Acta Mechanica Solida Sinica

, Volume 31, Issue 3, pp 290–309 | Cite as

Lithium Diffusion and Stress in a Polycrystalline Film Electrode

  • YanFei Zhao
  • Bo Lu
  • Junqian Zhang


In the present work, the two-dimensional analytical solution for Li diffusion in grains and grain boundaries of a polycrystalline film electrode is established with consideration of Li-segregation at the grain boundary. The stress field induced by the inhomogeneity of Li concentration, called chemical stress here for brevity, is analyzed via the finite element simulation. The effects of the grain boundary including its size, its diffusion coefficient as well as the segregation phenomenon on the solute concentration and the chemical stress are examined. It shows that grain boundaries can assist fast diffusion and significantly affect the stress profile in the whole film. It proves that tailoring the grain boundary size or other grain boundary-related parameters may provide an alternative strategy for improving the overall diffusivity and mechanical stability of battery electrodes.


Polycrystalline film Two-dimensional diffusion Grain boundary Segregation Chemical stress 

List of symbols

x (\(x'\)) and y





Film thickness


Half grain width


Half GB width

C and \(C^\prime \)

Li concentrations in grain and GB

D and \(D^\prime \)

Diffusion coefficients for grain and GB

\(\Delta G\)

Free energy of segregation


Segregation coefficient


Gas constant


Absolute temperature


Apparent diffusion flux

\(J_{0}\) and \(J_{0}^{\prime }\)

Diffusion fluxes entering grain surface and GB surface

\(\varDelta \)

Ratio of diffusion coefficients

\(\alpha _{mn}, {\alpha }_{mn}^{\prime }, \beta _n, A_{mn}, {A}_{mn}^{\prime }\)

Coefficients in solutions of concentration

\({\bar{x}} \, ({{\bar{x}}^{\prime }})\) and \({\bar{y}}\)

Dimensionless coordinates


Dimensionless time


Dimensionless GB size

\({\bar{\alpha }}_{mn}, {\bar{\alpha }}_{mn}^{\prime }, {\bar{\beta }}_n, {\bar{A}}_{mn}, {\bar{A}}_{mn}^{\prime }\)

Dimensionless coefficients in solutions of concentration

\({\bar{C}}\) and \({{\bar{C}}^{\prime }}\)

Dimensionless concentration

\(\sigma _{{\textit{ij}}}\) and \({\sigma }_{{\textit{ij}}}^{\prime }\)

Stress components

\(\varepsilon _{{\textit{ij}}}\) and \({\varepsilon }_{{\textit{ij}}}^{\prime }\)

Strain components

\(E (E^{\prime })\), \(\upsilon \, (\upsilon ^{\prime })\) and \({\varOmega }\, ({\varOmega }^{\prime })\)

Young’s modulus, Poisson’s ratio and partial molar volume


Displacement components

\({\bar{\sigma }}_{{\textit{ij}}}\) and \({\bar{\sigma }}_{{\textit{ij}}}^{\prime }\)

Dimensionless stress

\({\bar{{\varOmega }}}\) and \({\bar{{\varOmega }}}^{\prime }\)

Dimensionless partial molar volume

\(\gamma \)

Ratio of Young’s moduli



We would like to acknowledge the support of National Natural Science Foundation of China under Grant Nos. 11702164, 11702166, 11672168 and 11332005, the Science and Technology Commission of Shanghai Municipality under Grant No. 14DZ2261200 and Shanghai Sailing Program No. 17YF1406000.


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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringShanghai UniversityShanghaiChina
  2. 2.Materials Genome InstituteShanghai University, and Shanghai Materials Genome InstituteShanghaiChina
  3. 3.Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiChina
  4. 4.Department of MechanicsShanghai UniversityShanghaiChina
  5. 5.Shanghai Key Laboratory of Mechanics in Energy EngineeringShanghai UniversityShanghaiChina

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