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Handbook of Mechanics of Materials pp 1–54Cite as

Micro-mechanics in Electrochemical Systems

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Abstract

A framework is presented that treats the combined effects of nonlinear elastic deformation, lattice constraints, and electrochemical potentials. The electro-chemo-mechanical diffusion potential is derived, and the particular case where ionic species are subject to a crystal lattice constraint is also derived. By combining energy balance and local entropy production, the framework provides a consistent method to treat the evolution of charged species which also carry anelastic deformations in a crystal lattice. The framework is used to derive a finite element formulation that applies to general cases of interest for diffusion of active species in battery electrodes and in fuel cells. We demonstrate the application of the finite element formulation first with two cases: ambipolar diffusion and kinetic demixing. The predicted system response demonstrates how mechanical effects cannot be disregarded, even in the presence of dominant electrostatic forces acting on ion transport. A cation-rich surface layer is predicted when elastic forces, due to Vegard’s stress, participate in the migration of defects in multicomponent oxides. Simulations show how stress plays a central role in the cation segregation to interfaces, a phenomenon regarded as critical to power and durability of solid oxide fuel cells. Then, we analyze the interplay between electro-chemo-mechanics and fracture in battery electrodes. The presence of fracture in the electrode particles perturbs the stress field and results in stress concentration around the crack tips, which locally affects lithium concentration. This may prevent full lithiation and promote further fracture propagation. Finally, we simulate mechanical degradation of all-solid-state batteries via fracture within the solid electrolyte material. Such cracks would block Li diffusion and reduce the composite electrode’s effective ionic conductivity.

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Notes

  1. 1.

    In this paper, we use ϖi as notation instead of the Larché–Cahn notation MiK. The second dependent species K is treated explicitly.

  2. 2.

    Larché and Cahn’s work mostly focused on the effect of self-stress, i.e., stress induced by nonhomogeneous concentration. In [35] Larché and Cahn took into account problems, which require “the simultaneous solution of the equations of elasticity and those of chemical equilibrium.” However, they were able to decouple the chemo-mechanical problem under the assumption of constant diffusion potential. The use of numerical methods permits solutions to a fully coupled electrochemical–mechanical problem with external applied loads.

  3. 3.

    Notation: ∇ and Div denote the gradient and divergence with respect to the material point in the reference configuration; a superposed dot denotes the material time-derivative. Vectors, second-, and fourth-order tensors are marked with bold and blackboard bold symbols, to distinguish them from scalar values. Throughout, we assume Fe − 1 = (Fe)−1 and Fe − T = ((Fe)−1)T.

  4. 4.

    The rate of change of species i in \( \mathcal{P} \) is given by

    $$ \frac{d}{dt}{\int}_{\mathcal{P}}{c}_i dV=-{\int}_{\partial \mathcal{P}}{\mathbf{J}}_i\cdot \mathbf{N} dA $$

    Bringing the time derivative inside the integral and applying the divergence theorem,

    $$ {\int}_{\mathcal{P}}\left({\dot{c}}_i+\mathrm{Div}\;{\mathbf{J}}_i\right)\; dV=0 $$

    which leads to the local form of the mass balance for the species i.

  5. 5.

    In the proposed application B will be taken as zero. No gravitational effects are considered, and electrostatic forces are not treated as body forces.

    A different approach to electromechanical coupling takes into account electric forces into the linear momentum balance [15]. Those forces are included in the form of body forces or by defining a total stress, as the sum of the classical mechanical stress and the Maxwell stress. Several contributions using this approach can be found in recent literature in applications to elastomeric and polymeric materials wherein electric field produces very large deformations. The key point is that the mechanical properties of these materials can be changed rapidly and reversibly by externally applied electric fields.

    Our treatment of electrostatic forces is limited to their impact on the transport of ionic species. We neglect their effect on the mechanical properties of the material and, therefore, assume negligible the contribution of the Maxwell stress in the equilibrium equation.

  6. 6.

    The Piola–Kirchhoff stress tensor P is related to the Cauchy stress tensor σ via the Piola transformation

    $$ \mathbf{P}= J\sigma {\mathbf{F}}^{-T} $$
    (4)

    with J = det (F) being the determinant of the deformation gradient. The Cauchy stress is defined in the current (or deformed) configuration, while the Piola–Kirchhoff stress is defined in the reference one. We make use of the Piola–Kirchhoff stress in compatibility with the other quantities expressed in the reference configuration. In classical linear mechanics the distinction between reference and Eulerian configuration is not taken into account, and the Cauchy stress is the most commonly used stress quantity.

  7. 7.

    If there were fields that could perform work through variations in any other quantity (such as dipoles, magnetic moments, total charge) within \( \mathcal{P} \), then these should be included in Eq. (3). We ignore such fields in this formulation. If the fields were included, they would appear in the coupled differential equations that are derived below in a manner analogous to B.

  8. 8.

    It follows that the elastic deformation is also a function of the species concentration Fe(χ,ci) = F(χ)Fa − 1(ci).

  9. 9.

    The coefficients βi correspond to the ratio between the lattice parameter of the intercalating species i and the one of the hosting compound. The relative volume change upon intercalation can be computed as the determinant of the anelastic deformation tensor. In linear kinematics, it can be approximated as follows

    $$ \frac{V+ dV}{V}=\det \left({\mathbf{F}}^a\right)={\left(1+\frac{1}{\rho_h}\sum \limits_i{\beta}_i{c}_i\right)}^3\to 1+3\ast \left(\frac{1}{\rho_h}\sum \limits_i{\beta}_i{c}_i\right) $$
    (16)

    such approximation is valid in the limit βi → 0. The factor 3βi/ρh corresponds to the partial molar volume of the intercalating species with respect to the hosting compound.

  10. 10.

    Given the following identities

    $$ \frac{\partial }{\partial {n}_i}=\frac{\partial }{\partial {c}_i}\frac{\partial {c}_i}{\partial {n}_i}=\frac{\partial }{\partial {c}_i}\frac{1}{V_0} $$
    (23)

    the first term in the state equation Eq. (25) of the diffusion potential i (in the case of transport of species uncoupled from mechanical and electrostatic effects) is compatible with the classical definition of the chemical potential as derivative of the internal energy (integrated over the reference volume) with respect to the number of moles ni of species i; where entropy, volume and number of moles of species k ≠ i are held constant. The second term in Eq. (25) corresponds to the stress-dependent component of the diffusion potential introduced by Larché and Cahn in [38]. If we used the partial molar volume, instead of the relative lattice parameter βi, in the definition of the anelastic deformation, a \( \frac{1}{3} \) factor would appear in the expression.

  11. 11.

    Not to be confused with J, a vector indicating the flux. A prescribed flux on the boundary will be represented with the scalar valued quantity \( \overline{J} \).

  12. 12.

    Operators used in Eq. (45) have the following definition

    $$ {\displaystyle \begin{array}{r}{\left(A\otimes A\right)}_{ij kl}={A}_{ij}{A}_{kl}\\ {}{\left(A\odot A\right)}_{ij kl}=\frac{1}{2}\left({A}_{ik}{A}_{jl}+{A}_{il}{A}_{jk}\right)\end{array}} $$

    and the fourth-order identity tensor \( \mathbb{I} \) is

    $$ {\left(\mathbb{I}\right)}_{ijkl}={\delta}_{ik}{\delta}_{jl} \; \mathbb{I}A=A. $$
  13. 13.

    Not to be confused with F, a tensor values variable, representing the deformation gradient.

  14. 14.

    Notation: the symbol ∇× denotes the gradient with respect to the deformed configuration. The following transformation between reference and spatial configuration (coordinate system marked respectively with X and x) applies

    • ∇×f = F−T ∇ f, f being a generic scalar function;

    • u = FU, u being a vector defined in the spatial configuration and U the corresponding vector in the material configuration.

  15. 15.

    For a discussion on the validity of the electroneutrality condition Eq. (52), we refer the reader to Sect. 11.8 of [43].

  16. 16.

    The chemo-mechanical component of the diffusion potential appearing in Eq. (53) corresponds to the following definition

    $$ {\mu}_i^{CM}\left({C}^e,{c}_i\right)=\frac{\partial {\varUpsilon}^M\left({C}^e,{c}_i\right)}{\partial {c}_i}+ RT\ \ln \left(\frac{c_i}{c_{i_{max}}-{c}_i}\right)+ RT\ \ln\ {\overline{\gamma}}_i-\frac{\beta_i}{\rho_h}\mathrm{Tr}\;\left({\mathbf{F}}^{eT}P\right) $$
  17. 17.

    In Eq. (53), we made use of the relation E = FTe between the electrostatic field e(x,t) = ∇×ϕ defined in the deformed configuration, the corresponding field E(X,t) = ∇ϕ in the Lagrangian configuration (see [15] for the complete derivation).

  18. 18.

    In order to avoid confusion with other symbols, calligraphic notation is used within this section to indicate the vector of nodal forces and the stiffness and mass matrix.

  19. 19.

    The directional or Gâteaux derivative of a generic function f about \( {x}_{n+1}^{(k)} \) is defined as

    $$ {\left.<D\;{f}^{(k)}\right|}_{n+1},\varDelta x>=\frac{d}{d\varepsilon}f{\left.\left({x}_{n+1}^{(k)}+\varepsilon\;\varDelta x\right)\right|}_{\varepsilon =0} $$
    (86)

Abbreviations

B :

body force per unit of reference volume

C e :

elastic right Cauchy–Green strain tensor

ℂ:

elasticity tensor

D i :

diffusivity of species i

\( \mathcal{D} \) :

dissipation density

E :

electric field in the reference configuration

F :

Faraday’s constant

\( {\mathcal{F}}^{ext} \) :

vector of external forces in the finite element discretization

\( {\mathcal{F}}^{int} \) :

vector of internal forces in the finite element discretization

F :

deformation gradient

F e :

elastic deformation gradient

F a :

anelastic deformation gradient

G :

shear modulus or second Lamé constant

I 1 :

first invariant of the elastic right Cauchy–Green strain tensor I1 = tr(Ce)

J :

Jacobian, J = det(Fe)

\( \overline{J} \) :

imposed normal flux on the Neumann boundary

J i :

outward flux of species i, measured per unit of undeformed area per unit time

J Q :

outward heat flux, measured per unit of undeformed area per unit time

\( {J}_{q_i} \) :

outward flux of charge i per unit of undeformed area per unit time

\( \mathcal{K} \) :

stiffness matrix in the finite element discretization

L :

film thickness

M i :

mobility of species i

N :

finite element shape function

\( \mathcal{M} \) :

mass matrix in the finite element discretization

P :

first Piola–Kirchhoff stress tensor

P e :

equilibrium first Piola–Kirchhoff stress tensor

Q :

distributed heat source per unit of reference volume

R :

gas constant

\( \mathcal{R} \) :

vector of residuals in the finite element discretization

T :

absolute temperature

\( \overline{T} \) :

applied traction

U :

internal energy per unit of undeformed volume V0

c i :

number of moles of the chemical species i per unit reference volume V0

\( {c}_{i_{max}} \) :

total number of sites per unit reference volume V0 available to the species i

c 0 :

initial concentration

\( \overline{c} \) :

imposed concentration on the Dirichlet boundary

c 0 :

initial concentration

e :

electric field in the spatial configuration

n i :

number of moles of the chemical species i

q i :

charge of the chemical species i per unit volume

t :

time

u :

displacement

v :

finite element test function

W ext :

external power

z i :

valence of species i

ϒ M :

elastic free energy density

ϒ EC :

electrochemical free energy density

\( {\varUpsilon}_{II}^{EC} \) :

excess of electrochemical free energy density for nonideal solution

α :

parameter in the trapezoidal time stepping algorithm

β i :

relative lattice constant of species i intercalating into hosting material

γ :

activity coefficient

s :

entropy per unit of reference volume

λ :

first Lamé constant

μ Θ :

chemical potential at the system reference state

\( {\mu}_i^{EC} \) :

electrochemical potential of the i − th chemical species

\( {\mu}_i^M \) :

stress-dependent component of the diffusion potential of species i

v :

Poisson’s ratio

\( {\overline{\omega}}_i \) :

diffusion potential of the i – th chemical species

ρ h :

molar density of the hosting compound

ϕ :

electropotential

χ :

deformation mapping

\( \overline{\chi} \) :

imposed value of the deformation mapping on the Dirichlet boundary

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Authors and Affiliations

  1. Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

    Giovanna Bucci & W. Craig Carter

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  1. Giovanna Bucci
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  2. W. Craig Carter
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Correspondence to Giovanna Bucci .

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Editors and Affiliations

  1. Materials Science and Strength of Materials, University of Stuttgart Institute for Materials Testing, Stuttgart, Germany

    Siegfried Schmauder

  2. Department of Civil Engineering, National Taiwan University Department of Civil Engineering, Taipei, T´ai-pei, Taiwan

    Chuin-Shan Chen

  3. BEC 254, University of Alabama at Birmingham, Birmingham, Alabama, USA

    Krishan K. Chawla

  4. Fulton School of Engineering, Arizona State University, Tempe, Arizona, USA

    Nikhilesh Chawla

  5. Engineering Mechanics, Zhejiang University Engineering Mechanics, Hangzhou, China

    Weiqiu Chen

  6. Katayanagi Advanced Research Laboratorie, Tokyo University of Technology Katayanagi Advanced Research Laboratorie, Tokyo, Japan

    Yutaka Kagawa

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Bucci, G., Carter, W.C. (2018). Micro-mechanics in Electrochemical Systems. In: Schmauder, S., Chen, CS., Chawla, K., Chawla, N., Chen, W., Kagawa, Y. (eds) Handbook of Mechanics of Materials. Springer, Singapore. https://doi.org/10.1007/978-981-10-6855-3_63-1

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