Wetting of Silicon Wafers by n-Alkanes - ACS Publications - American

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Wetting of Silicon Wafers by n-Alkanes B. M. Law,*,† A. Mukhopadhyay,‡ J. R. Henderson,§ and J. Y. Wang| Condensed Matter Laboratory, Department of Physics, Kansas State University, Manhattan, Kansas 66506-2601, Department of Physics & Astronomy, Wayne State University, Detroit, Michigan 48201, Department of Physics & Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom, and Max-Planck-Institut fu¨ r Metallforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany Received June 9, 2003. In Final Form: July 10, 2003 We examine the wetting behavior of various n-alkanes on both oxide-coated and silane-coated Si wafers. n-Hexane and n-heptane completely wet a silane-coated Si wafer while n-octane exhibits a wetting transition at Tw = 60 °C. In the complete wetting region, the wetting layer thicknesses as a function of temperature and height are markedly thinner than that which would be predicted by a long-ranged nonretarded dispersion interaction in competition with a gravitationally determined chemical potential. By contrast, the isothermal pressure measurements of Beaglehole and Christenson (J. Phys. Chem. 1992, 96, 3395) for n-pentane on an oxide-coated Si wafer display a much thicker wetting film than would be predicted when the dispersion and pressure-induced chemical potential are taken into account. We resolve these disparate experimental results as a function of temperature, height, and pressure by considering the influence of a number of medium-ranged interactions.

1. Introduction Solid surfaces are usually coated by an adsorbed film generated by the influence of surface interactions upon surrounding vapor molecules.1 Surprisingly, this subject is not well understood despite many decades of research. For certain systems,2-4 the adsorbed film thickness is well described by the chemical potential difference from bulk liquid-vapor coexistence (determined by the undersaturated vapor pressure) which is balanced by a long-range dispersion interaction. However, there are also many other systems which exhibit large deviations from such behavior.4-8 As has been emphasized previously,5,4,7 but which was less well-recognized in the older literature, the prediction that the film thickness is governed by the dispersion interaction is strictly valid only in the complete wetting regime where a bulk liquid completely wets the substrate. In the partial wetting regime, where a liquid droplet on the solid substrate possesses a finite contact angle, the film thickness at a solid surface can only be understood if the short-range interactions are taken into account. In many prior studies, the wettability of the surface was not known and the terminology “adsorbed film” encompassed both partially wettable and completely wettable surfaces. We prefer the more accurate terminology “adsorbed film” or “wetting film” to represent, respectively, a film in the partial or complete wetting regime. This terminology will be used throughout the †

Kansas State University. Wayne State University. § University of Leeds. | Max-Planck-Institut fu ¨ r Metallforschung. ‡

(1) Adamson, A. W. Physical chemistry of surfaces, 4th ed.; Wiley: New York, 1982. (2) Sabisky, E. S.; Anderson, C. H. Phys. Rev. A 1973, 7, 790. (3) Tidswell, I. M.; Rabedeau, T. A.; Pershan, P. S.; Kosowsky, S. D. Phys. Rev. Lett. 1991, 66, 2108. (4) Panella, V.; Chiarello, R.; Krim, J. Phys. Rev. Lett. 1996, 76, 3606. (5) Beaglehole, D.; Radlinska, E. Z.; Ninham, B. W.; Christenson, H. K. Phys. Rev. Lett. 1991, 66, 2084. (6) Beaglehole, D.; Christenson, H. K. J. Phys. Chem. 1992, 96, 3395. (7) Lawnik, W. H.; Goepel, U. D.; Klauk, A. K.; Findenegg, G. H. Langmuir 1995, 11, 3075. (8) Batchelder, D. N.; Cheng, Y. L.; Evans, S. D.; Henderson, J. R. Mol. Phys. 2000, 12, 807.

remainder of this paper. It is important to distinguish between the partial and complete wetting regimes; hence, a profitable area to study adsorbed/wetting films is in the vicinity of a wetting transition where both regimes can be examined on the same substrate by a simple change in the temperature. In recent work we examined the behavior of the contact angle,9,10 contact angle hysteresis,9,10 and line tension9 for n-octane and 1-octene droplets upon a silane-coated silicon wafer in the vicinity of a first-order wetting transition. In this paper, we extend this work to an examination of the adsorbed/wetting film thickness for n-alkanes on a silanated silicon wafer in the vicinity of a first-order wetting transition. This is a particularly appropriate subject to study at this time with the advent of the “nanotechnology revolution”. Silicon wafers coated with either their native oxide or a self-assembled silane monolayer are frequently the surfaces of choice in this revolution because of their molecular smoothness, inertness, ready availability, and ease with which they can be fashioned into microelectromechanical systems (MEMS) using standard semiconductor lithographic techniques. Frequently it has been assumed that these silicon surfaces are ideal where one only needs to take into account the long-ranged dispersion interaction together with an appropriate chemical potential in order to predict both the adsorption and wetting behavior of liquids upon such a surface. In this paper, we demonstrate that the temperature, pressure, and height dependence of the wetting of n-alkanes on silicon wafers can only be understood if medium-ranged interactions are also taken into account. This paper is organized as follows. In section 2 we outline the theory for long-ranged first-order wetting transitions with an emphasis on the difference between adsorbed and wetting films. From this discussion, firm theoretical predictions can be drawn concerning the behavior of the wetting film thickness when the long-range dispersion interaction controls this film thickness. In sections 3 and (9) Wang, J. Y.; Betelu, S.; Law, B. M. Phys. Rev. Lett. 1999, 83, 3677. Wang, J. Y.; Betelu, S.; Law, B. M. Phys. Rev. E 2001, 63, 031601. (10) Wang, J. Y.; Crawley, M.; Law, B. M. Langmuir 2001, 17, 2995.

10.1021/la035011v CCC: $25.00 © 2003 American Chemical Society Published on Web 08/19/2003

Wetting of Silicon Wafers by n-Alkanes

Figure 1. Interfacial potential per unit area V at temperatures T < Tw, T ) Tw, and T > Tw where each of the curves have been displaced vertically for clarity. A dry substrate at thickness L ) 0 has energy σs while an adsorbed film of thickness La has energy σsv. For thicknesses greater than the repulsive barrier at L0, the potential decays as W/L2 (where W > 0) to a thick wetting film at Lw f ∞ with energy σsl + σlv. S is the spreading coefficient (1). For T > Tw, if the system is removed from bulk liquid-vapor coexistence, the potential V follows the heavy dashed line at large L and Lw (where V is a minimum) occurs at a finite thickness.

4 we discuss, respectively, the sample preparation and experimental techniques which allow us to study adsorbed/ wetting films and droplets in the vicinity of a first-order wetting transition. Our experimental results, together with earlier isothermal pressure measurements for npentane on an oxide-coated Si wafer,6 are discussed in section 5 and shown to be in disagreement with the longranged wetting predictions of section 2. In section 6, we resolve this discrepancy between theory and experiment by considering various medium-range potentials. The paper concludes with a summary of our findings in section 7. 2. Long-Range Wetting Transitions The traditional view11 of a thermally induced first-order wetting transition is schematically depicted in Figure 1 where the interaction potential per unit area V is plotted as a function of film thickness L at various temperatures in the vicinity of the wetting transition temperature Tw. In the partial wetting region, at temperatures below Tw (Figure 1, lower curve), a global minimum in the interfacial free energy exists at an adsorption thickness La corresponding to an adsorbed film at the solid-vapor interface with surface energy σsv. This global minimum is separated from a local minimum in the free energy, at large film thicknesses (L f ∞), by an energy barrier at thickness Lo (of order nanometers9). This local minimum describes the surface energy state as the film thickness increases to large values within a macroscopic liquid droplet deposited upon the solid substrate with contact angle θ (Figure 2a). As L f ∞, the surface energy asymptotically approaches σsl + σlv (the sum of the solid-liquid and liquid-vapor surface energies) proportional to W/L2 for nonretarded dispersion interactions.11 The adsorbed film, of thickness La, and macroscopic droplet are in mechanical equilibrium where the contact angle θ is related to the surface energy (11) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827.

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Figure 2. (a) Configuration used to study the contact angle in the vicinity of a wetting transition. An interferometric microscope measures the contact angle θ of a liquid droplet with lateral radius r0 and height h situated in the vapor phase on a homogeneous and molecularly flat solid substrate. The droplet is in mechanical equilibrium with an adsorbed film of thickness La. (b) Configuration used to study the film thickness on a solid substrate in the vicinity of a wetting transition. The solid substrate is suspended vertical above a bulk liquid phase. An ellipsometer measures the film thickness L at the Brewster angle θB as a function of temperature T and height H above the liquid surface.

σij between phases i and j via Young’s equation

cos θ )

σsv - σsl S )1+ σlv σlv

(1)

For convenience we have also introduced the spreading coefficient S ) σsv - (σsl + σlv) into this equation where this quantity provides a measure of the difference in surface energies between the global and local energy minima (Figure 1). An extensive discussion of the interrelationship between Young’s equation, the adsorbed film thickness, the interaction potential, and mechanical and thermodynamic equilibrium is provided in ref 12. As the wetting transition temperature Tw is approached both θ and S approach zero. At Tw, θ and S are equal to zero and the two minima possess identical energies (Figure 1, middle curve). Above Tw (Figure 1, upper curve) the adsorption minimum is a local rather than a global minimum and the surface is completely covered by a thick wetting film. The signature of a first-order wetting transition is therefore that θ approaches zero and, in particular, cos θ approaches 1 linearly with temperature T as T f Tw.13,14 Alternatively, a first-order wetting transition is also characterized by a discontinuous jump from an adsorption thickness La to a wetting thickness Lw at Tw. The first definitive paper exhibiting these two characteristics (at a liquid-vapor surface) is perhaps ref 15. If the system (12) Sharma, A. Langmuir 1998, 14, 4915. (13) Schick, M. in Les Houches, Session XLVIII, 1988 - Liquids at Interfaces; Charvolin, J., Joanny, J. F., Zinn-Justin, J., Eds.; NorthHolland: Amsterdam, 1990. (14) Indekeu, J. O. Int. J. Mod. Phys. B 1994, 8, 309. (15) Schmidt, J. W.; Moldover, M. R. J. Chem. Phys. 1983, 79, 379.

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is removed from bulk liquid-vapor coexistence by a chemical potential µ, which differs from its value at coexistence (µcx), then the wetting layer thickness Lw will be finite rather than macroscopic (Figure 1 (upper curve), dashed line). In general, the total interaction potential per unit area is given by16

Vtot ) σsl + σlv + V(L) + hbL

(2)

where hb ) ∆n|µ - µcx|, ∆n is the difference between the saturated liquid and vapor densities; hence, ∆n ) 1/v with v a molecular volume. In eq 2 and below V(L), which contains the effects of all short- and long-range interactions present, has been shifted such that V(∞) ) 0. A system can be removed from bulk liquid-vapor coexistence by, for example, controlling the pressure p relative to the saturated vapor pressure psat where17

hb ) -

( )

kBT p log v psat

(3)

Alternatively, for isothermal systems, the pressure at height H above a liquid-vapor surface is given by17

(

p ) psat exp -

)

mgH kBT

(4)

where mgH is the gravitational potential energy for a molecule of mass m; hence from eqs 3 and 4

hb ) FgH

(5)

in a gravitational field where F ) m/v is to a good approximation the mass density of the saturated liquid. For nonretarded dispersion interactions

Vdisp(L) ) W/L2

(6)

where the subscript disp indicates that V(L) only contains the long-range dispersion interaction and the coefficient W is related to the Hamaker constant A via W ) -A/ 12π.17 The wetting layer thickness Lw, where Vtot exhibits a global minimum (T g Tw), is determined by the condition that dVtot/dL ) 0 at Lw; hence

Lw )

(

)

2Wv kBT log(psat/p)

1/3

(7)

if the vapor pressure p is the control parameter or

Lw )

2W (FgH )

1/3

(8)

if the height H above a bulk liquid-vapor surface is the control parameter. It is important to note that the predictions contained within eqs 7 and 8 are only valid provided (i) T > Tw, (ii) the nonretarded dispersion interaction is the dominant interaction, and (iii) W > 0. In the partial wetting regime, at T < Tw, an adsorbed film of thickness La will cover the substrate when in thermodynamic equilibrium; however, the shape of the potential V(L) in the vicinity of the adsorption minimum (Figure 1) is not well understood as it is controlled by short-ranged interactions. Hence, there (16) Dietrich, S. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J., Eds.; Academic: London, 1987; Vol. 12. (17) Israelachvili, J. N. Intermolecular and surface forces, 2nd ed.; Academic: London, 1992.

are no general predictions for how the adsorption thickness La will vary as a function of vapor pressure p in this region. Sabisky and Anderson2 found excellent agreement with a more extended version of eq 8, based upon the full Dzyaloshinskii-Lifshitz-Pitaevskii (DLP) theory of dispersion interactions,18 for thick liquid helium films (1.230 nm) on various solid substrates at low temperatures; in this case it is known that helium wets these substrates. However, more recent experimental studies at higher temperatures indicate that perhaps other surface interactions, besides the long-range nonretarded dispersion interaction, may on occasion play an important role in determining the wetting film thickness. Both agreement4 and disagreement4-8 with eq 7 has been found, even when it is known that the system is in the complete wetting regime. Mecke and Krim19 suggest that constrained capillary wave fluctuations at the liquid-vapor surface of the wetting film may have an important impact in determining the film thickness and in explaining the discrepant results. A recent experiment20 examined the adsorption isotherms of hydrogen on gold at low temperatures and found agreement with these constrained capillary wave ideas. In the succeeding sections we examine the wetting film thickness as a function of temperature T, height H, and pressure p for n-alkanes on either an oxide-coated or a hexadecyltrichlorosilane-coated Si wafer. n-Pentane, nhexane, and n-heptane completely wet these surfaces at all temperatures studied while n-octane exhibits a wetting transition at Tw ∼ 60 °C. In the wetting regime, we compare the measurements with the predictions of eqs 7 and 8 and discuss them in relationship to the constrained capillary wave model.19 3. Sample Preparation All the liquids used in our experiments were purchased from Aldrich Chemical Co. and possessed better than 99% purity. They were used as received without any further purification. The (100) silicon wafers were purchased from Semiconductor Processing Company. The original wafers, of thickness 3 mm, were circles of radius 3.8 cm which had been diced into quarters for ease of handling. They were polished on one side and possessed n-type phosphorus doping with a resistivity of 1-10 Ω cm. The wafers were further reduced in size to ∼1 cm × 3 cm, using a diamond saw, after first protecting the polished surface with a strong adhesive tape. Great care was taken to prepare a homogeneous and atomically smooth silane-coated silicon wafer surface using the following procedure: 1. The wafers were first rinsed and ultrasonically cleaned in chloroform for 15-20 min to dissolve any gross organic impurities. 2. They were then immersed and occasionally stirred in a solution consisting of 30% H2O2 + 70% H2SO4 for 30 min at 120 °C. This cleaning procedure removes trace organic impurities from the surface.21 The wafers were then rinsed in 18 MΩ cm distilled, deionized water and dried using a warm jet of air. 3. After complete drying, the wafers were immersed for 2 h in a silane solution consisting of 70 mL of hexadecane, 30 mL of carbon tetrachloride, and 0.5 mL of n-hexade(18) Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, L. P. Adv. Phys. 1961, 10, 165. (19) Mecke, K. R.; Krim, J. Phys. Rev. B 1996, 53, 2073. (20) Vorberg, J.; Herminghaus, S.; Mecke, K. Phys. Rev. Lett. 2001, 87, 196105. (21) Kern, W.; Puotinen, D. A. RCA Rev. 1970, 31, 187.

Wetting of Silicon Wafers by n-Alkanes

cyltrichlorosilane.22 This procedure was performed inside a closed atmospheric bag which had been dried in the preceding 2 h using phosphorus pentoxide (P2O5). The temperature of the silane solution was maintained at 18 °C using a recirculating closed-loop water system. It is believed that the trichlorosilane molecules first physisorb on the oxide-coated Si wafer where hydrolysis eventually leads to a chemically anchored silane monolayer at this surface. 4. The wafers were finally removed, rinsed in chloroform, and ultrasonicated in chloroform for 15 min. This procedure dissolves away any unanchored silane molecules from the substrate. A monolayer with a low critical surface tension (σc ) 18.7 erg/cm2) and low contact angle hysteresis (∼1-2°) was formed on the silicon wafer through this silanization procedure.22 Contact mode atomic force microscopy indicates a surface roughness of ∼0.5 nm for this monolayer over an area of 10 × 10 µm2 where this value for the surface roughness conforms with the underlying surface roughness for a bare silicon wafer.9 4. Experimental Techniques Two experimental techniques have been used to detect the presence of a first-order wetting transition for nalkanes on the same silane-coated Si wafer. Ellipsometry and microscopic interferometry were used to measure, respectively, n-alkane films and the contact angle of n-alkane droplets on this substrate. We describe each technique in turn. A phase-modulated ellipsometer23 was used to measure the thickness of vapor-adsorbed n-alkane films on the substrate. The Si wafer was suspended vertically above a bulk liquid sample inside a cylindrical glass cell where a metal clip mechanically held the wafer against a chemically resistant stainless steel plate. The temperature of the sample cell was controlled by a two-stage thermostat, constructed from concentric metallic shells which were thermally isolated from each other. These two stages possess a combined thermal stability of ∼0.1 mK over 2 h and ∼1 mK over a day, as measured by two matched precision thermistors placed at either end of the sample cell. An ac bridge arrangement, which is described in detail in ref 24, ensured that thermal gradients within the sample cell were less than 50 µK/cm. A He-Ne laser beam from the ellipsometer, which was focused to a small spot (∼0.25 mm), is reflected off the Si wafer at a height H above the liquid-vapor surface at an angle of incidence equal to the Brewster angle, θB (∼75.5°), for the bare Si wafer (Figure 2b). The signal measured by the ellipsometer Fj is very sensitive to surface structure at this angle of incidence and can be readily interpreted in terms of film thickness.25 The extreme sensitivity of ellipsometry and its application in measurements of both the optical constants and the thickness of thin liquid films are well documented in the literature.26 Ellipsometry measures the ellipticity, Fj ) Im(rp/rs)θB, which is the imaginary component of the complex reflection amplitude ratio for p- and s-polarized light at an angle of incidence equal to the Brewster angle where θB is determined by the condition that Re(rp/rs)θB ) 0. In the (22) Brzoska, J. B.; Ben Azouz, I.; Rondelez, F. Langmuir 1994, 10, 4367. (23) Beaglehole, D. Physica B & C 1980, 100B, 163. (24) Mukhopadhyay, A.; Law, B. M. Phys. Rev E 2000, 62, 5201. (25) Azzam, R. M. A.; Bashara, N. M. Ellipsometry and Polarized Light; North-Holland: Amsterdam, 1987. (26) Beaglehole, D. In Fluid Interfacial Phenomena; Croxton, C. A., Ed.; Wiley: New York, 1986.

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Figure 3. Dielectric model used to obtain the film thickness L from the ellipsometric data. This model was also used in the calculation of the dispersion coefficient, Wi, where the dielectric properties of hexadecyltrichlorosilane were assumed to be similar to those of hexadecane.

absense of any capillary wave fluctuations, Fj is related to the optical dielectric profile (z) at depth z within the interface by the Drude equation27 1/2

π (1 + 3) Fj ) λ 1 - 3

∫

[(z) - 1][(z) - 3] (z)

dz

(9)

where 1 and 3 represent the optical dielectric constants of the incident (z f -∞) and the substrate medium (z f ∞), respectively. This formula assumes that (z) is locally isotropic and that the thickness of the film is small compared with the wavelength of light, λ ) 632.8 nm. A model for (z) is required in order to determine the adsorbed/wetting film thickness L from the ellipsometric data. We assume that (z) ) , where  is the dielectric constant of the n-alkane. Silicon wafers normally possess an oxide layer of thickness d1 (∼nm) and optical dielectric constant SiO2 ()2.123); therefore a reasonable model for the adsorbed film on a silane-coated Si wafer is the triple layer model exhibited in Figure 3. With this model, for d1, d2, L , λ, eq 9 implies that Fj is proportional to the film thickness L

Fj )

{

1/2 ( - 1)( - Si) π (1 + Si) L+ λ 1 - Si 

(SiO2 - 1)(SiO2 - Si) SiO2

d1 +

}

(silane - 1)(silane - Si) d2 silane

(10) Thicker films (L + d1 + d2 J 20 nm) exhibit significant deviations from this linearity in L.25 Maxwell’s equations must then be solved numerically for the p- (rp) and s-wave (rs) complex reflection amplitudes in order to interpret the Fj data.28 We found that Fj ) 20 × 10-3 for the bare Si wafer and Fj ) 42 × 10-3 for the silane-coated Si wafer. Application of eq 10 results in an oxide layer thickness and silane monolayer thickness of d1 ) 2.0 nm and d2 ) 2.2 nm, respectively, in agreement with the silane film thickness given in ref 29. In a typical ellipsometric measurement, the temperature is set and approximately 8 h is allowed for the system to attain thermal and diffusive equilibrium. Twenty Fj and T measurements are then collected over the next 2 h. From these 20 measurements the mean and standard deviation for Fj are determined. The large real component for the Si dielectric constant (Si ) 15.07 - j0.15) provides a strong dependence for the ellipticity Fj upon the film thickness L. We measure Fj with a sensitivity of ∼10-5 corresponding to a thickness sensitivity of ∼0.001 nm averaged over the (27) Drude, P. The Theory of Optics; Dover: New York, 1959. (28) Born, M.; Wolf, E. Principles of Optics, 6th (corrected) ed.; Pergamon: Oxford, 1980. Law, B. M.; Beaglehole, D. J. Phys. D 1981, 14, 115. (29) Bain, C. D.; Troughton, E. B.; Tau, Y. T.; Evall, J.; Whitesides, G. M.; Nuzzo, R. G. J. Am. Chem. Soc. 1989, 111, 321.

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size of the incident beam spot for organic liquids on silicon. The typical standard deviation of Fj for each data point was ∼3 × 10-4; however, different temperature scans on the same liquid exhibited a reproducibility for Fj of (3 × 10-3. We therefore took this later value (corresponding to an error in L of ∼(0.3 nm) as a conservative estimate for the error in Fj. An interferometric microscope was used to measure the contact angle of droplets deposited upon the silane-coated Si wafer.9 The substrate is placed horizontally on top of a thermoelectric cooler inside an oven containing the liquid of interest. The temperature of the oven and the thermoelectric cooler are controlled by two separate temperature controllers. If the substrate surface is smooth and homogeneous, a spherical cap-shaped droplet is formed on the substrate when the thermoelectric cooler temperature is slightly lower than the oven temperature. The droplet is illuminated by a mercury lamp where the light is limited by a band-pass filter with center wavelength λ ) 551 nm and width 10 nm. For droplets of small contact angle (θ j 10°), the reflected beam from the liquidsubstrate interface interferes with the reflected beam at the liquid-vapor surface. The resultant interference pattern is magnified and projected onto a CCD camera where real time images of droplets can be viewed on a TV screen. If ri and hi are the radius and the thickness of droplet corresponding to the ith interference fringe then

hi ) iλ/2n

Figure 4. Variation of cos θ with temperature T for a n-octane droplet situated upon a hexadecyltrichlorosilane-coated Si wafer. A first-order wetting transition occurs at Tw ≈ 51 °C where cos θ ) 1.

(11)

where the integral (half-integral) values of i correspond to the bright (dark) fringes, λ is the wavelength of the illuminating light, and n is the refractive index of the liquid. The fringe pattern can be fitted using a nonlinear least-squares method to the equation

[ ( )]

hi ) h 1 -

ri r0

2

(12)

to obtain values for the center thickness (h) and base radius (r0) of the spherical cap-shaped droplet (Figure 2a). Finally, the contact angle θ is determined from

θ ) arctan(2h/r0)

(13)

This technique enables us to study drops of small contact angle ( TwL one is in the complete wetting regime with a wetting film thickness of 4-5 nm. The wetting transition temperature estimated from the film thickness measurement (TwL ) 63 °C) is considerably higher than the corresponding wetting transition determined from the contact angle measurements (Twθ ) 51 °C) where the typical variation in the wetting transition temperature for both techniques is ∼1 °C. The difference in Tw for the two techniques could potentially be caused by slight changes in the surface energy due to contamination.31 Unfortunately it is no longer possible to check out the reason for this discrepancy. Both n-hexane and n-heptane completely wet the silanecoated Si wafer because σlv < σc. Figures 4 and 5 indicate that n-octane undergoes a first-order wetting transition at Tw ∼ 60 °C where, surprisingly, the wetting layer thickness Lw ∼ 5 nm is a factor of 10 smaller than that expected from long-ranged interactions. This discrepancy is particularly apparent in Figure 6 where the wetting layer thickness for n-octane at a temperature of T ) 70 °C (>TwL) on the silane-coated Si wafer is shown as a function of height H above the liquid-vapor surface. The wetting layer thickness is approximately constant with a value of ∼5 nm and does not exhibit the H-1/3 dependence (31) An increase in σlv by as little as ∼0.14 erg/cm2 can raise the wetting transition temperature by as much as 6 °C.9

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Figure 7. Variation of the wetting film thickness Lw with vapor pressure p relative to its saturation value, psat, at 18 °C for n-pentane on an oxide-coated Si wafer (triangles) from ref 6. The dotted, dashed, and solid lines show, respectively, the influence of the dispersion (a1 ) 0), medium-range (a1 ) 1), and optimal medium and long-range model (a1 ) 0.02 J/m2) using eqs 2 and 18.

resulting from a competition between the long-range nonretarded dispersion interaction and gravity (eq 8) indicated by the dotted line in Figure 6.32 Other groups have also observed discrepancies between the long-ranged wetting predictions and experimental results; particularly when examining the wetting behavior on solid substrates. In Figure 7 we show the wetting results of Beaglehole and Christenson6 for n-pentane on an oxidecoated Si wafer (triangles) as a function of pressure p relative to the saturated vapor pressure psat at a temperature T ) 18 °C. An oxide-coated Si wafer surface is expected to be a higher energy surface than a silane-coated Si wafer surface, and n-pentane is known to completely wet this surface.33 For this experiment, the measured wetting layer thicknesses are larger than would be predicted from long-ranged interactions (dotted line).32 The temperature, height, and pressure measurements of Figures 5-7 are therefore inconsistent with the longranged wetting predictions of section 2. These results also cannot be explained by the constrained capillary wave model19 which, for various chemical potential differences µ - µcx, gives wetting layer thicknesses that either agree with or exceed the long-range wetting predictions. The experimental measurements, however, are internally consistent with each other, thus suggesting a common origin. Specifically the n-hexane and n-heptane measurements in Figure 5 exhibit a similar thermal variation which also agrees with the high-temperature n-octane data at T ∼ 70 °C. In addition, if the n-hexane and n-heptane data (Figure 5) is extrapolated to T ) 18 °C, a wetting layer thickness of Lw ∼ 8 nm would be obtained; (32) In the modeling of the dispersion interaction, the presence of the oxide and silane layers should be taken into account. This is accomplished as follows: Vdisp(L) = W2/L2 + (W1 - W2)/(L + d2)2 + (W0 - W1)/(L + d1 + d2)2 where d1 and d2 are defined in Figure 3 with W0 ≡ Wair,alkane,Si = 4.2 × 10-21J, W1 ≡ Wair,alkane,SiO2 = 1.9 × 10-22 J (Table 11.3,17), and W2 ≡ Wair,alkane,silane = 2.2 × 10-22 J. W0 is the dominant contribution due to the large Si dielectric constant. (33) Merkl, C.; Pfohl, T.; Riegler, H. Phys. Rev. Lett. 1997, 79, 4625.

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this value is consistent with the 18 °C isothermal pressure data extrapolated to psat (Figure 7). 6. Medium-Ranged Wetting To understand the experimental results depicted in Figures 5-7, forces which play prominent roles at length scales of ∼5 nm must be examined. At these length scales structural forces frequently are important where these forces arise from the “hard-sphere” packing of molecules against each other and against any surfaces that may be present. Structural forces give rise to an oscillatory pair correlation function within the bulk fluid, as well as liquid layering effects at surfaces. For thin films on a solid substrate, the medium-ranged interaction potential (with subscript m), which we define as the leading order at large L in the absence of power-law forces, takes the form34,35,36

Vm(L) ) a1 exp(-2RL) + a2(Lo/σ)-ω × cos(2πL/σ) exp(-RL) (14) where a1 > 0, σ is an effective hard sphere molecular diameter, and from Figure 4 in ref 35 we can assume a linearly increasing temperature dependence to the inverse decay lengths of the exponential factors

R)

Ro(T - Ttr) σ(Tc - Ttr)

(15)

Ro ∼ 1 with Ttr (Tc) denoting the triple (critical) temperature for the fluid. Here the oscillatory exponentially decaying term arises from hard sphere packing of molecules against a surface where the prefactor (Lo/σ)-ω originates from linear renormalization due to capillary wave fluctuations, Lo is a lateral length (parallel to the surface) over which the fluctuations are averaged and the exponent

ω=

πkBT σlvσ2

(16)

In mean field theory, where long-wave fluctuations are ignored, ω ) 0 and hence layering effects are prominent at all surfaces. Layering effects will also be important (i.e., ω ∼ 0) for high surface tension liquids at low temperatures (e.g., liquid metals36). In other circumstances, for thin liquid films on solid substrates, averaging over the capillary wave fluctuations with a macroscopic length scale Lo at the liquid-vapor surface of the thin film will mask the oscillatory behavior. For sufficiently small Lo relative to σ (e.g., for small lateral-sized computer simulations), oscillatory layering will still be evident; however, for increasing observation length Lo (and ω > 0), the oscillatory layering will be damped out.35,36 In many situations, including the one studied here, the experimental measurements will not be limited in lateral scale and Lo is then determined by the capillary length of the system (∼1 mm); hence, Lo . σ and for a finite value of ω (∼1 for n-alkanes)(Lo/σ)-ω ≈ 0 and the oscillatory behavior will be absent. If an oscillatory potential of significant magnitude is present, discrete jumps in thickness of magnitude σ (=0.6 nm) would be evident as the film thickness increased. Discrete jumps are not observed in the film thickness measurements of n-hexane and (34) Chernov, A. A.; Mikheev, L. V. Phys. Rev. Lett. 1988, 60, 2488. (35) Henderson, J. R. Phys. Rev. E 1994, 50, 4836. (36) Tarazona, P.; Chaco´n, E.; Reinaldo-Falaga´n, M.; Velasco, E. J. Chem. Phys. 2002, 117, 3941.

Table 1. Interaction Parameters, Equation 18 a1 (J/m2) Ro σ (nm) b1 a

0.02 0.7 0.6 0.233

β (nm-1) Ttr (°C)a Tc (°C)

0.464 -90.6 267.1

Here it is assumed that Ttr = Tmelt.

n-heptane (Figure 5), which is a strong indicator that the oscillatory term is unimportant. The exponentially decaying term, or “structural term”, in eq 14 arises from averages over squared deviations in the density where a1 is expected to be positive and of order σlv.34 For the temperatures relevant to our data, the combination Vdisp + Vm cannot give rise to a global minimum in the potential at Lw ∼ 5 nm. Our data instead imply that an additional medium-ranged term must be present.37 In particular, we find that the data are consistent with a medium-range potential Vm(L) generalized to

Vm′(L) ) exp(-2RL) - b1 exp(-βL)

(17)

where β can be taken to be independent of T because R(T) already generates the desired temperature dependence, b1 > 0 determines the relative magnitudes of the two terms, and a possible origin of the medium-ranged term exp(βL) will be discussed in the summary.38 The full potential, including both long- and medium-range interactions, will therefore take the form

V(L) ) a1[exp(-2RL) - b1 exp(-βL)] + Vdisp(L)

(18)

where the precise form for Vdisp(L) is given in ref 32 and eq 18 is used within eq 2 to correctly account for any chemical potential offsets from coexistence. The parameter a1 determines the relative magnitude of the mediumranged potential relative to the long-range potential. A good desciption of the temperature (Figure 5), height (Figure 6), and pressure (Figure 7) data was obtained using the parameters listed in Table 1, as indicated by the heavy solid lines in each of these figures. The mediumrange terms primarily determine the wetting layer thickness Lw as a function of temperature T in Figure 5 as can be deduced from the dashed line in this figure, which represents the medium-ranged contribution in the absence of the dispersion and gravitational potentials. The decrease in Lw with increasing T arises primarily from the R ∼ T ( eq 15) dependence. The dominance of the medium-range interaction is also the reason for the relative independence of Lw on height H (Figure 6). For small values of the wetting layer thickness, the dispersion term plays an increasingly prominent role because it diverges as L-2 as L f 0. For this reason both the mediumand long-range terms are important in the isothermal pressure measurements of Figure 7. For a1 ) 0 the dispersion and disjoining pressure terms determine the wetting thickness (dotted line, Figure 7) in accord with (37) For ease of analysis, in fitting the data in Figures 5-7 (heavy solid lines), Ttr and Tc were fixed at the n-heptane values in order to enforce corresponding states through eq 15. All of the n-alkanes are expected to exhibit rather similar wetting behavior, both as a function of pressure (Bradberry, G. W.; Vukusic, P. S.; Sambles, J. R. J. Chem. Phys. 1993, 98, 651. Schlangen, L. J. M.; Koopal, L. K.; Cohen Stuart, M. A.; Lyklema, J. Langmuir 1995, 11, 1701) and as a function of height, because they all have similar molecular and interaction properties. (38) Pseudowetting has sometimes been modeled using the dispersion interaction together with its next-to-leading order term, namely, V(L) ) W/L2 + A/L3 with W < 0 and A > 0;16 however, this form is only relevant in the limit of small W when the next-to-leading order correction becomes important. In our case, W is always large and positive;32 thus, such a form cannot give rise to a minimum in the potential at Lw ∼ 5 nm.

Wetting of Silicon Wafers by n-Alkanes

the long-ranged wetting theory in section 2; conversely, for a1 ) 1 J/m2 the medium-ranged potential together with the disjoining pressure term primarily determines the wetting thickness (dashed line, Figure 7). The optimal fit to the n-pentane data, which represents a compromise between medium- and long-range interactions, occurs for a value of a1 ) 0.02 J/m2 (solid line), a value that is of the same order of magnitude as the surface tension for n-alkanes. The pressure data exhibits some minor deviations that may be caused by shorter ranged forces associated with the adsorption minimum, namely, in the theory considered thus far we have not attempted to incorporate the thin-thick wetting transition that would describe the first-order transition from the adsorption minimum at La ∼ 0.5 nm to the wetting minimum at Lw ∼ 5 nm that is observed for n-octane at a temperature of TwL ≈ 63 °C. In the inset to Figure 6, Vtot ( eq 2) is depicted for T ) 70 °C and H ) 10 mm with the parameters from Table 1. A global minimum occurs at L ∼ 4 nm, primarily due to the medium-ranged interactions. At larger distances, the potential exhibits a maximum at L ∼ 17 nm and a very shallow local minimum at L ∼ 46 nm due to the competition between the dispersion interaction and gravity (which is hardly visible, even when the potential is magnified by a factor of 10 (light solid line)). An adsorption minimum at La ∼ 0.5 nm has not yet been incorporated into this potential. This equation also describes the n-hexane temperature data (Figure 5, light solid line), but for slightly different values of b1 ()0.093) and β ()0.426 nm-1) where Ttr ) -95 °C and Tc ) 234.2 °C. 7. Summary Silicon surfaces coated with either their native oxide or a self-assembled monolayer (SAM) have found extensive utility in the field of nanotechnology. Micrometer-sized motors, oscillators, and complex channels have been fashioned out of silicon using standard semiconductor lithographic techniques in the emerging fields of MEMS and microfluidics. SAMs have been used to coat and control the wetting behavior of these substrates. It has frequently been assumed that the wetting properties of nonpolar liquids, such as n-alkanes, on such surfaces is controlled by a competition between the dispersion interaction and the local chemical potential of the fluid; strictly speaking this formulation only holds in the complete wetting regime where a droplet of the liquid exhibits zero contact angle. There is a lot of evidence in the literature that this simple picture for silicon surfaces, and for solid surfaces in general, does not necessarily hold. However, a problem with a number of these prior measurements, which exhibited disagreement with the long-ranged wetting result, was whether the fluid was truly in the complete wetting regime. In this publication, we have examined the wetting behavior of a number of n-alkanes on a silane-coated Si wafer. For n-octane both the droplet contact angle decreases to zero (Figure 4) and a discontinuous jump in film thickness (Figure 5, solid diamonds) occurs at a firstorder wetting transition temperature of Tw ∼ 60 °C; thus, n-octane completely wets this Si surface for T > Tw. The lower surface tension liquids, n-hexane and n-heptane, completely wet this surface for all temperatures that were examined (Figure 5, circles and squares, respectively). These thermal measurements are consistent with earlier isothermal pressure measurements for n-pentane on an oxide-coated Si wafer (Figure 7)6 where both measurements give a similar wetting layer thickness (Lw ∼ 8 nm) at a temperature of 18 °C at the saturated vapor pressure psat. The temperature measurements (Figure 5), as well

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as, height measurements for n-octane at 70 °C (Figure 6) are a factor of 10 smaller than the predictions from a nonretarded dispersion interaction (dotted line, Figure 6) while the isothermal pressure measurements resulted in films that are much thicker than the predicted values from the dispersion interaction (dotted line, Figure 7). These results provide conclusive evidence that the long-ranged dispersion interaction together with a local chemical potential are not providing the stabilizing forces for these measurements. Medium-ranged forces must therefore be playing an important role in determining the thickness of these wetting films. In section 6 we examined the influence of various structural forces on the wetting layer thickness. A consistent description of the temperature, height, and pressure data (heavy solid lines in, respectively, Figures 5-7) was obtained through eqs 2 and 18 with the dispersion term Vdisp(L) given in ref 32. The term a1 exp(-2RL) arose from structural forces while the exp(-βL) term arose from an as yet unspecified opposing interaction. A good description of the data was obtained for the values of the parameters listed in Table 1. Promising features of this surface potential are (i) the temperature dependence of Lw (Figure 5) arises quite naturally from R ∼ T, (ii) the amplitude Ro ) 0.7 is of similar magnitude to a square-well model density functional calculation35 where Ro ∼ 1, (iii) the amplitude of the structural term a1 is of order the surface tension, as expected, and (iv) by introduction of a single term -b1 exp(-βL) the mysterious pressure, temperature, and height data could be explained. An open question is, what is the origin of the exp(-βL) term that must be present for both the oxide-coated and silane-coated Si wafer? There seem to be a number of potential sources for this term, and further research is needed to determine its origin: (i) a strained layer occurs at the interface between Si and its native oxide,39 if this strained layer produces a dipolar layer of fixed orientation then an additional mediumranged exponential interaction will result (ref 17, p 256) where image dipoles must also be taken into account at all interfaces;40 (ii) a dilute distribution of dipolar impurity molecules at the solid surface may be able to produce a similar effect. It remains to be seen whether the medium-ranged forces considered in this publication can also account for the wetting behavior of other liquids studied by Beaglehole and Christenson on a Si substrate.6 If the origin for the exp(-βL) term can be determined, it could potentially be used to tune the surface potential. As mentioned in section 6, the interaction potential in eq 18 is still incomplete because it does not yet incorporate the adsorption minimum at very small thicknesses, La ∼ 0.5 nm. A complete potential,41 which we hope to present at some future date, should be consistent with (i) the n-octane wetting transition at TwL ≈ 63 °C (Figure 5), (ii) the n-octane line tension behavior that was measured in ref 9, (iii) the surface energy difference ∆σ ) σsv - σsl for (39) Daum, W.; Krause, H.-J.; Reichel, U.; Ibach, H. Phys. Rev. Lett. 1993, 71, 1234. Meyer, C.; Lu¨pke, G.; Emmerichs, U.; Wolter, F.; Kurz, H.; Bjorkman, C. H.; Lucovsky, G. Phys. Rev. Lett. 1995, 74, 3001. Arzate, N.; Mendoza, B. S. Phys. Rev. B 2001, 63, 113303. (40) Jo¨nsson, B.; Wennerstro¨m, H. J. Chem. Soc., Faraday Trans. 2 1983, 79, 19. (41) The adsorption minimum, at La ∼ 0.5 nm, can be generated by adding a term -c1 exp(-L/ξ) to eq 18 where the correlation length ξ ∼ t-ν with reduced temperature t ) (Tc - T)/Tc and critical exponent ν = 0.63. The minimum at La is generated via a competition between Vdisp and the exp(-L/ξ) term where, for the temperature range of interest to us, 1/ξ > 2R > β. (The parameter 1/ξ was denoted Ro in ref 35.) It is a fairly complicated process to obtain a self-consistent description of the n-alkane contact angle data (including the wetting transition), as a function of both temperature10 and chain length, as well as, quantitatively describe the n-octane line tension data;9 this task will be left to a later publication.

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both n-octane10 and the longer n-alkanes in the partial wetting regime, and (iv) possibly the unusual wetting observations of very long chain n-alkanes in ref 33. Acknowledgment. The authors thank Dr. H. K. Christenson for providing the isothermal n-pentane data from ref 6. B.M.L. thanks Drs. Jae-Hie Cho and Klaus

Law et al.

Mecke for detailed discussions on this work. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, and to the National Science Foundation through Grant Number DMR-0097119 for support of this research. LA035011V