The characteristic length (or penetration length), λ, which describes the envelope
of the damped oscillation, is equal to 1/k, the reciprocal of the propagation constant.
The characteristic length can be written in terms of density ρ L and viscosity η L of the
liquid as follows:
λ ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
η L
π f 0 ρ L
r
For an AT-cut quartz crystal vibrating at 10 MHz, with one face in contact with
pure water at T ¼ 20
C, ρ L ¼ 0.9982 g cm
À3
, and η L ¼ 1.0022 Â 10
À2 g cm
À1 s
À1 ,
the characteristic length is about 180 nm.
Assuming (1) the continuity of the velocity field at the separation surface (that is
the quartz surface transverse speed is equal to that of the adjacent fluid) and (2) the
force exerted by the liquid on the quartz surface is equal and opposite to the force
that the quartz exerts on the fluid, the difference Δf between the resonance frequency
of the unperturbed crystal f 0 and that in contact with the liquid is given by the
Kanazawa–Gordon equation:
Δ f ¼ f 0
3=2
ffiffiffiffiffiffiffiffiffiffiffiffi
η L ρ L
π μ q ρ q
r
According to this model, the quartz crystal does not transmit the vibration to the
entire liquid above the surface, since the transverse displacement decays with
exponential law with a characteristic decay length, λ, so that just a thin layer of
liquid produces the response of the quartz crystal. The effective mass of liquid Δm L
can be calculated using the following relation:
Δm L ¼ λρ L ¼
ffiffiffiffiffiffiffiffiffi
ρ L η L
πf 0
r
By considering the above equations, for an AT-cut quartz crystal vibrating at
10 MHz, with one face in contact with pure water at T ¼ 20
C, ρ L ¼ 0.9982 g cm
À3 ,
and η L ¼ 1.0022 Â 10
À2 g cm
À1 s
À1 , the frequency shift is Δf ¼ 2,020 Hz and the
effective mass of liquid is Δm L ¼ 17 Â 10
À6 g.
2.4 Small Load Approximation: The Electromechanical
Model
A more general description of the response of the quartz crystal resonator in contact
with a generic sample is given by the so-called small load approximation model
[24, 25]. It can be derived using the electromechanical model of a quartz crystal
resonator. In the approximation of small loads and small frequency variations close
Quartz Crystal Microbalance Sensors: New Tools for the Assessment of. . .
321
of the damped oscillation, is equal to 1/k, the reciprocal of the propagation constant.
The characteristic length can be written in terms of density ρ L and viscosity η L of the
liquid as follows:
λ ¼
ffiffiffiffiffiffiffiffiffiffiffiffi
η L
π f 0 ρ L
r
For an AT-cut quartz crystal vibrating at 10 MHz, with one face in contact with
pure water at T ¼ 20
C, ρ L ¼ 0.9982 g cm
À3
, and η L ¼ 1.0022 Â 10
À2 g cm
À1 s
À1 ,
the characteristic length is about 180 nm.
Assuming (1) the continuity of the velocity field at the separation surface (that is
the quartz surface transverse speed is equal to that of the adjacent fluid) and (2) the
force exerted by the liquid on the quartz surface is equal and opposite to the force
that the quartz exerts on the fluid, the difference Δf between the resonance frequency
of the unperturbed crystal f 0 and that in contact with the liquid is given by the
Kanazawa–Gordon equation:
Δ f ¼ f 0
3=2
ffiffiffiffiffiffiffiffiffiffiffiffi
η L ρ L
π μ q ρ q
r
According to this model, the quartz crystal does not transmit the vibration to the
entire liquid above the surface, since the transverse displacement decays with
exponential law with a characteristic decay length, λ, so that just a thin layer of
liquid produces the response of the quartz crystal. The effective mass of liquid Δm L
can be calculated using the following relation:
Δm L ¼ λρ L ¼
ffiffiffiffiffiffiffiffiffi
ρ L η L
πf 0
r
By considering the above equations, for an AT-cut quartz crystal vibrating at
10 MHz, with one face in contact with pure water at T ¼ 20
C, ρ L ¼ 0.9982 g cm
À3 ,
and η L ¼ 1.0022 Â 10
À2 g cm
À1 s
À1 , the frequency shift is Δf ¼ 2,020 Hz and the
effective mass of liquid is Δm L ¼ 17 Â 10
À6 g.
2.4 Small Load Approximation: The Electromechanical
Model
A more general description of the response of the quartz crystal resonator in contact
with a generic sample is given by the so-called small load approximation model
[24, 25]. It can be derived using the electromechanical model of a quartz crystal
resonator. In the approximation of small loads and small frequency variations close
Quartz Crystal Microbalance Sensors: New Tools for the Assessment of. . .
321