  Linear sweep and cyclic voltammetry techniques are important tools for
electrochemists and have a variety of applications in engineering. Voltammetry
has classically been treated with the Randles-Sevcik equation, which assumes an
electroneutral supported electrolyte. In this paper, we provide a comprehensive
mathematical theory of voltammetry in electrochemical cells with unsupported
electrolytes and for other situations where diffuse charge effects play a role,
and present analytical and simulated solutions of the time-dependent
Poisson-Nernst-Planck (PNP) equations with generalized Frumkin-Butler-Volmer
(FBV) boundary conditions for a 1:1 electrolyte and a simple reaction. Using
these solutions, we construct theoretical and simulated currentvoltage curves
for liquid and solid thin films, membranes with fixed background charge, and
cells with blocking electrodes. The full range of dimensionless parameters is
considered, including the dimensionless Debye screening length (scaled to the
electrode separation), Damkohler number (ratio of characteristic diffusion and
reaction times) and dimensionless sweep rate (scaled to the thermal voltage per
diffusion time). The analysis focuses on the coupling of Faradaic reactions and
diffuse charge dynamics, although capacitive charging of the electrical double
layers (EDL) is also studied, for early time transients at reactive electrodes
and for non-reactive blocking electrodes. Our work highlights cases where
diffuse charge effects are important in the context of voltammetry, and
illustrates which regimes can be approximated using simple analytical
expressions and which require more careful consideration.
