Estimation of the Stress Intensity Factors for Surface Cracks in Spherical Electrode Particles Subject to Phase Separation
Authors
Abstract
Experiments have frequently shown that phase separation in lithium-ion battery electrodes could lead to the formation of mechanical defects, hence causing capacity fading. The purpose of the present work has been to examine stress intensity factors for pre-existing surface cracks in spherical electrode particles during electrochemical deintercalation cycling using both analytical and numerical methods. To this end, we make use of a phase field model to examine the time-dependent evolution of the concentration and stress profiles in a phase separating spherical electrode particles. By using a geometrical approximation scheme proposed in the literature, stress intensity factors at the deepest point of the pre-existing surface cracks of semi-elliptical geometry are calculated with the aid of the well-established weight function method of fracture mechanics. By taking advantage of a sharp-interphase core-shell model, an analytical solution for the maximum stress intensity factors arising at the deepest point of the surface cracks during a complete deintercalation half-cycle is also developed. Numerical results for evolution of the concentration profile and the distribution of the hoop stresses in the particle are presented; further, the stress intensity factors found numerically based on the phase field model are compared with those predicted by the analytical core-shell model. The results of the numerical model suggest that the maximum stress intensity factor could significantly vary with changes in the surface flux, increasing potentially by a factor of two within the range of parameters considered here, when the concentration difference between the two phases is decreased.
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. 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, No. 9, pp. A137-A140, 2001.
3. 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.
4. 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.
5. 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.
6. 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.
7. Malav, V., Jangid, M. K., Hait, I., and Mukhopadhyay, A., “In Situ Monitoring of Stress Developments and Mechanical Integrity During Galvanostatic Cycling of LiCoO2 Thin Films”, ECS Electrochemistry Letters, Vol. 4, No. 12, pp. A148-A150, 2015.
8. Esmizade, S., Haftbaradaran, H., and Mossaiby F., “The Effect of Phase Separation on Diffusion Induced Stresses in Spherical and Cylindrical Electrode Particles”, Computational Methods in Engineering, Vol. 37, No. 1, pp. 29-50, 2018 (In Farsi).
9. Lee, H.-W., Muralidharan, P., Ruffo, R., Mari, C. M., Cui, Y., and Kim, D. K., “Ultrathin Spinel LiMn2O4 Nanowires as High Power Cathode Materials for Li-ion Batteries”, Nano letters, Vol. 10, No. 10, pp. 3852-3856, 2010.
10. Put, B., Vereecken, P. M., Labyedh, N., Sepulveda, A., Huyghebaert, C., Radu, I. P., and Stesmans, A., “High Cycling Stability and Extreme Rate Performance in Nanoscaled LiMn2O4 Thin Films”, ACS Applied Materials & Interfaces, Vol. 7, No. 40, pp. 22413-22420, 2015.
11. Liu, X. H., Zheng, H., Zhong, L., Huang, S., Karki, K., Zhang, L. Q., Liu, Y., Kushima, A., Liang, W. T., Wang, J. W. and Cho, J. H., “Anisotropic Swelling and Fracture of Silicon Nanowires During Lithiation”, Nano letters, Vol. 11, No. 8, pp. 3312-3318, 2011.
12. Ryu, I., Choi, J. W., Cui, Y., and Nix, W. D., “Size-Dependent Fracture of Si Nanowire Battery Anodes”, Journal of the Mechanics and Physics of Solids, Vol. 59, No. 9, pp. 1717-1730, 2011.
13. Liu, X. H., Zhong, L., Huang, S., Mao, S. X., Zhu, T., and Huang, J. Y., “Size-Dependent Fracture of Silicon Nanoparticles During Lithiation”, ACS Nano, Vol. 6, No. 2, pp. 1522-1531, 2012.
14. Christensen, J., and Newman, J., “A Mathematical Model of Stress Generation and Fracture in Lithium Manganese Oxide”, Journal of The Electrochemical Society, Vol. 153, No. 6, pp. A1019-A1030, 2006.
15. 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.
16. 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.
17. Zhang, J., and Wang, C., “Vibrating Piezoelectric Nanofilms as Sandwich Nanoplates”, Journal of Applied Physics, Vol. 111, No. 9, p. 094303, 2012.
18. Griffith, A. A., “The Phenomena of Rupture and Flow in Solids”, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Vol. 221, pp. 163-198, 1921.
19. Anderson, T. L., Fracture mechanics: Fundamentals and Applications, CRC press, 2017.
20. Newman, J., and Raju, I., “An Empirical Stress-Intensity Factor Equation for the Surface Crack”, Engineering Fracture Mechanics, Vol. 15, No. 1-2, pp. 185-192, 1981.
21. Petroski, H., and Achenbach, J., “Computation of the Weight Function from a Stress Intensity Factor”, Engineering Fracture Mechanics, Vol. 10, No. 2, pp. 257-266, 1978.
22. Woodford, W. H., Chiang, Y. -M., and Carter, W. C., “Electrochemical Shock of Intercalation Electrodes: a Fracture Mechanics Analysis”, Journal of the Electrochemical Society, Vol. 157, No. 10, pp. A1052-A1059, 2010.
23. Woodford, W. H., Chiang, Y. -M., and Carter, W. C., “Electrochemical Shock in Ion-Intercalation Materials with Limited Solid-Solubility”, Journal of the Electrochemical Society, Vol. 160, No. 8, pp. A1286-A1292, 2013.
24. Esmizadeh, S., Haftbaradaran, H., and Mossaiby, F., “An Investigation of the Critical Conditions Leading to Deintercalation Induced Fracture in Two-Phase Elastic Electrode Particles Using a Moving Interphase Core-Shell Model”, European Journal of Mechanics-A/Solids, Vol. 74, pp. 96-111, 2019.
25. 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.
26. 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.
27. Singh, G. K., Ceder, G., and Bazant, M. Z., “Intercalation Dynamics in Rechargeable Battery Materials: General Theory and Phase-Transformation Waves in LiFePO4”, Electrochimica Acta, Vol. 53, No. 26, pp. 7599-7613, 2008.
28. 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.
29. Crank, J., The Mathematics of Diffusion, Oxford University Press, 1979.
30. 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.
31. 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.
32. Cogswell, D. A., and Bazant, M. Z., “Coherency Strain and the Kinetics of Phase Separation in LiFePO4 Nanoparticles”, ACS Nano, Vol. 6, No. 3, pp. 2215-2225, 2012.
33. 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.
34. 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.
35. Bueckner, H., “Novel Principle for the Computation of Stress Intensity Factors”, Zeitschrift fuer Angewandte Mathematik & Mechanik, Vol. 50, No. 9, 1970.
36. Rice, J. R., “Some Remarks on Elastic Crack-Tip Stress Fields”, International Journal of Solids and Structures, Vol. 8, No. 6, pp. 751-758, 1972.
37. Mattheck, C., Munz, D., and Stamm, H., “Stress Intensity Factor for Semi-Elliptical Surface Cracks Loaded by Stress Gradients”, Engineering Fracture Mechanics, Vol. 18, No. 3, pp. 633-641, 1983.
38. Timoshenko, S., and Goodier, J., Theory of Elasticity, McGraw-Hill, 1951.
39. Huggins, R., and Nix, W., “Decrepitation Model for Capacity Loss During Cycling of Alloys in Rechargeable Electrochemical Systems”, Ionics, Vol. 6, No. 1, pp. 57-63, 2000.
2. 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, No. 9, pp. A137-A140, 2001.
3. 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.
4. 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.
5. 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.
6. 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.
7. Malav, V., Jangid, M. K., Hait, I., and Mukhopadhyay, A., “In Situ Monitoring of Stress Developments and Mechanical Integrity During Galvanostatic Cycling of LiCoO2 Thin Films”, ECS Electrochemistry Letters, Vol. 4, No. 12, pp. A148-A150, 2015.
8. Esmizade, S., Haftbaradaran, H., and Mossaiby F., “The Effect of Phase Separation on Diffusion Induced Stresses in Spherical and Cylindrical Electrode Particles”, Computational Methods in Engineering, Vol. 37, No. 1, pp. 29-50, 2018 (In Farsi).
9. Lee, H.-W., Muralidharan, P., Ruffo, R., Mari, C. M., Cui, Y., and Kim, D. K., “Ultrathin Spinel LiMn2O4 Nanowires as High Power Cathode Materials for Li-ion Batteries”, Nano letters, Vol. 10, No. 10, pp. 3852-3856, 2010.
10. Put, B., Vereecken, P. M., Labyedh, N., Sepulveda, A., Huyghebaert, C., Radu, I. P., and Stesmans, A., “High Cycling Stability and Extreme Rate Performance in Nanoscaled LiMn2O4 Thin Films”, ACS Applied Materials & Interfaces, Vol. 7, No. 40, pp. 22413-22420, 2015.
11. Liu, X. H., Zheng, H., Zhong, L., Huang, S., Karki, K., Zhang, L. Q., Liu, Y., Kushima, A., Liang, W. T., Wang, J. W. and Cho, J. H., “Anisotropic Swelling and Fracture of Silicon Nanowires During Lithiation”, Nano letters, Vol. 11, No. 8, pp. 3312-3318, 2011.
12. Ryu, I., Choi, J. W., Cui, Y., and Nix, W. D., “Size-Dependent Fracture of Si Nanowire Battery Anodes”, Journal of the Mechanics and Physics of Solids, Vol. 59, No. 9, pp. 1717-1730, 2011.
13. Liu, X. H., Zhong, L., Huang, S., Mao, S. X., Zhu, T., and Huang, J. Y., “Size-Dependent Fracture of Silicon Nanoparticles During Lithiation”, ACS Nano, Vol. 6, No. 2, pp. 1522-1531, 2012.
14. Christensen, J., and Newman, J., “A Mathematical Model of Stress Generation and Fracture in Lithium Manganese Oxide”, Journal of The Electrochemical Society, Vol. 153, No. 6, pp. A1019-A1030, 2006.
15. 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.
16. 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.
17. Zhang, J., and Wang, C., “Vibrating Piezoelectric Nanofilms as Sandwich Nanoplates”, Journal of Applied Physics, Vol. 111, No. 9, p. 094303, 2012.
18. Griffith, A. A., “The Phenomena of Rupture and Flow in Solids”, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Vol. 221, pp. 163-198, 1921.
19. Anderson, T. L., Fracture mechanics: Fundamentals and Applications, CRC press, 2017.
20. Newman, J., and Raju, I., “An Empirical Stress-Intensity Factor Equation for the Surface Crack”, Engineering Fracture Mechanics, Vol. 15, No. 1-2, pp. 185-192, 1981.
21. Petroski, H., and Achenbach, J., “Computation of the Weight Function from a Stress Intensity Factor”, Engineering Fracture Mechanics, Vol. 10, No. 2, pp. 257-266, 1978.
22. Woodford, W. H., Chiang, Y. -M., and Carter, W. C., “Electrochemical Shock of Intercalation Electrodes: a Fracture Mechanics Analysis”, Journal of the Electrochemical Society, Vol. 157, No. 10, pp. A1052-A1059, 2010.
23. Woodford, W. H., Chiang, Y. -M., and Carter, W. C., “Electrochemical Shock in Ion-Intercalation Materials with Limited Solid-Solubility”, Journal of the Electrochemical Society, Vol. 160, No. 8, pp. A1286-A1292, 2013.
24. Esmizadeh, S., Haftbaradaran, H., and Mossaiby, F., “An Investigation of the Critical Conditions Leading to Deintercalation Induced Fracture in Two-Phase Elastic Electrode Particles Using a Moving Interphase Core-Shell Model”, European Journal of Mechanics-A/Solids, Vol. 74, pp. 96-111, 2019.
25. 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.
26. 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.
27. Singh, G. K., Ceder, G., and Bazant, M. Z., “Intercalation Dynamics in Rechargeable Battery Materials: General Theory and Phase-Transformation Waves in LiFePO4”, Electrochimica Acta, Vol. 53, No. 26, pp. 7599-7613, 2008.
28. 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.
29. Crank, J., The Mathematics of Diffusion, Oxford University Press, 1979.
30. 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.
31. 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.
32. Cogswell, D. A., and Bazant, M. Z., “Coherency Strain and the Kinetics of Phase Separation in LiFePO4 Nanoparticles”, ACS Nano, Vol. 6, No. 3, pp. 2215-2225, 2012.
33. 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.
34. 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.
35. Bueckner, H., “Novel Principle for the Computation of Stress Intensity Factors”, Zeitschrift fuer Angewandte Mathematik & Mechanik, Vol. 50, No. 9, 1970.
36. Rice, J. R., “Some Remarks on Elastic Crack-Tip Stress Fields”, International Journal of Solids and Structures, Vol. 8, No. 6, pp. 751-758, 1972.
37. Mattheck, C., Munz, D., and Stamm, H., “Stress Intensity Factor for Semi-Elliptical Surface Cracks Loaded by Stress Gradients”, Engineering Fracture Mechanics, Vol. 18, No. 3, pp. 633-641, 1983.
38. Timoshenko, S., and Goodier, J., Theory of Elasticity, McGraw-Hill, 1951.
39. Huggins, R., and Nix, W., “Decrepitation Model for Capacity Loss During Cycling of Alloys in Rechargeable Electrochemical Systems”, Ionics, Vol. 6, No. 1, pp. 57-63, 2000.
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