Modeling of Electrochemical Characteristics of Graphite and Transition Metal Oxides Thin
MODELING THE ELECTROCHEMICAL CHARACTERISTICS OF GRAPHITE AND TRANSITION METAL OXIDE THIN FILM ELECTRODES: A QUASI-METALLIC APPROACH
M.D.Levi and D.Aurbach
Department of Chemistry, Bar-Ilan University, 52900 Ramat-Gan, ISRAEL
ABSTRACT
This paper reviews our last findings related to application of a Frumkin-type intercalation isotherm as a tool for a quantitative description of electrochemical insertion of Li-ions into various Li-insertion anodes and cathodes. Four major electroanalytical techniques, namely, slow-scan rate cyclic voltammetry (SSCV), potentiostatic intermittent titration (PITT), galvanostatic intermittent titration (GITT) and electrochemical impedance spectroscopy (EIS), are frequently used for the determination of chemical diffusion coefficient of Li-ions, D. We have defined their characteristic time-invariant functions F(E), specific of each technique (I n-1/2, I t1/2, dE /dt1/2 and Aw, respectively) in such a way that the diffusion time constant t can be expressed as a combination of the related function with the differential intercalation capacitance, Cint. Such form of presentation allows F(E) to be inter-related, thus demonstrating the equivalence in application of three differential techniques -PITT, GITT and EIS- for obtaining kinetic data. A common feature observed in the experimental plots, D vs. E , for a variety of Li-insertion compounds, appears in the form of deep minima corresponding to the peak values of the differential intercalation capacitance. This was elucidated on the basis of an analysis which combines the concept of Frumkin intercalation isotherm with a simple mechanistic diffusion model.
INTRODUCTION
Li-insertion cathodes and anodes, like lithiated transition metal oxides and graphites, respectively, have recently attracted much attention as being the most promising materials for high-energy Li-ion batteries. The first Li-ion battery appeared on the market in 1991. The principle on which the Li-ion battery works can be described as follows. The cathode material is usually one of the transition metal oxides of the general formulae LixMO2 where M stands for Mn, Ni or Co, whereas the anode is graphite or disordered hard carbon, LixC6. Here X=1 for the cathode materials and graphite, and X>1 for the disordered carbon. During charging, Li+ is extracted from the cathode and then transported through liquid or polymeric electrolyte solution and finely inserted into the carbonaceous anode. During discharge, the process is reverted: Li-ion is transferred from the anode to the cathode. It is not surprising that this kind of battery is often called the “Rocking Chair Battery” or the “Swing Battery”. Numerous attempts have been made to prepare materials with high specific capacity, good rechargeability and fast Li-ion transfer kinetics.
Although theoretical and methodological basis for quantitative thermodynamic and kinetic characterization of intercalation phenomena has been already reviewed (see an excellent treatise on solid-state electrochemistry (B. G. Bruce, Ed. [1]) there still remains the question related to the link between the lattice models, more precisely the quantity of dm/dX (m denotes the chemical potential of ions, whereas X is the intercalation level), and the solid-state chemical diffusion coefficient, D [2]. In our recent paper [3], we tried to take into account some kinetics effects, such as slow interfacial ion transfer and its connection to intercalation isotherms. Practical Li-insertion electrodes, as evidenced by their electrochemical impedance spectra measured in aprotic solvents, almost always reveal this interfacial kinetic contribution in the medium frequency domain of their spectra [4-6]. Moreover, the picture of intercalation processes becomes even more complicated if there is a preferential orientation of powder particles in the composite electrode coatings, or in the case of a broad particle size distribution [3].
The goal of this paper is to review Li insertion processes into thin-film electrodes, both with respect to their kinetics and thermodynamics. We show that a first-order phase transition during charge and discharge of Li-insertion compounds, complicated by slow Li-ion transfer kinetics, can be described by a simple Frumkin-type isotherm, taking into account the short-range interactions between the intercalation sites. The proposed description of insertion processes in terms of intercalation isotherm easily enables the comparison between the expressions for the solid-state diffusion time constant obtained for different electroanalytical techniques, and for finding the relationship between their characteristic time-invariant (but potential-dependent) functions.
RESULTS AND DISCUSSION
1. The Frumkin-Type Intercalation Isotherm
The approach used for the description of Li-intercalation into inorganic hosts will be illustrated here, using the insertion of Li into CoO2 as an example. A clear voltammetric peak (and hence, the corresponding plateau on the charge and discharge curves) is well documented [3,5] and relates to X ranging from 1 to 0.75 (referring to LixCoO2). The number of Li-ions per unit intercalation site is assumed to be one corresponding to the following stoichiometry:
4 LiCoO2 = 4 L 0.75CoO2 + e- + Li+ (1)
Definition of the stoichiometry of the intercalation reaction is an important step in the kinetic description. This is because the intercalation site plays the role of a molecule, if one formally compares reaction (1) with the conventional localized redox-species reactions [7]. The stoichiometry following Eq. 1 relates exactly to a two-phase co-existence region (3.6 – 4.0 V vs. Li/Li+), as is evident from the in-situ XRD characterizations [8]. Thus, advancement of the above insertion/deinsertion reaction can be realized in terms of occupation of intercalation sites with Li+. Concerning the graphite anode, a single intercalation site includes 6 carbon atoms, and thus, the occupation of sites is defined by X in LixC6.
A schematic view of a thin inorganic matrix, capable of incorporating of Li+ cations (anions are completely excluded) which thus exhibits permselective behavior is shown in Fig.1. One of the important principles of thermodynamic equilibrium applied to such an intercalation electrode demands that the total charge due to cations inserted into the matrix bulk from the solution be compensated (locally) by an equal amount of electronic species transferred from the substrate metal (see Fig. 1).
The presence of two kinds of species during intercalation results in a rather complicated picture of potential distribution in the host bulk and across the both current collector / intercalation electrode and intercalation electrode / solution interfaces, as compared with the distribution in the classical case of metal/solution interface. Two kinds of mobile species in the insertion compounds evidently create two possible kinetic limitations. These limitations are due to the transfer of the electronic species across the current collector / intercalation electrode interface and ionic species across the intercalation electrode / solution interface. Moreover, two additional mass-transport steps may appear due to the movement of both species from the electrode host’s boundaries to its interior.
Significantly, the equilibrium potential distribution may change during Li insertion processes, as intercalation proceeds. Fig. 2 contains a schematic view of a typical case of a dielectric film possessing different partition constants for the electronic equilibrium across the Me/host interface [9].
A dielectric film possessing a very low concentration of mobile electronic species reveals a potential distributed within the whole film. As the electronic partition constant becomes higher, the potential in the host starts to drop in close proximity to the electrode host/solution interface. This activity reflects behavior typical of inorganic semiconductors in contact with electrolyte solutions. Further advancement of the intercalation process with a variation of electrode potential results in equal potential drops across both interfaces (symmetrical “electron-counterion’ case [9]). This result appears as a consequence of the local electroneutrality in the host bulk, assuming a 1:1 ratio between the elementary charges of the electronic and ionic species [9,10], and an equilibrium for both the electronic species across the current collector / electrode host interface and the ionic species across the electrode host/solution interface. Finally, when the electronic partition constant is high, typical of a metal, the distribution of interfacial potential becomes similar to the classical distribution, with a single potential drop in the part of solution close to its boundary with the host material. We designate this case as a quasi-metallic approximation to intercalation electrodes. Mathematical expressions describing equilibrium and kinetic characteristics of these electrodes are then reduced to the simplest form (see below).
Once reaction products, stoichiometry and the type of the potential distribution across the interfaces are defined, one can proceed with formal kinetic analysis of intercalation reactions. The combination of the Frumkin-type isotherm with the Butler-Volmer equation for slow charge/discharge transfer (here we suggest Li+ transfer across the electrode host/solution interface) results in an equation which is valid for description of a quasi-equilibrium intercalation/deintercalation reaction [5,11,12]:
Idim=(ko/dfn){(1-X) exp[-(1-a)gX) exp [(1-a)f(E-Eo)]-X exp(agX) exp[-af(E-Eo)] (2)
Here Idim is the dimensionless current; k = (ko/dfn) is the dimensionless rate constant, with ko and n representing the standard heterogeneous rate constant (cm/s) and the potential scan rate (V/s), respectively, d is the thickness of the host matrix (cm); g is the dimensionless interaction parameter, and f is defined as f = F/RT (39.8 V-1 at room temperature). The charge-transfer coefficient a in Eq. (2) is taken symmetrically for the anodic and cathodic reactions, which is a reasonable initial approximation for many electrochemical reactions: a = 0.5. X in Eq. 2 denotes the Li content in Li1-X CoO2. The rate of anodic (deintercalation) reaction is proportional to X in LiXCoO2 (or (1-X) in Li1-X CoO2).
At equilibrium, the net current passed through the intercalation electrode should be zero. By equalizing both terms of Eq. 2, one immediately obtains the Frumkin isotherm:
X/(1-X) = exp [f (E - Eo)] exp (-gX) (3)
Frumkin isotherm (Eq. 3), in contrast to Langmuir one, includes the interaction term exp (-gX). This makes Frumkin isotherm steeper than the Langmuir one for attractive interactions (g < 0), or, on the contrary, flatter for repulsive interactions between the intercalation sites (g >0). There appears a critical value of gcrit = -4 such that as g < gcrit attractive interactions become extremely intensive, resulting in first-order phase transition. Thus gcrit is appropriate for distinguishing between monotonous and non-monotonous character of intercalation reactions.
The extent to which the intercalation reaction may deviate from equilibrium is is expressed by the dimensionless constant K = ko/dfn. The larger is K, the less is the deviation of the charge and discharge process from equilibrium conditions. For a given intercalation reaction (characterized by ko), the system is closer to equilibrium at lower scan rates, and for thinner coatings.
We have recently successfully applied kinetic equation (2) for the simulation of cyclic voltammetric response for Li intercalation reactions [3,5,12]. One important result is worth to be reviewed here. Fig. 3 shows a best fit of a SSCV curve of LixCoO2 electrode with Eq. 2, in the range of potentials from 3.5 to 4.4V. The scan rate was 10 mV/s, the fitted parameters were g = - 4.2 and ko = 8.0 10-7 cm/s. In this case, as is seen from the above figure, a satisfactory agreement between the experimental and theoretical curves is observed only for the anodic peak (see Fig. 3). The height of the theoretical cathodic peak is somewhat larger than that of the experimental peak, whereas the peak-potential separation agrees with the theoretical one. Simulation of a (capacitive-like) plateau on the SSCV curve located at higher anodic potentials is beyond the scope of Eq. 2.
Fig.4a presents as an example a family of theoretical SSCV curves simulated according to Eq.2 with g = - 4.2 and different effective scan rates (K between 0.4 and 400). Experimental SSCV curves for the same range of n are shown in Fig.4 b. Both families of curves are in broad agreement with each other in the range of n between 10 and 50 mV/s. We noted two limiting ranges of n covering n > 50 mV/s and possibly n <10mV/s in which a considerable difference between both sets of curves had been observed. At higher scan rates, experimental SSCV curves revealed a deviation towards a diffusion-controlled behavior. The deviation of experimental curves from the theoretical ones in the limit of low scan rates may be of principal importance: at g < - 4 , as discussed above, Li intercalation proceeds via first-order phase transition, thus the differential capacity peak starts to increase enormously approaching a delta-function behavior (see Fig. 4a). Deviation of the experimental curves from this limiting behavior can be explained in terms of large Ohmic potential drops developed as the current increases steeply (flattens the actual voltammetric peak).
2. Simultaneous Application of SSCV, PITT, GITT and EIS for Studying Li-Ion Solid-State Diffusion Kinetics
As mentioned above (see Introduction section), thin composite electrodes are not well-defined systems compared to conventional metal and semiconductor electrodes, thus in order to obtain reliable results one has to utilize for their characterization a variety of electroanalytical techniques. We recently demonstrated the advantage of simultaneous application of SSCV, PITT, EIS and GITT for characterization of lithiated graphite and several transition metal oxides with subsequent modeling of their electroanalytical responses [3,4,5,11,12]. Here we present a short review of the results obtained.
All four electroanalytical techniques under consideration are related to the same finite-space solid-state diffusion appearing in the host material after application of the input signal specific of the technique used: linear potential scan in SSCV, small potential steps in PITT, current pulse in GITT and small ac voltage in EIS. The fundamentals of one-dimensional finite-space diffusion problem and the routes for quantitative treatments of the output responses were developed by K Aoki et al. [13] (SSCV), W.Weppner and R.A.Huggins [14,15] (PITT and GITT), and C.Ho et al [16] (EIS). A primary diffusion parameter obtained by these techniques is the characteristic diffusion time t which is defined for one-dimensional case as
t = l2/D (4)
where l is the characteristic diffusion length, D denotes the chemical diffusion coefficient connected through thermodynamics to the Frumkin intercalation isotherm. Note that correct calculations of D through Eq. 4 depend on the adopted values of l which for the powdered composite electrode is expected to correlate with the average particle size rather than with the electrode’s thickness. This should be taken into account when comparing D for the same materials obtained by different authors.
Table 1 summarizes expressions for the characteristic time-invariant (but potential-dependent) functions F(E) and the differential intercalation capacity Cint that were derived for four basic electroanalytical techniques. In this Table, GITT 1 corresponds to a limiting case of short current pulses, whereas GITT 2 refers to small current pulses of longer duration (for further details see below).