Article pubs.acs.org/Langmuir
Liquid Drop Runs Upward between Two Nonparallel Plates Xin Heng and Cheng Luo* Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, 500 West First Street, Woolf Hall 226, Arlington, Texas 76019, United States S Supporting Information *
ABSTRACT: We have recently observed an interesting phenomenon: even under gravity, a microliter-scaled silicone oil drop was still able to run upward between two nonparallel plates that were approximately vertically placed. We also saw the same phenomenon in the case of isopropyl alcohol (IPA) drops. In this work, we developed simple models to interpret this phenomenon, followed by experimental validation. We demonstrated that, by changing the locations of drops or tilt and opening angles of plates, the moving directions of silicone oil, IPA, and water drops could be controlled. In the cases of silicone oil and IPA, we also found that the speed of a drop had a linear relation with the square of the drop location when the drop was far away from the corner of two nonparallel plates and that the drop moved faster as it became closer to this corner.
1. INTRODUCTION A capillary force on a liquid drop is related to surface tension and a characteristic length.1 When either factor is varied on a surface, a motion may be induced in the drop. For example, because of the difference in surface tensions at the rear and front of a drop, the drop may move on a surface with a thermal2 or chemical2,3 gradient. Meanwhile, a capillary force may also be created using asymmetric geometry of a structure, which produces different characteristic lengths at two opposite edges of a drop. For instance, a drop was found to move inside a conic capillary tube,4 on a conic fiber,1,5,6 on a spider silk,7 or between two nonparallel plates.8−10 Such a structure has varied radii1,4−7 or gaps along its longitudinal direction.8−10 Accordingly, a pressure difference is generated in a drop located on the structure, resulting in a directional motion of the drop. Furthermore, it is reported that, in some cases, a capillary force may overcome the gravity of a drop, making the drop run upward on an inclined surface3 or even on vertically oriented conic fibers.1,5 In a recent test (Figure 1a), we have also found that a 0.2 μL drop of silicone oil was able to run upward between two nonparallel plates that were approximately vertically oriented (the corresponding video is provided in the Supporting Information). The average speed was about 0.5 mm/s. Furthermore, in another test (Figure 1b), we found that the same phenomenon occurred as well to a 0.5 μL drop of isopropyl alcohol (IPA). To find the reason, we explore this phenomenon here.
Figure 1. Multi-exposed photography of (a) silicone oil drop and (b) IPA drop moving upward at three different time instants inside two nonparallel SiO2-coated Si plates that were approximately vertically oriented. Scale bars represent 2.5 mm. and “edge 2”, respectively. β is set to be the tilt degree of the middle plane between the two plates. We have 0° < β < 180°. o and α are used to denote the apex edge and opening angle of the two plates, respectively. a1 and b1 denote the two points that edge 1 intersects with the bottom and top plates, separately, and a2 and b2 are set to be the two intersecting points that edge 2 forms with the bottom and top plates, respectively. p1 and p2 are used to represent liquid pressures at edges 1 and 2, separately. θ1 represents the equilibrium contact angle11 at a1 and b1, and θ2 is used to stand for the equilibrium contact angle at a2 and b2. θa and θr are maximal and minimal values of both θ1 and θ2 at the equilibrium limit, respectively. It has been indicated in ref 12 that, in addition to surface properties, the experimentally measured
2. THEORETICAL MODELING Received: November 24, 2014 Revised: February 20, 2015 Published: February 20, 2015
2.1. Preliminaries. Figure 2 shows the cross-sectional schematic of a liquid drop placed between two tilted, nonparallel plates. For simplicity, the left and right edges of the liquid drop are called “edge 1” © 2015 American Chemical Society
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To make liquid drops move upward as in the observed cases of silicone oil and IPA (Figure 1), eq 4 should be met for β = 90°. This means that the right-hand side of this inequality should be greater than 1. Three different approaches have been proposed to realize it. The first approach is given as follows. As also shown in our previous work,9 if θa < (π/2 + α/2) and α ≥ (θa − θr), then cos(α/2 − θa) > cos(α/2 + θr) and cos(α/2 − θa) > 0, which mean that the right-hand side of eq 4 is positive. Also, as observed from eq 4, the right-hand side of this inequality approaches infinity when lp or ll approaches 0. In other words, if lp or ll is small enough, then the right-hand side of eq 4 should be greater than 1. In the first approach, it is required that α ≥ (θa − θr). In case this inequality does not hold, the second approach may be used. It is readily shown that, if (lp + ll)cos(α/2 − θa) > lp cos(α/2 + θr), then the right-hand side of eq 4 is also positive. Meanwhile, as in the first approach, if lp or ll is small enough, then the right-hand side of eq 4 should be greater than 1. Finally, as observed from eq 4, the third approach is to ensure that (lp + ll)cos(α/2 − θa) > lp cos(α/2 + θr) and α → 0°. In summary, when the sufficient conditions in any of these three approaches are satisfied, a liquid drop should keep moving upward until it fills the corner of two vertically oriented plates. Once eq 4 is met for β = 90°, it also holds true when β has any other value. This implies that, if a drop is capable of running up in two vertically oriented plates, then it can also do so when the plates are inclined at any other degree. 2.3. Conditions for Moving Upward in Special Cases. On SiO2-coated Si plates, the values of θr and θa for silicone oil were measured to be 2° and 3°, respectively, with an error of 1°, while these values for IPA were 13° and 18°, respectively. The difference between θr and θa is small for either liquid, indicating that θa may be approximated as θr. Furthermore, in our tests on silicone oil and IPA drops, α is always less than 6°. Accordingly, for example, sin α/2, cos(α/2 − θr), and cos(α/2 + θa) are simplified as α/2, cos θr, and cos θr, respectively. Thus, relations 2−4 can be, separately, approximated as
Figure 2. Cross-sectional schematic of a liquid drop placed between two tilted, nonparallel plates.
values of θa and θr may also depend upon the geometry of the experimental setup. In this work, to consider the geometric effect, θa and θr are not measured directly on a single plate. Also, from an experimental standpoint, it is difficult to know beforehand whether a drop is stationary, moves upward, or runs downward between two nonparallel plates at a particular tilt degree. Therefore, the values of θa and θr are determined by pressing and relaxing microliter-scaled drops between two nonparallel plates when β is 0°, respectively. During the pressing process, two limiting contact angles are found, at edges 1 and 2, when the two edges begin to move outward. Their average value is chosen as θa. Likewise, in the relaxing process, the average of two limiting contact angles, which are obtained when edges 1 and 2 start to move inward, is set to be θr. The corresponding values of θa and θr are also adopted as the values for any value of β. According to eqs 3 and 4 in our previous work,9 the average pressure gradient along the longitudinal direction of the drop is
(α
(α
)
)
γ cos 2 − θ1 γ cos 2 + θ2 Δp = − α α ll lllp sin 2 ll(lp + ll)sin 2
(1)
where Δp = p2 − p1 and γ denotes the surface tension of the liquid. Meanwhile, the gradient of gravitational pressure is ρg sin β along the longitudinal direction of the drop. Balancing Δp/ll with ρg sin β yields the following condition for a drop to be stationary between two nonparallel plates: 2
(
λ cos sin β =
α 2
lllp sin
)−
− θ1 α 2
2
(
λ cos
α 2
sin β =
sin β >
)
+ θ2
ll(lp + ll)sin
α 2
(2)
sin β
⎢ − α α lllp sin 2 ll(lp + ll)sin 2 ⎣
(
)
(
(3)
When Δp/ll > ρg sin β, the drop should run upward. Subsequently, θ1 and θ2 in eq 1 are changed to θa and θr, respectively. Accordingly, we should have α ⎡ λ 2 cos α − θ λ 2 cos 2 + θr a 2 ⎢ − sin β < ⎢ α α lllp sin 2 ll(lp + ll)sin 2 ⎣
(
)
(
) ⎤⎥ ⎥ ⎦
(static)
2λ 2 cos θr αlp(lp + ll)
(downward)
2λ 2 cos θr αlp(lp + ll)
(upward)
(5)
(6)
(7)
2.4. Dynamic Analysis. The Navier−Stokes equation is the governing equation for a viscous, incompressible flow, which is also our case. It is a vector equation and normally difficult to solve because of its nonlinear nature. To find a simplified solution in our case, the following two conditions are considered. First, the Reynolds number is assumed to be small and has a value much less than 1. Thus, the inertial term in the Navier−Stokes equation can be neglected in comparison to the viscous term. Second, a drop mainly moves along the x direction (the corresponding x−y coordinates are marked in Figure 2). Accordingly, the velocity components of the drop along both y and z directions may be neglected. On the basis of these considerations, it is readily shown that the Navier−Stokes equation is reduced to
) ⎤⎥ ⎥ ⎦
2λ 2 cos θr αlp(lp + ll)
μ (4)
As demonstrated in our previous work, if θa > (π/2 + α/2), then the right-hand side of eq 4 is less than zero when the drop is very close to the apex of the two plates. Accordingly, eq 4 is not met, because sin β considered here is positive. This implies that the drop cannot keep going up to fill the corner of the two plates. Therefore, to have a drop keep running up, we should at least have θa < (π/2 + α/2), which implies that the plate surface prefers to be lyophilic.
∂ 2ux ∂y
2
=
∂p + ρg sin β ∂x
(8)
where ux is the velocity component of the drop along the x direction, μ is the dynamic viscosity, μ(∂2ux/∂y2) represents a viscous force per unit volume along the x direction, p is the liquid pressure, and −(∂p/ ∂x + ρg sin β) is the gradient of the driving pressure along the x direction. Next, following a line of reasoning previously applied to explore the flow of a thin film on an inclined surface,13 we use a scaling law to find
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Figure 3. Relations between log(sin β) and log[lp(lp + ll)] for stationary (a) silicone oil and (b) IPA drops. Lines and scattered points represent the fitted and experimental results, respectively. an expression of the average speed ua along the x direction. The viscous term, which is given on the left-hand side of eq 8, has an order of magnitude of μ(ua/e2), where e denotes half thickness of the drop. Furthermore, ∂p/∂x is approximated as −Δp/ll, whose expression is given in eq 1. Accordingly, it follows from eq 8 that
ua ∝
⎞ e 2 ⎛ Δp − ρg sin β ⎟ ⎜ μ ⎝ ll ⎠
3. RESULTS AND DISCUSSION In this work, the maximum thickness of a drop between two plates was assumed to be less than its capillary length. That is, the maximum value for (lp + ll) is limited by λ/α. Accordingly, although the gravitational effect was still considered along the longitudinal direction of the drop, this effect was neglected across the drop thickness. Consequently, during the derivation of eq 1, p1 and p2 on edges 1 and 2 were considered to be uniform. The minimum value of α in the experimental tests was not less than 1°. To validate the derived relations, IPA, silicone oil, and water were tested in this work. Their values of λ are 1.7, 1.4, and 2.7 mm, separately. Thus, the maximum values of (lp + ll) for these three liquids should be 97, 80, and 155 mm, respectively, which also set limits for drop sizes and locations in our experiments. Four types of experiments were performed on SiO2-coated Si plates to validate relations 4−7 as well as relation 12. The tilt degree of two nonparallel plates was controlled by pushing the plates to move around a hinge using a micromanipulator. The first two types of experiments were performed on IPA and silicone oil. In the first type, three sets of tests were performed on either IPA or silicone oil to examine eq 5. The values of α in these tests were fixed to be 1°, 3°, and 5°. The sizes and locations of drops were varied in each set of tests. After a drop had been placed between two plates, they were slowly rotated in the vertical plane until the drop was stationary. Subsequently, the values of β, ll, and lp were measured; the corresponding data point (log[lp(lp + ll)], log(sin β)) was plotted in Figure 3. Accordingly, three sets of data points were generated for either IPA or silicone oil. In view of eq 5, log(sin β) has a linear relation with log[lp(lp + ll)], and the corresponding slope is −1. Thus, a straight line with a slope of −1 was used to fit each set of data points, and the intercept of this line with the vertical axis was employed to determine α. Theoretical and experimental results are considered to have a good match based on the following two points. First, as observed from Figure 3, the variation of each set of data points around the corresponding fitted line was small. Second, the fitted values of α were close to their experimental counterparts.
(9)
Subsequently, we consider the average speeds in the cases of IPA and silicone oil. As what was done in the previous subsection to obtain the right-hand sides of relations 5−7, Δp/ll may be simplified as (2γ cos θr)/αlp(lp + ll). Furthermore, the value of e in the entire drop is approximated as the value at the middle of the drop, which is α/2(lp + (ll/2)). Consequently, in the cases of IPA and silicone oil, we have
ua ∝
(
α 2 lp + μ
ll 2 2
) ⎡⎢
⎤ 2γ cos θr − ρg sin β ⎥ ⎢⎣ αlp(lp + ll) ⎥⎦
(10)
It is observed from this relation that, as lp → 0, i.e., when the drop is very close to the apex of the plates, we should have ua → ∞. Accordingly, the corresponding Reynolds number is also very large, because this number linearly increases with ua. This is against our assumption that the Reynolds number should be small. Therefore, to make it valid to apply relation 10, the considered drop should not be very close to the apex of the two plates. In case the drop is far away from the apex of the two plates in the sense that lp ≫ ll, relation 10 is then simplified as α ua ∝ (2γ cos θr − ρg sin βαlp2) μ (11) This relation can also be written as ua =
αk (2γ cos θr − ρg sin βαlp2) μ
(12)
where k is a constant that will be determined by experiments. Two points are observed from this equation. First, ua linearly increases with the decrease in lp2. This point means that the drop moves faster when it becomes closer to the apex of the two plates. Second, ua decreases with the increase in the tilt angle. This result implies that, as the inclination degree of the two plates increases, the drop runs slower. As will be detailed in the next section, these two points are validated in our experimental tests. 2745
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Figure 4. Oil drop first (a) moves upward, then (b) stops, (c) moves downward with the increase of β, and finally (d) runs upward again with the decrease of β. The opening angle α is fixed as 2°. Scale bars represent 5 mm. Panels a2−d2 are the optical images of the drop when tilt angles shown in panels a1−d1 are held for 2 or 3 s before the plates are further rotated, respectively.
Figure 5. Moving directions of drops for different combinations of lp and ll in cases of (a) silicone oil and (b) IPA. The blue line in either panel a or b represents the relation between lp and ll that is obtained from eq 5, when β and α are 90° and 3.5°, respectively. The downward and upward motion regions were determined using eqs 6 and 7. Data points represent experimental results for varied combinations of lp and ll.
Figure 6. Oil drop continuously moves toward the corner of two plates when β increases from 0° to 184°. The opening angle α is fixed to be 2°. Scale bars represent 5 mm.
The fitted values of α were 0.7°, 2.3°, and 5° for the three sets of tests in the case of silicone oil, while they were 0.7°, 2.5°, and 5.6° for IPA. The measured values of α in the three sets of tests were 1°, 3°, and 5°. When the measurement error of 0.5° was taken into consideration, the fitted values have a good match with the experimentally measured values.
The second type of experiments was employed to examine relations 5−7. Three points were observed from these relations. First, a drop may go upward, be stationary, or go downward depending upon the value of β. Second, for fixed β and α, the moving direction of a drop also depends upon lp and ll. Third, in case eq 7 is met for β = 90°, which implies that the capillary 2746
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Figure 7. Positions of a water drop at three different time instants, as it ran upward inside two vertically oriented plates for (a) α = 20° or (b) α = 10.5°. Scale bars represent 1 mm.
Figure 8. Plot of lp2 versus ua of (a) silicone oil and (b) IPA for β = 0°, 10°, and 20°. The straight lines are the relations predicted using eq 12.
the data point, instead of being stationary, the drop moved upward. During the process of simplifying eq 2 to obtain eq 5, an error should be induced in eq 5. The mismatch between the theoretical prediction and experimental result at the specific data point may be caused by this error or experimental error. In the third test, lp and ll were chosen to make eq 7 satisfied for β = 90°. Accordingly, as the plates were slowly rotated from 0° to 184°, the corresponding drop kept moving toward the corner of the two plates (Figure 6), which was different from that in the first test of this type of experiments. The third type of experiments was performed on water to validate eq 4. On a SiO2-coated Si plate, the values of θr and θa for water were measured to be 55° and 67°, separately. Because
force was always larger than the gravitational force, a drop should run toward the corner of the two plates for any tilt angle. Three tests were conducted in this type of experiments to validate these three points (Figures 4−6). As shown in the first test (Figure 4), the moving direction of a drop could be controlled by varying β. In the second test, β and α were fixed to be 90° and 3.5°, respectively. According to eq 5, when lp(lp + ll) = (2λ2 cos θr)/α, a liquid should be stationary between two plates. Otherwise, by eqs 6 and 7, the liquid should either move downward or upward. As shown in Figure 5, except one data point, all of the results in the second test match the corresponding theoretical predictions. This data point was on the curve of lp(lp + ll) = (2λ2 cos θr)/α (Figure 5a). However, at 2747
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4. SUMMARY AND CONCLUSION In this work, we explored the conditions for a liquid to run upward inside two nonparallel plates. According to the derived relations, the moving direction and speed of a liquid drop may be controlled by varying the location of the drop or changing the tilt and opening angles of the two plates. These relations were validated by four types of experiments. As observed from the tests on IPA, silicone oil, and water, our results provide new ways to manipulate a liquid drop.
of relatively large values of these angles and their difference, eq 4, instead of eq 7, should be applied to the case of water. Three approaches were presented in subsection 2.2 to make eq 4 hold true when β was 90°. Accordingly, these approaches were also experimentally validated. On the basis of the considerations of the first approach, when α, lp, and ll were chosen to be 20°, 1.10 mm, and 1.95 mm, respectively, as expected, a 2.5 μL water drop moved upward inside two plates that were vertically oriented (Figure 7a). According to the second approach, when α, lp, and ll were chosen to be 10.5°, 1.0 mm, and 2.9 mm, respectively, a 3 μL water drop moved upward inside two vertically oriented plates (see Figure 7b, and the corresponding video is also available in the Supporting Information). The third approach was actually validated by the second type of experiments for IPA and silicone oil. It was therefore not specifically examined for the case of water. The fourth type of experiments was explored to examine eq 12. Three sets of tests were performed for both silicone oil and IPA. The value of α was fixed to be 5°, while those of β in these tests were varied to be 0°, 10°, and 20°, respectively. The sizes (2 μL) and initial locations (13.3 and 13.2 mm for silicone oil and IPA, respectively) of drops were fixed in each set of tests. As the drop was placed between two plates, it moved toward the corner. Data points (ua, lp2) in each set of tests were marked in Figure 8. It is observed from this figure that, in all of the tests, the speed has a quick increase when lp2 becomes less than 70 mm2. For example, for β = 0°, the speed of the silicone oil drop suddenly increases from 1.8 to 2.0 mm/s, while the IPA drop moves too fast to accurately determine its speed (thus, the corresponding speed is not given in Figure 8b). Because Δp increases with the decrease in lp, the speed also keeps increasing when the drop becomes close to the apex of the plates. As lp2 is less than 70 mm2, the drop is less than 8.4 mm away from the apex of the plates. The ratio of this distance with the drop size (about 2.6 mm) is less than 3.2. Therefore, as discussed in subsection 2.4, eq 12 is better applied to the case when lp2 is larger than 70 mm2. This phenomenon of quickly changing the speed at a location close to the apex of the plates has also been reported in ref 14 when a silicone oil drop was moving inside two nonparallel plates that were horizontally placed. Two common points can be observed from the results of both silicone oil and IPA when lp2 is larger than 70 mm2 (Figure 8): (i) the drop speed approximately changes with lp2 in a linear manner, and (ii) at the same location (i.e., for the same value of lp2), ua decreases with increasing β. These two points have a good match with the corresponding results predicted in subsection 2.4 using eq 12. Furthermore, according to the results for β = 0°, the values of k in eq 12 are found to be 0.02 and 0.0024, respectively, for silicone oil and IPA. Subsequently, the lp2−ua relations that are predicted using eq 12 are drawn in Figure 8. As observed from this figure, the experimental results have a better match with the theoretical results in the case of silicone oil. For example, for β = 0°, the value of ua is 1.8 ± 0.05 mm/s for silicone oil, while it is 3.6 ± 0.3 mm/s for IPA, which has a larger variation around the average value. In addition, in the case of IPA, experimental data points vary on both sides of the corresponding theoretical curve. The relatively larger mismatch between theoretical and experimental results in the case of IPA may be due to the fact that IPA is easy to evaporate, which is not considered in the theoretical model. At room temperature, a 2 μL IPA drop visibly vanished on a SiO2-coated plate within 2 min, while a 2 μL silicone oil drop had almost no change in its volume within the same time period.
■
ASSOCIATED CONTENT
S Supporting Information *
Videos of Figures 1a and 7b. This material is available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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