Quartz Crystal Microbalance used to Characterize Electrochemical Metal Deposition

Thanks to the piezoelectric behaviour of quartz and its adoption in the quartz crystal microbalance (QCM), a technique is available which allows the minutest changes in mass to be followed as a function of time. Frequency shifts as a function of time provide valuable information, for example on the mass or thickness of an electroplated or chemically deposited coating. Growth or deposition rates as well as current efficiencies of electrochemical processes are likewise accessible in this way. Time resolution is of the order of a few milliseconds, mass resolution of the order of nanograms. Using this technique one can, for example, follow the change in current efficiency of copper deposition from an acid bath due to inhibition resulting from the adsorption of organic molecules or the formation of an electrochemical double layer. One application of the quartz crystal microbalance reported here is the study of very thin electroplated films in the range 5 to 20 nm.

1 Introduction

The Quartz Crystal Microbalance (QCM) is a very sensitive analytical method that permits the in-situ detection of interfacial processes, e.g. weight gain by surface electrodeposition. The measurement principle is based on the change in the natural frequency of a quartz resonator that can be excited to a resonant oscillation by AC voltage (piezoelectricity) [1-5].

The QCM can be used under vacuum, in gas phase and more recently in liquid environments. Under vacuum it can be used for monitoring the rate of thin film deposition. In liquids, for example, it can be effectively used to determine the affinity of bio-molecules, viruses or polymers to surfaces or to investigate corrosion phenomena. For its use in liquids the QCM is completed with an electrochemical set-up. In this case, the microbalance is called the Electrochemical Quartz Crystal Microbalance (EQCM) [6-8].

Generally, the EQCM can assist in providing information about the plating mechanism and methodology, essential for process and electrolyte development. The ratio between the mass deposited and the total charge measured at the electrode corresponds to the process current efficiency.

The QCM is a very surface sensitive technique and so the detection limit for mass deposition lies in the order of a few nanograms. Today it is used in chemistry, physics, materials and surface science as well as for maritime and medical research and development.

2 Theoretical Background of the QCM Technology

In 1880 the Curie brothers, Pierre and Jacques, demonstrated the first piezoelectric effect as the ability of some materials (notably crystals and certain ceramics) to generate an electric potential in response to applied mechanical stress. Conversely, a mechanical deformation (ability of a substance to shrink or expand) is produced when an electric field is applied. This effect is formed in crystals that crystallize into non-centro-symmetric space group, i.e. quartz. Only 20 of the overall 32 crystallographic classes show this piezoelectric phenomenon and from these, only a small minority was found suitable for practical applications.

There are two criteria that a suitable piezoelectric material has to fulfill:

  • The substance must either exist in nature or be artificially prepared as a monocrystal with a sufficiently large proportion of defect-free monocrystals or polycrystalline piezoelectric textures
  • The substance should possess outstanding piezoelectric properties, low internal friction loss coefficient and long-term high temperature stability

Today, quartz (α-quartz) is the most common piezoelectric material because it possesses extraordinarily low damping with strong piezoelectric properties and it can easily be synthesized into a pure, crystalline form with low defect density. The α-quartz belongs to the trigonal trapezoidal group of the trigonal crystal system. The α-quartz is stable to 573 °C, above which, the β-quartz occurs. The density of α-quartz is ρq = 2.648 gr/cm3 and the shear modulus is μq = 2.947·1011 gr/(cm · s2).

Piezoelectric resonators are prepared by cutting the desired component from a large single quartz crystal at a precise angle with respect to the crystalline axes. The AT-cut crystal represents the most commonly used for QCM applications and is fabricated by a plate cut from a crystal of quartz such that the plate contains the X-axis and makes an angle of approx 35° with the optic or Z-axis (Fig. 1).

Fig. 1: Quartz crystal and AT-cut with Z as optical axis and X as polar or piezo axis
Fig. 1: Quartz crystal and AT-cut with Z as optical axis and X as polar or piezo axis

The crystal possesses excellent temperature and frequency characteristics, providing close to zero values for both the temperature coefficient and the resonance frequency drift (induced by the alternating electric field) within the range 0 to 50 °C.

A piezoelectric device consists of a plate or a bar cut from a piezoelectric material, with two or more metal electrodes on opposing crystal sides, enabling crystal polarization and mechanical deformation of the piezoelectric material to occur (Fig. 2).

Fig. 2: Assembly of the crystal, showing polarization and mechanical deformation
Fig. 2: Assembly of the crystal, showing polarization and mechanical deformation

The electrode pads overlap in the centre of the crystal, with tabs extending from each to the edge of the crystal where electrical contacts are attached. Voltage applied to the electrodes causes a strain to occur within the resonator. For the AT-cut quartz crystal, mechanical strain will occur in the shear direction (x-axis). The strain induced in a piezoelectric material by an applied potential of known polarity will be equal and opposite in direction to that of the applied polarity.

If an alternating electric field (AC voltage) is applied, the quartz crystal will start to oscillate [9-11] and vibrate, creating transverse acoustic waves that propagate across the crystal, reflecting back into the crystal at the surface. The vibration amplitude is parallel to the crystal surface and the x- direction.

A standing wave condition can be established when the acoustic wavelength is equal to twice the combined thickness of the crystal and electrodes, d = λ/2. Thus, the resonance frequency can be related to the thickness of the crystal (Fig. 3).

Fig. 3: Schematic illustration of the shear deformation...
Fig. 3: Schematic illustration of the shear deformation of a quartz crystal after application of an external electric field

Mathematically, the piezoelectric effect can be expressed as follows:

where P is the polarization, x the elongation and e is the piezoelectric polarization modulus.

For the inverse piezoelectric effect, the relation can be used, where d is the elongation modulus.

The fundamental frequency f0 of the acoustic wave (also known as the thickness shear mode, TSM, or the fundamental mode) is given by the following equation:

in which vq is the velocity of the acoustic wave in AT cut quartz along the Y-axis (3.34·104 m/s, acoustic velocity), f0 is the resonance frequency of the quartz crystal prior to the mass change, tq is the combined thickness of the quartz resonator and electrodes. The oscillation mode or thickness-shear-mode (TSM) can be described as a lateral oscillation as illustrated in Figure 3. Typical operating frequencies of the QCM lie within the range of 5 MHz to 10 MHz [12].

Assuming infinite isotropic medium in which no deformation in the bulk plate occurs (i.e. the volume remains constant) during the shear vibration, the amplitude depends only on the plate thickness:

Where ξ(y,t) is the amplitude and t is the duration.

During layer formation onto the quartz plate, a maximum amplitude is observed for each layer. The frequency is found to depend only on the mass inertia and not on the elastic properties of the adsorbed layer. Thus, equation <3> can be written as:

where Δm̃Q is the change in mass of the quartz plate with a thickness change of Δl and where Δm̃ is the mass change of the adsorbed layer. From this, the following can be derived:

Cm is a constant with the relationship:

where n is the mode of vibration. Equation <7> is the basic Sauerbrey equation.

2.1 The Sauerbrey Equation

In 1959, Sauerbrey was the first to establish a linear relationship between the change in the resonant frequency Δf of a piezoelectric crystal and the adsorbed mass Δm deposited [13, 14]:

Where f0 is the resonant frequency of the quartz resonator, ρq and μq are the density and shear modulus of the quartz, A is the area and Δm the adsorbed mass of substance under investigation.

Inserting the Sauerbrey constant Sf allows equation <8> to simplify to:

This proportionality factor Sf is a material-specific parameter and is also referred to as the integral coating weighing sensitivity [15-18]. Increasing solution viscosity reduces the sensitivity to mass change at the resonator centre and increases field fringing. An increase in the rigid mass within the electrode area also confines the sensitivity towards the centre of the resonator.

The derivation of the mass frequency relationship assumes that all deposited material exists at the antinode of the standing wave, propagating outwards from the quartz crystal, i.e. all the forming deposit can be treated merely as an extension of the quartz crystal itself.

The Sauerbrey equation relies on the following main assumptions,

  • The layer on the electrode must be rigidly attached to the crystal surface, so that it moves with it. Such a condition will exist, for example, during monolayer adsorption of small molecules and for thin layer metal deposition
  • The increase in mass should be much less than the mass of the crystal itself Δm/mquartz ~ 2 %. If the film/deposit growth is thin enough, errors resulting from discrepancies between the acoustic propagation characteristics in the quartz crystal and within the film are negligible
  • The frequency change must be small (Δf/f < 0.05)

Modifications to the Sauerbrey equation involve the introduction of additional parameters. For example, if the frequency change is higher than 5 % (Δf/f > 0.05), the so called Z-match method must be applied [19]:

where fc is the resonance frequency of the composite resonator, formed from the crystal and the surface film, ff represents the resonant frequency of the developing film (Δf = fc – f0), and zf = ρq · vq = (ρq · μq)1/2 and zq = ρf · vf = (ρf · μf)1/2 are the acoustic impedance of the film and the quartz, respectively. The Z-match relates to the difference in the acoustic impedance of the quartz material and the deposit. When calculating Z-match, the density, ρf, and the shear modulus, μf, of the deposited material must also be known.

In liquid media, the frequency shift is also influenced by other parameters, such as the visco-elastic properties [20-30] of the layer close to the vibrating crystal, the pressure of the surrounding medium into which the EQCM is immersed [25], surface roughness changes [31-40], changes in temperature [41] and also the stress of the electrode under investigation [42].

Thus, the overall response of the EQCM is the sum of the frequency shift of all the contributions, i.e.

In 1985, Bruckenstein and Shay [20], together with Kanazawa and Gordon [23] were the first to develop theoretical models predicting the frequency change for the immersed crystal and incorporating the viscosity and density of the solution. Hence, the vibrations of the AT-quartz crystal parallel to the QCM liquid interface results in the radiation of a shear wave into the liquid. The decay of the shear wave velocity in the x direction can be described as an exponentially dampened cosine function:

in which k is the propagation constant, z the distance from the resonator surface, V0 the maximum amplitude of the shear wave, and ω the angular frequency. In Figure 4 a description of the dampened shear propagation in the liquid is illustrated [43].

Fig. 4: Description of the shear wave propagation...
Fig. 4: Description of the shear wave propagation in a newtonian fluid in terms of shear velocity in the x direction as a function of distance from this interface [40]

In Kanazawa and Gordon´s model [23] the quartz crystal resonator is treated as an inert elastic solid and the liquid a purely viscous medium:

Hence, the interaction of the vibrating quartz crystal with the viscous medium is expressed as the decrease in frequency, which is proportional to the square root of (solution viscosity x density).

Stockbridge could verify that the frequency of QCM increases linearly with the pressure caused by the hydrostatic pressure on the elastic modulus of quartz [25]:

Within the temperature range of 0 ºC to 80 ºC, the dependence is relatively small and can be linearized in the form:

The temperature dependence of the frequency of an AT cut quartz crystal is shown in Figure 5 [38].

Fig. 5: Frequency/temperature function of AT cut quartz crystal [38]
Fig. 5: Frequency/temperature function of AT cut quartz crystal [38]

The electrode surface roughness can cause frequency shifts due to the entrapment of liquid molecules at the surface, which may vibrate with the crystal motion. This additional vibration translates as a weight gain to the crystal. The quantity of entrapped molecules is determined by surface roughness, geometry and size of the crystal. A similar effect occurs at increased film thickness or for deposits of high porosity.

If the average roughness is not greater than the acoustic wavelength, the Sauerbrey equation will relate to an average thickness of the deposited layer. The frequency shift induced by surface roughness can be described as follows:

where the average height = h, the average lateral length = a, the average distance between inhomogeneities = L, the decay length = δ, and Ψ = scaling function.

Thus, the relationship between roughness and the decay length δ of the fluid viscosity will determine the dependency relationship of the frequency of the quartz crystal resonator to the fluid properties and the morphology of the interface.

2.2 Equivalent Circuit

The acoustic resonator can be modeled on a likeness to a mechanical oscillation. The mechanical oscillation circuit consists of the mass m, the spring constant k, and the damping constant α (Fig. 6). The system can be expressed by the following differential equation:

Cady [44], Van Dyke [45], Dye [46] and Crane [47] showed that the electric properties of piezoelectric resonators may be determined as an equivalent circuit represented by resistors, inductors and capacitors (Fig. 7).

Fig. 6: Oscillation equivalent circuit in which a mass m is...
Fig. 6: Oscillation equivalent circuit in which a mass m is connected to an unmovable wall via spring with the spring constant k and a damping unit with the damping constant α

Fig. 7: Equivalent circuit of a quartz...
Fig. 7: Equivalent circuit of a quartz crystal unit oscillates in air (a) and in liquid (b) [48]

C0 is the electrical capacitance of the quartz between the electrodes, R1 corresponds with the dissipation of the oscillation energy (caused by solution viscosity and the mounting of the crystal), C1 with the stored energy in the oscillation and L1 corresponds to the inertial component of the oscillation, i.e. changes of mass on the electrode.

If the quartz crystal is now immersed into a solution, the proposed model is illustrated in Figure 7b. Lsoln and Rsoln are the motional inductance and resistance due to liquid loading and Lm is the motional induct ance due to mass loading. Cx is the freeloading capacitance of the QCM and test fixture. In the following formulae:

DQ = is the dielectric constant of quartz
ε0 = the permittivity of free space
r = a dissipation coefficient corresponding with the energy losses during oscillation
ε = the piezoelectric stress constant
c = the elastic constant

Only C0 is independent of ε and does not participate directly in piezoelectricity. L1 depends on density and quantity, being equivalent to the mass per unit area in the Sauerbrey equation.

Figure 8 shows typical Bode diagrams for two different damping factors (R). fs is the frequency of zero phase, where the current flowing through the crystal is exactly in phase with the applied voltage.

Because of C0 which is located parallel to the oscillation circuit, two resonance frequencies exist: fs is the serial resonance frequency and fp is the parallel resonance frequency.

Fig. 8: Bode-diagrams after excitation...
Fig. 8: Bode-diagrams after excitation of a 5 MHz quartz crystal, A: the damping R = 10 Ω, B: R = 1000 Ω; fs is the serial resonance frequency of the complete equivalent circuit. fp is the parallel resonance frequency of the complete equivalent circuit [49]

From the resonance peak, the so called quality factor Q can be calculated.

Equation <20> shows the inverse relationship of the quality factor Q with R1. It can be predicted that a decrease in viscosity (causing an increase in R1), will result in a decrease of Q, whereas a linear dependence was found between R1 and (ρ1η1)1/2. From frequency changes (as a function of time), valuable information can be obtained, in particular: deposited mass or layer thickness, growth or deposition rate/ process current efficiency.

3 Experimental Set-up

Figure 9 shows schematically a typical experimental set-up of the quartz crystal microbalance (QCM), electrochemical quartz crystal microbalance (EQCM) respectively, used in liquids [2].

Fig. 9: Schematic experimental set-up of...
Fig. 9: Schematic experimental set-up of an EQCM; CE = Counter Electrode, RE = Reference Electrode, WE = Working Electrode

With this set-up, transient changes in frequencies, current and / or potential can be measured and monitored under static and dynamic conditions by using the system of three electrodes. Using the Sauerbrey and subsequentequations, deposited mass may be monitored in situ. Frequency sensitivity changes (time resolution) are in the millisecond range (typically 30 ms), whilst for mass difference the resolution is at the nanogram level (in the described set-up 23 ng/(Hz · cm²)). The AT-cut α-quartz crystal is typically vacuum deposited with a 100 nm layer of either gold or platinum. Auger- and ESCA studies can be used to qualify the purity of the metal coatings [50].

The quartz crystal used in this study is of 0.3 mm thickness with a diameter of 14 mm. The special designed electrode layout enables the generation of the stable shear oscillation with the electronic driver. The resonance frequency of the crystal is typically f0 = 5MHz. Figure 10 shows the details of the electrode set-up. The electrolyte is pumped at constant flow onto the oscillating quartz crystal (WE) via an injection tube.

Fig. 10: Electrode set-up as detail of the used EQCM
Fig. 10: Electrode set-up as detail of the used EQCM

3.1 Precision of the EQCM technique

The electro-deposition rate for the acid copper process was chosen to evaluate the precision and detection limits of the EQCM technique. Figure 11 shows the gradual thickness increase of the copper layer as a function of time at a fixed current density of 2A/dm2. A theoretical deposit growth rate of 0,074 μm/ 10 s at 2A/dm2 is based on a 100 % cathodic current efficiency.

Fig. 11: Copper thickness increase vs. time...
Fig. 11: Copper thickness increase vs. time for the acid copper electrolyte, measured using an EQCM

Figure 12 compares the EQCM-measured values to those obtained theoretically. After 80 s to 120 s the measured values come very close to the theoretical 100 % current efficiency.

Figure 12: Deposition rates of acid copper in time intervals of 10 s
Fig. 12: Deposition rates of acid copper in time intervals of 10 s

From this data, deviation from theoretical for the acid copper process can be expressed as in Figure 13. The initial 12 % deviation rapidly falls after 2 min to 3 min to 2 % to 2,5%. This initial high deviation can be explained as being due to strong inhibition caused by adsorbed organic molecules or the formation of the electro-chemical double layer. The deposition rate and thickness increase become linear after 2 min to 3 min, providing current efficiencies very close to 100 %, with a deviation of approx 2 % to 3%. This is the maximum percentage error associated with EQCM measurements.

Fig. 13: Error of EQCM-measured deposition rates for an acid copper electrolyte
Fig. 13: Error of EQCM-measured deposition rates for an acid copper electrolyte

4 Application – Ultrathin palladium / gold plating

Due to the trend towards plating thickness within the nanometer range for palladium (10 nm to 20 nm) and gold (3 nm to 5 nm), it is crucial to understand the deposition behavior of each process type over their entire operating windows. This is particularly important during precious metal deposition where nonuniform plating efficiency occurs over varying current density and operating conditions. If the process is not sufficiently robust, a slight variation of plating parameters may lead to a significant deviation from the target deposit thickness.

By using the EQCM as an advanced lab tool, the process deposition behavior over a wide range of parameters can be studied in more detail. The palladium (Pallacor® HT) and gold (Aurocor® PPF) processes have been specifically developed to provide consistent plating rates, dependent only on the applied current density, thus ensuring precise plating thickness. A typical layer combination is: nickel (0.7 μm), palladium (10 nm to 20 nm) and gold (5 nm to 10 nm) For this layer combination of nickel/palladium/gold, the frequency change measured using the EQCM is illustrated in Figure 14.

Fig. 14: Frequency change of the oscillating quartz crystal...
Fig. 14: Frequency change of the oscillating quartz crystal as a function of time for nickel, palladium and gold deposition

The frequency change during palladium deposition has been investigated in more detail for two different palladium concentrations, 2 g/l and 7 g/l (Fig. 15).

Fig. 15: Frequency change of the oscillating quartz crystal...
Fig. 15: Frequency change of the oscillating quartz crystal as a function of time for palladium at two different concentrations of palladium

From the frequency/ time diagrams measured using the quartz crystal microbalance it is possible to determine the deposition rates of the palladium electrolyte at varying palladium concentration (Fig. 16). The trends show a very linear and high deposition rate of 1 μm / 3 min, even for very short durations (1 s to 5 s) and for a relatively low palladium concentration of 5 g/l.

Fig. 16: Pallacor® HT deposition rates at...
Fig. 16: Pallacor® HT deposition rates at 1 A/dm2, room temperature for different palladium concentrations using the EQCM technique

Figure 17 depicts the current efficiency as a function of the palladium concentration, measured at room temperature and 1 A/dm2. The influence of temperature, brightener concentration and agitation has been investigated further for a palladium concentration of 5 g/l. Figure 18 summarizes the significant effect of agitation on deposition rate and current efficiency, respectively.

Fig. 17: Current efficiency at varying palladium...
Fig. 17: Current efficiency at varying palladium concentration (measured at RT and 1 A/dm2)

Fig. 18: Deposition characteristics of...
Fig. 18: Deposition characteristics of Pallacor® HT as function of temperature and agitation

In addition, comparison studies have been carried out for gold deposition at 0.5 A/dm2, 2 g/l gold, 20 °C. The gold deposit was applied to 1 μm nickel overplated with 10 nm palladium. Figure 19 illustrates the frequency change at two different electrolyte pHs (4 and 6.5).

Fig. 19: Frequency change of the oscillating...
Fig. 19: Frequency change of the oscillating quartz crystal as a function of time for gold at two different pH values

Further deposition investigations of the gold electrolyte as a function of gold concentration and current density have been carried out, focussing on the first 5 seconds duration. Figure 20 illustrates that at the lower pH 4.5, gold deposition commences after a kinetic inhibition period, whereas at higher pH value, inhibition does not occur.

Fig. 20: Deposition rate of the gold...
Fig. 20: Deposition rate of the gold...
Fig. 20: Deposition rate of the gold electrolyte (Aurocor® PPF) at pH=4 and pH=6,5

These examples demonstrate how the EQCM can be effectively used to investigate individual process parameters and their influence on the deposition characteristic of the electrolyte. Wheras the examples have thin layers, the usage of quartz crystal microbalance is not limited to such thin layers.

5 Summary

The quartz crystal microbalance can be used as an effective in-situ technique for the study of

  • the affinity of bio-molecules, viruses or polymers to surfaces
  • corrosion phenomena
  • electroless and electro-deposition / dissolving behavior and mechanism

Sauerbrey was the first to establish a linear relationship between the change in resonant frequency Δf and the mass change Δm of the surface layer / coating and was able to use the QCM as a mass censoring method. In liquid media, frequency changes measured by a QCM EQCM respectively may also be as a result of other phenomena and not solely due to mass change. Frequency shift may be influenced by parameters, such as the visco-elastic properties of the layer surrounding the vibrating crystal, the pressure due to the medium into which the EQCM is immersed, changes to the surface roughness, temperature and also stress of the electrode under investigation.

Many process properties can be accurately quantified by this technique including deposition rate/ current efficiency as shown from the ultrathin palladium and gold plating examples.


All used electrolytes are commercial products of Atotech Deutschland GmbH


  1. C. Lu (Ed.): Application of Piezoelectric Quartz Crystal Microbalances, Vol.7, Elsevier, Amsterdam, 1984
  2. R. Schumacher, Chemie in unserer Zeit, No.5, 1999
  3. R. Schumacher, Angew. Chemie. 1990, 102, 347
  4. M. Wünsche, H. Meyer, R. Schumacher, Galvanotechnik 87 (1996) Nr. 5
  5. D. A. Buttry, M. D.Ward, Chem. Rev. 92 1335 (1992)
  6. K. Doblhofer, K. G. Weil: Application of the Quartz Microbalance in Electrochemistry. Bunsen-Magazin 9, 162, (2007)
  7. C. Eickes, J. Rosenmund, S. Wasle, K. Doblhofer, K. Wang, K. G. Weil: The Electrochemical Quartz Crystal Microbalance (EQMW) in the Studies of Complex Electrochemical Reactions, Electrochim. Acta, 45, 3623-3628 (2000)
  8. K. Doblhofer, S. Wasle, D. M. Soares, K. G. Weil, G. Ertl: An EQMW Study of the Electrochemical Copper(II)/Copper(I)/Copper System in the Presence of PEG and Chloride Ions. J. Electrochem. Soc. 150, C657-C664 (2003)
  9. W. D. Callister: Materials Science and Engineering: An Introduction. Second Edition, 1985, John Wiley and Sons, Brisbane
  10. Arthur L. Ruoff: Materials Science, 1973, Prentice-Hall, New Jersey
  11. C. J .F. Bottcher: Theory of Electric Polarisation, 1952, Elsevier Publishing Company, London
  12. H. Bahadur, R. Parshad; in Physical Acoustics, Vol 16 (W.P. Mason and R.N. Thurston, eds.), Academic Press, New York, 1982, 37-171.
  13. G. Sauerbrey, G., Z. Physik, 1959, 155, 206-222
  14. G. Sauerbrey; Proc. Annu. Freq. Control Symp., 1967, 21, 63
  15. A. C. Hillier, M. D. Ward; Anal. Chem. 64:2539(1992)
  16. M. D. Ward, E. J. Delawski; Anal. Chem. 63:886(1991)
  17. Gabrielli, M. Keddam, R. Torresi; J. Electrochem. Soc. 138:2657(1991)
  18. A. Martin, H. R. Hager; J. Appl. Phys. 65:2630(1989)
  19. C. Lu, O. Lewis; J. Appl. Phys. 43:4385(1972)
  20. S. Bruckenstein, S. Swathirajan; Electrochim. Acta 30:851(1985)
  21. R. Normura, A. Minemura; Nippon Kagaku Kaishi, 1980:1621(1980)
  22. R. Normura, M. Okuhara; Anal. Chim Acta, 142:281(1982)
  23. K. K. Kanazawa, J. G. Gordon; Anal. Chem Acta, 175:99(1985)
  24. H. Hagar; Chem Eng Commun 43:25(1986). H. E. Hager, P. D. Verge, Sens. Actuators, :271(1985)
  25. C. D. Stockbridge; in Vacuum Microbalance Techniques, K. H. Behrndt ed., vol. 5. p. 147, plenum press, NY, 1996
  26. H. Muramatsu, E. Tamiya, I. Karube; Anal. Chem. 60:2142(1988)
  27. S. Z. Yao, T. A. Zhou; Anal. Chim. Acta. 212:61(1988)
  28. T. Zhou, L. Nie, S. Yao; J. Electroanak. Chem. 293:1(1990)
  29. Z. A. Shana, D. E. Radtka, U. R. Keller, F. Josse, D. T. Haworth; Anal. Chim. Acta, 231:314(1990)
  30. M. Thompson, G. K. Dhaliwal, C. L. Arthur; Anal. Chem., 58:1206(1986)
  31. R. Schumacher, G. Borges, K. K. Kanazawa; Surfaces Science 163:L621(1985)
  32. R. Schumacher, J. Gordon, O. Melory; J. Elecroanal. Chem., 216:127(1987)
  33. A. Muller, M. Wicker, R. Schumacher, R. N. Schindler; Ber. Bunsenges. Phys. Chem., 92:1395(1988)
  34. R. Beck, U. Pittermann, K. G. Weil; J. Electrochem Soc, 39:453(1992)
  35. D. A. Buttry, M. D. Ward; Chem. Rev., 92:1355(1992)
  36. M. Urbakh, L. Daikhin; Phys. Rev. B., 49:4866(1994)
  37. L. Daikhin, M. Urbakh; Faraday Discuss., 107:27(1997)
  38. M. Urbakh, L. Daikhin; Langmuir, 10:2836(1994)
  39. L. Daikhin, M. Urbakh; Langmuir, 12: 6354(1996)
  40. V. Tsionsky, L. Daikhin, M. Urbakh, E. Gileadi; Langmuir, 11:674(1995)
  41. J. Zelenka; in Piezoelectric Resonators and their applications, Elsevier NY,1986
  42. E. Gileadi, V. Tsionsky; J. Electrochem. Soc. 147:567(2000)
  43. M. D. Ward; in Physical Electrochemistry, P. 1, I. Rubinstien, Editor, Marcel Dekker, NY, 1991
  44. W. G. Cady; in Piezoelectrocity, W. G. Cady ed., Dover publication Inc., NY , 1964
  45. K. S. Van Dyke; Phys. Rev. 25:895(1925)
  46. D. W. Dye; in Piezoelectric Quartz Resonator and Equivalent Electrical Circuit, Proc. Phil. Soc. 38:399(1926)
  47. R. A. Crane, G. Fischer; J. Phys. D., 12:2019(1979)
  48. Interfacial Electrochemistry, A. Wieckowski ed., Marcel Dekker, Inc, NY 1999
  49. http://www.uni-mainz.de/FB/Chemie/fbhome/physc/Dateien/PCF_QCM_skript_060105.pdf
  50. M. Wünsche, Dissertation FU Berlin, 1995

PDF Version of the article

Epub Version of the article

Flash Version of the article
[qr-code size=”2″]


Comments are closed.