The Maximum Lithium Concentration that can be stored to or taken out of the active material. This is not the maximum lithium concentration of the active material itself, but the difference between the minimum and maximum reachable lithium concentration. Usually, the minimum reachable lithium concentration of common active materials used in cathodes is not zero. The reason is that often the last lithium ions cannot be removed or could only be removed with a cell potential that is so high that it could damage the battery. can be calculated from the maximum experimentally accessible specific capacity of a material when determining the open circuit voltage potential (see below), the material density , and Faraday’s constant :

Its unit is mol/m3, i.e. the concentration is provided as amount of accessible lithium ions per volume.

Measures a material’s ability to conduct electrons.

Measures the electrolyte’s ability to conduct ions (such as Li ions). GeoDict can consider the concentration dependence of ionic conductivity constants for electrolyte.

Measures a material’s ability to equilibrate differences in concentration. The ionic diffusion constant in active materials is often measured via Galvanostatic Intermittent Titration Technique (GITT), where the reaction of a half cell to a current pulse and the subsequent relaxation is recorded, see Weppner et al., 1977. It can also be measured by Electrochemical Impedance Measurements (EIS), see Ecker et al., 2015. The ionic diffusion constant of an active material can depend on the material SOC. GeoDict can consider the concentration dependence of ionic diffusion constants for electrolyte.

Since GeoDict 2025 this replaces the Butler-Volmer Rate Constant known from earlier GeoDict releases.

This new parameter has many advantages:

• It has a physical interpretation which is easier to understand,
• It can be determined by experiments much more easily,
• It has the unit of the current density, which is the same unit for all cases.

If you are used to the Butler Volmer Rate Constant from previous GeoDict releases, you may find it useful to find the Butler Volmer Rate Constant in the BatteryDict GUI directly below the Maximum Exchange Current Density as shown here. It is computed automatically from the entered value for Maximum Exchange Current Density.

The Maximum Exchange Current Density is not a material parameter of one material alone but depends on the active material as well as on the electrolyte. It is necessary to define the interface condition between voxels of active material and those of electrolyte. It characterizes “how easy” ions can be exchanged between active material and electrolyte.

First, the derivation of Butler-Volmer Interface Current Density, , as calculated in BatteryDict is shown. The Butler-Volmer equation found in literature is (see Ecker et al., 2015)

with the current , the contact area between electrode’s active material(s) and electrolyte, the exchange current density , the charge transfer coefficient , the charge number (which is for a lithium-ion battery), and the overpotential . Considering , the Butler-Volmer equation simplifies to

For insertion electrodes, literature provides a relation of the exchange current density to the lithium concentration in the electrolyte and the lithium concentration in the solid with considering the charge transfer coefficient (Ecker et al., 2015)

where the Butler-Volmer Rate Constant is defined as with a particle transfer rate and Faraday’s constant .

When this relation is inserted into equation (147) above and the overpotential is equal to , the equation describing the Butler-Volmer Interface Current Density in BatteryDict is obtained,

The can be computed from the Maximum Exchange Current Density as follows:

is the Maximum Exchange Current Density in equilibrium electrolyte concentration at 50% lithiation (material SOC = 50%). It replaces the used until GeoDict 2025.

To obtain a value of from the measured exchange current densities at different material SOCs of the electrode material, the experimental data for are fit to a modified equation (148) using :

Note that it might be necessary to convert literature data for from a function of lithium composition to a function of SOC as described in (158) . The next figure shows the experimental data as a function of SOC for NMC333 and a reference electrolyte (EC/DMC/EMC 1:1:1, salt LiPF6, concentration 1 mol/L) as reported in Schmalstieg et al., 2018, Part 1 and Schmalstieg et al., 2018, Part 2. The Maximum Exchange Current Density is now easy to determine as the maximum of this half circle, which is about 1.34 A/m² for an equilibrium concentration of 1000 mol/m² in the electrolyte. If you use an electrolyte with a different lithium concentration in equilibrium than the equilibrium concentration where the exchange current densities were measured, you need to adjust the value of the Maximum Exchange Current Density for this electrolyte equilibrium concentration. For example, the default electrolyte in GeoDict has an equilibrium concentration of and the corresponding Maximum Exchange Current Density needs to be calculated:

Figure 1: Measured data for of NMC333 at 15°C from the Supplementary Information of Schmalstieg et al., 2018, Part 2 as a function of electrode SOC and fit to the data using the above equation. The fit results in

This is also known as Open-Circuit Voltage, hence the abbreviation OCV. The open-circuit potential is directly related to the chemical potential of lithium within the active material. Experimentally, it is determined by measuring the potential difference between the respective electrode and a lithium counter electrode in equilibrium, i.e. without load applied to the electrode. To this end, either relaxation steps are introduced before measurement at a distinct SOC or a “quasi-OCV” function is measured at very low charging rates, e.g. of C/100 Schmalstieg et al., 2018, Part 1. The open circuit potential is generally path-dependent, i.e., it can vary between the charging or discharging process. This phenomenon is known as OCV hysteresis and is observed when the chemical potential of the electrode varies between lithium intercalation and lithium deintercalation (Barai et al., 2015). Additionally, it depends on the charging state of the electrode and is therefore a function of the state of charge (SOC) of the material (see also State Of Charge (SOC)).

In literature, the OCV function is often provided as a function of the specific capacity , e.g. in mAh g-1 against the potential vs Li/Li+ (Nitta et al., 2015). The maximum experimentally accessible specific capacity of a material is denoted as Cspec,max and the OCV curve ends at this specific capacity. Figure 4(e) from Nitta et al., 2015 shows an example from literature for various cathode materials. In this case, the x-axis of the OCV function must be transformed from the specific capacity to the material SOC by the equation:

GeoDict interpolates the OCV curve between the points provided for the OCV function of the material. For the interpolation it is necessary that the OCV function is monotonous, i.e., the OCV must increase with decreasing material SOC.

Sometimes, data in literature are also provided as a function of lithium composition. For instance, for cathode materials the lithium composition x ranges from the maximal lithium composition () for stoichiometric NMC333 (LixNi0.33Mn0.33Co0.33O2)) to the minimally reachable lithium composition . This minimally reachable lithium composition can be calculated via the maximum experimentally accessible specific capacity and the theoretical capacity , which is calculated with the equation:

where is the number of lithium atoms per unit cell in the maximally lithiated material, is the molar mass, and Faraday’s constant. For cathode materials, the minimum lithium composition is found by comparing to , for instance for NMC333:

The material SOC as a function of lithium composition x is then given via linear transformation. E.g. for a cathode material such as NMC333:

Concentration of lithium ions in the electrolyte solution. This is the lithium concentration of the electrolyte if it is in thermodynamic equilibrium. As soon as the battery starts charging or discharging, local concentration gradients will build up in the electrolyte.

In the absence of concentration gradients, the transference number measures the fraction of ionic current carried by lithium ions. GeoDict can consider the concentration dependence of transference number for the electrolyte.

The concentration expansion coefficient is needed to calculate the material expansion upon lithium intercalation using BatteryDict-Degradation. The expansion coefficient is calculated with the equation

where is the maximal volume change due to lithium intercalation and is the maximum (removable) lithium concentration, Wenzler et al., 2023.