Sauerbrey relation

The classical sensing application of quartz crystal resonators is microgravimetry [1,5]. Many commercial instruments are around. These devices exploit the Sauerbrey relation (Eq. 28). For thin films, the resonance frequency is—by and large—inversely proportional to the total thickness of the plate. The latter increases when a film is deposited onto the crystal surface. Monolayer sensitivity is easily reached. Flowever, when the film thickness increases, viscoelastic effects come into play, as was for instance recognized by Lu and Lewis, who derived a refined version of the Sauerbrey equation [6]. These authors mainly intended to improve the microweighing procedure. Actually measuring viscoelastic properties with the QCM was not a major issue... 

Before going into the details of the calculation for thin films, we briefly come back to a statement made earlier with regard to the proportionality of frequency shift and added mass (as opposed to film thickness). This propor-tionaUty is the essence of the Sauerbrey relation. The frequency shift-mass proportionality holds for all thin films, regardless of their viscoelastic properties. It even applies to laterally heterogeneous samples as long as these are so thin that viscoelasticity can be ignored. In the latter case, the areal mass density of course is an average mass density. 

In 1959, Prof. Gunter Sauerbrey at Physikalisch-Technische Bundesanstalt, Berlin, Germany [1] demonstrated that upon adding mass to a QCM sensor surface there is a frequency decrease, which is proportional to the added mass, provided that the mass is (i) small compared to the weight of the crystal, (ii) rigidly adsorbed, and (hi) evenly distributed over the active area of the crystal. The linear relation between changes in frequency. A/, and adsorbed mass. Am, is therefore often refereed to as the Sauerbrey relation ... 

In this section we will illustrate the model by theoretically comparing the situations when a viscoelastic film is sensed in a gaseous or liquid environment. Figure 5 shows the predicted changes in / and D (for a 5 MHz crystal) versus film thickness for a typical viscoelastic film with Pf=1.0x 10 kgm , f = 1.0 X 10 Ns m and p,f = 1.0 x 10 N m, probed in air and water. Also displayed in this plot is the predicted change in/ according to the Sauerbrey relation (Eq. 1) versus thickness for a rigid film with the same density. 

At not too high a film thickness (< 200 nm) in air (Fig. 5a), it is seen that direct conversion, via the Sauerbrey relation, of the induced change in / by this type of film would be reliable. For thicknesses 200-500 nm it would lead to a slight overestimation of the thickness (or mass). Although not obvious from this plot, the magnitude of this deviation depends critically on the viscoelastic properties and thickness of the film, as previously carefully treated by Lucklum et al. [23]. 

Adsorbed or tethered hpid vesicles induce relatively high damping (Figs. 6, 9, and 10). However, a Voigt-based analysis, in order to correct the apparent mass to obtain the correct mass, shows that the imderestimations using the Sauerbrey relation are generally less than 20%. In contrast, if multiple... 

The relatively recent technical and theoretical improvements (siunmarized above) and the associated imderstanding of the involved processes have broadly ehminated the early obscurity and skepticism connected with hquid-phase applications of the QCM technique. This skepticism was founded on examples of large deviations from the Sauerbrey relation, and strange (as perceived at that time) responses to mass loads with very viscous films. Concerns about the interpretation of QCM data were raised when operation of the technique in aqueous solutions was young [44]. Since then, the theoretical developments and associated deeper quantitative and conceptual understanding of more and more complex systems have removed most of the skepticism. 

For a 5 MHz crystal, a decrease of 1 Hz corresponds to the film deposition of 17.7 ng/cm according to eq,(3). Sauerbrey relation has been verified for the application of film deposition (or remo val) up to a mass load of 2% of the mass per unit area of the unloaded quartz resonator[10]. 

In this paper we discuss three issues related to our ability to exploit the undoubted attractions of the EQCM technique (a) the extent of mobile species uptake as a function of solution concentration (b) the use of transient measurements to obtain (additional) selectivity and (c) the need to establish that the criteria are satisfied for the Sauerbrey equation (equation [1]) to be used to convert measured frequency changes to mass changes. Of these, (a) and (c) have been demonstrated (see previous paragraph) to be directly relevant to QCM-based biosensors. The concept of using transient measurements in this context has not yet been explored, but is a natural development. 

Electrochemical quartz crystal microbalance (EQCM) is a powerful tool for elucidating interfacial reactions based on the simultaneous measurement of electrochemical parameters and mass changes at electrode surfaces. The microbalance is based on a quartz crystal wafer, which is sandwiched between two electrodes, used to induce an electric held (Fig. 2.21). Such a held produces a mechanical oscillation in the bulk of the wafer. Surface reactions, involving minor mass changes, can cause perturbation of the resonant frequency of the crystal oscillator. The frequency change (A/) relates to the mass change (Am) according to the Sauerbrey equation ... 

Sauerbrey [7] in 1959 related the change in resonance frequency of a piezo-electric quartz crystal with the mass deposited onto or removed from the crystal surface. This approach has been used to perform micro-gravimetric measurements in the gas phase like metal evaporation. For (Afo) the Sauerbrey equation states that ... 

Combining Equations 3.2, 3.3, and 3.9 gives the Sauerbrey equation commonly used to relate changes in TSM resonant frequency to surface mass density p, tl] ... 

Obviously the factor relating Im(ZL) to A/s is proportional to 1/dq and, at the first glance, independent of the crystal cut. A/s is independent of k and hence of the electrode diameter as well. As consequence of Vq = 2dq/o the frequency shift remains dependent on Vq. The effect of wave velocity on the sensor s frequency sensitivity is dependent upon whether a certain resonance frequency or a certain crystal thickness is the (experimentally) given value. It can be easily demonstrated in the simplest case of pure mass sensitivity (Zl =ja>pcdc holds). Following Sauerbrey, the frequency sensitivity can be rewritten as ... 

Here and in the following, /o is the resonance frequency of the crystal in the reference state (which usually is the uncoated state) mf and mq are the areal mass densities (mass per unit area) of the film and the crystal, respectively. The relation df/dq = mil mq evidently requires that the density of the film and the crystal are the same. It will turn out that the fractional frequency shift is the same as the ratio of mf and mq for all thin films, regardless of fheir acoustic properties. Therefore, one may memorize the relation A///o =- mf/mq right here. Note that this Sauerbrey limit only holds for films much thinner than the wavelength of sound, L. 

Even when used in vacuum or in an inert gas at ambient pressure, the QCM acts as a balance only under certain conditions, as discussed below. Under these conditions the change of mass caused by adsorption or deposition of a substance from the gas phase can be related directly to the change of frequency, by the simple equation derived by Sauerbrey [8]. 

QCM was extensively used as a mass sensitive detector in vacuum applications and has become an important tool for monitoring mass changes occuring at the crystal surface when it is immersed in a liquid, as reviewed recently [148]. Sauerbrey provided a description and experimental proof of the linear mass-frequency relation for... 

The deposition of noble metals onto oscillating quartz crystals of the thickness shear type, for fine adjustment of their frequency, has already been carried out for many years by frequency standard manufacturers. The idea of using the frequency decrease by mass deposition to determine the weight of the coating is comparatively new. Sauerbrey [35] and Lostis [36] were the first to propose the quartz-crystal microbalance. The AT-cut crystal oscillating in a thickness shear mode was found to be best suited for this purpose. The thickness xq of an infinite quartz plate is directly related to the wavelength A. of the continuous elastic transverse wave, the phase velocity vq of that wave and the frequency vq (i.e. the period xq) of the oscillating crystal, as shown in Fig. 4 ... 

Sauerbrey substituted the area mass density A mq of an additional quartz wafer by the area mass density mf of the deposited foreign material. He made the assumption that for small mass changes, the addition of foreign mass can be treated as an equivalent mass change of the quartz crystal itself. His relation for the frequency change of the loaded crystal is... 

In addition to deriving the equation that now bears his name, Sauerbrey also developed a method for measuring the characteristic frequency and its changes by using the crystal as the frequency-determining component of an oscillator circuit. It should be mentioned that the Sauerbrey equation was developed for oscillation in air and only applies to rigid masses attached to the crystal. However, Kanazawa and Gordon have shown that quartz crystal microbalance measurements can be performed in liquid, in which case a viscosity-related decrease in the resonant frequency is observed ... 

In the simplest format, usually only the series resonant frequency is measured. This method, known as quartz crystal microbalance (QCM), was developed by Sauerbrey (1959), who derived a relation between changes in resonant frequency, Afs, and changes in the surface mass density, p ... 

The most effective method to measure the adsorbed mass is the quartz crystal microbalance (QCMB). This method goes back to the work of Sauerbrey. ° The apphcation of this method is based on the following equation relating the shift A/of the resonance frequency /o of a quartz crystal to the change of the mass of the crystal Am divided by A (area)... 

In QCM-D, the change in resonance frequency is often directly related to the mass of the adsorbed layer according to the Sauerbrey equation (eq 1) ... 

In conclusion the EQCM is a powerful technique that can help us in understanding better electrodeposition processes and to characterize both the deposits and the properties of the adjacent media. Combining EQCM with other electrochemical and non-electrochemical techniques is not all the time a trivial thing, but it can help in gaining more information on the investigated systems. The authors of this entry encourage the scientists to further develop combinations of EQCM and other techniques but also to use the EQCM technique alone. However, one should be aware that the simple linear relation between frequency decrease and deposited mass (Sauerbrey equation) cannot be applied in all cases. More sophisticated evaluation schemes are available and they should be used if needed. 

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