The frequency of oscillation of the quartz crystal is partially dependent on the thickness of the crystal. During normal operation, all the other influencing variables remain constant; thus a change in thickness correlates directly to a change in frequency. As mass is deposited on the surface of the crystal, the thickness increases; consequently the frequency of oscillation decreases from the initial value. With some simplifying assumptions, this frequency change can be quantified and correlated precisely to the mass change using Sauerbrey's equation. Other techniques for measuring the properties of thin films include Ellipsometry, Surface Plasmon Resonance (SPR) Spectroscopy, and Dual Polarisation Interferometry.
Today, microweighing is one of several uses of the QCM. Measurements of viscosity and more general, viscoelastic properties, are of much importance as well. The “non-gravimetric” QCM is by no means an alternative to the conventional QCM. Many researchers, who use quartz resonators for purposes other than gravimetry, have continued to call the quartz crystal resonator “QCM”. Actually, the term "balance" makes sense even for non-gravimetric applications if it is understood in the sense of a force balance. At resonance, the force exerted upon the crystal by the sample is balanced by a force originating from the shear gradient inside the crystal. This is the essence of the small-load approximation.
Crystalline α–quartz is by far the most important material for thickness-shear resonators. Langasite (La3Ga5SiO14, “LGS”) and gallium-orthophosphate (GaPO4) are investigated as alternatives to quartz, mainly (but not only) for use at high temperatures. Such devices are also called “QCM”, even though they are not made out of quartz (and may or may not be used for gravimetry).
When the QCM was first developed, natural quartz was harvested, selected for its quality and then cut in the lab. However, most of today’s crystals are grown in the lab using seed crystals. The seed crystals serve as an anchoring point for crystal growth; encouraging growth in two directions and limiting growth in another. The crystals, AT or SC (discussed below) used in most applications operate in the thickness shear mode at a frequency in the 1-30 MHz range.
with u0 the amplitude of lateral displacement, n the overtone order, d the piezoelectric strain coefficient, Q the quality factor, and Uel the amplitude of electrical driving. The piezoelectric strain coefficient is given as d = 3.1·10‑12 m/V for AT-cut quartz crystals. Due to the small amplitude, stress and strain usually are proportional to each other. The QCM operates in the range of linear acoustics.
AT-cut crystals are singularly rotated Y-axis cuts in which the top and bottom half of the crystal move in opposite directions (thickness shear vibration) during oscillation. The AT-cut crystal is easily manufactured. However, it has limitations at high and low temperature, as it is easily disrupted by internal stresses caused by temperature gradients in these temperature extremes (relative to room temperature, ~25 °C). These internal stress points produce undesirable frequency shifts in the crystal, decreasing its accuracy. The relationship between temperature and frequency is cubic. The cubic relationship has an inflection point near room temperature. As a consequence the AT-cut quartz crystal is most effective when operating at or near room temperature. For applications which are above room temperature, water cooling is often helpful. Stress-compensated (SC) crystals are available with a doubly-rotated cut that minimizes the frequency changes due to temperature gradients when the system is operating at high temperatures, and reduces the reliance on water cooling. SC-cut crystals have an inflection point of ~92 °C. In addition to their high temperature inflection point, they also have a smoother cubic relationship and are less affected by temperature deviations from the inflection point. However, due to the more difficult manufacturing process, they are more expensive and are not widely commercially available.
The motional resistance of the resonator, R1, is also used as a measure of dissipation. R1 is an output parameter of some instruments based on advanced oscillator circuits. R1 usually is not strictly proportional to the bandwidth (although it should be according to the BvD circuit; see below). Also, in absolute terms, R1 – being an electrical quantity and not a frequency – is more severely affected by calibration problems than the bandwidth.
Equivalent circuits build on the electromechanical analogy. In the same way as the current through a network of resistors can predicted from their arrangement and the applied voltage, the displacement of a network of mechanical elements can predicted from the topology of the network and the applied force. The electro-mechanical analogy maps forces onto voltages and speeds onto currents. The ratio of force and speed is termed “mechanical impedance”. Note: Here, speed means the time derivative of a displacement, not the speed of sound. There also is an electro-acoustic analogy, within which stresses (rather than forces) are mapped onto voltages. In acoustics, forces are normalized to area. The ratio of stress and speed should not be called "acoustic impedance" (in analogy to the mechanical impedance) because this term is already in use for the material property Zac = ρc with ρ the density and c the speed of sound). The ratio of stress and speed at the crystal surface is called load impedance, ZL. Synonymous terms are "surface impedance" and "acoustic load." The load impedance is in general not equal to the material constant Zac = ρc = (Gρ)1/2. Only for propagating plain waves are the values of ZL and Zac the same.
The electro-mechanical analogy provides for mechanical equivalents of a resistor, an inductance, and a capacitance, which are the dashpot (quantified by the drag coefficient, ξp), the point mass (quantified by the mass, mp), and the spring (quantified by the spring constant, κp). For a dashpot, the impedance by definition is Zm=F / (du/dt)=ξm with F the force and (du/dt) the speed). For a point mass undergoing oscillatory motion u(t) = u0 exp(iωt) we have Zm = iωmp. The spring obeys Zm =κp/(iω). Piezoelectric coupling is depicted as a transformer. It is characterized by a parameter φ. While φ is dimensionless for usual transformers (the ratio of the number of loops an both sides), it has the dimension charge/length in the case electromechanical coupling. The transformer acts as an impedance converter in the sense that a mechanical impedance, Zm, appears as an electrical impedance, Zel, across the electrical ports. Zel is given by Zel = φ2 Zm. For planar piezoelectric crystals, φ takes the value φ = Ae/dq, where A is the effective area, e is the piezoelectric stress coefficient (e = 9.65·102 C/m2 for AT-cut quartz) and dq is the thickness of the plate. The transformer often is not explicitly depicted. Rather, the mechanical elements are directly depicted as electrical elements (capacitor replaces a spring, etc).
There is a pitfall with the application of the electro-mechanical analogy, which has to do with how networks are drawn. When a spring pulls onto a dashpot, one would usually draw the two elements in series. However, when applying the electro-mechanical analogy, the two elements have to be placed in parallel. For two parallel electrical elements the currents are additive. Since the speeds (= currents) add when placing a spring behind a dashpot, this assembly has to be represented by a parallel network.
The figure on the right shows the Butterworth-van Dyke (BvD) equivalent circuit. The acoustic properties of the crystal are represented by the motional inductance, L1, the motional capacitance, C1, and the motional resistance R1. ZL is the load impedance. Note that the load, ZL, cannot be determined from a single measurement. It is inferred from the comparison of the loaded and the unloaded state. Some authors use the BvD circuit without the load ZL. This circuit is also called “four element network”. The values of L1, C1, and R1 then change their value in the presence of the load (they do not if the element ZL is explicitly included).
ff is the frequency of the fundamental. Zq is the acoustic impedance of material. For AT-cut quartz, its value is Zq = 8.8·106 kg m-2 s-1.
The small-load approximation is central to the interpretation of QCM-data. It holds for arbitrary samples and can be applied in an average sense. Assume that the sample is a complex material, such as a cell culture, a sand pile, a froth, an assembly of spheres or vesicles, or a droplet. If the average stress-to-speed ratio of the sample at the crystal surface (the load impedance, ZL) can be calculated in one way or another, a quantitative analysis of the QCM experiment is in reach. Otherwise, the interpretation will have to remain qualitative.
The limits of the small-load approximation are noticed either when the frequency shift is large or when the overtone-dependence of Δf and Δ(w/2) is analyzed in detail in order to derive the viscoelastic properties of the sample. A more general relation is
This equation is implicit in Δf*, and must be solved numerically. Approximate solutions also exist, which go beyond the small-load approximation. The small-load approximation is the first order solution of a perturbation analysis.
The definition of the load impedance implicitly assumes that stress and speed are proportional and that the ratio therefore is independent of speed. This assumption is justified when the crystal is operated in liquids and in air. The laws of linear acoustics then hold. However, when the crystal is in contact with a rough surface, stress can easily become a nonlinear function of strain (and speed) because the stress is transmitted across a finite number of rather small load-bearing asperities. The stress at the points of contact is high, and phenomena like slip, partial slip, yield, etc. set in. These are part of non-linear acoustics. There is a generalization of the small-load equation dealing with this problem. If the stress, σ(t), is periodic in time and synchronous with the crystal oscillation one has
Angular brackets denote a time average and σ(t) is the (small) stress exerted by the external surface. The function σ(t) may or may not be harmonic. One can always test for nonlinear behavior by checking for a dependence of the resonance parameters on the driving voltage. If linear acoustics hold, there is no drive level-dependence. Note, however, that quartz crystals have an intrinsic drive level-dependence, which must not be confused with nonlinear interactions between the crystal and the sample.
η’ and η’’ are the real and the imaginary part of the viscosity, respectively. Zac = ρc =(G ρ)1/2 is the acoustic impedance of the medium. ρ is the density, c, the spead of sound, and G = i ωη is the shear modulus. For Newtonian liquids (η’ = const, η’’ = 0), Δf and Δ(w/2) are equal and opposite. They scale as the square root of the overtone order, n1/2. For viscoelastic liquids (η’ = η(ω), η’’≠ 0), the complex viscosity can be obtained as
Importantly, the QCM only probes the region close to the crystal surface. The shear wave evanescently decays into the liquid. In water the penetration depth is about 250 nm at 5 MHz. Surface roughness, nano-bubbles at the surface, slip, and compressional waves can interfere with the measurement of viscosity. Also, the viscosity determined at MHz frequencies sometimes differs from the low-frequency viscosity. In this respect, torsional resonators (with a frequency around 100 kHz) are closer to application than thickness-shear resonators.
If the density of the film is known, one can convert from mass per unit area, mF, to thickness, dF. The thickness thus derived is also called the Sauerbrey thickness to show that it was derived by applying the Sauerbrey equation to the frequency shift. The shift in bandwidth is zero if the Sauerbrey equation holds. Checking for the bandwidth therefore amounts to checking the applicability of the Sauerbrey equation.
The Sauerbrey equation was first derived by G. Sauerbrey in 1959 and correlates changes in the oscillation frequency of a piezoelectric crystal with mass deposited on it. He simultaneously developed a method for measuring the resonance frequency and its changes by using the crystal as the frequency-determining component of an oscillator circuit. His method continues to be used as the primary tool in quartz crystal microbalance experiments for conversion of frequency to mass.
Because the film is treated as an extension of thickness, Sauerbrey’s equation only applies to systems in which (a) the deposited mass has the same acoustic properties as the crystal and (b) the frequency change is small (Δf / f < 0.05).
If the change in frequency is greater than 5%, that is, Δf / f > 0.05, the Z-match method must be used to determine the change in mass. The formula for the Z-match method is:
kF is the wave vector inside the film and dF its thickness. Inserting kF = 2·π·f /cF = 2·π·f·ρF / ZF as well as dF = mF / ρF yields
Here ZF is the acoustic impedance of the film (ZF = ρFcF = (ρFGf)1/2), kF is the wave vector and dF is the film thickness.
The poles of the tangent (kF dF = π/2) define the film resonances. At the film resonance, one has dF = λ/4. The agreement between experiment and theory is often poor close to the film resonance. Typically, the QCM only works well for film thicknesses much less than a quarter of the wavelength of sound (corresponding to a few micrometres, depending on the softness of the film and the overtone order).
Note that the properties of a film as determined with the QCM are fully specified by two parameters, which are its acoustic impedance, ZF = ρFcF and its mass per unit area, mF = dF/ρF. The wave number kF = ω/cF is not algebraically independent from ZF and mF. Unless the density of the film is known independently, the QCM can only measure mass per unit area, never the geometric thickness itself.
The indices F and Liq denote the film and the liquid. Here, the reference state is the crystal immersed in liquid (but not covered with a film). For thin films, one can Taylor-expand the above equation to first order in dF, yielding
Apart from the term in brackets, this equation is equivalent to the Sauerbrey equation. The term in brackets is a viscoelastic correction, dealing with the fact that in liquids, soft layers lead to a smaller Sauerbrey thickness than rigid layers.
For thin films in liquids, there is an approximate analytical result, relating the elastic compliance of the film, JF’ to the ratio of Δ(w/2); and Δf. The shear compliance is the inverse of the shear modulus, G. In the thin-film limit, the ratio of Δ(w/2) and –Δf is independent of film thickness. It is an intrinsic property of the film. One has
For thin films in air an analogous analytical result is
Here JF’’ is the viscous shear compliance.
a) The QCM always measures an areal mass density, never a geometric thickness. The conversion from areal mass density to thickness usually requires the physical density as an independent input.
b) It is difficult to infer the viscoelastic correction factor from QCM data. However, if the correction factor differs significantly from unity, it may be expected that it affects the bandwidth Δ(w/2) and also that it depends on overtone order. If, conversely, such effects are absent (Δ(w/2) « Δf, Sauerbrey thickness same on all overtone orders) one may assume that (1-ZLiq2/ZF2)≈1.
c) Complex samples are often laterally heterogeneous.
d) Complex samples often have fuzzy interfaces. A "fluffy" interface will often lead to a viscoelastic correction and, as a consequence, to a non-zero Δ(w/2) as well as an overtone-dependent Sauerbrey mass. In the absence of such effects, one may conclude that the outer interface of film is sharp.
e) When the viscoelastic correction, as discussed in (b), is insignificant, this does by no means imply that the film is not swollen by the solvent. It only means that the (swollen) film is much more rigid than the ambient liquid. QCM data taken on the wet sample alone do not allow inference of the degree of swelling. The amount of swelling can be inferred from the comparison of the wet and the dry thickness. The degree of swelling is also accessible by comparing the acoustic thickness (in the Sauerbrey sense) to the optical thickness as determined by, for example, surface plasmon resonance (SPR) spectroscopy or ellipsometry. Solvent contained in the film usually does contribute to the acoustic thickness (because it takes part in the movement), whereas it does not contribute to the optic thickness (because the electronic polarizability of a solvent molecule does not change when it is located inside a film).
Often, the external object is so heavy that it does not take part in the MHz oscillation of the crystal due to inertia. It then rests in place in the laboratory frame. When the crystal surface is laterally displaced, the contact exerts a restoring force upon the crystal surface. The stress is proportional to the number density of the contacts, NS, and their average spring constant, κS. The spring constant may be complex (κS* = κS’ + iκS’’), where the imaginary part quantifies a withdrawal of energy from the crystal oscillation (for instance due to viscoelastic effects). For such a situation, the small-load approximation predicts
The QCM allows for non-destructive testing of the shear stiffness of multi-asperity contacts.
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