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where F is the measured frequency of quartz, AFm relates to mass loading, AFP is dependent on pressure,

Fig. 2 A typical magnetic suspension balance (MSB). Source: From Ref.[21].

AFZ changes with density and viscosity, and AFr is a function of surface roughness.

According to the well-known Sauerbrey equation^311 AFm is directly proportional to the mass loading on the crystal:

where mq is the shear modulus of quartz, rq is the density of the crystal, Dm is the film mass per real surface

Fig. 3 Schematic view of a piezoelectric QCM crystal (only one side shown). A typical 5-MHz QCM from International Crystal Manufacturing consists of a thin quartz crystal (0.85 cm diameter) sputtered with a metal electrode (0.35 cm diameter) on each side. Both sides should be exposed to scCO2 because of high pressure.

Fig. 3 Schematic view of a piezoelectric QCM crystal (only one side shown). A typical 5-MHz QCM from International Crystal Manufacturing consists of a thin quartz crystal (0.85 cm diameter) sputtered with a metal electrode (0.35 cm diameter) on each side. Both sides should be exposed to scCO2 because of high pressure.

area, n is the number of faces of the crystal exposed, and Cm is the mass sensitivity of QCM, which is a function of the characteristic properties (F0, mq, and rq) of the quartz crystal. Eq. (2) applies only if the adsorbed mass is much less than the mass of the crystal and it assumes this mass is firmly attached to the surface; hence the material moves together with the crystal. Such conditions are assumed to be fulfilled in cases of coated solid films and self-assembled monolayers.

The pressure dependence of frequency, AFP, increases with increasing pressure linearly, as shown by Stockbridge[32] for gases up to 15 psi. Susse[33] described a similar relationship for liquids up to 1.5 x 104 psi. Thus AFP can be written as:

where a is the proportionality constant and Cp is the pressure sensitivity of QCM crystal, both of which are independent of the type of fluid in contact with the crystal.

The viscosity and density contribution, DFZ, describes the interaction of the vibrating crystal with a Newtonian viscous fluid. The interaction leads to an additional loading of the crystal, causing a decrease in frequency. DFZ is expressed to be proportional to the square root of the product of viscosity and density of the surrounding fluid:[32,34,35]

A Fz

xl/2

where pf and zf are the absolute density and viscosity of the fluid, respectively.

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Power supply - Oscillator circuit

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Fig. 4 Schematic view of an experimental setup of QCM measurement system in scCO2. Source: From Ref.[50].

Roughness Effect

A common feature of the approaches to QCM theory as reviewed by Thompson et al.[27] is the neglect of microscopic properties of interfaces and surface rough-ness.[36] This leads to deviations of most experimental frequency shift data from the predictions by these approaches. Buttry and Ward[23] and Schumacher, Borges, and Kanazawa[37] observed that up to 80% of the observed frequency shifts for a crystal in contact with a liquid could be attributed to roughness effects.

As the microscopic properties of crystal-fluid interfaces change with the surface morphology, a predictive analytical expression for the roughness contribution to frequency, AFr, is not available. Urbakh and Dai-khin[36,38,39] formulated a model for surface roughness, assuming that a rough surface can be characterized by the average height (h), lateral length (a), and distance (l) between the inhomogeneities of the surface (Fig. 5). They incorporated this model into a framework of perturbation theory[36,38] and derived a general relationship for AFr as:

where C is a scaling function, related to three dimen-sionless factors, a/d, a/L, and h/a; d is the decay length of fluid velocities, d = [Zf / (pFoPf )]1/2 (6)

In particular, if one assumes a slowly varying roughness (h/a < 1 and h/d < 1) for the crystal surface with a limit of a/d ^ 1 or a/d ^ 1, the scaling function C is proportional to the ratio a/d.[36] Reformulating Eq. (5) yields:

where Cr = Cd is defined as the frequency-roughness correlation factor. This factor is only a function of sur-

Table 2 Comparison of two microweighing methods

Characteristics

Gravimetric: MSB[21]

Piezoelectric: QCM[24]

Mass sensitivity

10-5-10-6g13

<10-9g

Time resolution of

5 x 10~2 sec13 (need fine damping control system)

<10~3 sec

response

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