The load impedance is in general equal to the ratio between the applied stress and
the speed on the quartz crystal surface. Using the BVD equivalent circuit, it is
possible to derive an important relationship that binds the resonance frequency and
the dissipation variation as a function of the stress–speed ratio. So that if an explicit
form of Z L is known, it is possible to calculate both frequency and dissipation
variations no matter what is the sample in contact with the quartz crystal surface.
In this model it is useful to introduce a complex resonance frequency defined as
follows:
~
f ¼ f þ iΓ
where the real part f is the resonance frequency and the imaginary part Γ is “half
bandwidth at half maximum” of the resonance. Indeed, Γ is related to the dissipation
factor D, which is a dimensionless parameter defined as the ratio between the energy
loss and stored in each cycle:
D
E dissipated
2πE stored
through the following relation:
D ¼
2Γ
f
Dissipation is an important physical observable because it is related to the
viscoelastic properties of the sample in contact with the quartz crystal surface.
In the small load approximation one obtains the following relationship:
Δ ~
f
f f
¼
i
πZ q
Z L ¼
i
πZ q
σ
_
u
where Z q ¼ ρ q v q ¼ (ρ q μ q )
1/2
¼ 8.8 Â 10
5 g cm
À2 s
À1 is the acoustic impedance of an
AT-cut quartz crystal, σ is the mechanical stress, and u’ is the velocity on the quartz
crystal surface, respectively. This relationship is decisive for modeling QCM data
and will be used in the next paragraph to derive frequency and dissipation variations
for a layered system evenly distributed over the quartz crystal surface. These
equations are valid if the following conditions are fulfilled:
• the quartz crystal and the layered system are laterally homogeneous and infinite;
• the quartz crystal mechanical deformation is caused only by a transverse shear
wave with a wave vector perpendicular to the surface of the crystal (thicknessshear mode). There are no compression and no flexural waves;
Quartz Crystal Microbalance Sensors: New Tools for the Assessment of. . .
323
the speed on the quartz crystal surface. Using the BVD equivalent circuit, it is
possible to derive an important relationship that binds the resonance frequency and
the dissipation variation as a function of the stress–speed ratio. So that if an explicit
form of Z L is known, it is possible to calculate both frequency and dissipation
variations no matter what is the sample in contact with the quartz crystal surface.
In this model it is useful to introduce a complex resonance frequency defined as
follows:
~
f ¼ f þ iΓ
where the real part f is the resonance frequency and the imaginary part Γ is “half
bandwidth at half maximum” of the resonance. Indeed, Γ is related to the dissipation
factor D, which is a dimensionless parameter defined as the ratio between the energy
loss and stored in each cycle:
D
E dissipated
2πE stored
through the following relation:
D ¼
2Γ
f
Dissipation is an important physical observable because it is related to the
viscoelastic properties of the sample in contact with the quartz crystal surface.
In the small load approximation one obtains the following relationship:
Δ ~
f
f f
¼
i
πZ q
Z L ¼
i
πZ q
σ
_
u
where Z q ¼ ρ q v q ¼ (ρ q μ q )
1/2
¼ 8.8 Â 10
5 g cm
À2 s
À1 is the acoustic impedance of an
AT-cut quartz crystal, σ is the mechanical stress, and u’ is the velocity on the quartz
crystal surface, respectively. This relationship is decisive for modeling QCM data
and will be used in the next paragraph to derive frequency and dissipation variations
for a layered system evenly distributed over the quartz crystal surface. These
equations are valid if the following conditions are fulfilled:
• the quartz crystal and the layered system are laterally homogeneous and infinite;
• the quartz crystal mechanical deformation is caused only by a transverse shear
wave with a wave vector perpendicular to the surface of the crystal (thicknessshear mode). There are no compression and no flexural waves;
Quartz Crystal Microbalance Sensors: New Tools for the Assessment of. . .
323