2.3 Cantilever Bending Method for Measurement of Changes in Surface …
53
α = 4θ ≈ 0. The distance x between the clamped end and reflection point of the
cantilever is negligibly small as compared to the distance W between the clamped
end of the cantilever and the detector plane. Equation (2.18) leads to
θ ≈
a
4W
.
(2.19)
The substitution of Eq. (2.19) into Eq. (2.17) provides the following relationship [1]:
1
R
= κ ≈
a
2LW
.
(2.20)
Equation (2.20) means that the position of the reflected light spot on the detector
plane moves in response to the change in bending of the cantilever (i.e.,
1
R
or
κ). Therefore, the displacement a of the reflected light spot on the detector plane
due to the change in bending of the cantilever is given by [1, 24]:
1
R
= κ ≈
a
2LW
.
(2.21)
The optical detection of the cantilever bending has been used as a common technique for measurement of the surface stress change of a solid electrode in electrolyte
solution [18, 25–32], which needs an electrochemical cell with an optical window.
In an electrochemical system, a thin metal or semiconductor film coated on one side
of a rectangular cantilever plate consisting of a glass or Si wafer, mica, etc., is mostly
used as an electrode and the changes in surface stress can be measured from the
change in the curvature (reciprocal of the curvature radius) of the cantilever as a
function of the electrode potential or the surface charge density. Figure 2.11 shows
the schematic representation and optical configuration of the cantilever bending setup
in an electrochemical system [1, 33]. The problem in the electrochemical system is
that the refraction of a laser light occurs at the optical window due to the difference
in refractive index between air and solution.
As seen from Fig. 2.11, if the incident light is exactly normal to the optical window
plane (or to the air/solution interface), no refraction of the incident light occurs at
the optical window. Nevertheless, even if the normal incidence of the laser light,
the direction of the light reflected by the surface of the cantilever is not normal to
the optical window plane because of the bending of the cantilever. As a result, the
refraction of the reflected light occurs at the optical window and the direction of
the reflected light after passing the optical window shifts downward due to the low
refractive index of air as compared to solution. The position of the spot of the reflected
light on the position-sensitive detector also shifts downward (B
→ B). This means
that the refraction of the reflected light at the optical window has to be taken into
consideration for the exact determination of the curvature or curvature radius of the
cantilever in the electrochemical system [1, 33]. In Fig. 2.11, the distances between
the optical window and the cantilever and between the spots of the incident light
53
α = 4θ ≈ 0. The distance x between the clamped end and reflection point of the
cantilever is negligibly small as compared to the distance W between the clamped
end of the cantilever and the detector plane. Equation (2.18) leads to
θ ≈
a
4W
.
(2.19)
The substitution of Eq. (2.19) into Eq. (2.17) provides the following relationship [1]:
1
R
= κ ≈
a
2LW
.
(2.20)
Equation (2.20) means that the position of the reflected light spot on the detector
plane moves in response to the change in bending of the cantilever (i.e.,
1
R
or
κ). Therefore, the displacement a of the reflected light spot on the detector plane
due to the change in bending of the cantilever is given by [1, 24]:
1
R
= κ ≈
a
2LW
.
(2.21)
The optical detection of the cantilever bending has been used as a common technique for measurement of the surface stress change of a solid electrode in electrolyte
solution [18, 25–32], which needs an electrochemical cell with an optical window.
In an electrochemical system, a thin metal or semiconductor film coated on one side
of a rectangular cantilever plate consisting of a glass or Si wafer, mica, etc., is mostly
used as an electrode and the changes in surface stress can be measured from the
change in the curvature (reciprocal of the curvature radius) of the cantilever as a
function of the electrode potential or the surface charge density. Figure 2.11 shows
the schematic representation and optical configuration of the cantilever bending setup
in an electrochemical system [1, 33]. The problem in the electrochemical system is
that the refraction of a laser light occurs at the optical window due to the difference
in refractive index between air and solution.
As seen from Fig. 2.11, if the incident light is exactly normal to the optical window
plane (or to the air/solution interface), no refraction of the incident light occurs at
the optical window. Nevertheless, even if the normal incidence of the laser light,
the direction of the light reflected by the surface of the cantilever is not normal to
the optical window plane because of the bending of the cantilever. As a result, the
refraction of the reflected light occurs at the optical window and the direction of
the reflected light after passing the optical window shifts downward due to the low
refractive index of air as compared to solution. The position of the spot of the reflected
light on the position-sensitive detector also shifts downward (B
→ B). This means
that the refraction of the reflected light at the optical window has to be taken into
consideration for the exact determination of the curvature or curvature radius of the
cantilever in the electrochemical system [1, 33]. In Fig. 2.11, the distances between
the optical window and the cantilever and between the spots of the incident light