In the active material lithium ions recombine with an electron and are therefore electrically neutral. Hence, they are independent of electrical potentials and undergo only diffusive motion driven by gradients in the lithium concentration. The ion flux is given by

where is the diffusion constant of lithium in the solid material.

The diffusion of ions in the active material is therefore described by the equation

In the active material domain, the electrical current is carried by electrons. Therefore, the electronic current density is simply given by Ohm’s law

Where is the electronic conductivity and the electrical potential. Thus, it follows

In the electrolyte, the flow of charged species (lithium and counter ions) are responsible for the electrical current. Under the assumption of electroneutrality it is sufficient to consider only the transport of lithium ions. Since they are charged, their motion is influenced by two driving forces, the differences in the ion concentration and in the potentials, such that the ionic current density is given by

where is the ionic conductivity and the transference number which (in the absence of concentration gradients) describes the fraction of current carried by the positive lithium ions. is here the universal gas constant, the temperature and Faraday’s constant.

To simplify the complexity, this equation assumes a constant activity of lithium ions such that the activity does not appear in the model. Additionally, we note, that the second term sometimes can be found in the literature with an additional factor of 2. This needs to be taken into account consistently in the determination of the transference number.

The charge conservation and electroneutrality equation in the electrolyte is therefore

Similar as the current density, also the ion flux depends on concentration and potential gradients. With the ionic diffusion constant the flux in the electrolyte can be formulated as

This leads to the diffusion and migration equation in the electrolyte

where is the porosity.

Further explanation about the theoretical background for the ion transport in the electrolyte can be found in Latz et al., 2011.

For solid electrolytes the transference number is always 1. Therefore, there is no concentration gradient in the solid electrolyte. Thus, for solid electrolytes the equations in the electrolyte simplify to constant lithium concentration

and for the charge conservation and electroneutrality it follows

The Carbon Binder Domain (CBD Phase) is represented by respective voxels in the battery structure. The CBD phase can either be simulated as a non-porous material or as a porous material with non-resolved micropores (see Electrochemistry). In the GeoDict material database, the material of “PVDF Binder and Carbon Black” is shipped as an effective material representing the CBD properties.

For non-porous binder, the CBD phase is a pure electron conductor. So, only Ohm’s law is solved (see equation (154) for the active material). At interfaces to active materials, electrons can be transferred from CBD phase to active material and vice versa.

For porous binder, there is a solid fraction of the CBD phase and a pore fraction of the CBD phase. The solid fraction of the CBD phase conducts electrons and Ohm’s law is solved considering the effective electronic conductivity of the porous binder. The pore phase of the CBD phase is filled with electrolyte and conducts lithium ions. For this fraction, the equations for the electrolyte as stated above (diffusion and migration equation (158) Diffusion and migration and charge conversation and electroneutrality equation (156) Charge conservation and electroneutrality are solved.

For the interface of non-porous binder and active material, a (de)intercalation of lithium ions is not possible. For the interface of porous binder and active material, a (de)intercalation of lithium ions to/from the pore fraction of the porous binder that is filled with electrolyte is possible. BatteryDict considers that the interfacial area of active material to porous binder is only partly available for lithium intercalation.

See the sketch below. The ion exchange is calculated with the Butler-Volmer equation. See equation (162) Butler-Volmer interface current density.

On the interface between electrolyte and active material lithium ions move from the electrolyte into the active material on intercalation or vice versa on deintercalation. This flux of ions is continuous across the interface and its magnitude is given by the Butler-Volmer current density such that the interface conditions for the fluxes and current densities are given by

here, is the interface normal pointing from solid into electrolyte. Hence, a positive corresponds to deintercalation while a negative sign indicates an intercalation reaction. The Butler-Volmer interface current density depends on concentrations and potentials on both sides of the interface and is given by

Here, is the Maximum Exchange Current Density in equilibrium, the lithium concentration of the electrolyte in equilibrium (see below), and the open-circuit potential of the active material versus lithium.

For Li-ion concentration or , the square root in the Butler-Volmer interface current density gets 0, which can lead to numerical solver problems. As stated in State of Charge (SOC) and State of Lithiation (SOL), the SOC is related to the Li-ion concentration in the active materials. Therefore, the range of the cell and electrode state of charge can only be set between 5% and 95% in the Charge Battery and the Charge Electrode dialogs. A real battery in experimental conditions usually is set to operate in a certain voltage window (compare also the different stopping criteria) to avoid permanent damage of the cell. The relevant voltage window usually is in a SOC range smaller than the range between 5% and 95% SOC and therefore can be fully covered in the simulation.

Furthermore, during the intercalation of Li-ions from the electrolyte into the active material, these Li-ions attain one electron per ion and become neutral in charge. During the de-intercalation of Li from the active material into the electrolyte, every Li atom loses one electron and becomes a positively charged Li-ion.

Hence, in the solid material, the charge transport is performed solely by electrons, whereas in the electrolyte, the charge transport is performed solely by ions.

For solid electrolytes the term is always 1, because the Li-ion concentration in the electrolyte does not change and thus, is always .

For the simulation of a single electrode, the other electrode is modeled as a never-ending lithium reservoir, not limiting the battery performance. Therefore, there exist no and and the Butler-Volmer interface condition (equation (162) Butler-Volmer interface current density) simplifies to

The build-in default value for the Maximum Exchange Current Density of a lithium reservoir for the default electrolyte concentration of 1200 mol/m³ is:

This is based on an assumed exchange current density between the lithium reservoir and the electrolyte of 640 A/m² at an electrolyte concentration of 1000 mol/m³, see Kremer et al., 2020.

At interfaces between active material grains, exchange of Li ions is possible only if both grains are modeled with the same material ID in GeoDict. In case of different material IDs, no exchange of Li ions is possible between the grains. Especially, if different grains of the same active material are assigned to separate Material IDs in the GeoDict structure, no ion exchange happens between the IDs.

In case the separator is modeled as non-porous material, it is considered as filled with electrolyte throughout the simulation and the interface is modeled in the same way as the interface between electrolyte and active materials or the interface between electrolyte and lithium reservoir.

Also, in the case of a porous separator with effective properties, it is always assumed that the direct interface between separator and active material is completely filled with some electrolyte. The whole interface is therefore described by the equations for interaction between electrolyte and active material. The same is valid for the interface between separator and a lithium reservoir.