In order to understand the whole range of benefits of using tuning forks for fluid sensing, they are compared to piezoelectric disc sensors. The first application using a piezoelectric sensor for measuring the physical mass was described by Sauerbrey in 1959 [

23], who showed that the resonance frequency

${f}_{0}$ of thickness shear-vibrating piezoelectric quartz disks reacts very sensitively to a rigid attached mass layer (

$\Delta m$), following

$\Delta f/{f}_{0}=-\Delta m/{m}_{0}$ where

${m}_{0}$ is the resonator mass and

$\Delta m\ll {m}_{0}$. Frequencies can be measured by simple means with highest accuracy and therefore such sensors can be used to measure very small layer thicknesses (nm) in vapor deposition systems, or molecules binding to a functionalized surface [

9]. They are known as quartz crystal microbalances (QCM), and can also be used to measure the physical properties of fluids as was first described by Kanazawa and Gordon in 1985 [

24]. As may be wrongly suggested by the Sauerbrey equation (Equation (

2)), it is not the static fluid weight which is measured, but instead the dynamic drag of a thin layer of viscous fluid following the small vibrations (<nm) of the surface. If the vibrating surface excites a shear wave, the velocity of a plane shear wave propagating in the

x-direction decays exponentially with

$exp(-x/\delta )$, where the characteristic decay length

$\delta $ is given by

with

$\eta ,\rho $, and

$\omega $ denoting dynamic viscosity, mass-density, and angular frequency. The typical range is from hundreds of nanometers to a few microns.

In Equations (

2) and (

3) [

23,

24] the resonance parameter changes for rigid mass and viscous fluid are shown. Here, additional quantities

$A,{\rho}_{q}$, and

${\mu}_{q}$ are disc face area, density, and shear modulus of quartz, respectively. As two individual fluid parameters

$\rho $ and

$\eta $ are of interest, a second resonance parameter associated with the damping has to be considered. A natural choice is the

Q-factor, but also the damping ratio

$\zeta ={\left(2Q\right)}^{-1}$ could be used. However, as Equation

3 shows,

$\rho $ and

$\eta $ always appear as products, and can therefore not be measured individually. This is one fundamental drawback which is solved by using QTFs rather than QCMs [

25]. Another feature which makes QTFs particularly attractive and was, e.g., pointed out by Matsiev [

26] in 1998, is that operation frequencies are typically much lower. Commercial tuning forks that are used, for example, as clock reference in wrist watches, typically operate at 32.768 kHz and are well suited to fluid sensing, whereas the resonance frequencies of QCMs are typically not lower than 1 MHz. Often, it is preferable [

11] to measure viscosity at lower frequencies because, due to the onset of non-Newtonian behavior, high frequency responses are difficult to interpret and are less comparable to results provided by laboratory instruments. The low frequency shear resonators introduced by Reichel in [

27] follow this line of argumentation, for instance. We note additional benefits of using lower frequencies which are associated with the larger decay lengths

$\delta $ (Equation (

1)). For instance, sensors featuring small

$\delta $ are more likely to suffer from gradual contamination, e.g., with surface active additives, the deposition of particles, or the accumulation of gas micro-bubbles. Certain surface acoustic wave sensors using shear horizontal or Love waves are even worse off in this respect. Here, only a thin surface layer is vibrating and therefore the moved sensor mass

${m}_{0}$ is very low, which makes them extremely sensitive to fluid changes [

28], but also to contamination. Due to the high frequencies of such devices (e.g., 116 MHz in [

29] for a Love wave sensor) and their small decay lengths, the shear wave cannot pass through a contamination layer to sense the fluid. Also for fluids with a micro-structure such as suspensions or emulsions [

30], larger

$\delta $ are preferable [

11]. Despite being the most important characteristic for describing plane shear waves and thus the fluid structure interaction of QCMs,

$\delta $ does not fully describe the flow around the prongs of a vibrating tuning fork. Generally, when a convex body is vibrating transversely in liquid, there is also displaced fluid flowing around the body which can be described by potential flow theory. Heinisch [

31] has shown by experiments that hydrodynamic fluid forces acting on tuning forks of square and circular cross-sections are very similar. The rectangular cross-section in

Figure 1 (i.e., aspect ratio 0.57) can be assumed as square-like, which is substantiated by the tabulated values of hydrodynamic functions given in [

32]. We therefore base our arguments on the similarity to circular cross-sections, where the shear velocity of the potential flow contribution decays with with

$1/{r}^{2}$ instead of

$exp(-x/\delta )$ as would be the case for purely shear vibrating sensors. (It is mentioned for the sake of completeness that while

${v}_{y}\propto exp(-x/\delta )$ describes the decay of plane shear-waves, for cylinders the decay of the azimuthal velocity

${v}_{\theta}$ in the radial direction

r follows the real part of modified Bessel functions K, i.e.,

${v}_{\theta}\propto -\mathrm{Re}({\mathrm{K}}_{0}(\sqrt{2j}\phantom{\rule{0.166667em}{0ex}}r/\delta )+{\mathrm{K}}_{2}(\sqrt{2j}\phantom{\rule{0.166667em}{0ex}}r/\delta ))$, with

$j=\sqrt{-1}$, as is described in [

33]). It can therefore be concluded that the extension of the flow regime is larger, and a wider spatial averaging is achieved, which also reduces vulnerability to surface contamination. This theory is underpinned by the results in

Section 3.3, where a QTF is used for characterizing particles in suspension much larger in diameter than

$\delta $ [

21]. The above-mentioned benefits also apply to cantilever sensors which can also be used for viscosity sensing [

34,

35]. Indeed, tuning fork sensors are often considered as two counteracting cantilevers, as is indicated by the mode shape in

Figure 1. Furthermore, the hydrodynamic models derived for cantilevers can be used as well [

31]. Due to the high degree of similarity, methods established for cantilevers such as the estimation of mechanical stiffness [

36,

37] or using the QTF as a mass sensor is possible. However, benefits of using tuning fork sensors remain in the form of the insensitivity of the clamping condition at the base of the QTF, for instance. Due to symmetry, a rigid boundary condition is realized between the two prongs for the symmetric modes which is not the case for simple cantilevers where mounting (also termed

attachment or

anchor) losses require significant attention [

38]. Commercially available QTFs are also highly developed products, featuring low fabrication tolerances and high temperature stability at low costs.