pubs.acs.org/Langmuir © 2010 American Chemical Society
Hysteresis with Regard to Cassie and Wenzel States on Superhydrophobic Surfaces Neelesh A. Patankar* Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208 Received November 11, 2009. Revised Manuscript Received December 24, 2009 Cassie and Wenzel formulas have been used extensively to model the apparent contact angles on rough surfaces. Theoreticians and experimentalists, alike, have noted that such formulas that are based on homogenization theories, while useful, do not capture hysteresis which is determined by the details of contact line motion. Thus, it is of immediate interest to model hysteresis in the context of Cassie and Wenzel type formulas. To address this issue, we consider an ad-hoc generalization of the theory for hysteresis by Joanny and de Gennes.1 We establish its applicability by reanalyzing the contact angle data from literature for drops in Cassie states on pillar-type roughness geometries. Using this theoretical framework, it is possible to focus on advancing and receding contact line motions separately unlike the analyses of drop motion on inclined planes that quantify the combined effects of advancing and receding fronts. We show how information about the details of contact line motion, during advancing or receding, can be translated into the theoretical framework for hysteresis. The conclusions based on such analyses provide useful physical insights into the energetics of contact line motion and are consistent with experimental observations. We also show that the theoretical framework could be used as a useful guideline to hypothesize the possible mechanisms of pinning/depinning of contact lines that are implicit in the experimental data.
1. Introduction 2 3 Cassie and Wenzel formulas are commonly used to model the enhancement of hydrophobicity due to roughness. It is wellknown that these formulas are based on the energy minimization principle applied to the respective wetting configurations on rough surfaces.4-6 Specifically, the contact angle represents an average (or homogenized) measure of the minimum-energy state experienced by the moving contact line. These formulas provide useful guidance to design rough superhydrophobic surfaces since they provide a measure of the ground state of the liquid-surface contact.7 If one raises the ground state energy, then the surface would tend to be superhydrophobic. Superhydrophobicity is considered to imply large contact angles and low hysteresis of drops on rough surfaces. Hysteresis can be quantified in terms of the difference between the advancing and receding contact angles—greater difference implies higher hysteresis. It can also be quantified in terms of the ability of a drop to slide off an inclined surface. When the hysteresis is high, the drop tends to stick to the surface. This is typically undesirable. To that end, surfaces on which drops are in the Cassie state (i.e., drops that reside on top on roughness crests) and have large contact angles are preferred.8,9 In the Wenzel state the fluid impales the grooves of the roughness features, thus making the drops sticky. The Cassie and Wenzel formulas, together with the analysis of energy barriers between the two states, have been effective in identifying the type of roughness geometries that lead *E-mail:
[email protected]. (1) (2) (3) (4) (5) (6) (7) (8) (9)
Joanny, J. F.; Degennes, P. G. J. Chem. Phys. 1984, 81, 552–562. Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11–16. Wenzel, R. N. J. Phys. Colloid Chem. 1949, 53, 1466–1467. Patankar, N. A. Langmuir 2003, 19, 1249–1253. Marmur, A. Langmuir 2003, 19, 8343–8348. Marmur, A.; Bittoun, E. Langmuir 2009, 25, 1277–1281. Quere, D. Annu. Rev. Mater. Res. 2008, 38, 71–99. Lafuma, A.; Quere, D. Nat. Mater. 2003, 2, 457–460. He, B.; Lee, J.; Patankar, N. A. Colloids Surf., A 2004, 248, 101–104.
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to the Cassie state.7,10,11 This does not imply that drops in the Cassie state do not exhibit hysteresis—they do. However, the degree of hysteresis is typically lower in the Cassie state compared to the Wenzel state. It has been recognized that Cassie and Wenzel formulas do not capture hysteresis which is primarily caused by pinning and depinning type behavior of the moving contact line.1,12-21 Hence, there have been several studies focused on modeling the contact line motion on chemically heterogeneous or rough surfaces to understand and eventually develop predictive models for hysteresis.1,12-21 Experiments, especially those by McCarthy and co-workers, have shown that the advancing and receding contact angles on many superhydrophobic surfaces do not agree with the predictions of Cassie or Wenzel formulas.22 This has been recently discussed in the literature.6,23,24 Thus, the importance of contact lines to quantify the apparent contact angles has been recognized by theoreticians and experimentalists alike. It is therefore of interest to find whether hysteresis can be modeled by homogenized energy-based approaches that are used to derive Cassie and Wenzel formulas. This consolidation appears possible by using prior theoretical approaches to quantify hysteresis.1,19,20 (10) Patankar, N. A. Langmuir 2004, 20, 8209–8213. (11) Patankar, N. A. Langmuir 2004, 20, 7097–7102. (12) Anantharaju, N.; Panchagnula, M. V.; Vedantam, S.; Neti, S.; Tatic-Lucic, S. Langmuir 2007, 23, 11673–11676. (13) Extrand, C. W. J. Colloid Interface Sci. 1998, 207, 11–19. (14) Extrand, C. W. Langmuir 2002, 18, 7991–7999. (15) Extrand, C. W. Langmuir 2004, 20, 5013–5018. (16) Good, R. J. J. Am. Chem. Soc. 1952, 74, 5041–5042. (17) Johnson, R. E.; Dettre, R. H. J. Phys. Chem. 1964, 68, 1744–1750. (18) Kusumaatmaja, H.; Yeomans, J. M. Langmuir 2007, 23, 6019–6032. (19) Pomeau, Y.; Vannimenus, J. J. Colloid Interface Sci. 1985, 104, 477–488. (20) Raphael, E.; Degennes, P. G. J. Chem. Phys. 1989, 90, 7577–7584. (21) Shanahan, M. E. R.; Carre, A.; Moll, S.; Schultz, J. J. Chim. Phys. Phys.Chim. Biol. 1986, 83, 351–354. (22) Oner, D.; McCarthy, T. J. Langmuir 2000, 16, 7777–7782. (23) Gao, L. C.; McCarthy, T. J. Langmuir 2007, 23, 3762–3765. (24) Gao, L. C.; McCarthy, T. J. Langmuir 2009, 25, 7249–7255.
Published on Web 01/19/2010
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Past work by Good,16 de Gennes and co-workers,1,20 Extrand,13-15 and others9,12,17-19,21,25-28 have contributed toward the understanding and modeling of hysteresis. The objective of this work is to build upon the insights from the prior work and address the key challenge of developing of predictive models to quantify hysteresis by using homogenized energy-based approaches. To that end, we put past data for contact angles on rough surfaces within the framework of a generalization of the theory for hysteresis by de Gennes and co-workers1,20 and draw insights that could be useful in the broad goal of modeling hysteresis with energy-based methods. We will specifically focus on the Cassie state since that is more amenable to analysis according to the theory for hysteresis by de Gennes and coworkers.28 In section 2 we will briefly discuss the approach by de Gennes and co-workers.1,20 In section 3, analyses of past data and discussion will be presented. Conclusions will be presented in section 4.
2. Modeling Framework for Contact Angle Hysteresis We present the theoretical framework provided by Joanny and de Gennes1 (hereafter referred to as JD) and then present an adhoc conceptual generalization that will be used to analyze experimental data. Joanny-de Gennes Theory for Hysteresis. JD developed a theory for chemically or physically heterogeneous surfaces. A physical heterogeneity or defect would be a roughness feature on a base substrate. They assumed that the defects were dilute, i.e. their number density is low, so that their effect is additive. They also assumed that the defects were strong so that the contact line can be effectively pinned, which plays an important role in causing hysteresis. Under the aforementioned assumptions, JD showed that the advancing contact angle, θadv, deviates from the “equilibrium” contact angle θE on a surface according to the following equation γla ðcos θadv - cos θE Þ ¼ - Eadv
ð1Þ
where γla is the liquid-air surface tension and Eadv is the energy dissipated per unit area by an advancing contact line. The equilibrium contact angle is defined as the apparent contact angle at the contact line in the presence of thermal agitation or mechanical vibrations. This would be the most stable or ground state(s) of the liquid-solid contact. Thus, θE could be regarded as the contact angle according to Cassie or Wenzel formulas depending on the nature of the wetting configuration on the rough surface. This is consistent with the derivations of JD for dilute defects. Similarly, JD showed that the receding contact angle, θrec, deviates from θE because of the energy per unit area, Erec, dissipated by a receding contact line: γla ðcos θrec - cos θE Þ ¼ Erec
ð2Þ
They obtained expressions for Eadv and Erec assuming dilute defects. Equations 1 and 2 imply that γla ðcos θrec - cos θadv Þ ¼ Erec þ Eadv ¼ Wd
ð3Þ
where Wd is the total energy dissipated per unit area around a hysteresis cycle. (25) Anantharaju, N.; Panchagnula, M. V.; Vedantam, S. Langmuir 2009, 25, 7410–7415. (26) Dorrer, C.; Ruhe, J. Langmuir 2006, 22, 7652–7657. (27) Gao, L. C.; McCarthy, T. J. Langmuir 2006, 22, 6234–6237. (28) Reyssat, M.; Quere, D. J. Phys. Chem. B 2009, 113, 3906–3909.
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It is important to note that the energies Eadv and Erec can be formulated and quantified by looking for events during respective contact line motions where there is a discontinuous decrease in energy.1 This is the dissipated energy implied in the theory by JD.1 This can happen, e.g., during depinning of a contact line. Thus, in our analysis in section 3, we propose estimates for Eadv and Erec for various advancing and receding mechanisms by focusing on contact line events where there is a discontinuous decrease in energy. Ad-Hoc Generalization of Joanny-de Gennes Formulation for Hysteresis. Equations 1-3 were derived rigorously by JD for dilute defects. Additionally, the heterogeneous surfaces were assumed to be made of materials that do not exhibit hysteresis. To analyze realistic scenarios, both these assumptions can be restrictive. In many of the reported experiments, including those to be analyzed here, the typical area fractions of roughness features are not necessarily low and the materials used to make rough surfaces have unequal advancing and receding contact angles even when they are “flat”. To facilitate analysis, we assume an ad-hoc conceptual generalization of the formulation by JD. Specifically, we focus on drops in Cassie states. We do so because the drops in Cassie states are (i) more amenable to analysis like that by JD,28 (ii) desirable for superhydrophobicity, and (iii) studied more extensively to result in substantial data in literature. However, application to Wenzel states may also be envisaged. We focus on surfaces with pillar-type roughness features. Consider that the surface is made of a material with an advancing angle θA and a receding angle θR. When a drop is in a Cassie state, the contact area below the drop can be considered to be a surface of contact angle θs = 180°, i.e., a liquid-air interface, that is decorated with regions of different contact angles corresponding to the contact with the material of the top of the pillars. The surface energy for liquid contact with the pillar tops is characterized by a defect energy per unit area.1 Since the surface material has an advancing and receding contact angle, we quantify two defects energies per unit area: hA corresponding to an advancing contact line and hR corresponding to a receding contact line. Consequently, the defect energies are given by hA ¼ γla ðcos θA - cos θs Þ hR ¼ γla ðcos θR - cos θs Þ
ð4Þ
Corresponding to each defect energy, we follow the approach by JD and define “equilibrium” contact angles1 θEA, for an advancing contact line, and θER, for a receding contact line: γla ðcos θEA - cos θs Þ ¼ φhA ¼ φγla ðcos θA - cos θs Þ γla ðcos θER - cos θs Þ ¼ φhR ¼ φγla ðcos θR - cos θs Þ
ð5Þ
where φ is the area fraction of the top of the pillars in a horizontal plane. Equation 5 results in the Cassie formula cos θEA = φ cos θA þ φ - 1 after substituting cos θs = -1 for the advancing contact line. Thus, the equilibrium contact angle θEA during advancing is the angle obtained from the Cassie formula for the advancing contact line. This formula does not account for any effects of pinning or depinning of the advancing contact line due to the presence of the roughness. It is a measure of the ground or most stable state of the advancing contact line given the advancing contact angle of the surface material itself. Similarly, it can be verified that cos θER = φ cos θR þ φ - 1, where θER represents the ground state of a receding contact line. DOI: 10.1021/la904286k
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Figure 2. Advancing mechanism where the contact line instantly wets the tops of subsequent pillars.14,15,27,29
Figure 1. Cosine of the advancing contact angles measured by Oner and McCarthy22 for ODMCS surfaces (θA = 102°) compared with the theoretical prediction for θEA according to Cassie’s equation.
Generalizing eqs 1 and 2, we write expressions for the advancing and receding contact angles on the rough surface γla ðcos θadv - cos θEA Þ ¼ - Eadv ð6Þ γla ðcos θrec - cos θER Þ ¼ Erec where Eadv and Erec are the energies associated with additional dissipation by advancing and receding contact lines due to the presence of the roughness. It is nontrivial to obtain simple models for Eadv and Erec for nondilute area fractions of the pillars. Information about Eadv and Erec could be obtained from experimental data. Equations 5 and 6 can be used to obtain a generalized form of eq 3: γla ðcos θrec - cos θadv Þ ¼ γla φðcos θR - cos θA Þ þ Eadv þ Erec ð7Þ where γlaφ(cos θR - cos θA) represents the energy per unit area dissipated due to hysteresis of the surface material, and Eadv þ Erec represents the additional energy dissipated in the hysteresis cycle due to the presence of surface roughness. Thus, the extended formulation is consistent with the underlying physical processes. Drop roll-off experiments on an inclined plane can give quantitative information for Δ cos θ = cos θrec - cos θadv. For example, Reyssat and Quere28 reported such data for rough surfaces with cylindrical pillar geometries as a function of the area fraction φ of the pillars. In this work we specifically analyze the data for Cassie drops according to eq 6 instead of the combined eq 7 to gain insights into the separate mechanisms during advancing and receding. Data from different groups are analyzed in the same way to identify the key common features. We consider data for the area fraction of the pillars ranging from very low to high—thus not restricting to low area fractions.
3. Results and Discussion We consider two data sets from literature by Oner and McCarthy22 and Dorrer and Ruhe.26 These data are reanalyzed in the context of eq 6 in the following sections. 3.1. Advancing Contact Angle. Experimental Results. Figure 1 shows a plot of cos θadv vs φ measured experimentally by Oner and McCarthy22 for n-octyldimethylchlorosilane (ODMCS) surfaces with different pillar geometries (see ref 22 for details). In each of the plotted cases the drop was in a Cassie state. φ is the area fraction of the pillars. ODMCS has θA = 102° and θR = 94°. The Cassie formula gives the relevant equilibrium contact angle 7500 DOI: 10.1021/la904286k
θEA: cos θEA = φ cos θA þ φ - 1. cos θEA is also plotted in Figure 1. As expected, the advancing contact angle θadv is not in agreement with the angle θEA predicted by the Cassie formula. The Cassie formula does not agree with the advancing angle because it does not resolve the energy associated with depinning of the advancing contact line. θadv is found to be more than 170° and constant not only for this set of surfaces but also for all other cases reported by Oner and McCarthy.22 Constant contact angles with respect to the area fraction were also reported for drops in Cassie states by others.26,28 However, the magnitude of the advancing angle has differed in the range from 175° to 155° among data from different groups.22,26,28 Thus, there are two issues to consider: to understand the constant advancing angle in the context of the theory by JD and to speculate why the constant advancing contact angles reported by different groups may be different. This is discussed next. Discussion. Constant and high values of advancing angles for Cassie drops have been explained by noting that the advancing contact line gets pinned to the pillar edges. The contact line does not move but rather the liquid-air interface descends upon and instantly wets the tops of the next pillars (Figure 2).14,15,18,26,27,29 As a result the apparent contact angle is close to 180°. This observation can be translated into the theory for hysteresis by JD as follows. To get the advancing angle, it is necessary to obtain Eadv in eq 6. According to the mechanism of advancing depicted in Figure 2, the energy change associated with the advancing event corresponds to the wetting of the pillar tops as the liquid-air interface descends on it. This energy change is due to the removal of a liquid-air interface with area equal to that of the pillar tops, and the formation and removal of liquid-solid and solid-air interfaces, respectively, on the pillar tops. Thus, Eadv is given by Eadv ¼ φγla ðcos θA - cos θs Þ
ð8Þ
Equations 5, 6, and 8 imply that cos θadv = cos θs = -1, i.e., θadv = 180°, as expected. As noted earlier, in a Cassie state the contact area below the drop is regarded to be a liquid-air interface, with θs = 180°, which is decorated with defects corresponding to the pillar tops. The fact that the advancing angle θadv is equal to θs implies that the defects do not affect the resultant contact angle of the heterogeneous liquid-air interface below the drop. Equations 5, 6, and 8 can be used to explain this. These equations imply that the reduction in the effective surface energy according to eq 5 is exactly canceled by the pinning energy associated with the advancing mechanism (eqs 6 and 8). Thus, the advancing angle is equal to θs(=180°). Physically this implies that the contact line remains pinned until the liquid-air interface becomes horizontal. While the aforementioned trend for the advancing contact angle has been reported in prior experiments,22,26,28 in general, the advancing angle need not necessarily follow that trend. For example, advancing contact angles, for the pillar geometry, that are in agreement with the Cassie formula have been reported over Langmuir 2010, 26(10), 7498–7503
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Figure 3. Cosine of the receding contact angles measured by Oner and McCarthy22 and Dorrer and Ruhe26 compared with the theoretical
prediction for θER according to Cassie’s equation. Oner and McCarthy22 data for 8 μm square pillars for (a) dimethyldichlorosilane (DMDCS) surfaces (θR = 102°), (b) n-octyldimethylchlorosilane (ODMCS) surfaces (θR = 94°), and (c) heptadecafluoro-1,1,2,2tetrahydrodecyldimethylchlorosilane (FDDCS) surfaces (θR = 110°). (d) Data from Dorrer and Ruhe26 for square pillar of different sizes (as indicated in the legend in μm) for a surface material with θR = 71°. See original reference for further details.
a range of area fractions.12 The advancing angle could depend on variety of factors including geometry of the roughness and its size relative to the drop size, weight of the drop, presence of disturbances, etc. The state of the drop corresponding to θadv = 180° is a highenergy state compared to a drop in the ground state with an equilibrium contact angle θEA according to the Cassie formula.4 It is seen from Figure 1 that as φ increases the difference between θEA and θadv increases; i.e., the drop with the advancing contact angle is increasingly less favorable, energetically, compared to the ground state. Equivalently, Eadv increases as φ increases. Thus, any disturbance (e.g., vibrations in the system) could make the transition from the advancing state to a lower energy state more likely as φ increases. The advancing contact line may not remain pinned and may jump sooner, resulting in a lower apparent contact angle compared to 180°.18 To develop fully predictive energy-based models, additional considerations including the energy barriers to contact line motion will be necessary. Even when constant advancing angles are found, their reported magnitudes have differed. For example, Oner and McCarthy22 reported constant advancing angles in the range of 170°-175° whereas Dorrer and Ruhe26 reported ∼156° advancing angle for all their cases. Theoretically these values are expected to be close to 180°. This disagreement could be because of, among other reasons, errors caused by the failure to obtain a fit that precisely describes the shape of the drop during contact angle measurements.26 In this regard it should be noted that the weight of the drop can dominantly influence the shape of the drop near the substrate and consequently the measured contact angle. Consider, e.g., a 10 μL water drop on a rough substrate. This size is typical in many reported experiments. Gravity is not expected to significantly influence the shape of drops of this size (gravity and surface (29) Gao, L. C.; McCarthy, T. J. Langmuir 2006, 22, 2966–2967.
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tension forces are of same order for drops that are 82 μL or larger). However, when the contact angle is 180°, the shape of the drop is significantly distorted from a spherical shape in the vicinity of the substrate.30 This lowers the center of the drop and leads to a nonzero contact area of radius l with the substrate. For a 10 μL drop, it can be shown that l ∼ 600 μm.30 This is the size of the base of the drop that would be observed in contact angle measurement experiments even when the apparent contact angle is supposed to be 180°. If one incorrectly fits a spherically shaped cap on top of a contact area of radius 600 μm and volume 10 μL, then the apparent contact angle that would be measured is ∼156°—a value similar to that reported by Dorrer and Ruhe.26 This indicates that errors in shape estimation, caused by distortions due to gravity, could lead to uncertainties in the reported contact angles that are in the same range as the reported discrepancies in the literature. 3.2. Receding Contact Angle. Experimental Results. Figure 3 shows a plot of cos θrec vs φ measured experimentally by Oner and McCarthy22 and Dorrer and Ruhe.26 In all cases there are pillar roughness geometries and the drops are in the Cassie state (see Figure 3 and its caption for details). The Cassie formula gives the relevant equilibrium contact angle θER: cos θER = φ cos θR þ φ - 1, where θR is the receding contact angle of the surface material. cos θER is also plotted in Figure 3. It is seen that the measured receding contact angles do follow the overall trend according to the Cassie formula. The degree of disagreement quantifies the energy Erec associated with the receding contact line on the rough surface. Figure 4 shows a plot of Erec, calculated according to eq 6, for the same data as in Figure 3. These results are discussed below. Discussion. There are three potential scenarios that can be considered for the receding mechanism as shown in Figure 5.14,15,26,27,29 (30) Mahadevan, L.; Pomeau, Y. Phys. Fluids 1999, 11, 2449–2453.
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The first scenario is where the posterior pillar tops are dewetted without significant deformation of the liquid-air interface (Figure 5a); i.e., it lifts off like in a rolling motion. This is similar to the advancing mechanism considered in Figure 2, but in reverse. In this case Erec is given by Erec = -φγla(cos θR - cos θs). Using cos θs = -1, it is seen that Erec is negative, and eq 6 gives θrec = 180° corresponding to the rolling motion. The system would gain energy rather than dissipate it during receding motion. Thus, this receding mechanism is not expected. This is consistent with experimental observations.22,26 The second scenario for the receding mechanism is shown in Figure 5b. This mechanism has been discussed by Gao and McCarthy.27,29 In this case, as the liquid-air interface recedes, the posterior pillar tops are not immediately dewetted. Instead, it deforms and increases the liquid-air surface energy. Eventually, the force to deform the interface is not sustained by the pillars. Consequently, the entire pillar top dewets, and the liquid-air interface relaxes as shown in Figure 5b. Finally, the third scenario is shown in Figure 5c which was discussed by Extrand14,15 and Dorrer and Ruhe.26 In this case the pillar tops are gradually dewetted, and the sudden decrease in energy occurs when the liquid-air interface that is pinned at the inner edge of the pillar depins and relaxes (see Figure 5c). Any scenario that is intermediate between scenarios two and three may also be considered. Below we discuss scenarios two and three separately in the context of JD theory.
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In the second scenario for receding (Figure 5b) Erec can be written as Erec ¼ - φγla ðcos θR - cos θs Þ þ Edef;s2
ð9Þ
where subscript s2 implies scenario two. The terms involving cosines in eq 9 represent the energy associated with dewetting the tops of the pillars, and Edef,s2 is the energy associated with the extra deformation of the liquid-air interface corresponding to the pinned state. Equations 7-9 imply that γla ðcos θrec - cos θadv Þ ¼ Edef;s2
ð10Þ
for receding according to scenario two. Note that by using eq 8 we have assumed that the advancing mechanism is as depicted in Figure 2, i.e., θadv = 180°. Equation 10 implies that, in receding scenario two, the net energy loss during a hysteresis cycle is equal to the energy required to deform the liquid-air interface to the pinned state during receding. Scenario two for receding is implicit in the analysis by Reyssat and Quere,28 who measured the lefthand side of eq 10 from experiments on drop movement on an inclined plane and compared the data to theoretical estimates for the deformation energy during pinning of a receding contact line. In the third scenario for receding (Figure 5c) Erec can be written as Erec ¼ Edef;s3
ð11Þ
where subscript s3 implies scenario three. In this scenario since the contact line dewets the pillar tops continuously rather than discontinuously, the dewetting energy does not arise in eq 11, unlike in scenario two (eq 9). Thus, the receding energy is only due to Edef,s3 which is the energy associated with the extra deformation of the liquid-air interface corresponding to the pinned state shown in Figure 5c. Equations 7, 8, and 11 imply that γla ðcos θrec - cos θadv Þ ¼ φγla ðcos θR - cos θA Þ þ φγla ðcos θA - cos θs Þ þ Edef;s3
Figure 4. Erec/γla, computed according to eq 6, vs φ for the data by
Oner and McCarthy22 and Dorrer and Ruhe26 shown in Figure 3.
ð12Þ
for receding according to scenario three. As before, by using eq 8, we have assumed that θadv = 180°. Equation 12 implies that, in receding scenario three, the net energy loss during a hysteresis cycle is not solely due to the deformation of the liquid-air interface unlike in scenario two (eq 10). The analysis above can help differentiate which of these scenarios are likely to underlie experimental data. To illustrate, we consider the data depicted in Figures 3 and 4. Erec was computed according to eq 6. Given the trends shown in Figure 4 for Erec it is of interest to enquire which of the two scenarios— scenario two (eq 9) or scenario three (eq 11)—is a more appropriate description of the receding mechanism. If it is scenario two, then the deformation energy Edef,s2 can be computed according to eq 9, where all the remaining terms are known from experimental
Figure 5. Three scenarios for the receding mechanism:14,15,26,27,29 (a) the receding surface lifts off from posterior pillars; (b) the receding surface remains pinned (dotted line) until it instantly dewets the entire pillar top and relaxes the liquid-air interface (solid line); (c) the receding surface remains pinned (dotted line) at the inside edge of the pillar top until it relaxes the liquid-air interface (solid line).
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Erec with a low value (seen in all cases except one) at an intermediate area fraction. At low area fractions the trend is consistent with the pinning of the contact line on pillar tops as discussed above. The low values of Erec at intermediate area fractions are most likely due to collective pinning/depinning events that reduce the overall energy of pinning.28 While the data are insufficient to be conclusive, the possibility of a second peak at high area fractions in the data by Dorrer and Ruhe26 could be indicative of a different mechanism where some kind of pinning might occur on the voids. This is likely because, at large area fractions of the pillars, the contact line may recede continuously on the pillar tops rather be pinned on it.
Figure 6. Edef,s2/γla, computed according to eq 9, vs φ for the data
by Oner and McCarthy22 and Dorrer and Ruhe26 shown in Figure 3.
data. This is plotted in Figure 6. On the other hand, if it is scenario three, then the deformation energy Edef,s3 = Erec (eq 11). This implies that Figure 4 also represents the plot for Edef,s3. The plots for the deformation energy in Figures 6 and 4 are very different in character. By comparing these trends for the deformation energy with expected trends from theoretical consideration, it could be possible to suggest which of the two scenarios are more likely. To that end, it is noted that Reyssat and Quere28 did present theoretical estimates for the deformation energy during pinning. Their theoretical trends for the deformation energy of pinning appear consistent with the trends in Figure 6 and not in Figure 4, thus lending support to scenario two as the likely receding mechanism. It is also noted that computational simulations also show scenario two as the receding mechanism for drops in Cassie states.18 A more detailed study of contact line motion is necessary to fully resolve the issue of how the contact line recedes, i.e., according to Figure 5b or Figure 5c, or either depending on the parameters. Finally, few comments are in order pertaining to the overall trend of Erec with respect to the area fraction φ in Figure 4. Erec is the total energy of the pinning/depinning events of the receding contact line, irrespective of whether it occurs according to scenario two or three. The data are suggestive of two peaks in
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4. Conclusion In this paper we address the issue of modeling hysteresis in the context of Cassie and Wenzel formulas. To do so, we consider an ad-hoc generalization of the theory for hysteresis by Joanny and de Gennes.1 The theoretical framework is physically relevant and also a useful tool to analyze experimental data. It presents a way to separately quantify the energy dissipated by the advancing and receding contact lines. We reanalyzed the contact angle data from literature for drops in Cassie states on pillar-type roughness geometries. We showed that the information about the details of contact line motion can be translated into the theoretical framework for hysteresis. Specifically, we considered an advancing motion where the contact line does not move but rather the liquid-air interface descends upon and instantly wets the tops of the next pillars. Based on this mechanism, the energy associated with the advancing motion was formulated. Upon using this energy in the expression for the advancing contact angle, the experimentally observed constant and large value ∼180° was retrieved. Similarly, we considered the data for receding contact angles from literature and reanalyzed them by assuming various probable receding mechanisms. On the basis of the hypothesized receding mechanisms, we formulated the energy dissipated during receding motion. We used it in the theoretical framework to enquire which receding mechanisms are likely. It was found that a likely scenario for the receding mechanism is the one where the liquid-air interface is pinned to the pillar top and eventually dewets the entire pillar top and simultaneously relaxes the deformed liquid-air interface. These conclusions were consistent with evidence from prior numerical results in the literature. In summary, we establish a viable theoretical framework that can model hysteresis in the context of Cassie and Wenzel formulas based on homogenization theories. We also show that the theoretical framework for hysteresis could be used as a systematic guideline to hypothesize the possible mechanisms of pinning/ depinning of contact lines that are implicit in the experimental data. This can help gain better insights into designing surfaces with low hysteresis.
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