Quartz Crystal Microbalance

Theoretical background

The QCM is a mass sensing device with the ability to measure very small mass changes on a quartz crystal resonator in real-time. The sensitivity of the QCM is approximately 100 times higher than an electronic fine balance with a sensitivity of 0.1 mg. This means that QCMs are capable of measuring mass changes as small as a fraction of a monolayer or single layer of atoms. The high sensitivity and the real-time monitoring of mass changes on the sensor crystal make QCM a very attractive technique for a large range of applications as a sensor in:

1. Thin Film thickness monitoring in different deposition techniques, including both vacuum deposition and liquid plating and etching
2. Biochemistry and biotechnology
3. Drug delivery and drug research
4. In situ monitoring of lubricant and petroleum properties

Piezoelectric effect

The physical basis of operation of the QCM originates in the converse piezoelectric effect, in which the application of an electric field across a piezoelectric material induces a deformation of the material. The piezoelectric effect was discovered by Jacques and Pierre Curie in 1880. You can read more about the discovery of the piezoelectric effect here.

Piezoelectricity is observed in acentric materials (like Rochelle salt, tourmaline and quartz), which are materials that crystallize into non-centro-symmetric space groups. . A unit cell of an acentric crystal lattice possesses a dipole moment due to the arrangement of atoms in the unit cell. At equilibrium, all the dipoles in the bulk material are randomly oriented and there is no net dipole moment.

1. Piezoelectric effect: When stress is applied to the piezoelectric material, all the dipoles orient themselves parallel to the direction of the stress, generating an electric field. The magnitude of the difference in electric potentials across the material is proportional to the applied stress.
2. Converse piezoelectric effect: When an electric field is applied to a piezoelectric material, the dipoles orient themselves parallel to it, resulting a lattice strain and overall deformation. The direction of deformation depends on the applied potential, and the extent of the shear strain depends on the magnitude of the applied potential. The opposite electric field polarity produces an identical strain, but in the opposite direction.

Quartz Crystal

Quartz is the most stable form of silica (or silicon dioxide, SiO₂). Alpha-quartz crystals are mostly employed for QCM applications, because of their superior mechanical and piezoelectric properties. The cut-angle with respect to crystal orientation determines the mode of oscillation. The AT cut crystal, which is the most commonly used for QCM applications, is fabricated by slicing through a quartz rod with a cut angle of 35°10' with respect to the optical axis, as shown in Figure 2. This cut-angle produces shear displacement perpendicular to the resonator surface. The advantage of the AT cut quartz crystal is that it has nearly zero frequency drift with temperature around room temperature.

The QCM electrode and the resonance frequency

In QCM studies a thin disk sliced from single crystals of the alpha-quartz is used. The disk is sandwiched between two metal electrodes that are vapor deposited on either side of the crystal. Gold electrodes have been the most commonly used in QCM studies, because of the ease with which Au is evaporated. However, Cu, Ni, Pt and other metals have also been employed.

When an alternating electric field is applied over the electrodes, the quartz crystal starts to oscillate. The result of the vibration motion of the quartz crystal is the establishment of a transverse acoustic wave that propagates across the crystal, reflecting back into the crystal at the surface. A standing wave condition can be established when the acoustic wavelength is equal to twice the combined thickness of the crystal and electrodes. Thus, the resonance frequency is related to the thickness of the crystal by the following equation:

Where v_q is the velocity of the acoustic wave in AT cut quartz (3.34×10⁴ m/sec), f₀ is the resonance frequency of the quartz crystal prior to the mass change and t_q is the thickness of the quartz resonator. To learn more about resonance, see the literature section. This relation assumes that the velocities of sound in the electrodes and in quartz are identical, and that the thickness of the electrodes is small in comparison with that of quartz. Typical operating frequencies of the QCM lie within the range of 5 to 10 MHz.

The fundamental frequency of the QCM decreases with increasing mass, increasing viscosity of liquid, and with increasing roughness of its electrodes. These dependencies are shown in Figure 4:

Some times it is useful to present the quartz resonator by simplest equivalent circuit of a resonator, as shown in Figure 5.

QCM mass dependence

The change of mass per unit area, Δm, caused by adsorption or deposition of a substance on one side of the QCM is related directly to the change of frequency, Δf_m, by the simple equation (Sauerbrey, 1956):

where C_m is proportional to the squared fundamental frequency, f₀:

The constant (C_m⁰) is determined by the properties of quartz only. For AT-cut quartz resonators, C_m⁰ = 2.257×10^-6 cm² mg^-1 Hz^-1. In this experiment we will use an AT-cut QCM with f₀= 6 MHz, so that C_m = 0.0812 Hz cm² ng^-1. Pay attention that Δm is the change of mass per unit area, and it is usually measured in ng cm^-2.

QCM viscosity dependence

The term associated with the influence of the viscosity and density of liquid in can be written as:

Where η and ρ are the viscosity and density of the liquid.

QCM electrode roughness dependence

The simple relation given in equation 4 is justified only if the surface of the QCM in contact with the liquid is ideally flat. However, the real surface of the QCM is never ideally flat. The shape of the QCM response strongly depends on the morphology of the surface in contact with liquids. As a consequence, in order to use the given QCM as a viscometer one needs to calibrate it using liquids with known properties (η and ρ). Theoretical considerations of the interactions of vibrating rough surfaces with liquids show that the correct way of presenting data is by plotting Δf/ρ versus (η/ρ)^½ as shown in Figure 6.

Reminder: Electrochemical Deposition / Dissolution

The change of electrode mass during deposition or dissolution at constant current may be estimated using Faraday's law. Faraday's law states that the mass, Δm, of a substance precipitated in a charge transfer reaction is proportional to the charge, J·t, expended for this reaction. Here Δm, J and t are mass, current and time respectively. If J is expressed in units of current density, e.g. A/cm², then Δm represents mass per unit area: mg/cm².

Where M is the molecular weight of the deposited or dissolved material, n is the number of moles, F is Faraday's constant (F = 96485 C mol^-1) and t is time.

The reaction takes place in a three-electrode electrochemical cell, shown schematically in Figure 7. The cell consists of a working (W) and a counter (C) electrodes immersed in electrolyte, connected to a power supply. The reaction occurs between these electrodes. The third (reference) electrode (R) is an electrode with a stable electrochemical potential used in measuring the potential of the working electrode. This electrode also has contact with electrolyte but is surrounded by an insulating tube (Luggin tip or Luggin capillary) to prevent its polarization.

The potential distribution between the working and counter electrodes is shown in Figure 8a. Inside the metal electrodes there is no potential change. The sharp potential drops at points a and d describe the Galvani potentials at the metal/electrolyte interfaces. The potential change in the solution between points a and d arises from the current (J) passed through the electrolyte. Thus, the power supply creates an ohmic potential drop between the two Galvani potentials equal to the product J·R, where R is the resistance of electrolyte between working and counter electrodes.

Figure 8b shows the potential distribution between the working and reference electrodes. The sharp potential drops at points a and c describe, again, the Galvani potentials at the metal/electrolyte interfaces and the potential changes in the solution between points a and b an ohmic potential. Thus, voltage measured between W and R is the result of the difference between Galvani potentials and IR potential drop. This IR drop is always included in the measured potential, E. Therefore, it is important to minimize this error, and to place the Luggin tip as close as possible to the working electrode. When current does not pass through the cell the IR drop is always zero, and the position of the reference electrode is immaterial.

The relation between the current and the potential in an electrochemical cell is called the polarization curve. The general equation of the polarization curve is the Butler-Volmer equation:

There are two limits to equation 6: at the low overpotential limit, fη << 1 so that:

At the high overpotential limit |J_C| << J_A or |J_C| >> J_A (meaning that the overpotential is either large and negative or large and positive), so that the smaller one of them may be neglected. Equation 6 then gives:

Liquid viscosity

Viscosity is the measure of a material's resistance to flow. It is a result of the internal friction of the material's molecules. Figure 9 shows the viscosities of different liquids plotted against their molecular weight. It could be seen that:

In the case of cause of saturated hydrocarbons, Van der Waals forces are the only interactions between molecules. In the case of substances containing OH groups, hydrogen bonds result in much stronger interactions and the molecules form clusters containing different numbers of molecules. It means that the weight and sizes of particles moving, in the liquid flow, are larger than the molecular weight and size of unique molecule. If one suggests that interactions between clusters are similar to that acting between molecules of saturated hydrocarbon, then it is possible to estimate the weight of cluster, as is shown in Figure 9 by the blue arrow for water and the black arrows for alcohols. Thus a particle moving in water is 8-times heavier than an individual molecule (where the number 8 is a result of some kind of averaging). The reason that liquid water has only clusters of this size is not understood. There are a lot of different clusters, which are in dynamical equilibrium. Pay attention that corresponding average numbers for alcohols decrease with increasing molecular weight: the longer is the alkyl chain more difficult the association of molecules in a cluster due hydrogen bonding becomes.

Lastly, it is important to mention that the viscosity strongly depends on temperature. In our experiments, which are performed at room temperature (20°C - 30°C), the temperature dependence of density is negligible. The temperature dependence of the viscosity itself is given by equation 9. The constant of some common substances are given in table 1.

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