Kinetic overpotentials define the additional potential necessary for bringing the lithium ions from the electrolyte into the active materials of cathode or anode. They are computed from the Butler-Volmer expression.

Separately for anode and cathode interface the kinetic overpotentials are given by

where is the Butler-Volmer current density, the solid potentials, the electrolyte potential, the open-circuit potential, the voxel face area (voxel length)², and the total current.

The kinetic heat production is given by

Diffusive overpotentials result from the distribution of lithium ions in the active materials of an electrode that is not uniform during charging or discharging. E.g., during charging, lithium ions enter the anode active materials at the surface of the grains and diffuse into the grains. The slower this diffusion in the active materials, the higher the difference in the distribution of lithium and the higher the additional potential necessary for keeping a desired charge rate. If the simulation is started from an equilibrium situation, these concentration differences in the active materials increase during the simulation. The diffusive overpotentials can be time-dependent and might get higher the longer the charging process is running.

Diffusive overpotentials can even get negative in certain situations. The reason is that to compute the overpotential, the local surface concentration is compared to the concentration over the whole electrode. Two effects are therefore relevant for the diffusive overpotentials, diffusion in one grain and the possible inhomogeneity in the loading of particles in the electrode.

The diffusion overpotentials are given separately for anode and cathode active material by

where is the Butler-Volmer current density, the open-circuit potential as function of local surface concentration , the open-circuit potential as function of global electrode SOC in equilibrium, the voxel face area (voxel length)², and the total current.

For one active material, the global electrode SOC is calculated as follows:

where the values for all active material voxels are summed up. This means that the and therefore also the varies in the sum. , on the other hand, always has the same value. For more than one active material, this is more complicated.

The diffusive heat production for the active material is then given by

where is the Faraday constant (96485.33289 C/mol), is the local ion flux density in a cell, is the volume of a single voxel (voxel length)³, and is the local diffusivity.

A diffusive overpotential can also build up in the electrolyte due to concentration gradients that can form during charging in the electrolyte. Please note that overpotential due to concentration gradients in the separator is accounted to the diffusive overpotential of the electrolyte.

The diffusion overpotentials for the electrolyte are given by

where is the local current density, the universal gas constant, is the temperature, is the Faraday constant, lithium concentration in electrolyte, is the transference number, is the volume of a single voxel (voxel length)³, and the total current.

The diffusive heat production for the electrolyte is given by

Resistive overpotentials define the additional potential necessary to overcome bulk electrical resistivities in the charging process. These resistivities depend on the electrical bulk conductivities of the different battery materials. For electrodes, it relates from the electronic conductivities of the binder and active materials in this electrode, as well as the corresponding current collector. For the electrolyte, it relates to the ionic conductivity of the electrolyte and separator.

The resistive overpotentials are given separately for anode, cathode, and electrolyte by

where is the current density in the respective material, for anode and cathode resistive overpotential is the electrical conductivity in the respective material (ionic conductivity for the electrolyte‘s resistive overpotential and electronic conductivity for the electrodes‘ resistive overpotential), is the voxel volume (voxel length)³, and the total current.

The resistive heat production is given by