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A '''quartz crystal microbalance''' (QCM) measures a mass per unit area by measuring the change in [[frequency]] of a [[quartz crystal]] resonator.  The [[resonance]] is disturbed by the addition or removal of a small mass due to oxide growth/decay or film deposition at the surface of the acoustic resonator. The QCM can be used under vacuum, in gas [[phase (matter)|phase]] ("gas sensor", first use described by King<ref name=king64>{{cite journal | author= W.H. King, Jr.| title = Piezoelectric sorption detector | journal = Analytical Chemistry|issue=9|pages=1735––1739|year= 1964|volume= 36 | doi= 10.1021/ac60215a012}}</ref>) and more recently in liquid environments. It is useful for monitoring the rate of deposition in [[Thin-film deposition|thin film deposition]] systems under vacuum. In liquid, it is highly effective at determining the [[chemical affinity|affinity]] of molecules ([[protein]]s, in particular) to surfaces functionalized with recognition sites.  Larger entities such as [[virus]]es or [[polymer]]s are investigated, as well.  QCM has also been used to investigate interactions between biomolecules. Frequency measurements are easily made to high precision (discussed below); hence, it is easy to measure mass densities down to a level of below 1 μg/cm<sup>2</sup>. In addition to measuring the frequency, the [[dissipation]] is often measured to help analysis. The dissipation is a parameter quantifying the [[damping]] in the system, and is related to the sample's [[Viscoelasticity|viscoelastic]] properties.  The dissipation is equal to the ratio of bandwidth, w, (see below) and frequency, f.
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[[Quartz]] is one member of a family of [[crystals]] that experience the [[Piezoelectricity|piezoelectric effect]]. The piezoelectric effect has found applications in high power sources, sensors, actuators, frequency standards, motors, etc., and the relationship between applied [[voltage]] and mechanical deformation is well known; this allows probing an acoustic resonance by electrical means. Applying alternating current to the quartz crystal will induce oscillations. With an alternating current between the electrodes of a properly cut crystal, a standing [[shear wave]] is generated.  The [[Q factor]], which is the ratio of frequency and [[Bandwidth (signal processing)|bandwidth]], can be as high as 10<sup>6</sup>. Such a narrow resonance leads to highly stable oscillators and a high accuracy in the determination of the resonance frequency. The QCM exploits this ease and precision for sensing. Common equipment allows resolution down to 1&nbsp;Hz on crystals with a fundamental [[resonant frequency]] in the 4 – 6&nbsp;MHz range. A typical setup for the QCM contains water cooling tubes, the retaining unit, frequency sensing equipment through a microdot feed-through, an oscillation source, and a measurement and recording device.
 
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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 equation|Sauerbrey's equation]].<ref name=sauerbrey>{{cite journal | author = G. Sauerbrey | title = Verwendung von Schwingquarzen zur Wägung dünner Schichten und zur Mikrowägung| journal = Z. Phys.|pages= 206|year=1959 | volume=155 | doi =  10.1007/BF01337937|bibcode = 1959ZPhy..155..206S | issue =  2 }}</ref> 
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Other techniques for measuring the properties of thin films include [[Ellipsometry]], [[Surface Plasmon Resonance]] (SPR) [[Spectroscopy]], and [[Dual Polarisation Interferometry]].


== Gravimetric and non-gravimetric QCM ==
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The classical sensing application of quartz crystal resonators is microgravimetry.<ref name=lu>{{cite book | editor =  C. Lu, A. W. Czanderna | title = Applications of Piezoelectric Quartz Crystal Microbalances| publisher = Elsevier |place= Amsterdam|year= 1984}}</ref><ref name=arnau>{{cite book | editor =  A. Arnau| title= Piezoelectric Transducers and Applications| publisher = Springer|place= Heidelberg|year= 2004}}</ref><ref name=grate>{{cite journal | author=  J.W. Grate|title=Acoustic Wave Microsensor Arrays for Vapor Sensing| journal = Chem. Rev|pages= 2627–48|year=2000|volume= 100 | doi=  10.1021/cr980094j | pmid=  11749298 | issue=  7}}</ref><ref name=steinem>{{cite book| title =  Piezoelectric Sensors | editor= A.J.C. Steinem and A. Janshoff | publisher= Springer|place= Heidelberg|year= 2006}}</ref>  Many commercial instruments, some of which are called [[Thin-film thickness monitor|thickness monitors]], are available. These devices exploit the Sauerbrey relation.<ref name=sauerbrey /> For thin films, the resonance frequency is usually 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. However, when the film thickness increases, viscoelastic effects come into play.<ref name=lu1972>{{cite journal | author =  C.S. Lu and O. Lewis| title=Investigation of film-thickness determination by oscillating quartz resonators with large mass load| journal = J. Appl. Phys.| pages= 4385|year= 1972 | volume= 43 | doi =  10.1063/1.1660931|bibcode = 1972JAP....43.4385L | issue = 11 }}</ref> In the late 80’s, it was recognized that the QCM can also be operated in liquids, if proper measures are taken to overcome the consequences of the large damping.<ref name=bruckenstein>{{cite journal | author=  S. Bruckenstein and M. Shay|title=Experimental aspects of use of the quartz crystal microbalance in solution| journal = Electrochim. Acta|pages= 1295|year=1985 | volume = 30 | doi=  10.1016/0013-4686(85)85005-2 | issue= 10}}</ref><ref name=ward>{{cite journal | author=  M.D. Ward and D.A. Buttry| title=In Situ Interfacial Mass Detection with Piezoelectric Transducers|journal = Science|pages= 1000–7|year=1990|volume= 249 | doi=  10.1126/science.249.4972.1000 | pmid= 17789608 | issue= 4972|bibcode = 1990Sci...249.1000W }}</ref> Again, viscoelastic effects contribute strongly to the resonance properties.
 
 
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Today, microweighing is one of several uses of the QCM.<ref name=PCCP>{{cite journal | author=  D. Johannsmann|title=Viscoelastic, mechanical, and dielectric measurements on complex samples with the quartz crystal microbalance| journal = Physical Chemistry Chemical Physics|pages= 4516–34|year=2008 | volume = 10 | doi=  10.1039/b803960g | pmid= 18665301 | issue= 31|bibcode = 2008PCCP...10.4516J }}</ref>
 
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.
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Crystalline α–quartz is by far the most important material for thickness-shear resonators. [[Lanthanum gallium silicate|Langasite]] (La<sub>3</sub>Ga<sub>5</sub>SiO<sub>14</sub>, “LGS”) and [[Gallium phosphate|
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gallium-orthophosphate]] (GaPO<sub>4</sub>) are investigated as alternatives to quartz, mainly (but not only) for use at high temperatures.<ref name=fritze>{{cite journal | author=  H. Fritze and H.L. Tuller| title=Langasite for high-temperature bulk acoustic wave applications|journal = Appl. Phys. Lett.|pages= 976|year=2001|volume= 78 | doi=  10.1063/1.1345797|bibcode = 2001ApPhL..78..976F | issue=  7 }}</ref><ref name=elam>{{cite journal | author=  J.W. Elam and M.J. Pellin| title=GaPO4 Sensors for Gravimetric Monitoring during Atomic Layer Deposition at High Temperatures|journal = Anal. Chem.|pages= 3531–5|year=2005|volume= 77 | doi=  10.1021/ac050349a | pmid=  15924385 | issue=  11}}</ref>  Such devices are also called “QCM”, even though they are not made out of quartz (and may or may not be used for gravimetry).


== Surface acoustic wave-based sensors ==
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The QCM is a member of a wider class of sensing instruments based on acoustic waves at surfaces.  Instruments sharing similar principles of operation are shear horizontal [[surface acoustic wave]] (SH-SAW) devices,<ref name=martin>{{cite journal | author=  F. Martin, M.I. Newton, G. McHale, Ka.A. Melzak and E. Gizeli|title=Pulse mode shear horizontal-surface acoustic wave (SH-SAW) system for liquid based sensing applications| journal = Biosensors and Bioelectr.|pages= 627|year=2004|volume= 19 | doi=  10.1016/S0956-5663(03)00257-4 | issue=  6}}</ref><ref name=gulyaev>{{cite journal | author=  Y.V. Gulyaev|title=Review of shear surface acoustic waves in solids|journal= IEEE Trans. Ultrason. Ferroelectr. Freq. Control|pages= 935–8|year=1998|volume= 45 | doi=  10.1109/58.710563 | pmid=  18244248 | issue=  4}}</ref> [[love wave|Love-wave]] devices<ref name=gizeli>{{cite journal | author=  E. Gizeli, N. J. Goddard, C. R. Lowe, and A. C. Stevenson|title=A Love plate biosensor utilising a polymer layer| journal = Sensors and Actuators B: Chemical|pages= 131|year=1992|volume= 6 | doi=  10.1016/0925-4005(92)80044-X}}</ref> and [[torsion (mechanics)|torsional]] resonators.<ref name=mcskimin>{{cite journal | author=  H.J. McSkimin|title=Measurement of Dynamic Shear Viscosity and Stiffness of Viscous Liquids by Means of Traveling Torsional Waves| journal = Journal of the Acoustical Society of America|pages= 355|year=1952|volume= 24 | doi=  10.1121/1.1906904|bibcode = 1952ASAJ...24..355M | issue=  4 }}</ref><ref name=stokich>{{cite journal | author=  T. M. Stokich, D. R. Radtke, C. C. White, and J. L. Schrag| title=An instrument for precise measurement of viscoelastic properties of low viscosity dilute macromolecular solutions at frequencies from 20 to 500 kHz|journal = Journal of Rheology|pages= 1195|year=1994|volume= 38 | doi=  10.1122/1.550608|bibcode = 1994JRheo..38.1195S | issue=  4 }}</ref> Surface acoustic wave-based devices make use of the fact that the reflectivity of an acoustic wave at the crystal surface depends on the [[mechanical impedance|impedance]] (the stress-to-speed ratio) of the adjacent medium.  (Some acoustic sensors for temperature or pressure make use of the fact that the speed of sound inside the crystal depends on temperature, pressure, or bending.  These sensors do not exploit surface effects.)  In the context of surface-acoustic wave based sensing, the QCM is also termed “bulk acoustic wave resonator (BAW-resonator)” or “thickness-shear resonator”.  The displacement pattern of an unloaded BAW resonator is a standing shear wave with [[node (physics)|anti-nodes]] at the crystal surface.  This makes the analysis particularly easy and transparent.


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=== Resonator crystals ===
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[[Image:quartz resonators with front and back electrodes.jpg|thumb|400px|Photograph of typical quartz crystal resonators as used for QCM, metallised with gold electrodes (left: front electrode, right: back electrode) by vapor deposition.]]


When the QCM was first developed, natural quartz was harvested, selected for its quality and then cut in the [[laboratory|lab]]. However, most of today’s crystals are grown using [[seed crystal]]s. 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&nbsp;MHz range.   The "AT" or "SC" oriented cuts (discussed below) are widely used in applications.<ref name=techcrystal>{{cite web
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=== Electromechanical coupling ===
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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,<ref name=zhang>{{cite journal | author=  C. Zhang and J.F. Vetelino|title=Chemical sensors based on electrically sensitive quartz resonators| journal = Sens. Actuators B|pages= 320|year=2003|volume= 91 | doi=  10.1016/S0925-4005(03)00094-7}}</ref> 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.<ref name=iec>IEC standard 60444-1</ref> A π-network is an arrangement of [[resistor]]s, which almost [[short circuit|short-circuit]] the two electrodes.  This makes the device less susceptible to electrical perturbations.


=== Shear waves decay in liquids and gases ===
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Most acoustic-wave-based sensors employ shear (transverse) waves. Shear waves decay rapidly in liquid and gaseous environments. [[longitudinal wave|Compressional (longitudinal) waves]] would be radiated into the bulk and potentially be reflected back to the crystal from the opposing cell wall.<ref name=lin>{{cite journal | author= Z.X. Lin and M.D. Ward|title=The Role of Longitudinal Waves in Quartz Crystal Microbalance Applications in Liquids| journal = Anal. Chem.|pages= 685|year=1995|volume= 67 | doi=  10.1021/ac00100a001 | issue=  4}}</ref><ref name=eggers>{{cite journal | author=  F. Eggers and Th. Funck| title=Method for measurement of shear-wave impedance in the MHz region for liquid samples of approximately 1 ml|journal = J. Phys. E: Sci. Instrumen.|pages= 523|year=1987|volume= 20 | doi= 10.1088/0022-3735/20/5/011|bibcode = 1987JPhE...20..523E | issue=  5 }}</ref> Such reflections are avoided with transverse waves. The range of penetration of a 5&nbsp;MHz-shear wave in water is 250&nbsp;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.
 
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=== Modes of operation ===
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Economic ways of driving a QCM make use of oscillator circuits.<ref name=horowitz>{{cite book | author= P. Horowitz and W. Hill|title= The Art of Electronics|edition = 2|publisher= Cambridge University Press|place= New York|year= 1989}}</ref><ref name=arnau2002>{{cite journal | author=  A. Arnau, T. Sogorb and Y. Jimenez|title=Circuit for continuous motional series resonant frequency and motional resistance monitoring of quartz crystal resonators by parallel capacitance compensation| journal = Rev. Sci. Instr.|pages= 2724|year= 2002|volume= 73 | doi=  10.1063/1.1484254|bibcode = 2002RScI...73.2724A | issue=  7 }}</ref>  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<ref name=beck>{{cite journal | author=  R. Beck, U. Pittermann and K.G. Weil|title=| journal = Ber. Bunsen-Ges. Phys. Chem.|pages= 1363|year=1988|volume= 92}}</ref> and ring-down.<ref name=rodahl>{{cite journal | author=  M. Rodahl M and B. Kasemo|title=A simple setup to simultaneously measure the resonant frequency and the absolute dissipation factor of a quartz crystal microbalance| journal = Rev. Sci. Instr.|pages= 3238|year=1996|volume= 67 | doi=  10.1063/1.1147494|bibcode = 1996RScI...67.3238R | issue=  9 }}</ref><ref name=sittel>{{cite journal | author=  K. Sittel, P. E. Rouse, and E. D. Bailey|title=Method for Determining the Viscoelastic Properties of Dilute Polymer Solutions at Audio-Frequencies| journal = Journal of Applied Physics|pages= 1312|year=1954|volume=25 | doi=  10.1063/1.1721552|bibcode = 1954JAP....25.1312S | issue= 10 }}</ref>  In impedance analysis, the [[electrical conductance|electric conductance]] as a function of driving frequency is determined by means of a [[network analyzer (electrical)|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.


[[Image:QCM principle.gif|framed|Impedance analysis is based on electrical conductance curve. The central parameters of measurement are the resonance frequency f<sub>res</sub> and the bandwidth w.]]
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[[Image:QCM ringdown.gif|framed|Ring-down yields the equivalent information in time-domain measurements. The dissipation ''D'' is equal to ''Q''<sup>-1</sup>.]]
 
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=== Energy trapping ===
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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.  This confines the displacement field to the center of the crystal by a mechanism called energy trapping.<ref name=bottom>{{cite journal | author=  V.E. Bottom|title=Introduction to Quartz Crystal Unit Design|publisher= Van Nostrand Reinhold|place= New York|year= 1982}}</ref>  The crystal turns into an acoustic lens and the wave is focused to the center of the crystal.  Energy trapping is necessary in order to be able to mount the crystal at the edge without excessive damping.  Energy trapping slightly distorts the otherwise planar wave fronts.  The deviation from the plane thickness-shear mode entails flexural contribution to the displacement pattern.  Flexural waves emit compressional waves into the adjacent medium, which is a problem when operating the crystal in a liquid environment.


=== Overtones ===
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Planar resonators can be operated at a number of [[overtone]]s, typically indexed by the number of nodal planes parallel to the crystal surfaces. Only odd [[harmonic]]s 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.<ref name=goka>{{cite journal | author= S. Goka, K. Okabe, Y. Watanabe and H. Sekimoto|title=Multimode Quartz Crystal Microbalance| journal = Jpn. J. Appl. Phys. PART 1|pages= 3073|year=2000|volume= 39 | doi= 10.1143/JJAP.39.3073|bibcode = 2000JaJAP..39.3073G }}</ref>
 
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=== Amplitude of motion ===
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The [[amplitude]] of lateral displacement rarely exceeds a nanometer.  More specifically one has


<math>u_0=\frac 4{\left( n\pi \right) ^2}dQU_{\mathrm{el}}
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with ''u''<sub>0</sub> the amplitude of lateral displacement, ''n'' the overtone order, ''d'' the piezoelectric strain coefficient, ''Q'' the quality factor, and ''U''<sub>el</sub> the amplitude of electrical driving.  The piezoelectric strain coefficient is given as ''d''&nbsp;=&nbsp;3.1·10<sup>‑12</sup>&nbsp;m/V for AT-cut quartz crystals.  Due to the small amplitude, [[Stress (physics)|stress]] and [[Strain (materials science)|strain]] usually are proportional to each other.  The QCM operates in the range of linear acoustics.
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=== Effects of temperature and stress ===
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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.
 
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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)<ref name=epsontoyocom>{{cite web | author=|title=Who needs crystal devices |url= http://www.epsontoyocom.co.jp/english/special/crystal/timing_04.html|accessdate=2007-05-30|date=2007-03-22 |archiveurl = http://web.archive.org/web/20070718103339/http://www.epsontoyocom.co.jp/english/special/crystal/timing_04.html <!-- Bot retrieved archive --> |archivedate = 2007-07-18}}</ref><ref name=icmfg>{{cite web | author=|title=Quartz crystal FAQs from International Crystal|url= http://www.icmfg.com/crystalfaqs.html#q4|accessdate=2007-05-30|year=2007}}</ref> during oscillation. 
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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 function|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.<ref name=ballato>{{cite journal | author=  A. Ballato, T. J. Lukaszek, and E. P. Eernisse|title=| journal = Ieee Transactions on Sonics and Ultrasonics|pages= 163|year=1979|volume= 26}}</ref> 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.
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=== Electrochemical QCM ===
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The QCM can be combined with other surface-analytical instruments.  The [[electrochemistry|electrochemical]] QCM (EQCM) is particularly advanced.<ref name=schumacher>{{cite journal | author=  R. Schumacher|title=The Quartz Microbalance: A Novel Approach to the In-Situ Investigation of Interfacial Phenomena at the Solid/Liquid Junction[New Analytical Methods(40)]|journal= Angew. Chem. Int. Ed. Engl.|pages= 329|year=1990|volume= 29 | doi=  10.1002/anie.199003293 | issue=  4}}</ref><ref name=bruckenstein1985b>{{cite journal | author=  S. Bruckenstein and M. Shay|title=An in situ weighing study of the mechanism for the formation of the adsorbed oxygen monolayer at a gold electrode| journal = J. Electroanal. Chem. Interfacial Electrochem.|pages= 131|year=1985|volume= 188 | doi=  10.1016/S0022-0728(85)80057-7}}</ref><ref name=buttry>{{cite journal | author=  D.A. Buttry and M.D. Ward|title=Measurement of interfacial processes at electrode surfaces with the electrochemical quartz crystal microbalance| journal = Chem.  Rev.|pages= 1335|year=1992|volume= 92 | doi=  10.1021/cr00014a006 | issue=  6}}</ref>  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.


== Quantification of dissipative processes ==
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For advanced QCMs, both the resonance frequency, ''f''<sub>r</sub>, and the bandwidth, ''w'', are available for analysis. The latter quantifies processes which withdraw energy from the oscillation. These may include damping by the holder and [[ohm's law|ohmic]] losses inside the electrode or the crystal. In the literature some parameters other than ''w'' itself are used to quantify bandwidth. The Q-factor (quality factor) is given by ''Q''&nbsp;=&nbsp;''f''<sub>r</sub>/''w''. The “dissipation”, ''D'', is the inverse of the Q-factor: ''D''&nbsp;=&nbsp;''Q''<sup>−1</sup>&nbsp;=&nbsp;''w''/''f''<sub>r</sub>. The half-band-half-width, Γ, is Γ&nbsp;=&nbsp;''w''/2. The use of Γ is motivated by a complex formulation of the equations governing the motion of the crystal. A [[complex number|complex]] resonance frequency is defined as ''f''<sub>r</sub><sup>*</sup>&nbsp;=&nbsp;''f''<sub>r</sub>&nbsp;+&nbsp;iΓ, where the [[imaginary part]], Γ, is half the bandwidth at half maximum.  Using a complex notation, one can treat shifts of frequency, Δ''f'', and bandwidth, ΔΓ, within the same set of (complex) equations.
 
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The motional resistance of the resonator, ''R''<sub>1</sub>, is also used as a measure of dissipation.  ''R''<sub>1</sub> is an output parameter of some instruments based on advanced oscillator circuits.  ''R''<sub>1</sub> usually is not strictly proportional to the bandwidth (although it should be according to the BvD circuit; see below).  Also, in absolute terms, ''R''<sub>1</sub> – being an electrical quantity and not a frequency – is more severely affected by [[calibration]] problems than the bandwidth.
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Modeling of acoustic resonators often occurs with equivalent [[electrical circuits]].<ref name=thurston>{{cite book|author= R.N. Thurston|editor= C. Truesdell |title=Mechanics of Solids| chapter= 36|publisher=Springer|place=Heidelberg|year=1984|volume = 4|page=257}}</ref> Equivalent circuits are [[algebra]]ically equivalent to the [[continuum mechanics]] description<ref name=reed>{{cite journal| author=C.E. Reed, K.K. Kanazawa and J.H. Kaufmann|title=Physical description of a viscoelastically loaded AT-cut quartz resonator|journal= J. Appl. Phys.|pages= 1993|year=1990|volume= 68| doi=10.1063/1.346548|bibcode = 1990JAP....68.1993R| issue=5 }}</ref> and to a description in terms of acoustic reflectivities.<ref name=johannsmann>{{cite journal|author=D. Johannsmann, K. Mathauer, G. Wegner, and W. Knoll|title=Viscoelastic properties of thin films probed with a quartz-crystal resonator|journal= Phys. Rev. B|pages= 7808|year=1992|volume= 46|doi=10.1103/PhysRevB.46.7808|bibcode = 1992PhRvB..46.7808J|issue=12 }}</ref>  They provide for a graphical representation of the resonator’s properties and their shifts upon loading.  These representations are not just cartoons.  They are tools to predict the shift of the resonance parameters in response to the addition of the load.
 
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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 ''Z''<sub>ac</sub> = ρ''c'' with ρ the [[density]] and ''c'' the speed of sound).  The ratio of stress and speed at the crystal surface is called load impedance, ''Z''<sub>L</sub>.  Synonymous terms are "surface impedance" and "acoustic load."<ref name=arnau2002 /> The load impedance is in general not equal to the material constant ''Z''<sub>ac</sub> = ρ''c'' = (''G''ρ)<sup>1/2</sup>. Only for propagating plane waves are the values of ''Z''<sub>L</sub> and ''Z''<sub>ac</sub> the same.
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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]], ξ<sub>p</sub>), the point mass (quantified by the mass, ''m''<sub>p</sub>), and the [[spring (device)|spring]] (quantified by the [[hooke's law|spring constant]], κ<sub>p</sub>). For a dashpot, the impedance by definition is ''Z''<sub>m</sub>=''F'' / (d''u''/d''t'')=ξ<sub>m</sub> with ''F'' the force and (d''u''/d''t'') the speed). For a point mass undergoing oscillatory motion ''u''(''t'') = ''u''<sub>0</sub> exp(iω''t'') we have ''Z''<sub>m</sub> = iω''m''<sub>p</sub>. The spring obeys ''Z''<sub>m</sub> =κ<sub>p</sub>/(iω). Piezoelectric coupling is depicted as a [[transformer]]. It is characterized by a parameter φ.  While φ is dimensionless for usual transformers (the turns ratio), it has the dimension charge/length in the case of electromechanical coupling. The transformer acts as an impedance converter in the sense that a mechanical impedance, ''Z''<sub>m</sub>, appears as an electrical impedance, ''Z''<sub>el</sub>, across the electrical ports. '' Z''<sub>el</sub> is given by ''Z''<sub>el</sub> = φ<sup>2</sup> ''Z''<sub>m</sub>. For planar piezoelectric crystals, φ takes the value φ = ''Ae''/''d''<sub>q</sub>, where ''A'' is the effective area, ''e'' is the piezoelectric stress coefficient<ref name=beck /> (''e'' = 9.65·10<sup>−2</sup> C/m<sup>2</sup> for AT-cut quartz) and ''d''<sub>q</sub> 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.).
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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.
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[[File:QCM BvD equivalent circuit.gif|framed|right|Butterworth-van-Dyke (BvD) equivalent circuit. ''C''<sub>0</sub> is the electrical (parallel) capacitance across the electrodes. ''L''<sub>1</sub> is the motional inductance (proportional to the mass). ''C''<sub>1</sub> is the motional capacitance (inversely proportional to the stiffness) and ''R''<sub>1</sub> is the motional resistance (quantifying dissipative losses). ''A'' is the effective area of the crystal, ''Z''<sub>L</sub> is the load impedance, and φ converts between electrical and mechanical impedance.]]
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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, ''L''<sub>1</sub>, the motional capacitance, ''C''<sub>1</sub>, and the motional resistance ''R''<sub>1</sub>.  ''Z''<sub>L</sub> is the load impedance.  Note that the load, ''Z''<sub>L</sub>, 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 ''Z''<sub>L</sub>.  This circuit is also called “four element network”.  The values of ''L''<sub>1</sub>, ''C''<sub>1</sub>, and ''R''<sub>1</sub> then change their value in the presence of the load (they do not if the element ''Z''<sub>L</sub> is explicitly included).
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== Small-load approximation ==
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The BvD circuit predicts the resonance parameters.  One can show that the following simple relation holds as long as the frequency shift is much smaller than the frequency itself:
 
 
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<math>\frac{\Delta f^{*}}{f_f}=\frac i{\pi Z_q}Z_L</math>
 
 
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''f''<sub>f</sub> is the frequency of the [[fundamental frequency|fundamental]]. ''Z''<sub>q</sub> is the acoustic impedance of material. For AT-cut quartz, its value is ''Z''<sub>q</sub> = 8.8·10<sup>6</sup> kg m<sup>−2</sup> s<sup>−1</sup>.
 
 
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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.<ref name=heteorogeneous_note>Heterogeneous samples will, in general, lead to scattering of acoustic waves, which is not captured by just calculating the average stress.</ref><ref name=laschitsch>{{cite journal|author=A. Laschitsch and D. Johannsmann|title=High frequency tribological investigations on quartz resonator surfaces|journal= J. Appl. Phys.|pages= 3759|year=1999|volume= 85|doi=10.1063/1.369745|bibcode = 1999JAP....85.3759L|issue=7 }}</ref> Assume that the sample is a complex material, such as a [[cell culture]], a sand pile, a froth, an assembly of spheres or [[vesicle (biology)|vesicles]], or a droplet.  If the average stress-to-speed ratio of the sample at the crystal surface (the load impedance, ''Z''<sub>L</sub>) can be calculated in one way or another,<ref name=johannsmann2008>{{cite journal|author=D. Johannsmann, I. Reviakine, E. Rojas, and M. Gallego |title=Effect of sample heterogeneity on the interpretation of QCM data: comparison of combined quartz crystal microbalance/atomic force microscopy measurements with finite element method modeling|journal=Analytical Chemistry|pages= 8891|year=2008|volume= 80|doi=10.1021/ac8012829|pmid=18954080|issue=23}}</ref> a quantitative analysis of the QCM experiment is in reach. Otherwise, the interpretation will have to remain qualitative.
 
 
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The limits of the small-load approximation are noticed either when the frequency shift is large or when the overtone-dependence of Δ''f'' and Δ(''w''/2) is analyzed in detail in order to derive the viscoelastic properties of the sample.  A more general relation is
 
<math>Z_L=-iZ_q\tan \left( \pi \frac{\Delta f}{f_f}\right)</math>
 
This equation is [[implicit function|implicit]] in Δ''f''<sup>*</sup>, 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 theory|perturbation analysis]].<ref name=johannsmann2001>{{cite journal|author=D. Johannsmann|title=Derivation of the shear compliance of thin films on quartz resonators from comparison of the frequency shifts on different harmonics: A perturbation analysis|journal=Journal of Applied Physics|pages= 6356|year=2001|volume= 89|doi=10.1063/1.1358317|bibcode = 2001JAP....89.6356J|issue=11 }}</ref>
 
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 system|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
 
<math>\frac{\Delta f}{f_f}=\frac 1{\pi Z_q}\,\frac 2{\omega u_0}\left\langle
\sigma \left( t\right) \cos \left( \omega t\right) \right\rangle _t</math>
 
<math>\frac{\Delta (w/2) }{f_f}=\frac 1{\pi Z_q}\,\frac 2{\omega u_0}\left\langle
\sigma \left( t\right) \sin \left( \omega t\right) \right\rangle _t</math>
 
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.
 
== Viscoelastic modeling ==
 
=== Assumptions ===
For a number of experimental configurations, there are explicit expressions relating the shifts of frequency and bandwidth to the sample properties.<ref name=nakamoto>{{cite journal|author=T. Nakamoto and T. Moriizumi|title=A Theory of a Quartz Crystal Microbalance Based upon a Mason Equivalent Circuit|journal= Jpn. J. Appl. Phys.|pages= 963|year=1990|volume= 29|doi=10.1143/JJAP.29.963|bibcode = 1990JaJAP..29..963N }}</ref><ref name=bandey>{{cite journal|author= H.L. Bandey, S.J. Martin. R.W. Cernosek and A.R. Hillman|title=Modeling the Responses of Thickness-Shear Mode Resonators under Various Loading Conditions|journal= Anal. Chem.|pages= 2205|year=1999|volume= 71|doi= 10.1021/ac981272b|issue= 11}}</ref><ref name=lucklum>{{cite journal|author= R. Lucklum, C. Behling and P. Hauptmann|title=Role of Mass Accumulation and Viscoelastic Film Properties for the Response of Acoustic-Wave-Based Chemical Sensors|journal= Anal. Chem.|pages= 2488|year=1999|volume=71|doi= 10.1021/ac981245l|issue= 13}}</ref><ref name=benes>{{cite journal|author= E. Benes |title=Improved quartz crystal microbalance technique|journal=J. Appl. Phys.|pages= 608|year=1984|volume= 56|doi= 10.1063/1.333990|bibcode = 1984JAP....56..608B|issue= 3 }}</ref>  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|wave-vector]] perpendicular to the surface normal (thickness-shear mode). There are neither compressional waves<ref name=iec /><ref name=lin /> nor flexural contributions to the displacement pattern.<ref name=friedt>{{cite journal|author=J.M. Friedt, K.H. Choi, L. Francis and A. Campitelli|title=Simultaneous Atomic Force Microscope and Quartz Crystal Microbalance Measurements: Interactions and Displacement Field of a Quartz Crystal Microbalance|journal= Jpn. J. Appl. Phys. PART 1|pages= 3974|year= 2002|volume= 41|doi=10.1143/JJAP.41.3974|bibcode = 2002JaJAP..41.3974F }}</ref> There are no nodal lines in the plane of the resonator.
* All stresses are proportional to strain.  Linear viscoelasticity holds.<ref name=borovikov>{{cite journal|author=V.V. Borovikov, R.A. Dialnyan and I.M. Shmyt’ko|title=|journal= Sov. Phys. Tech. Phys.|pages= 325|year=1987|volume= 32}}</ref>
* Piezoelectric stiffening may be ignored.
 
=== Semi-infinite viscoelastic medium ===
For a semi-infinite medium, one has<ref name=mason>{{cite book|author=W.P. Mason|title=Piezoelectric Crystals and their Applications to Ultrasonics|publisher= Van Nostrand|place= Princeton|year=1948}}</ref><ref name=kanazawa>{{cite journal|author= K.K. Kanazawa and J.G. Gordon II|title=|journal= Anal. Chim. Acta|pages= 175|year= 1985|volume= 99}}</ref><ref name=borovikov1976>{{cite journal|author=A.P. Borovikov|title=|journal= Instruments and Experimental Techniques|pages= 223|year=1976|volume= 19}}</ref>
 
<math>\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 }</math>
 
<math>=\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) }</math>
 
η’ and η’’ are the [[real part|real]] and the [[imaginary part]] of the viscosity, respectively. ''Z''<sub>ac</sub> = ρ''c'' =(''G'' ρ)<sup>1/2</sup> is the acoustic impedance of the medium. ρ is the density, ''c'', the speed of sound, and ''G'' = i&nbsp;ωη is the [[shear modulus]].
For [[newtonian fluid|Newtonian liquids]] (η’ = const, η’’ = 0), Δ''f'' and Δ(''w''/2) are equal and opposite. They scale as the square root of the overtone order, ''n''<sup>1/2</sup>. For viscoelastic liquids (η’ = η(ω), η’’≠ 0), the complex viscosity can be obtained as
 
<math>\eta ^{\prime }=-\frac{\pi Z_q^2}{\rho _{\mathrm{Liq}}\,f}\,\frac{\Delta
f\Delta \left( w/2\right) }{f_f^2}</math>
 
<math>\eta ^{\prime \prime }=\frac 12\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}</math>
 
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&nbsp;nm at 5&nbsp;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<ref name=stokich /> (with a frequency around 100&nbsp;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 = -ω<sup>2</sup>''u''<sub>0</sub>''m''<sub>F</sub>, where ''u''<sub>0</sub> is the amplitude of oscillation and ''m''<sub>F</sub> is the (average) mass per unit area. Inserting this result into the small-load-approximation one finds
 
<math>\frac{\Delta f^{*}}{f_f}\approx \frac i{\pi Z_q}\frac{-\omega ^2u_0m_{%
\mathrm{F}}}{i\omega u_0}=-\frac{2\,f}{Z_q}m_{\mathrm{F}}</math>
 
If the density of the film is known, one can convert from mass per unit area, ''m''<sub>F</sub>, to thickness, ''d''<sub>F</sub>. 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 (Δ''f'' / ''f'' &lt; 0.05).
 
If the change in frequency is greater than 5%, that is, Δ''f'' / ''f'' &gt; 0.05, the Z-match method must be used to determine the change in mass.<ref name=lu1972 /><ref name=borovikov1976 /> The formula for the Z-match method is:
 
<math>\tan \left( \frac{\pi \Delta f}{f_f}\right) =\frac{-Z_{\mathrm{F}}}{Z_q}\tan
\left( k_{\mathrm{F}}d_{\mathrm{F}}\right)</math>
 
''k''<sub>F</sub> is the wave vector inside the film and ''d''<sub>F</sub> its thickness. Inserting ''k''<sub>F</sub> = 2·π·''f'' /c<sub>F</sub> = 2·π·''f''·ρ<sub>F</sub> / ''Z''<sub>F</sub> as well as ''d''<sub>F</sub> = ''m''<sub>F</sub> / ρ<sub>F</sub> yields
 
<math>\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)</math>
 
=== Viscoelastic film ===
For a viscoelastic film, the frequency shift is
 
<math>\frac{\Delta f^{*}}{f_f}=\frac{-1}{\pi Z_q}Z_{\mathrm{F}}\tan \left( k_{\mathrm{F}}d_{\mathrm{F}}\right)</math>
 
Here ''Z''<sub>F</sub> is the acoustic impedance of the film (''Z''<sub>F</sub> = ρ<sub>F</sub>''c''<sub>F</sub> = (ρ<sub>F</sub>''G''<sub>f</sub>)<sup>1/2</sup>)= (ρ<sub>F</sub>/''J''<sub>f</sub>)<sup>1/2</sup>), ''k''<sub>F</sub> is the wave vector and ''d''<sub>F</sub> is the film thickness. ''J''<sub>f</sub> is the film's viscoelastic compliance, ρ<sub>F</sub> is the density.
 
The poles of the [[tangent (trigonometric function)|tangent]] (''k''<sub>F</sub> ''d''<sub>F</sub> = π/2) define the film resonances.<ref name=granstaff>{{cite journal|author=V.E. Granstaff and S.J. Martin|title=Characterization of a thickness-shear mode quartz resonator with multiple nonpiezoelectric layers|journal= J. Appl. Phys.|pages= 1319|year=1994|volume= 75|doi=10.1063/1.356410|bibcode = 1994JAP....75.1319G|issue=3 }}</ref><ref name=martin1991>{{cite journal|author=Martin, S.; Granstaff, V. and Frye, G.|title=Characterization of a quartz crystal microbalance with simultaneous mass and liquid loading|journal= Anal Chem.|year= 1991|pages=2272–2281|volume= 63|doi=10.1021/ac00020a015|issue=20}}</ref> At the film resonance, one has ''d''<sub>F</sub> = 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''<sub>F</sub> = ρ<sub>F</sub>''c''<sub>F</sub> and its mass per unit area, ''m''<sub>F</sub> = ''d''<sub>F</sub>/ρ<sub>F</sub>. The wave number ''k''<sub>F</sub> = ω/''c''<sub>F</sub> is not algebraically independent from ''Z''<sub>F</sub> and ''m''<sub>F</sub>. 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<ref name=domak>{{cite journal|author=A. Domack, O. Prucker, J. Rühe and D. Johannsmann|title=Swelling of a polymer brush probed with a quartz crystal resonator|journal= Phys. Rev. E|pages= 680|year=1997|volume= 56|doi=10.1103/PhysRevE.56.680|bibcode = 1997PhRvE..56..680D }}</ref><ref name=voinova>{{cite journal|author= M.V. Voinova, M. Rodahl, M. Jonson and B. Kasemo|title=Viscoelastic Acoustic Response of Layered Polymer Films at Fluid-Solid Interfaces: Continuum Mechanics Approach|journal= Physica Scripta|pages= 391|year=1999 |volume=59|doi= 10.1238/Physica.Regular.059a00391|arxiv = cond-mat/9805266 |bibcode = 1999PhyS...59..391V|issue= 5 }}</ref>
 
<math>\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) }</math>
 
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 series|Taylor-expand]] the above equation to first order in ''d''<sub>F</sub>, yielding
 
<math>\frac{\Delta f^{*}}{f_f}=\frac{-m_{\mathrm{F}}}{\pi Z_q}\left( 1-\frac{Z_{
\mathrm{Liq}}^2}{Z_{\mathrm{F}}^2}\right)=\frac{-m_{\mathrm{F}}}{\pi Z_q}\left( 1-J_{\mathrm{F}}\frac{Z_{\mathrm{Liq}}^2}{\rho_{\mathrm{F}}}\right)</math>
 
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''<sub>F</sub>’ to the ratio of Δ(w/2); and Δ''f''. The shear compliance is the inverse of the shear modulus, ''G''. In the thin-film limit, the ratio of Δ(w/2) and –Δ''f'' is independent of film thickness.  It is an intrinsic property of the film. One has<ref name=du>{{cite journal|author=B. Y. Du and D. Johannsmann|title=Operation of the Quartz Crystal Microbalance in Liquids: Derivation of the Elastic Compliance of a Film from the Ratio of Bandwidth Shift and Frequency Shift|journal= Langmuir|pages= 2809–12|year=2004|volume= 20|doi=10.1021/la035965l|pmid=15835157|issue=7}}</ref>
 
<math>\frac{\Delta \left( \omega /2\right) }{-\Delta f}\approx \eta \omega
J_F^{\,\prime }</math>
 
For thin films in air an analogous analytical result is<ref name=johannsmann1999>{{cite journal|author=D. Johannsmann|title=Viscoelastic analysis of organic thin films on quartz resonators|journal=Macromolecular Chemistry and Physics|pages= 501|year=1999|volume= 200|doi=10.1002/(SICI)1521-3935(19990301)200:3<501::AID-MACP501>3.0.CO;2-W|issue=3}}</ref>
 
<math>\Delta \left( \omega /2\right) =\frac 8{3\rho _{\mathrm{F}}Z_q}f_f^{\,4}m_{
\mathrm{F}}^3n^3\pi ^2J^{\prime \prime }</math>
 
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) « Δ''f'', Sauerbrey thickness same on all overtone orders) one may assume that (1-''Z''<sub>Liq</sub><sup>2</sup>/''Z''<sub>F</sub><sup>2</sup>)≈1.
  </ul>
 
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).
 
== Point contacts ==
The equations concerning viscoelastic properties assume planar layer systems.  A frequency shift is also induced when the crystal makes contact with discrete objects across small, load-bearing asperities.  Such contacts are often encountered with rough surfaces.  It is assumed that the stress–speed ratio may be replaced by an average stress–speed ratio, where the average stress just is the lateral force divided by the active area of the crystal.
 
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, ''N''<sub>S</sub>, and their average spring constant, κ<sub>S</sub>.  The spring constant may be complex (κ<sub>S</sub><sup>*</sup> = κ<sub>S</sub>’ + iκ<sub>S</sub>’’), 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
 
<math>\frac{\Delta f^{*}}{f_f}=\frac{N_S}{\pi Z_q}\frac{\kappa _S^{*}}\omega</math>
 
The QCM allows for non-destructive testing of the shear stiffness of multi-asperity contacts.
 
==See also==
{{colbegin|3}}
*[[Sauerbrey equation]]
*[[Weighing scale]]
*[[Piezoelectricity]]
*[[Thin-film thickness monitor]]
*[[QCM-D]]
{{colend}}
 
== References ==
{{reflist|2}}
 
==External links==
*[http://www.yb.gmm.gu.se/method.php?ID=3 Quartz Crystal Microbalance with Dissipation monitoring]
*[http://www.pharmaceutical-int.com/categories/qcm-technology/quartz-crystal-microbalance-qcm.asp What is QCM?]
*[http://www.inano.dk/research/competences-and-facilities/nanotools/quartz-crystal-microbalance/ Quartz crystal microbalance]
*[http://www.tectra-gmbh.com/qmb.htm Quartz crystal microbalance for vacuum applications (HV and UHV) to monitor thin film growth]
*[http://www.pc.tu-clausthal.de/en/forschung/ak-johannsmann/qcm-modellierung/ Tutorial on modelling the behavior of the QCM]
*[http://sites.google.com/site/christopher3reed/qcmfaq QCM mini-FAQ]
 
[[Category:Weighing instruments]]

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"My theory was if these individuals could work this out, it would be a big step in the right direction, and so i would think you're likely to hear a lot more on immigration reform around the House side soon," he explained.Democratic group members are Reps. Luis Gutierrez of Illinois and Zoe Lofgren and Xavier Becerra of California. The Republicans are DiazBalart and Sam Johnson and John Carter of Texas."I am optimistic there are new voices in the Republican Party that want to get this done in the House of Representatives," Gutierrez said.Other lawmakers are also in touch with group members, including Rep.

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