pubs.acs.org/Langmuir © 2009 American Chemical Society
Contact Angles of Nanodrops on Chemically Rough Surfaces Gersh O. Berim and Eli Ruckenstein* Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received March 9, 2009. Revised Manuscript Received April 14, 2009 The experimental observations of Gao and McCarthy [Gao, L.; McCarthy, T. Langmuir, 2007, 23, 3762] that only the interfacial area near the leading edges of the drop on physically smooth but chemically rough solid surfaces affects the contact angle and that most of the contact area has no effect is checked for nanodrops on the basis of a density functional theory. The contact angle was calculated for three cases: (i) the leading edges of the drops are located on much higher or (ii) much lower hydrophobic surfaces than the remaining surface beneath the drop; (iii) the surface is composed of a periodic array of two kinds of stripelike solid plates. In the first two cases, if the distance between the leading edges and the region which has higher or lower hydrophobicity is sufficiently large, there is agreement with the experiments mentioned. However, when those distances are sufficiently small, the internal part affects the value of the angle. In the third case, we found that the internal part always affects the wetting angle. All these peculiarities, as well as the contact angle hysteresis, can be explained by accounting for the local conditions in the vicinity of the leading edges of the drop.
1. Introduction In a recent paper, Gao and McCarthy1 provided experimental evidence that the contact angle θ of a liquid drop on a chemically rough solid surface is determined by the interactions near the three-phase contact line, with the contribution of the remaining solid-liquid interfacial area inside the drop having no effect. They attempted to represent the data using the standard CassieBaxter equation2 cosθCB ¼ fA cos θA þ fB cos θB
ð1Þ
where θA and θB are the contact angles which the drop makes with the uniform surfaces A and B, and fA and fB are the area fractions of A and B surfaces beneath the drop. They found that there is no agreement and concluded that this equation is incorrect in general. In the discussion that followed in the literature (see refs 3-7), various arguments were brought pro and contra that conclusion. The main conclusion of refs 3-5 was that the area fractions fA and fB involved in the Cassie-Baxter equation are local quantities calculated in the vicinity of the contact line. However, the considerations of Whyman et al.6 regarding the equilibrium conditions of a drop on a surface, based on the minimization of the global free energy of the drop, revealed that the entire contact area between fluid and solid contributes to the value of the contact angle. Finally, Marmur and Bittoun7 have shown that the Cassie-Baxter as well as the Wentzel equation for a physically rough surface8 are valid when the drop size is much larger (about 3 orders of magnitude larger) than the wavelength of the chemical heterogeneity or physical *Author to whom correspondence should be addressed. E-mail: feaeliru@ acsu.buffalo.edu. Telephone: 716-645-1179. Fax: 716-645-3822. (1) (2) (3) (4) (5) (6) (7) (8)
Gao, L.; McCarthy, T. Langmuir 2007, 23, 3762. Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546. McHale, G. Langmuir 2007, 23, 8200. Nosonovsky, M. Langmuir 2007, 23, 9919. Panchagnula, M. V.; Vedantam, S. Langmuir 2007, 23, 13242. Whyman, G.; Bormashenko, E.; Stein, T. Chem. Phys. Lett. 2008, 450, 355. Marmur, A.; Bittoun, E. Langmuir 2009, 25, 1277. Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988.
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roughness. In the experiments of Gao and McCarthy, the above condition is not satisfied; consequently, the Cassie-Baxter equation is not applicable. All the abovementioned considerations were made in the framework of classical thermodynamics, which involves the surface tensions and the Young equation for the contact angle of the drop on smooth surfaces. Because of the inapplicability of the Cassie-Baxter equation (eq 1) to small drops (nanodrops) on rough surfaces, it is reasonable to employ another tool for the treatment of such a case. In refs 10 and 11, a nanodrop on a rough surface was treated on the basis of a density functional theory,9 which provided a rigorous microscopic approach to the problem. Such statistical mechanical considerations involve only the potentials of the fluid-fluid and fluid-solid interactions and are more appropriate for the examination of inhomogeneous surfaces than the classical approach that uses spatially dependent surface tensions, quantities which, in fact, are not clearly defined. In the present paper, we use a nonlocal density functional theory (DFT) to examine nanodrops on chemically inhomogeneous surfaces which are similar to those used in the experiment of ref 1. We restrict the calculations to nanodrops because the macrodrops contain a too large number of molecules to allow us to carry out with the available computers the extensive numerical calculations required by DFT. The system under consideration is presented in Figure 1 and consists of a box of infinite dimension in the y-direction (normal to the plane of the figure) and finite dimensions Lx and Lh in the x- and h-directions, respectively. The box is filled with a one-component fluid of constant average density Fav that is in contact with a chemically rough solid surface. The surface is composed of a sequence of two kinds of plates, A and B, which are made of different materials and interact differently with the fluid molecules. The plates can have, generally, different thicknesses and are infinite in the y-direction and semi-infinite (9) Tarazona, P. Phys. Rev. A 1985, 31, 2672. (10) Berim, G. O.; Ruckenstein, E. J. Chem. Phys. 2008, 129, 014708. (11) Berim, G. O.; Ruckenstein, E. J. Chem. Phys. 2008, 129, 114709.
Published on Web 05/06/2009
DOI: 10.1021/la900848e
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Figure 1. Schematic representation of a drop on a chemically rough surface in the x-h plane. Plates A and B are infinite in the y-direction, semi-infinite in the h-direction and have different values of the energy interaction parameters ɛfs,A and ɛfs,B (the y-axis is normal to the plane of the figure).
in the h-direction. Consequently, the fluid density distribution (FDD) F(r) is uniform in the y-direction and nonuniform in the x- and h-directions, that is, F(r) F(x,h). Periodic boundary conditions are employed in the x-direction, and the upper boundary of the box is treated as a hard wall without attractive interaction with the fluid molecules. The suitability of such a two-dimensional model for a three-dimensional drop was examined in ref 7. The interaction between the fluid molecules is described by the Lennard-Jones potential φff(r) = 4ɛff[(σff/r)12 - (σff/r)6] for r g σff and φff(r) = ¥ for r < σff, where r |r - r0 |; the coordinates r and r0 provide the locations of the fluid molecules, and σff and ɛff are the fluid hard core diameter and energy parameter, respectively. The same kind of potential is selected for the fluidsolid interactions, with the energy parameter ɛff and hard core diameter σff in φff(r) being replaced by ɛfs and σfs, respectively, and r being the distance between a molecule of fluid and that of the solid wall. For such a system, the Euler-Lagrange equation for the FDD F(x,h) has the form log½Λ3 Fðx, hÞ -Qðx, hÞ ¼
λ kB T
ð2Þ
where the first term represents the ideal gas contribution, Q(x,h) is a functional of F(x,h), Λ = hP/(2πmkBT)1/2 is the thermal de Broglie wavelength, kB and hP are the Boltzmann and Planck constants, respectively, T is the absolute temperature, m is the mass of a fluid molecule, and λ is a Lagrange multiplier arising because of the constraint of fixed average density of the fluid. The functional Q(x,h) accounts for a reference hard sphere system of fluid molecules, the fluid-solid interactions, and the fluidfluid interactions treated in the mean-field approximation. The explicit form of Q(x,h) and details of the derivation of eq 2 can be found in ref 10. The constraint of fixed average density leads to Fav ¼
1 V
Z V
dr FðrÞ
ð3Þ
where V is the fixed volume of the system and provides the following expression for λ "
1 λ ¼ -kB T log Fav VΛ3 9286 DOI: 10.1021/la900848e
#
Z dr e V
Qðx, hÞ
ð4Þ
Figure 2. Example of the fluid density distribution in the system (only its left-hand side is presented). The lighter areas correspond to higher fluid densities. The numbers indicate the values of the dimensionless fluid density Fσ3ff. The white line represents the drop profile extracted for Fdivσff3 = 0.383 as described in the text.
By eliminating λ between eqs 2 and 4, one obtains an integral equation for the FDD F(x,h), which can be solved by iterations. The details of the iteration procedure are provided in ref 10. In Figure 2, an example of the fluid density distribution obtained as a solution of the Euler-Lagrange equation is presented. The lighter areas correspond to higher fluid densities. Because of the absence of a sharp vapor-liquid interface, the contact angle for nanodrops is not clearly defined. To extract it from the given FDD, we used an approach employed in refs 10 and 12. First, the drop profile h = h(x) is determined inside the vapor-liquid interface as the line passing through the points where the local density F(x,h) has a constant value Fdiv. The latter quantity is considered equal to the fluid density for an equimolar dividing surface at some distance h0 from the solid surface (see refs 12 and 13) In the present calculations, this distance was selected to be 5σff. The determined profile is presented in Figure 2 as a white solid line. After the profile is determined, its part for h > 2.5σff was approximated by a circle and extended up to the solid surface. The contact angle was taken equal to that made by the circle with the solid surface. (Note that such a definition of the contact angle is similar to that used in the goniometric measurements of the contact angles of macroscopic drops.14) The intersections of the circle with the solid surface provided the locations of the contact lines, and the coordinates of intersections were used to evaluate the fractions of the contact area beneath the drop occupied by the A- and B-plates. To estimate the importance of the liquid-solid interfacial area inside the drop to the value of the contact angle, several specific cases will be examined below. In all of them, the leading edges of the drop are considered to be located on A-plates, at a fixed interaction energy parameter between them and the fluid. As for the B-plates, either we change the fraction of the surface area beneath the drop occupied by them by changing the width of the B-plates or we change the energy parameter ɛfs,B characterizing the interaction of the B-plates with the fluid molecules. In this way, situations are generated in which the conditions in the vicinity of the contact line are unchanged and those in the remaining solid liquid interface beneath the drop are varied. In all the considered cases, argon was selected as the fluid with the parameters of the Lennard-Jones interaction potential ɛff/kB = 119.76 K, σff = 3.405 A˚. For both kinds of plates, the hard core diameters and the number densities were taken the same (σfs = 3.727 A˚, Fs = 1.91 1028 m-3). The B-plates differ from 6 ɛfs,A only. More the A-plates by the energy parameter ɛfs,B ¼ details of the considered cases and the results obtained are presented in the next sections. (12) Giovambattista, N.; Debenedetti, P. G.; Rossky, P. J. J. Phys. Chem. B 2007, 111, 9581. (13) Porcheron, F.; Monson, P. A. Langmuir 2006, 22, 1595. (14) Johnson, R. E.; Dettre, R. H. Surf. Colloid Sci. 1969, 2, 85.
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Article Table 2. Contact Angles for an Ultrahydrophobic Area Inside a Hydrophobic Surface
Figure 3. Schematic representation of a hydrophilic area (B-plate) inside a hydrophobic surface (A-plates).
dB/σff
fB
Δl/σff
θ (deg)
θCB (deg)
10 16 16.4 16.8 17.2
0.389 0.620 0.640 0.660 0.680
7.8 4.8 4.6 4.3 4.0
100.6 101.6 102.0 102.0 103.6
120.0 133.6 134.9 136.3 137.6
Table 1. Contact Angles for a Hydrophilic Area Inside a Hydrophobic Surface dB/σff
fB
Δl/σff
θ (deg)
θCB (deg)
6 10 14 18 22 26
0.239 0.401 0.560 0.710 0.821 0.883
9.5 7.4 5.4 3.7 2.4 1.8
100.6 100.6 100.6 99.2 94.2 85.6
84.3 73.1 61.4 49.0 38.0 30.5
2. Case 1: Hydrophilic Area Inside a Hydrophobic Surface In this case, presented in Figure 3, the solid-liquid interface consists of three parts, with a plate B located completely beneath the drop. The energy parameter ɛfs,A of the A-plates is selected to be ɛfs,A/kB = 91.8 K. The surface of the A-plate is slightly hydrophobic with a contact angle of θA = 100.6. The energy parameter ɛfs,B of the B-plates is selected to be ɛfs,B/kB = 183.6 K and corresponds to a hydrophilic surface with a wetting angle θB = 0. The thickness dB of the B-plate was changed from 6σff to 26σff. For T = 85 K and Favσff3 = 0.16, the width of the contact area of the drop was varied from 25σff for the smallest thickness of the B-plate to 29σff for the largest one. The contact angles calculated on the basis of DFT are listed in Table 1 in the fourth column. The third column provides the distance Δl between the leading edge of the drop and the nearest boundary of a B-plate (see Figure 3). The contact angle θ which the drop makes with the surface is constant (θ = 100.6) for dB e 14σff (fB e 0.560, Δl > 5σff) and decreases to 85.6 with increasing dB up to dB = 26σff (fB = 0.88). For comparison purposes, the contact angle θCB calculated with eq 1 is provided in the last column of Table 1. This angle changes from 84.3 for dB = 6σff to 30.5 for dB = 26σff.
3. Case 2: Ultrahydrophobic Area Inside a Hydrophobic Surface In this case, the surface has the same structure as in the previous one (see Figure 3), but the B-plate is ultrahydrophobic with ɛfs,B/kB = 0 K and θB = 180. (Such a case can be imagined by assuming the presence of trapped air instead of a B-plate). The A-plates are as those considered in section 2. The width of the contact area of the drop with the solid surface is about 26σff. The results obtained are listed in Table 2. A drop, containing a B-plate beneath it, can exist only if the width dB of the B-plate is smaller than 17.2σff. For dB > 17.2σff, the EulerLagrange equation, eq 2, does not have a stable or metastable solution with the leading edge located on A-plates. The contact angle θ which the drop makes with the surface varies between 100.6 and 103.6 as dB increases from 0 to 17.2σff. (The contact angle θCB calculated with eq 1 changes from 100.6 to 137.6.) A visible change of the contact angle begins when the distance Δl between the leading edge and the nearest boundary of a B-plate becomes smaller than 5σff. Langmuir 2009, 25(16), 9285–9289
Figure 4. Two characteristic states of a nanodrop on a surface with periodic chemical roughness. The drop D1 (D2) is symmetrical with respect to the vertical plane passing through the middle of plate A (B).
4. Case 3: A Surface with Periodic Chemical Roughness In this case, the drop is located on a surface composed of plates A and B of equal thicknesses d, with the leading edges of the drop being located on the A-plates (Figure 4). The energy parameter ɛfs,A = 153 K and ɛfs,B varies from ɛfs,B = 0 K to ɛfs,B = 0.8ɛfs,A. The thickness of the plates is d = 3σff. The surface generates a periodic potential with a wavelength equal to 2d (6σff). The details regarding the calculations of the interaction potential can be found in ref 10. As shown previously,11 a nanodrop on the surface presented in Figure 4 can be symmetrical with respect to the middle of plate A (drop D1, Figure 4a) or plate B (drop D2, Figure 4b). For the same fluid average density in the system, one of these drops is stable (i.e., corresponds to a global minimum of the free energy) and the other one is metastable (corresponds to a local minimum of the free energy). Depending on specific conditions (chemical nature and sizes of the plates, temperature, fluid average density), the leading edges of the drop can be located on either A- or B-plates. Only those drops which have the leading edges on A-plates were selected by us for analysis. In those cases, the variation of ɛfs,B changes mainly the contribution to the free energy of that part of the area beneath the drop which is at some distance from the leading edges, with the contribution of the area near the leading edges remaining almost the same. Changing the energy parameter ɛfs,B of the liquid-solid interactions for the B-plates but keeping that of A-plates fixed, one can model the above situations. To check the role of the wavelength of the inhomogeneity, one can also change the thickness d of the plates. In Table 3, the contact angles θ extracted from the calculated fluid density distribution are listed for drops D1 and D2 together with the contact angle θCB calculated with eq 1. The contact angle θA = 49.3 and the values of θB and fB are provided in the second and third columns of Table 3, respectively. The first conclusion, which can be drawn from the data presented in Table 3, is that the change of the strength of interactions inside the internal area beneath the drop does affect the contact angle θ of nanodrops on rough surfaces. Such an effect occurs for both drops D1 and D2. The contact angle θ for DOI: 10.1021/la900848e
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Table 3. Contact Angles for a Surface with Periodic Chemical Roughness (dB = 3σff) ɛfs,B/ɛfs,A
θB (deg)
Δl/σff
fB
θ (deg)
θCB (deg)
1.5 0.1 0.8
66.7 109.8 117.6
65.3 91.4 95.1
0.2 1.2 2.5
80.7 90.3 103.3
75.8 90.6 105.8
drop D1 0.8 0.2 0.0
79.3 180 180
0.500 0.409 0.455 drop D2
0.6 0.4 0.2
107.2 136.8 180
0.43 0.48 0.56
Table 4. Contact Angles for a Surface with Periodic Chemical Roughness (dB = 5σff) ɛfs,B/ɛfs,A
θB (deg)
Δl/σff
fB
θ (deg)
θCB (deg)
drop D1 0.8 0.6 0.4 0.2
79.3 107.2 136.8 180
0.412 0.441 0.468 0.504
0.2 0.0
180 180
0.661 0.637
0.35 1.2 1.8 2.5
65.0 71.5 77.2 84.4
62.6 76.5 89.7 100.4
109.3 114.5
123.9 127.1
drop D2 0.1 0.6
drop D1 changes from 66.7 to 117.6 when δ ɛfs,B/ɛfs,A varies from δ = 0 to δ = 0.8. For the drop D2, θ varies from 80.7 to 103.3 when δ varies from δ = 0.6 to δ = 0.2. The change of the contact angle takes place along with that of the distance Δl between the leading edge of the drop (located on an A-plate) and the closest boundary between the A- and B-plates (see Table 3). A similar analysis was carried out for a surface with a larger width of the plates, d = 5σff. The corresponding results are listed in Table 4. Drop D1 is stable for δ = 0.8 but metastable at other values of δ, whereas drop D2 is stable at all listed values of δ.
5. Discussion The solid lines in Figure 5 present the contact angles obtained via DFT as functions of the fraction of B-plate beneath the drop for the cases considered in sections 2 and 3 (cases 1 and 2, respectively). For fB < 0.6, the contact angle calculated via DFT is almost independent of fB and remains the same independent of the nature of the B-plate (hydrophilic or hydrophobic) present beneath the drop. The distance Δl between the leading edges of the drop and the B-plates is in this case larger than 5σff. For the hydrophilic B-plate, the results are in agreement with the experimental observations of Gao and McCarthy1 which have shown that for a hydrophilic spot in a hydrophobic field (analogous to case 1 in our considerations) the contact angle is almost independent of fB for fB < 0.64. (Note that in their experiments surfaces with fB > 0.64 and hydrophobic surfaces have not been considered.) For fB > 0.6 (Δl < 5σff), our DFT results for cases 1 and 2 exhibit a change in the contact angle. One of the reasons for that change is that for fB > 0.6 the distance between the leading edges of the drop and the B-plates becomes very small, and hence, the local conditions near the leading edges are considerably changed compared to those on an uniform surface. From a microscopic point of view, those conditions are determined solely by the potential Ufs(x,h) of the fluid-solid interactions. In Figure 6, 9288 DOI: 10.1021/la900848e
Figure 5. Dependence of the contact angles calculated via DFT on the fraction of the surface of B-type beneath the drop for a hydrophilic area inside a hydrophobic surface (case 1) and for an ultrahydrophobic area inside a hydrophobic surface (case 2).
the change of this potential in the vicinity of the boundary between the A- and B-plates is plotted as a function of x for semi-infinite A-plates and dB = 10σff. The plateaus at the right and left had sides of the B-plate and one in the middle of the B-plate provide potentials Ufs(x, h) approximately equal to those for uniform infinite A- and B-plates, respectively. The transition between those values occurs within a distance of about 8σff and begins at a distance of 5σff when approaching the boundary between A- and B-plates from the A-plate side. Just at the latter distance begins the change of the calculated contact angle (see Tables 1 and 2). The results for cases 1 and 2 support the suggestion that the contact angle is determined by conditions near the leading edges of the drop and that the remaining area of the solid-liquid interface is irrelevant. The results obtained for case 3 seem to be in contradiction with the above conclusions. Indeed, whereas the fraction of B-plates beneath the drop is approximately equal to fB = 0.5, the change in the energy parameter ɛfs,B from ɛfs,B = 0 to ɛfs,B = 0.8ɛfs,A results in large changes of the contact angle θ. However, this can be explained if one takes into account that the leading edges of the drop which are located on A-plates are very close to the neighboring B-plates (see Tables 3 and 4) and, therefore, are strongly affected by the interactions with those plates. Even a microscopic displacement of the order of one molecular diameter of the leading edge due to changes in ɛfs,B can provide large changes in the contact angle. Note that, from a thermodynamic point of view, the role of the local conditions in the value of the contact angle was examined in detail by Marmur and Bittoun.7 Along with a qualitative analysis of the origin of the contact angle changes with changing of the location of the contact line, it was shown that a drop of a fixed (macroscopic) volume on a rough surface can have multiple metastable states possessing different contact angles (contact angle hysteresis). For the nanodrops considered in the present paper, which have very small volumes, either the metastable states are absent or their number is small. Among the surfaces examined in this paper, only those composed of periodic sequences of A- and B-plates (case 3) provided a metastable drop in addition to the stable one. For this reason, it is not possible to estimate the contact angle hysteresis from the analysis of the metastable states. However, the contact angle hysteresis for nanodrops can be detected in another way. Let us consider a drop on a vertical surface identical to that examined in section 4 (case 3) in the presence of gravity (see Figure 7a). As was shown previously in Langmuir 2009, 25(16), 9285–9289
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Figure 6. Potential of the fluid-solid interaction as function of x for the surface considered in section 3 (case 1) with dB = 10σff, calculated at a distance from the solid surface h = 0.5σff.
Figure 7. Schematic representation of a nanodrop of mass m on a chemically rough vertical surface in the presence of gravity. The considered surfaces correspond to case 3 [panel (a)] and case 1 [panel (b)].
Langmuir 2009, 25(16), 9285–9289
ref 11, the gravitational acceleration a has to be taken enormously large (a ∼ 1010 m/s2) to affect the shape of the nanodrop. In this case, the angles θ1 and θ2 can be considered as estimates for the advanced and receding contact angles of a nanodrop on a rough surface. Increasing the acceleration a up to the critical value ac at which the drop loses its mechanical equilibrium, one can find the largest (smallest) values of θ1 (θ2) which provide the contact angle hysteresis. For example, for the case dB = dA = 3σff examined in ref 11, the contact angle changes between 76.5 and 128.4 for drop D1 and between 102.6 and 136.2 for drop D2. Similar calculations performed for a surface corresponding to case 1 (see Figure 7b) with dB = 6σff provide a range of values between 58.7 and 117.6. Note that the value of the contact angle θ2 for the surface presented in Figure 7b depends on the width dB of plate B, that is, on the fraction of surface area beneath the drop occupied by the B-plate. For example, for dB = 6σff (fB = 0.23) and dB = 18σff (fB = 0.64), this angle has the values 93.5 and 81.7, respectively, for a gravitational acceleration a = 7.3 1010 m/s2.
DOI: 10.1021/la900848e
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