The use of the electroneutrality approximation (EN) to model electrochemical systems has been controversial for decades. The Newman school of thought typically uses EN.¹ EN has been extensively used in conjunction with the concentrated solution and porous electrode theories, such as in battery models.^2,3 However, EN is considered inadequate to capture dynamics at short time and length scales of the order of the Debye length. Several such studies of the physical validity and simulation utility of simplifying approximations have been performed for systems described by the highly stiff Poisson-Nernst-Planck (PNP) equations for charged species transport.^4–7 Notably, these efforts addressed the specification of appropriate boundary conditions for the electrostatic potential near electrode-electrolyte interfaces by modeling the structure of the electric double layer. The full PNP solution was then compared with those obtained with approximations such as EN or the zero-field (ZF) assumption for infinitesimally thin double layers. EN assumption does not guarantee improved ease of computation.^5,7

Mathematically, the electroneutrality assumption can be considered similar to the continuity condition in fluid flow. While the first principles model is often solved directly, many researchers solve the Pressure-Poisson equation, which requires additional boundary conditions.⁸ Interestingly, EN for electrochemical systems is approximate, whereas continuity for fluid flow is rigorous. On the other hand, Poisson’s equation is more rigorous than EN for electrochemical systems but is approximate for flow problems. Another subtle mathematical aspect is that the Nernst-Planck-EN (NPEN) equations as originally written are index-2 Differential Algebraic Equation (DAE) for potential, just as the time-dependent Navier Stokes equation is an index-2 DAE for local pressure. There is no explicit occurrence of potential in the NPEN equations, though this may usually be alleviated by eliminating one concentration variable and using that equation for potential, resulting in an index-1 DAE. In contrast, the dynamic PNP equations are an index-1 system. The discussion and controversy related to the choice of appropriate boundary conditions for pressure at the walls is another point of similarity.

This talk will discuss the relevance of EN relative to the PNP and PNPZF models at different length and time scales for four problems of interest – i) The Bazant model system, ii) Lithium deposition in symmetric cells iii) Liquid junction potentials and iv) Parallel plate reactors.^4,6,9,10 We will evaluate the effect of simplifying assumptions such as EN and ZF from a simulations perspective. This will be studied in terms of the modification of equation structure and the stiffness avoided or worse, introduced. We will also identify the models more prone to numerical artifacts and discuss the utility of simplifying assumptions in this context. For example, if finite difference methods (as opposed to finite-volume method) do not conserve charge or mass, which approximation is less erroneous? That is, if one cannot resolve the length scales to a given tolerance, which assumption gives more accurate results with a relatively coarse computational grid? We will also attempt to determine the extents of time and length scales that can be resolved with a given numerical method. Equations that will benefit from robust and adaptive time solvers will also be examined. We will present our efforts towards efficient real-time simulation of these models for control, design, and parameter estimation. We will also attempt to derive boundary conditions that are more accurate than ZF and EN but less stiff than the full PNP model, potentially allowing for the use of double layer physics sans the associated computational complexity.

Acknowledgments

The authors acknowledge financial support from the Battery500 Consortium. We acknowledge the influence of seminal works in this field from Prof. John Newman and Prof. Martin Bazant.

References

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