Faraday Discuss., 1997, 107, 91»104 Viscoelastic properties of thin �lms studied with quartz crystal resonators Oliver Wol� ,a Eberhard Seydelb and Diethelm Johannsmanna*§ a Max-Planck-Institute for Polymer Research, PO Box 3148, D-55021 Mainz, Germany b KV G»Quartz Crystal T echnology, GmbH, PO Box 61, D-74922 Neckarbischofsheim, Germany We discuss the possibility of determining shear moduli for thin polymeric –lms coated on quartz resonators by analysing the variation of normalized frequency shift d f/f with diÜerent harmonics.For sufficiently thick –lms, the elastic compliance can be deduced from the increase in d f/f with overtone order. However, a correction must be applied to account for a frequency dependence of the Sauerbrey factor relating frequency shift to –lm mass.Since the elastic eÜect scales with the square of the –lm mass, the correction is most important for thin –lms. Possible sources of frequency dependence of the Sauerbrey factor are the –nite thickness of the electrodes, lateral stress components, and insufficient control of the electrical boundary conditions. In order to obtain the shear compliance, the frequency dependence of the Sauerbrey factor was measured and subtracted from the data on polymeric thin –lms.The correction procedure can be avoided by using electrodeless quartzes and exciting the vibration with external electrodes across an air gap. Since, in this case, the Sauerbrey factor is approximately constant, the shear compliance can be obtained directly from the variation of d f/f with overtone order. The structure and dynamics of polymeric materials on the mesoscopic scale has become of increasing interest.This has been partly stimulated by the widespread tendency towards miniaturization in the semiconductor industry. On the other hand, there is a genuine scienti–c interest in polymers at this length scale because the mesoscopic scale coincides with many intrinsic length scales of polymers such as the radius of gyration, the hydrodynamic screening length, the domain size in block copolymers, the persistence length of stiÜ main chains and many others.1h4 Mechanical dynamical studies on this scale, however, are difficult.The standard mechanical dynamical equipment does not achieve the required mechanical precision. The most prominent approach is the surface forces apparatus (SFA),5 where two crossed cylinders are approached to each other. The dynamical behaviour of polymers con–ned in the space between the cylinders can be probed by either modulating their distance or by shearing them with respect to each other.6h10 A somewhat complementary approach consists of analysing the resonances of quartz crystal oscillators.In the context of quartz crystal microbalances,11,12 where frequency shifts of coated quartz resonators are converted to deposited mass, viscoelastic eÜects were widely perceived as an impediment to accurate mass determination. If, on the other hand, sufficient information can be gathered from the quartz resonator, the viscoelastic constants may actually be derived from § e.mail : johannsmann=mpip-mainz.mpg.de 9192 V iscoelastic properties of thin –lms quartz measurements.When resonance frequency and bandwidth are measured on several harmonics, this becomes feasible. In order to analyse many resonances it is advantageous to determine the acimpedance of the resonator passively with an impedance analyser instead of including the resonator in an oscillator circuit.Fig. 1 shows typical traces of the ac-conductance around a resonance. We –t Lorentzians to these curves and use the centre and bandwidth for further analysis. A second advantage of impedance analysis is that trouble shooting is much easier when the whole impedance spectrum, including all undesired modes, is available. The formalism to derive shear compliances starts out from a one-dimensional acoustic model.13h16 Neglecting the eÜect of electrodes, the frequency shift of a quartz resonator in contact with a viscoelastic medium is17h20 d f *\i ff p Z* Zq (1) where d f *\d f]idC is the shift of the complex resonance frequency f *\f]iC, with 2C the bandwidth.kg m~1 s~2 is the acoustic impedance of AT-cut Zq\8.8]106 quartz and the fundamental frequency.The generalized acoustic impedance Z* at an ff interface is de–ned as Z*\p/(du/dt) with p the shear stress and u the particle displacement. All quantities are complex. Eqn. (1) assumes an acoustically homogeneous, laterally in–nite quartz plate with Electromechanical eÜects are ignored. For a travelling acoustic wave in a bulk df */f@1. medium, the generalized impedance Z* is the same as the conventional acoustic impedance Z\ov\(oG)1@2 with o the density, v the speed of sound, and G\G@]iGA the shear modulus.When several waves contribute to the stress, p is the sum of the shear stress exerted by all waves, including the waves re—ected in the sample. The generalized impedance Z* clearly is not a material constant but depends on geometry.In a geometry with shear waves travelling in both directions, we get Z*\G ik(u`[u~) iu(u`]u~) \)(oG) 1[r 1]r\Z 1[r 1]r (2) Fig. 1 Ac electrical conductance of a quartz resonator around its third harmonic. The frequency and the bandwidth are obtained by –tting Lorentzians to the conductance.O. W olÜ et al. 93 with k\(o/G)1@2 the wave vector, and the amplitudes of waves travelling forward u` u~ and backward, the normalized amplitude of the re—ected wave at the quartz r\u~/u` surface, and Z\(oG)1@2 the acoustic impedance of the –lm at the quartz surface.Z is understood as a material constant, not as a generalized impedance in the sense of eqn. (1). An important feature appears when the re—ectivity r in eqn. (2) approaches [1. This is, for example, the case when the sample is a homogeneous –lm with a thickness d equal to a quarter of the wavelength of sound.In this case the bandwidth 2C becomes large. The frequency shift df decreases with increasing –lm thickness and can even change sign. This situation has been termed ì–lm resonanceœ because the –lm itself forms a resonator which has an eigenfrequency close to the driving resonatorœs frequency.When the two coupled resonators (quartz and –lm) have similar frequencies, transfer of energy into the –lm is most efficient and the damping of the quartz resonance reaches a maximum. For a rubbery polymer –lm with a thickness of a few lm, the –lm resonance typically occurs in the range of tens of MHz. The –lm resonance has, for example, been used to probe the solvent induced plasti–cation of a 1.6 lm –lm of poly(isobutylene).21 Upon swelling both the mass and the compliance of the polymer –lm increase.22 The changes in the elastic compliance –t well to a solvent-induced decrease in the glass-transition temperature Tg . For swollen polymeric layers in a liquid environment, –lm resonances can also be observed; via a –lm resonance, the ìacoustic thicknessœ of an adsorbate can be assessed.In ref. 23, the degree of swelling of a polymer brush24h26 has been acoustically measured in this way. Depending on solvent quality, the polymer chains stretch away from the surface to lower the osmotic pressure inside the brush.27,28 For the polystyrene» cyclohexane system, the solvent quality can be varied via temperature. As the degree of swelling changes, the diÜerent harmonics go through the condition for –lm resonance.Accurate modelling has to account for the smoothness of the viscoelastic pro–le. Interestingly, the equivalent thickness derived from the acoustic method is signi–cantly higher than the ellipsometric thickness because the acoustic technique is very sensitive to the dilute outer tails of the segment density distribution.If the acoustic load is a –lm of thickness d, the calculation of r in eqn. (2) is readily done analytically and we &ndas d f *\ i ff pZq Z*\[ ff pZq Z tan(kd)\[ ff pZq )(oG) tan[)(o/G)ud] (3) The pole of the tangent corresponds to the –lm resonance. For thin –lms eqn. (3) can be Taylor expanded around d\0. Using we –nd tan(x)Bx]13 x3 d f *B[ ff pZg Cmu]J(u) m3u3 3o D (4) with J\J@[iJA\1/G the shear compliance and m\od the mass per unit area.The frequency shift is not entirely due to the inertia of the deposited mass, there is an elastic correction. Eqn. (4) is a complex equation and can be used to determine both J@ and JA. For rubbery –lms with a thickness in the lm range, the elastic contribution is quite strong. If one neglects the dispersion of J(u), one can determine J@ by plotting d f/f vs.f 2 according to21,29 d f * f B[2 ff Zq mC1]J(u) 4p2m2 3o f 2D (5) The quantity can be considered as an ìapparent massœ. It equals the true [(d f/f )(Zq/2 ff) mass, if elastic corrections and/or all artifacts discussed below have been corrected for.94 V iscoelastic properties of thin –lms Fig. 2 Normalized frequency shifts d f/f (a) and normalized bandwidths dC/f (b) versus f 2 for a 1.6 lm –lm of poly(isobutylene).The mass m and the elastic compliance J@ can be determined from the oÜset and the slope in (a). The viscous compliance JA is determined from the slope in (b). With the elastic correction omitted, eqn. (5) is equivalent to the Sauerbrey equation.11 Fig. 2(a) shows a plot of d f/f vs.f 2 for a 1.6 lm –lm of poly(isobutylene). The mass m and the shear compliance J@ correspond to the oÜset and the slope, respectively. If dispersion cannot be neglected, one may assume a reasonable power law for J(u) and still obtain an estimate of J@. In practice, the in—uence of dispersion onto the derivation of J@ turns out to be rather minor. For JA the procedure is analogous [Fig. 2(b)]. For a small deposited mass, the elastic contribution in eqn. (5) can be neglected. If eqn. (5) were rigorously correct, the ratio d f/f would be constant and independent of the overtone order. The constant of proportionality relating relative frequency shift d f/f and mass m is and is sometimes called the ìSauerbrey constantœ. As we discuss Cs\2 f0/Zq below, is in fact not a constant but depends on overtone order.Therefore, we use the Cs term ìSauerbrey factor œ instead of ìSauerbrey constantœ. Experimentally, it turns out that d f/f does depend on the overtone order. The mass derived from the Sauerbrey equation depends on the harmonic used and diÜers between the diÜerent harmonics by up to some per cent. This shortcoming is not severe for mass determination.The determination of shear compliances, on the other hand, entirely rests on the frequency dependence of d f/f and yields seriously erroneous results, when the frequency dependence of the Sauerbrey factor is neglected. The case of negligible elastic contribution (i.e. low –lm mass) is the state of reference and needs to be accurately described. Because the elastic correction scales as m2 [eqn.(5)], this is particularly important for the investigation of thin –lms. Measurements of elasticity are more difficult than measurements of thin –lm viscosity because both the pure mass and the –nite shear compliance aÜect the frequency shift. Viscous phenomena can be traced down to the scale of molecularly thin –lms, because there is no problem of subtracting the eÜect of true mass.Any increase in bandwidth is known to originate from viscous dissipation. This fact has been exploited to investigate friction on the atomic scale.30O. W olÜ et al. 95 This paper is concerned with an improved description of the eÜect of pure mass loading on the frequency of quartz resonators. After the reference state of pure mass loading is accurately described, separation of the elastic contribution becomes feasible and the accuracy of shear compliances determination is increased.A second approach is to look for ways actually to eliminate the frequency dependence of the Sauerbrey factor. One –nds that, for electrodeless quartz blanks, the frequency dependence of the Sauerbrey factor is largely reduced.By exciting the blanks with external electrodes across an air gap, the resonances can still be measured. Using electrodeless quartz blanks is, therefore, an alternative route to the determination of shear compliances of thin –lms. Results A rough calculation shows that a –nite compliance J@ should be completely negligible for –lms thinner than ca. 10 nm, because the elastic contribution scales as the second power of mass.Langmuir»Blodgett (LB) layers of organic polymers are a convenient system for testing the eÜect of pure mass loading because they can easily be deposited layer by layer in a well de–ned way. We successively deposited LB layers of poly(cmethyl- L-co-n-octadecyl-L-glutamate), ìpolyglutamateœ for short, onto a quartz crystal. We used optically polished quartzes with planar surfaces.Polyglutamate forms good LB layers with a layer thickness of ca. 1.7 nm.31 Fig. 3 shows the normalized frequency shift d f/f on 20 harmonics as a function of f 2. Clearly d f/f is not constant. The straight lines are –ts to the function d f f \[2 ff Zq m(1]bS f 2) (6) While the functional form of eqn. (6) was certainly motivated by eqn.(5), it turns out that it is well con–rmed experimentally. When –tting a function with the a(1]bS f l) Fig. 3 Normalized frequency shifts d f/f vs. f 2 for a set of LB layers of polyglutamate. The observed positive slope is in disagreement with the simple one-dimensional model of quartz resonators. The straight lines are –ts to eqn. (6).96 V iscoelastic properties of thin –lms exponent l as a –t parameter, the exponent l is found to be between 1.8 and 2.The frequency dependence of the Sauerbrey factor is reasonably well characterized by the single parameter Eqn. (6) is ful–lled best on the high harmonics. On low harmonics bS . there is a correction due to the electrically induced stress. Also, energy trapping is more efficient on higher harmonics. In order to avoid in—uences from the mounting structure and E-–eld induced stress, it is preferable to work on harmonics higher than n\5.Because eqn. (5) and (6) have the same functional form, a plot of d f/f vs. f 2 always yields a straight line. However, the slope b has contributions from and the elastic bS correction : bfilm\bS]J@ 4p2m2 3o (7) with the experimentally observed slope. Once is known, it can simply be sub- bfilm bS tracted.can be determined in reference experiments by evaporating small amounts of bS gold onto the existing gold electrodes. If the added mass is small enough, one may assume that the additional gold layer behaves rigidly and use the frequency shifts to determine One may then deposit the polymeric layer of interest onto this quartz bS .resonator and subtract from to obtain J@. bS bfilm In Fig. 4 we plot versus relative electrode mass for three diÜerent quartzes. The bfilm electrode mass is not given as an absolute number because the frequency of the empty quartz is not known. In the –rst four experimental steps we evaporated additional gold onto the existing gold electrodes. These steps serve to measure The lines are –ts to a bS .parabola. After four steps of evaporation of gold, we deposited the polyglutamate LB –lms onto the electrodes. We investigated –lms of 40 and 80 layers of polyglutamate, corresponding to a thickness of 68 and 136 nm. These corresponding values for have bfilm been encircled in Fig. 4. They are signi–cantly displaced from the line. From the deviation, one derives J@\25 GPa~1 and 17~1GPa~1 for 40 and 80 layers, respectively.Subtracting clearly is an essential step in the derivation. To our knowledge, this is the bS –rst time that elastic properties have been obtained from quartz resonators for –lms thinner than 100 nm. Presumably, the scatter in the J@ values is due to systematic errors. We estimate the derived elastic constants to be accurate within a factor 2.Given the fact versus electrode mass for three diÜerent quartzes. The –rst four data points correspond Fig. 4 bS to the evaporation of gold onto the existing electrodes. The data are well described by a secondorder polynomial. The encircled data points correspond to the deposition of 40 and 80 LB layers of polyglutamate. After has been determined from the gold evaporation, the shear compliance bS of the LB layers can be determined from the change of slope indicated in the –gure.O.W olÜ et al. 97 that the elastic moduli of polymers can vary over many orders of magnitude, these error bars are not prohibitively large. We have also measured the shear compliance of this polyglutamate by coating a second metal layer on top of the LB –lm.32 In this ìsandwich geometryœ J@ becomes accessible.In these experiments we –nd values for J@ similar to those given above. When repeating the experiment with diÜerent quartzes, we realized that is an bS increasing function of the electrode mass. This raises the question whether goes to bS zero in the limit of electrodeless quartzes. In this case, using electrodeless quartz blanks could be a way to circumvent altogether the problem of the frequency-dependent Sauerbrey factor.It is indeed possible to determine the resonance frequencies of bare quartz blanks, when the blank is suspended on an air cushion and the oscillation is excited across an air gap. Fig. 5 illustrates this scheme. This scheme has the additional attractive feature that all eÜects of mounting or energy trapping33 are eliminated. The price to be paid for the conceptual simplicity is the experimental difficulty related to the air gap.The amplitude of oscillations decreases. Also, due to E-–eld induced stiÜening, the resonance frequency depends on the width of the air gap. Fig. 6 displays the frequencies and the amplitude of oscillation of an empty quartz blank as a function of the air gap.The frequency has been divided by the overtone order for the purpose of display. The amplitude decreases with increasing air gap because the exciting electric –eld decreases. The normalized frequencies also depend on overtone order and decrease by ca. 0.02% when going to high harmonics. Presumably, this eÜect is caused by lateral components in the stress –eld and can be accounted for by rigorous three-dimensional modelling.Interestingly, the resonance frequency depends on the width of the air gap. This behaviour is rooted in the piezoelectric stiÜening due to the electric –eld. The dependence of frequency on the air gap (i.e. the electric –eld) is important for two reasons : –rst, it shows that precise control of the geometry is essential for performing reproducible experiments.We believe that the much reduced data quality encountered when using the air gap is partly caused by the fact that the quartz plate slightly moves on the air cushion. Secondly, it demonstrates the in—uence which the electrical boundary conditions can have on the resonance frequency. If the electrode thickness aÜects the electrical boundary conditions, this may well be a source of frequency dependence of the Sauerbrey constant.It turns out that, while the accuracy of frequency determination has suÜered, the frequency dependence of the Sauerbrey constant is much reduced for empty quartz blanks. Fig. 7 shows as function of electrode mass. Owing to the experimental diffi- bS culties originating from the air gap, the scatter of the data is considerably larger than in the previous experiment. can be determined only for electrodes with a thickness of bS Fig. 5 Experimental set-up to measure the resonances of electrodeless quartz blanks suspended on an air cushion. In this con–guration all in—uences from energy trapping or mounting are eliminated. With electrodes a non-vanishing is still observed. Without electrodes is below the level bS bS of sensitivity.98 V iscoelastic properties of thin –lms Fig. 6 Resonance frequencies divided by overtone order (a) and amplitudes of oscillation (b) for a bare quartz blank excited across an air gap for various widths of the air gap some tens of nm. The line in Fig. 7 is –t to a parabola with bBb0m2 (8) The best –t gives Hz~1 kg~1 m2. b0B[(2.4^0.4)]10~13 Fig. 8 shows d f/f vs. f 2 for a 300 nm –lm of poly(isobutylene) spin-cast onto a bare quartz blank. can be neglected and the slope is entirely given by J@ 4p2m2/3o. bS bfilm as a function of electrode thickness obtained by successively evaporating gold layers Fig. 7 bS onto a quartz blank. The data were taken by exciting the vibration across an air gap.is bS negative and its modules increases with electrode thickness. The line is a –t to a parabola with The best –t gives Hz~2 kg~1 m2. bBb0m2. b0B[(2.4^0.4)]10~13O. W olÜ et al. 99 Fig. 8 Normalized frequency shifts d f/f vs. f 2 for a 300 nm –lm of poly(isobutylene) spin-cast onto a bare quartz blank. Since is negligible, the slope can be converted directly to elastic compli- bS ance J@.J@B3 GPa~1. Unfortunately, the data are much noisier than in conventional measurements with electrodes evaporated onto the quartz crystals, because of the low amplitude oscillation. From the linear –t we derive J@B3 GPa~1. This value is somewhat higher than that derived in ref. 21 for a 1.6 lm –lm of poly(isobutylene). There is a literature value of J@\1.7^0.2 GPa~1 derived with the temperature»frequency superposition principle.34 This value lies between the values derived in ref. 21 and in this work. The procedure using an air cushion is, although conceptually more appealing, less accurate. For a quantitative determination of shear compliances, conventional electrodes and a separate determination of the coefficient is superior. bS Discussion In the following, we discuss three possible sources for frequency dependence of the Sauerbrey factor in more detail : (a) the –nite thickness of the electrodes, (b) lateral stress components due to the three-dimensional nature of the quartz plates and (c) insufficient control of the electrical boundary conditions.(a) Finite thickness of the electrodes Presumably, a major contribution to the frequency dependence of the Sauerbrey factor is due to the electrodes.It has previously been recognized that the Sauerbrey factor decreases when very thick electrodes are used,35 some commercial quartz crystal microbalances have built-in software to correct for this eÜect. An analogous argument applies to the case of high harmonics. We consider the case of symmetric electrodes and use the continuity of stress and displacement at the quartz/electrode interface : Asin(kq h)\Bcos(ke d) (9a) AGq kqcos(kq h)\BGe kesin(ke d) (9b) where A and B are the amplitudes of oscillation in the quartz and the electrodes, k\ is the wave vector, the indices ìqœ and ìeœ denote the quartz and the electrode, u(o/Gq)1@2100 V iscoelastic properties of thin –lms h is half the thickness of the quartz blank and d is the electrode thickness.In eqn. (9) the stress due to the electric –eld is neglected. The determinant of eqn. (9) must vanish, which results in Zqcot(kq h)\Zetan(ke d) (10) where Z\Gk is the acoustic impedance. We express k as kn\k0, n(1]e)\np(1]e)/2h in order to focus on the small deviations of from the values for the empty k0, n\np/2h quartz.The small number e equals the normalized frequency shift d f/f, n\1, 3, 5, . . . is the overtone order. With cot[np(1]e)/2]\[tan[npe/2] we get [Zq tanAn p 2 eB\Ze tanCn p 2 vq ve (1]e) d hD (11) Expanding eqn. (11) to –rst order in e and d results in eB[ oe oq d h (12) which is the Sauerbrey equation for symmetric mass loading. For high harmonics, this linearization is not accurate enough.While the –lm deposited onto the electrodes may be very thin, the electrodes themselves are not. When increasing the thickness of the electrodes by a small amount dd, the viscoelastic properties of the new layer are of no importance to the frequency shift d f. To –rst order, d f is proportional to the deposited mass dm\odd. However, the viscoelastic properties of the electrodes do in—uence the Sauerbrey factor, which is the constant of proportionality.When the electrode thickness is increased by a layer of thickness dd, eqn. (11) reads : [ZqtanCn p 2 (e]de)D\ZetanCn p 2 vq ve (1]e]de)Ad h]dd h BD (13) To –rst order in de and dd this is : [ZqCn p 2 e]n3Ap 2B3 e3 3 ]n p 2 de]n3Ap 2B3e2deD\ ZeGn p 2 vq ve C(1]e) d h](1]e) dd h ]de d hD] 1 3 An p 2 vq veB3C(1]e)3Ad hB3]3(1]e)2Ad hB3 de]3(1]e)3Ad hB2 dd h DH (14) Eqn.(14) is ful–lled for dd and de\0. Terms of zeroth order in dd and de can therefore be subtracted. Collecting terms linear in de and dd we get deC[nZq p 2 [n3ZqAp 2B3e2[nZe p 2 vq ve d h[n3ZeAn 2 vq veB3 (1]e)2Ad hB3D \ dd h CnZe p 2 vq ve (1]e)]n3ZeAn 2 vq veB3 (1]e)3Ad hB2D (15) If both e and d are neglected, we again –nd the Sauerbrey equation.Keeping e and d to second order results in de\[dd h Ze Zq vq ve C1]e] Ze Zq vq ve d h]n2Ap 2 vq veB2Ad hB2[n2Ap 2B2e2D (16)O. W olÜ et al. 101 Inserting [eqn. (12)] yields eB([oe/oq)(d/h) de\[ oe oq dd h G1]n2Ap 2B2CAvq veB2[Aoe oqB2DAd hB2H (17a) or d f f \[dm ff Zq G1]f 2m2A p 2oh ffB2CAvq veB2[Aoe oqB2DH \[dm ff Zq [1]b0 f 2m2] (17b) The above calculation was carried out for the symmetric case for simplicity. The analogous calculation for an electrode deposited on just one side yields a prefactor of 1/2 on the right-hand side of eqn.(17). The coefficient remains unchanged. Eqn. (17) predicts b0 that the Sauerbrey factor should depend linearly on f 2 and on the square of the electrode mass m. It is in accordance with experiment as far as the functional form is concerned.Inserting numbers for gold electrodes on 4 MHz quartz, we obtain a value of Hz~1 kg~1 m2. This is about twice the experimental value of b0\[4.7]10~13 ([2.4^0.4)]10~13 Hz~1 kg~1 m2 (Fig. 7). If the small mass added onto the electrodes is of a diÜerent material (such as a polymer), a three-layer model is needed.Eqn. (10) is changed to16 Zqcot(kq h)\ Zetan(ke de)]Zftan(kf df) 1[Zf/Zetan(kf df)tan(ke de) (18) where the subscript f denotes the –lm on top of the electrodes. To –rst order in eqn. df (17) remains unchanged. The viscoelastic properties are only noticeable when considering terms of third order in Expanding the right-hand side of eqn. (18) to third order df. in and u, one –nds df , de , Zqcot(kq h)Bu(me]mf)] u3 3 Cmf2 Zf3]3mf2me Ze2 ]3mfme2 Ze2 ] me3 Ze2D (19) where the masses m\od have been used instead of thicknesses.The acoustic impedance of the –lm does not aÜect the term linear in u (i.e. overtone order n). It does, however, Zf in—uence the term proportional to u3. The diÜerence in induced by the viscoelastic bfilm properties of the –lm is *b\4p2 3 mf2 1[Zf2/Ze2 Zf2 (20) *b is indicated by arrows in Fig. 7. Since is much smaller than for polymer –lms Zf Ze on gold electrodes, eqn. (20) can be approximated by *bB4p2 3 mf2 Zf2\Jf 4p2mf2 3o (21) This relation has been used in the Results section to derive the shear compliances for a –lm of polyglutamate (Fig. 4). (b) Lateral stress components Owing to the –nite lateral width of the quartz blanks, the modes of oscillation are not pure thickness shear modes, but contain lateral shear components as well.Stevens and Tiersten36 have given an approximate analytical description of quartz resonators with102 V iscoelastic properties of thin –lms Fig. 9 Normalized frequencies (a) and frequency shifts (b) and (c) calculated according to ref. 36 for a bare quartz blank and a quartz blank covered with 2 [3.2 nm, (b)] and 8 [12.8 nm, (=) (Ö) (c)] LB layers of polyglutamate (>) convex surfaces.We have calculated the frequencies of the pure thickness shear modes with a commercial software package (KVG, Neckarbischofsheim) within this threedimensional formalism. The radius of curvature was set to 10 m, which comes close to a —at surface.The resonance frequencies for the empty quartz blanks are shown in Fig. 9(a). The program reproduces the experimental data qualitatively, as shown in Fig. 6(a). However, the decrease in f/n with overtone order is stronger in Fig. 6. The normalized resonance frequencies decrease with overtone order because the relative importance of the lateral stress components decreases. The vertical gradients of displacement scale as the overtone order, whereas the lateral gradients increase only slightly with overtone order.In a second step, we calculated the shift of frequency on deposition of LB –lms of polyglutamate [Fig. 9(b) and (c)]. d f/f depends on frequency. There is a somewhat irregular behaviour on low harmonics because the amplitude of oscillation at the rim of the quartz is still large enough to in—uence the resonance frequency. The quartz blanks are not contoured and there is no energy trapping. On higher harmonics, the pro–le of oscillation is entirely con–ned to the central area, and the dependence of normalized resonance frequency on overtone order becomes regular.Fitting eqn. (6) to the data given in Fig. 9(b) and (c) (excluding low harmonics) yields in the range of 2x10~20 bS Hzv2.does not increase with –lm thickness. The agreement of the data with the func- bS tional form of eqn. (6) is not very good. Since, according to experiment, is negative, bS lateral stresses cannot be the only source of frequency dependence of the Sauerbrey factor, however, they may contribute to some extent.O. W olÜ et al. 103 The software also calculates the displacement pattern of a given mode. For high harmonics, the oscillation is con–ned to the central area of the quartz. This statement holds even in the absence of bevelling or energy trapping. For high harmonics, quartzes mechanically mounted at their rim can be easily excited, even in the absence of energy trapping. From these calculations, one may conclude that the mounting should not aÜect the Sauerbrey factor on high harmonics.(c) Electrical boundary conditions In eqn. (9) the stress from the external electric –eld has been neglected. With this coupling included, eqn. 9(b) is changed to13,37 ACGq kq cos(kq h)[ e34 2 e33 h sin(kq h)D]e34 /0 h \BGe ke sin(ke d) (18) with C m~2 a piezoelectric constant, the relative permit- e34\9.65]10~2 e33\4.54 e0 tivity, and the external voltage./0 Neglect of piezoelectric coupling may be justi–ed because –rst, the piezoelectric coupling as given by the coupling coefficient is weak and, sec- i2\e34 2 /(e33Gq)\0.008 ondly, because the new term drops out when taking the derivatives of eqn. (18) with respect to e and d [cf. eqn. (11)]. It aÜects the frequency but not the shifts in frequency caused by the –lm.On the other hand, the dependence of resonator frequency on the air gap width (Fig. 6) shows that the electrical boundary conditions may be of importance. In eqn. (18) the voltage is treated as a –xed boundary condition. The current —owing /0 into the electrodes is supposed always to generate a charge density exactly compensating the shear-induced surface polarization.The charge density is For less A[e34/h sin(kqh)]. than perfect conductors, the current may not exactly match the surface polarization. This clearly is the case when external electrodes are used and there is a gap between the quartz surface and the electrodes. Formally, one could introduce a prefactor a(u) before the term in eqn. (18) to account for electrical imperfections. If a(u) depends on sin(kqh) both electrode thickness and frequency, a frequency dependence of the Sauerbrey factor may result.Conclusions Determining shear moduli for polymeric thin –lms from the shifts in resonance frequencies of quartz resonators remains a challenge. On high harmonics, the variation of normalized frequency shift with overtone order is described by d f/f\[2 ff/Zq(1]b f 2).The parameter b contains a contribution from the –nite elasticity of the deposited –lm. Elastic eÜects can easily be observed for –lms less than 100 nm thick. However, the parameter b is also in—uenced by other factors such as the –nite width of the electrodes, lateral stress components, and the electrical boundary conditions. We have proposed a scheme to subtract these other in—uences in order to derive the shear compliance J@ of the –lm from the experimentally determined parameter b.The approach rests on comparing the b parameters for an additional layer of gold and for the polymeric layer of interest. For electrodeless quartz blanks, the b parameter is dominated by the –lms elastic compliance. 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