2.3 Cantilever Bending Method for Measurement of Changes in Surface …
55
β ≈ tan β =
a − y
W − d
≈
a
W
.
(2.24)
Since the position B on the SPD is really measured as a spot of the reflected light, it is
necessary to know the relationship between α and β in order to obtain the curvature
radius from Eq. (2.22).
The relationship between α and β is derived from the law of refraction (i.e., Snell’s
law) at the interface between two media of different refractive indices:
sin α
sin β
=
n a
n s
,
(2.25)
where n a and n s are the refractive indices of air and solution, respectively. For small
deflections, Eq. (2.25) leads to
sin α
sin β
≈
α
β
≈
1
n s,a
,
(2.26)
where n s,a is the refractive index of the solution with respect to air, and it is practically
equal to n s because of n a = 1.0003. Furthermore, α can be obtained from Eqs. (2.24)
and (2.26):
α ≈
a
n s,a W
.
(2.27)
Thus, the curvature radius of the cantilever R is eventually given by [1, 33]:
R ≈
2n s,a LW
a
.
(2.28)
Equation (2.13) or Eq. (2.14) proves that the change in surface stress g is proportional to the change in reciprocal curvature radius
1
R
. If applying Eq. (2.28) to
two different deflections of the cantilever, the following relationship is obtained:
1
R
≈
a
2n s,a LW
,
(2.29)
where a is the displacement of the reflected light spot on the PSD due to the change
in bending of the cantilever electrode. The refractive indexes of aqueous solutions
are in the range of n s,a = 1.33–1.48. The neglect of the refraction of the reflected
light at the optical window in the electrochemical system causes an error of about
25–30% for the determination of
1
R
or g [1, 33].
55
β ≈ tan β =
a − y
W − d
≈
a
W
.
(2.24)
Since the position B on the SPD is really measured as a spot of the reflected light, it is
necessary to know the relationship between α and β in order to obtain the curvature
radius from Eq. (2.22).
The relationship between α and β is derived from the law of refraction (i.e., Snell’s
law) at the interface between two media of different refractive indices:
sin α
sin β
=
n a
n s
,
(2.25)
where n a and n s are the refractive indices of air and solution, respectively. For small
deflections, Eq. (2.25) leads to
sin α
sin β
≈
α
β
≈
1
n s,a
,
(2.26)
where n s,a is the refractive index of the solution with respect to air, and it is practically
equal to n s because of n a = 1.0003. Furthermore, α can be obtained from Eqs. (2.24)
and (2.26):
α ≈
a
n s,a W
.
(2.27)
Thus, the curvature radius of the cantilever R is eventually given by [1, 33]:
R ≈
2n s,a LW
a
.
(2.28)
Equation (2.13) or Eq. (2.14) proves that the change in surface stress g is proportional to the change in reciprocal curvature radius
1
R
. If applying Eq. (2.28) to
two different deflections of the cantilever, the following relationship is obtained:
1
R
≈
a
2n s,a LW
,
(2.29)
where a is the displacement of the reflected light spot on the PSD due to the change
in bending of the cantilever electrode. The refractive indexes of aqueous solutions
are in the range of n s,a = 1.33–1.48. The neglect of the refraction of the reflected
light at the optical window in the electrochemical system causes an error of about
25–30% for the determination of
1
R
or g [1, 33].