# Physics-based models (2): batteries

In our last blog post, we introduced what physics-based models and discussed their strengths in a general context. Now, let's narrow our focus and explore how these models specifically apply to lithium-ion batteries. In this case, we want our models to describe how lithium ions flow inside of the battery. To illustrate it we will focus on electrochemical models only, but thermal and degradation models follow the same ideas, just with a bit more of technical complexity.

So, what do we need to build our models? First we need to consider the geometry of our battery. We want to capture the ion behaviour both in the porous electrodes and the electrolyte, so we consider the two domains separately. We do not consider the separator as it is inactive material (we do consider the electrolyte filling the pores in the separator though). To capture how lithium flows in each phase, we need to combine two ingredients: conservation laws and constitutive relations. Conservation laws are principles that state that a certain physical property (i.e. a measurable quantity) does not change in the course of time within an isolated physical system, and they can be expressed as a mathematical equation. For example, conservation of mass relates the concentration of a given substance to the molar flux. For lithium-ion batteries we want to write equations for conservation of lithium and charge in both the electrode and the electrolyte. But conservation laws on their own are incomplete: they tell us that lithium and charge are conserved, but not how they move around. For that, we need to write constitutive relations.

Constitutive relations (or equations) relate two different quantities and allow us to complete our model. For example, Fickian diffusion states that the molar flux is proportional to the gradient of the concentration. Constitutive relations can be either phenomenological or derived from first-principles, which means that using different constitutive relations results in different equations. This can be quite confusing and is particularly true for lithium-ion battery models, where in the literature we can find several models, all quite similar but slightly different. A quite common choice for constitutive relations in lithium-ion battery models are Fickian diffusion for lithium in the particles, Ohm’s law for charge in the electrodes, Stefan-Maxwell equations for the electrolyte and Butler-Volmer equation for the intercalation reaction. Each of these could be a lengthy blog post on themselves, but a thorough review on these models can be found in the review paper by Brosa Planella et al.

So, how do all these conservation laws and constitutive relations connect to the well-known battery models, such as the Doyle-Fuller-Newman (DFN) model or the Single Particle Model (SPM)? In fact they are both based on the laws and relations stated above, but they also use some further clever assumptions than make them simpler, and that’s the key for their popularity. In the next blog post we will talk about them.

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