Basics of a Quartz Crystal Microbalance
Introduction
This document provides an introduction to the quartz crystal microbalance (QCM) which is an instrument that allows a user to monitor small mass changes on an electrode. The reader is directed to the numerous reviews^{1} and book chapters^{2} for a more indepth description concerning the theory and application of the QCM. A basic understanding of electrical components and concepts is assumed.
The two major points of this document are:
 Explanation of the Piezoelectric Effect
 Equivalent Circuit Models
Explanation of the Piezoelectric Effect
The application of a mechanical strain to certain types of materials (mostly crystals) results in the generation of an electrical potential across that material. Conversely, the application of a potential to the same material results in a mechanical strain (a deformation). Removal of the potential allows the crystal to restore to its original orientation. The igniters on gas grills are a good example of everyday use of the piezoelectric effect. Depressing the button causes the springloaded hammer to strike a quartz crystal thereby producing a large potential that discharges across a gap to a metal wire igniting the gas.
Quartz is by far the most widely utilized material for the development of instruments containing oscillators partly due to historical reasons (the first crystals were harvested naturally) and partly due to its commercial availability (synthetically grown nowadays). There are many ways to cut quartz crystals and each cut has a different vibrational mode upon application of a potential. The ATcut has gained the most use in quartz crystal microbalance applications due to its low temperature coefficient at room temperature. This means that small changes in temperature only result in small changes in frequency. It has a vibrational mode of thickness shear deformation as shown below in Figure 1.
Quartz Crystal No Applied Potential
Quartz Crystal Under Applied Potential
Figure 1. Graphical Representation of Thickness Shear Deformation.
The application of an alternating potential (a sine wave in nearly all cases) to the crystal faces causes the crystal to oscillate. When the thickness of the crystal (t_{q}) is twice the acoustical wavelength, a standing wave can be established where the inverse of the frequency of the applied potential is of the period of the standing wave. This frequency is called the resonant frequency, f_{0}, and is given by the equation
(1)
where μ_{q} is the shear modulus (a ratio of sheer stress to shear strain), ρ_{q} is the density, and t_{q} is the crystal thickness. The amount of energy lost during oscillation at this frequency is at a minimum. The ratio of peak energy stored to energy lost per cycle is referred to as the quality factor, Q, and is given by the equation
(2)
where f_{c} is the center frequency and f_{FWHM} is the full width at half max. This full width at half max is also called the bandwidth. For quartz crystals in air, Q can exceed 100,000 while in solution Q decreases to ~3000. This is because the crystal has been damped by the solution. This damping increases the amount of energy lost per cycle, decreasing Q as shown in Figure 2 below.
Figure 2. Comparison of High Q (solid line) and Low Q (dashed line).
Equivalent Circuit Model
The mechanical model (Figure 3) of an electroacoustical system consists of a mass (M), a compliance (C_{m}), and a resistance (r_{f}). The compliance represents energy stored during oscillation and the resistance represents energy dissipation during oscillation.
Figure 3. Quartz Crystal Microbalance Equivalent Mechanical Model.
The QCM mechanical model can be electrically modeled in several different ways. The easiest model to understand might be an RLC circuit as shown in Figure 4.
Figure 4. RLC Circuit.
Here R_{1} represents the energy dissipated during oscillation, C_{1} represents the energy stored during oscillation, and L_{1} represents the inertial component related to the displaced mass. At the resonant frequency, f_{s}, the impedance of the circuit is at a minimum and is equal in magnitude to R_{1} as shown in Figure 5.
Figure 5. Impedance Spectrum for a Series RLC Circuit.
Making electrical contact to a quartz crystal is most easily done be addition of an electrode to each face of the crystal. These electrodes introduce an additional capacitance (C0) in parallel with the series RLC as shown in Figure 6. This circuit is commonly referred to as the Butterworth van Dyke (BvD) model.
Figure 6. Butterworth van Dyke Equivalent Circuit Model.
The circuit shown above now has two resonant frequencies, f_{s} and f_{p}, which stand for the series resonant frequency (as in the original RLC circuit) and the parallel resonant frequency, respectively. The impedance spectrum for the BvD model is shown below with a minimum at f_{s} and a maximum at f_{p}.
Figure 7. Impedance Spectrum for the Butterworth van Dyke Equivalent Circuit Model.
Most commercial quartz crystal microbalances that rely upon a phase lock oscillator, manually cancel out C_{0}, and only report the series resonant frequency, f_{s} since f_{s} ~= f_{0 }and f_{s} is obviously dependent upon L_{1}. The eQCM 10M reports both frequencies, f_{s} and f_{p}, and a relative impedance spectrum.
Since these two resonant frequencies are dependent upon L_{1}, mass changes on the surface of the electrode will result in frequency changes. When a deposited film is thin and rigid, the decrease in frequency can be directly correlated to the increase in mass using the Sauerbrey^{4} equation.
(3)
where f_{0} is the fundamental frequency of the crystal as defined in Equation (1), m is the mass added, n is the harmonic number (e.g. n = 1 for a 5 MHz crystal driven at 5 MHz), and μ_{q} and ρ_{q} are as defined above. Equation (3) can be reduced to
(4)
where C_{f} is the calibration constant. The calibration constant for a 5 MHz Atcut quartz crystal in air is 56.6 Hz cm^{2} g^{1}.
The majority of electrochemical experiments will correspond to a lowloading case, allowing you to directly correlate changes in frequency with changes in mass using the Sauerbrey equation (4).
Once a film has been deposited and the quartz crystal is immersed in a liquid, the BvD model can be modified as shown below to include coupling to the liquid.
Figure 8. Equivalent Circuit Model for a Quartz Crystal Immersed in a Liquid
Three new components account for the mass loading of the film, L_{f}, and the liquid loading (based on L_{L} and R_{L}). Both new mass loadings L_{f} and L_{l} have the effect of reducing the frequency as indicated with the black arrow in the chart below. The original BvD model (black curve) shows resonance approximately 300 kHz higher than the modified BvD model (red curve). Notice that the shapes of the resonance curves are identical despite the addition of the film and liquid loading.
Figure 9. Comparison of BvD and Modified BvD Models.
Knowing the liquids viscosity and density allows one to calculate^{3}the expected frequency decrease upon immersing the crystal in that liquid using the equation
(5)
where f_{a} is the frequency of the crystal in air, _{L} is the viscosity of the liquid, _{L} is the density of the liquid, _{q} is the shear modulus of the quartz crystal, and _{q} is the density of the quartz crystal.
For example, upon immersing the crystal in pure water, a frequency decrease of approximately 800 Hz is expected.
Equation (4) holds as long as the film is assumed to be thin and rigid. When the film is no longer acoustically thin or is not rigid the BvD model can be modified further as shown below.
Figure 10. One Possible Equivalent Circuit Model for a Quartz Crystal Coated with a Polymer Film and Immersed in a Liquid
Two new features have been added Film Loading (ηfρf) and Elasticity (μ_{f}). A viscoelastic polymer will influence the resonant frequencies based on the films viscosity (ηf), density (ρf) and elasticity (μ_{f}). If the polymer is rigid or ηfρf does not change during the experiment, there will be no change in peak shape and contributions from ηfρf can be ignored. In cases where ηfρf does change during the experiment, the frequency, magnitude and shape of the peak will also change as shown in Figure 11.
Figure 11. Comparison of BvD and Viscous Polymer Modified BvD Models.
Another way to visualize if ηfρf is changing during an experiment is to look at the reduced quality factor, Q_{R} as a function of time.
(6)
Changes in Q_{R} correspond to changing ηfρf. Quantifying changes in ηfρf is challenging mathematically and will not be explored further here. The reader is directed to the literature^{5} for additional information at this time.
References

Hillman, A. R. The EQCM: electrogravimetry with a light touch, J. Solid State Electrochem., 15, 2011, 16471660.

Buttry, D. A. The Quartz Cystal Microbalance as an In Situ Tool for Electrochemistry, in Electrochemical Interfaces. Modern Techniques for InSitu Interface Characterization, H. D. Abruña, Ed., VCH Publishers, Inc., New York, 1991, 53166. Hillman, A. R. The Electrochemical Quartz Crystal Microbalance. In Instrumentation and Electroanalytical Chemistry; Bard, A. J., Stratmann, M., Unwin, P. R., Eds.; Encyclopedia of Electrochemistry; Wiley: New York, 2003; Vol. 3, 230289.

Glassford, A. P. M. J. Vac. Sci. Technol. 1978, 15, 1836. Kanazawa, K. K.; Gordon II, J. Anal. Chem. 1985, 57, 1770. Kanazawa, K. K.; Gordon II, J. Analytica Chimica Acta 1985, 175, 99105.

Muramatsu, H.; Tamiya, E.; Karube, I. Anal. Chem. 1988, 60, 21422146. Hillman, A. R.; Jackson, A.; Martin, S. J. Anal. Chem. 2001, 73, 540549. Martin; Hillman; Etchnique
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