A6 Dimensional analysis of linear sweep voltammetry at microdisc and hemispherical electrodes

A6.1 Microdisc electrode

The mass transport equation:

(A6.1)

may be converted into a dimensionless form:

(A6.2)

where:

, and

(A6.3)

The electrode surface concentration is given by:

a0 = [1 + exp(-Q)]-1 where

(A6.4)

hence the concentration is a function of two space variables, a time variable and the normalised potential (which itself is a function of time):

(A6.5)

In terms of dimensionless variables, the LSV waveform is:

(A6.6)

where:

(A6.7)

Hence the concentration is a function of four dimensionless variables:

(A6.8)

The current is given by:

(A6.9)

In terms of R and Z this becomes:

(A6.10)

Hence:

(A6.11)

where f_disc denotes an unknown function which depends only on t and v.

A6.2 Hemispherical electrode

The mass transport equation:

(A6.12)

may be rewritten in term of dimensionless space and time variables:

(A6.13)

where:

and

(A6.14)

Considering the electrode surface concentration, as for the microdisc electrode:

(A6.15)

The current is given by:

(A6.16)

or in terms of R:

(A6.17)

Hence:

(A6.18)

where f_hemi denotes an unknown function which depends only on t and v.

A6.3 Comparison of peak currents of microdisc and hemispherical electrodes

At the peak of the voltammogram :

(A6.19)

The time dependence of the normalised potential is given by equation (A6.6). Differentiating gives:

(A6.20)

Hence the peak is also defined by:

(A6.21)

Applying this constraint to equation (A6.11) for a microdisc gives:

(A6.22)

Similarly for the hemisphere, applying constraint (A6.21) to equation (A6.18) gives:

(A6.23)

The ratio of peak currents is therefore:

(A6.24)

Since:

(A6.25)

then:

(A6.26)

Hence the peak current ratio for 'equivalent' electrodes (where r_hemi = 2r_disc/p) is solely a function of dimensionless scan rate:

(A6.27)

where f₁, f₂ etc. denote arbitrary unknown functions.