International Journal of Electrochemistry

Volume 2015, Article ID 496905, 5 pages

http://dx.doi.org/10.1155/2015/496905

## Modeling the Lithium Ion/Electrode Battery Interface Using Fick’s Second Law of Diffusion, the Laplace Transform, Charge Transfer Functions, and a [4, 4] Padé Approximant

Missouri Southern State University, 3950 Newman Road, Joplin, MO 64801, USA

Received 29 April 2015; Accepted 3 June 2015

Academic Editor: Miloslav Pravda

Copyright © 2015 John H. Summerfield and Charles N. Curtis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work investigates a one-dimensional model for the solid-state diffusion in a LiC_{6}/LiMnO_{2} rechargeable cell. This cell is used in hybrid electric vehicles. In this environment the cell experiences low frequency electrical pulses that degrade the electrodes. The model’s starting point is Fick’s second law of diffusion. The Laplace transform is used to move from time as the independent variable to frequency as the independent variable. To better understand the effect of frequency changes on the cell, a transfer function is constructed. The transfer function is a transcendental function so a Padé approximant is found to better describe the model at the origin. Consider .

#### 1. Introduction

The Li-ion batteries in hybrid electric vehicles experience electric current pulses during their use. These transient electric pulses stress the crystal structure of the electrodes and cause lithium salt crystal growth, both of which inhibit easy ion diffusion. Also the lithium ion cells are attached in series of 72 cells. This added variable increases the chance of varying pulses. These stressors lead to shortened battery life. In order to build better batteries these pulses need to be better understood. The electric pulses have low frequency, between 0.01 Hz and two Hz [1].

The purpose of this work is to present a relatively simple mathematical model of the Li^{+} ions in the electrolyte of a LiC_{6}/LiMnO_{2} battery diffusing into the electrodes. Lithium ion battery electrolyte conductivity has been explored [2]. Software to help students better understand the chemistry of batteries is available as well [3]. The Li^{+} battery electrochemical reactions have also been addressed [4]. Most recently Li^{+} ion movement in the electrolyte has been investigated [5].

The model presented here is appropriate for Analytical Chemistry [6], Instrumental Analysis [7], Physical Chemistry [8], or Electrochemistry [9] classes. In Analytical Chemistry it could be presented as an example of stationary electrode chemistry. In Instrumental Analysis it could be presented as part of the introduction to disk electrode chemistry. In Physical Chemistry it could be presented as a diffusion example. In an Electrochemistry class it could be explored as an example of electrode chemistry or as an aspect of battery systems management. In all courses the model is presented as an in-class lecture. Also it could be extended into an independent study project. This topic will be discussed at the end of the paper.

Transport properties have been known for over a hundred years yet they continue to be relevant [10]. Specifically batteries rely on the movement of ions due to the difference in voltage from electrode to electrode. This motion is termed ion migration. The ions also move from spot to spot in the battery due to differences in their concentrations. This motion is characterized as solid-state diffusion when the ions move in and out of the electrodes.

Mathematical modeling of migration and diffusion within a lithium ion battery is quite involved. The voltages and the solution concentrations must be simulated for the surface of the electrodes and for the electrolyte between them. These equations are made more complex because the kinetics and transport parameters that describe the surfaces and the highly concentrated electrolyte are nonlinear [11].

Presently four differential equations are used to simulate the ion/electrode diffusion of a typical lithium ion battery that consists of a LiMnO_{2} cathode and a LiC_{6} anode, separated by the electrolyte, LiPF_{6} [12]. There is great interest in moving away from a graphite anode to a better charge carrying semiconductor and replacing the manganese in the cathode with less toxic silicon or sulfur. Both changes are made ultimately to increase electric current [13–16].

This paper relies on one differential equation to model the Li-ion diffusion into the solid-state electrode. The equation is a form of Fick’s second law of diffusion [17]. Diffusion is stressed because during the cell’s operation a mean electric field is created at the electrode’s surface so diffusion plays an important role. Also, simplification of solid-state diffusion as a whole is a current topic [18].

#### 2. Li-Ion/Electrode Model

There are three simple atomic models. The atom can be viewed as a plane, a cylinder, or a sphere. The lithium ion is best described as a sphere. The lithium ion’s concentration as a function of time is best described by the Laplacian in spherical coordinates:where is the distance from the origin—the center of the lithium ion— is the polar angle measured down from the North Pole, and is the azimuthal angle.

For the case of one-dimensional solid-state diffusion (1) becomeswhere is the solid-state diffusion coefficient. The value of is 2.0 10^{−12} cm^{2} s^{−1} for the LiC_{6} electrode and is 3.7 10^{−12} cm^{2} s^{−1} for the LiMnO_{2} electrode [12].

To simplify, the derivatives on the right side of (2) are solved. This yieldsThis partial differential equation has two boundary values. At , the center of the lithium ion, . At , the radius of the lithium ion, 1 10^{−4 }cm, , where for the negative LiC_{6} electrodeIn (4), is the total electrode plate area, is the electrode length, and is the current flowing through the electrode [19]. Returning to the original boundary condition, is the solid-state diffusion coefficient. The parameter is the electrode volume fraction, 0.580 for the LiC_{6} electrode and 0.500 for the LiMnO_{2} electrode. The Faraday constant is , 96,487 C/mol [12].

#### 3. Laplace Transform

The Laplace transform converts the independent variable of an equation from time to frequency. All other independent variables are unaffected. The Laplace transform is a set of rules that are applied depending on whether the function is a polynomial, exponential derivative, and so forth [20]. For example, consider the boundary condition at , . The Laplace transform is where the capital letter indicates that the function has undergone the transform. The independent variable is now , frequency, rather than time.

Turning to (3), a time derivative is transformed by multiplication by frequency plus a term that includes initial conditions. This change converts to . The second term is zero since if the frequency is 0 Hz then there is no current pulse and no change in concentration. The transformation of the entire equation can be written as

The solution to this differential equation is provided in the appendix. The final result at is

#### 4. Transfer Function

The interest is in the surface concentration of the ions on the electrode so a transfer function from to is sought. A transfer function is the ratio of output to input for a system. The roots of the numerator of a transfer function are the zeros of the function. Zeros will block the frequency transmission. The roots of the numerator are the singularities. Frequency goes to infinity at these points.

The transfer function shows that the Laplace transform of the output is the product of the transfer function of the system and the transform of the input. In the transform domain the action of a linear system on the input is simply a multiplication with the transfer function. The transfer function is a natural generalization of the concept of gain—the increase in frequencies in the operating frequency range—of a system [21].

The model yields the transfer functionThis transfer function is shown in Figure 1 for the anode and cathode.