INFICON RQCM - Quartz Crystal Microbalance Research System User Manual
Page 58
RQCM – RESEARCH QUARTZ CRYSTAL MICROBALANCE
THEORY OF OPERATION
5-6
When the QCM comes in contact with a liquid, there is a decrease in frequency that is dependent
upon the viscosity and density of the liquid. Kanazawa’s solution for the change in resonant
frequency of the crystal due to liquid loading is shown in Equation 7.
Equation 7
q
q
L
L
q
f
f
ρ
µ
π
ρ
η
⋅
⋅
⋅
−
=
∆
2
3
Where:
f
q
= Resonant frequency of unloaded crystal in Hz.
ρ
q
= Density of quartz = 2.648×103 kg/m
3
.
q
µ
= shear modulus of quartz = 2.947×10
10
Pa
L
ρ
= density of the liquid in contact with the electrode in kg/m
3
,
L
η
= viscosity of the liquid in contact with the electrode in N · Sec/m
2
Liquid loading also dampens the resonant oscillation of the crystal causing an increase in series
resonance resistance, R, of the crystal. ∆f and ∆R measurements are both routinely used as
independent indicators of mass loading and viscosity at the crystal-liquid interface of the QCM
resonator during chemical and electrochemical depositions in solution
A Butterworth-Van Dyke equivalent circuit model (Figure 16) was applied to derive a linear
relationship between the change in series resonance resistance, ∆R, of the crystal and
L
L
ρ
η
⋅
under liquid loading. Using the relations in this study the change in resistance, ∆R, can
be put in the form:
Equation 8
(
)
(
)
2
26
3
2
3
32
2
e
A
f
f
R
q
r
q
q
⋅
⋅
⋅
⋅
⋅
⋅
⋅
∆
=
∆
ρ
µ
ρ
π
Where:
R
∆
= change in series resonance resistance in Ω,
r
A
= active area of INFICON 1-inch crystal = 3.419×10
-5
m
2
26
e
= piezoelectric constant for an AT cut quartz = 0.095 kg/sec
2
/V
For example, moving the crystal from air to pure water @ 20°C, Equation 7 and Equation 8
predict a decrease in
f
of 714 Hz and an increase in R of 357.4 Ω, respectively. These values are
in agreement with the results observed with an RQCM using a 5 MHz, 1-inch diameter, polished,
gold coated mounted on a INFICON Crystal Holder. Note that pure water @ 20°C has a density
(ρ
L
) of 998.2 kg/m
3
, and a viscosity (
η
L
) of 1.002×10
-3
N · sec/m
2
.
Excellent agreement between the frequency and resistance equations and the experimental results
has been proved
, making the QCM an excellent tool for the evaluation of fluid properties.
Application examples include in-situ monitoring of lubricant and petroleum properties
tight correspondence between theory (Equation 7 and Equation 8) and the RQCM is clearly