While I was working with interdigitated array electrodes (IDA electrodes) for my previous research Real-time Impedimetric MUC1 Aptasensor using Microfluidic Symmetric Gold Electrode, I discovered a major problem with its impedance spectrum when viewing the Nyquist plot. Until then, there hadn’t been any studies that can explain the diffusion phenomenon of redox species between the band electrodes due to an applied sinusoidal voltage wave, and no circuit elements appear to be suitable for modeling the diffusion impedance of IDA electrodes. Therefore, I devised a theory for modeling an IDA electrodes impedimetric properties using conformal mapping and cylindrical finite length approximation methods, and succeeded to construct a circuit element that can replace the Warburg element, and extract important parameters from the corresponding equivalent circuit model. This can aid researchers in relevant fields to model their systems more accurately.

Here’s a video for visualizing the diffusion phenomenon between the band electrodes of an IDA electrode. COMSOL is used for simulating the time-dependent concentration field of redox species.

**Abstract**

An analytical problem for impedimetric sensing is usually encountered when using interdigitated array (IDA) electrodes. Finite diffusion of redox species dominates at low frequencies and confuses researchers, making incorrect understanding of underlying phenomena possible. In this work, an integral equation for calculating the diffusion impedance of IDA electrodes is derived using conformal mapping and cylindrical finite length approximation. Electrodes of different bandwidths and gap widths are fabricated, and their heights and symmetric electrochemical characteristics are verified. Simulations are performed to verify the predicted constant concentration contours. The calculated zero-frequency impedance showed high correlation with the reciprocal of limiting current calculated from literature study (R^{2} = 0.992) and from chronoamperometry experiments (R^{2} = 0.970). Further evidence for the correctness of theory is established due to the fact that experimental EIS data and calculated impedances are highly consistent (R^{2} ≥ 0.948 for real and imaginary part). This sheds some light on explaining the diffusion phenomenon of impedance using IDA electrodes in the low frequency spectrum. An equivalent circuit fitting program is further designed for fitting several elements including the IDA electrode diffusion impedance derived in the theory. The program succeeded to accurately fit the EIS data (average MSE = 0.611), which using the Warburg element failed (average MSE = 54.86). Parameters such as the ratio of electrode bandwidth to gap width and diffusion coefficient can also be determined by fitting the data from a single EIS experiment. Another impedance calculation program is also given, which can aid researchers in relevant fields to model their systems more accurately.

**Figures and Tables**

Figure 1. (a) Geometric definition of a unit cell (red translucent region). (b) Open finite-length diffusion elements link between the ICCB and the electrode surface within the left half of a unit cell.

Figure 2. Path of redox species that finitely diffuse between the ICCB and the electrode within a unit cell. The finite diffusion length at *x _{e}* is determined by the elliptic arc length.

Figure 3. The conformal mapping of the (a)*z*, (b)*t* and (c)*u* plane applied in this theory.

Figure 4. The finite diffusion length against the position on the electrode.

Figure 5. The diffusion region (gray area) of redox species between a certain length derivative of a point on the electrode (*dx _{e}*) and its corresponding length derivative on the ICCB (–

*dy*).Figure 5. The diffusion region (gray area) of redox species between a certain length derivative of a point on the electrode (

_{ICCB}*dx*) and its corresponding length derivative on the ICCB (–

_{e}*dy*).

_{ICCB}Figure 6. Theoretical normalized IDA diffusion impedance for different values of *w _{g}*/

*w*and constant number of bands (

_{e}*N*). The numbers beside the equi-frequency (dashed) lines indicate the reduced frequency

*w*/

^{2}ω*D*.

Figure 7. (a) Illustration and (b) photograph of the IDA electrode chip clipped with a microwell.

Table 1. Equivalent circuit elements for diffusion impedance.

Figure 8. The reciprocal of limiting current plotted against the calculated real part of IDA diffusion impedance at *ω*→ 0. |*I _{lim}*| is (a) the absolute value of limiting current, or the current at 10sec when applying a voltage of –0.2V (vs CE/RE) in a solution containing 0.1M KCl and (b) 5mM Fe(CN)

_{6}

^{3-/4-}or (c) 5mM Fe(CN)

_{6}

^{3-}in a chronoamperometry experiment.

Figure 9. (a) Nyquist plot of experimental EIS data of an IDA electrode (*w _{g}*–

*w*= 100-50(μm)) and its equivalent circuit fitted data. The frequency range of the experimental data is 10

_{e}^{-2}~ 10

^{5}Hz and the frequency range used for fitting is 1 ~ 10

^{5}Hz. The subfigure at the top-left is the circuit being fitted and is equivalent to

*R*(

*Q*(

*RQ*)), which is a typical Randles circuit with the Warburg element replaced by a constant phase element (CPE). (b ~ d) Nyquist plot of experimental (solid) and calculated (hollow) IDA diffusion impedances at (b)

*w*= 100 and (c)

_{g}*w*= 50. The frequency range of all the data is 10

_{g}^{-2}~ 10

^{5}Hz. The unit for widths is μm.

Figure 10. Raw and fitted EIS data of 3 bare IDA electrode chip with the Randles circuit using different elements for diffusion impedance modeling. (a) Warburg element, (b) open finite-length diffusion element, (c) CPE finite-length diffusion element and (d) IDA diffusion element. The frequency range of all the data is 10^{-2}~ 10^{5}Hz. (units: μm)

Figure 11. (a) *S*/*N _{f}* (or MSE) of the fitting results for the 9 IDA electrode chips using different diffusion impedance elements. (b) Calculated and fitted values of

*Y*from the 9 chips. (c) Fitted value of

_{0}*w*/

^{2}*D*against its calculated value. (d) Fitted value of

*w*/

_{e}*w*against its calculated value.

**Related Publications**

__C.-Y. Lai__, J.-H. Weng, W.-L. Shih, L.-C. Chen, C.-F. Chou, P.-K. Wei, Diffusion impedance modeling for interdigitated array electrodes by conformal mapping and cylindrical finite length approximation,*Electrochimica Acta*,**320**(2019) 134629.

https://doi.org/10.1016/j.electacta.2019.134629__C.-Y. Lai__, J.-H. Weng, W.-L. Shih, L.-C. Chen, C.-F. Chou, P.-K. Wei, Diffusion impedance modeling for interdigitated array electrodes by conformal mapping and cylindrical finite length approximation,*11th International Symposia on Electrochemical Impedance Spectroscopy*, (2019).

[abstract] [presentation pdf] [presentation clip]