Sauerbrey equation for

Finally, we rely on the Sauerbrey equation for the conversion of frequency data to mass data (and thus solution composition). This procedure is only valid for rigid polymer films. We therefore regard establishment of rigidity as vital. 

When the permeability length scale is the shortest length of the problem, <3 and L, the layer-induced shift, A/i, is proportional to the density of the liquid and does not depend on the viscosity. It has the form of the Sauerbrey equation for mass loading. This effect results from the inertial motion of the liquid trapped by the inhomogeneities in the interfacial layer ... 

The Sauerbrey equation for electrochemical deposition can be rewritten as follows ... 

The first application of the quartz crystal microbalance in electrochemistry came with the work of Bruckenstein and Shay (1985) who proved that the Sauerbrey equation could still be applied to a quartz wafer one side of which was covered with electrolyte. Although they were able to establish that an electrolyte layer several hundred angstroms thick moved essentially with the quartz surface, they also showed that the thickness of this layer remained constant with potential so any change in frequency could be attributed to surface film formation. The authors showed that it was possible to take simultaneous measurements of the in situ frequency change accompanying electrolysis at a working electrode (comprising one of the electrical contacts to the crystal) as a function of the applied potential or current. They coined the acronym EQCM (electrochemical quartz crystal microbalance) for the technique. 

The Sauerbrey equation predicts a mass sensitivity per unit area of 0.226 Hz cm 2 ng . For a typical crystal the exposed area is c. 0.25 cm2 and the absolute mass sensitivity is 0.904 Hz ng 1. The resolution of modern frequency counters is easily +0,1 Hz in 10 MHz, giving a theoretical mass resolution of c. 9 x 10 10 g in practice this is usually found to be closer to 2 ng. 

Although QCM has originally been devised as a mass sensor for operation in vacuum or gas, it also appeared to be suitable for measurements of the mass and visco-elastic changes at a solid-liquid interface. That was possible due to elaboration of the dedicated oscillator circuitry [115]. The Sauerbrey Equation (2) was derived for resonator oscillations in vacuum. However, it also holds for solution measurements provided that (1) the deposited film is rigid and (2) it is evenly... 

Can such an approach be used for verification of the Sauerbrey equation Discuss the origin of possible experimental artifacts. 

Mercury binding leads to an increase of mass of the gold layer which can be detected by electro-acoustic transducers based on quartz microbalance (QMB the abbreviation QCM = quartz crystal microbalance is also widely used), surface acoustic waves (SAW)—devices [20] or microcantilevers [21,22], Adsorption of mercury vapour increases resonance frequency of shear vibrations of piezoelectric quartz crystals (Fig. 12.2). This process can be described by Sauerbrey equation [23]. For typical AT-cut quartz, this equation is... 

The Electrochemical Quartz Crystal Microbalance (EQCM) The resonant frequency of a quartz crystal oscillator is perturbed from its base value (f ) by attached overlayers. For thin, rigid films the measured change in resonant frequency (Af) with attached mass (AM) is described by the Sauerbrey equation (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 ... 

Thin films of PVF exposed to non-swelling solvents are anticipated to behave as rigidly coupled masses, so that the Sauerbrey equation (equation (13.1)) accurately describes the EQCM responses. However, experimental conditions may not always conform to this situation. For example, an early EQCM study of PVF electroprecipitation yielded film mass data that were not consistent with coulometric assay of the film [39] non-rigidity of PVF films in the CH2C12 deposition solution was suggested. Also, Oyama et al. have studied thermoresponsive N-isopropylacrylamide-vinylferrocene copolymers [40] and found them to exhibit potential dependent viscoelastic characteristics. In the light of these observations, we use the crystal impedance technique (illustrated schematically in Fig. 13.1) to guide our study of PVF films. 

The buildup of multilayers (HRP/PSS) on (PEI/PSS)2-modified surfaces using the common LbL assembly technique that includes washing and drying the whole film after adsorption of each layer [1-3] was also investigated by the QCM method (Fig. 2). The thickness of the adsorbed film, estimated by means of the Sauerbrey equation in dry state, is 0.77 0.7 nm for one HRP/PSS bilayer. 

The quartz crystal microbalance (QCM) is an excellent tool for these investigations since the frequency change produced by the adsorption on the surface of a piezoelectric crystal can be used to assess the mass (to a few ng/cm ) of the adsorbent using the Sauerbrey equation. Since the adsorbed protein layers can have some degree of structural flexibility or viscoelasticity that is undetectable by the determination of the resonance frequency alone, the energy loss, or dissipation factor (D), due to the shear of the adsorbent on the crystal in aqueous solution must also be determined.The technique is termed QCM-D and as well as representing an improvement in the study of biomolecular-surface interactions, it presents an opportunity to observe the adsorption of AFP and PVP, on a model nucleator with a hydrophilic surface. 

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... 

Using Eq. 26 for the parameter mq, we arrive at the famous Sauerbrey equation ... 

Zq is the acoustic impedance of AT-cut quartz its value is 8.8 x 10 kg m s Strictly speaking, Zq is a complex quantity Z + iZ", where Z" accounts for internal friction. Zq is often considered to be real. When this happens, the fundamental frequency/f must also be a real number (see end of Sect. 2). The Sauerbrey equation fails to account for viscoelasticity and also, when applied in liquids, cannot distinguish between the adsorbed material itself and solvent trapped inside the adsorbed film. When a mass is derived by means of the Sauerbrey equation, the interpretation of this mass parameter is sometimes difficult. The terms Sauerbrey mass and Sauerbrey thickness are used in order to indicate that the respective parameters have been calculated by the simple Sauerbrey equation. 

The Sauerbrey equation shows that a thin uniform film rigidly attached to the quartz surface does not influence the width of the mechanical resonance. However, it was experimentally shown for a number of systems that adsorption on the quartz surface produced both a shifts of frequency and an increase of the width of the resonance [34-38]. This phenomenon can be explained, assuming sHppage at the adsorbate-substrate interface. 

It should be noted that for l/S 2> 1 the roughness-induced frequency shift includes a term that does not depend on the viscosity of the liquid, the first term in Eqs. 37 and 33. It reflects the effect of the non-imiform pressure distribution, which is developed in the liquid under the influence of a rough oscillating surface [80]. The corresponding contribution has the form of the Sauerbrey equation. This effect does not exist for smooth interfaces. The second term in Eq. 37 and Eq. 39 describe a viscous contribution to the QCM response. Their contribution to A/ has the form of the QCM response at a smooth liquid-solid interface, but includes an additional factor R that is a roughness factor of the surface. The latter is a consequence of the fact that for l/S 1 the liquid sees the interface as being locally fiat, but with R times its apparent surface area. 

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