1D Isothermal Lithium-Ion Battery

The ionic charge balances and material balances are modeled according to the equations for binary 1:1 electrolytes (Ref. 1). The material parameters for the electrolyte refer to a plasticized EC/DMC electrolyte residing in a polymer matrix, and the electrolyte volume fraction specified in the model hence refer to the sum of the liquid electrolyte and polymer matrix volume fractions.

The electric potential in the electron conducting phase,

, is calculated using a charge balance based on Ohm’s law where the charge transfer reactions result in source or sink term.

For the electrolyte phase of the porous electrodes, effective conductivities, σleff, are used that take porosity and tortuosity into account as given by the following expression:

where γ is the Bruggeman coefficient, using a value of 3.3. The diffusivity is treated similarly.

Fickian diffusion describes the transport in the spherical particles. The diffusion equation is expressed in spherical coordinates for the material balance of lithium in the particles.

Butler-Volmer electrode kinetics describes the local charge transfer current density in the electrodes. The Butler-Volmer expressions are introduced as source or sink terms in the charge balances and material balances.

Boundary Conditions

For the electronic current balance, a potential of 0 V is set on the negative electrode’s current collector/feeder boundary. At the positive electrode current collector/feeder, the current density is specified. In this model, the current density is cycled through a discharge, followed by an interval of zero current, and a final charging stage. The inner boundaries facing the separator are insulating for electric currents.

For the ionic charge balance in the electrolyte, the current collector/feeder boundaries are insulating. Insulation boundary conditions also apply to the material balances.

At the particle surface in the local particle model, the material flux is determined by the local electrochemical reaction rate.

Material Properties

The material properties are those of a typical lithium-ion battery. The electrolyte consists of 2 M LiPF6 salt in 1:2 EC:DMC (by volume) solvent and p(VDF-HFP). The electrode materials are carbon-based material for the negative electrode and LiyMn2O4 for the positive electrode.

The electrolyte conductivity and the equilibrium potential of the negative and positive electrodes are composition dependent through experimentally measured data. This data is tabulated in interpolating functions or the Material library in the model. The properties vary significantly during the charge and discharge phases due to the changes in composition.

The model specifies the electrolyte conductivity according to the function in Figure 2.

Figure 2: The model specifies the ionic conductivity of the electrolyte using an interpolation function according to this behavior with concentration.

Figure 3 displays the equilibrium potentials for the negative and positive electrodes as functions of the measured state of charge (SOC).

Figure 3: The equilibrium voltage of the electrode materials.

The model uses the following definition of SOC:

The initial SOC-values for the positive and negative electrodes are 0.17 and 0.56, respectively, with the concentrations selected initially in the model. This corresponds to an open circuit cell voltage of approximately 4.22 V, which corresponds to a fully charged battery.

For complete details on the material properties and constants, see Ref. 1.

Results and Discussion

Discharge Curves

The battery is initially at a fully charged state. A first modeling approach is to simulate discharge at various current densities and then display the discharge curves. The results show the capacity of the battery at different discharge rates. This model defines end-of-discharge as the time when the cell voltage drops below 3 V. The nominal discharge current density, corresponding to case 1C below (a current density corresponding to a theoretical full discharge in one hour), is 17.5 A/m2.

Figure 4: Discharge curves for various discharge rates.

Figure 4 shows that the maximum discharge capacity of 17.5 Ah/m2 is obtained for a current density of 1.75 A/m2 (0.1 C). It can also be seen that the 3 V discharge capacity decreases slightly when applying a 1C discharge current and dramatically when going above that. At 4C, the battery delivers approximately 50% of the theoretical capacity before it reaches a cell voltage of 3 V. The discharge curves are identical to those presented in Ref. 1.

Discharge and Charge Cycle

Figure 5 depicts the discharge-charge cycle applied in the next step of the simulation. The cycle applies 2000 s of discharge at nominal current density (case 1C above), 300 s at open circuit, then 2000 s of charge at nominal current density, and finally open-circuit conditions.

Figure 5: Cell voltage and current during the applied cycle.

During both discharge and charge, the cell voltage experiences ohmic losses of approximately 100 mV and a concentration overpotential of about 50 mV. These losses are clearly visible in the plot when the current is set to 0, where you can see an immediate relaxation of the voltage due to the relaxation of ohmic losses followed by a slower relaxation of the concentration overpotential.

It is possible to visualize the contributions of the different losses to the total overpotential. You can plot the contribution as shown in Figure 6 using the following procedure:

Plot the electrolyte potential profile at the initial stage of the discharge with a bias of 148 mV to get all the plots in the same range of potential.

Plot the electrolyte potential profile at the end of the discharge adding a bias of 558 mV, again in order to get the profile in the same scale as the overpotential.

Plot the reaction overpotential profile.

All plots are within similar range of potential.

Figure 6: Voltage losses in the battery during discharge.

Figure 6 shows that the two main losses are due to the reaction overpotential and electrolyte resistance. The difference between electrolyte potential profiles initially and toward the end of the discharge stage indicates the influence of the concentration polarization; it is approximately half the magnitude of the two previous losses and is thus also important to consider. The figure does not include the electronic potential profile in the solid phase, but the simulations show that contributions from the ohmic losses in the electronic conductors are negligible.

To further investigate the reason for the steep voltage decrease, you can plot the concentration profile in the electrolyte. Figure 7 depicts the profile at several stages during the discharge and charge cycle.

Figure 7: Electrolyte-phase concentration profiles at various times.

The cell experiences significant concentration polarization due to the low effective diffusion coefficient in the electrolyte, which in turn leads to variations in ionic conductivity and concentration overpotential.

The concentration in the solid phase at the surface of the particles also affects the current density. Figure 8 depicts the distribution of the concentration in the solid-phase particles just before the end of the discharge.

Figure 8: Concentration distribution of lithium in the solid particles during the discharge phase. (Dashed lines: Center of particles. Solid lines - Surface of particles.)

At 1800 s the concentration in the negative electrode varies from 9500 mol/m3 in the particle center to approximately 7000 mol/m3 at the surface, with moderate variation along the width of the cell. At the positive electrode, the variation along the width is much more pronounced with a surface concentration of approximately 13,000 mol/m3 at the electrolyte interface and only 8500 mol/m3 at the positive current collector/feeder. This large variation arises due to the nonuniform current distribution, which in turn is caused mainly by the poor electrolyte conductivity (due to low porosity) but also by the concentration polarization.

As the surface concentration changes, the equilibrium voltage also varies, causing a lower reaction overpotential and a decrease in the local current density. This effect tends to even out the local charge transfer current density to some extent but results in a larger overall voltage loss in the battery.

Reference

1. M. Doyle, J. Newman, A.S. Gozdz, C.N. Schmutz, and J.M. Tarascon, “Comparison of Modeling Predictions with Experimental Data from Plastic Lithium Ion Cells,” J. Electrochem. Soc., vol. 143, no. 6, pp. 1890–1903, 1996.

Battery_Design_Module/Batteries,_Lithium-Ion/li_battery_1d

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

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In the tree, select .

Click .

Click

In the tree, select .

(The will perform a time-dependent simulation, using a initialization study step to calculate the initial potentials in the cell.)

Click

Global Definitions

Parameters 1

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 li_battery_1d_parameters.txt.

Geometry 1

The geometry contains three domains. Create the geometry by specifying the lengths of the domains.

Interval 1 (i1)

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

In the table, enter the following settings:


Lengths (m)
L_neg
L_sep
L_pos

Click

Global Definitions

This model uses an interpolation function, imported from a text file, for the equilibrium potential of the negative electrode material.

Interpolation 1 (int1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Click

Browse to the model’s Application Libraries folder and double-click the file li_battery_1d_Eeq_neg.txt.

Click

In the text field, type Eeq_neg.

Locate the section. From the list, choose .

From the list, choose .

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


Function	Unit
Eeq_neg	V

Click

Piecewise 1 (pw1)

The load cycle is set up using a Piecewise function. The data is imported from a text file and smoothing is enabled.

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type t.

From the list, choose .

In the text field, type 10.

Find the subsection. Click

Browse to the model’s Application Libraries folder and double-click the file li_battery_1d_load_cycle.txt.

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

In the text field, type 1.

Click

Definitions

Variables (Positive Current Collector)

In the toolbar, click

and choose .

In the window for , type Variables (Positive Current Collector) in the text field.

Locate the section. From the list, choose .

Select Boundary 4 only.

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


Name	Expression	Unit	Description
i_app	pw1(t)*i_1C	A/m²	Applied battery cell current density

Materials

This model also uses battery materials available in the material library.

Add Material

LMO is selected as positive electrode material.

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

Note: In the Materials node, cEeqref denotes the maximum lithium concentration in the active material.

Materials

LMO, LiMn2O4 Spinel (Positive, Li-ion Battery) (mat1)

Select Domain 3 only.

In the window, expand the node.

Interpolation 1 (Eeq_int1)

In the window, expand the node, then click .

In the window for , click

Add Material

LiPF6 in 1:2 EC:DMC and p(VdF-HFP) is selected as electrolyte.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

LiPF6 in 1:2 EC:DMC and p(VdF-HFP) (Polymer, Li-ion Battery) (mat2)

Click in the window and then press Ctrl+A to select all domains.

In the window, expand the node.

Interpolation 1 (sigmal_int1)

In the window, expand the node, then click .

In the window for , click

Lithium-Ion Battery (liion)

Porous Electrode 1

In the window, under right-click and choose .

Select Domain 1 only.

In the window for , locate the section.

From the σs list, choose . In the associated text field, type Ks_neg.

Locate the section. In the εl text field, type epsl_neg.

In the εs text field, type epss_neg.

Locate the section. From the list, choose . In the fl text field, type epsl_neg^brugg.

From the list, choose .

From the list, choose . In the fDl text field, type epsl_neg^brugg.

Particle Intercalation 1

In the window, click .

In the window for , locate the section.

In the cs,init text field, type cs0_neg.

From the cs,max list, choose . In the associated text field, type csmax_neg.

Locate the section. From the Ds list, choose . In the associated text field, type Ds_neg.

In the rp text field, type rp_neg.

Click to expand the section. From the socmin list, choose . From the socmax list, choose .

Porous Electrode Reaction 1

In the window, click .

In the window for , locate the section.

From the Eeq list, choose . In the associated text field, type Eeq_neg(liion.cs_surface/csmax_neg).

Locate the section. From the list, choose .

In the k text field, type k_neg.

In the cl,ref text field, type cl_ref.

Click to expand the section. From the dEeq/dT list, choose .

Porous Electrode 2

In the toolbar, click

and choose .

Select Domain 3 only.

In the node, the electrolyte material has been assigned to all domains. You hence need to set the electrode material selection manually, which in this case is LMO electrode material.

In the window for , locate the section.

From the list, choose .

Locate the section. In the εl text field, type epsl_pos.

In the εs text field, type epss_pos.

Locate the section. From the list, choose . In the fl text field, type epsl_pos^brugg.

From the list, choose .

From the list, choose . In the fDl text field, type epsl_pos^brugg.

Particle Intercalation 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the cs,init text field, type cs0_pos.

Locate the section. From the Ds list, choose . In the associated text field, type Ds_pos.

In the rp text field, type rp_pos.

Changing to quadratic elements in the particle dimension improves accuracy.

Click to expand the section. In the Nord text field, type 2.

Porous Electrode Reaction 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the k text field, type k_pos.

In the cl,ref text field, type cl_ref.

Electric Ground 1

In the toolbar, click

and choose .

Select Boundary 1 only.

Electrode Current Density 1

In the toolbar, click

and choose .

Select Boundary 4 only.

In the window for , locate the section.

In the in,s text field, type -i_app.

Initial Values 1

In the window, click .

In the window for , locate the section.

In the cl text field, type cl_0.

Global Definitions

Default Model Inputs

Set up the temperature value used in the entire model.

In the window, under click .

In the window for , locate the section.

In the tree, select .

Find the subsection. In the text field, type T.

Study 1

First set up a study to investigate the defined discharge/charge cycle.

Step 2: Time Dependent

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,10,8000).

In the toolbar, click

Results

Boundary Electrode Potential with Respect to Ground (liion)

A number of plots are created by default. The first plot shows the electrode potential versus time on the boundary where you applied the current density condition. Since you grounded the other electrode, this equals the cell voltage. Modify this plot to also plot the battery current on a second y-axis. (Figure 5):

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Locate the section. Select the check box.

In the associated text field, type Cell voltage (V).

Select the check box.

Locate the section. From the list, choose .

Point Graph 1

In the window, expand the node, then click .

In the window for , click to expand the section.

Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
Voltage

Point Graph 2

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. Select the check box.

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


Legends
Current

In the toolbar, click

1D Plot Group 6

Now reproduce the rest of the plots in the model documentation for the discharge/charge cycle, starting with a plot for comparing the different voltage losses (Figure 6).

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Line Graph 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

In the list, select .

Locate the section. From the list, choose .

Locate the section. In the text field, type phil+0.148.

Click to expand the section. Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
phil+0.148

Line Graph 2

Right-click and choose .

In the window for , locate the section.

In the text field, type liion.eta_per1.

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


Legends
eta

Line Graph 3

Right-click and choose .

In the window for , locate the section.

In the list, select .

Locate the section. In the text field, type phil+0.558.

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


Legends
phil+0.558

1D Plot Group 6

In the window, click .

In the window for , locate the section.

Select the check box.

In the associated text field, type x.

Select the check box.

In the associated text field, type Voltage (V).

In the toolbar, click

Electrolyte Salt Concentration (liion)

Modify the default plot of the electrolyte concentration to compare the profile at various times during the discharge/charge cycle (Figure 7).

In the window, click .

In the window for , locate the section.

From the list, choose .

In the list, choose , , , , and .

Line Graph 1

In the window, expand the node, then click .

In the window for , locate the section.

Select the check box.

Electrolyte Salt Concentration (liion)

In the window, click .

In the window for , locate the section.

Select the check box.

In the associated text field, type x.

Select the check box.

In the associated text field, type Electrolyte concentration, cl.

In the toolbar, click

1D Plot Group 7

This plot shows the concentration profile within the particles in the electrodes (Figure 8).

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the list, choose , , and .

Locate the section. From the list, choose .

In the text area, type Electrode particle lithium concentration, surface (solid) and center (dashed).

Locate the section. Select the check box.

In the associated text field, type x.

Select the check box.

In the associated text field, type cs (mol//m<sup>3</sup>).

Line Graph 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Click in the upper-right corner of the section. From the menu, choose .

Locate the section. Select the check box.

Line Graph 2

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Click in the upper-right corner of the section. From the menu, choose .

Click to expand the section. Find the subsection. From the list, choose .

From the list, choose .

In the toolbar, click

Click the

button in the toolbar.

Global Definitions

To reproduce the discharge curves in Figure 4, modify the problem formulation slightly to use a constant discharge current only, and then set up a parametric study that solves for four different discharge currents. Use a stop condition to stop the solver if the cell voltage drops below 2.0 V.

Parameters 1

Begin by adding a global current variable.