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Theory for influence of uncompensated solution resistance on EIS of diffusion limited adsorption at rough electrode

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Abstract

Theory is developed for the electrochemical impedance spectroscopy (EIS) of the diffusion-limited adsorption process coupled with reversible charge transfer at rough electrodes under the influence of ubiquitous uncompensated solution resistance. This study quantitatively relates the impedance response of rough electrode to its phenomenological components, viz., diffusion limited adsorption, reversible charge transfer and uncompensated solution resistance. The random roughness of electrode is expressed by the surface statistical property, i.e., power spectrum of roughness. The fractal nature of roughness is characterized in terms of fractal dimension, lower cut-off length and topothesy length. The high-frequency regime is controlled by the uncompensated solution resistance whereas the low-frequency regime is governed by the adsorption process. The magnitude of impedance as well as phase decreases with rise in adsorption isotherm (length) parameter. The intermediate frequency regime is controlled by the coupling of adsorption and uncompensated solution resistance with the diffusion process. The fractal roughness parameters has quantitative influence on the magnitude of impedance over whole frequency regime while the phase plot shows qualitative difference in the intermediate frequency regime. The governing length scales which controls the characteristic crossover frequencies are: diffusion length, adsorption-ohmic coupling length and topothesy length (or width of interface). The three emergent crossover frequencies are: (i) ohmic reduced inner crossover frequency (ii) adsorption roughness topothesy dependent pseudo-quasireversibility characteristic frequency (iii) outer crossover frequency.

Graphic abstract

Synopsis: This study quantitatively relates the impedance response of rough electrode to its phenomenological components, viz., diffusion limited adsorption, reversible charge transfer and uncompensated solution resistance. The random roughness of electrode is expressed by the surface statistical property, i.e., power spectrum of roughness.

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Abbreviations

K :

Adsorption Isotherm Parameter

\(R_{\Omega }\) :

Uncompensated solution resistance

\(D_H\) :

Haussdorff Fractal Dimension

\(\ell \) :

Lower Cut-off Length

L :

Upper Cut-off Length

\(\ell _\tau \) :

Topothesy Length

\(L_{a\Omega }\) :

Adsorption-Ohmic coupling length

\(C_{ad}\) :

Adsorption capacitance

\(L_{D}\) :

Diffusion length(\(\sqrt{D/i\omega }\))

\(C_{\alpha }^{0}\) :

Bulk concentration of the species \(e.g.\;\alpha =O,R\)

\(\delta C_{\alpha }\) :

Change in concentration of species

\(\eta _0\) :

Small Amplitude of Perturbed Potential

\({\vec{K}_{\Vert }}\) :

Vector \((K_{x},K_{y})\)

\({\hat{n}}\) :

Normal vector drawn in outward direction

\(\vec{r_{\Vert }}\) :

Vector (x, y)

\(\Gamma _\alpha \) :

Excess Surface Concentration

\(\delta (\vec{K}_{\Vert })\) :

Dirac delta function in wave-vector \(\vec{K}_{\Vert }\)

\(\zeta (\vec{r_{\Vert }})\) :

Arbitrary surface profile

\({\hat{\zeta }}(\vec{K_{\Vert }})\) :

Fourier transform of the arbitrary surface profile

j(t):

Current density

\(y(\omega )\) :

Admittance density

\(Y(\omega )\) :

Total admittance

\(Y_{P}(\omega )\) :

Smooth electrode admittance

\(Y_{W}(\omega )\) :

Warburg admittance

\(Y_{ad}(\omega )\) :

Adsorption admittance

\(F_1 \) :

Appell’s Function

\(_2F_1 \) :

Hypergeometric Function

\({K}_{\Vert }\) :

\([K_{x}^{2}+K_{y}^{2}]^{1/2}\)

\(\delta \) :

\(D_H-5/2\)

\(\omega _{i}\) :

Inner cut-off angular frequency

\(\omega _{o}\) :

Outer cut-off angular frequency

\(\Gamma \) :

Specific volume capacitance

\(\mu \) :

Strength of fractality

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Acknowledgements

Authors show their gratitude towards University of Delhi, India and DST-SERB, India (EMR/2016/007779) for financial support.

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Authors and Affiliations

  1. Department of Chemistry, University of Delhi, Delhi, 110007, India

    Shruti Srivastav, Manish Kumar & Rama Kant

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  1. Shruti Srivastav
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Correspondence to Rama Kant.

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Srivastav, S., Kumar, M. & Kant, R. Theory for influence of uncompensated solution resistance on EIS of diffusion limited adsorption at rough electrode. J Chem Sci 133, 50 (2021). https://doi.org/10.1007/s12039-021-01901-w

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