Supplementary material: Piezoelectric-actuated, piezoresistive-sensed circular
micromembranes for label-free biosensing applications
T. Alava1, F. Mathieu1, P. Rameil2, Y. Morel2, C. Soyer3, D. Remiens3 and L. Nicu1
1
CNRS ; LAAS ; 7, avenue du Colonel Roche, F-31077 Toulouse, France
2
DGA Maitrise MNRBC, DGA/DT/CEB, 5 rue Lavoisier, 91710 Vert le Petit, France
3
University of Lille Nord de France, F-59000 Lille, France
Frequency scans on 5 different channels, each corresponding to one specific membrane,
were possible due to a dedicated analogue electronics specifically designed and assembled1
for this application (Figure 1).
Figure 1. (Color online) Zoom on micromembranes’ (0,0) resonance mode (440-µm radius,
purple curve, D; 430-µm radius red curve, B; 420-µm radius, green curve, C; 400-µm radius,
blue curve, A. The 137 kHz resonance peak appearing on the four curves is the (0,0) mode of
the 360-µm radius reference membrane.
1
Table I summarize the measured resonant frequencies in air and liquid (as well as the
corresponding quality factors) for the first three modes of vibration presenting a nodal circle,
i.e. modes (0,i) with i=0,2.
RM
Mode
f res _ air _ meas
(m,n)
(kHz)
360 µm
(0,0)
(0,1)
(0,2)
400 µm
(0,0)
(0,1)
(0,2)
420 µm
(0,0)
(0,1)
(0,2)
430 µm
(0,0)
(0,1)
(0,2)
440 µm
(0,0)
(0,1)
(0,2)
Qair
f res _ liq _ meas
Qliq
(kHz)
f res _ liq _ mod el
f liquid
(kHz)
(model vs. meas.)
32,3
127,9
108,3
38,9
66,6
(±2,2)
(± 9,5)
(±0,7)
(± 8,4)
524,5
142,6
213,3
92
(±2,8)
(± 16,2)
(±1,4)
(± 2,6)
1133,5
74,6
526
(±2.6)
(± 16,9)
(±3,6)
109,8
96,6
32,2
47,3
(±1,1)
(± 9,8)
(±0,9)
(± 20,9)
417
169
168,4
96,3
(±7)
(± 26)
(±4,5)
(± 0,4)
884,2
188,5
Unavailable
Unavailable
Unavailable
(±20)
(± 11,5)
105,1
95,3
30,2
76
24,7
(± 2,3)
(± 7,1)
(±0,7)
(± 2)
392,7
150
156,4
97,6
(± 6.5)
(± 19)
(±3,1)
(± 8,9)
826,9
122,6
280,6
Unavailable
Unavailable
(± 19,7)
(± 23,1)
100,4
92
29,36
79,6
22,9
(± 2,1)
(± 6,6)
(±0,4)
(± 20,4)
377,2
138
150,5
102,6
(± 3,7)
(± 24)
(± 2,9)
(± 14,9)
794,2
118,7
376,4
(± 8,2)
(± 22,4)
(±47,5)
97,8
100,3
28,1
71
(± 1,1)
(± 4,4)
(±0,6)
(± 10,7)
363,8
129
143,2
94,7
(± 1,8)
(± 22)
(± 2)
(± 3,1)
761,9
168,6
Unavailable
Unavailable
(± 2,6)
(± 35,6)
270
292
194
515,45
25,97
148,4
135,8
129,1
20%
9,7%
1,5%
23,7
13,3
Unavailable
24%
14,6%
Unavailable
27,7%
16%
332,8
1%
22,1
26,7%
123,3
Unavailable
15,7%
Unavailable
2
Table I. Measured (average values for three devices per type of micromembrane and
corresponding precisions) and calculated resonant frequencies for all dimensions of circular
micromembranes in case of modes (0,i) with i integer from 0 to 2. The corresponding
measured quality factors are equally reported.
We used the extended Lamb’s model2 to analytically determine the theoretical values
(seventh column of Table I) of the membranes’ resonant frequencies in liquid. The model is
based on the seminal works of H. Lamb who proposed in 1920 a theoretical model3 that
allows estimating the shift of the resonant frequency from vacuum to water of a thin circular
plate filling an aperture in a plane (and rigid) wall which is in contact on one side with an
unlimited mass of water
In this model, the kinetic energy TL of the liquid is calculated and given by:
 dw 
TL  0.21   L  R  

 dt 
2
3
(1)
where  L is the liquid density, R is the membrane radius and w(t,r) is the normal
displacement of the membrane at a distance r from the centre.
Adding TL to the kinetic energy of the vibrating membrane and applying the energetic
Rayleigh-Ritz method3 provide an additional factor for the calculation of the resonant
frequency in liquid:
fL 
f vac
1 
(2),
where  , usually called added virtual mass incremental (AVMI) factor, is equal to:
3
  0.67 
L R
M h
(3),
with M and h respectively the density and the thickness of the membrane.
References
1
T. Alava, N. Berthet-Duroure, C. Ayela, E. Trevisiol, M. Pugniere, Y. Morel, P.
Rameil, and L. Nicu, Sens. Act. Chem. B 138 (2), 532 (2009).
2
M. Amabili and M. K. Kwak, J. Fluids Struct. 10, 743 (1996).
3
H. Lamb, Proc. Royal Soc. London A 98, 205 (1920).
4