Mechanical analysis and optimization of a microcantilever sensor
coated with a solid receptor film
Genki Yoshikawa*
International Center for Materials Nanoarchitectonics (MANA),
National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, 305-0044, Japan
Supplementary Material
Dependence of cantilever deflection on the thickness and Young’s modulus of a coating film
The deflection of a cantilever varies according to the given parameters of the cantilever and
coating film. Figure S1 shows the dependence of cantilever deflection on the thickness and Young’s
modulus of coating film with various parameters. Note that all figures in Fig. S1 are plotted with
same scales in both horizontal and vertical axes. It is found that the position of optimum coating
thickness, with which the deflection peak is observed, is not affected by the length, width, or the
amount of induced strain in the coating film as confirmed in Figs. S1(a)~(d) and equation (S2),
whereas these parameters determine the amount of deflection. As seen in Figs. S1(e) and (f), the
thickness of cantilever and Poisson’s ratio of coating film affect both optimum thickness and
deflection although the influence of Poisson’s ratio is rather small in the present case. By changing
the cantilever material (having different Young’s modulus or Poisson’s ratio), the optimum coating
thickness and amount of deflection are also varied as shown in Figs. S1(g)~(i).
1
Calculation and analysis of the optimum coating thickness
In order to find the optimum coating thickness values, the derivative of the deflection with
respect to the coating thickness is calculated as follows:
1


  E w t (1  )

 2  Ec wc tc (1  f )
 2
c
 f f f

 4tf  
 4  tc  6t f tc

 E w t (1  )

  Ec wc tc (1  f )

f
f f
c





d z
2
3
2






E
w
t
(1


)
E
w
t
(1


)
E
w
t
(1


)
 3l  f
f
f f
c
c
f
 4  f f f
 c cc2
 6tc  
 (t f  tc )  2t f 
dt f
 E w t (1  ) E w t (1  )



c c c
f
f
f f
c
 Ec wc tc (1  f )






2
  E f w f t f (1  c )

 2  Ec wc tc (1  f )
 2


 4tf  
 4  tc  6t f tc 
 



 E w t (1  )

 Ec wc tc (1  f )

c

 f f f





(S1)
By solving (dΔz/dtf) = 0 for tf, five solutions are obtained. Among them, the following one yields
the optimum coating thickness (tf-op) with reasonable values:
t f op 
tc
 X 1/3  X 1/3  1
2
(S2)
where
X
2 Ec wc 1  f   E f w f 1  c   2 Ec wc 1  f
E f w f 1  c 
  E w 1    E
c
c
f
f
w f 1  c 

(S3)
which can be simplified in the case of wc = wf, using Uc = Ec(1-νf) and Uf = Ef(1-νc):
X
2U c  U f  2 U c U c  U f 
Uf
(S4)
In the case of νc ~ νf, X can be approximately simplified further as follows:
X
2 Ec  E f  2 Ec  Ec  E f 
Ef
(S5)
Note that equations (S2) and (S4) are same as equations (5) and (6) in the text, respectively.
Equations (S2) ~ (S5) indicate several important aspects of the optimum coating thickness (tf-op) as
follows:
1) tf-op is proportional to the cantilever thickness (tc).
2) tf-op is determined by widths (wc and wf), Young’s moduli (Ec and Ef), and Poisson’s ratios (νc and
νf) of the cantilever and coating film as well as tc.
3) tf-op does not depend on the length (l) or the amount of the induced strain in the coating film (εf).
4) In the case of wc = wf, tf-op is no longer dependent on the width and determined only by Young’s
2
moduli (Ec and Ef), Poisson’s ratios (νc and νf), and tc.
5) Further, in the case of νc ~ νf, tf-op depends only on the two parameters; Young’s moduli (Ec and
Ef) and tc.
6) In the case of Uf ~ Uc, X approaches ~1, resulting in tf-op ~ tc/2.
7) The smaller (larger) value of Ef leads to the smaller (larger) value of X, resulting in the higher
(lower) value of tf-op owing to the dominant contribution of X-1/3 term in equation (S2).
Using equations (S2) as well as (S3) ~ (S5), one can readily find the optimum coating film
thickness (tf-op) under given conditions and can check its dependence on the relevant parameters.
For example, the dependence of optimum coating thickness on Young’s modulus of coating film
(Ef) is plotted in Fig. S2. It is found that the optimum coating thickness is higher (lower) for a
coating film with smaller (larger) Young’s modulus (cf. Figs. 4, S1, S2, and the above point 7).
Thus, the optimum coating thickness can be attributed to a specific point where the additional
stiffness due to the additional thickness of coating film becomes dominant over the effective force
induced by the entire coating film.
It should be noted that equation (S2) results in a non-numerical tf value in the case of
Ecwc(1-νf) > Efwf(1-νc) or Uf > Uc or Ef > Ec because of negative values appearing in the square roots
in equations (S3) ~ (S5). In such a case, one has to plot the deflection of cantilever as those shown
in Fig. S1 to find the optimum coating thickness.
3
4
FIG. S1. The dependence of cantilever deflection on the thickness and Young’s modulus of coating
film with various parameters. All figures are plotted with same scales in both horizontal and vertical
axes. The legend is shown only in (a) and (g), whereas the same color corresponds to the same
Young’s modulus in all cases. The other parameters used in each case are summarized in Table SI.
The assumed material of cantilever is indicated in each title with the parameter varied from those in
the case of (a) in the parenthesis. The widths of the cantilever and coating film are set at the same
value w (= wc = wf). (a) ~ (f) are based on the parameters of Si cantilever (Ec = 170 [GPa] and νc =
0.28). (g) is based on a different Poisson’s ratio (νc = 0.40) than that of Si. (h) and (i) are based on
the parameters of SU-8 cantilever (Ec = 5 [GPa] and νc = 0.30).
5
TABLE SI. The parameters used in Fig. S1. The parameters varied from those in the case of (a) are
described in bold characters with gray highlight.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Ec (GPa)
170
170
170
170
170
170
170
5
5
νc
0.28
0.28
0.28
0.28
0.28
0.28
0.40
0.30
0.30
νf
0.30
0.30
0.30
0.30
0.30
0.40
0.30
0.30
0.30
tc (μm)
1
1
1
1
7
1
1
1
7
l (μm)
500
2000
500
500
500
500
500
500
500
w (μm)
100
100
500
100
100
100
100
100
100
εf (ppm)
10
10
10
100
10
10
10
10
10
6
FIG. S2. Dependence of optimum coating thickness (tf-op) on Young’s modulus of coating films.
Pink, blue, and light blue lines correspond to cantilevers made of silicon (Ec = 170 [GPa], νc = 0.28),
while yellow line to the cantilever made of SU-8 (Ec = 5 [GPa], νc = 0.30). Poisson’s ratio of
coating film (νf) is set at 0.40 for blue line and at 0.30 for pink, light blue, and yellow lines. The
optimum thicknesses of PMMA, PU, and CMC found in Fig. 4 are indicated as black, red, and
green dots.
7