pubs.acs.org/Langmuir © 2010 American Chemical Society
Consolidation of Hydrophobic Transition Criteria by Using an Approximate Energy Minimization Approach Neelesh A. Patankar* Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, B224, Evanston, Illinois 60208-3111 Received December 16, 2009. Revised Manuscript Received January 21, 2010 Recent experimental work has successfully revealed pressure induced transition from Cassie to Wenzel state on rough hydrophobic substrates. Formulas, based on geometric considerations and imposed pressure, have been developed as transition criteria. In the past, transition has also been considered as a process of overcoming the energy barrier between the Cassie and Wenzel states. A unified understanding of the various considerations of transition has not been apparent. To address this issue, in this work, we consolidate the transition criteria with a homogenized energy minimization approach. This approach decouples the problem of minimizing the energy to wet the rough substrate, from the energy of the macroscopic drop. It is seen that the transition from Cassie to Wenzel state, due to depinning of the liquid-air interface, emerges from the approximate energy minimization approach if the pressure-volume energy associated with the impaled liquid in the roughness is included. This transition can be viewed as a process in which the work done by the pressure force is greater than the barrier due to the surface energy associated with wetting the roughness. It is argued that another transition mechanism, due to a sagging liquid-air interface that touches the bottom of the roughness grooves, is not typically relevant if the substrate roughness is designed such that the Cassie state is at lower energy compared to the Wenzel state.
1. Introduction The quest to make superhydrophobic surfaces for low drag or self-cleaning properties1 has been guided by two primary requirements. First, a drop deposited on the surface should have a large contact angle, and second, the drop should move easily relative to the surface, i.e., the hysteresis should be low. Both these requirements are typically satisfied by drops that reside on top of roughness grooves, i.e. drops in Cassie states.2-4 Drops that impale the roughness grooves, i.e., in Wenzel states,5 tend to be “sticky” and have high hysteresis.3,4 Thus, it is of interest to understand when a drop will transition to the Wenzel state.6 This can help design surfaces on which such transition does not occur or is delayed. Various mechanisms for transition to the Wenzel state, on pillar type roughness geometries, have been identified. In this work, we focus on the mechanisms discussed below. Transition Due To Laplace Pressure. This pertains to the transition of a sessile drop from a Cassie state to a Wenzel state. There are two ways in which transition can be induced (see Figure 1). The first is a depinning mechanism and the second is a sag mechanism.8,10,12,14 In a Cassie state the liquid-air interface hangs between pillars. The interface is curved due to the Laplace pressure inside the drop.10,12,14 If the hanging interface is such that it cannot remain pinned at the pillar tops then it proceeds downward into the roughness grooves and fully wets the surface. Lack of pinning occurs if the contact angle formed by the liquid-air interface is greater than the maximum contact angle *Corresponding author. (1) Barthlott, W.; Neinhuis, C. Planta 1997, 202, 1–8. (2) Cassie, A. B. D. Discuss. Faraday Soc. 1948, 3, 11–16. (3) He, B.; Lee, J.; Patankar, N. A. Colloids Surf. A: Physicochem. Eng. Asp. 2004, 248, 101–104. (4) Lafuma, A.; Quere, D. Nat. Mater. 2003, 2, 457–460. (5) Wenzel, R. N. J. Phys. Colloid Chem. 1949, 53, 1466–1467. (6) Patankar, N. A. Langmuir 2004, 20, 7097–7102. (7) Bartolo, D.; Bouamrirene, F.; Verneuil, E.; Buguin, A.; Silberzan, P.; Moulinet, S. Europhys. Lett. 2006, 74, 299–305.
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Figure 1. Side views of the transition from Cassie to Wenzel state due to depinning and sag mechanisms.7-14 A pillar-type roughness geometry is considered.
that can be sustained at a corner.6,15 This is the depinning mechanism. Even when a liquid-air interface can remain pinned at the pillar tops, transition to the Wenzel state is possible. This happens if the sag in the curved liquid-air interface is such that it touches the bottom of the roughness groove.10,14 Transition Due To Impact. It has been shown that drops deposited from some height16 or impacted on the surface with some velocity can induce transition to the Wenzel state.7,8,11,13 In this case the drop reaches the surface at some velocity and the subsequent collision with the surface gives rise to high pressure at the base of the drop. This high pressure can cause depinning or sag based transition discussed above.8,10-14 (8) Deng, T.; Varanasi, K. K.; Hsu, M.; Bhate, N.; Keimel, C.; Stein, J.; Blohm, M. Appl. Phys. Lett., 2009, 94, 133109. (9) Extrand, C. W. Langmuir 2004, 20, 5013–5018. (10) Jung, Y. C.; Bhushan, B. Scr. Mater. 2007, 57, 1057–1060. (11) Jung, Y. C.; Bhushan, B. Langmuir 2008, 24, 6262–6269. (12) Kusumaatmaja, H.; Blow, M. L.; Dupuis, A.; Yeomans, J. M. EPL 2008, 81, 36003. (13) Reyssat, M.; Pepin, A.; Marty, F.; Chen, Y.; Quere, D. Europhys. Lett. 2006, 74, 306–312. (14) Reyssat, M.; Yeomans, J. M.; Quere, D. EPL 2008, 81, 26006. (15) Oliver, J. F.; Huh, C.; Mason, S. G. J. Colloid Interface Sci. 1977, 59, 568–581. (16) He, B.; Patankar, N. A.; Lee, J. Langmuir 2003, 19, 4999–5003.
Published on Web 02/16/2010
DOI: 10.1021/la9047424
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remains constant. This assumption implies that chemical equilibrium (i.e., equating the chemical potentials of all component species in the phases) is not imposed across liquid-air interfaces. Using 2 in 1, the stable equilibrium states are obtained by minimizing X ðPo -Pφ ÞV φ þ ðT φ -To ÞS φ þ N φ μφ ð3Þ Uσ þ φ
Figure 2. Schematic of a drop on a pillar-type roughness geometry.
Formulas based on geometric considerations have been proposed to identify the criteria for depinning or sag based transitions (see Figure 1).8-10,12,14 Transition has also been considered as a process of overcoming the energy barrier between the Cassie and Wenzel states.6 It is of interest to enquire how the key criteria for transition correlate in the context of energy minimization theory. In this work, we consider the transition from Cassie to Wenzel state by using the energy minimization approach.
2. Energy Minimization We consider the static or equilibrium angle θe of the substrate material which is assumed to be hydrophobic, i.e., θe > 90°. The consequent analysis will represent the so-called “ground states” of the drop on the rough substrate. Consider a liquid drop deposited on a rough substrate with a pillar-type roughness geometry as shown in Figure 2. The drop size is considered large compared to the length scale of the roughness. Assume that the ambient air is at constant temperature To and pressure Po. Let the air phase be denoted by R. Let the liquid drop phase be denoted by β. The different interfaces between the solid, liquid, and air will be collectively denoted by phase σ which will be assumed to be sharp. Stable equilibrium states of this system can be obtained by minimizing the availability which is the relevant energy function17 Uσ þ
X ðU φ þ Po V φ -To Sφ Þ
ð1Þ
φ
where U φ ¼ T φ Sφ -Pφ V φ þ N φ μφ
ð2Þ
In eqs 1 and 2, superscripts φ denote phases R or β. The summation with respect to φ in eq 1 represents addition of the quantity in the bracket for phases R and β. Uφ is the total internal energy, Vφ is the total volume, Sφ is the total entropy, and μφ is the chemical potential of phase φ. Tφ and Pφ are the temperature and pressure of phase φ. Nφ are the number of moles of phase φ. Uσ is the total surface energy of all the interfaces. It is obtained by summing up the surface tension (energy) times the area of all the interfaces. Contact angle measurements are assumed to be made on time scales much shorter than the evaporation time scale. Therefore, in eq 1, it is assumed that each of the phases R and β are single components and that the mass of each individual phase (17) Modell, M.; Reid, R. C., Thermodynamics and Its Applications: PrenticeHall: Englewoods Cliff, NJ., 1983.
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Gravitational potential is ignored by assuming that the drop sizes are less than the capillary length scale of 2.7 mm for water. By imposing thermal equilibrium, each phase is assumed to be at temperature To. Thus, the temperature terms in eq 3 make no contribution. Since NR is assumed constant the corresponding chemical potential term has no effect on the minimization of availability. Phase β is considered incompressible and of constant mass. Consequently, its pressure-volume and chemical potential terms are reformulated in terms of a Lagrange multiplier corresponding to the volume constraint. The availability A to be minimized becomes A ¼ Po -Pβ V β -Voβ þ U σ
ð4Þ
which is the Gibbs free energy of the system. The first term is due to the volume constraint on phase β. The volume Vβ of phase β will be equal to its fixed volume Voβ after the minimization. The corresponding Lagrange multiplier is Po-Pβ. Pβ is the mechanical pressure of phase β; thermodynamic pressure is not defined for an incompressible fluid. Consider the drop in a Cassie state with the liquid-air interface hanging between the pillar tops as shown in Figure 2. The hanging liquid-air interface is curved. For the purposes of quantifying its surface energy we will assume it to be flat (see Figure 2) since the actual shape is not known unless a more complex problem is solved. Upon substituting the expressions for surface energy in eq 4, the availability AC corresponding to the Cassie state is given by
where R is the radius of the drop (Figure 2), Scap is the area of the spherical cap of the drop, θ is the apparent contact angle, Abase is the base area of the drop projected on the horizontal plane, Atot is the total solid-air area of the dry surface, and σsg is the solid-air surface energy. θrc is the apparent contact angle based on Cassie formula:2 cos θcr = φ cos θe þ φ - 1, where φ is the area fraction of the pillar tops in the horizontal plane. The third and fourth terms on the right-hand side of eq 5 denote the changes in energy of an initially dry substrate that is wetted in the region corresponding to Abase. The question of whether the availability AC of the Cassie state corresponds to an equilibrium state or not, and if so, whether it is the global minimum or not will be explored below. Consider that the interface at the top of the pillars, in the Cassie state, impales the grooves to some depth h (Figure 2). The availability A of this general state can be written as A ¼ ðPo -Pβ ÞðV β -Voβ -Vimp Þ þ ðPo -Pβ ÞVimp þ σlg 2πR2 ð1 -cos θÞ -πR2 sin2 θσ lg cos θcr -ðr -1ÞπR2 sin2 θσ lg cosθe
)
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where Vimp is the volume of the fluid that has impaled the grooves. (Po - Pβ)Vimp is added and subtracted because it is the pressurevolume energy associated with the liquid that has impaled the roughness. It plays an important role in quantifying the depinning transition, which will be discussed in section 3.1. The area of the pillar sides relative to the horizontal area is given by
a known value of h. It can be shown that this equilibrium solution is in fact the stable equilibrium with respect to θ, R, and Pβ.19,20 The only thing that remains to be verified is the minimum (i.e., stable) energy states with respect to h, which will be discussed through section 3.
3. Transition from Cassie to Wenzel state r -1 ¼
4ah ða þ bÞ2
where a, b, and h are as defined in Figure 2. The term σsgAtot is not included since it is constant and plays no role in the minimization. Minimizing the availability in eq 6 will give the stable equilibrium states of the system. The volume of the liquid phase Vβ is given by
where Vup is the volume of the drop shape above the substrate. The square bracket denotes ‘function of’ and the expression for the volume Vimp of the liquid that has impaled the roughness is also indicated in eq 7. The availability to be minimized can be written as i h 9 A Pβ , R, θ, h ¼ Po -Pβ Vup ½R, θ -Voβ þ σ lg Scap ½R, θ> > = ! β 4ah P -Po > > cos θe þ hð1 - φÞ -σlg Abase ½R, θ cos θcr þ ; σlg ða þ bÞ2 ð8Þ where Scap and Abase are as defined in eq 5, and expressions for r - 1 and Vimp have been substituted. Equation 8 shows that the availability should be minimized with respect to four parameters: Pβ, R, θ, and h. The macroscopic parameters of the drop are represented by Pβ, R, and θ, whereas h represents the microscopic parameter that specifies the location of the liquid-air interface in the roughness geometry. The term multiplying Abase in eq 8 represents the surface energy per unit area associated with the wetting of the rough substrate. Thus, it represents the energy of the microscopic state of the substrate. The others terms quantify the energy of the macroscopic drop. The microscopic state can be analyzed independent of the macroscopic problem resulting in a decoupling in the energy analysis. Equation 8 shows that the availability is linear with respect to h under the approximations considered here. It will be discussed in sections 3.2 and 3.3 that the possible stable equilibrium values of h are 0 and H, where H is the pillar height (see Figure 2). Once h is known, A can be extremized with respect to Pβ, R and θ.18 Setting ∂A/∂θ = 0, ∂A/∂Pβ = 0, and ∂A/∂R = 0 give equilibrium conditions. It follows that at equilibrium, the apparent contact angle θ is given by cos θ ¼ cos θcr ¼ φ cos θe þ φ -1,
for h ¼ 0
ð9Þ
which is the Cassie formula. A Wenzel-type formula (eq 18) can also be obtained when h=H, which will be discussed in Section 3.3. Additionally, the extremization conditions imply that Vβ[R,θ,h] = Vβo and Pβ - Po =2σlg/R. These two equations and the apparent contact angle formula (eq 9 or 18) can be solved for θ, R, and Pβ for (18) Patankar, N. A. J. Adhes. Sci. Technol. 2009, 23, 413–433.
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3.1. Depinning Transition. To understand the variation of A with respect to h we re-write eq 8 as follows 9 !> πR3 h i > > > 2 -3 cos θ þ cos3 θ -Voβ > A Pβ , R, θ, h ¼ Po -Pβ > > 3 > > > > > 2 c = þ σlg 2πR2 ð1 - cos θÞ -πR2 sin θσlg cos θr 0 1 > > Pβ -Po C B Cb > 1 b 2πR2 sin2 θσlg > B > > cos θe þ C 2þ -hB > > 2 @ A > b a a 2σlg ða þ bÞ > > b 1þ > ; 2a ð10Þ Note that, on the right-hand side, the term involving h represents the contribution to the availability due to the state of the liquidair interface in the roughness grooves relative to the Cassie state. If the Cassie state, where the liquid-air interface hangs between pillar tops, is to be a stable equilibrium position (i.e., a local energy minimum) then the availability A should increase as h increases. On the other hand, if the Cassie state is not possible then the availability A should decrease as h increases. It would mean that the liquid-air interface will readily wet the roughness grooves and the drop will proceed to the Wenzel state. This represents the depinning of the liquid-air interface from the pillar tops. It will happen when the term multiplying h in eq 10 is negative, i.e., when the following condition is satisfied Pβ -Po >
-4σlg cos θe b 2b 1 þ 2a
ð11Þ
A similar condition was also reported earlier by Zheng and Zhao.21 The accuracy of this expression for the depinning condition is affected by the approximations made in theoretical analysis above. An equivalent expression can be derived from geometric considerations. To do so consider a spherical liquid-air interface hanging between diagonally opposite pillars and impose the condition that the interface will impale the grooves if the contact angle at the pillar tops is greater than the equilibrium contact angle. The corresponding condition for depinning or impalement is given by8,10,12,14 Pβ -Po >
-4σlg cos θe pffiffiffi 2b
ð12Þ
Another condition can derived if the interface is considered hanging between adjacent pillars. In that case the depinning condition is given by Pβ -Po >
-4σlg cos θe 2b
ð13Þ
(19) Marmur, A. Langmuir 2003, 19, 8343–8348. (20) Patankar, N. A. Langmuir 2003, 19, 1249–1253. (21) Zheng, Q. S.; Yu, Y.; Zhao, Z. H. Langmuir 2005, 21, 12207–12212.
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All the three criteria in eqs 11-13 are based on approximations. Even the geometric criteria in eqs 12 and 13 are derived by assuming a shape of the liquid-air interface which is not known exactly unless the complex shape of the interface is solved. The expression from energetic considerations in eq 11 is equal to that in eq 13 for small values of b/a. This is reasonable since the flat interface approximation (see Figure 2) used in the energy-based approach is increasingly accurate for smaller values of b/a. This transition condition can also be consolidated with the concept of surface energy barrier between Cassie and Wenzel states introduced earlier.6 To do so, consider the two terms involving h in eq 8 that led to the transition criterion in eq 11. The first term -4ahσlg cos θe represents the increase in surface energy caused by wetting the sides of the pillars, in one periodic cell of the roughness geometry, as the liquid-air interface impales the groove. This energy is maximum at h = H. Thus, -4aHσlg cos θe is the surface energy barrier, in one periodic cell, corresponding to the transition from Cassie to Wenzel state.6 The energy to overcome this barrier comes from the work done by pressure which is the second term involving h in eq 8. It shows that the work done by the pressure force, in one periodic cell, to move the liquid-air interface by a distance H is given by (Pβ - Po)(1 - φ) (a þ b)2H. For transition to occur, the work done by pressure should be greater than the surface energy barrier (greater because the extra work done goes into overcoming the viscous resistance to motion): β
2
ðP -Po Þð1 -φÞða þ bÞ H > -4aHσ lg cos θe
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! b2 H < R 1- 1- 2 2R
(22) Kwon, Y.; Patankar, N.; Choi, J.; Lee, J. Langmuir 2009, 25, 6129–6136. (23) Sbragaglia, M.; Peters, A. M.; Pirat, C.; Borkent, B. M.; Lammertink, R. G. H.; Wessling, M.; Lohse, D. Phys. Rev. Lett., 2007, 99.
ð15Þ
where R is the radius of curvature based on the pressure, i.e.
ð14Þ
which leads to the transition criterion in eq 11. The above discussion shows that the criteria for transition based on energy, geometric, and surface energy barrier considerations are consolidated into a single framework as desired. Specifically, we see that the transition criterion based on depinning emerges from energy considerations only if the pressure-volume energy associated with the liquid that has impaled the roughness is also considered along with the surface energy of wetting the roughness. The availability associated with the microscopic problem (i.e., the terms involving h in eq 8) has as input the liquid pressure Pβ which is obtained from the macroscopic problem above the substrate. The transition condition in eq 11 is unaffected by how the liquid achieves this pressure. Pβ could be due to the Laplace pressure of a drop (explicitly considered in this work),8,10,12,14 or due to some imposed pressure due to squeezing,22 or the water hammer pressure due to sudden deceleration of a colliding drop,8 or the dynamic pressure due to an impacting drop,7,8,11,13 and so on. Whatever the situation may be, the pressure Pβ should be estimated and used in the transition criterion to check if the depinning based transition will occur or not. A few comments are to be noted. In this work, the transition is assumed to be a result of the liquid-vapor interface invading the roughness grooves from the top to the bottom. This is shown to be relevant in prior studies discussed above. There are cases23 where it has been shown that the Cassie to Wenzel transition begins by a top-down transition of the liquid-vapor interface below the center of the drop; however, the remaining transition appears to proceed sideways within the roughness grooves. The discussion in this work does not resolve this mode of transition. Additionally, transitions caused solely by contact line dynamics have been
8944 DOI: 10.1021/la9047424
proposed in the past.24 That mode of transition is also not considered here. In a different scenario, the transition on surfaces with cavities is additionally resisted by compression of air in the cavities. This makes the Cassie state more robust.25 That analysis is discussed elsewhere.18 3.2. Sag Transition. The Laplace pressure Pβ leads to a sag in the liquid-air interface between pillars.10,12,14 If the sag is such that the liquid-air interface touches the base of the roughness groove then the drop transitions to a Wenzel state from the Cassie state (see Figure 1).10,14 This mechanism of transition is different from the depinning transition considered above. In fact in this case the parameters are such that the depinning criterion (eqs 11-14) is not satisfied; i.e., the liquid-air interface can remain pinned between the pillar tops, but it still transitions because the pillars are not sufficiently tall for the sag to remain above the bottom of the grooves. This is a geometric criterion that would also be resolved within the energy based approach if the detailed shape of the hanging liquid-air interface is considered.26 In our approximate theory that is not considered. The sag criterion for transition from Cassie to Wenzel state is given by10,14
2σlg R ¼ β P -Po Here, we will compare this criterion for transition with another criterion for H based on energy considerations. This is discussed below. The impalement of the liquid-air interface in the roughness grooves leads to a change in availability per unit area, which is given by the terms involving h in eq 8. When the liquid-air interface reaches the bottom of the pillars, the base of the roughness grooves has to be wetted to complete the transition to the Wenzel state. The associated wetting energy per unit area (=-σlg(1 - φ)(1 þ cos θe)) should be added to that in eq 8. Thus, the net change in availability per unit area ΔA0 C-W for Cassie to Wenzel transition in the substrate is given by ΔA0 C - W ¼ -σlg
Pβ -Po cos θe þ Hð1 -φÞ þ ð1 - φÞð1 þ cos θe Þ 2 σlg ða þ bÞ 4aH
!
ð16Þ Note that the wetting of the bottom of the grooves leads to a discontinuous decrease in the availability, at h = H, by an amount equal to σlg(1 - φ)(1 þ cos θe). This implies that the wetted or Wenzel state is a stable (local minimum energy) state. Additionally, if we assume that eq 11 is not satisfied, it implies that the Cassie state is also a possible stable equilibrium position. Thus, in this case, two stable equilibrium positions corresponding to h = 0 (Cassie state) and h = H (Wenzel state) are possible on the rough substrate. If we now impose a condition that the Cassie state be (24) Bormashenko, E.; Pogreb, R.; Whyman, G.; Erlich, M. Langmuir 2007, 23, 6501–6503. (25) Bahadur, V.; Garimella, S. V. Langmuir 2009, 25, 4815–4820. (26) Dupuis, A.; Yeomans, J. M. Langmuir 2005, 21, 2624–2629.
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the global minimum, i.e., ΔA0 C-W > 0, it gives a condition on the height H of the pillars -bð1 þ cos θe Þ 1 H> 0 B bC B cos θe C þ C 2B @ b RA 1þ 2a
ð17Þ
where R ¼
2σ lg -Po Þ
energy will scale as σlgH/R sinθ and it will oppose spreading due to pressure; i.e, the corresponding line tension will be positive. This contribution to the apparent contact angle is expected to counteract the effect of the H/R term in eq 18. Appropriate modeling of the line tension type contributions is essential to describe the lateral invasion of the liquid within the grooves. There could be different final static equilibrium of the liquid front within the roughness grooves;e.g., circular, square, etc.;depending on the dynamic process during lateral invasion of the grooves by the liquid.23,27 This issue is not considered in the present work. Here we assumed a circular front as was assumed in the models by Cassie and Wenzel.
ðPβ
4. Conclusion
has been substituted. Substrates satisfying the condition in eq 17 are deemed to be robust against Cassie to Wenzel transition. If the drop radius is large compared to the roughness geometry then the b/R term corresponding to the pressure-volume energy can be neglected and eq 17 reduces to that considered in our earlier work.6 It can be verified that substrates that satisfy eq 17 do not typically satisfy the sag transition condition in eq 15. Thus, sag transition is usually feasible only when the Wenzel state is lower energy. If substrates are designed such that the Cassie state is lower energy, then typically the sag transition will not occur. 3.3. Wenzel State. It was stated before that minimizing eq 8 with h = 0 gives the Cassie formula (eq 9). As seen above, h = H is also a possible stable equilibrium solution corresponding to the Wenzel state. In this case, an additional term = -σlgAbase[R,θ] (1 - φ)(1 þ cos θe) must be added to the availability function to account for the wetting of the bottom of the grooves. Upon minimizing the availability function in eq 8 with this change, the apparent contact angle is given by
In this work, we consider the minimization of an approximate availability function to obtain the various states of a drop on a rough hydrophobic substrate. The theoretical formulation is developed from first principles so that all the underlying assumptions are explicitly identified. The availability function has two parts. The first part corresponds to the energy of the drop shape above the substrate, i.e., the “macroscopic problem.” The second part corresponds to the wetting of the roughness grooves, i.e., the “microscopic state,” which is relevant to understand the transition from Cassie to Wenzel state. The consideration of the microscopic problem is decoupled from the macroscopic problem. It is found that in order to resolve the pinning or depinning of the liquid-air interface between the pillar tops, it is essential to account not only for the surface energy associated with wetting the roughness but also the pressure-volume energy of the liquid that impales the roughness during transition. The transition due to depinning can be viewed as a process in which the work done by the pressure force is greater than the barrier due to the surface energy associated with wetting the roughness. Transition criterion obtained from energy considerations compared well with criterion obtained from geometric considerations. The criterion for the transition due to a sagging liquid-air interface that touches the bottom of the roughness grooves is also considered. It is argued that this transition will not be relevant if the substrate roughness is designed such that the Cassie state is at lower energy compared to the Wenzel state. It is also found that the liquid at high pressure that impales the roughness in the Wenzel state gives rise to a tendency of the liquid to spread laterally through the rough substrate. This effect is like a negative line tension term in a modified Wenzel formula for the apparent contact angle of the drop. This effect will be opposed by the interfacial energy of the liquid front within the roughness geometry, which is not resolved in our theory. In summary, a consolidated understanding of the various considerations of transition is reported by using the energy minimization framework.
cos θ ¼
4aH
H cos θe þ 2ðφ -1Þ , R ða þ bÞ 2
for h ¼ H
ð18Þ
As expected, the first term on the right-hand side of eq 18 is the same as Wenzel’s formula.5 The term involving H/R in eq 18, where R is the drop radius, is an additional term associated with the pressure-volume energy of the liquid that has impaled the roughness. It is negligible for large drops or small pillar heights. Physically, it implies the tendency of the liquid at high pressure to spread laterally through the rough substrate. Consequently, its effect is to reduce the apparent contact angle (see eq 18). This term is equivalent to a negative line tension τ given by τ = -σlg(1-φ)H sin θ. It is noted, however, that there is an opposing effect that is not resolved in our approximate formulation. While accounting for the surface energy associated with the substrate in a Wenzel state, the interfacial energy of the liquid front within the roughness is not included in our theory or in the Wenzel formula. This contribution to the
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(27) Courbin, L.; Denieul, E.; Dressaire, E.; Roper, M.; Ajdari, A.; Stone, H. A. Nat. Mater. 2007, 6, 661–664.
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