# Physics-based models (1): what are they?

Physics-based models for lithium-ion batteries are quite popular these days, and they have literally seen an exponential growth in the past decades. According to Google Scholar results, the number of articles published each year (since 2000) matching the search «"physics-based" model "lithium" battery» have been doubling every 3.3 years. But what are physics-based models? The term is used to describe models which are built upon fundamental laws of physics (as opposed to empirical models), and their uses extend way beyond lithium-ion batteries. Their main advantage with respect to empirical models is that they provide a lot more information on the processes they describe.

Let’s consider a very simple example, much simpler than batteries: dropping a ball from the top of a tower. We could run a lot of experiments, dropping a ball from different heights and collecting data on the time it takes to reach the ground. After analysing the data, we could decide to fit a function to the data (maybe a parabola?) and that would give us an empirical model. Of course, this model would give us quite accurate predictions but presents three main shortcomings:

- We would have no clue on why the result is a parabola.
- If we changed some settings of the experiment (e.g. do the experiment on the moon) we would not know if our model would work unless we tested that new situation.
- If we wanted to “fix” our model for that new situation we would have to start from scratch.

A (very simple) physics-based model would be to combine Newton’s second law and Newton’s law of universal gravitation to write a differential equation for the position of the ball (and in fact we can use PyBaMM to solve it). Solving the equation we would find that the trajectory vs time is indeed a parabola, hence overcoming shortcoming 1. Our physics-based model would also offer insight on what to change when testing another situation and also we would be aware of when the modelling assumptions break down. For example, we would be aware that we neglected air friction and considered the acceleration of gravity to be constant, so we could judge if the model would work in a new situation (overcoming shortcoming 2). Finally, if we wanted to extend our model to this new situation (e.g. account for air friction) we would not have to start from scratch, but just introduce a new force into the model and compute the solution to the new equation.

The advantages of physics-based models that this simple example demonstrated, also apply to lithium-ion battery models. For example, a Single Particle Model with electrolyte (SPMe) would clearly show that it breaks down when electrolyte depletion occurs, and therefore we would know in advance which are the maximum discharge rates that we can simulate. Or degradation mechanisms can be incorporated much more easily into the physics-based model rather than on an equivalent-circuit one. In our next blog post we will discuss how physics-based models for lithium-ion batteries are built and which are the most popular ones.

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