Influence of Roughness on the Admittance of the Quartz Crystal

Jan 4, 2002 - The effect of surface roughness on the response of the QCM has been considered, both theoretically and experimentally. A new theoretical...
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Anal. Chem. 2002, 74, 554-561

Influence of Roughness on the Admittance of the Quartz Crystal Microbalance Immersed in Liquids Leonid Daikhin, Eliezer Gileadi, Galina Katz, Vladimir Tsionsky,* Michael Urbakh, and Dmitrij Zagidulin†

School of Chemistry, Raymond and Beverly Sackler, Faculty of Exact Sciences, Tel-Aviv University, Ramat-Aviv 69978, Israel

The effect of surface roughness on the response of the QCM has been considered, both theoretically and experimentally. A new theoretical approach to the description of the effect of roughness on the response of the QCM is proposed that accounts for the multiscale nature of roughness. Performing experiments in liquids having a wide range of viscosity and density made it possible to understand, for the first time, what characteristics of roughness influence the QCM experiments. The most important conclusion of the current study is that, to understand the experimental data, one has to take into account at least two types of roughness: slight and strong. We found that measurements of the frequency shift observed are not sufficient for the interpretations of the experimental data observed, and a full analysis of the impedance spectrum is called for. A steady increase in the application of the quartz crystal microbalance (QCM) for studies of the solid/liquid interface was evidenced in the past two decades. It was successfully demonstrated that the QCM can serve as a sensitive tool to probe bulk liquid viscosity and density,1-4 interfacial friction,5 thin-film viscoelasticity,6,7 and polymer film properties.8-10 The application of the QCM in electrochemistry is so widespread that any single citation would be incomplete and misleading. It was used to study adsorption phenomena,8,11-13 double-layer structure,8,11,14-17 metal * Corresponding author. E-mail: [email protected]. Fax: +972 3 6409293. † Permanent address: Institute of Chemistry, Gostauto 9, Vilnius, LT-2600, Lithuania. (1) Bruckenstein, S.; Shay, M. Electrochim. Acta 1985, 30, 1295-1300. (2) Kanazawa, K. K.; Gordon, J. G., II. Anal. Chim. Acta 1985, 175, 99-105. (3) Nomura, T.; Okuhara, M. Anal. Chim. Acta 1982, 142, 281-284. (4) Zeng, X.; Moon, S.; Bruckenstein, S. Anal. Chem. 1998, 70, 2613-2617. (5) Watts, E. T.; Krim, J.; Widom, A. Phys. Rev. B 1990, 41, 3466-3472. (6) Johannsmann, D.; Mathauer, K.; Wegner, G.; Knoll, W. Phys. Rev. B 1992, 46, 7808-7815. (7) Reed, C. E.; Kanazawa, K. K.; Kaufman, J. H. J. Appl. Phys. 1990, 68, 19932001. (8) Buttry, D. A. Inb Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker Inc.: New York, 1991; Vol. 17, p 1. (9) Buttry, D. A.; Ward, M. D. Chem. Rev. 1992, 92, 1355-1379. (10) Glidle, A.; Hillman, A. R.; Bruckenstein, S. J. Electroanal. Chem. 1991, 318, 411-419. (11) Hepel, M. Electrode Solution Interface Studied With Electrochemical Quartz Crystal Nanobalance. In Interfacial Electrochemistry; Wieckowski, A., Ed.; Marcel Dekker: New York, 1999; p 599-630. (12) Daikhin, L.; Gileadi, E.; Tsionsky, V.; Urbakh, M.; Zilberman, G. Electrochim. Acta 2000, 45, 3615-3621.

554 Analytical Chemistry, Vol. 74, No. 3, February 1, 2002

deposition processes,8,18-20 etc. However, there are still certain problems, which have not been solved, that make it difficult to interpret experimentally measured QCM responses correctly. One of these problems, which has already been addressed in the literature,21-29 is the influence of roughness of the vibrating surface of the QCM on its response. When the surface of the resonator is rough, the liquid motion generated by its oscillation becomes much more complicated than for a smooth surface. A variety of additional mechanisms of coupling between the acoustic waves and the motion of the liquid can arise, such as the generation of a nonlaminar motion, the conversion of the in-plane surface motion into motion in the direction perpendicular to the surface, and trapping of liquid by cavities and pores. It has been experimentally demonstrated22-24,26,27 that a roughness-induced shift of the resonant frequency includes both the inertial contribution, due to the mass of the liquid rigidly coupled to the surface, and the contribution due to the additional viscous energy dissipation caused by the nonlaminar motion in the liquid. Measurements of the complex shear mechanical impedance26 have been used to analyze different contributions to the roughness-induced response of the quartz resonator and to correlate the experimental results with the device roughness. Nevertheless, this subject has been poorly developed and the (13) Zilberman, G.; Tsionsky, V.; Gileadi, E. Electrochim. Acta 2000, 45, 34733482. (14) Tsionsky, V.; Daikhin, L.; Gileadi, E. J. Electrochem. Soc. 1996, 143, 22402245. (15) Tsionsky, V.; Daikhin, L.; Gileadi, E. J. Electrochem. Soc. 1995, 142, L233L234. (16) Jusys, Z.; Bruckenstein S. Electrochem. Commun. 2000, 2, 412-416. (17) Kern, P.; Landolt D. J. Electroanal. Chem. 2001, 500, 170-177. (18) Bund A.; Schwitzgebel, G. Electrochim. Acta 2000, 45, 3703-3710. (19) Bruckenstein, S.; Kanige, K.; Hillman, A. R. J. Appl. Electrochem. 1996, 26, 171-173. (20) Gileadi, E.; Tsionsky, V. J. Electrochem. Soc. 2000, 147 (2), 567-574. (21) Yang, M.; Thompson, M. Anal. Chem. 1993, 65, 1158-1168. (22) Beck, R.; Pittermann, U.; Weil, K. G. J. Electrochem. Soc. 1992, 139, 453461. (23) Yang, M.; Thompson. M. Langmuir 1993, 9, 1990-1994. (24) Schumacher, R.; Borges, G.; Kanazawa, K. K. Surf. Sci. 1985, 163, L621L626. (25) Schumacher, R.; Gordon, J. G.; Melroy, O. Electroanal. Chem. 1987, 216, 127-135. (26) Martin, S. J.; Frye, G. C.; Ricco, A. J.; Senturia, S. D. Anal. Chem. 1993, 65, 2910-2922. (27) Bruckenstein, S.; Fensore, A.; Li, Z. F.; Hillman, A. R. J. Electroanal. Chem. 1994, 370, 189-195. (28) Urbakh, M.; Daikhin, L. Phys. Rev. B 1994, 49, 4866-4870. (29) Urbakh, M.; Daikhin, L. Langmuir 1994, 10, 2836-2841. 10.1021/ac0107610 CCC: $22.00

© 2002 American Chemical Society Published on Web 01/04/2002

interpretation of experimental results is ambiguous. This limits the use of the QCM technique as a widely employed tool for studies of the solid/liquid interface. In the present work, we concentrated on the effect of roughness on the QCM response in a series of liquids having a wide range of viscosity and density. The conditions were chosen so that the contribution of the effect of roughness is much larger than other effects mentioned above (double-layer structure, adsorption phenomena, etc.). A new theoretical approach to the description of the effect of roughness on the response of the QCM that accounts for the multiscale nature of roughness is proposed. Comparison with experimental data unambiguously demonstrates that this consideration is critical for understanding of the response of the QCM in liquids. This interpretation is supported by STM images of the interfaces studied. Models for the Interpretation of QCM Response at Rough Surfaces. The dependence of the QCM response (the resonance frequency shift, f, and the width of the resonance, Γ) on the interface morphology is determined by the relationship between the characteristic sizes of roughness and the length scales of the shear modes in the liquid and the quartz resonator. The length scales in the liquid (the velocity decay length, δ) and in the crystal (wavelength of the shear-mode oscillations, λ) are defined by the Navier-Stokes equation and by the wave equation for elastic displacement, respectively. For the frequencies used in QCM experiments, f ≈ 1-10 MHz, the lengths δ ) (η/πfF)1/2 and λ ) (µq/Fq)1/2f - 1 are of the order 0.1-1 µm and 0.1 cm, respectively where F and η are the density and the viscosity of the liquid, respectively, and Fq and µq are the density and shear modulus of the quartz crystal. The surface profile may be specified by a single valued function z ) ξ(R)of the lateral coordinates R that gives a local height of the surface with respect to a reference plane (z ) 0). The latter is chosen so that the average value of ξ(R) equals zero. Surfaces used in QCM experiments may have various scales of roughness. To clarify this point, let us consider two limiting cases: slight and strong roughnesses, which are schematically shown in Figure 1. For the slight roughness (cf. Figure 1a), the “amplitude” of deviation from the reference plane z ) 0 is much less than the lateral characteristic length. For strong roughness (cf. Figure 1b), the “amplitude” and “period” of repetitions are of the same order of magnitude. To stress the multiscale nature of roughness, the profile function can be written as the sum of the functions that characterize the profile of the specific scale i

ξ(R) )

∑ξ (R)

Figure 1. Schematic representations of slight (a) and strong (b) roughness. (c) represents a combination of both types of roughness. See text for details.

i and 〈 〉 represents averaging over the lateral coordinates. Usually the correlation function 〈ξi(R′)ξi(R′ - R)〉 has a Gaussian form 〈ξi(R′)ξi(R′ - R)〉 ) hi2 exp(- |R|2/li2), where hi is the root-meansquare height of the roughness and li is the lateral correlation length. Thus, the morphology of the rough surface can be characterized by the set of lengths {hi, li}. Below we limit our consideration to the case of two-scale interfaces (cf. Figure 1c), one corresponds to slight roughness and the other to strong roughness. This is admittedly a simplified model of a real surface, but it serves to explain the variation of the QCM response to roughness modification. Slight Roughness. The case of slight roughness was considered in our previous publications.28,29 The problem was solved in the framework of the perturbation theory with respect to the parameters |∇ξ(R)| , 1 and h/δ , 1. The shift in resonance frequency and the width of the resonance can be written in the following form28,29

∆f ) -

(1)

i

i

Γ) For the calculation of the response of the QCM, the height-height pair correlation function is needed.29 When rough structures having different scales do not correlate, the total correlation function can be written in the form

〈ξ(R′)ξ(R′ - R)〉 )

∑ 〈ξ (R′)ξ (R′ - R)〉 i

i

(2)

i

where 〈ξi(R′)ξi(R′ - R)〉 is the correlation function for the scale

[

f 2Fδ h2 1 + 2 F(l/δ) 1/2 (Fqµq) l

[

h2 2f 2Fδ 1 + Φ(l/δ) (Fqµq)1/2 l2

]

]

(3)

(4)

The scaling functions F(l/δ) and Φ(l/δ) are expressed through Fourier components of the pair correlation function of the roughness g(K).28 This correlation function g(K)can be defined as

h2g(K) )

∫dR exp(- iKR) 〈ξ(R′)ξ(R′ - R)〉

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(5) 555

In the present work, we perform STM measurements of g(K) in parallel to QCM experiments. The correlation function yields the most detailed characterization of the surface structure. Usually, in electrochemistry, the surface roughness is described in terms of an integral parameter, the roughness factor, R, that is the trueto-apparent surface area ratio. For the slight roughness, the roughness factor is expressed through the correlation function28 as

R)1+

h2 2

dK g(K)K ∫(2π)

2

2

(6)

For the Gaussian random roughness we have R ) 1 + 2h2/l2. The first terms in brackets in eqs 3 and 4 define the shift and the broadening of the resonance at the smooth crystal/liquid interface2. The surface roughness leads to additional decrease of frequency and broadening of the QCM response. The particular form of the scaling functions F(l/δ) and Φ(l/ δ) is determined by the morphology of the surface. However, the asymptotic behavior of these functions for l/δ . 1and l/δ , 1 is universal28 and has the form

F(l/δ) ) (l/δ)[R1 + R2δ/l] at l/δ . 1

(7)

F(l/δ) ) (l/δ)[β1 + β2l/δ] at l/δ , 1

(8)

Φ(l/δ) ) γ1 at l/δ . 1

(9)

Φ(l/δ) ) (l/δ)2γ2 at l/δ , 1

(10)

For random Gaussian roughness, the parameters are R1 ) π1/2, R2 ) (R - 1)l2/h2, β1 ) 3π1/2, β2 ) - 2 and γ1 ) (R - 1)l2/h2, γ2 ) 2. It should be stressed that, for slight roughness, the roughness-induced width of the resonance is always smaller than the corresponding shift in frequency. It should be noted that for l/δ . 1 the roughness-induced frequency shift (see eqs 3 and 7) includes a term that does not depend on the viscosity of liquid, the first term in eq 7. It reflects the effect of the nonuniform pressure distribution, which is developed in the liquid under the influence of a rough oscillating surface.28 The corresponding contribution has the form of the Sauerbrey equation. This effect does not exist for flat interfaces. The second term in eq 7, and eq 9, describe a viscous contribution to the QCM response. Their contribution to ∆f has a form of the QCM response at a flat liquid/solid interface but includes an additional factor R that is a roughness factor of the surface. The latter is a consequence of the fact that for l/δ . 1 the liquid “sees” the interface as being locally flat, but with R times higher surface area. Strong Roughness. The perturbation theory cannot be applied to describe the effect of strong roughness. An approach based on Brinkman’s equation is used instead to describe the hydrodynamics in the interfacial region.30 The flow of a liquid through a nonuniform surface layer is treated as the flow of a liquid through a porous medium.31-33 The morphology of the interfacial layer is (30) Daikhin, L.; Urbakh, M. Langmuir 1996, 12, 6354-6360. (31) Brinkman, H. C. Appl. Sci. Res. A 1947, 1, 27-32. (32) Sahimi, M. Rev. Mod. Phys. 1993, 65, 1393-1534.

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characterized by a local permeability, ξH, that depends on “porosity” of the layer, φ. A number of equations for the permeability have been suggested. For instance, the empirical Kozeny-Carman equation32 yields a relationship between ξH2 and the porosity ξH2 ∼ r2φ 3/(1 - φ)2, where r is the characteristic size of inhomogeneities. We considered an interfacial layer of thickness, L, that is in contact both with the quartz crystal and the bulk liquid. The liquid flow through the interfacial layer is described by the following equation30

d2 i2πfFv(z,f) ) η 2v(z,f) + ηξH-2[V0 - v(z,f)] dz

(11)

where V0 is the amplitude of the quartz surface velocity and v(z,t) ) v(z,f) exp(i2πft) is the velocity of the liquid in the layer. In this equation, the effect of the solid phase on the flow of liquid is given by the resistive force, which has a Darcy-like form, ηξH-2[V0 v(z,f)]. In the case of high porosity, the resistive force is small and eq 11 is reduced to the Navier-Stokes equation, describing the motion of the liquid in contact with a smooth quartz surface. The resistive force increases with the decrease of porosity and strongly influences the liquid motion. At very low “porosity”, all liquid located in the layer is trapped by the roughness and moves with a velocity equal to the velocity of the crystal surface. Brinkman’s equation represents a variant of the effective medium approximation, which does not describe explicitly the generation of nonlaminar liquid motion and conversion of the inplane surface motion into the motion normal to the interface. These effects result in additional channels for energy dissipation, which are effectively included in the model by introduction of the Darcy-like resistive force. The liquid-induced frequency shift and width of the resonance have the following form30

∆f ) -

(

2f 2F L 1 Re + 2 21/2 q (µqFq) ξH q1 0

{

2q0 1 1 [cosh(q1L) - 1] + sinh(q1L) W ξ 2q 2 q1 H 1 Γ)-

(

1 4f 2F L Im + 2 21/2 q (µqFq) ξH q 1 0

{

2q0 1 1 [cosh(q1L) - 1] + sinh(q1L) W ξ 2q 2 q1 H 1

})

(12)

})

(13)

Here q0 ) (i2πfF/η)1/2, q12 ) q02 + ξH-2, and W ) q1cosh(q1L) + q0sinh (q1L). The first terms on the right-hand side of eqs 12 and 13 describe the QCM response for the smooth quartz crystal/ liquid interface.2 The additional terms represent the shift in frequency and the increase in width of the QCM response, caused by the interaction of the liquid with a nonuniform interfacial layer. When the permeability length scale is the shortest length of the problem, ξH , δ and ξH , L, the layer-induced shift, ∆fl, is proportional to the density of the liquid and does not depend on (33) Jones, L. G.; Marques, C. M.; Joanny, J.-F. Macromolecules 1995, 28, 136142.

the viscosity. It has the form of the Sauerbrey equation for a mass loading. This effect results from the inertial motion of the liquid trapped by the inhomogeneities in the interfacial layer.

∆fl ) -

2f 2F (L - ξH) (µqFq)1/2

(14)

The effective thickness of the liquid film rigidly coupled to the oscillating surface is equal to L - ξH and is less than the thickness of the inhomogeneous layer, L. The increase of the permeability ξH leads to the enhancement of the velocity gradient in the layer, which results in a decrease of the frequency shift due to mass loading and in an increase of the width caused by the energy dissipation. When the layer thickness is the shortest length of the problem, L , δ, L , ξH, and ξH , δ, the frequency shift is also proportional to the density of the liquid, and does not depend on viscosity

∆fl ) -

2f 2FL (L/ξH)2 3(µqFq)1/2

(15)

However, in contrast to the previous case, it cannot be related to the mass of trapped liquid. The correction to the width of the resonance depends on the viscosity and is substantially less than the layer-induced shift. We would like to stress that, in both limiting cases discussed above, the corrections to the shift and to the width differ considerably. EXPERIMENTAL SECTION Measurements were performed in a tubelike glass cell (diameter of 12.5 mm), closed at the bottom by the quartz crystal resonator (AT-cut type, diameter 15 mm). The crystal was cradled between Viton O-rings, resting on a soft bronze bellows. Leakage was prevented by applying gentle pressure, forcing the crystal upward. This construction allows keeping the outside surface of the resonator (or both sides if needed) in a dry hydrogen atmosphere. The cell was situated in an air thermostat, which served also as Faraday cage. The accuracy of temperature control (25 ( 0.1 °C) was very important, in particularly when highviscosity liquids were tested, because they usually have very strong dependence of viscosity on temperature. The quartz crystal resonator was cleaned with cyclohexane in a Soxhlet apparatus for about 1 h. This was followed by a 30-min treatment in concentrated sulfuric acid and thorough rinsing in Milli-Q water. All chemicals were reagent grade. We used gold-coated quartz resonators, purchased from Intellemetrics and Inficon (plano-convex, f0 ) 6 MHz). The gold surface on these resonators was obtained by vacuum deposition, which will be referred to as VAu. Following a series of measurements in hydrogen and in different liquids, the cell was filled with a 10 mM aqueous solution of HAuCl4. Rough surfaces were obtained by electroplating of gold at a current density close to the limiting current (∼1 mA/cm2). These roughened gold surfaces will be referred to as EAu. After the washing procedure (with water, methanol, and diethyl ether and drying in hydrogen without dismantling the cell), the above measurements were repeated. Repeated measurements were performed to ensure that the results

Table 1

H2 diethyl ether methanol water sucrose 10% sucrose 20% sucrose 40% sucrose 50% sucrose 60%

density, g/cm3

viscosity, mPa

velocity decay length, µm

89.8 × 10-6 34 0.714 35 0.787 35 0.997 35 1.037 35 1.080 35 1.174 35 1.227 35 1.256 35

9 × 10-3 34 0.222 34 0.544 34 0.890 34 1.170 36 1.70 36 5.19 36 12.3 36 43.90 36

0.0023 0.129 0.191 0.231 0.259 0.290 0.485 0.817 1.349

obtained in different liquids were not affected by the sequence of measurements. The properties of the liquids (at 25 °C) used in this work are summerized in Table 1.34-36 A Hewlett-Packard Network analyzer model 5100A was used for impedance measurements. Lab-View 5.0 software was employed for data acquisition and processing. Three-term calibration: “open”, “short”, and “load” (80 Ω) was used. A total of 1400 pair of data points of the real and the imaginary parts of admittance were collected in each sweep. The network analyzer allows using its “internal” data processing. However, we did not use this option, because it assumes an a priori equivalent circuit to describe the behavior of the resonator. To obtain the resonance frequency (f) and the width of resonance (Γ), the rough data of frequency dependence of admittance were approximated by polynomials (of second degree for the real part and third degree for the imaginary part) in narrow regions near the relevant extrema, as shown in Figure 2. Network analyzer model 5100A allows programming of not only the linear frequency sweep. To increase the accuracy in obtaining f and Γ, it was programmed in such way that the densities of points obtained were larger in the regions of the extrema than on the wings of the resonance curves. The accuracy of f and Γ obtained depends on the quality of the resonator and hence on the surrounding medium. In a hydrogen atmosphere, the mean error never exceeded 0.5 Hz for both f and Γ, but in a 50% solution of sucrose, for example, it increased up to a few tens of hertz. It has to be emphasized that when using “internal” data processing of the Network Analyzer the accuracy decreases remarkably, because the internal procedure takes into account all points obtained including points on the “wings” of resonance, where the relative precision for every individual point is lower. Usually one set of data with all solutions was obtained during ∼1 h. The QCM drift was less than 10 Hz/h; thus it did not introduce any remarkable error. STM images were obtained ex situ with a Digital Instrument Co. Nanoscope II. RESULTS AND DISCUSSION Preliminary Results. The fundamental equation governing the behavior of the quartz resonator is the well-known Sauerbrey equation,37 which relates linearly the frequency shift observed to (34) Lide, D. R., Ed. Handbook of Chemistry and Physics, 77th ed.; CRC Press: Boca Raton, FL, 1996-1997; pp 6-208, 8-99. (35) Landolf-Bornstein Numerical Data and Relationship in Physics, Chemistry, Astronomy, Geophysics and Technology, Chemistry, 6th ed.; Synowietz, C. Ed.; 1977; Vol. IV/1b, Part 1.3, pp 2, 117, 155, 202. (36) Hatschek, E. The Viscosity of Liquids; Bell, E. G. and Sons, Ltd.: London, 1928; Chapter 8, p 118. (37) Sauerbrey, G. Z. Phys. 1959, 155, 206-222.

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Figure 3. Changes of the QCM properties during cathodic deposition of gold on gold substrate from 10 mM aqueous solution of HAuCl4 at current densities: 20 (open circles) and 500 µA/cm2 (solid circles). (a) Ratio of the observed shift of the fundamental frequency to the shift predicted for mass loading alone. (b) Width of the resonance (Γ).

Figure 2. Typical rough data on the QCM impedance (a, real part; b, imaginary part) obtained from the Network Analyzer. Insets show both experimental points and corresponding approximations, the lines (see text).

the added mass per unit area of crystal cross section, Ft(0),

∆f ) -

2f 2Ff(0)

xFqµq

(16)

In our previous publications,20,38,39 we described deviations from the Sauerbrey equation during deposition of silver and copper, which originate from several phenomena: (i) changes in the structure of the interface when one metal is replaced by another; (ii) stress caused by mismatch between the crystal structure of substrate and deposited metal; (iii) loss of an intermediate (Cu+ in the case of Cu deposition) from the reaction zone due to diffusion into the bulk of the solution. Small deviations from the Sauerbrey equation were also observed in the case of discharge of Au3+ on Au, where Au+ could be a stable intermediate. All the effects mentioned above were observed in experiments conducted at very low current densities (i < 100 µA/cm2) during the early stages of deposition of a metal on a foreign substrate, when only a few monolayers were deposited. The aim of the present work was to generate various types of roughnesses on the same quartz resonator in the course of the same experiment and to study their effect on the response of the QCM. Here we (38) Tsionsky, V.; Daikhin, L.; Zilberman, G.; Gileadi, E. Faraday Discuss. 1997, 107, 337-350. (39) Tsionsky, V.; Gileadi, E. Mater. Sci. Eng. A 2001, 302 (1), 120-127.

558 Analytical Chemistry, Vol. 74, No. 3, February 1, 2002

used a well-known fact that metal plating under diffusion limitations leads to surface roughening. The shift of frequency during deposition with a current close to the limiting current density is presented in Figure 3a. Curve 1 shows the ratio of the observed frequency shift (∆fexp) to the shift predicted by eq 16, (∆fcalc). The latter value was calculated under the assumption that Au deposition is a result of Au3+ discharge with 100% efficiency. At short times, the ratio ∆fexp/∆fcalc is much larger than unity and decreases rather sharply with increasing time, leveling off at a value of ∼1.25. The response of the QCM is evidently larger than that predicted for mass loading alone. Correspondingly, the resonance width (Γ) increases monotonically and does not seem to approach an asymptotic value, as seen in curve 1 of Figure 3b. The results obtained during deposition at much lower current densities (i/iL , 1) show quite different behavior, as seen in curves 2 in Figure 3a and b: the ratio ∆fexp/ ∆fcalc is very close to 1.0 and there are practically no changes in Γ. The details of the behavior of the curve presented of the type shown in Figure 3 will be discussed in a future publication. When deposition was interrupted, no remarkable changes of f or Γ were observed. Thus, the process described by curves 1 in Figure 3 could be terminated at any point during the experiment to obtain a resonator with the desired properties. Figure 4 shows typical cross-sectional profiles of STM images obtained at different scales for two types of surfaces. Every profile shown in Figure 4 is a result of an independent STM measurement. To emphasize the features of roughness at small scale (Figures 2 and 3), a smooth long-scale background was subtracted from the original image. It should be noted that the difference between the two types of surfaces is particularly striking at the intermediate scale, showing that the electrochemically prepared surface (EAu) is more rough than that deposited by vacuum

Figure 4. Profiles of STM images of the gold surfaces of the QCM at different scales: (a) as purchased (VAu) and (b) after electrochemical deposition of additional gold layers at current density close to limiting current (EAu).

(VAu). The STM images as well as images obtained with an optical microscope show that both surfaces, before and after electrochemical deposition, can be classified as having slight roughness, with h/l , 1, and hence do not have inhomogeneities that could result in splitting of the QCM resonance, which justifies the use of the Sauerbrey equation when discussing the experimental data. DISCUSSION Equations describing the influence of roughness on the QCM impedance show that to characterize the response in a liquid we need the absolute value of Γ and the shift of the frequency ∆f, which corresponds to a transfer of the resonator from a nonviscous gas phase to a viscous liquid. In other words, we need a reference point. The resonance frequency in hydrogen at 1 atm was chosen. As shown in our previous work,40 under these conditions, even resonators with rough surfaces do not exhibit significant dissipation due to the very low viscosity of hydrogen. Figure 5 shows a typical set of raw data of the real part of admittance obtained in the experiment in different fluids, including gaseous H2. It is impossible to present data obtained for hydrogen (1) and in liquids (2-5) on the same scale. For the scale chosen here, the admittance in hydrogen looks like a δ function (a vertical line). Whereas, on another scale (cf. inset to Figure 5), it looks like very narrow (Γ 10 Hz, as compared to several kilohertz in liquids). Note also that the admittance in H2 is much higher than in the liquids shown in Figure 5. Substituting hydrogen with different liquids decreases the quality of the resonator by 4 orders of magnitude. The increase of viscosity and density of the liquid broadens the resonance curve. The usual form of presenting the experimental data in liquids is to represent the real and imaginary components of the QCM response as a function of the density of the liquid F and of the parameter (Fη)1/2. However, these (40) Tsionsky, V.; Daikhin, L.; Urbakh, M.; Gileadi, E. Langmuir 1995, 11, 674678.

Figure 5. Dependencies of the real part of the admittance of the QCM on frequency at 25 °C: (1) hydrogen, (2) dimethyl ether, (3) water, (4) 40% and (5) 50% aqueous solutions of sucrose.

parameters are the natural variables only for the flat interfaces. Equations 3, 4 and 12, 13 show that for rough surfaces it is more convenient to consider the quantities ∆f/f 2F and Γ/f 2F, which are the functions of the velocity decay length in the liquid, δ, as shown in Figures 6 and 7. The dependence of these two parameters on δ is linear for the ideally smooth surface of quartz resonator loaded in one side, as shown in Figures 6 and 7(1). Points represent experimental data for two surfaces with different roughness. The stronger the roughness, the larger the deviation of the experimental data from the lines for the ideally smooth surface. The experimental dependence of the quantity Γ/f 2F on the velocity decay length exhibits a sharp increase from zero to some finite value at small δ followed by gentle growth at larger values of δ. This effect is particularly pronounced for very rough surfaces. In Figure 6a, the theoretical dependence of the Γ/f 2F on the decay length is shown for various values of correlation lengths (curves 2-4). These curves were calculated within the model of the strong roughness (eq 13). The curves show a sharp increase of Γ/f 2F at small δ, but it is accompanied by an abrupt decrease for larger δ, where this parameter approaches the line expected for the ideal surface (line 1). To demonstrate the effect of slight roughness, we show a curve 5, calculated according to eqs 4 and 9. We choose the parameter R so that the straight line 5 connects the origin with the experimental point for the largest value of the velocity decay length. The behavior of Γ/f 2F for different film thicknesses is shown in Figure 6b. Here curves 2-4 were calculated in accordance with eq 13. Lines 1 and 5 are the same as the lines 1 and 5 in Figure 6a. From Figure 6b, one can see that the increase of the film thickness, L, results in increase of the width. Figure 6 Analytical Chemistry, Vol. 74, No. 3, February 1, 2002

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Figure 6. Dependencies of the imaginary part of the admittance of the QCM on the decay length: Points show experimental data. Lines 1a and 1b show the QCM response for ideally smooth surface; 2a4a show the influence of strong roughness in accordance to eq 13 for different values of ξH (2, 69 nm; 3, 172 nm; 4, 276 nm) and L ) 506 nm; 2b-4b show the same for different values of L (2, 460 nm; 3, 506 nm; 4, 690 nm) and ξH ) 506 nm; 5a and 5b show the influence of slight roughness (roughness factor R ) 1.3, eqs 4 and 9).

shows that there is no way to describe the experimental data under the assumption that only one type of roughness, strong or slight, characterizes the surface. Thus, we are forced to assume that in our experimental system we have a multiscale roughness, which could be described as a combination of the strong and the slight rough structures (cf. Figure 1c). Under these assumptions, it is possible to use eqs 3, 4, 12, and 13 to calculate the resonance frequency shift and the change in the width of the resonance. The results of the fitting are shown in Figure 7. The calculations demonstrate that the correlation length, l, of slight roughness is much larger than the velocity decay lengths for both surfaces: 690 nm for VAu and 1334 nm for EAu, and the ratio h/l equals 0.18 and 0.39, respectively. After fitting the Γ/f 2F versus δ plots, we have almost all the parameters needed to describe the QCM response. The only quantity that we still have to find is the pressure-induced part of the slight roughness contribution to the frequency shift that does not depend on δ (see eqs 3 and 7). This quantity can be determined from the fitting of the dependence ∆f/f 2F on δ. The results are shown in Figure 7b. STM measurements (Figure 4) clearly demonstrate the multiscale nature of roughness. The characteristic lateral lengths observed in profile images vary from 0.7 to 7 µm. The heights of these structures vary from 20 to 200 nm. Thus, the only slight roughness of different scales is seen in the STM data. Some of the scales have a very small ratio (h/l). For example, Figure 4, panel 1a, 1b, shows that for the lateral coordinate lying in the 560 Analytical Chemistry, Vol. 74, No. 3, February 1, 2002

Figure 7. Dependencies of imaginary (a) and real (b) parts of the admittance of the QCM on the decay length in different liquids for the ideally smooth surface (lines 1) and experimental data for two real surfaces: EAu (open circles) and VAu (closed circles). Lines 2 and 3 represent results of the fitting; see text.

range between 5 and 8 µm the slope (h/l) equals 0.07. The contribution of an interfacial roughness with such a slope to the response of the QCM is negligible. However, the same figure indicates that there are regions where the slope (h/l) equals to 0.3. The latter agrees with the slope used in our fitting calculation. Thus, not all scales of the roughness seen in the STM images are expected to yield a significant contribution to the response of the QCM. It should also be noted that STM measurement of roughness is a very complicated task and not all roughness scales could be presented on STM images.41 The parameters (ξH and L) of the strongly rough film, which we obtained from the fitting, show the scale of the rough structure of the interface that influences the QCM response. For this scale, the correlation length, ξH, equals to 207 and 172 nm, and L ) 368 and 506 nm for EAu and VAu, respectively. Thus, the structure of roughness strongly depends on the procedure that was used for the preparation of the interface. Unfortunately, STM images do not include information on very narrow cavities or pores. Thus, the above parameters cannot be associated with the length scales shown in Figure 4. CONCLUSION We have studied the effect of surface roughness on the response of the QCM in contact with a liquid. Performing experiments in liquids having a wide range of viscosities and densities made it possible to understand, for the first time, what characteristics of roughness influence the QCM measurements. (41) Hiesgen, R.; Meissner, D. Ultramicroscopy 1992, 42, 1403-1411.

It is clearly demonstrated that, to describe the experimental data, one has to account for the multiscale nature of roughness. In this paper, we used a simplified model that includes two type of roughness: slight and strong. Suitable roughness parameters could be found for each type of surface tested that would fit the experimental data to the models developed. Even though the fit is not perfect, this can serve as very strong support for the validity of the models employed, in view of the large range of solvents tested. Some of solvents are totally different chemically and have viscosities that differ by 2 orders of magnitude. Measurements of the frequency shift observed are clearly not sufficient for interpretation of the experimental data observed, and a full analysis of the impedance spectrum is called for. It should also be noted that the information on roughness, obtained in QCM experiments, is different from that found from STM. QCM measurements do not see weak roughness with very small slopes, h/l , 1, which give the main contribution to the STM images. A word of caution is needed, in particular for the common use of the QCM in the electrochemical mode. Considering Figure 6 and Table 1, we note that most of the liquids one would use for

electrochemical and analytical studies would appear in the lower end of Figure 6, where the velocity decay length is in the range of 100-300 nm. This is the region where the interplay between the two types of roughness is strongest and it is most difficult to fit the data to either model. This inherent difficulty should be borne in mind whenever an attempt is made to interpret the impedance response of the QCM, when operated with simple liquids (notably water, alcohols, and most of the polar solvents used in nonaqueous electrochemistry). ACKNOWLEDGMENT Financial support for this work by the Israel Science Foundation of the Israel Academy of Science and Humanities is gratefully acknowledged. D.Z. thanks Tel-Aviv University for providing a Postdoctoral Fellowship.

Received for review July 9, 2001. Accepted October 30, 2001. AC0107610

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