EIS Data Fitting using Different Diffusion Impedance Elements

From the above results, one can be confident that the theory can accurately approximate the limiting current (section 3.4.4) and the impedance (section 3.4.5) for diffusion of redox species of IDA electrodes at wg/we > 0.1. In this section, an equivalent circuit element is constructed for EIS data fitting of the diffusion impedance of IDA electrodes and its accuracy and effectiveness with other circuit elements are compared. A program for equivalent circuit fitting is written in advance. The program implements a complex nonlinear least square (CNLS) process (Lasia, 2002), which the weighted sum of squares of error, S, is defined as

𝑆𝑆 = � �𝑤𝑤𝑑𝑑,Re�Re�𝑍𝑍𝑑𝑑,𝑐𝑐𝑖𝑖𝑐𝑐� − Re(𝑍𝑍𝑑𝑑,𝑒𝑒𝑥𝑥𝑒𝑒)�²+ 𝑤𝑤𝑑𝑑,Im�Im�𝑍𝑍𝑑𝑑,𝑐𝑐𝑖𝑖𝑐𝑐� − Im(𝑍𝑍𝑑𝑑,𝑒𝑒𝑥𝑥𝑒𝑒)�²� imaginary parts of the impedance of the i–th frequency. The subscript exp indicates experimental value and the subscript cal indicates the calculated value while fitting. A weighting of wi,Re = 1/|Re(Zi,exp)| and wi,Im = 1/|Im(Zi,exp)| is chosen for fitting of parameters.

A Levenberg-Marquardt algorithm is used for minimization of S, which is provided by ALGLIB.

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Three other diffusion elements are compared with the IDA diffusion element as in Table 3-1. The open finite-length diffusion element is common for modeling 1D planer finite diffusion systems such as the rotating ring-disk electrode (RRDE) (Barnes et al., 2012). This element exhibits a 45° line at high frequencies and a semi-circle at low frequencies on the Nyquist plot. Because 2D finite diffusion exists in an IDA electrode system, so the initial phase angle would be slightly lower that 45° and a somehow imperfect semi-circle would appear (cf. Figure 3-8). Thus a parameter n is introduced into the open finite-length element for better fitting, and this new element is arbitrarily named as the “CPE finite-length diffusion” element. This n has the same effect as in the CPE and can further help reducing the minimum S when fitting. For convenience, the symbol for the CPE finite-length diffusion element is set as “X” and the IDA diffusion element is set as “I”.

Table 3-1 Equivalent circuit elements for diffusion impedance.

Element Name Symbol Value Parameters

Warburg W Z_W= 1

All nine IDA electrode chips are washed thoroughly, clipped with microwells and added with the solution used for electrochemical characterization. EIS experiments are performed and each chip is analyzed using a single EIS data. Nyquist plots of three of them (wg-we = 100-25, 50-50 and 2-100(μm)) along with fitted results using the four elements are shown in Figure 3-14. Obviously, the Warburg element fails to work, making the fitted data yield parameters with large errors (Figure 3-14a). The open finite-length diffusion element obtains a better fit (Figure 3-14b). The semi-circle can approximately match the down-bending curve of the IDA electrode’s diffusion impedance. However, the IDA diffusion impedance isn’t really a perfect semi-circle (cf. Figure 3-8). It is somehow longer in the horizontal direction, especially when wg/we becomes larger. Nevertheless, this element can be regarded as the minimum required one for characterizing an IDA electrode. The CPE finite-length diffusion element fits the data even better (Figure 3-14c).

Using this element, one can be confident that parameters such as Rct, Qdl and Rs are correctly fitted to a generally satisfying degree. However, it is still unsure what can be told from the two parameters 𝛿𝛿/√𝐷𝐷 and n. The fitting result of the IDA diffusion element from the theory is also accurate (Figure 3-14d), and the parameters Y0, we/w and w²/D can be extracted from it.

Figure 3-14 Raw and fitted EIS data of three 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 ~ 10⁵Hz. (units: μm)

Table 3-2 Equivalent circuit fitting results for nine IDA chips using four different diffusion impedance elements.

R(Q(RW)) (W : Warburg element) R(Q(RO)) (O : Open finite-length diffusion element)

R_s 1.14μ 88μ 0.649μ 1.72μ 0.284μ 0.117μ 6.98m 202μ 93.7μ R_s 34 4.1 6.28 9.83 8.57 4.77 7.44 21.8 6.15 R(Q(RX)) (X : CPE finite-length diffusion element) R(Q(RI)) (I : IDA diffusion element)

R_s 30.9 21.5 13.4 28.6 16.7 11.9 15.3 20.6 19 R_s 29.3 19.5 13.1 26.6 16 12.4 15.3 19.3 17.8

The fitted parameters obtained using all four elements inside the Randles circuit are shown in Table 3-2. The results using the IDA diffusion element are consistent with the experimental plot in that all parameters hold a reasonable range that possess their own physical meanings. The average weighted sum of squares (S/𝑁𝑁_𝑑𝑑) for each element and chip are shown in Figure 3-15a. This is also the mean square error (MSE) of the fitting result. Different elements have different magnitudes of errors and all fall in a certain range.

The IDA diffusion element has an average MSE of 0.611. Compared to the Warburg element (average MSE = 54.86), this proves the effectiveness of the theory applied for equivalent circuit fitting. The coefficient Y0 is calculated in advance using theoretical values, and it is compared with the fitted values (Figure 3-15b). The errors have an average of 3.77×10^-5s^1/2/Ω and the standard deviation is 8.11×10^-5s^1/2/Ω. These errors might be caused by temperature fluctuation, uneven active electrode areas, bulk concentration changes, or the fitting bias itself. Fitted values of w²/D and we/w are plotted against calculated values (Figure 3-15c and d). The fitted values for w²/D show a high correlation with the calculated values (R² = 0.940). With a known w or D, one is able to determine the other value. The correlation between the fitted values and calculated values of we/w has a lower value (R² = 0.869). This suggests a more difficult attempt for finding this ratio using EIS data. The reason might arise from the fact that the shapes of the diffusion impedance look more similar and get closer to each other as wg/we becomes smaller (cf. Figure 3-8).

Figure 3-15 (a) S/Nf (or MSE) of the fitting results for the nine IDA electrode chips using different diffusion impedance elements. (b) Calculated and fitted values of Y0 from the nine chips. (c) Fitted value of w²/D against its calculated value. (d) Fitted value of we/w against its calculated value.

The results above show the accuracy and the effectiveness of the theory applied for equivalent circuit modeling for the EIS data of IDA electrodes. The data from a single EIS experiment is enough for finding Y0, w²/D and we/w. This solves the problem for erroneous parameter acquisition when using elements that cannot express the diffusion nature of IDA electrodes. This also proves that it is able to verify certain parameters in situ and even predict unknown geometries.