Phase-field model and its application in electrochemical energy storage materials

Zhang Geng; Wang Qiao; Sha Li-Ting; Li Ya-Jie; Wang Da; Shi Si-Qi

doi:10.7498/aps.69.20201411

Abstract

With the rapid progress of computer technology, computational research exhibits significant advantages in investigating microstructure evolution of material systems. As a computational research method of material dynamics, increasing attention has been paid to the phase-field model because of its avoidance of complicated interface tracking and convenience of dealing with applied fields. Theoretical framework of the phase-field model and three current phase-field models for multicomponent multiphase systems (the Carter, Steinbach, and Chen models) are introduced and reviewed in terms of interpretation of phase-field variables, way of coupling thermodynamic database, way of constructing the free energy density, and evolution equations. This review only focuses on the application of the phase-field model in electrochemical energy storage materials, and introduces its existing phase-field simulation results, which demonstrates that the phase-field model has tremendous potential in describing the microstructure evolution (anisotropic transport and phase separation, elastic and plastic deformation, crack propagation and fracture, dendrite growth, etc) and improving the performance of electrochemical energy storage materials. Finally, from two aspects of improving phase-field theory and extending application, future development trend and problems to be solved of phase-field simulations in electrochemical energy storage materials are discussed and looked ahead.

Keywords:

phase-field model /
electrochemical energy storage materials /
stress evolution /
plastic deformation /
dendrite growth

Authors and contacts

1.
Physical Science and Engineering Division (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia
2.
Materials Genome Institute, Shanghai University, Shanghai 200444, China
3.
School of Materials Science and Engineering, Shanghai University, Shanghai 200444, China

Corresponding author: Li Ya-Jie, liyajiejuly@shu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11874254) and Shanghai Pujiang Talents Program, China (Grant No. 2019PJD016)

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References

[1]	Gogotsi Y, Simon P 2011 Science 334 917 Google Scholar
[2]	Ceder G 2010 MRS Bull. 35 693 Google Scholar
[3]	Wang A P, Kadam S, Li H, Shi S Q, Qi Y 2018 NPJ Comput. Mater. 4 15 Google Scholar
[4]	Andersen H C 1980 J. Chem. Phys. 72 2384 Google Scholar
[5]	Stramer O 2001 J. Am. Stat. Assoc. 96 342
[6]	Shi S, Gao J, Liu Y, Zhao Y, Wu Q, Ju W, Ouyang C, Xiao R 2016 Chin. Phys. B 25 018212 Google Scholar
[7]	Zou Z Z, Li Y Y, Lu Z H, Wang D, Cui Y H, Guo B K, Li Y J, Liang X M, Feng J W, Li H, Nan C W, Armand M, Chen L Q, Xu K, Shi S Q 2020 Chem. Rev. 120 4169 Google Scholar
[8]	Biner S B 2017 Programming Phase-Field Modeling (Berlin: Springer)
[9]	Steinbach I, Pezzolla F, Nestler B, Sedklberg M, Ptieler R, Schmitz G J, Rezende J L L 1996 Physica D 94 135 Google Scholar
[10]	De Groot S R, Mazur P 2013 Non-Equilibrium Thermodynamics (Courier Corporation)
[11]	Wheeler A A, Boettinger W S, McFadden G B 1992 Phys. Rev. A 45 7424 Google Scholar
[12]	Kazaryan A, Wang Y, Dregia S A, Patton B R 2000 Phys. Rev. B 61 14275 Google Scholar
[13]	Karma A, Rappel W J 1998 Phys. Rev. E 57 4323 Google Scholar
[14]	Simmons J P, Shen C, Wang Y 2000 Scr. Mater. 43 935 Google Scholar
[15]	Granasy L, Pusztai T, Saylor D, Warren J A 2007 Phys. Rev. Lett. 98 035703 Google Scholar
[16]	Chakrabarti D J, Laughlin D E 2004 Prog. Mater. Sci. 49 389 Google Scholar
[17]	Xu Z, Meakin P 2008 J. Chem. Phys. 129 014705 Google Scholar
[18]	Hötzer J, Seiz M, Kellner M, Rheinheimer W, Nestler B 2019 Acta Mater. 164 184 Google Scholar
[19]	Wang Y U 2006 Acta Mater. 54 953 Google Scholar
[20]	Koslowski M, Cuitino A M, Ortiz M 2002 J. Mech. Phys. Solids 50 2597 Google Scholar
[21]	Wang Y U, Jin Y M, Cuitin A M, Khachaturyan A G 2001 Acta Mater. 49 1847 Google Scholar
[22]	Boussinot G, Le Bouar Y, Finel A 2010 Acta Mater. 58 4170 Google Scholar
[23]	Yu P, Hu S Y, Chen L Q, Du Q 2005 J. Comput. Phys. 208 34 Google Scholar
[24]	Stefanovic P, Haataja M, Provatas N 2009 Phys. Rev. E 80 046107 Google Scholar
[25]	Takaki T, Tomita Y 2010 Int. J. Mech. Sci. 52 320 Google Scholar
[26]	Hakim V, Karma A 2009 J. Mech. Phys. Solids 57 342 Google Scholar
[27]	Karma A, Kessler D A, Levine H 2001 Phys. Rev. Lett. 87 045501 Google Scholar
[28]	Cahn J W 1961 Acta Metall. Sin. 9 795 Google Scholar
[29]	Allen S M, Cahn J W 1972 Acta Metall. Sin. 20 423 Google Scholar
[30]	Kim S G, Kim W T, Suzuki T 1999 Phys. Rev. E 60 7186 Google Scholar
[31]	Zhang L, Steinbach I 2012 Acta Mater. 60 2702 Google Scholar
[32]	Steinbach I, Zhang L, Plapp M 2012 Acta Mater. 60 2689 Google Scholar
[33]	Cogswell D A, Carter W C 2011 Phys. Rev. E 83 061602
[34]	Chen L Q 2002 Annu. Rev. Mater. Res. 32 113 Google Scholar
[35]	Yu H C, Wang F, Amatucci G G, Thornton K 2016 J. Phase. Equilib. Diffus. 37 86 Google Scholar
[36]	Garcı́a R E, Bishop C M, Carter W C 2004 Acta Mater. 52 11 Google Scholar
[37]	Guyer J E, Boettinger W J, Warren J A, McFadden G B 2004 Phys. Rev. E 69 021603 Google Scholar
[38]	Guyer J E, Boettinger W J, Warren J A, McFadden G B 2004 Phys. Rev. E 69 021604 Google Scholar
[39]	Han B C, Ven A V d, Morgan D, Ceder G 2004 Electrochim. Acta 49 4691 Google Scholar
[40]	Cogswell D A, Bazant M Z 2012 ACS Nano 6 2215 Google Scholar
[41]	Singh G K, Ceder G, Bazant M Z 2008 Electrochim. Acta 53 7599 Google Scholar
[42]	Bazant M Z 2012 arXiv preprint arXiv 1208 1587
[43]	Bai P, Cogswell D A, Bazant M Z 2011 Nano. Lett. 11 4890 Google Scholar
[44]	Hong L, Li L, Chen-Wiegart Y K, Wang J, Xiang K, Gan L, Li W, Meng F, Wang F, Wang J, Chiang Y M, Jin S, Tang M 2017 Nat. Commun. 8 1194 Google Scholar
[45]	Li Y Y, Chen H G, Lim K, Deng H D, Lim J, Fraggedakis D, Attia P M, Lee S C, Jin N, Moškon J, Guan Z, Gent W E, Hong J, Yu Y S, Gaberšček M, Islam M S, Bazant M Z, Chueh W C 2018 Nat. Mater. 17 915 Google Scholar
[46]	Fleck M, Federmann H, Pogorelov E 2018 Comput. Mater. Sci. 153 288 Google Scholar
[47]	Zhang X Y, Hao F, Chen H S, Fang D N 2015 Mech. Mater. 91 351 Google Scholar
[48]	Zuo P, Zhao Y P 2015 Phys. Chem. Chem. Phys. 17 287 Google Scholar
[49]	Lim C, Yan B, Yin L, Zhu L 2012 Electrochim. Acta 75 279 Google Scholar
[50]	Gao F, Hong W 2016 J. Mech. Phys. Solids 94 18 Google Scholar
[51]	Chen L, Fan F, Hong L, Chen J, Ji Y Z, Zhang S L, Zhu T, Chen L Q 2014 J. Electrochem. Soc. 161 F3164 Google Scholar
[52]	Zhang X, Krischok A, Linder C 2016 Comput. Method Appl. M. 312 51 Google Scholar
[53]	Zhao K, Pharr M, Vlassak J J, Suo Z 2011 J. Appl. Phys. 109 016110 Google Scholar
[54]	Zhao K, Pharr M, Cai S, Vlassak J J, Suo Z 2011 J. Am. Ceram. Soc. 94 S226 Google Scholar
[55]	Bower A F, Guduru P R, Sethuraman V A 2011 J. Mech. Phys. Solids. 59 804 Google Scholar
[56]	Walk A C, Huttin M, Kamlah M 2014 Eur. J. Mech. A.Solids 48 74 Google Scholar
[57]	Aranson I S, Kalatsky V A, Vinokur V M 2000 Phys. Rev. Lett. 85 118 Google Scholar
[58]	Marconi V I, Jagla E A 2005 Phys. Rev. E 71 036110 Google Scholar
[59]	Spatschek R, Muller-Gugenberger C, Brener E, Nestler B 2007 Phys. Rev. E 75 066111 Google Scholar
[60]	Miehe C, Hofacker M, Welschinger F 2010 Comput. Method Appl. M. 199 2765 Google Scholar
[61]	Miehe C, Welschinger F, Hofacker M 2010 Mech. Phys. Solids 58 1716 Google Scholar
[62]	Hofacker M, Miehe C 2013 Int. J. Numer. Meth. Eng. 93 276 Google Scholar
[63]	Bhandakkar T K, Gao H 2010 Int. J. Solids Struct. 47 1424 Google Scholar
[64]	Bhandakkar T K, Gao H 2011 Int. J. Solids Struct. 48 2304 Google Scholar
[65]	Woodford W H, Chiang Y M, Carter W C 2010 J. Electrochem. Soc. 157 A1052 Google Scholar
[66]	Zhu M, Park J, Sastry A M 2012 J. Electrochem. Soc. 159 A492 Google Scholar
[67]	Gao Y F, Zhou M 2013 J. Power Sources 230 176 Google Scholar
[68]	Klinsmann M, Rosato D, Kamlah M, McMeeking R M 2016 Mech. Phys. Solids 92 313 Google Scholar
[69]	Zhao K, Pharr M, Vlassak J J, Suo Z 2010 J. Appl. Phys. 108 073517 Google Scholar
[70]	Huttin M, Kamlah M 2012 J. Appl. Phys. 101 133902
[71]	Liang L, Chen L Q 2014 Appl. Phys. Lett. 105 263903 Google Scholar
[72]	Miehe C, Dal H, Schänzel L M, Raina A 2016 Int. J. Numer. Meth. Eng. 106 683 Google Scholar
[73]	Zhao Y, Xu B X, Stein P, Gross D 2016 Comput. Method. Appl. M. 312 428 Google Scholar
[74]	Zhuang Y, Zou Z Y, Lu B, Li Y J, Wang D, Avdeev M, Shi S Q 2020 Chin. Phys. B 29 068202 Google Scholar
[75]	Lu B, Ning C Q, Shi D X, Zhao Y F, Zhang J Q 2020 Chin. Phys. B 29 026201 Google Scholar
[76]	Hong Z J, Viswanathan V 2020 ACS Energy Lett. 5 3254 Google Scholar
[77]	Li Y, Hu S, Sun X, Stan M 2017 NPJ Comput. Mater. 3 1 Google Scholar
[78]	Yamaki J I, Tobishima S I, Hayashi K, Saito K, Nemoto Y, Arakawa M 1998 J. Power Sources 74 219 Google Scholar
[79]	Jana A, Woo S I, Vikrant K S N, García R E 2019 Energy Environ. Sci. 12 3595 Google Scholar
[80]	Li G, Liu Z, Huang Q, Gao Y, Regula M, Wang D, Chen L Q, Wang D 2018 Nat. Energy 3 1076 Google Scholar
[81]	Li G, Liu Z, Wang D, He X, Liu S, Gao Y, AlZahrani A, Kim S H, Chen L Q, Wang D 2019 Adv. Energy Mater. 9 1900704 Google Scholar
[82]	Xu F D, Graff G L, Zhang i, Sushko M L, Chen X, Shao Y, Engelhard M H, Nie Z, Xiao J, Liu X, Sushko P V, Liu J, Zhang J G 2013 J. Am. Chem. Soc. 135 4450 Google Scholar
[83]	Monroe C, Newman J 2003 J. Electrochem. Soc. 150 A1377 Google Scholar
[84]	Shibuta Y, Okajima Y, Suzuki T 2007 Sci. Technol. Adv. Mater. 8 511 Google Scholar
[85]	Okajima Y, Shibuta Y, Suzuki T 2010 Comput. Mater. Sci. 50 118 Google Scholar
[86]	Liang L, Qi Y, Xue F, Bhattacharya S, Harris S J, Chen L Q 2012 Phys. Rev. E 86 051609 Google Scholar
[87]	Chen L, Zhang H W, Liang L Y, Liu Z, Qi Y, Lu P, Chen J, Chen L Q 2015 J. Power Sources 300 376 Google Scholar
[88]	Tan J, Tartakovsky A M, Ferris K, Ryan E M 2016 J. Electrochem. Soc. 163 A318 Google Scholar
[89]	Yurkiv V, Foroozan T, Ramasubramanian A, Shahbazian-Yassar R, Mashayek F 2018 Electrochim. Acta 265 609 Google Scholar
[90]	Yurkiv V, Foroozan T, Ramasubramanian A, Shahbazian-Yassar R, Mashayek F 2018 MRS Commun. 8 1285 Google Scholar
[91]	Yan H H, Bie Y H, Cu X Y, Xiong G P, Chen L 2018 Energy Convers. Manage. 161 193 Google Scholar
[92]	Hong Z J, Viswanathan V 2018 ACS Energy Lett. 3 1737 Google Scholar
[93]	Hong Z J, Viswanathan V 2020 ACS Energy Lett. 5 2466 Google Scholar
[94]	Hong Z J, Viswanathan V 2019 ACS Energy Lett. 4 1012 Google Scholar
[95]	Mu W, Liu X, Wen Z, Liu L 2019 J. Energy Storage 26 100921 Google Scholar
[96]	Zhang R, Shen X, Cheng X B, Zhang Q 2019 Energy Storage Mater. 23 556 Google Scholar

Cited By

图 1 相变界问题的尖锐界面和扩散界面的对比

Figure 1. Comparison between sharp and diffusive interfaces for phase boundary problems.

DownLoad: Full-Size Img PowerPoint

图 2 相场变量与Steinbach模型、Chen模型中的相权重函数的比较

Figure 2. Comparison of phase field and phase-weight functions in Steinbach and Chen models.

DownLoad: Full-Size Img PowerPoint

图 3 相场模型中研究电化学储能材料微结构演化的步骤

Figure 3. Steps for investigating microstructure evolution of electrochemical energy storage materials in phase-field model.

DownLoad: Full-Size Img PowerPoint

图 4 (a) 锂离子嵌入到FePO₄时活跃面(010)快速扩散(蓝色箭头)和相分离示意图^[43]; (b) LiFePO₄相变过程中的三种可能的扩散路径: 体扩散、表面扩散和电解液扩散^[45]

Figure 4. (a) Schematic diagram of diffusion of active surface (010) (blue arrows) and phase separation when Li-ion intercalates FePO₄^[43]; (b) three potential migration paths in phase transition of LiFePO₄: bulk, surface, and electrolyte diffusions^[45].

DownLoad: Full-Size Img PowerPoint

图 5 硅颗粒的弹性和塑性变形^[50-52]

Figure 5. Elastic and plastic deformations of Si particles^[50-52].

DownLoad: Full-Size Img PowerPoint

图 6 锂化过程中的裂纹扩展　(a)初始状态; (b)相偏析导致裂纹扩展; (c)相界面赶上裂纹尖端; (d)裂纹尖端在相界面处开始分叉; (e)相界面离开裂纹尖端向中心移动; (f)反应停止^[73]

Figure 6. Crack propagation during lithiation process: (a) Initial state; (b) phase segregation generates crack propagation; (c) the phase interface catches up with the crack tip; (d) the crack tip starts to branch at the phase interphase; (e) phase interface leaves crack tip and moves towards center; (f) the reaction stops ^[73].

DownLoad: Full-Size Img PowerPoint

图 7 相场模型模拟的Li枝晶生长　(a) 序参量, (b) Li⁺浓度和(c)电势的演化^[87]

Figure 7. Dendrite growth of Li simulated by phase-field model: evolution of (a) order parameter, (b) concentration of Li⁺, and (c) electric potential ^[87].

DownLoad: Full-Size Img PowerPoint

[1]	Gogotsi Y, Simon P 2011 Science 334 917 Google Scholar
[2]	Ceder G 2010 MRS Bull. 35 693 Google Scholar
[3]	Wang A P, Kadam S, Li H, Shi S Q, Qi Y 2018 NPJ Comput. Mater. 4 15 Google Scholar
[4]	Andersen H C 1980 J. Chem. Phys. 72 2384 Google Scholar
[5]	Stramer O 2001 J. Am. Stat. Assoc. 96 342
[6]	Shi S, Gao J, Liu Y, Zhao Y, Wu Q, Ju W, Ouyang C, Xiao R 2016 Chin. Phys. B 25 018212 Google Scholar
[7]	Zou Z Z, Li Y Y, Lu Z H, Wang D, Cui Y H, Guo B K, Li Y J, Liang X M, Feng J W, Li H, Nan C W, Armand M, Chen L Q, Xu K, Shi S Q 2020 Chem. Rev. 120 4169 Google Scholar
[8]	Biner S B 2017 Programming Phase-Field Modeling (Berlin: Springer)
[9]	Steinbach I, Pezzolla F, Nestler B, Sedklberg M, Ptieler R, Schmitz G J, Rezende J L L 1996 Physica D 94 135 Google Scholar
[10]	De Groot S R, Mazur P 2013 Non-Equilibrium Thermodynamics (Courier Corporation)
[11]	Wheeler A A, Boettinger W S, McFadden G B 1992 Phys. Rev. A 45 7424 Google Scholar
[12]	Kazaryan A, Wang Y, Dregia S A, Patton B R 2000 Phys. Rev. B 61 14275 Google Scholar
[13]	Karma A, Rappel W J 1998 Phys. Rev. E 57 4323 Google Scholar
[14]	Simmons J P, Shen C, Wang Y 2000 Scr. Mater. 43 935 Google Scholar
[15]	Granasy L, Pusztai T, Saylor D, Warren J A 2007 Phys. Rev. Lett. 98 035703 Google Scholar
[16]	Chakrabarti D J, Laughlin D E 2004 Prog. Mater. Sci. 49 389 Google Scholar
[17]	Xu Z, Meakin P 2008 J. Chem. Phys. 129 014705 Google Scholar
[18]	Hötzer J, Seiz M, Kellner M, Rheinheimer W, Nestler B 2019 Acta Mater. 164 184 Google Scholar
[19]	Wang Y U 2006 Acta Mater. 54 953 Google Scholar
[20]	Koslowski M, Cuitino A M, Ortiz M 2002 J. Mech. Phys. Solids 50 2597 Google Scholar
[21]	Wang Y U, Jin Y M, Cuitin A M, Khachaturyan A G 2001 Acta Mater. 49 1847 Google Scholar
[22]	Boussinot G, Le Bouar Y, Finel A 2010 Acta Mater. 58 4170 Google Scholar
[23]	Yu P, Hu S Y, Chen L Q, Du Q 2005 J. Comput. Phys. 208 34 Google Scholar
[24]	Stefanovic P, Haataja M, Provatas N 2009 Phys. Rev. E 80 046107 Google Scholar
[25]	Takaki T, Tomita Y 2010 Int. J. Mech. Sci. 52 320 Google Scholar
[26]	Hakim V, Karma A 2009 J. Mech. Phys. Solids 57 342 Google Scholar
[27]	Karma A, Kessler D A, Levine H 2001 Phys. Rev. Lett. 87 045501 Google Scholar
[28]	Cahn J W 1961 Acta Metall. Sin. 9 795 Google Scholar
[29]	Allen S M, Cahn J W 1972 Acta Metall. Sin. 20 423 Google Scholar
[30]	Kim S G, Kim W T, Suzuki T 1999 Phys. Rev. E 60 7186 Google Scholar
[31]	Zhang L, Steinbach I 2012 Acta Mater. 60 2702 Google Scholar
[32]	Steinbach I, Zhang L, Plapp M 2012 Acta Mater. 60 2689 Google Scholar
[33]	Cogswell D A, Carter W C 2011 Phys. Rev. E 83 061602
[34]	Chen L Q 2002 Annu. Rev. Mater. Res. 32 113 Google Scholar
[35]	Yu H C, Wang F, Amatucci G G, Thornton K 2016 J. Phase. Equilib. Diffus. 37 86 Google Scholar
[36]	Garcı́a R E, Bishop C M, Carter W C 2004 Acta Mater. 52 11 Google Scholar
[37]	Guyer J E, Boettinger W J, Warren J A, McFadden G B 2004 Phys. Rev. E 69 021603 Google Scholar
[38]	Guyer J E, Boettinger W J, Warren J A, McFadden G B 2004 Phys. Rev. E 69 021604 Google Scholar
[39]	Han B C, Ven A V d, Morgan D, Ceder G 2004 Electrochim. Acta 49 4691 Google Scholar
[40]	Cogswell D A, Bazant M Z 2012 ACS Nano 6 2215 Google Scholar
[41]	Singh G K, Ceder G, Bazant M Z 2008 Electrochim. Acta 53 7599 Google Scholar
[42]	Bazant M Z 2012 arXiv preprint arXiv 1208 1587
[43]	Bai P, Cogswell D A, Bazant M Z 2011 Nano. Lett. 11 4890 Google Scholar
[44]	Hong L, Li L, Chen-Wiegart Y K, Wang J, Xiang K, Gan L, Li W, Meng F, Wang F, Wang J, Chiang Y M, Jin S, Tang M 2017 Nat. Commun. 8 1194 Google Scholar
[45]	Li Y Y, Chen H G, Lim K, Deng H D, Lim J, Fraggedakis D, Attia P M, Lee S C, Jin N, Moškon J, Guan Z, Gent W E, Hong J, Yu Y S, Gaberšček M, Islam M S, Bazant M Z, Chueh W C 2018 Nat. Mater. 17 915 Google Scholar
[46]	Fleck M, Federmann H, Pogorelov E 2018 Comput. Mater. Sci. 153 288 Google Scholar
[47]	Zhang X Y, Hao F, Chen H S, Fang D N 2015 Mech. Mater. 91 351 Google Scholar
[48]	Zuo P, Zhao Y P 2015 Phys. Chem. Chem. Phys. 17 287 Google Scholar
[49]	Lim C, Yan B, Yin L, Zhu L 2012 Electrochim. Acta 75 279 Google Scholar
[50]	Gao F, Hong W 2016 J. Mech. Phys. Solids 94 18 Google Scholar
[51]	Chen L, Fan F, Hong L, Chen J, Ji Y Z, Zhang S L, Zhu T, Chen L Q 2014 J. Electrochem. Soc. 161 F3164 Google Scholar
[52]	Zhang X, Krischok A, Linder C 2016 Comput. Method Appl. M. 312 51 Google Scholar
[53]	Zhao K, Pharr M, Vlassak J J, Suo Z 2011 J. Appl. Phys. 109 016110 Google Scholar
[54]	Zhao K, Pharr M, Cai S, Vlassak J J, Suo Z 2011 J. Am. Ceram. Soc. 94 S226 Google Scholar
[55]	Bower A F, Guduru P R, Sethuraman V A 2011 J. Mech. Phys. Solids. 59 804 Google Scholar
[56]	Walk A C, Huttin M, Kamlah M 2014 Eur. J. Mech. A.Solids 48 74 Google Scholar
[57]	Aranson I S, Kalatsky V A, Vinokur V M 2000 Phys. Rev. Lett. 85 118 Google Scholar
[58]	Marconi V I, Jagla E A 2005 Phys. Rev. E 71 036110 Google Scholar
[59]	Spatschek R, Muller-Gugenberger C, Brener E, Nestler B 2007 Phys. Rev. E 75 066111 Google Scholar
[60]	Miehe C, Hofacker M, Welschinger F 2010 Comput. Method Appl. M. 199 2765 Google Scholar
[61]	Miehe C, Welschinger F, Hofacker M 2010 Mech. Phys. Solids 58 1716 Google Scholar
[62]	Hofacker M, Miehe C 2013 Int. J. Numer. Meth. Eng. 93 276 Google Scholar
[63]	Bhandakkar T K, Gao H 2010 Int. J. Solids Struct. 47 1424 Google Scholar
[64]	Bhandakkar T K, Gao H 2011 Int. J. Solids Struct. 48 2304 Google Scholar
[65]	Woodford W H, Chiang Y M, Carter W C 2010 J. Electrochem. Soc. 157 A1052 Google Scholar
[66]	Zhu M, Park J, Sastry A M 2012 J. Electrochem. Soc. 159 A492 Google Scholar
[67]	Gao Y F, Zhou M 2013 J. Power Sources 230 176 Google Scholar
[68]	Klinsmann M, Rosato D, Kamlah M, McMeeking R M 2016 Mech. Phys. Solids 92 313 Google Scholar
[69]	Zhao K, Pharr M, Vlassak J J, Suo Z 2010 J. Appl. Phys. 108 073517 Google Scholar
[70]	Huttin M, Kamlah M 2012 J. Appl. Phys. 101 133902
[71]	Liang L, Chen L Q 2014 Appl. Phys. Lett. 105 263903 Google Scholar
[72]	Miehe C, Dal H, Schänzel L M, Raina A 2016 Int. J. Numer. Meth. Eng. 106 683 Google Scholar
[73]	Zhao Y, Xu B X, Stein P, Gross D 2016 Comput. Method. Appl. M. 312 428 Google Scholar
[74]	Zhuang Y, Zou Z Y, Lu B, Li Y J, Wang D, Avdeev M, Shi S Q 2020 Chin. Phys. B 29 068202 Google Scholar
[75]	Lu B, Ning C Q, Shi D X, Zhao Y F, Zhang J Q 2020 Chin. Phys. B 29 026201 Google Scholar
[76]	Hong Z J, Viswanathan V 2020 ACS Energy Lett. 5 3254 Google Scholar
[77]	Li Y, Hu S, Sun X, Stan M 2017 NPJ Comput. Mater. 3 1 Google Scholar
[78]	Yamaki J I, Tobishima S I, Hayashi K, Saito K, Nemoto Y, Arakawa M 1998 J. Power Sources 74 219 Google Scholar
[79]	Jana A, Woo S I, Vikrant K S N, García R E 2019 Energy Environ. Sci. 12 3595 Google Scholar
[80]	Li G, Liu Z, Huang Q, Gao Y, Regula M, Wang D, Chen L Q, Wang D 2018 Nat. Energy 3 1076 Google Scholar
[81]	Li G, Liu Z, Wang D, He X, Liu S, Gao Y, AlZahrani A, Kim S H, Chen L Q, Wang D 2019 Adv. Energy Mater. 9 1900704 Google Scholar
[82]	Xu F D, Graff G L, Zhang i, Sushko M L, Chen X, Shao Y, Engelhard M H, Nie Z, Xiao J, Liu X, Sushko P V, Liu J, Zhang J G 2013 J. Am. Chem. Soc. 135 4450 Google Scholar
[83]	Monroe C, Newman J 2003 J. Electrochem. Soc. 150 A1377 Google Scholar
[84]	Shibuta Y, Okajima Y, Suzuki T 2007 Sci. Technol. Adv. Mater. 8 511 Google Scholar
[85]	Okajima Y, Shibuta Y, Suzuki T 2010 Comput. Mater. Sci. 50 118 Google Scholar
[86]	Liang L, Qi Y, Xue F, Bhattacharya S, Harris S J, Chen L Q 2012 Phys. Rev. E 86 051609 Google Scholar
[87]	Chen L, Zhang H W, Liang L Y, Liu Z, Qi Y, Lu P, Chen J, Chen L Q 2015 J. Power Sources 300 376 Google Scholar
[88]	Tan J, Tartakovsky A M, Ferris K, Ryan E M 2016 J. Electrochem. Soc. 163 A318 Google Scholar
[89]	Yurkiv V, Foroozan T, Ramasubramanian A, Shahbazian-Yassar R, Mashayek F 2018 Electrochim. Acta 265 609 Google Scholar
[90]	Yurkiv V, Foroozan T, Ramasubramanian A, Shahbazian-Yassar R, Mashayek F 2018 MRS Commun. 8 1285 Google Scholar
[91]	Yan H H, Bie Y H, Cu X Y, Xiong G P, Chen L 2018 Energy Convers. Manage. 161 193 Google Scholar
[92]	Hong Z J, Viswanathan V 2018 ACS Energy Lett. 3 1737 Google Scholar
[93]	Hong Z J, Viswanathan V 2020 ACS Energy Lett. 5 2466 Google Scholar
[94]	Hong Z J, Viswanathan V 2019 ACS Energy Lett. 4 1012 Google Scholar
[95]	Mu W, Liu X, Wen Z, Liu L 2019 J. Energy Storage 26 100921 Google Scholar
[96]	Zhang R, Shen X, Cheng X B, Zhang Q 2019 Energy Storage Mater. 23 556 Google Scholar

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Vol. 69, Issue 22 - 2020-11-20

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