# The quartz crystal microbalance

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### Early History Of The Quartz Crystal Microbalance

Piezoelectricity is defined as electric polarization produced by mechanical strain in certain crystals, the polarization being proportional to the strain. The Curies first observed piezoelectricity in 1880 as a potential difference generated across two surfaces of a quartz crystal under strain. The converse piezoelectric effect, the deformation of a piezoelectric material by an applied electric field, was predicted by Lippman. Thus, when a thin wafer from a piezoelectric crystal such as quartz is placed in an alternating electric field of the right frequency it will oscillate in a mechanically resonant mode of the wafer. The resonance frequency depends upon the angles with respect to the optical axis at which the wafer was cut from a single crystal and inversely on the crystal thickness. Figure 1 shows typical cuts used to produce such wafers from a quartz single crystal. The angle most commonly chosen is referred to as the AT cut, 35º 15' from the Z or optic axis of the crystal. The AT-cut angles are chosen so that the temperature dependence of the resonant frequency is essentially zero at 25°C. These thin quartz plates with attached electrodes are called transverse shear mode (TSM) resonators.

The development and applications of quartz plate TSM resonators is a venerable field in electrical engineering. In the early 1920's the National Bureau of Standards (U.S.) began studies of quartz-crystal oscillators as frequency standards. To meet the growing demand for better accuracy, NBS sought outside partners, and began collaboration on oscillators with the Naval Research Laboratory and Bell Telephone Laboratories. In 1929 Bell Labs delivered four complete temperature-controlled 100 kHz oscillators to NBS, and these oscillators quickly became the national primary standard of radio frequency. By 1952 the facility involved a larger number of oscillators and the measurement uncertainty had been reduced to about 2 parts in 108.

An entertaining account of the history of the history of the quartz crystal industry in the USA indicates the critical role that quartz resonators played in the development of radio communications during World War II. Quartz resonators are presently found in many commercial products from quartz timepieces to ultra-stable frequency counters.

Quartz plate resonators have been used as sensitive microbalances for thin adherent films since the late 1950's, following the pioneering work of Sauerbrey, who coined the term quartz crystal microbalance. A widely used acronym is QCM.

### The Literature Of Thermal Analysis And Of The Quartz Crystal Microbalance

It is useful to compare the history of the development of thermal analysis with the history of the development of the quartz crystal microbalance. Scopus(www.Scopus.com)) is the largest online abstract and citation database of scientific and technical research literature and quality web sources. We have searched on the term “thermal analysis” and the term “QCM” or “quartz crystal microbalance” in Scopus, and have found all references dating Final Draft of “The Quartz Crystal Microbalance”, by Allan Smith, a chapter for Volume 5 of the Handbook of Thermal Analysis and Calorimetry, Elsevier to 1966. A Scopus literature search on the terms “QCM” or “quartz crystal microbalance” shows about 4500 citations since 1966. Figure 3 shows the number of publications per year for thermal analysis and for the quartz crystal microbalance. The QCM publications remain at 1-4% of the thermal analysis publications until about 1990, when they gradually increase to 15-18% by 2005 (reasons for this expansion are discussed below).

Yet the number of publications involving BOTH thermal analysis and QCM is miniscule (5, all in the last five years). The number involving both QCM and calorimetry is not much larger (28, again in the last few years). These data show that the techniques of thermal analysis and the quartz crystal microbalance have developed independently of each other until very recently.

The following figures present a breakdown of the fields of the published references in thermal analysis and in QCM and how they have changed since 1971. Figures 4-7 show the professional fields assigned to all thermal analysis publications in the time intervals 1966-1980, 1981-1990, 1991-2000, and 2001-2003. Figures 8-11 show the fields assigned for QCM in the same time intervals.

For thermal analysis, materials science has the largest number of publications in each time interval. The next largest category is engineering in 1966-1980, but this shifts to chemistry and chemical engineering in 2001-2003.

For the quartz crystal microbalance, however, the pattern of dominant fields is quite different. 78% of the early literature (1966-80) is in engineering (predominantly electrical) and physics and astronomy, with only 18% in materials science and chemistry.

The 61 references before 1980 are in engineering and physics journals, and deal with applications of the QCM in vacuum science and technology to determine the mass and thickness of deposited metallic films. The 1980's show a significant increase in materials science and chemistry (to 33%) and in chemical engineering (to 11%), at the expense of engineering. In the 1990's the field with the largest number of publications on QCM is chemistry, with materials science second. This trend is continued in 2001-2003. As we discuss below, during the mid-to-late 1980's it was shown that the QCM can function as a sensor when immersed in water or other liquids or solutions. This discovery led to rapid increases both in the total number of publications (Figure 3) and the number in chemistry and chemical engineering in such sub-disciplines as electrochemistry, analytical chemistry, and surface chemistry.

Table I shows the journals with the most thermal analysis publications, 1965-2005, and Table II shows the Journals with the most QCM publications, 1965-2005. The complete lack of overlap between these two lists is another indication that the quartz crystal microbalance is unknown to the thermal analysis and calorimetry community.

### Principles Of Operation Of The QCM

The resonant frequency of the fundamental acoustic mode of vibration of a quartz TSM resonator of thickness hq is Final Draft of “The Quartz Crystal Microbalance”, by Allan Smith, a chapter for Volume 5 of the Handbook of Thermal Analysis and Calorimetry, Elsevier f0 = (ìq/ñq)1/2/2hq (1) where ìq and ñq are the shear modulus and density of quartz(7). The shift in frequency due to deposition of a film of the same acoustic impedance as quartz is proportional to the deposited mass per unit area of the film, Äm/A, a relationship first given by Sauerbrey:

Äf = -(2f0

2 /(ìqñq)1/2)Äm/A = -(2f0

2 /(ìqñq)1/2) hfñf = -C hfñf = -CÄm/A (2)

In Eq. (2) ñf and hf are the density and thickness of the deposited film. For an AT-cut 5 MHz crystal at room temperature, C = 56.6 Hz/(ìg/cm2). Because it is easy to measure frequencies to ±.01Hz, changes in mass per unit area of < 1ng/cm2 are measurable with the QCM.

The electrical characteristics of a QCM are well represented by a simple RLC damped resonator equivalent circuit (7), termed the Butterworth-Van Dyke equivalent circuit (Figure 12). The series resonant frequency f of an RLC resonant circuit is given by ! fs =1 2" 1 L Cs (3)

Here L is the dynamic inductance, a measure of the oscillating mass of the quartz, and Cs is the dynamic capacitance, a measure of the elasticity of the oscillating body (8). The resistance of the RLC circuit is related to the quality factor Q (the width of the resonance), the dissipation D, and the full width at half maximum äf of the resonance by the relationship Rmot = 2ðf L D = 2ðf L /Q = 2ð L äf where Rmot is the motional or dynamic resistance of the quartz resonator. For an uncoated resonator, Rmot is a measure of the internal frictional damping of the quartz; coatings provide additional damping. The width of the resonance for an uncoated 5 MHz resonator is about 50 Hz, (i.e. Q = 105), and the damping within the quartz that gives rise to this broadening can be determined by measuring the motional resistance R of the uncoated resonator, typically 10 ohms. When thin, stiff films are deposited on the QCM surface the increase in R is small, but softer, thicker films (i.e., rubbery polymers 5-20 microns thick) can increase R by hundreds or even thousands of ohms.

Since the mid 1980's it has been recognized that TSM resonators can also operate in fluid media if electronic oscillator drivers of suitable gain are employed to excite the resonator and to offset the losses due to damping of the resonator by the fluid. When immersed in water, the motional resistance of a 5 MHz TSM resonator increases to ~360 ohms. For an infinite viscoelastic liquid in contact with the TSM, the frequency shift is Final Draft of “The Quartz Crystal Microbalance”, by Allan Smith, a chapter for Volume 5 of the Handbook of Thermal Analysis and Calorimetry, Elsevier Äfliq = - (2ñqhq)-1 (ñlçlf0/Ã°)1/2 where ñl and çl are the density and viscosity of the liquid. For a 5 MHz QCM immersed in water at 25°C, Äfliq = -710 Hz.

More complete theories of the operation of transverse shear mode resonators have been given by Kanazawa (10,11), Martin et al , Lucklum and Hauptmann , Voinova, Jonson and Kasemo, Johannsmann , Tsionsky and Arnau . In these theories the electrical impedance of the QCM Zq is complex. The impedance of a TSM resonator damped by a finite viscoelastic film can be described as the sum of two complex impedances:

### Z Zq ZL (5)

where the acoustic load impedance due to the film ZL contains both an inductive and a resistive part. Equations are given relating the complex impedance of a TSM resonator damped by a finite viscoelastic film to four parameters characterizing the film: the thickness hf, the density ñf, the shear storage modulus Gf' and the shear loss modulus Gf”.

Shear moduli are functions of the frequency at which they are measured, so for TSM resonators G' and G” are determined at the QCM resonant frequency f0 or one of its overtones.

The acoustic load impedance ZL of a film measured at frequency ù = 2ðf, is ZL = (iùhf ñf) tan ϕ / ϕ , (6)

where ϕ is the (complex) acoustical phase shift ,

ϕ = ùhf √ (ñf/ G) (7)

Here the modulus Gf = Gf' + iGf” is also complex.

Two convenient experimental measures of the film properties are

(a) the difference in resonant frequency Äf = f(crystal + film) - f(crystal)

(b) the difference in motional resistance ÄR = R(crystal + film) - R(film).

For thin films (the gravimetric region) the frequency shift Äf is proportional to ñfhf, the mass per unit area of the film (the Sauerbrey relation, Eq. 2). It is possible to define an “ideal rigid mass layer” with small acoustic phase shifts , for which the acoustic load impedance is purely imaginary and the Sauerbrey limit is reached.

The relationship between ZL and the measured quantities is

Äf/f0 = -Im(ZL)/ ðZcq (8)

ÄR/4ðf0Lq = Re(ZL)/ ðZcq (9)

Final Draft of “The Quartz Crystal Microbalance”, by Allan Smith, a chapter for Volume 5 of the Handbook of Thermal Analysis and Calorimetry, Elsevier Here f0, Zcq, and Lq are the resonant frequency, acoustic impedance, and motional inductance of the bare quartz crystal.

Both Äf and ÄR are zero in the limit of zero film thickness. Voinova et al and Johannsmann present equations for both quantities in a power series expansion in the thickness hf. The result for Äf and ÄR, to third order in hf, is The compliance can be used instead of the modulus to quantify storage and loss behavior in viscoelastic solids. The shear storage compliance is defined as!

J'=G'

(G'2 +G"2 )

and the shear loss compliance is defined as

!

J"=G"

(G'2 +G"2 )

Thus, the thin film limit equations can be rewritten as

! "f

f 0

=

# f hf

#qhq

1+

4hf

2$f 0

2# f J'

3

%

&

'

(

)

*

and

Eq. is useful in estimating the thickness of compliant films at which deviations from the Sauerbrey equation are noticeable. Eq. is useful in interpreting motional resistance measurements of thin films. In the thin film limit, the motional resistance change is proportional to the square of the film density, the cube of the film thickness,

!

"f

f0

=

# f hf

#qhq

1+

hf

2

3

4$f 0

2# fG'

G'2 +G"2

%

&

'

(

)

*

!

"R =

2Lq

3#Zq

$4% f

2hf

3 G"

G'2 +G"2

!

"R =

2Lq

3#Zq

$4% f

2hf

3J"

Final Draft of “The Quartz Crystal Microbalance”, by Allan Smith, a chapter for Volume 5 of the Handbook of Thermal Analysis and Calorimetry, Elsevier and the loss compliance of the film. For a 5 MHz QCM, typical values for Lq and Zq are 0.0402 Henry and 8.84x106 Pa s/m, respectively.

### Detection Electronics

### Simple QCM driving circuits

Simple circuits to drive quartz resonators were developed in the 1960's and 70's. The most common and least expensive detection electronics involves such a driving circuit and a frequency counter. It is the series resonance frequency that is recorded. For Figure 12 the series resonant frequency is given by Eq. (3), whereas the parallel resonance frequency is given by

!

fp =

1

2" L1C *

(16)

where

!

C* =

CsCp

Cs +Cp

(17)

The static parallel capacitance Cp is the capacitance between the quartz electrodes, the crystal holder, and the leads to the driving circuit. Typical values are between 4 pF and 30 pF. For a typical 5 MHz quartz oscillator with Cp = 20 pF, L1 = 0.033 H, C1 = 0.0307 pF, and R1 = 10 Ù, the series resonant frequency fs = 5.0000 MHz, the quality factor is 96,500, the difference between parallel and series resonant frequencies is 3.30 kHz, and the resonance line width äf is 52 Hz.

When QCM's began to be used with liquids there was a need to develop more specialized circuits. The damping produced by liquids causes a decrease in Q by 2-3 orders of magnitude and a corresponding broadening of the resonance. Eichelbaum et al give references to Handbooks that present standard crystal oscillator circuits. They discuss the developments needed for interface circuits to operate with a fluid in contact with one of the crystal faces.

### Frequency And Damping Measurements

Beginning in the 1990's, driving circuits that produce both resonance frequency outputs and analog outputs that measure the motional resistance began to be commercially available (see, for example, www.maxtekinc.com). A list of companies now providing such electronics is given in Appendix I.

### Impedance Analysis

Using an impedance analyzer, it is possible to measure both the real and imaginary parts of Zq for TSM resonators at the resonant frequency and many of its overtones. With Final Draft of “The Quartz Crystal Microbalance”, by Allan Smith, a chapter for Volume 5 of the Handbook of Thermal Analysis and Calorimetry, Elsevier suitable analysis software is in possible to determine both the shear modulus G' and the loss modulus G” of the QCM.

### Is The Transverse Shear Mode Resonator A True Microbalance?

This is actually still a controversial question, after many years of discussion. We give verbatim quotes from some of the leaders in the field:

### Comments In The Literature

### V. Tsionsky, in Electroanalytical Chemistry, Vol. 22, Ed. A.J.

### Bard And I. Rubenstein, 2003

“ Is the quartz crystal microbalance really a microbalance? For one thing, it should rightly be called a nano-balance, considering that the sensitivity of modern-day devices is on the order of 1-2 ng/cm2 and could be pushed further, if necessary. More importantly, calling it a balance implies that the Sauerbrey equation applies strictly, namely that the frequency shift is the sole result of mass loading. It is well known that [in the case of operation of the resonator in a liquid] this is not the case, and the frequency shift observed could more appropriately be expressed by a sum of terms of the form

Äf = Äfm + Äfç + ÄfP + ÄfR + Äfsl + ÄfT (18)

where the different terms on the right hand side of this equation represent the effects of mass loading, viscosity and density of the medium in contact with the vibrating crystal, the hydrostatic pressure, the surface roughness, the slippage effect, and the temperature, respectively, and the different contributions can be interdependent. It should be evident from the above arguments that the term quartz crystal microbalance is a misnomer, which could (and indeed has) led to erroneous interpretation of the results obtained by this useful device. It would be helpful to rename it the quartz crystal sensor (QCS) which describes what it really does - it is a sensor that responds to its nearest environment on the nano-scale. However, it may be too late to change the widely used name. The QCM or its analogue in electrochemistry, the EQCM, can each act as a nano-balance under specific conditions, but not in general.”

### R. Lucklum, Anal. Bioanal. Chem. Comm. 2006, 384, 667-682

“The name microbalance implies that acoustic sensors measure mass or mass changes only. Indeed, in many applications acoustic sensors are used to convert a mass accumulated on the surface into a frequency shift. In chemical and, especially, biochemical applications, however, this basic understanding of the sensor principle can easily lead to misintepretation of experimental results, especially when working in a liquid environment. It also hinders recognition by the experimenter of the outstanding capabilities of quartz crystal resonators, sensors, and other acoustic devices not available to other sensor principles. In a more general view acoustic sensors enable sensitive probing of changes within films attached to the transducer surface and at solid-solid and solid-liquid interfaces…”

Final Draft of “The Quartz Crystal Microbalance”, by Allan Smith, a chapter for Volume 5 of the Handbook of Thermal Analysis and Calorimetry, Elsevier V. Mecea, Sensors and Actuators 2006, 128, 270-277

“This article reveals that the local mass sensitivity of the quartz crystal microbalance (QCM) depends on the local intensity of the inertial field developed on the crystal surface during crystal vibration. …The maximum intensity of this field in the center of the quartz resonator is a million times higher than the intensity of the gravitational field on the Earth. Experimental results reveal that the product of the minimum detectable mass and the intensity of the field acting on that mass is a constant for both QCM and beam balances, explaining thus why QCM is more sensitive than conventional analytical balances. It is show that the apparent effect of liquid viscosity on the frequency response of a quartz crystal resonator is, in fact, the result of the field intensity dependency of the mass sensitivity, being thus clear than QCM is really a mass sensor. “

M. Thompson, Phys. Chem. Chem. Phys. 2004, 6, 4928-4938

“For some considerable period of time it was assumed that the TSM structure employed in water simply responded to added or lost material on the sensor surface… In recent times it become clear that the device is exquisitely sensitive to changes in interfacial conditions. This observation is of great significance when biochemical interactions are detected using the TSM device. This technology involves the placement of biochemical receptors, such as antibody or nucleic acid species, on surfaces where the coupling processes described in the present paper area prevalent. Accordingly, modulation of such coupling by biochemical interactions instigated at the sensor surface constitute a new and highly sensitive detection strategy.”

### Calibration

For a film interacting with a gas, Eq. 14 provides a quantitative means of determining if any calibration is needed for the mass measurement. The numerator of the factor outside the brackets can be expressed as mfilm/Afilm, the mass per unit area of the film. If the second term inside the brackets is << 1, then the Sauerbrey equation is obeyed and the QCM is a true gravimetric device. The correction factor is proportional to the square of the film thickness, the film density and the film's shear compliance. Since virtually all polymers at room temperature are below their glass transition temperature at frequencies of 5 MHz, the time-temperature superposition principle (25,26) can be used to estimate the magnitude of J' and thus the thickness at which the Sauerbrey equation becomes inaccurate. At 20 °C for a rubbery polymer such as polyisobutylene, the correction is 1% for a film 4 ìm thick. For a glassy polymer such as polystyrene, the mass correction factor is 1% for a thickness of 45ìm.

For QCM's interacting with liquid films, the problem is much more complex. Tsionsky et al(20)discusses thoroughly the various limiting cases in which it is still possible to apply the Sauerbrey equation, but they are much more limited than in the case of film/gas interaction.

### Comparison Of Gravimetric And Sauerbrey Masses

As long as the correction factor in Eq. (14) relating frequency shift to change in mass per unit area is small, the QCM functions as a true balance with nanogram sensitivity. To Final Draft of “The Quartz Crystal Microbalance”, by Allan Smith, a chapter for Volume 5 of the Handbook of Thermal Analysis and Calorimetry, Elsevier prove this fact, we determined the resonant frequencies of 20 uncoated QCM crystals, then weighed the crystals with a 5-place analytical balance (Mettler 261). We spraycoated each crystal uniformly with a multi-component polymer coating used in the electronics industry, attempting to obtain films of thickness ~ 1 ìm. Each crystal was dried at 200°C for 2 minutes to drive off residual solvents. We reweighed the coated crystal and divided each mass difference by the area of the coated crystal, 5.06 cm2 to determine a gravimetric film mass per unit area for each sample. From the measured frequency of each coated crystal and Eq. (2) we determined the Sauerbrey film mass per unit area. Comparison of the two results is shown in Figure 14. The mean difference in mass between gravimetric and Sauerbrey masses was 2.4 ± 4.6 %, well within the experimental error of the less accurate gravimetric mass determination. Nonuniformity in film coating and incomplete drying also may contribute to the scatter in the data.

### Determination Of Shear And Loss Modulus At QCM Frequencies

The determination of shear storage and loss moduli of thin viscoelastic films with TSM resonators has been reviewed . Even though the basic physics of damped TSM resonators is well understood, the effort to determine Gf' and Gf” from measurements of frequency shift and motional resistance change has been fraught with problems. For very thin, rigid films, the frequency shift contains no information on either Gf' or Gf” because the Sauerbrey limit (Eq. 2) is reached. For thicker and/or lossier films the frequency shift and motional resistance depend on Gf' and Gf” in a complex manner not obvious by examining the equations.

### Sample Preparation

In order for both mass and heat flow sensors to operate, the thin film sample must adhere to the top surface of the QCM and be of uniform thickness. The mechanical behavior of films on the quartz microbalance has been modeled by Kanazawa, who examined the amplitude of the shear displacement in the quartz crystal and in the overlying film for several cases. For a 1 volt peak RF applied voltage typical of the Stanford Research Systems oscillator driver, the amplitude of the shear wave of a bare crystal is 132 nm.

Mecea has calculated the inertial acceleration at the center of a similar quartz resonator, and finds that it is roughly 106 g, where g is the gravitational constant. At these extremely high accelerations, powder or polycrystalline samples cannot be used because they do not adhere to the surface and follow the transverse motion of the QCM surface.

Many methods have been used to prepare thin film samples for the QCM. For metallic and inorganic materials, vacuum or electrochemical deposition has been traditionally used, but methods such as sol-gel formation of films can also be employed. Uniform films of polymers can be made by dip coating, drop coating, spray coating, or spin coating. To achieve sample homogeneity and uniformity the best of these methods is spin-coating, if the film material is amenable to this treatment. Self-assembled monolayer (SAM) chemistry can be used to construct planar micro- and nano-structures. The effect of non-uniform thickness of films on the operation of the QCM has been treated by several authors.

Final Draft of “The Quartz Crystal Microbalance”, by Allan Smith, a chapter for Volume 5 of the Handbook of Thermal Analysis and Calorimetry, Elsevier Since the measured frequency difference Äf = f(crystal + film) - f(crystal) is proportional to the mass per unit area of the film, the total sample mass is obtained from Eq. 2 as (Äm/A)Afilm where Afilm is the area of the film exposed to the gas.