Rethinking Balance Sheets: A Poisson-Nernst-Planck Based Approach for Modeling Concentration and Flux Profiles Inside an Electrochemical Cell

TL;DR

This work critiques the Bard Faulkner White balance sheet analysis of ion transport in electrochemical cells by solving the one-dimensional Poisson-Nernst-Planck equations. It shows that the balance sheet assumptions—uniform bulk concentrations and a spatially constant electromigration—lead to inconsistencies, especially with mass conservation and realistic flux profiles, even when current is kept divergence-free. The PNP framework yields spatially varying concentration gradients and nonuniform fluxes throughout the cell, while maintaining current conservation, thereby providing a physically grounded, first-principles method that can serve as a teaching and research tool. The study also demonstrates extensions to systems with supporting electrolytes and transient hydrogen evolution, highlighting the limitations of simplified balance-sheet analyses and the importance of incorporating more detailed physics in electrochemical transport modeling.

Abstract

Electrochemical cells serve as a building block for producing and storing electrical energy from chemical reactions. The analysis of ion transport in these systems forms the foundation for understanding more complex electrochemical systems that are becoming increasingly present in the broader societal energy infrastructure. From a pedagogical perspective, the ``balance sheets" introduced in Chapter 4 of Electrochemical Methods: Fundamentals and Applications by Alan J. Bard, Larry R. Faulkner and Henry S. White (hereafter referred to as BFW) provides a first-pass approach to analyze ion transport in electrochemical cells. However, the balance sheet approach lacks first-principles justifications from the underlying equations that describe the transport processes in electrochemical cells. In this work, we compare a first-principles approach via the Poisson-Nernst-Planck equations to describe ion transport in electrochemical cells to that of the balance sheet approach. By re-working the examples presented in BFW, we illustrate that the balance sheet approach is only valid in limited scenarios. Furthermore, we show that the PNP equations provide a more physical route to analyze ion transport in electrochemical systems. We hope the approach outlined here will be adopted by electrochemical engineering researchers and instructors.

Figures (16)

• Figure 1: Schematic representation of copper redox cell presented, as given in Section 4.3 of Bard, Faulkner and White bard2022electrochemical. An electrolyte solution composed of $10^{-3}$ M Cu(NH$_3)_4^{2+}$, $10^{-3}$ M Cu(NH$_3)_2^{+}$, and $3 \times 10^{-3}$ M Cl$^-$ in $0.1$ M NH$_3$ is placed in an electrochemical cell with a cathode at $x = -\ell$ and an anode at $x = \ell$. A current is produced via the reduction of Cu(II) at the cathode and the oxidation of Cu(I) at the electrode.
• Figure 2: Recreation of Figure 4.3.3 from Bard, Faulkner and White bard2022electrochemical. Depiction of electromigrative and diffusive rate contributions as determined by the balance sheet procedure.
• Figure 3: Concentration and flux profiles of Cu(I), Cu(II), and Cl$^-$ for the steady state copper redox cell. The a) concentration profiles, b) electromigrative flux, and c) diffusive flux for each ionic species. d) Total flux and e) current with $\mathcal{J} = 1/6$, $\gamma = 36$, and the balance sheet approach (dashed or dotted line) for $k = 6$.
• Figure 4: Concentration profiles with increasing $\mathcal{J}$ for Cu(I), Cu(II), Cl$^-$. The concentration profiles of a) Cu(I), b) Cu(II), and c) Cl$^-$ for $\mathcal{J} =$ 0.3, 0.5, and 0.7 using the PNP simulations. The concentration of Cu(II) reaches zero at $\tilde{x} = -1$ for $\mathcal{J} =$ 0.7.
• Figure 5: Flux and current profiles at $\mathcal{J} = 0.7$ for Cu(I), Cu(II), Cl$^-$. The a) electromigrative, b) diffusive, and c) total flux for each ionic species and d) current from the solution of eqn. \ref{['eqn:flux_final_SS']} (solid line) for $\mathcal{J} = 0.7$, $\gamma \approx 8.7$, and the balance sheet approach (dashed or dotted line) for $k=6$.