Transport and Kinetic Aspects of Alkali Metal Ions Intercalation into AVPO₄F Framework

KVPO₄F cathode materials were prepared using the two-step freeze-drying assisted solid-state technique as it was reported earlier.¹⁵ At first step, vanadium phosphate VPO₄ was synthesized. For this, NH₄VO₃ (>99%) and NH₄H₂PO₄ (>99.5%) was dissolved in distilled H₂O under vigorous stirring at 70°C with a drop-wise addition of an ascorbic acid solution as a carbon source and reducing agent. The resulting blue solution was pulverized into liquid nitrogen and objected to a sublimation at a low pressure (10⁻² bar) within 72 hours. The obtained powder was annealed at 825°C for 7 hours. The residual carbon content in the VPO₄ sample was evaluated by a thermogravimetric method in air (80% Ar:20% O₂) to be ∼4%. At the second stage, the prepared VPO₄ was subsequently mixed with an equivalent amount of KHF₂ (>99.5%), grinded in a mortar and fired under a steady Ar flow at 650°C for 1h followed by quenching to room temperature to prevent the formation of by-products and/or decomposition of the target material.

An appropriate amount of ethylene carbonate, EC, (99%, anhydrous, Aldrich) was dissolved in diethyl carbonate, DEC, (99%, anhydrous, Aldrich) to achieve 1:1 volume ratio. After the complete dissolution of EC, the mixture was dried with activated 3 Å molecular sieves overnight. KPF₆ (0.5 M K⁺ electrolyte, >99.9%, Aldrich) or NaClO₄ (1 M Na⁺ electrolyte, >99.9%, Aldrich) salts were dried at room temperature in vacuum (p(O₂) <10⁻² atm) for 12–16 h. The electrolyte salts were dissolved in EC/DEC solution under Ar atmosphere. Commercial 1 M LiPF₆ in EC/DEC (battery grade, Aldrich) was used as a Li⁺ electrolyte.

Composite electrodes were fabricated by mixing 70% KVPO₄F, 13% Super C carbon black (Timcal) and 17% polyvinylidenefluoride (PVDF) as a binder. The electrode mass was suspended in an appropriate amount of N-methylpyrrolidone and spread over a platinum support. The resulting electrodes were dried at 120°C in vacuum (p <10⁻² atm) for 7 h. The typical loading of the cathode material was 0.3–0.5 mg·cm⁻².

X-ray powder diffraction patterns were collected by a Bruker D8 Advance powder diffractometer (Cu-Kα_1,2 radiation) operating at 35 kV and 40 mA. According to the XRD phase analysis, all KVPO₄F materials exploited in this work for electrochemical measurements contained less than 3% of admixture phases, and can be considered as single-phase.

The morphology, particle size and chemical composition of the initial material as well as the composition of the recovered electrodes were investigated using Carl Zeiss NVision 40 scanning electron microscope (field-emission LaB₆-cathode, In-Lens-detector, 10÷20 kV) equipped with an energy-dispersive X-ray spectroscopy attachment (INCA Energy 350X-Max 80, Oxford, Si-(Li)-detector). The KVPO₄F material consists of well-defined round-shaped or shapeless particles of 100÷300 nm size at the longest direction (Figure 1).

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The bond valence summation (BVS) method and its derivatives such as bond valence energy landscapes (BVEL) are recognized as a quick and simple tool to analyze the possible shape and dimensionality of diffusion paths for a given mobile ion. The BVEL method is considered to be more reliable because in comparison to the parent BVS method, it takes into account both similar and opposite charge interactions in the lattice as well as an asymmetry of ion coordination environments. In contrast to BVS, the BVEL approach operates with specific energy values, E(A). The details and mathematical procedure of E(A) calculations can be found in References 16,17. It should be noted that these energy values do not have direct physical meaning still they are usually in a good correlation with those provided by MD or DFT approaches. The algorithm of BVEL calculations is implemented in the 3DBVSMAPPER program,¹⁸ which automatically produces a three-dimensional grid of E(A) values for alkali ions (Li⁺, Na⁺, K⁺).

Since this method deals with a static model of the crystal structure not considering possible lattice distortions or atoms displacements upon the ion migration, it provides only a tentative view on the ion migration pattern within the structure. A more accurate analysis of the ion migration including calculations of energy profiles and activation barriers could be performed using much more resource-intensive density functional theory (DFT) simulations. However, they require reliable data on structure transformations taking place during ion de/intercalation, which could be retrieved, for instance, from operando diffraction experiments. In this work, the BVEL method was utilized for the preliminary investigation of alkali ion migration topology and dimensionality to be used in the further analysis of electrochemical data.

Electrochemical measurements in Li⁺, Na⁺ and K⁺ electrolytes were performed in a three electrode glass cells with working volumes of ca. 5 mL (cylindrical cell, 3 cm diameter). Counter and reference electrodes were prepared by spreading metal foil (Li or Na) over copper plates (ca. 1 cm²). The working electrode (Pt foil, ca. 0.5 cm²) was placed opposite to the counter electrode at a distance of ca. 0.7–1 cm, to ensure optimal current line distribution. The reference electrode was located in close proximity to the working electrode (ca. 1 mm) to minimize the effect of the uncompensated ohmic resistance in solution.

The initial KVPO₄F material was used as a working electrode in all electrochemical experiments. A partial substitution of K ions for Li or Na ions was observed in the course of cyclic voltammetry scans. It was found that cycling the electrode in a wide potential range (at potentials more positive than 4.4 V vs Li⁺/Li) results in a slow degradation of the material, which takes place in the large excess of the electrolyte presumably due to dissolving of V⁺⁴ at higher potentials.^19,20 Although it is possible to remove up to 0.85 K from the host structure¹⁵ in this study the anodic potential limit did not exceed the value, which corresponds to the extraction of less than 0.6 K from the host structure.

Large volumes of electrolyte were necessary to achieve low concentrations of K ions in the Li⁺ and Na⁺ electrolytes to prevent backward intercalation of the deintercalated potassium ions. In what follows, for the sake of simple notations, we refer to the substitution of K ions for Li and Na ions as to the intercalation into the AVPO₄F host, implying only a partial substitution of K⁺ for different cations. More precise notation of the lithium and sodium intercalated materials would correspond to the formula M_xK_yVPO₄F, where M is Li or Na.

All the reported voltammograms were recorded in the third potential scan (after stabilization of the electrode's electrochemical response).

Na and Li metal foils on a copper support were used as counter and reference electrodes in the respective electrolytes. In the K⁺ electrolyte, due to the high risk associated with using of metallic potassium, an Ag⁺/Ag reference electrode, separated from the working and counter electrodes compartment, was utilized. Graphite electrode was applied as a counter electrode for measurements in the K⁺ electrolyte. The potentials of Ag⁺/Ag, Li⁺/Li and Na⁺/Na reference electrodes were calibrated against the potential of the Fc⁺/Fc (ferocenium/ferrocene) redox couple in EC/DEC solvent to provide a common reference potential scale in all the electrolytes under study. The cells were assembled in an argon-filled glove box (MBraun, c(H₂O) < 0.1 ppm, c(O₂) < 50 ppm). Electrochemical data were registered with Biologic VMP3 potentiostat/galvanostat.

Potentiometric intermittent titration technique (PITT) was employed to determine the potential dependence of Li⁺, Na⁺ and K⁺ apparent diffusion coefficients. The potential was scanned in 10 mV steps. At the end of each potential step, the residual current did not exceed the background value (typically 1–3 μA). Diffusion coefficient calculations were performed assuming the mean particle diameter of 0.2 μm.

Impedance spectroscopy measurements (EIS) were performed to determine charge transfer resistance values of Li⁺, Na⁺ and K⁺ intercalation reactions. Impedance spectra were registered in steps of 10 mV, after equilibrium state was reached for each potential value (usually 1 h in single-phase and up to 3 h in phase-transforming regions). The frequency range was 10 kHz – 10 mHz, with a 5 mV peak-to-peak alternating voltage. For the evaluation of the equivalent circuit parameters, experimental impedance spectra were analyzed using MEISP program package.²¹

A conventional equivalent circuit with the account for the ion transition through CEI (R_CEI C_CEI contour) adsorbate layer was employed to model the intercalation reaction (Figure 2).^22–25

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As it was discussed in a number of publications,²⁶ application of Cottrell equation, which assumes the limiting step of semi-infinite planar diffusion without any additional current-limiting complications, gives erroneous results for evaluation of diffusion coefficients for intercalating materials, which demonstrate slow charge-transfer kinetics and non-zero ohmic resistance. In this work we used the model formulated by C. Montella to describe a small-step chronoamperometric response for the case of a non-negligible ohmic resistance and slow charge-transfer kinetics for planar,²⁷ cylindrical and spherical diffusion geometries.²⁸ The model is derived for the case of single-phase monodisperse intercalating system for small-step chronoamperometric experiments. In our systems multiple phase transitions are possible (as it is outlined in Cyclic voltammetry section), which should result in artificial current minima in the vicinity of the potentials of phase transitions. However, the diffusion coefficient values in the regions of single phase behavior should be considered reliable. The magnitude of potential steps in the PITT procedure was 10 mV to ensure the applicability of the model. Narrow material particles size distribution also allows using the model for quantitative estimates.

Within this approach, the current vs time relation I(t) for planar geometry is given in the form of infinite series:

where ΔQ is the intercalation/deintercalation charge passed during potential step ΔE, b_n – n-th positive root of the equation b·tan b − Λ = 0, τ – time constant for the diffusion process, D – diffusion coefficient, L – diffusion layer thickness.

Λ is a key model parameter reflecting relative contributions of diffusion and others limiting steps:

where R_diff, R_ct, R_Ω - effective resistance of diffusion, charge transfer and ohmic resistances, respectively.

In case of cylindrical or spherical diffusion the current vs time dependence is given by:

where J₀, J₁ are Bessel functions of zeroth and first order, respectively.

Experimental current vs. time transients were approximated by the Equations 1, 4 and 5 (depending on the appropriate diffusion geometry) using least-squares method. τ, Λ and ΔQ values were varied in the course of the fitting procedure. The background current value I_bkg was evaluated by averaging the long-time domain regions of the chronoamperograms. The initial value of ΔQ was estimated by integrating chronoamperograms followed by the subtraction of background current contribution. During calculation of I(t) summation was carried out until m-th series member was less than 10⁻⁶ of the sum of the previous members:

where p = 1, 2, 3 for planar, cylindrical, and spherical geometries of diffusion. To provide equal contributions from different time domains, logarithmic timescale was used during approximation. Experimental data were interpolated to a fixed number of equidistant points on log(t) scale (usually 50 points per order).

The crystal structure of the initial KVPO₄F material features the chains of the corner-sharing VO₄F₂ octahedra bridged by the PO₄ tetrahedra to form a rigid framework with a 3D network of continuous spacious cavities. Potassium accommodates two 9-coordinated sites, designated as K1 and K2, which are located in the open tunnels along the a and b axes respectively (Figure 3). As it was shown in our previous work,¹⁵ both deintercalated K_xVPO₄F and Li-substituted Li_xK_1-xVPO₄F materials preserve the initial framework displaying an extremely small volume variation within 2.2%. The intercalated Li ions in Li_xK_1-xVPO₄F fill the K2 position as well as form a specific 4-coordinated Li3 position. In case of Na, the structural data for Na_xK_1-xVPO₄F are not established so far. Still, the stability of the "VPO₄F" framework, which does not undergo any significant distortions or changes in interatomic distances during the de/intercalation of both small Li and much larger K ions, allows to exclude emerging of severe lattice transformations upon Na de/intercalation. In other words, it was assumed that the possible topology and dimensionality of the migration pathways in Na_xK_1-xVPO₄F could be tentatively analyzed based on the initial structural model of KVPO₄F, with K atoms being manually replaced by Na and unit cell parameters being decreased so that the cell volume is reduced by 2.5%.

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The migration maps of three alkali ions in AVPO₄F obtained using BVEL calculations are demonstrated in Figure 3. According to the BVEL results, all alkali metal positions are to be involved into the diffusion process. The migration network dimensionality varies depending on the nature of the alkali metal: from 3D in case of Li to 2D and 1D for Na and K respectively (Figure 3). The Li⁺ migration map comprises all three geometrically accessible channels along the a, b and c axes. Na⁺ ions seem to follow a two-dimensional pathway in the (010) plane while K⁺ ions exhibit a curved one-dimensional diffusion profile along the c axis. A preferable 1D ionic transport along the [001] direction in the KTP-type structures was repeatedly confirmed by direct ionic conductivity measurements.^29–31

Figure 4 shows cyclic voltammograms of AVPO₄F electrodes in Li⁺, Na⁺ and K⁺ EC/DEC electrolytes. In all the three electrolytes, the shape of the curves is different. In the Li⁺ electrolyte, (Figure 4a) two pairs of peaks at 3.63 and 3.68 V vs Li⁺/Li are observed, with the peak-to-peak separations of 20–40 mV. The second set of unresolved peaks is located at ca. 3.93 V. Another anodic peak is observed at 4.13 V, with the cathodic peak being merged with the peaks at more negative potentials. Previous studies¹⁵ showed that up to the potential of 4.25 V ca. 0.5 K ions are substituted for Li ions. As further increase in the anodic potential limit results in a slow degradation of the material, the electrochemical measurements in the Li⁺ electrolyte were restricted to 4.25 V implying the maximal deintercalation level in this study of 0.5 Li ions per formula unit. The results of EDX analysis of the electrode cycled in lithium electrolyte (Table I) confirm this consideration.

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Sodium intercalation follows different patterns as compared to the Li system (Figure 4b). The first pair of sharp peaks with a peak-to-peak separation of 20 mV is observed at 3.43 V vs Na⁺/Na. At 3.70 V two more broad peaks appear, followed by another two pairs of peaks at 3.88–4.10 V. For all the pairs of peaks the peak-to-peak separations are in the range of 20–40 mV, which suggests that at low scan rates the intercalation primarily proceeds under thermodynamic control. According to crystallochemical considerations as well as the results of the BVEL analysis, larger Na⁺ ions are likely to accommodate the spacious K1 and K2 sites. The EDX analysis of the electrode revealed that 0.7 K⁺ ions are replaced by Na⁺ ions upon cycling up to 4.1 V vs Na⁺/Na.

The cyclic voltammogram of the AVPO₄F electrode in the potassium electrolyte (Figure 4c) is similar to that in the sodium electrolyte, except for the first pair of peaks, which is present only in the Na⁺ electrolyte. Two pairs of sharp well-resolved peaks appear at 1.16–1.21 V vs Ag⁺/Ag followed by two pairs of more complex shaped peaks at 1.37–1.52 V. The peak-to-peak separations are again small (20–40 mV), pointing to the similar thermodynamic reaction rate control, characteristic for all three alkali metal intercalates into AVPO₄F. Ca. 0.55 K ions per formula unit are extracted at the potential of 1.6 V vs Ag⁺/Ag (Table I).

The potentials of Li⁺/Li, Na⁺/Na and Ag⁺/Ag reference electrodes were calibrated against the potential of the Fc⁺/Fc redox couple (3.26 V vs Li⁺/Li) to provide a common potential scale for all three intercalation reactions under study (Figure 4d). It is only possible to compare the redox potentials for the processes of ion insertion/deinsertion reactions for ions, which reside in the similar positions from the viewpoint of interactions with the host lattice. Thus, the comparison of the potential for lithiation of AVPO₄F with the potentials of sodium and potassium intercalation/deintercalation reactions would not be adequate, as in the AVPO₄F structure, Li⁺ ions additionally occupy a distinct Li3 site. Due to a less capacious coordination environment this site is not capable to accommodate large Na⁺ and K⁺ ions, which results in a completely different Li⁺ ions de/intercalation behavior from the crystal structure viewpoint. However, we can compare the formal redox potential of the first intercalation/deintercalation pair of peaks in the K electrolyte (0.65 V vs Fc⁺/Fc) with the similar peaks in the Na electrolyte (0.59 V vs Fc⁺/Fc). The difference in the formal potentials amounts to 60 mV, while the difference in the solvation energies of K⁺ and Na⁺ in carbonate-based electrolyte should amount to ca. 60 kJ·mol⁻¹.^32,33 The redox potential of the ion insertion/extraction reaction is determined by the difference between the ion solvation energy, G^A+_solv, and its energy in the host lattice, G^A+_cryst:

Thus, a 60 kJ·mol⁻¹ difference in solvation energy would imply a ca. 0.6 V difference in the redox potential values, if the energies of Na⁺ and K⁺ in the crystal lattice were close. If the formal potential of the first pair of peaks in the Na⁺ electrolyte is assumed to correspond to the first de/intercalation process in the K⁺ electrolyte, the difference between the two processes is ca. 275 mV, which is still significantly lower than the calculated 0.6 V potential shift. This apparently contradicts the experimental observations, indicating that G^Na+_cryst is significantly more negative than the value for K⁺. This is in a qualitative agreement with the results of BVEL analysis, which indicates that the accommodation and migration of the Na⁺ ions within the KVPO₄F lattice are ca. 50 kJ·mol⁻¹ more energetically favorable in comparison to the K⁺ ones (see AVPO₄F crystal structure and diffusion pathways section). Obviously, quantitative estimates of the potential differences for all three intercalation reactions can be attained in the framework of higher-level computational approaches such as DFT.

Apparent diffusion coefficient values for Li⁺, Na⁺ and K⁺ were determined from PITT data. As for all the three intercalation reactions under study the shape of the peaks does not correspond to the simple semi-infinite solid state diffusion control,³⁴ the data cannot be analyzed by using simple forms of Cottrell's equation. Based on the BVEL results, planar, cylindrical and spherical diffusion geometries were adapted for modeling K⁺, Na⁺ and Li⁺ diffusion in AVPO₄F host, respectively. Kinetic parameter Λ, evaluated from the fitting of experimental current transients, can be used for a qualitative comparison of the ratios of diffusion and charge transfer resistance values for the reaction.²⁷

Figure 5 shows apparent diffusion coefficients, calculated using Montella's methodology, as well as incremental charges Q at each potential step. For the lithium intercalation reaction (Figure 5a) two sharp peaks in the Q(E) dependence are observed at 3.63 and 3.68 V vs Li⁺/Li, followed by a complex broad peak in the potential region 3.85–4.05 V. The anodic peak at 4.1 V in the cyclic voltammogram is absent in the Q(E) plot, which is indicative of a very slow process, which does not attain an equilibrium state under the conditions of a slow potential scan. The diffusion coefficient varies greatly with the composition, showing distinct minima, which coincide with the first peaks in Q(E). In the potential region of 3.85–4.10 V an increase in the D values is observed (from 3·10⁻¹⁴ to 5·10⁻¹³ cm²·s⁻¹). The kinetic parameter of Montella's model is in the order of 10, which indicates the predominantly diffusional reaction rate control. This is not surprising given that the low apparent diffusion coefficient values possibly originate from stronger "pinning" interactions of Li⁺ ions with the lattice, which might severely hinder their diffusion within the structure.

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In the sodium electrolyte (Figure 5b) the incremental charge vs potential dependence shows a series of maxima at the potentials, corresponding to phase transitions or ordering phenomena upon Na⁺ deintercalation. The diffusion coefficient follows the trends in Q(E) plot, showing pronounced minima in the vicinity of corresponding CV peaks. We stress that this is a purely artificial result, originating from the application of the formalism, derived only for single-phase systems, and the corresponding minimum D values in the two-phase coexistence regions should be regarded as unphysical.^27,28 In the regions of presumable single-phase behavior, the diffusion coefficients are in the order of 1–2·10⁻¹² cm²·s⁻¹. The kinetic parameter Λ values, evaluated from the fitting of the current transients, are close to the value of 6, indicating a rather fast charge-transfer kinetics and mainly diffusion control of the intercalation reaction at high scan rates under cyclic voltammetry conditions or high charge-discharge rates under the conditions of galvanostatic cycling.

In the K⁺ electrolyte (Figure 5c) the incremental charge vs potential dependence is similar to that in the sodium electrolyte. The apparent diffusion coefficient values in the regions between the deintercalation maxima are in the range 6·10⁻¹²–4·10⁻¹¹ cm²·s⁻¹. The Λ value for K⁺ intercalation amounts to 1.5–2.0, which points to the slower charge-transfer kinetics, as compared to the sodium system, and mixed (diffusion and kinetic) reaction rate control.

The comparison of the average values of diffusion coefficients for Li⁺, Na⁺ and K⁺ in the AVPO₄F material suggests that the lowest diffusion coefficient value is observed for Li ion, while the diffusion coefficient of K⁺ is 0.5 – 1 orders of magnitude higher than that of Na⁺ ion. A similar effect of the intercalating cation nature on the D value was observed in Ref. 9 for nickel hexacyanoferrate Prussian blue analogue.

If absolute D values are addressed, the value of 10⁻¹³ cm²·s⁻¹ for lithium ion diffusion in AVPO₄F matrix is significantly lower, than the values for LiCoO₂ (10⁻⁹ – 10⁻¹¹ cm²·s⁻¹) and LiMn₂O₄ (10⁻¹⁰ – 10⁻¹² cm²·s⁻¹) conventional cathode materials, nevertheless, it is still higher than in LiFePO₄ material (10⁻¹³ – 10⁻¹⁵ cm²·s⁻¹).³⁵ Given the low values of the particles size, low diffusion coefficients for Li⁺ ion should not limit the performance of the material even at substantially high scan rates.

Sodium diffusion coefficients (10⁻¹² cm²·s⁻¹) in AVPO₄F are substantially higher than the values reported for a promising Na intercalating material Na_2/3Fe_1/2Mn_1/2O₂ (10⁻¹³ – 10⁻¹⁴ cm²·s⁻¹),³⁶ and comparable to the D value for Na₃V₂(PO₄)₂F₃ (10⁻¹² – 10⁻¹³ cm²·s⁻¹).³⁷ Potassium diffusion in AVPO₄F (D ∼ 10⁻¹¹ cm²·s⁻¹) is slower than in the Prussian blue cathode (D ∼ 10⁻¹⁰ – 10⁻¹¹ cm²·s⁻¹),³⁸ but still is fast compared to diffusion of Li⁺ and Na⁺ in conventional cathode materials, which makes this system very attractive for technological applications.

Estimating charge transfer resistance values for an ion intercalation reaction provides a direct way to assess the rate of charge transfer. For an ion transfer reaction, in the simplest consideration, the value of the charge transfer resistance, R_ct, should reflect the contributions from the ion desolvation and the transition of the ion through EEI. These two steps were shown to provide the highest contribution to the activation barrier of the reaction in computational studies of ion transfer processes in the course of metal deposition.^39,40 Obviously, more complicated kinetic mechanisms can take place in an ion intercalation reaction (preceding ion adsorption or slow insertion of the ion into the oxide matrix). However, given the scarcity of experimental information, consideration of complex schemes can hardly be advantageous.

R_ct values were determined from the impedance spectra, registered in the course of the ion deintercalation reaction in the Li⁺, Na⁺, and K⁺ electrolytes. Analogously to the treatment of PITT data, different diffusion geometries were applied to model the spectra. Figure 6 collects representative impedance spectra, registered for all three alkali ion electrolytes.

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The impedance spectra in the Li⁺ electrolyte show a large potential-dependent semicircle at medium frequencies (Figure 6a), and a smaller semicircle in the high-frequency limit (Figure 6b). The high-frequency semicircle was attributed to the resistance of surface layers (R_CEI), while the medium frequency semicircle is considered to reflect the reaction charge transfer resistance, according to the conventional treatment in literature.^22–24 While the radius of the high frequency semicircle remained almost constant in the course of the deintercalation process, the radius of the lower frequency semicircle decreases significantly with the increase in the potential values (from ca. 40 kOhm·cm² at 3.550 V to 6 kOhm·cm² at 4.000 V). At the potentials over 3.9 V vs Li⁺/Li, Warburg and capacitive parts of the impedance spectra can be detected in the selected frequency range.

For sodium and potassium, CEI and charge transfer resistance semicircles are not well-resolved at all potentials during the deintercalation. However, the indications of the presence of two partially overlapping semicircles can be clearly observed (Figures 6c, 6d). Good fits of the experimental data to the selected equivalent circuit can be achieved with the constant radius of the first semicircle, while the radius of the second semicircle demonstrates significant variation with the potential. The R_CEI values in the K⁺ and Na⁺ electrolytes were found to be close to the values, observed in the Li⁺ electrolyte (20–30 Ohm for ca. 0.4 mg of active material in the electrodes), while the R_ct values were at least one order of magnitude lower than those in the Li⁺ electrolyte.

Figure 7 shows the potential dependence of charge transfer resistances for three de/intercalation reactions in the common Fc⁺/Fc potential scale. The R_ct values are normalized per true surface area of the electrode, which was calculated from the coulometric mass of the material and the average particle size. For all the intercalating cations the R_ct values exhibit a significant dependence on the composition. In the Butler-Volmer kinetic model, the charge transfer resistance value is given by:

where n is the number of transferred electrons, i₀ – exchange current density, k_s – heterogeneous rate constant of charge transfer, α – transfer coefficient, c_R, c_O – concentration of ion involved in the reaction and vacancy concentration, respectively. The main reason of the R_ct variation with the potential, if a simple one-step charge-transfer model is assumed, originates from the k_sc^α_Rc^{1 − α}_O factor, which leads to a very sharp increase in R_ct in the beginning of the de/intercalation, when the ions/vacancy concentration is very low. Such a variation can be found in literature data for LiCoO₂ electrodes.²⁴ However, the charge transfer resistance values for the deintercalation of Li ions from the AVPO₄F material do not exhibit a sharp initial rise in R_ct. This might suggest that the limiting step for the lithium intercalation/deintercalation involves a chemical step of ion desolvation. More conclusive information could be obtained by evaluating the potential dependence of the intrinsic rate constant, k_s, which can also show a variation in the different phases of the material. However, the rate constant cannot be estimated without the information on the intercalation isotherm and the ranges of the stability for all the phases, which are formed in the course of intercalation. The detailed structural study of the AVPO₄F material upon the intercalation of different cations will be the scope of our further studies.

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In the Na⁺ and K⁺ electrolytes at the beginning of the deintercalation the variation in the charge-transfer resistance is higher, than that in the Li⁺ electrolyte. However, this variation is still low in comparison to the predictions from Butler-Volmer kinetics. Thus, the preceding desolvation step is also likely to affect the rate of the Li⁺ and Na⁺ intercalation reaction.

If the charge transfer resistance values for Li⁺, Na⁺ and K⁺ cations are compared in the regions far from the beginning of the deintercalation, where these values are relatively constant, it can be seen that the R_ct values for lithium deintercalation are one order of magnitude higher than those for the Na⁺ and K⁺ intercalation. This can be tentatively attributed to the higher desolvation energy of lithium (ca. 110 and 170 kJ/mol more negative than the values for Na⁺ and K⁺, respectively), which should result in a higher activation barrier and lower reaction rate. The charge transfer resistance values for Na⁺ and K⁺ intercalation reaction are in the range of 150–300 kOhm·cm², and given the variation of R_ct in the K⁺ electrolyte it is hard to draw a solid conclusion on the relation of the charge transfer rate for the two intercalation reactions. However, it is clear that for the suggested limiting step of the ion desolvation, the barrier should be considerably higher for sodium ions, and the charge transfer resistance should be higher. These considerations imply that for the potassium ion, which is the largest ion in the series, the transition through the CEI/electrode interface involving the displacement of adsorbate layers from the oxide surface, might be rate limiting.

The results of our study are in agreement with the previous findings of Z. Ogumi's group. It was demonstrated that the intercalation reaction activation energies for a range of materials (LiCoO₂, LiMn₂O₄, Li_4/3Ti_5/3O₄) are very close in carbonate electrolytes (propylene carbonate PC, EC/DEC, EC/DMC).^1,3,5,41 This is a very strong indication that the overall rate of the intercalation reaction is controlled by the solvent properties. Later it was found that the R_ct value shows a pronounced dependence on the presence of EEI-forming additives.² This finding showed that the transition of the ion through the EEI, rather than desolvation alone, contributes to the reaction activation barrier. Our results suggest that the interplay between the desolvation energy and the energy for ionic transition through EEI results in different limiting steps for ions of different size.

The presented kinetic analysis is based on a very crude assumption of the ion intercalation mechanism, which comprises ion desolvation and transition through EEI as rate limiting steps. However, this apparently simplified treatment allowed us to rationalize the experimental observations on the rate of different cations intercalation into the AVPO₄F material, suggesting factors, which primarily determine the reaction rates for Li⁺, Na⁺ (desolvation step) and K⁺ (transition through adsorbate layer/electrode interface).