Comment pubs.acs.org/Langmuir
Cite This: Langmuir XXXX, XXX, XXX-XXX
Reply to Comment on “Solid−Liquid Work of Adhesion” ABSTRACT: Extrand’s interpretation in his “Comment on “Solid-Liquid Work of Adhesion” by Tadmor and Coworkers” may lead to an important discussion and physical understanding of the problem. Below, we compare the two approaches and elucidate the differences to put them in the right perspective.
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xtrand1 and Tadmor et al.2 consider a centrifugal adhesion balance (CAB) experiment3,4 in which the CAB induces a slowly growing normal force that pulls on a drop. The force causes the drop to be heavier and heavier, until it eventually flies away from the surface on which it was placed. Tadmor et al. obtain the work of adhesion from the point at which the triple line starts shrinking spontaneously, which happens at the later stages, though not at the end, of the CAB experimental run. The same event is also considered by Extrand for the work of adhesion. The values of the drop diameter and work of adhesion are therefore identical in the two approaches. The difference is related to the way the two approaches back calculate the contact angle that should represent the work of adhesion.5−7 Extrand developed a work of adhesion equation in a similar logic as that described in refs 8 and 9, but for different geometries, and is as follows: WExtrand = γ cos θR
second represents a normal adhesion in a planar geometry. It is important to realize that normal and lateral adhesion are different. Comparing these contact angles to experimental contact angle measurements, we note that the actual contact angles measured experimentally varies within the contact angle hysteresis, which is between 79° and 107.5° for stationary drops that are inflated and deflated. However, as the drop is moving, or as the triple line is sliding, much lower (and higher) contact angles are possible. For example, the minimal work of adhesion value suggested, in ref 16, eq 5, depends on the method of obtaining the maximal advancing contact angle. Hence, it depends on parameters such as the gravitational constant (which can be varied by CAB experiments). Figure 1 shows such a contact angle in a series of drops from a CAB work of adhesion experimental run as described below.
(1)
while Tadmor et al. used the Young−Dupré equation: WY−D = γ(1 + cos θ )
2,10−15
(2)
Both Extrand and Tadmor et al. calculate the work of adhesion to be the force divided by the perimeter at the spontaneous onset of triple line shrinking:2 WExtrand = WY − D =
FD πD P
Figure 1. Pictures of water drops during a CAB run of an increasing effective gravity field which pulls on the drop from a silanized (C18) silicon surface as the effective gravity is slowly and continuously enhanced according to eqs 2, 3, and 5 in ref 2, and at a constant centrifugal jerk of 0.5 rpm/sec. The image highlighted with the red box represents the moment of onset of detachment considered by Tadmor et al; whereas, the one highlighted with the yellow box shows the last image captured before the droplet snapped. The numbers under the frames represent the gravitational accelerations, which was subjected to the drop, at the time the frames were taken.
(3)
where θR is the receding contact angle, θ is the thermodynamic contact angle, γ is the surface tension, FD is the pull-off force, or the Dupré force, and DP is the pull off diameter of the triple line, and the work of adhesion is noted by WExtrand and WY−D to reflect the fact that there are two ways to relate it to contact angles according to Extrand and to Young−Dupré, respectively. Note that not only do the relations to the contact angles differ, the contact angles are also different (θR in Extrad’s equation and θ in the Young−Dupré equation). Correspondingly, they get the same work of adhesion, but different contact angles: Extrand gets 45°, while Tadmor et al. get 106°. These angles refer to the contact angles for water on silanized (C18) silicon surface and were calculated using eqs 1 and 2, respectively, using the experimental work of adhesion values, which were same for both approaches. While the Young−Dupré equation is well established, and we believe it is the correct theoretical representation of the work of adhesion, Extrand’s equation deserves attention, especially since it resulted in a good fit in refs 8 and 9. We believe that Extrand’s equation is correct for capillary rise, and Dupré’s equation is correct for a liquid’s normal detachment from a solid. The first represents a lateral adhesion in a tube geometry, while the © XXXX American Chemical Society
We marked with a red square the frame from which the spontaneous triple line shrinkage starts (see Figure 1). This is the point whose force and drop diameter are considered for the calculation of the work of adhesion. We marked the last frame before detachment with a yellow square. This frame is irrelevant for work of adhesion calculations, though sometimes it is mistakenly considered as such. This frame represents the onset of spontaneous shrinking of the liquid−liquid bridge which happens after the onset of spontaneous solid−liquid area shrinking marked by the red frame. Following the g values underneath the frames helps one notice that the drop starts shrinking without an increase in the effective gravity (spontaneous shrinking) at the red frame. Received: September 24, 2017 Published: November 12, 2017 A
DOI: 10.1021/acs.langmuir.7b03350 Langmuir XXXX, XXX, XXX−XXX
Comment
Langmuir
Figure 2. A typical force versus circumference plot for a work of adhesion experimental CAB run is shown along with snapshots of various stages a droplet goes through as the normal force increases. Note that the obtained work of adhesion is independent of drop size or gravitational acceleration.2
Table 1. Stages in the Drop-Solid Detachment Process stage
description
notes
1
Initially as the pulling force increases, the triple line remains pinned and only the contact angle changes.
2 3
At higher forces, an increase in the force causes both the triple line and the contact angle to change. The higher is the force, the lower is the incremental force change for the same triple line shrinkage. At some critical force, the triple line spontaneously starts shrinkinga
4 5 6
The actual process of the triple line spontaneous shrinkage. The triple line stopsb its spontaneous shrinkage. Further liquid−liquid elongation causes a liquid−liquid snap.
Force increase is required for this process. Force increase is required for this process. 3.1. Tadmor et al. consider this event. 3.2. Extrand considers this event. 3.3. No force increase is required from here on. No force increase is required here.
a
The entire process from stage 1 until the beginning of stage 3 can be called as the region of contact line shrinkage, whereas only at the beginning of stage 3 does the spontaneous contact line shrinkage occur, and it continues until the drop is peeled off (fully or partly) from the surface. bA notable exception to this stage is the case when the triple line proceeds to shrink until the drop is completely pealed from the solid surface: The drop is either pinched off tearing a liquid−liquid contact, or is pealed completely from the solid leaving a dry surface behind.
contact angle using eq 1, and Tadmor et al. calculated it using eq 2. Correspondingly, they get different contact angles of 45° and 106°, respectively. Equation 1 has a good fit to the experimental results in refs 8 and 9, while eq 2 is the wellestablished Young−Dupré equation. We believe both equations have merit, where eq 1 represents lateral adhesion for the geometry of refs 8 and 9, and the Young−Dupré eq 2 represents the normal work of adhesion. It is important to realize that normal and lateral adhesions are different. To put this in context, we also mention other approaches to evaluate lateral adhesion, such as 1. that of ElSherbini17 and others, who related the lateral adhesion force to the difference between the advancing and receding angle, and, of course, 2. direct measurement of lateral adhesion that is decoupled from the normal force as performed by CAB experiments.3
The frame marked yellow has the lowest contact angle recorded, which is 68°. This is still greater than Extrand’s calculated contact angle, but the fact that CAB experiments can produce contact angles that are beyond the spectrum of regular advancing and receding contact angles shows that there can be yet more extreme advancing and receding contact angles, because those commonly measured are true just for a gravitational field of 1 g, whereas a CAB can create conditions that span between −5 g to +5 g. Therefore, they do not represent an absolute extreme state, but just a local maximum or minimum that corresponds to the order of magnitude of earthly gravity. Finally, we present some of the frames shown in Figure 1, also as a function of the force induced on the drop by the CAB. This is shown in Figure 2, while in Table 1 we describe the stages considered in Figure 2. The overall process can be described in the following six stages, whose numbers correspond to Figure 2: Figure 2 and Table 1 demonstrate the event used for the determination of the work of adhesion, namely, the moment spontaneous shrinkage commences. In summary, as noted in Table 1 (c.f. Figures 1 and 2), both Extrand and Tadmor et al. calculate the work of adhesion using eq 3 and considering stage 3 in Table 1. This stage represents the moment where spontaneous shrinkage of the solid−liquid contact occurs, namely, the onset of detachment. This also corresponds to the image that is highlighted with a red box in Figure 1. From the work of adhesion, Extrand back-calculated a
S. Gulec† S. Yadav† R. Das† R. Tadmor*,†,‡ †
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The Department of Chemical Engineering, Lamar University, Beaumont Texas 77705, United States ‡ Ben Gurion University, Beer-Sheva 8410501, Israel
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] and
[email protected]. B
DOI: 10.1021/acs.langmuir.7b03350 Langmuir XXXX, XXX, XXX−XXX
Comment
Langmuir ORCID
R. Das: 0000-0003-1008-0086 R. Tadmor: 0000-0002-6300-4645 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This study was supported by NSF grants CMMI-1405109 and CBET-1428398 and CBET-0960229. REFERENCES
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DOI: 10.1021/acs.langmuir.7b03350 Langmuir XXXX, XXX, XXX−XXX
