On the impedance response of interdigitated electrodes

Journal Pre-proof On the impedance response of interdigitated electrodes Arthur R. Dizon, Mark E. Orazem PII:

S0013-4686(19)31871-7

DOI:

https://doi.org/10.1016/j.electacta.2019.135000

Reference:

EA 135000

To appear in:

Electrochimica Acta

Received Date: 23 July 2019 Revised Date:

18 September 2019

Accepted Date: 2 October 2019

Please cite this article as: A.R. Dizon, M.E. Orazem, On the impedance response of interdigitated electrodes, Electrochimica Acta (2019), doi: https://doi.org/10.1016/j.electacta.2019.135000. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Published by Elsevier Ltd.

On the Impedance Response of Interdigitated Electrodes Arthur R. Dizona , Mark E. Orazema,∗ a Department

of Chemical Engineering, University of Florida, Gainesville, FL, 32611, USA

Abstract Finite-element impedance simulations were used to provide a relationship between the characteristic dimension and the physical dimensions of the interdigitated electrodes. Inclusion of electric and displacement currents allowed simulation of the capacitive loop associated with the geometric capacitance. An error analysis was used to quantify the influence of mesh and domain sizes on the numerical accuracy of the simulations. The interdigitated electrode geometry is shown to induce a frequency dispersion, dependent on electrode digit width, height, and separation, that can be characterized in terms of a complex ohmic impedance with real asymptotic limits for ohmic resistance at high and low frequencies. Characteristic dimensions, calculated from the primary ohmic resistance, the geometric capacitance, and the high-frequency ohmic resistance were in agreement for all geometries considered. The characteristic dimension calculated from the low-frequency ohmic resistance deviated under conditions that led to frequency dispersion. The Havriliak-Negami equation is shown to provide a good representation of the complex ohmic impedance. The present work is applicable to the analysis and interpretation of experimental data obtained using interdigitated electrodes. Keywords: interdigitated electrodes; impedance spectroscopy; frequency dispersion

1

1. Introduction

2

Interdigitated electrodes are microelectrodes deposited onto a substrate such that the working and counter

3

electrodes have the appearance of interlaced fingers separated by a small distance. The micron-scale sep-

4

aration between the working and counter electrodes serves to increase the sensitivity for electrochemical

5

impedance spectroscopy measurements and to facilitate electrochemical analyses with small sample sizes.

6

The primary use of interdigitated electrodes is as electrochemical sensors. Bueno and Paix˜ao1 studied the

7

use of copper interdigitated electrodes to detect adulteration of ethanol with water by dielectrometry, which is

8

the measurement of the relative permittivity using alternating current. The method was found to be accurate

9

for detecting adulteration in mixtures of 10 to 25 vol.% of water in ethanol. Staginus2 studied the application

10

of interdigitated electrodes for the detection of organic pollutants. Mohamad et al.3 studied the detection

11

of triglycerides in hexane. Matylitskaya et al.4 studied the kinetics on nano-scale interdigitated electrodes ∗ Corresponding

author Email address: [email protected] (Mark E. Orazem)

Preprint submitted to Electrochimica Acta

October 3, 2019

12

for lab-on-a-chip applications and observed reduced sensitivity resulting from surface films. Interdigitated

13

electrodes are commonly employed for label-less detection of cancer cells and viruses, in which the electrode

14

surfaces are functionalized to detect specific cell types.5, 6, 7, 8, 9, 10

15

Zaretsky et al.11 developed a mathematical framework for the relationship between surface capaci-

16

tance and the periodic interdigitated electrode microstructure for dielectrometry. An early example of

17

interdigitated-electrode fabrication was demonstrated by Chidsat et al.,12 where platinum interdigitated

18

electrodes were made with a digit width of 3.5 µm and a digit spacing of 2.5 µm. Zou et al.13 demonstrated

19

the fabrication of nanometer-scale electrodes designed for bioimpedance measurements. Skjolding et al.14

20

concluded that optimal signal sensitivity occurred with electrode widths less than 1 µm through kinetic

21

studies using nano-interdigitated electrodes. Pliquett et al.15 demonstrated an increase in the noise of impe-

22

dance measurements with reduced digit width. Ibrahim et al.16 optimized interdigitated electrode sensitivity

23

through finite-element simulations and concluded that the optimal ratio of digit spacing and width was 0.66.

24

Jeon et al.17 fabricated nano-gap interdigitated aluminum electrodes and proposed that fabrication defects

25

at electrode edges contributed to deviations from predicted results.

26

Hong et al.18 attributed experimental departures from theoretically predicted impedance to fringing,

27

or edge effects. Chen et al.19 developed a dual-channel interdigitated electrode to mitigate the effect

28

of the substrate on capacitive measurements. Tomˇc´ık20 accounted for overlapping diffusion layers of the

29

interdigitated electrodes and the effect on impedance. Blume et al.21 provided a mathematical framework for

30

the effect of multiple layers deposited on the surface of an interdigitated electrode using conformal mapping.

31

MacKay et al.22 studied increasing the sensitivity of interdigitated electrodes by performing simulations of

32

gold nano-particles adhered to the surface of gold interdigitated electrodes. The simulations predicted that

33

the nano-particles adhered to the electrode surface disrupted the screening effect of the ions in solutions. De

34

La Rica et al.23 studied the viability of polysilicon interdigitated electrodes for electrochemical impedance

35

spectroscopy measurements and demonstrated the detection of urea with polysilicon interdigitated electrodes.

36

Olthuis et al.24 studied interdigitated electrodes coated with a resistive Ta2 O5 film in the determination

37

of the interdigitated electrode cell constant. Using a frequency range of 20 Hz to 500 kHz, experimental

38

cell constants were found to be approximately 10-20% smaller than cell constants calculated from conformal

39

mapping. The disparity between the experimental and calculated cell constants was attributed to edge

40

effects, i.e., geometry-induced nonuniform current and potential distributions.

41

The influence of edge effects on impedance response has been studied extensively for disk electrodes,

42

starting with Newman25 in 1970. Brichzin et al.26 performed oxygen-reduction kinetic studies using Sr-

43

doped LaMnO3 disk microelectrodes and demonstrated the relationship between the disk-electrode geometry

44

and subsequent nonuniform current distribution on polarization resistance. Huang et al.27, 28, 29 showed that

45

the frequency dispersion associated with relaxation of nonuniform current and potential distributions can be

46

expressed in terms of a complex ohmic impedance. Their approach was extended to rectangular electrodes

2

47

by Davis et al.30 More recently, Gharbi et al.31 showed that the Havriliak–Negami model32 provided a good

48

fit to the complex ohmic impedance for a disk electrode and identified the high- and low-frequency limits of

49

the complex ohmic impedance as Re,HF and Re,LF , respectively.

50

Patterned electrodes, used in the development of solid-oxide fuel cells (SOFC) with a geometry simi-

51

lar to interdigitated electrodes, have nonuniform current distributions and are subject to geometry-induced

52

frequency dispersion. Mizusaki et al.33 performed kinetic studies of hydrogen oxidation using patterned

53

electrodes, an early example of the use of the patterned-electrode geometry in the development of SOFC.

54

Goodwin et al.34 developed a computational kinetic model informed by the work of Mizusaki et al.33 for

55

hydrogen oxidation on Ni–yttria-stabilized zirconia patterned anodes. Chen at al.35 studied the influence

56

of interfacial capacitances and cell geometry on the impedance response of a SOFC and concluded that the

57

pattern spacing strongly influenced the impedance response. Ciucci et al.36 performed hydrogen-oxidation

58

kinetic studies that accounted for the influence of electrode geometry and ionic transport through a mixed-

59

ionic electron conductor (MIEC) and identified two reaction mechanisms associated with multiple current

60

pathways. Chueh at al.37 demonstrated increased electrochemical activity using patterned CeO2 –metal

61

anodes that showed good agreement between the simulated and experimental impedance responses of hy-

62

drogen oxidation. Nenning et al.38 developed an SOFC that utilized an interdigitated-electrode geometry

63

that was operated both as a patterned anode and as an interdigitated-electrode cell. Comparison of exper-

64

imental measurements in the two operational modes yielded kinetic parameters and electrical properties of

65

the MIEC.

66

Liu at al.39 performed finite-element simulations and demonstrated good agreement with previous re-

67

ported experimental data of SOFC. They concluded that the patterned electrode geometry has a strong

68

influence on the impedance response. A geometry-dependent high-frequency feature was observed in the

69

simulated impedance responses, but a physical interpretation was not provided.39 Such a high-frequency

70

feature could be consistent with the frequency dispersion reported, for example, by Davis et al.30 and Gharbi

71

et al.31

72

The objective of the present work is to explore the extent to which the discrepancy reported in the

73

literature between cell constants extracted from simulations and those extracted from experiments can be

74

attributed to the influence of geometry-induced frequency dispersion. Previous work on the influence of

75

electrode geometry on impedance response has emphasized isolated disk and rectangular electrodes. The

76

close proximity of interdigitated working and counter electrodes may be expected to distort the current and

77

potential distributions. The objective of the present work was, therefore, to investigate the influence of

78

geometry on the impedance response of interdigitated electrodes. Emphasis is placed on the relationship

79

between the high- and low-frequency limits of the complex ohmic impedance and the high-frequency loop

80

associated with the geometric capacitance.

3

Counter

L

S

Working

W

Working

W

H

Counter

S

(a)

W

(b)

Figure 1: Schematic representation of an interdigitated electrode from a (a) top-down view and (b) a cross-section side view. The working electrode is in gray and the counterelectrode is in black. The electrode digit dimensions are length L, width W , spacing S, and height H. In (a), the gray dashed box contains the repeating unit cell of the electrode. In (b), the dotted line contains the simulation domain and the dashed line is the plane of symmetry between the working and counterelectrode.

81

2. Methods

82

An interdigitated microelectrode comprises interlaced fingers, as shown in Figure 1(a). The electrode

83

dimensions are the height H, width W , length L, and spacing between digits S. The number of fingers or

84

digits can vary depending on design. The total area of the electrode is given by A = nd L (W + 2H)

(1)

85

where nd is the number of digit electrode pairs. The area of the collector wires was not included. A side-view

86

cross section of an interdigitated electrode is shown in Figure 1(b). Shibata40 presented a scanning trans-

87

mission electron microscope image of a commercially available interdigitated electrode (Metrohm DropSens

88

G-IDEAU5), yielding an estimate of 120 nm for the height of an electrode.

89

The impedance of interdigitated electrodes in media was obtained by a numerical solution using COMSOL TM

90

R Multiphysics version 5.0. The computer used to perform the calculations was a Dell R

91

workstation with two E5620 2.4 GHz Intel

92

2.1. Mathematical Model Development

93

Xeon

R

processors and 96 GB of RAM.

The governing equation was conservation of current density ∇·i=0

94

Precision T7500

(2)

The current density vector i comprised both electrical and displacement currents, i.e., i = κE +

4

dD dt

(3)

95

where κ is the electrical conductivity, E is the electric field vector, and D is the electrical displacement

96

vector. The electrical displacement vector was obtained from the constitutive relationship D = r 0 E

97

and the electric-field vector was related to the electrical potential Φ by E = −∇Φ

98

(5)

Equations (2)–(5) are straightforward applications of Maxwell’s equations.41 The electric potential and current were represented in phasor notation as

99

100

(4)

e jωt } Φ = Φ + Re{Φe

(6)

i = i + Re{eiejωt }

(7)

and

101

where ω is the angular frequency, Φ and i are the steady-state electrical potential and current density,

102

e and ei are complex potential and current density phasors, respectively. The phasors Φ e respectively, and Φ

103

and ei are functions of frequency and position, but are independent of time. Under assumptions of a steady state and uniform properties, substitution of equations (3) to (7) into

104

105

equation (2) yields ∇ · i = −κ∇2 Φ = 0

106

107

(8)

The conservation of current under a sinusoidal steady-state may be expressed as e=0 ∇ · ei = − (κ + jωr 0 ) ∇2 Φ

(9)

e=0 (1 + jωr 0 ρ) ∇2 Φ

(10)

or

108

which are equivalent forms. As the permittivity of vacuum 0 = 8.8542 × 10−14 C/V cm is a very small

109

number, equation (10) shows that, for electrolytes of modest resistivity, the contribution of the displacement

110

current can be neglected for impedance measurements in the usual 10 mHz–100 kHz range. The displacement

111

current plays an important role for measurements performed in high-resistivity electrolytes such as organic

112

liquids.42, 43

113

2.1.1. Primary Ohmic Resistance

114

The boundary condition at the electrode surface for calculation of the primary resistance was that the

115

electrolyte potential just outside the diffuse part of the double layer was uniform, i.e., Φ0 = Φm . The

116

boundary condition for the insulating surfaces and the plane of symmetry of the working electrode was

117

n · ∇Φ = 0, where n is the outwardly directed unit normal vector. The boundary condition far from the 5

118

electrode and for the plane of symmetry between the working and counterelectrode shown in Figure 1(b)

119

was Φ = 0. The solution to the steady-state equation (8) yielded a value for the primary ohmic resistance

120

Re,p = Φm /I, where I was obtained by integration of the steady-state current density over the surface of

121

the electrode.

122

2.1.2. Impedance Spectroscopy

123

124

The boundary condition for normal current density at the electrode surface may be expressed for a pure capacitance in the time domain as ∂ (Φm − Φ0 ) ∂t

(11)

  em − Φ e0 = jωCdl Φ

(12)

n · κ∇Φ|electrode = Cdl 125

and in the frequency domain, using phasor notation, as e n · κ∇Φ

electrode

126

e m is the potential difference of the metal and a reference, and Cdl is where n is the unit normal vector, Φ

127

the double-layer capacitance. The approach used in the present work follows that of Huang et al.27 and

128

e m due to symmetry. The boundary condition for Gharbi et al.31 The amplitude of the perturbation was 2Φ

129

e = 0. The boundary the insulating surfaces and the plane of symmetry of the working electrode was n · ∇Φ

130

condition far from the electrode and for the plane of symmetry between the working and counterelectrode

131

e = 0. The impedance is a frequency-dependent transfer function given by shown in Figure 1(b) was Φ Z (ω) =

em Φ eI

(13)

132

e m is a complex phasor representing the potential difference between the electrode and a reference where Φ

133

located far from the electrode surface, and eI is a complex phasor representing the integral of ei over the

134

electrode surface.

135

2.2. Simulation Parameters, Domain, and Meshing

136

R Simulations were performed using the COMSOL Multiphysics finite-element software with triangular-

137

element discretization and a Lagrange quadratic shape function. A multi-frontal massively parallel sparse

138

direct solver (MUMPS) was used with a relative tolerance set at 10−4 . To achieve the refined mesh size

139

required at the electrode surface, two semi-circular sub-domains were defined that contained the electrode

140

corners. The sub-domains were contained within a quarter circle domain of radius Hdom .

141

The simulation domain and meshing parameters used for the simulation of an isolated rectangular elec-

142

trode are shown in Table 1. The meshed domain of an isolated rectangular electrode is shown in Figure 2.

143

For an interdigitated electrode, the mirror-image counterelectrode was simulated by the addition of a vertical

144

plane of symmetry located between the working and counterelectrode. The interdigitated electrode digits of

145

width W were separated by a distance S. Therefore, a vertical plane of symmetry was added at the position 6

Table 1: The meshing parameters used for the simulation of an isolated rectangular electrode.

Simulation Parameter

value

Hdom

1500W

sdom

250W

se

2.25×10−6 W

ss

750se

(b)

(a)

(c)

Figure 2: Image of the 2-D simulation domain and meshing of an isolated interdigitated-electrode digit with a width of 10 nm using the simulation domain and meshing parameters prescribed in Table 1. The simulation domain and meshing was identical to one used for a 2-D axisymmetric disk electrode with a radius of 5 nm embedded in an insulated plane. (a) Full simulation domain. (b) Enlarged image of the electrode surface. (c) Enlarged image of the edge of the embedded electrode.

7

(b) (a) (c)

Figure 3: Image of the 2-D simulation domain and meshing of an interdigitated-electrode digit pair with an electrode width of 10 nm and an electrode spacing of 9.9 µm. The truncation of the hemispherical domain represented a plane of symmetry between the working and counterelectrode. (a) Full simulation domain. (b) Enlarged image of the electrode surface. (c) Enlarged image of the edge of the embedded electrode.

146

of (W + S) /2. The simulation domain and meshing for a working electrode in the presence of a mirror-image

147

counterelectrode is shown in Figure 3. Simulations with a finite-height used the same simulation domain

148

and meshing parameters used for a flat electrode. The boundary conditions for the electrode were assigned

149

to the vertical electrode surface.

150

The accuracy of the IDE simulations was assessed by evaluating the primary and low-frequency ohmic

151

resistances as functions of the domain size and maximum element size at the electrode corners and edges.

152

The primary and low-frequency ohmic resistances approached an asymptotic values, Re,p,0 and Re,LF,0 ,

153

respectively, as the domain size was increased for a maximum element size at the electrode periphery given

154

by Table 1. The results are presented in Figure 4(a) for four electrode configurations. The values of W , S,

155

and H for the electrode configurations, presented in Table 2, represent extremes for the calculations reported

156

in the present work. The values of ohmic resistance followed −n Re,k = Re,k,0 + aHdom

(14)

157

The resulting values for Re,p,0 and Re,LF,0 , obtained by nonlinear Levenberg–Marquardt algorithm44 imple-

158

mented in Origin 2019br , are presented in Table 2. The relative percent error, calculated as ε = 100

|Re,k − Re,k,0 | Re,k,0

(15)

159

is presented in Figure 4(b). The simulations reported in the present work, shown as filled symbols in

160

Figure 4(b), had errors that ranged from 2 × 10−4 % to 0.04%. 8

101 100

1.000

10-1 0.995

10-2 10-3

0.990 0.05

0.10

0.15

10-4

0.20

10-4

(a)

10-3

10-2

10-1

(b)

Figure 4: Results of a parametric study showing the influence of domain size on the calculated primary (black) and low-frequency (grey) ohmic resistance: a) ohmic resistance scaled by the asymptotic ohmic resistance (see Table 2) as a function of inverse domain size. The line represents the fit of equation (14). (b) The absolute percent error as a function of inverse domain size. The closed symbols represented the dimensions used for the simulations reported in the present work.

Table 2: Asymptotic values of ohmic resistance obtained changing the domain radius for a fixed value of maximum element size at the electrode periphery.

Case

W /µm

S/µm

H/µm

Re,p,0 /Ωcm2

Re,LF,0 /Ωcm2

a

1

9

0

12.6256±0.00065

13.5077±0.00034

b

5

5

0

50.0070±0.00033

54.2745±0.00028

c

9.9

0.1

0

28.0794±0.00030

54.7878±0.00021

d

5

5

0.16

48.83633±0.0011

53.714±0.0019

9

101 1.06

100

1.04

10-1

1.02

10-2

1.00 0

2

4

0

6

(a)

1

2

3

4

5

(b)

Figure 5: Results of a parametric study showing the influence of maximum element size at the electrode periphery on the calculated primary (black) ohmic resistance: a) ohmic resistance scaled by the asymptotic ohmic resistance (see Table 3) as a function of the logarithm of maximum element size. The line represents the fit of equation (16). (b) The absolute percent error as a function of logarithm of maximum element size. The closed symbols represented the dimensions used for the simulations reported in the present work.

Table 3: Asymptotic values of ohmic resistance obtained changing the maximum element size at the electrode surface for a fixed value of domain size.

Case

W /µm

S/µm

H/µm

Re,p,0 /Ωcm2

Re,LF,0 /Ωcm2

a

1

9

0

12.619±0.010

13.502

b

5

5

0

49.988±0.041

54.27404

c

9.9

0.1

0

28.087±0.010

54.78589

d

5

5

0.16

48.7840±0.0033

53.70734

161

A similar study was performed to explore the influence on maximum element size at the electrode edge

162

for the fixed domain size given in Table 1. For the range of maximum element sizes considered, the low-

163

frequency ohmic resistance was found to be independent of maximum element size. The primary ohmic

164

resistance, however, was sensitive to the mesh size used and followed n

Re,p = Re,p,0 + a [log (se )]

(16)

165

The results are presented in Figure 5(a) The resulting values for Re,p,0 , obtained by nonlinear Levenberg–

166

Marquardt algorithm44 implemented in Origin 2019br , and and Re,LF,0 taken from the simulations with the

167

smallest maximum element size, are presented in Table 3. The relative percent error, calculated as equation

168

(15), is presented in Figure 5(b). The simulations reported in the present work, shown as filled symbols in

169

Figure 5(b), had errors on that ranged from 0.0075% to 0.091%.

170

The asymptotic values of ohmic resistance reported in Table 2 for a fixed maximum element size and 10

171

Table 3 for a fixed domain size were in agreement within 0.055%. Thus, the parametric studies reported in

172

Figures 4 and 5 suggest a maximum calculation uncertainty of 0.09%.

173

2.3. Process Model

174

Nonlinear regression using the Levenberg–Marquardt algorithm44 was performed on the synthetic impe-

175

dance data generated by the finite-element simulations. The process model comprised three contributions,

176

i.e., Z = Zg + Ze + Z0

(17)

177

The impedance associated with the geometric capacitance Zg was modeled as a resistor and constant-phase

178

element connected in parallel given by Zg =

Re,HF 1 + (jω)αg Qg Re,HF

(18)

179

where Re,HF was the high-frequency ohmic resistance, and αg and Qg were parameters associated with the

180

geometric capacitance.

181

While a CPE was used to provide a general fitting to the simulation results, the mesh refinement described

182

in Section 2.2 yielded almost ideal capacitive loops with αg ≈ 0.999. The constant-phase-element parameters

183

Qg and αg were used to calculate the effective capacitance according to the equation developed by Brug et

184

al.45 given as (1−α )/αg

g Re,HF g Cg = Q1/α g

185

(19)

The ohmic resistance can be used to extract the resistivity of the sample media following Re,HF = ρδHF

(20)

186

where ρ is resistivity of the medium and δHF is the characteristic electrode dimension associated with the high-

187

frequency ohmic resistance. The dielectric constant r of the medium may be obtained from the geometric

188

capacitance, i.e., Cg =

r 0 δHF

(21)

189

where 0 is the permittivity of vacuum. Use of equations (20) and (21) requires knowledge of the interdigitated

190

electrode characteristic dimension δHF .

191

The ohmic impedance Ze was represented by the use of up to four Voigt elements46 as Ze =

n RC X k=1

Rk 1 + jωRk Ck

(22)

192

A series of Voigt elements yielded the lowest standard error in the regressed parameters of the geomet-

193

ric capacitance in comparison to other equivalent circuits such as an RCPE. Following equation (12), the

194

interfacial impedance Z0 was that of a capacitor, i.e., Z0 =

1 jωCdl

11

(23)

195

The regression of equation (17) to the synthetic data under statistical weighting yielded parameters with

196

95.4% confidence intervals that did not include zero.

197

3. Results

198

Simulations of interdigitated electrodes were performed to determine the influence of the electrode digit

199

width, spacing, and height on the on the impedance response. Parameters were extracted from the simulated

200

data by nonlinear regression of an impedance process model given by equation (17).

201

The simulation frequency range of 1 mHz to 1 GHz was sufficient to display the high-frequency loop

202

associated with geometric capacitance. The simulations were performed using an assumed solution resistivity

203

of ρ = 105 Ωcm and a dielectric constant of 11.294, representative of an organic non-polar liquid. The results

204

of the simulations, presented in dimensionless format, are independent of the electrolyte properties.

205

The width of an interdigitated-electrode digit W was varied from 1 to 9.9 µm. The sum of the electrode

206

width W and gap S was held constant at 10 µm. As a result, the electrode spacing was given by S =

207

10 µm − W . Simulations were also performed to account for the influence of the finite height of the electrode

208

digits. The height of the electrode was varied from 0 to 0.160 µm; whereas, for these simulations, the width

209

and spacing were held constant at 5 µm. The effective ohmic resistance and geometric capacitance were

210

extracted by nonlinear regression of the process model given in equation (17).

211

A typical calculated impedance response for the Metrohm DropSens G-IDEAU5 interdigitated electrode is

212

presented in Figure 6, in which the contributions of the geometric and interfacial impedances are delineated.

213

The high-frequency ohmic resistance Re,HF obtained from the impedance response is shown to be equal

214

to the primary ohmic resistance Re,p obtained from steady-state calculations and the ohmic resistance Re

215

obtained from the analytic solution to Laplace’s equation, which does not account for the displacement

216

current. The low-frequency ohmic impedance Re,LF can be seen as the extrapolation of the low-frequency

217

impedance response to the real axis. The value of Re,HF was obtained from nonlinear regression of equation

218

(17), and the value of Re,LF was obtained from the real part of the impedance at a frequency of 1 mHz,

219

which was sufficiently small to yield the asymptotic value of the real part of the impedance.

220

221

Two characteristic frequencies can be identified. The characteristic frequency corresponding to the maximum of the geometric-capacitance loop is given by fc,g =

222

1 2πρr 0

Thus, the dimensionless frequency associated with the geometric capacitance is Kg = ωρr 0

223

(24)

(25)

The frequency associated with geometry-induced frequency dispersion can be expressed as fc,LF =

1 2πRe,LF Cdl 12

(26)

-1.5

-1.0

-0.5

0.0 0.0

0.5

1.0

Figure 6: Calculated impedance response for the Metrohm DropSens G-IDEAU5 electrode scaled by the calculated primary ohmic resistance. The digit width and separation was 5 µm and the electrode height was 120 nm. The calculated impedance is shown by the dashed line. The impedance associated with the geometric capacitance is given as a solid line, and the interfacial impedance is given as a vertical dotted line.

224

Thus, the dimensionless frequency associated with geometry-induced frequency dispersion was given by KLF = ωRe,LF Cdl

(27)

225

The onset of geometry-induced frequency dispersion may be expected to be visible for frequencies larger

226

than KLF = 0.1.

227

The results are presented in the subsequent sections for isolated flat rectangular electrodes embedded in

228

an insulating plane, for flat rectangular electrodes embedded in an insulating plane and in close proximity,

229

and for the raised rectangular electrodes in close proximity. The case involving raised rectangular electrodes

230

in close proximity represents the typical interdigitated electrode; whereas, the case for flat rectangular

231

electrodes embedded in an insulating plane and in close proximity represents an idealized interdigitated

232

electrode.

233

3.1. Impedance of an Isolated Flat Electrode

234

The impedance response of isolated rectangular electrodes, scaled by the low-frequency ohmic resistance,

235

is shown in Figure 7 in Nyquist format. Results were superposed for electrode widths from 1 to 9.9 µm. The

236

ohmic resistances obtained from the primary current distribution, high-frequency ohmic resistance, and the

237

low-frequency ohmic resistance were found to be linearly dependent on the electrode width, i.e., Re = ηρW

13

(28)

1.5

1.0

0.5

0.0 0.0

0.5

1.0

Figure 7: The impedance response in Nyquist format for an isolated flat rectangular electrode scaled by the low-frequency ohmic resistance. The dimensionless characteristic frequencies Kg and KLF are labeled with circles and squares, respectively. Results are superposed for electrode widths from 1 to 9.9 µm.

238

where the average geometric factors η for the seven simulations (for W values of 1, 2.5, 5, 7.5, 9, 9.5, and

239

9.9 µm) were 5.53912±0.00029 for Re,p , 5.538260±0.000052 for Re,HF , and 5.61037000±0.00000025 for Re,LF .

240

The primary and high-frequency ohmic resistances were equal to within 0.016%. As Re,LF /Re,HF = 1.013,

241

i.e., a difference of 1.3%, the contribution of frequency dispersion for the isolated electrode is much less than

242

is shown in Figure 6 for an interdigitated electrode, for which Re,LF /Re,HF = 1.096 (a difference of 9.6%).

243

The scaled modulus and phase angle are shown in Figures 8(a) and 8(b). The curves for scaled modulus

244

|Z|/Re,HF plotted as a function of Kg superposed at high frequencies. The impedance response for Kg > 0.01

245

corresponded to the capacitive loop associated with the geometric capacitance, and the phase at Kg = 1 has

246

a value of −45◦ . This point is shown as a circle in Figure 7. The curves for scaled modulus |Z|/Re,LF plotted

247

as a function of KLF superposed at low frequencies. The impedance response for KLF < 100 corresponded

248

to the capacitive behavior associated with the interfacial impedance. The phase at KLF = 1 has a value of

249

−45◦ , and this point is represented as a square in Figure 7 where −Zj /Re,LF = Zr /Re,LF = 1.

250

3.2. Influence of Counterelectrode Proximity

251

The influence of counterelectrode proximity on impedance response is shown in Nyquist format in Figure 9.

252

The sum of the electrode width and spacing was fixed at 10 µm, thus, the electrode spacing decreased as the

253

electrode width increased. As the spacing decreased, the ratio between the low-frequency ohmic resistance

254

and high-frequency ohmic increased from 1.012 for W = 0.1 µm and S = 9.9 µm to 1.92 for W = 9.9 µm

255

and S = 0.1 µm. Thus, for a gap to electrode with ratio of S/W = 99, the electrode behaved as an 14

10-8 102

10-4

100

104

108 102

101

101

100

100

10-1

10-1

10-2 -8 10

10-4

100

10-8 -90 -75 -60 -45 -30 -15 0 10-8

10-2 108

104

(a)

10-4

100

104

108

10-4

100

104

108

(b)

Figure 8: The impedance response in Bode format for an isolated flat interdigitated-electrode digit with the electrode width as a parameter: a) Modulus scaled by either the high-frequency ohmic resistance or the low-frequency ohmic resistance as a function of dimensionless frequencies, Kg and KLF , respectively and b) phase angle as a function of dimensionless frequencies, Kg and KLF .

1.5

1.0

0.5

a b c d

0.0 0.0

e

0.5

1.0

Figure 9: The impedance of an interdigitated electrode scaled by the low-frequency ohmic resistance in Nyquist format with the electrode width as a parameter. These results show the influence of working and counter electrode proximity. The labels for individual spectra are: a) W = 0.1 µm and S = 9.9 µm; b) W = 1.0 µm and S = 9.0 µm; c) W = 5.0 µm and S = 5.0 µm; d) W = 9.01 µm and S = 1.0 µm; and e) W = 9.9 µm and S = 0.1 µm. The dimensionless characteristic frequencies of Kg = 1 and KLF = 1 are indicated by circles and squares, respectively.

15

10-6 102

10-3

100

103

101

101

100

100

-1

10

d c

10-2 -6 10

b

a

103

100

104

108

10-4

100

104

108

-60 -45 -30 -15

10 100

10-4

-75

-1

e

10-3

10-8 -90

106 102

0 10-8

10-2 106

(a)

(b)

Figure 10: The impedance response in Bode format for a flat interdigitated-electrode digit with the electrode width as a parameter: a) Modulus scaled by either the high-frequency ohmic resistance or the low-frequency ohmic resistance as a function of dimensionless frequencies, Kg and KLF , respectively and b) phase angle as a function of dimensionless frequencies, Kg and KLF . The labels for individual spectra are those shown in Figure 9.

256

isolated electrode; whereas, for smaller values of S/W , the influence of the adjacent electrode increased the

257

contribution of the ohmic impedance.

258

The scaled modulus and phase are shown in Figures 10(a) and 10(b) as a function of dimensionless

259

frequencies Kg and KLF . The extent of frequency dispersion in Figure 3.2 can be compared to that shown

260

in Figure 8 for isolated electrodes. The curves for scaled modulus |Z|/Re,HF plotted as a function of Kg

261

superposed at high frequencies, and the curves for scaled modulus |Z|/Re,LF plotted as a function of KLF

262

superposed at low frequencies. The phase at Kg = 1 and KLF = 1 has values of −45◦ , which correspond to

263

symbols shown in Figure 9.

264

3.3. Impedance Response of a Finite-Height Electrode

265

The influence of electrode height is shown in the impedance responses in Figure 11, in which the electrode

266

height varied from 0 to 160 nm for W = S = 5 µm. The impedance response was not very sensitive

267

to electrode height in the range of simulated electrode heights. The impedance response for an electrode

268

height of 120 nm, corresponding to the Metrohm DropSens G-IDEAU5 electrode, was almost identical to

269

the response shown in Figure 11 for an electrode height of 160 nm.

270

4. Discussion

271

A cell constant is required to employ measurements on interdigitated electrodes to extract electrical

272

properties of adjacent media. The discrepancy reported in the literature between cell constants extracted 16

1.5 1.0 0.5 0.0 0.0

0.5

1.0

Figure 11: The impedance of an interdigitated electrode scaled by the low-frequency ohmic resistance in Nyquist format with the electrode height as a parameter. The electrode width and spacing was fixed at W = S = 5 µm. Results for electrode heights of 0 and 160 nm are displayed. The dimensionless characteristic frequencies of Kg = 1 and KLF = 1 are indicated by circles and squares, respectively.

273

from simulations and those extracted from experiments can be attributed to the influence of frequency

274

dispersion on ohmic impedance described in the present work. The cell constant is given by Kcell =

Re ρA

(29)

275

where A is the electrode area. The cell constant shown in equation (29) has units of cm−1 and can be related

276

to the characteristic dimension of the cell δ following δ = Kcell A

(30)

277

The value of the cell constant depends on whether the ohmic resistance used is that corresponding to high

278

or low frequency. The assessment of characteristic dimension and the characterization of ohmic impedance

279

are presented in the following section.

280

4.1. Characteristic Dimension of an Interdigitated Electrode

281

The characteristic dimension of interdigitated electrodes may be obtained from values for ohmic resistance

282

and geometric capacitance. The characteristic dimension based on the steady-state primary ohmic resistance

283

is given by δp =

Re,p ρ

(31)

284

where Re,p was the ohmic resistance obtained from the steady-state calculation of the primary current

285

distribution. The characteristic dimension based on the high-frequency ohmic resistance is given by δHF =

Re,HF ρ

17

(32)

100 100

10-1 10-1

10-2 10-2

10-3 10-2 10-1 100 101 102

10-3 10-2 10-1 100 101 102

(a)

(b)

Figure 12: The scaled characteristic dimension of an interdigitated electrode as a function of the ratio between the electrode width and spacing with method of calculation as a parameter: a) The ratio of the characteristic dimensions for flat interdigitated electrodes δ0,k to that of isolated rectangular electrodes δ0,k,iso and b) the characteristic dimension of flat interdigitated electrode scaled by the sum of the electrode width and spacing.

286

where Re,HF was the high-frequency ohmic resistance obtained by nonlinear regression of the process model

287

shown in equation (17) to the simulated impedance response. The characteristic dimension based on the

288

low-frequency ohmic resistance is given by δLF =

Re,LF ρ

(33)

289

where the value of Re,LF was obtained from the limiting value of the real part of the impedance at frequencies

290

sufficiently small that the influence of ohmic impedance was negligible. The value was extracted from a

291

frequency of f = 1 mHz, for which KLF < 10−5 . The characteristic dimension based on the geometric

292

capacitance is given by δCg =

r 0 Cg

(34)

293

where Cg was the capacitance associated with the high-frequency loop. The value of Cg was obtained by

294

nonlinear regression of the process model shown in equation (17) to the simulated impedance response. The

295

effective capacitance Cq was obtained from the regressed values of Re,HF , Qg , and αg by use of the Brug

296

formula given as equation (19).

297

The ratio of the characteristic dimensions for flat interdigitated electrodes δ0,k to that of isolated rectan-

298

gular electrodes δ0,k,iso is presented in Figure 12(a) as functions of the width to spacing ratio. The subscript

299

k is used as an identifier of the method of calculation, i.e., equation (31), (32), (33), or (34). The ratio of

300

characteristic dimensions approached a value of unity as W/S decreased, which suggests that as the spacing

301

was increased, the electrode behaved like an isolated electrode. As W/S increased, the ratio of character-

302

istic dimensions appeared to approach a constant value. Excellent agreement was observed between the

18

1.00 0.99 0.98 0.97

10-3

10-2

10-1

Figure 13: The low and high frequency characteristic dimensions as a function of interdigitated electrode height with method as a parameter. The dimensions were extracted from an interdigitated electrode where the electrode width was equal to the electrode spacing. The characteristic dimensions were scaled by the characteristic dimension of a flat interdigitated electrode.

303

characteristic dimensions obtained from the primary ohmic resistance (equation (31)) high-frequency ohmic

304

resistance (equation (32)), and geometric capacitance (equation (34)). For small values of W/S, the charac-

305

teristic dimension obtained from the low-frequency ohmic resistance (equation (33)) was in agreement with

306

those obtained by equations (31), (32), and (34). This result is consistent with the observation that fre-

307

quency dispersion was less evident for isolated rectangular electrodes. For larger values of W/S, a substantial

308

deviation was observed.

309

The scaled characteristic dimensions of flat interdigitated electrodes are presented in Figure 12(b) as

310

functions of the ratio of electrode width and spacing. The maximum in the high-frequency characteristic

311

dimension was observed when the electrode width and spacing were approximately equal, in agreement

312

with the observations of Ibrahim et al.16 As the W/S increased, the difference between the low-frequency

313

(equation (33)) and high-frequency (equations (31), (32), and (34)) characteristic dimensions increased,

314

which suggested that a wider electrode and small electrode spacing yielded greater frequency dispersion.

315

For small W/S, the scaled characteristic dimensions showed a power-law dependence on W/S, which was a

316

result of the scaling of the ohmic resistance with the width of the electrode. This observation is in agreement

317

with the conclusions of Skjolding et al.,14 where optimal sensitivity was observed with thinner interdigitated

318

electrode digits.

319

The ratio of the characteristic dimension of an interdigitated electrode with a finite height to that of a flat

320

electrode is presented in Figure 13 as a function of dimensionless height. The electrode width and spacing

321

were equal, i.e., W/S = 1 for the results presented in Figure 13. The ratio of characteristic dimensions

322

approached a value of unity as the height of the electrode was decreased. As the height of the electrode was

323

increased, the characteristic dimension of an electrode with a finite height decreased, which was attributed

19

-0.04

-0.04

-0.02

-0.02

0.00 1.00

1.02

1.04

1.06

1.08

1.10

0.00 1.00

1.02

(a)

1.04

1.06

1.08

1.10

(b)

Figure 14: Calculated complex ohmic impedance, scaled by the primary ohmic resistance, in Nyquist format with the electrode height-to-width ratio as a parameter: (a) results for H/W values ranging from 0 to 0.04 and (b) results for H/W = 0 and H/W = 0.024. Synthetic data are represented by circles, and the dotted line was obtained by regression of equation (35).

324

to the additional surface area of the vertical sides. In the range of electrode heights simulated, the high-

325

frequency characteristic dimension was reduced by 2.4%; whereas, the low-frequency characteristic dimension

326

was reduced by 1%.

327

4.2. Ohmic Impedance of an Interdigitated Electrode

328

The equivalent-circuit process model utilized Voigt elements in the nonlinear regression. The use of Voigt

329

elements yielded an excellent representation of the ohmic impedance, but required up to eight regression

330

parameters. Gharbi et al.31 showed that a model proposed by Havriliak and Negami32 provided an excellent

331

representation of the complex ohmic impedance for a disk electrode, i.e., Ze = Re,p + 

Re,LF − Re,HF 1−β 1−ν 1 + (jωτ )

(35)

332

where τ was the characteristic time constant of the ohmic impedance. The exponents ν and β dictate the

333

limiting phase angles of the ohmic impedance at high and low frequency.

334

The ohmic impedance was obtained by regression of equation (17) to synthetic data. The ohmic impe-

335

dance of an interdigitated electrode with equal width and spacing is shown in Figure 14(a) with the ratio

336

between the electrode height and width as a parameter.

337

The results for nonlinear regression of equation (35) to synthetic ohmic impedance data are shown in

338

Figure 14(b) for interdigitated electrodes with H/W = 0 and H/W = 0.024. The height to width ratio of

339

0.024 corresponds to a height of 120 nm for an interdigitated electrode with a width and spacing of 5 µm,

340

consistent with the Metrohm DropSens G-IDEAU5. The resulting parameters are shown in Table 4. The

341

regressed parameters were statistically significant and suggest that the Havriliak–Negamii equation, shown

342

by Gharbi et al.31 to provide an acceptable fit to the ohmic impedance of a disk electrode, provides an

343

acceptable fit for the ohmic impedance of interdigitated electrodes.

20

Table 4: Regressed parameters of the ohmic impedances shown in Figure 14(b) using the Havriliak-Negami equation shown in equation (35)

Parameter

H/W = 0

H/W = 0.024

Re,HF /Re,p

1.00008±0.00024

0.998220±0.000092

Re,LF /Re,p

1.09918±0.00019

1.085370±0.000022

τ [ms]

0.1195±0.0012

0.1423±0.0010

ν

0.0280±0.0023

0.0190±0.0020

β

0.01277±0.0059

0.3326±0.0043

344

The ohmic impedance in the present work was developed for an ideally polarizable interdigitated electrode.

345

Further work is suggested for interdigitated electrodes under the influence of films and faradaic reactions.

346

Gharbi et al.31 demonstrated that interfacial CPE behavior on a disk electrode yielded a depressed ohmic

347

impedance but did not influence the values of the low- and high-frequency ohmic resistance. Orazem and

348

Tribollet showed in Figure 13.13 of their book,47 for example, that the value of Re,LF /Re,HF is reduced under

349

the influence of faradaic reactions for a disk electrode geometry.

350

4.3. Interdigitated Electrode Cell Constant

351

Simulations were performed that replicated commercially available interdigitated electrodes with a width

352

and spacing of 5 µm. The characteristic dimension obtained from calculation of the primary ohmic resis-

353

tance had a value of 5.001 µm. The low-frequency characteristic dimension was found to be 5.4274 µm.

354

When an electrode height of 120 nm was included in the simulations, the low and high frequency charac-

355

teristic dimensions were reduced to the values of 5.3835 µm and 4.900 µm, respectively. The reduction was

356

attributed to the additional electrode surface area and accompanying frequency dispersion on the vertical

357

faces of the electrode. The inclusion of a finite height was shown to reduce the characteristic dimension

358

and, subsequently, the cell constant Kcell . For conductive media, where the measured frequencies are below

359

KLF = 1, measurement of an ohmic resistance yields the low-frequency ohmic resistance. The cell constant

360

Kcell calculated using the low-frequency ohmic resistance will be larger than the cell constant obtained from

361

numerical simulations of the primary resistance. The characteristic dimension based on the low-frequency

362

ohmic resistance was shown to be roughly 10% larger than the characteristic dimension based on the high-

363

frequency ohmic resistance. The present work demonstrated that for accurate extraction of a electrical

364

properties of a medium using interdigitated electrodes requires use of the applicable characteristic dimension

365

in the appropriate measured frequency range.

366

5. Conclusions

367

The nonuniform current and potential distribution associated with the interdigitated electrode yields a

368

frequency dispersion similar to that reported for a disk electrode. This contribution can be expressed as an 21

369

ohmic impedance with asymptotic values for ohmic impedance at high and low frequencies. The contribution

370

of the ohmic impedance was found to be strongly dependent on the electrode width and, to a lesser extent,

371

on the electrode height. For a large value of electrode width with a narrow gap between electrodes, the low-

372

frequency ohmic resistance had a value that was roughly twice that of the primary or high-frequency ohmic

373

resistance. For the configuration of a Metrohm DropSens G-IDEAU5 electrode with equal 5 µm electrode

374

width and gap and a 120 nm electrode height, the low-frequency ohmic resistance had a value that was 10

375

percent of the primary or high-frequency ohmic resistance.

376

The finite-element simulations reported in the present work for a capacitive electrode surface provide an

377

upper bound for the contribution of the ohmic impedance. As discussed by Orazem and Tribollet47 for a

378

disk electrode, the value of Re,LF /Re,HF can be reduced under the influence of faradaic reactions for a disk

379

electrode geometry. As reported by Gharbi et al.31 for a disk electrode, the presence of constant-phase-

380

element behavior on the disk electrode does not affect the value of Re,LF /Re,HF , though the corresponding

381

Nyquist plots show depressed loops. These observations were confirmed for the interdigitated electrode in

382

work not reported here. As reported by Gharbi et al.31 for a disk electrode, the Havriliak-Negami equation

383

provided a good representation for the ohmic impedance for interdigitated electrodes.

384

Two characteristic frequencies were identified in the present work: one associated with the geometric

385

capacitance (equation (24)) and one associated with the onset of geometry-induced frequency dispersion

386

(equation (26)). The characteristic frequency of the geometric capacitance was independent of the charac-

387

teristic dimension of the interdigitated electrode. The characteristic frequency associated with frequency

388

dispersion was dependent on the cell geometry. The ohmic impedance described in the present work will not

389

be evident for measurements performed for frequencies smaller than the characteristic frequency reported as

390

equation (26).

391

The characteristic dimensions obtained from the geometric capacitance, the primary ohmic resistance,

392

and the high-frequency ohmic resistance were equal for all electrode geometries studied. The characteristic

393

dimensions obtained from the low-frequency ohmic resistance deviated from the values obtained from the

394

geometric capacitance, the primary ohmic resistance, and the high-frequency ohmic resistance for geometric

395

configurations under which the ratio Re,LF /Re,HF deviated from unity.

396

6. Acknowledgments

397

The authors express appreciation for the support of Mr. Michael J. Kubicsko, Metrohm USA, Inc. Mark

398

Orazem acknowledges financial support from the University of Florida Foundation Preeminence and the Dr.

399

and Mrs. Frederick C. Edie term professorships.

22

400

7. List of Symbols

Symbol

Description

A

electrode area, cm2

a

coefficient for extrapolation of the ohmic resistance

Cdl

double-layer capacitance, F/cm2

Ck

capacitance for Voigt-element used to model ohmic impedance, F/cm2

Cg

geometric capacitance, F/cm2

D

electrical displacement vector, C/cm2

E

electric field vector, V/cm

fc,g

characteristic frequency for the geometric capacitance loop, Hz

fc,LF

characteristic frequency based on low-frequency ohmic resistance, Hz

H

electrode digit height, cm

Hdom

domain radius, cm

I

total steady-state current, A

i

current density vector, A/cm2

Kcell

cell constant, cm−1

Kg

dimensionless frequency based on geometric-capacitance

KLF

dimensionless frequency based on low-frequency ohmic resistance

L

electrode digit length, cm

n

exponent for extrapolation of the ohmic resistance

nd

number of electrode digit pairs

n

normal unit vector

Qg

high-frequency CPE parameter, Fsα /cm

r0

radius of disk electrode, cm

Re

ohmic resistance, Ωcm2

Re,HF

high-frequency ohmic resistance, Ωcm2

Re,LF

low-frequency ohmic resistance, Ωcm2

Re,p

primary ohmic resistance obtained from steady-state primary current distribution, Ωcm2

Rk

resistance for Voigt-element used to model ohmic impedance, Ωcm2

sdom

maximum element size in the simulation domain, cm

se

maximum element size at electrode corners, cm

ss

maximum element size on the electrode surface, cm

W

electrode digit width, cm

X

steady-state variable X

e X

complex phasor variable X

23

X0

interfacial variable X

Xm

variable X at the electrode surface

Z

total impedance, Ωcm2

Ze

ohmic impedance, Ωcm2

Zg

impedance associated with geometric capacitance, Ωcm2

Z0

interfacial impedance, Ωcm2

Greek Symbol

Description

αg

regressed high-frequency CPE parameter

β

exponent used in the Havriliak–Negami equation

δ

characteristic dimension, cm

δCg

characteristic dimension obtained from geometric capacitance, cm

δHF

characteristic dimension obtained from high-frequency ohmic resistance, cm

δLF

characteristic dimension obtained from low-frequency ohmic resistance, cm

δp

characteristic dimension obtained from the primary ohmic resistance, cm

0

permittivity of vacuum, 8.8542×10−14 C/Vcm

r

relative dielectric constant

ε

absolute percent error

κ

electrical conductivity, S/cm

ν

exponent used in the Havriliak–Negami equation

Φ

electrical potential, V

ρ

electrolyte resistivity, Ωcm

τ

time constant used in the Havriliak–Negami equation, s

ω

angular frequency, rad/s

24

401

402

403

404

405

References [1] L. Bueno, T. R. L. C. Paix˜ ao, A copper interdigitated electrode and chemometrical tools used for the discrimination of the adulteration of ethanol fuel with water, Talanta 87 (2011) 210–215. [2] J. Staginus, Polydimethylsiloxane-coated interdigitated electrodes for capacitive detection of organic pollutants in water: A systematic guide, Ph.D. thesis, Delft University of Technology (2015).

406

[3] F. Mohamad, M. A. Sairin, N. N. A. Nizar, S. A. Aziz, D. M. Hashim, S. Misbahulmunir, F. Z. Rokhani,

407

Investigation on interdigitated electrode design for impedance spectroscopy technique targeting lard

408

detection application, in: 9th International Conference on Sensing Technology, Piscataway, 2015, pp.

409

739–744.

410

[4] V. Matylitskaya, S. Kasemann, G. Urban, C. Dincer, S. Partel, Electrochemical characterization of

411

nanogap interdigitated electrode arrays for lab-on-a-chip applications, Journal of the Electrochemical

412

Society 165 (3) (2018) B127–B134.

413

414

[5] L. Yang, Y. Li, G. F. Erf, Interdigitated array microelectrode-based electrochemical impedance immunosensor for detection of escherichia coli O157:H7, Analytical Chemistry 76 (4) (2004) 1107–1113.

415

[6] R. Maalouf, C. Fournier-Wirth, J. Coste, H. Chebib, Y. Sa¨ıkali, O. Vittori, A. Errachid, J.-P. Cloarec,

416

C. Martelet, N. Jaffrezic-Renault, Label-free detection of bacteria by electrochemical impedance spec-

417

troscopy: comparison to surface plasmon resonance, Analytical Chemistry 79 (13) (2007) 4879–4886.

418

[7] X. Guo, A. Kulkarni, A. Doepke, H. B. Halsall, S. Iyer, W. R. Heineman, Carbohydrate-based label-

419

free detection of escherichia coli ORN 178 using electrochemical impedance spectroscopy, Analytical

420

Chemistry 84 (1) (2011) 241–246.

421

[8] M. Labib, A. S. Zamay, O. S. Kolovskaya, I. T. Reshetneva, G. S. Zamay, R. J. Kibbee, S. A. Sattar,

422

T. N. Zamay, M. V. Berezovski, Aptamer-based viability impedimetric sensor for bacteria, Analytical

423

Chemistry 84 (21) (2012) 8966–8969.

424

[9] C. Hu, D.-P. Yang, Z. Wang, L. Yu, J. Zhang, N. Jia, Improved EIS performance of an electrochemical

425

cytosensor using three-dimensional architecture Au@BSA as sensing layer, Analytical Chemistry 85 (10)

426

(2013) 5200–5206.

427

428

429

430

431

432

[10] L. Han, P. Liu, V. A. Petrenko, A. Liu, A label-free electrochemical impedance cytosensor based on specific peptide-fused phage selected from landscape phage library, Scientific Reports 6 (2016) 22199. [11] M. C. Zaretsky, L. Mouayad, J. R. Melcher, Continuum properties from interdigital electrode dielectrometry, IEEE Transactions on Electrical Insulation 23 (6) (1988) 897–917. [12] C. E. Chidsey, B. Feldman, C. Lundgren, R. W. Murray, Micrometer-spaced platinum interdigitated array electrode: fabrication, theory, and initial use, Analytical Chemistry 58 (3) (1986) 601–607. 25

433

[13] Z. Zou, J. Kai, M. J. Rust, J. Han, C. H. Ahn, Functionalized nano interdigitated electrodes arrays on

434

polymer with integrated microfluidics for direct bio-affinity sensing using impedimetric measurement,

435

Sensors and Actuators A: Physical 136 (2) (2007) 518 – 526.

436

437

[14] L. Skjolding, C. Spegel, A. Ribayrol, J. Emn´eus, L. Montelius, Characterisation of nano-interdigitated electrodes, in: Journal of Physics: Conference Series, Vol. 100, IOP Publishing, 2008, p. 052045.

438

[15] U. Pliquett, D. Frense, M. Sch¨ onfeldt, C. Fr¨atzer, Y. Zhang, B. Cahill, M. Metzen, A. Barthel, T. Nacke,

439

D. Beckmann, Testing miniaturized electrodes for impedance measurements within the beta-dispersion–

440

a practical approach, Journal of Electrical Bioimpedance 1 (1) (2010) 41–55.

441

[16] M. Ibrahim, J. Claudel, D. Kourtiche, M. Nadi, Geometric parameters optimization of planar inter-

442

digitated electrodes for bioimpedance spectroscopy, Journal of Electrical Bioimpedance 4 (1) (2013)

443

13–22.

444

[17] D.-Y. Jeon, S. Jeong Park, Y. Kim, M.-J. Shin, P. Soo Kang, G.-T. Kim, Impedance characterization

445

of nanogap interdigitated electrode arrays fabricated by tilted angle evaporation for electrochemical

446

biosensor applications, Journal of Vacuum Science & Technology B 32 (2) (2014) 021803.

447

448

[18] J. Hong, D. S. Yoon, S. K. Kim, T. S. Kim, S. Kim, E. Y. Pak, K. No, AC frequency characteristics of coplanar impedance sensors as design parameters, Lab on a Chip 5 (3) (2005) 270–279.

449

[19] Z. Chen, A. Sepulveda, M. Ediger, R. Richert, Dielectric spectroscopy of thin films by dual channel

450

impedance measurements on differential interdigitated electrode arrays, European Physical Journal B

451

85 (8).

452

453

454

455

456

457

458

459

[20] P. Tomˇc´ık, Microelectrode arrays with overlapped diffusion layers as electroanalytical detectors: theory and basic applications, Sensors 13 (10) (2013) 13659–13684. [21] S. O. Blume, R. Ben-Mrad, P. E. Sullivan, Modelling the capacitance of multi-layer conductor-facing interdigitated electrode structures, Sensors and Actuators B: Chemical 213 (2015) 423–433. [22] S. MacKay, P. Hermansen, D. Wishart, J. Chen, Simulations of interdigitated electrode interactions with gold nanoparticles for impedance-based biosensing applications, Sensors 15 (9) (2015) 22192–22208. [23] R. De La Rica, C. Fern´ andez-S´ anchez, A. Baldi, Polysilicon interdigitated electrodes as impedimetric sensors, Electrochemistry Communications 8 (8) (2006) 1239–1244.

460

[24] W. Olthuis, W. Streekstra, P. Bergveld, Theoretical and experimental determination of cell constants

461

of planar-interdigitated electrolyte conductivity sensors, Sensors and Actuators B: Chemical 24 (1-3)

462

(1995) 252–256.

463

464

[25] J. S. Newman, Frequency dispersion in capacity measurements at a disk electrode, Journal of the Electrochemical Society 117 (1970) 198–203. 26

465

[26] V. Brichzin, J. Fleig, H.-U. Habermeier, G. Cristiani, J. Maier, The geometry dependence of the polar-

466

ization resistance of Sr-doped LaMnO3 microelectrodes on yttria-stabilized zirconia, Solid State Ionics

467

152 (2002) 499–507.

468

[27] V. M.-W. Huang, V. Vivier, M. E. Orazem, N. P´eb`ere, B. Tribollet, The apparent CPE behavior of

469

an ideally polarized blocking electrode: A global and local impedance analysis, Journal of the Electro-

470

chemical Society 154 (2007) C81–C88.

471

472

[28] 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.

473

[29] V. M.-W. Huang, V. Vivier, I. Frateur, M. E. Orazem, B. Tribollet, The global and local impedance

474

response of a blocking disk electrode with local CPE behavior, Journal of the Electrochemical Society

475

154 (2007) C89–C98.

476

477

[30] K. Davis, A. Dizon, C. L. Alexander, M. E. Orazem, Influence of geometry-induced frequency dispersion on the impedance of rectangular electrodes, Electrochimica Acta 283 (2018) 1820–1828.

478

[31] O. Gharbi, A. Dizon, M. E. Orazem, M. T. T. Tran, B. Tribollet, V. Vivier, From frequency dispersion to

479

ohmic impedance: A new insight on the high-frequency impedance analysis of electrochemical systems,

480

Electrochimica Acta 320 (2019) 134609.

481

482

[32] S. Havriliak, S. Negami, A complex plane representation of dielectric and mechanical relaxation processes in some polymers, Polymer 8 (1967) 161–210.

483

[33] J. Mizusaki, H. Tagawa, T. Saito, T. Yamamura, K. Kamitani, K. Hirano, S. Ehara, T. Takagi, T. Hikita,

484

M. Ippommatsu, et al., Kinetic studies of the reaction at the nickel pattern electrode on YSZ in H2 -H2 O

485

atmospheres, Solid State Ionics 70 (1994) 52–58.

486

487

488

489

[34] D. G. Goodwin, H. Zhu, A. M. Colclasure, R. J. Kee, Modeling electrochemical oxidation of hydrogen on Ni–YSZ pattern anodes, Journal of the Electrochemical Society 156 (9) (2009) B1004–B1021. [35] C. Chen, D. Chen, W. C. Chueh, F. Ciucci, Modeling the impedance response of mixed-conducting thin film electrodes, Physical Chemistry Chemical Physics 16 (23) (2014) 11573–11583.

490

[36] F. Ciucci, W. C. Chueh, D. G. Goodwin, S. M. Haile, Surface reaction and transport in mixed conductors

491

with electrochemically-active surfaces: a 2-D numerical study of ceria, Physical Chemistry Chemical

492

Physics 13 (2011) 2121–2135.

493

494

[37] W. C. Chueh, Y. Hao, W. Jung, W. Haile, High electrochemical activity of the oxide phase in model ceria–Pt and ceria–Ni composite anodes, Nature Materials 11 (2) (2012) 155.

27

495

[38] A. Nenning, A. K. Opitz, T. M. Huber, J. Fleig, A novel approach for analyzing electrochemical prop-

496

erties of mixed conducting solid oxide fuel fell anode materials by impedance spectroscopy, Physical

497

Chemistry Chemical Physics 16 (2014) 22321–22336.

498

499

500

501

[39] J. Liu, F. Ciucci, Modeling the impedance spectra of mixed conducting thin films with exposed and embedded current collectors, Physical Chemistry Chemical Physics 19 (2017) 26310–26321. [40] S. Shibata, Thermal atomic layer deposition of lithium phosphorus oxynitride as a thin-film solid electrolyte, Journal of the Electrochemical Society 163 (13) (2016) A2555–A2562.

502

[41] J. C. Maxwell, A Treatise on Electricity and Magnetism, Vol. 1, Clarendon Press, Oxford, 1873.

503

[42] K. N. Allahar, D. P. Butt, M. E. Orazem, H. A. Chin, G. Danko, W. Ogden, R. E. Yungk, Impedance

504

of steels in new and degraded ester-based lubricating oil, Electrochimica Acta 51 (2006) 1497–1504.

505

[43] V. F. Lvovich, M. F. Smiechowski, AC impedance investigation of conductivity of automotive lubricants

506

using two- and four-electrode electrochemical cells, Journal of Applied Electrochemistry 39 (12) (2009)

507

2439.

508

509

[44] K. Levenberg, A method for the solution of certain non-linear problems in least squares, Quarterly of Applied Mathematics 2 (2) (1944) 164–168.

510

[45] G. J. Brug, A. L. G. van den Eeden, M. Sluyters-Rehbach, J. H. Sluyters, The analysis of electrode

511

impedances complicated by the presence of a constant phase element, Journal of Electroanalytical

512

Chemistry 176 (1984) 275–295.

513

[46] P. Agarwal, M. E. Orazem, L. H. Garc´ıa-Rubio, Measurement models for electrochemical impedance

514

spectroscopy: 1. Demonstration of applicability, Journal of the Electrochemical Society 139 (1992)

515

1917–1927.

516

517

[47] M. E. Orazem, B. Tribollet, Electrochemical Impedance Spectroscopy, 2nd Edition, John Wiley & Sons, Hoboken, 2017.

28

Highlights:   

Finite-element impedance simulations provide relationship between characteristic dimension and dimensions of interdigitated electrodes. Model accounted for the loop associated with geometric capacitance. Frequency dispersion characterized in terms of a complex ohmic impedance with real asymptotic limits for ohmic resistance at high and low frequencies.