INFICON RQCM - Quartz Crystal Microbalance Research System - Manual (Page 54)

RQCM – RESEARCH QUARTZ CRYSTAL MICROBALANCE
THEORY OF OPERATION 5-2
It is important to note that under these assumptions, the change in frequency is a function of mass
per unit area. Therefore, in theory, the QCM mass sensor does not require calibration. However,
keep in mind that the Sauerbrey equation is only strictly applicable to uniform, rigid, thin-film
deposits 21 . Vacuum and gas phase thin-film depositions which fail to fulfill any of these
conditions actually exhibit more complicated frequency-mass correlations and often require some
calibration to yield accurate results.
5.2 Z-MATCH EQUATION
Sauerbrey’s original assumptions were of course, questionable and indeed work with crystals
heavily loaded with certain materials showed significant and predictable deviations between the
measured mass and that predicted by Equation 3. Lu and Lewis 22 analyzed the loaded crystal as a
one-dimensional composite resonator of quartz and the deposited film which led to the equation
shown below which is also referred to as the Z-Match equation.
Equation 4
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
⋅ ⋅ ⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ ⋅
⋅
= ∆ −
f
f f
R
f R
N
m
q
z
z
q q π
π
ρ
tan tan 1
where:
m
∆ = change in mass per unit area in g/cm 2 ,
N
q
= Frequency Constant for AT-cut quartz crystal = 1.668 x 10 5 Hz x cm =
q
q q
ρ
µ ρ
2
ρ
q
= Density of quartz = 2.648 g/cm 3 .
f
q
= Resonant frequency of unloaded crystal in Hz.
f
= Resonant frequency of loaded crystal in Hz.
R
z
= Z-Factor of film material =
f f
q q
µ ρ
µ ρ
⋅
⋅
= Acoustic Impedance Ratio
f
ρ = Density of material g/cm 3
q
µ = shear modulus of quartz = 2.947×10 11 g · cm -1 · s -2
f
µ = shear modulus of film material in g · cm -1 · s -2 .
This equation introduces another term into the relationship which is the ratio of the acoustic
impedance of quartz to the acoustic impedance of the deposited film. The acoustic impedance is
associated with the transmission of a shear wave in the deposited mass. Notice that the units of
the frequency constant for quartz is length/time or velocity. Also note that if the acoustic
impedance ratio is equal to one, quartz on quartz, then Equation 4 reduces to Equation 3.