pubs.acs.org/Langmuir © 2010 American Chemical Society
Multimode Dynamics of a Liquid Drop over an Inclined Surface with a Wettability Gradient A. K. Das and P. K. Das* Department of Mechanical Engineering, IIT Kharagpur, 721302, India Received January 12, 2010. Revised Manuscript Received May 2, 2010 A liquid drop placed over a solid surface with a wettability gradient self-propels to minimize its surface energy. It can also climb an inclined plane if the applied gradient strength is high enough. We investigate the motion of liquid drops over an inclined gradient surface using a unique 3D computational technique. The technique combines diffuse interface in a smoothed particle hydrodynamics simulation to study the internal fluid structure and the contact line dynamics. Simulation results reveal that drop motion is dependent on its volume, surface inclination, and the strength of the wettability gradient. It has been demonstrated that, depending on these parameters, a drop can experience upward or downward motion or can remain stationary on the inclined plane. Finally, drop mobility maps which give an idea about the regimes of uphill and downhill movement of a drop over gradient surfaces have been proposed.
Introduction Drop manipulation over a solid surface is an emerging field of research due to its growing applications in microfluidic devices.1,2 Controlled transportation of drops over a solid surface can be used for the cooling of hot spots in microscale applications. Drop manipulation can be utilized for micropumping, in the design of valve and seals, and in micromixing and microreaction. A drop can move over a solid surface by the effect of gravitational force, thermal gradient,3,4 or chemical wettability gradient.5,6 The velocity of microdrops due to chemical wettability gradient could be a few orders faster compared to that produced by Marangoni flow and gravitational field. As a result, a surface with a gradient of surface energy has emerged as a potential candidate in MEMS applications.7,8 A surface with a wettability gradient basically employs an imbalance of surface force on the circumferential plane of a droplet base. This can propel a droplet on a horizontal plane in the direction of increasing gradient. With the increase in wettability gradient, it is possible even to make the drop climb upward against gravity. Chowdhury and Whitesides9 demonstrated the uphill movement of a water drop over a treated silicon wafer (7.2°/mm). They have polished the silicon wafer by decyltrichlorosilane to obtain a velocity on the order of few millimeters per second for drop of volume 1-2 μL. There exists a volume of literature5,7-10 which report investigations on the motion of drops over a gradient surface. This problem has been studied through theoretical simulations as well as experiments. Unfortunately, not *Author for correspondence. Prof. P. K. Das, Indian Institute of Technology, Kharagpur-721302, India. Tel.: þ91-03222-282916. Fax: þ9103222-282278. E-mail address:
[email protected]. (1) Gau, H.; Herminghaus, S.; Lentz, P.; Lipowsky, R. Science 1999, 283, 46–49. (2) Gallardo, B. S.; Gupta, V. K.; Eagerton, F. D.; Jong, L. I.; Craig, V. S.; Shah, R. R.; Abbott, N. L. Science 1999, 283, 57–60. (3) Ford, M. L.; Nadim, A. Phys. Fluids 1994, 6, 3183–3185. (4) Chen, J. Z.; Troian, S. M.; Darhuber, A. A.; Wagner, S. J. Appl. Phys. 2005, 97, 014906.1–014906.9. (5) Greenspan, H. P. J. Fluid Mech. 1978, 84, 125–143. (6) Hitoshi, S.; Satoshi, Y. Langmuir 2003, 19, 529–531. (7) Daniel, S.; Chaudhury, M. K. Langmuir 2002, 18, 3404–3407. (8) Moumen, N.; Subramanian, R. S.; McLaughlin, J. B. Langmuir 2006, 22, 2682–2690. (9) Chaudhury, M. K.; Whitesides, G. M. Science 1992, 256(6), 1539–1541. (10) Liao, Q.; Wang, H.; Zhu, X. J. Eng. Thermophys. 2007, 28, 134–136.
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much effort has been made to explore the dynamics of a droplet over an inclined surface with a gradient of surface energy. Thiele et al.11 identified a reaction limited zone below the droplet moving over a gradient surface and proposed that the change of reaction rate causes a different driving force for droplet movement. Later, Pismen and Thiele12 developed an asymptotic solution for drop dynamics over a gradient surface using lubrication theory. A hybrid boundary-finite element method has been used by Iliev13 for the study of drop movement over horizontal gradient surface. Subramanian et al.14 made some approximation of the drop shape over a gradient surface by collection of wedges and applied lubrication theory to evaluate the quasi steady speed of drop translation over a horizontal surface. Huang et al.,15 for the first time, employed a numerical technique based on lattice Boltzmann method for the investigation of wettability controlled planar movement of a liquid drop in lab-on-a-chip systems. Recently, Liao et al.16 numerically simulated the equilibrium shape of a liquid drop on a surface having a surface energy gradient applying a finite element method. However, until now, numerical techniques have not been employed to investigate the internal fluid structures of a liquid drop during uphill movement over an inclined gradient surface. In this work, we have used a hybrid numerical scheme of diffused interface (DI)17,18 based smoothed particle hydrodynamics (SPH)19,20 to simulate the uphill movement of a liquid drop over an inclined gradient surface. Though DI was used in the simulation of contact line dynamics, its incorporation in SPH is of (11) Thiele, U.; John, K.; B€ar, M. Phys. Rev. Lett. 2004, 93(2), 027802.1– 027802.4. (12) Pismen, L. M.; Thiele, U. Phys. Fluids 2006, 18, 042104.1–042104.10. (13) Iliev, S. D. Comput. Methods Appl. Mech. Eng. 1995, 126(3-4), 251–265. (14) Subramanian, R. S.; Moumen, N.; McLaughlin, J. B. Langmuir 2005, 21 (25), 11844–11849. (15) Huang, J. J.; Shu, C.; Chew, Y. T. J. Colloid Interface Sci. 2008, 328, 124– 133. (16) Liao, Q.; Shi, Y.; Fan, Y.; Zhu, X.; Wang, H. Appl. Therm. Eng. 2009, 29, 372–379. (17) Jacqmin, D. J. Comput. Phys. 1996, 155, 96–127. (18) Anderson, D. M.; McFadden, G. B.; Wheeler, A. A. Ann. Rev. Fluid Mech. 1998, 30, 139–165. (19) Monaghan, J. J. J. Comput. Phys. 1994, 110, 399–406. (20) Liu, G. R.; Liu, M. B. Smoothed Particle Hydrodynamics - a mesh free particle method; World Scientific: Singapore, 2003.
Published on Web 05/19/2010
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recent origin. Xu et al.21 combined DI with SPH using mass density as the ordered parameter and investigated precipitation and dissolution kinetics. Tartakovsky et al.22 introduced manybody repulsive and viscous force, two-body attractive force, and other body forces for multiphase multicomponent fluid flow to combine the momentum conservation equation with the modified van der Waals equation. In one of our earlier efforts,23 we forged the DI concept into Lagrangian formulation to simulate drop dynamics over an inclined surface. Using the same approach, efforts have been made to simulate the dynamics of a threedimensional drop over an inclined surface with a wettability gradient. Study of the internal flow pattern of the drop reveals the physics behind the upward and downward motion of the drop. We have also prepared a drop mobility map by varying the volume of the drop, inclination angle, and strength of the wettability gradient. Mathematical Model and Its Numerical Implementation. A combined DI based SPH methodology is used to capture complex interfaces and contact line dynamics generated due to uphill movement of a drop on an inclined gradient surface. Das and Das23 developed the DI-SPH technique and implemented it to simulate the motion of a droplet down an inclined plane. The same methodology has been used in the present investigation. Only a brief mention of the governing equations and an outline of its numerical implementation are provided below. For further details, one may refer to Das and Das.23 The continuity and the momentum conservation equations for both the liquid drop and the surrounding air can be written as follows using the Lagrangian formulation: DF ¼ - Fr:v Dt F
Dv C ¼ - rp þ r:τ þ Fgz rφ Dt CaCn
ð1Þ
inclination angle
wettability gradient
5°
3°/mm
5°
5°/mm
10°
4°/mm
10°
6°/mm
Fi ¼
To implement the DI-SPH scheme, initially, the domain is discretized into a number of ordered particles initially which are characterized by mass mi, velocity vi, and density Fi. SPH (21) Xu, Z.; Meakin, P.; Tartakovsky, A. M. Phys. Rev. E 2009, 79, 036702.1– 036702.7. (22) Tartakovsky, A. M.; Ferris, K. F.; Meakin, P. Comput. Phys. Commun. 2009, 180, 1874–1881. (23) Das, A. K.; Das, P. K. Langmuir 2009, 25, 11459–11466.
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100Δx 120Δx 130Δx 110Δx 120Δx 130Δx 90Δx 100Δx 120Δx 100Δx 110Δx 120Δx
26 0 0 38 0 0 26 12 0 32 10 0
j¼1 N P j¼1
mj Wij ! mj Wij Fj
ð5Þ
where, W(|xi - xj|, h) is the smoothing function between particles i and j, and h is the smoothing length around a particle. Following the same discretization methodology, eqs 2-4 can be written as ! Rβ N N X DvRi σRβ DWij X i þ σi ¼ mj þ ξij þ mj g Dt Fi Fj Dxβi j¼1 j¼1 ! Ci Cj ðφi - φj Þ þ N X m: j Cai Cni Caj Cnj DWij jr F F j Drij ij j¼1 i j or ! Rβ N N X DvRi σRβ σ DWij X ¼ mj i2 þ i2 þ ξij þ mj g β Dt Fi Fi Dxi j¼1 j¼1 ! Ci Cj ðφi - φj Þ þ N X m: j Cai Cni Caj Cnj DWij jrij j FF Drij j¼1 i j
ð3Þ
ð4Þ
number of fluid particles crossing the solid boundary
N P
The first part of the chemical potential is the surface gradient energy of the domain, and the last part signifies the bulk energy of fluid mixtures. The detail of the formulation is provided elsewhere.23 To update the interface locations and color field, the following equation is used along with no flux condition near the wall and diffusively controlled local equilibrium at the interface DC ¼ kr2 φ Dt
D
expresses the property of one particle and its spatial derivative as a cumulative weighted summation of that property or its derivative of its neighbors.20 Accordingly, after posing kernel approximation and particle approximation in SPH, the density of a particle can be approximated as20
ð2Þ
It may be noted that the color code C is used to differentiate the two phases and Ca is the Capillary number (Ca = νμ/σ). The last term in the right-hand side of eq 2 denotes the contribution due to diffuse interface. Cn defines the mean interfacial thickness in terms of the characteristic length of the system.23 Chemical potential φ for each phase can be defined as φ ¼ ½rðC þ 1Þ2 ðC - 1Þ2 - ðCn2 r2 CÞ
Table 1. Effect of Empirical Constant D for Various Inclination Angles and Wettability Gradients
φi ¼
ð6Þ
N X m: j ððCi þ 1Þ2 ðCi - 1Þ2 - ðCj þ 1Þ2 ðCj - 1Þ2 Þ DWij jrij j FF Drij j¼1 i j
-
2 2 N X m: j ðCni þ Cnj ÞðCi - Cj Þ D2 Wij jrij j FF Dr2ij j¼1 i j
N X dCi m: j ðki þ kj Þðφi - φj Þ D2 Wij ¼ dt FF jrij j Dr2ij j¼1 i j
ð7Þ
ð8Þ
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Figure 1. Effect of numerical parameters on drop shape.
Stress tensor σRβ i can be defined by taking care about the in situ pressure ( pi) and viscous stresses in particle-based lagrangian domain as follows: ! Dvi β Dvi R 2 Rβ Rβ Rβ ð9Þ σ i ¼ - pi δ þ μi þ - ðrvi Þδ Dxi R Dxi β 3 To appropriately model the forces near the interfaces, the particles near it are treated specially. Several interparticle forces are applied across the diffuse interface. Continuum models of such forces are adopted in diffuse interface concept. DI-SPH forms of such forces are tested satisfactorily for simpler case studies (Das and Das23). The interactive forces across the interface are described below. As the phenomenon involves the dynamic evolution of the interfacial configurations, surface tension force across the interface is to be modeled with utmost care. Surface tension force (Fsi) is applied to the fluid particles using the continuum surface force method (CSF).24 Fsi can be written as N X m: j ðGi - Gj Þ DWij n δs ð10Þ Fsi ¼ - σ lv F F jrij j Drij jnjs j¼1 i j where Gi ¼
N X m: j ðCi - Cj Þ DWij F F jrij j Drij j¼1 i j
ð11Þ
and δs is the interface delta function. n can be calculated as follows: n ¼
N X m: j ðCi - Cj Þ DWij F F jrij j Drij j¼1 i j
ð12Þ
(24) Morris, J. P.; Fox, J. P.; Yi, Z. J. Comput. Phys. 1997, 136, 214–226.
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The impermeable solid substrate is modeled by two layers of virtual particles which resist the fluid particles penetrating the wall by applying the resistive force (Frij) of the following form: 2 Frij
r0 ¼ D4 rij
!12
r0 rij
!6 3 5xij r2ij
for rij < r0 ¼ 0 for rij gr0
ð13Þ
The minimum distances of the virtual and the fluid particles at every time instant are expressed as r0. The value of the numerical constant D depends on the maximum speed of the particles near the solid-liquid interface. After a rigorous parametric variation for a wide range of inclination angle and wettability gradient, the value of D has been chosen as 120Δx.23 Some representative situations of parametric variations are shown in Table 1. Interfacial resistive force (Frij) and surface force (Fsi) are applied to each particle after repositioning, based on the local momentum equation including surface tension force. Time step is determined considering the imposed diffusion limit due to the Lagrangian nature of the scheme.20 To handle the change of contact angle due to the change of drop volume near the triple line, a particle redistribution scheme is imposed. Local contact angle (θ0 ) can be determined as " θi0
¼ cos
-1
# dθ σ cos θint þ ðxi cos R - xint Þ dx Rσ lv
ð14Þ
Effect of line tension on the contact line is taken care of in eq 14 following Vafaei and Podowski.25 Spatial variation of in situ contact angle due to wettability gradient is combined in the same (25) Vafaei, S.; Podowski, M. Z. Nucl. Eng. Des. 2005, 235, 1293–1301.
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Figure 2. Uphill movement of a liquid drop due to wettability gradient. 9550 DOI: 10.1021/la100145e
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Figure 3. Comparison of drop shape with experimental observation of Chaudhury and Whitesides9.
equation. Here, θint is the contact angle at location xint and dθ/dx is the strength of the wettability gradient. R is the angle of inclination of the gradient surface from horizontal. In the above equation, σ is the line tension and R is the equivalent circular contact radius of the base area A. After the momentum equation is solved, the locations of the particles at the vicinity of the solid boundary are checked. They are repositioned to satisfy the local contact angle θ0 . This procedure is repeated iteratively until a stable solution for the drop shape is achieved. For the computation of a three-dimensional drop, initially, the liquid domain over the inclined gradient surface is taken as a rectangular prism. The dimension of the prism depends on the drop size. A three-dimensional gaseous domain around the liquid prism is also considered to simulate the interfacial forces appropriately. A 200 mm 50 mm 50 mm domain is used for the span of the drop movement and its shielding periphery. To start with, the liquid and the air particles are uniformly spaced in the domain. The spacing between the particles is chosen after a rigorous optimality test. The effect of particle spacing on the interface configuration is shown in Figure 1a. A 2 μL drop over a 15° inclined surface having 1°/mm wettability gradient is simulated. On the basis of a compromise between the computational effort and accuracy, 0.1 mm uniform spacing between the particles has been chosen for the present work. During the investigation, different drop volumes were considered; accordingly, the number of particles varies between 104 and 106. Choice of particle spacing for the diffuse interface is another critical step of the simulation. Figure 1b shows the change of shape for the previous conditions with different span of diffuse interface. On the basis of this study, four particle spacing is allowed as the diffuse interfacial region. The simulation starts with a zero velocity of the particles. Due to the fluid motion internal to the drop, particles attain nonzero velocities and reach their respective equilibrium positions under the action of potential and surface energies. The simulation progresses through steady dynamic solutions of the drop shapes for corresponding surface inclination and wettability gradient. Though the formulation is applicable for any fluid pair, all the case studies have been made for water drops in an environment of air. Langmuir 2010, 26(12), 9547–9555
Results and Discussion Figure 2 shows the simulated particle positions for a water drop of 2 μL volume over a surface having 15° inclination with the horizontal plane. A wettability gradient of 1°/mm is considered in the upward direction along the surface. It can be observed from the figure that the drop has translated up due to the presence of a strong wettability gradient. Change of surface topology and contact angles are also visible from the drop shape at different time instants. Peristaltic motion of the drop can also be visible from the particle positions as the drop moves up. The simulation results have been compared against the pioneering experimental observation of Chowdhury and Whitesides9 in Figure 3. The water drop shape and its position at three different time levels, after positioning the drop over the surface (15° upward inclination with 7.2°/mm wettability gradient), matches very well with the experimental observations of Chowdhury and Whitesides.9 The above exercise shows the suitability of the DI-SPH hybrid scheme for the simulation of drop dynamics over a gradient surface. The fluid dynamics associated with the motion of a drop over a solid surface is rather complex. The classical no-slip condition of the wall is apparently violated at the three-phase contact line. This poses a problem to both the theoretical formulation as well as its computational implementation. DI is a successful scheme26 to handle this anomaly. Further, the fluid movement internal to the drop plays a very crucial role in determining the magnitude and direction of its bulk movement. The internal fluid flow of a moving droplet resembles rolling or sliding or a combination of both.27 Das and Das23 captured this fluid motion successfully using a hybrid DI-SPH scheme for a drop sliding down a plane with uniform wettability. The existence of a wettability gradient renders the drop dynamics more complex. To elucidate this, we have simulated the fluid motion inside a drop for three different (7°/mm, 4°/mm, and 1°/mm) wettability gradients of the solid surface making an angle 10° with the horizontal. Results are shown in Figure 4. The simulation for the three different wettability gradients represents (26) Ding, H.; Spelt, P. D. M. J. Fluid Mech. 2007, 576, 287–296. (27) Gao, L.; McCarthy, T. J. Langmuir 2006, 22, 6234–6237.
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Figure 4. Velocity vectors of the particles in an inclined plane.
three completely different phenomena. For a wettability gradient of 7°/mm, a very strong clockwise circulation of the fluid particles is seen within the drop. This clearly indicates upward movement of the drop against gravity. On the other hand, for a wettability gradient of 1°/mm the circulation is anticlockwise. This signifies a downward motion of the drop in spite of a nonfavorable wettability in the direction of motion. It is obvious that the dynamics of a liquid droplet over an inclined surface depend on the complex interplay between the 9552 DOI: 10.1021/la100145e
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gravitational force and the surface force. A drop resting on an inclined plane can start sliding down when the gravitational pull can overcome the pinning force at the triple line and the resistive force due to fluid friction. This can be achieved either by increasing the volume of the drop or by tilting the surface more toward the vertical. The drop dynamics becomes more involved when the inclined surface possesses a wettability gradient with the wettability increasing in the upward direction. For a given drop volume and inclination, at a low value of wettability gradient the gravitational pull will dominate and will make the drop slide down the plane (Figure 4c). With the increase of the wettability gradient, obviously the surface force opposing the downward movement will start increasing. Within a range of wettability gradient, the gravity will no longer be able to pull down the drop. The drop will remain stationary. This is depicted by Figure 4b. The internal fluid particles have negligible velocity. No welldeveloped circulation cell is visible. With a further increase of the wettability gradient, a distinct clockwise circulation cell develops inside the drop (Figure 4a) and it starts moving up. It is a well-known fact that the contact angle hysteresis is responsible for the movement of a drop over a surface. Therefore, it would be prudent to examine the change of contact angle during the motion of a drop over an inclined plane with wettability gradient. Such a study is depicted in Figure 5. A 2 μL water drop over a 15° inclined surface is considered. Two extreme contact angles are subtended at A and B, by the cross-sectional profile of the drop taken at the plane of symmetry. Variations of these two angles are studied for the variation of the wettability gradient when the nominal contact angle of the surface is taken as 85°. It may be noted though that the wettability at A is always higher than that at B. At a low wettability gradient of the surface, the contact angle at A is smaller compared to that at B. This is a case of the downward movement of the drop due to the dominance of gravitational pull. With the increase of gradient strength, the difference between the contact angles at A and B reduces. At some point, these two angles are almost equal. Such a situation indicates the static state of the drop. With a further increase in the wettability gradient, though the contact angle increases at both the ends, the angle at A increases at a faster rate signifying upward movement of the drop. While the angle at A shows a monotonic increase with the wettability gradient, interestingly enough, the angle at B initially decreases and then increases at a slower rate. The combined effect of these two angles makes the drop slide downward, stay stationary, and climb upward over a range of wettability gradient. We have also calculated the vertical component of the drop velocity for different drop volumes (50 μL, 5 μL, and 100 nL) climbing over a surface inclined at 15° with the horizontal. In order to estimate the velocity, a reference point over the surface, equidistant from the advancing and the receding ends, has been considered. In Figure 6, the velocity of the liquid drop has been plotted as a function of wettability gradient. The velocity increases gradually with the wettability gradient. It may be noted that a finite value of wettability gradient is needed for the mobility of a drop in the upward direction. This value increases with the drop volume. Further, for the same wettability gradient, a smaller drop will have a larger velocity. This trend can be easily explained, as a small drop has a low inertia and a small footprint. In Figure 7, drop mobility curve has been reported for water drop of 50 μL volume over gradient inclined surface. It has been observed that the drop shows three different kinds of behavior as far as its movement is concerned over inclined gradient surfaces. At a very small inclination, the drop initially remains static but it moves uphill as the wettability gradient increases. As the Langmuir 2010, 26(12), 9547–9555
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Figure 5. Contact angle hysteresis of a drop over an inclined gradient surface.
Figure 6. Vertical component of drop velocity on a 15° inclined plane as a function of wettability gradient.
Figure 7. Drop mobility map for a water drop of 50 μL volume. Langmuir 2010, 26(12), 9547–9555
inclination increases, the drop climbs down the surface due to the action of gravitational force. With the increase in wettability gradient, the downward movement of the drop ceases, and at a particular gradient strength, the drop again remains static over the surface. Further increase in gradient strength causes an upward movement. To describe different regimes in the mobility map, we have marked one vertical line on the plot which eventually signifies variation of wettability gradient for a surface having constant inclination. In the region where the drop moves downward (region AB), an anticlockwise circulation cell is generated inside the drop. As we move up from point A to point B, circulation strength decreases. At some wettability gradient over point B, the drop become static though anticlockwise circulation still remains. In this region (region BO), the anticlockwise circulation cell is not strong enough to overcome the drop inertia to move. At point O, the direction of circulation cell DOI: 10.1021/la100145e
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changes and a clockwise rotation starts. This helps in opposing the gravitational pull. However, in region OC gravity still dominates and keeps the drop pinned over the surface. Beyond point C, the strength of the clockwise circulation can overcome the resistance against the upward motion and the drop starts moving up. By varying the angle of inclination, three regimes can
Figure 8. Drop mobility map for a water drop of 5 μL volume.
Figure 9. Drop mobility map for a water drop of 100 nL volume.
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be clearly identified for mobility of liquid drop. Most importantly, one narrow region has been identified in the curve where the drop will remain static over the surface. It separates two mobile regimes of drop movement, namely, upward and downward. It has also been observed that the span of the static zone is maximized over moderate angles of inclination, but it narrows down for both low and high inclination. Over slightly inclined gradient surface, upward movement can be achieved at a low-level wettability gradient, which enhances the change of circulation direction and causes the static regime to be minimized. On the other hand, on a highly inclined surface the downward pull is dominant, and to generate the upward motion, a large magnitude of wettability gradient is necessary. It eventually narrows down the static zone at high angle of inclination, but at intermediate angle of inclination (8-12° for 50 μL drop), both forces are dominant. So, shift from one regime to other regime cannot be abrupt, as this widens the static region. The drop mobility curves have also been reported in Figures 8 and 9 for drops of volume 5 μL and 100 nL, respectively. Three different regimes of upward movement, static drop, and downward movement could also be observed in these figures. It can be noted that the static zone is reduced with the decrease in drop size. This is due to the reduction in drop inertia. Further, a study has also been made to investigate the hydrodynamics of a static drop over an inclined surface. It has been observed from the mobility map that a drop remains static over a large variation of wettability gradient strength at a moderate inclination of the solid surface. Numerically estimated drop profile shows that, though the drop remains static, the drop spreads as the gradient strength increases for a fixed inclination angle of the solid surface. We have shown one representative simulation in Figure 10 where a drop of 50 μL is placed over a 10° inclined gradient surface. It can be observed that the drop foot radius is more for 5°/mm strength compared to 3°/mm. On the other hand, drop height decreases as the gradient strength increases. As the drop becomes flattened at a higher gradient strength, the height of its center of gravity is reduced compared to that of a drop on a surface with a lower strength of gradient. Ultimately, surface force dominates and creates an internal circulation to propel the drop upward.
Figure 10. Spreading of a static drop as wettability with the increase in gradient strength. 9554 DOI: 10.1021/la100145e
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In summary, we report the dynamics of three-dimensional liquid drops over an inclined gradient surface (wettability gradient increasing in the upward direction) using a combined DI-SPH formulation. In general, the dynamics of the drop depends on parameters like liquid properties, drop volume, surface inclination, and wettability gradient of the surface. The combination of these parameters will decide one of the three possible dynamic states of the drop, namely, climbing up, moving down, and static. These states can be adequately explained by two parallel approaches. A distinct switchover in the direction of circulation cells take place between climbing (28) Abdelgawad, M; Wheeler, A. R. Proceedings of Transducers and Eurosensors 07; Lyon, France, 2007.
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up and moving down, while there is literally no internal circulation inside a static drop. On the other hand, the contact angle hysteresis also changes its sign between upward and downward movement of the drop. In between, there is a range in which contact angles at the extreme ends of the drop profile are almost equal, signifying no movement. 3D droplet actuation or all-terrain droplet actuation28 is a fascinating proposition for manipulating droplet motion. It can have enormous potential in microfluidics, as it can increase the capacity of the device substantially. It has been shown that electrowetting possesses the possibility of such manipulation. The possibility of employing gradient surface for the same purpose is demonstrated by the present investigation.
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