pubs.acs.org/Langmuir © 2009 American Chemical Society
Effects of Capillary Condensation in Adhesion between Rough Surfaces Jizeng Wang, Jin Qian, and Huajian Gao* Division of Engineering, Brown University, Providence, Rhode Island 02912 Received February 5, 2009. Revised Manuscript Received June 8, 2009 Experiments on the effects of humidity in adhesion between rough surfaces have shown that the adhesion energy remains constant below a critical relative humidity (RHcr) and then abruptly jumps to a higher value at RHcr before approaching its upper limit at 100% relative humidity. A model based on a hierarchical rough surface topography is proposed, which quantitatively explains the experimental observations and predicts two threshold RH values, RHcr and RHdry, which define three adhesion regimes: (1) RHRHcr, water menisci freely form and spread along the interface between the rough surfaces.
1. Introduction Most surfaces are not smooth when viewed near the atomic scale. Capillary effects on rough surfaces play critically important roles in the fields of nanolithography,1 nanotribology,2 microelectromechanical systems,3 biochemistry4 and colloidal physics.5 Recently, DelRio et al.6 performed a set of microcantilever experiments to measure the adhesion energy between micromachined surfaces as a function of the surface roughness (rms) and relative humidity (RH). Here rms is defined as the root-meansquare deviation of the surface from its average height within the sampling area. The results of DelRio et al. for surface roughness from 2.6 to 10.3 nm showed that the adhesion energy remains constant below a critical relative humidity (RHcr) and then jumps to a higher value at RHcr before approaching the upper limit 2γLV cos θ, where γLV = 0.073 N/m is the liquid-vapor surface energy, and θ is the contact angle between the meniscus and the surface. These experimental findings are critically important to the understanding of the effects of capillary condensation in adhesion between rough surfaces. However, there are currently no theoretical explanations why the adhesion energy would remain constant before it abruptly jumps to a higher value at the critical humidity RHcr. There is also no theoretical relationship between RHcr and the surface roughness. Here we show that these interesting results can be explained from the mechanics of wet adhesion taking into account the hierarchical structure of surface roughness. We proceed by considering different regimes of RH in the following.
2. Regime RH < 23%: No Stable Water Bridges between Rough Surfaces Wet adhesion between rough surfaces is characterized by the formation of water menisci between asperities due to capillary condensation, as shown in Figure 1. The pressure *Corresponding author. Tel: þ1 401 863-2626. Fax: þ1 401 863-9025. E-mail address:
[email protected]. (1) Piner, R. D.; Zhu, J.; Xu, F.; Hong, S.; Mirkin, C. A. Science 1999, 283, 661. (2) Adams, M. J.; Briscoe, B. J.; Law, J. Y.; Luckham, P. F.; Williams, D. R. Langmuir 2001, 17, 6953. (3) Syms, R. R. A.; Yeatman, E. M.; Bright, V. M.; Whitesides, G. M. Microelectromech. Syst. 2003, 12, 387. (4) Shao, Z. F.; Mou, J.; Czajkowsky, D. M.; Yang, J.; Yuan, J. Y. Adv. Phys. 1996, 45, 1. (5) de Lazzer, A.; Dreyer, M.; Rath, H. J. Langmuir 1999, 15, 4551. (6) DelRio, F. W.; Dunn, M. L.; Phinney, L. M.; Bourdon, C. J.; de Boer, M. P. Appl. Phys. Lett. 2007, 90, 163104.
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difference between inside and outside of a meniscus is given by the Young-Laplace equation:7 ΔP ¼ -
γLV RK
ð1Þ
where RK is the mean radius of curvature. At thermodynamic equilibrium, the relative vapor pressure is described by the Kelvin equation:7 RT γ lnðRHÞ ¼ - LV ð2Þ Vm RK where R is the gas constant, T is the temperature, and Vm is the molar volume. Combining eqs 1 and 2 yields Vm ΔP ð3Þ RH ¼ exp RT which indicates that the water meniscus is subjected to a negative pressure, and cavitation failure may occur as the negative pressure becomes sufficiently large. Theories on the tensile properties of liquids predict that the theoretical limit of the negative pressure of water is about -140 MPa from homogeneous nucleation theory8,9 and about -200 MPa from spinodal breakdown theory.10-12 When the negative pressure approaches -200 MPa, the Laplace equation would indicate that the radius of curvature has reached 0.37 nm, the size of a single water molecule. This fact also supports that there should be a theoretical negative pressure of around -200 MPa for water. Despite many experimental investigations since the 1840s,13-15 there have been no experimental reports of laboratory samples of water stretching to a negative pressure comparable to the theoretical limit of -200 MPa. A recent experiment16 has shown that the negative pressure (7) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1985. (8) Fisher, J. C. J. Appl. Phys. 1948, 19, 1063. (9) Blander, M.; Katz, J. Am. Inst. Chem. Eng. J. 1975, 21, 883. (10) Trevena, D. H. Contemp. Phys. 1976, 17, 109. (11) Trevena, D. H. Cavatation and Tension in Liquids; Adam Hilger: Bristol, England, 1978. (12) Speedy, R. J. J. Phys. Chem. 1982, 86, 982. (13) Berthelot, M. Ann. Chem. 1850, 30, 232. (14) Apfel, R. J. Acoust. Soc. Am. 1971, 49, 145. (15) Henderson, S.; Speedy, R. J. J. Phy. E 1980, 13, 778. (16) Yang, S. H.; Nosonovsky, M.; Zhang, H.; Chung, K. H. Chem. Phys. Lett. 2008, 451, 88.
Published on Web 08/28/2009
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distinguish between three regimes as shown in Figure 1. Figure 1a shows a schematic of the first regime, RH < RHdry, in which there are no water bridges. Figure 1b illustrates the second regime in which water bridges can form along the edges of the initial solid-solid contact regions and can even partially spread into such region until pinned by some subscale asperities. We introduce c as the area fraction of contact before capillary condensation occurs, i.e., the total area of solid-solid contact divided by the total area of the interface, and cw as the area fraction occupied by water bridges during the detachment process. In the regime RHcr > RH > RHdry, we assume that the water bridges can form along the edges of the initial solid-solid contact regions and can even spread partially into these regions. In this process, the surface energy change per unit area in separating the two solids can be expressed by the Dupre equation7 Δγ ¼ ðγ1L þγ2L -γ12 Þcw þðγ1V þγ2V -γ12 Þðc -cw Þ
Figure 1. Schematic illustrations of different regimes of RH for adhesion between rough surfaces. (a) In the first regime, RH < RHdry, no stable water bridges can form; (b) In the second regime, RHdry < RH < RHcr, water bridges can form along the edge of and partially spread into a dry contact region until pinned by some subscale defects; (c) In the third regime, RH > RHcr, water menisci can freely spread along the interface to form water bridges between surface asperities.
can reach -160 MPa, close to the theoretical limit of -200 MPa. Setting ΔP = -200 MPa in eq 3 yields a theoretical lower bound for the RHdry of RHdry = 23%, below which no stable water bridges can form, in which case only dry adhesion exists. We are aware that capillary forces at RH values as low as 15% were detected for silicon tips in atomic force microscopy (AFM) experiments.17 However, these experiments were not performed under thermodynamic equilibrium17 and the Kelvin equation does not apply in such situations.
3. Regime RHcr > RH > RHdry: Water Bridges Confined to Initial Solid-Solid Contact Regions DelRio et al.6 found that the adhesion energy of their rough surface samples remains constant prior to an abrupt jump at a critical relative humidity RH = RHcr. More recently, Kim et al.18 confirmed this finding but reported that the pull-off force varies significantly with humidity in this regime. Prior to these experimental observations, Qian and Gao19 theoretically investigated the scaling effects of wet adhesion mediated by a liquid bridge between a fiber and a solid surface and found that the pull-off force depends on the size of the liquid bridge, while the adhesion energy remains constant. The results by Qian and Gao were obtained based on the mechanical equilibrium of a liquid bridge with an externally applied pull-off force without considering thermodynamic equilibrium of the water-vapor interface. To extend the result of Qian and Gao19 to the case of thermodynamic equilibrium, we consider a rough surface adhering to a flat substrate subjected to capillary condensation. Each asperity on the rough surface is assumed to be structured with multiscale roughness, i.e., subscale asperities located on top of larger asperities. During the pulling process, the Kelvin radius remains constant and so does the Laplace pressure. We (17) Sirghi, L.; Szoszkiewicz, R.; Riedo, E. Langmuir 2006, 22, 1093. (18) Kim, D. I.; Grobelny, J.; Pradeep, N.; Cook, R. F. Langmuir 2008, 24, 1873. (19) Qian, J.; Gao, H. Acta Biomater. 2006, 2, 51.
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ð4Þ
where subscripts 1 and 2 denote the two solids, subscript L represents the liquid, V the vapor and γij is the relevant surface energy (e.g., γ1L stands for the surface energy of the interface between solid 1 and liquid). The attractive force per unit area contributed from the meniscus can be written as19 F≈ -cw ΔP
ð5Þ
where we have assumed that the contact angle θ is very small. Since the maximum stable meniscus height is limited by 2RK cos θ, simple multiplication of the meniscus height and the attractive force per unit area, F, gives the work done per unit area by meniscus as Wm ¼ 2FRK cos θ≈2cw γLV cos θ
ð6Þ
where we have applied the Laplace equation, and the wetted area is assumed to remain unchanged due to the pinning effect of subscale roughness (see further analysis below). Inserting the Young equation for the newly formed solidvapor-liquid interfaces γ1L ¼ γ1V -γLV cos θ,
γ2L ¼ γ2V -γLV cos θ
ð7Þ
into eq 4, we have the surface energy change as Δγ ¼ -2γLV cw cos θþðγ1V þγ2V -γ12 Þc
ð8Þ
The total work done to separate the contact of unit area in the presence of menisci is then Wt ¼ Wm þΔγ ¼ ðγ1V þγ2V -γ12 Þc
ð9Þ
which is the same as that in the case of complete dry adhesion. This explains why the adhesion energy is independent of RH, while the capillary force itself depends sensitively on the liquid bridge according to eq 5. Similar conclusions have been reached by Kim et al.18 We note that the work by Kim et al.18 makes two implicit assumptions about the water bridges: (1) they are confined to the initial solid-solid contact regions, and (2) they can completely penetrate inward into these regions, i.e., c = cw. We find these assumptions may be unnecessarily strong and our analysis takes into account the possibility that the water bridges only partically penetrate into the initial solid-solid contact regions during the detachment process, i.e., c g cw. Instead, we hypothesize that the typical Langmuir 2009, 25(19), 11727–11731
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hierarchical structure of a rough surface may actually prevent the water menisci from continuous expansion into the initial solidsolid contact regions. To substantiate the above assumption and also explain the experimental observation that the adhesion energy is independent of RH for a wide range of RH (from RHdry to RHcr) until there is a sudden jump, let us consider a two-level sinusoidal rough surface. We define the interfacial separation as hðxÞ ¼ h0 ½1þsinð2πx=LÞþh1 ½1þsinðnπx=LÞ
ð10Þ
where L is the length of each period, R h0 . h1. For this surface, the rms roughness Ris Hrms {(1/a) a0[h(x) - HAVG]2dx}1/2, where HAVG = (1/a) a0h(x)dx is the average separation, and a is the width of the interface. It follows from eq 10 that Hrms ≈ 0.7h0. We assume that the contact angles of both surfaces are zero. At a given RH, a meniscus can form between the rough surface and the flat surface only when its two edges are tangent to the upper and lower surfaces. Taking h0 = 10 nm, h1 = 0.67 nm, n = 42, and L = 10 nm, Figure 2a shows the configuration of the rough surface, and Figure 2b gives the necessary RH value for meniscus with upper edge location at (x, h(x)). It can be seen from Figure 2b that the edge coordinate of the meniscus increases smoothly with RH when h1 = 0. However, even for h1 = 0.67 nm, the change is no longer smooth; instead the solution becomes fragmented into different segments with large gaps between the segments. For segment A in Figure 2b, the meniscus size exhibits almost no change, even though RH varies from 0 to around 75%. This indicates that an energy barrier must be overcome in order for the meniscus to jump from a state on segment A to one on segment B. For the water meniscus to jump from AC to BD under constant temperature and vapor pressure (Figure 2c), we assume that there are already water monolayers on surfaces AB and CD, but water molecules need to further condense from the vapor phase into the gap region ABCD. The energy barrier for this process is estimated as the energy needed to condense water molecules into ABCD. If we define μl and μv as the chemical potentials per mole in the liquid and vapor phases, respectively, and V as the volume of ABCD with thickness t, the change in the Gibbs free energy of the system due to vapor condensation is ΔE = V(μl - μv)/Vm = -(VRT/Vm) ln(p/ps) = -(VRT/Vm) ln RH, where Vm is the molar volume, P is the partial pressure in vapor, ps is the saturation vapor pressure, and RH = p/ps is the relative humidity. It follows from the Kelvin equation (eq 2) that the energy barrier for the water meniscus to jump from AC to BD is ΔE = γLVV/ RK. Assuming a meniscus thickness of 1.0 nm in the out-of-plane direction, the energy barrier for the jump from AC to BD in Figure 2c is about 6 kBT at room temperature. A possible explanation of the experimental observation that the adhesion energy is independent of RH for a wide range of RH (from RHdry to RHcr) until a sudden jump may thus lie with the fact that the surface is not smooth near the atomic scale. The subscale roughness can effectively pin the water meniscus until a critical RH, as shown in Figures 1b and 2b. The water menisci formed in the second regime can only spread inward into smaller surface gaps within the adhesion regions but not outward into open gaps.
4. RH = RHcr: Initial Jump of the Adhesion Energy Since Hrms describes the fluctuations of surface heights around its average value, it can be expected that water bridges can form between the surface and most asperities when the maximum height of a water bridge 2RK cos θ approaches Hrms. When this happens, the wetted area will have an abrupt change so that cw Langmuir 2009, 25(19), 11727–11731
Figure 2. Wet adhesion on a two-level sinusoidal rough surface. (a) Configurations of a water meniscus between a two-level sinusoidal rough surface and a flat substrate. (b) The equilibrium x-coordinate of the upper edge of the meniscus as a function of the RH. (c) An energy barrier required for the water meniscus to jump over the subscale roughness from AC to BD.
becomes much larger than the solid-solid contact area c. The critical condition for this sudden change is 2RK cos θ ¼ Hrms
ð11Þ
Applying Kelvin’s equation to eq 11 gives a critical value for the relative humidity RHcr as Vm γLV cos θ ð12Þ RHcr ¼ exp -2 RT Hrms DOI: 10.1021/la900455k
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Figure 3. The RHcr as a function of the rms roughness (Hrms).
DelRio et al.6 have performed a set of microcantilever experiments to measure the adhesion energy between micromachined surfaces as a function of RH and the rms roughness. The cantilevers have landing pad roughness ranging from 2.6 to 10.3 nm. Recently, DelRio et al.20 discussed that the conformal nature of the sacrificial oxide layer makes the rms roughness of the bottom of the cantilever to be about 2.4 nm. Assuming that lower and upper surfaces have different Gaussian random roughness profiles hl and hu, we can approximate the effective rms roughness of the contacting surfaces as the standard deviation of hl þ hu. If the rms roughness of hu is 2.4 nm and rms roughnesses of hl are 2.6, 4.4, 5.6, and 10.3 nm, respectively, the corresponding effective rms interfacial roughness would be 3.5, 5.0, 6.1, and 10.6 nm, respectively. In addition, adsorbed water layers may appear on the lower and upper surfaces. Taking the water layer thickness as two monolayers of water of thickness of about 0.56 nm,20 the rms heights can be further adjusted to 2.3, 3.8, 4.9, and 9.4 nm, respectively. Figure 3 shows that the RHcr values predicted by eq 12 with θ = 0 are in reasonable agreement with the experimental observations by DelRio et al.1 for cantilevers with landing pad roughness ranging from 2.6 to 10.3 nm, even when the roughness of the lower surface and/or water monolayers have been taken into account.
5. Regime RH > RHcr: Water Menisci Freely Form and Spread Extensively along the Interface When RH exceeds RHcr, as shown in Figure 1c, we expect that water menisci can no longer be pinned by subscale roughness and can freely spread both inward and outward into the open gaps along the interface; in the limit of RH approaching 100%, cw would approach 1. In this regime, the surface energy change can be rewritten as Δγ ¼ ðγ1L þγ2L -γ12 Þc
ð13Þ
where not only all of the solid-solid contact regions will be occupied by water, but the water bridges can also spread into some of the open gaps along the interface. The work done by water menisci is then Wm ≈2cw γLV cos θ
ð14Þ
where cw is now much larger than c. The total work done per unit area becomes Wt ¼ ΔγþWm ¼ ðγ1V þγ2V -γ12 Þcþ2cw0 γLV cos θ
ð15Þ
(20) DelRio, F. W.; Dunn, M. L.; de Boer, M. P. Scr. Mater. 2008, 59, 916.
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Figure 4. Comparison of the adhesion energy from the experiments by DelRio et al.6 and the prediction for Hrms = 2.6, 4.4, 5.6, and 10.3 nm, respectively.
where we have used eq 7, and c0w = cw - c is the additional adhesion area created by water bridges outside the initial solidsolid contact regions. Although it seems difficult to determine c0w theoretically, we find the expression ( 0, RH < RHcr ð16Þ cw0 ¼ a1 RH12 þa0 , RHgRHcr
a1 ¼ ½1 -Wd =ð2γLV cos θÞ=ð1 -RH12 cr Þ 12 a0 ¼ ½ -RH12 cr þWd =ð2γLV cos θÞ=ð1 -RHcr Þ
where Wd = (γlV þ γ2V - γ12)c; a1 and a2 are found from asymptotical matching with the limiting behaviors c0w= 0 when RH = 0 and c0w ≈ 1 when RH = 100%. Figure 4 shows that inserting the above expression for c0w into eq 15 gives reasonable prediction for the adhesion energy as a function of RH for different surface roughnesses.
6. Conclusions We have developed a model based on a hierarchical rough surface topography to quantitatively explain relevant experimental observations on the effects of capillary condensation in adhesion between rough surfaces. Our model predicts two threshold RH values, RHcr and RHdry, which define three distinct adhesion regimes: (1) RH< RHdry, there are no stable water bridges and adhesion is governed by dry adhesion; (2) RHdry< RH< RHcr, water bridges are confined to initial solid-solid contact regions, and the adhesion energy is independent of the wetting area; (3) RH > RHcr, water menisci freely form and spread into some of the open gaps along the interface, and the adhesion energy depends on the wetting area. In the second regime, we envision that the water bridges are nucleated only along the edges of the initially solid-solid contact regions and cannot spread outward into the open gaps along the interface, while taking into account the possibility that the water menisci may partially spread into the initial solid-solid contact regions as the surfaces become detached. At the critical humidity RHcr, both these restrictions are removed, and substantial water bridges form along the interface, and the area of dry adhesion becomes small compared to that of wet adhesion. In other words, as long as the area of wet adhesion is contained within the area of dry adhesion Langmuir 2009, 25(19), 11727–11731
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and unable to spread freely into the open gaps along the interface, the adhesion energy is not affected by the capillary effect. We have shown that subscale asperities can pose significant energy barriers to the nucleation of large water bridges along the interface.
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Acknowledgment. The authors are grateful to Prof. M. L. Dunn of University of Colorado for thoughtful and inspirational comments during the preparation of this work.
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