The Effect of Phase Separation on Diffusion Induced Stresses in Spherical and Cylindrical Electrode Particles
Authors
Abstract
Experiments have frequently shown that phase separation in lithium-battery electrodes could lead to mechanical failure, poor cycling performance, and reduced capacity. Here, a phase-field model is utilized to investigate how phase separation affects the evolution of the concentration and stress profiles within the spherical/cylindrical electrode particles, during both insertion and extraction half-cycles. To this end, the governing equations are derived and then discretized using the central finite difference method. The resulting algebraic equations are solved numerically with the aid of the Newton-Raphson method to determine both the concentration and stress fields in the electrode particles. For further verification, the results are compared against predictions of an analytical core-shell model. The results suggest that, within the range of parameters considered here, phase separation could lead to a more than five-fold increase in the maximum tensile stress at the particles surface.
Keywords
1. Tarascon, J. -M., and Armand, M., “Issues and Challenges Facing Rechargeable Lithium Batteries”, Nature, Vol. 414, No. 6861, pp. 359-367, 2001.
2. Kasavajjula, U., Wang, C., and Appleby, A. J., “Nano-and Bulk-silicon-based Insertion Anodes for Lithium-ion Secondary Cells”, Journal of Power Sources, Vol. 163, No. 2, pp. 1003-1039, 2007.
3. Winter, M., Besenhard, J. O., Spahr, M. E., and Novak, P., “Insertion Electrode Materials for Rechargeable Lithium Batteries”, Advanced Materials, Vol. 10, pp. 725-763, 1998.
4. Beaulieu, L., Eberman, K., Turner, R., Krause, L., and Dahn, J., “Colossal Reversible Volume Changes in Lithium Alloys”, Electrochemical and Solid-State Letters, Vol. 4, pp. A137-A140, 2001.
5. Go, J. -Y., and Pyun, S. -I., “Investigation of Stresses Generated During Lithium Transport Through the RF Sputter-Deposited Li1- δ CoO2 Film by a DQCR Technique”, Journal of the Electrochemical Society, Vol. 150, No. 8, pp. A1037-A1043, 2003.
6. Wang, D., Wu, X., Wang, Z., and Chen, L., “Cracking Causing Cyclic Instability of LiFePO4 Cathode Material”, Journal of Power Sources, Vol. 140, No. 1, pp. 125-128, 2005.
7. Arora, P., White, R. E., and Doyle, M., “Capacity Fade Mechanisms and Side Reactions in Lithium‐ion Batteries”, Journal of the Electrochemical Society, Vol. 145, No. 10, pp. 3647-3667, 1998.
8. Cheng, Y. -T., and Verbrugge, M. W., “Diffusion-induced Stress, Interfacial Charge Transfer, and Criteria for Avoiding Crack Initiation of Electrode Particles”, Journal of the Electrochemical Society, Vol. 157, No. 4, pp. A508-A516, 2010.
9. Zhao, K., Pharr, M., Vlassak, J. J., and Suo, Z., “Fracture of Electrodes in Lithium-ion Batteries Caused by Fast Charging”, Journal of Applied Physics, Vol. 108, No. 7, p. 073517, 2010.
10. Gabrisch, H., Wilcox, J., and Doeff, M., “TEM Study of Fracturing in Spherical and Plate-like LiFePO4 Particles”, Electrochemical and Solid-State Letters, Vol. 11, No. 3, pp. A25-A29, 2008.
11. Xia, Y., and Yoshio, M., “An Investigation of Lithium Ion Insertion into Spinel Structure Li‐Mn‐O Compounds”, Journal of the Electrochemical Society, Vol. 143, No. 3, pp. 825-833, 1996.
12. Sun, C. -F., Karki, K., Jia, Z., Liao, H., Zhang, Y., Li, T., Qi, Y., Cumings, J., Rubloff, G. W., and Wang, Y., “A Beaded-string Silicon Anode”, Acs Nano, Vol. 7, No. 3, pp. 2717-2724, 2013.
13. Lin, Y., Yang, Y., Ma, H., Cui, Y., and Mao, W. L., “Compressional Behavior of Bulk and Nanorod LiMn2O4 under Nonhydrostatic Stress”, The Journal of Physical Chemistry C, Vol. 115, No. 20, pp. 9844-9849, 2011.
14. Laffont, L., Delacourt, C., Gibot, P., Yue Wu, M., Kooyman, P., Masquelier, C. and Marie Tarascon, J., “Study of the LiFePO4/FePO4 Two-phase System by High-resolution Electron Energy Loss Spectroscopy”, Chemistry of Materials, Vol. 18, No. 23, pp. 5520-5529, 2006.
15. Weichert, K., Sigle, W., van Aken, P. A., Jamnik, J., Zhu, C., Amin, R., Acartürk, T., Starke, U., and Maier, J., “Phase Boundary Propagation in Large LiFePO4 Single Crystals on Delithiation”, Journal of the American Chemical Society, Vol. 134, No. 6, pp. 2988-2992, 2012.
16. Bai, P., Cogswell, D. A., and Bazant, M. Z., “Suppression of Phase Separation in LiFePO4 Nanoparticles During Battery Discharge”, Nano Letters, Vol. 11, No. 11, pp. 4890-4896, 2011.
17. Tang, M., Huang, H. -Y., Meethong, N., Kao, Y. -H., Carter, W. C., and Chiang, Y. -M., “Model for the Particle Size, Overpotential, and Strain Dependence of Phase Transition Pathways in Storage Electrodes: Application to Nanoscale Olivines”, Chemistry of Materials, Vol. 21, No. 8, pp. 1557-1571, 2009.
18. Golmon, S., Maute, K., Lee, S. -H., and Dunn, M. L., “Stress Generation in Silicon Particles During Lithium Insertion”, Applied Physics Letters, Vol. 97, No. 3, p. 033111, 2010.
19. Haftbaradaran, H., Song, J., Curtin, W., and Gao, H., “Continuum and Atomistic Models of Strongly Coupled Diffusion, Stress, and Solute Concentration”, Journal of Power Sources, Vol. 196, No. 1, pp. 361-370, 2011.
20. Hao, F., and Fang, D., “Reducing Diffusion-induced Stresses of Electrode-Collector Bilayer in Lithium-ion Battery by Pre-strain”, Journal of Power Sources, Vol. 242, pp. 415-420, 2013.
21. Cheng, Y. -T., and Verbrugge, M. W., “Evolution of Stress Within a Spherical Insertion Electrode Particle under Potentiostatic and Galvanostatic Operation”, Journal of Power Sources, Vol. 190, No. 2, pp. 453-460, 2009.
22. Cheng, Y. -T., and Verbrugge, M. W., “The Influence of Surface Mechanics on Diffusion Induced Stresses Within Spherical Nanoparticles”, Journal of Applied Physics, Vol. 104, No. 8, p. 083521, 2008.
23. Deshpande, R., Cheng, Y. -T., and Verbrugge, M. W., “Modeling Diffusion-induced Stress in Nanowire Electrode Structures”, Journal of Power Sources, Vol. 195, No. 15, pp. 5081-5088, 2010.
24. Zhang, X., Shyy, W., and Sastry, A. M., “Numerical Simulation of Intercalation-induced Stress in Li-ion Battery Electrode Particles”, Journal of the Electrochemical Society, Vol. 154, No. 10, pp. A910-A916, 2007.
25. Deshpande, R., Cheng, Y.-T., Verbrugge, M. W., and Timmons, A., “Diffusion Induced Stresses and Strain Energy in a Phase-transforming Spherical Electrode Particle”, Journal of The Electrochemical Society, Vol. 158, No. 6, pp. A718-A724, 2011.
26. Park, J., Lu, W., and Sastry, A. M., “Numerical Simulation of Stress Evolution in Lithium Manganese Dioxide Particles Due to Coupled Phase Transition and Intercalation”, Journal of the Electrochemical Society, Vol. 158, No. 2, pp. A201-A206, 2011.
27. Han, B., Van der Ven, A., Morgan, D., and Ceder, G., “Electrochemical Modeling of Intercalation Processes with Phase Field Models”, Electrochimica Acta, Vol. 49, No. 26, pp. 4691-4699, 2004.
28. Cahn, J. W., and Hilliard, J. E., “Free Energy of a Nonuniform System. I. Interfacial Free Energy”, The Journal of Chemical Physics, Vol. 28, No. 2, pp. 258-267, 1958.
29. Huttin, M., and Kamlah, M., “Phase-field Modeling of Stress Generation in Electrode Particles of Lithium Ion Batteries”, Applied Physics Letters, Vol. 101, No. 13, p. 133902, 2012.
30. Song, Y., Li, Z., Soh, A., and Zhang, J., “Diffusion of Lithium Ions and Diffusion-induced Stresses in a Phase Separating Electrode under Galvanostatic and Potentiostatic Operations: Phase Field Simulations”, Mechanics of Materials, Vol. 91, pp. 363-371, 2015.
31. Haftbaradaran, H., Maddahian, A., and Mossaiby, F., “A Fracture Mechanics Study of the Phase Separating Planar Electrodes: Phase Field Modeling and Analytical Results”, Journal of Power Sources, Vol. 350, pp. 127-139, 2017.
32. Crank, J., The Mathematics of Diffusion, Oxford University Press, 1979.
33. Levi, M., and Aurbach, D., “Frumkin Intercalation Isotherm-a Tool for the Description of Lithium Insertion Into Host Materials: A Review”, Electrochimica Acta, Vol. 45, No. 1, pp. 167-185, 1999.
34. Burch, D., and Bazant, M. Z., “Size-dependent Spinodal and Miscibility Gaps for Intercalation in Nanoparticles”, Nano Letters, Vol. 9, No. 11, pp. 3795-3800, 2009.
35. Burch, D., “Intercalation Dynamics in Lithium-ion Batteries”, Ph.D. Desertation Massachusetts Inst of Tech Cambridge Dept of Mathematics, 2009.
36. Doyle, M., Fuller, T. F., and Newman, J., “Modeling of Galvanostatic Charge and Discharge of the Lithium/polymer/insertion Cell”, Journal of the Electrochemical Society, Vol. 140, No. 6, pp. 1526-1533, 1993.
37. Bazant, M. Z., “Theory of Chemical Kinetics and Charge Transfer Based on Nonequilibrium Thermodynamics”, Accounts of Chemical Research, Vol. 46, No. 5, pp. 1144-1160, 2013.
38. Timoshenko, S., and Goodier, J., Theory of Elasticity, McGraw-Hill book Company, 1951.
2. Kasavajjula, U., Wang, C., and Appleby, A. J., “Nano-and Bulk-silicon-based Insertion Anodes for Lithium-ion Secondary Cells”, Journal of Power Sources, Vol. 163, No. 2, pp. 1003-1039, 2007.
3. Winter, M., Besenhard, J. O., Spahr, M. E., and Novak, P., “Insertion Electrode Materials for Rechargeable Lithium Batteries”, Advanced Materials, Vol. 10, pp. 725-763, 1998.
4. Beaulieu, L., Eberman, K., Turner, R., Krause, L., and Dahn, J., “Colossal Reversible Volume Changes in Lithium Alloys”, Electrochemical and Solid-State Letters, Vol. 4, pp. A137-A140, 2001.
5. Go, J. -Y., and Pyun, S. -I., “Investigation of Stresses Generated During Lithium Transport Through the RF Sputter-Deposited Li1- δ CoO2 Film by a DQCR Technique”, Journal of the Electrochemical Society, Vol. 150, No. 8, pp. A1037-A1043, 2003.
6. Wang, D., Wu, X., Wang, Z., and Chen, L., “Cracking Causing Cyclic Instability of LiFePO4 Cathode Material”, Journal of Power Sources, Vol. 140, No. 1, pp. 125-128, 2005.
7. Arora, P., White, R. E., and Doyle, M., “Capacity Fade Mechanisms and Side Reactions in Lithium‐ion Batteries”, Journal of the Electrochemical Society, Vol. 145, No. 10, pp. 3647-3667, 1998.
8. Cheng, Y. -T., and Verbrugge, M. W., “Diffusion-induced Stress, Interfacial Charge Transfer, and Criteria for Avoiding Crack Initiation of Electrode Particles”, Journal of the Electrochemical Society, Vol. 157, No. 4, pp. A508-A516, 2010.
9. Zhao, K., Pharr, M., Vlassak, J. J., and Suo, Z., “Fracture of Electrodes in Lithium-ion Batteries Caused by Fast Charging”, Journal of Applied Physics, Vol. 108, No. 7, p. 073517, 2010.
10. Gabrisch, H., Wilcox, J., and Doeff, M., “TEM Study of Fracturing in Spherical and Plate-like LiFePO4 Particles”, Electrochemical and Solid-State Letters, Vol. 11, No. 3, pp. A25-A29, 2008.
11. Xia, Y., and Yoshio, M., “An Investigation of Lithium Ion Insertion into Spinel Structure Li‐Mn‐O Compounds”, Journal of the Electrochemical Society, Vol. 143, No. 3, pp. 825-833, 1996.
12. Sun, C. -F., Karki, K., Jia, Z., Liao, H., Zhang, Y., Li, T., Qi, Y., Cumings, J., Rubloff, G. W., and Wang, Y., “A Beaded-string Silicon Anode”, Acs Nano, Vol. 7, No. 3, pp. 2717-2724, 2013.
13. Lin, Y., Yang, Y., Ma, H., Cui, Y., and Mao, W. L., “Compressional Behavior of Bulk and Nanorod LiMn2O4 under Nonhydrostatic Stress”, The Journal of Physical Chemistry C, Vol. 115, No. 20, pp. 9844-9849, 2011.
14. Laffont, L., Delacourt, C., Gibot, P., Yue Wu, M., Kooyman, P., Masquelier, C. and Marie Tarascon, J., “Study of the LiFePO4/FePO4 Two-phase System by High-resolution Electron Energy Loss Spectroscopy”, Chemistry of Materials, Vol. 18, No. 23, pp. 5520-5529, 2006.
15. Weichert, K., Sigle, W., van Aken, P. A., Jamnik, J., Zhu, C., Amin, R., Acartürk, T., Starke, U., and Maier, J., “Phase Boundary Propagation in Large LiFePO4 Single Crystals on Delithiation”, Journal of the American Chemical Society, Vol. 134, No. 6, pp. 2988-2992, 2012.
16. Bai, P., Cogswell, D. A., and Bazant, M. Z., “Suppression of Phase Separation in LiFePO4 Nanoparticles During Battery Discharge”, Nano Letters, Vol. 11, No. 11, pp. 4890-4896, 2011.
17. Tang, M., Huang, H. -Y., Meethong, N., Kao, Y. -H., Carter, W. C., and Chiang, Y. -M., “Model for the Particle Size, Overpotential, and Strain Dependence of Phase Transition Pathways in Storage Electrodes: Application to Nanoscale Olivines”, Chemistry of Materials, Vol. 21, No. 8, pp. 1557-1571, 2009.
18. Golmon, S., Maute, K., Lee, S. -H., and Dunn, M. L., “Stress Generation in Silicon Particles During Lithium Insertion”, Applied Physics Letters, Vol. 97, No. 3, p. 033111, 2010.
19. Haftbaradaran, H., Song, J., Curtin, W., and Gao, H., “Continuum and Atomistic Models of Strongly Coupled Diffusion, Stress, and Solute Concentration”, Journal of Power Sources, Vol. 196, No. 1, pp. 361-370, 2011.
20. Hao, F., and Fang, D., “Reducing Diffusion-induced Stresses of Electrode-Collector Bilayer in Lithium-ion Battery by Pre-strain”, Journal of Power Sources, Vol. 242, pp. 415-420, 2013.
21. Cheng, Y. -T., and Verbrugge, M. W., “Evolution of Stress Within a Spherical Insertion Electrode Particle under Potentiostatic and Galvanostatic Operation”, Journal of Power Sources, Vol. 190, No. 2, pp. 453-460, 2009.
22. Cheng, Y. -T., and Verbrugge, M. W., “The Influence of Surface Mechanics on Diffusion Induced Stresses Within Spherical Nanoparticles”, Journal of Applied Physics, Vol. 104, No. 8, p. 083521, 2008.
23. Deshpande, R., Cheng, Y. -T., and Verbrugge, M. W., “Modeling Diffusion-induced Stress in Nanowire Electrode Structures”, Journal of Power Sources, Vol. 195, No. 15, pp. 5081-5088, 2010.
24. Zhang, X., Shyy, W., and Sastry, A. M., “Numerical Simulation of Intercalation-induced Stress in Li-ion Battery Electrode Particles”, Journal of the Electrochemical Society, Vol. 154, No. 10, pp. A910-A916, 2007.
25. Deshpande, R., Cheng, Y.-T., Verbrugge, M. W., and Timmons, A., “Diffusion Induced Stresses and Strain Energy in a Phase-transforming Spherical Electrode Particle”, Journal of The Electrochemical Society, Vol. 158, No. 6, pp. A718-A724, 2011.
26. Park, J., Lu, W., and Sastry, A. M., “Numerical Simulation of Stress Evolution in Lithium Manganese Dioxide Particles Due to Coupled Phase Transition and Intercalation”, Journal of the Electrochemical Society, Vol. 158, No. 2, pp. A201-A206, 2011.
27. Han, B., Van der Ven, A., Morgan, D., and Ceder, G., “Electrochemical Modeling of Intercalation Processes with Phase Field Models”, Electrochimica Acta, Vol. 49, No. 26, pp. 4691-4699, 2004.
28. Cahn, J. W., and Hilliard, J. E., “Free Energy of a Nonuniform System. I. Interfacial Free Energy”, The Journal of Chemical Physics, Vol. 28, No. 2, pp. 258-267, 1958.
29. Huttin, M., and Kamlah, M., “Phase-field Modeling of Stress Generation in Electrode Particles of Lithium Ion Batteries”, Applied Physics Letters, Vol. 101, No. 13, p. 133902, 2012.
30. Song, Y., Li, Z., Soh, A., and Zhang, J., “Diffusion of Lithium Ions and Diffusion-induced Stresses in a Phase Separating Electrode under Galvanostatic and Potentiostatic Operations: Phase Field Simulations”, Mechanics of Materials, Vol. 91, pp. 363-371, 2015.
31. Haftbaradaran, H., Maddahian, A., and Mossaiby, F., “A Fracture Mechanics Study of the Phase Separating Planar Electrodes: Phase Field Modeling and Analytical Results”, Journal of Power Sources, Vol. 350, pp. 127-139, 2017.
32. Crank, J., The Mathematics of Diffusion, Oxford University Press, 1979.
33. Levi, M., and Aurbach, D., “Frumkin Intercalation Isotherm-a Tool for the Description of Lithium Insertion Into Host Materials: A Review”, Electrochimica Acta, Vol. 45, No. 1, pp. 167-185, 1999.
34. Burch, D., and Bazant, M. Z., “Size-dependent Spinodal and Miscibility Gaps for Intercalation in Nanoparticles”, Nano Letters, Vol. 9, No. 11, pp. 3795-3800, 2009.
35. Burch, D., “Intercalation Dynamics in Lithium-ion Batteries”, Ph.D. Desertation Massachusetts Inst of Tech Cambridge Dept of Mathematics, 2009.
36. Doyle, M., Fuller, T. F., and Newman, J., “Modeling of Galvanostatic Charge and Discharge of the Lithium/polymer/insertion Cell”, Journal of the Electrochemical Society, Vol. 140, No. 6, pp. 1526-1533, 1993.
37. Bazant, M. Z., “Theory of Chemical Kinetics and Charge Transfer Based on Nonequilibrium Thermodynamics”, Accounts of Chemical Research, Vol. 46, No. 5, pp. 1144-1160, 2013.
38. Timoshenko, S., and Goodier, J., Theory of Elasticity, McGraw-Hill book Company, 1951.
)