Diffusion-Induced Stress in a Lithium-Ion Battery

Figure 2 shows the normalized lithium concentration in a particle at a particular position (center) in the negative electrode, at various times during discharge and relaxation. Note in all subsequent plots, the solid and dashed lines refer to the discharge and relaxation periods, respectively. During discharge, which involves de-intercalation of lithium ions from the negative electrode particles, it can be seen that the concentration in the particle decreases with time. At the end of the relaxation period, the concentration settles at a final value.

Figure 2: Normalized lithium concentration in a particle at the center of the negative electrode, at various times during discharge (solid lines) and relaxation (dashed lines).

The radial component of the diffusion-induced stress in a particle at the center of the negative electrode, at various times during discharge and relaxation, is shown in Figure 3. During discharge (de-intercalation of lithium ions from the particle), the radial stress is compressive inside the spherical particle. At any given time, the compressive radial stress is highest at the center and decreases monotonically to 0 at the surface of the particle. During the relaxation period, the compressive radial stress inside the particle relaxes to 0.

Figure 3: Radial component of the diffusion-induced stress in a particle at the center of the negative electrode, at various times during discharge (solid lines) and relaxation (dashed lines).

Similarly, Figure 4 shows the tangential component of the diffusion-induced stress in a particle at the center of the negative electrode, at various times during discharge and relaxation. During discharge, the tangential stress is compressive at the center and tensile at the surface of the spherical particle. Note that at the center of the particle, the tangential and radial stresses always have the same magnitude so that the stress state at the center of the particle is purely hydrostatic. The crossover from compressive to tensile occurs at

, after the stress reaches steady state.

Figure 4: Tangential component of the diffusion-induced stress in a particle at the center of the negative electrode, at various times during discharge (solid lines) and relaxation (dashed lines).

Figure 5 shows the von Mises stress in a particle at the center of the negative electrode, at various times during discharge and relaxation. During discharge, the von Mises stress is 0 at the center of the particle and reaches a maximum value at the surface of the spherical particle. During the relaxation period, the stress inside the particle relaxes to 0.

Figure 5: von Mises stress in a particle at the center of the negative electrode, at various times during discharge (solid lines) and relaxation (dashed lines).

Figure 6 and Figure 7 shows the radial and tangential components, respectively, of the diffusion-induced strain at surface of particles along the length of the negative electrode, at various times during discharge and relaxation. During de-intercalation of lithium ions from the particle, the surface strain decreases with time in the negative electrode. Also, the surface value of strain is not uniform across the electrode because of variation in particle concentration across the electrode. The maximum value of surface strain is observed at the current collector end. At the end of the relaxation period, the strain eventually reaches a value corresponding to the final particle concentration.

Figure 6: Radial component of diffusion-induced surface strain along the length of the negative electrode, at various times during discharge (solid lines) and relaxation (dashed lines).

Figure 7: Tangential component of diffusion-induced surface strain along the length of the negative electrode, at various times during discharge (solid lines) and relaxation (dashed lines).

The total elastic strain energy density that is stored in the negative electrode material particles provides the driving force for particle fracture. This quantity is plotted along the length of the negative electrode in Figure 8, at various times during discharge. It can be seen that the total elastic strain energy density increases with time across the electrode. The value is initially higher at the separator end. The maximum value gradually moves across the electrode toward the current collector end at the end of discharge.

Figure 8: Total elastic strain energy density stored in the negative electrode material particles, plotted along the length of the negative electrode, at various times during discharge.

All the above plots show a quantity plotted either across the particle dimension or along the length of the negative electrode, at various times during discharge. Alternatively, it is useful to plot the time evolution of any quantity at different positions in the negative electrode. The von Mises stress at the surface of particles at two positions in the negative electrode (current collector end and separator end) is plotted versus time in Figure 9. The plot also includes the applied current for reference. The von Mises stress contains contribution from both the radial and tangential components of the diffusion-induced stress. As seen from Figure 3 and Figure 4, the surface value of the radial component of the diffusion-induced stress is 0. So, the surface von Mises stress essentially mimics the behavior of the tangential component of the diffusion-induced surface stress. In Figure 9, it can be seen that the von Mises stress initially increases at the separator end and subsequently at the current collector end. The stress induced in the electrode particles varies at different positions along the negative electrode due to concentration gradients in the porous electrode. Similarly, it can be expected that the total elastic strain density, that provides the driving force for fracture, also varies at different positions along the negative electrode. It can be seen in Figure 9 that the surface von Mises stress is higher at the current collector end as compared to the separator end, toward the end of the discharge period. Finally, the stress in the particles relaxes to 0 during the relaxation period.

Figure 9: Time evolution of the von Mises stress at the surface of the particles at two positions in the negative electrode. The applied current is also plotted for reference.

References

1. Y-T. Cheng, and M.W. Verbrugge, “Evolution of stress within a spherical insertion electrode particle under potentiostatic and galvanostatic operation,” Journal of Power Sources, vol. 190, pp. 453–460, 2009.

2. V. Malave, J.R. Berger, and P.A. Martin, “Concentration-dependent chemical expansion in lithium-ion battery cathode particles,” Journal of Applied Mechanics, vol. 81, pp. 091005 1–9, 2014.

Battery_Design_Module/Batteries,_Lithium-Ion/diffusion_induced_stress

Modeling Instructions

Start this tutorial by opening a template file that contains a 1D battery model. Alternatively, you may build the template model yourself by following the instructions found in the template model.

From the menu, choose .

Browse to the model’s Application Libraries folder and double-click the file li_battery_seed.mph.

Global Definitions

Parameters 1

The template model already contains a few parameters. Change the value of the initial cell voltage and add a few parameters for the Young’s modulus and reference exchange current densities.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Ecell_init	4.15[V]	4.15 V	Initial cell voltage
E	10[GPa]	1E10 Pa	Young's modulus
i0ref_neg	0.96[A/m^2]	0.96 A/m²	Reference exchange current density, negative electrode
i0ref_pos	0.70[A/m^2]	0.7 A/m²	Reference exchange current density, positive electrode

Step 1 (step1)

Define a step function that will be used later to set up the applied current.

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 1.

In the text field, type 0.

Relative volume change as function of soc

Define an interpolation function that defines the relative expansion of graphite material as a function of state-of-charge.

In the toolbar, click

and choose .

In the window for , type Relative volume change as function of soc in the text field.

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


t	f(t)
0	0
0.5	0.05
0.75	0.05
1	0.1

In the text field, type fvol.

Locate the section. From the list, choose .

Locate the section. In the text field, type 1.

In the text field, type 1.

Definitions (comp1)

Load the variables for this model from a text file.

Variables 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

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

Lithium-Ion Battery (liion)

The physics has already been set up in the template model. Next, set up the inputs required for stress and strain calculations in the node of the porous electrode domain corresponding to the negative electrode (graphite material). Also, specify the reference exchange current density for the electrode kinetics in the nodes of the negative and positive electrodes. Finally, specify the applied current (that corresponds to a discharge current of 4C for 850 s followed by a relaxation period for the next 250 s) in the node.

Particle Intercalation 1

In the window, expand the node, then click .

In the window for , click to expand the section.

Select the check box.

In the E text field, type E.

In the ΔV/V0 text field, type fvol(liion.cs_pce1/liion.csmax).

It is essential to have a finer resolution along the graphite particle dimension and this can be done by setting a linear distribution with 30 elements.

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

In the Nel text field, type 30.

Porous Electrode Reaction 1

In the window, click .

In the window for , locate the section.

In the i0,ref(T) text field, type i0ref_neg.

Porous Electrode Reaction 1

In the window, expand the node, then click .

In the window for , locate the section.

In the i0,ref(T) text field, type i0ref_pos.

Electrode Current 1

In the window, click .

In the window for , locate the section.

In the Is,total text field, type Iapp.

Study 1

Step 2: Time Dependent

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type range(0,10,1100).

In the window, click .

In the window for , locate the section.

Clear the check box.

In the toolbar, click

Results

Battery cell voltage

The probe plot show the battery cell voltage (Figure 1).

In the window, under click .

In the window for , type Battery cell voltage in the text field.

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

Locate the section. Select the check box.

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

Locate the section. Clear the check box.

In the toolbar, click

Study 1/Solution 1 (sol1)

To plot along the graphite particle dimension, you need to create a Solution dataset that refers to the extra dimension that is set up by the Porous Electrode node corresponding to the negative electrode.

Study 1/Solution 1 (4) (sol1)

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Negative particle concentration

Next, follow the steps below to plot the normalized lithium concentration in a particle at a particular position (say center) in the negative electrode, at various times during discharge and relaxation. Note in all the subsequent plots in the model, the solid and dashed lines refer to the discharge period and the relaxation period, respectively. (Figure 2).

In the toolbar, click

and choose .

In the window for , type Negative particle concentration in the text field.

Line Graph 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 0,100,300,400,500,600,700,850.

Locate the section. From the list, choose .

Locate the section. In the text field, type comp1.atxd1(15e-6,liion.cs_pce1/liion.csmax).

Click to expand the section. Select the check box.

Line Graph 2

Right-click and choose .

In the window for , locate the section.

In the text field, type 860,875,900,1100.

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

From the list, choose .

Negative particle concentration

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. Select the check box.

In the associated text field, type Normalized particle dimension (1).

Select the check box.

In the associated text field, type Normalized lithium concentration (1).

In the toolbar, click

Now, create a plot of the radial component of the diffusion-induced stress in a particle (Figure 3) at the center of the negative electrode, at various times during discharge and relaxation. This can be done by duplicating the previous plot.

Diffusion-induced stress, radial component

Right-click and choose .

In the window for , type Diffusion-induced stress, radial component in the text field.

Locate the section. In the text field, type Diffusion-induced stress, radial component (Pa).

Line Graph 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type 100,300,400,500,600,700,850.

Locate the section. In the text field, type comp1.atxd1(15e-6,liion.sr_pce1).

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

Line Graph 2

In the window, click .

In the window for , locate the section.

In the text field, type comp1.atxd1(15e-6,liion.sr_pce1).

Locate the section. From the list, choose .

Diffusion-induced stress, radial component

In the window, click .

In the window for , locate the section.

Select the check box.

In the text field, type 0.05.

In the text field, type 1.

In the text field, type -1.5E7.

In the text field, type 0.2E7.

Locate the section. From the list, choose .

In the toolbar, click

Now, create a plot of the tangential component of the diffusion-induced stress in a particle (Figure 4) at the center of the negative electrode, at various times during discharge and relaxation. This can be done by duplicating the previous plot.

Diffusion-induced stress, tangential component

Right-click and choose .

In the window for , type Diffusion-induced stress, tangential component in the text field.

Locate the section. In the text field, type Diffusion-induced stress, tangential component (Pa).

Line Graph 1

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type comp1.atxd1(15e-6,liion.stheta_pce1).

Line Graph 2

In the window, click .

In the window for , locate the section.

In the text field, type comp1.atxd1(15e-6,liion.stheta_pce1).

Diffusion-induced stress, tangential component

In the window, click .

In the window for , locate the section.

In the text field, type 1.2E7.

In the toolbar, click

Now, create a plot of the von Mises stress in a particle (Figure 5) at the center of the negative electrode, at various times during discharge and relaxation. This can be done by duplicating the previous plot.

von Mises Stress