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Influence of Geometry-Induced Frequency Dispersion on the Impedance of Ring Electrodes
Accepted Manuscript Title: Influence of Geometry-Induced Frequency Dispersion on the Impedance of Ring Electrodes Author: Yu-Min Chen Christopher L. Alexander Christopher Cleveland Mark E. Orazem PII: DOI: Reference:
S0013-4686(17)30510-8 http://dx.doi.org/doi:10.1016/j.electacta.2017.03.044 EA 29078
To appear in:
Electrochimica Acta
Received date: Revised date: Accepted date:
18-1-2017 3-3-2017 6-3-2017
Please cite this article as: Yu-Min Chen, Christopher L. Alexander, Christopher Cleveland, Mark E. Orazem, Influence of Geometry-Induced Frequency Dispersion on the Impedance of Ring Electrodes, (2017), http://dx.doi.org/10.1016/j.electacta.2017.03.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Research Highlights
Highlights:
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Finite-element simulations of the impedance of ring electrodes. Characteristic dimensions for ring electrodes identified Characteristic frequencies found at which ring electrodes cause time-constant dispersion
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*Revised Manuscript (including Abstract) Click here to view linked References
Influence of Geometry-Induced Frequency Dispersion on the Impedance of Ring Electrodes Yu-Min Chena , Christopher L. Alexandera , Christopher Clevelanda , Mark E. Orazema,∗ of Chemical Engineering, University of Florida, Gainesville, FL, 32611, USA
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Abstract
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Finite-element simulations for the impedance response of ring electrodes were used to identify the characteristic frequency associated with the influence of electrode geometry on impedance response. An approximate expression for the characteristic ring-electrode dimension was found to be adequate for a wide ring, and have an error of 20 percent for a thin ring. A refined expression for the characteristic dimension is presented. The characteristic frequency associated with the influence of ring-electrode geometry is always larger than that associated with the geometry of a disk. These results guide the design of ring-shaped sensors that employ impedance measurements. Keywords: sensors, numerical simulation, impedance spectroscopy, frequency dispersion
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1. Introduction
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The ring-electrode geometry has been widely used in electrochemical systems with electrochemical impedance spectroscopy (EIS) as an analysis technique. Hsieh et al. [1] conducted impedance measurements on ring-shaped interdigitated electrodes to characterize the concentration change of glycated hemoglobin. Besio and Prasad[2] used a concentric ring electrode to analyze the skin– electrode impedance. They performed single-frequency impedance measurements at 1 kHz and concluded that copper is the best material among several metals for their biosensor. Li et al. [3] conducted impedance measurements on a rotating ring electrode to determine the mechanism of the chlorine evolution reaction. Their work suggested that, on a Pt surface, the rate of chloride discharge and simultaneous chlorine adsorption is first order with respect to chloride concentration; whereas, the rate of the adsorption and desorption process is second order with respect to chloride concentration. The current and potential distribution for the ring electrode geometry has been widely studied. The primary current distribution associated with the ring electrode geometry is nonuniform. Pierini and Newman [4] presented an analytic solution for the primary and secondary current distribution on ring electrodes. Datta et al. [5] used the finite-element method to calculate the steady-state current and potential distribution for concentric ring-ring electrodes and double concentric (ring-ring-ring) electrodes used for transcranial current stimulation. Their work showed that the electric field on a ring electrode decreases rapidly in the radial direction. Mansor and Ibrahim [6] also performed simulations indicating that the electric field on a ring interdigitated electrode reaches a maximum at the ring edges and a minimum at the center of the rings. The term frequency or time-constant dispersion is used to describe the broadening of the impedance response associated with a distribution of time constants. Frequency dispersion associated with an electrode geometry originates from the nonuniform current and potential distribution, which changes as a function of frequency. Newman [7] showed, by solution of Laplace’s equation, that frequency dispersion is observed for disk electrodes for frequencies larger than a characteristic value.
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∗ Corresponding
author Email address: [email protected] (Mark E. Orazem)
Preprint submitted to Electrochimica Acta
March 3, 2017
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A dimensionless frequency may be expressed as [7–11] K=
ωC0 `c,disk . κ
(1)
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where ω is the angular frequency, κ is the electrolyte conductivity, C0 is the capacitance of the disk electrode, and `c,disk is the characteristic length for a disk electrode. Huang et al. [8] showed that, for `c,disk = r0 (2)
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the characteristic frequency for a disk electrode occurs near K = 1; thus, the associated characteristic dimension for the disk electrode is the radius of the disk. Alexander et al. [11] showed that the characteristic dimension for a rough disk is given by `c,rough disk = fr r0
(3)
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where fr is the rugosity or roughness factor defined to be the ratio of the actual surface area to the superficial surface area of the disk. Alexander et al. [11] showed further that the characteristic dimension for roughness is `c,roughness = fr2 P (4)
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2. Mathematical Development
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where P is the period associated with roughness. Equation (1) facilitates estimation of the characteristic frequency at which dispersion influences the impedance response for a given electrode. For K > 1, frequency dispersion is associated with the nonuniform current distribution. Therefore, frequency dispersion may be eliminated in a desired frequency range by choosing the appropriate parameters `c and κ to ensure that K < 1. The objective of this work is to find the characteristic dimension associated with the impedance response of a ring electrode.
The ring serves as the working electrode, which is embedded in an infinite, insulated plane with a hemispherical counterelectrode located at infinity. For the present work, the geometric parameters r1 and r2 represent the inner and outer radii of the ring electrode, respectively. The potential distribution for an electrolyte with uniform conductivity is governed by Laplace’s equation, which may be expressed in cylindrical coordinates as ∂Φ ∂2Φ 1 ∂ r + =0 (5) r ∂r ∂r ∂y 2
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where axial symmetry was assumed. e components as The potential can be expressed in terms of steady-state Φ and oscillating Φ e jωt } Φ = Φ + Re{Φe
(6)
Similarly, the potential applied at the electrode surface can be expressed as V = V + Re{Ve ejωt }
(7)
where V is the steady-state value and Ve is the oscillating value. The steady-state solution for the current distribution at a blocking electrode shows that the potential is uniform and the current is equal to zero. As the problem is linear, solution for the oscillating variables does not require a solution for the steady-state equation.[9] The impedance response of the ring requires solution of ! e e 1 ∂ ∂Φ ∂2Φ r + =0 (8) r ∂r ∂r ∂y 2 2
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subject to boundary conditions that the oscillating potential is equal to zero for distances far away from the disk, i.e., e → 0 as r2 + y 2 → ∞ Φ (9) and that, on the insulating plane surrounding the ring, e ∂Φ =0 ∂y y=0
for r < r1
and r > r2
(10)
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For a blocking electrode, Huang et al. [8] showed that the flux condition at the surface of a disk electrode can be expressed as ∂ (V −Φ(0)) ∂Φ (11) = −κ C0 ∂t ∂y y=0
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where Φ(0) is the potential outside the diffuse part of the double layer. The corresponding expression in terms of oscillating variables is given by e ∂ Φ e (12) jωC0 Ve − Φ(0) = −κ ∂y y=0
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The impedance was calculated as
Z (ω) =
Ve Ie
(13)
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for a specified range of frequencies, where Ie is the oscillating current at the working electrode obtained by integrating the local current density over the surface of the electrode.
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3. Numerical Method
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The finite-element-analysis solver and simulation software COMSOL Multiphysicsr was used to solve Laplace’s equation in cylindrical coordinates for the ring electrode. Triangular elements were employed with quadratic interpolation. Due to the singularities that arise near the edge of the ring electrode, a nonuniform meshing scheme was implemented near the surface of the ring electrode. A direct linear solver was used. The meshing was refined manually to ensure that the impedance response of a disk electrode yielded the correct high-frequency primary resistance, i.e., Re κr0 = 1/4.[12] Similar mesh sizes were then used for the ring electrodes. In contrast to the frequency-dependent adaptive mesh algorithm employed by Michel et al.,[13] the same meshing was then employed for all frequencies. Calculation of 80 frequencies (10 points per decade from 10−2 Hz to 109 Hz) required less than 30 min on a 64-bit Dell Precision T7400 workstation with dual Xeon E5410 2.33 GHz processors and 32G Byte of RAM. A schematic representation of the finite element mesh used for the ring electrode simulations is provided in Figure 1 detailing: a) the entire domain and b) an enlarged region showing the ring electrode embedded in an insulating plane. The counterelectrode in Figure 1(a) was placed at a distance 500-1000 times that of the outer ring radius to minimize the influence of the counterelectrode placement. The exposed ring electrode surface was coplanar to the inner and outer insulating surfaces.
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4. Results Finite-element (COMSOLr ) simulations were performed to explore the influence geometry has on the impedance response for a ring electrode. Values of r1 were varied from 0 to 0.995 cm, and values of r2 ranged from 0.102 to 2 cm, yielding values of r1 /r2 ranging from 0 to 0.995. The capacitance was fixed at C0 = 1 µF/cm2 , and the conductivity was fixed at κ = 0.02 S/cm. These
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b
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b
a
Insulator
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Ring electrode
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Insulator
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Figure 1: Schematic representation of a sample finite element mesh used for ring electrode simulations: a) entire domain and b) an enlarged region showing the two insulating surfaces and the ring electrode.
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105 Hz
r1/r2 0.40 0.80 0.91 0.98
0.4
-Zj r2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Disk r1=0
0.2
104 Hz 105 Hz
105 Hz
0.0 0.0
107 Hz
0.2
0.4
0.6
Zr r2 Figure 2: Scaled calculated impedance presented in Nyquist format with r1 /r2 as a parameter. The dashed line represents the impedance of a disk electrode. Simulations were performed for 10 logarithmically spaced frequencies per decade, and symbols are used to highlight every 2 decades. Ring dimensions are presented in Table 1.
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Table 1: Ring electrode geometries used for the simulations presented in Figures 2 and 3.
r1 , cm 0.1 0.2 0.4 0.1 0.2
r2 , cm 1 0.5 0.5 0.11 0.205
fc , kHz 3.32 10.9 46.1 507 1050
ip t
r1 /r2 0.1 0.4 0.8 0.9091 0.9756
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values could correspond to an oxide film on an electrode surface, but, as the results are presented in dimensionless form, they may be applied to other systems. A Nyquist plot is presented in Figure 2 for some representative calculations. At low frequencies, the results indicate a vertical line corresponding to a capacitive behavior. The deviation from a vertical line observed at high frequencies has been attributed for a disk electrode, shown by dashed lines, to frequency dispersion associated with electrode geometry. The frequencies indicated in Figure 2 do not show a repeating pattern as the value of r2 was not the same for the different simulations. The ring dimensions are presented in Table 1. The dimensionless modulus of the impedance is presented in Figure 3(a) as a function of frequency. At a given frequency, the dimensionless modulus was smaller for the wider ring electrodes, and this effect is most visible at low frequency. The smaller impedance of the wider ring electrodes can be attributed to the larger surface area. The modulus for the ring electrode approaches that of the disk as r1 /r2 → 0. The imaginary-impedance-derived phase angle can be expressed as dlog Zj × 90◦ . dlog f
(14)
d
ϕdZj =
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Alexander et al. [11] showed that the imaginary-impedance-derived phase angle is very sensitive to the onset of frequency dispersion for capacitive electrodes. The calculated imaginary-impedancederived phase angle is presented in Figure 3(b) as a function of frequency. Frequency dispersion, seen as a deviation of the phase angle from −90◦ , is dependent on the geometric parameter r1 /r2 . At frequencies less than 1 kHz, the geometry of the ring does not introduce frequency dispersion for the parameters used in the simulation. As the ring width decreases, the onset of frequency dispersion occurs at a higher frequency.
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4.1. Approximate Dimensionless Frequency Based strictly on the observation that the onset of frequency dispersion varies according to the ring electrode width divided by a factor accounting for curvature, an approximate expression for the dimensionless characteristic frequency is ωC0 r2 − r1 K= (15) κ 1 + r12 /r22 where the characteristic dimension for the ring electrode is given by `c,ring =
r2 − r1 1 + r12 /r22
(16)
A measure of the suitability of equation (15) is found by seeking the superposition of the results shown in Figure 3(b) when plotted as functions of K. The results are shown in Figure 4 with r1 /r2 as a parameter. The value r1 /r2 = 0 corresponds to a disk electrode. The results shown in Figure 4 are approximately superposed as compared to the results presented in Figure 3(b).
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1 0
3
1 0
2
1 0
1
1 0
0
r1 /r 0 .1 0 .4 0 .8 0 .9 0 .9
1 0
2
0 0 0 1 8
D is k r 1 = 0
-1
1 0
-2
1 0
-1
1 0
0
1 0
1
2
1 0
3
1 0
1 0
4
1 0
5
6
1 0
1 0
7
1 0
f / H z
r1 /r 0 .1 0 .4 0 .8 0 .9 0 .9
9
1 0
1 0
2
0 0 0 1 8
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-8 5
1 0
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D is k r 1 = 0
j
/ d e g re e
-9 0
8
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(a)
-8 0
-7 5 0
1 0
1
1 0
2
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1 0
d
d Z
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4
cr
2
|Z | r
1 0
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1 0
3
1 0
4
1 0
5
1 0
6
f / H z (b)
Figure 3: Scaled calculated impedance presented in Bode format as a function of frequency with r1 /r2 as a parameter: a) scaled modulus of the impedance and b) imaginary-impedance-derived phase angle. The ring dimensions are presented in Table 1. The dashed line represents the impedance of a disk electrode. Simulations were performed for 10 logarithmically spaced frequencies per decade, and the line represents the interpolation between calculated values.
r1 /r 0 0 .1 0 .4 0 .8 0 .9 0 .9
-8 5
2
D is k 0 0 0 1
K = 1 8
j
/ d e g re e
-9 0
d Z
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
-8 0
-7 5 1 0
-2
1 0
-1
1 0
0
1 0
1
Figure 4: Ring electrode imaginary-impedance-derived phase angle as a function of dimensionless frequency with r1 /r2 as a parameter.
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1 .1
1 .0 0 .2
0 .4
0 .6
r 1/r 2
0 .8
1 .0
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0 .0
cr
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g
1 .2
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Figure 5: The correction factor g as a function of the geometric ratio r1 /r2 .
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The superposition presented in Figure 4, however, is not perfect. In the frequency domain close to K = 1, the ring electrodes have an imaginary-impedance-derived phase angle greater than that of the disk electrode (r1 /r2 = 0). Frequency dispersion for the ring electrodes is seen for values of dimensionless frequency that are smaller than K = 1. In contrast, as is seen in Figure 3(b), frequency dispersion for the ring electrodes is seen at frequencies that are larger than that for the disk electrode. 4.2. Correction to the Dimensionless Frequency
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A correction to the dimensionless frequency was obtained by requiring that the corrected dimensionless frequency K ∗ = Kg = 1 at the phase angle ϕdZj = −88.366◦ for which K = 1 for the disk electrode. The correction factor g can be expressed as a fifth-order polynomial as 2 3 4 5 r1 r1 r1 r1 r1 g = 1 + 1.0742 − 4.2227 + 8.9244 − 9.4462 + 3.8806 (17) r2 r2 r2 r2 r2
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The correction factor g is shown in Figure 5 as a function of geometric ratio r1 /r2 . When the geometric ratio is extremely close to 0, the ring electrode can be regarded to be a disk, and the correction factor approaches unity. When the geometric ratio is greater than 0, the correction factor increases with the geometric ratio, reaching a maximum value of 1.2 as r1 /r2 → 1. The corrected dimensionless frequency may be expressed as ωC0 r2 − r1 ∗ g (18) K = κ 1 + r12 /r22 and the characteristic length can be expressed as π r2 − r1 ∗ `c,ring = g 4 1 + r12 /r22
(19)
A comparison is presented in Figure 6 among the approximate characteristic length given in equation (16), the simulated values, and the refined expression given in equations (17) and (19). The characteristic length approaches r2 as r1 /r2 → 0 and zero as r1 /r2 → 1. The imaginary-impedance-derived phase angle is provided in Figure 7 as a function of dimensionless frequency K ∗ given in equation (18) with r1 /r2 as a parameter. The results for different values of r1 /r2 are superposed near K ∗ = 1, and frequency dispersion is observed at K ∗ = 1. 7
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COMSOL
1.0
Approximation Polynomial fit
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0.8
cr
c
/ r2
0.6 0.4
0.0
0.2
0.4
0.6
0.8
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0.0
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0.2
1.0
r1/r2
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Figure 6: Characteristic length as a function of the geometric ratio r1 /r2 . COMSOLr simulation results are shown as blue triangles, the empirical approximation given as equation (16) is shown as red squares, and polynomial fit is given as a black line.
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r1 /r 0 0 .1 0 .4 0 .8 0 .9 0 .9 2
D is k 0 0 0 1 8
j
/ d e g re e
-9 0
d Z
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
K
-8 0
-7 5 1 0
-2
1 0
-1
*
= 1
1 0
0
1 0
1
Figure 7: Ring electrode imaginary-impedance-derived phase angle as a function of dimensionless frequency K ∗ (equation (18)) with r1 /r2 as a parameter.
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1 0
1 0 .1
1
2
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/ r
cr
0 .0 1
ip t
fc
,r in g
/ fc
,d is k
1 0 0
Figure 8: Ratio of the characteristic frequency of the ring electrode fc,ring to the characteristic frequency of the disk electrode fc,disk as a function of the corrected characteristic length `∗c,ring scaled by the outer radii of the disk, r2 .
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5. Discussion
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In order to eliminate the influence of frequency dispersion in a desired frequency range, the ring dimension may be selected to ensure that K ∗ < 1, i.e., `∗c,ring ≤
κ
2πfmax C0 g
(20)
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where fmax is the maximum desired frequency. Similar expressions were developed by Huang et al. [9] for a disk electrode. For a given electrolyte conductivity and electrode capacity, the frequency at which the ring electrode geometry influences the impedance response can be compared to that associated with the disk electrode geometry as r2 1 + (r1 /r2 )2 fc,ring (21) = fc,disk (r1 − r2 ) g
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The results are presented in Figure 8 as a function of `∗c,ring /r2 . The characteristic frequency for a ring electrode is always greater than that for a disk electrode with radius equal to r2 . For a thin ring, the characteristic frequency can be several orders of magnitude larger than that for the disk electrode. As the value of r1 approaches zero, the characteristic dimension given in equation (19) approaches r2 , the radius of a disk. As the value of r1 approaches r2 , the characteristic dimension approaches 1.2(r1 − r2 ), which is slightly larger than the width of the electrode. This value may be compared to the characteristic dimension of d1 d2 /(d1 + d2 ) obtained by Davis et al. [14, 15] for a rectangular electrode of dimension d1 × d2 . For a long narrow electrode, the characteristic dimension for the rectangle approaches the width of the rectangle. The slight factor of 1.2 discrepancy between the limiting values for the ring and the rectangular electrode may be attributed to the observation that a long thin rectangle has, nevertheless, edges that do not exist for a thin ring. 6. Conclusions The frequency dispersion associated with a ring electrode geometry may be associated with a characteristic electrode dimension. The approximate formula presented in equation (16) is shown to be adequate for a wide ring, and have an error of 20 percent for a thin ring. A refined expression for the characteristic dimension is presented as equation (19). 9
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Selection of an appropriate ring dimension, as indicated by equation (20), can ensure that the impedance measured throughout a desired frequency range is unaffected by the dispersion caused by electrode geometry. This work shows further that the characteristic frequency associated with the influence of ring-electrode geometry with inner and outer radii r1 and r2 , respectively, is always larger than that associated with the geometry of a disk of radius r2 . These results may serve to guide the design of ring-shaped sensors that employ impedance measurements.
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7. Acknowledgments
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The support of Medtronic Diabetes (Northridge, CA), Rui Kong and Andrea Varsavsky, program monitors, is gratefully acknowledged.
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8. References [1] K.-M. Hsieh, K.-C. Lan, W.-L. Hu, M.-K. Chen, L.-S. Jang, M.-H. Wang, Glycated hemoglobin (hba1c) affinity biosensors with ring-shaped interdigital electrodes on impedance measurement, Biosensors and Bioelectronics 49 (2013) 450–456.
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[2] W. Besio, A. Prasad, Analysis of skin-electrode impedance using concentric ring electrode, in: Engineering in Medicine and Biology Society, 2006. EMBS’06. 28th Annual International Conference of the IEEE, IEEE, 2006, pp. 6414–6417.
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[3] F.-B. Li, A. R. Hillman, S. D. Lubetkin, A new approach to the mechanism of chlorine evolution: Separate examination of the kinetic steps using ac impedance on a rotating thin ring electrode, Electrochimica acta 37 (15) (1992) 2715–2723.
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[4] P. Pierini, J. S. Newman, Ring electrodes, Journal of the Electrochemical Society 125 (1) (1978) 79–84.
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[5] A. Datta, M. Elwassif, F. Battaglia, M. Bikson, Transcranial current stimulation focality using disc and ring electrode configurations: Fem analysis, Journal of neural engineering 5 (2) (2008) 163.
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[6] A. F. M. Mansor, S. N. Ibrahim, Simulation of ring interdigitated electrode for dielectrophoretic trapping, in: 2016 IEEE International Conference on Semiconductor Electronics (ICSE), 2016, pp. 169–172. doi:10.1109/SMELEC.2016.7573618. [7] J. S. Newman, Frequency dispersion in capacity measurements at a disk electrode, Journal of the Electrochemical Society 117 (1970) 198–203.
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[8] V. M.-W. Huang, V. Vivier, M. E. Orazem, N. P´eb`ere, B. Tribollet, The apparent CPE behavior of a disk electrode with faradaic reactions, Journal of the Electrochemical Society 154 (2007) C99–C107. [9] V. M.-W. Huang, V. Vivier, M. E. Orazem, N. P´eb`ere, B. Tribollet, The apparent CPE behavior of an ideally polarized blocking electrode: A global and local impedance analysis, Journal of the Electrochemical Society 154 (2007) C81–C88.
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[10] C. Blanc, M. E. Orazem, N. P´eb`ere, B. Tribollet, V. Vivier, S. Wu, The origin of the complex character of the ohmic impedance, Electrochimica Acta 55 (2010) 6313–6321. [11] C. L. Alexander, B. Tribollet, M. E. Orazem, Contribution of surface distributions to constantphase-element (CPE) behavior: 1. influence of roughness, Electrochimica Acta 173 (2015) 416– 424. [12] J. S. Newman, Resistance for flow of current to a disk, Journal of the Electrochemical Society 113 (5) (1966) 501–502. [13] R. Michel, C. Montella, C. Verdier, J.-P. Diard, Numerical computation of the faradaic impedance of inlaid microdisk electrodes using a finite element method with anisotropic mesh adaptation, Electrochimica Acta 55 (2010) 6263–6273. [14] K. Davis, C. L. Alexander, M. E. Orazem, Influence of geometry-induced frequency dispersion on the impedance of rectangular electrodes (2017) in preparation. [15] M. E. Orazem, B. Tribollet, Electrochemical Impedance Spectroscopy, 2nd Edition, John Wiley & Sons, Hoboken, 2017.
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List of Figures
2
7
4
4
6 6 7
8 8
9
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8
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5 6
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4
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3
Schematic representation of a sample finite element mesh used for ring electrode simulations: a) entire domain and b) an enlarged region showing the two insulating surfaces and the ring electrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scaled calculated impedance presented in Nyquist format with r1 /r2 as a parameter. The dashed line represents the impedance of a disk electrode. Simulations were performed for 10 logarithmically spaced frequencies per decade, and symbols are used to highlight every 2 decades. Ring dimensions are presented in Table 1. . . . . . . . . Scaled calculated impedance presented in Bode format as a function of frequency with r1 /r2 as a parameter: a) scaled modulus of the impedance and b) imaginaryimpedance-derived phase angle. The ring dimensions are presented in Table 1. The dashed line represents the impedance of a disk electrode. Simulations were performed for 10 logarithmically spaced frequencies per decade, and the line represents the interpolation between calculated values. . . . . . . . . . . . . . . . . . . . . . . . . . . Ring electrode imaginary-impedance-derived phase angle as a function of dimensionless frequency with r1 /r2 as a parameter. . . . . . . . . . . . . . . . . . . . . . . . . The correction factor g as a function of the geometric ratio r1 /r2 . . . . . . . . . . . . Characteristic length as a function of the geometric ratio r1 /r2 . COMSOLr simulation results are shown as blue triangles, the empirical approximation given as equation (16) is shown as red squares, and polynomial fit is given as a black line. . . . . . . . Ring electrode imaginary-impedance-derived phase angle as a function of dimensionless frequency K ∗ (equation (18)) with r1 /r2 as a parameter. . . . . . . . . . . . . . Ratio of the characteristic frequency of the ring electrode fc,ring to the characteristic frequency of the disk electrode fc,disk as a function of the corrected characteristic length `∗c,ring scaled by the outer radii of the disk, r2 . . . . . . . . . . . . . . . . . .
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1
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S0013-4686(17)30510-8 http://dx.doi.org/doi:10.1016/j.electacta.2017.03.044 EA 29078
To appear in:
Electrochimica Acta
Received date: Revised date: Accepted date:
18-1-2017 3-3-2017 6-3-2017
Please cite this article as: Yu-Min Chen, Christopher L. Alexander, Christopher Cleveland, Mark E. Orazem, Influence of Geometry-Induced Frequency Dispersion on the Impedance of Ring Electrodes, (2017), http://dx.doi.org/10.1016/j.electacta.2017.03.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Research Highlights
Highlights:
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Finite-element simulations of the impedance of ring electrodes. Characteristic dimensions for ring electrodes identified Characteristic frequencies found at which ring electrodes cause time-constant dispersion
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*Revised Manuscript (including Abstract) Click here to view linked References
Influence of Geometry-Induced Frequency Dispersion on the Impedance of Ring Electrodes Yu-Min Chena , Christopher L. Alexandera , Christopher Clevelanda , Mark E. Orazema,∗ of Chemical Engineering, University of Florida, Gainesville, FL, 32611, USA
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a Department
Abstract
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Finite-element simulations for the impedance response of ring electrodes were used to identify the characteristic frequency associated with the influence of electrode geometry on impedance response. An approximate expression for the characteristic ring-electrode dimension was found to be adequate for a wide ring, and have an error of 20 percent for a thin ring. A refined expression for the characteristic dimension is presented. The characteristic frequency associated with the influence of ring-electrode geometry is always larger than that associated with the geometry of a disk. These results guide the design of ring-shaped sensors that employ impedance measurements. Keywords: sensors, numerical simulation, impedance spectroscopy, frequency dispersion
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1. Introduction
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The ring-electrode geometry has been widely used in electrochemical systems with electrochemical impedance spectroscopy (EIS) as an analysis technique. Hsieh et al. [1] conducted impedance measurements on ring-shaped interdigitated electrodes to characterize the concentration change of glycated hemoglobin. Besio and Prasad[2] used a concentric ring electrode to analyze the skin– electrode impedance. They performed single-frequency impedance measurements at 1 kHz and concluded that copper is the best material among several metals for their biosensor. Li et al. [3] conducted impedance measurements on a rotating ring electrode to determine the mechanism of the chlorine evolution reaction. Their work suggested that, on a Pt surface, the rate of chloride discharge and simultaneous chlorine adsorption is first order with respect to chloride concentration; whereas, the rate of the adsorption and desorption process is second order with respect to chloride concentration. The current and potential distribution for the ring electrode geometry has been widely studied. The primary current distribution associated with the ring electrode geometry is nonuniform. Pierini and Newman [4] presented an analytic solution for the primary and secondary current distribution on ring electrodes. Datta et al. [5] used the finite-element method to calculate the steady-state current and potential distribution for concentric ring-ring electrodes and double concentric (ring-ring-ring) electrodes used for transcranial current stimulation. Their work showed that the electric field on a ring electrode decreases rapidly in the radial direction. Mansor and Ibrahim [6] also performed simulations indicating that the electric field on a ring interdigitated electrode reaches a maximum at the ring edges and a minimum at the center of the rings. The term frequency or time-constant dispersion is used to describe the broadening of the impedance response associated with a distribution of time constants. Frequency dispersion associated with an electrode geometry originates from the nonuniform current and potential distribution, which changes as a function of frequency. Newman [7] showed, by solution of Laplace’s equation, that frequency dispersion is observed for disk electrodes for frequencies larger than a characteristic value.
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∗ Corresponding
author Email address: [email protected] (Mark E. Orazem)
Preprint submitted to Electrochimica Acta
March 3, 2017
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A dimensionless frequency may be expressed as [7–11] K=
ωC0 `c,disk . κ
(1)
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where ω is the angular frequency, κ is the electrolyte conductivity, C0 is the capacitance of the disk electrode, and `c,disk is the characteristic length for a disk electrode. Huang et al. [8] showed that, for `c,disk = r0 (2)
cr
the characteristic frequency for a disk electrode occurs near K = 1; thus, the associated characteristic dimension for the disk electrode is the radius of the disk. Alexander et al. [11] showed that the characteristic dimension for a rough disk is given by `c,rough disk = fr r0
(3)
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where fr is the rugosity or roughness factor defined to be the ratio of the actual surface area to the superficial surface area of the disk. Alexander et al. [11] showed further that the characteristic dimension for roughness is `c,roughness = fr2 P (4)
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2. Mathematical Development
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where P is the period associated with roughness. Equation (1) facilitates estimation of the characteristic frequency at which dispersion influences the impedance response for a given electrode. For K > 1, frequency dispersion is associated with the nonuniform current distribution. Therefore, frequency dispersion may be eliminated in a desired frequency range by choosing the appropriate parameters `c and κ to ensure that K < 1. The objective of this work is to find the characteristic dimension associated with the impedance response of a ring electrode.
The ring serves as the working electrode, which is embedded in an infinite, insulated plane with a hemispherical counterelectrode located at infinity. For the present work, the geometric parameters r1 and r2 represent the inner and outer radii of the ring electrode, respectively. The potential distribution for an electrolyte with uniform conductivity is governed by Laplace’s equation, which may be expressed in cylindrical coordinates as ∂Φ ∂2Φ 1 ∂ r + =0 (5) r ∂r ∂r ∂y 2
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where axial symmetry was assumed. e components as The potential can be expressed in terms of steady-state Φ and oscillating Φ e jωt } Φ = Φ + Re{Φe
(6)
Similarly, the potential applied at the electrode surface can be expressed as V = V + Re{Ve ejωt }
(7)
where V is the steady-state value and Ve is the oscillating value. The steady-state solution for the current distribution at a blocking electrode shows that the potential is uniform and the current is equal to zero. As the problem is linear, solution for the oscillating variables does not require a solution for the steady-state equation.[9] The impedance response of the ring requires solution of ! e e 1 ∂ ∂Φ ∂2Φ r + =0 (8) r ∂r ∂r ∂y 2 2
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subject to boundary conditions that the oscillating potential is equal to zero for distances far away from the disk, i.e., e → 0 as r2 + y 2 → ∞ Φ (9) and that, on the insulating plane surrounding the ring, e ∂Φ =0 ∂y y=0
for r < r1
and r > r2
(10)
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For a blocking electrode, Huang et al. [8] showed that the flux condition at the surface of a disk electrode can be expressed as ∂ (V −Φ(0)) ∂Φ (11) = −κ C0 ∂t ∂y y=0
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where Φ(0) is the potential outside the diffuse part of the double layer. The corresponding expression in terms of oscillating variables is given by e ∂ Φ e (12) jωC0 Ve − Φ(0) = −κ ∂y y=0
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The impedance was calculated as
Z (ω) =
Ve Ie
(13)
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for a specified range of frequencies, where Ie is the oscillating current at the working electrode obtained by integrating the local current density over the surface of the electrode.
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3. Numerical Method
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The finite-element-analysis solver and simulation software COMSOL Multiphysicsr was used to solve Laplace’s equation in cylindrical coordinates for the ring electrode. Triangular elements were employed with quadratic interpolation. Due to the singularities that arise near the edge of the ring electrode, a nonuniform meshing scheme was implemented near the surface of the ring electrode. A direct linear solver was used. The meshing was refined manually to ensure that the impedance response of a disk electrode yielded the correct high-frequency primary resistance, i.e., Re κr0 = 1/4.[12] Similar mesh sizes were then used for the ring electrodes. In contrast to the frequency-dependent adaptive mesh algorithm employed by Michel et al.,[13] the same meshing was then employed for all frequencies. Calculation of 80 frequencies (10 points per decade from 10−2 Hz to 109 Hz) required less than 30 min on a 64-bit Dell Precision T7400 workstation with dual Xeon E5410 2.33 GHz processors and 32G Byte of RAM. A schematic representation of the finite element mesh used for the ring electrode simulations is provided in Figure 1 detailing: a) the entire domain and b) an enlarged region showing the ring electrode embedded in an insulating plane. The counterelectrode in Figure 1(a) was placed at a distance 500-1000 times that of the outer ring radius to minimize the influence of the counterelectrode placement. The exposed ring electrode surface was coplanar to the inner and outer insulating surfaces.
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4. Results Finite-element (COMSOLr ) simulations were performed to explore the influence geometry has on the impedance response for a ring electrode. Values of r1 were varied from 0 to 0.995 cm, and values of r2 ranged from 0.102 to 2 cm, yielding values of r1 /r2 ranging from 0 to 0.995. The capacitance was fixed at C0 = 1 µF/cm2 , and the conductivity was fixed at κ = 0.02 S/cm. These
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b
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b
a
Insulator
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Ring electrode
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Insulator
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Figure 1: Schematic representation of a sample finite element mesh used for ring electrode simulations: a) entire domain and b) an enlarged region showing the two insulating surfaces and the ring electrode.
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105 Hz
r1/r2 0.40 0.80 0.91 0.98
0.4
-Zj r2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Disk r1=0
0.2
104 Hz 105 Hz
105 Hz
0.0 0.0
107 Hz
0.2
0.4
0.6
Zr r2 Figure 2: Scaled calculated impedance presented in Nyquist format with r1 /r2 as a parameter. The dashed line represents the impedance of a disk electrode. Simulations were performed for 10 logarithmically spaced frequencies per decade, and symbols are used to highlight every 2 decades. Ring dimensions are presented in Table 1.
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Table 1: Ring electrode geometries used for the simulations presented in Figures 2 and 3.
r1 , cm 0.1 0.2 0.4 0.1 0.2
r2 , cm 1 0.5 0.5 0.11 0.205
fc , kHz 3.32 10.9 46.1 507 1050
ip t
r1 /r2 0.1 0.4 0.8 0.9091 0.9756
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values could correspond to an oxide film on an electrode surface, but, as the results are presented in dimensionless form, they may be applied to other systems. A Nyquist plot is presented in Figure 2 for some representative calculations. At low frequencies, the results indicate a vertical line corresponding to a capacitive behavior. The deviation from a vertical line observed at high frequencies has been attributed for a disk electrode, shown by dashed lines, to frequency dispersion associated with electrode geometry. The frequencies indicated in Figure 2 do not show a repeating pattern as the value of r2 was not the same for the different simulations. The ring dimensions are presented in Table 1. The dimensionless modulus of the impedance is presented in Figure 3(a) as a function of frequency. At a given frequency, the dimensionless modulus was smaller for the wider ring electrodes, and this effect is most visible at low frequency. The smaller impedance of the wider ring electrodes can be attributed to the larger surface area. The modulus for the ring electrode approaches that of the disk as r1 /r2 → 0. The imaginary-impedance-derived phase angle can be expressed as dlog Zj × 90◦ . dlog f
(14)
d
ϕdZj =
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Alexander et al. [11] showed that the imaginary-impedance-derived phase angle is very sensitive to the onset of frequency dispersion for capacitive electrodes. The calculated imaginary-impedancederived phase angle is presented in Figure 3(b) as a function of frequency. Frequency dispersion, seen as a deviation of the phase angle from −90◦ , is dependent on the geometric parameter r1 /r2 . At frequencies less than 1 kHz, the geometry of the ring does not introduce frequency dispersion for the parameters used in the simulation. As the ring width decreases, the onset of frequency dispersion occurs at a higher frequency.
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4.1. Approximate Dimensionless Frequency Based strictly on the observation that the onset of frequency dispersion varies according to the ring electrode width divided by a factor accounting for curvature, an approximate expression for the dimensionless characteristic frequency is ωC0 r2 − r1 K= (15) κ 1 + r12 /r22 where the characteristic dimension for the ring electrode is given by `c,ring =
r2 − r1 1 + r12 /r22
(16)
A measure of the suitability of equation (15) is found by seeking the superposition of the results shown in Figure 3(b) when plotted as functions of K. The results are shown in Figure 4 with r1 /r2 as a parameter. The value r1 /r2 = 0 corresponds to a disk electrode. The results shown in Figure 4 are approximately superposed as compared to the results presented in Figure 3(b).
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1 0
3
1 0
2
1 0
1
1 0
0
r1 /r 0 .1 0 .4 0 .8 0 .9 0 .9
1 0
2
0 0 0 1 8
D is k r 1 = 0
-1
1 0
-2
1 0
-1
1 0
0
1 0
1
2
1 0
3
1 0
1 0
4
1 0
5
6
1 0
1 0
7
1 0
f / H z
r1 /r 0 .1 0 .4 0 .8 0 .9 0 .9
9
1 0
1 0
2
0 0 0 1 8
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-8 5
1 0
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D is k r 1 = 0
j
/ d e g re e
-9 0
8
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(a)
-8 0
-7 5 0
1 0
1
1 0
2
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1 0
d
d Z
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4
cr
2
|Z | r
1 0
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1 0
3
1 0
4
1 0
5
1 0
6
f / H z (b)
Figure 3: Scaled calculated impedance presented in Bode format as a function of frequency with r1 /r2 as a parameter: a) scaled modulus of the impedance and b) imaginary-impedance-derived phase angle. The ring dimensions are presented in Table 1. The dashed line represents the impedance of a disk electrode. Simulations were performed for 10 logarithmically spaced frequencies per decade, and the line represents the interpolation between calculated values.
r1 /r 0 0 .1 0 .4 0 .8 0 .9 0 .9
-8 5
2
D is k 0 0 0 1
K = 1 8
j
/ d e g re e
-9 0
d Z
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
-8 0
-7 5 1 0
-2
1 0
-1
1 0
0
1 0
1
Figure 4: Ring electrode imaginary-impedance-derived phase angle as a function of dimensionless frequency with r1 /r2 as a parameter.
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1 .1
1 .0 0 .2
0 .4
0 .6
r 1/r 2
0 .8
1 .0
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0 .0
cr
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g
1 .2
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Figure 5: The correction factor g as a function of the geometric ratio r1 /r2 .
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The superposition presented in Figure 4, however, is not perfect. In the frequency domain close to K = 1, the ring electrodes have an imaginary-impedance-derived phase angle greater than that of the disk electrode (r1 /r2 = 0). Frequency dispersion for the ring electrodes is seen for values of dimensionless frequency that are smaller than K = 1. In contrast, as is seen in Figure 3(b), frequency dispersion for the ring electrodes is seen at frequencies that are larger than that for the disk electrode. 4.2. Correction to the Dimensionless Frequency
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A correction to the dimensionless frequency was obtained by requiring that the corrected dimensionless frequency K ∗ = Kg = 1 at the phase angle ϕdZj = −88.366◦ for which K = 1 for the disk electrode. The correction factor g can be expressed as a fifth-order polynomial as 2 3 4 5 r1 r1 r1 r1 r1 g = 1 + 1.0742 − 4.2227 + 8.9244 − 9.4462 + 3.8806 (17) r2 r2 r2 r2 r2
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The correction factor g is shown in Figure 5 as a function of geometric ratio r1 /r2 . When the geometric ratio is extremely close to 0, the ring electrode can be regarded to be a disk, and the correction factor approaches unity. When the geometric ratio is greater than 0, the correction factor increases with the geometric ratio, reaching a maximum value of 1.2 as r1 /r2 → 1. The corrected dimensionless frequency may be expressed as ωC0 r2 − r1 ∗ g (18) K = κ 1 + r12 /r22 and the characteristic length can be expressed as π r2 − r1 ∗ `c,ring = g 4 1 + r12 /r22
(19)
A comparison is presented in Figure 6 among the approximate characteristic length given in equation (16), the simulated values, and the refined expression given in equations (17) and (19). The characteristic length approaches r2 as r1 /r2 → 0 and zero as r1 /r2 → 1. The imaginary-impedance-derived phase angle is provided in Figure 7 as a function of dimensionless frequency K ∗ given in equation (18) with r1 /r2 as a parameter. The results for different values of r1 /r2 are superposed near K ∗ = 1, and frequency dispersion is observed at K ∗ = 1. 7
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COMSOL
1.0
Approximation Polynomial fit
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0.8
cr
c
/ r2
0.6 0.4
0.0
0.2
0.4
0.6
0.8
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0.0
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0.2
1.0
r1/r2
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Figure 6: Characteristic length as a function of the geometric ratio r1 /r2 . COMSOLr simulation results are shown as blue triangles, the empirical approximation given as equation (16) is shown as red squares, and polynomial fit is given as a black line.
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r1 /r 0 0 .1 0 .4 0 .8 0 .9 0 .9 2
D is k 0 0 0 1 8
j
/ d e g re e
-9 0
d Z
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
K
-8 0
-7 5 1 0
-2
1 0
-1
*
= 1
1 0
0
1 0
1
Figure 7: Ring electrode imaginary-impedance-derived phase angle as a function of dimensionless frequency K ∗ (equation (18)) with r1 /r2 as a parameter.
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1 0
1 0 .1
1
2
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/ r
cr
0 .0 1
ip t
fc
,r in g
/ fc
,d is k
1 0 0
Figure 8: Ratio of the characteristic frequency of the ring electrode fc,ring to the characteristic frequency of the disk electrode fc,disk as a function of the corrected characteristic length `∗c,ring scaled by the outer radii of the disk, r2 .
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5. Discussion
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In order to eliminate the influence of frequency dispersion in a desired frequency range, the ring dimension may be selected to ensure that K ∗ < 1, i.e., `∗c,ring ≤
κ
2πfmax C0 g
(20)
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where fmax is the maximum desired frequency. Similar expressions were developed by Huang et al. [9] for a disk electrode. For a given electrolyte conductivity and electrode capacity, the frequency at which the ring electrode geometry influences the impedance response can be compared to that associated with the disk electrode geometry as r2 1 + (r1 /r2 )2 fc,ring (21) = fc,disk (r1 − r2 ) g
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The results are presented in Figure 8 as a function of `∗c,ring /r2 . The characteristic frequency for a ring electrode is always greater than that for a disk electrode with radius equal to r2 . For a thin ring, the characteristic frequency can be several orders of magnitude larger than that for the disk electrode. As the value of r1 approaches zero, the characteristic dimension given in equation (19) approaches r2 , the radius of a disk. As the value of r1 approaches r2 , the characteristic dimension approaches 1.2(r1 − r2 ), which is slightly larger than the width of the electrode. This value may be compared to the characteristic dimension of d1 d2 /(d1 + d2 ) obtained by Davis et al. [14, 15] for a rectangular electrode of dimension d1 × d2 . For a long narrow electrode, the characteristic dimension for the rectangle approaches the width of the rectangle. The slight factor of 1.2 discrepancy between the limiting values for the ring and the rectangular electrode may be attributed to the observation that a long thin rectangle has, nevertheless, edges that do not exist for a thin ring. 6. Conclusions The frequency dispersion associated with a ring electrode geometry may be associated with a characteristic electrode dimension. The approximate formula presented in equation (16) is shown to be adequate for a wide ring, and have an error of 20 percent for a thin ring. A refined expression for the characteristic dimension is presented as equation (19). 9
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Selection of an appropriate ring dimension, as indicated by equation (20), can ensure that the impedance measured throughout a desired frequency range is unaffected by the dispersion caused by electrode geometry. This work shows further that the characteristic frequency associated with the influence of ring-electrode geometry with inner and outer radii r1 and r2 , respectively, is always larger than that associated with the geometry of a disk of radius r2 . These results may serve to guide the design of ring-shaped sensors that employ impedance measurements.
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7. Acknowledgments
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The support of Medtronic Diabetes (Northridge, CA), Rui Kong and Andrea Varsavsky, program monitors, is gratefully acknowledged.
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8. References [1] K.-M. Hsieh, K.-C. Lan, W.-L. Hu, M.-K. Chen, L.-S. Jang, M.-H. Wang, Glycated hemoglobin (hba1c) affinity biosensors with ring-shaped interdigital electrodes on impedance measurement, Biosensors and Bioelectronics 49 (2013) 450–456.
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[2] W. Besio, A. Prasad, Analysis of skin-electrode impedance using concentric ring electrode, in: Engineering in Medicine and Biology Society, 2006. EMBS’06. 28th Annual International Conference of the IEEE, IEEE, 2006, pp. 6414–6417.
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[3] F.-B. Li, A. R. Hillman, S. D. Lubetkin, A new approach to the mechanism of chlorine evolution: Separate examination of the kinetic steps using ac impedance on a rotating thin ring electrode, Electrochimica acta 37 (15) (1992) 2715–2723.
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[4] P. Pierini, J. S. Newman, Ring electrodes, Journal of the Electrochemical Society 125 (1) (1978) 79–84.
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[5] A. Datta, M. Elwassif, F. Battaglia, M. Bikson, Transcranial current stimulation focality using disc and ring electrode configurations: Fem analysis, Journal of neural engineering 5 (2) (2008) 163.
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[6] A. F. M. Mansor, S. N. Ibrahim, Simulation of ring interdigitated electrode for dielectrophoretic trapping, in: 2016 IEEE International Conference on Semiconductor Electronics (ICSE), 2016, pp. 169–172. doi:10.1109/SMELEC.2016.7573618. [7] J. S. Newman, Frequency dispersion in capacity measurements at a disk electrode, Journal of the Electrochemical Society 117 (1970) 198–203.
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[8] V. M.-W. Huang, V. Vivier, M. E. Orazem, N. P´eb`ere, B. Tribollet, The apparent CPE behavior of a disk electrode with faradaic reactions, Journal of the Electrochemical Society 154 (2007) C99–C107. [9] V. M.-W. Huang, V. Vivier, M. E. Orazem, N. P´eb`ere, B. Tribollet, The apparent CPE behavior of an ideally polarized blocking electrode: A global and local impedance analysis, Journal of the Electrochemical Society 154 (2007) C81–C88.
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[10] C. Blanc, M. E. Orazem, N. P´eb`ere, B. Tribollet, V. Vivier, S. Wu, The origin of the complex character of the ohmic impedance, Electrochimica Acta 55 (2010) 6313–6321. [11] C. L. Alexander, B. Tribollet, M. E. Orazem, Contribution of surface distributions to constantphase-element (CPE) behavior: 1. influence of roughness, Electrochimica Acta 173 (2015) 416– 424. [12] J. S. Newman, Resistance for flow of current to a disk, Journal of the Electrochemical Society 113 (5) (1966) 501–502. [13] R. Michel, C. Montella, C. Verdier, J.-P. Diard, Numerical computation of the faradaic impedance of inlaid microdisk electrodes using a finite element method with anisotropic mesh adaptation, Electrochimica Acta 55 (2010) 6263–6273. [14] K. Davis, C. L. Alexander, M. E. Orazem, Influence of geometry-induced frequency dispersion on the impedance of rectangular electrodes (2017) in preparation. [15] M. E. Orazem, B. Tribollet, Electrochemical Impedance Spectroscopy, 2nd Edition, John Wiley & Sons, Hoboken, 2017.
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List of Figures
2
7
4
4
6 6 7
8 8
9
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5 6
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3
Schematic representation of a sample finite element mesh used for ring electrode simulations: a) entire domain and b) an enlarged region showing the two insulating surfaces and the ring electrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scaled calculated impedance presented in Nyquist format with r1 /r2 as a parameter. The dashed line represents the impedance of a disk electrode. Simulations were performed for 10 logarithmically spaced frequencies per decade, and symbols are used to highlight every 2 decades. Ring dimensions are presented in Table 1. . . . . . . . . Scaled calculated impedance presented in Bode format as a function of frequency with r1 /r2 as a parameter: a) scaled modulus of the impedance and b) imaginaryimpedance-derived phase angle. The ring dimensions are presented in Table 1. The dashed line represents the impedance of a disk electrode. Simulations were performed for 10 logarithmically spaced frequencies per decade, and the line represents the interpolation between calculated values. . . . . . . . . . . . . . . . . . . . . . . . . . . Ring electrode imaginary-impedance-derived phase angle as a function of dimensionless frequency with r1 /r2 as a parameter. . . . . . . . . . . . . . . . . . . . . . . . . The correction factor g as a function of the geometric ratio r1 /r2 . . . . . . . . . . . . Characteristic length as a function of the geometric ratio r1 /r2 . COMSOLr simulation results are shown as blue triangles, the empirical approximation given as equation (16) is shown as red squares, and polynomial fit is given as a black line. . . . . . . . Ring electrode imaginary-impedance-derived phase angle as a function of dimensionless frequency K ∗ (equation (18)) with r1 /r2 as a parameter. . . . . . . . . . . . . . Ratio of the characteristic frequency of the ring electrode fc,ring to the characteristic frequency of the disk electrode fc,disk as a function of the corrected characteristic length `∗c,ring scaled by the outer radii of the disk, r2 . . . . . . . . . . . . . . . . . .
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1
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