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Faraday Discuss., 1997, 107, 1»13 Introductory Lecture Acoustic interactions from Faradayœs crispations to MEMS Richard M. White Department of Electrical Engineering and Computer Sciences, and the Berkeley Sensor and Actuator Center, University of California, Berkeley, CA, USA, 94720 In an 1831 paper, Michael Faraday described observations of the interaction of a vibrating solid with a liquid supported by the solid.The motions induced on the surface of the liquid, which he termed crispations, resulted from the creation of capillary waves, which are still a subject of research. The contributions to the present Faraday Discussion concern chie—y the use of piezoelectric crystals to study the properties of –lms and of liquids at liquid/solid interfaces. In this Introductory Lecture, the characteristics of piezoelectric acoustic devices used for sensing are reviewed and contrasted.Recent developments in the fabrication of micromechanical structures to make microelectromechanical systems (MEMS) are reviewed. Some applications of this technology to make ultrasonic piezoelectric devices that sense and actuate liquids and gases are described. 1 From Faraday to the present The topic of acoustic interactions is quite appropriately the subject of a Faraday Discussion, since Faraday himself contributed to this –eld. Faraday published in 1831 a long paper on the interactions of vibrating solid surfaces with particles and liquids.1 Faraday described many ingenious experiments performed on metal, glass and wooden plates that supported piles of granular media such as sand, or which were in contact with water or other liquids such as ìwhite of egg, ink and milkœ.The solid members were excited by rubbing, bowing or being struck. In the articleœs long appendix, titled ìOn the Forms and States assumed by Fluids in contact with vibrating elastic surfaces œ, Faraday wrote that ìCrispations appear on the surface of the water .. . The crispation presents the appearance of small concoidal elevations . . . œ. In other words, the surface has been ìcurledœ as a result of the acoustic interaction. Faraday studied the relationship of the frequency of the vibrations of the liquid surface to that of the underlying solid member. In order to be able to make these measurements, he had to lower the frequencies of the vibrations to values low enough so that he could count them.This entailed making his apparatus larger and larger, and he ultimately worked with a board ìeighteen feet long . . . , the layer of water being now three fourths of an inch in depth and twenty-eight inches by twenty inches in extentœ. A key –nding was that ìEach heap [of liquid] . . . recurs or is re-formed in two complete vibrations of the sustaining surface ; but as there are two sets of heaps, a set occurs for each vibration.œ In other words, the frequency of the disturbance at a point on the liquid surface is one-half that of the supporting vibrating surface. As the support rises, alternate peaks appear on the liquid surface ; as the support falls, the liquid forming those peaks —ows sideways, so that on the next upward motion of the support, 12 Introductory L ecture peaks appear on the liquid surface between the original peaks.Any peak rises only once for every two upward motions of the support. This conclusion was the subject of discussion for the next –fty years, until, in 1883, Lord Rayleigh published a paper titled, ìOn the Crispations of Fluid Resting Upon a Vibrating Supportœ that settled the matter.2 Incidentally, Rayleigh noted in his initial paragraph that ììSimilar crispations are observed on the surface of liquid in a large wine-glass .. . which is caused to vibrate in the usual manner by carrying the moistened –nger round the circumference.œœ, thus opening this experimental –eld to us all. By the time Rayleigh studied these disturbances, the experimental arts had advanced substantially, so he was able to excite the vibrating members electromagnetically and measure the frequency with a motor-driven apertured disk that rotated at a known rate and through which he could observe the motion of the liquid surface and the underlying support.From such observations made with liquid on a bar vibrating at 31 Hz, Rayleigh was able to determine that ìthere are two complete vibrations of the support for each complete vibration of the water, in accordance with Faradayœs original statement.œ Rayleigh proceeded to analyze the vibrations and identify the role of surface tension, which governs the velocity of wave propagation for small wavelengths where the eÜect of gravity is negligible.Incidentally, such capillary waves are still the subject of research. It is interesting to compare Faradayœs 1831 paper with scienti–c articles today. His paper is quite long by modern standards, and it is enjoyable to read. Faraday describes his ingenious experiments, which he illustrates with small drawings, but he uses no equations. This is probably due to Faradayœs lack of mathematical training.3 Todayœs journal articles are much shorter, often less readable, and they typically employ mathematical techniques ranging from classical analysis to numerical modelling and simulations.Experimental observations are made using sophisticated electronic, mechanical, optical, chemical and biochemical techniques, with instruments that are often controlled by a computer.We now take for granted our ability to ìseeœ individual atoms and to design molecules. These substantial changes are re—ected in the papers that follow in the present Discussion. But for experimentalists, some problems remain from the old days, such as how to glue one part onto another to form a room temperature bond that survives in a humid atmosphere. 2 Acoustics The range of frequencies involved in acoustical phenomena is today enormous, as Fig. 1 shows.Useful frequencies range from roughly 0.01 Hz for terrestrial phenomena to several terahertz, the highest frequency at which coherent elastic waves have been generated piezoelectrically, by means of light incident upon a piezoelectric crystal. The frequencies used in the papers that follow range from a few to hundreds of MHz for the acoustic sensors indicated in boxes in Fig. 1, and up to 1 GHz for the acoustic microscope. The choice of materials that can be used for acoustic transduction is now large. In the early 1900s, the only practical piezoelectric material was natural quartz that had been cut and polished. Today, properly oriented crystalline quartz is still important, but many other piezoelectrics are also used, such as lithium niobate and tantalate, zinc oxide, aluminium nitride, lead-zirconate-titanate and gallium arsenide.Some of these piezoelectrics are cut from a larger crystal, an example being the familiar quartz crystal shear-mode crystal employed in the quartz crystal microbalance (QCM). Films of these piezoelectrics are also important.They are produced by a variety of means: sputtering (from the pure element in a reactive chemical gas or from a compound source), pulsed laser deposition (ablation from a compound target upon which intense laser pulses are incident), chemical vapor deposition (CVD), spun-on polymeric piezoelectrics, and the sol»gel technique (spin-on deposition of chemical precursors in anR.M. W hite 3 Fig. 1 Acoustic (elastic) wave spectrum. The frequencies of waves that have been excited or detected range over roughly fourteen orders of magnitude. The chief acoustic devices used for studying acoustic interactions are boxed. organic binder that is later vaporized). Magnetic, electrostatic and thermal acoustic excitation means are also available.Fig. 2 shows the con–gurations, electrodes and particle motions of the acoustic sensors that have been used most frequently to study acoustic interactions with thin –lms and interfaces.5 In Fig. 2 the arrows indicate the particle motion in the solid, and the heavy black regions represent the conducting electrodes used to excite the devices. Fig. 2 Structures and properties of selected acoustic devices that may be used to study interactions with thin –lms and liquids.See text for details. (Reprinted with permission from ref. 4 ( 1993 American Chemical Society.)4 Introductory L ecture Fig. 3 Pictorial representations of elastic waves in solids. Motions of groups of atoms are depicted in these cross-sectional views of plane elastic waves propagating to the right.Vertical and horizontal displacements are exaggerated for clarity. Typical wave speeds, are shown below each vp , sketch. The familiar QCM, used in the majority of the papers that follow, is identi–ed here as the thickness-shear mode device (TSM), emphasizing the mode rather than the crystalline material or its gravimetric capability. The particle motion at the boundary of the piezoelectric is parallel to the plane face of the crystal ; coupling into an adjacent liquid occurs because of the zero-slip condition : the molecules of liquid must follow the motion of the solid.(Detailed consideration of this boundary condition appears in several papers in this Discussion.) The acoustic energy is distributed throughout the thicknessshear crystal, which is usually operated at resonance and so is an odd integral number of half-wavelengths thick.Because of the mechanical fragility of very thin crystals, the operating frequencies of TSM devices are typically no higher than tens of MHz. As a result, the acoustic energy in the solid is contained in a relatively thick region (the crystal thickness is 100 lm or more). Since the gravimetric response decreases as the volume-to-surface ratio of a device increases, the gravimetric sensitivity, of the TSM Sm , is low in comparison with that of the other devices illustrated.6 The gravimetric sensitivity is de–ned as where f is the resonant frequency of the device and *f Sm\(*f/f )/*m, is the shift of resonant frequency produced when a mass per unit area *m is added to the surface of the device.In the surface acoustic wave device (SAW), the acoustic energy in the solid is concentrated within a region near the free surface whose thickness is a small fraction of a wavelength. SAW operation at hundreds of MHz is possible and the attainable gravimetric sensitivity can be an order of magnitude greater than that of the typical TSM device. Since the particle motion of the simple SAW has components that are both normal and parallel to the free surface, and since the SAW phase velocity greatly exceeds the speed of sound in most liquids, acoustic energy is radiated into an adjacent liquid, limiting the usefulness of the SAW for studying acoustic interactions with liquids. An alternative SAW device, the surface transverse wave device,7 employs a contoured or coated free surface and an oriented crystalline substrate that permits generation of aR.M. W hite 5 surface-bound acoustic wave that has purely transverse particle motion, permitting its use in studies of acoustic interactions with liquids. The third device in Fig. 2, the —exural plate wave (FPW) device, is made by micromachining processes discussed below.A piezoelectric –lm located on one side of a thin supporting membrane is excited by interdigitated conducting electrodes, as with SAW excitation. Because of its location to one side of the neutral plane of the composite membrane, the electrically induced deformation of the piezoelectric excites a propagating wave that involves —exure of the membrane, as shown by the cross section in the FPW view in Fig. 2. The gravimetric sensitivity of the FPW device may be quite large. Because the phase velocity for the FPW decreases as the ratio of membrane thickness to wavelength decreases, the FPW typically operates in the low MHz range. In practical devices, the FPW phase velocity may be as low as a few hundred m s~1 (see Fig. 3), a value smaller than the speed of sound in water, and even in air at STP.Hence, even though the FPW membrane moves both perpendicularly and parallel to its surface, as does the surface of a SAW device, the FPW does not radiate a propagating wave into an adjacent —uid. Instead, an evanescent acoustic disturbance is generated in the —uid. The fourth structure shown in Fig. 2 is the acoustic plate mode (APM) device.Its particle motions are parallel to the top surface, resulting from the orientation of that surface relative to the crystalline axes of the single-crystal piezoelectric substrate. The APM device is rugged mechanically, as is the SAW, but its gravimetric sensitivity is relatively low because the acoustic energy is distributed throughout the relatively thick crystal. 3 Microfabrication and MEMS Starting in the 1980s, technologists have learned how to fabricate small mechanical structures using techniques, adaptations and augmentations of the photolithographic processes developed for making integrated circuits.8,9 Culminating in the active –eld called MEMS, for microelectromechanical systems, these developments are relevant to the present Discussion for two reasons : (1) one can use micromachining to make acoustic sources that can be used in research studies of acoustic interactions and (2) it is likely that some of the acoustic interactions being studied may –nd their way into commercial characterization systems made by the MEMS approach. Here, brie—y, are some of the present MEMS fabrication capabilities.Fig. 4 shows the two chief approaches to micromachining.In bulk micromachining, one fabricates structures via thin-–lm deposition and photolithography atop a singlecrystal silicon wafer. Ultimately, by etching through an aperture in a protectively coated region on the bottom of the wafer, one removes the majority of the underlying silicon. As an example, this process is used to make the FPW device of Fig. 2; etching of the silicon leaves a membrane that must be thin in order to support low-velocity plate waves. The other approach is surface micromachining. Here one builds the desired structures on an easily etched sacri–cial layer that is dissolved after patterning of the structures. This is the technique used to make movable structures such as the rotor of a micromotor or structures that are suspended only a few microns above a substrate (Fig. 5). Wet chemical etching used in such fabrication is classi–ed as either isotropic (etch rate independent of crystalline plane) or anisotropic (orientation dependent). Wet etchants whose etch rate is strongly aÜected by boron doping of crystalline silicon are sometimes also used to produce desired features ; an example is shown later in Section 4 (Fig. 12) where this technique was used to fabricate the 45° re—ectors in the corners of the diamond-shaped FPW device. Dry (plasma) etching is often preferred over wet etching as it avoids the stiction (sticking of the structures to the underlying surface)6 Introductory L ecture Fig. 4 Bulk and surface micromachining processes. associated with the removal of —exible microstructures from a liquid etch or rinse bath.Techniques for dealing with stiction include supercritical carbon dioxide drying10 and the application of special organic surface coatings.11 Many MEMS devices employ polycrystalline silicon (polysilicon) –lms that are only a few lm thick and many lm long. For stiÜer members, high-aspect-ratio structures have been made by electroplating into cavities produced photolithographically in a very thick photoresist (the LIGA process9) or by forming polysilicon structures produced by chemical vapor deposition in deeply etched cavities made in reusable crystalline silicon molds (the Hexsil process12).Newer micromechanical structures of note include the following : (1) hinged structures made by surface micromachining.13 A planar structure formed on a sacri–cial layer is provided with a staple-like polysilicon part that forms a hinge after removal of the sacri–cial layer.When released, such structures can be rotated upwards and locked into position at right angles to the supporting substrate. Members having lengths of a mm orR. M. W hite 7 Fig. 5 Scanning electron micrographs of surface micromachined polysilicon structures suspended one to two microns above underlying silicon substrate.Top: spring. Bottom: interleaved electrodes of a torsional oscillator driven electrostatically. (Photographs courtesy of Berkeley Sensor and Actuator Centre.) more and a thickness of only a few lm have been made for use as mirrors and, when properly interconnected, as structural members. (2) Electronic or mechanical parts made on separate wafers by many diÜerent processes combined to make rugged hybrid devices.For example, one can attach —uid-tight caps and channels over micromechanical parts, and can locate and bond III»V laser dies onto chips containing micromechanical devices such as movable mirrors. (3) Inexpensive polymeric MEMS devices made using reusable etched silicon molds.Precise replication of structures on the lm scale is possible.9 4 Flexural plate wave devices The FPW devices described in Section 2 can be used for sensing and as actuators for liquids and gases.8 Introductory L ecture 4.1 Sensing FPW devices have shown the ability to measure liquid or gas density, liquid viscosity, the concentration of chemical vapors absorbed in a polymer –lm on the membrane, and the deposition of proteins from solution.5 Recently, two additional gravimetric uses of these devices have been demonstrated. In the –rst, the FPW was used to follow the growth of the bacterium Pseudomonas putida, which has been proposed for the in situ remediation of toxic organic waste deposits in the ground.A suspension of non-adherent bacteria was made and placed in a gas-tight chamber containing an FPW device (shown in Fig. 6), connected in a delay-line oscillator circuit.A bolus of toluene was then injected into the chamber. After a one hour period of adjustment (the so-called lag phase for the bacteria), the frequency of the FPW oscillator steadily decreased as the bacteria —ourished.This was expected, since fractional oscillator frequency changes, *f/f, are equal to the fractional phase velocity changes, *v/v, caused by increase of the bacterial concentration in the chamber. Changes ceased at the time expected for exhaustion of the toluene. The advantage of using this acoustic interaction is that one obtains real-time information about the growth without having to remove samples for measurement in a microscope. The second gravimetric interaction observed was the shift of phase velocity in an FPW device due to the reaction of antibodies in an immunoassay for an antigen present Fig. 6 Typical —exural wave device. Thickness of the surrounding silicon frame, nominally 500 lm, is reduced here for clarity. P, the periodicity of the interdigital transducers (IDT), which equals the wavelength, is typically 100 lm.(Reprinted with permission from ref. 14.)R. M. W hite 9 in breast cancer patients.15 In this case, a novel mass ampli–cation step was used to increase the detection sensitivity, as we shall now describe. These experiments were carried out in an enclosed —ow cell like that shown in Fig. 7. A sandwich immunoassay developed at the Cancer Research Fund of Contra Costa was employed (Fig. 8). First, the membrane surface was coated with a fusion protein, NP5. Next, a solution containing a mixture of serum with a monoclonal antibody, Mc5, was added in an incubation step. Mc5 can bind either to the NP5 fusion protein on the surface or to the antigen BrE-Ag in solution. After this the cell was —ushed (wash step).If the breast cancer antigen BrE-Ag was present in the serum, it would compete with the NP5 for binding to the Mc5 and leave the surface less densely covered with antibody than in the case where no antigen is present. (The wash step would carry away any Mc5 that was bound to the antigen.) The next step was to present a second antibody (goat anti-mouse IgG) that could react with the NP5-bound Mc5 and which was conjugated to 10 nm gold spheres.This produced mass loading of the membrane that was proportional to the amount of NP5-bound Mc5 present. The mass loading will be greater when the antigen is not present, as there is no competition with the NP5 for binding of Mc5. In order to increase sensitivity, a solution was added that plated silver onto the gold spheres to which the IgG was conjugated.The resultant silver plating increased the sphere diameters by about 15 times and produced a clear gravimetric indication of the presence of the antigen (Fig. 9). This test was run with the antigen at disease levels ; detection at a level at least ten times smaller appears possible. The advantages of using this acoustic interaction are the avoidance of the need for radioactive techniques and certi–cation, and the possibility of being able to detect at even lower concentrations than are possible with the present radioactive technique.Fig. 7 Exploded schematic view of a —ow-cell FPW liquid sensor. (Figure courtesy of Dr. Ben Costello, Berkeley MicroInstruments, Inc.)10 Introductory L ecture Fig. 8 Schematic summary of breast epithelial antigen competitive assay performed on an FPW device. The breast cancer antigen is denoted BrE-Ag. (Illustration courtesy of Amy Wang.) 4.2 Actuation Owing to its small volume-to-surface ratio and its generation of an evanescent disturbance in a —uid to which it is coupled, the FPW amplitude can be large and the wave can produce signi–cant non-linear acoustic eÜects in —uids.In particular, the phenome- Fig. 9 Results of gravimetric detection in competitive assay for breast epithelial antigen. Top curve: shift of FPW oscillator frequency when antigen is present at concentration 12 lg ml~1. Middle curve: larger oscillator frequency shift observed in absence of antigen. Bottom curve: net frequency shift (magnitude of the diÜerence between the larger shift observed in the absence of antigen and that observed when antigen is present), showing clear evidence of the presence of the antigen.(Illustration courtesy of Amy Wang).R. M. W hite 11 non of acoustic streaming can induce steady —ow in either a liquid or gas adjacent to the membrane.5 Two manifestations of this streaming will be described brie—y.Fig. 10 shows how one may produce and observe —uid motion generated by acoustic streaming. Water seeded with 2 lm diameter polystyrene marker spheres is placed on an FPW membrane. When the FPW transducer is driven (for example, at 3 MHz with 10 V amplitude), the spheres observed with an optical microscope move in the direction of wave propagation. The steady velocity of the liquid motion is proportional to the square of the wave amplitude, which is measured by laser diÜraction.5 In an 18 lm high channel closed at the ends [Fig. 10(a)], recirculation was observed (Fig. 11) in good agreement with analysis.16 Similarly made observations on liquid behavior in a diamond-shaped re-entrant closed chamber that had a unidirectional transducer Fig. 10 Set-up for measuring velocity of liquid —ow produced by acoustic streaming.(a) Crosssection view of —ow cell having an FPW transducer on the bottom membrane and glass slide to cover top of well. (b) Cell mounted on stage of microscope with video camera and VCR for recording images (left), and sketch of liquid containing marker spheres (right). (Reproduced with kind permission from ref. 16. 1995, IEEE.) ( Fig. 11 Flow velocity vs. height above the membrane showing good agreement of measured (data points) and calculated (solid curve) values for the —ow cell shown in Fig. 10. Below 170 lm, the liquid —ows to the right in the direction of wave propagation; above that height it —ows left due to recirculation in the closed cell. (Reproduced with kind permission from ref. 16. 1995, IEEE.) (12 Introductory L ecture beneath the membrane showed that marker spheres that were initially randomly distributed were forced into discrete channels, evidently as a result of wave interference at the 45° re—ecting boundaries (Fig. 12). This channelling may be applicable to performing —ow cytometry on a small chip. The agitation near the membrane caused by the FPW action may have applications other than transport along the membrane.We noted in an earlier publication17 that the stirring produced near the membrane of an FPW device that formed part of an electrochemical cell increased current —ow by aiding diÜusion in bringing reactants to the electrode deposited on the membrane. In more recent work, we have found18 that a biochemical reaction at a surface can be stimulated or suppressed by the FPW. Fig. 13 shows frequency shifts measured in the immunoassay illustrated in Fig. 8 when the incubation of Mc5 on the membrane was performed while the FPW transducer was being excited at diÜerent amplitudes. With no ultrasonic excitation, incu- Fig. 12 Top view (right) and magni–ed corner view (left) of diamond-shaped re-entrant —ow cell. Unidirectional transducers beneath the diamond-shaped membrane produce steady —ow in the clockwise direction.Near the 45° re—ectors at top and bottom, the 2 lm diameter polystyrene marker spheres are forced into discrete channels 70 lm apart ; this distance equals the wavelength/ 21@2. (Reproduced with kind permission from ref. 16 1995, IEEE.) ( Fig. 13 FPW oscillator frequency shifts vs.wave amplitude as antibody is allowed to attach to the membrane while liquid is stirred by the —exural plate wave. With an intermediate level of excitation (20 nm amplitude) more antibody binds than when agitation is absent. For 40 nm amplitude, virtually no antibody binds. (Unpublished data provided by Amy Wang.).R. M. W hite 13 bation produced some surface coverage. With a 20 nm wave amplitude, the amount deposited on the NP5-coated surface increased.At twice the wave amplitude, which would result in four times as much —uid motion due to acoustic streaming, binding was almost entirely suppressed. It appears possible to use this eÜect to enhance or suppress the attachment of molecules at surfaces. 5 Concluding remarks The study of acoustic interactions at interfaces has a long history.The papers that follow show what a wealth of information can be gleaned from such research. One may hope that the growth of micromachining will contribute both tools for research in this –eld and to realizing devices that exploit these interactions for real-world applications. References 1 M. Faraday, Philos. T rans. R. Soc. (L ondon), 1831, 299. 2 Lord Rayleigh, Philos. Mag., 1883, 16, 50. 3 C. C. Gillispie, Dictionary of Scienti–c Biography, Charles Scribnerœs Sons, New York, 1991, pp. 527» 540. 4 J. W. Grate, S. J. Martin and R. M. White, Anal. Chem. A, 1993, 65, 940 5 D. S. Ballantine, R. M. White, S. J. Martin, A. J. Rico, E. T. Zellers, G. C. Frye and H. Wohltjen, Acoustic W ave Sensors : T heory, Design and Physico-Chemical Applications, Academic Press, New York, 1997. 6 S. W. Wenzel and R. M. White, Appl. Phys. L ett., 1989, 54, 1976. 7 R. L. Baer, B. J. Costello, S. W. Wenzel and R. M. White, Proc IEEE Ultrason. Symp., 1991, 293. 8 S. M. Sze, Semiconductor Sensors, John Wiley & Sons, New York, 1994. 9 M. Madou, Fundamentals of Microfabrication, CRC Press, Boca Raton, FL, 1997. 10 G. T. Mulhern, D. S. Soane and R. T. Howe, T echnical Digest, 7th International Conference on Solid- State Sensors and Actuators, T ransducer œ93, Yokohama, Japan, June 7»10, 1993, p. 296. 11 M. R. Houston, R. Maboudian and R. T. Howe, T echnical Digest, 1996 Solid-State Sensor and Actuator W orkshop, Hilton Head, SC, June 1996, p. 42. 12 C. G. Keller and R. T. Howe, L ate News Digest, 1996 Solid-State Sensor and Actuator W orkshop, Hilton Head, SC, June 1996, p. 31. 13 K. S. J. Pister, M. Judy, S. Burgett and S. Fearing, Sens. Actuators, 1992, 33(3), 249. 14 S. W. Wenzel, Doctoral dissertation, EECS, Dept. University of California, Berkeley, CA, 1993. 15 A. W. Wang, R. Kiwan, R. M. White and R. Ceriani, 9th International Conference on Solid-State Sensors and Actuators, Chicago, IL, June 16»19, 1997, p. 191. 16 C. E. Bradley and R. M. White, Proc. IEEE Ultrason. Symp., 1994, 593; C. E. Bradley, J. M. Bustillo and R. M. White, Proc. IEEE Ultrason. Symp., 1995, 505. 17 T. R. Tsao, R. M. Moroney, B. A. Martin and R. M. White, Proc. IEEE Ultrason. Symp., 1991, 181. 18 A. W. Wang, unpublished data. Paper 7/07747E; Received 27th October, 1997

ISSN:1359-6640    

DOI:10.1039/a707747e

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 107, 15»26 Interaction of surface acoustic waves with viscous liquids Glen McHale,* Michael I. Newton, Markus K. Banerjee and S. Michael Rowan Department of Chemistry and Physics, T he Nottingham T rent University, Clifton L ane, Nottingham, UK NG11 8NS A small amount of a viscous oil deposited on a high energy surface adopts a spherical cap cross-section with a dynamic contact angle that evolves following simple power laws of t~3@10 for a droplet and t~2@7 for a stripe.If the surface being wet forms part of a surface acoustic wave (SAW) device the rate of spreading can be monitored using changes in both the attenuation and the phase of the surface wave. This system provides a probe of both the SAW»viscous liquid interaction and the solid»liquid interaction.In this work we report simultaneous optical and SAW measurements on the spreading of poly(dimethyl)siloxane oil of viscosity 100 000 cS. Experiments using pulsed SAWs of frequency 170 MHz show an overall exponential type decrease in transmitted signal amplitude as the oil spreads. The decreasing amplitude of the SAW is also accompanied by distinct maxima and minima indicating transmission resonances.A SAW signal re—ected from the stripe of oil is observed and is seen to oscillate in amplitude as the advancing front of the —uid reduces the acoustic pathlength. The optical data allows the pro–le of the liquid to be constructed and con–rms the simple power law behaviour of the dynamic contact angle. The geometrical information on dynamic contact angle, contact width and height of —uid is correlated with the SAW signal and possible mechanisms for the transmission and re—ection resonances are discussed.Understanding the wetting of solid surfaces at macroscopic and microscopic length scales is extremely important for a number of practical applications. In the agrochemical industry, controlling the –lm formation properties in pesticide formulations and the drainage properties of containers to ensure little residue remains is of practical importance in minimising environmental pollution.The manufacturing industry also has a strong motivation to understand wetting due to the use of lubrication and surface treatments for the manufacture of composite materials and the spin-coating of –lms. The solid»liquid interaction determines properties, such as adhesion strength, and can be investigated by measuring the contact angles of various —uids on the surface.Traditional studies have concentrated on equilibrium properties, but over the last decade the dynamics of wetting and dewetting processes have become a focus of interest. Prior to 1985 dynamic wetting was the subject of steady, but unspectacular, research activity. With the exception of Tannerœs 1979 hydrodynamic theory1 relating to the complete spreading of non-volatile, viscous oils on high energy surfaces, theory did not accord well with experimental data.The intervention of de Gennes2 signi–cantly altered this perspective. It gave a clear exposition of the role of interfacial energies and energy dissipation in determining the rate of spreading and quasi-equilibrium shapes of microdroplets (see, for example, the review by Leç ger and Joanny3).Fundamental aspects of liquid»solid surface interactions, polymer slippage,4 layering in precursor –lms,5 the transition from complete to partial wetting regimes6 and —uid interface stability7 have since been elucidated. 1516 Interaction of SAW s with viscous liquids In equilibrium studies of wetting, a drop is placed on a solid surface and, provided its volume is small, it forms a spherical cap with a –nite contact angle.If the surface is of high energy, a thin –lm forms rather than a macroscopic shape. Dynamic wetting is related to the approach to the equilibrium and its relationship with the interfacial energies in the system.For oils the spreading rate is determined by a balance between the change in surface free energy and viscous dissipation, and for high energy surfaces with vanishing equilibrium angles, this leads to simple power law relations.2,8,9 Quantitative methods for examining macroscopic droplets draining into microscopic –lms have tended to use pro–le or plan views of the liquid ; examples of these approaches include optical pro–ling, laser interferometry,10 X-ray re—ectivity,11 ellipsocontrast12 and ellipsometry13 and each method oÜers advantages in terms of speed, ease of use and horizontal and vertical resolution.Optical re—ection microscopy is able to measure vertical resolutions of 770 using blue light, whereas ellipsocontrast enables vertical ” resolutions of 50 but with only a qualitative observation of the dynamics.Ellip- ”, sometry allows the vertical resolution to be improved to the a- ngstroé m level whilst retaining a good lateral resolution of ca. 20 lm. Similarly X-ray re—ectivity provides 1 ” vertical resolution, but only at the expense of an averaged lateral resolution. These methods are extensions of ones used in static equilibrium studies and all use plan or pro–le views of the spreading liquid.In contrast, Lin and co-workers14,15 have recently used a quartz crystal microbalance (QCM) to measure the spreading rate. The mass sensitivity of this device allowed the authors to examine whether condensation to create a thin precursor –lm of water was necessary for the superspreading of trisiloxane surfactants.However, this is one of very few attempts to use acoustics for the study of dynamic wetting processes of spatially localised liquids. Surface acoustic wave (SAW) devices have been used to study liquids, but such experiments have generally concerned changes in quantities, such as the pH, viscosity, conductivity and permittivity, for a –xed pool of liquid.16 In SAW devices the acoustic waves propagate along the solid/vapour interface and have over 90% of their energy con–ned to within one acoustic wavelength of the surface.Such devices have been extensively used in gas sensing studies because the mass sensitivity of SAW devices is orders of magnitude superior to the QCM, with adsorbate concentrations in the region of parts per billion detectable.17 In the case of a —uid dynamically wetting the surface of a SAW device, we can anticipate changes in the acoustic signal due to the SAW»liquid interaction. Indeed, such changes in the SAW signal have been demonstrated in a preliminary study of the SAW-localised wetting liquid interaction.18 Given the SAW deviceœs extreme sensitivity to mass loading, such a technique may provide a unique detection method capable of detecting sub-monolayer —uid coverages. SAW devices could be potentially used to monitor both the early stages of wetting, when the liquid has a macroscopic shape, and the later stages when the —uid forms a –lm of thickness less than one acoustic wavelength, The development of the SAW tech- jSAW .nique requires a greater understanding of surface wave»localised —uid interactions than currently exists, but would complement other techniques and extend the use of acoustic wave methods.The SAW approach is not an extension of the techniques used in equilibrium studies of wetting, but is a new in-plane method of studying wetting. In this paper we describe the –rst simultaneous optical and surface wave observations of the wetting dynamics of a viscous —uid.The experiments show the eÜect of a spreading stripe of oil located in the SAW path on the amplitude of transmitted and re—ected signal pulses. The signals are progressively attenuated by the spreading of the oil, but also show a distinct, and rich, structure of maxima and minima. Interferometry is used to follow the evolution of the pro–le of the —uid directly at the location interacting with the surface wave.The changes in the SAW signals are correlated with the geometric changes in the pro–le of the oil. The time evolution of the contact width, do , contact angle, h, height, and spherical radius, R, of the oil are accurately described ho ,G. McHale et al. 17 by power law relationships, which have been predicted theoretically.9 All speci–c data presented in this paper relate to SAW devices working at ca. 170 MHz and oils of 100 000 cS, although a range of devices and oils has been used in the study. Method A SAW is a coupled elastic and electric –eld disturbance and can therefore be generated by distorting the surface of a piezoelectric material in a periodic manner.19 To study dynamic wetting it is convenient to choose a system that gives complete wetting, i.e.a vanishing equilibrium contact angle, and has a low characteristic speed of wetting, v*\c/g where c is the liquid surface tension and g is the viscosity. The analysis of wetting rates is also simpli–ed if the length scale of the —uid is much less than a capillary length, i~1\(c/og)1@2, so that gravity can be ignored and a spherical cap crosssectional pro–le of the —uid is obtained.The solution of the equations governing the dynamic change of the pro–le is possible if mass conservation is satis–ed and this suggests the use of a non-volatile —uid. These criteria are well satis–ed for a system using small amounts of poly(dimethyl)siloxane (PDMS) oils on lithium niobate.PDMS is available in a wide range of viscosities, thereby allowing a choice of timescales. Moreover, PDMS is well characterised for dynamic wetting experiments with the spreading of droplets on silicon widely reported in the literature.3 The SAW devices were produced on 128° rotated y-cut lithium niobate (Yamaju Ceramics) by lithography from masks designed according to the procedure given by Smith et al.20 Two interdigital transducers (IDTs) were located 2 cm apart and each consisted of 13 pairs of –ngers with equal width –ngers and spacing, giving a periodicity of 22 lm; the aperture was 260 lm.These devices had a resonant frequency of 170 MHz giving a surface wave speed of 3700 ms~1 and a nominal 50 ) matching impedance. The choice of structure represents a compromise between a short acoustic wavelength and a high frequency.A short acoustic wavelength is desirable since it provides a horizontal length scale, but the consequent higher frequency operation imposes greater constraints on both the electronics and the accuracy of the lithography. Experiments were also performed using other high frequency designs and a lower frequency (60 MHz) multistrip coupling type device supplied by RACAL-MESL.The SAW devices were operated in a variety of arrangements with the most complex allowing both pulse transit and re—ection measurements to be obtained simultaneously (Fig 1). Our previous system18 was less —exible and did not include either optical observations or simultaneous measurements of the re—ection and single transit signals. A signal generator (SG) provides a 170 MHz continuous wave (cw) radiofrequency (rf) signal, which is combined using two cascaded double balanced mixers (DBMs) with a 1 kHz sequence of 400 ns wide pulses from a pulse generator (PG).The DBMs act as switches and so create rf pulses. Two cascaded DBMs are necessary to suppress completely the cw signal outside the pulse duration and this is a signi–cant diÜerence to the system in our previous work.18 These rf pulses are supplied to the transmitting IDT via a hybrid junction (HJ).The IDT converts the electrical signal into SAWs using the piezoelectric coupling. When the surface wave arrives at the second IDT the inverse piezoelectric eÜect generates an electrical signal that can be ampli–ed and detected.The purpose of the hybrid junction is to allow any re—ected surface wave that arrives back at the transmitting IDT to be directed to an ampli–cation and detection circuit. The hybrid junction therefore acts as a circulator, directing rf pulse signals to the transducer and signals from the transducer to the detecting circuit. This use of a pulse system and a circulator allows the transmitting IDT to act also as a receiver of surface wave re—ections.The DBM in the detection circuit for re—ection signals is simply used to gate out transients caused by the switching of the rf signal. The bandwidth of the oscilloscope is18 Interaction of SAW s with viscous liquids Fig. 1 Schematic diagram showing the setup of the SAW system. By using interferometry, changes in the amplitude of pulsed SAWs can be monitored with simultaneous observation of the liquid pro–le. For de–nitions see text. 100 MHz and so in both detection circuits the pulsed rf signal is recti–ed using diode detectors (Ts) before being displayed on a digital storage oscilloscope (DSO). When no oil is on the surface of the device the oscilloscope trace for the single transit signal consists of a periodic pulse signal at the 1 kHz repetition rate.The time between pulses gives the speed of the surface wave in travelling the 2 cm between the transducers. The area beneath each pulse on the oscilloscope is a measure of the SAW amplitude and changes in this area represent changes in attenuation of the surface wave. When a stripe of oil is on the surface, three pulses, rather than one, are observable, which correspond to a single transit, a double transit and a re—ection from the oil ; these paths are indicated in Fig 2.The double transit signal is due to the surface wave travelling from the –rst IDT being re—ected by the second IDT and so passing back along the liquid-loaded surface for a second time. The re—ection from the oil can be clearly identi–ed by the measured time delay between transmission and reception of the pulse by the –rst transducer. Several simpler con–gurations were also used to examine the attenuation and re—ection of surface waves.Rf pulses from the cascaded DBMs were supplied directly to the –rst IDT of the SAW device and the signal observed at the second IDT was then ampli–ed and detected, thereby omitting the electronics for the re—ection and double transit circuit.Similarly, in some re—ection experiments the ampli–ers and detection system for the single transit signal was omitted from the circuit. The oil was deposited across the surface wave path using the edge of a razor blade to create a stripe on the surface. The ease of deposition varies with the viscosity of the oil.A 100 000 cS oil (Aldrich), with a characteristic speed of v*\0.022 mm s~1 and a capillary length of i~1\1.5 mm, was used. The deposited stripe typically had an initial width of ca. 0.1 mm which increased to 0.4 mm after 6»8 h. Since the width of oil is much less than the capillary length we would expect the —uid to be reasonably approximated by a spherical cap cross-section during its spreading (Fig. 2). However, it is possible that small deviations from this shape will occur towards the late stages of spreading. A given stripe of oil always varies in both width and height over its length, but these are relatively slow variations over the width of the transducer aperture. The precise geometry of the stripe is determined using the simple interferometry arrangement shown in Fig 1.Sodium light (j\589 nm) is directed onto the stripe and is re—ected by the top surface of the oil at the air/oil interface and by the bottom surface of the oil atG. McHale et al. 19 Fig. 2 The three acoustic signals measured for each surface wave pulse generated by the transducer. The single transit signal passes directly between the transducers.The double transit signal uses a re—ection from the opposite transducer so giving a double passage along the —uid-loaded interface. The third signal is a direct re—ection from the —uid. the oil/lithium niobate surface. If the diÜerence in pathlengths is a half integer number of optical wavelengths, destructive interference occurs, whereas if the diÜerence is an integer number, constructive interference occurs.The image is viewed through a camera and recorded onto videotape (VCR). After the experiment selected images are captured onto a personal computer (PC), the contrast stretched, a simple 3]3 mean average applied and the pro–les (x,y,gray) obtained using an image processing package. Programs have been written in ìCœ to calibrate the pro–les to real world coordinates, identify the positions of the maxima and minima in the light intensity, construct the cross-section shape and –t the best possible circle. The width of stripe is identi–ed from the images by –nding the edge separating the stripe from the background light intensity.The width can be clearly identi–ed even when the fringes towards the edge cannot be separately distinguished. Fig. 3 shows the interference pattern observed after approximately 3.5 h of spreading of the oil (corresponding to the data shown in Fig 5). The dashed lines in Fig. 3 indicate the approximate position of the SAW transducer aperture, and its centre, along the stripe of oil. The diÜerence in height between adjacent light and dark fringes is j/4n where the refractive index, n, of the oil is 1.402 and this gives a vertical resolution of 105 nm.The slow variation in height along the crest of the stripe, compared to the variation across the stripe, can be seen by the greater spacing of the fringes. The horizontal resolution is determined by the magni–cation of the system, which was chosen to ensure the full width of stripe would be observable over the complete spreading time.This meant that a full fringe pro–le across the stripe could only be obtained towards the end of the experiment. However, the existence of just a few identi–able fringes near the crest of the stripe of oil is sufficient to enable the spherical radius of the cross-section to be measured. The inset in Fig. 3 shows the extracted pro–le along a line corresponding to the centre of the SAW transducer aperture.The circles are the data points for each of 11 maxima and minima light intensities either side of the crest of the stripe ; the solid curve is the spherical –t to the data. Since it was also possible at all times in any experiment to identify the stripe width the contact angle can be determined. For the images near the20 Interaction of SAW s with viscous liquids Fig. 3 The parallelism of the edges and slow variation of the crest height along the stripe of the oil after 3.5 h of spreading is impressive (image corresponds to data shown in Fig. 5). The relative location of the aperture of the SAW transducers is indicated by the dashed lines. The inset shows the measured pro–le using 22 central fringes and this is well described by a circular –t (solid line).end of the experiment the contact angle computed from the full pro–le obtained from the fringes was compared to the angle deduced from using the central fringes and the measured width. These two methods give contact angles agreeing to ca. ^0.05°. In principle, a higher magni–cation could be used and measurements on only one side of the stripe used.Where a partial fringe pro–le is used the accuracy of the contact angle measurement is primarily limited by the accuracy in the width determination, which is ca. ^2 lm. Results For consistency all the data presented relate to experiments using the same viscosity oil and two devices constructed from the same mask design. The time variation in the SAW signals and the optical data shown by these experiments have been observed in many other experiments.Fig. 4 shows the change in the transmitted SAW signal over an 8 h period obtained using a simple single transit con–guration. The upper curve used a device with a resonant frequency of 167 MHz and an rf level of 27 mV (peak to peak), whilst the lower curve used a device with a resonant frequency of 170 MHz and an rf level of 43 mV.The upper curve has been displaced vertically upwards by 6.75 to displayG. McHale et al. 21 Fig. 4 Single transit data show an overall attenuation but with a rich oscillatory structure (A»G and A@»D@) and some subsidiary peaks (X and Y). The upper curve (167 MHz device operated at 27 mV) has been displaced vertically upwards by 6.75 to –t on the same graph as the data from a 170 MHz device (operated at 43 mV).it on the same graph as the lower curve. Both curves show the same trend with two or three large oscillations, labelled A»G and A@»D@, superimposed on an overall decay of the amplitude. The extent of the oscillations is signi–cant giving a reduction of the signal levels by 90% or more. In addition to these oscillations some smaller scale structure can be seen on the signals. This can be seen as edges on the negative slope from the third maxima to third minima of the upper curve (E»F).On the lower curve, these edges appear as peaks, labelled X and Y, on the negative slope between the second main maxima and second main minima of the lower curve (B@»C@). It is interesting to note that the increase in width of the stripe between the data at X and Y is 12 lm and, to within the accuracy of the optical measurements, is The optical data for the upper curve jSAW/2.shows that the contact width increases from 208 lm at B to 429 lm at G. To conserve volume a compensating decrease in height from 15 to 8 lm also occurs. The contact angle decreases from 16.3° to 4.1° during this time.The period between B@ and D@ on the lower curve corresponds to an increase of the contact width from 140 to 229 lm with a corresponding reduction in height from 9 to 5 lm. In this case, the contact angle reduces from 14.8° to 4.8°. The major maxima to maxima changes in these single transit signals therefore correspond to increases in width of greater than a single acoustic wavelength and decreases in height of less than one half of an acoustic wavelength. Experiments using the full SAW system allow the single, double transit and re—ection signals to be compared.The form of the double transit signal is identical to the single transit signal with the same overall decrease in amplitude and structure of maxima and minima. Fig. 5 shows the data from a re—ection experiment. The upper curve, which has been shifted up by 10.5, is the double transit signal. The subsidiary peak structure occurring in the –rst 40 min is very similar to that in Fig. 4. The use of time rather than a logarithmic time axis shows more clearly the overall decay in amplitude of the transit signal. The lower curve in Fig. 5 demonstrates the time dependence of the re—ection22 Interaction of SAW s with viscous liquids Fig. 5 Re—ection from the stripe (lower curve) showing signi–cant oscillations as the oil spreads ; the numbers labelling the oscillations are the measured width, in lm, of the stripe. The upper curve is the double transit signal. The inset shows the early change in both signals in the early stages of spreading indicated by the vertical dashed line on the main graph.from the stripe of oil. Some structure can be seen during the same period (t\2500 s) as the subsidiary peak structure in the double transit signal occurs. However, the most striking feature is the much larger oscillations in the re—ection signal occurring later in the experiment. The contact width for each of the main minima and maxima in the re—ection are indicated in lm in Fig. 5. Optically it is the contact width, and the spherical radius, R, which are measured d0 , and these have a theoretical time dependence given by,9 d0\C567V 3c(t]c) L 0 3 gJw D1@7 and R\C83349V 2c3(t]c)3 16384L 0 2 g3Jw 3 D1@7 where is the volume of —uid per unit length, is a cut-oÜ parameter and c is a V /L 0 Jw constant determined by the initial cross-sectional shape of the oil.These formulae are valid for systems with conserved masses of —uid, a vanishing equilibrium contact angle and a dynamic contact angle less than ca. 60°. The values of contact width and spherical radius corresponding to the SAW data in Fig. 5 are shown in Fig. 6. Simultaneously –tting to both sets of data using the theoretical 1/7th and 3/7th power laws gives, d0\95.2(t[41)1@7 and R\45.9(t[41)3@7 where and R are in lm and t is in seconds.These –ts used 33 images, chosen using a d0 logarithmic time scale from data between 5 min after the start of the experiment and the end time of 8 h. An image was included for each maxima and minima in the SAW re—ection data. The –ts for and R are accurate to within 1% and 2%, respectively, for d0 all data points and are shown by the solid lines in Fig. 6. To show the data for R on the same graph they have been divided by a factor of ten. For a spherical cross-sectionG. McHale et al. 23 Fig. 6 (a) Contact width, and (b) spherical radius (divided by ten, i.e., R/10) for the experiment d0 , in Fig. 5 are well –tted by the predicted power laws geometry, knowledge of and R gives the contact angle, h, and height, using, d0 h0 , d0\2R sin h and d0\2h0/tan h/2 and for angles less than ca. 60° give time dependences of, h0\C 27V 4gJw 896L 0 4 c(t]c)D1@7 and h\C 128V g2Jw 2 147L 0 c2(t]c)2D1@7 The –tted time dependence of the contact angle and height can be obtained from the –ts to the width and spherical radius. In the –rst 40 min of this experiment, the width increases from \214 lm to 299 lm with a corresponding decrease in height from [11 lm to 8 lm.During this time the contact angle decreases from [12° to ca. 6°. Subsequently, the width increases to 415 lm whilst the height and contact angle reduce to 5.5 lm and 3°, respectively. The increase in the width of the stripe over the last –ve peaks in the SAW re—ection signal is 91 lm giving an average peak-to-peak change in width of ca. 23 lm. This value is close to the SAW wavelength of 22 lm, although the precise peak-to-peak change in the width varies considerably. The percentage accuracy of the width measurement is extremely high, but the diÜerences in widths are only accurate to ca. ^3 lm. The inset in Fig. 5 shows the structure in the initial stage of spreading, which is indicated by the vertical dashed line on the main graph.The double transit signal in the inset is remarkably similar in form to the signal on the main graph and is not an error. The subsidiary peaks on the double transit signal cannot be clearly identi- –ed with either peaks or troughs of the re—ection signal, although this is not surprising given that the re—ection is weak in the early stages of spreading.It is interesting to note that the two larger initial peaks shown in the inset correspond to a change in width of 21 lm, which is close to the acoustic wavelength of 22 lm.24 Interaction of SAW s with viscous liquids Discussion The attenuation in the single and double transit signals can be expected to depend on the mass loading per unit area of the surface.However, as the —uid spreads the increasing surface coverage also requires a reduction in height of the —uid previously covering the surface. The overall progressive attenuation of the transit signals suggests that the increasing coverage is the dominant factor. It is likely that a full description of any damping will need to allow for a penetration depth into the —uid.For example, the shear wave penetration depth, d, of a 170 MHz wave for the 100 000 cS oil would be d\(g/ nof )1@2\1.4 lm. Providing the thickness of the oil is greater than the penetration depth, the attenuation would depend on the time dependent contact area and hence the width of the stripe. In these experiments, the height of —uid is larger than 1.4 lm and this appears to be consistent with the observed overall attenuation of the signal.The eÜect of mass loading would tend to increase the phase speed, and hence the SAW wavelength, along the —uid/solid interface. However, the pathlength over which this occurs is small compared to the 2 cm separation between transducers and we would not expect to observe a measurable shift in the position of the pulses on the oscilloscope.It is also possible that some of the SAW energy would be lost due to mode conversion and this would result in greater attenuation. At the interface between a solid and liquid two surface waves are possible, a leaky surface wave and a non-leaky Stoneley wave. Separate to the attenuation there are the oscillations that occur as the liquid spreads across the surface.The strong maxima and minima in the single transit signal, and even the subsidiary peak structure, mirror the oscillatory structure in the double transit signal. However, it is not possible to identify a precise correlation between the oscillations in the transit signals and the peaks and troughs in the re—ection signals. Nonetheless, Fig. 5 does suggest that the overall amplitude of the transit signals provides an envelope modulating the magnitude of the re—ection peaks. To within the accuracy of our optical system, the strong re—ection peaks do seem to correlate with changes in the contact width of one SAW wavelength. There is also some evidence that the subsidiary peak structure seen on the transit signals corresponds to changes in width of one-half a SAW wavelength, although this conclusion is based on very few peaks.The strong maxima and minima in the transit signals (Fig. 4 and 5) represent much greater changes in the contact width of the oil than a SAW wavelength. The height of the —uid is changing by 3»4 lm between consecutive maxima or minima and does not suggest a clear pattern. It is therefore likely that the oscillatory structure in the SAW signals has more than one source.One possibility is that the spreading —uid is eÜectively mapping out the diÜraction pattern due to the transducer aperture. The near –eld distance to the last diÜraction maxima is approximately mm, where a is the aperture width. Comparing a2/4jSAW\0.8 this with the central location of the liquid between the IDTs, we do not believe this to be important.Several physical mechanisms can be envisaged that would cause transmission and re—ection resonances. The localised liquid has a de–nite width and height and these introduce two natural length scales that could result in resonances at speci–c values by allowing standing wave conditions to be matched. In the case of the width the stripe may act as a resonant cavity.The pulse length used in the experiments is 400 ns, which equates to a distance of 1.5 mm on the unloaded surface and this is far less than the width of the stripe. It is then possible that a partial re—ection from the oil/air interface as the pulse emerges from the stripe could interfere with the part of the pulse still entering the region covered by the oil.A standing wave would then occur when a resonant condition was matched. Allowing for the damping introduced by the oil, we would expect a transmitted energy with an exponential decay moderated by an oscillatory factor dependent on A simple model of this kd0 . is given by a wave propagating along a string of variable density. The density is givenG. McHale et al. 25 one value for and a second value for a denser region, which x[o d0/2 o , x\ o d0/2 o , causes the damping. The model is then equivalent to a freely propagating wave outside and a damped harmonic oscillator with an equation, o x o\d0/2 d2t dx2]2b dt dx]k1 2 t\0 for The solution of this problem requires three parts, A o x o\d0/2. exp(ikx)]B exp([ikx) for [C exp(ikx)]D exp([ikx)]exp([bx) for x\[d0/2, o x o\ and a transmitted wave of F exp(ikx) for and these must be matched at d0/2 x[d0/2, Solving gives a transmission coefficient for the energy of, x\^d0/2.KF AK2\ exp([2bd0) 1]As 2B2 sin2(k0 d0) where and This form for the transmitted k0\)(k12[b2) s\(k4]k14[2k2k0 2 )/k2k0 2 . energy would then show an exponential damping with superimposed oscillations as the stripe width, increased.d0 , Separate to whether the width acts as a resonant cavity it would also be possible to have transmission resonances due to the changing height of the stripe. The transmission coefficient for a surface wave incident on a –xed width step of height on a substrate is h0 known to have a form dependent on where and show oscillations as a kh0 , k\2n/jSAW , function of It is therefore possible that simply regarding the stripe as two con- kh0.19 secutive steps would lead to a transmission coefficient dependent on the stripe height relative to the SAW wavelength and hence an oscillation. Since the height decreases much more slowly than the width increases, such an oscillation would be relatively slow.A more complex problem than a step is a substrate in the shape of a solid wedge and this has been considered in the literature.21,22 In this case, extremely strong oscillations, including almost total re—ection, were found to occur in the re—ection and transmission coefficients as the wedge angle was systematically decreased.Maxima in the re—ection coefficient were found to correspond to minima in the transmission coefficient and vice versa.Krylov and Mozhaev22 have ascribed this behaviour to the diÜering phase velocities of the lowest symmetric (longitudinal) and –rst antisymmetric (—exural) plate modes propagating in a plate of variable thickness. It is possible that similar mode interactions may take place near the three-phase contact line within the liquid layer and the controlling variable might then be the contact angle.The stripe geometry for the small contact angles in our experiments could be approximated by two wedges. The existence of contact angles at both the front and the back of the stripe would make such a calculation more difficult. Further work is required to determine the exact form of the attenuation of the transit signals and the physical processes causing the oscillations in the transit and re—ection signals.The spatial resolution of the optical part of the system needs to be increased to enable greater precision in measurements of the contact width of the oil. The number of major maxima and minima observed in the transit signals is small and makes determining a correlation with the geometric information more difficult.Since the spreading process slows according to a t~1@7 law, this suggests that the experimental system needs to be developed to enable the –rst few minutes of spreading to be analysed. Alternatively, either a longer timescale of several days could be used to allow more oscillations or the viscosity of oil could be reduced to increase the intrinsic spreading rate. Signi–cant bene–ts could also be derived from increasing the frequency of operation of the SAW devices as this would reduce the horizontal length scale which we believe determines the oscillations.In principle, the SAW approach used in this work is not restricted to a26 Interaction of SAW s with viscous liquids system of a stripe of oil spreading on a high energy surface. A –nite equilibrium angle could be achieved by chemical treatments to control the polarisation of the surface, for example, by silanisation. It also seems likely that the SAW device could be used to observe the spreading of a drop of oil rather than a stripe.Indeed, if the drop was located to the side of the SAW transmission path, changes in the transit signal might be observed as the droplet spread into the acoustic path.Such changes may then indicate the presence of a precursor –lm spreading ahead of the bulk of the drop. Conclusion Simultaneous optical and surface wave measurements have been performed on the spreading of small stripes of high viscosity oils. The changes in contact width, contact angle, height and spherical radius of the oil are in agreement with theoretical power laws.As the oil spreads it interacts with the pulsed surface waves and produces a rich structure in the transit signals. Strong maxima and minima are seen in these signals together with possible subsidiary peaks. The major oscillations occur slowly as the —uid width increases by more than one acoustic wavelength, but the subsidiary peaks indicate smaller changes of around one-half of a wavelength.Re—ections are also observed from the —uid and these show maxima and minima that are consistent with optically observed changes in width of the oil of one surface acoustic wavelength. The transit signal is progressively attenuated as the oil wets the surface and this attenuation appears to modulate the amplitude of the re—ection signal. The authors would like to acknowledge the Central Microstructure Facility at the Rutherford Appleton Laboratory for funding and production of the lithographic masks for the 170 MHz devices and RACAL-MESL who provided the 60 MHz devices. One of the authors (G.M.) is grateful to Professor V.V. Krylov for discussions on the interpretation of the experimental data. Image processing was performed using the UTHSCSA ImageTool program (developed at the University of Texas Health Science Center at San Antonio, TX, and available from the Internet by anonymous FTP from maxrad6.uthscsa.edu). References 1 L. Tanner, J. Phys. D: Appl. Phys., 1979, 12, 1473. 2 P. G. de Gennes, Rev. Mod. Phys., 1985, 57, 827. 3 L. Leç ger and J. F. Joanny, Rep. Prog. Phys., 1992, 55, 431. 4 F. Brochard-Wyart, P. G. de Gennes, H. Hervet and C. Redon, L angmuir, 1994, 10, 1566. 5 F. M. Tiberg and A. M. Cazabat, L angmuir, 1994, 10, 2301. 6 P. Silberzan and L. Leç ger, Phys. Rev. L ett., 1991, 66, 185. 7 P. Carles and A. M. Cazabat, J. Colloid Interface Sci., 1993, 157, 196. 8 G. McHale, M. I. Newton and S. M. Rowan, J. Phys. D: Appl. Phys., 1994, 27, 2619. 9 G. McHale, M. I. Newton, S. M. Rowan and M. Banerjee, J. Phys. D: Appl. Phys., 1995, 28, 1925. 10 D. Chen and N. Wada, Phys. Rev. L ett., 1989, 62, 3050. 11 J. Daillant, J. J. Benattar and L. Leç ger, Phys. Rev. A, 1990, 41, 1963. 12 D. Ausserreç , A. M. Picard and L. Leç ger, Phys. Rev. L ett., 1986, 57, 2671. 13 F. Heslot, A. M. Cazabat and P. Levinson, Phys. Rev. L ett., 1989, 62, 1286. 14 Z. X. Lin, R. M. Hill, H. T. Davis and M. D. Ward, L angmuir, 1994, 10, 4060. 15 Z. X. Lin, T. Stoebe, R. M. Hill, H. T. Davis and M. D. Ward, L angmuir, 1996, 12, 345. 16 J. Kondoh, J. Saito, S. Shiokawa and H. Suzuki, Jpn. J. Appl. Phys., 1996, 35, 3093. 17 J. Ricco and S. J. Martin, Sens. Actuators B, 1993, 10, 123. 18 M. I. Newton, G. McHale, S. M. Rowan and M. Banerjee, J. Phys. D: Appl. Phys., 1995, 28, 1930. 19 A. A. Oliner, Acoustic Surface W aves, Springer-Verlag, Berlin, 1978. 20 W. R. Smith, H. M. Gerard, J. H. Collins, T. M. Reeder, J. Shaw, IEEE T rans. Microwave T heory T ech., 1969, 17, 867. 21 I. A. Victorov, Sov. Phys. Dokl. (Engl. T ransl.), 1958, 3, 304. 22 V. V. Krylov and V. G. Mozhaev, Sov. Phys. Acoust. (Engl. T ransl.), 1985, 31, 457. Paper 7/03015K; Received 2nd May, 1997

ISSN:1359-6640    

DOI:10.1039/a703015k

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 107, 27»38 Influence of surface roughness on the quartz crystal microbalance response in a solution New con–guration for QCM studies Leonid Daikhin and Michael Urbakh* School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, T el-Aviv University, 69978, Ramat-Aviv, T el-Aviv, Israel The eÜect of surface roughness on a quartz crystal microbalance (QCM) response in contact with a liquid has been investigated.A perturbation theory was used for the description of the eÜect of slight roughness and an approach based on Brinkmanœs equation was used for strongly rough surfaces. In the latter approach, one treats the —ow of a liquid through a nonuniform surface layer as the —ow of a liquid through a porous medium. Our description includes both the eÜect of viscous dissipation in the interfacial layer and the eÜect of the liquid mass rigidly coupled to the surface.A relation between QCM response and the interface geometry has been found. We suggested a new approach to study the QCM response in liquids that provides an eÜective way to analyse the eÜect of roughness. The models discussed here can be used for the treatment of QCM response of rough electrode surfaces, of porous deposited –lms, and of surface polymer –lms.Recently, there has been increased attention to the application of QCM for studies of solid/liquid interfaces.1,2 It has been successfully demonstrated that QCM can serve as a sensitive tool to probe interfacial friction,3 thin –lm viscoelasticity,4,5 polymer –lm properties2 and bulk liquid viscosity and density.6,7 The combination of the QCM technique with electrochemical methods has made possible in situ measurements of minute mass changes which take place during adsorption, underpotential deposition, dissolution of surface –lms, and other electrochemical processes.2,8,9 In most of these investigations, frequency changes were interpreted in terms of rigid mass changes, based on the Saurbrey equation.10 In liquids, however, the QCM response depends on various factors such as electrode microstructure, morphology of surface –lms, interfacial liquid properties, and on the solid»liquid coupling at the interface.In many cases, frequency changes are not consistent with the predictions given by Saurbrey equations.1,11h13 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.A smooth resonator operating in contact with a Newtonian liquid generates planeparallel laminar —ow in the liquid, causing a decrease in the resonant frequency and an increase in the resonator damping proportional to (og)1@2, where o and g are the liquid density and viscosity.6,7,14 The in—uence of surface microstructure on the QCM response in contact with liquids has only begun to be investigated.15h25 When the surface of the resonator is rough [see Fig. 1(a)], the liquid motion generated by the oscillating surface becomes much more complicated than for the smooth surface. A variety of additional mechanisms of coupling between acoustic waves and liquid motion can arise, such as the generation of non-laminar motion, the conversion of in-plane surface motion into surface-normal liquid motion and trapping of liquid by cavities and 2728 In—uence of surface roughness on the QCM response in a solution Fig. 1 Schematic sketch of interfacial geometries : (a) traditional QCM con–guration, (b) liquid layer con–ned between a rough immobile surface and a quartz resonator with a —at interface pores.It has been demonstrated experimentally16h18,20h25 that a roughness-induced shift of the resonant frequency includes both an inertial contribution and a contribution due to the additional viscous energy dissipation caused by non-laminar motion in the liquid.Measurements of the complex shear mechanical impedance20 have been used to analyse diÜerent contributions to the roughness-induced response of the quartz resonator and to correlate the experimental results with the device roughness. Nevertheless, this subject is highly undeveloped and the interpretation of experimental results is ambiguous, preventing the QCM technique being widely employed as an analytical tool for studies of solid/liquid interfaces.Here, we suggest a new approach to study the eÜect of roughness on QCM response. We consider a con–guration where a liquid layer is con–ned between a rough immobile solid surface and a quartz resonator with a —at interface [see Fig. 1(b)]. This geometry allows us to study the coupling of damped waves in a liquid with a rough surface, which is responsible for roughness-induced changes of the frequency and the width of the resonance.The important advantage of the proposed geometry is the existence of a new parameter, the distance between the quartz crystal and the rough surface, variation of which provides an eÜective way of studying the roughness-induced response. The experimental data obtained in this geometry will allow us to test theoretical predictions, to establish the dependences of QCM response on the characteristic sizes of roughness and on the properties of liquids and to use these results for the evaluation of the eÜect of roughness in the traditional QCM geometry.We also hope that the use of an immobile rough substrate as a working electrode will make it possible to separate the roughnessinduced contribution to QCM from other possible contributions, in particular from mass loading.Theoretical models of QCM response at rough surfaces The dependence of the resonance frequency on the properties of the liquid and on the interface morphology is determined by the relations between the characteristic sizes of surface roughnesses and the length scales de–ned by the Navier»Stokes equation for liquid velocities and by the wave equation for the elastic displacement in the crystal.These length scales are the decay length of liquid velocities, d\(2g/uo)1@2, and the wavelength of the shear-mode oscillations in the quartz crystal, j\2n(kq/oq)1@2u~1; where and are the shear modulus and the density of the quartz crystal and u is the kq oq frequency of oscillations.For the frequencies used in QCM, uB1»10 MHz, the lengths d and j are of the order of 0.1»1 lm and 0.1 cm, respectively.L . Daikhin and M. Urbakh 29 The solid surface pro–le may be speci–ed by a single valued function m(x, y) giving the local height of the solid with respect to a reference plane [see Fig. 1(a)]. The STM measurements show that the characteristic heights of the roughness of metallic –lms on the quartz crystal are ca. 10»100 nm.15,17,24 The lateral scales of roughness can change over a wide range, from 10~1 nm to 102»103 nm. As a result, we are confronted with both ì slight œ roughness, for which the characteristic height is less than the lateral sizes, and ìstrongœ roughness (cavities, pores and bumps) for which the height is of the same order, or larger, than the lateral size.Slight roughness It is impossible, at present, to provide a uni–ed description of the QCM response for non-uniform solid/liquid interfaces with an arbitrary geometrical structure. The limiting case, of slightly rough surfaces, has been considered in our recent papers.22,23 The problem has been solved in the framework of the perturbation theory with respect to the parameters and For the roughness described by a one-scale corre- o +m(R) o@1 h/d@1.lation function the —uid-induced shift and the width of the resonance frequency has been written in the form22,23 *u\ur[u0\[(u0)3@2(og)1@2 n(2oq kq)1@2 G1] h2 l2 F(l/d)H (1) *C\C[C0\(u0)3@2(og)1@2 n(2oq kq)1@2 G1] h2 l2 U(l/d)H (2) where h is the root mean square (rms) height, and l is the lateral scale length, which has the meaning of the correlation length for a random roughness and the period for a periodic corrugation ; and are the frequency and the width of the u0\(n/d)(kq/oq)1@2 C0 resonance of the free quartz crystal –lm.The scaling functions F(l/d) and U(l/d) are expressed through Fourier components of the pair correlation function of roughness g(K).22,23 The correlation function g(K) is related to the Fourier component of the surface pro–le function m(K) by o m(K) o2\Sh2g(K) (3) where S is the area of crystal surface.It should be mentioned that the roughness factor R, i.e. the true-to-apparent surface ratio, is proportional to the square of the ìslopeœ of the roughness, h/l, and can be written as R\1] h2 2l2 P d2K (2n)2 g(K)K2 (4) The –rst terms in braces in eqn.(1) and (2) de–ne the shift, and the broadening, *us , of the shear resonance at the smooth crystal/liquid interface.7 The presence of *Cs , surface roughness leads to the additional decrease, and broadening, of the *urh , *Crh , resonance frequency. Our calculations demonstrate the generation of the liquid motion normal to the surface oscillations and the non-uniform distribution of the liquid pressure in the interfacial layer.Both inertial and viscous components contribute to the roughness-induced QCM response. We see that the roughness-induced shift and width are proportional to the square of the slope of the roughness, h/l, and to the one-parameter scaling functions, F(l/d) and U(l/d).The scaling parameter, l/d, is the ratio of the characteristic lateral length of the roughness to the decay length of the liquid velocity. The particular form of the scaling functions F(l/d) and U(l/d) is determined by the morphology of the interface.22,23 However, the asymptotic behaviour of these functions30 In—uence of surface roughness on the QCM response in a solution for and are universal.In both limits the roughness-induced shift of the l/dA1 l/d@1 resonance frequency is proportional to the liquid density and depends slightly on the viscosity. Only in the region where l is of the order of d we found the strong dependence of the resonance frequency on g. We also remark that in these asymptotic regions, l/dA and the shift is proportional to the square of the frequency of the free quartz 1 l/d@1, crystal, as in the case of mass loading.1 However, our results show that the in—uence u0 2 , of slight surface roughness on the frequency shift cannot be explained in the terms of the mass of —uid ìtrappedœ by surface cavities as was proposed in previous papers.7,8 This statement can be illustrated by consideration of the sinusoidal roughness pro–le.The mass of the liquid ìtrappedœ by the sinusoidal grooves does not depend on the slope of the roughness, h/l, and is equal to hS/2. However, eqn. (1) demonstrates that the frequency shift, increases with the increase in the slope. For all values of the param- *urh , eters (o, g, the roughness-induced width of the resonance is less than the u0) roughness-induced shift.Eqn. (2) shows that in the regions and the width, l/dA1 l/d@1 is proportional to the factors and respectively. *Crh, (og)1@2u03@2 (o)3@2(g)~1@2u05@2, Fig. 2 presents the dependences of the frequency shift on a liquid viscosity calculated for the Gaussian correlation function of roughness, g(K)\nl2 exp([l2K2/4). We see that the roughness leads to new viscosity dependences which do not appear in the case of a smooth surface.The eÜect of roughness is most pronounced in the low viscosity limit when the liquid-induced shift, at a smooth surface is small. The roughness- *us , induced shift tends to a constant in the high viscosity limit when g[l2u0 o/2. The results obtained allow us to estimate the eÜect of roughness on the QCM response if the surface pro–le function m(R) can be found from the independent measurements, for instance via scanning tunnelling microscopy (STM).Strong roughness In the case of strong roughness, the theoretical approaches based on perturbation theory cannot be applied and we have to use a simpli–ed model. Here, we propose the coarse- Fig. 2 Dependence of the resonance frequency shift on the viscosity of a liquid for a randomly rough surface.Roughness-induced shifts, for l\(»») 1000, (» » ») 2500 and (… … … …) 5000 [*urh , —uid-induced shift, [*u]10~1, for a smooth interface. The calculations were carried out ”; (K) for dyn cm~2, gcm~3, MHz, o\1 g cm~3, kq\2.947]1011 oq\2.648 f0\u0/2n\5 h\30 nm.L . Daikhin and M. Urbakh 31 grained description based on Brinkmanœs equation for the velocity –eld in the interfacial region.25 In this approach, one treats the —ow of a liquid through a non-uniform surface layer as the —ow of a liquid through a porous medium.26h30 The morphology of the interfacial layer is characterized by a local permeability that depends on the ìporosityœ of the layer. For hydrodynamic purposes we treat the interfacial layer [see Fig. 1(a)] as a twophase (solid»liquid) porous medium26h33 with a permeability of The physical mH 2 . meaning of the permeability length scale, depends on the nature of the interfacial mH , layer. For instance, in the case of rough surface layers, is related to their porosity and, mH for an entangled polymer layer, is of the same magnitude as the equilibrium corre- mH lation length.34 The liquid —ow through the interfacial layer is described by Brinkmanœs equation26h33 iuovx(z, u)\g d2 dz2 vx(z, u)]gmH~2[V0[vx(z, u)] (5) where is the velocity of the crystal surface and u)exp(iut) is the V0 exp(iut) vx(z, t)\vx(z, velocity –eld in the liquid.In this equation the eÜect of the solid phase on the liquid —ow is given by the resistive force, that has a Darcy-like form, gmH~2[V0[vx(z, u)]. Brinkmanœs equation presents a variant of the eÜective medium approximation which does not describe explicitly the generation of non-laminar liquid motion and conversion of the in-plane surface motion into the normal-to-interface liquid motion.These eÜects result in additional channels of energy dissipation which are eÜectively included into the model by the introduction of the Darcy-like resistive force.Note that the QCM response discussed here is sensitive to energy dissipation, rather than to the details of the velocity –eld in the liquid. While the limitation of Brinkmanœs equation are apparent, there is no alternative equation in the literature which has been accepted unconditionally. Eqn. (5) shows that the eÜect of the interfacial layer is most pronounced for small values of The velocity gradient in the layer decreases with the decrease of mH/L .mH/L , and for the majority of the layer moves with a velocity equal to the velocity of mH/L @1 the quartz surface, This behaviour re—ects the trapping of the liquid by the surface V0 . roughness. For higher values of the parameter the in—uence of the interfacial mH/L P1, layer goes down and the velocity –eld approaches the velocity –eld at the smooth solid/ liquid interface. The liquid-induced shift, *u, and the width, *C, of the resonance frequency may be written in the form25 *u\[ u0 2 o n(kq oq)1@2 ReA1 q0 ] L mH 2 q1 2 [ 1 W 1 mH 2 q1 2 G2q0 q1 [cosh(q1L )[1]]sinh(q1L )HB (6) *C\[ u0 2 o n(kq oq)1@2 ImA1 q0 ] L mH 2 q1 2 [ 1 W 1 mH 2 q1 2 G2q0 q1 [cosh(q1L )[1]]sinh(q1L )HB (7) Here and The –rst terms q0\(iuo/g)1@2, q1\q02]mH~2 W \q1cosh(q1L )]q0sinh(q1L ).in the right-hand side of eqn. (6) and (7) describe the QCM response for the smooth quartz crystal/bulk liquid interface.6 It should be mentioned that, in this case, the liquidinduced shift and width are equal. The additional terms present the shift and the width of the QCM resonance caused by the interaction of the liquid with the non-uniform interfacial layer.When the permeability length scale is the shortest length of the problem, and mH@d the layer-induced shift is proportional to the liquid density and does not depend mH@L , on the viscosity. It has the form of the frequency shift due to a mass loading.The eÜect32 In—uence of surface roughness on the QCM response in a solution results from the inertial motion of the liquid trapped by the inhomogeneities in the interfacial layer. The eÜective thickness of the liquid –lm rigidly coupled to the oscillating surface is equal to and is less than the thickness of the inhomogeneous L [mH layer, L . This point must be taken into account in interpreting the QCM data.The increase of the permeability leads to the enhancement of the velocity gradient in the mH 2 layer, which results in the decrease of the mass loading shift and in the increase of the width caused by the energy dissipation. When the layer thickness is the shortest length of the problem, and L @d, L @mH *u is also proportional to the liquid density and does not depend on viscosity.mH@d, 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 of the resonance diÜer considerably. The usual form of the description of the experimental data in liquids is the representation of the real and imaginary components of QCM response as functions of the liquid density o and of the parameter (og)1@2.Fig. 3 and 4 show the dependences of the layerinduced shift and the width of the resonance frequency on o and (og)1@2, which have been calculated according to eqn.(6) and (7). The common features of all curves are the Fig. 3 Layer-induced shift of the resonance frequency vs. (a) (go)1@2 and (b) o. The calculations were carried out for MHz, nm, decay length of shear waves in water, and f0\u0/2n\5 d0\250 layer thicknesses : L \0.5 (1) (2) o\1 g cm~3 in (a) and g\1 cP in d0 ; mH/L \0.5, mH/L \0.2 ; (b). Fig. 4 Layer-induced width of the resonance frequency vs.(a) (go)1@2 and (b) o. The calculations were carried out for MHz, nm, the decay length of shear waves in water, f0\u0/2n\5 d0\250 and layer thicknesses, (1) (2) o\1 g cm~3 in (a) and g\1 cP L \0.5d0 ; mH/L \0.5, mH/L \0.2 ; in (b).L . Daikhin and M. Urbakh 33 increase in the QCM response with the increase in –lm thickness and with the decrease in the ratio The frequency shift changes rapidly with (og)1@2 in the region of mH/L .where the limiting value is of the order of unity, and slowly (og)1@2\t * (L /d, mH/L ), t * increases with increase in the parameters L /d and For higher values of (og)1@2 the mH/L . function *u [(og)1@2] tends to a constant [see Fig. 3(a)]. This behaviour re—ects the fact that, in the high-viscosity limit, when the gradient of the velocity in the inter- L /d@1, facial layer is small and the layer in—uences only slightly the velocity –eld in the bulk liquid.As a result, the viscosity-dependent contribution to the frequency shift, which is proportional to the velocity gradient, would remain the same as for the smooth solid/ liquid interface. This point explains also the decrease in the layer-induced width of the resonance with increasing (og)1@2 for [see Fig. 4(a)]. We remark (og)1@2[t * (L /d, mH/L ) that the plots *u vs. (og)1@2 presented in Fig. 3(a) closely resemble the plots in Fig. 2 obtained for slightly rough surfaces within the perturbation approach. Both the perturbation approach and the Brinkman description lead to similar dependences of QCM response on o and g.Fig. 3(b) shows that the frequency shift is linear in the liquid density for small thicknesses of the interfacial layer, and deviations from the linear proportionality L /d@1, arise as the thickness increases. The deviations are caused by an increase in the contribution of the energy dissipation processes to the QCM response with increases in the thicknesses.Our calculations predict the interesting feature of the layer-induced width of the resonance as a function of (og)1@2 [see Fig. 4(a)]. For L Pd the curves for C vs. (og)1@2 have maxima located at The appearance of the maxima results (og)1@2Bt * (L /d, mH/L ). from the fact that the energy dissipation induced by the interfacial layer diminishes for both high and low viscosities.This eÜect has already been discussed. The dependences of the resonance width on the liquid density shown in Fig. 4(b) re—ect the in—uence of both the mass loading and the energy dissipation contributions. For thick layers, the curves *C vs. o have quadratic form, as predicted by eqn. (2). New geometry for QCM measurements We now consider a model for the coupling of shear waves in a quartz crystal bounded by a —at surface with damped waves in a liquid layer con–ned between a quartz crystal and a rough immobile surface.We take the planes z\0 and z\d being coincident with the unconstrained and constrained faces of a quartz resonator, respectively, and the z axis pointing toward the liquid. Here, d is the thickness of the quartz crystal –lm. We assume that the rough interface between the wall and the liquid is described by the equation z\d]H]m(x, y).The plane z\d]H is chosen such that the average value of the function m(x, y) over the surface is equal to zero. Here, H is the average thickness of the liquid –lm [see Fig. 1(b)]. The elastic displacements in the crystal u(r, t)\u(r, u)exp(iut) are described by the following wave equation [u2u(r, u)\ kq oq +2u(r, u) (8) The liquid velocity u) is the solution of the linearized Navier»Stokes equation ø(r, iuø(r, u)\[1 o +P(r, u)] g o +2ø(r, u) (9) where P(r, u) is the pressure in the liquid.The velocity must also satisfy the incompressibility condition.34 In—uence of surface roughness on the QCM response in a solution Boundary conditions for the elastic displacements and the liquid velocity include : (a) the absence of forces acting on the unconstrained crystal surface z\0; (b) the equality of crystal and liquid velocities at the interface z\d; and (c) the equality of liquid velocity to zero at the interface z\d]H]m(x, y).We have solved the problem by perturbation theory,22,23 which is valid for slightly rough interfaces.In order to solve eqn. (9) it is convenient to Fourier transform the liquid velocities, the pressure and the roughness pro–le function m(x, y) from the tangential coordinates R\(x, y) to the corresponding wave vectors according to K\(Kx , Ky) the equation m(K)\PdR m(R)exp([iKR) (10) The solution of eqn. (9) has the form P(K, z, u)\P1(K)exp[[K(z[d)]]P2(K)exp[K(z[d[H)] (11) va(K, z, u)\Aa(K)exp[[qK(z[d)]]Ba(K)exp[qK(z[d[H)][ 1 ou Ka P1(K) ]exp[[K(z[d)][ 1 ou Ka P2(K)exp[K(z[d[H)], a\x, y (12) vz(K, z, u)\ i qK ;a KaAa(K)exp[[qK(z[d)][ i qK ;a Ka Ba(K)exp[qK(z[d[H)] [ i ou KP1(K)exp[[K(z[d)]] i ou KP2(K)exp[K(z[d[H)] (13) Here K\oKo, and z, u) and z, u) are the projections of qK\(iuo/g]K2)1@2 va(K, vz(K, the vector of velocity v(K, z, u) on the axes a\x, y and z, correspondingly.The coefficients and are obtained from the boundary con- Aa(K), Ba(K) P1(K), P2(K) ditions. The explicit expressions for these coefficients are presented in the Appendix. After determination of and the resonance frequency measured Aa(K), Ba(K), P1(K) P2(K) in QCM experiments can be found from the equality of the shear stresses on two sides of the interface z\d k d dz ua(z\d, u)\g d dz va(z\d, u), a\x, y (14) Substitution of the expressions (A1)»(A12) into eqn.(14) gives the equation for the resonance frequency which is valid up to the second order in the rms height of roughness ur , tan(kd)\[ o oq k q0 Ccoth(q0H)]1 S q0 sinh(q0H) P dK (2n)2 o m(K) o2K cos2/ D(K)D (15) where D(K)\exp([qKH) ]M[Y1(K)exp([KH)]Y2(K)]cosh(qKH)[[Y1(K)]Y2(K)exp([KH)]N (16) where S is the area of a crystal surface, and / is the angle between the k\u(oq/kq)1@2 direction of shear oscillations and the vector K.It is often assumed that random roughness obeys the Gaussian distribution, which is characterized by two parameters: the rms height, h, and the lateral correlation length, l. In this case o m(K) o2\Sh2l2n exp([l2K2/4).L .Daikhin and M. Urbakh 35 Now, the shift and the width of the resonance frequency may be rewritten in the form *u\[(u0)3@2(og)1@2 n(2oq kq)1@2 AReG(1[i)cothC(1]i) H d DH] h2 l2 F(H/d, l/d)B (17) *C\[(u0)3@2(og)1@2 n(2oq kq)1@2 AImG(1[i)cothC(1]i) H d DH] h2 l2 U(H/d, l/d)B (18) and F(H/d, l/d)\ l 2d ReG 1 sinh[(1]i)H/d] P0 = dx x2D(x, H/d, l/d)exp([x2/4)H (19) U(H/d, l/d)\ l 2d ImG 1 sinh[(1]i)H/d] P0 = dx x2D(x, H/d, l/d)exp([x2/4)H (20) The –rst terms in the large curved brackets in eqn.(17) and (18) de–ne the shift and the broadening of the shear resonance for a liquid layer con–ned between a smooth quartz crystal and a smooth wall. The presence of roughness at the immobile surface leads to additional change in the resonance frequency and broadening. Fig. 5 shows the dependences of the resonance frequency shift on the liquid viscosity for diÜerent values of the correlation length of roughness.The presence of roughness leads to an increase in compared to the case of smooth surfaces. The in—uence of ur roughness is most pronounced for HOd. For a given roughness factor, the eÜect decreases with increase in the correlation length. This behaviour re—ects the fact that the presence of roughness reduces the eÜective thickness of the con–ned liquid layer.As a result the resistance force acting on the quartz crystal decrease and the resonance frequency increases. Fig. 6 presents the roughness-induced shift as a function of the distance between the quartz crystal and the rough surface. Again, the in—uence of roughness is most pronounced in the region HOd.The eÜect is proportional to the roughness factor and depends strongly on the correlation length of roughness. We believe that a comparison of experimental data on H dependence of *u with eqn. (17) and (18) will allow us to check the prediction of the theory and to estimate the roughness factor and the correlation length. The use of the new variable parameter, H, which is absent in the traditional QCM con–guration, gives an eÜective way of studying the eÜect of roughness.Fig. 5 Dependence of the shift of the resonance frequency in con–ned geometry on the viscosity of a liquid. The calculations were carried out for dyn cm~2, gcm~3, kq\2.947]1011 oq\2.648 o\1 g cm~3, nm, the decay length of shear waves in water, f0\u0/2n\5 MHz, d0\250 roughness factor R\1.5 ; (1) smooth immobile surface ; (2) and (3) rough surface with correlation lengths, (2) and (3).l\d0/2 d0/536 In—uence of surface roughness on the QCM response in a solution Fig. 6 Dependence of the roughness-induced shift of the resonance frequency in con–ned geometry on the thickness of the liquid layer. The calculations were carried out for kq\2.947 ]1011 dyn cm~2, gcm~3, MHz, o\1 g cm~3, g\1 cP, 250 oq\2.648 f0\u0/2n\5 d0\ nm, roughness factor R\1.5 ; (1) and (2).l\d0/2 l\d0/5 Note that in both geometries discussed here [Fig. 1(a) and (b)] the roughness-induced response of QCM is determined by the same parameters. It allows us to predict the eÜect of roughness in the traditional QCM con–guration on the basis of the results obtained in the con–ned geometry.Summary We have considered the eÜect of surface roughness on the QCM response in contact with a liquid. The problem has been studied within the perturbation theory approach and with the use of Brinkmanœs equation for the velocity in the non-uniform interfacial region. We have found that the frequency changes owing to both the inertial motion of a liquid rigidly coupled to the surface and to the additional viscous energy dissipation induced by roughness. We suggest a new geometry of QCM measurements which enables one to separate the eÜect of roughness from other contributions.The results obtained are as follows. (1) For the slightly rough surfaces the roughness-induced shift and the width are proportional to the square of the ìslopeœ of the roughness and depend on the height» height correlation function.The eÜect of the roughness is most pronounced for low liquid viscosity. The rms height and the lateral correlation length of the slight roughness can be estimated from the dependences of the shift and the width on the liquid viscosity and on the frequency of the free quartz crystal.(2) For the very rough surfaces the roughness-induced shift and width of the resonant frequency are determined by two parameters: d/L and They are the ratios of the mH/L . decay of the shear wave in the liquid and the permeability length scale in the interfacial layer to the thickness of this layer. (3) Our calculations demonstrate that, while for smooth surfaces the liquid-induced shift and width of the resonance are equal, they diÜer essentially for rough surfaces.Our results show that empirical representation of the QCM response in a liquid as a linear combination of a (og)1@2 and a o term20 can be applied for the description of the shift of the resonant frequency in the case of thin rough layers, L \d. Both the perturbation approach and the Brinkman description lead to similar dependences of QCM response on o and g. (4) The new con–guration for QCM experiments is considered where a liquid layer is con–ned between a rough immobile surface and a quartz resonator with a —at interface.It is shown that the roughness-induced shift and width of the resonance frequency are determined by two parameters H/d and l/d. In this con–guration the distance betweenL .Daikhin and M. Urbakh 37 the quartz resonator and the immobile surface could be easily changed thus providing an eÜective way to study the eÜect of roughness on QCM response in liquids. Appendix In order to derive the equations for the coefficients and we Aa(K), Ba(K) P1(K), P2(K) follow the perturbation scheme developed in ref. 22 and 23. To the lowest order in rms height, h, we arrive at results for a liquid layer con–ned between a smooth surface of quartz crystal and a smooth immobile wall Aa(0)(K)\(2n)2 d(K)iCau cos(kd) exp(qKH) sinh(qKH) , P1(0)(K)\0 (A1) Ba(0)(K)\[(2n)2 d(K)iCau cos(kd) 1 sinh(qKH) , P2(0)(K)\0 (A2) To the –rst order in h we obtain P1(1)(K)\ ou qK iu K cos(kd) ;b KbCb q0 m(K)Y1(K), b\x, y (A3) P2(1)(K)\ ou qK iu K cos(kd) ;b KbCb q0 m(K)Y2(K), b\x, y (A4) Aa(1)(K)\iuq0 cos(kd)m(K) 1 1[exp([2qKH) A Ka KqK ;b KbCb[Y1(K)]Y2(K)exp([KH)] [exp([qKH)GKa KqK ;b KbCb[Y1(K)exp([KH)]Y2(K)] ];b KbCb cos(kd) sinh(q0H) [1]exp(q0H)]HB (A5) Ba(1)(K)\iuq0 cos(kd)m(K) 1 1[exp([2qKH) A Ka KqK ;b KbCb[Y1(K)exp([KH)]Y2(K)] [exp([qKH)GKa KqK ;b KbCb[Y1(K)]Y2(K)exp([KH)]H ];b KbCb cos(kd) sinh(q0H) [1]exp(q0H)]B (A6) Here, we introduced functions and which have the following form Y1(K) Y2(K) Y1(K)\ 1]exp(q0H) [1[exp([2KH)]sinh(q0H)(a12[a2 2) ]Ma1[exp([KH)[exp([qKH)][a2[1[exp([qKH[KH)]N (A7) Y2(K)\ 1]exp(q0H) [1[exp([2KH)]sinh(q0H)(a12[a2 2) ]Ma1[1[exp([qKH[KH)][a2[exp([KH)[exp([qKH)]N (A8)38 In—uence of surface roughness on the QCM response in a solution where a1\1[ K qK[1[exp([2KH)] ]M1[exp([qKH[KH)]exp([KH)[exp([KH)[exp([qKH)]N (A9) a2\exp([qKH)[ K qK[1[exp([2KH)] ]Mexp([KH)[exp([qKH)]exp([KH)[1[exp([qKH[qKH)]N (A10) To the second order in h we have Aa(2)(0)\[ exp([q0H) 1[exp([2q0H) P dK (2n)2 qK m([K)[Aa(1)(K)exp([qKH)[Ba(1)(K)] (A11) Ba(2)(0)\ 1 1[exp([2q0H) P dK (2n)2 qK m([K)[Aa(1)(K)exp([qKH)[Ba(1)(K)] (A12) References 1 M.Thompson, L.A. Kipling, W. C. Duncan-Hewitt, L. V. Rajakovic and B. A. Cavic-Vlasak, Analyst, 1991, 116, 881. 2 D. A. Buttry and M. D. Ward, Chem. Rev., 1992, 92, 1355. 3 E. T. Watts, J. Krim and A. Widom, Phys. Rev. B, 1990, 41, 3466. 4 D. Johannsmann, K. Mathauer, G. Wegner and W. Knoll, Phys. Rev. B, 1992, 46, 7808. 5 C. E. Reed, K. K. Kanazawa and J. H. Kaufman, J. Appl. Phys., 1990, 68, 1993. 6 S. Bruckenstein and M. Shay, Electrochim. Acta, 1985, 30, 1295. 7 K. K. Kanazawa and J. G. Gordon II, Anal. Chim. Acta, 1985, 175, 99. 8 D. A. Buttry, in Electroanalytical Chemistry, ed. A. J. Bard, Marcel Dekker, New York, 1991, vol. 17, p. 1. 9 R. Schumacher, Angew. Chem., Int. Ed. Engl., 1990, 29, 329. 10 G. Sauerbrey, Z. Phys., 1959, 155, 206. 11 A. L. Kipling and M. Thompson, Anal. Chem., 1990, 62, 1514. 12 L. V. Rajakovic, B. A. Caviv-Vlasak, V. Ghaemmaghami, M. R. K. Kallury, A. L. Kipling and M. Thompson, Anal. Chem., 1991, 63, 615. 13 V. Tsionsky, L. Daikhin and E. Gileadi, J. Electrochem. Soc., 1995, 143, 2240. 14 R. Beck, U. Pittermann and K. G. Weil, Ber. Bunsen-Ges. Phys. Chem., 1988, 92, 1363. 15 M. Yang, M. Thompson and W. C. Duncan-Hewitt, L angmuir, 1993, 9, 802. 16 R. Beck, U. Pittermann and K. G. Weil, J. Electrochem. Soc., 1992, 139, 453. 17 M. Yang and M. Thompson, L angmuir, 1993, 9, 1990. 18 R. Schumacher, G. Borges and K. K. Kanazawa, Surf. Sci., 1985, 163, L621. 19 R. Schumacher, J. G. Gordon and O. Melroy, J. Electroanal. Chem., 1987, 216, 127. 20 S. J. Martin, G. C. Frye, A. J. Ricco and S. D. Sentaria, Anal. Chem., 1993, 65, 2910. 21 S. Bruckenstein, A. Fensore, Z. Li and A. R Hillman, J. Electroanal. Chem., 1994, 370, 189. 22 M. Urbakh and L. Daikhin, Phys. Rev. B, 1994, 49, 4866. 23 M. Urbakh and L. Daikhin, L angmuir, 1994, 10, 2839. 24 V. Tsionsky, L. Daikhin, M. Urbakh and E. Gileadi, L angmuir, 1995, 11, 674. 25 L. Daikhin and M. Urbakh, L angmuir, 1996, 12, 6354. 26 H. C. Brinkman, Appl. Sci. Res. A, 1947, 1, 27. 27 M. Sahimi, Rev. Mod. Phys., 1993, 65, 1393. 28 R. Hilfer, Adv. Chem. Phys., 1996, 92, 299. 29 C. K. M. Tam, J. Fluid. Mech., 1969, 38, 537. 30 T. S. Lundgren, J. Fluid. Mech., 1972, 51, 273. 31 P. G. de Gennes, Macromolecules, 1981, 14, 1637. 32 G. H. Fredrickson and P. Pincus, L angmuir, 1991, 7, 786. 33 L. G. Jones, C. M. Marques and J-F. Joanny, Macromolecules, 1995, 28, 136. 34 For entangled polymer the correlation length is the average distance between entanglement points. Paper 7/03124F; Received 7th May, 1997

ISSN:1359-6640    

DOI:10.1039/a703124f

出版商:RSC    

年代:1997

数据来源: RSC

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Faraday Discuss., 1997, 107, 39»52 Acoustic waves at the solid/liquid interface Hans-Dieter Liess, Aleksandar Knezevic, Martin Rother and Josef Muenz University of the Federal Armed Forces Munich, D-85577 Neubiberg, Germany The coupling of acoustic waves across interfacial boundaries depends upon the morphology of the surface and the orientation of the waves at the boundary. The dependence of the wave propagation at the boundary as a function of the physical properties of the liquid is described.The primary in—uence comes from the density and viscosity. However, the electrical conductivity of the liquid may also contribute to the interactions at such boundaries. The structure of the interface is an important parameter in discriminating the in—uences of the density and viscosity of the liquid.It is possible to optimise the surface structure for sensitivity to either density or viscosity and to achieve a high sensitivity to liquid densities despite samples with a broad range of viscosities. The in—uence of surface morphology of the thickness shear mode resonator on the liquid properties is studied by using diÜerent structured electrodes to separate the density and viscosity of the liquid.Finally, a sensor was developed to measure antifreeze concentration in water. 1 Introduction It is well known that AT-cut quartz resonators oscillating in the thickness shear mode have mechanical properties and temperature behaviour which are favourable for microbalance applications. Since 1980, such quartz microbalances have also been used in liquids. The basic theory1 describes the frequency behaviour for rigid, viscous and viscoelastic overlayers.It was shown that the resonance frequency of a microbalance having gold electrodes with one side exposed to the liquids depends primarily on their viscosity»density product.2 It has also been shown that the mass sensitivity is identical in air and liquid.3 If, however, the surface roughness is also considered, impedance analysis of the microbalance shows an additional frequency shift due to viscous-coupled liquid and trapped mass.4h6 Using two resonance circuits, one with a smooth surface and one with a surface structured perpendicular to the quartz oscillation, it is possible to measure diÜerentially the density value of the liquid.7 Applying this value to the frequency shift of a resonator with a smooth surface, the viscosity can be calculated. The work presented here displays the in—uence of the electrode surface morphology on the behaviour on the resonator, as measured by impedance analysis.This dependence on morphology is used in sensor applications to determine the concentration of —uid mixtures by measuring the liquid density and compensating the viscosity. 2 Experimental design In order to study the dependence of a resonator on the morphology of the interface between a surface and a liquid, diÜerent structured surfaces were made and characterised. For this characterisation, the resonators were investigated in diÜerent viscous liquids and their impedance and resonance frequency measured. 3940 Acoustic waves at the solid/liquid interface 2.1 Interfaces Four diÜerent electrode structures were studied with the following characteristics (Table 1). Smooth surface made by evaporation of metal on polished crystals. Grooves oriented perpendicular to the shear movement. Grooves oriented parallel to the shear movement. Rough surface made by electrochemical deposition. The structures were placed on the entire surface of the crystal which was exposed to the liquid.The thickness, depth and spacing of the grooves were varied in order to study the in—uence of the diÜerent structures on the resonance behaviour of the quartz in liquid. 2.2 Fabrication of devices, grooves and characterisation The polished AT-cut quartz crystals were purchased from Quarzkeramik, Stockdorf, Germany.After cleaning the crystal, a 30 nm thick layer of chromium was deposited onto both sides of the resonator as a glue metal, followed by a vapour-deposited gold layer. These electrodes were structured in the following two diÜerent ways. Grooves were etched into a 2.5 lm thick gold surface in a stepwise manner achieving diÜerent depths. Alternatively, the grooved structures were built up from a thinner gold layer of Table 1 Structure and oscillation direction of the applied resonator electrodesH-D.L iess et al. 41 0.3 lm by analogous photolithographical pre-treatment and electrochemical deposition. For rough surfaces, diÜerent morphologies were achieved by changing the deposition parameters, such as temperature, current density and ratio of the sizes of the working to the counter-electrode. The dimensions of the surface structure were studied using scanning electron microscopy.The groove pro–les were exposed by breaking the crystal perpendicular to the groove direction. 2.3 Set-up The crystals were placed in a custom-made liquid cell which allowed the application of diÜerent temperatures and diÜerent test —uids.The data were collected in either impedance mode or oscillation mode (Fig. 1). In the impedance mode, the complex admittance is measured and the various elements of the equivalent circuit are calculated. In the oscillation mode, only the resonance frequency of the resonator is measured. This mode, however, allows an equilibrium state to be reached. 2.4 Electrical equivalent circuit It is generally known that the equivalent circuit consists of two parallel branches.The serial branch contains all the elements representing the mechanical properties of the crystal near its resonance frequency. Their values are expressed as a resistance, an inductance and a capacitance. The parallel branch contains only the parallel capacitance of the electrodes, connecting wires and the stray capacitance.The additional in—uence of the liquid on the resonator changes the components of the equivalent circuit by adding an additional series resistance and a series inductance (Fig. 2). The additional series resistance results from the power losses caused by (R2) viscous coupling. The additional series inductance consists of two parts and The (L 2 L 3). series inductance is generated by the mass movement due to viscous coupling. L 2 However, the series inductance consists of mass loading caused by trapped liquid on L 3 the surface.7 Fig. 1 Experimental set-up42 Acoustic waves at the solid/liquid interface Fig. 2 Equivalent circuit of a liquid loaded quartz. and represent the viscous coupling, R2 L 2 L 3 represents the mass loading.7 The change in the series resonance frequency due to the in—uence of the liquid is calculated as follows : From *f\fLiquid[fAir\ 1 2p)[(L 1]L 2]L 3)C1] [ 1 2p)(L 1C1) it follows that for L 1AL 2]L 3 *f\[ L 2]L 3 2L 1 fAir 3 Results The impedance analysis measurements were made with various structures and various test liquids with diÜerent viscosities and densities. 3.1 Measurements on structures perpendicular to the shear movement Fig. 3 shows the values for the inductance measured on resonator structures grooved perpendicular to the shear movement as a function of the viscosity»density product of the test liquid. The groove dimensions and (as de–ned in Fig. 4) are 5 lm each, and b1 b2 the height of each groove is 1.1 lm. The test liquids are a series of ethylene glycol and water mixtures.The measured inductance, corresponding to the trapped mass loading, is linearly L 3 , dependant on test liquid density (Fig. 5). This results in a change in the resonance frequency which is, in the same way, dependent on the density of the test liquid. The measured value of 840 Hz is lower than the theoretical value of 1319 Hz. This diÜerence can be explained by assuming that a part of the liquid trapped within the grooves does not move in phase with the oscillating device and is eÜectively part of the bulk solution.H-D. L iess et al. 43 Fig. 3 Equivalent circuit elements and vs. the square root of the (|) *L (\L 2]L 3) (L) L 2 viscosity»density product of the test liquid (water, 30, 50, 60 and 100% antifreeze) measured by impedance analysis. Quartz electrode : structured perpendicular to the oscillation, lm, b1\b2\5 h\1.1 lm.In agreement with theory7 the groove depth also in—uences the inductance, and L 3 , the resonance frequency. It has been shown (Fig. 6 and 7) that, as groove depth increases, the resonance frequency change increases as well. This can be viewed as resulting from the deeper grooves trapping more liquid. 3.2 EÜect of rough surfaces Our previous measurements are in agreement with published results7 describing the in—uence of the test liquid density on the resonator.These measurements show that the viscosity eÜect on these grooved, yet otherwise smooth, surfaces is similar to the viscosity eÜect at smooth surfaces. This eÜect changes with the introduction of surface Fig. 4 Perpendicular-to-the-oscillation structured quartz electrode44 Acoustic waves at the solid/liquid interface Measured increase of the inductance and the resulting frequency decrease rep- Fig. 5 (K) L 3 (=) resenting the increase in the moved mass caused by trapped liquid on the electrode surface vs. the density of the test liquid (water, 30, 50, 60 and 100% antifreeze).Quartz electrode (see Fig. 3) : structured perpendicular to the oscillation, lm, h\1.1 lm. b1\b2\5 Fig. 6 Shift (*f ) in the series resonance frequency (maximum of the real part of the admittance) vs. the square root of the viscosity»density product of the test liquid (water, 30, 50 and 70% antifreeze) measured by impedance analysis. Quartz electrode : structured perpendicular to the oscillation, lm.The resonance frequency of the unperturbed quartz is given with b1\b2\5 (L) h\0.25 lm: 7114.3 kHz, h\0.8 lm: 7168.8 kHz; h\1.1 lm: 7200.25 kHz, h\1.55 (Ö) (K) (>) lm: 7244.25 kHz.H-D. L iess et al. 45 Fig. 7 Shift (*f ) in the series resonance frequency (maximum of the real part of the admittance) vs. the structure depth of the water and antifreeze loaded quartz measured by impedance (|) (K) analysis.Quartz electrode : structured perpendicular to the oscillation, lm. A and C b1\b2\5 are expected curves if liquid trapping is assumed, B and D are expected curves if no liquid trapping is assumed. roughness (Fig. 8). For our experiments rough surfaces (Fig. 9) have been created by electrochemical gold deposition with a current density of more than 3 mA cm~2.As the size of the gold particles deposited onto the surface increases, the sensitivity to the liquid density increases by a much larger extent (Fig. 10) relative to the smooth structured electrodes. This can be explained by trapping of the liquid between the gold grains. Our rough surfaces showed a grain size of ca. 1 lm. Hereby, the increase in surface area leads, however, to an increase in the viscous coupling.For the measurement of the density, the viscosity has to be compensated as described in Section 4. 3.3 Measurements on structures parallel to the shear movement Measurements on resonator structures grooved parallel to the shear movement show a higher dependence on the liquid viscosity owing to the enlarged surface area (Fig. 11 and 12). As expected, this structure does not lead to a additional sensitivity term to the density. This structure is therefore useful for the compensation of the viscosity dependence in the rough structure. 4 Application The thickness shear mode resonator was developed to measure the concentration of liquid mixtures by their viscosity and their density. The determination of these values46 Acoustic waves at the solid/liquid interface Fig. 8 Equivalent circuit elements and vs. the square root of the (|) *L (\L 2]L 3) (L) L 2 viscosity»density product of the test liquid (water, 30, 50, 60, and 100% antifreeze) measured by impedance analysis. Quartz electrode : structured perpendicular to the oscillation, lm, b1\5 lm, h\3.5 lm, rough. b2\15 was achieved from the resonance frequency of the resonators in oscillation mode.As a test —uid for the resonator application, mixtures of ethylene glycol and water (antifreeze) have primarily been used. Fig. 13 shows the viscosity and density of the —uid relative to the antifreeze concentration in water. 4.1 Sensor arrangement The sensor arrangement with measurement and reference resonators is shown schematically in Fig. 14. The frequency diÜerence between the two oscillators is proportional to Fig. 9 SEM photograph of a used rough electrode surface (magni–ed 20 000 times)H-D. L iess et al. 47 Measured increase in the inductance and the resulting frequency decrease Fig. 10 (K) L 3 (Ö) representing the increase in the moved mass caused by trapped liquid on the electrode surface vs.the density of the test liquid (water, 30, 50, 60 and 100% antifreeze). Quartz electrode (see Fig. 8) : structured perpendicular to the oscillation, lm, lm, h\3.5 lm, rough. b1\5 b2\15 Fig. 11 Shift (*f ) in the series resonance frequency (maximum of the real part of the admittance) of the water and antifreeze loaded quartz vs. the stepwise-etched structure depth of the (>) (=) quartz electrode measured by impedance analysis.Quartz electrode : structured parallel to the oscillation, lm. (»») expected for antifreeze, (» » ») expected for water. b1\b2\548 Acoustic waves at the solid/liquid interface Fig. 12 Equivalent circuit element representing the viscous coupling between oscillating elec- L 2 trode surface and the —uid measured for several diÜerent surface areas.Liquids: water 30% (|), and 50% antifreeze g cm~1 s~1, q\1.083 g cm~3; g (L) (K). (K) fdyn\0.043 (L) fdyn\0.024 cm~1 s~1, q\1.052 g cm~3; gcm~1 s~1, q\0.998 g cm~3. (|) fdyn\0.010 the density or the viscosity»density product. This frequency diÜerence depends on the combination of the selected structures as follows. Resonators with a perpendicular-tothe- motion structured electrode vs.resonators with a smooth electrode. Resonators with a perpendicular-to-the-motion structured electrode vs. resonators with a parallel-to-the- Fig. 13 Density and viscosity (l) of the antifreeze used (Glycoshell) for 20 °CH-D. L iess et al. 49 Fig. 14 Sensor set-up Fig. 15 Admittance magnitude and phase of the quartz loaded with water»antifreeze mixtures (»») water, (… … …) 30% antifreeze, (»») 70% antifreeze, (» ») 100% antifreeze. Top: structured parallel to the oscillation with h\100 nm, lm.Bottom: structured perpendicular to b1\b2\5 the oscillation with h\1.5 lm, lm. b1\b2\550 Acoustic waves at the solid/liquid interface Fig. 16 Oscillation frequencies of the two quartz resonators in Fig. 14 parallel and per- (|) (=) pendicular to the oscillation structured quartz motion structured electrode.Resonators with a rough-structured electrode vs. resonators with a parallel-to-the-motion structured electrode. 4.2 Admittance spectra Fig. 15 shows the admittance spectra of two resonators for both parallel and perpendicular structures, loaded with water»antifreeze mixtures. Perpendicular structures show higher frequency shifts compared to the parallel structured surfaces due to trapped Fig. 17 DiÜerence in frequency changes of the two quartz resonators in Fig. 14H-D. L iess et al. 51 Fig. 18 Top: resonance frequencies of two quartz resonators loaded with antifreeze»water mixtures measured by impedance analysis. Structured parallel to the oscillation, h\3 lm, (K) b1\9 lm, lm, rough.Structured parallel to the oscillation, h\1.25 lm, lm, b2\11 (>) b1\5 b2\15 lm. Bottom: frequency diÜerence of these two quartz resonators follows the non-linear dependence of the antifreeze concentration. The in—uence of the —uid viscosity must be compensated. Therefore, both quartz resonators need equal sensitivity to the viscosity (*L 2\L 2 Fluid A [L 2 Fluid B).liquids. Fig. 16 shows the oscillation frequencies of the quartz resonators and Fig. 17 shows the diÜerence in their frequency shifts. 4.3 Separation of the density and viscosity parameters The application of both perpendicular and smooth resonators and the subtraction of both resonance frequencies to separate the liquid density and viscosity has already been52 Acoustic waves at the solid/liquid interface Fig. 19 DiÜerence in the frequencies of the two quartz resonators in Fig. 18 vs. the density of the test liquids and over the entire antifreeze»water con- Table 2 *L 2 *L 3 centration range (0»100 vol.%) for two quartz resonators with the same viscous coupling quartz 1 quartz 2 *L 2\L 2 100ÜAF[L 2 water/lH 31 31.2 *L 3\L 3 100ÜAF[L 3 water/lH 27.7 9.8 Quartz electrode 1: structured parallel to the oscillation, h\3 lm, lm, lm, rough.Quartz electrode b1\9 b2\11 2: structured parallel to the oscillation, h\1.25 lm, b1\ lm, lm. 5 b2\15 published.7 However, in the case of a broad test —uid viscosity range and a small density range, the diÜerent in—uence of the viscosity on diÜerent structured resonators has to be compensated in order to distinguish between density and viscosity. Only if this in—uence of the viscosity is the same, as for the two quartz resonators in Table 2, can the density be found from the diÜerence between the resonance frequencies (Fig. 18 and 19). References 1 C. E. Reed, K. K. Kanazawa and J. H. Kaufman, J. Appl. Phys., 1990, 68, 1993. 2 K. K. Kanazawa and J. G. Gordon, Anal. Chim. Acta, 1985, 175, 99. 3 S. Bruckenstein and M. Shay, Electrochim. Acta, 1985, 30, 295. 4 R. Schuhmacher, J. Electroanal. Chem., 1987, 219, 311. 5 M. Thompson, A. L. Kipling, W. C. Duncan-Hewitt, L. V. Rajakovic and B. A. Cavic-Vlasak, Analyst, 1991, 116, 881. 6 D. M. Soares, Meas. Sci. T echnol., 1993, 4, 549. 7 S. J. Martin, G. C. Frye, K. O. Wessendorf, Sens. Actuators A, 1994, 44, 209. Paper 7/03205F; Received 9th May, 1997

ISSN:1359-6640    

DOI:10.1039/a703205f

出版商:RSC    

年代:1997

数据来源: RSC

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Faraday Discuss., 1997, 107, 53»60 Scanning electrode quartz crystal analysis Application to metal coatings Tetsu Tatsuma, Shuichiro Yamaguchi, Isao Wakabayashi, Kyoko Mori and Noboru Oyama* Department of Applied Chemistry, Faculty of T echnology, T okyo University of Agriculture and T echnology, Naka-cho, Koganei, T okyo 184, Japan Scanning electrode quartz crystal analysis (SEQCA) has been applied to the monitoring of quartz surfaces that are partially coated with gold.The lateral resolution for gold coatings is better than that for insulating coatings. The mass sensitivity in the centre region is better than that in the edge region. On the other hand, the lateral resolution is independent of the area and location of the gold coating. It may be possible to distinguish a conducting coating from an insulating one.Gold coatings can be located in NaCl solutions if the concentration is O10 mM. The possibility of further improvement in lateral resolution is discussed. A quartz crystal resonator is commonly used to monitor changes in solid surface properties, such as mass,1h3 rheology,4h7 roughness8 and solvophilicity.9 We envisage two-dimensional mapping of those properties.10 With this end in mind, we attempted to oscillate (or resonate) the quartz crystal only at a speci–c, small region.We used a small electrode which is separated from the quartz crystal as one of the two electrodes of a resonator, and scanned it over the quartz crystal. It is known that an electrodeseparated quartz crystal resonator,11h16 whose electrode(s) is separated from the quartz plate, can be oscillated.Fig. 1 illustrates our concept of SEQCA. There are three diÜerent modes; an over-scanning mode (Fig. 1A), under-scanning mode (Fig. 1B) and dualprobe mode (Fig. 1C). In the over-scanning mode, a normal size electrode is set on the opposite side of a quartz crystal plate to the ìmonitoredœ surface, and the small electrode is scanned over the ìmonitoredœ surface as a probe.In the under-scanning mode, an electrode is sputtered on the ìmonitoredœ side of the quartz crystal, and the small electrode is scanned under the opposite side to the ìmonitoredœ surface. In the dual-probe mode, two small scanning electrodes are synchronized, and a quartz plate is set between them. In our recent study,10 we constructed a prototype system for SEQCA.We found that qualitative mapping of the mass distribution is possible with a lateral resolution of between 1 (9 MHz quartz) and 2 (5 MHz) mm. Viscoelastic properties of the load on the quartz surface can also be measured qualitatively. However, we have studied only insulating loads. In what follows, we study the behaviour of SEQCA in the case where the load is electrically conducting. Additionally, the possibility of improving the lateral resolution is discussed.Experimental Materials 5 MHz AT-cut quartz crystal plates, the edges of which are bevelled (12.5 mm diameter, —at region : 10 mm diameter), were obtained from Hokuto Denko (Japan). A gold wire 5354 SEQCA of metal coatings Fig. 1 Schematic illustration (side view) for SEQCA in : A, the over-scanning mode; B, the underscanning mode; and C, the dual-probe mode mm) was used as a scanning electrode.Partial coating of a quartz crystal (diameterD1.0 plate with gold was conducted by sputtering with an appropriate mask. Instruments An impedance analyser (4194A, Hewlett-Packard, USA) and a semi-automatic prober (AWP-1050HR, Wentworth Laboratory) on a —oat desk, covered with a shield box, were used for SEQCA measurements.The probe is immobilized and the sample stage is mobile. The sample surface can be visually monitored in situ with a microscope and a CCD camera. Both the impedance analyser and the semi-automatic prober were controlled by a computer. Measurements The over-scanning mode (Fig. 1A) was used in the present SEQCA experiments.A quartz plate was set on a large copper electrode. The probe was set ca. 10 lm above the quartz plate. The probe was scanned across the quartz crystal, and an impedance spectrum was obtained every 100 lm. The data obtained were transferred to a computer for data processing.T . T atsuma et al. 55 Results and Discussion SEQCA measurement of a gold coating As an example of an electrically conducting material, gold was sputtered onto the 5 MHz quartz plate, and SEQCA measurements were performed. The diameter of the gold coating was ca. 2, 4 or 9 mm, and its thickness was ca. 60, 120 or 240 nm. A probe of 1 mm diameter was set ca. 10 lm above the crystal. Fig. 2A and B show the distributions of the resonance frequency (F) and the maximum value of conductance (Gmax), respectively, for a crystal carrying a gold coating of ca. 2 mm diameter. The lateral resolution was much better than that for the insulating coatings. This may be because Fig. 2 Two-dimensional distributions of : A, [F; and B, after mass loading (gold coating, Gmax , ca. 2 mm diameter) on a 5 MHz quartz plate. A scanning probe of 1.0 mm diameter was set ca. 0.01 mm above the quartz plate.56 SEQCA of metal coatings Fig. 3 Proposed electrically equivalent circuit for the SEQCA system with a metal coating on a quartz crystal the scanning electrode and the gold coating form a capacitor, and a resonator is formed between the gold coating and the normal-size electrode (Fig. 3). That is, the eÜective electrode area is larger than the area of the scanning electrode.Since the value Gmax increases10 and the resonance frequency decreases10,17 with increasing electrode area, when the thickness of the quartz crystal is not negligible compared to the diameter of the electrode, a quartz crystal resonates more stably when the probe is located above the gold coating. The value was higher in the gold-coating region than the bare region.This can Gmax also be explained in terms of the increased eÜective electrode area. As mentioned above, the value increases with increasing electrode area,10 and hence the value in Gmax Gmax the coating region is higher when the area of the coating is larger than that of the probe. Observed poor dependence of the value on the thickness of coatings is in line *Gmax with this explanation.When the diameter of the coating was almost the same as that of the probe (1 mm) the value in the coating region was close to that in the bare Gmax region. Dimension of gold coating Fig. 4 shows the dependence of the resonant frequency change on the –lm thickness. The dashed line represents the ideal case where Sauerbreyœs relationship holds. When the quartz crystal is fully coated with an insulating layer,10 the mass sensitivity ([47 Hz cm2 lg~1) is lower than that for the ideal case ([56.5 Hz cm2 lg~1).However, in the present case, the mass sensitivity is very close to the ideal one if the –lm area is large (diameter\9 mm). As mentioned above, the resonance frequency decreases as the eÜective electrode area increases ; this can cause excess frequency decreases if the coating is electrically conducting (i.e.the coating behaves as an electrode) and is much larger than the area of the probe. On the other hand, the lateral resolution showed no signi–cant dependence on the diameter of the gold coating. Location of gold coating Next, we examined the dependence of the resolution and the mass sensitivity on the location of the gold coating. The diameter and thickness of the gold coatings were ca. 2 mm and 120 nm, respectively. Fig. 5 shows the distribution of the *F values. As can be seen, the resolution does not depend appreciably on the location of the gold coating. On the other hand, the mass sensitivity depends signi–cantly on the location of the goldT . T atsuma et al. 57 Fig. 4 Dependence of the resonance frequency change after mass loading [gold coating, ca. 2, (L) 4 or 9 mm diameter] on a 5 MHz quartz plate (|) (») coating. The sensitivity towards the gold coating in the centre region was higher than that towards the coating in the edge region. This is in line with previous reports18h20 in which mass is loaded to diÜerent points of a normal quartz crystal resonator.Thus, we must take account of this sensitivity gradient when we map mass distribution. A strange result is that the frequency decrease in the edge region was larger than that in the centre region for each sample. For instance, for the coating located at x\0.2»2.3 mm (Fig. 5, the frequency decrease ([*F) at x\2.3 mm was larger than that at >) x\0.2 mm. This seems to contradict the fact that the mass sensitivity is best at the centre of the quartz crystal.This can be explained using Fig. 6. As can be seen, the distribution of the resonance frequency for a bare quartz is concave.10 This is also the case for the bare region of the crystal carrying a gold coating. However, in the goldcoating region, the coating behaves as an immobilized electrode, as mentioned above and hence the resonance frequency is almost constant in this region.Thus, subtraction of the resonance frequencies for a bare quartz from those for the gold-carrying crystal Fig. 5 Distributions of the change in resonance frequency after mass loading (gold coating, ca. 2 mm diameter, ca. 120 nm thick) on 5 MHz quartz plates. A scanning probe of 1.0 mm diameter was set ca. 0.01 mm above the quartz plate. DiÜerent symbols represent diÜerent samples.58 SEQCA of metal coatings Fig. 6 Schematic illustration of the distribution of the resonance frequency before and after mass loading (gold) results in the *F distribution plots shown in Fig. 5. The same thing holds for the *Gmax distribution. These eÜects should not occur with insulating coatings.On the basis of the unique SEQCA behaviour of a conducting material, it is probably possible to determine whether the load on the crystal is electrically conducting or insulating. Fig. 7 Distributions of : A, the resonance frequency ; and B, for bare quartz (5 MHz) in air Gmax , and for a crystal carrying a gold coating (ca. 2 mm diameter, ca. 120 nm thick) in 0, 1, 10 or 100 mM NaCl aqueous solutionsT .T atsuma et al. 59 Measurements in electrolytes Measurements were also conducted in aqueous NaCl solutions (0, 0.1, 1, 10 and 100 mM) (Fig. 7). The diameter and thickness of the gold coating were ca. 2 mm and 120 nm, respectively. In solution, the lateral resolutions were worse than in air. However, the gold coating could be located from the distribution of the resonance frequency in O1 mM NaCl solutions.From the distribution of the value, it could be located even in Gmax a 10 mM NaCl solution. When the electrolyte concentration is high, the solution behaves as an electrode owing to its high ionic conductivity,21,22 so that the over-scanning mode does not work. In such a case, the under-scanning mode may be eÜective because the probe is not in the solution in this mode, however, this is beyond the scope of the present work.Future developments Here, we found that the lateral resolution of SEQCA for a conducting material is better than that for insulating materials. As we have reported previously,10 although it is possible to locate insulating coatings by means of SEQCA, the resolution is too low for practical mass mapping.However, we found that the lateral resolution obtained for an insulating coating with a 9 MHz quartz crystal (ca. 1 mm) was better than that obtained with a 5 MHz quartz (ca. 2 mm). Thus, it may be possible to improve the lateral resolution by using thinner quartz crystals. This is currently being investigated. To improve the resolution, we have to make the crystal resonate only in the region just below the probe.In other words, the oscillation should not propagate from the expected resonating region to other regions. In view of this, we may have to seek out the best piezoelectric material on which a vibrating region can be well separated from a quiescent region. That may not necessarily be a quartz crystal, we may have to prepare an array of minute piezoelectric chips, which are separated from each other.Conclusions The lateral resolution of SEQCA for gold coatings on a quartz crystal is better than that for insulating coatings. The mass sensitivity depends on the area and location of the gold coating, although the lateral resolution is independent of these factors. It may be possible to distinguish a conducting coating from an insulating one on the basis of their diÜerent SEQCA behaviour.SEQCA measurements are possible in aqueous electrolytes, even in over-scanning mode. Gold coatings can be located in NaCl solutions if the concentration is O10 mM. This work was supported in part by a Grant-in-Aid for Scienti–c Research A-1 (No. 07305038) from the Ministry of Education, Science, Sports and Culture of Japan.References 1 G. Sauerbrey, Z. Phys., 1959, 155, 206. 2 D. A. Buttry and M. D. Ward, Chem. Rev., 1992, 92, 1355. 3 N. Oyama and T. Ohsaka, Prog. Polym. Sci., 1995, 20, 761. 4 R. Borjas and D. A. Buttry, J. Electroanal. Chem., 1990, 280, 73. 5 A. Glidle, A. R. Hillman and S. Bruckenstein, J. Electroanal. Chem., 1991, 318, 411. 6 H. Muramatsu and K. Kimura, Anal.Chem., 1992, 64, 2502. 7 T. Tatsuma, K. Takada, H. Matsui and N. Oyama, Macromolecules, 1994, 27, 6687. 8 R. Beck, U. Pittermann and K. G. Weil, J. Electrochem. Soc., 1992, 139, 453. 9 H. Inaba, M. Iwaku, T. Tatsuma and N. Oyama, J. Electroanal. Chem., 1995, 387, 71. 10 N. Oyama, T. Tatsuma, S. Yamaguchi and M. Tsukahara, Anal. Chem., 1997, 69, 1023. 11 Y. Watanabe, Elektr. Nachr. T ech., 1928, 5, 45.60 SEQCA of metal coatings 12 W. G. Cady, Physics, 1936, 7, 237. 13 T. Nomura and F. Tanaka, Bunseki Kagaku, 1990, 39, 773. 14 T. Nomura, T. Yanagihara and T. Mitsui, Anal. Chim. Acta, 1991, 248, 329. 15 Z. Mo, L. Nie and S. Yao, J. Electroanal. Chem., 1991, 316, 79. 16 K. Takada, T. Tatsuma, N. Oyama and T. Nomura, J. Electroanal. Chem., 1994, 370, 103. 17 H. Bahadur and R. Parshad, Phys. Acoustics, 1982, 16, 37. 18 C. Gabrielli, M. Keddam and R. Torresi, J. Electrochem. Soc., 1991, 138, 2657. 19 M. D. Ward and E. J. Delawski, Anal. Chem., 1991, 63, 886. 20 A. C. Hillier and M. D. Ward, Anal. Chem., 1992, 64, 2539. 21 T. Nomura, F. Tanaka, T. Yamada and H. Itoh, Anal. Chim. Acta, 1991, 243, 273. 22 T. Nomura, Y. Ohno and Y. Takaji, Anal. Chim. Acta, 1993, 272, 187. Paper 7/03195E; Received 27th June, 1997

ISSN:1359-6640    

DOI:10.1039/a703195e

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss. 1997 107 61»76 General Discussion Dr. Johannsmann opened the discussion of the Introductory Lecture To what extent is the microfabricated pump you have described a peristaltic pump? Prof. White responded In our case only the lower surface of the two that form the channel has transducer-generated —exural waves propagating in it. Our upper boundary is thick and relatively rigid in contrast to the usual peristaltic pump whose entire cylindrical boundary is deformed by a propagating disturbance. I believe that we could make a true peristaltic pump by driving both boundaries. Depending upon the distance between them one would get partial or full peristaltic pumping. In the latter case —uid should move very rapidly at the velocity of sound in the —uid-loaded membranes.Dr. Ricco asked What are the limits on pressures that might be created using the —exural plate wave (FPW) as the basis for a peristaltic pump as a consequence of the ìthinnessœ of the membrane and the resulting limitations in its mechanical properties ? Prof. White responded These are still open questions. We know from experiments that our membranes can withstand a static pressure diÜerence of at least 1 atm across them. Calculations also suggest that with the acoustic streaming eÜect we could generate a longitudinal pressure drop of nearly 1 atm. Prof. Hillman asked In the case of the —exural plate mode devices what is the role of surface roughness? Prof. White responded The surface of the silicon nitride membrane that contacts the —uid is quite smooth as it was formed by chemical vapor deposition on the polished surface of a silicon wafer.In pumping surface roughness seems to be of little importance. In the regime where the thickness-to-wavelength ratio is as small as a few percent the component of particle motion normal to the membrane is an order of magnitude larger than that parallel to the membrane so acoustic coupling into the –eld is primarily due to compressional rather than shear forces which one would expect to be highly dependent upon surface roughness. Local roughness may cause particles to stick to the surface and thus interfere with —uid —ow there. Dr. Liron commented It was very attractive to watch the ìpumpingœ eÜect of the bacteria in the FPW sensor presented by the video –lm.However from a biosensor point of view if you wish to detect a bacterium e.g. by immobilizing an antibody on the sensor surface the pumping eÜect (shearing force) seems to act in the opposite direction to a binding eÜect that is mandatory for obtaining biosensor response to the interacting bacterium»antibody pair yet you claim that sensitivity of the FPW sensor for detecting bacteria was increased by the pumping eÜect. How do you explain this contradiction ? Prof. White responded The bacteria shown in the video which were being agitated ultrasonically were non-adherent as were the bacteria whose growth we monitored over time by measuring the increase in density of the liquid as the bacteria grew and divided. In experiments with antibodies binding to proteins deposited on the membrane of our device we have found that gentle —exural plate wave agitation at 20 nm amplitude increases the binding that occurs in a given time; we suspect this is the result of gentle stirring that brings the antibodies more frequently into contact with binding sites.With 61 62 General Discussion agitation at twice the wave amplitude (four times the acoustic streaming force) we found that binding was almost entirely suppressed presumably because of the larger acoustic streaming shear forces near the surface. Prof. Hillman opened the discussion on Dr. McHaleœs paper You discuss the possible participation of two types of surface wave a leaky surface wave and a non-leaky Stonely wave. Could you comment further on the circumstances in which each might dominate? v is larger or smaller than the surface wave st which depends Dr.McHale replied I mention that the surface acoustic wave when it impinges on the three phase line of contact between the solid vacuum and liquid can be transformed into two separate waves a generalized Rayleigh wave (leaky surface wave) and a pseudo-Stoneley (or Scholte) wave. The –rst of these loses energy as it propagates whilst the second propagates without loss.1 Both travel along the solid»liquid interface. The speed of the leaky surface wave is likely to be similar to that of the surface wave on the unloaded surface. However the pseudo-Stoneley wave has a speed v on whether the speed of sound in the —uid l v speed saw .2 The speed of the pseudo-Stoneley wave is approximately the same as the speed of sound in the —uid when vl@vsaw but tends towards vsaw when vl[vsaw .This corresponds to the pseudo-Stoneley wave having its energy predominantly on the liquid side of the interface or on the solid side of the interface. In our experiments we have no direct way of knowing how much of the energy might be transformed into each of these components nor how these two components might be recombined into a surface wave on leaving the —uid loaded region of our devices. We have now measured the speed of sound in our poly(dimethyl)siloxane (PDMS) oil and it is around 1020^50 m s~1 which on comparison to the surface wave speed suggests that any pseudo-Stoneley wave occurring would travel at around 1020 m s~1.So two possible situations arise in which a pseudo-Stoneley wave might dominate the transmitted signal. The –rst is if the path along the —uid loaded region is sufficiently long i.e. a wide enough stripe such that the leaky surface wave becomes negligible (owing to its damping). The second is if the path is sufficiently long that the diÜering transit times of the two waves (arising from the diÜering speeds of propagation) allows the 400 ns wide pulses to become separated. However it is more likely that the second situation would lead to a dephasing between the waves rather than a clean separation of the pulses so that when they recombine on exiting the —uid loaded region an interference could occur. Indeed such interference is an alternative explanation to those given in our paper for some of the oscillations observed in the surface wave signals as the stripes of oil spread.1 M. de Billy and G. Quentin J. Appl. Phys. 1983 54 4314. 2 S. N. Guzhev Sov. Phys. Acoust. 1991 37 594. Dr. Johannsmann asked Do the wetting dynamics of the drop depend on whether the substrate is oscillating or not? Can you detect a precursor –lm? Dr. McHale responded We have no evidence that at our input power levels the wetting dynamics we observe are in—uenced by the presence of the surface wave. However we have so far only concentrated on the macroscopic droplet dynamics and have not systematically investigated the in—uence of the input power. It is clearly possible that the dynamics may be in—uenced by the presence of the surface wave and now we have our optical system in place we can start to investigate this question more thoroughly.The relatively few experiments we have with simultaneous optical observations show that the drops follow the expected 1/7th power laws for the various geometrical parameters. Clearly the possibility of the surface wave in—uencing the wetting 63 General Discussion dynamics would be a more critical question if the experiment were to observe a thin spreading –lm. The idea of detecting the precursor –lm using the acoustic technique was the motivation for starting our experiments although we are far from realising it in practice. The surface wave technique has an intrinsic ability to detect sub-monolayer coverages and should therefore have sufficient sensitivity to detect any precursor –lm present.We envisage that a phase shift technique would be used with a suitably clean and highenergy surface (e.g. silicon cleaned of oxide layers) and with a droplet outside the surface wave transmission path spreading into the path. However virtually no work has been performed on the interaction of surface waves with localised —uids and we have therefore concentrated on building our understanding of the surface wave interaction with a macroscopic spreading liquid. We do not have clean enough surfaces of the correct type and have not looked for the precursor –lm. I think the use of an acoustic device to detect the precursor –lm and its dynamics would truly be an exciting development. Dr. Ricco asked How much of the decay in peak-to-valley diÜerence (amplitude plot in Fig.5 of your paper) can be attributed to deterioration in the straight-crested (ìcoherentœ) character of the advancing oil edge and how much is a result of the thinning of the –lm as it spreads ? Dr. McHale replied The quality of the stripe as a parallel-sided shape with a uniform circular cross-section can be seen from Fig. 3 in the paper. Over the region de–ned by the transducer aperture the height variation of the stripe is only around 200 nm and this becomes less as the stripe spreads. The contact width variation of the stripe across the aperture is negligible compared with the measurement accuracy. One difficulty with providing surface wave signal»geometrical correlations is that although the absolute contact width measurement is very accurate diÜerences in contact widths over one surface wave oscillation are relatively inaccurate (^2 lm).Nonetheless these data do provide evidence of a systematic reduction in the change in contact width corresponding to an oscillation in the re—ected signal (as you indicate). I do not believe it is due to a deterioration in the character of the advancing oil edge or thinning of the –lm as it spreads. We really need sufficient data to con–rm that such a systematic trend exists rather than it being our experimental accuracy. At the present time our more recent experiments do seem to con–rm this trend and this may indicate that the correlation is only approximately with the change in contact width for example it may involve the contact angle.Dr. Kanazawa said I am presuming that your analysis was based on the behaviour of a Newtonian —uid. At these frequencies it is likely that the —uid properties contain an appreciable elastic component. Could you comment on the possible eÜects of such elasticity in your analysis ? If elasticity does play a role then would it be possible to get some information regarding its value by comparing the results from a simple re—ection measurement with that of a double passage? Dr. McHale said I think it is probably generous to say that we have an analysis based on a Newtonian —uid. We have a model for the spreading rate of the liquid that is based on the surface free energy changes and viscous losses due to the —ow. However we havenœt really given a model based on —uid properties that links the surface wave signal to the liquid geometry.It should be possible to make such links and so obtain physical properties of the —uid. For example if we treat the surface wave as a damped harmonic oscillator and consider the extra damping induced by the force on the surface due to the shear force in 64 General Discussion the liquid we can produce an estimate of the changes in dissipation and resonant frequency. This is a model similar to that of Rodahl and Kasemo.1 This gives an attenu- (d ation in dB of 27.3 0/jSAW)(ol/os)(d/ts) where d0 is the contact width the os are the density of the liquid and substrate d is the penetration depth (1.4 lm for our liquid) and t is the thickness of the substrate which is oscillating.Given that the surface wave energy is con–ned to within one wavelength of the surface we can give a lower limit of s 1.5 dB per 0.1 mm of contact width for the attenuation of a transit signal. This appears to be consistent with the overall decrease in the transit signals as the stripe spreads although the very large oscillations make it difficult to measure the attenuation precisely (see Fig. 4 in our paper). A better model should be able to reverse the argument and so allow physical properties to be deduced from the attenuation change as the liquid geometry evolves notwithstanding my comments on the oscillations. I think that before we can extract information on elasticity we need to progress much further in understanding the source and factors in—uencing the re—ections.The understanding and theoretical analysis of such re—ections is very much underdeveloped in the literature. I therefore –nd it difficult to give a clear answer on the information on elasticity that may be obtained by comparing results from a re—ection and a double passage of the signals. 1 M. Rodahl and B. Kasemo Sens. Actuators A 1996 54 448. Prof. Lewis commented I agree that mode conversion from the surface wave into a pressure (sound) wave propagating away from the surface into the oil stripe is very likely to occur. However the oil stripe is likely to be a very poor resonator by reason of its shape and boundary conditions. In these circumstances it is likely to be a scattering absorber rather than a resonator efficient enough to produce clear resonance features.Dr. McHale responded I commented that the large oscillations in the transit signals such as B D and F in Fig. 4 may be due to a resonant conversion of surface wave energy into sound waves propagating into the bulk liquid. This idea is similar to the eÜect observed for quartz crystal microbalances (QCM) for example by yourselves at this discussion meeting.1 Indeed the path of a particle in our substrate surface undergoes an elliptical motion and I would expect this to be more efficient at generating sound waves in the liquid than in the case of the QCM. The vertical direction might then act as a cavity for standing sound waves in the liquid. If a sound wave is generated it would propagate with an angle of at least sin~1 (v from the vertical.However for a sound wave of speed 1020 m s~1 and a surface l/vsaw) wave of speed 3700 m s~1 this gives an angle of 16° which is not too great. I can also imagine the shape of the stripe is not too much of a problem. The contact angles are fairly low (\20°) and the central region of the stripe therefore has quite a low curvature. For example in Fig. 3 in our paper the crest of the stripe constant to within 800 nm (i.e. two bright fringes) covers one quarter of the stripe width. Compared to the wavelength of sound in the liquid of around 6.2 lm the height variation in the centre of the stripe might still allow it to act as a resonant absorber. Of course the main evidence to suggest a resonant conversion of SAW energy to sound waves in the liquid is the measured variation between successive minimas in the transit signals and other explanations may be possible.1 S. M. Reddy J. P. Jones and T. J. Lewis Faraday Discuss. 1997 107 177. Prof. White asked Have the polymer precursors been observed experimentally ? Does the presence or absence of the SAW observably alter the motion of the polymer layer ? General Discussion 65 Dr. McHale replied Polymer precursors have been observed experimentally. For example Silberzan and Leç ger1 used ellipsocontrast for qualitative observation and ellipsometry for quantitative information on the precursor –lms. This paper also gives references to earlier experimental observations of the precursor –lm and its dynamics.Several techniques can be used but these tend to work from the plan view and involve an average measurement over for example a beam spot size. X-Ray re—ectivity was used by Benattar et al.2 to observe the spreading of PDMS on strongly oxidised silicon. The central droplet was found to be surrounded by a long almost —at tongue of nm scale thickness. Re—ection microscopy shows only thicknesses of the order of 1 lm for the –rst black fringe. Qualitatively a breath pattern can be used to show the existence of a –lm around the macroscopic droplet. The pattern is due to the condensation of water droplets on the oil and the precursor –lm turns out to be as thin as a few ” at the edge.3 For more precise measurements they used ellipsometry for a system of PDMS on silicon.Other reports exist in the literature and there is some discussion on whether precursor –lms are speci–c to the long chain polymeric liquids such as PDMS. The presence of a surface wave may alter the motion of a precursor –lm as I discussed in my earlier response to Dr. Johannsmann. However as acoustic waves have not been used to detect precursor layers it remains an open question. 1 P. Silberzan and L. Leç ger Macromolecules 1992 25 1267. 2 J. J. Benattar J. Daillaint L. Bosio and L. Leç ger Coll. Phys. C7 Suppl. 10 1989 50 39. 3 F. Heslot A. M. Cazabat N. Fraysse and P. Levinson Adv. Coll. Int. Sci. 1992 39 129. Prof. Lewis opened the discussion on Prof. Urbakhœs paper First your suggested arrangement for studying the properties of rough layers is ingenious and can yield important information.However it might be very difficult to interpret it when applied to the usual practical arrangement of a rough crystal/electrode interface for the following reason The system has three elements bulk crystal a rough layer and a liquid layer. In a conventional arrangement the rough layer is adjacent to the smooth (bulk) crystal and the liquid is outermost. If the elements are changed in order so that the rough layer is outermost a diÜerent overall impedance results. The expression for the impedance especially when attenuation is present is complicated and it will be difficult to interpret it to yield information about the conventional liquid outermost problem. Secondly would the shear wave properties of the material of the rough layer have to be taken into account since the shear wave may penetrate into it indeed must penetrate into it signi–cantly for the method to be eÜective ? Prof.Urbakh responded In reply to your –rst question the important advantage of the proposed con–guration where a liquid layer is con–ned between a rough immobile solid surface and a quartz resonator is the existence of a new external parameter the distance between the quartz crystal and the rough surface. Variation of this parameter provides an eÜective way to separate diÜerent contributions to the QCM response. We derived the expressions which allow analysis of the impedance in the proposed con–guration. In reply to your second question elastic properties of a rough layer should be taken into account only if the characteristic sizes of roughness are of the order or larger than the wavelength of shear-mode oscillations in the corresponding materials.For the frequencies used in QCM experiments the latter length is of the order of 10~2»10~1 cm. Usually sizes of roughness are much smaller than this wavelength and the elastic properties of the rough layer could be ignored. Prof. Liess commented I propose one should not exclude slippage between the solid phase and the liquid. At least any phase transition has the potential for unexpected behaviour. 66 General Discussion Prof. Weil said I wonder whether the same information cannot be obtained by using several overtones to obtain information over a wide range of penetration depths of the shear wave.This appears to be simpler than the proposed technique. Prof. Urbakh responded I agree that in principal similar information could be obtained by using a wide range of frequencies. However the number of overtones which are used in the QCM measurements is very limited and the sensitivity diÜers for diÜerent overtones. Another important advantage of the suggested con–guration is the possibility of studying interfacial properties of massive electrodes rather than thin –lms deposited on quartz as in the case of the traditional con–guration. Dr. Martin asked You mentioned in your introduction that QCM measurements and calculations have shown that slip does not exceed 10%. Can you describe the evidence that shows that slip is occurring (at the solid/liquid interface) at all ? Prof.Urbakh responded For complex —uids there is experimental evidence that slippage occurs at the —uid/solid interface. For instance direct optical measurements of slippage of a polymer melt at solid surfaces have been reported recently.1 Theoretical calculations of slippage have also been reported either through molecular dynamics simulations2 or more phenomenological studies.3 1 K. B. Migler H. Hervet and L. Leç ger in Dynamics in Small Con–ning Systems MRS Proceedings ed. J. M. Drake J. Klafter R. Kopelman and D. D. Awschalom 1993 vol. 290 p. 13; K. B. Migler H. Hervet and L. Leç ger Phys. Rev. L ett. 1993 70 287. 2 P. A. Thompson and M. O. Robbins Phys. Rev. A 1990 41 6830. 3 P. G. de Gennes C. R. Acad. Sci.Paris B 1979 288 219. Dr. Johannsmann commented It may be possible to glue mica sheets to the quartz surface and in this way obtain a very —at surface. Although the experiment certainly is difficult because it is hard to obtain large sheets of smooth mica and the in—uence of the glue is uncertain the experiment may serve to test the theories about roughness. Prof. Krim asked You have presented a theory for the response of a QCM in the presence of a rough immobile surface. Has such an experimental geometry ever been successfully constructed ? Many surfaces are characterized by self-affine fractal roughness where there are no characteristic length scales apart from the upper and lower lateral cut-oÜs to scaling. Does your theory apply to this category of roughness or could it be adapted to such surfaces ? Prof.Urbakh responded Here we reported the results of the theoretical analysis of a new geometry for QCM studies. So far the suggested con–guration has not been constructed experimentally. In the case of slight roughness our theory could be easily adapted to self-affine fractal roughness. Initial work in this direction has already been done by G. Palasantzas.1 In the case of strong roughness the situation is much more complicated and additional work is needed. 1 G. Palasantzas Phys. Rev. E 1994 50 1682. Prof. Crooks commented An earlier discussion invoked the possibility of using single crystal surfaces for quartz oscillators. Analogy was made to the surface forces apparatus (SFA). I indicated that I thought the analogy wasnœt valid because the contact area in SFA experiments is very small (too small to be practical for oscillators).Prof. Urbakh indicated that much higher area SFA experiments had been reported by others. 67 General Discussion Prof. Urbakh responded SFA experiments with a high contact area have been performed recently by Dr. J. M. Drake from Exxon Research and Engineering Company Annandale NJ. Dr. Lucklum commented One has to take the existence of shear waves and compressional waves in liquids into account with diÜerent characteristic values (wavelength decay length). If the suggested sensor design can recognise trapped liquid at the cap the distance from the quartz surface must be smaller than the wavelength of the compressional wave.On the other hand the compressional waves propagate through the liquid (simple liquid without signi–cant damping) and penetrate into the cap hence even the geometry of the cap must be carefully designed. One can get sufficiently smooth surfaces by spin-coating a polymer e.g. polystyrene (adapted from semiconductor technology). Prof. Urbakh responded Our calculations take into account the propagation of pressure waves in the con–ned liquid layer. I prefer to use the term pressure waves rather than compressional waves because here we discuss the case of incompressible liquids. Our calculations show that the decay length of pressure waves in a liquid is of the order of the lateral length of roughness (see also ref. 1 and 2). Depending on the geometry of the interface this decay length could be as large as or shorter than the decay length of the velocity –eld in liquids d\(2g/uo)1@2 where o and g are the liquid density and viscosity and u is the frequency of quartz oscillation.1 M. Urbakh and L. Daikhin Phys. Rev. B 1994 49 4866. 2 Z. Lin and M. D. Ward Anal. Chem. 1995 67 685. Mr. Etchenique opened the discussion of Prof. Liessœs paper In your paper Fig. 11 shows the change in resonance frequency *f when the height of the structure depicted in Fig. 4 is increased. For the movement parallel to the oscillation I expected there to be an eÜective increase in area only when the characteristic height of the structure is larger than the decay length of the velocities in the liquid (d\2g/uo)1@2. The decay length for water is about 300 nm and in Fig.11 it is possible to see that there is an increase in *f only after the height reaches 300 nm. However for the antifreeze data the *f length is about 1.5 nm so it should be impossible to increase the eÜective area by means of a structure shorter than that length. Could your model explain the behaviour of those plots ? Prof. Liess responded Fig. 11 shows a structure which has been etched parallel to the direction of the movement. Therefore the remark is correct that the area increases only eÜectively if the height h of the grooves exceeds the eÜective decay length d. This implies however that the surface inside the grooves is even. In this case the frequency shift shows in principle the behaviour shown in Fig.1. For water Fig. 11 shows a change in the slope approximately at the decay length (d) of 300 nm. In reality however this is d negative frequency shift –D f h d height of groove h Fig. 1 Shift in the series resonance frequency vs. structure depth h and decay length d 68 General Discussion not the case for grooves which are made by electrochemical etching. They have a rough surface on their sides and base. The roughness increases the deeper the grooves are etched. This second element of in—uence leads to an additional change in frequency due to an increased mass load. For antifreeze the decay length d is 1.5 lm. This exceeds the maximal depth h of the grooves. The observed slope of the frequency shift *f vs. the structure depth (of the grooves) is therefore more the result of an increased roughness than of the etched groove structure depth.Dr. Lucklum commented Experiments undertaken in our laboratory with a chessboard structure indicate that the liquid does not move in a similar line structure if the quartz surface vibrates perpendicular to the orientation of the lines. Prof. Urbakh commented The important conclusion of your work is that the application of resonators with grooves perpendicular and parallel to the motion of the QCM allow separation of the eÜects of liquid density and viscosity. A few years ago we studied theoretically the eÜect of grooves perpendicular and parallel to the motion the quartz crystal on the QCM response (see ref. 1). We found that this eÜect is determined by relations between the period and height of grooves and the decay length of liquid velocity d\(2g/uo)1@2 where o and g are the liquid density and viscosity and u is the frequency of quartz oscillations.Our calculations show that the suggested method of separation of the eÜects of liquid density and viscosity could work only in a limited range of the parameters. 1 M. Urbakh and L. Daikhin Phys. Rev. B 1994 49 4886; L angmuir 1994 10 2839. Prof. Liess said The comment is correct. As a general rule for an efficient sensor application the dimensions of the grooves should be as follows (1) the depth h of the grooves should be greater than the decay length d of the liquid and (2) the width b of the grooves should be about or less than twice the decay length 2d of the liquid.(See Fig. 2.) Dr. Martin asked You mentioned that the criterion for liquid trapping by crevices is that the crevices be small compared with the liquid decay length (and have steep side walls). You also mentioned that periodic grooves have a decay length that is determined by the groove periodicity (rather than liquid properties). Does the second observation relax the criterion for trapping by periodic grooves? That is does periodicity help trap a —uid that would not be trapped by features of the same size that are randomly distributed on the surface ? Prof. Urbakh responded Shear waves with various decay lengths are excited in a liquid due to oscillations of a quartz crystal with a rough surface and/or with periodic d h d > b d >2 Fig.2 Shift in the series resonance frequency vs. structure depth h and decay length d 69 General Discussion grooves. These decay lengths could be estimated as dn\[d~2](2nn/l)2]~1@2 where d\(2g/uo)1@2 l is the lateral size of roughness (period of periodic grooves) n\0 1 2 . . . o and g are the liquid density and viscosity and u is the frequency of quartz oscillations. This equation shows that decay lengths are determined by both liquid properties and by sizes of roughness and/or by the groove periodicity. As a result the criterion for liquid trapping has the same form for both periodic grooves and random roughness. Dr. Lucklum asked Have you performed experiments about the wetting of the structured surfaces with the liquid and the relation between geometrically de–ned mass and eÜectively trapped mass? Prof.Liess responded Measurement results of water»antifreeze mixtures with structured electrodes show that the equivalent circuit inductance is proportional to the density of the liquid and therefore to the antifreeze concentration. We attribute this to the mass (of the liquid caught in the grooves) which is moved with the crystal. Some organic solvents however deviate from this expected behaviour. There might be some evidence to hold incomplete wetting (eÜectively trapped mass) responsible for the observed reaction but unfortunately this does not exclude slippage as a possible explanation either. Prof. Garrell commented The lower part of Fig. 18 shows the diÜerence in the resonance frequencies of the two oscillators vs.antifreeze concentration. Is the curvature real ? That is does the precision of the data allow you to draw a curve rather than a straight line or does a straight line (perhaps not going through the –rst and last points) –t the data well ? If the curvature is real does it follow the trend expected based on viscosity vs. concentration (shown in Fig. 13) ? Can it be related to the trend in g vs. concentration o vs. concentration or (go)1@2 vs. concentration ? Prof. Liess responded The curvature in Fig. 18 (lower part frequency shift vs. concentration) is real. The straight line in this –gure was only drawn to underline the deviation from the linear dependency. The reason for this behaviour is that the two components penetrate each other.A similar curvature is observed in Fig. 13 (density shift vs. concentration). The elimination of the concentration results in a linear relationship between the frequency shift and the density of the liquid. This shows that the measurement is in accordance with well known results. Dr. Kanazawa commented I wonder if there is another physical dimension involved here. Speci–cally is it possibly the distance involved in laminar —ow particularly when the channels running parallel to the direction of oscillation could have an important eÜect on your analysis ? For example if the —uid properties were such that the boundary layer were much thinner than either the width or the depth this might have a much smaller eÜect than if that layer were comparable to or even thicker than those dimensions.Mr. Ostanin asked Is the mixture of antifreeze and water an emulsion or a solution ? If it is an emulsion (mixture of nm sized particles) then the eÜect of curvature (nonlinear) can be understood. In the case of a true solution I do not understand the result. Prof. Liess responded The mixture of water and antifreeze is a solution. The reason for the curvature (see Fig. 18) is that the two components penetrate each other as discussed in my response to Prof. Garrellœs question. A similar curvature is observed in Fig. 13 (density shift vs. concentration). 70 General Discussion Dr. Rapp commented In Fig. 15 of your paper you compared the admittance magnitude and phase of the diÜerent structured quartz with diÜerent water»antifreeze mixtures.It is suggested that an ìautomatedœ readout of the diÜerent behaviours (i.e. the separation of viscosity and density in—uences in Fig. 16»19) is obtained with a dual quartz oscillator con–guration and a corresponding diÜerence frequency signal generated as an output (Fig. 14). I would like to comment that within an oscillator the phase position is –xed and the measured frequency change will correlate to the frequency change at this –xed position. In Fig. 15 the frequency change at a certain phase position would also be predictable e.g. the upper plot a phase position of 40° will lead to a frequency shift of about 400 Hz (thin solid curve changes to long dashed curve) while at a phase position of 60° (along the suggested dotted lines) this will lead to a frequency shift of about 700 Hz for the same curves ! Hence it is obvious that at diÜerent set phase positions within your oscillators you would obtain diÜerent values of frequency change.In the light of this I would like to ask How have you been able to obtain the relevant data i.e. the separation of viscosity and density in—uences in Fig. 16»19 with the simple dual quartz oscillator con–guration and its uncertainty of inaccurately set phase positions ? Or in other words at which set phase position have you considered the data to be valid ? During the discussion you mentioned that you have calculated the frequency response at zero phase (corresponding to the theoretical ideal point of resonance?) but the strong suggestion in your paper is that you obtained the data with the oscillator con–guration in Fig.14. Also in Fig. 15 it is obvious that at zero phase (your answer) you will not get an oscillation after treating both devices with antifreeze over 30%! Prof. Liess responded This comment is correct. In order to get the quartz oscillating especially at higher antifreeze concentrations (70»100%) two identical oscillator circuits at a phase position of 60° (and not at 0°) had to be used. Unfortunately there was a misunderstanding in the discussion the phase diÜerence (not the phase itself) of the two oscillator circuits was kept at 0°. This was only valid for Fig. 15»17. These measurements were made with two balanced oscillators. All other measurements (Fig. 3 5»8 10»12 18 and 19) were recorded with the impedance analyser.In those cases the frequency shift was not determined for a certain phase position but by the shift in the maximum of the real part of the admittance. Mr. Drake asked Did you look at the edge eÜects of the electrodes and the response of the modi–ed electrode at diÜerent points on the electrodeœs surface ? Prof. Liess responded No we didnœt look at electrode edge eÜects. But some other groups have already studied these eÜects for smooth electrode surfaces.1~3 1 A. C. Hillier and M. D. Ward Anal. Chem. 1992 64 2539. 2 M. D. Ward and E. J. Delawski Anal. Chem. 1991 63 886. 3 D. M. Ullevig J. F. Evans and M. G. Albrecht Anal. Chem. 1982 54 2341. Prof. Calvo opened the discussion of Prof. Oyamaœs paper I would like to refer to the ìstrange eÜectœ you mention on p.57 where there is a diÜerence in radial sensitivity depending on whether the layer is conducting or insulating. In a practical case the conductivity of such a layer may change in the course of an experiment either by uptake of some component from a vapour in the gas phase or by electrochemical uptake of ions in the liquid phase. In those cases do you see a way in which you can make use of this ìstrange eÜectœ to probe the changes in the surface layer conductivity ? 71 General Discussion Secondly with reference to the capacitive origin of this eÜect (abnormal radial sensitivity for metal coatings) it would make a big diÜerence to excite the quartz crystal from below the quartz crystal or from above since the parallel plate capacitor will be quite diÜerent in either case.Prof. Oyama responded In reply to your –rst question a change in the surface layer conductivity results in changes not only of mass sensitivity but also of lateral resolution. We may be able to monitor changes in the conductivity from both of these changes. If you wish to see a mass change alone you can use the underscanning mode. In response to your second point we agree entirely. We are currently studying single electrode quartz crystal analysis (SEQCA) in the under-scanning mode and have observed big diÜerences. Prof. Lewis commented Using a scanning electrode rather than one –xed directly on the crystal surface introduces a subtle change in con–guration whereby the interfacial region instead of being only in the acoustic shear wave –eld is now also in the electrical –eld.In these circumstances the electrical properties of the interfacial region its capacitance and resistance become important since they in—uence the drive voltage across the piezoelectric quartz. The interface will be structured and may have a signi–cant double-layer and will certainly be diÜerent for insulating and conducting layers. These eÜects will be accentuated when a liquid is present but may also exist when adsorbed (water) layers occur. Prof. Oyama responded SEQCA is sensitive not only to mass changes but also to conductivity permittivity viscosity and density of substances present between the probe and quartz. This is a characteristic of electrode-separated quartz crystals (see ref.13»16 of our paper). Although we have not yet examined SEQCA of an interface with an adsorbed layer we basically think that an adsorbed layer may aÜect SEQCA results. However this eÜect may be negligible if the conductivity of the layer is not high. In a liquid phase if the adsorbed layer changes the structure of the double layer especially in a solution with low ionic strength it may aÜect SEQCA results. Dr. Johannsmann commented I would like to report on a –nding which correlates with the results by Prof. Oyama. In order to get rid of all eÜects of mounting and electrodes we have designed an apparatus where bare quartz blanks can be suspended on an air cushion and excited across an air gap (see Fig. 3). Naturally the amplitude of oscillation decreases which is a problem.At some point we inserted quartz blanks with metal coatings into the apparatus. The metal coatings would correspond to the electrodes in the usual con–guration. In this case they were not actually electrically connected. Interestingly the amplitude for these quartzes was much higher than for electrode-less quartzes. At this point we only have a vague understanding of why that is the case. The reason should be connected to the electrical boundary conditions. When one draws the Fig. 3 Apparatus showing bare quartz blanks suspended on an air cushion and excited across an air gap 72 General Discussion Butterworth»van Dyke equivalent circuit one implicitly assumes that the source excitr ing the quartz has a low output impedance.The source supplies a voltage 0(u) to the electrodes regardless of what the state of oscillation is. If the oscillation is large there will be a large current into the electrodes to compensate the large dielectric dipole moment of the quartz. One assumes that the current is always just large enough to keep the voltage r0(u) –xed. This picture obviously breaks down when there is an air gap between the quartz and the electrode. Any charge in the electrodes is separated from the quartz surface by the air gap. Prof. Gabrielli asked Did you try to change the diameter of the probe to improve the lateral resolution ? Prof. Oyama responded Yes we tried to use 0.5 and 0.2 mm probes. For a 0.5 mm probe the lateral resolution was almost the same as that for a 1 mm probe since the resolution is –ner than the probe diameter due to the electrical contact between the probe and the gold coating via a ìcapacitorœ as shown in Fig.3 of our paper. When a 0.2 mm probe is used the measurement suÜered from a low signal to noise ratio due to low maximum values of conductance G (the reason for this is described in our previous max paper1). The signal to noise ratio might be improved by using a thinner quartz plate,1 this has not yet been examined. 1 N. Oyama T. Tatsuma S. Yamaguchi and M. Tsukahara Anal. Chem. 1997 69 1023. Prof. Bartlett asked In Fig. 2 the results show that the lateral resolution of the scanning probe technique is high even though the scanning probe electrode is quite large (in this case 1.0 mm diameter) when compared with the scan distance.Why is the lateral resolution so high in this experiment? Prof. Oyama responded As we have described in the text and Fig. 3 of our paper the probe is electrically contacted with the gold coating via a ìcapacitorœ when the probe is set over the gold coating. This results in much better lateral resolution than that for insulating materials. Dr. Rapp commented Prof. Bartlett has pointed out that with a 1 mm diameter probe it is not possible to probe a 9 mm electrode on the quartz plate as accurately as you have depicted in your talk. My comment on the fact that you have observed such sharp edges is that electrostatic eÜects will in—uence a voltage loading (at the applied frequency) at the moment when you begin to pass the electrode.With this assumption the sharp edges can be explained by some sort of on»oÜ switching between the –rst and the last overlap of the probe towards the scanned electrode. Prof. Lewis commented The abrupt rather than gradual change in response as the probe is moved over the electrode edge is at –rst surprising but it has to be remembered that resonance has to be established. The onset of a resonance response may be quite abrupt and will require minimum conditions concerning the Q-value of the system. This may not occur until the probe is well over the electrode edge. Dr. Rodahl commented In response to the previous question regarding the step-like behaviour of Fig. 2 isnœt it simply that since the oscillator has two eÜective thicknesses (one where there is no electrode and one where there is) it has two distinct resonance frequencies (which are then the ones observed in Fig.2) ? In this case the probe excites one or the other (depending on where it is) and there are no intermediate resonant frequencies. 73 General Discussion Mr. Etchenique asked Have you tried to load the crystal far from the probe? By this I mean by putting a small drop of liquid 1 or 2 mm away to test that there is no response. If there were a response it could be that the crystal is resonating as a whole regardless of the exact position of the probe. Prof. Oyama responded Actually the quartz crystal resonates in any region but the resonance is stronger around the position of the probe.Thus we can locate a load on the quartz. This has been discussed in a previous paper.1 1 N. Oyama T. Tatsuma S. Yamaguchi and M. Tsukahara Anal. Chem. 1997 69 1023. Dr. Lucklum commented Results from 2D FEM analyses of a quartz disc with a mass-less electrode and a real electrode with mass shows signi–cant energy trapping due to a foreign mass and the quartz crystal surface. It shows also the electrical potential and relation between in-plane and out-of-plane displacement.1 1 S. Schratz and P. Hauptmann Chemnitzer Fachtagung Mikrosystemtechnik Chemnitz Germany 1997 p. 9. Dr. Johannsmann asked When you move the probe towards the centre of the quartz plate there is a very sudden jump in the frequency as you enter the area covered by the electrode. As other participants have pointed out this is somewhat surprising.Is it possible that you in some cases excite ìanharmonic sidebandsœ and jump to the fundamental when the probe approaches the metal layer ? In other words do you possibly observe mode jumping? Prof. Oyama responded In the present case we believe that we do not observe ìmodulated jumpingœ. The frequency jump can be explained by the electrode edge eÜect (mentioned above). However as you point out we can observe some sidebands since we measure impedance as a function of frequency. They are much weaker than the fundamental mode when the Gmax value is high enough though it is not necessarily negligible when the Gmax value is as low as 10~6 S. In the present case the Gmax value is of the order of 10~5 or 10~4 S.Dr. Kanazawa commented There are two areas on which I would like to comment. First I refer to the question regarding the abruptness of the lateral transitions. While this may be a simplistic view it seems to me that your conducting electrode must be an equipotential so that as the probe makes capacitive contact with that electrode the frequency changes can be abrupt. Secondly with regard to Prof Lewisœ question regarding the double layer. I am in agreement with the notion that the charging time for the double layer can be much longer than the radio frequency period. However the data showing the transition in resolution between the 0.001 M and the 0.01 M solution relative to the 0.1 M solution was striking and I wonder if this could be related to the increased concentration decreasing the charging time? Prof.Oyama responded We have addressed your –rst point in our previous responses. I refer to your second point this behavior can simply be explained in terms of increasing conductivity. If you dip a whole QCM (normal type) into a concentrated electrolyte solution you cannot perform any measurement due to short-circuiting between the two electrodes. In our case the same thing happens between the probe and the gold coating. Therefore the lateral resolution decreases with increasing concentration (i.e. increasing conductivity) and –nally the impedance no longer depends on the location of the probe. 74 General Discussion Prof. Chesters asked This is probably a trivial question but is the end of your 1 mm gold wire probe perfectly —at? This is not mentioned in your paper.Prof. Oyama responded The probe has been well polished so the surface is smooth but it is hard to say that it is perfectly —at. However in our previous study1 the resonance properties were found to depend strongly on the electrode area. Therefore we do not believe that only a point on the surface functions as an eÜective probe. 1 N. Oyama T. Tatsuma S. Yamaguchi and M. Tsukahara Anal. Chem. 1997 69 1023. Prof. Thompson asked Prof. Urbakh and Prof. Liess mentioned in their presentations that they expect that about 10»20% of the device signal will be related to slip phenomena. Is this prediction speci–ed for a particular surface free energy? If not what diÜerence if any would they expect for the extremes of high hydrophilicity to high hydrophobicity? g Prof.Urbakh responded Molecular dynamics simulations1 predict that slippage at simple liquid/solid interfaces is not large the diÜerence between velocities of the solid and the –rst layer of liquid molecules usually does not exceed a few percent. This eÜect depends on the commensurability between liquid and solid structures. Slippage could be stronger at complex liquid/solid interfaces.2 However even very small slippage could essentially modify the eÜect on liquid on the QCM because in this case the –rst layer of the liquid connects the solid surfaces with the thick liquid –lm. The in—uence of slippage on the resonance frequency could be estimated as *fsl\(*v/v)*fg where *v is the diÜerence between velocities of the solid surface v and of the –rst layer of liquid molecules and *f is the viscosity eÜect on QCM in the case of non-slip boundary conditions.The eÜect of slippage is determined by the lateral corrugation of the energy of interaction between solid and liquid molecules rather than by the energy itself. The corrugation prevents slippage. I expect that slippage will be stronger in the extreme of high hydrophobicity where liquid»solid interaction could be less sensitive to the lateral structure of the solid surface. 1 P. A. Thompson and M. O. Robbins Phys. Rev. A 1990 41 6830; L. Bocquet and J-L. Barrat Phys. Rev. E 1994 49 3079. 2 K. B. Migler H. Hervet and L. Leç ger in Dynamics in Small Con–ning Systems MRS Proceedings eds.J. M. Drake J. Klafter R. Kopelman and D. D. Awschalom 1993 vol. 290 p. 13; K. B. Migler H. Hervet and L. Leç ger Phys. Rev. L ett. 1995 70 287. Prof. Liess responded We consider that both eÜects slippage and incomplete mass transport might be possible. However we have not yet determined which dominates. We also do not know the relationship to surface free energy hydrophilicity or hydrophobicity. It is however an interesting question. For the liquids we used in our experiments slippage should have played a minor role as the velocities were quite small. Prof. Hayward asked The amount of —uid trapped in roughness cavities will depend on secondary —ow which does not move with the crystal. I calculate a vibration amplitude of 3 nm for a 9 MHz AT-cut crystal operating in water with 1 V drive.This is much smaller than the typical cavity size. Will changing the amplitude by varying the driving voltage applied to a crystal yield additional useful information? Prof. Liess replied For the quartzes in our experimental set-up we calculated an oscillation amplitude of about 2 nm. We also did some tests varying the voltage drive but we were not able to observe a correlation between the applied voltage and the 75 General Discussion resonance behaviour of the quartzes although one would have expected to see an in—uence. Dr. Johannsmann commented People working with the surface forces apparatus have frequently encountered situations where slip was accompanied by stick»slip. Slip only occurred at a critical stress level.This kind of slip is a non-linear phenomenon. It should be easily identi–ed if it occurs on quartz resonators because it should depend strongly on amplitude. Also if slip occurs at the bottom of a droplet sitting on a quartz plate the oscillation should speed up the wetting dynamics. Prof. Urbakh responded It should be stressed that there is no direct relation between the phenomenon of slippage at liquid/solid interfaces and a stick»slip motion observed in experiments with the surface forces apparatus (SFA). The stick»slip motion is a non-linear phenomenon which is a manifestation of a diÜerence between static and kinetic friction. In contrast theoretical and experimental studies show that the eÜect of slippage could be a linear phenomenon with respect to a surface stress.1 1 P.G. De Gennes C. R. Acad. Sci. Paris B 1979 288 219; B. N. J. Persson Phys. Rev. B 1993 48 18 140; J. Krim and A. Widom Phys. Rev. B 1988 38 12 184. Dr. McHale commented It is certainly possible that the presence of a surface wave will help a liquid edge overcome local energy minima and so speed up the wetting dynamics. I am unsure whether it would be observable from the overall dynamics of the macroscopic —uid. However the precursor –lm is thought to evolve by a diÜusion process and I can imagine it would be important in this regime. Mr. Ostanin said Slippage on surfaces of single crystals was observed for krypton. This is due to incommensurability of the adsorbate on the surface so it can also be true for water.It is interesting to plot slippage of water against amplitude of vibration. Prof. Krim commented The work on Kr/Au that Mr. Ostanin has just referred to is that which I published in 1991.1 We detected slippage at the interface of Kr with Au(111) and observed that the solid Kr/Au(111) interface was about –ve times more slippery than the liquid Kr/Au(111) interface. The slipperiness of the solid was indeed attributable to the highly incommensurate nature of the interface. For liquid Kr the slippage was so slight that the Sauerbrey mass loading assumptions were largely adequate. For solid Kr the –lm was partly decoupled from the motion of the oscillator. 1 J. Krim D. H. Solina and R. Chiarello Phys. Rev. L ett. 1991 66 181. Prof. Kasemo commented As a general comment on slip I would argue that slip is a rare event in the chemical systems we are usually discussing (see Prof.Urbakhœs comment on p. 74). The reason is that the (binding) energy variation along the surface» except e.g. in some noble gas monolayers»varies quite considerably. This is usually referred to as the surface corrugation. Even on a perfect surface the corrugation is usually large enough to prevent slip in most systems. However surfaces are rarely perfect and the electrode –lms and surfaces we are talking about here are polycrystalline with grain boundaries steps etc. which give rise to pinning which in turn prevents slip even more. I am not denying the possibility of slip in very special systems such as noble gas monolayers and especially in incommensurate overlayers on single crystals but in my opinion slip events are rare exceptions to the general case which is negligible slip.Dr. Martin commented We were one of the –rst to report the observation of slip at the solid/liquid interface with an acoustic device. Further experiments have led us to 76 General Discussion conclude that slip does not occur (with water) even when the surface has been treated to make it hydrophobic. We made measurements similar to those by Thompson and coworkers1 in which we compared the liquid-loading response of QCMs with surfaces treated to be hydrophilic and hydrophobic. We noted that with rough devices the hydrophobic treatment gave diminished response (in comparison with hydrophilic treatment) in agreement with Thompson and co-workers. However with smooth devices (having features much smaller than the liquid decay length) no diÜerence arose due to surface treatment. We concluded that these surface treatments were modifying the trapping of —uid at the surface i.e. the penetration of —uid into surface asperities rather than causing slip to occur. 1 B. A. Cã avicç F. L. Chu L. M. Furtado G. L. Hayward D. P. Mack M. E. McGovern H. Su and M. Thompson Faraday Discuss. 1997 107 159. Dr. Doblhofer commented D. C. Grahame1 was the –rst to point out that water forms an ì ice-like œ structure at the polarized metal/electrolyte interface. This view has been supported since.2 Considering the surface roughness present even on well prepared metal electrodes deposited on quartz I –nd it difficult to imagine a ìslippingœ process taking place at the interface. 1 D. C. Grahame J. Chem. Phys. 1995 23 1725. 2 M. F. Toney J. N. Howard J. Richer G. L. Borges J. G. Gordon O. R. Melroy D. G. Wiesler D. Yee and L. B. Sorensen Surf. Sci. 1995 335 326. Dr. Johannsmann commented Slip may be a rare phenomenon but it has been observed and proven to exist for some special situations. It should be interesting to reproduce these conditions on a quartz resonator.

ISSN:1359-6640    

DOI:10.1039/FD107061

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 107, 77»90 Mechanical behaviour of –lms on the quartz microbalance K. Keiji Kanazawa Department of Chemical Engineering, Stanford University, Stanford, CA 94305-5025, USA The in—uence of –lms on the resonance character of the quartz resonator has been modelled from a variety of points of view. Purely mechanical models, using the notion of the piezoelectrically stiÜened modulus, have been very successful in describing the in—uence of elastic –lms.Viscoelastic behaviour required another approach and two equivalent methods of dealing with it are possible. The ultimate objective of these is to relate the observed electrical behaviour of the resonance (obtained by impedance analysis, for example) to the physical properties of the –lm. The focus has then, understandably, been on the description of the electrical parameters of the resonator, as in—uenced by the materials properties of the –lm.Here, we focus instead on the mechanical behaviour of the –lms, using the physical model rather than the equivalent circuit approach. The amplitude of the shear displacements in the quartz and in the overlayer are examined for several cases ; the unloaded resonator, the resonator loaded with elastic overlayers, the resonator loaded with a Newtonian —uid and the resonator loaded with a viscoelastic medium.The details of the mathematical analysis are given. Early studies relating the frequency changes of the quartz resonator loaded with an overlying –lm used a mechanical description of the quartz motion. By understanding the physical nature of the amplitudes of the motion of the quartz resonator, Sauerbrey1 was able to infer the change of frequency that would be caused by a thin overlayer on that resonator.Miller and Bolef2 were also concerned with the physical behaviour of the shear waves re—ecting from interfaces, to extend the Sauerbrey relation to elastic –lms of arbitrary thickness.These results were put into a simpli–ed form, usable in deposition monitors, by Lu and Lewis.3 In the foregoing, the amplitudes of motion were based on a mechanical model, the piezoelectric and dielectric properties of the quartz being included by adding a small correction term to the bulk elastic modulus of the quartz, described as the ì piezoelectrically stiÜenedœ shear modulus.These studies, however, were limited to the behaviour of elastic overlayers. The resonant frequencies were determined by –nding the poles in the amplitude»frequency relation. More recently, there has been increasing interest in examining not only the frequency changes caused by the overlayer, but also the losses incurred. See, e.g. the review by Buttry and Ward.4 The interest in lossy media was prompted, in part, by the discovery5 that the resonator could also function as a deposition monitor in liquids, prompting a closer look at the detailed physics of the resonant behaviour.For example, Hager6 recognized that a clear understanding of the loading caused by a Newtonian liquid could be used to extract physical —uid properties. Eggers and Funck,7 in a continuation of their long interest in acoustic sensors, not only demonstrated the relation between their measured acoustic impedance and —uid properties, but also estimated the amplitude of shear vibration in the liquid using a relationship by Sauerbrey.8 The close relation between the behaviour of the loading –lm, as determined by its physical properties, and 7778 Mechanical behaviour of –lms the measured properties of the compound resonator, was clearly recognized.These connections were seized upon to study interfacial slip,9 lossy adsorbates,10 –lm rheology,11 dispersion in colloidal –lms12 conductive —uid13 conducting-polymer –lms14 and protein multilayers.15 These references represent only a very few applications. The use of the resonator with viscoelastic media has led to a surge in the use of the impedance/ admittance analysis, where the spectrum of electrical impedance/admittance of the compound resonator is recorded as a fucntion of the exciting frequency. This new ability to extract additional information has changed the focus of study from the physical behaviour of the compound resonator to its electrical behaviour.The electrical equivalent circuit of the resonating crystal, as described by the Butterworth»van Dyke circuit16 has served very eÜectively as a method of electrically characterizing the crystal. It will be recalled that the equivalent circuit consists of four components, three of which are coupled to the physical properties of the quartz and overlayer. This equivalent circuit consists of two branches, with one branch containing a simple capacitor and the other branch consisting of a series connection of an inductance, a capacitance and a resistance.These components in the series branch of this circuit are in—uenced by the physical properties of the loading –lm. The series resistance, in particular, is conveniently measured and is easily interpretable as re—ecting the mechanical losses in the overlayer.Based on interpreting this equivalent circuit, a number of researchers have looked at gases,17 liquids,18h24 interfaces,25 viscoelastic media26h29 and sensors.30 In addition to this equivalent circuit approach, is another electrical analogue approach which is more complete, in the sense that it models the layered compound resonator, including speci–cally the piezoelectric, dielectric and mechanical properties of the quartz.The electrical behaviour of a compound resonator formed by the quartz and a loading overlayer was established,31 where the physical properties of the –lm were represented as a transmission line. Benes32 provided the –rst treatment of this kind, and includes the viscoelastic case, although this was not speci–cally treated.Others33h36 have taken a similar approach to relating observed impedance/admittance spectra to overlayer and interface properties. These electrical studies are yielding valuable methods for the study of viscoelastic –lms loading the quartz resonator. However, with the focus having shifted to electrical analogues, it is useful to recall the comment of Hayward and Jackson,37 that often these analogues can mask the physical insight into the detailed nature of the loading mechanisms.It is in this spirit that the following results are presented. The focus will be on the correlations between the motion of the quartz and overlayer with the observed electrical behaviour. The hope is that such correlations may add to an understanding of how the mechanical behaviour of the quartz and its overlayer is re—ected in the observed electrical properties.In so doing, it may be of assistance, not only in interpreting the observed behaviour, but also in generating new insights which might lead to fresh ways to use the AT-cut quartz resonator. Overview of theoretical treatment The amplitude variations at various points in the compound resonator will be described both in terms of peak magnitudes and displacements as a function of time at various places in the resonator.Hence, it is necessary to include a minimal amount of discussion on the method by which these calculations were made. The treatment is basically identical to that discussed in an earlier publication.33 It was found possible to simplify the mathematics using a modi–ed geometry, and this is indicated in Fig. 1. Taking the origin at the interface between the quartz and the overlayer considerably simpli–es the arithmetic. The overlayer is characterized by its density o, its shear modulus k and its viscosity g. The quartz parameters include its density its shear oQ ,K. K. Kanazawa 79 Fig. 1 Geometry used for the analysis of the compound resonator. The interface between the quartz and the overlayer de–nes the zero for the y axis.The quartz thickness is and the over- lQ layer thickness is e. modulus its appropriate piezoelectric constant its relative permittivity and c66 , e26 , e22 a –ctitious viscosity This viscosity is included for the convenience of being able to set gQ . the Q of the unloaded resonator to a convenient value. For all the calculations given, we have chosen the properties of the quartz such that its equivalent circuit re—ects a series resonance at exactly 5 MHz with a Q of 100 000.The free face at is referred to y\[lQ as the BOTTOM, the quartz/overlayer interface at y\0 is called the INTERFACE and the free surface at the top of the overlayer will be called the TOP.For those interested in reproducing these calculations, the details are given in the Appendix. For the purposes of the main body of this paper, it is sufficient to note the following : The solutions were obtained for shear waves that are varying harmonically taking the form exp iut. The shear wave solutions in both the quartz and the overlayer are framed in terms of the sum of a wave travelling in the ]y direction and another in the [y direction.For the quartz, the amplitude of the wave travelling in the ]y direction is A and in the [y direction is B. Similarly, the wave amplitudes in the overlayer are C and D. These wave amplitudes are complex quantities. The wave vectors for the shear waves in the quartz and overlayer are and k, respectively.The voltage applied kQ across the quartz has peak magnitude there being applied at the INTER- /0 , [/0/2 FACE and at the BOTTOM. The current density resulting from the application of /0/2 this voltage is j. The unknown constants A, B, C, D, E and F, where E and F are descriptive of the electical behaviour, are determined by solving consistently to satisfy six boundary conditions : the vanishing stress at the BOTTOM and at the TOP, the continuity of stress and displacement across the INTERFACE and the application of the applied potentials at the BOTTOM and INTERFACE.The solution is purely physical, without use of any electrical analogues. From a knowledge of j, however, the electrical behaviour of the compound resonator under impedance/admittance analysis can be derived.The shear wave in the quartz then takes the form: u(y, t)\A exp ikQ y]B[exp([ikQ y)]exp iut (1) and the shear wave in the overlayer takes an analogous form. Using the appropriate value of y, this solution can be particularized to any plane in the compound resonator. When we are interested in the magnitude of the shear wave displacements, then it is sufficient to take the magnitude of eqn.(1). When we are interested in the displacement of the wave at any time t, then it is necessary to take the real part of eqn. (1). Results Unloaded quartz The simplest case is that of the unloaded quartz resonator and the description of the mechanical shear wave behaviour for that case can serve as a reference point for further80 Mechanical behaviour of –lms discussions.Some con–dence in the calculations can be obtained from a study of the spectral dependence of the admittance of the resonator, since the calculation of the current density –rst requires the calculation of the quartz shear wave amplitudes. The admittance of the resonator was calculated for the case when the peak rf voltage applied was 1.0 V.This voltage is presumed to be applied from a voltage source of zero source impedance. The resulting magnitude of the admittance and its phase are shown in Fig. 2. The magnitude of the admittance behaviour, illustrated on the left-hand side, shows a rise to a sharp peak in the admittance at the series resonant frequency at exactly 5 MHz and a corresponding sharp dip in the admittance at the anti-resonant (parallel resonant) frequency.The phase behaviour also correctly mirrors the expected behaviour for the resonator, going through a negative decrease of 180° as one proceeds from low frequency through the series resonance and returning to 0° in going through the antiresonance. These results give some con–dence in the validity of the calculations. Further evidence for the reliability of the calculations is obtained by plotting the admittance diagram in polar fashion, a circle representing the behaviour of the series resonant circuit.This is shown in Fig. 3. It will be recalled that the diameter of the circle corresponds to the maximum conductance and that the centre of the circle lies slightly above the abcissa, proportional to the relative permittivity of the quartz.The frequency at which the conductance is a maximum is the intrinsic resonant frequency of the resonator, corresponding to the resonance in the series arm of the equivalent circuit. The ì series resonanceœ occurs at the point of zero phase, at a frequency slightly higher than the intrinsic resonance. The observance of this admittance circle serves to substantiate further the correctness of the calculations, giving con–dence in examining the magnitude of the shear wave displacements at the TOP surface, as shown in Fig. 4. The peak amplitude of the shear wave at the TOP surface is seen to be 1329 for ” this crystal, which is resonant at 5 MHz, has an unloaded Q of 100 000 and is excited by a 1 V peak rf voltage. The phase dependence is also interesting, re—ecting the known behaviour of mechanically resonant systems, with the displacement being in phase with the excitation at frequencies below resonance and out of phase at frequencies above resonance.At resonance, the shear wave displacement is in quadrature with the excitation. The full width at the 3 db points (45° phase shifts) is seen to be ca. 50 Hz, corresponding to a Q of 100 000.Denoting the magnitude of the shear wave amplitude by U, the peak velocity of the surface displacement is given by uU, which is ca. 4.2 m s~1. The acceleration of a point on the TOP surface is given by the relation u2U, an astounding Fig. 2 Magnitude of the admittance spectrum is shown on the left. Both the series resonance and the anti-resonance are seen.The phase, shown on the right, also exhibits the known behaviour.K. K. Kanazawa 81 Fig. 3 Real part of the admittance along the abcissa and the imaginary part along the ordinate, showing the well known circular plot characterizing the resonance 1.3]108 m s~2. This corresponds to an acceleration of 13]106 G! It is no wonder that the quartz crystal resonator is so sensitive to mass deposited on its surface.Judging by the impedance behaviour seen in Fig. 2, wave amplitude evidence for the occurrence of the anti-resonance at the higher frequency near 5.016 MHz was anticipated. It is puzzling that no evidence for this behaviour was seen in examining the shear amplitude data in this frequency range. Elastic overlayers Until rather recently, the major use of the quartz microbalance was for the monitoring of the deposition of elastic overlayers, such as metal –lms.It is well known that the decrease in resonant frequency for thin elastic –lms is linear with increasing mass density of deposit, according to the Sauerbrey relation.1 When the Miller»Bolef2 formulation Fig. 4 Maximum displacement of the TOP surface indicated at the peak of this spectral curve for the displacement is ca. 1300 The corresponding phase is shown by the dotted line. ”.82 Mechanical behaviour of –lms Fig. 5 Non-linear behaviour of the frequency is visible here when the frequency»mass relations are plotted for tungsten (W), magnesium (Mg) and graphite (C) was put into a mathematically tractable form by Lu and Lewis,3 it was found that the behaviour was not only non-linear, but was strongly dependent on the physical properties of the overlayer.For those studies, the physical properties were summarized in the acoustic impedance of the –lm. Films having a small acoustic impedance showed a frequency»mass density behaviour that was concave down, and those with a large acoustic impedance showed a behaviour that was concave up. This is illustrated in Fig. 5, where the behaviour for graphite, magnesium and tungsten is shown. Tungsten, with its large acoustic impedance, shows a decidedly upward curvature. Graphite, on the other hand, with a low acoustic impedance shows a fairly strong downward curvature. Magnesium, with its intermediate acoustic impedance shows an almost linear behaviour over this whole frequency range.Note that the frequency change is large, [1.5 MHz from the unloaded frequency. Another way to describe these observations is that the frequency sensitivity in terms of frequency change per mass density decreases with increasing deposition for stiÜ –lms and increases with increasing deposition for soft –lms. In order to examine the dependence of this behaviour on the mechanical –lm properties, we have calculated the amplitude displacements for these three –lms.We have done this at a mass density of 0.15 kg m~2 for all three –lms. This corresponds to thicknesses of 7.772 lm for tungsten, 66.67 lm for graphite and 86.21 lm for magnesium. The displacements at the BOTTOM, INTERFACE and TOP surfaces are illustrated in Fig. 6. We have found it useful to take the BOTTOM displacement (solid lines) as a reference.In the case of the stiÜ tungsten –lm, shown in the upper-left graph, there appear to be only two curves, however, this is an artifact, resulting from the fact that the INTERFACE and TOP displacements are virtually identical. These planes are seen to be displaced by a smaller magnitude than the bottom surface.It is very suggestive that the smaller displacement has given rise to the decreased mass sensitivity. If we now look at the soft graphite –lm, the INTERFACE moves a distance somewhat less than the BOTTOM surface. However, the TOP surface moves a distance considerably larger by almost a factor of three ! Again, this is consistent with the notion that the increased mass sensitivity is a result of the magni–ed displacement of the TOP surface.Finally, in the case of the magnesium –lm, the INTERFACE again moves a distance smaller than the BOTTOM, but the top surface moves an almost equal amount greater than the BOTTOM. These two eÜects tend to cancel, with the result that the mass sensitivity remains almost unchanged. These results clearly indicate the close relation between the observed behaviour of the resonator and the mechanical behaviour of the –lms.K.K. Kanazawa 83 Fig. 6 Displacements of the BOTTOM (»»), INTERFACE (» … » … ») and TOP (… … … … … …) interfaces for 0.15 kg m~2 of deposit are quite distinct for W (upper left), C (upper right) and Mg (bottom) Newtonian overlayers Nomuraœs use of the quartz crystal microbalance for analytical studies in liquids triggered a large body of work in areas like electrochemistry.It would be of interest then, to study the behaviour of the shear waves in this compound resonator. We have chosen a water-like liquid for study, a liquid having a density of 1000 kg m~3 and a frequencyindependent viscosity of 0.001 N s m~2. While it is possible to study the general problem of arbitrary thickness, we have chosen to look at a thick overlayer, corresponding to immersion.The thickness of the ìwaterœ layer has been taken to be 0.1 mm. A shear wave in a Newtonian liquid is strongly damped in the direction of its propagation. In fact, for water, the decay length is of the order of 2400 The damping of the ”. shear wave has a number of important consequences.First, it insures that the quartz crystal can operate in the liquid, the losses in the liquid being limited by the –nite depth of penetration. Secondly, a small portion of the —uid is coupled to the crystal motion and a frequency decrease is observed. Thirdly, the viscous nature of the motion in the region near the crystal gives rise to losses which are sensed by the resonator.In fact, the Q of the resonant circuit is profoundly decreased. The surface displacements are illustrated in Fig. 7. The BOTTOM surface of the crystal is seen to have a 45 displacement. The dis- ” placement of the INTERFACE cannot be seen because its motion is virtually identical to that of the BOTTOM. The TOP interface, being many decay lengths away from the crystal, has a zero displacement.The frequency has decreased from its 5 MHz unloaded frequency by 722 Hz. Using a crystal area of 0.342 cm2, corresponding to one of our electrodes having a diameter of 1/4 in. with a small ìhandleœ, we can calculate a resistance of 364 ). This compares with a value of 12.2 ) for the unloaded crystal. The damping of the shear wave can be seen in Fig. 8, where the magnitudes of the shear waves in the quartz and in the water are shown.The cosine-like wave anticipated for the quartz is visible. The very strong attenuation of the shear wave in the water can be seen. The amplitude reverses phase for a short interval in the region around 0.4 lm; this is worthy of closer examination. The phase of the shear waves in quadrature with the excitation is shown in Fig. 9.As in the84 Mechanical behaviour of –lms Fig. 7 Displacements of the BOTTOM, INTERFACE and TOP surfaces for a 5 MHz crystal having one face covered with 10 lm of water reveals a peak displacement of ca. 45 ” Fig. 8 Variation of the displacement magnitude as a function of distance in the crystal and overlayer. The distances for y[0 (the overlayer) have been multiplied by 100 for clarity.Fig. 9 Changes in the phase of the shear waves with distance in the quartz and overlayer. Taking the zero phase to represent the quartz region, the phase in the overlayer increases linearly with distance.K. K. Kanazawa 85 previous –gure, the distance in the water regime is multiplied by 100. The phase in the quartz region is very close to zero.The spike occurring at ca. [170 lm is believed to be a computational artifact resulting from the fact that it occurs when the shear wave displacement is vanishingly small. Interestingly, the phase in the water regime increases linearly with distance. It is reminiscent of the uniformly increasing phase along distributed transmission lines, such as for the Warburg type equivalent circuits38 which have been used to represent diÜusion phenomena.This, perhaps, should not be too surprising, since the equations describing the shear wave in a Newtonian liquid have the same form as the diÜusion equation, as indicated below. For the shear wave, Lv/Lt\(g/o)(L2v/Ly2) while for linear diÜusion, LC/Lt\D(L2C/Ly2). The velocity v of the shear wave has the same space»time description as the concentration C for diÜusion.Viscoelastic media This case is the most complex. We shall consider the case of a polymer-like material. Its physical properties are chosen to illustrate certain features of the solution which appear interesting. We shall –x the thickness of the –lm at 6 lm. The shear modulus is chosen such that, if the –lm were lossless, the wavelength would be ca. 27 lm. The –lm is, therefore, still less than a quarter wavelength in thickness, but it is approaching the critical quarter wavelength thickness. The shear modulus is taken to be 2]107 N m~2. We shall vary the viscosity of the –lm so that it may be possible to judge the values of g for which the –lm is elastic-like and for which it is liquid-like. It is noted that, for frequencies near 5 MHz, the ratio of ug to k is unity when g\0.7 N s m~2.The change in frequency with increasing viscosity is informative and is shown in Fig. 10. The resonant frequency for this –lm is seen to increase rather abruptly with increasing viscosity. The frequencies both below and above this step are relatively independent of viscosity. It is interesting that this transition occurs in the neighborhood where ug/kB1.At low viscosities, where this ratio is much less than unity, the –lm behaves more as an elastic –lm. At high viscosities, the behaviour of the –lm is closer to that of a liquid. The behaviour of the shear displacement of the TOP surface will provide some insight into this. The displacement of the TOP surface of the –lm is plotted in logarithmic fashion vs.the log of the –lm viscosity in Fig. 11. Starting from the least viscous (nearly elastic) –lm, we see that, as for the soft graphite, the top displacement is extremely large. As the viscosity increases and the –lm takes on more liquid-like properties, the top surface displacement decreases as would be Fig. 10 Resonant frequency of the resonator coated with a 6 lm –lm having a density of 1200 kg m~3 and a shear modulus of 2]107 N m~2 shows a sharp s-shaped rise as the viscosity is increased86 Mechanical behaviour of –lms Fig. 11 Displacement of the TOP of the –lm is non-monotonic and shows a minimum as the –lm viscosity is increased expected. Recall from Fig. 7 that the TOP surface of water had a zero displacement. In this case, however, the displacement goes to a minimum and then proceeds to increase again.This results from the contribution of the viscosity to the real part of the propagation constant for the shear wave. The expression for the square of the real part of this propagation constant in the –lm is expressed below. kR2\1 2 u2ok k2]u2g2 [1])(1]u2g2)] (2) For small values of viscosity in the elastic regime, the real part of the propagation constant is dominated by the shear modulus of the –lm .The frequency behaviour in Fig. 10 can now be understood. At low viscosities, the –lm is like that of a very soft elastic material. Like graphite, the frequency»mass density curve is concave down and the frequency has decreased to a value just above 4.82 MHz. In fact, using the Lu and Lewis formalism, a –lm of 6 lm with a density of 1200 kg m~3 and a shear modulus of 2]107 N m~2 would have a resonance at 4.8273, exactly that calculated ! As the viscosity increases above 0.7 N m~2, eqn.(2) shows that the viscosity begins to dominate the real part of the propagation constant. The propagation constant drops to a small value. In this limit, the thickness of 6 lm is negligible compared with the shear wavelength and the frequency of the resonator rises to 4.959 MHz, exactly that predicted by the Sauerbrey relation for a –lm of this mass density.It is useful to include the variation of the resistance of the compound resonator as the viscosity increases from negligible levels to very high levels. This is shown in Fig. 12 below. The resistance is seen to start at a value close to the 12.2 ) of the unloaded resonator.It quickly rises, with increasing viscosity, to extremely large values, of the order of 30 000 ). However, as the –lm passes from elastic-like to liquid-like, the resistance decreases again and, for very viscous –lms, the resistance reverts again to values close to that for the unloaded resonator.The behaviour for the viscoelastic –lm is relatively complex, and depends on the interplay between the –lmœs mechanical properties, the shear wavelength and the –lm thickness. For the reader interested in looking at more details of the behaviour of the compound resonator, a table of the variation of the resonant frequency, the resonator resistance, the BOTTOM displacement, the INTERFACE displacement and the TOP displacement are given as a function of the –lm viscosity in Table 1.Perhaps an interesting point to note is the ratio of the TOP displacement to the BOTTOM displacement as the –lm goes from elastic-like to liquid-like. In the elasticK. K. Kanazawa 87 Fig. 12 Series resistance increases as viscosity increases at low –lm viscosities, but reaches a maximum and then decreases with further increases in the –lm viscosity limit, the TOP displacement is much larger than the BOTTOM, accounting for the high mass sensitivity there.In the liquid limit, the displacements are all nearly equal, as would be the case for –lms in the Sauerbrey limit. Conclusions The relationships between the resonator shear displacements and the observed electrical behaviour of loaded AT-cut quartz resonators have been illustrated.The large magnitude of the displacements for the unloaded resonator subject any rigidly coupled material to large G forces. The dependence of the frequency»mass relation of elastic media on their acoustic impedance has been shown to be closely related to the relative Table 1 Variations in frequency, resistance and displacements of the BOTTOM, INTERFACE and TOP surfaces as a function of the overlayer viscosity viscosity frequency resistance BOT/ INT/ TOP/ /Pa s /MHz /) ” ” ” 0 4.827 306 11.86 1422 1413 8807 7]10~5 4.827 307 47.14 357.7 355.4 2215 2]10~4 4.827 306 112.7 149.6 148.7 926.7 7]10~4 4.827 308 364.7 46.2 45.9 286.3 2]10~3 4.827 320 1020 16.53 16.43 702.4 7]10~3 4.827 485 3535 4.77 4.74 29.5 2]10~2 4.828 650 9958 1.697 1.686 10.41 7]10~2 4.844 000 29830 0.572 0.570 3.200 2]10~1 4.914 400 34110 0.488 0.487 1.745 7]10~1 4.955 050 12450 1.316 1.315 1.935 2]100 4.958 668 4466 3.664 3.663 3.917 7]100 4.959 140 1289 12.70 12.69 12.76 2]101 4.959 178 459 35.64 35.63 35.65 7]101 4.959 185 139.8 117.0 117.0 117.0 2]102 4.959 184 56.82 288.0 287.9 287.9 7]102 4.959 183 24.88 658 657 657 2]103 4.959 184 16.58 986.9 986.4 986.4 7]103 4.959 184 13.39 1222 1222 122288 Mechanical behaviour of –lms displacements of the quartz and the overlayer.The strong damping of the shear wave in Newtonian liquids permits the use of the resonator in liquid media by limiting the losses. The phase behaviour of the shear wave in the liquid is consistent with the diÜusionequation- like form of the de–ning shear equation. The behaviour of viscoelastic media is complex, with the frequency sensitivity going from Lu and Lewis type behaviour to Sauerbrey behaviour as the –lm losses are increased.The relation between the –lm viscosity and resonator resistance is not necessarily monotonic. A knowledge of the physical behaviour of the overlying –lm is very useful in interpreting the results of electrical measurments. The continued encouragement of this work by the Department of Chemical Engineering at Stanford and of CPIMA (Center for Polymer Interfaces and Macromolecular Assemblies) is gratefully acknowledged.Appendix The geometry used in this analysis was illustrated in Fig. 1. The peak ac potential across the crystal is with applied at the upper quartz face and at the bottom.U0 , [U0/2 U0/2 The wave equation for the quartz shear waves is : c� 66 d2u dy2\[uoQ u (A1) where c� 664c66] e26 e22 ]iugQ (A2) Similarly, in the overlayer k 6 d2u dy2\[u2ou (A3) where k 6 4k]iug (A4) As before, we integrate the equation coupling the shear displacement to electrical –eld twice to obtain the relation : /\ e26 e22 u]Ey]F (A5) where E and F are undetermined constants.The shear waves in the quartz satisfying eqn. (A1) are given by u\A exp ikQ y]B exp([ikQ y) (A6) where kQ4u)(oQ/c� 66) . And in the overlayer, u\C exp iky]D exp([iky) (A7) where k4u)(o/k 6 ) . The six undetermined constants, A, B, C, D, E and F are evaluated by satisfying the six boundary conditions :K.K. Kanazawa 89 At the stress is zero. y\[lQ , At the potential is y\[lQ , ]U0/2. Across the y\0 interface, the stress is continuous. Across the y\0 interface, the displacements are continuous. At y\0, the potential is [U0/2. At y\e, the stress is zero. These lead to the following six equations : ikQ c� 66A exp([ikQ lQ)[ikQ c� 66 B exp(ikQ lQ)]e26 E\0 (A8a) e26 e22 A exp([ikQ lQ)] e26 e22 B exp(ikQ lQ)[ElQ]F\U0 2 (A8b) ikQ c� 66A[ikQ c� 66B[ikk 6 C]ikk 6 D]e26E\0 (A8c) A]B[C[D\0 (A8d) e26 e22 A] e26 e22 B]F\[U0 2 (A8e) C exp(ike)[D exp([ike)\0 (A8f) We have solved these equations explicitly for the constant A.A\[U0 1 a]bd (A9) where a4 e26 e22 [1[exp([ikQ lQ)][ikQ c� 66 lQ e26 exp([ikQ lQ) (A10) b4 kQ c� 66[1[exp([ikQ lQ)][kk 6 1[exp(i2ke) 1]exp(i2ke) kQ c� 66[1[exp(ikQ lQ)]]kk 6 1[exp(i2ke) 1]exp(i2ke) (A11) d4 e26 e22 [1[exp(ikQ lQ)]]ikQ c� 66 lQ e26 exp(ikQ lQ) (A12) The other constants are then evaluated from the relations : B\bA (A13a) C\ 1 1]exp(i2ke) (A[B) (A13b) D\C exp(i2ke) (A13c) E\[ikQ c� 66 e26 [A exp([ikQ lQ)[B exp(ikQ lQ)] (A13d) F\[U0 2 [ e26 e22 (A]B) (A13e)90 Mechanicas Finally, the current density J contains the information required for the determination of the electrical properties.This is obtained from the relation : J\[iue22 E (A14) References 1 G. Sauerbrey, Z. Phys., 1959, 155, 206. 2 J. G. Miller and D. I. Bolef, J. Appl. Phys., 1968, 39, 5815. 3 C. Lu and O. Lewis, J. Appl. Phys., 1972, 43, 4385. 4 D. A. Buttry and M. D. Ward, Chem. Rev, 1992, 92, 1355. 5 T. Nomura and A. Minemura, Nippon Kagaku Kaishi, 1980, 1621. 6 H. E. Hager, Chem. Engin. Commun., 1986, 43, 25. 7 F. Eggers and Th Funk, J. Phys. E, 1987, 20, 523. 8 G. Sauerbrey, Archiv. Elek. Ubertragung, 1964, 18, 624. 9 J. Krim and A. Widom, Phys. Rev. B, 1988, 38, 12184. 10 Yong-gui Dong, Guan-ping Feng and Cheng-qun Gui, Sens. Actuat., 1993, 13ñ14, 551. 11 H. Muramatsu, X.Ye and T. Ataka, J. Electroanal. Chem., 1993, 347, 247. 12 J. S. Graham and D. R. Rosseinsky, J. Chem. Soc., Faraday T rans., 1994, 90, 3657. 13 S. Bruckenstein, M. Michalski, A. Fensore, Z. Li and A. R. Hillman, Anal. Chem., 1994, 66, 1847. 14 H. Daifuku, T. Kawagoe, N. Yamamoto, T. Ohsaka and N. Oyama, J. Electroanal. Chem., 1989, 274, 313. 15 J. Rickert, A. Brecht and W. Gopel, Anal.Chem., 1997, 69, 1441. 16 K. S. Van Dyke, Proc. I.R.E. , 1928, 16, 742. 17 M. Rodahl, F. Hook, A. Krozer, P. Brzezinski and B. Kasemo, Rev. Sci. Instrum., 1995, 66, 3924. 18 H. Muramatsu, E. Tamiya and I. Karube, Anal. Chem., 1988, 60, 2142. 19 R. Beck, U. Pitterman and K. G. Weil, Ber. Bunsen-Ges. Phys. Chem., 1988, 92, 1363. 20 C. Barnes, Sens. Actuat., 1992, 30, 197. 21 Z. Lin, C. M. Yip, I. S. Joseph and M. D. Ward, Anal. Chem., 1993, 65, 1546. 22 M. Yang and M. Thompson, Anal. Chem., 1993, 65, 1158. 23 S. J. Martin, K. O. Wessendorf, C. T. Gebert, G. C. Frye, R. W. Cernosek, L. Casaus and M. A. Mitchell, Proc. 47th Annual Frequency Control Symp., IEEE, Piscataway, NJ, USA, 1993, p. 603. 24 Z. Shana and F. Josse, Anal. Chem. 1994, 66, 1955 25 M. Yang, M. Thompson and W. C. Duncan-Hewitt, L angmuir, 1993, 9, 802. 26 T. Okajima, H. Sakurai, N. Oyama, K. Tokuda and T. Ohsaka, Electrochim. Acta, 1993, 6, 747. 27 D. M. Soares, Meas. Sci. T echnol, 1993, 4, 549. 28 P. A. Topart, M. A. M. Noeel and H-D. Liess, T hin Solid Films, 1994, 239, 196. 29 S. J. Martin, A. J. Ricco and G. C. Frye, T ech. Digest, 90 IEEE Solid State Sensor Actuator Workshop, 98, Piscataway, NJ, USA, 1990. 30 J. Wang, M. D. Ward, R. C. Ebersole and R. P. Foss, Anal. Chem., 1993, 65, 2553. 31 R. Krimholtz, D. A. Leedom and G. L. Mathaei, Electron. L ett., 1970, 6, 398. 32 E. Benes, J. Appl. Phys., 1984, 56, 608. 33 T. Nakamoto and T. Moriizumi, Jpn. J. Appl. Phys., 1990, 29, 963. 34 D. Johannsmann, K. Mathauer, G. Wegner and W. Knoll, Phys. Rev. B, 1992, 46, 7808. 35 F. Josse, Z. Shana, C.E. Radtke and D. T. Haworth, IEEE T rans. UFFC, 1990, 37, 359. 36 C. E. Reed, K. K. Kanazawa and J. H. Kaufman, J. Appl. Phys., 1990, 69, 1993. 37 G. Hayward and M. N. Jackson, IEEE T rans. UFFC, 1986, 33, 41. 38 J. C. Wang, J. Electrochem. Soc. , 1987, 134, 1715. Paper 7/02998E; Received 1st May, 1997

ISSN:1359-6640    

DOI:10.1039/a702998e

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

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. Depositing the polymer –lm onto bare quartz blanks and exciting the vibration via external electrodes across an air gap is, therefore, is a second possibility to determine shear moduli of thin –lms.We thank T. Jaworek and G. Wegner for providing samples and for helpful discussions.104 V iscoelastic properties of thin –lms References 1 P. G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1979. 2 G. Reiter, Europhys. L ett., 1993, 23, 579. 3 J. L. Keddie, R. A. Jones, and R. A. Cory, Europhys. L ett., 1994, 27, 59. 4 C. L. Jackson and G. B. McKenna, J. Chem. Phys., 1990, 93, 9002. 5 J. N. Israelachvili, Intermolecular and Surface Forcs, Academic Press, London, 2nd edn., 1991. 6 J. Klein, Annu. Rev. Mater. Sci., 1996, 26, 581. 7 J.Klein and E. Kumacheva, Science, 1995, 269, 816. 8 J. Klein, E. Kumacheva, D. Perahia, D. Mahalu and S. Warburg, Faraday Discuss., 1994, 98, 173. 9 J. Van Alsten and S. Granick, Phys. Rev. L ett., 1988, 61, 2570. 10 G. Hadzioannou, S. Patel, S. Granick and M. Tirell, J. Am. Chem. Soc., 1986, 108, 2869. 11 G. Sauerbrey, Arch. Elektrotech. 1964, 18, 617. Uè bertragung, 12 Applications of Piezoelectric Quartz Crystal Microbalances, ed.C. Lu and A. W. Czanderna, Elsevier, Amsterdam, 1984. 13 R. N. Thurston, in Mechanics of Solids, ed. C. Truesdell, Springer-Verlag, Heidelberg, 1984, vol. 4, ch. 36, p. 257. 14 W. P. Mason, Piezoelectric Crystals and their Applications to Ultrasonics, Van Nostrand, Princeton, NJ, 1948. 15 C. E. Reed, K. K. Kanazawa and J. H. Kaufman, J. Appl. Phys., 1990, 68, 1993. 16 V. E. GranstaÜ and S. J. Martin, J. Appl. Phys., 1994, 75, 1319. 17 D. Johannsmann, PhD thesis, Universitaé t Mainz, 1990. 18 T. Nakamoto and T. Moriizumi, Jpn. J. Appl. Phys., 1990, 29, 963. 19 H. F. Tiersten, J. Acoust. Soc. Am., 1963, 35, 234. 20 F. Eggers and Th. Funck, J. Phys. E, 1987, 20, 523. 21 A. Domack and D. Johannsmann, J. Appl. Phys., 1996, 80, 2599. 22 A. Katz and M. D. Ward, J. Appl. Phys., 1996, 80, 4153. 23 A. Domack, O. Prucker, J. Rué he and D. Johannsmann, Phys. Rev. A, 1997, 56, 680. 24 P. G. deGennes, J. Phys. (Paris), 1976, 37, 1443. 25 G. J. Fleer, M. A. Cohen Stuart, J. M. H. M. Scheutjens, T. Cosgrove and B. Vincent, Polymers at Interfaces, Chapman and Hall, London, 1993. 26 J. Rué he, Habil.-thesis, Universitaé t Bayreuth, 1995. 27 A. Halperin, J. Phys. (Paris), 1988, 49, 547. 28 E. B. Zhulina, O. V. Borisov, V. A. Pryamitsyn and T. M. Birshtein, Macromolecules, 1991, 24, 140. 29 D. Johannsmann, K. Mathauer, G. Wegner and W. Knoll, Phys. Rev. B, 1992, 46, 7808. 30 J. Krim, D. H. Solina and R. Chiarello, Phys. Rev. L ett., 1991, 66, 181. 31 G. Duda, A. J. Schouten, T. Arndt, G. Lieser, G. F. Schmidt, C. Bubeck and G. Wegner, T hin Solid Films, 1988, 159, 221. 32 O. WolÜ, T. Jaworek, G. Wegner and D. Johannsmann, in preparation. 33 V. E. Bottom, Introduction to Quartz Crystal Unit Design, Van Nostrand Reinhold, New York, 1982. 34 J. D. Ferry, V iscoelastic Properties of Polymers, Wiley, New York, 1980. 35 C. Lu and O. Lewis, J. Appl. Phys., 1972, 43, 4385. 36 S. D. Stevens and H. F. Tiersten, J. Acoust. Soc. Am., 1986, 79, 1811. 37 E. Benes, K.-C. Harms and G. Thorn, Proceedings of the 39th Annual Frequency Control Symposium, 1985, p. 556. Paper 7/03017G; Received 16th July, 1997

ISSN:1359-6640    

DOI:10.1039/a703017g

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 107, 105»121 Viscoelastic characterization of electroactive polymer films at the electrode/solution interface Helen L. Bandey,a A. Robert Hillman,a Mark J. Browna and Steven J. Martinb a Department of Chemistry, University of L eicester, L eicester, UK L E1 7RH b Sandia National L aboratory, Albuquerque, NM 87185, USA We describe and explore a new general transmission line model that describes the viscoelastic characteristics of thin –lms in terms of their shear moduli.The model contains within it the commonly used lumped-element model as a limiting case, and we delineate the conditions under which the simpler lumped-element model can be applied without signi–cant error. We apply our analysis to the interpretation of crystal impedance spectra acquired dynamically during the electrodeposition of poly(2,2@-bithiophene) (PBT) conducting polymer –lms.In the very early stages of deposition, PBT entrapped within surface features behaves as a rigidly coupled mass. Beyond this, it is a viscoelastic material behaving as a Maxwell —uid. PBT storage and loss moduli are initially polymer coverage dependent, rising to limiting values G@BGAB5]106 dyn cm~2.These values imply an acoustic decay length of ca. 0.3 lm. Accordingly, –lms signi–cantly thicker than this behave as bulk material. For thick –lms there is also evidence for resonance eÜects. Monitoring the frequency response of a quartz crystal oscillator is a well established technique for the characterization of thin –lms.1 The –rst quantitative description of this situation was given by Sauerbrey,2 who showed that the resonant frequency change (*f ) of a quartz crystal oscillator in response to a change in the –lm areal density (*M/g cm~2) is given by *f\[(2/oq vq) *Mf 0 2 (1) where is the density of the crystal, is the wave velocity within it, and is the initial oq vq f0 frequency (in our case, 10 MHz).The Sauerbrey equation has formed the basis for the interpretation of quartz crystal microbalance (QCM) data associated with the deposition and subsequent manipulation of a wide variety of –lms at the gas/solid interface.1 Until recently, it was generally perceived that exposure of the resonator to a liquid, rather than gaseous, medium would result in such severe energy loss (damping) as to prevent oscillation of the crystal.However, the work of Kanazawa and Gordon3 and of Bruckenstein and Shay4 showed that the crystal would launch a shear wave into the liquid, but that the energy loss was not so severe as to preclude oscillation. The decay length in the liquid, d, is given by where and are the liquid viscosity (gL/noL f0)1@2, gL oL and density. This work opened the way for the development of a wide range of in situ studies, including the eÜect of the contacting solution per se.5h7 Most notable among these developments was the electrochemical quartz crystal microbalance (EQCM), in which the exciting electrode of the QCM is also under potential control in an electrochemical cell.The sub-nmol sensitivity and the time resolution of the EQCM technique 105106 Electroactive polymer –lms at the electrode/solution interface have allowed the study of electrochemically driven processes in terms of the overall reaction (under thermodynamic control) and of its mechanism (under kinetic control).Processes studied include electrodeposition or dissolution of polymer, metal and metal oxide/hydroxide –lms, and their exchange of ions and solvent with the bathing electrolyte during electrochemical manipulation; these applications have been reviewed elsewhere. 8h11 Early QCM and EQCM studies generally presumed that the –lm was rigidly coupled to the underlying crystal, allowing interpretation of resonant frequency changes via the Sauerbrey equation. For polymers in particular, this assumption is questionable. As the temperature is raised above the glass-transition temperature, –lm viscoelastic characteristics will change abruptly, as exempli–ed by the temperature-dependent impedance responses of polyisobutylene –lms.12,13 Even below the glass-transition temperature, exposure to solvent vapours can result in softening of polymer –lms; this can form the basis of a gas-phase sensor for organic solvents.12 Exposure to liquid media (as in the electrochemical context) will almost invariably result in solvent partition, so plasticization of the –lm by solvent to yield a non-rigidly coupled –lm must be considered. Furthermore, one can use simple thermodynamic arguments to show that the solvent population of a –lm is dependent upon both electrolyte composition and –lm charge (redox) state.14 Thus, –lm rigidity under one set of conditions implies no guarantee of rigidity under diÜerent conditions.In interpreting EQCM data, it therefore becomes vital to determine whether or not the –lm is rigidly coupled to the crystal. One technique for doing this, the crystal impedance method, involves determination of the complete frequency response in the vicinity of resonance.Purely gravimetric changes result in frequency shifts (described by the Sauerbrey equation) with no change in peak admittance. Viscoelastic changes result in changes in peak admittance, as a consequence of the change in energy loss to the ambient —uid. This approach has been used to study the response of quartz crystal resonators (and other piezoelectric devices) immersed in liquids, both with and without viscoelastic coatings. 15h19 Crystal impedance responses have been studied for immersed quartz crystals loaded with polypyrrole,20,21 Na–on,22 montmorillonite/polyvinylalcohol composite,23 thermoresponsive N-isopropylacrylamide»vinylferrocene copolymer,24 tungsten trioxide25 and Langmuir»Blodgett26,27 –lms. The method has also been used to observe rheological changes during protein denaturing28 and deposition/redox switching of conducting polymers.29,30 In the majority of these cases, crystal impedance is used primarily as a qualitative diagnostic of –lm (non-) rigidity.20,21,23h25,27,29 In some cases, changes in –lm characteristics have been parametrized through equivalent circuit models, such as the Butterworth »van Dyke (BVD)17h19,22,26,28,29 or transmission line15 models (see below).In such descriptions –lm mass changes and energy loss processes, respectively, are represented by inductive and resistive elements ; nevertheless, this has been a largely empirical approach, not yielding direct physical insight into the nature of the –lms. The next step in this area, and the issue upon which this paper focuses, is the quantitative interpretation of crystal impedance responses in terms of physically meaningful parameters, in particular shear moduli.Martin et al. have used an equivalent circuit approach to characterize polymer-–lm-coated resonators in air and extract shear storage and loss moduli.13 They subsequently derived models for quartz crystals loaded with both rigidly coupled mass and liquid.31,32 Wegner et al.have used a continuum electromechanical model to extract shear moduli and compliance for a poly(c-methyl-Lglutamate- co-c-n-octadecyl-L-glutamate) Langmuir»Blodgett»Kuhn –lm coated on a quartz crystal and exposed to air.33 Here, we consider the case of –nite and semi-in–nite viscoelastic –lms exposed to Newtonian —uids. This situation is likely to be a common one for surface-con–ned elec-H.L . Bandey et al. 107 troactive polymers. Additionally, we take into account the eÜect of the underlying electrode surface being rough; this is likely to be a feature of most practical situations. We describe a general transmission line model for this situation, show how the commonly used Butterworth»van Dyke lumped-element model is a special case within it, and explore the regimes under which the simpler lumped element model can be employed without signi–cant error.We then use our new model to interpret crystal impedance measurements on growing PBT –lms. PBT is a member of the thiophene-based family of conducting polymers with potential applications in a range of electrochemical, electronic and optical devices.34,35 In a preliminary study,29 we made crystal impedance measurements on PBT –lms, but were able only to show qualitatively that they were non-rigid and to –t the data empirically to a lumped-element equivalent circuit model.Using our new model, we are now able to extract complex shear moduli, for the polymer Gf\G@]iGA, –lms. Under dynamic conditions, during electropolymerization and deposition, one can then use shear moduli data to map progressive failure of –rst the Sauerbrey equation and subsequently the Butterworth»van Dyke model, as the physical nature of the –lm changes with increasing –lm thickness.Theory The thickness-shear mode resonator consists of a thin disk of AT-cut quartz with electrodes coated on both sides. Owing to the piezoelectric properties and crystalline orientation of the quartz, the application of a voltage between these electrodes results in a shear deformation of the crystal.The crystal can be electrically excited into a number of resonant modes (harmonics), each corresponding to a unique standing shear wave pattern across the thickness of the crystal. For each of these resonant modes, displacement maxima occur at the crystal faces.If a medium contacts one or both of the resonator surfaces, the oscillating surface(s) interact mechanically with the contacting medium. Owing to the electromechanical coupling that occurs in the quartz, the mechanical properties of the contacting medium are re—ected in the electrical properties of the resonator. The object of this section is to relate the electrical properties of the resonator to the mechanical properties of the contacting medium in order that the latter may be extracted from measurement of the former.A one-port electrical device is characterized by its input electrical impedance, Z, measured over a range of excitation frequencies, f. In practice, this is commonly accomplished using a network analyser that excites the device with an incident voltage and measures the re—ected one.The ratio of the re—ected to incident voltages is denoted by the ì scattering parameterœ this is a complex quantity, representing both the magni- S11; tude ratio and phase relation between the incident and re—ected signals. The input impedance is found from by: S11 Z( f )\Z0C1]S11( f ) 1[S11( f )D (2) where is the characteristic impedance of the measurement system (typically 50 )) Z0 The most accurate electrical representation to date, for a piezoelectric bulk-wave resonator is given by the Mason model,36 shown in Fig. 1(a). This contains two acoustic ports connected by a transmission line that represents the phase shift undergone by an acoustic wave in propagating across the quartz thickness.A transformer, representing the electromechanical coupling between the applied voltage and shear displacement in the quartz, couples the acoustic ports to an electrical port. In practice, the acoustic ports are terminated by a mechanical impedance, that characterizes the mechanical Zl ,108 Electroactive polymer –lms at the electrode/solution interface Fig. 1 Equivalent circuits : (a) transmission line model; (b) modi–ed BVD (lumped-element) model properties of the medium contacting the resonator surface(s). Provided that can be Zl related to properties of the contacting medium, electrical measurements made through the electrical port can be interpreted to provide information about the contacting medium. From the distributed (Mason) model [Fig. 1(a)] the input electrical impedance of the quartz resonator is :32,37 Z\ 1 juC0 C1[ K2 / 2 tan(//2)[j(Zl/Zq) 1[j(Zl/Zq)cot(/) D (3) where is the static capacitance of the resonator, K2 and are the quartz electro- C0 Zq mechanical coupling factor and characteristic mechanical impedance, respectively, and / is the acoustic phase shift across the resonator ; where u\2nf, h is the quartz /\uh/vq , thickness, and is the shear-wave velocity for quartz.Eqn. (3) can be represented as a vq static capacitance, in parallel with a motional impedance, which arises from C0 , Zm , mechanical resonance, where is given by37 Zm Zm\ 1 juC0 C //K2 2 tan(//2) [1D]/(Zl/Zq) 4K2uC0 C1[ j(Zl/Zq) 2 tan(//2)D~1 (4) Eqn. (4) can be written as a sum of two terms: The –rst term, Zm\Zm(1)]Zm(2).Zm(1), describes the motional impedance for an unperturbed (without surface loading) quartz resonator and can be represented as a series combination of an inductance, capac- L 1, itance, and resistance, [shown in Fig. 1(b)] : C1 R1, Zm(1)\R1]juL 1] 1 juC1 (5) The second term, in eqn. (4) arises from surface loading. When surface loading is Zm(2), small, so that37 oZl o Zq @2 tanA/ 2B (6)H. L .Bandey et al. 109 then we can approximate as13,38 Zm(2) Ze\Zm(2)\ Nn 4K2usC0 AZl ZqB (7) where N is the harmonic number (1, 3, 5, . . .) and where is the series reso- us\2nfs , fs nant frequency. This shows that under certain conditions the distributed model [Fig. 1(a)] is approximated by the considerably simpler ìlumped-elementœ model shown in Fig. 1(b). This approximation is good provided the condition speci–ed by eqn. (6) is satis–ed. Since /BNn at resonances, tan(//2) is large. With an unspeci–ed loading on the resonator, however, it is difficult to say, a priori, how large and, consequently, what restriction on applies. We can be conservative by assuming that tan(//2)P1 and Zl requiring that in order to apply the lumped-element model.We will oZl o@2Zq , examine this approximation in more detail below. The mechanical impedance arising from loading the surface(s) of the quartz res- Zl , onator, is de–ned as : Zl\ Txy vx Ky/0 (8) where is the surface shear stress (force per area) and is the in-plane surface particle Txy vx velocity. It is evident from eqn. (8) that, in the absence of a surface load, leading Txy\0, to In this case, eqn.(7) indicates that and the motional branch of Fig. 1(b) Zl\0. Ze\0 contains only and This is, the model reduces to the familiar BVD model L 1, C1, R1. derived for an unperturbed resonator. To explain the experimental results, we need to determine the mechanical impedance imposed by the two composite-medium con–gurations shown in Fig. 2. In Fig. 2(a) we consider an ideal mass layer overcoated with a semi-in–nite Newtonian —uid.In Fig. 2(b), we consider a stack consisting of (1) an ideal mass layer, (2) a viscoelastic layer of –nite thickness and (3) a semi-in–nite Newtonian —uid. In the case of a composite medium, the resulting impedance at the resonator surface is generally not the sum of the individual layers considered separately.Intermediate layers in which there is an acoustic Fig. 2 Physical models used to describe the data: (a) ideal mass layer covered by semi-in–nite Newtonian —uid; (b) ideal mass layer covered by –nite viscoelastic layer then semi-in–nite Newtonian —uid110 Electroactive polymer –lms at the electrode/solution interface phase shift cause a transformation of the impedance contributed by upper layers.The general approach is to start with the uppermost layer and work downwards, determining the impedance ìlooking intoœ each subsequent layer, until we arrive at the impedance at the resonator surface (Zl). Newtonian —uid The mechanical impedance contributed by a semi-in–nite Newtonian —uid is :38 Zl\Auog 2 B1@2 (1]j) (9) where o and g are the density and viscosity of the —uid.Mass layer An ideal mass layer is considered to be in–nitesimally thick, yet impose a –nite mass per area on the resonator surface. This condition is approximated by a layer that is sufficiently thin and rigid so that negligible acoustic phase shift occurs across the layer thickness. For such a mass layer alone, the mechanical impedance is :38 Zl\juos (10) where is the mass per area contributed by the layer.os Ideal mass layer plus semi-in–nite Newtonian —uid Since an ideal mass layer, assumed in–nitesimally thick, does not have a phase shift associated with displacements applied to it, the impedance of the composite [Fig. 2(a)] consisting of a mass layer plus semi-in–nite —uid is the sum of the impedances of each considered separately [eqn.(9) and (10)] : Zl\juos]Auog 2 B1@2 (1]j) (11) Finite viscoelastic –lm The mechanical impedance ìlooking intoœ a viscoelastic –lm with a mechanical impedance of above is :32 Zl @ Zl\ZcCZl @ cosh(chf)]Zc sinh(chf) Zc cosh(chf)]Zl @sinh(chf)D (12) where and c are the characteristic mechanical impedance and shear-wave propaga- Zc tion constant in the –lm, respectively : where and Zc\(ofGf)1@2 ; c\ju(of/Gf)1@2, of Gf are the density and complex shear modulus respectively, and is –lm (Gf\G@]jGA), hf thickness.In the case of a –lm plus a semi-in–nite Newtonian —uid layer, is given by Zl @ eqn. (9). If we consider the composite medium shown in Fig. 2(b), the in–nitesimal thickness of the ideal mass layer allows us simply to add the mechanical impedance of this layer [eqn.(10)] to that of the –lm plus liquid determined above: Zl\juos]ZcCZl @ cosh(chf)]Zc sinh(chf) Zc cosh(chf)]Zl @ sinh(chf)D (13a)H. L . Bandey et al. 111 with Zl @ \Auog 2 B1@2 (1]j) (13b) Model agreement The transmission line and lumped-element models give essentially the same result as long as is sufficiently small. It is difficult to determine analytically when this oZl o/Zq condition is satis–ed, however.Cernosek et al. have compared the changes in resonant frequency predicted by the two models under various loading conditions.39 They found that with mass loading, the models give agreement within 1% when mg cm~2; osO5 at the upper limit, With liquid loading, agreement within 1% was oZl o/ZqB0.17. obtained for ogO1000 g2 cm~4 s~1, corresponding to With a visco- oZl o/ZqB0.20.elastic –lm, the number of combinations of variables is very large, making it difficult to examine all possibilities. The heaviest loading arises, however, with low-loss –lms (GA@ which are near resonance (chBn/2). In all these cases, agreement between models in G@) the prediction of resonant frequency change is within 1% as long as This oZl o/ZqO0.2.thus serves as a good guideline as to when the transmission line model can be replaced by the simpler lumped-element model. Experimental Au electrodes were deposited on 10 MHz AT-cut polished quartz crystals (International Crystal Manufacturing, Oklahoma City, OK). A (3-mercaptopropyl)trimethoxysilane underlayer was used to provide good Au adhesion to the quartz.This underlayer was prepared by re—uxing the quartz substrates for 15 min in a solution containing 1 : 1 : 50 (3-mercaptopropyl)trimethoxysilane : water : propan-2-ol. The coated crystals were then dried at 108 °C. Au electrodes were evaporated onto the crystals through a mask; the piezoelectric and electrochemically active Au areas were 0.21 and 0.23 cm2, respectively. Crystal impedance measurements were made using a Hewlett-Packard HP8751A network analyser in re—ectance mode, as described previously.29 The quartz crystal microbalance and network analyser were connected via a 50 ) coaxial cable and an HP8512A transmission/re—ection unit.The admittance data acquisition was computer controlled by a HP BASIC program running on the network analyser in-built computer.PBT deposition and characterization were carried out in a conventional threeelectrode electrochemical cell with one electrode of the quartz crystal acting as the working electrode. Potentials were measured with respect to an Ag`(0.01 mol dm~3)/Ag0 reference electrode ; for comparison with other work, potentials are reported with respect to a saturated calomel electrode (SCE).The counter-electrode was a platinum gauze. Measurements were made at room temperature (20^2 °C) under potential control using either a Thompson Electrochem. Autostat potentiostat and Miniscan potential sweep generator, or an Oxford Electrodes potentiostat. Current responses during deposition and cyclic voltammetry were recorded on an Advance Bryans series 60 000 X»Y recorder.PBT –lms were deposited by potentiostatic electrochemical polymerization of 2,2@- bithiophene. The polymerization solution contained 5 mmol dm~3 BT (Aldrich) and 0.1 mol dm~3 tetraethylammonium tetra—uoroborate (TEAT) (Aldrich, [99%) in acetonitrile. At t\0, the applied potential was stepped from 0.0 V to a selected potential (Epol) in the range Film thickness was varied via deposition time, which 1.1OEpol/VO1.2.was in the range 5»40 min (according to and the desired coverage). At the end of the Epol deposition, the potential was scanned slowly (2 mV s~1) back to 0 V. The integrated112 Electroactive polymer –lms at the electrode/solution interface current response was used as a coulometric measure of polymer coverage. Transfer of the –lms to monomer-free background electrolyte gave voltammetric responses in which the peak currents were linear with scan rate (and charges independent of scan rate) in the range 1\v/mV s~1\10.Results Overview of raw data Fig. 3 shows representative crystal impedance spectra acquired dynamically during the electropolymerization of BT. Spectra were taken at 30 s intervals but, for presentational purposes, we show only selected spectra.Prior to application of the potential step (when E\0 V), peak admittance values corresponded to a resonant resistance of ca. 130»140 ). This is consistent with other data,29 and re—ects resistive components of ca. 10 and 130 ) associated with the crystal itself and energy losses to the solution. Upon application of the potential step into the polymerization range, the resonance moves to lower frequency and the peak admittance decreases. Qualitatively, this is consistent with the deposition of a –lm in which there is substantial energy loss, i.e.a viscoelastic PBT –lm. Data for analogous experiments at other deposition potentials (see below) were qualitatively similar. The primary eÜect of polymerization visible in the raw data was that the shift in peak position (decrease in resonant frequency and admittance) was more rapid at higher potentials.In a previous (preliminary) communication,29 we had been able to demonstrate departure from rigid coupling of the –lm to the crystal at a qualitative level, but had Fig. 3 Crystal impedance spectra for the electrodeposition of PBT. Solution : 5 mmol dm~3 bithiophene»0.1 mol dm~3 V.Times marked on spectra are referred TEAT»CH3CN. Epol\1.125 to application of potential step.H. L . Bandey et al. 113 been unable to proceed further. What we are now able to do is to interpret the data in terms of a physical model that leads to shear moduli. With the greater level of sophistication aÜorded by our new model (described in the Theory section) we are also able to make subtle distinctions between the physical characteristics of –lms prepared under diÜerent polymerization conditions (here, polymerization potential). We suggest that this is a valuable exercise, given the anecdotal evidence that the variability of conducting polymer characteristics is a signi–cant impediment to their exploitation in a range of electronic and optical devices.Equivalent circuit descriptions Fig. 4 shows the results of –tting crystal impedance data for PBT deposition (at three diÜerent polymerization potentials) to the modi–ed Butterworth»van Dyke equivalent circuit of Fig. 1(b). Fig. 4(a) and (b), respectively, display the inductive and resistive Fig. 4 Fits of crystal impedance data as a function to time during PBT deposition to modi–ed BVD equivalent circuit model of Fig. 2. (a) Inductive component of Z in Fig. 2, (b) Resistive uL 2 ; component of Z in Fig. 2, Polymerization potential : 1.125 V (experiment of Fig. 3), R2. (L) 1.150 V 1.200 V (|), (K).114 Electroactive polymer –lms at the electrode/solution interface components of the impedance, representing the inertial mass and energy loss associated with the depositing –lms.The more rapid increases with time of both components with increasing polymerization potential allow us to place the qualitative statements of the previous section regarding Fig. 3 and analogues on a quantitative, albeit at this stage empirical, footing. The rate of polymerization (represented by the current) increases with potential, as has been previously reported in detail.40 It therefore makes more sense to compare the resonant resistance data of Fig. 4(b) for –lms of equal thickness. To the extent that polymer and solvent densities are similar (see below), this can be done via a plot of R vs. uL , since uL represents inertial mass. The data of Fig. 4 are displayed in this format in Fig. 5. The dispersion (with potential) of Fig. 4 is, within experimental scatter, removed. The implication is that there is a common (uL , R) line along which the system moves during PBT deposition ; the primary eÜect of polymerization potential then is to aÜect the rate at which this line is traversed. An important feature of the data in Fig. 5 is the upward curvature. This suggests that the energy loss per unit mass of –lm increases with polymer coverage.In the following section we place this on a quantitative basis by using our new model to determine –lm shear moduli as a function of coverage and deposition potential. Determination of –lm shear moduli We –rst consider the data for –lm deposition at low potentials, exempli–ed by the experiment of Fig. 3. In doing so, it is necessary to explain in a little detail the model we eventually employed to provide what we feel to be the most physically realistic description of the system.Initially, we attempted to –t the data to a two-component model involving a smooth quartz resonator loaded with a –nite viscoelastic –lm (the PBT) and, beyond that, a semi-in–nite Newtonian —uid (the acetonitrile electrolyte solution).Particularly at short times (thin –lms) the –ts were poor. We then recognized that the Au surface of the quartz resonator, even though smooth to the eye (mirror –nish) is rough on a molecular scale. Thus some polymer will be present in ì valleys œ on the surface. It has been suggested41 that liquid present within surface corrugations behaves as mass rigidly vs. for the data of Fig. 4 Fig. 5 R2 uL 2H. L . Bandey et al. 115 Fig. 6 Experimental data (points) and –tted curve (line) for crystal impedance spectrum at t\3 min in Fig. 3. Left-hand panel: magnitude data; right-hand panel: phase data. Lumped-element and transmission line –ts were indistinguishable. coupled to the resonator. Although there may be some debate about the details of this coupling,41,42 there is no doubt that the general picture is supported by the observation of immersion-induced frequency shifts greater than predicted by the modulation layer model of Kanazawa and Gordon.3 On this basis, we reasoned that a similarly entrapped viscoelastic polymer would also behave as a rigidly coupled mass layer.We therefore employed a three-component model involving the quartz resonator loaded with a –nite ideal mass layer (PBT entrapped in surface corrugations), a –nite viscoelastic layer (PBT beyond the surface corrugations) and a semi-in–nite Newtonian —uid (the electrolyte solution).This more sophisticated, and physically realistic, model involves an additional parameter, the thickness of the ideal mass layer, In order to hs . characterize the viscoelastic layer fully, we therefore require an independent measure of This was accomplished by comparing the responses for a bare crystal in air and hs .exposed to the electrolyte, as follows. Eqn. (9) shows that the ìbulkœ liquid (that beyond the surface features) is associated with resistive and inductive components of equal magnitude, i.e. a plot of vs. would have unit slope.However, the rigidly coupled R2 uL 2 material in the surface corrugations has, by de–nition, no resistive component. The difference between the measured inductive and resistive components can therefore be ascribed solely to the entrapped, rigidly coupled, material. This concept is well known and has been used by Cernosek et al. to acquire simultaneous viscosity and density data for lubricant —uids.43 For the ìsmoothœ crystals we used in this work, we found that the ì rigid œ layer thickness, i.e.surface feature depth, was in the range 10»30 nm. This is consistent with their visually smooth appearance. In passing, we point out that we made preliminary measurements employing as-supplied crystals having 0.5 lm –nish, but had considerable difficulty obtaining consistent shear modulus data. We attribute this to the fact that a larger, less uniform and less reproducible fraction of the polymer is entrapped in the surface features.Hence the fraction of the overall signal attributable to and eÜectively –tted as the viscoelastic component is smaller and less well de–ned. SEM observations of our ìsmoothœ crystals support the notion of surface features at the 30 nm level.Thus, we employed a model in which the viscoelastic –lm, whose properties we sought, was located between the ideal mass layer (10»30 nm of rigidly coupled PBT) and the electrolyte (whose properties were then known). Raw data and –ts for amplitude and phase data in the vicinity of resonance at short and long times (thin and thick –lms)116 Electroactive polymer –lms at the electrode/solution interface during the experiment of Fig. 3 are shown in Fig. 6 and 7, respectively. In both cases, and at all intermediate times, the transmission line and lumped-element (BVD) models [see Fig. 1(a) and (b), respectively] gave indistinguishable –ts. This is both interesting and practically important. There is no question that the lumped-element model is considerably simpler to use than the transmission line model (see Theory section).However, it is clear that the simple additivity of the lumped-element model will be inappropriate at sufficiently high loading of the crystal. It is, therefore, important to know how far one can load the crystal and still obtain an adequate description with the lumped-element model.We recently44 discussed qualitatively the progressive failure of the Sauerbrey equation and then the lumped-element model with increasing –lm thickness, but were unable to predict how far one could use the lumped-element model as a reasonable description. Clearly, for the –lms studied here, the lumped-element model suffices. In assessing this result in the context of the predictions of the Theory section, it is necessary to know the –lm thickness. The polymer coverage at the end of each experiment was determined by a slow potential sweep from the polymerization potential back to 0 V.On the basis of a doping level of 0.32 charges per thiophene ring,40 the reductive charge was converted to a coverage, expressed in terms of moles of electroactive sites, C/mol cm~2.Based on the approximation40 of constant polymerization/deposition efficiency, the charge passed at any point during the deposition could then be used to estimate –lm thickness at the time of each impedance spectrum acquired during the deposition experiment. For the data of Fig. 6 and 7, respectively, we then have C\11 and 255 nmol cm~2. These, and the determined –lm thicknesses, 0.014 and 0.209 lm, are consistent with the known density of bithiophene and minor solvent penetration.The storage and loss moduli resulting from the –ts of Fig. 6 and 7 are G@\0.0325]106 and 4.85]106 dyn cm~2, respectively, and GA\0.238]106 and 4.46]106 dyn cm~2, respectively. These values are signi–cantly diÜerent, and the high quality of the –ts in Fig. 6 and 7 could only be achieved within a relatively narrow and non-overlapping band of G@ and GA values for the two –lm thicknesses.The full set of storage and loss moduli for this growing –lm as a function of polymerization charge (representing polymer coverage) are shown in Fig. 8. Although there is some uncertainty associated with the –rst data point in each case (see below), there is unequivocal evidence for increasing shear modulus with polymer coverage during the early stages of deposition, rising to limiting values of G@BGAB5]106 dyn cm~2.These values are Fig. 7 Experimental data (points) and –tted curve (line) for crystal impedance spectrum at t\40 min in Fig. 3. Left-hand panel: magnitude data; right-hand panel: phase data. Lumped-element and transmission line –ts were indistinguishable.H. L .Bandey et al. 117 Fig. 8 Storage (G@, and loss (GA, moduli for PBT deposition as a function of charge passed. L) Ö) Data from Fig. 3. Fitting details as for Fig. 6 and 7. consistent with a polymer behaving rather like a Maxwell —uid. The two points which remain to be discussed are the source of the uncertainty in the early time data and the reason for the variation of modulus with coverage.Taking the –rst of these, we look at the total polymer coverage and that part of it present in surface features. As stated above, the charge passed implies CB11 nmol cm~2. Surface features ca. 30 nm deep could accommodate ca. 15 nmol cm~2 of thiophene rings, i.e. ca. 4 nmol cm~2 of electroactive sites (ca. 40% of the total present).Clearly the amount of polymer contributing to the viscoelastic component of the system is small and will be difficult to characterize. (We note that this is entirely analogous to the problems experienced in our early eÜorts to characterize thicker –lms using rougher electrodes.) In order to explain the variation of shear modulus with coverage, we appeal to the literature on the deposition of a wide range of pyrrole- and thiophene-based polyheterocycle –lms.40,45h47 Despite some variation in absolute rates and mechanistic details, the general picture is a nucleation and growth mechanism that initially involves independent and spatially separated nuclei which grow and eventually overlap.Viewed through the prism of a macroscopic probe, such as the present experiment, this will appear as a slab of material in which solvent is progressively replaced by polymer, until a constant ìbulkœ composition is reached.The data of Fig. 8 are consistent with this model, as the measured shear modulus will increase upon the replacement of solvent in the surface layer by polymer. The point at which ìbulkœ properties are achieved will be dependent upon the number and growth mechanism of the nuclei. In the present case, this is reached after passage of ca. 6 mC cm~2 of charge, which corresponds to a –lm of thickness ca. 0.24 lm. Bulk polymer and –lm resonance eÜects We followed the procedure described in the previous section to interpret the data from deposition experiments carried out at higher potentials, exempli–ed by the data of Fig. 4 for polymerization potentials of 1.15 and 1.20 V. When the –lms were relatively thin, the118 Electroactive polymer –lms at the electrode/solution interface Table 1 Summary of end of deposition data for PBT –lms of Fig. 4 polymerization potential/V 1.125 1.150 1.200 C/nmol cm~2 a 356 774 1760 hf/lm based on coulometric coverage 0.292 0.635 1.44 range which model can accommodate 0.206»0.362 0.242»0.411 0.220»0.392 best –t to model 0.284 0.326 0.306 entrapped (ì rigid œ) polymer os/lg cm~2 1.03 1.59 1.45 hs/nm 10.3 15.9 14.5 gL oL/g2 cm~4 s~1 –tted value for bulk liquidb 0.002 898 0.003 022 0.002 820 a Coulometrically determined by a cathodic sweep following deposition ; see text.b Literature value (20 °C): 0.002 859 g2 cm~4 s~1. results were essentially as described above.However, in these experiments, thick –lms were obtained in relatively short times and we had considerable difficulties in obtaining consistent –ts. The essential problem was that we could obtain what appeared to be a physically reasonable –t in terms of shear moduli, but that the –tted –lm thickness was way below what was physically reasonable on the basis of the coulometric coverage data.The problem is illustrated by Table 1, most obviously with respect to the data at 1.200 V, where the range of –lm thickness values which could be accommodated by the model was a factor of three to four below that based on the coulometric estimate. We take the view that the shear moduli for –lms deposited at higher potentials are not dramatically diÜerent from those deposited at low potentials.This is supported by the data for –lms grown at high potentials early on in the deposition process (when the –lms are still thin). In other words, we pursue the line that this is a –lm thickness issue, rather than one of polymerization potential per se. On the basis of this we can calculate the decay length for the acoustic wave in the growing polymer –lm.We –nd that dB0.3 lm. Thus, in the latter stages of deposition at higher potentials (the data of the last two columns in Table 1), the –lm thickness (based on coulometric evidence) is somewhat greater than the decay length, The measurement is thus not sensitive to hf[d. polymer beyond this distance from the exciting electrode and the resonator sees this as a ìbulkœ –lm, i.e.we have moved from the –nite to the semi-in–nite viscoelastic material case, and –lm thickness is no longer a measurable quantity. We note that the observed value of –lm thickness at which the transition between these two cases occurs lends support to the –tted values of shear moduli. Had the shear moduli been markedly larger (or smaller), the transition would have occurred at rather higher (or lower) –lm thicknesses.Interestingly, for selected spectra during the deposition of –lms at these higher potentials, we were able to obtain apparently reasonable –ts in terms of shear modulus for several –lm thicknesses. For example, for the –lm deposited at 1.200 V, after a deposition time of 6 min, we found G@B6]106 and GAB4]106 dyn cm~2 were achieved with lm, but storage and loss moduli of the same order could be –tted with hfB0.301 or 1.70 lm.On the basis of the acoustic wave decay length in the –lm, we hfB1.31 suggest that this is the consequence of resonance eÜects in the –lm. A major factor in our detection of this eÜect was the acquisition of a series of spectra during the deposition process. The anomaly was highlighted by the apparent –t to a parameter set implying aH.L . Bandey et al. 119 Fig. 9 Cartoon depicting PBT electrodeposition onto visually ìsmoothœ Au-coated quartz resonators. Black shaded areas represent entrapped (ì rigid œ) PBT, dark grey represents viscoelastic PBT, and light grey represents liquid (electrolyte). Curves schematically indicate maximum displacement during oscillation.decrease in –lm thickness with increasing deposition time. An isolated measurement might have been vulnerable to a fortuitous mathematically reasonable –t with a physically unreasonable parameter set. Conclusions Crystal impedance measurements can provide considerable insight into the characteristics of electroactive polymer –lms during their electropolymerization and deposition.We have developed a new transmission line model that describes the viscoelastic characteristics of thin –lms in terms of their shear moduli. This is a general model that contains within it the commonly used lumped element model as a limiting case. Our analysis provides conditions under which the simpler lumped element model can be applied without signi–cant error.We have applied our analysis to the interpretation of crystal impedance spectra acquired dynamically during the electrodeposition of PBT conducting polymer –lms. In applying the model to experimental data, it was necessary to take into account the –nite120 Electroactive polymer –lms at the electrode/solution interface surface roughness even of visually ìsmoothœ crystals.Viscoelastic polymer entrapped within surface features can be described as a rigidly coupled layer, as has been proposed for similarly entrapped liquid. Beyond this, electrodeposited PBT behaves as a viscoelastic layer. Fitted –lm thickness values are consistent with coulometric estimates of polymer coverage. This layer has storage and loss moduli, G@ and GA, respectively, that are initially polymer coverage dependent, rising to a limiting value of G@BGAB5]106 dyn cm~2.We attribute the variations at low coverage to progressive replacement of solvent with polymer in a nucleation and growth process. The limiting shear moduli at higher coverage are characteristic of a Maxwell —uid. This picture is summarized schematically in Fig. 9. For very thick –lms, here realised in polymerization experiments at higher potentials, we encountered two problems.First, the shear moduli imply an acoustic wave decay length of ca. 0.3 lm, so thicker –lms appear as ìbulkœ material and –lm thickness cannot be determined. Second, we occasionally found multiple –ts in terms of –lm thickness, which we attribute to resonance eÜects. H.L.B. thanks the University of Leicester for a studentship and Sandia National Laboratories for hospitality during the early stages of this work.M.J.B. thanks the EPRSC for a studentship (95308787). References 1 Methods and Phenomena, ed. C. Lu and A. W. Czanderna, Elsevier, Amsterdam, 1984, vol. 7. 2 G. Z. Sauerbrey, Z. Phys., 1959, 155, 206. 3 K. K. Kanazawa and J. G. Gordon, Anal. Chem., 1985, 57, 1770. 4 S. Bruckenstein and M. Shay, Electrochim. Acta, 1985, 30, 1295. 5 G. L. Hayward and G. Z. Chu, Anal. Chim. Acta, 1994, 288, 179. 6 M. Yang and M. Thompson, Anal. Chem., 1993, 65, 1158. 7 S. Kurosawa, H. Kitajima, Y. Ogawa, M. Muratsugu, E. Nemoto and N. Kamo, Anal. Chim. Acta, 1993, 274, 209. 8 M. D. Ward and D. A. Buttry, Science, 1990, 249, 1000. 9 R. Schumacher, Angew. Chem. Int.Ed. Eng., 1990, 29, 329. 10 D. A. Buttry, in Electroanalytical Chemistry, ed. A. J. Bard, Marcel Dekker, New York, 1991, vol. 17. p. 1. 11 S. Bruckenstein and A. R. Hillman, in Handbook of Surface Imaging and V isualization, ed. A. T. Hubbard, CRC Press, Boca Raton, FL, 1995, p. 101. 12 S. J. Martin and G. C. Frye, Appl. Phys. L ett., 1990, 57, 1867. 13 S. J. Martin and G. C. Frye, IEEE Ultrasonics Symp.Proc., IEEE, New York, 1991, p. 393. 14 S. Bruckenstein and A. R. Hillman, J. Phys. Chem., 1988, 92, 4837. 15 C. Filia� tre, G. Barde` che and M. Valentin, Sens. Actuators A, 1994, 44, 137. 16 L. Tessier, M. Lethiecq, D. Certon and F. Patat, J. Phys., 1994, 4, 1205. 17 M. Yang and M. Thompson, Anal. Chim. Acta, 1993, 282, 505. 18 R. Beck, U. Pittermann and K.G. Weil, Ber. Bunsen-Ges. Phys. Chem., 1988, 92, 1363. 19 J. Auge, P. Hauptmann, F. Eichelbaum and S. Roé sler, Sens. Actuators B, 1994, 18ñ19, 518. 20 H. Muramatsu, X. Ye, M. Suda, T. Sakahura and T. Ataka, J. Electroanal. Chem., 1992, 322, 311. 21 P. A. Topart, M. A. M. Noeé l and H-D. Liess, T hin Solid Films, 1994, 239, 196. 22 H. Muramatsu, X. Ye and T. Ataka, J. Electroanal.Chem., 1993, 347, 247. 23 T. Okajima, H. Sakurai, N. Oyama, K. Tokuda and T. Ohsaka, Bull. Chem. Soc. Jpn., 1992, 65, 1884. 24 N. Oyama, T. Tatsuma and T. Takahashi, J. Phys. Chem., 1993, 97, 10504. 25 H. Inaba, M. Iwaku, T. Tatsuma and N. Oyama, J. Electroanal. Chem., 1995, 387, 71. 26 H. Muramatsu and K. Kimura, Anal. Chem., 1992, 64, 2502. 27 N. Oyama, S. Ikeda, O. Hatozaki, M. Shimomura, K. Mishima and S. Nakamura, Bull. Chem. Soc. Jpn., 1993, 66, 1091. 28 M. Yang, F. L. Chung and M. Thompson, Anal. Chem., 1993, 65, 3713. 29 A. Glidle, A. R. Hillman and S. Bruckenstein, J. Electroanal. Chem., 1991, 318, 411. 30 P. A. Topart and M. A. M. Noeé l, Anal. Chem., 1994, 66, 2926. 31 S. J. Martin, V. E. GranstaÜ and G. C. Frye, Anal. Chem., 1991, 63, 2272. 32 S. J. Martin and V. E. GranstaÜ, J. Appl. Phys., 1994, 75, 1319. 33 D. Johannsmann, K. Mathauer, G. Wegner and W. Knoll, Phys. Rev. B, 1992, 46, 7808. 34 T. J. Skotheim, Handbook of Conducting Polymers, Marcel Dekker, New York, 1986.H. L . Bandey et al. 121 35 J. Roncali, Chem. Rev., 1992, 92, 711. 36 J. F. Rosenbaum, Bulk Acoustic W ave T heory and Devices, Artech House, Boston, 1988. 37 R. Lucklum, C. Behling, R. W. Cernosek and S. J. Martin, J. Phys. D., 1997, 30, 346. 38 S. J. Martin, G. C. Frye, A. J. Ricco and S. D. Senturia, Anal. Chem., 1993, 65, 2910. 39 R. W. Cernosek, S. J. Martin, A. R. Hillman and H. L. Bandey, Frequency Control Symp., Orlando, FL, May 28»30, 1997. 40 A. R. Hillman and M. J. Swann, Electrochim. Acta, 1988, 33, 1303. 41 R. Beck, U. Pittermann and K. Weil, J. Electrochem. Soc., 1992, 139, 453. 42 M. Urbakh and L. Daikhin, Phys. Rev. B., 1994, 49, 4866. 43 R. Cernosek, S. J. Martin, J. J. Spates, A. N. Rumpf and K. O. Wessendorf, paper 1241, 189th Electrochemical Society Meeting, Los Angeles, May 1996. 44 H. L. Bandey, M. Gonsalves, A. R. Hillman and S. J. Martin, paper 934, 190th Electrochemical Society Meeting, San Antonio, October 1996. 45 A. Asavapiriyanont, G. K. Chandler, G. A. Gunawardena and D. Pletcher, J. Electroanal. Chem., 1984, 177, 229. 46 A. Asavapiriyanont, G. K. Chandler, G. A. Gunawardena and D. Pletcher, J. Electroanal. Chem., 1984, 177, 245. 47 A. R. Hillman and E. F. Mallen, J. Electroanal. Chem., 1987, 220, 351. Paper 7/04278G; Received 18th June, 19

ISSN:1359-6640    

DOI:10.1039/a704278g

出版商:RSC    

年代:1997

数据来源: RSC

摘要:

Faraday Discuss., 1997, 107, 123»140 Determination of polymer shear modulus with quartz crystal resonators Ralf Lucklum* and Peter Hauptmann Otto-von-Guericke-University Magdeburg, Institute for Measurement T echnology and Electronics (IPE), P.O. Box 4120, D-39016 Magdeburg, Germany The dependence of the electrical response of polymer-coated acoustic wave sensors on the viscoelastic properties of the coating material is used for the determination of the complex shear parameter of thin polymer –lms.We discuss the acoustic behaviour and the electrical response of a coated quartz crystal vibrating in the thickness shear mode at its fundamental frequency as well as its third and –fth harmonic. The changes in material properties were induced by temperature changes. Both a glassy and a rubbery polymer consistency were investigated.We also discuss the concept of impedance analysis and parameter –tting to extract the coating shear parameters and we introduce a double layer concept for the direct calculation of the polymer shear parameters. Quartz resonators are commonly used as the frequency normal due to their very high quality factor (Q-factor).As sensor elements the high sensitivity of their mechanical eigenfrequency to surface mass change is exploited. In almost all cases an AT-cut quartz crystal vibrating in the thickness shear mode is used. A well known application is the measurement of mass deposition (rates) in vacuum deposition technology (gravimetric principle). In chemical sensor applications, analyte sorption in the sensitive layer, which covers one or both surfaces of the quartz disc, results in a measurable surface mass change.In all these applications the quartz resonator works as the frequencydetermining element of an electrical oscillator and the oscillating frequency is measured. The Q-factor of a quartz resonator with a foreign layer is still high, hence the oscillating frequency is very stable and can be measured with high resolution.The frequency change can be directly related to deposition rates, mass or thickness changes of the coating or analyte concentrations. Quartz resonators have become popular for the determination of the liquid properties1h3 or of the material parameters of thin –lms4h6 that cover quartz crystal surfaces and interfaces.7h9 Whereas the density and viscosity of a semi-in–nite liquid can be derived from frequency measurements,3 the characterisation of thin –lms requires a more complex analysis of the electrical behaviour of the resonator near its resonant frequency.Impedance analysis is typically performed with a network analyser. The values of the shear parameters cannot be mathematically extracted from the electrical sensor response without some approximations.Hence, the theoretical impedance curves are –tted to the experimental data set by shear parameter variation. Both the geometric and material parameters of the –lm in—uence the vibrational behaviour in certain ways and it is not possible to separate the diÜerent variables. Even impedance analysis cannot 123124 Determination of polymer shear modulus with quartz crystal resonators overcome the ambiguity between several solutions with diÜerent shear parameter values for slightly diÜerent coating thicknesses.We present the results obtained from a –tting procedure that uses a separately determined coating thickness and a literature value for its density. We also determine the in—uence of errors in the diÜerent parameters and measurements on the extracted shear parameter values.Finally, we introduce a double layer arrangement that allows the determination of the –lm shear parameters without knowledge of the –lm thickness. Theory Acoustics To calculate the shear parameters of a thin layer from the electrical response of a vibrating quartz crystal we must consider the sensor as a composite resonator consisting of a single piezoelectric layer with multiple nonpiezoelectric layers in which the acoustic wave propagates.The piezoelectric quartz substrate is used for both generating and detecting the acoustic wave. The characteristics of wave propagation within the multilayer arrangement can be altered by the geometric and intrinsic material parameters of all of the layers.Changes in wave amplitude and phase in the quartz crystal indicate physical property changes occurring in the contacting nonpiezoelectric layers, including changes in the shear modulus of the –lm. When using an AT-cut quartz crystal vibrating in thickness shear mode, the vibrational behaviour of the quartz crystal and the wave propagation in a multilayer arrangement can be derived in a one-dimensional (1D) model from the solution of the wave equation.This is a –rst commonly accepted approximation. The high aspect ratio between the diameter of the quartz disc and the thickness of the crystal makes this assumption reasonable. However, it is known that the quartz disc exhibits vibrational patterns, which are at a maximum in the centre of the circular electrode and which decay towards the edge.10 An important consequence of the shear amplitude distribution is the deviation of some of the quartz disc parameters calculated from electrical measurements from either geometric or literature values.The most obvious of these is the piezoelectrically active area, which is not identical to the electrode area. In consequence, the geometric area is replaced by an eÜective area, which is used to calculate the electrical capacitance, and is necessary in order to calculate the elements of the C0 , motional arm of the Butterworth»van Dyke equivalent circuit.The identities of the electrically eÜective area and the mechanically eÜective area are still questionable. A second eÜective value is the quartz viscosity, It is much higher than the literature value and gq .includes other damping mechanisms. A reasonable determination of this value is even more involved. In contrast to the calculations in the 1D model with only pure thickness shear modes (fundamental and higher harmonics), a quartz crystal with –nite lateral dimensions also generates a lot of other modes. The amplitudes are much smaller, e.g. the resulting out-of-plane displacement at fundamental frequency is about two orders of magnitude smaller than the in-plane displacement.11 In consequence, more dissipation mechanisms are activated, which have not yet been taken into account in detail.We also substitute the thickness of the quartz crystal with an eÜective value, which includes hq , the contributions from both electrodes ; this is done for simplicity.Both electrodes can also be considered as separate layers. Since the electrode works toward a shear stressed layer when the quartz is coated with the –lm a noticeable change is expected in the contribution from this layer and, consequently, a change in the eÜective quartz thickness. The eÜective values are determined from the impedance curve of an uncoated quartz disc in a preceding step.Owing to a lack of information about their change after the coating procedure we assume the changes are negligible. The validity limits of thisR. L ucklum and P. Hauptmann 125 Fig. 1 Shear amplitude distribution of a 5 MHz quartz crystal vibrating in thickness shear mode at fundamental frequency with diÜerent surface mass loads, calculated with FEM second approximation have not been de–ned, yet.We analysed the shear amplitude distribution with the –nite element method (FEM) (CAPA§). Fig. 1 depicts the results for a 5 MHz quartz crystal of 25 mm width and a gold electrode width of 12 mm for a massless electrode (0 nm Au), a gold electrode thickness of 175 nm and the same electrode (175 nm) with an additional 1 lm thick polymer coating. The change in the bandwidth is obvious, especially the diÜerence between the massless and the real electrode.This eÜect is sometimes also called acoustic energy trapping and indicates both a change in the eÜective area and in the eÜective quartz viscosity. In contrast, the polymer coating does not signi–cantly change the shear amplitude distribution.Fig. 2 shows the relation between the geometric (electrode) area and the eÜective area. The mass respective to a typical electrode thickness of 150»200 nm results in an about 7% reduced eÜective area. Thicker gold layers result in a further decrease of the eÜective area but the eÜect slowly vanishes, the eÜective area seems to converge to a constant value.The more interesting in—uence of the polymer coating is analysed starting from the 175 nm gold layer. Owing to the smaller density of the material (a polymer density of 1000 kg m~3 was assumed) the eÜect is much smaller but still 0.4%. If the coating thickness is reduced so that the density»thickness product is equivalent to the density»thickness product of the gold layer, the calculated values for the eÜective area are similar (within numerical accuracy).11 A detailed analysis of the –nite lateral dimensions of the quartz crystal on the 1D model based on –nite element simulations is in progress.The transmission line technique is convenient for modelling 1D multilayer structures. There are no restrictions on the number of layers, layer thickness or complex shear moduli of the layer material.Assuming two acoustic waves travelling in opposite directions, the superposition of these waves must satisfy the acoustic boundary conditions on § CAPA is a commercially distributed –nite element program (Prof. Lerch, University Linz, Austria) with special algorithms to model piezoelectric elements. The simulation of a quartz crystal resonator for chemical applications and material parameter determinations de–nes certain requirements that the FEM analyses have to meet.It must provide various types of elements to model piezoelectric, elastomechanical and viscoelastic materials. The mesh design must consider the high aspect ratio of the quartz disc as well as the large ratio between the quartz thickness and the coating thickness. Finally, an extremely high frequency resolution is required to be comparable with experimental results.126 Determination of polymer shear modulus with quartz crystal resonators Fig. 2 EÜective vibrating area of a 5 MHz quartz crystal under diÜerent loads calculated from the shear amplitude distribution each interface. The similarity and the mathematical unity present in the descriptions of an electromagnetic –eld and an acoustic –eld results in a treatment based on an electrical transmission line.The following analogy relates the acoustic and electrical properties to each other : mechanical tension, T electrical voltage, u particle velocity, v electrical current, I acoustic impedance, Za\T /v electrical impedance, Ze\u/I The impedance concept in acoustic wave propagation problems uses a chain matrix technique.All layers of the composite resonator act as transmission lines of –nite length and are usually acoustically impedance mismatched with respect to the adjacent layers. We de–ne a propagation matrix, and a transfer matrix, for each layer (i) of Pi , Ti , thickness hi . Pi\Ce~ci hi 0 0 eci hiD Ti\c1 Zi 1 1 Zi [1d (1) with c\j u (G/o)1@2 and Za\(oG)1@2 where c is the complex wave propagation constant, u is the angular frequency, G is the shear modulus, o is the density of the layer and The layers are assumed to be j\J[1.isotropic and uniform. This is a third approximation. Under many experimental conditions the –lm thickness is not perfectly uniform and is therefore replaced by an eÜective value.The –lm might be a composite consisting of an attached layer (quartz surface roughness) and a bulk layer. However, we found the eÜect of this to be negligible. Con-R. L ucklum and P. Hauptmann 127 sidering the layer i as the quadrupole transformation matrix, relates the mechanical Mi , tension and particle velocity at the acoustic ports of the layer : Mi\Ti~1Pi~1Ti\3cosh(ci hi) 1 Zi sin(ci hi) Zi sin(ci hi) cosh(ci hi) 4 (2) For the piezoelectric quartz layer the electrical port must be included as the third port.The relation between the mechanical vibration amplitude and the driving voltage can be derived from the resonatorœs equivalent circuit.12 This model consists of two cascaded transmission lines and is composed of distributed and lumped elements.The piezoelectricity is modelled by the transformer of turn ratio 1 : ' and by the element jX.12 The complete transmission line model used for the description of our experimental arrangement is shown in Fig. 3. The bottom quartz surface at port GH is stress free, corresponding to a short-circuited acoustic port. The upper surface is coated with the material to be investigated.In the double layer arrangement the upper surface of the coating is loaded by an additional layer. In that case is the resulting impedance ZEF from both layers. The electrical impedance, Ze, can be obtained from the following expression (f\ for other abbreviations see Appendix): ZL a /Zq , Ze\ZAB\ 1 juC0 ]jX] 1 '2 ZCD\ 1 juC0 C1[ Kq2 aq 2 tan(aq/2)[jf 1[jf cot(aq) D (3) Ze can be transformed into a parallel circuit consisting of a static capacitance, and C0 , the so called motional impedance, Zm e : Ze\ 1 juC0 pZme \ Zm e 1]juC0Zm e (4) can be split into two parts, the –rst representing the uncoated quartz and the Zm e (Zqm) second the transformed acoustic load (ZLm).Zme \ 1 juC0 C aq/Kq2 2 tan(aq/2) [1D] 1 uC0 aq 4Kq2 f 1[jf/2 tan(aq/2) (5) The additive character of the motional quartz impedance and the motional load impedance is useful for the determination of the shear parameters and is exact within 1D models.Eqn. (4) and (5) can be applied without restrictions to the load on the quartz Fig. 3 Transition line model of a composite resonator, consisting of a piezoelectric layer and two nonpiezoelectric layers at one surface128 Determination of polymer shear modulus with quartz crystal resonators crystal.13 The quartz parameters (index q) can be obtained from the impedance of the unperturbed quartz [f\0] by using eqn.(5). The acoustic load of a viscoelastic –lm with –nite thickness, and without an hp , additional load at the surface can be calculated from eqn. (6) as follows : Zpa\j(opG 1 p)1@2 tanCuSAop G 1 pBhpD\jZcp tanCuSAop G 1 pBhpD (6) is the intrinsic acoustic impedance. Under sinusoidal deformation, the shear Zcp modulus of a viscoelastic material is de–ned as a complex quantity : G 1 p\G@]jGA (7) The real part (Re) presents the component stress in-phase with strain and gives rise to energy storage.The imaginary part (Im) represents the component of stress 90° out-ofphase with strain and is a measure of energy dissipation.Therefore, G@ and GA are referred to as shear storage modulus and shear loss modulus, respectively. Fig. 4 illustrates the Re and the Im parts of for a 1 lm thick –lm in a range of shear Zpa\ZL parameters typically for polymers, valid for a 5 MHz quartz resonator. The right-hand range represents the polymer in the glassy state.This area is rather —at, hence changes in the shear parameters are not re—ected very well in the acoustic impedance. The small diagram on the right hand side of the upper part shows a detail of the same diagram on a diÜerent scale. As one can see, falls slightly when G@ or GA decrease. The experi- Im(Zp a ) ment must be carefully designed and the eÜective values must be exactly determined to avoid huge errors in G.As an example of error sources the bar at G@\109 Pa and GA\107 Pa indicates the shift of if the –lm thickness is determined 0.1% larger. Im(Zp a ) G@ and GA of the thinner –lm had to change to 250 and 0.6 MPa, respectively, to get the same acoustic impedance as the thicker –lm. The lower left-hand range in Fig. 4, where the polymer is in the rubbery state, shows signi–cant in—uences on both shear parameters.The imaginary part of the impedance has a deep minimum, the real part has a maximum. This eÜect is an acoustic phenomenon and can be observed when the upper coating surface lags behind the motion of the quartz-coating interface. Because this eÜect is related to acoustic resonance, we call this range of strong shear parameter in—uence on the acoustic impedance the region of acoustic resonance.Although changes rapidly with G, measurement is almost impos- Zp a sible when exceeds several 10 000 Pa s m~1. Quartz oscillation is highly damped Re(Zp a ) and quartz resonance cannot be observed. In the case of an additional load at the upper surface of the coating eqn. (8) must be applied : Zpg a \Zcp Z2a ]jZcp tanCuSAop G 1 pBhpD Zcp]jZ2 a tanCuSAop G 1 pBhpD (8) If this additional acoustic load results from a second layer of thickness then in hg Z2 a eqn.(7) must be substituted by eqn. (6) with the respective values. Eqn. (8) can be transformed into : Zpg a \Zcp jZcg tanCuSAog GgBhgD]Zp a Zcp]jZp a Zcg Zcp tanCuSAog GgBhgD (9)R. L ucklum and P. Hauptmann 129 Fig. 4 Complex acoustic load of a 5 MHz quartz resonator from a 1 lm thick –lm at diÜerent complex shear moduli. G1\G@ and G2\GA.130 Determination of polymer shear modulus with quartz crystal resonators Note, that the unknown –lm thickness of the –rst coating cancels out. can then be Zcp separated out: Zcp\A Zp a Zg a Zpg a Zpa]Zga[Zpg a B1@2 (10) Eqn. (10) describes a new experimental procedure to determine the characteristic impedance of the –rst layer and thus its complex shear modulus, without prior knowledge of the –lm thickness.The complex electrical impedance of a quartz resonator with a single layer of the material under investigation must be measured –rst and the respective complex acoustic impedance must be calculated [eqn. (4) and (5)].Next this –lm must be loaded with a known acoustic load. However, if the additional load is realised with a layer of a thin rigid material the characteristic impedance of this material and the thickness of the deposited –lm must be known. The acoustic impedance of this double layer arrangement must be determined again from the complex electrical impedance. The quartz parameters necessary to calculate the acoustic impedance from the electrical impedance must have been determined previously from impedance analysis of the unperturbed quartz resonator.This method has the capability to simplify the shear parameter determination of thin –lms. The optimisation of the additional acoustic load necessary for sufficient accuracy and sensitivity of the double layer method is in progress.The procedure described includes the assumption that both the quartz and the coating parameters do not change during the deposition of the additional load and that the characteristic (bulk) impedance of the deposited material is also valid for this –lms. In principle, even a liquid load should be possible if the material under investigation does not interact with the liquid.Material properties Beside the acoustic phenomena, a material eÜect has also to be taken into account. Many coating materials, especially polymers, are subjected to the so-called WLF formalism. 14 The WLF equation is a scaled temperature»frequency relation of characteristic relaxation processes : (T [T=)logAX uB\(T0[T=)logAX u0B (11) Eqn. 11 de–nes a set of hyperbolas with common asymptotes at (Vogel temperature) T= and log(X) (XB1012»1015 Hz for simple glass formers).This set is parameterised by one value, at a given mobility or at a given temperature T0 [log(u0)\log(X)] u0 (T0[T=).15 From the molecular point of view, the dynamic properties of a polymer depend on its segmental mobility. The acoustic wave is a mechanical perturbation of the polymer –lm.The polymer chain segments try to relax back to equilibrium conditions. As long as the characteristic relaxation time, q, is much longer than the period of the oscillation the relaxation is ìfrozenœ. This is the glassy state. The rubbery state is characterised by a deformation on the timescale larger than the relaxation time. The temperature of the transition from glassy to rubbery consistency is the glass transition temperature, At Tg .this temperature uqB1. The dependence of on the measurement frequency is appar- Tg ent. As a consequence, one has to distinguish between the static glass transition temperature, determined, e.g., by diÜerential scanning calorimetry (DSC), and the Tg , so-called dynamic glass transition temperature, which must be applied for acoustic Ta , measurements.At typical probing frequencies for acoustic devices, is a few tens of Ta degrees higher than Tg .R. L ucklum and P. Hauptmann 131 Note, that the changes in the intrinsic polymer properties do not appear directly as changes in the electrical properties of the composite resonator. They become eÜective via acoustic properties of the coating through the eÜective surface impedance experienced by the resonator at its interface to the coating.A possible way of how the complex shear parameter changes during glass transition and hence in—uences the acoustic impedance is indicated by the dots in Fig. 4. Experimental details Poly(isobutylene) (PIB) with a medium molecular weight of 380 000 and a density of 0.92 g cm~3, obtained from Aldrich, has been examined. PIB is a rubbery material at room temperature (Tg\[68 °C).All experiments were performed with a polished 5 MHz AT-cut quartz crystal (Maxtek, Torrance, CA) of 2.54 cm diameter. The electrode geometry contains a 12.9 mm diameter earthed electrode on one side while the other side contains a 6.6 mm diameter electrode, which is at rf potential.The electrode pattern was created by vacuum-evaporating a 15 nm chromium adhesion layer, followed by a 160 nm gold layer. The quartz crystal was mounted in a measuring cell and was in electrical contact with the pin contacts. The parasitic contact resistance and capacitance have been taken into account during the calibration and –tting of the measured impedance data. A constant —ow of dry nitrogen was supplied to prevent any dependence on humidity.The whole cell was mounted in a temperature test chamber (Tenney Environmental, Parsippany, NJ, USA). The temperature was controlled with a thermocouple in the measuring cell adjacent to the quartz crystal. The temperature was increased from [50 °C to 150 °C in 5 K steps. An HP 8752A (Hewlett»Packard) network analyser measured the complex re—ection coefficient, directly at the quartz contacts.Measurements were S11, performed three times at 801 points centred about the fundamental, third and –fth harmonic resonant frequencies. The whole run took about 14 h. The polymers were prepared by spin-casting onto the resonator at 1500 rpm from a solution in chloroform»toluene mixture. The quartz surface had been cleaned previously with chloroform at 2000 rpm. The solvent was subsequently driven oÜ during annealing at 150 °C for at least 30 min.The resulting –lm thickness was about 1 lm. The additional load for our double layer arrangement tests was realised with a gold layer that was deposited by vacuum evaporation of 99.999 Au at 10~6 Torr. The source» target distance was about 8 cm, resulting in a deposition rate of 0.9 s~1.The coating ” thickness was determined from the frequency change of a second quartz crystal. Each experiment started with a scan of the uncoated quartz crystal at every temperature for all three frequencies. It was followed by the coating procedure. The coated device was then scanned using the same procedure as for the uncoated quartz.The measuring program was automatically controlled and recorded using HP VEE for Windows (Hewlett-Packard). Results and Discussion Extraction of shear parameters The extraction of the shear storage and the shear loss moduli of the polymer –lm was performed by –tting the theoretically calculated curves directly with the re—ection co-132 Determination of polymer shear modulus with quartz crystal resonators efficient set.This value is related to the impedance by Z\Z0 1]S11 1[S11 (12) where is 50 ). The electrical impedance, Ze, from eqn. (4) was transformed into the Z0 complex re—ection coefficient using eqn. (12). First, the quartz parameters were determined using eqn. (3) with f\0. Although the experimental setup was carefully calibrated, it was necessary to include an external capacitance, parallel to into the equivalent circuit to account for parasitic capac- Cex , C0 itance.In contrast to the external capacitance does not in—uence the motional ele- C0 , ments of the equivalent circuit. The shear parameter set was determined from the best-–t values of G@ and GA using eqn. (4)»(8). The coating thickness necessary for an unambiguous determination of the shear modulus was calculated from the absolute frequency shift at [50 °C (after coating) using eqn.(13), which was derived from eqn. (4) and (5) :16 hp\3 2 tanAaq( fb) 2 B[ 4Kq02 aq( fb)4 Zcq 2pfb op (13) As a second simple method Sauerbreyœs equation17 can be applied. The –lm density is known. Although our equation and Sauerbreyœs equation imply that the material is a rigid thin –lm and the internal quartz losses are negligible, this is still the method with the highest accuracy when compared with the other methods available, as long as the experimental conditions ful–l the assumptions made.This is outlined in Fig. 5. The error in the –lm thickness is determined from the diÜerent absolute frequency shifts after coating the Fig. 5 Error in the coating thickness, calculated from diÜerent resonant frequency shifts with diÜerent approximationsR. L ucklum and P. Hauptmann 133 Fig. 6 Part of the –tting –eld where the –tting algorithm searches for the global error minimum between the experimental values from the impedance measurements and the theoretical values calculated from the full transmission line model by varying G@\G1 and GA\G2 quartz crystal with the –lm.The following notation is used in this and the following section : g\glassy G@\G1\109 Pa, GA\G2\107 Pa t\transition G@\G1\107 Pa, GA\G2\107 Pa r\rubbery G@\G1\106 Pa, GA\G2\105 Pa 1\thin hp\200 nm 2\medium hp\600 nm 3\thick hp\1 lm fb series zero phase resonant frequency used for eqn. 13 fm resonant frequency at impedance minimum fs series zero phase resonant frequency fp parallel zero phase resonant frequency fn resonant frequency at impedance maximum The calculated thicknesses diÜer less than 0.3% from the values calculated with the full transmission line model in the glassy state even for the 1 lm thick coating.The best results are obtained from eqn. (13). The error is less than 0.03%.The same results are found also for a glassy polymer with a higher GA value [e.g. poly(isobutylmethacrylate] of about 108 Pa (not shown). If the material is in the transition area at the chosen temperature, the thickness error is still less than 1% up to a coating thickness of 600 nm. Eqn. (13) still gives the best results. The method does not work properly if the material is in the rubbery state.In analytical applications of acoustic sensors an error in the calculated –lm thickness or an equivalent error in the calculated sorbed mass might be acceptable and extends the validity of the microbalance assumptions. In all other cases the shear parameters must be taken into account.18134 Determination of polymer shear modulus with quartz crystal resonators Fig. 6 presents part of the –tting –eld. The algorithm minimises the average –tting error between the theoretical and the experimental values of the complex impedance by changing G@ and GA for every frequency within the measuring range. Even though the program starts its search at a certain G@ and GA, the –nal result for both parameters does not depend on the starting point.This is an important criterion in checking whether the algorithm is able to –nd the global minimum within the –tting range or whether it stops at one of the local minima. Fig. 6 depicts another source of error. While the minimum of G@ is well de–ned along a huge range of GA, the GA dependence is rather weak. Under the experimental conditions of the chosen example, the glassy state, the –tting error landscape looks more like a moat than a crater.Fig. 7(a) shows the error curves on a logarithmic scale along the three diÜerent lines marked in Fig. 6. G@ has a signi–cant minimum at both the –tted Fig. 7 Fitting error dependence at the best –t values for the shear storage and shear loss modulus at two diÜerent temperatures corresponding to : (a) the glassy state (0 °C) and (b) the rubber state (75 °C).The expected value for the shear loss modulus (GA\G2\105 Pa) is added for the low temperature experiment.R. L ucklum and P. Hauptmann 135 Fig. 8 In—uence of selected parameter errors on the computed values for the complex shear modulus of the coating (for abbreviations see text)136 Determination of polymer shear modulus with quartz crystal resonators Fig. 8 (Continued) value (GA\51) and the expected value of at least 105 Pa. Even the computed value for G@ is almost the same at both values of GA. In contrast, GA has no clear minimum. The –tting error at GA\105 Pa is only slightly higher than that at the global minimum. This behaviour indicates a limitation of the method in calculating the shear parameters from the electrical response of the quartz resonator coated with the material under investigation. However, our –tting algorithm has no physical background and looks only for the global minimum.A practical but highly subjective method is to calculate a reasonable combination of G@ and GA and to value the resulting –tting error of this combination. If the result is satisfactory the experimental setup should be optimised to shift the acoustic phenomenon more in the direction of the expected shear parameter values, as discussed above.An increase of the –lm thickness should be sufficient. Fig. 7(b) shows the same analysis at a higher temperature, where the material is in the rubbery state. In this case both G@ and GA exhibit signi–cant minima. The –tting algorithm works with high accuracy. Fig. 8 summarises our analyses of the in—uence of the coating thickness (h) and the eÜective quartz resonator values of parallel capacitance, (from the eÜective area), C0 external capacitance, and quartz viscosity, g\n. It bears out that even the 1 lm Cex , thick coating is not well adapted to the measuring principle, when using a probing frequency of 5 MHz, if the material is in the glassy state.While the shear storage modulus is of the correct order of magnitude (as long as the coating thickness is correctly determined), the shear loss modulus is completely out of range (even with the correct values for the coating thickness and the other parameters). Note, that the G range shown in the –gure is doubled. The situation is much improved in the transition area.An errorR. L ucklum and P. Hauptmann 137 in the coating thickness is again the strongest in—uence on the computed shear parameters, especially if the –lm is only 600 nm thick. Parallel and external capacitance as well as quartz viscosity inaccuracies do not in—uence the shear parameters signi–cantly. Thus the second approximation, to leave the eÜective quartz parameters constant after coating, is sufficiently uncritical (at least for the eÜective area).If the material is of a rubbery consistency the sensitivity of the method to parameter errors is also small. Only when the coating is thin is an exact value required for the coating thickness. On the other hand the principle cannot be used for the determination of the –lm thickness under those conditions, as shown in Fig. 5. Owing to the high damping of the resonator the shear storage modulus calculated from the 1 lm thick coating deviates slightly from thinner coatings even when using the exact values for all the parameters. A complete data set of the complex shear modulus of PIB is shown in Fig. 9. Two diÜerent error limits were prede–ned in the –tting procedures: 1 ppm, the standard value, and 100 ppm for a rough estimation.The error bars from the –tting procedure are added for the standard case. The –tting algorithm meets the limit value in almost every case, consequently the error bars are smaller than 1 ppm. Only at low temperatures does the above discussed minimum shape for GA result in a higher –tting error. At the same time the shear loss modulus leaves the physically acceptable range.The measuring principle has therefore reached its upper application limit at the coating thickness and resonant frequency used in the experiment. In contrast, the shear storage values are still useful. The diÜerence of this limitation between G@ and GA is a result of the diÜerent contributions of the real and imaginary parts of the complex shear modulus on the resonant frequency shift and the magnitude of the real and imaginary parts of the complex electrical impedance.The values can be determined with a diÜerent accuracy. Shear parameter sets Fig. 10 shows the parameter sets of PIB for the fundamental frequency at 5 MHz and the 3rd and 5th harmonic at 15 MHz and 25 MHz, respectively. The material undergoes Fig. 9 Fitting result of the complex shear modulus with diÜerent –tting error limits of the algorithm and –nal –tting error for the standard settings138 Determination of polymer shear modulus with quartz crystal resonators Fig. 10 (a) Shear storage and shear loss modulus and (b) loss factor of PIB at fundamental frequency (fun) and the 3rd and 5th harmonic, calculated from the electrical impedance of a PIBcoated 5 MHz quartz crystal a phase transition within the measurement range.The shear storage modulus decreases by two orders of magnitude at the fundamental frequency and one order of magnitude at the 5th harmonic. The shear loss modulus has a maximum at about 50 °C. As expected from WLF, the curves deviate from each other at higher temperatures.The gradient of the decline in the shear modulus on a logarithmic scale with respect to temperature is much lower for the higher harmonics. The loss factor has a maximum (fundamental frequency) or at least a shoulder (5th harmonic) referring to a phase transition at 65, 75 or 80 °C. The higher the probing frequency the higher the transition frequency. The transition area is very broad when running through a temperature regime.The diÜerence between the dynamic transition temperature and the static glass transition temperature is more than 130 °C. The temperature values are slightly lower than the transition temperatures determined for a high molecular weight PIB in a previous study.6 In that paper the characteristic relaxation frequencies determined by other methods were compared with the acoustically determined data and are plotted in an Arrheç nius diagram.Both the published and the new values –t better to the trace of the so-called b-relaxation, which is related to the local motion of side-groups and is characterised by a diÜerent activation energy and a diÜerent temperature»frequency relation. However, a correct interpretation of these results needs systematic investigation over a broader range of frequencies. The diÜerences in GA below 50 °C seem to be more related to the experimental method than to material eÜects.Conclusions The quartz crystal resonator is a device suitable for the determination of the shear storage and shear loss moduli of thin polymer –lms at high probing frequencies. Changes in the material properties result in changes in the electrical impedance of theR.L ucklum and P. Hauptmann 139 coated quartz crystal that can be measured with a network analyser. The physical background of this phenomenon is the dependence of the acoustic wave propagation constants on the material parameters of the coating. The experimental setup must consider the relation between the probing frequency (i.e. the wavelength of the probing acoustic wave) and the coating thickness, as well as the coating thickness itself, the shear storage shear loss moduli and the region of acoustic resonance in order to minimise the errors in the computed material parameters from the –tting procedure or determined coating thickness. Simply speaking, coating thickness (or surface mass) determination with high accuracy and –lm shear modulus determination are two applications of quartz crystal resonators which exclude each other.The eÜective quartz parameter errors contribute less to the computed complex shear modulus and can be determined from an analysis of the uncoated quartz. The authors are grateful to Stefan Schranz and Carsten Behling of the Otto-von- Guericke-University for providing assistance in program development and numerical calculations and for helpful discussions.This work was supported by the German Research Foundation under contract Lu605/2-1 and by the Ministry for Education and Research (BMBF) under contact 16 SV070/3. Part of the work was performed at Sandia National Laboratories and supported by the United States Department of Energy under Contract DE-AC04-94AL85000.Sandia is a multiprogramme laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy. Appendix Quartz constants oq\2.651]10~3 kg m~3 density eq\3.982]10~11 A2 s4 kg~1 m~3 permittivity eq\9.53]10~2 A s m~2 piezoelectric constant gq\3.5]10~4 kg m~1 s~1 viscosity cq\2.947]1010 N m~2 piezoelectric stiÜened elastic constant Kq02\eq2/eq cq electromechanical coupling factor for lossless quartz Kq2\eq2/[eq(cq]jugq)] electromechanical coupling factor for lossy quartz C0\eq(A/hq) static quartz capacitance u\2nf angular frequency a\u(hq/vq) wave phase shift in quartz Zq\oq vq\J(oq cq) speci–c quart impedance Relations for any medium (index m) vm\)(Gm/om) km\(u/vm) wave velocity and wavenumber Zm\om vm\)(omGm) speci–c acoustic impedance cm\j(u/vm)\ju)(om/Gm) complex wave propagation constant References 1 S.J. Martin, V. E. GranstaÜ and G. C. Frye, Anal. Chem., 1991, 63, 2272. 2 Z. Lin, C. M. Yip, S. Joseph and M. D. Ward, Anal. Chem., 1993, 65, 1546. 3 S. J. Martin, G. C. Frye and K. O. Wessendorf, Sens. Actuators A, 1994, 44, 209. 4 S. J. Martin and G. C. Frye, Ultrason. Symp. Proc., 1991, 393. 5 A. Katz and M. D. Ward, J. Appl. Phys., 1996, 80, 4153.140 Determination of polymer shear modulus with quartz crystal resonators 6 R. Lucklum, C. Behling, R. W. Cernosek and S. J. Martin, J. Phys. D: Appl. Phys., 1997, 30, 346. 7 Z. Lin, R. M. Hill, H. T. Davis and M. D. Ward, L angmuir, 1994, 10, 4060. 8 R. L. Garrell and J. E. Chadwick, Colloids Surf. A, 1994, 93, 59. 9 Z. Lin and M. D. Ward, Anal. Chem., 1996, 68, 1285. 10 E. Benes, M. Schmidt and V. Kravchenko, J. Acoust. Soc. Am., 1991, 90, 700. 11 S. Schranz and P. Hauptmann, 3. Chemnitzer Fachtagung Mikrosystemtechnik, Chemnitz, October 1997, proceedings, in press. 12 R. Krimholtz, D. A. Leedom and G. L. Mathaei, Electron. L ett., 1970, 6, 398. 13 C. Behling, R. Lucklum and P. Hauptmann, in Sensors and their applications V II, ed. A. T. Augousti, Inst. Physics, Bristol, 1995, 370; C. Behling, R. Lucklum and P. Hauptmann, Sen. Actuators A, 1997 in press. 14 M. L. Williams, R. F. Landel and J. D. Ferry, J. Am. Chem. Soc., 1955, 77, 3701. 15 E. J. Donth, Relaxation and T hermodynamics in Polymers: Glass T ransition, Akademie Verlag, Berlin, 1992. 16 C. Behling, R. Lucklum and P. Hauptmann, Sens. Actuators B, in press. 17 G. Sauerbrey, Z. Phys., 1995, 155, 206. 18 R. Lucklum, in Electrochemical Society Proceedings, Electrochemical Society, Pennington, NJ, USA, 1997, 97-19, 236»247. Paper 7/03127K; Received 7th May, 1997

ISSN:1359-6640    

DOI:10.1039/a703127k

出版商:RSC    

年代:1997

数据来源: RSC