Analytical study and modelling of a MEMS based
gas sensor for detection of gases by the
frequency shift method
Anik Mandal
Centre for Nanoscience and
Engineering, Indian Institute of
Science, Bangalore
Abstract— A gas sensor is a device which can detect a
particular type of gas and its concentration in parts per
million accurately. It finds applications in various fields
like monitoring of air quality index in polluted areas, to
analyze combustion gases at the exhaust of cars, to detect
potential harm due to leakage of some toxic / combustible
gas in industries, and in medical science. In the recent
days, due to improvements in miniaturization science
(which includes microfabrication and advanced
lithography techniques) we have now access to MEMs
based gas sensors which have several advantages over the
conventional gas sensors. In the present work, we have
attempted to design and derive a mathematical model of a
MEMS based gas sensor which absorbs a certain gas at a
cantilever tip and then the change in the resonant
frequency gives an indication of the concentration of gases
present .
Keywords—MEMS, adsorption, frequency shift, cantilever,
sensitivity.
Several researchers[3] have suggested the use of
unconventional fabrication processes for the fabrication of
MEMS based gas sensors.
II. MODELING
If we consider a simple cantilever beam, we know that
there is a deflection at the beam tip if we put a load on the
tip. If we consider this deflection as δ , then the resonant
frequency is giveb by Rayleigh’s formula as ω = (g/ δ)0.5 .
However, things get a bit complicated when we deal
with distributed loadings . In reality , there will be a
deflection at the cantilever tip due to the self-weight of the
cantilever as well .
Standard MEMS based gas sensors detect gases by the
phenomenon of adsorption. Adsorption is a surface
phenomenon in which molecules of a gas or liquid forms a
superficial monolayer on a surface. In order to simulate the
adsorption effect, we can use multiple approaches. For
example, we may consider an extremely thin layer of some
insulating material being placed near the cantilever tip
which acts as the monolayer of the adsorbed molecules and
also superficially creates a mass loading effect. However, in
this present work we have considered the cantilever beam as
being made up of two parts – one part is the part from the
base to near the tip and another small part is there near the
tip which actually serves as the gas adsorbing part. It is
important to consider the gas adsorption part at the tip only,
because this pronounces the cantilever deflection which in
turn serves the purpose of the resonant frequency shift
which is actually the parameter which gives a measure of
the gas concentration we are interested to measure.
I. INTRODUCTION
Primarily , mechanical (acoustic type) MEMS based gas
sensors are of two types. One is the Surface Acoustic Wave
type gas sensor and the other is a quartz crystal
microbalance (QCM) gas sensor. The basic principle is
same for both the two types of sensors where the signal
comes from the mass loading effect of the gas molecules
onto the sensitive layer which in turn produces a shift in the
resonant frequency of the structure [1]. The physics of
surface wave was studied extensively by Lord Rayleigh [2] .
The basic principle in propagation of surface acoustic waves
is that the waves die out in the out-of-plane direction and the
dominant direction of the wave is along the surface. The
SAW based gas sensors works on the principle of change in
frequency at the receiving electrodes due to the mass
loading effect at the sensing layer. Various types of SAW
gas sensors are there some of which has the sensitive layer
over the electrodes and some of which has the sensitive
layer as a delay-line midway between the two sets of IDTs
(interdigital transducers which are the two comb like
structures).
Fig. 1 : The two parts of the cantilever are as shown above . The
left end is fixed while the right end (blue) is the tip where we
consider the adsorption effects.
In order to simulate the mass loading effect due to adsorption,
we have considered that as if the density of the material at the
tip part changes (increases). There might be some error in this
estimation but since we are attempting to derive a simplified
mathematical model, the above assumption might be
reasonable. We have considered 5 runs, where the densities at
the cantilever tip have been taken to be 3200 kg/m3, 3600
kg/m3, 3900 kg/m3, 4100 kg/m3, 4600 kg/m3.
The modified version of the Rayleigh’s equation[4] is given as
𝑙
ω2 = 𝐸𝐼
∫0 (
2
𝑑2𝑦
) 𝑑𝑥)
𝑑𝑥 2
𝑙
𝑚′ ∫0 𝑦12 𝑑𝑥 + ∆𝑚𝑦22
Fig. 3 : Second mode undamped natural frequency
Where y is the deflection at any point, x is the distance
between the point and the end tip of cantilever, y1 is the
deflection caused by the homogenous cantilever mass, ∆𝑚 is
the lumped mass caused due to the mass loading effect by the
gas, & y2 is the deflection caused by this attached mass.
The first five modes of natural frequency of an undamped
cantilever can be derived from this equation. Numerically, they
are given as :
𝐸𝐼
ω1 = 3.516√
𝜌𝐴𝐿4
ω2 = 22.0345√
ω3 = 66.6792√
𝐸𝐼
𝜌𝐴𝐿4
Fig. 4: Third mode undamped natural frequency
𝐸𝐼
𝜌𝐴𝐿4
ω4 = 120.0902√
ω5 = 199.8600√
𝐸𝐼
𝜌𝐴𝐿4
𝐸𝐼
𝜌𝐴𝐿4
All the simulations has been done using Comsol
Multiphysics©
First , we have run the simulation for undamped conditions in
order to validate and ensure that the solver provides corrects
results .
Fig. 5 : Fourth mode undamped natural frequency
Fig. 6: Fifth mode undamped natural frequency
Fig. 2 : First mode undamped natural frequency
After simulating to get the undamped natural frequency modes ,
we have created the damping effect by intentionally modifying the
density differently . The data obtained are as given below:
ωn
ω
(3200
kgm3
)
ω (3600
kgm-3)
ω (3900
ω
ω (4600
kgm-3)
(4100
kgm-3)
kgm-3)
1st
mode
33351
31171
30309
29708
29325
28430
2nd
mode
208430
200220
197350
195450
194280
191680
3rd
mode
300940
288340
2883030
279210
276730
270790
4th
mode
320300
299900
291800
286140
282530
0
274080
5th mode 582280
567280
562270
559000
557000
552590
6th mode 921200
886850
873620
864530
858840
845760
Fig. 10: Fourth mode resonant frequency for density = 3200kg/m3
TABLE I. CONSIDERED DATA FOR THE PRESENT STUDY
Fig. 11: Fifth mode resonant frequency for density = 3200kg/m3
Fig. 7: First mode resonant frequency for density = 3200kg/m3
Fig. 12: Sixth mode resonant frequency for density = 3200kg/m3
Fig. 8: Second mode resonant frequency for density = 3200kg/m3
Fig. 9: Third mode resonant frequency for density = 3200kg/m3
We have then plotted our obtained results to get the figure
given below . :
III. CONCLUSION
As expected , the frequency decreases with increase in the
damping effect . We have plotted the variation of shift in
resonant frequency modes with increase in gas
concentration( i.e, increase in density of tip material in our
present case) .
In reality , this cantilever might not be helpful in getting
results as the adsorption phenomena is much more
complicated and the loading effect varies with several other
factors like temperature , etc . However, this is for the sake
of a simple mathematical analysis only.
REFERENCES
Johnson et Al : Design and Analysis of SAW Based MEMS
Gas Sensor for the Detection of Volatile Organic
Gases(2014).
[2] Rayleigh, L., "On waves propagated along the plane surface
of an elastic solid". Proc. London Math. Soc, 1885. 1(1): p. 411.
[3] Johnson et Al : Design and Analysis of SAW Based MEMS
Gas Sensor for the Detection of Volatile Organic
Gases(2014)
[4] Zhang et Al : Modeling and Analysis of Multiple Attached
Masses Tuning a Piezoelectric Cantilever Beam Resonant
Frequency (2020)
.
[1]