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Voltage-step transient on circular electrodes

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Abstract

Using the well-known Cottrell asymptote, voltage-step transient (VST) experiments in a two-electrode cell may provide data on the bulk concentration and diffusion coefficient of a working depolarizer, essential for interpretation of liming-diffusion-current (LDC) experiments. However, the Cottrell asymptote is never applicable in the early stages of the voltage-step transient process. There are three additional transport resistances to the basic diffusion transport that cannot be neglected at extremely high initial currents: Faradaic resistance at the working electrode surface, overall Ohmic losses in the electrochemical cell, and a possible galvanometric constraint in the outer circuit, involving the galvanometer, current follower, potentiostat, etc. The effect of these additional transport resistances on the transient current at higher polarization voltage is non-linear. In the present analysis, a 1D approximation is used, assuming tacitly uniform accessibility of the working electrode and negligible transport resistances at the counter electrode. The Faradaic resistance of a redox couple O + ne − = R according to Butler–Volmer electrode kinetics is considered. The results of 1D theory are corrected for edge effects using the Oldham asymptote. The effect of convection at longer times is included using the recent model of a transient process for circular working electrodes in the LDC regime.

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Abbreviations

A :

Effective area of working electrode, m2

A ∞ :

area of ideally smooth working electrode, m2

B :

≡ \( D_{O}^{1/2} c_{O}^{B} /D_{R}^{1/2} c_{R}^{B} \), transport parameter

C :

Cottrell coefficient (m s−1/2)

c, \( c_{O,R}^{B,W} \):

Molar concentration of depolarizers and their boundary values (mol m−2 s−1)

D, D O,R :

Coefficient of diffusion; values for the components (m2 s−1)

\( d_{T}^{ - 1/2} \) :

Semi-integral operator

E :

= Exp[P]

f :

= (E − 1)/(E+B), Nernst–Petersen factor

G :

= k G /k S , normalized galvanometric constraint

I :

Current (A)

J O,R :

Diffusion fluxes of depolarizers at surface (mol m−2 s−1)

K :

= \( k_{BV} (c_{O}^{B} )^{1 - \beta } (c_{R}^{B} )^{\beta } \), modified rate constant (mol m−2 s−1)

k :

= I/(nFA \( c_{work}^{B} \)), mass-transfer coefficient (m s−1)

k BV :

Rate constant factor in Butler–Volmer kinetics (m s−1)

k C :

= U/(nFAR C \( c_{work}^{B} \)), Ohmic limit for k(t) (m s−1)

k G :

= I G /(nFA \( c_{work}^{B} \)), galvanometric constraint (m s−1)

k S :

= μk C , starting (upper) limit for k(t) (m s−1)

L C :

Equivalent resistor length (m)

M :

≡ K/k C

nF :

The charge transfer per 1 mol of the working depolarizer (C mol−1)

N = N(T):

≡ k/k S , normalized flux

N L :

Solution to the linearized transport equation

O, R :

Oxy- and Red- forms of the redox couple, O + ne − ↔ R

P :

≡ −U nF/RT, normalized voltage; F = 95608 C mol−1, Faraday constant; R = 8.313 J K−1 mol−1, universal gas constant; T, absolute temperature

R C :

Ohmic resistance of the cell (Ω)

t :

Time from the initial voltage step

t S :

≡ (C/k S )2, characteristic lag time (s)

T :

≡ t/t S , normalized time

U :

Constant voltage between working and counter electrodes (V)

W = W(T):

≡ 1 − \( c_{O}^{W} /c_{O}^{B} \), normalized concentration driving force (for cathodic depolarizer)

z :

Normal distance to the working electrode (m)

β :

Symmetry coefficient in Butler–Volmer kinetics

Λ[N]:

United boundary-value constraint for the concentration driving force

∂ T :

Partial derivative with respect to T

μ :

≡ k S /k C

θ :

= π/4, the asymptotic divide time

η :

= U − IR C , overvoltage (V)

κ :

Specific conductivity of the bulk solution (S m−1)

δ :

≡ D/|k|, Nernst diffusion layer thickness (m)

c B,W :

Bulk or Wall compositions of the solution

C :

Ohmic resistance of the two-electrode cell

G :

Galvanometric constraint

NP :

Reversible electrodes according to the Nernst–Petersen rule

c O,R :

Oxy- or Red- form of redox couple

S :

Start of the process

ED:

Electrodiffusion

LDC:

Limiting diffusion current

VST:

Voltage-step transient

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Acknowledgment

Support by the Czech Science Foundation (GACR) through the projects 104/08/0428 and 104/09/0972 is acknowledged.

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Authors and Affiliations

  1. Institute of Chemical Process Fundamentals of ASCR, v.v.i., Rozvojova 135, 16502, Prague 6-Suchdol, Czech Republic

    Ondrej Wein & Valentin V. Tovchigrechko

Authors
  1. Ondrej Wein
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  2. Valentin V. Tovchigrechko
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Correspondence to Ondrej Wein.

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Wein, O., Tovchigrechko, V.V. Voltage-step transient on circular electrodes. J Appl Electrochem 41, 1065–1075 (2011). https://doi.org/10.1007/s10800-011-0288-x

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