Graphite negative electrodeHigh-crystallinity synthetic graphite exhibits superior capacity retention to low-crystallinity graphite in K-ion cells43. Therefore, a commercial, highly crystalline, synthetic graphite was utilised (SGP5, SEC Carbon). A larger particle size was utilised to minimise finite-size effects in the Kang-Chueh GITT measurements25. Pawley refinement of synchrotron X-ray diffraction (XRD) data confirms the highly crystalline, phase-pure nature of the synthetic graphite (Fig. 2a). Graphite particles comprise of highly oriented layers, shown in a single graphite particle in the the SEM image in Fig. 2b, where the basal planes are parallel to the graphene layers while the edge planes expose the end faces of the graphene layers44. Intercalation of K+ into graphite is similar to Li+ and only occurs through these edge planes where graphene layers are exposed as shown clearly in Fig. 2b, with the diffusion process advancing into the particle centre along the basal plane44,45,46,47. Therefore, the graphite particle S is only this edge area (Fig. 2b). Supplementary Fig. 1a and b show other SEM images of the graphite particles in powder and in a cast electrode, respectively, showing the flake-like morphology expected. The graphite average particle size distribution from SEM analysis is shown in Fig. 2c, with an average graphite particle size of 5.28 μm and thickness of 0.17 μm (Supplementary Fig. 2).Fig. 2: K graphite characterisation.Material and kinetic characterisation of the potassium synthetic graphite. a Pawley refinement of synchrotron XRD pattern (data in black, fit in teal, data-fit difference in grey, and reflection positions below in green). b SEM image of a single particle. c Particle size distribution. d Open-circuit voltage (OCV) profile from the Kang-Chueh GITT during depotassiation at 20 °C. e Effective diffusivity over composition from Kang-Chueh GITT analysis at 20 °C. Greyed out points indicate data which may be affected by SEI formation. f Exchange current density over composition at 20 °C. Error bars depict the standard error in the mean from at least two repeat measurements.Based on the active edge area and the SEM average particle size and thickness geometric analysis, the graphite particles were approximated as discs (Supplementary Fig. 3) with S determined from this disc edge geometric shaded area where K+ intercalation occurs. Levi et al. and Yang et al. also accounted for this edge intercalation area in graphite in their determination of Li graphite \(\widetilde{D}\)45,46. Adsorption methods such as Brunauer-Emmett-Teller (BET) are most commonly used to characterise S. However, in addition to being a limited and inaccurate proxy for S, for instance being unclear whether very fine pores measured by BET are wetted by the electrolyte25,48, it is highly unsuitable for determining S of graphite as it would include the large area of inactive basal plane regions of the graphite particles. The graphite reversible capacity is close to the theoretical capacity (260 vs. 279 mA h g−1, respectively, Supplementary Fig. 4), this indicates there is very little inactive region of the cast graphite electrode, supporting our method of estimation of S.Figure 2d shows the open-circuit voltage (OCV) profile of the graphite and Fig. 2e shows the effective \({\widetilde{D}}_{{{{\rm{g}}}}}\) results over stoichiometry from the Kang-Chueh GITT and determined S. The results show the median \({\widetilde{D}}_{{{{\rm{g}}}}}\) from both potassiation and depotassiation is 2.32 × 10−13 cm2 s−1. This appears to be two to three orders of magnitude lower than the values reported for Li+ in graphite18,36,45, suggesting slower diffusion for K graphite. However, the \({{{{\rm{Li}}}}}^{+}\,{\widetilde{D}}_{{{{\rm{g}}}}}\) should be measured using the Kang-Chueh GITT for a more meaningful comparison.Studies have shown using operando XRD and Raman49,50 that K+ intercalation into graphite progresses through several two-phase transformations and Onuma et al. proposed the following staging evolution49:$${{{{\mbox{Graphite}}} \rightarrow {{{\mbox{Disorderly}}} \, {{\mbox{stacked}}} \, {{\mbox{high}}} \, {{\mbox{stage}}}\atop{{{\mbox{Graphite}}}-{{\mbox{KC}}}_{96}}} \rightarrow {{{\mbox{Stage}}}\,4{{\mbox{L}}}-3{{\mbox{L}}}\atop{{{\mbox{KC}}}_{96}-{{{\mbox{KC}}}_{24}}}} \rightarrow {{{\mbox{Stage}}}\,2{{\mbox{L}}}\atop{{{\mbox{KC}}}_{28}-{{{\mbox{KC}}}_{24}}}} \rightarrow {{{{\mbox{Stage}}}\,1}\atop{{{\mbox{KC}}}_{8}}}}}$$Stages 4L, 3L and 2L exhibit “liquid-like” in-plane potassium distributions and Daumas-Hérold defects are generated during the phase transformations. The phases therefore have variable compositions and a high concentration of defects, causing their potentials to change during intercalation and explaining the lack of clear plateaus in Fig. 2d for x < 0.4. In contrast, stage 1 forms with a fixed composition through the complete filling of the graphene layers with potassium, eliminating Daumas-Hérold defects and resulting in a constant potential, as evident by the flat voltage plateau for x ~ 0.4 to 0.8 in Fig. 2d as KC8 forms. This appears to coincide with a two to three orders of magnitude drop in the effective \({\widetilde{D}}_{{{{\rm{g}}}}}\) between x ~ 0.4 to 0.8 (Fig. 2e). However, we exercise caution in analysing regions where \(\frac{\partial {V}_{{{{\rm{eq}}}}}}{\partial x}\) approaches zero (Supplementary Fig. 5), which in turn leads to \(\widetilde{D} \sim\) zero (Eq. (1)), which is not physical31.Onuma et al. further observed hysteresis between the intercalation and deintercalation processes, with a stage 2 (KC16) structure able to form from KC8 initially before Daumas-Hérold defects are again necessary for further deintercalation49. These differences may explain some of the directional differences in the \({\widetilde{D}}_{{{{\rm{g}}}}}\) profile during potassiation and depotassiation, particularly since composition-dependent overpotentials are avoided here through the relaxation-only GITT analysis25. However, autocatalysis effects may also contribute to direction-dependencies; therefore it is advisable to refrain from assigning excessive physical meaning to the direction-dependency25.We further note that \({\widetilde{D}}_{{{{\rm{g}}}}}\) appears to be significantly greater at low x during potassiation and at high x during depotassiation, which is not expected physically. Both of these inflated regions correspond to the first few pulses after switching the current direction, and may therefore be caused by SEI formation. Although the GITT measurements were performed with the current leading K-ion electrolyte (KFSI:TEP), continuous SEI formation is still a common issue, even after numerous cycles42,51. A recent study further evidences that the SEI is partially soluble52 and its composition can change dynamically during cycling14,53. Therefore, there may be a restructuring of the SEI that takes place when the direction of the current pulse is changed, consuming capacity until a sufficiently passivating structure is formed. This is supported by evidence that the SEI composition on graphite in K-ion cells changes considerably between charge and discharge54. The necessarily short current pulses applied here mean that this may influence the \({\widetilde{D}}_{{{{\rm{g}}}}}\) results over several pulses. We therefore believe the potassiation data to give a more reliable \({\widetilde{D}}_{{{{\rm{g}}}}}\) for x ~ 1, and depotassiation for x ~ 0, as highlighted in Fig. 2e.The impact of finite-size effects on the Kang-Chueh GITT \({\widetilde{D}}_{{{{\rm{g}}}}}\) can be assessed through evaluating the dimensionless pulse time (\(\hat{\tau }\) = \(\widetilde{D}\tau /{L}^{2}\))25. Supplementary Fig. 6 shows the dimensionless pulse time for the majority of the graphite Kang-Chueh GITT data is within the ideal valid semi-infinite region for 3D geometries, minimising finite-size effects25,34. Diffusion in graphite particles can also be considered 2D along the graphene planes, providing further mitigation against finite-size effects. Supplementary Fig. 7 shows the \({\widetilde{D}}_{{{{\rm{g}}}}}\) evaluated using PITT agrees reasonably well with the Kang-Chueh GITT results (Supplementary Note 1), though with less sensitivity to composition as expected from the limitations of potential step size in plateau regions, as mentioned previously. However, PITT results are inherently limited compared to relaxation-only Kang-Chueh GITT as described in the introduction and Supplementary Note 1. The PITT \({\widetilde{D}}_{{{{\rm{g}}}}}\) minima are shallower than for Kang-Chueh GITT, matching the findings from Markevich et al. who found that PITT is more susceptible to parasitic current contributions than GITT, resulting in overestimated and less accurate D28.Figure 2f shows j0,g of the graphite over stoichiometry in 2 m KFSI:TEP electrolyte fitted to the Butler-Volmer equation (Eq. (3)). The mean j0,g over the composition is 3.42 × 10−5 A cm−2. j0,g is similar to that found for Li graphite18,20 indicating similar charge-transfer reaction kinetics between Li+ and K+ and graphite. A critical challenge with accurately determining Rct from EIS is the fact that constantly evolving SEI and passivation layer formation occur at similar frequency ranges to charge transfer55. The equivalent circuit used to determine Rct and an example impedance spectrum for K graphite are shown in Supplementary Figs. 8 and 9, respectively. Rct was represented by R2 in the equivalent circuit, with R1 representing the SEI due to evidence that SEI impedance has a higher characteristic frequency than Rct56,57. From the Butler-Volmer fit (Eq. (3)) the reference current for the reaction k0,g is 8.07 × 10−5 A cm−2. The poor Butler-Volmer fit for the graphite j0,g matches the findings of Ecker et al. and Schmalstieg et al. for Li graphite18,58, though O’Regan et al. achieved a good fit for Li graphite20. These results indicate that the SEI interferes with the impedance measurements for the graphite electrode. Overlapping time constants for SEI formation and charge-transfer at the graphite electrode may make it difficult to correctly isolate Rct and thus very accurately determine j0,g, however, this provides a reasonable order of magnitude for j0,g, as required for the model.Potassium manganese hexacyanoferrate positive electrodeFigure 3a and b show the large, highly crystalline, non-agglomerated and cuboid KMF material—synthesised via a citrate-assisted co-precipitation (‘Methods’). Supplementary Fig. 1c and d show the material is homogeneous and monodisperse. Synchrotron XRD measurement of the KMF sample confirmed that the high degree of crystallinity achieved through more traditional synthesis had been retained (Fig. 3a). Further, Rietveld refinement revealed that a near identical structure and higher potassium concentration (1.871(3) per formula unit) compared to previous studies was produced (Supplementary Note 2)5,59,60. Elemental analysis by inductively coupled plasma mass spectrometry (ICP-MS) also indicated a low-vacancy/high potassium content from the Fe:Mn ratio of 0.98(5), giving a chemical formula of \({{{{\rm{K}}}}}_{1.871(3)}{{{\rm{Mn}}}}{[{{{\rm{Fe}}}}{({{{\rm{CN}}}})}_{6}]}_{0.98(5)}\) from the combined XRD/ICP-MS analysis. The material was synthesised to have as large particles as possible while maintaining performance to minimise finite-size effects and ensure D limitation in the Kang-Chueh GITT measurements. From SEM analysis the average KMF particle size was identified as 1.02 μm (Fig. 3c), significantly larger than other KMF materials synthesised42,59.Fig. 3: KMF characterisation.Characterisation of the potassium manganese hexacyanoferrate \({{{{\rm{K}}}}}_{2}{{{\rm{Mn}}}}[{{{\rm{Fe}}}}{({{{\rm{CN}}}})}_{6}]\) (KMF). a Rietveld refinement of synchrotron XRD pattern (data in black, fit in peach, data-fit difference in grey, and reflection positions below in green). b SEM image of a particle. c Particle size distribution. d Open-circuit voltage (OCV) profile from Kang-Chueh GITT during depotassiation at 20 °C. e Effective diffusivity over composition from Kang-Chueh GITT at 20 °C. Greyed out points indicate data which may be affected by CEI formation. f Exchange current density over composition at 20 °C. Error bars depict the standard error in the mean from at least two repeat measurements.There has been a historical mischaracterisation of \(\widetilde{D}\) for Prussian blue analogue (PBA) materials, with many studies characterising in the order of 10−8 to 10−11 cm2 s−1 35. This is due to several sources of error. First, as PBAs are frequently synthesised as agglomerated nanoparticles, S or the diffusion length, L, (\(L\propto \frac{n{V}_{{{{\rm{m}}}}}}{S}\) for a cube, where n is the number of moles) is often mischaracterised based on the agglomerate rather than the primary particle size (or for electrodeposited PBA films the film thickness rather than the individual nanoparticle size) resulting in significant overestimates of \(\widetilde{D}\) since \(\widetilde{D}\) is inversely proportional to S2 or proportional to L2 35. Inhomogeneous material, poor morphological characterisation and poor diffusivity analysis have also led to mischaracterisation35. However, recently Komayko et al. conducted a detailed analysis of several PBA materials35 using PITT analysis and improved characterisation of the materials, finding \(\widetilde{D}\) of various PBA materials assessed was in fact around four orders of magnitude lower than conventionally measured and of the order 10−12 to 10−15 cm2 s−1 35. However, they did not characterise \(\widetilde{D}\) for the KMF material.The highly uniform morphology and lack of agglomeration of the KMF synthesised here enables significantly increased accuracy in the determination of S, which has been a significant source of error in the determination of \(\widetilde{D}\) in the past35. Due to this morphological homogeneity, and the 3D framework structure of PBAs enabling K+ insertion through all exposed facets61, the geometric surface area of the KMF particle is a good measure of S. Given the reversible capacity is very close to the theoretical capacity (141 vs. 155 mA h g−1, Supplementary Fig. 10), this indicates there is very little inactive region of the cast KMF electrode, supporting our method of estimation of S.The OCV profile in Fig. 3d shows the two well-defined plateaus, which are characteristic of the low-vacancy KMF. Multiphase behaviour in PBA-positive electrodes is a consequence of maximising the theoretical capacity by reducing the vacancy content, which induces highly correlated distortions in the structure61. From previous in situ XRD studies we understand there to be three dominant phases; the Jahn-Teller distorted (x = 0) and slide distorted (x = 1) phases with an apparently undistorted phase at the intermediate x = 0.5 composition62. We rationalise results from the Kang-Chueh GITT experiment in this context. The results from the Kang-Chueh GITT (Fig. 3e) give a median \({\widetilde{D}}_{{{{\rm{KMF}}}}}\) from both potassiation and depotassiation of 5.50 × 10−14 cm2 s−1, showing \({\widetilde{D}}_{{{{\rm{KMF}}}}}\) is ~ four times lower than the \({\widetilde{D}}_{{{{\rm{g}}}}}\), indicating slower K+ transport in the KMF. The drops in \({\widetilde{D}}_{{{{\rm{KMF}}}}}\) appear to align with the two plateaus between x ~ 0.95 to 0.75 and x ~ 0.4 to 0.1, the former of which corresponds to an apparent drop of three to four orders of magnitude in the effective \({\widetilde{D}}_{{{{\rm{KMF}}}}}\). However, we once again exercise caution in analysing regions where \(\frac{\partial {V}_{{{{\rm{eq}}}}}}{\partial x}\) approaches zero (Supplementary Fig. 11), and without reliable quantification of the phase behaviour from in situ structural techniques one cannot comment further on this result.The \({\widetilde{D}}_{{{{\rm{KMF}}}}}\) results obtained here are within the lower range identified by Komayko et al. for other PBA materials evaluated with improved materials and analysis35, thus supporting the accuracy of the \({\widetilde{D}}_{{{{\rm{KMF}}}}}\) results ascertained here. Using the Kang-Chueh GITT relaxation analysis also avoids the current-related overpotential errors from PITT analysis in their study25,35. Authors He and Nazar found that the KMF analogue K2Fe[Fe(CN)6] (KFF) displayed notably inferior rate capability to its sodium equivalent when their crystallites are micron-sized63. This suggests that K+ diffusion is slower for K-PBA materials compared with Na equivalents.Similar to the graphite case, \({\widetilde{D}}_{{{{\rm{KMF}}}}}\) appears to also be enhanced for the first few pulses after the direction of the current pulse is changed, which could also indicate regions where cathode electrolyte interphase (CEI) formation is influencing the results. We therefore again believe the potassiation data to give a more reliable \({\widetilde{D}}_{{{{\rm{PBA}}}}}\) for x ~ 1, and depotassiation for x ~ 0, as highlighted in Fig. 3e. Supplementary Fig. 12 assesses the dimensionless pulse time for the KMF, also showing the majority of data is within or close to the ideal valid semi-infinite region for 3D geometries, again minimising finite-size effects25,34. Supplementary Fig. 13 shows the PITT \({\widetilde{D}}_{{{{\rm{KMF}}}}}\) results, again showing they match the Kang-Chueh GITT results in the general trend and average magnitude, however, again with limited composition resolution in the two-phase regions as identified in other works28,29.Figure 3f shows the KMF exchange current density, j0,KMF, over stoichiometry in 2 m KFSI TEP electrolyte, measured in a three-electrode cell, and again fitted to a form of the Butler-Volmer (Eq. (3) and ‘Methods’). The data fits the Butler-Volmer trend well, matching good Butler-Volmer j0 fits found for Li-ion lithium nickel manganese cobalt oxide (NMC) materials18,58. For the fits the charge-transfer coefficients (αa and αc) are maintained as 0.5, as is conventionally assumed17,18,58. O’Regan et al. allow αa and αc to be a free fitting parameter to improve the Butler-Volmer fit20. However, since αa and αc are highly difficult to measure accurately and validate17, and as 0.5 gave a good fit for the KMF, this conventional assumption was maintained. From the Butler-Volmer fit (Eq. (3)) k0,KMF was determined as 0.93 × 10−5 A cm−2. The equivalent circuit used to determine the KMF Rct and an example KMF impedance spectrum are shown in Supplementary Fig. 14 and 15, respectively. Again, Rct is represented by R2 in the equivalent circuit as R2 results in j0,KMF clearly fitting the Butler-Volmer trend, and also due to evidence that passivation layer formation occurs at higher frequencies than Rct57. The results show a mean j0,KMF of 0.34 × 10−5 A cm−2 over the composition which is approximately an order of magnitude lower than j0,g, indicating notably faster kinetics for the graphite negative electrode.The sluggish j0,KMF charge-transfer kinetics identified could finally explain the poor rate capability found by other studies assessing the KMF electrochemical performance59,63, where the underlying cause had not been identified5. For context j0,KMF is almost two orders of magnitude lower than that found for NMC positive electrode materials indicating significantly less efficient reaction kinetics to the high energy density Li-ion metal oxides18,20,58. However, the j0,KMF appears around half an order of magnitude higher than LFP (from 4.7–16.7 × 10−7 A cm−2) determined from fitting to commercial LFP electrode experimental data23 and to other models calibrated against experimental LFP testing64. Therefore, the kinetics appear to be similar or slightly better than that for LFP. Given LFP Li-ion would be the competitor for K-ion, rather than the high energy density high nickel positive electrodes5, the comparable j0 for KMF to LFP is promising for the competitiveness of K-ion.Full-cell potassium-ion modellingTo understand the potential of KIBs for fast-charging, we developed a KIB full-cell DFN model in a hypothetical cell based on the commercial LG M50 cylindrical cell format20,36, ensuring realistic electrode thicknesses, parameters and loadings were used. As mentioned, since there is no current K-ion electrolyte which provides practical coulombic efficiencies, two K-ion cells were modelled with alternative electrolytes. First, using KFSI:TEP, which is considered the current leading K-ion electrolyte in the K-ion research community, and is also the electrolyte we used in our experimental investigation42,65. Second, using a hypothetical electrolyte with equivalent properties to KFSI:DME (DMEe), the only other non-aqueous K-ion electrolyte fully characterised until now15. KFSI:DME properties were used as a model electrolyte to indicate full-cell K-ion performance once a potential suitable electrolyte has been developed and optimised. It is important to note KFSI:DME will not be the electrolyte utilised in commercial KIBs unless additives are developed which mitigate its tendency for cointercalation into graphite5. The K-ion models were developed using the electrode properties characterised here (\({\widetilde{D}}_{{{{\rm{g}}}}}\), j0,g, \({\widetilde{D}}_{{{{\rm{KMF}}}}}\) and j0,KMF) in combination with the KFSI:TEP electrolyte properties characterised by Zhao et al. (Supplementary Note 3 and Supplementary Table 3)66 or the KFSI:DME transport properties fully characterised in our previous work (Supplementary Note 3)15. The cells were modelled using the battery modelling package PyBaMM67. The total energy of each K-ion cell was simulated to be 7.3 Wh by adjusting the electrode thicknesses. The half-cell OCV profiles for the graphite negative electrode and KMF positive electrode determined here were implemented in the model (Figs. 2d and 3d, respectively). All electrode properties, including positive and negative electrode material particle size and electrode porosities were kept constant for both cases, and the negative/positive electrode capacity ratios (NP ratios) were set to be 1.1 as typical in commercial Li-ion cells68. Full details of the model and parameters are described in the ‘Methods’ and Supplementary Table 3.The median \({\widetilde{D}}_{{{{\rm{g}}}}}\) and \({\widetilde{D}}_{{{{\rm{KMF}}}}}\) (Supplementary Figs. 16 and 17) are being used in the model to represent the Kang-Chueh GITT data characterised in the regions where potassiation and depotassiation match, while minimising exposure to the extreme values—the high values where SEI/CEI formation and finite-size effects are likely to have some impact due to being above the ideal semi-infinite regime (Supplementary Figs. 6 and 12), and also avoiding impact from the potential plateau regions when \(\frac{\partial {V}_{{{{\rm{eq}}}}}}{\partial x}\) approaches zero (Supplementary Figs. 5 and 11) as mentioned previously.In commercial LFP LIBs, LFP particles need to be nanosized to increase the surface area available for reaction and decrease the diffusion length due to the substantially lower j0 and D of LFP than Li graphite. In the K-ion cell, \({\widetilde{D}}_{{{{\rm{KMF}}}}}\) is approximately four times lower than \({\widetilde{D}}_{{{{\rm{g}}}}}\), and j0,KMF is around one order of magnitude lower than j0,g. Consequently, the KMF particles must also be nanosized to match the faster kinetics of the graphite negative electrode. In the model, the KMF particle sizes are set to 500 nm, consistent with the commercial LFP particle size69.Figure 4a depicts the K-ion cell being modelled with the two electrolytes. The KMF positive electrode is 66% thicker than the graphite negative electrode due to the lower capacity and bulk density of the KMF material5. Figure 4b shows the specific energy and energy density of the K-ion chemistry based on the positive and negative electrode theoretical capacities and the simulated galvanostatic profile (Fig. 4c) using the stack-level model developed previously5,70. Figure 4c displays the DFN simulated galvanostatic profiles of the two chemistries during a 1 C charge, demonstrating the higher overpotentials experienced by the TEP cell. Finally, Fig. 4d shows the fast-charging performance comparison for the two K-ion chemistries, demonstrating the DMEe shows significantly higher rate capability than the TEP cell, achieving significantly higher accessible capacities at all rates simulated. Even at the fast-charging rate of 5 C the DMEe cell can access 34% capacity compared to 8% for the TEP cell. To understand the reasons for the significant difference in rate capability, Fig. 4e and f plot the overpotential components for each K-ion chemistry during a 5 C charge until the upper cut-off voltage is reached, beyond which cell degradation processes may take place13. Figure 4e shows the electrolyte concentration overpotentials are exceptionally high and growing quickly for the TEP K-ion cell early in the charge cycle at SOC < 0.1, causing the upper-cutoff voltage to be quickly reached. This is the result of significant electrolyte concentration gradient formation in the TEP cell due to its lower salt diffusivity (~ one order of magnitude lower than DMEe (Supplementary Table 3)15), thus limiting the transport of K+ to the graphite negative electrode during charge. The ionic conductivity of the TEP electrolyte is also ~ five times lower than that of the DMEe electrolyte (Supplementary Table 3)15, resulting in larger electrolyte ohmic overpotentials. The significantly slower electrolyte transport properties are a result of highly viscous nature of the TEP electrolyte (for context ~ five times higher viscosity than a commercial Li-ion carbonate electrolyte LP3051,71). Therefore, though KFSI:TEP is the current leading electrolyte for the K-ion research community using low loading in coin cells, this electrolyte is unsuitable for even moderately high charge rates using commercial cell electrode thicknesses and loading, and an electrolyte with faster transport properties must be developed.Fig. 4: K-ion full cell simulations.Full-cell Doyle-Fuller-Newman (DFN) simulations of two K-ion cells with different electrolytes in a cell based on the commercial LG M50 cylindrical cell format. The state-of-the-art modelled K-ion cell is graphite (G) ∣∣ \({{{{\rm{K}}}}}_{2}{{{\rm{Mn}}}}[{{{\rm{Fe}}}}{({{{\rm{CN}}}})}_{6}]\) (KMF) with 2 m KFSI TEP electrolyte (denoted as TEP). The other cell is a hypothetical electrolyte case using the characterised electrolyte properties of the 2 m KFSI:DME15 to simulate K-ion performance with an equivalent electrolyte (DMEe). The cell energy is 7.3 Wh. NP ratios were kept constant at 1.1 with constant electrode porosities, and properties for both chemistries. Modelled using PyBaMM67 at 20 °C. (1 C = 1.91 mA cm−2 for both). a Schematic of simulated K-ion cell. b Energy density and specific energy of the K-ion chemistry at the stack level using the stack-level model developed previously5,70. c Simulated galvanostatic profiles for the K-ion cells. d Accessible capacities at increasing C-rate. e Overpotential components during a 5 C charge for the TEP electrolyte. f Overpotential components during a 5 C charge for the DMEe electrolyte.Exploring the model DMEe K-ion chemistry further, where the electrolyte concentration overpotentials are not the limiting factor in determining rate, Fig. 4f shows the largest overpotential components are the negative and positive electrode concentration overpotentials with similar magnitudes. This reflects the transport limitations within the particles as the most significant factor limiting rate, though it is important to note the \({\widetilde{D}}_{{{{\rm{g}}}}}\) and \({\widetilde{D}}_{{{{\rm{KMF}}}}}\), utilised are likely underestimates as noted previously. Initially, the graphite concentration overpotential is most significant, yet, after SOC ~ 0.25 the KMF concentration overpotential becomes more dominant reflecting the KMF leaving the two-phase OCV plateau between 0.75 < x < 0.95 (Fig. 3d), approaching the apparently undistorted phase at 0.4 < x < 0.75 and thus the surface OCV increasing more rapidly. Outside of the KMF OCV plateau region the difference between the OCV at the KMF particle surface K+ concentration, \(U({c}_{{{{\rm{KMF}}}},{{{{\rm{K}}}}}^{+}}^{{{{\rm{s}}}}}\! )\), compared to the OCV at the bulk KMF K+ concentration, \(U({c}_{{{{\rm{KMF}}}},{{{{\rm{K}}}}}^{+}}\! )\), is larger, driving greater concentration overpotentials, ηc. Thus at this point during charging, with decreasing KMF K+ concentration, \({\eta }_{{{{\rm{c,KMF}}}}}=U({c}_{{{{\rm{KMF}}}},{{{{\rm{K}}}}}^{+}}\! )-U({c}_{{{{\rm{KMF}}}},{{{{\rm{K}}}}}^{+}}^{{{{\rm{s}}}}}\! )\) becomes larger than \({\eta }_{{{{\rm{c,g}}}}}=U({c}_{{{{\rm{g}}}},{{{{\rm{K}}}}}^{+}}\! )-U({c}_{{{{\rm{g}}}},{{{{\rm{K}}}}}^{+}}^{{{{\rm{s}}}}}\! )\) causing the KMF concentration overpotentials to start dominating over the graphite. The positive electrode reaction overpotential is significantly larger than the negative reaction overpotential throughout the charge reflecting the slower j0,KMF than j0,g, despite the smaller KMF particles and greater surface area for reaction. The electrolyte and solid ohmic overpotentials are very small contributions. An additional comparison using the electrolyte properties of the commercial Li-ion electrolyte LP5772 (Supplementary Fig. 18 and Supplementary Note 4) showed negligible difference in rate performance with the DMEe electrolyte, showing the electrode material properties are rate limiting with a considerably slower transport electrolyte than DMEe.Finally, since the LFP LIB is the Li-ion chemistry that K-ion would be competing with5, an LFP model in the same LG M50 cylindrical cell was also simulated for comparison (Supplementary Note 5 and Supplementary Fig. 19). Again the total cell energies were matched by adjusting the electrode thicknesses, and the particle sizes for each chemistry’s positve and negative electrode materials were kept constant between both models. KFSI and LiFSI in DME electrolytes were used for the K-ion and LFP Li-ion cell, respectively, for additional fair comparison and because both electrolytes were fully characterised over concentration in the same conditions previously15. Supplementary Fig. 19b shows the K-ion chemistry displays a comparable specific energy to an LFP chemistry (1% higher) though a significantly lower energy density (24% lower), due to the significantly lower bulk density of the KMF material compared to LFP (2.22 g cm−3 vs. 3.45 g cm−3 68, respectively (Supplementary Table 3)), again showing K-ion is competitive particularly where mass is a more critical constraint than space. Supplementary Fig. 19d shows the K-ion cell displays very similar rate performance to the LFP Li-ion cell (35% vs. 37% at 5 C, respectively), indicating K-ion is indeed competitive with LFP, should a suitable electrolyte be developed. In fact, given the conservative K-ion median \(\widetilde{D}\) assumptions, it is likely K-ion would exhibit superior rate performance to LFP from these results. The \({\widetilde{D}}_{{{{\rm{LFP}}}}}\) has been tuned and estimated to experimental data22, being ~ two to three orders of magnitude higher than an apparent median \({\widetilde{D}}_{{{{\rm{LFP}}}}}\) from GITT or EIS results in other studies33,73. It is clear that the K-ion cell can experience larger overpotentials before reaching the upper cut-off voltage due to the KIB OCV profile with a more gradual increase compared to that of the LFP Li-ion cell (Supplementary Fig. 19e). One of the most significant reasons for this is the higher intercalation potential of K+ compared with Li+ into graphite (~0.3 V and 0.1 V, vs. K+/Li+, respectively)5. Supplementary Note 4 further explores the overpotential components of the chemistries.Overall, the modelling draws two critical conclusions. First, an electrolyte with faster transport properties than the leading research KFSI:TEP electrolyte is required to realise the potential fast-charging capabilities of K-ion. Second, if an electrolyte is developed with transport comparable with KFSI:DME, the electrode material transport and kinetics become limiting in rate capability, not the electrolyte. To address the low \({\widetilde{D}}_{{{{\rm{KMF}}}}}\) and \({\widetilde{D}}_{{{{\rm{g}}}}}\) requires smaller particle sizes, making electrolyte stability even more important to maintain high coulombic efficiencies. The slower j0,KMF than j0,g reiterates the importance for researchers to focus on understanding the charge-transfer kinetics of the KMF positive electrode and improving them to match the faster kinetics at the graphite negative electrode.
