Mechano-Electrochemical Interaction in Solid-State Lithium Batteries

In this section, the governing equations for solving ion concentration, electric potential, and stress are presented. Given that diffusion induced volume strain of the cathode electrode can be directly given in terms of the applied current density and time, Li-ion transport in the cathode is not considered. Herein, Li-ion transport in the polymer electrolyte separator and Li plating at the electrode-electrolyte interface are focused on. For mechanics, the polymer electrolyte separator and cathode electrode are assumed to undergo linear elastic deformations, whereas Li metal has an elastic-plastic strain due to its low yield stress.³³ The above assumption is valid for relatively stiff polymer electrolyte with a large yield strain of elasticity.^34–36 Subsequently, the coupled model of mechanics-electrochemistry is applied to a one-dimensional (1D) problem under constant current condition.

Li-ion transport in the polymer electrolyte separator

Common lithium salts include LiPF₆, LiClO₄, and LiTFSI. Thus, monovalent ions are assumed to exist in the polymer electrolyte below. The flux of cations (Li⁺) and anions (denoted by X⁻) can be given by^37,38

$$\begin{eqnarray}&&{{\boldsymbol{N}}}_{+}=-{D}_{+}{\rm{\nabla }}{c}_{+}-{u}_{+}F{c}_{+}{\rm{\nabla }}\varphi -{u}_{+}{{\rm{\Omega }}}_{+}{c}_{+}{\rm{\nabla }}{p}_{{\rm{E}}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{{\boldsymbol{N}}}_{-}=-{D}_{-}{\rm{\nabla }}{c}_{-}+{u}_{-}F{c}_{-}{\rm{\nabla }}\varphi -{u}_{-}{{\rm{\Omega }}}_{-}{c}_{-}{\rm{\nabla }}{p}_{{\rm{E}}},\end{eqnarray} \tag{ 2 }$$

On the right-hand side, the first term describes the ion diffusion, and the second term represents the ion migration under electric field, which are consistent with the Nernst-Planck equation used in electrochemical systems.³⁹ The third term represents the pressure gradient-driven flux, and the previous theoretical studies have demonstrated that the pressure gradient has an impact on ion flux through polymer electrolytes.^37,38 It should be noted that if the Nernst-Einstein equation D = μRT is used, Eqs. 1 and 2 can be rewritten to the forms of those in previous studies.^37,38 D₊ (D₋) is the diffusion coefficient of Li⁺ ion (${X}^{-}$), c₊ (c₋) is the molar concentration of Li⁺ ion (${X}^{-}$), u₊ (u₋) is the mobility of Li⁺ ion (${X}^{-}$), Ω₊ (Ω₋) is the partial molar volume of Li⁺ ion (${X}^{-}$) in the polymer electrolyte, $\varphi$ is the electric potential, F is the Faraday's constant, and p_E is the hydrostatic stress in the polymer electrolyte calculated from the stress tensor σ_E

$$\begin{eqnarray}&&{p}_{E}=-\displaystyle \frac{1}{3}tr\left({{\boldsymbol{\sigma }}}_{{\rm{E}}}\right).\end{eqnarray} \tag{ 3 }$$

According to the Nernst–Einstein equation, the mobility is related to the diffusion coefficient by

$$\begin{eqnarray}&&{D}_{+}={u}_{+}RT,\,{D}_{-}={u}_{-}RT,\end{eqnarray} \tag{ 4 }$$

where R is the gas constant and T is the temperature.

If the solution is assumed to be electrically neutral, we have

$$\begin{eqnarray}&&{c}_{+}={c}_{-}=c.\end{eqnarray} \tag{ 5 }$$

Using the above equation, the electrical current density is given as

$$\begin{eqnarray}&&\begin{array}{l}{\boldsymbol{i}}=F\left({{\boldsymbol{N}}}_{+}-{{\boldsymbol{N}}}_{-}\right)=-F\left({D}_{+}-{D}_{-}\right){\rm{\nabla }}c\\ \,\,\,-\,\displaystyle \frac{{D}_{+}+{D}_{-}}{RT}{F}^{2}c{\rm{\nabla }}\varphi -\displaystyle \frac{{D}_{+}{{\rm{\Omega }}}_{+}-{D}_{-}{{\rm{\Omega }}}_{-}}{RT}Fc{\rm{\nabla }}{p}_{{\rm{E}}},\end{array}\end{eqnarray} \tag{ 6 }$$

Upon the substitution of Eq. 6 into Eqs. 1 and 2 by eliminating the potential gradient, it can be deduced

$$\begin{eqnarray}&&{{\boldsymbol{N}}}_{+}=-D{\rm{\nabla }}c+{t}_{+}\displaystyle \frac{{\boldsymbol{i}}}{F}-\displaystyle \frac{1}{2}\displaystyle \frac{D}{RT}{{\rm{\Omega }}}_{{\rm{E}}}c{\rm{\nabla }}{p}_{{\rm{E}}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{{\boldsymbol{N}}}_{-}=-D{\rm{\nabla }}c-\left(1-{t}_{+}\right)\displaystyle \frac{{\boldsymbol{i}}}{F}-\displaystyle \frac{1}{2}\displaystyle \frac{D}{RT}{{\rm{\Omega }}}_{{\rm{E}}}c{\rm{\nabla }}{p}_{{\rm{E}}},\end{eqnarray} \tag{ 8 }$$

Where

$$\begin{eqnarray}&&D=\displaystyle \frac{2{D}_{+}{D}_{-}}{{D}_{+}+{D}_{-}},\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{{\boldsymbol{\Omega }}}_{{\rm{E}}}={{\boldsymbol{\Omega }}}_{+}+{{\boldsymbol{\Omega }}}_{-},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{t}_{+}=\displaystyle \frac{{D}_{+}}{{D}_{+}+{D}_{-}}=\displaystyle \frac{{u}_{+}}{{u}_{+}+{u}_{-}}.\end{eqnarray} \tag{ 11 }$$

D is the electrolyte diffusion coefficient, Ω_E is the partial molar volume of LiX in the polymer electrolyte, and t₊ is the transference number, which represents the capability of current transport carried by Li⁺ ions.

In the polymer electrolyte, there is no production of Li⁺ ions. Thus, the accumulation of cations is caused by the net flux, which is expressed as

$$\begin{eqnarray}&&\displaystyle \frac{\partial {c}_{+}}{\partial t}+{\rm{\nabla }}\cdot {{\boldsymbol{N}}}_{+}=0.\end{eqnarray} \tag{ 12 }$$

Substituting Eq. 7 into Eq. 12, we have

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial c}{\partial t}+{\rm{\nabla }}\cdot \left(-D{\rm{\nabla }}c-\displaystyle \frac{1}{2}\displaystyle \frac{D}{RT}{{\rm{\Omega }}}_{{\rm{E}}}c{\rm{\nabla }}{p}_{{\rm{E}}}\right)\\ \,\,+\,\displaystyle \frac{{\rm{\nabla }}\cdot {t}_{+}}{F}\cdot {\boldsymbol{i}}-\displaystyle \frac{\left(1-{t}_{+}\right)}{F}{\rm{\nabla }}\cdot {\boldsymbol{i}}=0.\end{array}\end{eqnarray} \tag{ 13 }$$

For the conservation of charge, it gives

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{i}}=0.\end{eqnarray} \tag{ 14 }$$

If the transference number is independent of Li-ion concentration, the substitution of Eq. 14 into Eq. 13 yields

$$\begin{eqnarray}&&\displaystyle \frac{\partial c}{\partial t}+{\rm{\nabla }}\cdot {\boldsymbol{N}}=0,\end{eqnarray} \tag{ 15 }$$

where

$$\begin{eqnarray}&&{\boldsymbol{N}}=-D{\rm{\nabla }}c-\displaystyle \frac{1}{2}\displaystyle \frac{D}{RT}{{\rm{\Omega }}}_{{\rm{E}}}c{\rm{\nabla }}{p}_{{\rm{E}}}=0.\end{eqnarray} \tag{ 16 }$$

It should be noted that in addition to Eq. 12, the mass conservation of anions can lead to the same equations in 13 and 15.

The current density in Eq. 6 can be rephrased to

$$\begin{eqnarray}&&{\boldsymbol{i}}=-\kappa {\rm{\nabla }}\varphi -{\kappa }_{D}{\rm{\nabla }}\,\mathrm{ln}\,c-\displaystyle \frac{{D}_{+}{{\rm{\Omega }}}_{+}-{D}_{-}{{\rm{\Omega }}}_{-}}{RT}Fc{\rm{\nabla }}{p}_{{\rm{E}}},\end{eqnarray} \tag{ 17 }$$

Where

$$\begin{eqnarray}&&\kappa =\displaystyle \frac{{F}^{2}c}{RT}\left({D}_{+}+{D}_{-}\right),{\kappa }_{D}=\kappa \displaystyle \frac{RT}{F}\left(2{t}_{+}-1\right).\end{eqnarray} \tag{ 18 }$$

κ is the concentration-dependent conductivity, and κ_D is diffusion-induced conductivity. In the framework of concentrated solution theory, κ_D is modified as

$$\begin{eqnarray}&&\begin{array}{l}{\kappa }_{D}=\kappa \displaystyle \frac{2RT}{F}\left({t}_{+}-1\right)\left(1+\displaystyle \frac{\partial {f}_{\pm }}{\partial c}\right)\,{\rm{or}}\,\kappa \displaystyle \frac{RT}{F}\left({t}_{+}-1\right)\\ \,\,\times \,\left(1+\displaystyle \frac{\partial {f}_{\pm }}{\partial c}\right),\end{array}\end{eqnarray} \tag{ 19 }$$

where ${f}_{\pm }$ is the mean activity coefficient of the electrolyte.³⁹ Thus, the governing equation of charge transport in the electrolyte can be obtained by substituting Eq. 17 into Eq. 14

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \left(-\kappa {\rm{\nabla }}\varphi -{\kappa }_{D}{\rm{\nabla }}\,\mathrm{ln}\,c-\displaystyle \frac{{D}_{+}{{\rm{\Omega }}}_{+}-{D}_{-}{{\rm{\Omega }}}_{-}}{RT}Fc{\rm{\nabla }}{p}_{{\rm{E}}}\right)=0.\end{eqnarray} \tag{ 20 }$$

Interface

At the two electrolyte-electrode interfaces, the flux of anions is zero, which requires that

$$\begin{eqnarray}&&{{\boldsymbol{N}}}_{-}=-D{\rm{\nabla }}c-\left(1-{t}_{+}\right)\displaystyle \frac{{\boldsymbol{i}}}{F}-\displaystyle \frac{1}{2}\displaystyle \frac{D}{RT}{{\rm{\Omega }}}_{{\rm{E}}}c{\rm{\nabla }}{p}_{{\rm{E}}}=0.\end{eqnarray} \tag{ 21 }$$

Meanwhile, the current density is equal to the applied current density

$$\begin{eqnarray}&&i={i}_{{\rm{appl}}}.\end{eqnarray} \tag{ 22 }$$

At the Li metal-electrolyte interface, the electrochemical potential difference in the electrons, induced by stress, is expressed by Ref. 26

$$\begin{eqnarray}&&\begin{array}{l}{\rm{\Delta }}{\mu }_{{{\rm{e}}}^{-}}=-\displaystyle \frac{1}{2}\left[{{\rm{\Omega }}}_{{\rm{Li}}}+\left(1-{t}_{+}\right){{\rm{\Omega }}}_{{\rm{E}}}\right]\\ \,\,\,\,\times \,\left\{-\gamma {\rm{\nabla }}\cdot {{\bf{e}}}_{{\rm{n}}}+{{\bf{e}}}_{{\rm{n}}}\cdot \left[{{\bf{e}}}_{{\rm{n}}}\cdot \left(-{{\boldsymbol{\sigma }}}_{{\rm{Li}}}^{{\prime} }+{{\boldsymbol{\sigma }}}_{{\rm{E}}}^{{\prime} }\right)\right]\right\}\\ \,\,\,\,+\,\displaystyle \frac{1}{2}\left[{{\rm{\Omega }}}_{{\rm{Li}}}-\left(1-{t}_{+}\right){{\rm{\Omega }}}_{{\rm{E}}}\right]\left({\rm{\Delta }}{p}_{{\rm{Li}}}+{\rm{\Delta }}{p}_{{\rm{E}}}\right).\end{array}\end{eqnarray} \tag{ 23 }$$

where $\gamma$ is the surface energy, e_n is the unit vector normal to the interface, Δp_Li and Δp_E are the pressure changes of Li metal and polymer electrolyte at the interface, and the deviatoric stress tensors is defined as

$$\begin{eqnarray}&&{\boldsymbol{\sigma }}^{\prime} ={\boldsymbol{\sigma }}+p{\boldsymbol{\delta }}.\end{eqnarray} \tag{ 24 }$$

The electrical potential across the Li-electrolyte interface takes the form

$$\begin{eqnarray}&&{\varphi }_{I}={U}_{{\rm{Li}}}+\eta -\displaystyle \frac{{\rm{\Delta }}{\mu }_{{{\rm{e}}}^{-}}}{F},\end{eqnarray} \tag{ 25 }$$

where U_Li is the equilibrium potential, η is the overpotential on the Li metal surface. Using Eq. 25, the modified Butler–Volmer equation for the electrochemical reaction kinetics is thus deduced as³²

$$\begin{eqnarray}&&\begin{array}{l}{i}_{{\rm{BV}}}=F{k}_{{\rm{a}}}^{\beta }{\left({k}_{{\rm{c}}}{c}_{{\rm{I}}}\right)}^{\alpha }\exp \left(\displaystyle \frac{\alpha {\rm{\Delta }}{\mu }_{{{\rm{e}}}^{-}}}{RT}\right)\\ \,\,\,\times \,\left[\exp \left(\displaystyle \frac{\alpha F\eta }{RT}\right)-\exp \left(-\displaystyle \frac{\beta F\eta }{RT}\right)\right],\end{array}\end{eqnarray} \tag{ 26 }$$

Where i_BV is the current density generated by the electrochemical reactions at the Li-electrolyte interface, k_a and k_c are the anodic and cathodic rate constants, and α and β are the corresponding charge transfer coefficients.

Mechanics

Given that mechanical deformation is much faster than ion transport, the mechanical equilibrium equation reads

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{\sigma }}=0.\end{eqnarray} \tag{ 27 }$$

The constitutive equation of the polymer electrolyte is

$$\begin{eqnarray}&&{{\boldsymbol{\varepsilon }}}_{{\rm{E}}}={{\boldsymbol{M}}}_{{\rm{E}}}:{{\boldsymbol{\sigma }}}_{{\rm{E}}}+{\varepsilon }_{{\rm{E}}}^{0}{\boldsymbol{\delta }},\end{eqnarray} \tag{ 28 }$$

where ε_E is the strain tensor, δ is the Kronecker delta, and M_E is a forth-order tensor of elastic constants. In the regime of isotropic linear elasticity, we have

$$\begin{eqnarray}&&{{\boldsymbol{M}}}_{{\rm{E}}}=\displaystyle \frac{1}{{E}_{{\rm{E}}}}\left[\left(1+{\nu }_{{\rm{E}}}\right){\boldsymbol{I}}-{\nu }_{{\rm{E}}}{\boldsymbol{\delta }}{\boldsymbol{\delta }}\right].\end{eqnarray} \tag{ 29 }$$

Where E_E and v_E are the Young's modulus and Poisson's ratio of the polymer electrolyte, and I is the identity tensor.

In Eq. 28, ${\varepsilon }_{E}^{0}$ is the eigenstrain caused by ion concentration change, which is described as

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{E}}}^{0}={\left(1+{V}_{{\rm{E}}}\right)}^{\displaystyle \frac{1}{3}}-1\approx \displaystyle \frac{1}{3}{V}_{{\rm{E}}}=\displaystyle \frac{c-{c}_{0}}{3}{{\rm{\Omega }}}_{{\rm{E}}},\end{eqnarray} \tag{ 30 }$$

where V_E is the volumetric change and c₀ is the initial ion concentration.

For the cathode electrode, the constitutive equation is the same as the one in Eq. 27. For the Li metal anode, the constitutive equation is written as

$$\begin{eqnarray}&&{{\boldsymbol{\varepsilon }}}_{{\rm{Li}}}={{\boldsymbol{\varepsilon }}}_{{\rm{Li}}}^{{\rm{e}}}+{{\boldsymbol{\varepsilon }}}_{{\rm{Li}}}^{{\rm{p}}}={{\boldsymbol{M}}}_{{\rm{Li}}}:{{\boldsymbol{\sigma }}}_{{\rm{Li}}}+{{\boldsymbol{\varepsilon }}}_{{\rm{Li}}}^{{\rm{p}}},\end{eqnarray} \tag{ 31 }$$

where the strain is comprised of elastic strain and plastic strain. For the elastic deformation, the elastic constant tensor is expressed in terms of the Young's modulus E_Li and Poisson ratio v_Li

$$\begin{eqnarray}&&{{\boldsymbol{M}}}_{{\rm{Li}}}=\displaystyle \frac{1}{{E}_{{\rm{Li}}}}\left[\left(1+{\nu }_{{\rm{Li}}}\right){\boldsymbol{I}}-{\nu }_{{\rm{Li}}}{\boldsymbol{\delta }}{\boldsymbol{\delta }}\right].\end{eqnarray} \tag{ 32 }$$

Assuming that plastic deformation does not produce the volume change of Li metal, the trace of the plastic strain tensor is equal to zero, i.e.

$$\begin{eqnarray}&&{\rm{tr}}\left({{\boldsymbol{\varepsilon }}}_{{\rm{Li}}}^{{\rm{p}}}\right)=0.\end{eqnarray} \tag{ 33 }$$

For the threshold of plastic deformation, the yield surface is defined as

$$\begin{eqnarray}&&f\left({{\boldsymbol{\sigma }}}_{{\rm{Li}}},Y\right)=0,\end{eqnarray} \tag{ 34 }$$

where Y is a scalar function of plastic deformation history related to the current yield surface, which can be expressed as

$$\begin{eqnarray}&&{\rm{d}}Y\left({\bar{{\varepsilon }}}_{{\rm{Li}}}^{{\rm{p}}}\right)={E}_{{\rm{p}}}{\rm{d}}{\bar{{\varepsilon }}}_{{\rm{Li}}}^{{\rm{p}}},\end{eqnarray} \tag{ 35 }$$

where E_p is physically interpreted as the plastic modulus, and ${\bar{\varepsilon }}_{{\rm{Li}}}^{{\rm{p}}}$ is the effective plastic strain, defined as

$$\begin{eqnarray}&&{\bar{{\varepsilon }}}_{{\rm{Li}}}^{{\rm{p}}}=\displaystyle {\int }_{{\rm{history}}}\sqrt{\displaystyle \frac{2}{3}{\rm{d}}{{\boldsymbol{\varepsilon }}}_{{\rm{Li}}}^{{\rm{p}}}:{\rm{d}}{{\boldsymbol{\varepsilon }}}_{{\rm{Li}}}^{{\rm{p}}}},\end{eqnarray} \tag{ 36 }$$

Using the von Mises criterion, J₂ flow theory of plasticity gives the yield function

$$\begin{eqnarray}&&f\left({J}_{2},Y\right)={J}_{2}-\displaystyle \frac{{Y}^{2}}{2},\end{eqnarray} \tag{ 37 }$$

where J₂ is the second stress invariant, related to the deviatoric part of stress tensor

$$\begin{eqnarray}&&{J}_{2}=\displaystyle \frac{1}{2}{{\boldsymbol{\sigma }}}_{{\rm{Li}}}^{{\prime} }:{{\boldsymbol{\sigma }}}_{{\rm{Li}}}^{{\prime} }.\end{eqnarray} \tag{ 38 }$$

According to the consistency condition derived from Eq. 37, it can be deduced that

$$\begin{eqnarray}&&{\rm{d}}{{\boldsymbol{\varepsilon }}}_{{\rm{Li}}}^{{\rm{p}}}=\displaystyle \frac{1}{h}\displaystyle \frac{\partial f}{\partial {{\boldsymbol{\sigma }}}_{{\rm{Li}}}}\displaystyle \frac{\partial f}{\partial {{\boldsymbol{\sigma }}}_{{\rm{Li}}}}:{\rm{d}}{{\boldsymbol{\sigma }}}_{{\rm{Li}}}=\displaystyle \frac{9}{4{E}_{{\rm{p}}}{Y}^{2}}{{\boldsymbol{\sigma }}}_{{\rm{Li}}}^{{\prime} }{{\boldsymbol{\sigma }}}_{{\rm{Li}}}^{{\prime} }:{\rm{d}}{{\boldsymbol{\sigma }}}_{{\rm{Li}}},\end{eqnarray} \tag{ 39 }$$

where h designates a scalar parameter. Equation 39 builds the relationship between the plastic strain increment and the stress increment.

If the left boundary is fixed in Fig. 1, we have the mechanical boundary conditions in the y-axis direction

$$\begin{eqnarray}&&{{U}}_{1}=0,\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&{{U}}_{2}=\displaystyle \frac{{\sigma }_{yy}}{{K}},\end{eqnarray} \tag{ 41 }$$

where U₁ and U₂ are the y-axis displacements on the left and right boundaries, respectively, ${\sigma }_{yy}$ is the stress in the y-axis direction, and K is defined as the external stiffness which represents the buffering effect on the deformation of the multilayer structure.

In this section, the above model is applied to the layered structure in solid-state Li batteries, as shown in Fig. 1, which could be simplified to a one-dimensional (1D) problem. First, ion concentration in the polymer electrolyte separator is numerically solved. Then, the electric potential and stress can be analytically obtained.

Decoupled ion transport equation

For the Li metal, polymer electrolyte separator, and cathode electrode, the thickness of each layer is much smaller than the other two dimensions. Hence, we assume that Li-ion flux and electric field are along the y-axis direction (thickness direction), and that

$$\begin{eqnarray}&&{{\boldsymbol{\varepsilon }}}_{xx}={{\boldsymbol{\varepsilon }}}_{zz}=0.\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}&&{{\boldsymbol{\sigma }}}_{xx}={{\boldsymbol{\sigma }}}_{zz}.\end{eqnarray} \tag{ 43 }$$

If the two interfaces between electrodes and separator are intact, without delamination and slip, ${{\boldsymbol{\sigma }}}_{xx},$ ${{\boldsymbol{\sigma }}}_{yy}$ and ${{\boldsymbol{\sigma }}}_{zz}$ are the three principal stresses in each layer. In our macroscale model, interfacial excess stress between Li metal and polymer electrolyte separator is not considered.⁴⁰

In the thickness direction, the equilibrium equation of 27 implies that ${\sigma }_{yy}$ is constant, which follows that

$$\begin{eqnarray}&&{\sigma }_{{\rm{Li}}-yy}={\sigma }_{{\rm{E}}-yy}={\sigma }_{{\rm{C}}-yy}={\sigma }_{yy}.\end{eqnarray} \tag{ 44 }$$

For the polymer electrolyte, the substitution of Eqs. 42 and 43 into the constitutive equation of 28 leads to

$$\begin{eqnarray}&&{\sigma }_{yy}=\displaystyle \frac{{E}_{{\rm{E}}}}{1-2{\nu }_{{\rm{E}}}}\left[\displaystyle \frac{1-{\nu }_{{\rm{E}}}}{1+{\nu }_{{\rm{E}}}}{\varepsilon }_{{\rm{E}}-yy}-\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{E}}}}{3}\left(c-{c}_{0}\right)\right],\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&{\sigma }_{{\rm{E}}-xx}={\sigma }_{{\rm{E}}-zz}=\displaystyle \frac{{E}_{{\rm{E}}}}{1-2{\nu }_{{\rm{E}}}}\left[\displaystyle \frac{{\nu }_{{\rm{E}}}}{1+{\nu }_{{\rm{E}}}}{\varepsilon }_{{\rm{E}}-yy}-\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{E}}}}{3}\left(c-{c}_{0}\right)\right].\end{eqnarray} \tag{ 46 }$$

Thus, the solution of the mechanical problem could also be reduced to the y-axis direction, which is addressed in the sub-section of 3.2.

Because of the constant ${\sigma }_{yy}$ in Eq. 45, we have

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\sigma }_{yy}}{\partial y}=\displaystyle \frac{{E}_{{\rm{E}}}}{1-2{\nu }_{{\rm{E}}}}\left[\displaystyle \frac{1-{\nu }_{{\rm{E}}}}{1+{\nu }_{{\rm{E}}}}\displaystyle \frac{\partial {\varepsilon }_{{\rm{E}}-yy}}{\partial y}-\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{E}}}}{3}\displaystyle \frac{\partial }{\partial y}\left(c-{c}_{0}\right)\right]=0.\end{eqnarray} \tag{ 47 }$$

Using Eqs. 45–47, it gives

$$\begin{eqnarray}&&\displaystyle \frac{\partial {p}_{{\rm{E}}}}{\partial y}=-\displaystyle \frac{\partial }{\partial y}{\rm{tr}}\left({{\boldsymbol{\sigma }}}_{{\rm{E}}}\right)=\displaystyle \frac{2{E}_{{\rm{E}}}{{\rm{\Omega }}}_{{\rm{E}}}}{9\left(1-{\nu }_{{\rm{E}}}\right)}\displaystyle \frac{\partial c}{\partial y}.\end{eqnarray} \tag{ 48 }$$

Although σ_yy is constant, the pressure gradient exists in the y-axis direction through the polymer electrolyte separator due to the varying stresses of σ_xx and σ_zz. Equation 48 indicates that the pressure gradient is proportional to the Li-ion concentration gradient in the y-axis direction. Upon the substitution of Eq. 48 into Eq. 15, we have

$$\begin{eqnarray}&&\displaystyle \frac{\partial c}{\partial t}-\displaystyle \frac{\partial }{\partial y}\left[D\left(1+\theta \displaystyle \frac{c}{{c}_{0}}\right)\displaystyle \frac{\partial c}{\partial y}\right]=0.\end{eqnarray} \tag{ 49 }$$

where

$$\begin{eqnarray}&&\theta =\displaystyle \frac{{E}_{{\rm{E}}}{c}_{0}{{\rm{\Omega }}}_{{\rm{E}}}^{2}}{9\left(1-{\nu }_{{\rm{E}}}\right)RT}.\end{eqnarray} \tag{ 50 }$$

θ is a non-dimensional parameter, named as the coupling coefficient in this study. Correspondingly, the boundary condition of Eq. 21 is rewritten as

$$\begin{eqnarray}&&-D\left(1+\theta \displaystyle \frac{c}{{c}_{0}}\right)\displaystyle \frac{\partial c}{\partial y}=-\left(1-{t}_{+}\right)\displaystyle \frac{{i}_{{\rm{appl}}}}{F},\end{eqnarray} \tag{ 51 }$$

For the initial condition, ion is uniformly distributed through the polymer electrolyte

$$\begin{eqnarray}&&c(y,t=0)={c}_{0}.\end{eqnarray} \tag{ 52 }$$

Upon the substitution of Eq. 48 into Eq. 17, we have

$$\begin{eqnarray}&&\begin{array}{l}-\kappa \displaystyle \frac{\partial \varphi }{\partial y}-{\kappa }_{D}\displaystyle \frac{\partial \left(\mathrm{ln}\,c\right)}{\partial y}-2\theta \left({D}_{+}\displaystyle \frac{{{\rm{\Omega }}}_{+}}{{{\rm{\Omega }}}_{{\rm{E}}}}-{D}_{-}\displaystyle \frac{{{\rm{\Omega }}}_{-}}{{{\rm{\Omega }}}_{{\rm{E}}}}\right)\\ \,\times \,F\displaystyle \frac{c}{{c}_{0}}\displaystyle \frac{\partial c}{\partial y}=-{i}_{{\rm{appl}}}.\end{array}\end{eqnarray} \tag{ 53 }$$

As a result, Li-ion concentration distribution in the polymer electrolyte separator can be directly obtained from Eqs. 49, 51 and 52. Then, the electric potential is analytically solved from Eq. 53. Because of the pressure effect on ionic flux in Eqs. 1 and 2, the coupling coefficient is present in Eqs. 49 and 53, which indicates that pressure gradient could affect the distributions of ion concentration and electric potential in the polymer electrolyte.

In this work, the effects of pressure on electrochemical performance have two aspects. First, Eq. 1 or Eq. 48 shows that the pressure gradient affects ionic transport, which has been demonstrated in previous studies.^37,38 Second, Eqs. 23 and 26 shows that pressure affects electrochemical reaction kinetics, which has been addressed in previous studies.^27,32

Solution for mechanics

In the following, the analytical solution of stress is provided, focusing on the stress component in the y-axis direction.

From the constitutive equation, the y-axis strain in the cathode electrode is

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{C}}-yy}=\displaystyle \frac{\left(1-2{\nu }_{{\rm{C}}}\right)\left(1+{\nu }_{{\rm{C}}}\right)}{{E}_{{\rm{C}}}\left(1-{\nu }_{{\rm{C}}}\right)}{\sigma }_{yy}+\displaystyle \frac{1+{\nu }_{{\rm{C}}}}{1-{\nu }_{{\rm{C}}}}\displaystyle \frac{c-{c}_{0}}{3}{{\rm{\Omega }}}_{{\rm{C}}}.\end{eqnarray} \tag{ 54 }$$

For the polymer electrolyte separator, we have the strain in the y-axis direction as

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{E}}-yy}=\displaystyle \frac{\left(1-2{\nu }_{{\rm{E}}}\right)\left(1+{\nu }_{{\rm{E}}}\right)}{{E}_{{\rm{E}}}\left(1-{\nu }_{{\rm{E}}}\right)}{\sigma }_{yy}+\displaystyle \frac{1+{\nu }_{{\rm{E}}}}{1-{\nu }_{{\rm{E}}}}\displaystyle \frac{c-{c}_{0}}{3}{{\rm{\Omega }}}_{{\rm{E}}}.\end{eqnarray} \tag{ 55 }$$

Li metal has an elastic-plastic strain, and the parameter Y in Eq. 34 needs to be first determined from uniaxial tensile test. For simplicity, the linear isotropic hardening model is used. Thus, the yield stress can be expressed as

$$\begin{eqnarray}&&{\sigma }_{{\rm{u}}}=\displaystyle \frac{{E}_{{\rm{Li}}}{E}_{{\rm{t}}}}{{E}_{{\rm{Li}}}-{E}_{{\rm{t}}}}{\varepsilon }_{{\rm{Li}}}^{{\rm{p}}}+{\sigma }_{{\rm{s}}},\end{eqnarray} \tag{ 56 }$$

Where ${\varepsilon }_{{\rm{u}}}^{{\rm{p}}},$ ${\sigma }_{{\rm{s}}},$ and E_t are the plastic strain, the initial yield stress, and the post-yield tangent modulus from the stress-strain curve in uniaxial tensile test. Substituting Eq. 56 into Eq. 37 and noting that ${\varepsilon }_{{\rm{u}}}^{{\rm{p}}}={\bar{\varepsilon }}_{{\rm{Li}}}^{{\rm{p}}}$ deduced from Eq. 36, we have

$$\begin{eqnarray}&&Y={E}_{{\rm{p}}}{\bar{\varepsilon }}_{{\rm{Li}}}^{{\rm{p}}}+{\sigma }_{{\rm{s}}}.\end{eqnarray} \tag{ 57 }$$

where the plastic modulus E_p = E_LiE_t/(E_Li−E_t).

The Li metal in Fig. 1 has a stress state of (${\sigma }_{xx},$ ${\sigma }_{yy},$ ${\sigma }_{zz}$). Using Eqs. 37 and 43, we obtain

$$\begin{eqnarray}&&\left|{\sigma }_{yy}-{\sigma }_{Li-xx}\right|=Y.\end{eqnarray} \tag{ 58 }$$

Using Eqs. 31, 33, 57, 58, we can deduce that

$$\begin{eqnarray}&&\begin{array}{l}{\varepsilon }_{{\rm{Li}}-yy}=\displaystyle \frac{1-2{\nu }_{{\rm{Li}}}}{{E}_{{\rm{Li}}}}\left\{2\left(1+2\displaystyle \frac{{E}_{{\rm{p}}}}{{E}_{{\rm{Li}}}}{\nu }_{{\rm{Li}}}\right){\left[1+2\displaystyle \frac{{E}_{{\rm{p}}}}{{E}_{{\rm{Li}}}}\left(1-{\nu }_{{\rm{Li}}}\right)\right]}^{-1}\right.\\ \left.\,\,\,\,+\,1\right\}{\sigma }_{yy}+2\displaystyle \frac{1-2{\nu }_{{\rm{Li}}}}{{E}_{{\rm{Li}}}}{\left[1+2\displaystyle \frac{{E}_{{\rm{p}}}}{{E}_{{\rm{Li}}}}\left(1-{\nu }_{{\rm{Li}}}\right)\right]}^{-1}{\sigma }_{{\rm{s}}}.\end{array}\end{eqnarray} \tag{ 59 }$$

Thus, the y-axis strains in the three layers are expressed in terms of the stress ${\sigma }_{yy}.$ In the y-axis direction, we have the compatibility of deformation from Eq. 41

$$\begin{eqnarray}&&\begin{array}{l}{l}_{0}+\displaystyle {\int }_{-\left({l}_{{\rm{Li}}}+{l}_{0}\right)}^{0}{\varepsilon }_{{\rm{Li}}-yy}{\rm{d}}y+\displaystyle {\int }_{0}^{{l}_{{\rm{E}}}}{\varepsilon }_{{\rm{E}}-yy}{\rm{d}}y+\displaystyle {\int }_{{l}_{{\rm{E}}}}^{{l}_{{\rm{E}}}+{l}_{{\rm{C}}}}{\varepsilon }_{{\rm{C}}-yy}{\rm{d}}y\\ \,+\,\displaystyle \frac{{\sigma }_{yy}}{K}=0,\end{array}\end{eqnarray} \tag{ 60 }$$

where l_Li, l_E, and l_C are the thicknesses of the Li metal, polymer electrolyte separator, and cathode electrode, respectively, K is the stiffness applied to the layered structure, and l₀ is the thickness of the newly deposited Li. For a rigid constraint to the structure, K is infinite. The coordinate origin is set at the Li metal-separator interface, which is movable during plating.

At a time of t, it readily follows that

$$\begin{eqnarray}&&{l}_{0}=\displaystyle \frac{{i}_{{\rm{appl}}}t}{F}{{\rm{\Omega }}}_{{\rm{Li}}}.\end{eqnarray} \tag{ 61 }$$

Substituting Eqs. 54, 55, and 59 into Eq. 60, it gives

$$\begin{eqnarray}&&\begin{array}{l}{\sigma }_{yy}=\left[{C}_{1}\left({l}_{{\rm{Li}}}+{l}_{0}\right)+\displaystyle \frac{\left(1-2{\nu }_{{\rm{C}}}\right)\left(1+{\nu }_{{\rm{C}}}\right)}{{E}_{{\rm{C}}}\left(1-{\nu }_{{\rm{C}}}\right)}{l}_{{\rm{C}}}\right.\\ {\left.+\displaystyle \frac{\left(1-2{\nu }_{{\rm{E}}}\right)\left(1+{\nu }_{{\rm{E}}}\right)}{{E}_{{\rm{E}}}\left(1-{\nu }_{{\rm{E}}}\right)}{l}_{{\rm{E}}}+\displaystyle \frac{1}{K}\right]}^{-1}{C}_{2},\end{array}\end{eqnarray} \tag{ 62 }$$

where

$$\begin{eqnarray}&&{C}_{1}=\displaystyle \frac{1-2{\nu }_{{\rm{Li}}}}{{E}_{{\rm{Li}}}}\left\{2\left(1+2\displaystyle \frac{{E}_{{\rm{p}}}}{{E}_{{\rm{Li}}}}{\nu }_{{\rm{Li}}}\right){\left[1+2\displaystyle \frac{{E}_{{\rm{p}}}}{{E}_{{\rm{Li}}}}\left(1-{\nu }_{{\rm{Li}}}\right)\right]}^{-1}+1\right\},\end{eqnarray} \tag{ 63 }$$

$$\begin{eqnarray}&&\begin{array}{l}{C}_{2}=-2\displaystyle \frac{1-2{\nu }_{{\rm{Li}}}}{{E}_{{\rm{Li}}}}{\left[1+2\displaystyle \frac{{E}_{{\rm{p}}}}{{E}_{{\rm{Li}}}}\left(1-{\nu }_{{\rm{Li}}}\right)\right]}^{-1}{\sigma }_{{\rm{s}}}\left({l}_{{\rm{Li}}}+{l}_{0}\right)-{l}_{0}\\ \,\,\,+\,{\rm{\Delta }}{l}_{{\rm{C}}0}.\end{array}\end{eqnarray} \tag{ 64 }$$

Other stress components can be obtained from the constitutive equation. For the deduction of Eq. 62, two equations are used

$$\begin{eqnarray}&&{\rm{\Delta }}{l}_{{\rm{E}}0}=-\displaystyle {\int }_{0}^{{l}_{{\rm{E}}}}\displaystyle \frac{1+{\nu }_{{\rm{E}}}}{1-{\nu }_{{\rm{E}}}}\displaystyle \frac{c-{c}_{0}}{3}{{\rm{\Omega }}}_{{\rm{E}}}dy=0,\end{eqnarray} \tag{ 65 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{l}_{{\rm{C}}0}=-\displaystyle {\int }_{{l}_{{\rm{E}}}}^{{l}_{{\rm{E}}}+{l}_{{\rm{C}}}}\displaystyle \frac{1+{\nu }_{{\rm{C}}}}{1-{\nu }_{{\rm{C}}}}\displaystyle \frac{c-{c}_{0}}{3}{{\rm{\Omega }}}_{{\rm{C}}}{\rm{d}}y=\displaystyle \frac{1+{\nu }_{{\rm{C}}}}{1-{\nu }_{{\rm{C}}}}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{C}}}}{3}\displaystyle \frac{{i}_{{\rm{appl}}}t}{F}.\end{eqnarray} \tag{ 66 }$$

The first equation implies that ion transport in the polymer electrolyte separator does not induce a thickness change in the y-axis direction when the average ion concentration is kept at c₀. In contrast, Δl_C0 in Eq. 66 is the thickness shrinkage of the cathode electrode because of delithiation.

Equation 62 holds when Li metal is subject to elastic-plastic strain. If no plastic strain is generated in Li metal, and C₁ and C₂ are revised to

$$\begin{eqnarray}&&{C}_{1}=\displaystyle \frac{\left(1-2{\nu }_{{\rm{Li}}}\right)\left(1+{\nu }_{{\rm{Li}}}\right)}{{E}_{{\rm{Li}}}\left(1-{\nu }_{{\rm{Li}}}\right)},{C}_{2}=-{l}_{0}+{\rm{\Delta }}{l}_{{\rm{C}}0}.\end{eqnarray} \tag{ 67 }$$

A special situation is that the thickness increased in Li metal is equal to the thickness decreased in cathode electrode, i.e.

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{l}_{{\rm{C}}0}}{{l}_{0}}=\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{C}}}}{3{{\rm{\Omega }}}_{{\rm{Li}}}}\displaystyle \frac{1+{\nu }_{{\rm{C}}}}{1-{\nu }_{{\rm{C}}}}=1.\end{eqnarray} \tag{ 68 }$$

Then, all the stress components in Li metal are zero, and the polymer electrolyte separator and cathode electrode are not subject to the stress in the y-axis direction. Conversely, ${\sigma }_{yy}$ is generated when Eq. 68 is not satisfied.

As mentioned in the above section, ion concentration in the polymer electrolyte separator is can be numerically solved for the layered structure. Using the ion concentration profile, the electric potential and stress can be analytically obtained.

If the mechano-electrochemical coupling is neglected, i.e. the coupling coefficient θ = 0, the system can maintain a maximum current density, which is deduced from Eq. 51

$$\begin{eqnarray}&&{i}_{\mathrm{lim}\,{\rm{it}}}^{0}=\displaystyle \frac{2FD{c}_{0}}{\left(1-{t}_{+}\right){l}_{{\rm{E}}}}.\end{eqnarray} \tag{ 69 }$$

${i}_{\mathrm{lim}\,{\rm{it}}}^{0}$ is the limiting current density widely used in the literature. Here, we introduce the non-dimensional current density

$$\begin{eqnarray}&&\chi =\displaystyle \frac{{i}_{{\rm{appl}}}}{{i}_{\mathrm{lim}\,{\rm{it}}}^{0}}.\end{eqnarray} \tag{ 70 }$$

Furthermore, we introduce the non-dimensional ion concentration, coordinate, and time

$$\begin{eqnarray}&&\bar{c}=\displaystyle \frac{c}{{c}_{0}},\,\bar{y}=\displaystyle \frac{y}{{l}_{{\rm{E}}}},\,\bar{t}=\displaystyle \frac{Dt}{{l}_{{\rm{E}}}^{2}}.\end{eqnarray} \tag{ 71 }$$

Thus, the governing Eq. 49, the boundary condition 51, and the initial condition 52 can be rewritten in the non-dimensional forms. Unless otherwise indicated, the material parameters in Table I are used, leading to a coupling coefficient of θ = 0.3.

Figures 2a and 2b show the profiles of ion concentration and x-axis stress in the polymer electrolyte separator under χ = 0.5 at two times, respectively. Two cases are considered: θ = 0 (without stress effect on ion transport) and θ = 0.3. In comparison to the case of θ = 0, ion concentration is more uniform for θ = 0.3, leading to a relatively smaller concentration gradient. It indicates that stress can facilitate ion transport in the electrolyte. At the time of $\bar{t}$ = 0.6, ion concentration reaches the steady state. For θ = 0, the concentration is linearly distributed in the thickness direction, forming a constant gradient. For θ = 0.3, the concentration gradient continuously decreases along the y-axis direction in a gentle way, which can be confirmed by Eq. 49. Corresponding to the ion concentration distribution, non-linear ${\sigma }_{xx}$ distribution is generated at the time of $\bar{t}$ = 0.03 in Fig. 2b. For the two cases, the stress is always asymmetrical in the polymer electrolyte separator.

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Figures 2c and 2d show the concentration ${c}_{{\rm{I}}}$ at the Li metal-electrolyte interface and y-axis stress ${\sigma }_{yy}$ with time under the non-dimensional current density χ = 1. As time increases, the interface concentration decreases and approaches a steady-state value. For θ = 0, the interface concentration will eventually approach zero, whereas the interface concentration will be larger than zero for the case of θ = 0.3. Figure 2d shows the evolution of stress ${\sigma }_{yy},$ consisting of two stages. The transition distinguishing the two stages occurs at a very early plating stage. In the second stage, Li metal experiences elastic-plastic deformation. It can be seen that the stress linearly varies with time, which is in good agreement with previous studies.^30,31 For instance, it was observed that the stress in the thickness direction approximately has a linear relationship with time for all-solid-state batteries.³¹

To further explore the mechano-electrochemical coupling, Fig. 3 shows the dependence of the coupling coefficient on transport characteristics. Sand's time is the required time for ion depletion in the vicinity of the electrode-electrolyte interface. Physically, ion concentration is reduced to zero when t > t_Sand, and the system could not support the applied current due to diffusion limitation in the electrolyte. For θ = 0, Sand's time is expressed as⁴¹

$$\begin{eqnarray}&&{t}_{{\rm{Sand}}}=\pi D{\left[\displaystyle \frac{F{c}_{0}}{2{i}_{{\rm{appl}}}\left(1-{t}_{+}\right)}\right]}^{2}.\end{eqnarray} \tag{ 72 }$$

Using Eqs. 68–72, we have the non-dimensional Sand's time

$$\begin{eqnarray}&&{\bar{t}}_{{\rm{Sand}}}=\displaystyle \frac{D}{{l}_{{\rm{E}}}^{2}}{t}_{{\rm{Sand}}}=\displaystyle \frac{\pi }{16{\chi }^{2}}.\end{eqnarray} \tag{ 73 }$$

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In the case of θ = 0, Fig. 3a shows that the calculated Sand's time is consistent with the theoretical prediction by Eq. 73. In contrast, Sand's time is delayed for θ = 0.3. Delayed Sand's time helps to avoid dendrite formation during electrodeposition, which indicates that stress could increase the onset time of ion transport-driven instability at high currents. In addition, a linear trend is shown in the log-log plot for Sand's time and the current density, which is in good agreement with that of earlier experiments.⁴²

Figure 3b illustrates the limiting current density with varying coupling coefficient, which demonstrates that a stronger mechano-electrochemical coupling leads to a higher limiting current density. It is evident that a higher critical current density could suppress the tendency to dendrite formation. Correspondingly, Fig. 3c shows the phase map of electrodeposition stability. For θ = 0, unstable electrodeposition is caused when the non-dimensional applied current density χ ≥ 1. Increasing the coupling coefficient enhances the stability, pushing unstable electrodeposition to the stable regime, and the line depicts the transition boundary for the two regimes. Thus, stress could improve electrochemical performance by increasing the threshold of applied current to initiate dendrite formation and subsequent penetration through the separator.

Figure 4a shows the dimensionless surface overpotential on Li metal surface under χ = 0.5, which is calculated by using i_appl = i_BV. Equation 26 indicates that the overpotential relies on the ion concentration ${c}_{{\rm{I}}}$ and stress induced electrochemical potential difference ${\rm{\Delta }}{\mu }_{{{\rm{e}}}^{-}}$ at the electrode-electrolyte interface. In Fig. 4b, the two effects are compared for θ = 0 and θ = 0.3, which provides the reason why the overpotential is lower for θ = 0.3. Figures 4c and 4d explain the trend of the overpotential with time. In the early stage of plating, the overpotential dramatically increases, primarily attributed to the interface concentration decrease in Fig. 4c. Even after the interface concentration ${c}_{{\rm{I}}}$ reaches a steady-state value, the overpotential still has a very gentle increasing trend with time, which is completely due to the decreasing ${\rm{\Delta }}{\mu }_{{{\rm{e}}}^{-}}$ with time in Fig. 4d. For the flat electrode-electrolyte interface, the contribution from surface energy is zero, and Eq. 23 is reduced to

$$\begin{eqnarray}&&\begin{array}{l}{\rm{\Delta }}{\mu }_{{{\rm{e}}}^{-}}=-\displaystyle \frac{1}{2}\left[{{\rm{\Omega }}}_{{\rm{Li}}}+\left(1-{t}_{+}\right){{\rm{\Omega }}}_{{\rm{E}}}\right]\left(-{\sigma }_{{\rm{Li}}-yy}^{{\prime} }+{\sigma }_{{\rm{E}}-yy}^{{\prime} }\right)\\ \,\,\,\,\,+\,\displaystyle \frac{1}{2}\left[{{\rm{\Omega }}}_{{\rm{Li}}}-\left(1-{t}_{+}\right){{\rm{\Omega }}}_{{\rm{E}}}\right]\left({\rm{\Delta }}{p}_{{\rm{Li}}}+{\rm{\Delta }}{p}_{{\rm{E}}}\right).\end{array}\end{eqnarray} \tag{ 74 }$$

Thus, ${\rm{\Delta }}{\mu }_{{{\rm{e}}}^{-}}$ consists of two terms, originating from the deviatoric stress and the hydrostatic pressure at the interface, respectively. Their effects are illustrated in Fig. 4d, where the green circle denotes the plasticity transition of Li metal. Deviatoric stress always induces a positive electrochemical potential difference ${\rm{\Delta }}{\mu }_{{e}^{-}},$ which contributes to reducing the overpotential $\bar{\eta }$ in Fig. 4a. In contrast, ${\rm{\Delta }}{\mu }_{{{\rm{e}}}^{-}}$ caused by hydrostatic pressure continues to decrease with time after an initial increase. By comparing the two slopes when $\bar{t}$ > 0.4, it can be concluded that the total ${\rm{\Delta }}{\mu }_{{{\rm{e}}}^{-}}$ decreases with time, which accounts for the gentle increasing trend of $\bar{\eta }$ in Fig. 4a.

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Figure 5 shows the K-dependent stresses. A larger K denotes a higher external elastic stiffness to the layered structure. The extreme of K = +∞ represents a rigid constraint, and the corresponding stresses are given in Fig. 5. As expected, the compressive stress ${\sigma }_{yy}$ in the thickness direction increases and converges to the value for K = +∞. Hence, a flexible constraint can buffer the volumetric changes of two electrodes. Two electrodeposition thicknesses are considered, i.e. l₀ = 7.5 μm and l₀ = 15 μm, and Fig. 5a indicates that the stress for l₀ = 15 μm is almost twice that of the case l₀ = 7.5 μm, which complies with the linear trend in Fig. 2d.

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For Li metal and the cathode electrode, von Mises stress ${\sigma }_{{\rm{M}}}$ is used

$$\begin{eqnarray}&&{\sigma }_{{\rm{M}}}^{2}=3{J}_{2}=\displaystyle \frac{3}{2}\sigma ^{\prime} :\sigma ^{\prime} .\end{eqnarray} \tag{ 75 }$$

The von Mises stress in Li metal is uniform, while it varies in the cathode electrode along the y-axis direction due to non-uniform Li-ion concentration. Here, the average von Mises stress is calculated for the cathode electrode. To obtain the average von Mises stress, the average stresses in the x-axis and z-axis directions are deduced as

$$\begin{eqnarray}&&{\sigma }_{{\rm{C}}-xx}^{{\rm{A}}}={\sigma }_{{\rm{C}}-zz}^{{\rm{A}}}=\displaystyle \frac{{\nu }_{{\rm{C}}}}{1-{\nu }_{{\rm{C}}}}{\sigma }_{yy}+\displaystyle \frac{{E}_{{\rm{C}}}}{1-{\nu }_{{\rm{C}}}}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{C}}}}{3{{\rm{\Omega }}}_{{\rm{Li}}}}\displaystyle \frac{{l}_{0}}{{l}_{{\rm{C}}}}.\end{eqnarray} \tag{ 76 }$$

Figures 5b and 5c illustrate the von Mises stresses in Li metal and the cathode electrode, respective. Figure 5b demonstrates that Li metal only undergoes elastic deformation when K is very small. Because of the plastic strain, the von Mises stress in Li metal for l₀ = 15 μm is lower than twice of the stress for l₀ = 7.5 μm. Figure 5c shows that the von Mises stress in cathode electrode decreases with decreasing K. The elastic stiffness K directly releases the y-axis stress ${\sigma }_{yy}$ in our model, while the second term in Eq. 76 is independent of K. Therefore, the dependence of K on the von Mises stress in cathode electrode is not as strong as that in Fig. 5a.

The aforementioned results demonstrate the importance of mechanics in solid-state batteries. A larger coupling coefficient θ improves electrochemical performance, resulting in a higher limiting current density, a longer Sand's time, and a lower surface overpotential on Li metal, which contribute to the suppression of dendrite formation due to transport limitation. Based on Eq. 50, strategies towards the increase of θ include: increasing the Young's modulus E_C, bulk ion concentration c₀, partial molar volume Ω_C, and Poisson's ratio ${{\boldsymbol{\nu }}}_{{\rm{C}}}.$ From a mechanics standpoint, a flexible constraint to the layered structure can release the volumetric changes of electrodes and thus reduce stresses, which helps to avoid mechanical failures in solid-state batteries. In this work, the role of mechano-electrochemical coupling in suppressing the onset of dendrite formation is discussed under the assumption that Li plating is uniform at the electrode-electrolyte interface. However, uneven Li plating, originating from spatial heterogeneity, allows for a curved growing interface, and the role of mechano-electrochemical coupling in suppressing dendrite growth will be our future focus.

In our model, the key assumptions are summarized below.

• (1)  
  The model is developed for binary solutions under the framework of dilute solution theory, in which migration is independent of diffusion, as indicated in Eqs. 1 and 2. For concentrated solutions, the equation for ion flux needs to be modified to consider the interactions between diffusing ions, while the equations for mass balances, current flow, and electroneutrality remain valid for concentrated solutions.³⁹
• (2)  
  The film thicknesses of the Li anode, separator, and cathode are assumed to be much smaller than the sizes in other dimensions, i.e. the x-axis and z-axis directions in our model. Thus, the strain components are assumed to be 0 in the x-axis and z-axis directions, while the stress components exist in those two directions. If the film thickness is comparable to the sizes in other dimensions, Eq. 42 will be affected.
• (3)  
  The linear isotropic hardening model is used for the elastic-plastic behavior of Li metal anode. During plastic deformation, the yield stress is assumed to linearly varies with the strain based on a previous experiment.³³
• (4)  
  The cathode is assumed to be a uniform and isotropic electrode. Given that we aim to develop a model at the macroscale, the microstructure of cathode is not considered. Thus, our model could not describe the physical field and electrochemical behavior of various phases in the composite electrode.

In summary, a coupled model of mechanics-electrochemistry is developed for polymer electrolyte based solid-state batteries during Li plating, aimed at addressing the mechanism of stress tuned electrochemical performance. In the confined space of battery, the volumetric changes of two electrodes generate significant stresses in the Li metal film, solid electrolyte separator, and cathode electrode. For the polymer electrolyte separator, stress is found to enhance ion transport, resulting in a delayed Sand's time and increased critical current density, which contributes to averting the formation of Li dendrite formation, and this effect can be described by a dimensionless coupling coefficient. Stress at the Li metal-electrolyte interface is demonstrated to affect the electrochemical reaction kinetics, and the influences on the overpotential are different for the two parts from deviatoric stress and hydrostatic pressure. In addition, the external stiffness applied to the layered structure also determines the value of stress generated during Li plating, and a low stiffness can effectively buffer the volumetric changes. This fundamental study underlines the role of mechanics in the electrochemical performance of solid-state batteries, which provides guidance for the battery design towards stable electrodeposition and robust layered structure.

The information, data, or work presented herein was funded in part by the Office of Energy Efficiency and Renewable Energy (EERE), U.S. Department of Energy, under Award DE-EE0007766, and Purdue University faculty research grant.