Electrochemical systems are essential to modern life. They encompass energy technologies like batteries and fuel cells as well as systems for desalination, mineral extraction, and chemical sensors. The fundamental physics of these systems relies on two components: electrodes and an electrolyte. You might be familiar with electrolytes in the context of sports drinks. Fundamentally, a liquid electrolyte, whether in a drink or an electrochemical device, is an ionic salt solution like dissolved sodium chloride.
There is experimental evidence that using electrolytes where the ions have asymmetric valences (charges), like magnesium chloride which is made of +2 and -1 charges, can have significant effects on the performance of electrochemical systems. However, most existing mathematical models of these systems consider symmetric valence electrolytes like sodium chloride (+1 and -1 charge) because the symmetry dramatically simplifies the governing equations.
In our recent paper published in the Journal of The Electrochemical Society [1], we set out to examine analytically how asymmetric ion valences in an electrolyte impact the processes inside an electrolytic cell, a simple electrochemical device. We find that the ratio between the positive and negative ion charge magnitudes (which we called r) controls the cell's electrochemical behaviour, particularly near the electrodes, and we fully describe when qualitatively different regimes emerge.
In an electrolytic cell, illustrated in Figure 1, an applied voltage from a battery drives cations (positive ions) towards the negative electrode and drives anions (negative ions) towards the positive electrode. The ions are created and consumed at the electrodes through chemical reactions, allowing a continual flow of ions. This movement of charge sustains an electric current throughout the electrolyte. The bulk of the electrolyte is electrically neutral, but near the electrode surfaces there are thin regions of charge imbalance called the ‘electric double layer’. The charge-storing properties of the electric double layer are the basis for many energy technologies.
Figure 1: A schematic of an electrolytic cell
The underlying asymptotic framework of our model can be adapted to other modelling choices. As an example, we also provide analytic solutions for asymmetric-valence electrolytes with the boundary conditions that are often used to model biological ion channels. We notably provide explicit analytic solutions for the r=2 and r=½ cases in terms of elementary functions, which removes the need for numerical solving to model these common electrolytes.
Our work shows that under the assumption of mass-conservation for ions, the valence ratio r determines the qualitative behaviour of the ionic concentrations and electric potential. This is exemplified in Figure 2, where we show plots of the cation concentration, anion concentration, and electric potential as r varies for a fixed dimensionless voltage and current. Looking at the steep sections of the plots near x=0, as the valence ratio increases, the system switches between regimes where there are more anions or cations in the electric double layer. This is a transition between two well-described regimes in electrochemistry - ‘Gouy-Chapman theory’ and approaching the ‘limiting current’. In between, we identify a valence ratio-dependent point where the boundary layer structure vanishes.
Figure 2: The cation concentration, anion concentration and electric potential throughout the electrolytic cell for varying valence ratios r at a fixed dimensionless current and voltage.
We analytically describe this smooth transition in terms of the valence ratio, voltage, and current and summarise our results on a phase diagram. This diagram allows modellers to predict how each ion species will behave near the electrodes. Crucially, despite the ubiquity of the symmetric electrolyte assumption in modelling of electrochemical systems, our analysis shows that effects of valence asymmetry are significant and should be incorporated into mathematical models of electrochemical devices where there is evidence that valence impacts performance.
Georgina Ryan is a postgraduate student in Oxford Mathematics.
[1] Ryan, G. C., Dalwadi, M. P. & Griffiths, I. M. (2026). Modelling Intermediate-Current Transitions in Asymmetric-Valence Binary Electrolytes. Journal of The Electrochemical Society, 173(16), 166502. https://doi.org/10.1149/1945-7111/ae8b2d