Article pubs.acs.org/jchemeduc
Modeling the Lithium Ion Battery John Summerfield* Department of Chemical and Physical Sciences, Missouri Southern State University, Joplin, Missouri 64801, United States S Supporting Information *
ABSTRACT: The lithium ion battery will be a reliable electrical resource for many years to come. A simple model of the lithium ions motion due to changes in concentration and voltage is presented. The battery chosen has LiCoO2 as the cathode, LiPF6 as the electrolyte, and LiC6 as the anode. The concentration gradient and voltage gradient is linearized in two forms of Fick’s first law of diffusion. The voltage gradient increases the motion of the lithium ions by a factor of 100 compared to the concentration gradient. Possible extensions of this simple model are also addressed. KEYWORDS: Second-Year Undergraduate, Upper-Division Undergraduate, Analytical Chemistry, Physical Chemistry, Computer-Based Learning, Electrochemistry, Mathematics/Symbolic Mathematics
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charge carrying semiconductor and replacing the cobalt in the cathode with less toxic sulfur. Both changes are made ultimately to increase electric current.9−12 In this work, two simpler equations are used to model the lithium ions in the battery’s electrolyte. Both equations are a linear form of Fick’s first law of diffusion.13 Diffusion of the Li+ ions depends on the change in concentration between the two electrodes. The migration of the Li+ ions depends on the change in the voltage between the electrodes. This work focuses on the electrolyte because modeling the surface of the electrodes is beyond the scope of an undergraduate class.14 The physical parameters of a common lithium ion battery are scattered throughout the literature, but are collected and presented in this article. Hopefully by having the specific numbers all in one place, it will be easier for an instructor to present this timely topic of lithium ion battery processes.
relatively simple mathematical model is presented of the Li+ ions in the electrolyte of a typical lithium ion battery. Lithium ion battery electrolyte conductivity has been explored in this Journal1 and software to help students understand the chemistry of batteries is available as well.2 The Li+ battery electrochemical reactions have also been addressed in this Journal.3 The simple mathematical model presented here is appropriate for analytical chemistry,4 instrumental analysis,5 or physical chemistry classes.6 In analytical chemistry, it could be presented as a rationale for voltammetry. In instrumental analysis, it could be presented as part of the introduction to the measurement of the conductivity of electrolyte solutions. In physical chemistry, it could be presented as a transport property example or as an electrochemistry example. In all courses, the model is presented as an in-class lecture. It could also be extended into an independent study project; this topic will be discussed toward the end of the article. Transport properties have been known for over a hundred years yet they continue to be relevant.6 Specifically, batteries rely on the movement of ions due to the difference in voltage from electrode to electrode. This motion is termed ion migration. Batteries are designed to maximize migration. The ions also move from spot to spot in the battery due to differences in their concentrations. This motion is characterized as ion diffusion. Battery designers work to minimize diffusion.7 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. Presently, 49 differential equations are used to simulate the charging and discharging of a typical lithium ion battery that consists of a LiCoO2 cathode, a LiC6 anode, separated by the electrolyte, LiPF6.8 There is great interest in moving away from a graphite anode to a better © XXXX American Chemical Society and Division of Chemical Education, Inc.
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LI-ION BATTERY MODEL
Diffusion
The ions in the electrolyte diffuse because there are small changes in the electrolyte concentration. Linear diffusion is only considered here. The change in concentration, c, as a function of time, t, and distance, x, is, D ∂c ∂c =− ∂t ε ∂x
(1)
The negative sign indicates the ions are flowing from high concentration to low concentration. In this equation, D is the diffusion coefficient for the lithium ion. It has a value of 7.5 × 10−10 m/s in the LiPF6 electrolyte. The value for ε, the porosity of the electrolyte, is 0.724.15 Equation 1 is derived in Appendix I of the Supporting Information.
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dx.doi.org/10.1021/ed300533f | J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Article
A lithium ion battery has a lifetime of 2,000 charge/discharge cycles16 and for at least the first 800 cycles, that is, before electrode degradation and solubility effects begin to take their toll, computer models of lithium ion battery aging processes suggest that the concentration changes linearly throughout the electrolyte.16 Relying on this approximation, eq 1 becomes ∂c /∂x=(c 2 − c1)/S
Diffusion rate = −4.0 × 10−5
m·L Δc s·mol
(2)
where S is the length between the electrodes, 2.5 × 10−5 m.15 Substituting eq 2 into eq 1
∂c D (c 2 − c1) =− ∂t ε S
(3)
The original partial differential equation has been simplified to a linear equation. The concentration at the LiC6 anode is c1; the lithium ions move into the electrolyte and toward the cathode during discharge. The concentration at the LiCoO2 cathode is c2 during discharge. When the battery is recharged, the ion motion is reversed. Using eq 3, the concentration change can be graphed as a straight line from zero up to its maximum, 1.0 M.15 Phenomenologically, [Li+] = 0.0 M corresponds to a fully charged battery. At [Li+] = 1.0 M, the battery is fully discharged. These boundary values are relied on for direct in situ measurements and computational models.17
Figure 1. Rate of diffusion of Li+ ions as a function of change in concentration.
The y intercept is zero for this process. Discharge of the battery begins and a concentration gradient forms. The Li+ ion concentration is 1.0 M in the electrolyte so the diffusion rate reaches its maximum at that value. The negative sign indicates the ions are moving from high concentration to low concentration. The magnitude of the diffusion rate increases as the magnitude of the change in concentration increases. Migration rate is now examined. As before, to begin, the lithium ion battery is fully charged and the migration rate is zero as shown in Figure 2. The equation for the line is m·L Migration rate = −6.0 × 10−3 Δc s·mol
Migration
A change in the voltage also influences the movement of Li+ ions in the electrolyte. Fick’s first law takes the form zFc D ∂φ ∂c1 =− 0 ∂t RTε ∂x
(4)
The negative sign indicates the ions are flowing from high voltage to low voltage, just as a boulder rolls down a hill. In eq 4, the concentration at the LiC6 anode is c1, and z is the charge of the ion, +1, for Li+. Faraday’s constant is denoted by the F. Its value is 9.64853 × 104 C/mol. The initial electrolyte concentration is denoted by c0. The diffusion coefficient, D, continues to be 7.5 × 10−10 m/s. The ideal gas constant is R, with a value 8.3145 J/(mol K). The temperature, T, is chosen to be 298 K. The value for ε, the porosity of the electrolyte, continues to be 0.724.15 The change in voltage as a function of position is ∂φ/∂x. The nominal voltage of this cell is 3.7 V.18 Equation 4 is derived in Appendix II of the Supporting Information. The voltage will depend on the device in which it is installed and the running temperature. If the voltage, φ, is considered constant, which is in error of 0.1 V for the first 800 charge− discharge cycles,19 then eq 4 can be expressed as
zFφD ∂c ∂c =− ∂t RTε ∂x
Figure 2. Rate of migration of Li+ ions as a function of change in concentration.
The y intercept is zero for this process as well. Discharge of the battery begins and a voltage gradient forms. The Li+ ion concentration becomes 1.0 M in the electrolyte so the migration rate reaches its maximum at that value. The negative sign indicates the ions are moving from high voltage energy to low voltage energy. The magnitude of the diffusion rate increases as the magnitude of the change in concentration increases.
(5)
Linearization of this equation follows the same scheme as that used for diffusion. Using eq 2, the concentration change due to voltage is
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zFφD (c 2 − c1) ∂c =− ∂t RTε S
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DISCUSSION Intuitively, students believe the migration rate is greater than the diffusion rate for the Li+ ions in a common lithium ion battery electrolyte. However, because the specific battery data is not easily available, the quantitative difference is unknown. Figures 1 and 2 boldly show that to the first approximationan ideal electrolyte solutionmigration is on the order of 100 times faster. The graphs also illustrate that a complex
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RESULTS OF THE MODELING Diffusion rate is examined first. To begin, the lithium ion battery is fully charged. The initial diffusion rate is zero as shown in Figure 1. The equation for the line is B
dx.doi.org/10.1021/ed300533f | J. Chem. Educ. XXXX, XXX, XXX−XXX
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mathematical expressiona partial differential equationcan be simplified to a much easier to understand math expression a straight lineif one of the differentials describes a small physical change. In this case, the distance traveled by the ions is quite small.
ASSOCIATED CONTENT
* Supporting Information S
Derivation of eq 1, derivation of eq 4, answers to In the Classroom questions. This material is available via the Internet at http://pubs.acs.org.
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POSSIBLE EXTENSIONS There are a number of possible extensions to this work. The electrolyte is not an ideal solution. One way to incorporate concentrated solution theory into the simple model is to add a transference number term to eq 1.7 The starting equation would be D ∂c ∂c + (1 − t+) =− (7) ∂t ε ∂x + where t+ is 0.364 for Li in this battery. More complex transference terms are also possible.15 The original graphs could be created for the anion in the electrolyte, PF6−. The only new number is the diffusion coefficient, 3.0 × 10−11 m/s.20 However, with a bit of thought, the sign for the diffusion rate will need to be positive rather than negative since the anions are moving in the opposite direction of the cations due to the voltage. Doing this calculation would elucidate the inner workings of the electrolyte. The calculations could be carried out for a nickel cadmium battery. The needed data is available.21,22 This exercise would be instructive because the separator is typically paper or a polymer rather than a simple salt solution. Also, the electrolyte is a much more concentrated solution, typically about 7 mol/L. Another extension would be to use Fick’s second law of diffusion in place of the first law. Inside a real battery the concentration is changing with time as a typical Li+ battery has a 2,000 charge−discharge cycle life. Mathematically, a secondorder differential equation is used to describe this concentration change. That is, eq 1 becomes ∂c D ∂ 2c =− ∂t ε ∂x 2
Article
AUTHOR INFORMATION
Corresponding Author
*Email: summerfi
[email protected]. Notes
The author declares no competing financial interest.
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REFERENCES
(1) Compton, C. O.; Egan, M.; Kanakaraj, R.; Higgins, T. B.; Nguyen, S. T. J. Chem. Educ. 2012, 89, 1442. (2) Yang, E-M; Greenbowe, T. J.; Andre, T. J. Chem. Educ. 2004, 10, 587. (3) Treptow, R. S. J. Chem. Educ. 2003, 80, 1015. (4) Nicholson, R. S.; Shain, I. Anal. Chem. 1964, 36, 706. (5) Prasad, M. A.; Sangaranarayanan, M. V. Electrochim. Acta 2004, 49, 445. (6) Levine, I. N. Physical Chemistry, 6th ed.; McGraw-Hill: Boston, MA, 2009; pp 426−431, 493−509. (7) Newman, J., Thomas-Alyea, K. E. Electrochemical Methods, 3rd ed.; Wiley Interscience: New York, 2004; pp 8−18. (8) Subramanian, V. R.; Boovaragavan, V.; Diwakar, V. D. Electrochem. Solid-State Lett. 2007, 10, A255. (9) Doyle, M.; Fuller, T. F.; Newman, J. J. Electrohem. Soc. 1993, 140, 1526. (10) Ulldemolins, M.; Le Cras, F.; Pecqenard, B.; Phan, V. P.; Martin, L.; Martinez, H. J. Power Sources 2012, 206, 245. (11) Tan, L. P.; Lu, Z.; Tan, H. T.; Zhu, Rui, X. J.; Yan, Q.; Hng, H. H. J. Power Sources 2012, 206, 253. (12) Barchasz, C.; Mesguich, F.; Dijon, J.; Leprêtre, J.-C.; Patoux, S.; Alloin, F. J. Power Sources 2012, 211, 19. (13) Fick, A. Philos. Mag. 1855, 10, 30. (14) Ramadass, P.; Haran, B.; Gomadam, P.M., R.; White, Popov, B. N. J. Electrohem. Soc. 2004, 151, A196. (15) Northrop, P. W. C.; Ramadesigan, V.; De, S.; Subramanian, V. R. J. Electrohem. Soc. 2011, 158, A1461. (16) Zhang, Q.; Guo, Q.; Lui, S.; Dougal, R. A.; White, R. E. J. Power Sources 2005, 141, 359. (17) Harris, S. J.; Timmons, A.; Baker, D,R.; Monroe, C. Chem. Phys. Lett. 2010, 485, 265. (18) Lithium Ion Batteries, Sony, Inc., 2012; retrieved July 24 , 2012, http://www.sony.com.cn/products/ed/battery/download.pdf (accessed March 2013). (19) Ning, G.; Popov, B. J. Electrohem. Soc. 2004, 151, A1584. (20) Danilov, D.; Notten, P. H. L. Electrochim. Acta 2008, 53, 5569. (21) Pan, H.; Chen, Y.; Wang, C.; Wang, X.; Chen, C.; Wang, Q. J. Alloys Compd. 1999, 293−295, 680. (22) Ni-Cd Separators: Celgard, Inc., 2010. http://www.celgard.cn. com/pdf/library/Celgard_Overview_Brochure_10003.pdf (accessed March 2013). (23) Abramowitz, M., Stegun, I. A., Eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Dover:: New York, 1965.
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As an independent study project, students could utilize Mathematica or MATLAB to solve this differential equation. The solution is usually presented as a special function, the error function.23
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IN THE CLASSROOM After the lecture presenting the model, the following questions can be used in the classroom for discussion, in-class explorations, or written assignments. 1. Why is Li+ utilized in batteries? Why not K+, potassium is higher on the activity series? 2. Which products rely on lithium ion batteries? 3. Why is the rechargeable Li+ battery better than any other such as nickel cadmium or nickel metal hydride? 4. What occurs in the lithium battery when it charges and discharges? 5. What are important physical properties for the electrolyte? 6. Considering the traits of a good electrolyte, what are the two methods the lithium ions rely on to move from electrode to electrode? 7. How would migration and diffusion be described mathematically? The answers are included in the Supporting Information. C
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