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Onset of sliding motion in sessile drops with initially non-circular contact lines
Colloids and Surfaces A: Physicochem. Eng. Aspects 498 (2016) 146–155
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Onset of sliding motion in sessile drops with initially non-circular contact lines Nachiketa Janardan, Mahesh V. Panchagnula ∗ Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, India
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
• Impending motion in nearly elliptical drops is dependent on drop orientation. • Experiments and computations are used to determine moving and sliding angles. • Drop profile width is an important characteristic dimension.
a r t i c l e
i n f o
Article history: Received 29 December 2015 Received in revised form 14 March 2016 Accepted 16 March 2016 Available online 19 March 2016 Keywords: Contact angle hysteresis Inclined plane Non-circular contact line Profile width
a b s t r a c t The onset of motion of a drop with an initially non-circular three phase contact line was studied experimentally and numerically. Two drops of volume 10 l were made to coalesce and form a composite 20 l drop. The contact line of this drop was approximately elliptical and the local contact angle along the contact line was not a constant (as would have been the case with a circular contact line). The orientation of the drop to the impending direction of motion was varied. Inclined plate experiments were performed and the moving and sliding angles were noted in each case. It was observed that the moving and sliding angles of the drop were strongly dependent on this orientation. Specifically, the local conditions on the contact line at the front and back edges of the drop as well as the drop profile width were found to be the determining parameters. Surface evolver simulations were performed to understand the results of the experiments. It was found that the evolution of the contact line for non-circular drops was rather counter-intuitive when compared to the results from a drop with a circular contact line and resulted from a competition between gravity and the local contact angle hysteresis forces. © 2016 Elsevier B.V. All rights reserved.
1. Introduction
∗ Corresponding author. E-mail address: [email protected] (M.V. Panchagnula). http://dx.doi.org/10.1016/j.colsurfa.2016.03.046 0927-7757/© 2016 Elsevier B.V. All rights reserved.
The physics of sessile drops and the onset of their motion down an inclined plane have been studied extensively over the past
N. Janardan, M.V. Panchagnula / Colloids and Surfaces A: Physicochem. Eng. Aspects 498 (2016) 146–155
Fig. 1. Schematic of the drop with a non-circular contact line.
couple of decades [1–19]. The focus in this area of research, however, has been on sessile drops with an initially circular three phase contact line [5,7,20,21]. A sessile drop with an initially circular contact line has no concept of orientation associated with it. In this initial state, the profile of the drop is the same when viewed from any direction. This also implies that the distribution of the local contact angle () along the contact line is uniform. Therefore, one section of the three phase contact line is indistinguishable from the other. The onset of motion of such a drop is a problem that has been studied before by us and others [2,3,5,10,22–24] and is relatively well understood. However, when the contact line is noncircular, the contact angle distribution is no longer uniform; the contact angle varies from one section of the three phase contact line to another [25–27]. The onset of motion of a section of the contact line is dependent on this local contact angle (l ). Therefore, it is anticipated that the onset of motion of a drop with an initially non-circular contact line will be different from the onset of motion of a drop with an initially circular contact line even if the volumes of the two drops were to be the same. The orientation () of such a non-circular drop to the direction of motion will begin to play a key role in such onset of motion. In this manuscript, we present the results of a study on the onset of motion of drops with three phase contact lines that are nearly elliptical. The terminology and nomenclature that was introduced in our previous work [5] will be used here too. The interior angle that is subtended by the substrate and the free surface of the drop at the contact line is called the local contact angle l . If the surface is ideal, then this value is unique and is obtained from Young’s two limiting equation. A real surface, however, manifests contact angles that are called the advancing a and receding r contact angles. These contact angles are the maximum and minimum values that the contact angle can take and all the values in between r < < a are admissible as possible local contact angle values. This phenomenon is referred to as Contact Angle Hysteresis (CAH). Throughout this study, we will work with real surfaces. Fig. 1 shows a schematic of the contact line of the drop for purposes of representation. The direction of motion of the drop is along the axisX. The direction along which the camera observes the drop is the axis Y . The contact line is initially non-circular in shape. The contact line (which is nearly elliptical) can be described by a major axis of length, 2a and a minor axis of length, 2b. The major axis of the drop is along the axis X . The orientation of the drop is defined as the orientation of the major axis to the direction of motion, i.e. the angle between X and X , which will henceforth be referred to as . The profile length Lp is the width of the drop that is visible
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from the direction of motion. is the azimuthal angle of a point on the three phase contact line. It is defined with respect to the axis Y , and is the angle between the axis Y and the line joining the aforementioned point to the origin. Extrand and Kumagai [1] were the first to study the behavior and the contact angle distribution of a sessile drop with a non-circular contact line. They assumed that the contact angle distribution was linear in cos l with cos ˛ and cos r being the bounds of the distribution. They also assumed that a normal to the local segment of the contact line was along the radial direction, but this has been shown to be incorrect later [26]. Recently, Antonini et al. [25] performed studies on measuring the adhesion force that resists the motion of the sessile drop. They developed a methodology to measure the contact angle distribution around the three phase contact line. Chini and Amirfazli [26] have also developed a methodology to determine the adhesion force for drops with an arbitrarily shaped contact line. This adhesion force is the same as the hysteresis force that has been referred to in Janardan and Panchagnula [5]. An important point to be noted was the perspective error that arises while measuring the contact angle of drops with an elliptical contact line. When the major axis was not perpendicular to the camera (the direction of viewing), it was seen that the point of measurement on the contact line was incorrectly determined. The radial position of the point of measurement (as determined from the images of the drop) from the origin is incorrect and leads to errors in the reconstruction of the contact line. However, there was no perspective error in the measurement of the contact angle itself. In our experiments and simulations, we have not measured the radial positions of the points of measurement. We have merely measured the contact angle at the extremes of the drop profile, and compared the same with results from Surface Evolver (SE). Therefore, the question of perspective error in contact angle measurement does not arise in our case. The inclination angle (˛) is the angle through which the substrate is tilted. The front edge and the back edge of the drop are those parts of the drop, which are downstream and upstream respectively, with respect to the direction of motion of the drop. As detailed in our earlier work [5], the motion of the drop is gov→
erned by the equilibrium between the gravitational force F g and → force F h .
the hysteresis This hysteresis force is the magnitude of the total impeding force that acts on the contact line as a result of CAH. The gravitational force is the driving force for the motion and the hysteresis force resists said motion. →
→
m g sin (˛) = Fg
(1)
Here, m is the mass of the drop and ˛ is the angle of inclination of the substrate of the drop. In a case involving drops with a circular contact line, the contact line of the drop begins to deform locally at either the back edge or front edge at a critical angle of inclination of the inclined plane called the moving angle (˛m ). When a second critical angle of inclination (at which the other edge starts to deform) is reached the drop begins to display the onset of motion. This angle is called the sliding angle (˛s ). The resisting force due to CAH is given by → Fh
=
ˆ LV | cosY − cos |ndl
(2)
where l is the local contact angle of the drop on the contact line, is the surface tension of the liquid-air interface and dl is an infinitesimal element of the contact line with the dimension of length. The closed integral is performed along the contact line in order to determine the sum of the forces that are acting on all such infinitesimal elements of the contact line.
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A model for CAH [5] is given by
Fh =
mg sin ˛
kL cosr − cos˛
for ˛ < ˛c for ˛ = ˛c
(3)
Here k is a non-dimensional parameter that is determined experimentally and L is a length parameter, which was taken to be the length of the contact line. As the CAH and k are both nondimensional, the length parameter is required to ensure that the right hand side of Eq. (3) has units of force. The global motion of the drop is initiated when the gravitational force exceeds the hysteresis force at an inclination angle ˛ = ˛c . Berejnov and Thorne [10] have studied the behavior of sessile drops on inclined planes. They suggested the existence of three critical angles of inclination. The first critical angle corresponds to the loss of local equilibrium of the contact line at the front edge and the second corresponds to the loss of local equilibrium of the contact line at the back edge. The third critical angle corresponds to the case when global motion of the whole drop is initiated. It can be construed that between the second and third critical angles, loss of local equilibria at the front and back edges of the drop is responsible for the slow sliding motion of the drop. Further increase in the tilt angle will result in an imbalance between the forces acting on the →
drop, mg sin ˛ and F h , resulting in rapid accelerated motion of the drop down the inclined plane. This will occur at the third critical angle ␣ = ␣c . In our case, we have terminated our experiments at the second critical angle since we are interested only in the onset of motion. Therefore, we will refer to the Berejnov and Thorne [10] second critical angle as the sliding angle. This is the inclination angle at which both the front edge and back edge have lost local equilibrium and are both moving. However, we have continued the tilting process in a limited set of experiments to discover the Berejnov and Thorne [10] third critical angle, where global motion is observed. In our previous publication, [5] we noted that the value of the non-dimensional parameter (k) was dependent on L, which in that case was chosen to be the initial three phase contact line length. The k value in that case was 0.29 and is consistent with the literature [1,10,20,24,28,29]. However, all these results have been obtained from simulations and experiments on drops with an initially circular three phase contact line. It would be interesting to investigate similar drops with a non-circular three phase contact line, as they are ubiquitous. This forms the core motivation of the current work. 2. Experimental apparatus A Holmarc© contact angle goniometer used to perform the experiments is shown in Fig. 2. The goniometer consisted of a platform for a substrate, a backlight, a CCD camera and a syringe pump to inject the working liquid on the substrate. The platform could be tilted, allowing us to vary the inclination angle. The substrate was able to translate in a horizontal plane by means of a screw. A CCD camera was used to capture the motion of the drop and the video thus captured was analyzed for contact angles and drop position. Holmarc© CA software was used to measure the contact angles as well as to estimate the critical angles of inclination. All our experiments were performed in a controlled atmosphere at a constant temperature. The entire experiment lasted less than 7–8 min. This ensured that the system was not affected by evaporation. 3. Materials and methods The substrate used was a thin strip of Polydimethylsiloxane (PDMS). The liquid used in the experiments was distilled water. A syringe pump was used to accurately meter out a 10 l drop on to the substrate. The substrate was then translated by a set number
Fig. 2. Experimental set up.
of rotations of the screw. This enabled the substrate to move forward by a set distance. Another 10 l drop was then deposited on the surface. This drop was placed as close as possible to the original drop such that they were on the verge of coalescing. A liquid bridge was then constructed between the two drops with the help of a stainless steel needle and they were allowed to coalesce to create a 20 l drop with a non-circular contact line. Photographs were taken of the major and minor axis of the drop to ensure that the dimensions of the drops used in multiple experiments were the same or within a reasonable margin of error (±2%). The camera views the drop from the side (with the camera axis perpendicular to the direction of motion) and is in a unique position to capture the dimensions of the drop. The drop was first aligned with its major axis parallel to the direction of motion and photographed to obtain the major axis length. It was then aligned with its minor axis along the direction of motion and was photographed to obtain the minor axis length. The average ratio of the major axis to the minor axis in all experiments was 1.47. Efforts were made to keep it to a value as close to 1.5 as possible. This was done to ensure experimental repeatability. The rate of the tilting was less than 3◦ per minute, so as to achieve quasistatic conditions. r and a for the surface liquid air combination were measured prior to the tilting plate experiment using the goniometer. a was measured to be 99◦ , while r was found to be 67◦ . An infusionwithdrawal experiment [30] was performed to measure r and a where fluid is injected and withdrawn from a surface by means of a needle. The receding angle was measured on a drop with a large enough volume so that the needle did not affect the receding interface. The infusion/withdrawal experiments were repeated multiple times to ensure reproducibility of the measured values. The advancing and receding angles were repeatable to within 3◦ . The drop was rotated around an axis perpendicular to the substrate, on the substrate to allow experiments to be performed in situations when the drop was oriented differently to the direction of the moving force, i.e., the component of gravity along the inclined plane of the substrate. The point of reference for various cases was the orientation of the major axis () of the drop with respect to this moving force. Thus, we performed experiments for varying angles of orientation, from the case where the major axis of the drop was oriented along the force of gravity (0◦ ), to the case where the major axis was perpendicular to the force of gravity (90◦ ). was varied in intervals of 15◦ . The substrate was tilted until the drop was observed to move. This process was filmed using the CCD camera and the subsequent video analyzed to determine the angle of inclination and the contact angles of the front edge and the back edge of the drop. The front and back edges are small sections of the contact line that were seen
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to start moving at the point of incipient motion. The contact angle at the front edge is designated as the front angle f and the con-
tact angle at the back edge is designated as the back angle b . These two angles were also measured to understand the state of the drop prior to incipient motion. 4. Surface evolver model Surface Evolver (SE) [31], an open source software, was used to simulate the behavior of the non-circular drops on a substrate as it is being inclined. SE works on the principle of energy minimization and volume conservation subject to constraints. A detailed description of the model can be found in Brakke [31,32], Janardan and Panchagnula [5] and Prabhala et al. [33]. The model developed by Prabhala et al. [33] simulated the process of two sessile drops coalescing. This simulation was used to generate the initial condition for the tilting plate simulation. It begins with a condition where two drops were just allowed to touch. The interfacial tension of the common surface shared by the two drops was then set to zero, effectively removing the interface between the two drops and allowing them to coalesce. The contact angle distribution was not specified and the SE iterations were continued until the drop shape and the contact line, both reached minimum energy configurations. This provided the initial condition at ˛ = 0◦ . Janardan and Panchagnula [5] developed a model which enabled them to simulate the quasistatic behavior of a sessile liquid drop on an inclined plane. The inclination of the substrate was increased until the drop showed signs of impending motion. The drop formed from the coalescence process (as described above) was subjected to this inclination process. The orientation angle was set to a desired value prior to starting the simulation. The simulation was terminated when both the front and back edges of the drop were initiated into motion. The results are recorded in terms of the drop shape, contact line shape as well as the critical angles of inclination. We have performed successive mesh refinements to ensure that these results are grid independent. This was achieved at about 10000 elements [5]. The capillary number Ca = /, where v is the velocity associated with a moving section of the contact line. Typically, v ∼1 mm/s from our drop sliding experiments. Therefore, Ca < 10−6 indicating that viscous effects in the rearrangement of the contact line can be neglected. Similarly, the Bond number, Bo = ((gsin (␣)V2/3 )/) was always less than 1 from our experiments. From these estimates, we can conclude that the onset of motion in these drops can be treated as a quasistatic process where gravitational and interfacial tension effects dominate. As a measure of validation between the experimental data and data from SE for the case of non-circular drops, we present Figs. 3 and 4 which show the evolution of the normalized front and back contact angles as a function of normalized inclination, respectively. In Fig. 3, the quantitative match between our experiments and simulations is good up to an ␣/␣s ratio of 0.4. At this point (termed as the moving angle) it was observed that the back edge started moving in both SE and experimental data. Beyond this moving angle, the data from SE is seen to be constant while the experimental data displays variation that is a characteristic of stick slip motion [34–36]. SE is unable to model this stick slip motion because of strict definitions on the conditions required for motion. However, it is able to capture the average motion during this phase. We refer the reader to Janardan and Panchagnula [5] for further validation exercises focused on initially circular drops. In Fig. 4, the trends match quantitatively over the entire range of the inclination. From these two figures, we conclude that the SE model is capable of capturing the essential physics of incipient drop motion.
Fig. 3. Comparison of back angle b trends for experiments and simulations. Here ◦
=0 .
5. Results and discussion The primary qualitative difference between initially circular and non-circular drops is that the distribution of contact angles on the contact line is not uniform in the case of a non-circular drop as will be seen later. In other words, different points on the contact line may have different l values. This effect is manifested in the form of ˛m and ˛s being dependent on (orientation of the drop with respect to the direction of incipient motion), for the same mass of the drop. ˛m and ˛s were identified based on the incipience of motion on either the front edge or the back edge. This motion was determined by the value of the contact angle around that small part of the contact line. If the local contact angle on either the front or back edge exceeded a or was below r , the edge was observed to move. Thus the motion of the contact line, and that of the drop as a whole, was affected by . 6. Experimental results Fourteen experiments were performed with seven different experimental cases. The seven experimental cases were only distinguished by their values. The major and minor axis lengths of the initially non-circular drops were also measured. The experimental cases were seen to have similar major and minor lengths. This was done to ensure experimental repeatability. The profile length (Lp ) in this case was the linear dimension that was visible to a camera that was placed with its axis parallel to the direction of motion. In other words, it is the width of the drop. The profile length thus varied from a minimum to a maximum as was increased. In the case of = 0◦ , the profile length was equal to the length of the minor axis, while in the case of = 90◦ , it was the major axis. This profile length is analogous to the diameter in circular drops. We will next report detailed contact angle data from three of the seven experiments in Figs. 5–7. Fig. 5 is a plot of the local contact angles at the front f and
back b edges versus the inclination angle (˛). f and b , the front and back angles were measured from analyzing the experimental images. The contact angle data was obtained by fitting a tangent to the image at the contact point. The data was smoothed in time, using five values of the contact angles from five successive images and employing a smoothing algorithm. These images were all within a time frame of 1.5 s (Please note that the tilting process was performed at approximately 3◦ per minute). The
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Fig. 4. Comparison of front angle f
Fig. 5. Front angle f
trends for experiments and simulations. Here = 0◦ .
and back angle b
initial condition for this experiment was obtained by coalescing two drops placed on the substrate, where the line joining the centers of the two drops is parallel to the direction of motion ( = 0◦ ), along the inclined plane. For the case of ( = 0◦ ) (see Fig. 5), the back edge was observed to move first and accordingly b was seen
to decrease until ˛m was reached. Beyond˛m , b increased and decreased alternately. This is related to stick slip behavior that is a characteristic of contact line motion when the size of the defect less than the size of the drop. During this experiment, is much f continued to increase monotonically until ˛s was reached. At
˛ = ˛s , f approached the advancing angle value. At ˛s , a small portion of the contact line at the front of the drop was seen to start
vs angle of inclination (˛) for the case = 0◦ .
moving in addition to the back edge which was already in motion. Thus the drop was initiated into motion down the inclined plane. Fig. 6 is similar to Fig. 5 except that it was for the case of = 45◦ . It can be seen in this figure that the contact angle at the front edge of the drop f is observed to increase while b was observed to decrease. The moving angle in Fig. 6 is also greater than that in Fig. 5. In contrast to the case of = 0◦ , the front edge was seen to move first at ˛ = ˛m , following which f is observed to increase and decrease alternately. This appears to be the contact angle signature of a moving edge (be it front or back). During this time, b decreased monotonically until ˛ = ˛s . At ˛s , the drop reached the point of impending motion down the inclined plane.
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Fig. 6. Front angle f
Fig. 7. Front angle f
and back angle b
and back angle b
Fig. 7 is similar to Figs. 5 and 6, except that it was for = 90◦ . In contrast to both Figs. 5 and 6, ˛m is much lower while ˛s is higher. At this condition, the initial f was observed to be close to a and thus the front edge was seen to start moving nearly immediately upon starting the inclination process. During the entire tilting process, f was seen to vary around an average value, while b decreased monotonically. As mentioned before, this appears to be the signature of a moving edge. The key observation from the experimental results presented here is that three identical drops with similar dimensions and sessile on the same surface and oriented in different directions, will exhibit different behavior due to the difference in the distribution
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vs angle of inclination (˛) for the case = 45◦ .
vs angle of inclination (˛) for the case = 90◦ .
of the local contact angles along the contact line. The front and back angles in each case are different and thus the onset of motion behavior of the drop is different. The critical angles too are functions of this variance in local contact angle (see Fig. 8 below). These results will be further elaborated upon when the data from SE is discussed. Fig. 8 is a plot of the moving (˛m ) and sliding (˛s ) critical angles as a function of for both experimental data and SE data. It can be seen from this figure that there as increases, ˛s increases. ˛m however, increased up to a value of = 45◦ following which it decreased. It must be pointed out here that up to = 30◦ , it was the back edge that was observed to move first. For > 30◦ , it was the front edge that was seen to move first.
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Fig. 8. Moving angle (˛m ) and Sliding angle (˛s ) as functions of ).
Fig. 9. Plot of l distribution as a function of the azimuthal angle along the contact line. In this case, = 0◦ .
7. Results from surface evolver simulations Figs. 9–12 show results obtained from SE simulations. As mentioned before, the model requires the advancing and receding angles as constitutive inputs. The advancing and receding angles were taken to be 99◦ and 67◦ respectively, as obtained from the goniometer measurements on the same substrate as was used in the tilting plate experiments. Fig. 8 also shows values of ˛s and ˛m as a function of for data taken from SE. It can be seen from this figure that ˛s increases with increasing . ˛m also increases with up to = 82.5◦ and then decreases sharply. It was seen that when ˛m was increasing along with an increase in , the back edge determined the moving angle. ˛m exhibits a maximum at = 82.5◦ . However, when ˛m decreased, the front edge was seen to move first. These
results are qualitatively similar to the corresponding experimental cases. However, quantitatively, at = 75◦ and beyond, the drop does not slide off the substrate even when ˛ = 90◦ . This prediction does not match our experiments. A part of the reason for this mismatch could be that we do not allow for stick slip motion in our SE model. Stick-slip is a dynamical phenomenon and is therefore beyond the scope for inclusion into a quasi-static model. Another observation from these figures is that SE consistently overestimated the values of ˛m and ˛s in relation to the experiments. The maxima in the ˛m trend also shifted rightwards. The qualitative match however, was good and it is seen that the behavior of the drop in SE and the experiment is similar. The information that can be obtained from SE is significantly more detailed than those from experiments. For example, the actual shape of the
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Fig. 10. Evolution of the contact line for = 0◦ . Data is extracted from SE.
Fig. 11. Evolution of the contact line for = 90◦ . Data is extracted from SE.
Fig. 12. Evolution of the contact line for = 45◦ . Data is extracted from SE.
contact line can be obtained to yield a more complete picture of the sliding drop. In addition, the local contact angle distribution is obtained from the SE simulations. See Fig. 9. In this case, the ends of the major axis correspond to the case where = 90◦ and = 270◦ . The ends of the minor axis correspond to the case where = 0◦ and = 180◦ . We can also study the evolution of the contact lines as seen in Figs. 10–12. It must be noted that the SE model imposes a limit on the maximum and minimum angle that the liquid air interface is allowed to subtend on the substrate. These are a and r , respectively. Krasovitski and Marmur [37] studied the metastable shapes of a drop on an inclined plane in two dimensions theoretically, and showed that the maximum and minimum angle at the point of impending motion need not simultaneously equal a and r for a substrate where the defect length scale is comparable to the drop size. However, this statement does not hold valid for a substrate where the advancing and receding angles are simultaneously defined at each point, by assuming that the intrinsic defects that cause the hysteresis occurs at a much smaller scale, such as in the current work. Thus, in this case the advancing and receding angles must be the maximum and minimum angles on the substrate. It would be an instructive exercise to track the evolution of the three phase contact line as the substrate is inclined during the simulation. A point to be noted is that the value of ˛m and ˛s from the SE simulation was determined exactly as they were determined from the experiment. ˛m was the inclination angle at which one edge of the drop (back edge or front edge, depending on the initial condition) was seen to move. ˛s is the inclination angle at which both the front and back edge were seen to be in motion simultaneously.
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In the subsequent discussion, we will present data from three representative cases where = 0◦ , = 90◦ , and = 45◦ , in Figs. 10–12 respectively. Fig. 10 is a plot of the instantaneous three phase contact line at four different values of ˛ for = 0◦ . Firstly, it can be seen that the three phase contact line for ˛ = 0◦ and ˛ = ˛m (in this case, ∼20◦ ) are coincident. This is a corroboration of the fact that the moving angle was accurately identified. As ˛ increases, the back edge is seen to recede until ˛ = ˛s when the onset of motion of the drop is initiated. An important constituent motion associated with the onset of motion is evolution of the three phase contact line at points not near the front or back edge. For example, in this figure, one would notice that as the back edge recedes, the sides of the drop begin to advance outwards. This arises purely due to the local contact angle in that region of the three phase contact line exceeding the advancing angle. We will discuss this in detail later in the context of Fig. 11. Fig. 11 is similar to Fig. 10 except that = 90◦ . As ˛ increases, the front edge of the drop moves first marking the first critical angle,˛m . However a further increase in ˛ produces results that is relatively counter-intuitive. During this phase, the back edge of the drop remains pinned to the substrate while the back flanks of the drop have been initiated into receding motion. This motion is symmetric about the axis determining the direction of motion. One could construe this result as a generalization of the case investigated by Berejnov and Thorne [10] to determine the first critical angle (herein referred to as moving angle). In other words, it is not necessary that only the front edge of the drop determines the moving angle. Fig. 12 is a plot of the three phase contact line shape for four different values of ˛ for = 45◦ . It can be seen that the drop exhibits asymmetric dewetting in relation to the major axis. A small portion of the contact line along the sides of the drop, along the direction of motion, has advanced while the complementary portion of the contact line, opposite to it, has receded before the motion of the drop is initiated. The back edge of the drop also exhibits asymmetric dewetting in the sense that a larger portion of the dewetted area is to one side of the major axis. The interplay between the orientation angle, which determines the action of gravity and contact angle hysteresis, which determines the local action of the substrate on the drop, gives rise to these complex and counter-intuitive three phase contact line shapes. It can be seen from Figs. 10–12 that the same drop exhibits varying behavior based on its orientation. The question that then arises is whether a single equation can describe the onset of motion in such a drop under all orientations. In our earlier work [5], we had noted that the value of k associated with the hysteresis force in Eq. (3) was dependent on a length factor L. The value of k should be calculated using Eq. (3) and the inclination angle, ˛c where global motion of the drop is initiated. However, as was pointed out by Berejnov and Thorne [10] and was observed in our own experiments, the difference between ˛c and ˛s is small (<5◦ ). For this reason, we will use ˛s to calculate the value of k. For the case of initially circular drops, the choices of length factors, viz. initial contact line length, diameter or profile length are all linearly related. Many different values have been proposed in literature [1–3,5,7,10,24,28,38,39] based on various length factors, entirely based on experiments and computations on initially circular drops. In the case of a drop with an initially circular contact line, the length factor we chose was the initial contact line length of the drop. The k value in this case was 0.285. This k value is consistent with other works in the literature [1,20,24]. It would be interesting to see if this result can be extended to the case of initially non-circular drops. We have already presented experimental results as well as results from simulations in SE where the length of the contact line
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Table 1 Contact length = 12.57 mm for all these cases. S. No.
Profile length (Lp ) (mm)
(◦ )
SC–1
1.22 1.16 1.10 1.05 1.02 1.01 1 0.99 0.98 0.95 0.91 0.86 0.82
4.38 4.29 4.20 4.10 4.04 4.02 4 3.98 3.96 3.9 3.8 3.7 3.6
90◦ 90◦ 90◦ 90◦ 90◦ 90◦ Undefined because the CL is a circle 0◦ 0◦ 0◦ 0◦ 0◦ 0◦
Table 2 Contact length = 12.57 mm for all these cases. S. No.
(◦ )
SC–2 SC–3 SC–4 SC–5
0.82 0.86 0.91 0.95
0◦ < < 90◦ 0◦ < < 90◦ 0◦ < < 90◦ 0◦ < < 90◦
was maintained constant while was varied. While those results helped point us in the direction that plays a role in determining k, it was difficult to isolate the choice of length scale. Specifically, we would like to investigate the individual contributions of the profile length Lp as well as a shape parameter such as the ratio of the two axes of the non-circular drop. For this purpose, we have defined as the ratio of the axis oriented with X to the axis oriented with Y (see Fig. 1 for a description of the coordinate system). The first set of simulations allowed to vary while the initial length of the contact line was kept constant. Table 1 lists the simulation parameters. Initially, = 1 (i.e., the drop had a circular contact line). The second set of simulations allowed to vary for constant value of (see Table 2 for these values). The objective in performing these additional sets of simulations was to investigate whether k is a function of explicitly or if it is only a function of the profile length Lp . The profile length, as we have already defined, is the length dimension associated with the drop when it is viewed from the direction of motion at the initial condition. It is to be noted that the simulations in these cases were performed on strictly elliptical drops (initially), as opposed to the non-circular drops formed from a twodrop coalescence event. The initial contact line of the drop was made elliptical by imposing a constraint in SE only to determine the initial state. Subsequently, this constraint was relaxed to allow the drop to evolve as ˛ increases. r and a as well as other simulation parameters were maintained the same as in the previous cases. Fig. 13 is a plot of k versus Lp for all the cases listed in Tables 1 and 2. k is calculated from Eq. (3) with the initial contact line length as the length factor (L). It can be seen from this figure that k increases with increasing profile length until a critical value of Lp (≈4.2 mm) is reached. Beyond this value, it appears that k is relatively insensitive to Lp . One can therefore conclude from this figure that k is a function of Lp and not just the total contact line length. 8. Conclusion We have investigated the onset of motion in non-circular drops using both experiments as well as Surface Evolver simulations. We have created non-circular drops from coalescence of two sessile drops on a substrate. The orientation of this non-circular drop was
Fig. 13. Variation of as a function of profile length (Lp ).
varied and its effect studied. During this study it was found that profile length, which is the width of the drop as seen from the direction of impending motion, has a significant effect. It was observed that as the profile length increases, the sliding angle increased. We therefore, conclude that the motion of a drop, with a non-circular three phase contact line is dependent on the orientation of the drop through the profile length. We have also studied the process of onset of motion of such non-circular drops. We found that a small section of the contact line is locally initiated into motion prior to the rest of the drop. The location of the section where motion is initiated locally depends, rather counter-intuitively on the orientation of the non-circular drop as well as its specific shape. References [1] C.W. Extrand, Y. Kumagai, Liquid drops on an inclined plane: the relation between contact angles, drop shape, and retentive force, J. Colloid Interface Sci. 170 (1995) 515–521. [2] A.I. ElSherbini, A.M. Jacobi, Liquid drops on vertical and inclined surfaces: i. An experimental study of drop geometry, J. Colloid Interface Sci. 273 (2004) 556–565. [3] A.I. ElSherbini, A.M. Jacobi, Retention forces and contact angles for critical liquid drops on non-horizontal surfaces, J. Colloid Interface Sci. 299 (2006) 841–849. [4] F.J. Pierce, Understanding of sliding and contact angle results in tilted plate experiments, Colloids Surf. A: Physicochem. Eng. Aspects 323 (2008) 73–82. [5] N. Janardan, M.V. Panchagnula, Effect of the initial conditions on the onset of motion in sessile drops on tilted plates, Colloids Surf. A: Physicochem. Eng. Aspects 456 (2014) 238–245. [6] M.J. Santos, J.A. White, Theory and simulation of angular hysteresis on planar surfaces, Langmuir 27 (2011) 14868–14875. [7] H. Samuel, Study of wetting and adhesion interactions between water and various polymer and superhydrophobic surfaces, J. Phys. Chem. C 115 (2011) 14852–14861. [8] R. Thampi, Do liquid drops roll or slide on inclined surfaces? Langmuir 29 (2013) 3339–3346. [9] S.P. Thampi, R. Govindarajan, Minimum energy shapes of one-side-pinned static drops on inclined surfaces, Phys. Rev. E 84 (2011) 046304. [10] V. Berejnov, R.E. Thorne, Effect of transient pinning on stability of drops sitting on an inclined plane, Phys. Rev. E 75 (2007) 066308. [11] M. Musterd, C.R. van Steijn, M.T. Kreutzer, Droplets on inclined plates: local and global hysteresis of pinned capillary surfaces, Phys. Rev. Lett. 113 (2014) 066104. [12] C. Semprebon, M. Brinkmann, On the onset of motion of sliding drops, Soft Matter 10 (2014) 3325–3334. [13] S. t Mannetje, R. Ghosh, S. Lagraauw, A. Otten, C. Pit, J. Berendsen, D. van den Ende, F. Mugele, Trapping of drops by wetting defects, Nat. Commun. 5 (2014). [14] D.P. Gnanappa, E. Papageorgiou, A. Gogolides, A.G. Tserepi, A.G. Papathanasiou, Plasma nanotextured, robust superamphiphobic polymeric surfaces structurally stabilized through a Wetting–drying cycle, Plasma Processes Polym. 9 (2012) 304–315. [15] D.A. ’t Mannetje, H. Banpurkar, M.H.G. Koppelman, D. van den Ende, F. Mugele, Electrically tunable wetting defects characterized by a simple capillary force sensor, Langmuir 29 (2013) 9944–9949.
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Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Onset of sliding motion in sessile drops with initially non-circular contact lines Nachiketa Janardan, Mahesh V. Panchagnula ∗ Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, India
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
• Impending motion in nearly elliptical drops is dependent on drop orientation. • Experiments and computations are used to determine moving and sliding angles. • Drop profile width is an important characteristic dimension.
a r t i c l e
i n f o
Article history: Received 29 December 2015 Received in revised form 14 March 2016 Accepted 16 March 2016 Available online 19 March 2016 Keywords: Contact angle hysteresis Inclined plane Non-circular contact line Profile width
a b s t r a c t The onset of motion of a drop with an initially non-circular three phase contact line was studied experimentally and numerically. Two drops of volume 10 l were made to coalesce and form a composite 20 l drop. The contact line of this drop was approximately elliptical and the local contact angle along the contact line was not a constant (as would have been the case with a circular contact line). The orientation of the drop to the impending direction of motion was varied. Inclined plate experiments were performed and the moving and sliding angles were noted in each case. It was observed that the moving and sliding angles of the drop were strongly dependent on this orientation. Specifically, the local conditions on the contact line at the front and back edges of the drop as well as the drop profile width were found to be the determining parameters. Surface evolver simulations were performed to understand the results of the experiments. It was found that the evolution of the contact line for non-circular drops was rather counter-intuitive when compared to the results from a drop with a circular contact line and resulted from a competition between gravity and the local contact angle hysteresis forces. © 2016 Elsevier B.V. All rights reserved.
1. Introduction
∗ Corresponding author. E-mail address: [email protected] (M.V. Panchagnula). http://dx.doi.org/10.1016/j.colsurfa.2016.03.046 0927-7757/© 2016 Elsevier B.V. All rights reserved.
The physics of sessile drops and the onset of their motion down an inclined plane have been studied extensively over the past
N. Janardan, M.V. Panchagnula / Colloids and Surfaces A: Physicochem. Eng. Aspects 498 (2016) 146–155
Fig. 1. Schematic of the drop with a non-circular contact line.
couple of decades [1–19]. The focus in this area of research, however, has been on sessile drops with an initially circular three phase contact line [5,7,20,21]. A sessile drop with an initially circular contact line has no concept of orientation associated with it. In this initial state, the profile of the drop is the same when viewed from any direction. This also implies that the distribution of the local contact angle () along the contact line is uniform. Therefore, one section of the three phase contact line is indistinguishable from the other. The onset of motion of such a drop is a problem that has been studied before by us and others [2,3,5,10,22–24] and is relatively well understood. However, when the contact line is noncircular, the contact angle distribution is no longer uniform; the contact angle varies from one section of the three phase contact line to another [25–27]. The onset of motion of a section of the contact line is dependent on this local contact angle (l ). Therefore, it is anticipated that the onset of motion of a drop with an initially non-circular contact line will be different from the onset of motion of a drop with an initially circular contact line even if the volumes of the two drops were to be the same. The orientation () of such a non-circular drop to the direction of motion will begin to play a key role in such onset of motion. In this manuscript, we present the results of a study on the onset of motion of drops with three phase contact lines that are nearly elliptical. The terminology and nomenclature that was introduced in our previous work [5] will be used here too. The interior angle that is subtended by the substrate and the free surface of the drop at the contact line is called the local contact angle l . If the surface is ideal, then this value is unique and is obtained from Young’s two limiting equation. A real surface, however, manifests contact angles that are called the advancing a and receding r contact angles. These contact angles are the maximum and minimum values that the contact angle can take and all the values in between r < < a are admissible as possible local contact angle values. This phenomenon is referred to as Contact Angle Hysteresis (CAH). Throughout this study, we will work with real surfaces. Fig. 1 shows a schematic of the contact line of the drop for purposes of representation. The direction of motion of the drop is along the axisX. The direction along which the camera observes the drop is the axis Y . The contact line is initially non-circular in shape. The contact line (which is nearly elliptical) can be described by a major axis of length, 2a and a minor axis of length, 2b. The major axis of the drop is along the axis X . The orientation of the drop is defined as the orientation of the major axis to the direction of motion, i.e. the angle between X and X , which will henceforth be referred to as . The profile length Lp is the width of the drop that is visible
147
from the direction of motion. is the azimuthal angle of a point on the three phase contact line. It is defined with respect to the axis Y , and is the angle between the axis Y and the line joining the aforementioned point to the origin. Extrand and Kumagai [1] were the first to study the behavior and the contact angle distribution of a sessile drop with a non-circular contact line. They assumed that the contact angle distribution was linear in cos l with cos ˛ and cos r being the bounds of the distribution. They also assumed that a normal to the local segment of the contact line was along the radial direction, but this has been shown to be incorrect later [26]. Recently, Antonini et al. [25] performed studies on measuring the adhesion force that resists the motion of the sessile drop. They developed a methodology to measure the contact angle distribution around the three phase contact line. Chini and Amirfazli [26] have also developed a methodology to determine the adhesion force for drops with an arbitrarily shaped contact line. This adhesion force is the same as the hysteresis force that has been referred to in Janardan and Panchagnula [5]. An important point to be noted was the perspective error that arises while measuring the contact angle of drops with an elliptical contact line. When the major axis was not perpendicular to the camera (the direction of viewing), it was seen that the point of measurement on the contact line was incorrectly determined. The radial position of the point of measurement (as determined from the images of the drop) from the origin is incorrect and leads to errors in the reconstruction of the contact line. However, there was no perspective error in the measurement of the contact angle itself. In our experiments and simulations, we have not measured the radial positions of the points of measurement. We have merely measured the contact angle at the extremes of the drop profile, and compared the same with results from Surface Evolver (SE). Therefore, the question of perspective error in contact angle measurement does not arise in our case. The inclination angle (˛) is the angle through which the substrate is tilted. The front edge and the back edge of the drop are those parts of the drop, which are downstream and upstream respectively, with respect to the direction of motion of the drop. As detailed in our earlier work [5], the motion of the drop is gov→
erned by the equilibrium between the gravitational force F g and → force F h .
the hysteresis This hysteresis force is the magnitude of the total impeding force that acts on the contact line as a result of CAH. The gravitational force is the driving force for the motion and the hysteresis force resists said motion. →
→
m g sin (˛) = Fg
(1)
Here, m is the mass of the drop and ˛ is the angle of inclination of the substrate of the drop. In a case involving drops with a circular contact line, the contact line of the drop begins to deform locally at either the back edge or front edge at a critical angle of inclination of the inclined plane called the moving angle (˛m ). When a second critical angle of inclination (at which the other edge starts to deform) is reached the drop begins to display the onset of motion. This angle is called the sliding angle (˛s ). The resisting force due to CAH is given by → Fh
=
ˆ LV | cosY − cos |ndl
(2)
where l is the local contact angle of the drop on the contact line, is the surface tension of the liquid-air interface and dl is an infinitesimal element of the contact line with the dimension of length. The closed integral is performed along the contact line in order to determine the sum of the forces that are acting on all such infinitesimal elements of the contact line.
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A model for CAH [5] is given by
Fh =
mg sin ˛
kL cosr − cos˛
for ˛ < ˛c for ˛ = ˛c
(3)
Here k is a non-dimensional parameter that is determined experimentally and L is a length parameter, which was taken to be the length of the contact line. As the CAH and k are both nondimensional, the length parameter is required to ensure that the right hand side of Eq. (3) has units of force. The global motion of the drop is initiated when the gravitational force exceeds the hysteresis force at an inclination angle ˛ = ˛c . Berejnov and Thorne [10] have studied the behavior of sessile drops on inclined planes. They suggested the existence of three critical angles of inclination. The first critical angle corresponds to the loss of local equilibrium of the contact line at the front edge and the second corresponds to the loss of local equilibrium of the contact line at the back edge. The third critical angle corresponds to the case when global motion of the whole drop is initiated. It can be construed that between the second and third critical angles, loss of local equilibria at the front and back edges of the drop is responsible for the slow sliding motion of the drop. Further increase in the tilt angle will result in an imbalance between the forces acting on the →
drop, mg sin ˛ and F h , resulting in rapid accelerated motion of the drop down the inclined plane. This will occur at the third critical angle ␣ = ␣c . In our case, we have terminated our experiments at the second critical angle since we are interested only in the onset of motion. Therefore, we will refer to the Berejnov and Thorne [10] second critical angle as the sliding angle. This is the inclination angle at which both the front edge and back edge have lost local equilibrium and are both moving. However, we have continued the tilting process in a limited set of experiments to discover the Berejnov and Thorne [10] third critical angle, where global motion is observed. In our previous publication, [5] we noted that the value of the non-dimensional parameter (k) was dependent on L, which in that case was chosen to be the initial three phase contact line length. The k value in that case was 0.29 and is consistent with the literature [1,10,20,24,28,29]. However, all these results have been obtained from simulations and experiments on drops with an initially circular three phase contact line. It would be interesting to investigate similar drops with a non-circular three phase contact line, as they are ubiquitous. This forms the core motivation of the current work. 2. Experimental apparatus A Holmarc© contact angle goniometer used to perform the experiments is shown in Fig. 2. The goniometer consisted of a platform for a substrate, a backlight, a CCD camera and a syringe pump to inject the working liquid on the substrate. The platform could be tilted, allowing us to vary the inclination angle. The substrate was able to translate in a horizontal plane by means of a screw. A CCD camera was used to capture the motion of the drop and the video thus captured was analyzed for contact angles and drop position. Holmarc© CA software was used to measure the contact angles as well as to estimate the critical angles of inclination. All our experiments were performed in a controlled atmosphere at a constant temperature. The entire experiment lasted less than 7–8 min. This ensured that the system was not affected by evaporation. 3. Materials and methods The substrate used was a thin strip of Polydimethylsiloxane (PDMS). The liquid used in the experiments was distilled water. A syringe pump was used to accurately meter out a 10 l drop on to the substrate. The substrate was then translated by a set number
Fig. 2. Experimental set up.
of rotations of the screw. This enabled the substrate to move forward by a set distance. Another 10 l drop was then deposited on the surface. This drop was placed as close as possible to the original drop such that they were on the verge of coalescing. A liquid bridge was then constructed between the two drops with the help of a stainless steel needle and they were allowed to coalesce to create a 20 l drop with a non-circular contact line. Photographs were taken of the major and minor axis of the drop to ensure that the dimensions of the drops used in multiple experiments were the same or within a reasonable margin of error (±2%). The camera views the drop from the side (with the camera axis perpendicular to the direction of motion) and is in a unique position to capture the dimensions of the drop. The drop was first aligned with its major axis parallel to the direction of motion and photographed to obtain the major axis length. It was then aligned with its minor axis along the direction of motion and was photographed to obtain the minor axis length. The average ratio of the major axis to the minor axis in all experiments was 1.47. Efforts were made to keep it to a value as close to 1.5 as possible. This was done to ensure experimental repeatability. The rate of the tilting was less than 3◦ per minute, so as to achieve quasistatic conditions. r and a for the surface liquid air combination were measured prior to the tilting plate experiment using the goniometer. a was measured to be 99◦ , while r was found to be 67◦ . An infusionwithdrawal experiment [30] was performed to measure r and a where fluid is injected and withdrawn from a surface by means of a needle. The receding angle was measured on a drop with a large enough volume so that the needle did not affect the receding interface. The infusion/withdrawal experiments were repeated multiple times to ensure reproducibility of the measured values. The advancing and receding angles were repeatable to within 3◦ . The drop was rotated around an axis perpendicular to the substrate, on the substrate to allow experiments to be performed in situations when the drop was oriented differently to the direction of the moving force, i.e., the component of gravity along the inclined plane of the substrate. The point of reference for various cases was the orientation of the major axis () of the drop with respect to this moving force. Thus, we performed experiments for varying angles of orientation, from the case where the major axis of the drop was oriented along the force of gravity (0◦ ), to the case where the major axis was perpendicular to the force of gravity (90◦ ). was varied in intervals of 15◦ . The substrate was tilted until the drop was observed to move. This process was filmed using the CCD camera and the subsequent video analyzed to determine the angle of inclination and the contact angles of the front edge and the back edge of the drop. The front and back edges are small sections of the contact line that were seen
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to start moving at the point of incipient motion. The contact angle at the front edge is designated as the front angle f and the con-
tact angle at the back edge is designated as the back angle b . These two angles were also measured to understand the state of the drop prior to incipient motion. 4. Surface evolver model Surface Evolver (SE) [31], an open source software, was used to simulate the behavior of the non-circular drops on a substrate as it is being inclined. SE works on the principle of energy minimization and volume conservation subject to constraints. A detailed description of the model can be found in Brakke [31,32], Janardan and Panchagnula [5] and Prabhala et al. [33]. The model developed by Prabhala et al. [33] simulated the process of two sessile drops coalescing. This simulation was used to generate the initial condition for the tilting plate simulation. It begins with a condition where two drops were just allowed to touch. The interfacial tension of the common surface shared by the two drops was then set to zero, effectively removing the interface between the two drops and allowing them to coalesce. The contact angle distribution was not specified and the SE iterations were continued until the drop shape and the contact line, both reached minimum energy configurations. This provided the initial condition at ˛ = 0◦ . Janardan and Panchagnula [5] developed a model which enabled them to simulate the quasistatic behavior of a sessile liquid drop on an inclined plane. The inclination of the substrate was increased until the drop showed signs of impending motion. The drop formed from the coalescence process (as described above) was subjected to this inclination process. The orientation angle was set to a desired value prior to starting the simulation. The simulation was terminated when both the front and back edges of the drop were initiated into motion. The results are recorded in terms of the drop shape, contact line shape as well as the critical angles of inclination. We have performed successive mesh refinements to ensure that these results are grid independent. This was achieved at about 10000 elements [5]. The capillary number Ca = /, where v is the velocity associated with a moving section of the contact line. Typically, v ∼1 mm/s from our drop sliding experiments. Therefore, Ca < 10−6 indicating that viscous effects in the rearrangement of the contact line can be neglected. Similarly, the Bond number, Bo = ((gsin (␣)V2/3 )/) was always less than 1 from our experiments. From these estimates, we can conclude that the onset of motion in these drops can be treated as a quasistatic process where gravitational and interfacial tension effects dominate. As a measure of validation between the experimental data and data from SE for the case of non-circular drops, we present Figs. 3 and 4 which show the evolution of the normalized front and back contact angles as a function of normalized inclination, respectively. In Fig. 3, the quantitative match between our experiments and simulations is good up to an ␣/␣s ratio of 0.4. At this point (termed as the moving angle) it was observed that the back edge started moving in both SE and experimental data. Beyond this moving angle, the data from SE is seen to be constant while the experimental data displays variation that is a characteristic of stick slip motion [34–36]. SE is unable to model this stick slip motion because of strict definitions on the conditions required for motion. However, it is able to capture the average motion during this phase. We refer the reader to Janardan and Panchagnula [5] for further validation exercises focused on initially circular drops. In Fig. 4, the trends match quantitatively over the entire range of the inclination. From these two figures, we conclude that the SE model is capable of capturing the essential physics of incipient drop motion.
Fig. 3. Comparison of back angle b trends for experiments and simulations. Here ◦
=0 .
5. Results and discussion The primary qualitative difference between initially circular and non-circular drops is that the distribution of contact angles on the contact line is not uniform in the case of a non-circular drop as will be seen later. In other words, different points on the contact line may have different l values. This effect is manifested in the form of ˛m and ˛s being dependent on (orientation of the drop with respect to the direction of incipient motion), for the same mass of the drop. ˛m and ˛s were identified based on the incipience of motion on either the front edge or the back edge. This motion was determined by the value of the contact angle around that small part of the contact line. If the local contact angle on either the front or back edge exceeded a or was below r , the edge was observed to move. Thus the motion of the contact line, and that of the drop as a whole, was affected by . 6. Experimental results Fourteen experiments were performed with seven different experimental cases. The seven experimental cases were only distinguished by their values. The major and minor axis lengths of the initially non-circular drops were also measured. The experimental cases were seen to have similar major and minor lengths. This was done to ensure experimental repeatability. The profile length (Lp ) in this case was the linear dimension that was visible to a camera that was placed with its axis parallel to the direction of motion. In other words, it is the width of the drop. The profile length thus varied from a minimum to a maximum as was increased. In the case of = 0◦ , the profile length was equal to the length of the minor axis, while in the case of = 90◦ , it was the major axis. This profile length is analogous to the diameter in circular drops. We will next report detailed contact angle data from three of the seven experiments in Figs. 5–7. Fig. 5 is a plot of the local contact angles at the front f and
back b edges versus the inclination angle (˛). f and b , the front and back angles were measured from analyzing the experimental images. The contact angle data was obtained by fitting a tangent to the image at the contact point. The data was smoothed in time, using five values of the contact angles from five successive images and employing a smoothing algorithm. These images were all within a time frame of 1.5 s (Please note that the tilting process was performed at approximately 3◦ per minute). The
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Fig. 4. Comparison of front angle f
Fig. 5. Front angle f
trends for experiments and simulations. Here = 0◦ .
and back angle b
initial condition for this experiment was obtained by coalescing two drops placed on the substrate, where the line joining the centers of the two drops is parallel to the direction of motion ( = 0◦ ), along the inclined plane. For the case of ( = 0◦ ) (see Fig. 5), the back edge was observed to move first and accordingly b was seen
to decrease until ˛m was reached. Beyond˛m , b increased and decreased alternately. This is related to stick slip behavior that is a characteristic of contact line motion when the size of the defect less than the size of the drop. During this experiment, is much f continued to increase monotonically until ˛s was reached. At
˛ = ˛s , f approached the advancing angle value. At ˛s , a small portion of the contact line at the front of the drop was seen to start
vs angle of inclination (˛) for the case = 0◦ .
moving in addition to the back edge which was already in motion. Thus the drop was initiated into motion down the inclined plane. Fig. 6 is similar to Fig. 5 except that it was for the case of = 45◦ . It can be seen in this figure that the contact angle at the front edge of the drop f is observed to increase while b was observed to decrease. The moving angle in Fig. 6 is also greater than that in Fig. 5. In contrast to the case of = 0◦ , the front edge was seen to move first at ˛ = ˛m , following which f is observed to increase and decrease alternately. This appears to be the contact angle signature of a moving edge (be it front or back). During this time, b decreased monotonically until ˛ = ˛s . At ˛s , the drop reached the point of impending motion down the inclined plane.
N. Janardan, M.V. Panchagnula / Colloids and Surfaces A: Physicochem. Eng. Aspects 498 (2016) 146–155
Fig. 6. Front angle f
Fig. 7. Front angle f
and back angle b
and back angle b
Fig. 7 is similar to Figs. 5 and 6, except that it was for = 90◦ . In contrast to both Figs. 5 and 6, ˛m is much lower while ˛s is higher. At this condition, the initial f was observed to be close to a and thus the front edge was seen to start moving nearly immediately upon starting the inclination process. During the entire tilting process, f was seen to vary around an average value, while b decreased monotonically. As mentioned before, this appears to be the signature of a moving edge. The key observation from the experimental results presented here is that three identical drops with similar dimensions and sessile on the same surface and oriented in different directions, will exhibit different behavior due to the difference in the distribution
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vs angle of inclination (˛) for the case = 45◦ .
vs angle of inclination (˛) for the case = 90◦ .
of the local contact angles along the contact line. The front and back angles in each case are different and thus the onset of motion behavior of the drop is different. The critical angles too are functions of this variance in local contact angle (see Fig. 8 below). These results will be further elaborated upon when the data from SE is discussed. Fig. 8 is a plot of the moving (˛m ) and sliding (˛s ) critical angles as a function of for both experimental data and SE data. It can be seen from this figure that there as increases, ˛s increases. ˛m however, increased up to a value of = 45◦ following which it decreased. It must be pointed out here that up to = 30◦ , it was the back edge that was observed to move first. For > 30◦ , it was the front edge that was seen to move first.
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Fig. 8. Moving angle (˛m ) and Sliding angle (˛s ) as functions of ).
Fig. 9. Plot of l distribution as a function of the azimuthal angle along the contact line. In this case, = 0◦ .
7. Results from surface evolver simulations Figs. 9–12 show results obtained from SE simulations. As mentioned before, the model requires the advancing and receding angles as constitutive inputs. The advancing and receding angles were taken to be 99◦ and 67◦ respectively, as obtained from the goniometer measurements on the same substrate as was used in the tilting plate experiments. Fig. 8 also shows values of ˛s and ˛m as a function of for data taken from SE. It can be seen from this figure that ˛s increases with increasing . ˛m also increases with up to = 82.5◦ and then decreases sharply. It was seen that when ˛m was increasing along with an increase in , the back edge determined the moving angle. ˛m exhibits a maximum at = 82.5◦ . However, when ˛m decreased, the front edge was seen to move first. These
results are qualitatively similar to the corresponding experimental cases. However, quantitatively, at = 75◦ and beyond, the drop does not slide off the substrate even when ˛ = 90◦ . This prediction does not match our experiments. A part of the reason for this mismatch could be that we do not allow for stick slip motion in our SE model. Stick-slip is a dynamical phenomenon and is therefore beyond the scope for inclusion into a quasi-static model. Another observation from these figures is that SE consistently overestimated the values of ˛m and ˛s in relation to the experiments. The maxima in the ˛m trend also shifted rightwards. The qualitative match however, was good and it is seen that the behavior of the drop in SE and the experiment is similar. The information that can be obtained from SE is significantly more detailed than those from experiments. For example, the actual shape of the
N. Janardan, M.V. Panchagnula / Colloids and Surfaces A: Physicochem. Eng. Aspects 498 (2016) 146–155
Fig. 10. Evolution of the contact line for = 0◦ . Data is extracted from SE.
Fig. 11. Evolution of the contact line for = 90◦ . Data is extracted from SE.
Fig. 12. Evolution of the contact line for = 45◦ . Data is extracted from SE.
contact line can be obtained to yield a more complete picture of the sliding drop. In addition, the local contact angle distribution is obtained from the SE simulations. See Fig. 9. In this case, the ends of the major axis correspond to the case where = 90◦ and = 270◦ . The ends of the minor axis correspond to the case where = 0◦ and = 180◦ . We can also study the evolution of the contact lines as seen in Figs. 10–12. It must be noted that the SE model imposes a limit on the maximum and minimum angle that the liquid air interface is allowed to subtend on the substrate. These are a and r , respectively. Krasovitski and Marmur [37] studied the metastable shapes of a drop on an inclined plane in two dimensions theoretically, and showed that the maximum and minimum angle at the point of impending motion need not simultaneously equal a and r for a substrate where the defect length scale is comparable to the drop size. However, this statement does not hold valid for a substrate where the advancing and receding angles are simultaneously defined at each point, by assuming that the intrinsic defects that cause the hysteresis occurs at a much smaller scale, such as in the current work. Thus, in this case the advancing and receding angles must be the maximum and minimum angles on the substrate. It would be an instructive exercise to track the evolution of the three phase contact line as the substrate is inclined during the simulation. A point to be noted is that the value of ˛m and ˛s from the SE simulation was determined exactly as they were determined from the experiment. ˛m was the inclination angle at which one edge of the drop (back edge or front edge, depending on the initial condition) was seen to move. ˛s is the inclination angle at which both the front and back edge were seen to be in motion simultaneously.
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In the subsequent discussion, we will present data from three representative cases where = 0◦ , = 90◦ , and = 45◦ , in Figs. 10–12 respectively. Fig. 10 is a plot of the instantaneous three phase contact line at four different values of ˛ for = 0◦ . Firstly, it can be seen that the three phase contact line for ˛ = 0◦ and ˛ = ˛m (in this case, ∼20◦ ) are coincident. This is a corroboration of the fact that the moving angle was accurately identified. As ˛ increases, the back edge is seen to recede until ˛ = ˛s when the onset of motion of the drop is initiated. An important constituent motion associated with the onset of motion is evolution of the three phase contact line at points not near the front or back edge. For example, in this figure, one would notice that as the back edge recedes, the sides of the drop begin to advance outwards. This arises purely due to the local contact angle in that region of the three phase contact line exceeding the advancing angle. We will discuss this in detail later in the context of Fig. 11. Fig. 11 is similar to Fig. 10 except that = 90◦ . As ˛ increases, the front edge of the drop moves first marking the first critical angle,˛m . However a further increase in ˛ produces results that is relatively counter-intuitive. During this phase, the back edge of the drop remains pinned to the substrate while the back flanks of the drop have been initiated into receding motion. This motion is symmetric about the axis determining the direction of motion. One could construe this result as a generalization of the case investigated by Berejnov and Thorne [10] to determine the first critical angle (herein referred to as moving angle). In other words, it is not necessary that only the front edge of the drop determines the moving angle. Fig. 12 is a plot of the three phase contact line shape for four different values of ˛ for = 45◦ . It can be seen that the drop exhibits asymmetric dewetting in relation to the major axis. A small portion of the contact line along the sides of the drop, along the direction of motion, has advanced while the complementary portion of the contact line, opposite to it, has receded before the motion of the drop is initiated. The back edge of the drop also exhibits asymmetric dewetting in the sense that a larger portion of the dewetted area is to one side of the major axis. The interplay between the orientation angle, which determines the action of gravity and contact angle hysteresis, which determines the local action of the substrate on the drop, gives rise to these complex and counter-intuitive three phase contact line shapes. It can be seen from Figs. 10–12 that the same drop exhibits varying behavior based on its orientation. The question that then arises is whether a single equation can describe the onset of motion in such a drop under all orientations. In our earlier work [5], we had noted that the value of k associated with the hysteresis force in Eq. (3) was dependent on a length factor L. The value of k should be calculated using Eq. (3) and the inclination angle, ˛c where global motion of the drop is initiated. However, as was pointed out by Berejnov and Thorne [10] and was observed in our own experiments, the difference between ˛c and ˛s is small (<5◦ ). For this reason, we will use ˛s to calculate the value of k. For the case of initially circular drops, the choices of length factors, viz. initial contact line length, diameter or profile length are all linearly related. Many different values have been proposed in literature [1–3,5,7,10,24,28,38,39] based on various length factors, entirely based on experiments and computations on initially circular drops. In the case of a drop with an initially circular contact line, the length factor we chose was the initial contact line length of the drop. The k value in this case was 0.285. This k value is consistent with other works in the literature [1,20,24]. It would be interesting to see if this result can be extended to the case of initially non-circular drops. We have already presented experimental results as well as results from simulations in SE where the length of the contact line
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Table 1 Contact length = 12.57 mm for all these cases. S. No.
Profile length (Lp ) (mm)
(◦ )
SC–1
1.22 1.16 1.10 1.05 1.02 1.01 1 0.99 0.98 0.95 0.91 0.86 0.82
4.38 4.29 4.20 4.10 4.04 4.02 4 3.98 3.96 3.9 3.8 3.7 3.6
90◦ 90◦ 90◦ 90◦ 90◦ 90◦ Undefined because the CL is a circle 0◦ 0◦ 0◦ 0◦ 0◦ 0◦
Table 2 Contact length = 12.57 mm for all these cases. S. No.
(◦ )
SC–2 SC–3 SC–4 SC–5
0.82 0.86 0.91 0.95
0◦ < < 90◦ 0◦ < < 90◦ 0◦ < < 90◦ 0◦ < < 90◦
was maintained constant while was varied. While those results helped point us in the direction that plays a role in determining k, it was difficult to isolate the choice of length scale. Specifically, we would like to investigate the individual contributions of the profile length Lp as well as a shape parameter such as the ratio of the two axes of the non-circular drop. For this purpose, we have defined as the ratio of the axis oriented with X to the axis oriented with Y (see Fig. 1 for a description of the coordinate system). The first set of simulations allowed to vary while the initial length of the contact line was kept constant. Table 1 lists the simulation parameters. Initially, = 1 (i.e., the drop had a circular contact line). The second set of simulations allowed to vary for constant value of (see Table 2 for these values). The objective in performing these additional sets of simulations was to investigate whether k is a function of explicitly or if it is only a function of the profile length Lp . The profile length, as we have already defined, is the length dimension associated with the drop when it is viewed from the direction of motion at the initial condition. It is to be noted that the simulations in these cases were performed on strictly elliptical drops (initially), as opposed to the non-circular drops formed from a twodrop coalescence event. The initial contact line of the drop was made elliptical by imposing a constraint in SE only to determine the initial state. Subsequently, this constraint was relaxed to allow the drop to evolve as ˛ increases. r and a as well as other simulation parameters were maintained the same as in the previous cases. Fig. 13 is a plot of k versus Lp for all the cases listed in Tables 1 and 2. k is calculated from Eq. (3) with the initial contact line length as the length factor (L). It can be seen from this figure that k increases with increasing profile length until a critical value of Lp (≈4.2 mm) is reached. Beyond this value, it appears that k is relatively insensitive to Lp . One can therefore conclude from this figure that k is a function of Lp and not just the total contact line length. 8. Conclusion We have investigated the onset of motion in non-circular drops using both experiments as well as Surface Evolver simulations. We have created non-circular drops from coalescence of two sessile drops on a substrate. The orientation of this non-circular drop was
Fig. 13. Variation of as a function of profile length (Lp ).
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