Sauerbrey thickness

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 correct interpretation of the frequency shift from QCM experiments in liquids is a challenge. Practitioners often just apply the Sauerbrey equation (Eq. 28) to their data and term the resulting areal mass density Sauerbrey mass and the corresponding thickness Sauerbrey thickness . Even though the Sauerbrey thickness can certainly serve to compare different experiments, it must not be naively identified with the geometric thickness. Here is a fist of considerations ... 

QCM data. However, if the correction factor is significantly different from unity, it may be expected that it also affects the bandwidth, AF, and also that it depends on overtone order. If, conversely, such effects are absent (AT A/, Sauerbrey thickness same on all overtone orders) one may assume (1 - Zj /Z ) 1. 

To the experimentalist, slip looks like a negative Sauerbrey mass, where the slip length is equal to the negative Sauerbrey thickness. This model ignores roughness and lateral heterogeneities, which presumably play a role in most practical situations where sHp is observed. 

A Frequency shift Af on the 5 overtone (25 MFIz), B current density j, and C Sauerbrey thickness ds measured when depositing polyaciyllc add on the front electrode of a quartz crystal. A voltage 1/ of -0.8 V was applied for 45 minutes, as indicated by the arrows. The resulting film had a Sauerbrey thickness of about 75 nm in the reactant solution. 

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. 

Let us first consider the quartz-metal interface. In deriving the Sauerbrey equation, a pivotal assumption was made, namely that the shear velocities in the crystal and in the film are equal. By this assumption we perform a virtual transformation from thickness to mass. This is the weakest point of Sauerbrey derivation. Here we do not make the same assumption. The shear velocities in the crystal and in the film are... 

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. 

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 ... 

Obviously a small wave velocity in the crystal improves the mass sensitivity of the sensor for a given mechanical resonance frequency/), whereas a large wave velocity increases the mass sensitivity if thickness of the crystal must not fall below a specific value. Table 1 illustrates these basic findings for AT-cut (exemplarily for a small Vq) and BT-cut quartz (exemplarily for a high Vq) for two cases a lOOnm rigid film (Sauerbrey case) and a semi-infinite hquid with a viscosity of 1 cP (Kanazawa case). 

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. 

The term in brackets in Eq. 77 is a viscoelastic correction to the Sauerbrey equation. The viscoelastic correction is independent of film thickness in a li-qiud environment. This is in contrast to films in air or vacuum, where the viscoelastic correction scales as the square of the mass (Eq. 72). In air, the film surface is stress-free. The film only shears under its own inertia and in the hmit of vanishing film thickness, the shear strain goes to zero. As a conse-... 

For a homogeneous thin film with a thickness smaller than the wavelength of the shear oscillations, the shift of the resonance frequency can be expressed in terms of the change in surface mass density of the film, Anzf, (in units g cm ). This was given by Sauerbrey [8] as ... 

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]. 

Fig. 5 Changes in / (solid line) and D (dashed line) at = 1 for a 5 MHz crystal in contact with a viscoelastic film characterized by a homogeneous thickness, viscosity, and elasticity of 1 gcm , 30mPas, and 1 MPa, respectively, and a thickness varying between 0 and 1 p.m. In a, the medium is air. In b, the medium is water. Also shown is the frequency shift according to the Sauerbrey equation (Eq. 1) for the same film (open diamonds)...

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 ... 

The use of piezoelectric (PZ) devices as potential analytical chemistry sensors was not realized until Sauerbrey (3) described the relationship between the resonant frequency of an oscillating piezoelectric crystal and the mass deposited on the crystal surface. The relationship between the weight of an evenly distributed film of a metal and the resonant frequency of an AT cut crystal vibrating in the thickness shear mode can be expressed by ... 

QCM can be described as a thickness-shear mode resonator, since weight change is measured on the base of the resonance frequency change. The acoustic wave propagates in a direction perpendicular to the crystal surface. The quartz crystal plate has to be cut to a specific orientation with respect to the ciystal axis to attain this acoustic propagation properties. AT-cut crystals are typically used for piezoelectric crystal resonators[7]. The use of quartz crystal microbalances as chemical sensors has its origins in the work of Sauerbrey[8] and King [9] who... 

If the film thickness is negligibly small compared to the resonator thickness, the film has the same acoustic property as the quartz resonator. Then the change of resonance frequency due to surface film foimation can be written as the following equation, which is called the Sauerbrey relationship [8],... 

Equation (14) is useful in estimating the thickness of compliant films at which deviations from the Sauerbrey equation are noticeable. Equation (15) is useful in interpreting motional resistance measurements of thin films. In the thin-film limit, the motional resistance change is proportional to the square of the film density, the cube of the film thickness, and the loss compliance of the film. For a 5 MHz QCM, typical values for q and Zq are 0.0402 Henry and 8.84 x 10 Pa s/m, respectively. 

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