Quartz crystal microbalance and synchrotron X-ray reflectivity study of water and liquid xenon adsorbed on gold and quartz

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ELSEVIER

Surface

Science 306 (1994) 359-366

Quartz crystal microbalance and synchrotron X-ray reflectivity study of water and liquid xenon adsorbed on gold and quartz J. Krim a, C. Thompson b R.P. Chiarello *7a71, ’ Department of Physics, Northeastern University, Boston, hL4 02115, USA b Brooklyn Polytechnic Institute, Brooklyn, NY 1120, USA (Received

3 March

1993; accepted

for publication

29 November

1993)

Abstract The adsorption of water on gold, liquid xenon on gold, and liquid xenon on quartz was studied by use of a combined quartz crystal microbalance and synchrotron X-ray reflectivity technique. This technique can simultuneozdy provide the quantity of adsorbed mass, the adsorbed layer thickness, and the real-space electron density profile of the adsorbed layer/vapor interface. These systems were chosen to cover a range of wetting scenarios, based on simple theoretical predictions for their wetting behavior.

1. Introduction Equilibrium film growth on solid surfaces is a topic of longstanding interest for fundamental, as well as technological reasons [l-3]. In systems which exhibit “complete wetting,” or Frank-van der Merwe growth [4], the film thickens monotonically and remains flat up to macroscopic dimensions. Solid films in this category grow in a layerby-layer manner, and liquid films are characterized by a zero contact angle, 8, [5,6]. In systems which exhibit “incomplete wetting,” or StranskiKrastonov growth [4], uniform film growth is limited in extent, and bulk liquid droplets (0 < 8, < 180”) or solid crystallites form at the macroscopic

* Corresponding author. Fax: + 1 (708) 252 9373. ’ Present address: Argonne National Laboratory, science/GMT 205, Argonne, IL 60439, USA. 0039-6028/94/$07.00 0 1994 SSDZ 0039-6028(93)E1025-U

Geo-

level. An extreme form of the latter growth type or Volmer-Weber is termed “nonwetting,” growth [4], where virtually no surface adsorption occurs, and all condensation is directly in the form of bulk droplets (19,= 180”) or crystallites

[2,7,81. A simple rule for predicting wetting behavior involves the adsorbate self-attraction u and its affinity u for the substrate [9-111. If the ratio u/v is larger (smaller) than one, then complete (incomplete) wetting is expected. Solid films grown on amorphous substrates, or substrates with lattice spacings far from commensurability with the film, are frequently predicted by this rule to grow in a layer-by-layer fashion, but the associated strain in the film may cause “clustering” after a few layers (Stranski-Krastonov growth). The wetting behavior of liquid films, the focus of the present work, is perhaps simpler to predict, since strain effects do not enter into consideration.

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It is standard assumption that an adsorption isotherm, which depicts film adsorption at the microscopic level, is also characteristic of its macroscopic wetting behavior [2,7]. An isotherm that diverges as the pressure (p> asymptotically approaches the bulk saturation point (pa) is presumed to correspond to complete wetting, while one which approaches a finite value corresponds to incomplete wetting. Although the condensation of gases and vapors onto solid surfaces has been studied for nearly half a century [l], experimental investigations that have specifically attempted to relate macroscopic wetting behavior to adsorption at the microscopic level are virtually nonexistent. This is because the majority of investigations involving gas adsorption and phase transitions of adsorbed films have been performed on high-surface-area substrates (powders, exfoliated graphite, etc.), i.e. surfaces that are inaccessible to contact area measurements [12151. We examine here, for an open experimental geometry, liquid film wetting behaviors for three systems, which span a u/v range that should include both complete (xenon/gold, u/v = 1.04) and incomplete wetting (water/quartz, u/v = 0.31; xenon/quartz, u/v = 0.12). The xenon/gold and xenon/quartz interaction energies were determined from a generalized van der Waals potential [16]. The bulk cohesive energies used for xenon and water were 1908 K [17] and 5720 K [ 181, respectively. The water/gold interaction energy was scaled from the water/graphite [19] value and should be correct within 50%. Our purpose is to determine the extent to which the shape of the liquid/solid adsorption isotherm, specifically the asymptotic approach of p to pa with increasing adsorbed mass, is a reliable indication of zero contact angle on the macroscopic level. We conclude that this is a reliable gauge of wetting behavior for only the very simplest of experimental systems. Powders and other high-surface-area substrates have limited usefulness for wetting studies since they are inaccessible to contact angle measurements. They are, moreover, susceptible at the microscopic level to interparticle capillary condensation effects which change the shape of the isotherm for film thicknesses exceeding approxi-

Science 306 (1994) 359-366

mately four layers [15,20-221. Quartz crystal microbalances (QCM) [23], as well as the techniques of synchrotron X-ray reflectivity (SXR) [241 and ellipsometry [25-271, have been successfully employed to probe microscopic wetting behavior of films adsorbed on open surface geometries. The QCM measures the mass of the adsorbed film but provides no direct information on film surface geometry. Ellipsometry is a sensitive measure of film thickness for flat surface geometries but provides little quantitative information on the nonuniform surface geometries associated with incomplete wetting. X-ray reflectivity yields the density profile of the film and, as such, provides quantitative information for both planar and nonuniform surface geometries. The density profile is obtained only after the data have been modeled. Knowledge of the film thickness under study (or whether a film has, in fact, condensed on the point of interest) is therefore lacking while the data are being recorded. Since reflectivity measurements are routinely performed at synchrotron radiation sources under severe recording time constraints, the latter issue is an important consideration. It can be eliminated by combining reflectivity with QCM so as to have film thickness information readily available. Since the two techniques have not previously been combined, we describe in some detail how this can be accomplished in the following sections.

2. Experimental

details

Quartz crystal disks (1.2 cm in diameter by 0.03 cm thick) were obtained from the ValpeyFisher Co. (Hopkinton, MA). The 5 MHz AT-cut (transverse-shear mode oscillation) crystals had optically polished planar faces. A Pierce oscillator circuit [28,29] was used to drive the crystals, and the quality factor was typically 10’. Operation of the QCM necessitates that an electrical conductor be either attached to or, in very close proximity to, the quartz crystal. A standard practice is to deposit thin metal films ontv each of the quartz crystal faces. Thin (500-700 A) gold films were vapor-deposited onto the polished planar faces and served as the QCM electrode and the

RP. Chiarello et al. /Surface

SXR surface for water and liquid xenon adsorbed onto gold. Before gold film deposition, quartz crystals were ultrasonically cleaned in succession in acetone, methanol, and deionized water. They were then baked in an ultrahigh vacuum (UHV) chamber (pressure during bake N lop9 Torr) to 600 K. The crystals were cooled to room temperatyre, and 99.995% gold was vapor-deposited at 0.5 A/s to film thicknesses between 500 and 700 A. The gold film thickness was determined during deposition by a quartz crystal rate monitor. The gold thickness was accurately determined after deposition by using SXR. The samples were transferred either in vacuum or in an environment of flowing argon gas (research grade) to a cryostat designed for simultaneous QCM/SXR measurements. The cryostat sample chamber was an X-ray-transparent UHV-compatible aluminum cube (4.2 cm on edge). The cube was equipped with 0.025 cm thick beryllium windows that allowed for incident X-ray angles up to 15”. Cooling was accomplished by attaching a liquid nitrogen reservoir to the cube by means of a weak-link copper strap. The temperature was controlled to &I.05 K. The cube also had a gas inlet and vacuum line and electrical feedthroughs for a Pt thermistor and for connection to the QCM driving circuit. The cube vacuum system included a 50 L/s turbomolecular pump and a 2 L/s ion pump. The base pressure of the sample chamber was N lo-’ Torr. A stainless steel gas-handling system was used for introducing xenon gas and water vapor into the sample chamber. The equilibrium vapor pressure was measured above the adsorbed film by a capacitance pressure gauge. Water from a Millipore/Milli-Q double filtration system was distilled five times before use, and the researchgrade xenon gas was also distilled before use. The measurements were performed at the National Synchrotron Light Source on beamlines X16B and X20A. At beamline X16B a bent, asyrnmetrica~y cut Ge(ll1) monochromator provided 1.69 A wavelength X-rays. The incident beam spot size was 0.5 mm horizontal by 1.0 mm vertical. Beamline X20A had a Si(ll1) singlecrystal monochromator, which provided 1.3 A wavelength X-rays. The incident beam was fo-

Science 306 (1994) 359-366

361

cused to a 1 X 1.5 mm spot by means of a platinum-coated silica mirror.

3. Experimental

results

3.1. QCM adsorption isotherms The QCM-measured adsorption isotherms for water adsorbed on gold at 293 K and xenon adsorbed on gold at 165 K are shown in Figs. l(a) and (b), respectively. The data are plotted as QCM frequency shift (Sf) versus equilibrium vapor pressure (p) above the film. Hydrostatic and viscous drag corrections to the data have been taken into consideration. Changes in the QCM resonant frequency due to mass loading may be written in terms of adsorbed film mass according to the relation:

Sf= -

f 02Pfilm

2.21 x lo5 ’

(1)

where f. is the QCM resonant frequency in Hz, and pfilm is the adsorbed film mass per unit area in g/cm’. The constant in the denominator is determined by the crystal cut. One water monolayer corresponds to approximately 3 Hz shift in resonant frequency. The maximum frequency shift of the water/gold isotherm was 48 Hz, corresponding to a water film whose thickness was equivalent to 16 water monolayers at pO. The saturated vapor pressure was determined by continuously adding water vapor to the sample chamber until the QCM stopped oscillating due to damping by the large quantity of adsorbed water. The shape of the water/gold adsorption isotherm is consistent with previously published adsorption isotherms of water/pyrolytic carbon [30] and water/ polytetrafluoroethylene [31]. In Fig. l(b) the QCM adsorption isotherm of xenon/gold measured at 165 K (3.6 K above the bulk xenon triple point) is shown. At this temperature, condensation of liquid xenon was expected. From Eq. (1) a liquid xenon monolayer corresponds to a 14.5 Hz frequency shift. The maximum xenon/gold QCM frequency shift was 167 Hz, equivalent to approx-

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R.P. Chiarello et al. /Surface

imately 11.5 liquid xenon monolayers. A signature of complete wetting in the adsorption isotherm may be that the adsorbed film mass asymptotically approaches p0 with increasing p. The QCM isotherms in Figs. l(a) and (b) both exhibit this behavior, indicating that both water and liquid xenon completely wet gold.

3.2. Synchrotron X-ray reflectivity The reflection amplitude for X-ray radiation incident upon an ideally terminated interface can be calculated from classical optics [32,33]. The reflected intensity observed from real interfaces is less than that predicted by classical optics. Accordingly, the reflected intensity from a rough

Science 306 (1994) 359-366

interface separating two media with indices of refraction no and n, may be written as [34-361: R = A,,,/__

= lAoJ2

+m 1 dp(z) p,

FeeiQZ

e -QoQ~uf= 1A,,,( a,) 12,

(a) water/gold 293 K 25 6

10

-5 ‘i;i

X . ~

7

0

175

5

10

15

2(

145 115 85

25

0

100

200

300

400

500

600

70(

p(torr) Fig. 1. Adsorption xenon/gold measured frequency shift versus line is a guide for the

isotherms of (a) water/gold and (b) by QCM. The data arc plotted as QCM equilibrium vapor pressure. The dashed eye.

(2)

where A, 1 is the Fresnel reflection amplitude of classical o’ptics, P,,, is the density in the bulk, z is a height in the direction normal to the surface plane, p(z) is the real-space electron density distribution, (pi is the interfacial roughness or width, and Q, and Q, are the perpendicular momentum transfers inside media “0” and “l”, respectively. The reflected intensity from a multilayer may be calculated recursively from the expression [331: Aj_,,j(q)

+Aj,j+l(T+l)

R= 1 +Aj_l,j(ai)Aj,j+l(~+l) 40

dz 2

epiQJdl2 ewiQJdJ ’

(3)

where uj represents the interfacial roughness, and d, is the thickness of the jth layer. The specular SXR data from gold/quartz (QCM) and from water deposited onto the same gold surface (water/QCM> are shown in Fig. 2(a). The SXR of the QCM surface was measured in - 10K7 Torr vacuum. In addition to the specular component of the SXR, there is a diffuse scattering component. This diffuse scattering contributes to the background and was subtracted from all the SXR data presented here. The oscillations in the SXR are due to interference effects between the quartz/gold and gold/vacuum interfaces. The SXR scans shown in Figs. 2(b) and (c) were normalized by dividing the data by the calculated reflectivity of a flat semi;infinite gold slab. In the region near Q = 0.3 A-’ an additional oscillation is superimposed on the SXR of the QCM measured in vacuum after exposure of this surface to water vapor. As shown in the figure, the frequency of this oscillation increased with increased p, indicating that the adsorbed water layer thickness increased with increased p. The solid lines in Fig. 2(a) represent the best-fit models of the SXR using Eq. (3). The best-fit model parameters were estimated using the Marquandt method, and the fit quality was determined by a x2 value [37]. An instrumental

R.P. Chiarello et al. /Surface

Science 306 (1994) 359-366

363

yH,o = 8.5 A for p = 0.7p,, as well as dHIO = 60.4 A and un o = 12.4 A for p = 0.95p,. Fig. 3(a) shows the SXR of a QCM surface in contact with low7 Torr vacuum and the SXR of the same QCM surface in contact with a thick xenon film (xenon/QCM). The SXR of xenon/QCM was measured at 165 K and p = 0.95~~. A comparison of the normalized SXR is shown in the inset. The SXR of xenon/QCM exhibits similar features to the SXR of water/ QCM previously discussed. The best-fit model results are shown as the solid lines and were obtained in the same manner detailed above. The parameters o,f the best-fit model were as follow:: d,, = 485.9 A, a,, = 17.5 A, and d,, = 66.0 A. Because of the relatively large roughness of the gold/vacuum interface, uXe could not be reliably determined. The gold electron density was 0.91 of the value for bulk gold. The xenon layer electron density was held fixed at the bulk liquid xenon

weighting factor was used. The gold/quartz SXR was modeled first, and two layers were required to adequately describe the SXR. The fit parameters were the gold electron density, the gold/ vacuum interfacial roughness, and the gold film thickness. The qtartz/gold interfacial roughness was fixed to 3.4 A (determined from the SXR of QCM in vacuum), and the quartz electron density was fixed to 0.8 e-/A3 (bulk quartz). The gold electron density was 0.91 of that expected for bulk gold (4.6 e-/A3>, and the gold/vacuum inter-facial width (ok”) was 10.lOA. The gold film thickness (dAu) was 696 A. The SXR of water/QCM was modeled by adding a water layer to the gold/quartz model. The fit parameters were the water layer thickness (dHZo) and the water layer/water vapor inter-facial roughness (c~u+,). The water layer electron density was fixed at the bulk water value (0.33 e-/A3>. The ;esults of the best-fit model were dn,o = 20.2 A and

(a)

-0.7p/p,

12 0.06

0.1

0.14

0.8, 8, 0.22 Q( -)

0.26

0.3

I

4

0

0.1

0.2

0.4 t Q(A-‘,o’3 Fig. 2. (a) Specular X-ray reflectivity (logarithmic scale) as a function of Q for QCM gold surface in contact with vacuum ( * lo-’ Torr) and of the same QCM gold surface in contact with water vapor (0.7p/p,, and 0.95 p/p& The data are offset for clarity and every other data point is removed. The solid lines represent the best-fit model results (see text for details). Normalized reflectivity (logarithmic scale) of the QCM gold surface in vacuum (solid line) is compared to the normalized reflectivity of the same QCM gold surface in contact with water vapor at 0.7 p/p0 (b) and 0.95 p/p,, (c).

364

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Science 306 (1994) 359-366

of the RSED also increased with increasing p, consistent with an increase in the water/vapor interfacial width. This is inconsistent with a film growing monotonically thick with increasing p, i.e. complete wetting. If that is the case, the effect of the underlying substrate roughness on the roughness of the film/vapor interface (conformal roughness) dissipates as the adsorbed film thickness increases [24]. An optical window was placed above the QCM surface to visually examine the wetting behavior of water/gold. At p = po, bulk-like droplets were observed on the QCM surface. The RSED profiles (of the xenon/vapor interface) of xenon condensed onto gold and quartz are shown in Figs. 4(b) and (c), respectively. The RSED of xenon/gold is consistent with the condensation of a thick liquid xenon film on gold. In contrast, the RSED profile of xenon condensed onto quartz contains a sharp peak and a tail which slowly decays with increasing z. The peak

value. Fig. 3(b) shows the SXR and best-fit model results (solid lines) of a quartz surface in contact with lo-’ Torr vacuum and the same quartz surface in contact with xenon at 165 K and p = 0.95~~. The data were normalized by the calculated reflectivity of a flat semi-infinite quartz slab. The results of the best-fit model for xenon/ quartz are best summarized in terms of the realspace electron density profile of the xenon/gas interface described below. The real-space electron density (RSED) profiles were calculated from the best-fit model parameters of the SXR and the complementary error function 134,361. In Figs. 4(a)-(c), RSED profiles are plotted as a function of z for water/QCM, xenon/QCM, and xenon/ quartz, respectively. The substrate contribution to the RSED was subtracted. The RSED profiles for the two water coverages are shown in Fig. 4(a). The film thickness increased from 20.2 (0.7p/p,) to 60.4 A (0.95p/p,J with increased p. The width

(a)

(b) lo-’ Torr

0.05

0.1

0.15

0.2

Q&‘) Fig. 3. (a) X-ray reflectivity (logarithmic scale) as a function of Q for QCM gold surface in contact with vacuum (- lo-’ Torr) and xenon gas at p = 0.95po. Both reflectivity scans were recorded at a temperature of 165 K. The scans are offset and every other data point is removed. The solid lines are the best-fit model results (see text for details). In the inset the normalized reflectivity (logarithmic scale) of the QCM in contact with vacuum (dashed line) is compared to the normalized reflectivity of the QCM in contact with xenon gas at p = 0.95~~ (solid line). (b) Normalized reflectivity (logarithmic scale) of quartz surface in contact with vacuum (lo-’ Torr) and the same quartz surface in contact with xenon gas at p = 0.95~~. Both scans were recorded at a temperature of 165 K. The solid lines are the best-fit model results.

R. P. Chiarello et al. /Surface

Science 306 (1994) 359-366

365

4. Conclusions

680

720

760 (c)

800

xenon/quarK

0.8

Ga0.6 =; 5 CL

0.4

0.2

20

40

1

z(.b

Fig. 4. Real-space electron density profiles (RSED) determined from specular X-ray reflectivity of condensed layers of (a) water on gold (QCM), (b) xenon on gold (QCM), and (c) xenon on quartz. In (a), RSED profiles of water on gold for water vapor pressures of 0.7p/p, (dashed line) and 0.95p/p, (solid line) are shown. The RSED profiles were normalized by the electron density determined by SXR.

position corresponds to approximately 3-4 equivalent liquid xenon monolayers. This profile may be interpreted as a thin liquid xenon film (sharp peak) coexisting with bulk xenon droplets (nonzero slowly decreasing tail).

The above results demonstrate the feasibility of combining QCM and SXR to simultaneously determine the thermodynamic and structural properties of liquids condensed onto solid planar substrates. The QCM was used to measure the liquid/ solid adsorption isotherms and to monitor the quantity of adsorbed mass during SXR studies. SXR was used to determine the adsorbed film thickness and the real-space electron density profile of the (adsorbed) liquid/vapor interface. Both QCM and SXR have submonolayer coverage sensitivity, and SXR is routinely used to determine Angstrom-scale topographical features of surfaces and interfaces. These combined measurements were performed by using a single open-surface substrate that served as both the QCM electrode and the SXR surface. This allowed us to study thick liquid films by eliminating the possibility of capillary condensation phenomena. The liquid/solid systems studied here (water/ gold, liquid xenon/ gold, and liquid xenon/ quartz) were selected because they cover a range of wetting scenarios. The measurements were performed above the bulk melting points for water and xenon to ensure condensation of the liquid phase. The shapes of the adsorption isotherms of water/gold and liquid xenon/gold were consistent with both water and liquid xenon completely wetting the gold surface. The QCM adsorption isotherm was not measured for xenon/quartz. The SXR results for xenon/gold indicated that a thick liquid xenon film had condensed onto gold at p = 0.95~~. This result and the adsorption isotherm of liquid xenon/gold indicate that liquid xenon completely wetted gold, as expected. The SXR results for liquid xenon/ quartz were consistent with a thin liquid film coexisting with bulk droplets at p = 0.95~~. This observed incomplete wetting behavior of quartz by liquid xenon and the complete wetting of gold by liquid xenon were consistent with the simple theoretical calculations used to predict wetting behavior. The SXR measurements were performed for water/gold at two vapor pressures, p = 0.7~~ and

R.P. Chiarello et al. /Surface

366

It was determined that the adsorbed Fater/vapor interface increased from 8.5 to 12.4 A with increased quantity of water adsorbed onto the gold surface. For a thin water film, the water/vapor interfacial roughness may be expected to conform to the rooughness of the underlying gold substrate (10.1 A). However, as the water film approaches bulk, the effect of the underlying substrate on the roughness of the water/vapor interfacial width is expected to diminish. It was not possible to determine the exact nature of the water/vapor interface from the limited SXR data presented. However, these results may be interpreted as evidence for early stages of the formation of water droplets. This interpretation of the SXR results was substantiated by visual inspection of the gold surface at p =pO. At this saturated vapor pressure, bulk water droplets were observed on the gold surface, indicating that water incompletely wets gold after first forming a thick film. This result for the incomplete wetting of gold by water indicates that caution should be exercised when referring to the asymptotic approach to p,, in adsorption isotherms as a signature of complete wetting behavior.

p = 0.95p,.

5. Acknowledgements This work was supported by the National Science Foundation under grants DMR8657211 and DMR9204022. We gratefully acknowledge M.F. Toney and D. Weisler for collaboration in the early stages of this work.

6.

References

[l] S. Brunauer, The Adsorption of Gases and (Oxford Univ. Press, London, 1943). [2] A.W. Adamson, Physical Chemistry of Surfaces Interscience, New York, 1976). [3] See, for example, Proc. 4th Int. Conf. on Solid and 3rd European Conf. on Surface Science, France (1980), and references therein. [4] E. Bauer, Z. Kristallogr. 110 (1958) 372. [5] T. Young, Miscellaneous Works, Vol. 1, Ed. G. (Murray, London, 18.55).

Vapours (WileySurfaces Cannes,

Peacock

Science 306 (1994) 359-366

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DuprC, Theorie Mecanique de la Chaleur (Paris, 1869). Physics, 3rd [71 L.D. Landau and E.M. Lifshitz, Statistical ed., Part 1 (Pergamon, Oxford, 1980). k31R. Pandit, M. Schick and M. Wortis, Phys. Rev. B 26 (1982) 5112. [91 J.G. Dash, Phys. Rev. B 15 (1977) 3136. 1101D.E. Sullivan, Phys. Rev. B 20 (1979) 3991. [ill M. Bienfait, J.L. Seguin, J. Suzanne, E. Lerner, J. Krim and J.G. Dash, Phys. Rev. B 29 (1984) 983. 1121J.G. Dash, Films on Solid Surfaces (Academic Press, New York, 1975). of Gases with Solid Sur[131 W.A. Steele, The interaction faces (Pergamon, Oxford, 1974). [141 M. Bienfait, Surf. Sci. 162 (1985) 411. in Surface Films 2, Eds. H51 G.B. Hess, in: Phase Transitions H. Taub et al. (Plenum, New York, 1991). I161E. Cheng and M.W. Cole, Phys. Rev. B 38 (1988) 987. [171 G. Kane, J. Chem Phys. 7 (1939) 603. I181D. Eisenberg and W. Kauzman, The Structure and Properties of Water (Oxford Univ. Press, New York, 1969). (1984). [191 J. Krim, PhD Thesis, University of Washington M.A. La Madrid and M.J. Lysek, in: DO1 D.L. Goodstein, Phase Transitions in Surface Films 2, Eds. H. Taub et al. (Plenum, New York, 1991). M. Yamamoto and I. OhDll K. Morishige, K. Kawamura, fuji, Langmuir 6 (1990) 1417. WI J.Z. Larese, Q.M. Zhang, J.M. Hastings, L. Passell, J.R. Dennison and H. Taub, Phys. Rev. B 40 (1989) 4271. [231 J. Krim, J. Suzanne and J.G. Dash, Phys. Rev. Lett. 52 (1984) 635. P.S. Pershan and SD. [241 I.M. Tidswell, T.A. Rabedeau, Kosowsky, Phys. Rev. Lett. 66 (1991) 2108. 1251 H.S. Youn and G.B. Hess, Phys. Rev. Lett. 64 (1990) 918. [261 U.G. Volkmann and K. Knorr, Surf. Sci. 221 (1989) 379. E.Z. Radlinska, B.W. Ninham and H.K. [271 D. Beaglehole, Christenson, Phys. Rev. Lett. 16 (1991) 2084. [281 R.A. Heising, Quartz Crystal for Electric Circuits (Van Nostrand, New York, 1946). Eds., Applications of Piezo1291 C. Lu and A.W. Czanderna, electric Quartz Crystal Microbalances (Elsevier, Amsterdam, 1984). P. Hu and A.W. Adamson, J. Colloid [301 M.E. Tadros, Interface Sci. 49 (1974) 184. J. Colloid Interface Sci. 59 [311 P. Hu and A.W. Adamson, (1977) 605. [321 L.G. Parrat, Phys. Rev. 95 (1954) 359. 1331 M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1970). [341 L. Nevot and P. Croce, Rev. Phys. Appl. 15 (1980) 761. 1351 S.K. Sinha, E.B. Sirota, S. Garoff and H.B. Stanley, Phys. Rev. B 38 (1988) 2297. [36] A. Braslau, P.S. Pershan, G. Swislow and J. Als-Nielsen, Phys. Rev. A 38 (1988) 2457. [37] P.R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).