3.4 Surface Stress Versus Surface Charge Density or Potential …
85
3.4.2 Determination of ζ g,q by Dynamic Stress Analysis
Combined with Electrochemical Impedance
Spectroscopy
Surface stress–surface charge density coefficient ζ g,q can be determined as a function
of potential by a dynamic stress analysis (DSA) combined with electrochemical
impedance spectroscopy (EIS) using a cantilever bending [39, 40]. The principle
of DSA combined with EIS is briefly explained in Sect. 2.3.2 of Chap. 2. In a
conventional EIS experiment, the system responds to the application of a sinusoidal
potential, E = E dc + E o exp( jωt), where E dc is a dc potential, E o is the signal
amplitude, and ω is the angular frequency. If E o is sufficiently small, the current
response is linear and is formulated by i = i dc + i o exp[ j(ωt + ψ e )], where ψ e is
the phase angle between current and potential. On the other hand, the corresponding
surface stress response is formulated by g = g dc + g o exp[ j(ωt + ψ s )], where ψ s is
the phase angle between g and E. The ac component g ac of the total surface stress is
g o exp[ j(ωt + ψ s )].
By the way, the complementary explanation of stress impedance in addition to
electrochemical impedance is needed to understand the results obtained by DAS
combined with EIS. The electrochemical impedance Z e is given by [39, 40]:
Z e =
E o exp( jωt)
i o exp[ j(ωt + ψ e )]
=
E o
i o
exp(− jψ e ).
(3.6)
The absolute value of Z e is |Z e | =
E o
i o
. In analogy with the electrochemical
impedance, the stress impedance Z s is formulated by [39, 40]:
Z s =
E o exp( jωt)
g o exp[ j(ωt + ψ s )]
=
E o
g o
exp(− jψ s ).
(3.7)
The reciprocal of Z s is called the stress admittance, Y s (=Z
−1
s ). The ac component
q ac of the total surface charge density can be obtained by integrating the ac density
component i o exp[ j(ωt + ψ e )] with respect to time t:
q ac =
i o exp[ j(ωt + ψ e )]
jω
=
E o exp( jωt)
jωZ e
.
(3.8)
The ratio of g ac to q ac is equivalent to ζ g,q =
∂g
∂q
and is finally derived from Eqs. (3.6),
(3.7), and (3.8) as follows [39, 40]:
ζ g,q =
g ac
q ac
= jω
g o
i o
exp[ j(ψ s − ψ e )]
= ω
g o
i o
cos
ψ s − ψ e +
π
2
+ j sin
ψ s − ψ e +
π
2
= jωY s Z e
(3.9)
85
3.4.2 Determination of ζ g,q by Dynamic Stress Analysis
Combined with Electrochemical Impedance
Spectroscopy
Surface stress–surface charge density coefficient ζ g,q can be determined as a function
of potential by a dynamic stress analysis (DSA) combined with electrochemical
impedance spectroscopy (EIS) using a cantilever bending [39, 40]. The principle
of DSA combined with EIS is briefly explained in Sect. 2.3.2 of Chap. 2. In a
conventional EIS experiment, the system responds to the application of a sinusoidal
potential, E = E dc + E o exp( jωt), where E dc is a dc potential, E o is the signal
amplitude, and ω is the angular frequency. If E o is sufficiently small, the current
response is linear and is formulated by i = i dc + i o exp[ j(ωt + ψ e )], where ψ e is
the phase angle between current and potential. On the other hand, the corresponding
surface stress response is formulated by g = g dc + g o exp[ j(ωt + ψ s )], where ψ s is
the phase angle between g and E. The ac component g ac of the total surface stress is
g o exp[ j(ωt + ψ s )].
By the way, the complementary explanation of stress impedance in addition to
electrochemical impedance is needed to understand the results obtained by DAS
combined with EIS. The electrochemical impedance Z e is given by [39, 40]:
Z e =
E o exp( jωt)
i o exp[ j(ωt + ψ e )]
=
E o
i o
exp(− jψ e ).
(3.6)
The absolute value of Z e is |Z e | =
E o
i o
. In analogy with the electrochemical
impedance, the stress impedance Z s is formulated by [39, 40]:
Z s =
E o exp( jωt)
g o exp[ j(ωt + ψ s )]
=
E o
g o
exp(− jψ s ).
(3.7)
The reciprocal of Z s is called the stress admittance, Y s (=Z
−1
s ). The ac component
q ac of the total surface charge density can be obtained by integrating the ac density
component i o exp[ j(ωt + ψ e )] with respect to time t:
q ac =
i o exp[ j(ωt + ψ e )]
jω
=
E o exp( jωt)
jωZ e
.
(3.8)
The ratio of g ac to q ac is equivalent to ζ g,q =
∂g
∂q
and is finally derived from Eqs. (3.6),
(3.7), and (3.8) as follows [39, 40]:
ζ g,q =
g ac
q ac
= jω
g o
i o
exp[ j(ψ s − ψ e )]
= ω
g o
i o
cos
ψ s − ψ e +
π
2
+ j sin
ψ s − ψ e +
π
2
= jωY s Z e
(3.9)