A Boundary Condition for Porous Electrodes

Electrochemical Solid-State Letters, 7 9 A59-A63 004 0013-4651/004/79/A59/5/$7.00 The Electrochemical Society, Inc. A Boundary Condition for Porous Electrodes Venkat R. Subramanian, a, *,z Deepak Tapriyal, a Ralph E. White b, ** a Department of Chemical Engineering, Tennessee Technological University, Cookeville, Tennessee 38501, USA b Department of Chemical Engineering, University of South Carolina, Columbia, South Carolina 908, USA A59 A boundary condition for electrolyte concentration at the porous electrode/separator interface is developed. This boundary condition helps predict the electrolyte concentration profile in the porous electrode without having to solve for the concentration profile in the separator. 004 The Electrochemical Society. DOI: 10.1149/1.1773751 All rights reserved. Manuscript submitted October 8, 003; revised manuscript received January 1, 004. Available electronically July 3, 004. Electrochemical models that predict the performance of batteries accurately are usually complex because of the nonlinear coupling of the dependent variables in the governing equations. 1, These models have been used by various researchers to optimize the cell design, study the effect of system parameters thermal behavior. Modeling of electrochemical behavior of secondary batteries like lithiumion batteries involves solving electrolyte concentration electrolyte potential in the separator; electrolyte concentration, electrolyte potential, solid-state concentration, solid-state potential in the porous electrode. 1, Even when one-dimensional transport in x is considered, these models involve two coupled nonlinear partial differential equations in x, t in the separator three coupled nonlinear partial differential equations in x, t in the porous electrode. 1, In addition, solid-state diffusion should be solved in the pseudodimension r, t in the porous electrode. For predicting the thermal behavior, one must add an additional equation for temperature in both the separator the porous electrode. Analytical solutions for the mathematical models of Li-ion batteries are available for very few limiting cases. 3,4 Doyle Newman 3 presented a few limiting cases analyzed the electrochemical behavior of Li-ion batteries using these simplified models under certain operating conditions. Complexity of the models hence the need for numerical solution arises due to one or more of the following reasons: i nonlinear coupling between the electrolyte potential electrolyte concentration concentrated solution theory; ii nonlinear Butler-Volmer kinetics, iii nonlinear dependence of exchange current on the electrolyte or solid-state concentration, (iv) nonlinear dependence of open-circuit potential on the solid-state concentration, (v) dependence of electrolyte conductivity or electrolyte diffusion coefficient on the electrolyte concentration, (vi) dependence of solid-state diffusion coefficient on the solidstate concentration, (vii) dependence of transfer number on the electrolyte concentration. Typically, Li-ion battery models are complicated because of a combination of more than one of the above reasons. A Li-ion cell swich consists of a lithium foil, separator, a porous electrode. 1, To obtain the concentration profiles in the porous electrode, one must solve for the concentration in the profiles in both the separator the porous electrode. Even when the potential drop in the porous electrode dominates the net voltage of the cell swich, one must solve for the equations in both separator porous electrode. This is true because the boundary condition for the electrolyte concentration at the porous electrode/separator interface is not known. In this article, we arrive at this boundary condition by modeling galvanostatic discharge behavior of a Li-ion cell swich under solution-phase diffusion limitations. * Electrochemical Society Active Member. ** Electrochemical Society Fellow. z E-mail: vsubramanian@tntech.edu Theoretical Development The geometry modeled is shown in Fig. 1. The cell consists of a lithium-ion foil, a microporous separator, a porous electrode e.g., carbon or LiMn O 4 ). The following assumptions are made: i discharge behavior is dominated by solution-phase diffusion limitations, 3 ii kinetic diffusion-phase limitations are negligible, iii diffusion coefficient transfer number are independent of the electrolyte concentration. In the absence of potential gradients under uniform current distribution, the electrolyte concentration is governed by simplified equations. 3,4 The geometry modeled consists of two regions, a separator (0 x L s ) a porous electrode (L s x L s L c ). The concentration of electrolyte in the separator (c 1, mol/m 3 is governed by Fick s law of diffusion c 1 t D c 1 x where D is the diffusion coefficient m /s of the electrolyte. A material balance governs the concentration of electrolyte (c, mol/m 3 in the porous electrode c t 1 D 1.5 c x aj n 1 t where is the porosity of the electrode, a is the specific interfacial area m 1, j n is the pore wall flux of lithium ions mol/m /s. Note that the Bruggeman expression has been used for obtaining the effective diffusivity in the electrolyte. 5 Uniform initial conditions are assumed throughout the swich c 1 c c 0 at t 0 3 For a galvanostatic discharge, the rate of discharge governs the mass flux at x 0 Li foil c 1 x I1 t at x 0 4 ndf where I is the current density A/m, n is the number of electrons transferred in the electrochemical reaction (n 1 here, F is the Faraday constant. The mass flux is zero at the current collector (x L s L c ) c x 0 at x L s L c 5 Concentration mass flux are continuous at the separator-porous electrode interface (x L s ) c 1 c at x L s 6

A60 Electrochemical Solid-State Letters, 7 9 A59-A63 004 Table I. Values of parameters used in the simulation. Parameters Value Parameters Value D.6 10 10 m /s 0.35 I app 60 A/m L s 5 m F 96,487 L c 15 m t 0.0 c 0 1000 mol/m 3 J I1 t L s FDL c c 0 13 The dimensionless initial condition is C 1 C 1 at 0 14 The dimensionless boundary conditions are Figure 1. Lithium-ion cell swich, consisting of lithium-foil, separator, porous electrode. c 1 x c 3/ x at x L s Equations 1-7 govern the concentration distributions in the cell swich separator porous electrode. When the open-circuit potential depends strongly on the state of charge of the system or when kinetic resistances dominate ohmic resistances, 3,4 it is possible to assume that j n is given by its average value everywhere in the porous electrode j n I 8 afl c When Eq. 8 is substituted in Eq. we get c t D 1.5 c x I1 t nfl c The following dimensionless variables are introduced C 1 c 1 c 0 C c c 0 X x L s Dt L s 7 9 C 1 X Jr at X 0 15 X 0 at X 1 r 16 C 1 C at X 1 17 C 1 X C 3/ X at X 1 18 Equations 11 1 can be solved analytically using the separation of variables method. 3,6 Typical values of parameters are used for the simulation are listed in Table I. 7 Approximate Model We can predict the concentration profile in the porous electrode by solving for the concentration equation in the porous electrode alone. To do this, we need the boundary condition at the porous electrode/separator interface. We arrive at this boundary condition in this section. The dimensionless concentration in the separator is assumed to be a parabolic polynomial in X 8 C 1 a bx ex 19 where a, b, e are functions of time are obtained from the governing equations boundary conditions. The governing equation is satisfied in an average sense. That is Eq. 19 is substituted into the equation for C 1 Eq. 11 integrated from 0 to 1 to obtain da dt db dt de 3dt e 0 The average concentration in the separator (C 1) is obtained by integrating Eq. 19 from 0 to 1 r L c L s 10 C 1 a b e 3 1 The governing equations boundary conditions are converted to dimensionless form using Eq. 10 as follows C 1 C 1 X for 0 X 1 11 C X J for 1 X r 1 where J is the dimensionless current density given by Using the boundary condition at X 0 Eq. 15 Eq. 19 we get Substituting Eq. 19 in 17 we get b rj a b e C at X 1 3 Using Eq. 0,, 3 we can calculate a, b, e. Equation 18 is simplified as

Electrochemical Solid-State Letters, 7 9 A59-A63 004 A61 1.5 X rj 3C 3C 1 4 Table II. Values of eigenvalues coefficients for rä5, Ä0.35. where C 1 is obtained by substituting a, b, e in Eq. 1 dc 1 d 3rJ 3C 3C 1 5 Because the boundary condition at the interface is known, the concentration profile in the porous electrode is obtained by solving the following governing equation C Z J for Z 0 to r 6 where Z X 1 the boundary conditions are 1.5 Z rj 3C 3C 1 at Z 0 7 Z 0 at Z r 8 In addition, C 1 is governed by Eq. 5. The initial conditions are C 1 C 1 at 0 9 Next, to simplify the governing equations further we introduce C 1 Ju C 1 1 Jū 1 to obtain the following equations u u Z 1 30 1.5 u Z r 3u 3ū 1 at Z 0 31 dū 1 d u Z 0 at Z 1 3 3r 3u 3ū 1 ū 1 u 0 at 0 33 34 Equation 30 is solved in the Laplace domain with the boundary condition at Z 0 Eq. 31 sr Z u s q cosh 1/4 1 s 35 where q 3/ sr 61 r s sinh 5/4 s1/ r 1/4 s 3 3s cosh s1/ r 1/4 36 Equation 35 is inverted to the time domain by using the Heaviside expansion theorem 9 as follows u w n1 A n cos nr Z 1/4 exp n 37 where w is the steady-state profile given by w r3 3r 3r 3/ 31 rr Z 61 r 38 the eigenvalues are calculated using the transcendental equation sin 1/4 nr 5/4 n 3 3 cos 1/4 nr n 0 for n 1,,... 39 The coefficient A n in Eq. 37 is obtained by using the method of residues as 6,9 A n P dq ds s n for n 1,,... 40 where P is the numerator Q is the denominator of q defined in Eq. 36. Numerical values of A n n for 0.35, r 5 are given in Table II. Note that both eigenvalues coefficients are independent of J, the dimensionless current density. Substituting the value of u in C 1 Ju we get C 1 J w n1 A n cos nr Z 1/4 exp n 41 The steady-state profile in the porous electrode may be obtained by neglecting the transient term in Eq. 41 as C, 1 Jw 1 J r3 3r 3r 3/ 31 r1 r X 61 r 4 Average Concentration Steady-State Concentration Profiles Unlike common transient diffusion models, 10 the exact model Eq. 11-18 cannot be solved to obtain the steady-state profile by equating the transient terms to zero. This is true because the solution of steady-state versions of Eq. 11 1 involves four constants of integration of which only three can be found using the boundary conditions defined in Eq. 15-18. 3,6 To find the fourth constant of integration, Eq. 11 1 are integrated as n Eq. 39 n A n Eq. 40 0.3414 0 13.8967 0.767 1 1.7773 1.5 0.5095 1.696 3 0.14.164 4 0.119.6375 5 0.07799 1 0 1r 1 C 1 dx C 1 X X1 C 1 X X0 dx X X1r X X1 Jr 43 44 If we denote the average concentration in the separator the electrode as C 1 C, respectively, Eq. 43 44 can be combined to obtain

A6 Electrochemical Solid-State Letters, 7 9 A59-A63 004 Figure. Dimensionless concentration at the current collector is plotted against dimensionless time for different rates of discharge. The solid lines represent the exact model Eq. 11-18 the dotted lines represent the approximate solution developed Eq. 41. 1C rate corresponds to 60 A/m. Figure 3. Dimensionless concentration at the electrode-separator interface is plotted against dimensionless time for different rates of discharge. The solid lines represent the exact model Eq. 11-18 the dotted lines represent the approximate solution developed Eq. 41. 1C rate corresponds to 60 A/m. dc 1 d dc r d 0 45 Equation 45 can be integrated using the initial condition Eq. 14 to get C 1 rc 1 r 46 Steady-state versions of Eq. 11 1 can be solved with Eq. 15, 17, 18, 46 to obtain the steady-state concentration profiles as 1 C 1, 1 Jr 1 X 1 r r 31 r 47 C, 1 J r3 3r 3r 3/ 31 r1 r X 61 r 48 We observe that the steady-state concentration profile in the porous electrode obtained using the exact model Eq. 48 matches exactly with the steady-state profile obtained with the approximate model Eq. 4. Results Discussion The profiles obtained from the simplified model are compared with the exact model Eq. 11-18 in Fig. 3. The dimensionless concentration at the current collector is plotted as a function of dimensionless time for 30, 60, 10 A/m rates using both the approximate model developed the exact model Eq. 11-18 in Fig.. We observe that the approximate model predicts the concentration profiles accurately. We observe that the dimensionless concentration decreases with time for a given rate of discharge, approaches a steady state. In addition, we observe that the dimensionless concentration decreases faster for higher rates of discharge. The dimensionless concentration at the porous electrode/ separator interface is plotted as a function of dimensionless time for 30, 60, 10 A/m in Fig. 3. We observe that the approximate model predicts the concentration profiles accurately. We observe that the dimensionless concentration at the electrode/separator interface increases with time for a given rate of discharge, approaches a steady state. In addition, we observe that the dimensionless concentration increases faster for higher rates of discharge. The dimensionless concentration at the porous electrode/ separator interface is plotted as a function of dimensionless time for different values of r ratio of electrode length to the separator length in Fig. 4. We observe that for higher values of r, the dimensionless concentration depletes faster. From Fig., 3, 4 we conclude that the approximation is good may be used with no loss of accuracy. The methodology boundary condition developed for concentration at the separator/porous electrode interface in this article can be extended to more realistic models. This is true because even for complicated models the governing equation for concentration in the separator is very similar to Eq. 1. 1, Hence, the boundary condition developed for electrolyte concentration at the porous electrode/ separator interface Eq. 31 may be used for complicated models also. In addition, the boundary conditions for the electrolyte potential solid-phase potential at the porous electrode/separator interface are known accurately because the entire current is carried by the electrolyte phase at the interface. 1, In our next paper, we plan to predict the discharge behavior of the cell swich by solving for governing equations in the porous electrode alone without solving for the profiles in the separator. The boundary condition developed in this paper should also find use in theoretical analysis of porous Figure 4. Dimensionless concentration at the current collector plotted against dimensionless time for different values of r ratio of electrode length to separator length at the 1C rate of discharge 60 A/m. The solid lines represent the exact model Eq. 11-18 the dotted lines represent the approximate solution developed Eq. 4.

Electrochemical Solid-State Letters, 7 9 A59-A63 004 A63 electrodes impedance simulation of porous electrodes. We plan to publish this later. Acknowledgments The authors are grateful for the financial support of the project by National Reconnaissance Organization NRO under contract NRO 000-03-C-01. The authors also acknowledge the Center for Electric Power, Tennessee Tech University, for providing a research assistantship to D.T. The University of South Carolina assisted in meeting the publication costs of this article. List of Symbols a interfacial area, m 1 c concentration of the electrolyte, mol/m 3 t time, s x distance, m D diffusion coefficient of the electrolyte, m /s j n pore wall flux of Li ions, mol/m /s t transfer number I current density, A/m L c, L s length of the electrode, m, length of the separator, m F Faraday s law constant, 96,487 C/g equiv C,u dimensionless concentration, c/c 0 X,Z dimensionless distance, X x/l, Z X 1 J dimensionless current density see Eq. 13 r ratio of thickness of the electrode to the thickness of the separator, L c /L s n C 1, ū 1 Greek number of electrons transferred (n 1 for the simulation dimensionless average concentration see Eq. 1 porosity of the electrode n eigenvalues dimensionless time, Dt/L c ubscripts 0 initial condition 1 separator porous electrodes References 1. M. Doyle, T. F. Fuller, J. Newman, J. Electrochem. Soc., 140, 156 1993.. G. G. Botte, V. R. Subramanian, R. E. White, Electrochim. Acta, 45, 595 000. 3. M. Doyle J. Newman, J. Appl. Electrochem., 7, 846 1997. 4. J. S. Newman, Electrochemical Systems, pp. 454-495, Prentice Hall, Englewood Cliffs, NJ 1991. 5. D. A. G. Bruggeman, Appl. Phys., 4, 636 1935. 6. V. R. Subramanian R. E. White, J. Power Sources, 96, 385 001. 7. G. G. Botte R. E. White, J. Electrochem. Soc., 148, A54 001. 8. V. R. Subramanian, J. A. Ritter, R. E. White, J. Electrochem. Soc., 148, E444 001. 9. A. Varma M. Morbidelli, Mathematical Methods in Chemical Engineering, pp. 568-574, Oxford University Press, New York 1997. 10. J. Crank, Mathematics of Diffusion, pp. 44-47, Oxford University Press, New York 1975.