Langmuir 1994,10, 2836-2841
2836
Influence of the Surface Morphology on the Quartz Crystal Microbalance Response in a Fluid Michael Urbakh* and Leonid Daikhin School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, 69978 Ramat-Aviv, Tel-Aviv, Israel Received February 25, 1994. In Final Form: May 16, 1994@ The effect of surface roughness on a quartz crystal microbalance (QCM) response in contact with a fluid has been investigated. We have combined a perturbation theory approach to the description of the effect of slowly varying interfacial profiles with the scaling analysis for the interfaces includinginhomogeneities of a small radius of curvature (bumpsand cavities). Arelation between the shift ofthe resonance frequency and the interface geometry has been found. We have discussed the dependencies of the QCM response on the fluid viscosity and on the frequency of the free quartz oscillator. The effect of roughness on the results of adsorption measurements in the QCM experiments has also been studied. It has been demonstrated that the results of QCM measurements can provide valuable information about the interface morphology.
Introduction In the last few years the quartz crystal microbalance (QCM) has been successfully employed as a chemicalsensing device for the studies of solid-fluid interface^.'-^ The high sensitivity of the QCM has been especially useful for the investigation of adsorption processes, underpotential deposition, dissolution of surface films, anion adsorption, and other electrochemical p r o c e s s e ~ . ~How?~ ever, the theory of the QCM operated in contact with the fluid is yet highly undeveloped, and the most experimental results for the frequency shift ofthe quartz resonator show deviations from the theoretical prediction^.^-' A common feature of the theoretical approaches is the neglect of microscopicproperties of interfaces and surface roughness. In order to be able to make use of the high resolution of QCM measurements, the relation between the frequency changes and the changes of surface microstructure must be studied. The present work has focused on the theoretical description ofthe influence of surface roughness at a solidfluid interface on the QCM response. It has been shown experimentally that surface roughness can drastically affect the resonance f r e q u e n ~ y . ~ The - l ~ authors of refs 8 and 9 attributed the effect of roughness to the additional mass of the solvent trapped in surface cavities. They concluded that changes in surface roughness during the electrochemical oxidation are the dominant contributors to the observed frequency shift. However, it has also been demonstrated6,' that the frequency shift at the rough interface cannot be reduced to the effect of the mass rigidly coupled to the surface. In order to overcome this problem, the authors ofrefs 5-7 and 10 assumed that the roughness can also increase the viscous energy dissipation in the @Abstractpublished in Advance A C S Abstracts, July 1, 1994. (1)Thompson, M.;Kipling, L. A.; Duncan-Hewitt, W. C.; Rajakovic, L. V.; Cavic-Vlasak, B. A.Analyst (London) 1991,116,881. (2) Yang, M.; Thompson, M. Anal. Chem. 1993,65, 1158. (3) Buttry, D. A.;Ward, M. D. Chem. Rev. 1992,92,1355. (4)Schumacher, R.Angew. Chem., Int. Ed. Engl. 1990,29,329. (5)Yang, M.; Thompson, M.; Duncan-Hewitt, W. C. Langmuir 1993, 9,802. (6) Beck, R.; Pittermann, U.; Weil, K. G. J . Electrochem. SOC.1992, 139,453. (7) Yang, M.;Thompson, M. Langmuir 1993,9,1990. (8)Schumacher, R.;Borges, G.; Kanazawa, K. K. Surf. Sci. 1985, 163,L621. (9) Schumacher, R.;Gordon, J. G.; Melroy, 0.J . Electroanal. Chem. 1987,216,127. (10)Beck, R.; Pittermann, U.; Weil, K. G. Ber. Bunsen-Ges. Phys. Chem. 1988,92,1363.
fluid. The network analysis of the effect of roughness and interfacial fluid structure was presented in ref 5. The quantitative relations between the resonance frequency shift and the surface morphology have been derived in our recent workll for the slowly varying interfacial profiles. In the present paper we have applied the previous approach to the consideration of the resonance broadening and the effect of adsorption on a rough interface. We have combined a perturbation theory approach" to the description of the effect of slowly varying interfacial profiles with the scaling analysis for the interfaces including inhomogeneities of a small radius of curvature. The effects of a randomly rough surface and a periodical corrugation have been considered here. A relation between the QCM response and the interface geometry (the average height and the correlation function of roughness, the size and the surface density of cavities and bumps) has been found. We have discussed the dependencies of the QCM response on the fluid viscosity and on the frequency of the free quartz oscillator. The effect ofthe roughness on adsorption (mass)measurements in the QCM in situ experiments has also been investigated. It has been demonstrated that the results of QCM measurements can provide valuable information about the interface morphology.
Description and Discussion We now consider a model for the coupling of shear waves in a piezoelectric crystal bounded by the rough surface with damped waves in a fluid. The solid surface profile may be specified by a single-valued function & y ) giving the local height of the solid with respect to a reference plane, which we take to be the plane z = d. We take the z axis pointing toward the fluid and the plane z = 0 being coincident with the unconstrained face of the quartz resonator. Here d is the average thickness of the quartz crystal film. For convenience, we choose the position of the reference plane z = d so that the spatial average of the solid profile function & y ) vanishes. In order to determine the effect of roughness on the resonance frequency of the quartz crystal, we have to find the solutions of a wave equation for elastic displacements in the quartz crystal and of the linearized Navier-Stokes equation for fluid velocities.12 The solutions must satisfy (11)Urbakh, M.; Daikhin, L. Phys. Rev. B 1994,49,4866.
(12) Landau, L. D.;Lifshitz, E. M. Fluid Mechanics, 2nd ed.; Pergamon: New York, 1987.
Q743-7463/94l24lQ-2836$04.5QlQ 0 1994 American Chemical Society
Quartz Crystal Microbalance Response in a Fluid the boundary conditions which include (a) the absence of forces acting on the unconstrained crystal surface z = 0, (b) the equality of crystal and fluid velocities at the interface z = t(x,y), and (c) the equality of the absolute values and the opposite directions of the shear stress on the fluid side of the interface and of the shear stress on the quartz side. The nonslip boundary condition b is one of the fundamental assumptions in fluid mechanics.12 While experiments on macroscopic scales are consistent with this condition, the recent measurements which probe molecular scales indicate that the boundary conditions may be different.13J4 Molecular dynamics simulations15J6 have shown that the degree of slip at the interface is related to the fluid structure induced by the solid. At large interactions between fluid molecules and the substrate the first one or two fluid layers became locked to the wall. We will focus on this case only. For the determination of the resonance frequency of the quartz crystal bounded by a rough interface it is convenient to substitute condition c by the energy balance in the system under consideration.“ The energy balance constitutes that the rate of the change of total energy of the crystal and the kinetic energy of the fluid should be equal to the rate of the energy dissipation in the fluid. At present, it does not seem possible to obtain a general solution of the problem for the interfaces with arbitrary roughnesses. Therefore, the main task is to select an approximation which takes into account the physical peculiarities of a particular problem. The form of the dependence of the resonance frequency on the properties of the fluid and on the interface morphology is determined by the relations between the characteristic sizes of surface roughnesses and the length scales defined by the Navier-Stokes equation for fluid velocities and by the wave equation’ for the elastic displacement in the crystal. These length scales are the and the decay length of fluid velocities, 6 = (2q/Q~f)l/~, wavelength of the shear-mode oscillations in the quartz crystal, il= 2n(u/~,)l/~Q-~, where q and @farethe viscosity and the density ofthe fluid,p and Q , are the shear modulus and the density of the quartz crystal, and Q is the frequency of oscillations. For the frequencies used in QCM, !2 FZ 1-10 MHz, the lengths 6 and ilare on the order of 0.1-1 pm and 0.1 cm, respectively. The STM measurements show that the characteristic heights of the roughness of metallic films on the quartz crystal are about tens or hundreds of nanometer^.^,^ Here we will consider only the surfaces for which the heights of roughness, h, are smaller than the decay length, 6. It is just these surfaces that are of greatest interest from the standpoint of various applications. The lateral scales of roughness can change over a wide range, from tenths to hundreds and even thousands of nanometers. As a result we are confronted with both “slight”roughness, for which the characteristic height is less than the lateral sizes, and “strong” roughnesses (cavities, pores, and bumps), for which the height is of the same order or larger than the lateral size. We remark that for the slight roughnesses the lateral sizes can be both less and more than the decay length 6 , but for the strong roughnesses the lateral sizes are always less than 6. The wavelength ofthe shear-mode oscillations in the quartz crystal is much larger than the size of roughness. As a result the presence of roughness discussed in this work does not influence the (13)Watts, E.T.; Krim, J.; Widom, A. Phys. Reu. B 1990,41,3466. (14)Israelachvili, J. N.;McGuiggan, __ P. M.: Homola, A. M. Science 1988,240,189. (15)Thompson, P. A.; Robbins, M. 0. Phys. Rev. A 1990,41,6830. (16) Thompson, P.A.; Robbins, M. 0. Science 1990,250,792.
Langmuir, Vol. 10,No. 8, 1994 2837 solution of the wave equation for the elastic displacements in the crystal. 1. Slight Roughness. First we discuss the effect of slight roughnesses on the QCM response. This problem has been recently considered by us in the framework of the perturbation theory1’ with respect to the parameters IVt(R)I 1 and 116
