Resonator crystals

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 using seed crystals. A seed crystal serves as an anchoring point and template for crystal growth. Grown crystals are subsequently cut and polished into hair-thin discs which support thickness shear resonance in the 1-30 MHz range. The "AT" or "SC" oriented cuts (discussed below) are widely used in applications.^[21]

Electromechanical coupling

The QCM consists of a thin piezoelectric plate with electrodes evaporated onto both sides. Due to the piezo-effect, an AC voltage across the electrodes induces a shear deformation and vice versa. The electromechanical coupling provides a simple way to detect an acoustic resonance by electrical means. Otherwise, it is of minor importance. However, electromechanical coupling can have a slight influence on the resonance frequency via piezoelectric stiffening. This effect can be used for sensing,^[22] but is usually avoided. It is essential to have the electric and dielectric boundary conditions well under control. Grounding the front electrode (the electrode in contact with the sample) is one option. A π-network sometimes is employed for the same reason.^[23] A π-network is an arrangement of resistors, which almost short-circuit the two electrodes. This makes the device less susceptible to electrical perturbations.

Shear waves decay in liquids and gases

Most acoustic-wave-based sensors employ shear (transverse) waves. Shear waves decay rapidly in liquid and gaseous environments. Compressional (longitudinal) waves would be radiated into the bulk and potentially be reflected back to the crystal from the opposing cell wall.^[24][25] Such reflections are avoided with transverse waves. The range of penetration of a 5 MHz-shear wave in water is 250 nm. This finite penetration depth renders the QCM surface-specific. Also, liquids and gases have a rather small shear-acoustic impedance and therefore only weakly damp the oscillation. The exceptionally high Q-factors of acoustic resonators are linked to their weak coupling to the environment.

Modes of operation

Economic ways of driving a QCM make use of oscillator circuits.^[26][27] Oscillator circuits are also widely employed in time and frequency control applications, where the oscillator serves as a clock. Other modes of operation are impedance analysis,^[28] QCM-I, and ring-down,^[29][30] QCM-D. In impedance analysis, the electric conductance as a function of driving frequency is determined by means of a network analyzer. By fitting a resonance curve to the conductance curve, one obtains the frequency and bandwidth of the resonance as fit parameters. In ring-down, one measures the voltage between the electrodes after the exciting voltage has suddenly been turned off. The resonator emits a decaying sine wave, where the resonance parameters are extracted from the period of oscillation and the decay rate.

Energy trapping

To avoid dissipation of vibration energy (damping the oscillation) by the crystal holder, which touches the crystal at the rim, the vibration should be confined to the center of the crystal platelet. This is known as energy trapping.

For crystals with high frequencies (10 MHz and higher), the electrodes at the front and the back of the crystal usually are key-hole shaped, thereby making the resonator thicker in the center than at the rim. The mass of the electrodes confines the displacement field to the center of the crystal disk.^[31] QCM crystals with vibration frequencies around 5 or 6 MHz usually have a planoconvex shape; at the rim the crystal is too thin for a standing wave with the resonance frequency. Thus, in both cases the thickness-shear vibration amplitude is greatest at the center of the disk. This means that the mass-sensitivity is peaked at the center also, with this sensitivity declining smoothly to zero towards the rim (For high-frequency crystals, the amplitude vanishes already somewhat outside the perimeter of the smallest electrode.^[32]) The mass-sensitivity is therefore very non-uniform across the crystal surface, and this non-uniformity is a function of the mass-distribution of the metal electrodes (or in the case of non-planar resonators, the quartz crystal thickness itself).

Energy trapping slightly distorts the otherwise planar wave fronts. The deviation from the plane thickness-shear mode entails flexural contribution to the displacement pattern. If the crystal is not operated in vacuum, flexural waves emit compressional waves into the adjacent medium, which is a problem when operating the crystal in a liquid environment. Standing compressional waves form in the liquid between the crystals and the container walls (or the liquid surface); these waves modify both the frequency and the damping of the crystal resonator.

Overtones

Planar resonators can be operated at a number of overtones, typically indexed by the number of nodal planes parallel to the crystal surfaces. Only odd harmonics can be excited electrically because only these induce charges of opposite sign at the two crystal surfaces. Overtones are to be distinguished from anharmonic side bands (spurious modes), which have nodal planes perpendicular to the plane of the resonator. The best agreement between theory and experiment is reached with planar, optically polished crystals for overtone orders between n = 5 and n = 13. On low harmonics, energy trapping is insufficient, while on high harmonics, anharmonic side bands interfere with the main resonance.

Amplitude of motion

The amplitude of lateral displacement rarely exceeds a nanometer. More specifically one has

${\displaystyle u_{0}={\frac {4}{\left(n\pi \right)^{2}}}dQU_{\mathrm {el} }}$

with u₀ the amplitude of lateral displacement, n the overtone order, d the piezoelectric strain coefficient, Q the quality factor, and U_el 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.

Effects of temperature and stress

The resonance frequency of acoustic resonators depends on temperature, pressure, and bending stress. Temperature-frequency coupling is minimized by employing special crystal cuts. A widely used temperature-compensated cut of quartz is the AT-cut. Careful control of temperature and stress is essential in the operation of the QCM.

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)^[33][34] 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.^[35] 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.

Electrochemical QCM

The QCM can be combined with other surface-analytical instruments. The electrochemical QCM (EQCM) is particularly advanced.^[36][37][38] Using the EQCM, one determines the ratio of mass deposited at the electrode surface during an electrochemical reaction to the total charge passed through the electrode. This ratio is called the current efficiency.

Assumptions

For a number of experimental configurations, there are explicit expressions relating the shifts of frequency and bandwidth to the sample properties.^{[46][47][48][49]} The assumptions underlying the equations are the following:

• The resonator and all cover layers are laterally homogeneous and infinite.
• The distortion of the crystal is given by a transverse plane wave with the wave-vector perpendicular to the surface normal (thickness-shear mode). There are neither compressional waves^[23][24] nor flexural contributions to the displacement pattern.^[50] There are no nodal lines in the plane of the resonator.
• All stresses are proportional to strain. Linear viscoelasticity holds.^[51]
• Piezoelectric stiffening may be ignored.

Semi-infinite viscoelastic medium

For a semi-infinite medium, one has^[52][53][54]

${\displaystyle {\frac {\Delta f^{*}}{f_{f}}}={\frac {i}{\pi Z_{q}}}\,{\frac {\sigma }{\dot {u}}}={\frac {i}{\pi Z_{q}}}Z_{\mathrm {ac} }={\frac {i}{\pi Z_{q}}}{\sqrt {\rho i\omega \eta }}}$

${\displaystyle ={\frac {1}{\pi Z_{q}}}\,{\frac {-1+i}{\sqrt {2}}}{\sqrt {\rho \omega \left(\eta ^{\prime }-i\eta ^{\prime \prime }\right)}}={\frac {i}{\pi Z_{q}}}{\sqrt {\rho \left(G^{\prime }+iG^{\prime \prime }\right)}}}$

η’ and η’’ are the real and the imaginary part of the viscosity, respectively. Z_ac = ρc =(G ρ)^1/2 is the acoustic impedance of the medium. ρ is the density, c, the speed 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, n^1/2. For viscoelastic liquids (η’ = η(ω), η’’≠ 0), the complex viscosity can be obtained as

${\displaystyle \eta ^{\prime }=-{\frac {\pi Z_{q}^{2}}{\rho _{\mathrm {Liq} }\,f}}\,{\frac {\Delta f\Delta \left(w/2\right)}{f_{f}^{2}}}}$

${\displaystyle \eta ^{\prime \prime }={\frac {1}{2}}{\frac {\pi Z_{q}^{2}}{\rho _{\mathrm {Liq} }\,f}}\,{\frac {\left(\left(\Delta \left(w/2\right)\right)^{2}-\Delta f^{2}\right)}{f_{f}^{2}}}}$

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^[20] (with a frequency around 100 kHz) are closer to application than thickness-shear resonators.

Inertial loading (Sauerbrey equation)

The frequency shift induced by a thin sample which is rigidly coupled to the crystal (such as a thin film), is described by the Sauerbrey equation. The stress is governed by inertia, which implies σ = -ω²u₀m_F, where u₀ is the amplitude of oscillation and m_F is the (average) mass per unit area. Inserting this result into the small-load-approximation one finds

${\displaystyle {\frac {\Delta f^{*}}{f_{f}}}\approx {\frac {i}{\pi Z_{q}}}{\frac {-\omega ^{2}u_{0}m_{\mathrm {F} }}{i\omega u_{0}}}=-{\frac {2\,f}{Z_{q}}}m_{\mathrm {F} }}$

If the density of the film is known, one can convert from mass per unit area, m_F, to thickness, d_F. 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 Günter 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.^[9][54] The formula for the Z-match method is:

${\displaystyle \tan \left({\frac {\pi \Delta f}{f_{f}}}\right)={\frac {-Z_{\mathrm {F} }}{Z_{q}}}\tan \left(k_{\mathrm {F} }d_{\mathrm {F} }\right)}$

k_F is the wave vector inside the film and d_F its thickness. Inserting k_F = 2·π·f /c_F = 2·π·f·ρ_F / Z_F as well as d_F = m_F / ρ_F yields

${\displaystyle \Delta f=-{\frac {f_{f}}{\pi }}\left(\arctan {\frac {Z_{\mathrm {F} }}{Z_{q}}}\tan \left({\frac {2\pi f}{Z_{\mathrm {F} }}}m_{\mathrm {F} }\right)\right)}$

Viscoelastic film

For a viscoelastic film, the frequency shift is

${\displaystyle {\frac {\Delta f^{*}}{f_{f}}}={\frac {-1}{\pi Z_{q}}}Z_{\mathrm {F} }\tan \left(k_{\mathrm {F} }d_{\mathrm {F} }\right)}$

Here Z_F is the acoustic impedance of the film (Z_F = ρ_Fc_F = (ρ_FG_f)^1/2)= (ρ_F/J_f)^1/2), k_F is the wave vector and d_F is the film thickness. J_f is the film's viscoelastic compliance, ρ_F is the density.

The poles of the tangent (k_F d_F = π/2) define the film resonances.^[55][56] At the film resonance, one has d_F = 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, Z_F = ρ_Fc_F and its mass per unit area, m_F = d_F/ρ_F. The wave number k_F = ω/c_F is not algebraically independent from Z_F and m_F. Unless the density of the film is known independently, the QCM can only measure mass per unit area, never the geometric thickness itself.

Viscoelastic film in liquid

For a film immersed in a liquid environment, the frequency shift is^[57][58]

${\displaystyle {\frac {\Delta f^{*}}{f_{f}}}={\frac {-Z_{\mathrm {F} }}{\pi Z_{q}}}{\frac {Z_{\mathrm {F} }\tan \left(k_{\mathrm {F} }d_{\mathrm {F} }\right)-iZ_{\mathrm {Liq} }}{Z_{\mathrm {F} }+iZ_{\mathrm {Liq} }\tan \left(k_{\mathrm {F} }d_{\mathrm {F} }\right)}}}$

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 d_F, yielding

${\displaystyle {\frac {\Delta f^{*}}{f_{f}}}={\frac {-\omega m_{\mathrm {F} }}{\pi Z_{q}}}\left(1-{\frac {Z_{\mathrm {Liq} }^{2}}{Z_{\mathrm {F} }^{2}}}\right)={\frac {-\omega m_{\mathrm {F} }}{\pi Z_{q}}}\left(1-J_{\mathrm {F} }{\frac {Z_{\mathrm {Liq} }^{2}}{\rho _{\mathrm {F} }}}\right)}$

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.

Derivation of viscoelastic constants

The frequency shift depends on the acoustic impedance of the material; the latter in turn depends on the viscoelastic properties of the material. Therefore, in principle, one can derive the complex shear modulus (or equivalently, the complex viscosity). However, there are certain caveats to be kept in mind:

• The viscoelastic parameters themselves usually depend on frequency (and therefore on the overtone order).
• It is often difficult to disentangle effects of inertia and viscoelasticity. Unless the film thickness is known independently, it is difficult to obtain unique fitting results.
• Electrode effects can be of importance.
• For films in air, the small-load approximation must be replaced by the corresponding results from perturbation theory unless the films are very soft.

For thin films in liquids, there is an approximate analytical result, relating the elastic compliance of the film, J_F’ 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^[59]

${\displaystyle {\frac {\Delta \left(\omega /2\right)}{-\Delta f}}\approx \eta \omega J_{F}^{\,\prime }}$

For thin films in air an analogous analytical result is^[60]

${\displaystyle \Delta \left(\omega /2\right)={\frac {8}{3\rho _{\mathrm {F} }Z_{q}}}f_{f}^{\,4}m_{\mathrm {F} }^{3}n^{3}\pi ^{2}J^{\prime \prime }}$

Here J’’ is the viscous shear compliance.

Interpretation of the Sauerbrey thickness

The correct interpretation of the frequency shift from QCM experiments in liquids is a challenge. Practitioners often just apply the Sauerbrey equation to their data and term the resulting areal mass (mass per unit area) the "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. Worthwhile considerations are the following:

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-Z_Liq²/Z_F²)≈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). The difference in dry and wet mass is shown with QCM-D and MP-SPR for instance in protein adsorption on nanocellulose^[61][62] and in other soft materials.^[63]