Drops Sitting on a Tilted Plate: Receding and Advancing Pinning

Feb 28, 2012 - ABSTRACT: The wetting behavior of a liquid drop sitting on an inclined plane is investigated experimentally and theoretically. Using. S...
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Drops Sitting on a Tilted Plate: Receding and Advancing Pinning Tung-He Chou,† Siang-Jie Hong,‡ Yu-Jane Sheng,*,† and Heng-Kwong Tsao*,‡,§ †

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 106, Republic of China Department of Chemical and Materials Engineering, and §Department of Physics, National Central University, Jhongli, Taiwan 320, Republic of China

‡

S Supporting Information *

ABSTRACT: The wetting behavior of a liquid drop sitting on an inclined plane is investigated experimentally and theoretically. Using Surface Evolver, the numerical simulations are performed based on the liquid-induced defect model, in which only two thermodynamic parameters (solid−liquid interfacial tensions before and after wetting) are required. A drop with contact angle (CA) equal to θ is first placed on a horizontal plate, and then the plate is tilted. Two cases are studied: (i) θ is adjusted to the advancing CA (θa) before tilting, and (ii) θ is adjusted to the receding CA (θr) before tilting. In the first case, the uphill CA declines and the downhill CA remains unchanged upon inclination. When the tilted drop stays at rest, the pinning of the receding part of the contact line (receding pinning) and the depinning of the advancing part of the contact line (advancing depinning) are observed. The free energy analysis reveals that upon inclination, the reduction of the solid−liquid free energy dominates over the increment of the liquid−gas free energy associated with shape deformation. In the second case, the downhill CA grows and the uphill CA remains the same upon inclination. Advancing pinning and receding depinning are noted for the tilted drop at rest. The free energy analysis indicates that upon inclination, the decrease of the liquid−gas free energy compensates the increment of the solid−liquid free energy. The experimental results are in good agreement with those of simulations.

I. INTRODUCTION The shape of a liquid drop sitting on a tilted plane is different from that on a horizontal plane. When the gravity effect can be neglected, the droplet on a horizontal plane exhibits a spherical cap. Its thermodynamic equilibrium of the three interfacial tensions can be described by the Young’s equation1 for the static meniscus with the contact angle θ, i.e., cos θ = (γSG − γSL)/γLG, where γLG, γSG, and γSL represent liquid−gas, solid− gas, and solid−liquid interfacial tensions, respectively. Different from the axisymmetric shape with a unique contact angle (CA), the shape of a droplet on a tilted plane is asymmetric in the front-rear meniscus but symmetric with respect to the midplane on the major axis. Since the axisymmetry is broken due to gravity, a drop resting on an inclined plane has a distribution of CA and is typically characterized by the angle at the uphill side θu and that at the downhill side θd. Tilted drops which either get stuck or slide down the inclined surface are frequently observed in daily life, such as raindrops on car windshields and plant leaves. Moreover, an important example in structural genomics applications is the extraction of protein crystals from a hanging drop onto glass or plastic substrates. The pinning behavior and shape change of protein solution drops during the process of flipping a sitting drop plate upside down affect greatly the solvent evaporation rate, crystal growth rate, and the ultimate quality of the protein structure.2 The shape of a tilted drop can be related to the phenomenon of contact angle hysteresis (CAH), for which the initial contact angle of a drop on a horizontal plane, the advancing CA (θa), is different from the final CA after a cycle of inflation and © 2012 American Chemical Society

deflation of the droplet volume, the receding CA (θr). Just before a drop sitting on an inclined plane starts to move owing to gravity, the downhill CA of this drop should reach a maximum value (θd = θa) while the uphill CA would decrease to a minimum value (θu = θr). In general, the CAH is expressed in terms of the differences between the advancing and receding contact angles (Δθ = θa − θr). In order to oppose the sliding motion driven by gravitational force, there must exist resistance force, which is generally related to surface roughness, inhomogeneity, defects, or liquid adhesion.3−5 Since the contact line (the three-phase line) of the static drop is observed pinned on the inclined plane, resistance force is also referred to as the pinning force. In the theoretical studies about the tilted drop, the pinning behavior is typically modeled by the fixation of the contact line to the solid surface with a prespecified shape of the base (e.g., circle, ellipse, or both combined), and the drop shape is obtained by solving the Young−Laplace equation.4,6−9 Recently, the pinning of only one end of the drop (front or rear end) is assumed to investigate the CAH of a two-dimensional drop by free energy minimization.10 The critical angle of inclination at which the drop movement begins can be determined by balancing the gravitational force and the pinning force due to surface tension. Besides the contact line pinning, the frictional resistance to the interior liquid offered by the solid−liquid interaction is also Received: January 18, 2012 Revised: February 22, 2012 Published: February 28, 2012 5158

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tilted drop on a Plexiglas microscope slide are explored experimentally and theoretically. The comparison between simulations and experiments are illustrated. The simulation results compared to the experimental results of Berejnov and Thorne’s work are also made. Finally, the conclusions are presented in section IV.

proposed to simulate the drop movement over an inclined surface by smoothed particle hydrodynamics.11 When the tilted angle of the plane (α) reaches a critical limit, called the sliding angle αc, the drop with a specified volume begins to move downward. It is an important property for depicting the CAH on a superhydrophobic surface.12,13 On the other hand, for a given tilted angle, the drop with the volume exceeding a critical value cannot remain stuck. Quéré et al.12 calculated the critical volume of the spherical-cap drop on the inclined plane with small CAH. Recently, Berejnov and Thorne14 investigated experimentally the transient pinning behavior of a tilted water drop on a siliconized flat glass slide. It was found that the contact line exhibits three transitions of partial depinning: depinning of the advancing part of the contact line, depinning of the receding part of the contact line, and depinning of the entire contact line leading to the translational motion. It is observed that the downhill edge creeps forward but the uphill edge remains fixed before the uphill CA declines to θr. This phenomenon is consistent with the so-called receding pinning. As the angle of inclination is increased, the downhill part of the contact profile is elongated and the uphill part remains essentially unchanged. The maximum width is kept the same as the base diameter on the uninclined plane, and this minor axis separates the uphill and downhill parts of the contact profile. Evidently, these experimental observations are not explored in previous theoretical studies. The pinning behavior which is responsible for the tilted drop against the gravity is closely related to the CAH. There has been a long controversy over the origin of the CAH. Generally, there are two different mechanisms proposed: the mechanical pinning by localized defects15,16 and the manifestation of adhesion hysteresis.17−20 In the localized defect mechanism, the CAH is assumed to originate from some intrinsic blemishes, and these blemishes are more wettable than the other areas of the surface. As a result, the CAH is influenced by the size, strength, and density of the localized defects. In contrast, the adhesion mechanism states that the defects can be induced by the liquid due to the restructuring of the adhesive surface over a short period of time. The rearrangement of the molecules in contact with the liquid drop on the surface leads to the decrease of the interfacial energy. On the basis of the liquid adhesion mechanism, a simple model20 has been proposed to describe the CAH by introducing only two thermodynamic parameters: the original solid−liquid interfacial tension (γSL) and the induced solid−liquid interfacial tension (γ′SL) with γSL > γ′SL. In order to understand the experimental observation of Berejnov and Thorne’s work, depinning of the advancing and receding parts of the contact line and depinning of the entire contact line, experimental works and numerical simulations by Surface Evolver (SE), based on the liquid-induced defect model for the CAH, are adopted to investigate the behavior of a tilted drop, including the shape, contact profile, and sliding angle. A drop with CA equal to θ is first placed on a horizontal plate (α = 0°), and then the plate is tilted. Two cases are studied: (i) θ is initially adjusted to be the advancing CA (θa) before tilting and (ii) θ is initially adjusted to be the receding CA (θr) before tilting. Receding and advancing pinning/depinning associated with the tilted drop (α > 0°) are carefully investigated. This paper is organized as follows. In section II, the details of SE incorporating the liquid-induced defect model are presented. The experimental work of the tilted drop is described as well. In section III, both receding pinning and advancing pinning of the

II. SIMULATION DETAILS AND EXPERIMENTAL METHOD A. Simulation Details. There are two equivalent approaches used to obtain the shape of a tilted drop in the theoretical analysis: the Young−Laplace equation and energy minimization method. The Young−Laplace equation is a nonlinear partial differential equation derived from the hydrostatic pressure across the interface between two fluids (e.g., liquid and gas) due to surface tension. On the other hand, the energy minimization method is based on the minimization of the total free energy (e.g., the surface energy and the gravitational energy) subject to constraints. For a specified system, both methods will give essentially identical results within a statistically uncertainty. In this study, Surface Evolver (SE), which is an interactive program with finite element method developed by Brakke,21 is used to simulate the liquid droplet sitting on an inclined flat surface. Because SE possesses an efficient numerical algorithm to evolve the surfaces shaped by total free energy toward a local minimum energy, it has been applied for the study of various wetting phenomenon.22−24 In our SE simulation, a tilted liquid droplet is represented by 6 065 vertices, 18 064 edges, and 12 000 facets. Two constraints are used: the vertices of the contact line are required to lie on the surface, and the droplet volume is kept the same during iteration. Each equilibrium shape of a tilted drop is determined by searching the local minimum energy estimated by the gradient descent method. Our calculation was done until the total free energy difference converges to within the acceptable tolerance of 10−5, and generally it took about 1 h of CPU time on a PC Intel Core i7-2600 3.4 GHz running Windows 7. The

Figure 1. (a) Schematic of a liquid drop sitting on an inclined plane with the tilted angle α. The droplet shape is characterized by the contact angles at downhill edge (θd) and uphill edge (θu). (b) In our simulation, the experimental setup is equivalent to a horizontal drop subject to the external forces, g sin(α) parallel to the plane and g cos(α) perpendicular to the plane. schematic of a tilted drop is shown in Figure 1a, where g represents the gravitational acceleration and α denotes the angle of inclination. The tilted drop in Figure 1a is equivalent to a drop sitting on a horizontal plate subject to an external force of g sin(α), as shown in Figure 1b. The increase of the degree of inclination corresponds to the increment of the external force acting on the drop. The total free energy (F) associated with the liquid drop with the volume V consists of both the surface free energy and the gravitational energy, and it can be expressed as23

F = γLGALG + (γSL − γSG)ASL + 5159

∭V (ρgz) dV

(1)

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where ALG and ASL depict the liquid−gas and solid−liquid interfacial areas, respectively. The third term denotes the gravitational energy. Using the Young’s equation, eq 1 becomes

F = γLG[ALG − ASL cos(θ)] +

∭V (ρgz) dV

perpendicular to the plane of the Plexiglas slide (corresponding to the top view). On the other hand, the apparent contact angles at the downhill and uphill edges and the inclination angle of the plane are observed from the side view of the deformed droplet by using the Optem 125C optical system. The enlarged images are analyzed by the software MultiCam Easy 2007 (Shengtek, Taiwan).

(2)

The energy contribution of gravitation can be resolved into the vertical component g cos(α) and the horizontal component g sin(α), and the latter term is the major force to drive the deformation of the tilted drop. It can be expressed as F = γLGALG − γLG cos(θ)ASL + (g cos(α) dV + g sin(α)

∭V (ρx) dV )

III. RESULTS AND DISCUSSION A. Comparison between SE Simulations and Berejnov and Thorne’s Work. In Berejnov and Thorne’s work, the contact line of the statistically pinned drop exhibits three transitions of partial depinning as the degree of inclination is increased. The water drop deposited on a siliconized flat glass slide shows a contact angle 90°−92°. The uphill and downhill angles have been measured at different degrees of inclination for a specified drop volume, which varies approximately from 10 to 95 μL. Moreover, the onsets of the instability, including quasistatic displacement of the advancing and receding parts of the contact line and continuous motion of the whole contact line, are expressed in terms of the critical tilted angles as αa, αr, and αc, respectively, and it was found that αa < αr < αc. SE calculations are performed to simulate Berejnov and Thorne’s work, and the validity of the liquid-induced defect model can be examined. Only two thermodynamic parameters are required, and they are determined from the experimental data, θa = 92° and θr = 72°. The receding CA is taken from the onset of the local instability of the receding part of the contact line in their experiment. Figure 2 shows the variation of θd and θu with the degree of inclination for water drops of different volumes (20, 30, 50, and

∭V (ρz) (3)

Note that the z- and x-axes are normal and parallel to the tilted plane, respectively. All lengths are scaled by the capillary length of water (a = (ρg/γLG)1/2 = 0.27 cm) and the energy by γLGa2. Therefore, the dimensionless form can be expressed as Ft = FLG + FSL + FG. The zero point of the gravitational energy is set at the center of the drop in the absence of the external force. The droplet morphology associated with the local minimum of Ft corresponds to the equilibrium shape. The adhesion hysteresis of solid−liquid contact is used to elucidate the contact angle hysteresis, which relates to the spontaneous change in solid−liquid interfacial tension.17−20 Upon wetting, the restructuring of the solid−liquid interface over a short period of time, which can be recognized as liquid-induced defects with higher wettability, leads to the decrease of the solid−liquid interfacial tension from γSL to γ′SL. That is, during simulations, the solid−liquid tension within the wetted region changes to γ′SL while that in the exterior region remains unchanged, γSL. Since there exist two solid−liquid interfacial tensions, one can define two contact angles associated with droplet wetting: the advancing angle θa with cos(θa) = (γSG − γSL)/γLG and the receding angle θr with cos(θr) = (γSG − γ′SL)/γLG according to the Young’s equation. The θa corresponds to γSL because the advancing edge of the contact line is adhesion-like. Similarly, the θr corresponds to γ′SL because the receding edge of the contact line is separation-like. Consequently, the solid−liquid energy is FSL = −cos(θa)ASL* before rearrangement of surface molecules and FSL′ = −cos(θr)ASL* after restructuring of the surface molecules where ASL* is the dimensionless area of the solid−liquid interface. It is worth mentioning that there exists a subtle difference between the solid−liquid adhesion and the wetting of a liquid droplet on the solid surface. That is, while the liquid−gas contact is not considered in the former, it plays an important role in the droplet wetting. On the basis of the liquidinduced defect model, we shall show that the uphill contact angle of a sitting drop with θr < θu < θa reveals the frustration of the surface free energy and can be thermodynamically defined. B. Materials and Experimental Methods. The wetting behavior of a tilted drop is studied by depositing a purified water drop on the Plexiglas microscope slide, which is used as the tilted plate and purchased from Kwo-Yi Co. (Taiwan). It is cleaned by sonication with alcohol in an ultrasonic cleaning tank for 5 min. Note that the surface of the substrate is highly smooth and possesses a rms roughness of about 2 nm determined by AFM. After sonication, the Plexiglas slide is rinsed with deionized water, sonicated again in deionized water for 5 min, and then dried in a stream of nitrogen gas. In order to verify the stability of the substrate, the advancing and receding contact angles of the Plexiglas are determined by the needle-syringe method, which is performed with a contact angle goniometer, drop shape analysis system DSA10-MK2 (Krüss, Germany). Both of them are reproducible (advancing angle of 75°−78°, receding angle of 49°−52°). The CA of the drop sitting on the horizontal plate can be adjusted by inflation for the advancing angle and deflation for the receding angle. The Plexiglas microscope slide is then attached to a home-built inclined device by means of double-sided tap. Inclined plane experiments are conducted by raising the tilted angle manually after placing the water droplet on the surface of Plexiglas slide. A highresolution digital camera is mounted on the inclined device and aimed

Figure 2. Variation of θd and θu with the degree of inclination plotted for water drops of different volumes. The experimental data points of Berejnov and Thorne’s work can be reasonably represented by the simulation results depicted by dashed lines (θa = 92° and θr = 72°).

80 μL). The experimental data points of Berejnov and Thorne’s work can be reasonably represented by the simulation results depicted by empty symbols. The dashed lines are drawn to guide the eyes. As the tilted angle (α) is increased for a given drop volume, the downhill CA remains the same as the advancing CA while the uphill CA declines toward the receding CA. In our simulation, the critical tilted angle αr in which the drop starts to move can be determined by the condition that the tilted drop is unable to remain stuck on the plane numerically when the jiggling perturbation21 (a random permutation) is given. It is observed that the uphill CA becomes θr when the tilted angle reaches αr. The change of θu with α is more significant for a larger drop because of its larger driving force. Figure 3 shows the plot of the critical tilted angle against the drop volume. Although there are three critical angles 5160

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Figure 3. Critical angle α plotted against the droplet volume V. Simulation results are represented by the solid curve, and Berejnov and Thorne’s work is depicted by the symbol square, which denotes the onset of depinning of the receding part of the contact line.

reported from the experiments,14 all of them decline generally with increasing drop volume. Again, the experimental data points associated with the onset of the quasistatic displacement of the receding parts can be reasonably described by our SE simulation results depicted by the solid curve. Note that our SE simulation is purely based on the thermodynamic model, and the hydrodynamics inside the water drop associated with the sliding motion is not considered. As a consequence, the onset of the numerical instability associated with the continuous decay of the free energy due to gravitational energy loss, i.e., movement of the contact line along the gravity direction, provides the boundary between sticking and sliding of the drop. The static contact profile of a water drop with 90 μL at different tilted angles was also reported in Berejnov and Thorne’s work, as shown in Figure 4a. It is clearly shown that the advancing parts of the contact line move downhill when the degree of inclination is increased. This outcome reveals the depinning behavior of the advancing parts during inclination. When α exceeds the critical tilted angle (αr), the receding parts of the contact line become unstable and start to displace. The above features have been observed in our SE simulations, as depicted in Figure 4b. Moreover, the quantitative comparison between the experimental profiles in Figure 4a and the simulation results in Figure 4b is reasonably good. In the experimental work, the length of the major axis (la) is elongated by 9.0% and that of the minor axis (lb) is increased by 1.1% when α rises from 0° to αr (13°). In our simulations, the same αr is obtained and the expansion of la is 6.5% and that of lb is 0.3%. Because of the conservation of volume, the drop height is lowered by 4.1%. The difference between the simulation and experimental results may be attributed to their differences at α = 0°. The contact profile of the experimental work is not circular at α = 0° and (la − lb)/lb = 1.6%. Moreover, the diameter of the base is 7.65 mm in our simulation while lb in the experimental work is 7.31 mm. Our results indicate that depinning of the advancing part of the contact line takes place as α > 0°. However, the displacement extent of the advancing part of the contact line for the small degree of inclination may be too small to be observed in the experiments. Based on the liquid-induced defect model, the wettability of the wetted region (CA = θr) occupied by the drop is higher than that of the unwetted region (CA = θa). As a consequence, the displacement of the advancing part of the contact line can be related to the invasion of the contact line across the boundary between two regions with different wettabilities (θr < θa).16 It is generally believed

Figure 4. Top views normal to the substrate plane of the static contact lines for 90 μL of water drop, achieved as the substrate inclination is increased. The numbers on each contact line indicate the substrate inclination in degrees. (a) Berejnov and Thorne’s work; (b) SE simulations. The comparison of the lengths of major and minor axes between experiments and simulations is shown in the inset.

that the invasion occurs when the CA associated with the advancing part of the contact line (θ) is equal to or greater than the CA of the region with lower wettability, θ ≥ θa. Since the downhill CA always equals the CA of the unwetted area (θa = 92°), it is not surprising that the presence of the external force parallel to the plane due to gravity drives the invasion of the advancing part of the contact line, once the plane is inclined. Although the advancing part of the contact line is always displaced in the simulation, the receding part remains stuck and the uphill CA is always greater than θr, as α ≤ αr. The jiggling perturbation does not affect the equilibrium contact profile. That is, the wettability difference between the wetted and unwetted regions is able to hold the tilted drop against gravity. However, as α > αr, the receding part of the contact line is displaced and the tilted drop cannot maintain the same position. Therefore, the continuous movement of the contact line takes place in our simulation, and the free energy continues 5161

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decaying as the drop moves downhill. Note that the onset of the displacement of the receding part of the contact line signifies the translational motion of the tilted drop in our simulations, i.e., αr = αc. B. A Tilted Drop Beginning with the Advancing CA at α = 0°: Advancing Depinning and Receding Pinning. Based on liquid-induced defect model, we have successfully simulated Berejnov and Thorne’s experimental work performed on siliconized flat glass slides, which are smooth and chemically homogeneous. In order to apply our CAH simulations further to typical polymeric surfaces, the CAH experiments of a tilted drop with the volume 30 μL are conducted first for later comparison with the simulation work. The experiments are performed on Plexiglas slides with a rms roughness about 2 nm determined by AFM. The advancing and receding angles for a water drop on such a substrate are θa = 76° and θr = 52°, which are the only two parameters required in our thermodynamic CAH model. The downhill and uphill contact angles can then be determined as a function of the degree of inclination, θd(α) and θu(α). Note that the experiment is always started with θd = θu = θ(α = 0°). The validity of the liquid-induced defect model can be further examined by changing the initial CA of the drop on the horizontal slide θ(α = 0°). The case θ(α = 0°) = θa is studied first, and the case θ(α = 0°) = θr is investigated later. The comparison between these two cases will be made later. The change of the drop shape (side view) with the tilted angle (α) of the experiment is depicted in the lefthand column of Figure 5a. The top views are shown in Figure S1. Similar to Berejnov and Thorne’s result, the downhill CA remains the same as θa while the uphill CA decreases from θa toward θr as α is increased. When the drop starts to slide, the critical tilted angle is decided and it is found that αr = 29°. The CAH simulations are also performed, and the equilibrium shapes are consistent with the experimental images as shown in the righthand column of Figure 5a. Note that in our simulations, the tilted drop is equivalent to a drop on a horizontal plane with the drop subject to an external force, g sin(α), parallel to the plane as shown in Figure 1. The advancing depinning and receding pinning can be easily observed from this simulation layout. Moreover, the numerical instability associated with the continuous downward movement of the drop occurs also at α = 29°. The agreement between experimental and simulation results provides a justification of our theoretical model. The quantitative results are shown in Figure 5b for θd(α) and θu(α). The variations of the major length (la), minor length (lb), and height (h) with α are illustrated in the inset. Evidently, the experimental data points of θd and θu can be well described by the simulation outcomes. When the tilted angle rises to αr, the uphill CA reaches θr and the depinning behavior of the uphill edge begins. Consistent with Berejnov and Thorne’s work, the increment of the major length with α is clearly observed, and this result indicates that the depinning of the downhill edge due to gravity takes place once the plane is inclined (α > 0°). On the other hand, the variation of the minor length is insignificant, less than 2%. Since the area of the contact profile grows with α, the height of the drop declines because of the volume conservation. Moreover, the apex moves downward along the major axis as α is increased. Note that the experimental results of la, lb, and h are also in good agreement with the simulation outcomes. The above consequences clearly show the depinning of the downhill edge and the pinning of the uphill edge (receding pinning) for the case of a tilted drop with θ(α = 0°) = θa.

Figure 5. (a) Variation of the droplet shape with the tilted angle. The critical angle is αr = 29°. The depinning of the contact line at the downhill edge is clearly shown. (b) Variation of θd and θu with the tilted angle plotted for a drop with V = 30 μL on acrylic glass. The variations of the major length (la), minor length (lb), and height (h) with the tilted angle are illustrated in the inset. The experimental data points can be well described by the simulation outcomes depicted by open symbols. Dashed lines are a guide to the eye.

The SE simulation has successfully depicted the experimental observation of the shape of the tilted drop. As a result, more detailed information which is difficult to be extracted from experimental measurements,25 such as CA distribution and surface free energy, can be easily obtained from SE simulations. The CA distribution can be used to evaluate the capillary force against gravity, and Figure 6a shows the variation of the CA with the azimuth angle (ϕ) at different degrees of inclination. Note that the uphill and downhill edges correspond to ϕ = 0° (360°) and 180°, respectively. At α = 0°, i.e., the plate is horizontal, a nearly uniform CA, θa, is seen. As α is increased, the contact angles decay continuously from the downhill edge (ϕ = 180°) to the uphill edge (ϕ = 0° and 360°). However, there is about 1/6 of the contact line at the downhill edge, still 5162

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energy decays with increasing α reveals that the equilibrium tilted drop is thermodynamically favorable. It is worth noting that the change of total surface free energy (ΔFLG + ΔFSL′) actually declines with increasing α, as demonstrated in the inset of Figure 6b. This consequence implies that the tilted drop possesses lower surface free energy than that of the drop on a horizontal plane. It is somewhat surprising because the axisymmetric shape, such as the spherical cap, at α = 0° supposedly has minimal surface energy. Nonetheless, the above statement is true if one evaluates the change of total surface free energy (ΔFLG + ΔFSL) based on γSL associated with θa instead of γ′SL associated with θr. As shown by the dashed line in the inset, (ΔFLG + ΔFSL) grows with the tilted angle. The reason for the decay of the change of total surface free energy (ΔFLG + ΔFSL′) is due to the energy decrease from the wetted area with γ′SL. In fact, the lowest surface energy of (ΔFLG + ΔFSL′) is corresponding to the axisymmetric shape with the CA equals to θr, as pointed out by the arrow shown in the inset. Unfortunately, the expansion of the contact (wetted) area has to overcome an energy barrier associated with contact line pinning. The gravitational force parallel to the inclined plane provides the work to cross the barrier. C. A Tilted Drop Beginning with the Receding CA at α = 0°: Advancing Pinning and Receding Depinning. A typical experiment of a drop sitting on an inclined surface begins with the horizontal drop having its CA close to the advancing CA. Upon inclination, depinning of the advancing part of the contact line and pinning of the receding part of the contact line are observed. The downhill CA is kept at the advancing CA. The onset of the receding part of the contact line instability occurs when the uphill CA decreases to the receding CA. The above experimental observations have been confirmed by our SE simulations based on the liquid-induced defect model. What will happen if the tilted drop experiment begins with a horizontal drop possessing the receding CA? This experimental setup may provide another means to examine the validity of our theory. A 30 μL water drop with the receding CA (θr = 52°) on the horizontal Plexiglas slide is prepared by deflation of a water drop of volume 100 μL with a syringe. The variation of the drop shape (side view) with the tilted angle is shown in Figure 7a. The top views are illustrated in Figure S2. The uphill CA remains the same as θr whereas the downhill CA rises from θr toward θa, as α is increased. The critical tilted angle is determined to be α′r ≈ 35° as the drop starts to move. The CAH simulations are conducted, and the equilibrium shapes agree with the experimental images. Again in our simulations, the tilted drop is equivalent to a drop on a horizontal plane with the drop subject to an external force, g sin(α), parallel to the plane as shown in Figure 1. The advancing pinning and receding depinning can be easily observed from this simulation layout. In addition, the numerical instability associated with continuous movement takes place also at α′r = 35°. Note that the critical tilted angle of the case θ(α = 0°) = θr is greater than that of the case θ(α = 0°) = θa, i.e., α′r = 35° > αr = 29°, for a drop with the same volume (30 μL). It seems that the energy barrier for resisting drop sliding is higher for the former. The quantitative results are illustrated in Figure 7b, consisting of θd(α) and θu(α). Evidently, the experimental data points of θd and θu can be well depicted by the simulation results. As α is increased, the uphill CA remains at the receding CA (θr) but the downhill CA grows. When the tilted angle rises

Figure 6. (a) Variation of the CA with the azimuth angle (ϕ) at different degrees of inclination. Note that the uphill and downhill edges correspond to ϕ = 0° (360°) and 180°, respectively. (b) Variation of the free energy change (ΔF) with the degree of inclination including ΔFLG, ΔFSL′, ΔFSL, and ΔFG is plotted for the case θ(α = 0°) = θa.

exhibiting the CA close to the advancing angle (θa = 76°). This is associated with depinning of the advancing part of the contact line. On the other hand, there is only 1/12 of the contact line at the uphill edge, showing the CA near the receding angle (θr = 52°). It was shown that there exist parallel sides associated with a sliding drop.26 However, this feature is not clearly observed before the onset of drop sliding (α < αr = 29°) in our simulation. SE simulation can provide the energy change between the tilted drop at α and that at α = 0°, ΔF = F(α) − F(α = 0°), besides the information associated with the drop shape. Figure 6b shows the variation of the free energy change with the degree of inclination. Because ΔF is the sum of the liquid−gas interfacial energy, solid−liquid interfacial energy, and gravitational energy, the variations of these terms with α (ΔFLG, ΔFSL′, and ΔFG) are plotted to compare their contributions. As anticipated, the deformation of the drop shape from spherical cap leads to the increment of the liquid−gas area, and therefore the liquid−gas interfacial energy (ΔFLG) grows with increasing α. However, as the degree of inclination rises, the gravitational energy (ΔFG) declines because both contributions (parallel and normal to the plane) are reduced. Since the sum of ΔFLG and ΔFG is slightly greater than zero, the contribution of ΔFSL′ plays an important role in lowering the surface free energy. Evidently, as α rises, the solid−liquid interfacial energy (ΔFSL′) decreases owing to the increment of the area of the contact profile by depinning of the downhill edge, as illustrated in Figure S1. Note that ΔFSL′ is calculated by the product of the interfacial tension difference (γ′SL − γSG) and the contact area, i.e., −cos(θr)ASL*. The interfacial tension difference is negative because cos(θr) > 0. Obviously, the fact that the system’s free 5163

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Figure 8. (a) Variation of the CA with the azimuth angle (ϕ) for the tilted drop beginning with the receding angle. (b) Variation of the free energy change (ΔF) with α including ΔFLG, ΔFSL′, ΔFSL, and ΔFG plotted for the case θ(α = 0°) = θr.

edge. It is interesting to note about 1/3 of the contact line at the uphill edge, still displaying the CA close to the receding CA. It should be noted that in contrast to the case of θ(α = 0°) = θa, the area of the contact profile declines with increasing tilted angle for θ(α = 0°) = θr. The area at α′r = 35° is reduced to 87% of that at α = 0°. This is associated with depinning of the receding part of the contact line upon inclination. On the other hand, there is continuous growing of the CA at the downhill edge due to the advancing pinning. Note that the CA distribution is not required in total free energy calculation by eq 3. Based on the CA distribution, the calculated capillary force for the tilted drop associated with the case θ(α = 0°) = θr, (ρVg)sin α′r, is higher than that with the case θ(α = 0°) = θa, (ρVg)sin αr, and one has α′r > αr. Although the shape deformation of the tilted drop is driven by gravity, the change of the interfacial energy is distinctly different for the two cases: θ(α = 0°) = θa and θ(α = 0°) = θr. In the former, the change of the liquid−gas interfacial energy (ΔFLG) rises while the change of the solid−liquid interfacial energy (ΔFSL′) decays with increasing α. In the latter case, however, the change of the liquid−gas interfacial energy (ΔFLG) declines while the change of the solid−liquid interfacial energy (ΔFSL′) grows as α is increased. Figure 8b shows the variation of the free energy change with the degree of inclination. In general, the increment of ΔFSL′ due to the loss of the area of the contact profile is compensated by the decrease of ΔFLG due to the reduction of the liquid−gas interfacial area. The change of total surface free energy (ΔFLG + ΔFSL′) declines first and then rises gradually with increasing α, as demonstrated in the inset of Figure 8b. This result reveals that the inclination actually leads to an increase of the surface free energy based on γ′SL. It is worth noting that the inclination

Figure 7. (a) Variation of the droplet shape with the tilted angle. The critical angle is αr = 35°. The depinning of the contact line at the uphill edge is clearly shown. (b) Variation of θd and θu with the tilted angle plotted (la, lb, and h in the inset) for the case θ(α = 0°) = θr.

to α′r, the downhill CA reaches the advancing CA (θa) and the depinning behavior of the downhill edge begins. The decrease of the major length (la) with α is clearly observed, and this consequence indicates the depinning behavior of the uphill edge due to gravity upon inclination. Note that the major axis is parallel to the direction of the external force. The variation of the minor length (lb) is not substantial, but the area of the contact profile declines with increasing α. Consequently, the height of the drop rises due to the volume conservation, about 10% increment at α′r. The comparisons of la, lb, and h between experiment and simulation are in reasonable agreements. The aforementioned consequences show depinning of the uphill edge and pinning of the downhill edge. This phenomenon is referred to as advancing pinning. These observations totally differ from those of a tilted drop beginning with CA = θa at α = 0°. Figure 8a illustrates the CA distribution, which is often used to calculate the capillary force against gravity. At α = 0°, an almost uniform CA, θr, is observed. As α rises, the contact angles grow continuously from the uphill edge to the downhill 5164

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for the case θ(α = 0°) = θr is somewhat similar to the reverse process of the CAH and the change of total surface free energy (ΔFLG + ΔFSL) decreases toward the free energy of the axisymmetric shape with the CA equal to θa, as pointed out by the arrow shown in the inset of Figure 8b. Nonetheless, the drop will begin to move once θd reaches θa.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (Y.-J.S.); [email protected] (H.-K.T). Notes

The authors declare no competing financial interest.

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IV. CONCLUSIONS

ACKNOWLEDGMENTS This research work is financially supported bythe NCU/ITRI Joint Research Center and the National Science Council of Taiwan.

A liquid drop sitting on an inclined plane is closely related to the phenomenon of CAH. Just before a tilted drop starts to move owing to gravity, the uphill CA would decrease to the receding CA while the downhill CA remains at the advancing CA. Recently, Berejnov and Thorne’s work showed depinning of the advancing part of the contact line before the translational motion of the droplet. In this study, the wetting behavior of a tilted drop is investigated experimentally and theoretically. Two cases are studied: (i) θ is adjusted to the advancing CA (θa) before tilting and (ii) θ is adjusted to the receding CA (θr) before tilting. Using Surface Evolver, the numerical simulations are performed based on the liquid-induced defect model, in which only two thermodynamic parameters (solid−liquid interfacial tensions before and after wetting) are required. The experimental observations in Berejnov and Thorne’s work can be reasonably depicted by SE simulations. In the first case θ(α = 0°) = θa, the uphill CA declines and the downhill CA remains unchanged upon inclination. When the tilted drop stays at rest, pinning of the receding part of the contact line (receding pinning) and depinning of the advancing part of the contact line (advancing depinning) are observed. The experimental results are in good agreement with those of simulations. Moreover, the analyses of surface free energy given by SE simulations reveal that upon inclination, the reduction of the solid−liquid free energy (ΔFSL′) dominates over the increment of the liquid−gas free energy (ΔFLG) associated with shape deformation. However, there exists an energy barrier of the contact profile expansion, which can be overcome by gravity. Because of the increment of the area of the contact profile upon inclination, the first case is analogous to the forward process of the CAH, e.g., inflation of a drop. In the second case θ(α = 0°) = θr, the downhill CA grows and the uphill CA remains the same upon inclination. While the tilted drop stays at rest, pinning of the advancing part of the contact line (advancing pinning) and depinning of the receding part of the contact line (receding depinning) are noted. Again, the experimental results agree quite well with the simulation results. Furthermore, the free energy analyses indicate that upon inclination, the decrease of the liquid−gas free energy compensates the increment of the solid−liquid free energy. Since the area of the contact profile declines with increasing α, the second case is somewhat similar to the reverse process of the CAH, e.g., deflation of a drop.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Top views (normal to plane) of a tilted water drop (30 μL) on an acrylic glass slide for the case θ(α = 0°) = θa (Figure S1); top views (normal to plane) of a tilted water drop (30 μL) on an acrylic glass slide for the case θ(α = 0°) = θr (Figure S2). This material is available free of charge via the Internet at http://pubs.acs.org. 5165

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