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

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.

💡 Research Summary

This paper critically evaluates the widely used “balance sheet” approach (BSA) presented in Bard, Falkner & White’s textbook for analyzing ion transport in electrochemical cells, by comparing it with a first‑principles formulation based on the Poisson‑Nernst‑Planck (PNP) equations. The authors begin by outlining the pedagogical appeal of BSA: it assumes a bulk region with uniform ion concentrations and zero concentration gradients, so that electromigration alone carries the current, while diffusion is confined to thin boundary layers adjacent to the electrodes. This simplification leads to simple arithmetic expressions for the contributions of each ionic species to the total current, using equivalent conductance and transference numbers. However, the authors point out that BSA rests on several ad‑hoc assumptions—absence of bulk gradients, discontinuous diffusive fluxes at the electrode interface, and a constant electromigrative flux despite the presence of concentration gradients—that lack justification from the governing transport equations.

To test the validity of BSA, the authors solve the coupled PNP equations for three canonical electrochemical systems: (i) a copper redox cell without supporting electrolyte, (ii) the same copper cell with a supporting electrolyte (0.1 M NH₃), and (iii) a hydrogen evolution cell. The PNP framework treats each ionic species with a Nernst‑Planck flux (diffusion + electromigration) and couples these to Poisson’s equation for the electric potential, thereby enforcing electroneutrality (or the appropriate space‑charge distribution) and mass conservation everywhere in the cell.

In the copper cell without supporting electrolyte, BSA predicts that the bulk current is carried solely by electromigration, assigning transference numbers based on the assumption of zero concentration gradients. The PNP simulations, however, reveal pronounced concentration gradients near both electrodes, a non‑uniform electric field, and a significant diffusive contribution even in the bulk when the system approaches the limiting current. The authors show that the atomic fluxes derived from BSA do not satisfy continuity across the cell, whereas the PNP solution yields spatially constant total ionic fluxes and correctly reproduces the limiting‑current behavior. Transient PNP calculations further demonstrate how the system evolves from an initially uniform state to the steady‑state profile, a dynamic that BSA cannot capture.

When a supporting electrolyte is added, the overall conductivity increases and the electric field in the bulk is reduced. The PNP results indicate that diffusion becomes the dominant transport mechanism for the copper ions, while electromigration plays a minor role. BSA, still anchored to its original assumption of bulk electromigration dominance, overestimates the electromigrative contribution and mispredicts the current‑voltage relationship. This discrepancy underscores the importance of accounting for the shielding effect of supporting ions, which is naturally included in the PNP formalism.

The hydrogen evolution cell illustrates a more complex scenario involving multiple ionic species (H⁺, OH⁻, and supporting ions) and the formation of an electric double layer. PNP captures the coupled evolution of the potential and concentration profiles, including the non‑linear variation of the electric field within the diffuse layer. BSA, which treats each species independently and assumes a constant bulk field, fails to reproduce the observed current distribution and cannot address the impact of the double layer on reaction rates.

From a computational perspective, the authors implement a one‑dimensional finite‑difference discretization of the PNP equations and solve the resulting nonlinear system using Newton‑Raphson iterations. They provide Python scripts and Jupyter notebooks, demonstrating that the approach is accessible to students and researchers without requiring high‑performance computing resources. The paper argues that incorporating such first‑principles simulations into curricula would give learners a concrete appreciation of the limitations of heuristic methods like BSA and a deeper understanding of ion transport physics.

In conclusion, the study shows that the balance sheet method, while pedagogically convenient, is only valid under very restrictive conditions (e.g., perfectly mixed bulk, negligible concentration gradients, and low current densities). The Poisson‑Nernst‑Planck approach offers a physically consistent, quantitatively accurate framework that can handle a wide range of operating regimes, including transient behavior, limiting‑current phenomena, and the presence of supporting electrolytes. The authors recommend that electrochemical engineers and educators adopt PNP‑based modeling as a standard tool for both research analysis and classroom instruction, and they suggest future extensions to incorporate ion‑size effects, dielectric decrement, and ion‑ion correlations for even more realistic simulations.