Transport in Lithium Ion Batteries: Reconciling Impedance and Structural Analysis Raphael Zahn, Marie Francine Lagadec, and Vanessa Wood* Department of Information Technology and Electrical Engineering, ETH Zurich, 8092 Zurich, Switzerland S Supporting Information *
Here, we show that the values obtained by geometric analysis of 3D microstructures and those determined by EIS are in fact self-consistent. The EIS-derived MacMullin number, NM,EIS, and the diffusion simulation-derived MacMullin number, NM,DS, differ from each other because they describe different electrochemical situations. However, a MacMullin number extracted from a fractal analysis and random walks (RWs) of the 3D microstructure, NM,FA, shows good agreement with NM,EIS. First, we run diffusion simulations on a 3D reconstruction of a commercially available polyethylene separator (PE16A) obtained using focused ion beam scanning electron microscopy (FIB-SEM) tomography7,8 (Figure 1a). EIS measurements of PE16A give a MacMullin number that is almost twice as large as the diffusion simulation-derived one (Table 1). One reason for this discrepancy is that in steady-state diffusion simulations a significant part of the network, such as dead-end pores, does not contribute significantly to ion transport through the structure,9 while EIS probes the entire pore network. Indeed, Figure 1b shows an example of how an ion gradient does not develop in a dead-end. A skeleton analysis of the separator’s pore network (SI) shows that 13.0% of junctions lead to dead-ends making up 7.5% of the pore network’s length (Figure 1c). Furthermore, pores that run perpendicular to the overall concentration gradient do not carry diffusive flux to the same extent as pores running parallel to the concentration gradient. In contrast, the NM,EIS of the separator is found from the highfrequency intercept of the impedance spectra at 300 kHz (Figure 1d). At this frequency, the ions move short distances (∼130 nm) comparable to the average pore diameter. The entire pore structure, including dead-end paths, contributes to ion transport, resulting in a different NM (Table 1). NM,EIS can be larger or smaller than NM,DS depending on the τ and ε of the dead-end paths (SI). RWs explore the entire space of a porous structure. Previously, τ has been determined by comparing the mean-square displacements of RWs in the pore space to the mean-square displacement of unbound RWs.6 However, heterogeneous media typically show subdiffusion only on small length scales and normal Fickian diffusion on macroscopic scales.10 Therefore, limits for these diffusion regimes have to be known to obtain a meaningful analysis of the RW data.
ABSTRACT: Diffusion simulations, electrochemical impedance spectroscopy, and fractal-based random walk analysis can give different values for ion transport in porous media. Comparing results from these different techniques can be used to understand the impact of electrolyte selection, ion−surface interactions, and specific structural features on rate performance of electrochemical devices so that the microstructure and chemistry can be systematically optimized.
F
ast charge and discharge of lithium ion batteries requires fast ion transport through the electrolyte-filled cavities of the porous electrodes and the separator. If ion transport is slow, ion gradients build up and lead to high overpotentials and low power output.1 Diffusive transport through electrodes and separators is commonly characterized by two structural parameters, the porosity, ε, and the tortuosity, τ. The ratio of τ and ε, called the MacMullin number, NM, (or its inverse, the effective transport δ) relates the conductivity (or diffusion coefficient) of the pure electrolyte, σ0 (or D0), to that of the electrolyte in the separator or electrode, σeff (or Deff): NM =
σ D τ 1 = = 0 = 0 δ ε σeff Deff
(1)
While ε can easily be determined experimentally, calculating τ is not trivial. Equation 1 points out two common approaches for determining Nm and τ: (i) the ionic conductivities σ0 and σeff can be measured using electrochemical impedance spectroscopy (EIS),2,3 or (ii) the diffusion coefficients D0 and Deff can be determined from steady-state Fickian diffusion simulations using 3D microstructural data.4,5 These two methods generally yield different results,3 which are often attributed to experimental challenges.6 © 2017 American Chemical Society
Received: August 14, 2017 Accepted: September 13, 2017 Published: September 13, 2017 2452
DOI: 10.1021/acsenergylett.7b00740 ACS Energy Lett. 2017, 2, 2452−2453
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ACS Energy Letters
determined according to eq 2 agrees well with the one determined by EIS (Table 1). This work emphasizes the importance of having accurate 3D reconstructions of the porous structure under investigation and a multipronged approach to analysis. Diffusion simulations can be used to obtain information on ionic transport relevant for battery operating conditions, with skeletonization of the structure enabling identification of dead-ends and pore directionality that do not contribute to diffusive transport through the pore structure. NM,DS should not be compared to NM,EIS. Rather, a fractal and RW analysis can be applied to obtain transport values (NM,FA) that can be directly compared to EIS. Differences between NM,FA and NM,EIS can be used to quantify the extent of surface interactions of battery materials with the ions or solvent molecules in the electrolyte. Obtaining all three values for transport (N M,DS , N M,FA , and N M,EIS ) is critical for a comprehensive understanding of how structure and chemistry play a role in electrochemical performance.
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsenergylett.7b00740. Experimental section (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the ETH Energy Science Center Seed Fund (2-72013-14), an ETH Research Grant (0-20978-14), and an ERC Starting Grant (680070).
Figure 1. (a) 3D rendering of PE16A.7 (b) Steady-state ion gradient obtained by diffusion simulations. (c) 3D rendering of the skeletonized pore structure. Dead-end pores are marked in magenta. (d) AC impedance spectrum of PE16A. (e) Box count plot to determine the fractal dimension of the separator’s pore space. The exponent of a power law fit (magenta line) equals the fractal dimension of the pore space. The vertical lines indicate the limits of fractal behavior at small and large length scales. The blue line is the limiting case of nonfractal behavior. (f) RWs in the pore space are used to determine the spectral dimension. The graph shows the number of newly visited sites versus the number of steps. The exponent of the power law fit equals Dspec,pore/2.
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Table 1. Ion Transport Properties of PE16A
Here, we apply fractal analysis and a RW-based method to the tomographic reconstructed microstructure (Figure 1a) and calculate11 NM,FA = ε−1·(l1/l 2)2(1 − (Df / Dspec))
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(2)
where ε is the porosity of the structure, Df and Dspec are its fractal and spectral dimensions, respectively, and l1 and l2 are the lower and upper cutoff lengths for fractal behavior. Df and the cutoff lengths are determined by box-counting (see Figure 1e and SI), and Dspec are obtained with a RW (Figure 1f). The NM,FA 2453
DOI: 10.1021/acsenergylett.7b00740 ACS Energy Lett. 2017, 2, 2452−2453
