Analytical expressions for feedback currents at the scanning electrochemical microscope

Steady-state currents of the scanning electrochemical microscope (SECM), which can be used to convert approach curves to topographical information, are considered in four cases, corresponding to the combination of positive and negative feedback on the substrate surface with either a finite or infinite glass insulator around the disc electrode. If the substrate is an insulator and both substrate and insulator extend to infinity (case I), steady-state cannot be attained. When the bulk concentrations can be assumed to be restored at a finite distance (the glass radius), both the negative (case II) and positive (case IV) feedback steady-state currents can be analytically found by using properties of the Bessel functions in the solution of a dual series equation. The strong influence of the glass radius on the current in case II can be taken into account in the exact computation and in simple approximate expressions. Positive feedback with an infinite domain (case III) can be solved with a dual integral equation. All solutions can be easily computed with modern software. Moreover, very simple and accurate approximate expressions (from the literature and new suggestions) are assessed and compared. A consistent way of linking the theoretical and experimental normalized currents for finite glass radii is suggested in terms of the values of the currents corresponding to an infinite distance between the electrode and the substrate. An expression to estimate the electrode radius from the insulator radius and the current with no feedback is suggested.