Diffusion aspects of lithium intercalation as applied to the development of electrode materials for lithium-ion batteries

Abstract

The creation of new electrode materials and the modification of existing ones are important trends in the development of lithium-ion batteries. Of special significance is to evaluate their diffusivity, i.e., the ability of providing transfer of the electroactive component. Such electrochemical techniques as cyclic voltammetry, electrochemical impedance spectroscopy, potentiostatic intermittent titration technique, and galvanostatic intermittent titration technique are used for this purpose. The values of chemical diffusion coefficient D estimated in similar electrode materials are shown to scatter by several orders of magnitude. Principal causes of this rather considerable scattering are discussed, including the uncertainty of diffusion area estimations and the use of various approaches to deriving equations to calculate D. Our conclusions are illustrated by examples of D estimations in the electrode materials Li _x C₆, Li _x Sn, Li _x TiO₂, Li _x WO₃, LiM _y Mn_2−y O₄, and LiFePO₄.

Appendix

When the diffusion medium has a flat geometry or the thickness of the diffusion layer L is small in comparison with the particle radius, the phenomenological task reduces to solving Fick's equations for flat diffusion in a homogeneous solid with retardation R _s on the external boundary. As a result, the theoretical PITT equations, i.e., the dependence of the current density j on the electrode polarization ΔE was obtained [17, 18]:

$$ j=\frac{2 nFD\varDelta E}{L}\frac{\mathrm{d}{c}_{\mathrm{s}}}{\mathrm{d}E}{\displaystyle \sum_{\mathrm{k}=1}^{\infty }}\frac{\left({\alpha}_{\mathrm{k}}^2+{(hL)}^2\right){ \sin}^2{\alpha}_{\mathrm{k}}}{\alpha_{\mathrm{k}}^2+ hL+{(hL)}^2} \exp \left(-\frac{\alpha_{\mathrm{k}}^2 Dt}{L^2}\right) $$

(A1)

where dE/dc _s is the derivative of the electrode potential with respect to the surface concentration c _s of the potential-determining particles (coinciding, by sense, with the experimental derivative of the electrode potential with respect to the total concentration dE/dc, though perhaps differing numerically), h = |dE/dc _s|/nFDR _s is a characteristic parameter to determine the shape of the j(t) curves, R _s is the electrode surface resistance, α _k the kth positive root of the characteristic equation α tan α = hL. Clearly, hL ≡ Λ in Montella's designations [62]. Equation A1 significantly simplifies for short t:

$$ \begin{array}{c}\hfill j=\frac{\Delta E}{R_{\mathrm{s}}} \exp \left({h}^2 Dt\right)\mathrm{erfc}\left(h\sqrt{ Dt}\right),\kern0.5em t<<{\tau}_{\mathrm{d}}\hfill \\ {}\hfill j=\frac{\Delta E}{R_s}\left(1-\left|\frac{ dE}{d{c}_{\mathrm{s}}}\right|\frac{2}{ nF{R}_{\mathrm{s}}}\sqrt{\frac{t}{\pi D}}\right)\kern1.5em t<<{\left(D{h}^2\right)}^{-1}\hfill \end{array} $$

(A2)

when t ≪ τ _d (τ_d being the characteristic diffusion time). When GITT is used, the theoretical ΔE vs. j equation reads as [17, 18]:

$$ \varDelta E=\varDelta {E}_{\mathrm{s}}-\frac{j}{ nF}\frac{\mathrm{d}E}{\mathrm{d}{c}_{\mathrm{s}}}\left[\frac{t}{L}-\frac{L}{D}\left\{\frac{1}{3}-\frac{2}{\pi^2}{\displaystyle \sum_{\mathrm{k}=1}^{\infty }}\frac{1}{k^2} \exp \left(-\frac{k^2{\pi}^2 Dt}{L^2}\right)\right\}\right] $$

(A3)

where ΔE _s is a rapid jump of potential owing to the Ohmic and surface resistance and proportional to current at not too big j. In the semi-infinite approximation, Eq. A3 takes the form

$$ \varDelta E=\varDelta {E}_s-\frac{2j}{ nF}\frac{\mathrm{d}E}{\mathrm{d}{c}_{\mathrm{s}}}\sqrt{\frac{t}{\pi D}} $$

(A4)

which is valid with an error lesser than 5 % when t < 0.5τ _d.

Each of these equations can be used to calculate the chemical diffusion coefficient D. The simple Eqs. A2 and A4 are used for linear approximation of the initial fragments of the transients or current or potential in the j vs. t ^0.5 or ΔE vs. t ^0.5 coordinates, respectively. Equations A1 and A3 are used for computer processing of full transients. Some model parameters (S, L, R _s, dE/dc _s, and D) have to be determined independently and set, while the others become a result of processing. The use of Eqs. A2 and A4 requires no knowledge of L, but an exact estimation of the diffusion area A becomes crucial. On the contrary, Eqs. A1 and A3 are insensitive to errors in the A values but require the exact knowledge of the geometry and thickness of diffusion layer.

To estimate D from EIS data, the generalized formula of Warburg's diffusion impedance is applied [52, 53]. Then, the diffusion coefficient at the measurement point c ₀ can be calculated from the equation of semi-infinite diffusion

$$ D=\frac{1}{2}{\left(\frac{{\left|\mathrm{d}E/\mathrm{d}c\right|}_{c={c}_0}}{ nFS{W}_{\mathrm{D}}}\right)}^2 $$

(A5)

or from that of finite-length diffusion

$$ D={T}_{\mathrm{D}}{\left(\frac{{\left|\mathrm{d}E/\mathrm{d}c\right|}_{c={c}_0}}{ nFS{R}_{\mathrm{D}}}\right)}^2, $$

(A6)

where W _D the semi-infinite Warburg factor, R _D the diffusion resistance, T _D the characteristic diffusion time constant. Equation A5 goes over to the usual expression ${W}_{\mathrm{D}}= RT/{(nF)}^2{c}_0\sqrt{2D}$ when E(c) concentration equation by Eq. 4 is substituted into Eq. A5, i.e., when the activity of the potential-determining component is replaced by its volume concentration c ₀.

If the diffusional medium is an aggregative (two-layered) solid body with different diffusion coefficients D ₁ and D ₂ in each layer and different permeability parameters R ₁ and R ₂ for the two borders, this task can be simplified. As in the electrochemical experiment the electrode potential is determined by the surface concentration c _s, in order to find the latter it is enough to only consider the task of transfer in the external phase layer bounded by two borders. Assuming R ₁ = R ₂ = R _s, the theoretical PITT and GITT equations were derived [29]:

$$ j(t)= nFDh\varDelta E\left|\frac{\mathrm{d}{c}_{\mathrm{s}}}{\mathrm{d}E}\right|\left(\frac{1}{2+ hL}+4{\displaystyle \sum_{k=1}^{\infty}\left[\frac{ \exp \left(-{\alpha}_{\mathrm{k}}^2\frac{4 Dt}{L^2}\right)}{2+ hL+\frac{4{\alpha}_{\mathrm{k}}^2}{ hL}}+\frac{ \exp \left(-{\beta}_{\mathrm{k}}^2\frac{4 Dt}{L^2}\right)}{2+ hL+\frac{4{\beta}_{\mathrm{k}}^2}{ hL}}\right]}\right) $$

(A7)

$$ \varDelta E=\varDelta {E}_{\mathrm{s}}+\frac{j}{ nFDh}\left|\frac{\mathrm{d}E}{\mathrm{d}{c}_{\mathrm{s}}}\right|\left(1+ hL-{\displaystyle \sum_{k=1}^{\infty}\frac{2 hL\left({\alpha}_k^2+{(hL)}^2\right)}{\alpha_k^2\left( hL+{(hL)}^2+{\alpha}_k^2\right)}} \exp \left[-\frac{\alpha_k^2 Dt}{L^2}\right]\right), $$

(A8)

where α _k and β _k are the kth positive roots of the characteristic equations α tan α = hL and β cot β = − hL, respectively. At rather short t, when the diffusion front does not reach the internal boundary, Eqs. A7 and A8 reduce down to Eqs. A2 and A4, respectively.

To describe the E vs. c dependence with a Frumkin-type intercalation isotherm in the framework of the lattice gas model [54–60], the differential capacity C _dif(x) is used in accordance with the expression:

$$ {C}_{\mathrm{dif}}(x)={\left(g+\frac{1}{x}+\frac{1}{1-x}-0.5\gamma \alpha \left[1- \tanh {\left(\alpha \left(x-0.5\right)\right)}^2\right]\right)}^{-1}, $$

(A9)

where g is the interaction parameter of attraction (g < 0) and repulsion (g > 0) between species, γ appears as the distortion parameter characterizing the change in the interlayer distance, α represents the rigidity parameter. D as related to the self-diffusion coefficient D ₀ can be expressed by:

$$ \frac{D}{D_0}=\frac{ Fx\left(1-x\right){Q}_{\mathrm{m}}}{ RT{C}_{\mathrm{dif}}}, $$

(A10)

where Q _m is the maximum charge upon intercalation. The second version of the model comprises a more sophisticated expression for C _dif(x)

$$ {C}_{\mathrm{d}\mathrm{if}}(x)={\left(g+\frac{1}{x}+\frac{1}{1-x}-2\gamma \frac{\mathrm{d}p}{\mathrm{d}x}+\lambda {\left(\frac{\mathrm{d}p}{\mathrm{d}x}\right)}^2+\left(\lambda p-\gamma x\right)\frac{{\mathrm{d}}^2p}{\mathrm{d}{x}^2}\right)}^{-1} $$

(A11)

involving the interaction parameter λ between the intercalation particles and the host matrix, and the modulating function p(x), proportional to the interlayer spacing:

$$ p(x)=0.5\left[1+{\displaystyle \sum_m}{p}_m \tanh \left[{\alpha}_m\left(x-{x}_m^0\right)\right]\right], $$

(A12)

where m is the number of intercalation phases. To calculate the electrode potential from the intercalation isotherm by numerical integration, the following formula should be used

$$ \left(E-{E}_0\right)={Q}_{\mathrm{m}}\underset{0}{\overset{1}{\int }}{C}_{\mathrm{dif}}^{-1}\mathrm{d}x $$

(A13)