COMSOL 6.4 - Voltage Hysteresis in a Lithium Iron Phosphate (LFP) Electrode

Voltage Hysteresis in a Lithium Iron Phosphate (LFP) Electrode

Lithium iron phosphate (LFP) is a common positive electrode material in lithium-ion batteries. Specific for the LFP electrode material is that its equilibrium (open circuit) potential, when defined as a function of the lithiation state, features a large flat plateau with a more or less constant potential. The plateau also exhibits voltage hysteresis; that is, a difference in the plateau potential between lithiation and delithiation cycling, also referred to as a zero-current voltage gap.

This tutorial demonstrates a simplified model to include voltage hysteresis in a porous LFP electrode.

Model Definition

origin of the voltage hysteresis

The voltage hysteresis in an electrode material is commonly attributed (Ref. 1) to a nonmonotonic equilibrium potential (or Gibbs free energy) dependency with respect to the intercalated lithium concentration (state of lithiation). The solid blue line of Figure 1 shows an example of a nonmonotonic equilibrium potential of a single electrode particle as function of the state of lithiation. When lithiation starts in point 1, the equilibrium potential decreases until the inflection point 2 is reached, after which further lithiation results in an increase in potential, until the derivative changes sign again at a second inflection point.

The distance between the two inflection points (the “miscibility” gap) is believed to decrease with the particle size, and the magnitude of the voltage offset between them is also assumed to vary with size (Ref. 2), based on experimental observations (Ref. 3).

Figure 1: Schematic representation of the lithiation process in a particle subject to a nonmonotonic equilibrium potential function. Lithiation starts in point 1, proceeding via the inflection point 2 to point 3. Note: The voltage scale is arbitrary and only serves as an example.

The solid blue line in Figure 1 is however seldom observed experimentally, and the reason for this is related to the difficulties of recording the equilibrium potential of a single electrode particle. For electrodes comprising multiple particles, material inhomogeneities will result in the equilibrium potential exhibiting a different shape, which can be understood by considering an ensemble of many similar, but not identical, particles subject to a constant reduction current as follows: When lithiation starts in point 1 for the ensemble, all particles are initially at the same potential. However, as lithiation continues, small differences in particle properties result in one of the particles reaching point 2 first, and as this particle passes the inflection point, increasing cathodic overpotentials will favor a fast and accelerated continued reduction of this single particle up to the second inflection point, and then continued reduction until it reaches the same potential as the rest of the ensemble in point 3. During the process when the first particle gets reduced to point 3, all other particles in ensemble will remain at a position left of point 2, where they undergo slight oxidation in order to maintain the overall charge balance of the ensemble. Once the first particle has reached point 3, a subsequent particle will undergo the fast transition to point 3, and so on. During the fast transitions, the mixed potential of the ensemble will stay close to the dashed line in Figure 1, and not until all particles have reached point 3 will the equilibrium potential proceed to decrease from the common equilibrium potential level of point 2 and 3.

The process described above is also referred to as a “mosaic” transition between lithium-poor and lithium-rich phases. For smaller ensembles of particles it should be possible to observe these transitions as voltage ripples using a high-precision potentiostat when operating at a constant current, but practically for any ordinary battery electrode, the continuum limit of the mosaic transitions results in a smooth voltage plateau.

Reversing the process with delithiation starting in point 4, as shown in Figure 2, results in a similar behavior, with a fast transition occurring between point 5 and 6. The difference in the equilibrium potential levels at the inflection points 2 (in Figure 1) and 5 (in Figure 2) results in voltage hysteresis.

Figure 2: Schematic representation of the delithiation process in a particle subject to a nonmonotonic equilibrium potential function. Delithiation starts in point 4, proceeding via the inflection point 5 to point 6. Note: The voltage scale is arbitrary and only serves as an example.

LFP Particle Ensemble Model

The model defines an ensemble of particles by the usage of three dependent variables:

•

The concentration of intercalated lithium in the lithium-rich phase, cs,rich (mol/m3)

•

The concentration of intercalated lithium in the lithium-poor phase, cs, poor (mol/m3)

•

The average concentration of lithium in the particle ensemble, cs,avg (mol/m3)

The three variables above are in turn used to define the volume fractions of the lithium-poor and lithium-rich phases, ϕpoor (1) and ϕrich (1), using the relations

(1)

and

(2)

Diffusion is neglected within the ensemble. The material balances for the average intercalated concentration is defined as

(3)

where εs (1) is the electrode volume fraction and iv (mol/m3) the volumetric current density stemming from charge transfer reactions in the whole ensemble.

The volumetric charge transfer current densities at the electrolyte-electrode interface between the particle ensemble and the electrolyte from both the lithium-rich and lithium-poor phases are used to define iv according to

(4)

where iv,i (mol/m3) is the volumetric current density of the respective phase. (In the following, the subscript index i refers to either the lithium-rich or the lithium-poor phase.)

Figure 3: Schematic representation of the lithium-poor and lithium-rich equilibrium potential functions used by the ensemble model. Note: The voltage scale is arbitrary and only serves as an example.

Figure 3 shows a schematic plot of the equilibrium potential functions used for defining the overpotential (see Equation 11 below) of the lithium-rich and lithium-poor phases. Here the state of lithiation is defined as cs /cs,max. In Figure 3 the locations of the two inflection points have also been indicated, which define the two concentration levels cs,poor,max (mol/m3) and cs,rich,min (mol/m3) in terms of cs,max.

For the lithium-poor phase concentration, any cathodic charge transfer current density is assumed to be counterbalanced by fast phase transitions whenever the concentration level exceeds the concentration level of the inflection point, cs,poor,max (mol/m3), prohibiting further intercalation into the lithium-poor phase. The material balances for the lithium-poor phase concentration is hence defined as

(5)

(6)

The above way of conditionally adding the charge transfer current density to the phase concentration time derivatives implicitly defines the phase transition rate. This can be seen by differentiation with respect to time in Equation 1:

(7)

which may be rearranged into

(8)

When any of the conditionals in Equation 5 or Equation 6 sets the time derivative of either cs,richor cs,rich to zero, respectively, this may result in a nonzero time derivative of ϕpoor, which in turn results in changing the volume fractions of the lithium-rich and lithium-poor phases.

Charge transfer Kinetics

For each phase, charge transfer occurs according to Butler–Volmer expressions defined as:

(9)

where Av (m2/m3) is the specific surface area of the electrolyte-electrode interface, i0,i is the exchange current density, and ηi (V) is the overpotential.

The exchange current density is defined as

(10)

where cl (mol/m3) is the electrolyte ion concentration, i0,ref is the exchange current density for a reference state corresponding to the reference conditions cl = cl,ref and

, and cs,max (mol/m3) is the maximum intercalation concentration.

The overpotential is defined as

(11)

where ϕs (V) is the electrode phase potential, ϕl (V) the electrolyte phase potential, and Eeq, i (V) the equilibrium potential as defined in Figure 4 below.

COMSOL Implementation

The interface is used to define a half-cell model consisting of a Lithium-metal counter electrode, a separator, and a positive LFP porous electrode. The > node, with the option for defining the species concentration model, is used to define the cs,avg variable.

The lithium-poor and lithium-rich equilibrium potential curves, as shown in Figure 4, and some additional modeling parameters, were taken from Ref. 4.

Figure 4: Equilibrium potential for the lithium-poor (left) and the lithium-rich (right) particles used in the simulation. (Based on experimental data from Ref. 4).

A interface is added to solve for cs,poor and cs,rich on the porous electrode domain.

To illustrate the hysteresis loop, the interface is used to define a load cycle using 1C lithiation/delithiation current pulses with 10 minute resting periods in between. The simulation starts from a lithiated electrode. The first current pulse delithiates the electrode to around 50%, followed by a rest, followed by another delithiation pulse to reach almost full delithiation. After yet another rest period, the current direction is reversed in order to lithiate the electrode to around 50%, followed by another 10-minute rest. Finally, the electrode is once again almost completely delithiated.

A interface is added to integrate the current over time in order to compute the cycled capacity.

Results and Discussion

Figure 5 shows the cell voltage versus time during the load cycle. During each 10 minute rest period, a short rapid voltage relaxation, which stems from relaxation of gradients in the lithium-ion concentration in the cell, is followed by a longer plateau featuring a constant potential of either 3.452 V (after a delithiation pulse) or 3.412 V (after a lithiation pulse). These voltage levels correspond to the left-most data point of the lithium-rich particles, and the right-most data point of the lithium-poor particles in Figure 4, respectively.

Figure 5: Cell voltage versus time.

Figure 6 shows a corresponding voltage versus capacity plot, where the different voltages at around 50% lithiation (0.88 mA/cm2) illustrate the resulting voltage hysteresis at rest. A small hysteresis also in polarization is seen between the second and last delithiation step (above 0.92 mA/cm2). This is an effect of the current distribution in the porous electrode, which predominantly will favor lithiation/delithiation in the region close to the separator in the beginning of a current pulse. The same behavior (qualitatively) is reported in Ref. 4.

Figure 6: Cell voltage versus capacity for the delithiation-lithiation cycle.

References

1. W. Dreyer and others, “The thermodynamic origin of hysteresis in insertion batteries,” Nature Mater., vol. 9, pp. 448–453, 2010; doi.org/10.1038/nmat2730.

2. P. Ombrini and others, “Kinetically induced memory effect in Li-ion batteries,” EES Batteries, vol. 1, no. 3, pp. 437–449, 2025; dx.doi.org/10.1039/D5EB00014A.

3. N. Meethong and others, “Size-Dependent Lithium Miscibility Gap in Nanoscale Li1−x FePO4,” Electrochem. Solid-State Lett., vol. 10, no. 5, A134, 2007; doi.org/10.1149/1.2710960.

4. V. Srinivasan and J. Newman, “Existence of Path-Dependence in the LiFePO4 Electrode,” Electrochem. Solid-State Lett., vol. 9, no. 9, A110, 2006; doi.org/10.1149/1.2159299.

Battery_Design_Module/Lithium-Ion_Batteries,_Performance/lfp_voltage_hysteresis

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

In the tree, select > > .

Click .

In the tree, select > > .

Click .

In the tree, select > > .

Click .

Click

In the tree, select > > .

Click

Global Definitions

Parameters 1

Add a list of parameter definitions from a text file as follows:

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file lfp_voltage_hysteresis_parameters.txt.

Some definitions are highlighted in red, indicating missing operators. Add these next.

Interpolation - Eeq poor (lithiation from fully delithiated)

In the toolbar, click

and choose > .

In the window for , type Interpolation - Eeq poor (lithiation from fully delithiated) in the text field.

Locate the section. In the text field, type Eeq_poor.

Click

Browse to the model’s Application Libraries folder and double-click the file lfp_voltage_hysteresis_Eeq_poor.txt.

Locate the section. In the table, enter the following settings:


Function	Unit
Eeq_poor	V

In the table, enter the following settings:


Argument	Unit
t	mol/m^3

Click to expand the section. Clear the checkbox.

Clear the checkbox.

Click to expand the section. Select the checkbox.

In the text field, type Eeq_poor_inv.

Click

Interpolation - Eeq rich (delithiation from fully lithiated)

In the toolbar, click

and choose > .

In the window for , type Interpolation - Eeq rich (delithiation from fully lithiated) in the text field.

Locate the section. In the text field, type Eeq_rich.

Click

Browse to the model’s Application Libraries folder and double-click the file lfp_voltage_hysteresis_Eeq_rich.txt.

Locate the section. In the table, enter the following settings:


Function	Unit
Eeq_rich	V

In the table, enter the following settings:


Argument	Unit
t	mol/m^3

Click to expand the section. Clear the checkbox.

Clear the checkbox.

Locate the section. Select the checkbox.

In the text field, type Eeq_rich_inv.

Click

Analytic 1 (an1)

In the toolbar, click

and choose > .

Parameters 1

Check the parameter list again. The error indicators should now have vanished.

Analytic 1 - Eeq_avg

Also add a function for an averaged equilibrium potential function for the whole particle ensemble. This function is strictly not required in the model, but is convenient to have in order to assess a thermodynamically consistent equilibrium potential that can be used, for instance, to compute heat sources.

In the window, under click .

In the window for , type Analytic 1 - Eeq_avg in the text field.

In the text field, type Eeq_avg.

Locate the section. In the text field, type if(cs<cs_poor_max, Eeq_poor(cs), if(cs>cs_rich_min, Eeq_rich(cs), Eeq_poor(cs_poor_max)+(Eeq_rich(cs_rich_min)-Eeq_poor(cs_poor_max))*(cs-cs_poor_max)/(cs_rich_min-cs_poor_max))).

In the text field, type cs.

Locate the section. In the text field, type V.

In the table, enter the following settings: