Mei-Chin Pang^{a},
Yucang Hao^{a},
Monica Marinescu^{a},
Huizhi Wang^{a},
Mu Chen^{b} and
Gregory J. Offer*^{a}
^{a}Department of Mechanical Engineering, Imperial College London, Exhibition Road, South Kensington Campus, London, SW7 2AZ, UK. E-mail: gregory.offer@imperial.ac.uk
^{b}Beijing Institute of Aeronautical Materials (BIAM), P.O. Box 81-83, Beijing 100095, China
First published on 13th September 2019
Solid-state lithium batteries could reduce the safety concern due to thermal runaway while improving the gravimetric and volumetric energy density beyond the existing practical limits of lithium-ion batteries. The successful commercialisation of solid-state lithium batteries depends on understanding and addressing the bottlenecks limiting the cell performance under realistic operational conditions such as dynamic current profiles of different pulse amplitudes. This study focuses on experimental analysis and continuum modelling of cell behaviour under pulse operating conditions, with most model parameters estimated from experimental measurements. By using a combined impedance and distribution of relaxation times analysis, we show that charge transfer at both interfaces occurs between the microseconds and milliseconds timescale. We also demonstrate that a simplified set of governing equations, rather than the conventional Poisson–Nernst–Planck equations, are sufficient to reproduce the experimentally observed behaviour during pulse discharge, pulse charging and dynamic pulse. Our simulation results suggest that solid diffusion in bulk LiCoO_{2} is the performance limiting mechanism under pulse operating conditions, with increasing voltage loss for lower states of charge. If bulk electrode forms the positive electrode, improvement in the ionic conductivity of the solid electrolyte beyond 10^{−4} S cm^{−1} yields marginal overall performance gains due to this solid diffusion limitation. Instead of further increasing the electrode thickness or improving the ionic conductivity on their own, we propose a holistic model-based approach to cell design, in order to achieve optimum performance for known operating conditions.
Various efforts in the literature have been undertaken to improve different aspects of solid-state lithium batteries. Many researchers have focused on synthesising solid electrolytes and optimising the ionic conductivity. Hagman et al.^{7} first identified the NASICON-type of solid electrolyte in 1968 and showed how the cations move within the structure of the solid electrolyte. The ionic conductivity of this type of solid electrolyte was significantly improved when the size of the cations was reduced by substituting Ti^{4+} ions with Al^{3+} ions.^{8} Thangurai and Weppner proposed the garnet type of solid electrolyte in 2005.^{9} This type of solid electrolyte has a maximum ionic conductivity of 10^{−5} S cm^{−1} at room temperature when the solid electrolyte was doped with Ba^{2+} ions.^{10} Recently, Kato and co-workers^{11} proposed a new type of solid electrolyte using sulfide superionic conductors that exhibits an ionic conductivity of 2.5 × 10^{−2} S cm^{−2}. Apart from improving the ionic conductivity, reducing the interfacial impedance has also been a focal study in the literature. For example, Han et al.^{12} introduced an ultrathin Al_{2}O_{3} coating on garnet-based solid electrolytes and they have found that the interfacial impedance decreases from 1710 Ω cm^{−2} to 1 Ω cm^{−2}. Besides improving the ionic conductivity and reducing the interfacial impedance, Talin et al.^{13} have also modified the geometry of solid-state cells from planar to conical/cylindrical microcolumns and demonstrated that the cell performance of the modified geometry lags significantly behind the cell performance of the planar geometry. Different types of electrode materials have also been proposed as the positive electrode in solid-state lithium cells. For instance, Martha et al.^{14} demonstrated the electrochemical performance of a cell, in which Lithium Phosphorus Oxynitride (LiPON) was deposited on Li_{1.2}Mn_{0.525}Ni_{0.175}Co_{0.1}O_{2} (lithium-rich NMC) positive electrode by using the RF-magnetron sputtering method. A prototype design of all-solid-state lithium cell using Lithium Cobalt Oxide (LiCoO_{2}) as the positive electrode and LiPON as the solid electrolyte was commercialised by STMicroelectronics.^{15}
While there is an array of different possibilities to improve the rate performance of solid-state lithium batteries, there is a lack of systematic design principles to focus on the critical aspects of the cell design. The performance limiting mechanism in solid-state batteries under realistic operating conditions has received little attention. Danilov et al.^{16} used the Nernst–Planck governing equation to model the ionic flux due to diffusion and migration in the solid electrolyte. The two main drawbacks of this model lie the description of the transport mechanisms in the solid electrolyte and the estimation of the model parameters. The transference number calculated with Danilov's parameters is 0.15. This value is smaller than the transference number in standard liquid electrolytes (0.3–0.5)^{17–19} and deviates significantly from the material description of the ceramic-based solid electrolyte. Fabre et al.^{20} proposed a one-dimensional charge/discharge mathematical model, in which Ohm's law was used to relate the electric potential to the current density in the positive electrode and the solid electrolyte. Although the modelling results were validated with experimental measurements, the performance limiting mechanisms due to different cell components have not been quantified and compared. Moreover, batteries are not always subjected to constant current charge and discharge, and the current profiles are often dynamic with varying pulse amplitudes. Therefore, an understanding of the cell's response to pulse current operations can contribute to the optimal cell design for next-generation solid-state lithium batteries. We address this research question in this work by first analysing the cell response under pulse discharge and charge experimentally and then constructing a physical model to help reveal the performance-limiting factors. We show that mathematical models can provide valuable insights into the mechanism limiting cell behaviour, and thus form an essential tool in the process of cell design and upscaling for applications. By using the experimentally parameterised and validated model, we identify a maximum thickness of the positive electrode to avoid unnecessary voltage loss and the increase in internal resistance under pulse operating conditions. We also propose a maximum ionic conductivity, beyond which no further performance gains are obtained. Section 2 illustrates the governing model equations. Section 3 describes the electrochemical test rig and the experimental procedures. Section 4 presents the experimental results and the data analysis for model parameterisation. Finally, in Section 5, we use the validated model to identify the rate-limiting mechanisms under different application-relevant operating conditions and perform model-based cell design optimisation.
Fig. 1 The model domain, where the origin of the x-axis is defined at the interface between the negative lithium electrode and the LiPON solid electrolyte. |
During discharge, lithium is oxidised to form Li^{+} ions. The Li^{+} ions migrate through the solid electrolyte due to potential gradients. Reduction takes place at the interface between LiPON and LiCoO_{2}, and lithium atoms diffuse into the positive electrode. The reverse processes occur during charging.
LiPON is a glassy inorganic solid electrolyte developed by Bates et al.^{27} at Oak Ridge National Laboratory. The acronym LiPON denotes an inorganic compound with the generic composition Li_{x}PO_{y}N_{z}, in which the stoichiometric coefficients are related to each other via the charge neutrality condition: x = 2y + 3z − 5.^{28} Unlike conventional liquid electrolyte LiPF_{6}, where both Li^{+} and PF_{6}^{−} are mobile, only the Li^{+} ions are mobile in the solid electrolyte. Deng et al.^{29} presented a detailed analysis of lithium ions conduction mechanism in the solid electrolyte with the Molecular Dynamics simulation method. Their simulation results propose a cooperative knock-on like mechanism: the migrating interstitial Li^{+} displaces another Li^{+} towards a neighbouring site, which subsequently leads to a further Li^{+} migration. Hence, the inorganic solid electrolyte used in lithium batteries is often known as a single cation ionic conductor. The transference number for Li^{+}, T^{+} can be defined as the amount of lithium per Faraday of charge carried by migration across the solid electrolyte.^{30} The transference number for solid electrolyte is established by Patterson to be greater than 0.99.^{31} When T^{+} ≈ 1, only the flux of Li^{+} constitutes the electric current. At the continuum level, we can, therefore, assume that the rate of migration is equal to the rate of oxidation or reduction. This assumption implies that there will be no concentration gradients of Li^{+} and therefore no concentration polarisation in the solid electrolyte. This assumption has been shown valid through the operando neutron depth profiling measurement of lithium concentration profiles in the solid electrolyte, where a constant concentration has been observed in the bulk solid electrolyte (LiPON) at different times.^{32}
Based on the single-cation conduction mechanism and the unity transference number of the inorganic solid electrolyte, the Nernst–Planck equation (eqn (1)), which represents the flux of species in the solid electrolyte, can be simplified by removing the diffusion term. Because the solid electrolyte does not move at any point of time, the convection term can also be eliminated. Generally, the Nernst–Planck equation is strictly applicable to describe the mass transfer mechanism in a dilute solution only, i.e., when the ionic species do not interact with each other in the electrolyte.^{26} This assumption is deemed valid because only lithium ions are the only mobile ions in the inorganic solid electrolyte and no reactions occur between these mobile cations and the neighbouring atoms. Therefore, the flux density of the only mobile species in the electrolyte, Li^{+}, is given by:
(1) |
The flux of charged particles constitutes the current, i in the electrolyte. This mechanism can be represented as:
(2) |
Combining eqn (1) and (2), the current in the electrolyte becomes
(3) |
By defining the ionic conductivity as the sum in front of the differential , eqn (3) becomes
(4) |
Eqn (4) is equivalent to Ohm's law, in which the current density is proportional to the potential gradient. This expression was used for solid-state lithium battery modelling by Fabre et al.^{20} to simulate a constant current charge and discharge profile in the solid-state battery, and by Thomas-Alyea^{25} to describe an electrochemical model for a porous solid anolyte, in which lithium could deposit on the surface of the electrolyte to reduce the risk of cracks due to dendrites. The current work focuses instead on the effects of pulse current profiles on cell behaviour and on identifying the rate-limiting mechanism for different pulse operating conditions.
Usually, the Poisson equation in the PNP expression is used to describe the non-constant electric field at the electrode–electrolyte interface (≤10 nm) due to unequal distribution of cations and anions.^{22,24} Instead, this model conserves charge through a one Li^{+} in and a one Li^{+} out in the Butler–Volmer's boundary conditions. Uniform current is imposed throughout the electrolyte:
(5) |
(6) |
ϕ_{el}|_{x=0} = 0, | (7) |
(8) |
The Butler–Volmer equations were used to describe the rate of lithium flux between the domains. By using the Butler–Volmer equation, it is assumed only one reaction is occurring at the interface, and that the current is proportional to the net rate of the reaction given by the difference between the rate of anodic reaction (i.e., the first term in eqn (9) and (10)) and the rate of cathodic reaction (i.e., the second term):^{26}
(9) |
(10) |
i_{0,pos} = Fk_{pos}(c_{LCO,max} − c_{Li})^{αpos}(c_{Li})^{1−αpos}, | (11) |
η_{pos/neg} is the overpotential defined as the potential difference between the respective surface electrode potential, ϕ_{s} and that of the solid electrolyte, ϕ_{el}, minus the open-circuit potential of the electrodes, U:^{26}
η_{pos/neg} = ϕ_{s} − ϕ_{el} − U. | (12) |
Solving the governing equation in the positive electrode required two additional boundary conditions. The first boundary conditions (at the LiPON–LiCoO_{2} interface) describes the lithium flux density into the positive electrode and can be represented by Fick's first law as follows:
(13) |
The bulk and surface concentrations of lithium are correlated by solving eqn (6) and (13) together, and the concentration overpotential due to the mass transport from the surface to the bulk or vice versa can be deduced.^{63} The second boundary condition in the positive electrode domain stipulates a zero-flux condition at the electrode and current collector interface, as lithium cannot diffuse into the current collector:
(14) |
The system of DAEs in this continuum model was solved using the finite element method in COMSOL Multiphysics® V5.4. Table 1 summarises the governing equations with a list of required parameters in each computational domain.
PEIS was implemented at the open-circuit voltage of each pulse in order to characterise the impedance of the cell as a function of SOC. From the impedance spectra, the ionic conductivity of the solid electrolyte, exchange current densities and the concentration-dependent diffusion coefficient of lithium in the positive electrode were extracted. OCV as a function of lithium content was also measured.
In this study, an equivalent circuit model fitted to experimentally measured impedance data was constructed in order to extract electrochemical parameters required by the continuum model. Equivalent circuit model fits can be ambiguous since the different arrangement of the circuit elements can be fitted to obtain similar impedance curves.^{42} As the cell polarisation processes often appear convoluted in the Nyquist or Bode plots,^{43} we used the Distribution Function of Relaxation Times (DRT) analysis^{44} to distinguish the electrochemical processes in the solid-state lithium cell via
(15) |
The time constant of each process can be determined from the characteristic frequency, f_{c} corresponding to the local maxima of the peaks in Fig. 3(c) via τ_{DRT} = 1/2πf_{c}. While the DRT analysis is useful to separate the polarisation processes in the Nyquist plots, this empirical analysis has one limitation when applied to batteries. The boundary conditions for the DRT calculation require the complex impedance to tend to zero at high and low-frequency extremes.^{46} As shown in Fig. 3(a), the spectra show an increasing impedance modulus due to diffusion at low frequencies. Pre-processing of the impedance data is, therefore, necessary in order to subtract the low-frequency branch due to the diffusive behaviour of the cell before evaluating the spectra with the DRT method.^{46} Due to this limitation, we instead used a simple expression to estimate the characteristic time due to diffusion, τ_{diff} in the positive electrode:^{47}
(16) |
Fig. 3(c) shows the comparison between Z_{imag} to the DRT spectrum. While only two semi-circles are apparent from Fig. 3(a), the DRT analysis reveals the third semi-circle at 162 Hz. Fig. 3(d) illustrates DRT spectra at different SOCs. From Fig. 3(d), we can see that the peaks of the DRT spectra vary only marginally with the SOC. A small fourth peak can be observed towards the end of discharge. Because this peak has a maximum value (R_{pol} < 10 Ω cm^{2}), the contribution of this polarisation resistance is not considered in the present study.
Iriyama et al.^{48} has previously assigned the first semi-circle to the electrolyte resistance in LiPON and the second semi-circle to the charge-transfer resistance at the Li|LiPON interface, based on the impedance measurement of the Pt|LiPON|Pt symmetric blocking cell and the Li|LiPON|Li symmetric non-blocking cell. They attributed the third semi-circle with the diffusion tail to the charge transfer at the LiPON|LiCoO_{2} interface and the diffusion tail in the LiCoO_{2} by using the full cell Li|LiPON|LiCoO_{2} impedance measurement. Here, we make the same association. Table 2 summarises the frequency range and time-constant corresponding to each electrochemical process. The calculated time-constant due to charge transfer at both interfaces show that this phenomenon occurs between the microseconds and milliseconds timescale. This result implies that the effects due to the double-layer capacitance are negligible for pulses above 0.2 s and do not need to be considered into the governing equations for the continuum modelling. This experimental finding can be correlated to the modelling result by Ong and Newman,^{49} who have modelled the double-layer capacitance in a dual lithium-ion insertion cell. They showed that the double-layer capacitance shorts only the surface overpotential and film resistance at milliseconds. After this short time, the potential continues to change due to the strong dependence of U on the local concentration of lithium in the electrode.
Frequency range [Hz] | Time-constant, τ_{DRT} [s] | Possible cause |
---|---|---|
10^{6}–10^{4} | 2.0 × 10^{−6} | Ionic conductance in the solid electrolyte |
10^{4}–630 | 7.5 × 10^{−5} | Charge-transfer at the Li–LiPON interface |
630–1 | 9.8 × 10^{−4} | Charge-transfer at the LiPON–LiCoO_{2} interface |
The complex non-linear least squares (CNLS) algorithm was used in conjunction with the DRT analysis to fit the measured impedance spectra. The resulting equivalent-circuit model has three R-CPE pairs (Fig. 4(a)), in which the time constants and frequency ranges of each R-CPE pair correspond approximately to the characteristic frequency and range of the DRT peaks. After estimating the size of the semi-circle, these were used as the starting conditions for the equivalent-circuit model to calculate the resistance of each semi-circle. By using the values obtained from the CNLS fitting, an impedance simulation was also performed to ensure that the semi-circles representing each electrochemical process do not cross (Fig. 4(a)). The parameters estimated with the PEIS measurements are the exchange current densities, the ionic conductivity of the solid electrolyte and diffusion coefficients in the positive electrode. These parameters were calculated with eqn (17)–(22). The calculated parameters are shown in Fig. 4c and Table 3.
Parameters | Values | Estimated by | Literature values |
---|---|---|---|
Thickness of the solid electrolyte, d_{elec} | 2 μm | Value from literature^{a} | 2–4 μm |
Thickness of the positive electrode, d_{pos} | 6 μm | Value from literature^{a} | 5–8 μm |
Exchange current density at the Li|LiPON interface, i_{0,neg} | 0.48 ± 0.72 mA cm^{−2} | PEIS measurement | 1.5 mA cm^{−2b} |
Ionic conductivity, σ_{elec} | (2.31 ± 0.52) × 10^{−6} S cm^{−1} | PEIS measurement | 1.2 × 10^{−6} S cm^{−1b} |
Theoretical molar density of LiCoO_{2}, c_{LCO,max} | 51599.06 mol m^{−3} | Calculation | 50000 mol m^{−3b} |
By measuring the upper and lower voltage limit of the cell, the minimum and maximum lithium uptake in the positive electrode were estimated. The experimental result (see S1 in ESI†) shows that upon charging the voltage reaches approximately 4.5 V before increasing sharply, and that upon discharge, the voltage drops at 3.0 V and exhibits a voltage plateau at around 1.25 V before reaching zero. Based on this initial estimation, the cell's upper limit and the lower limit were set initially at 4.5 V and 3.0 V respectively. The initial voltage–capacity curves obtained by cycling a new cell between the voltage limits of 4.2 V and 3.0 V and 4.5 V and 3.0 V at a C-rate of C/40 show that a hysteresis effect can be observed between the discharge and charge curves at 4.5 V and 3.0 V (see S2 in ESI†). Due to the hysteresis effect observed after the first cycle, the voltage windows for the following experiments were set to 4.2 V and 3.0 V. These two voltage–capacity curves were then aligned, and the lithium content of the cell for the voltage window between 4.2 V and 3.0 V could be estimated. Fig. 4(b) shows the aligned and normalised OCV curve as a function of lithium content. Fabre et al. assumed the positive electrode operates between 0.5 and 1.^{20} This experimental measurement has, however, showed that the bulk positive electrode in a thin-film configuration could operate between 0.27 and 1 for the voltage window between 3 V and 4.2 V. This relationship between the OCV and the lithium content was included in the continuum model.
(17) |
(18) |
(19) |
(20) |
(21) |
When the diffusion coefficient is small (i.e., ω ≫ 2D_{Li}/d_{pos}^{2}), the phase difference between the current and the voltage is equal to β = 45°. In this case, the diffusion coefficient can be obtained from:
(22) |
Diffusion coefficients can also be determined with the GITT measurement from^{52}
(23) |
The definitions of τ_{pulse}, ΔV_{s} and ΔV_{t} can be found in Fig. 2(b). This equation is, however, only valid for short time constants (τ_{pulse} ≪ d_{pos}^{2}/D_{Li}).^{52} For the present cell, the smallest d_{pos}^{2}/D_{Li} is 184 s. Therefore, another set of GITT experiments with τ_{pulse} = 70 s was conducted for the measurement of the diffusion coefficients in the positive electrode.
Fig. 4(c) shows the experimental measurements of diffusion coefficients in the positive electrode using the PEIS (red marker) and the GITT method (dark blue marker). The experiments were repeated at a different cell with same methods under the same conditions. The error bar, as shown in Fig. 4(c) represents the deviations measured on two different cells and the instrumental errors of approximately 4%. From Fig. 4(c), we can observe that the diffusion coefficients in the positive electrode is not constant, as assumed by Danilov et al.,^{16} but decreases as the SOC decreases, as found also by Fabre et al.^{20} The difference in the diffusion coefficients measured by the PEIS and GITT methods can be explained by considering the conditions, under which the experiments were conducted. As the PEIS experiment was carried out at the open-circuit voltage of each pulse, the diffusion coefficient estimated with this method was measured under quasi-equilibrium conditions, during which no net current passed through. On the other hand, the GITT method was implemented with a small current. Hence, the difference, as observed in Fig. 4 implies that the diffusion behaviour in the positive electrode has a more complex dependence on the cell history.^{53–55}
As this work focuses on dynamic solid-state cell behaviour, diffusion coefficients estimated by the GITT method are chosen. The input interpolation diffusion was first fitted to the experimental pulse discharge data and subsequently compared with the measurement obtained by the GITT method. The concentration-dependent diffusion coefficient agrees very well with the GITT measurement for low SOCs. However, a deviation smaller than one magnitude of orders can be observed at high SOCs. The calculated characteristic time due to diffusion, τ_{diff}, as shown in Fig. 4(d), increases sharply as the SOC decreases. This experimental analysis indicates that solid diffusion in LiCoO_{2} could be the rate-limiting step for low SOCs.
The decrease in the diffusion coefficients in Fig. 4(c) can be correlated to the structural change of the LCO electrode observed by Matsuda et al.^{47} in the in situ Raman spectra. They attributed their experimental observation to five phases: hexagonal I (H1) phase (1 > x > 0.95), two-phase region (0.95 > x > 0.75), hexagonal II (H2) phase (0.75 > x > 0.55), monoclinic phase (0.55 > x > 0.45) and hexagonal II′ (H2′) phase (0.45 > x > 0.3). The relationship between the phase change and diffusive behaviour of solid lithium in LiCoO_{2} suggests that the phase transformation needs to be controlled to improve bulk mass transport. For instance, Bazant and co-workers have proposed that lowering the surface diffusivity could suppress the phase separation behaviour in Li_{x}FePO_{4}, another type of positive electrode that undergoes a phase change during lithiation and delithiation, though they have investigated the lithium diffusion mechanism at the solid/fluid interface.^{56}
Table 3 summarises the parameters used in this model. The charge transfer coefficients, α_{pos} and α_{neg} were assumed to be 0.5 in this model, assuming that the kinetic resistance between the charge and discharge of the cell is symmetric.^{25} The variation of this parameter between 0 and 1 does not affect the simulation results significantly.
Pulse charging at C/10 was also simulated to analyse the relaxation behaviour after charging. Fig. 5(b) shows the simulated pulse charging profile at C/10 compared to the experimentally measured curves and Fig. S3(b) in ESI† shows the voltage deviation in percentage between the simulation and measurement.
Fig. 5(c) shows the simulated and measured dynamic voltage profile of pulses at different C-rates (0.5C, 1C and 2C). The maximum voltage deviation between the simulation and experimental measurement is ±4% (see Fig. S4 in ESI†). The small error between the measured and simulated voltage profiles suggests that the simplified governing equations proposed here are sufficient to describe the relevant mechanisms at play in the dynamic operations of solid-state lithium batteries.
Danilov et al.^{16} have proposed four overpotentials: charge-transfer overpotentials at the positive electrode and the solid electrolyte interface, diffusion and migration overpotentials in the solid electrolyte and the concentration overpotential in the positive electrode. There are two significant shortcomings in their analysis. Firstly, the transference number of the solid electrolyte corresponding to Danilov's parameter values is 0.15, lower than the transference number in standard liquid electrolytes (≈0.3–0.5),^{17–19} where both cations and anions are mobile. A low transference number implies that the anion mostly provides the ionic conductivity. The lithium cation concentration gradient in the electrolyte would then occur due to the mobility of anions and their accumulation at the electrode surface.^{57} As a result of this low transference number, the mass-transfer overpotential across the solid electrolyte in Danilov's model includes a significant contribution from the diffusion overpotential. The transference number of 0.15 is in contradiction with the definition established by Patterson^{31} for solid electrolyte and the material behaviour of ceramic-based solid electrolytes, in which only the lithium-ions are mobile. This description corresponds to a transference number of 1, with no concentration overpotential in the solid electrolyte. Secondly, in Danilov's model, a critical model parameter – the diffusion coefficient of solid lithium in the positive electrode – was not estimated based on experimental data, and was assumed to be concentration-independent. The experimental measurement of solid diffusion in this work as well as the measurements by Fabre et al.^{20} show that the solid diffusion in the intercalation electrode is concentration-dependent. A constant diffusion coefficient, as assumed in the Danilov's model, leads to an underestimate of the role of lithium diffusive behaviour in the electrode to the cell dynamic performance. These two discrepancies (non-unity transference number and non-concentration dependent solid diffusion) have been corrected in their most recent publication.^{58} However, the implications of these inconsistencies on the results of the overpotential analysis in their previous works were not assessed.
In this work, voltage losses due to charge transfer at both interfaces, the ohmic drop in the solid electrolyte and the solid diffusion overpotential in the positive electrode are considered. Unlike Danilov's overpotential analysis, on the basis of unity transference number in the solid electrolyte,^{31} there is no concentration overpotential in the solid electrolyte, and therefore, only the overpotential due to the ohmic drop is considered in the solid electrolyte of this model. The solid diffusion overpotential was calculated based on the concentration-dependent solid diffusion estimated from our experimental measurements. The overpotentials were calculated as follows:^{16,59}
η^{ct}|_{x=0,x=delec} = ϕ_{s} − ϕ_{elec} − U, | (24) |
(25) |
η^{diff} = U_{LCO}(c_{Li}) − U_{LCO}(_{Li}), | (26) |
Fig. 6(a and b) show the overpotential profiles for SOCs 80% and 30% at a current density of 0.69 A m^{−2} (corresponding to 0.1C-rate). The overpotentials have similar magnitudes at high SOCs, with the solid diffusion overpotential slightly larger than the other overpotentials, followed by the ohmic drop and the charge transfer overpotentials. However, as the SOCs decreases, solid diffusion overpotential in the positive electrode increases significantly from 35% to 75% (Fig. 6(b)). Based on these results, we conclude that the solid diffusion overpotential is the rate-limiting mechanism for lower SOCs. This conclusion reflects the trend visible in Fig. 3(a) between low and high SOCS.
Fig. 6(c) shows the computed overpotentials as a function of current densities at 80% SOC under pulse operating conditions. Following the physical convention to denote the current flow, the anodic charge transfer overpotential during discharge is positive, whereas the cathodic charge transfer overpotential is negative. In Fig. 6(c and d), the positive sign of the anodic charge transfer overpotential was reverted to compare the absolute voltage loss with the other negative overpotentials and therefore do not represent the physical convention that denotes the flow of current. The model prediction shows that solid diffusion in the positive electrode has the most significant contribution in the power loss (27.87 mW m^{−2}) followed by the ohmic drop in the solid electrolyte (16.71 mW m^{−2}). The predicted dependence of overpotentials on current densities at low SOCs is shown in Fig. 6(d). At low SOCs, the range of allowed pulse currents without reaching the cut-off voltage decreases. Within this range of current densities, power losses are clearly dominated by solid diffusion in the positive electrode (36.48 mW m^{−2}). When more lithium is intercalated into the LiCoO_{2}, solid diffusion becomes the rate-limiting mechanism, followed by charge transfer at the positive electrode interface.
Likewise, the influence of the electrolyte ionic conductivity on the ohmic drop in the solid electrolyte is also analysed (Fig. 7(c and d)). Various materials and designs of solid electrolytes with improved ionic conductivity have been proposed in the literature.^{62} Table 4 illustrates selected examples of solid electrolytes. As the ohmic drop in the solid electrolyte was shown to be dependent on the current density, the ohmic drop as a function of ionic conductivity was evaluated at two different current densities in this model (0.69 A m^{−2} and 2.08 A m^{−2}). Improving the ionic conductivity from 10^{−6} S cm^{−1} to 10^{−5} S cm^{−1} would reduce the ohmic drop by approximately 90% and further improvement of the ionic conductivity from 10^{−5} S cm^{−1} to 10^{−4} S cm^{−1} would reduce the ohmic drop by another 90%. However, the simulation result shows that the ohmic drop in the solid electrolyte is no longer limiting the cell performance for both pulse current densities if the ionic conductivity is improved beyond 10^{−4} S cm^{−1}. Hence, we show that there is no benefit in using inorganic solid electrolytes with the ionic conductivity above 10^{−4} S cm^{−1} if bulk LiCoO_{2} continues to be used as the positive electrode.
Types of solid electrolytes | Composition | Ionic conductivity in S cm^{−1} |
---|---|---|
Argyrodite (Li_{6}PS_{5}Cl) | Li_{6}PS_{5}Br | 10^{−3}–10^{−2} |
Li_{6}PS_{5}Cl | 1.33 × 10^{−3} | |
Garnet (Li_{7}La_{3}Zr_{2}O_{12}) | Li_{6.75}La_{3}(Zr_{1.75}Nb_{0.25})O_{12} | 8 × 10^{−4} |
Perovskite (Li_{3x}La_{2/3−x}TiO_{3}) | Li_{3/8}Sr_{7/16}Ta_{3/4}Hf_{1/4}O_{3} | 3.8 × 10^{−4} |
NASICON (LiTi_{2}(PO_{4})_{3}) | LiTi_{2}(PO_{4})_{3}–0.2Li_{3}BO_{3} | 3 × 10^{−4} |
Sulfide (Li_{4}GeS_{4}) | 75Li_{2}S–25P_{2}S_{5} glass | 2 × 10^{−4} |
Li_{4.25+δ}(Ge_{0.75−δ'}Ga_{0.25})S_{4} | 6.5 × 10^{−5} | |
LISICON (γ-Li_{3}PO_{4}) | Li_{4−x}Si_{1−x}P_{x}O_{4}(x = 0.5, 0.6) | 1 × 10^{−6} |
We also show that the simplified governing equations and the parameterisation procedures developed in this work are sufficient to reproduce the experimental pulse discharge, pulse charging and dynamic pulse measurements. These validations show that the model can be expanded to study other solid electrolyte materials or with a different combination of positive electrode materials and electrolytes. This model can be used as a basis of the reduced-order model in the battery management system for solid-state lithium batteries in the future to predict cell behaviour under dynamic load conditions.
In the model-based overpotential analysis section, we demonstrate that solid diffusion in the LiCoO_{2} is the rate-limiting mechanism followed by the ohmic drop at high SOCs. For low SOCs, the electrode solid diffusion and charge transfer overpotential at the positive electrode interface are the two dominant voltage losses. Based on these rate-limiting mechanisms at different pulse operating conditions, our simulations using different thicknesses of the positive electrode show that the solid diffusion overpotential would increase by more than 150% if the electrode thickness were increased from the critical thickness to 18000 nm. This result implies that if decreasing diffusion coefficients at high lithium concentrations in the positive electrode are not addressed in the future cell design, increasing the thickness does not necessarily lead to increases in the usable capacity. Lastly, we propose that an ionic conductivity of solid electrolyte of 10^{−4} S cm^{−1} is good enough for applications if bulk LiCoO_{2} continues to be used as the positive electrode, as further improvement in the ionic conductivity is shown to yield a marginal overall cell performance gain. This study demonstrates that two popular trends in solid-state batteries research are not as productive as they might seem. Increasing the positive electrode thickness, or increasing the ionic conductivity of the solid electrolyte as much as possible, on their own, can only give limited performance increase. Instead, we suggest that a holistic model-based approach to cell design for optimum performance would be beneficial.
N_{i} | Flux density of species Li^{+} (mol m^{−2} s^{−1}) |
ΔV_{s} | Steady-state voltage difference (V) |
ΔV_{t} | Transient voltage difference (V) |
A_{elec} | Surface area of the electrolyte (m^{2}) |
A_{pos} | Surface area of the positive electrode (m^{2}) |
D_{Li} | Diffusion coefficient of Li in the positive electrode (m^{2} s^{−1}) |
D_{Li+} | Diffusion coefficient of Li^{+} in the solid electrolyte (m^{2} s^{−1}) |
F | Faraday's constant = 96487 (C mol^{−1}) |
R_{gas} | Universal gas constant (J mol^{−1} K^{−1}) |
R_{ohmic} | Ohmic resistance (Ω) |
R_{ct,neg} | Charge transfer resistance (Ω) |
R_{elec} | Electrolyte resistance (Ω) |
Res | Real and imaginary impedance residuals (%) |
T | Temperature (K) |
U | Open-circuit voltage (V) |
V | Voltage (V) |
V_{m} | Molar volume of the positive electrode (m^{3} mol^{−1}) |
Z | Impedance (Ω) |
Z_{pol} | Polarisation resistance (Ω) |
i | Current density (A m^{−2}) |
i_{pulse} | Externally applied pulse current density (A m^{−2}) |
_{Li} | Average lithium concentration in the positive electrode (mol m^{−3}) |
a | Specific interfacial area (m^{−1}) |
c_{Li} | Lithium concentration in the positive electrode (mol m^{−3}) |
c_{LCO},_{max} | Theoretical maximum lithium concentration in the positive electrode (mol m^{−3}) |
c_{Li+} | Concentration of Li^{+} in the solid electrolyte (mol m^{−3}) |
d_{elec} | Thickness of the solid electrolyte (m) |
d_{pos} | Thickness of the positive electrode (m) |
f_{c} | Characteristic frequency (Hz) |
i_{0,neg} | Exchange current densities (mA cm^{−2}) |
u_{Li+} | Mobility of Li^{+} in the solid electrolyte (m^{2} mol J^{−1} s^{−1}) |
v | Bulk velocity of the solid electrolyte (m s^{−1}) |
z_{Li+} | Number of proton charges carried by an ion (—) |
α_{neg} | Li–LiPON charge transfer coefficient (—) |
α_{pos} | LiPON–LiCoO_{2} charge transfer coefficient (—) |
β | Phase difference between the current and the voltage (°) |
η_{ct} | Charge transfer overpotential (V) |
η_{diff} | Overpotential due to solid diffusion (V) |
η_{ohm} | Overpotential due to ohmic drop (V) |
ω | Radian frequency (Hz) |
ϕ_{s} | Surface electrode potential (V) |
ϕ_{elec} | Electrolyte potential (V) |
σ_{Li+} | Conductivity of Li^{+} in the solid electrolyte (S cm^{−1}) |
τ | Time constant (s) |
τ_{diff} | Time constant due to diffusion in the positive electrode (s) |
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9cp03886h |
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