Analytical modeling and simulation of electrochemical charge/discharge behavior of Si thin film negative electrodes in Li-ion cells
Introduction
Lithiation/delithiation characteristics of a-Si solid electrodes in an electrochemical cell are ideally suited, for a number of reasons, for the development of physically-based models that provide a robust description of the electrochemical cell or battery behavior and performance. Firstly, other than Li, Si is the anode material that offers the highest energy density (∼3590 mAh g−1) [1]. Secondly, amorphous Si (a-Si) thin film electrodes, in particular, show better cycling characteristics than crystalline Si (c-Si) electrodes [2], [3], primarily due to the continuous and single-phase reaction (LixSi with x < 3.75) without phase transformation and with more homogeneous volume expansions. A further advantage is the relatively fast Li transport in the amorphous phase — the diffusion coefficient of Li in a-Si is ∼10−13 cm2 s−1, which is higher than that in c-Si (∼10−14 cm2 s−1) [4], [5]. More importantly, a-Si films remain fully amorphous during lithiation/delithiation cycling, especially if the film thickness is less than 1–2 μm [6], avoiding problems associated with crystalline-amorphous phase transitions. The modeling is also practically relevant because Si thin-film based Li-ion batteries are attracting a lot of attention as effective energy storage devices in microelectromechanical systems (MEMS), implantable medical devices, and in other such applications [5].
While it is clear that practically useful insights into the mass transfer and charge transfer aspects during cell cycling can be obtained by modeling, there is no specific modeling work in literature focusing on the charge/discharge processes in solid, or specifically, a-Si thin film electrode. There are several modeling works examining various aspects of cell/battery behavior of porous electrodes, which do not apply to the solid electrode situation considered here. Chandrasekaran et al. [1], adopting the well-known porous electrode models of Newman et al. [7], [8], analyzed lithiation/delithiation of spherical Si electrode particles within a porous electrode. Sethuraman et al. [9] analyzed the Tafel kinetics of lithiation/delithiation and the consequent irreversibilities in crystalline Si thin films but the transient Li concentration profiles within the electrode were not considered. The transient Li concentration profiles are essential for the accurate determination of activation overpotentials and hence, the overall cell potential. In general, the porous electrode approach is not applicable for solid thin film electrodes, and, as noted in Newman's review [10], a planar diffusion problem is to be solved for dealing with the planar reaction front characteristics of solid non-porous electrodes. It should be noted that even in C/graphite electrodes (which are modeled sometimes using porous-electrode-spherical-particle framework) fabricated by binder burnout, lithiation was optically confirmed [11] to proceed by the movement of a planar reaction front, suggesting that the planar reaction front and the models based on that notion are perhaps more relevant to the Li-ion battery electrodes in general, than the spherical particle based models. Several reviews well documenting the porous electrode theory and modeling approaches have been published [12], [13].
The objective of this research is to construct an analytical framework which can help to simulate easily the electrochemical charge/discharge behavior during lithiation/delithiation of a solid electrode — we demonstrate the modeling approach here using a-Si thin film electrode as an exemplar. Although numerical or COMSOL-type 3D modeling are prevalent in battery modeling, to generate a clear understanding of cell behavior and how it is impacted by the transient/interface mass transfer effects, physically meaningful analytical models will be needed. More importantly, analytic solutions that provide closed-form relationships between dependent and independent parameters can greatly help to examine parametrically the effects of the key variables with minimal numerical effort. In this study, we have obtained exact analytical solutions for the mass transport equations describing Li transport through the planar Si electrode, while satisfying appropriate boundary conditions. The analytical modeling framework, comprising these solutions, is used to closely simulate the Li-ion cell behavior and to illustrate how diffusion limitations and activation overpotentials affect the electrochemical performance. It is shown here that (i) the available experimental charge/discharge data, especially for a wide range of C-rates, can be simulated quite well and (ii) the hysteretic effects that amplify at higher C-rates can be largely attributed to the electrode-diffusion limitations. Further, using the modeling framework, we have examined the effect of the Li diffusion coefficient and the standard rate constant for charge transfer on the lithiation/delithiation capacities at different C-rates.
Section snippets
The a-Si/Li half-cell
Fig. 1(a) and (b) shows the modeling configuration of the electrochemical half-cell during discharge and charge cycles, respectively. Here, a-Si is the positive electrode and Li is the negative electrode with a liquid electrolyte transporting Li+ ions between the two electrodes. The lithiation/delithiation is assumed to occur under a steady state flux of Li+ to and from the a-Si electrode. It is also assumed that the continuous insertion of Li occurs by solid-state diffusion within the
Analytical models of lithiation/delithiation
Schematic transient Li concentration profiles within the a-Si electrode, during arbitrary lithiation and delithiation steps, are illustrated in Fig. 2(a) and 2(b), respectively. At any time, the transient profiles should satisfy Fick's second law:where y is the position along the thickness direction of electrode, C (y, t) is the Li concentration at location y within the electrode at time t, and D is the diffusivity of Li in LixSi (assumed to be constant for all x).
First, lithiation
Butler–Volmer kinetics
An electron transfer in a specific direction (oxidation/reduction) and at a specific rate can be facilitated at the electrode/electrolyte interface only if an excess potential, beyond the equilibrium electrode potential is provided [17]. The quantitative relationship between this overpotential and the oxidation/reduction current is expressed by the Butler–Volmer (B–V) equation. Accurate modeling of the Li-ion cell charge/discharge processes requires the calculation of activation overpotentials
Calculation procedure
Fig. 4 illustrates the sequence of calculations involved in simulating the charge/discharge behavior using the analytical equations in previous sections. First, the values of input parameters (current, film thickness, constants related to diffusion and kinetic processes) are specified. Then, the transient Li concentration profiles within the electrode during the lithiation and delithiation periods are obtained with the help of Equations (16), (17). The surface Li concentrations, C (0, t), are
Comparison of the simulated and the experimental cell behavior
In order to validate the model, three sets of experimental data, [19], [20], [21], corresponding to the first cycle charge/discharge behavior of a-Si film electrodes with varying thicknesses, cycled at different C-rates, were chosen. Table 1 shows a list of the values of the parameters used in the simulation. The value of diffusion coefficient of Li in a-Si films was determined to be about 10−13 cm2 s−1 [5], [19], [22], hence this value was used. The simulated charge/discharge potentials (solid
Hysteresis between charge-discharge curves
The lithiation/delithiation processes of an electrode, when simulated at different C-rates also enable us to study the hysteretic behavior during cell cycling. This effect has been analyzed before for crystalline Si electrodes, but those simulations were either done on the basis of the porous electrode theory [1] or for a thin film electrode without considering electrode Li diffusion [9], which strictly do not apply to the solid non-porous a-Si electrodes. To investigate hysteretic behavior,
Effect of diffusion coefficient on lithiation/delithiation capacities
The modeling framework also allows us to systematically examine the effect of electrode/interface parameters such as diffusion coefficient and standard rate constants on the lithiation/delithiation capacities. Fig. 8 shows the lithiation and the subsequent delithiation capacities simulated for a 200 nm thick a-Si electrode at different C-rates for the choice of three Li diffusivity values: 2 × 10−13, 1 × 10−13, and 5 × 10−14 cm2 s−1. Generally in literature [25], [26] the estimated diffusion
Effect of standard rate constant on lithiation/delithiation capacities
The standard rate constant, k, is a charge transfer parameter that indicates the kinetic facility of the charge transfer at the electrode-electrolyte interface. For lithiation/delithiation to occur smoothly, its values should be of the right magnitude and the rate of charge transfer should be commensurate with the Li transport within the electrode, if the electrode storage/discharge capacities are to be maximized. Fig. 9 shows the first cycle lithiation and delithiation capacities of a 200 nm
Conclusions
- (1)
A complete analytical modeling framework for predicting the charge/discharge potential behavior of a-Si thin film electrodes during lithiation/delithiation processes has been developed. The model utilizes two important steps: (i) explicit consideration of transient Li concentrations within the electrode determined by solving diffusive mass transport equations and (ii) the determination of time-dependent activation overpotentials, as calculated from the Butler–Volmer equation.
- (2)
The simulated
Acknowledgment
The research was partially supported through a grant from the DOE Office of Science, U.S. Department of Energy, DE-SC0008681.
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