Comparison of the lateral retention forces on sessile, pendant, and inverted sessile drops
Rafael de la Madrid, Fabian Garza, Justin Kirk, Huy Luong, Levi Snowden, Jonathan Taylor, Benjamin Vizena
aa r X i v : . [ phy s i c s . f l u - dyn ] F e b Comparison of the lateral retention forceson sessile, pendant, and inverted sessiledrops
Rafael de la Madrid, ∗ Fabian Garza, † Justin Kirk, Huy Luong, ‡ Levi Snowden, Jonathan Taylor, † Benjamin Vizena § Department of Physics, Lamar University, Beaumont, TX 77710
February 19, 2019
Abstract
We compare the lateral retention forces on sessile drops (which are drops thatare placed on top of a solid surface), pendant drops (which are drops that areplaced on the underside of the surface), and inverted sessile drops (which aredrops that are first placed on top and then on the underside of the surface byflipping the surface). We have found experimentally that the retention force ona truly pendant drop is always smaller than that on a sessile drop. However, theretention force on an inverted sessile drop is comparable to, and usually largerthan, that on a sessile drop. Thus, the retention force on a drop depends notonly on whether it is placed on top or on bottom of a surface, but also on thehistory of drop deposition, since such history affects the width, the shape andthe contact angles of the drop. ∗ E-mail: [email protected] † Current address: Department of Physics, Texas A&M University, College Station, TX 77843 ‡ Current address: Sage Automation, Beaumont, TX 77705 § Current address: METECS, Houston, TX 77289 Introduction
The study of liquid drops on solid substrates has attracted a great deal of attention [1–3], because of both its scientific interest and its industrial applications. However,there are many aspects of wetting and dewetting phenomena that are still not wellunderstood. One such aspect was reported in Ref. [4], where Tadmor et al. presenteda very counterintuitive property of the lateral retention force on liquid droplets at themoment the droplets start to slide on a solid surface. Contrary to the solid-solid frictioncase, a drop hanging from a solid surface experiences a larger retention force than adrop resting on the surface [4]. In spite of the attention it drew [5, 6], the origin of thiseffect remains unknown.The purpose of this paper is to put forward an explanation of the effect observedin Ref. [4]. We will argue that the origin of such effect is rooted on how the drops areformed. We will distinguish between sessile drops (which are drops that are formedby placing a small amount of liquid on top of a uniform, flat, solid surface), pendant drops (which are drops that are formed by placing the liquid on the underside of thesurface), and inverted sessile (or simply inverted ) drops (which are drops that areformed by placing the liquid on top of the surface, and then placing it on the undersideby flipping the surface). We have found that the retention force on a truly pendantdrop is smaller than that on a sessile drop, just as naive intuition suggests. We havealso found that the retention force on an inverted sessile drop is comparable to, andusually larger than, that on a sessile drop. This result may explain the effect observedin Ref. [4], because the drops placed on the underside of the solid substrate in Ref. [4]were not truly pendant drops, but rather inverted sessile drops.
Our experimental apparatus is a simplified version of the Centrifugal Adhesion Bal-ance [4] and the Kerberos drop accelerator [7–9]. It consists of a metallic frame withdimensions 90 cm ×
90 cm ×
120 cm inside of which the rotary unit is mounted, seeFig. 1. The four legs of the frame are bolted to the floor to reduce mechanical vibra-tions. The rotary unit consists of a servo motor [10], a shaft, and two pairs of aluminumrails that are attached perpendicularly to the shaft. The motor is connected to a powersupply and a computer, whose software controls the motor.A metallic box is mounted on the aluminum rails. The box has two cameras placedon top and on the side, which provide top and side views of the drops, see Fig. 2.A remote-controlled LED is used to set a common starting time in the videos of thecameras. On the door of the metallic box, we mounted a poly methyl methacrylate(PMMA) sheet (Optix R (cid:13) , by Plaskolite) such that, when a drop is placed on the sheetand the door is closed, the cameras have side and top views of the drop. Illuminationfor the side camera is provided by the remote-controlled LED. Lighting for the top2 Figure 1: Drop accelerator: -aluminum frame; -motor; -shaft; -box. Figure 2: Box: -top camera; -side camera; -PMMA sheet with drop; -remote-controlled LED light; -smart phone with accelerometer. Inside the box (not shownin the picture) there is an LED panel that provides the necessary illumination for thetop camera.camera is provided by an LED panel and an optical gradient [11]. To remove anyremnants of their protective films, the PMMA sheets were initially washed with hotwater and soap. Afterward, before each run, the PMMA sheets were cleaned with 70%isopropyl alcohol and paper tissue [12], and dried with lamplight.To place the water droplets on the PMMA sheet, we used a micropipette [13]. Asessile (pendant) drop was formed by slowly releasing the contents of the micropipetteon top (bottom) of the PMMA sheet. An inverted sessile drop was formed by slowlyreleasing the contents of the micropipette on top of the PMMA sheet, and then flippingthe sheet, so the drop ends up on the underside of the sheet. Figure 3 shows 30- µ Lsessile, pendant, and inverted sessile drops at rest. Clearly, an inverted sessile drop isnot the same as a pendant drop. In particular, the width of the inverted sessile drop3s larger than that of a pendant drop, but comparable to the width of the sessile drop.As we will see, the width is the main factor that differentiates the retention forces onsessile, pendant, and inverted sessile drops.
Figure 3: Side view of 30- µ L sessile , pendant , and inverted sessile drops at rest.To make the drops slide on the PMMA sheet, we rotated the box with a constantangular acceleration of about 0.086 rad/s . As the angular velocity of the drop in-creases, the centrifugal force increases [14]. This eventually makes the advancing edgeof the drop crawl forward in the radial direction, although the receding edge of thedrop stays pinned to the surface [8]. While the advancing edge crawls forward, thetriple line deforms slightly from its initial circular shape. At some point, when the cen-trifugal force is large enough, the receding edge of the drop also starts moving in theoutward, radial direction. When that happens, the whole drop moves in the outward,radial direction [8]. This is why we identify the onset of the motion of the drop withthe instant at which the receding (i.e., trailing) edge of the drop starts moving.We used the videos of the side camera and custom-made software to determine theinstant when the receding edge of the drop starts moving. At such instant, we obtainedthe contact angles from the videos of the side-view camera using ImageJ [15], and thewidth from the videos of the top-view camera using PixelZoomer [16].4 Results and Discussion
We measured the time it took each drop to start sliding on the PMMA sheet for eachtype of drop (sessile, pendant, and inverted sessile), for each volume (15, 20, 25, and30 µ L), and for twelve different PMMA sheets. For volumes larger than 30 µ L, it ishard to produce inverted drops, because big drops slide on the surface during the flip.Hence, for 40-100 µ L volumes, we analyzed only sessile and pendant drops at intervalsof 10 µ L. Using the times at which the onset of the motion occurs, we obtained thelateral retention force from the centrifugal force, F = mrω = ρV rα t , (3.1)where m is the mass of the drop, ρ is its density, V is its volume, r is the distance fromthe center of the drop to the axis of rotation (about 170 mm in our experiment), α isthe angular acceleration, and t is the time since the rotation of the motor started.For each volume and for each type of drop, we calculated the average time of about24 runs, and we obtained that t pendant < t sessile < t inverted . (3.2)Hence, F pendant < F sessile < F inverted . (3.3)The best way to visualize Eq. (3.3) is by plotting F sessile F pendant (see Fig. 4) and F inverted F sessile (see Fig. 5), which according to Eq. (3.3) should be greater than one. It is clear fromFig. 4 that F sessile is always greater than F pendant . It is also clear that as the volumedecreases, F sessile and F pendant become closer to each other. Indeed, for 100- µ L drops, F sessile is 18.9% larger than F pendant , but for 15- µ L drops, F sessile is only 2.4% largerthan F pendant . This is not surprising, because the influence of gravity compared to thatof surface tension decreases as the volume decreases, and therefore the drops becomemore alike as they become smaller.We can see in Fig. 5 that F inverted is comparable to, and usually larger than, F sessile .We can also see that, unlike Fig. 4, Fig. 5 does not show a steady decrease of the ratio F inverted F sessile as the volume decreases. What is more, for 25 µ L, F inverted is essentially thesame as F sessile . The reason is that it is difficult to prepare inverted sessile drops byflipping a surface. In fact, unless it is done properly [12], the drop tends to slide onthe surface during the flip, which may deform the drop and make it loose some of itsadhesion to the surface.Another way to rephrase Eq. (3.3) is by comparing the centrifugal accelerations atthe onset of the motion, see Table 1. As can be seen in Table 1, sessile drops startsliding at a larger acceleration than pendant drops, but at a smaller acceleration thaninverted drops, and hence Eq. (3.3) holds. 5
20 40 60 80 100
Volume H Μ L L F sessile F pendant Figure 4: Ratio F sessile F pendant for 15-100 µ L drops. Error bars are only statistical. The dashedline represents unity.
Volume H Μ L L F inverted F sessile Figure 5: Ratio F inverted F sessile for 15-30 µ L drops. Error bars are only statistical. The dashedline represents unity. Centrifugal Acceleration (m/s ) Volume ( µ L) 100 90 80 70 60 50 40 30 25 20 15
Sessile
Pendant
Inverted
To exclude systematic effects as the source of our results, we performed two additionalexperiments. One experiment was similar to that of Ref. [12], and it yielded the same6esults as the present paper. The other one, which can be very easily reproduced, wasa simple tilted-plate experiment. We placed a sessile and a pendant drop on a PMMAsurface and then slowly tilted the surface until the drops started to slide. We visuallyobserved that, on average, the pendant drops started to slide before the sessile drops.However, when the same experiment was done with sessile and inverted sessile drops, weobserved that, on average, the sessile drops started to slide before the inverted sessiledrops. We also did the tilted-plate experiment with water drops on polycarbonate(Lexan) and obtained the same results.The most important factor affecting the retention force in our experiments is howthe drop is formed. While preparing the drops, the pipette needs to be perpendicularto the surface and above the same point of the surface. Otherwise, the shape and thewidth of the drop may change, which will affect the retention force of the drop. Inaddition, the procedure of Ref. [12] to build inverted drops must be followed. Overall,it is not difficult to build sessile and pendant drops that are consistently similar, butinverted sessile drops require some skill and practice.
To really understand our theoretical explanation of the above experimental obser-vations, it is useful to first understand the derivation of the theoretical retentionforce [17–26]. In this section, we provide such derivation. Our derivation is an im-proved version of the derivation provided by Dussan and Chow [22].Let us consider a drop on a flat surface subject to an external force parallel to thesurface. Three surface tensions act on a given infinitesimal section ds of the triple lineat point P (see Fig. 6): The solid-liquid surface tension γ sl , the solid-vapor surfacetension γ sv , and the liquid-vapor surface tension γ ≡ γ lv . SolidLiquid θ γ Gas or Vapor sl γ sv P γ Figure 6: Surface tensions and contact angle at P. Point P is a generic point along thetriple line, not necessarily the advancing or receding edge.The solid-liquid and solid-vapor surface tensions act parallel to the surface, andthe liquid-vapor surface tension acts at an angle θ with respect to the surface, where θ is the contact angle at that point. Thus, in the direction parallel to the surface7ut perpendicular to the triple line, the surface tensions acting on an element ds oftriple line located at P are γ cos θ , γ sl and γ sv . When the contact angle is the Young,equilibrium angle θ Y , the net force on the element ds is zero, which leads to the Youngequation: γ cos θ Y + γ sl − γ sv = 0 . (3.4)When the contact angle θ at ds is not θ Y , the forces due to the surface tensions do notcancel each other, γ cos θ + γ sl − γ sv = 0 , (3.5)and therefore there is a nonzero capillary force per unit of length acting on the element ds in the direction parallel to the surface. If we denote by ˆn the unit vector perpen-dicular to the triple line and pointing outwardly (see Fig. 7), then such capillary forceper unit of length is given by ( − γ cos θ − γ sl + γ sv ) ˆn . (3.6)However, if the infinitesimal element ds at point P does not move, there must be a forcethat cancels the force in Eq. (3.6), much like when we push a solid resting on a surfacebut the solid doesn’t move, we say that our push is canceled by the static frictionalforce. The force that cancels that in Eq. (3.6) is the retention force per unit of lengthand is presumably due to solid-liquid-vapor interactions (compare with Eq. (6.1) inRef. [1], Eq. (4.1) in Ref. [3], and Eqs. (11) and (12) in Ref. [24]), d ~f k ds = ( γ cos θ + γ sl − γ sv ) ˆn . (3.7)Figure 7 shows the direction of the retention force at the infinitesimal element of tripleline located at point P.The net retention force on the whole drop is obtained by adding the infinitesimalretention forces acting on each infinitesimal element of the triple line C : ~f k = I C ( γ cos θ + γ sl − γ sv ) ˆn ds . (3.8)Because γ , γ sl and γ sv are constant [27], and because H C ˆn ds = 0, Eq. (3.8) simplifiesto ~f k = γ I C cos θ ˆn ds . (3.9)This is the exact equation that provides the retention force on a drop resting on auniform solid. When the contact area is symmetric with respect to the x -axis, the y -component of this force is zero, and therefore one only needs to worry about the x -component, f k = f k x = γ I C cos θ ˆn · ˆx ds . (3.10)To obtain the exact expression of f k , one needs to know8 θ cos df/dsn w i d t h +x advancing half receding half Pds
Figure 7: Triple line and contact area of a drop at the onset of lateral motion. Thecentrifugal force points in the + x -direction. The width w is the length of the dashedline.( i ) the exact shape of the triple line and, in particular, the variation along C of theangle between ˆn and the centrifugal force, i.e., the variation of ˆn · ˆx along thetriple line,( ii ) how the contact angle θ varies along the triple line.Because it is very difficult to obtain ( i )-( ii ) either experimentally or theoretically, itis common to make some approximations to obtain an effective, approximate expres-sion for the retention force that captures the essence of the exact one. One commonapproximation is to assume that in the “advancing half” (“receding half”) of the tripleline, the contact angle remains constant and equal to the contact angle at the advanc-ing (receding) edge of the drop, θ a ( θ r ). Within this approximation, and denoting by C a ( C r ) the contour associated with the advancing (receding) half of the triple line,Eq. (3.10) becomes f k = γ (cid:18)Z C a cos θ ˆn · ˆx ds + Z C r cos θ ˆn · ˆx ds (cid:19) (3.11) ≈ γ (cid:18) cos θ a Z C a ˆn · ˆx ds + cos θ r Z C r ˆn · ˆx ds (cid:19) (3.12)= γw (cos θ r − cos θ a ) (3.13)where in the last step we have used [22, 26] that R C r ˆn · ˆx ds = − R C a ˆn · ˆx ds = w , w being the width of the drop in the direction perpendicular to the motion of the drop,see Fig. 7.It is clear that, in going from Eq. (3.11) to Eq. (3.12), we are making an approx-imation. It is also clear that what we call the “advancing” and “receding” halves9s somewhat arbitrary. To correct for these approximations and still have a usefulformula, it is common to introduce a shape factor k in the retention force, f k = kγw (cos θ r − cos θ a ) . (3.14)The shape factor k carries the information lost in the approximation, that is, k accountsfor ( i )-( ii ). From a practical point of view, the shape factor is simply a fitting parameterthat makes the theoretical retention force f k be equal to the experimental one F . Hence,one can calculate k in terms of experimentally measurable quantities as k = Fγw (cos θ r − cos θ a ) . (3.15)One can improve the above approximation by assuming a specific variation of thecontact angle θ along the triple line. This is what, for example, Extrand and Gent [23]and ElSherbini and Jacobi [25] did, who obtained k = 2 /π and k = 24 /π , respectively.It should be noted however that Refs. [23, 25] assumed that the angle between ˆx andˆn is the same as the polar angle that locates each point on the triple line. Using anargument similar to ours, Carre and Shanahan derived Eq. (3.14) with a shape factorof π/ θ a ) and receding ( θ b ) edges of the drop for the situationat hand, independently of whether such angles are different in other situations. According to Eq. (3.14), the three main factors that affect the retention force are theshape of the drop ( k ), its width ( w ), and contact angle hysteresis (cos θ r − cos θ a ).In this section, we are going to elucidate the role that each of these factors plays indifferentiating the retention forces on sessile, pendant, and inverted drops.To elucidate whether the width of the drop plays an essential role in the differencebetween F sessile , F pendant and F inverted , we calculated the centrifugal force per unit ofwidth, Fw = (30 . ± .
2) mN / m sessile, 15-100 µ L , (29 . ± .
9) mN / m pendant, 15-100 µ L , (31 . ± .
9) mN / m inverted, 15-30 µ L . (3.16)Because F/w is essentially the same for sessile, pendant and inverted drops [29], weconclude that the retention force is proportional to the width of the drop. Thus, since10or each volume w pendant < w sessile < w inverted (see Table 2), it follows that F pendant Sessile Pendant Inverted θ r − cos θ a is very similar for sessile, pendant, and inverted drops,cos θ r − cos θ a = . ± . 017 sessile,0 . ± . 013 pendant,0 . ± . 007 inverted. (3.17)This result, coupled to the fact that θ r and θ a remain fairly constant as the volumeof the drops decreases, leads us to conclude that it is unlikely that contact anglehysteresis is essential in differentiating the retention forces on sessile, pendant andinverted drops [30].As mentioned above, the shape factor k is, for practical purposes, a free parameterthat allows one to fit the theoretical retention force f k to the experimental one F .Thus, it is difficult to determine the influence of the shape of the drop on the retentionforce. One can nevertheless use Eq. (3.15) to obtain a value of the shape factor interms of experimentally measurable quantities. In our experiment, we obtained that k = . ± . 08 sessile,0 . ± . 04 pendant,1 . ± . 05 inverted. (3.18)Because these shape factors are fairly similar, it is unlikely that the shape of the tripleline and the variation of the contact angle along such line are critical in determiningwhich type of drop gets more pinned to the surface.Thus, among the three factors that make up the retention force in Eq. (3.14), thewidth seems to be the one that most critically determines which kind of drop gets morepinned to the surface. Because the width of a sessile drop is larger than the width ofa pendant drop (see Table 2), and because the retention force is proportional to the11idth, we have that F sessile is larger than F pendant . However, when we flip a sessile dropand make it an inverted sessile drop, at the onset of the motion the width is slightlylarger than what it would had been for the original sessile drop (as long as the flip isdone as explained in Ref. [12]), and that is what makes F inverted larger than F sessile .One can prepare drops of a given type and volume, e.g., 80- µ L sessile drops, indifferent ways so that the widths of the drops are different. Because it is proportionalto the width, the retention force should be larger for those drops with a larger width,even though they all are sessile drops of the same volume. Indeed, an experimentwas done in Ref. [12] with 80- µ L sessile drops that were formed with different widthsrelative to the radial direction of the motion. It was found that the retention forceincreased with the width, even though the volume and the type of drop were the same.That the experiment in Ref. [12] with 80- µ L drops can be easily explained by sayingthat the retention force is proportional to the width further suggests that the width isthe most critical parameter in differentiating F sessile , F pendant and F inverted , and that theretention force depends not only on whether the drop is placed on top or bottom of asolid surface, but also on the history of droplet deposition.The lack of volume dependence in Fig. 5 can also be understood from the propor-tionality between the retention force and the width. Due to the way we build inverteddrops, the ratio w inverted w sessile is close to unity for any volume, and therefore F inverted F sessile shouldalways be close to unity, independently of the volume of the drop.It is important to mention that the influence of the history of drop deposition onthe retention force has been studied recently by R´ıos-L´opez et al. [8, 9], who foundthat one can still use Eq. (3.14) even when the drop history contains an initial stageof tilting. In their case, the relevant geometrical feature that enters Eq. (3.14) is theinitial length of the drop, whereas in our case it is the width at the onset of the motion. Using a rotating-platform experiment, we have observed that the lateral retentionforce on sessile drops is larger than on pendant drops, but smaller than on invertedsessile drops. We have identified the width of the drop as the critical parameter thatdetermines this result, even though the contact angles and the shape of the triple linealso affect the retention force. Because the retention force is proportional to the width,and because we built the drops such that w pendant < w sessile < w inverted , we have that F pendant < F sessile < F inverted . In general, the retention force on a drop does not dependonly on whether it is placed on top or on bottom of a surface, but also on how the dropwas formed. Our results can be easily reproduced, by using either a simple tilted-plateexperiment or other drop accelerators [4, 7].We would like to note that the experiment in Ref. [4] was done with a differentliquid-solid combination, and for smaller volumes. We believe that our results wouldalso be applicable to many other liquid-solid combinations for large volumes. However,12or very small volumes, it might be possible that F sessile is smaller than F pendant , aslong as cos θ r − cos θ a remains constant as the volume of the drops decreases [30]. Ifsuch was the case, then contact angle hysteresis should suffice to explain why F sessile becomes smaller than F pendant . Our experiment, however, does not address the effectof the resting time on the retention force [4]. Acknowledgments The authors thank Rafael Tadmor for many enlightening discussions. Additionalthanks are due to Sage Automation, Thomas Michel, Thomas Podgorski, Taylor White-head, Jason Dark, Jon Klipfel, Aaron Burlew, David Jackson, Miles Stone, JaredRichards, and Al Sauerman. Financial support from a Lamar Presidential Fellowshipis gratefully acknowledged. References [1] de Gennes, P. G.; Brochard-Wyart, F.; Qu´er´e, D. Capillarity and Wetting Phe-nomena , Springer, New York, 2004.[2] Erbil, H.Y. Surface Chemistry of Solid and Liquid Interfaces , Wiley-Blackwell,2006.[3] Bormashenko, E. A. Wetting of Real Surfaces , Walter de Gruyter, 2013.[4] Tadmor, R.; Bahadur, P.; Leh, A.; N’guessan, H. E.; Jaini, R.; Dang, L. Mea-surement of Lateral Adhesion Forces at the Interface Between a Liquid and aSubstrate. Phys. Rev. Lett. , 266101 (2009).[5] Minkel, J. R. Focus: Hanging Droplets Feel More Friction. Phys. Rev. Focus ,21 (2009).[6] Fitzgerald, R. J. A New Look at Friction. Phys. Today , 16 (2010).[7] Evgenidis, P.; Kali´c, K.; Kostoglou, M.; Karapantsios, T. D. Kerberos: A ThreeCamera Headed Centrifugal/Tilting Device for Studying Wetting/Dewettingunder the Influence of Controlled Forces. Colloids and Surfaces A: Physic-ochem. Eng. Aspects , 38-48 (2017).[8] R´ıos-L´opez, I.; Evgenidis, S.; Kostoglou, M.; Zabulis, X.; Karapantsios, T. D. Ef-fect of Initial Droplet Shape on the Tangential Force Required for Spreading andSliding Along a Solid Surface. Colloids and Surfaces A: Physicochem. Eng. As-pects , 164-173 (2018). 139] R´ıos-L´opez, I.; Karamaoynas, P.; Zabulis, X.; Kostoglou, M.; Karapantsios, T.D. Image Analysis of Axisymmetric Droplets in Wetting Experiments: A NewTool for the Study of 3D Droplet Geometry and Droplet Shape Reconstruction. Colloids and Surfaces A: Physicochem. Eng. Aspects Phys. Rev. Lett. , 036102 (2001).[12] de la Madrid, R.; Whitehead, T.; Irwin, G. Comparison of the Lateral RetentionForces on Sessile and Pendant Water Drops on a Solid Surface. Am. J. Phys. ,531-538 (2015).[13] Even though syringes are slightly more precise than micropipettes, we did not usethem. The reason is that when we built pendant drops with a syringe, there wasmore liquid left in the syringe as compared to when we built sessile and inverteddrops. With a micropipette, you can visually check that all the liquid has beendeposited on the surface.[14] As the angular velocity increases, small vibrations of the apparatus produce asmall trembling motion on surface of the drops. The triple line however does notseem to be significantly affected by such vibrations.[15] https://imagej.nih.gov/ij/[16] http://pixelzoomer.com[17] Macdougall, G.; Ockrent, C. Surface energy relations in liquid/solid systems I.The adhesion of liquids to solids and a new method of determining the surfacetension of liquids. Proc. Royal Soc. Lond. A , 151-173 (1942).[18] Frenkel, Y. I. On the behavior of liquid drops on a solid surface 1. The sliding ofdrops on an inclined surface. J. Exp. Theo. Phys. (USSR), , 659-668 (1948).[19] Kawasaki, K. Study of wettability of polymers by sliding of water drop. J. ColloidSci. , 402-407 (1960).[20] Furmidge, C. G. L. Studies at Phase Interfaces I. The Sliding of Liquid Dropson Solid Surfaces and a Theory of Spray Retention. J. Coll. Int. Sci. , 309-324(1962).[21] Brown, R. A.; Orr, Jr., F. M.; Scriven, L. E. Static Drop on an Inclined Plate.Analysis by the Finite Element Method. J. Coll. Int. Sci. , 76-87 (1980).[22] Dussan, E. B.; Chow, R. T.-P. On the Ability of Drops and Bubbles to Stick toNon-Horizontal Surfaces of Solids. J. Fl. Mech. , 1-29 (1983).1423] Extrand, C. W.; Gent, A. N. Retention of Liquid Drops by Solid Surfaces. J. Coll. Int. Sci. , 431-442 (1990).[24] Carre, A.; Shanahan, M. E. R. Drop Motion on an Inclined Plane and Evaluationof Hydrophobic Treatments to Glass. J. Adhesion , 177-185 (1995).[25] ElSherbini, A. I.; Jacobi, A. M. Liquid Drops on Vertical and Inclined Surfaces; I.An Experimental Study of Drop Geometry. J. Coll. Int. Sci. , 556-565 (2004).[26] De Coninck, J.; Fernandez Toledano, J. C.; Dunlop, F.; Huillet, T. Pinning of aDrop by a Junction on an Incline. Phys. Rev. E , 042804 (2017).[27] If γ sl and γ sv were not constant along the triple line (for example, if the drop strad-dled the boundary between two different solids), then H C ( γ sl − γ sv )ˆn ds would notbe zero, and the retention force would be provided by Eq. (3.8), not by Eq. (3.9).[28] Krasovitski, B.; Marmur, A. Drops Down the Hill: Theoretical Study of LimitingContact Angles and the Hysteresis Range on a Tilted Plane. Langmuir , 3881-3885 (2005).[29] Although it remains fairly constant, Fw has a slight tendency to increase as thevolume decreases.[30] Due to gravity and the deformability of liquid water, the contact angles of sessiledrops are about 6 ◦ smaller than those of pendant and inverted drops. Thus, cos θ r − cos θ a is always a bit smaller for sessile drops. Such difference in cos θ r − cos θ a ishowever not enough to upset Eq. (3.3), at least in the volume range we havestudied. 15 Sessile Pendant Inverted Figure 8: Table of Contents Graphic: µµ