assuming a weakly curved substrate. However, this assumption is valid only when the size of the droplet is far smaller than the curvature radius of the substrate. In this paper, we derive a more accurate analytical model for describing the driving force by exploring the geometric characteristics of a spherical droplet on a cylindrical substrate. In contrast to the perturbative solution, our model is valid under a much weaker condition, i.e., the contact region between the droplet and the substrate is small compared with the curvature radius of the substrate. Therefore, we show that for superhydrophobic surfaces, the derived analytical model is applicable even if the droplet is very close to the apex of a conical substrate. Our approach opens an avenue for studying the behavior of droplets on the tip of the conical substrate theoretically and could also provide guidance for the experimental design of curved surfaces to control the directional motion of droplets.

Introduction

The transportation of a droplet has drawn great attention in both academia and industry. Inspired by nature such as the self-cleaning of lotus leaves,[1,2] water or fog collection by spider silk,[3] Namib desert beetles,[4] and cactus,[5] directional water transportation on the peristome surface of Nepenthes alata(6) and on the wings of butterfly[7] and cicada,[8] the selectivity for liquid transportation at different surface tensions on Araucaria leaf,[9] and the geometric tip-induced flipping for droplets on the needles of Sabina chinensis,[10] a large number of technologies and strategies have been developed for water purification,[11] water collection,[12−16] and controlled transport of droplets.[17,18] The general way to control the directional transport of droplets is by introducing a wettability gradient,[19,20] a roughness gradient,[21,22] or a structure gradient-induced vapor layer gradient.[23] Interestingly, experiments and molecular dynamics (MD) simulations have found that the shape gradient of a substrate typically such as conical substrates can also lead to directional motion of droplets.[24−34] It is believed that the surface free energy gradient is the main driving force in the spontaneous movement of droplets toward the region with a lower curvature.[32−43] In general, the free energy of a droplet–substrate system in a steady state can be quantified as[44]U = γ(ALV – ALS cos θ), where ALV and ALS denote the contact area of liquid–vapor and liquid–substrate interface, respectively. Therefore, theoretical analysis of the shape gradient-induced driving force requires to accurately calculate the curvature-dependent ALV and ALS since the surface tension of a liquid γ and the contact angle θ[45] are almost constant.[43,46,47] The simplest approximate model to obtain curvature-dependent free energy is to treat both the droplet and the substrate as spheres.[30,39,42] Recently, Galatola[43] and McCarthy et al.[29] investigated the dynamics of a droplet on a conical substrate by performing an approximate calculation of a spherical droplet on a weakly curved cylinder, where the radius of the cylinder corresponds to the local curvature radius of the conical substrate that the droplet is in contact, and the theoretical predictions show agreement with experiments.[14,15,39] However, the analytical solution is based on a perturbative analysis for a substrate close to a plane,[43] and it is applicable only when the radius of the droplet is sufficiently small with respect to that of the substrate so that the variation of the droplet radius is almost independent of the curvature radius of the substrate.

In this work, we theoretically derive the free energy of a droplet on the outside of a conical substrate with consideration of the variation of the droplet radius. The accurate analytical expressions of ALV and ALS for a spherical droplet in contact with a cylindrical substrate are first obtained by exploring the geometric characteristics. We reveal that the ratio of the liquid–substrate contact size over the radius of the substrate is more suitable to be used as a small quantity in the approximation theory. As a result, by comparison with the Surface Evolver (SE) simulation results, our approximate analytical solutions are valid in a wider range than the previous perturbation method. Especially for superhydrophobic cones, it can effectively predict the behavior of a droplet close to the conical apex. We found that near the conical apex, the curvature-induced driving force increases significantly with the increase in cone angle, while far away from the conical apex, the curvature-induced driving force decreases with the increase in cone angle.

The outline of this paper is as follows: In Section , the mathematical model of a droplet on different substrates will be given, and the approximate solutions will be discussed. In Section , we will show the system free energy and the curvature-induced driving force and compare them with those obtained by Galatola,[43] Li et al.,[34] and by Lv et al.,[39] and we will show the dynamic behavior of a droplet moving on a conical substrate under the action of the curvature-induced driving force and the resistance force from contact angle hysteresis. Finally, we conclude with a brief summary in Section .

Mathematical Models

To analytically calculate the curvature-dependent ALV and ALS, there are two approximate models: a spherical droplet on a spherical substrate (S-S model) and a spherical droplet on a cylindrical substrate (S-C model), where the radius of the spherical and the cylindrical substrates corresponds to the local curvature radius of the conical substrate in contact with the droplet. The half-apex angle of the conical substrate is denoted by α.

A Spherical Droplet on a Spherical Substrate (S-S Model)

Considering a spherical droplet on a spherical substrate (S-S model), as shown in Figure , the interfacial areas of the liquid–substrate interface and the liquid–vapor interface arei.e.,where φ1 and φ2 can be determined aswhere Rs and Rd are the radii of the substrate and the droplet, respectively, and h is the distance from the center of the droplet to the vertex of the spherical substrate. The parameters of φ1 and φ2 as shown in Figure can be determined by minimizing the system free energy, which will be presented in the next section. The volume of the droplet can be calculated aswhere Vr = π/3(2Rs + h1)(Rs – h1)2 + π/3(2Rd + h2)(Rd – h2)2, h1 = Rs cos φ1, and h2 = Rd cos φ2. Substituting the above expressions into eqs , 4, and 7, we have

Figure 1

A spherical droplet on a spherical substrate with a contact angle of θ.

On the other hand, the contact angle θ satisfies

A Spherical Droplet on a Cylindrical Substrate (S-C Model)

For the model of a spherical droplet on a cylindrical substrate (Figure ), the equations of the spherical droplet and the cylindrical substrate in the rectangular coordinate system as shown in Figure a arewhere h0 is the distance between the center of the sphere and the axis of the cylinder and Rs and Rd are the radii of the substrate and the droplet, respectively.

Figure 2

(a) Sketch of a spherical droplet on a cylindrical substrate with a contact angle of θ, where the y-axis is along the axis of the cylindrical substrate and the z-axis goes through the center of the spherical droplet. (b) In the unfolded view along the generatrix AB in (a), the liquid–substrate contact area can be approximated as an ellipse as discussed in the main text.

Then, the interface areas ALS and ALV readandwhere

The volume of the droplet can still be formulated using eq , and Vr can be calculated aswhere x1(y, z) and x2(y, z) are given by

Approximate Analytical Solution of the S-C Model

To accurately determine ALS and ALV, numerical integration of eqs , 15, and 20 is generally required. However, an approximate analytical solution is highly desired since it is more convenient to guide the experimental design. To obtain the analytical expressions of ALS and ALV, we first derive the equations of the spherical droplet and the cylindrical substrate in the cylindrical coordinate system (Figure a):Then, eqs and 13 can be rewritten aswhere h0 = h + Rs. The equation of the contact line between the spherical droplet and the cylindrical substrate is thusTaking the Taylor expansion of cos φ aswhere o(φ2) can be neglected when φ is very small, then eq can be simplified as

With the definitions of Rsφ = x′ and y = y′, the equation of the contact line can be rewritten as (Figure b)where a and b arewhere Rs – Rd < h0 < Rs + Rd. Based on eq , the liquid–substrate interface area ALS can be calculated as

Based on eq and dividing Vr with a series of concentric cylindrical surfaces as shown in Figure a, dVr can be readily formulated asandi.e.,

Figure 3

The removed volume of a spherical droplet on a cylinder can be regarded as either the sum of many slices of the sphere cut by cylindrical shells with different radii (a) or as the sum of many slices of the cylinder cut by spherical shells with different radii (b).

Based on eqs and 35, the volume of the droplet can be written aswhere and .

It is noted that the infinitesimal volume dVr can be easily correlated with the liquid–vapor interface area ALV because ALV = (4πRd2 – Ar|) and

Based on eqs and 37, we obtain

Based on eq , we obtain

Results and Discussion

The System Free Energy of a Spherical Droplet on a Cylindrical Substrate

The system free energy of a liquid droplet on a solid substrate can be quantified as[44]

The contact angle θ satisfies[45]where γ, γSV, and γLV are the liquid–vapor, solid–vapor, and liquid–solid interfacial tension, respectively. When considering the line tension, the system free energy readswhere τ is the line tension and L is the perimeter of the solid–liquid contact area. Taking typical values for the spherical droplet on a smooth substrate (without microstructures): τ = 10–9 – 10–6 N,[48−50] a liquid radius of >100 μm, a contact angle of 90°, and the surface energy of water γ = 0.072 N/m, we have τL ≤ 6.28 × (10–13 – 10–10) N·m ≪ (γALV)|min = 4.52 × 10–9 N·m. Therefore, the line tension effect can be safely neglected on a smooth substrate.

Then, under the conditions of a constant droplet volume and contact angle, h can be determined by minimizing the system free energyi.e.,whereand

The system free energy can thus be calculated by solving eqs and 44 under the condition of

If the contact size between a droplet and a conical substrate is small by comparison with the local curvature radius of the conical substrate, then the free energy of a droplet–conical substrate system can be approximated by that of a droplet–cylindrical substrate system with the radius of (Figure )where s is the coordinate along the generatrix of the conical substrate. Then, by substituting eqs and 39 into eq , we obtain the system free energy of a droplet on a conical substrate as

Figure 4

The model of a spherical droplet on a conical substrate with a half-apex angle α.

For simplicity, we introduce the nominal radius of the droplet as and then define the dimensionless system free energy and the local radius of Rs as U* = U/γr02 and Rs* = Rs/r0, respectively. The dimensionless free energy of a droplet on a conical substrate is plotted in Figure , where the perturbative solution to the S-C model by Galatola (hereafter abbreviated as “Galatola’s approximation”)[43] is also shown for comparison. It is obvious that the S-S model shows a consistent trend by comparison with the exact numerical solution of the S-C model, but there is a large deviation for the S-S model in ref (39) and the Galatola’s approximation[43] (the double dot-dashed line and the green dotted line in Figure ), where Galatola’s approximation shows good agreement at a large radii (corresponding to a weakly curved cylinder) but considerable deviation exists at small Rs* values, while our approximate analytical solution (eq ) agrees very well with the exact numerical solution in the entire range of Rs* (red solid line in Figure ).

Figure 5

The dimensionless system free energy U/γr02 versus the dimensionless local radius Rs/r0 of the conical substrate. The double dot-dashed line is the approximate solution in ref (39). The green dotted line is the result based on Galatola’s approximation.[43] The blue dashed line is the numerical solution of the S-C model, and the red solid line is the approximate analytical solution of the S-C model based on eq . The contact angle of cosθ = – 0.25 and the volume of the droplet Vd = 30 mm3 are used in the calculations.

Based on eqs and 54, the curvature gradient-induced driving force Fcurv can be readily calculated by

Defining the dimensionless curvature gradient as Fcurv* = Fcurv/γr0, we have

The curvature gradient-induced force, Fcurv*, is plotted in Figure , where the results obtained by the S-S model,[39] perturbative analytical solution,[43] and the Surface Evolver (SE) simulation in ref (34) are also shown for comparison. To compare with the literatures, the contact angle θ in Figure a is varied form θ = 90° to θ = 120°, and the half-apex angle of the conical substrate α is 19.5°,[39] where the blue dashed lines are taken from ref (39). It is clear that both the S-S model and our approximate S-C model show similar trends on Rs/r0, that is, the curvature-induced force decreases drastically with the increase in Rs/r0 and tends to zero at positions far away from the apex, but significant deviation can appear at small Rs/r0 values. Similarly, by comparison with the perturbative analytical solution (black dashed line in Figure b),[43] our approximate analytical solution (red line in Figure b) shows much better agreement with the SE simulation results[34] (blue dashed line in Figure b). The curvature-induced force decreases with increasing Rs/r0 and tends to 0 on the position far away from the apex.

Figure 6

The dimensionless curvature gradient-induced force Fcurv* versus the dimensionless local radius Rs/r0 of a conical substrate. (a) Conical substrate with a half-apex angle α = 19.5°. The red solid lines are calculated based on eq , and the blue dashed lines are the results based on the S-S model in ref (39). (b) Conical substrate with a half-apex angle α = 5° and a contact angle θ = 80°; the volume of the droplet Vd = 30 mm3 is used. The blue dashed line is obtained by the Surface Evolver simulation in ref (34), the black dot-dashed line is the approximation result from ref (43), and the red solid line is our calculation based on eq .

In addition, when plotting the dimensionless curvature gradient-induced force Fcurv* versus the dimensionless coordinate s/r0 (i.e., along the generatrix of the conical substrate) with a varied half-apex angle α (Figure ), we observed that the driving force decreases drastically as s/r0 increases, and the smaller the θ, the higher the driving force, which agrees well with MD simulations.[38,39] Remarkably, our model also predicts that far away from the conical apex, the driving force decreases as α increases, but near the conical apex, the driving force drastically grows as α increases. Such a prediction suggests that a larger apex angle can lead to a faster water collection speed near the conical apex, which is consistent with experimental observation.[14,15,39] It is noteworthy that our approximate formula is valid as long as the size of the liquid–solid contact area is small by comparison with the curvature radius of the substrate. Therefore, our approximation method can predict the behavior of a droplet in the region very close to the conical apex in the hydrophobic (superhydrophobic) case. This is very different from the perturbative approximation method, typically such as Galatola’s approximation,[43] which is derived based on the condition that the droplet radius is far smaller than the curvature radius of the substrate. In other words, even in the case of superhydrophobic with an almost zero contact area, if the droplet is large, then the perturbative approximation method will fail even at the region far away from the apex of a conical substrate.

Figure 7

The curvature gradient-induced force Fcurv* versus the coordinate s along the generatrix of conical substrates with different half-apex angles (α = 30 (green double dot – dashed line),45 (blue dashed line), and 60° (red solid line)), where the contact angle and the volume of the droplet are set as θ = 90 and 120° and Vd = 30 mm3, respectively.

Dynamic Analysis of the Motion of a Droplet on a Conical Substrate

When a droplet moves on a conical substrate, it suffers from both the curvature gradient-induced driving force (eq ) and the resistance force from the contact angle hysteresis (Fh)[51]where θa and θr are the advancing and receding contact angles, respectively, and w is a characteristic length of the contact area, where for the elliptical contact interface shown in Figure b, w = a, i.e.,andwhere β = b/a = λs–1. The resultant force on the droplet is thus Fa = Fcurv – Fh. Then, the equation of motion of the droplet in the steady state readswhere ρ is the density of the liquid and v is the velocity of the droplet. Integration of eq gives

Substituting eq into eq gives

Based on eq , the velocity v of the droplet versus the generatrix of the conical substrate can be obtained (Figure ). It is observed that the velocity of a droplet on a conical substrate increases first and then decreases and finally goes to zero as the distance from the conical apex increases, and the larger the half-apex angle, the faster the average velocity of the directional movement of a droplet on a conical substrate. This prediction also agrees well with the experimental observations by Gurera and Bhushan.[14,15]

Figure 8

The stead-state velocity v of a droplet moving along the generatrix of conical substrates with half-apex angles of α = 5,10,15,30, and 60°, where the contact angle and the volume of the droplet are set as θ = 90° and Vd = 30 mm3, respectively. The advancing and receding contact angles are θa = 95° and θr = 85°, respectively. The liquid–solid interface tension is γ = 1 N/m, and the density of the liquid is ρ = 1.0 × 103 kg/m3.

Conclusions

In summary, we present a theoretical model for describing the curvature gradient-induced directional motion of a droplet on a conical substrate. By exploring the geometric characteristics of a sphere droplet on a cylindrical substrate and formulating the contact interface area of the liquid–substrate and the liquid–vapor, we derived a new approximate analytical expression of the system free energy and the curvature gradient-induced driving force. By comparison with the approximate analytical solution based on the perturbation method,[43] our analytical solution shows much better agreement with the exact numerical solution, which we attributed to the fact that our method only requires that the contact size of the droplet–solid substrate is smaller than the curvature radius of the substrate, which is much weaker than the condition required by the used perturbation method in the literature. We further show that our theoretical calculations agree well with the results obtained by the Surface Evolver (SE) simulations,[34] the molecular dynamics simulations,[38,39] and experimental observations.[14,15,39] Considering that the developed new analytical model is valid at the region close to the apex of a conical substrate, especially for superhydrophobic substrates, we anticipate that our method should provide a simple but practical guide to experimental design of curved surfaces for studying and controlling the directional motion of droplets.