Experimental and numerical analysis to identify the performance limiting mechanisms in solid-state lithium cells under pulse operating conditions

Mei-Chin Panga, Yucang Haoa, Monica Marinescua, Huizhi Wanga, Mu Chenb and Gregory J. Offer*a
aDepartment of Mechanical Engineering, Imperial College London, Exhibition Road, South Kensington Campus, London, SW7 2AZ, UK. E-mail: gregory.offer@imperial.ac.uk
bBeijing Institute of Aeronautical Materials (BIAM), P.O. Box 81-83, Beijing 100095, China

Received 10th July 2019 , Accepted 13th September 2019

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 LiCoO2 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.

1 Introduction

Conventional lithium-ion batteries use aprotic organic electrolyte (e.g., propylene carbonate or dimethyl carbonate) or polymer liquid electrolyte (e.g., polyvinylidene fluoride or polymethyl methacrylate) to transfer the ions between the positive and the negative electrodes.1 One of the main challenges facing this type of battery is the heat generation and gas formation within the cell. Each type of cell has a specified critical temperature, exceeding it will lead to a breakdown of the Solid Electrolyte Interphase (SEI) at the negative electrode, adverse chemical reactions between the lithiated carbon and the electrolyte, melting of the separator (polyethene and polypropylene), collapse of the liquid electrolyte and eventually disintegration of the positive electrode.2 These reactions and decomposition of the constituents are termed thermal runaway, which may be triggered by overheating (due to the presence of an external heat source),2 high external ambient temperature, excessive currents, overcharging or short circuit within the cell. Solid-state lithium batteries could reduce these safety concerns because solid electrolyte materials are non-flammable.3,4 Also, due to the high gravimetric and volumetric capacity of lithium (i.e., 3860 mA h g−1 and 2060 mA h cm−3) compared to graphite (i.e., 372 mA h g−1 and 837 mA h cm−3),5 the use of lithium metal as the negative electrode in the solid-state lithium batteries could potentially improve the cell gravimetric energy density by 40% and volumetric energy density by 70% in comparison to conventional lithium-ion batteries.6

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 Ti4+ ions with Al3+ 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 Ba2+ ions.10 Recently, Kato and co-workers11 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 Al2O3 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 Li1.2Mn0.525Ni0.175Co0.1O2 (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 (LiCoO2) 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.

2 Model formulation

2.1 Computational domain

In general, three transient phenomena occur when the external current is interrupted: relaxation of the double-layer capacitance, local equalisation of the state-of-charge and reduction of concentration gradients in the solid electrode.21 This work focuses on the continuum modelling of pulse behaviour and relaxation phenomena in solid-state lithium cells. Fig. 1 illustrates the domain modelled in this work. Assuming isotropic material properties in the y- and z-direction, the three-dimensional model (xyz) was reduced to a one-dimension model (in the x-direction). As the negative electrode is a pure lithium metallic thin film, and no ohmic losses were assumed across the thin film such that the model considers the solid electrolyte and the positive electrode only. Therefore, the origin of the x-axis was defined at the interface between the metallic lithium and the solid electrolyte. The electrons transport was not considered in this model, and the electrons were assumed to be instantaneously available.22
image file: c9cp03886h-f1.tif
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 LiCoO2, and lithium atoms diffuse into the positive electrode. The reverse processes occur during charging.

2.2 Governing equations in the solid electrolyte

In the literature, the Poisson–Nernst–Plack (PNP) equation has been used at the microscopic23 and the macroscopic level22,24 to depict the transport mechanism in the solid electrolyte. However, the mass transfer mechanism in the solid electrolyte can also be represented with a simple Ohm's law.20,25 The relationship between the comprehensive PNP model and the Ohm's law is, however, not well understood and has not been described clearly in the literature. This relationship will be illustrated in this section with the derivations based on the dilute solution theory.26

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 LixPOyNz, in which the stoichiometric coefficients are related to each other via the charge neutrality condition: x = 2y + 3z − 5.28 Unlike conventional liquid electrolyte LiPF6, where both Li+ and PF6 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:

image file: c9cp03886h-t1.tif(1)
where DLi+ is the diffusion coefficient of lithium-ions in the solid electrolyte, cLi+ is the concentration of lithium-ions in the solid electrolyte, ∂cLi+/∂x is the concentration gradient due to diffusion in the electrolyte, zLi+ is the number of proton charges carried by an ion, uLi+ is the mobility of lithium-ions in the electrolyte due to potential gradients ∂ϕel/∂x, F is the Faraday's constant (F = 96[thin space (1/6-em)]487 C mol−1) and v is the bulk velocity of the solid electrolyte.

The flux of charged particles constitutes the current, i in the electrolyte. This mechanism can be represented as:

image file: c9cp03886h-t2.tif(2)

Combining eqn (1) and (2), the current in the electrolyte becomes

image file: c9cp03886h-t3.tif(3)

By defining the ionic conductivity as the sum in front of the differential image file: c9cp03886h-t4.tif, eqn (3) becomes

image file: c9cp03886h-t5.tif(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-Alyea25 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:

image file: c9cp03886h-t6.tif(5)

2.3 Governing equation in the positive electrode

The governing equation describing the lithium transport in the positive LiCoO2 electrode is the Fick's second law of diffusion (eqn (6)). As our experimental characterisation results show that the diffusion impedance changes with the State-of-Charge (SOC), the diffusion coefficient in the positive electrode is modified to be dependent on the amount of (de-)intercalated lithium:
image file: c9cp03886h-t7.tif(6)
where cLi is the lithium concentration, DLi is the solid diffusion and xLi is the lithium stoichiometry in the positive electrode. The concentration-dependent solid diffusion mechanism will be further elucidated in Section 4.3. Depending on the SOC, LiCoO2 has an electronic conductivity of 14–31 S cm−1 at 25 °C.33 As this conductivity is 107 magnitude of orders higher than the ionic conductivity in the solid electrolyte (see Table 3), the potential drop in the positive electrode due to the mobility of electrons is negligible. Therefore, the cell voltage in the model can be determined by solving the surface electrode potential, ϕs across the solid electrolyte.

2.4 Boundary conditions

The two boundary conditions required for solving the governing equations in the solid electrolyte domain are described as follows:
ϕel|x=0 = 0, (7)
image file: c9cp03886h-t8.tif(8)
where ipulse is the externally applied pulse current density.

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

image file: c9cp03886h-t9.tif(9)
image file: c9cp03886h-t10.tif(10)
where αneg and αpos are the charge transfer coefficients, which denote the fraction of applied potential that favours the respective reactions;26 Rgas is the universal gas constant (Rgas = 8.314 J mol−1 K−1), T is the cell temperature, i0,neg and i0,pos are the exchange current densities at the respective interface. While i0,neg is determined from the experimental measurement, i0,pos is dependent on the lithium concentration at the interface between the solid electrolyte and the positive electrode:20
i0,pos = Fkpos(cLCO,maxcLi)αpos(cLi)1−αpos, (11)
where kpos is a fitting parameter in this model and cLCO,max is the theoretical maximum molar density of lithium in the positive electrode.

ηpos/neg is the overpotential defined as the potential difference between the respective surface electrode potential, ϕs and that of the solid electrolyte, ϕel, minus the open-circuit potential of the electrodes, U:26

ηpos/neg = ϕsϕelU. (12)

Solving the governing equation in the positive electrode required two additional boundary conditions. The first boundary conditions (at the LiPON–LiCoO2 interface) describes the lithium flux density into the positive electrode and can be represented by Fick's first law as follows:

image file: c9cp03886h-t11.tif(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:

image file: c9cp03886h-t12.tif(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.

Table 1 Summary of governing equations and required model parameters
Computational domain Governing equations Boundary conditions Required model parameters
Solid electrolyte image file: c9cp03886h-t13.tif image file: c9cp03886h-t14.tif  
image file: c9cp03886h-t15.tif image file: c9cp03886h-t16.tif σLi+, i0,neg, αneg,T
image file: c9cp03886h-t17.tif αpos
Positive electrode image file: c9cp03886h-t18.tif image file: c9cp03886h-t19.tif DLi(xLi)
image file: c9cp03886h-t20.tif

3 Experimental procedures for parameterisation and model validation

Pulse current experiments were carried out to characterise the cell behaviour at different SOCs and 25 °C ambient temperature followed by the impedance experiments to measure the model parameters. As this model focuses on the understanding of the cell response to the pulse operations, the experimental pulse discharge and charging measurements were also used as model validations in this work. This section describes the experimental setup (Section 3.1), followed by experimental procedures to characterise the relaxation behaviour and measure the model parameters (Section 3.2).

3.1 Experimental setup

Experiments were conducted on solid-state microcells from ST Microelectronics (Enfilm™ EFL700A39, nominal capacity 0.7–1.0 mAh and nominal voltage window 3.0–4.2 V) with lithium negative electrode, LiPON solid electrolyte and LiCoO2 positive electrode. The cells were placed inside a transparent housing on top of a Peltier element thermoelectric cooler (European Thermodynamics APH-127-10-25-S: 17 W, 1.9 A, ΔT = 68 K), which was used to control the cell's temperature throughout the experiments. The thermocouples type K were fixed on the surface of the cell. The Peltier element inside the transparent housing was attached to an aluminium cooling block, which was used as a heat sink with a through-hole for water flowing. The water flow through the aluminium blocks was maintained via a water pump. A radiator was also used to dissipate the heat into the environment. Six identical rigs were constructed to run parallel experiments under different operating conditions. The electrochemical experiments were conducted with Gamry Interface 1000 (frequency range: 10 μHz–1 MHz; current range: 10 nA–1 A).

3.2 Experimental procedures

A combination of Galvanostatic Intermittent Titration Technique (GITT) and Potentiostatic Electrochemical Impedance Spectroscopy (PEIS) techniques were deployed to characterise cell behaviour and measure the model parameters. The experimental procedures from Birkl et al.34 were adapted to include PEIS measurements at the open-circuit voltage of each pulse. The cell was first equilibrated at 25 °C by using the temperature control with the Peltier element before being charged with a CC-CV procedure to 4.2 V and relaxed for three hours. Complex impedance as a function of frequency from 1 mHz to 1 MHz with an AC signal of 5 mV was then performed to measure the initial impedance of the cell. After the initial PEIS characterisation, the cell was discharged with a current of C/10 to remove 10% of the nominal cell capacity. After the current was interrupted, the cell was relaxed for 3 h before the next PEIS measurement. The cell was subsequently discharged at C/10 followed by the voltage relaxation and PEIS measurement until the cell voltage reached the lower limit of 3 V.

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.

4 Experimental characterisation results and parameters analysis

This section first describes the experimentally measured pulse behaviour (Section 4.1) and the PEIS impedance spectra obtained as a function of SOC (Section 4.2). As the impedance spectra are subsequently used to estimate the values of model parameters, the linear Kramers–Kronig validity test was used to quantify the experimental reliability in our measurements. Distribution of relaxation times analysis is used to help identify the electrochemical processes in the solid-state lithium cell. Finally, Section 4.3 discusses how the model parameters were obtained from the experimental data measurements.

4.1 Pulse behaviour

Fig. 2(a) shows the voltage measurement during the pulse discharge at C/10. Fig. 2(b) shows how information is extracted from the experimental pulse measurements. Transient voltage difference, ΔVt can be calculated by determining the voltage difference of each pulse discharge, and steady-state voltage difference, ΔVs can be calculated from the voltage difference between two consecutive relaxation points. The pulse time constant, τpulse is the period of each pulse discharge, whereas the voltage drop, IRdrop is the instantaneous voltage drop at the beginning of each pulse or at the end of each pulse. Fig. 2(c) illustrates the calculated cell internal resistance for all pulses. The internal resistance of the solid-state lithium cells obtained from GITT is several orders of magnitude higher than lithium-ion cells, which is usually in the mΩ range.35,36 If solid-state lithium batteries would be used in the future for higher C-rates and no significant efforts are made to reduce the magnitude of the internal resistance, the voltage drop would be significant and affect the cell performance. The internal resistance increases marginally at the beginning of SOC but rises drastically to ≈450 Ω cm2 at the end of SOC. The calculated ΔVt and ΔVs for each pulse is displayed in Fig. 2(d). Both ΔVt and ΔVs first decrease from the first pulse discharge towards the sixth pulse discharge but increase drastically at the end of SOC due to the rise in the cell internal resistance.
image file: c9cp03886h-f2.tif
Fig. 2 (a) Experimentally measured pulse discharge and OCV with the GITT method. The red dot indicates the point in time, at which the PEIS measurement was taken. (b) An illustration on how ΔVt, ΔVs, τpulse and IRdrop are extracted from the experimental data. (c) The calculated resistance for all pulses: The resistance have similar values from 90% SOC to 30% SOC, but a significant increase can be observed for SOC less than 30%. (d) The increase in ΔVt and ΔVs at the end of SOC can be correlated to the increase in the cell internal resistance.

4.2 Impedance and distribution relaxation times analysis

Fig. 3(a) shows the impedance spectra at the 90% and 10% SOC. The possible assignment of the semi-circles to the corresponding physical mechanism will be discussed along with the DRT analysis. The tail of the impedance spectra is attributed to diffusion and varies most between the two SOCs. The diffusion impedance at 90% SOC has a nearly 90° slope, whereas the diffusion impedance at 10% SOC has a nearly 45° slope. As SOCs decreases, the gradient of the diffusion slopes decreases and the magnitude of the impedance increases. The near 90° angle diffusion as observed at 90% SOC can be associated to a large ratio of DLi/dpos2 (i.e., a thin electrode or a large diffusion coefficient) whereas the near 45° angle diffusion at 10% SOC can be correlated to a small ratio of DLi/dpos2 (i.e., a thick electrode or a small diffusion coefficient).37 In our case, as the impedance spectroscopy was carried out under the same frequency sweep on the same cell and assuming a negligible change of thickness during the impedance measurement, the most probable explanation for the difference in the experimental measurement of the impedance spectra is the decrease in the effective diffusion coefficient as the depth-of-discharge increases. The linear Kramers–Kronig validity test38 was used to quantify the experimental reliability in our measurements. Fig. 3(b) shows the residual plots of the impedance spectra at 90% SOC and 10% SOC. The Kramers–Kronig relation tests the stability, causality and linearity of the impedance measurement.39–41 A high residual denotes the failure of the experimental measurements.41 In this work, the residual plots of the impedance measurement indicate an acceptable deviation of approximately ±0.5%.
image file: c9cp03886h-f3.tif
Fig. 3 (a) Experimentally measured PEIS at 90% SOC (blue marker) and 10% (red marker) SOC. (b) Relative errors of the real and imaginary experimental data: The experimental data have residuals of ≈±0.5% and a maximum residual of 0.57%, which means the measurements are compliant with the KK-relations. (c) Imaginary part of the PEIS compared to the DRT spectrum: the DRT analysis reveals a third semi-circle within the convoluted semi-circles from the PEIS measurement. (d) Overlay of the DRT spectrum as a function of SOC: ionic conductance and charge transfer at both interfaces change marginally with the SOC within the voltage window of 3.0 V and 4.2 V.

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) analysis44 to distinguish the electrochemical processes in the solid-state lithium cell via

image file: c9cp03886h-t21.tif(15)
where Rohmic is the frequency-independent ohmic resistance, and Zpol(ω) denotes the polarisation part of the impedance.42,45

The time constant of each process can be determined from the characteristic frequency, fc corresponding to the local maxima of the peaks in Fig. 3(c) via τDRT = 1/2πfc. 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, τdiff in the positive electrode:47

image file: c9cp03886h-t22.tif(16)
where dpos is the thickness of the positive electrode and DLi(xLi) is the experimentally measured diffusion coefficient.

Fig. 3(c) shows the comparison between Zimag 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 (Rpol < 10 Ω cm2), 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|LiCoO2 interface and the diffusion tail in the LiCoO2 by using the full cell Li|LiPON|LiCoO2 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.

Table 2 The attribution of semi-circles to the possible physical mechanisms: the calculated time-constants show that the two processes identified by Iriyama et al.48 as ionic conductance and charge-transfers happen between the microseconds and milliseconds timescale
Frequency range [Hz] Time-constant, τDRT [s] Possible cause
106–104 2.0 × 10−6 Ionic conductance in the solid electrolyte
104–630 7.5 × 10−5 Charge-transfer at the Li–LiPON interface
630–1 9.8 × 10−4 Charge-transfer at the LiPON–LiCoO2 interface

4.3 Model parameterisation

Most of the electrochemical parameters used in this model to describe the pulse behaviour in the solid-state lithium cell were estimated based on experimental measurements.

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.

image file: c9cp03886h-f4.tif
Fig. 4 Experimental parameterisation of the continuum model: (a) the proposed equivalent circuit model comprises three R-CPE pairs corresponding to the three peaks observed in the DRT analysis at PEIS 90% SOC. The black dashed line indicates the simulated impedance curve to ensure that the semi-circles representing the electrochemical processes do not intersect with each other. (b) Aligned and normalised OCV curves to estimate the lithium content in the cell. The black dashed solid line indicates the OCV curve for the voltage window 4.2 V and 3.0 V and the blue solid line denotes the voltage window 4.5 V and 3.0 V. (c) Experimental measurement of diffusion coefficients in the positive electrode. The dark blue plot indicates the measurement by the GITT method, whereas the red plot denotes the measurement using the PEIS procedure. (d) The characteristic time, τchar due to concentration-dependent solid diffusion in the positive electrode.
Table 3 Summary of parameters (a = Larfaillou et al.,15 b = Fabre et al.20)
Parameters Values Estimated by Literature values
Thickness of the solid electrolyte, delec 2 μm Value from literaturea 2–4 μm
Thickness of the positive electrode, dpos 6 μm Value from literaturea 5–8 μm
Exchange current density at the Li|LiPON interface, i0,neg 0.48 ± 0.72 mA cm−2 PEIS measurement 1.5 mA cm−2[thin space (1/6-em)]b
Ionic conductivity, σelec (2.31 ± 0.52) × 10−6 S cm−1 PEIS measurement 1.2 × 10−6 S cm−1[thin space (1/6-em)]b
Theoretical molar density of LiCoO2, cLCO,max 51599.06 mol m−3 Calculation 50[thin space (1/6-em)]000 mol m−3[thin space (1/6-em)]b

4.3.1 OCV as a function of lithium content. Because lithium is used as the negative electrode, and its voltage is 0 V vs. Li/Li+, the equilibrium voltage of the cell is the equilibrium voltage of the positive electrode. At equilibrium, the rate of the forward reaction is equal to the rate of the backward reaction. The electrode potential is then the electrochemical potential of electrons in equilibrium with the reactant and the product species.26 When the open-circuit voltage of the cell is measured, the amount of lithium intercalated into the positive electrode can be deduced.

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.

4.3.2 Ionic conductivity. The ionic conductivity of the solid electrolyte was determined from the electrolyte thickness delec, surface area of the electrolyte Aelec and electrolyte resistance Relec:50
image file: c9cp03886h-t23.tif(17)
4.3.3 Exchange current density. The exchange current density at the interface between the negative electrode and the solid electrolyte was estimated with:20
image file: c9cp03886h-t24.tif(18)
where Rct,neg is the charge transfer resistance determined from fitting an equivalent circuit model to PEIS measurements. A high exchange current density implies that the surface overpotential is small and the reaction is fast.26
4.3.4 Theoretical molar density of LCO. The theoretical molar density of LiCoO2 was estimated as follows:
image file: c9cp03886h-t25.tif(19)
where ρLCO is the density of the positive electrode (ρLCO = 5050 kg m−3)51 and mLCO is the molar mass of LiCoO2 (mLCO = 0.09787 kg mol−1).
4.3.5 Diffusion coefficients in the positive electrode. The diffusion coefficients were determined from PEIS data based on the diffusion equations derived by Ho et al.37 When the diffusion coefficient is large (i.e., ω ≪ 2DLi/dpos2), the expressions for complex impedance follow:
image file: c9cp03886h-t26.tif(20)
image file: c9cp03886h-t27.tif(21)
where Vm is the molar volume of LiCoO2, U is the open-circuit voltage, x is the lithium content in LiCoO2, ω is the radian frequency (ω = 2πf), dpos is the electrode thickness and Apos is the surface area of the positive electrode. The cell encapsulation has a surface area of 4 cm2. However, the graphical representation of the internal structures published by STMicroelectronics reveals a different size of the interfacial area between the two electrodes and the electrolyte within the encapsulation.15 The surface area was therefore approximated as a square with a length and width of 1.4 ± 0.2 cm. The parameters calculated with this approximation (Table 3) were compared with literature values20 and show a reasonable agreement. The determination of DLi based on eqn (20) and (21) assumes that the current is 90° out of phase with the voltage (i.e., β = 90°). The simultaneous solutions of these two equations yield the values for DLi and dU/dx.

When the diffusion coefficient is small (i.e., ω ≫ 2DLi/dpos2), the phase difference between the current and the voltage is equal to β = 45°. In this case, the diffusion coefficient can be obtained from:

image file: c9cp03886h-t28.tif(22)

Diffusion coefficients can also be determined with the GITT measurement from52

image file: c9cp03886h-t29.tif(23)

The definitions of τpulse, ΔVs and ΔVt can be found in Fig. 2(b). This equation is, however, only valid for short time constants (τpulsedpos2/DLi).52 For the present cell, the smallest dpos2/DLi is 184 s. Therefore, another set of GITT experiments with τpulse = 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, τdiff, as shown in Fig. 4(d), increases sharply as the SOC decreases. This experimental analysis indicates that solid diffusion in LiCoO2 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 LiCoO2 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 LixFePO4, 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, αpos and αneg 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.

5 Model-based overpotential analysis and cell design

5.1 Model validation

Fig. 5(a) shows the simulated cell voltage for a pulse discharge at C/10 compared to experimentally measured voltages. The pulse discharge curves change gradually from a linear profile at high SOCs to a concave profile at low SOCs. The change in the shapes of the voltage profiles, as observed in the experimental measurement, can only be reproduced in the model by using a concentration-dependent solid diffusion coefficient in the positive electrode. As the diffusion coefficient of lithium in the positive electrode decreases for low SOCs, this mechanism can be the bottleneck in the cell performance when increasing the thickness of the positive electrode to achieve a higher cell capacity. The voltage deviations between the predicted and measured curves can be found in Fig. S3(a) in ESI.
image file: c9cp03886h-f5.tif
Fig. 5 Comparison between model predictions (dashed lines) and experimental measurements (solid lines): (a) pulse discharge (b) pulse charging. (c) Dynamic pulses with different C-rates: the current profile (red) and the current profile (blue).

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.

5.2 Quantification of overpotentials

Once parameterised and experimentally validated, the model was used to analyse the rate-limiting mechanisms inside the cell at different operating conditions. The rate-limiting mechanism was studied by analysing the overpotentials that caused the voltage losses in the cell.

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 Patterson31 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

ηct|x=0,x=delec = ϕsϕelecU, (24)
image file: c9cp03886h-t30.tif(25)
ηdiff = ULCO(cLi) − ULCO([c with combining macron]Li), (26)
in which ηct indicates the charge transfer overpotentials, ηohm denotes the ohmic drop in the solid electrolyte and ηdiff represents the overpotential due to the concentration-dependent solid diffusion in the positive electrode. [c with combining macron]Li denotes the average lithium concentration in the positive electrode.

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.

image file: c9cp03886h-f6.tif
Fig. 6 (a) Comparison of calculated voltage losses for SOCs more than 50% in (a) and for SOCs less than 30% in (b) based on the experimental pulse discharge profiles (ipulse = [thin space (1/6-em)]0.69 A m−2 and τpulse[thin space (1/6-em)] = [thin space (1/6-em)]1 h). (c) The impact of operating current density on the various contributions to power loss due to overpotentials at SOC 80% and (d) at SOC 30%.

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 LiCoO2, solid diffusion becomes the rate-limiting mechanism, followed by charge transfer at the positive electrode interface.

5.3 Model-based cell design

Based on the performance-limiting mechanisms discussed in Section 5.2, critical thickness of the positive electrode, as well as a critical ionic conductivity that minimises the voltage loss due to the solid diffusion and ohmic drop was calculated for the pulse operating conditions. Fig. 7 shows the influence of different cell designs on cell performance and losses. A thicker electrode is commonly suggested in the literature as a solution to increasing the capacity of the cell.60,61 In a cell in which bulk LiCoO2 is used as the positive electrode, increasing its thickness will achieve a higher capacity only if the operating current is low.20 The model prediction shows that the critical thickness of the positive electrode under the pulse operating condition (ipulse = 0.694 A m−2, τpulse = 1[thin space (1/6-em)] h, σelec = 10−6 S cm−1) is 4400 nm. After the interruption of discharge current at 3600 s, solid lithium continues to diffuse into the inner part of the electrode. Fig. 7(a) shows the diffusive flux in the positive electrode across the electrode at time t[thin space (1/6-em)] = [thin space (1/6-em)]3600 s (solid lines) and t[thin space (1/6-em)] = [thin space (1/6-em)]3650 s (dotted lines) respectively. The x-axis is normalised to the electrode thickness. The position between the positive electrode and the solid electrolyte is denoted by x[thin space (1/6-em)] = [thin space (1/6-em)]0. When the electrode thickness increases beyond the critical thickness, the diffusive peaks at t[thin space (1/6-em)] = [thin space (1/6-em)]3600 s and t[thin space (1/6-em)] = [thin space (1/6-em)]3650 s shift towards the interface with the solid electrolyte. This result indicates that less percentage of the electrode will be utilised, as shown in Fig. 7(b). When the electrode thickness is increased to 18[thin space (1/6-em)]000 nm, the model prediction shows that less than 90% of the electrode is used after the cell relaxes and reaches the equilibrium. As a result, the solid diffusion overpotential increases from 9.6 mV to 26.3 mV and the cell internal resistance increases from 194 Ω cm2 to 433 Ω cm2 when the electrode thickness is increased from 4400 nm to 18[thin space (1/6-em)]000 nm. This analysis demonstrates the importance of using a model-based procedure to calculate the optimum thickness of the electrode given the operating conditions. Making the electrode as thick as possible does not necessarily lead to an increase in the usable capacity of the cell.
image file: c9cp03886h-f7.tif
Fig. 7 (a) Diffusive flux in the positive electrode as a function of normalised electrode thickness: the solid lines indicate the diffusive flux at t = 3600 s whereas the dotted lines indicate the diffusive flux at t = 3650 s (ipulse = 0.694 A m−2, τpulse = 1 h, σelec = 10−6 S cm−1). (b) Ratio of electrode utilisation as a function of normalised electrode thickness: the solid lines indicate model evaluation at t = 3600 s whereas the dashed lines indicate the evaluation at time after the electrode achieves the equilibrium (t = 14[thin space (1/6-em)]340 s) (ipulse = 0.694 A m−2, τpulse = 1 h, σelec = 10−6 S cm−1). (c) Calculation of ohmic drop in the solid electrolyte with different ionic conductivities for ipulse = 0.694 A m−2. (d) Ohmic drop in the solid electrolyte with different ionic conductivities for ipulse = 2.083 A m−2.

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 LiCoO2 continues to be used as the positive electrode.

Table 4 Other selected examples of solid electrolytes (reproduced from Zheng et al. with permission from Elsevier62)
Types of solid electrolytes Composition Ionic conductivity in S cm−1
Argyrodite (Li6PS5Cl) Li6PS5Br 10−3–10−2
Li6PS5Cl 1.33[thin space (1/6-em)] × [thin space (1/6-em)]10−3
Garnet (Li7La3Zr2O12) Li6.75La3(Zr1.75Nb0.25)O12 8[thin space (1/6-em)] × [thin space (1/6-em)]10−4
Perovskite (Li3xLa2/3−xTiO3) Li3/8Sr7/16Ta3/4Hf1/4O3 3.8[thin space (1/6-em)] × [thin space (1/6-em)]10−4
NASICON (LiTi2(PO4)3) LiTi2(PO4)3–0.2Li3BO3 3[thin space (1/6-em)] × [thin space (1/6-em)]10−4
Sulfide (Li4GeS4) 75Li2S–25P2S5 glass 2[thin space (1/6-em)] × [thin space (1/6-em)]10−4
Li4.25+δ(Ge0.75−δ'Ga0.25)S4 6.5[thin space (1/6-em)] × [thin space (1/6-em)]10−5
LISICON (γ-Li3PO4) Li4−xSi1−xPxO4(x = 0.5, 0.6) 1[thin space (1/6-em)] × [thin space (1/6-em)]10−6

6 Conclusions

In summary, we have used experimental and numerical methods to study the pulse behaviour observed in the solid-state lithium cell. The experimental pulse voltage measurements show that solid-state lithium cell has a high internal resistance of ≈200 Ω cm2 and it increases for low SOCs to ≈450 Ω cm2. This magnitude of internal resistance could affect cell performance significantly if the cell were enlarged to achieve a higher capacity. By comparing the PEIS measurements at different SOCs, the significant increase in the internal resistance at the end of SOC is attributed to the decrease in the solid diffusion coefficient. Based on the model presented here, future work aiming to improve solid-state battery performance should focus on the design of the positive electrode for solid-state lithium cells to improve the mass transport of solid lithium in the electrode and hence reduce the diffusion impedance and internal resistance. By calculating the time constant of different electrochemical processes using the DRT method, our experimental results show that ionic conductance and charge transfer kinetics happen in the timescale between microseconds and milliseconds. The comparison of these time constants to the pulse discharge and charging duration show that if the pulse duration exceeds milliseconds, charge transfer at either interface is not the primary mechanism governing the pulse voltage response of solid-state lithium cells.

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 LiCoO2 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 18[thin space (1/6-em)]000 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 LiCoO2 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.


List of symbols and physical constants

NiFlux density of species Li+ (mol m−2 s−1)
ΔVsSteady-state voltage difference (V)
ΔVtTransient voltage difference (V)
AelecSurface area of the electrolyte (m2)
AposSurface area of the positive electrode (m2)
DLiDiffusion coefficient of Li in the positive electrode (m2 s−1)
DLi+Diffusion coefficient of Li+ in the solid electrolyte (m2 s−1)
FFaraday's constant = 96[thin space (1/6-em)]487 (C mol−1)
RgasUniversal gas constant (J mol−1 K−1)
RohmicOhmic resistance (Ω)
Rct,negCharge transfer resistance (Ω)
RelecElectrolyte resistance (Ω)
ResReal and imaginary impedance residuals (%)
TTemperature (K)
UOpen-circuit voltage (V)
VVoltage (V)
VmMolar volume of the positive electrode (m3 mol−1)
ZImpedance (Ω)
ZpolPolarisation resistance (Ω)
iCurrent density (A m−2)
ipulseExternally applied pulse current density (A m−2)
[c with combining macron]LiAverage lithium concentration in the positive electrode (mol m−3)
aSpecific interfacial area (m−1)
cLiLithium concentration in the positive electrode (mol m−3)
cLCO,maxTheoretical maximum lithium concentration in the positive electrode (mol m−3)
cLi+Concentration of Li+ in the solid electrolyte (mol m−3)
delecThickness of the solid electrolyte (m)
dposThickness of the positive electrode (m)
fcCharacteristic frequency (Hz)
i0,negExchange current densities (mA cm−2)
uLi+Mobility of Li+ in the solid electrolyte (m2 mol J−1 s−1)
vBulk velocity of the solid electrolyte (m s−1)
zLi+Number of proton charges carried by an ion (—)
αnegLi–LiPON charge transfer coefficient (—)
αposLiPON–LiCoO2 charge transfer coefficient (—)
βPhase difference between the current and the voltage (°)
ηctCharge transfer overpotential (V)
ηdiffOverpotential due to solid diffusion (V)
ηohmOverpotential due to ohmic drop (V)
ωRadian frequency (Hz)
ϕsSurface electrode potential (V)
ϕelecElectrolyte potential (V)
σLi+Conductivity of Li+ in the solid electrolyte (S cm−1)
τTime constant (s)
τdiffTime constant due to diffusion in the positive electrode (s)

Conflicts of interest

There are no conflicts of interest to declare.


Much appreciated is the strong support received from Beijing Institute of Aeronautical Materials (BIAM). The research was performed at the BIAM-Imperial Centre for Materials Characterisation, Processing and Modelling at Imperial College London. This work was partially carried out with funding from the Faraday Institution (faraday.ac.uk; EP/S003053/1), grant number FIRG003. M.-C. Pang would like to thank Dr S. Cooper for discussions during the early stage preparation of this work; Y. Zhao, O. Shaw and WK. Yoong for discussions on the experimental setup; Dr G. Madabattula and Dr S. O'Kane for reviewing the model formulations.


  1. K. Xu, Chem. Rev., 2004, 104, 4303–4418 CrossRef CAS PubMed.
  2. D. P. Finegan, M. Scheel, J. B. Robinson, B. Tjaden, I. Hunt, T. J. Mason, J. Millichamp, M. Di Michiel, G. J. Offer, G. Hinds, D. J. L. Brett and P. R. Shearing, Nat. Commun., 2015, 6, 6924 CrossRef CAS PubMed.
  3. N. Kamaya, K. Homma, Y. Yamakawa, M. Hirayama, R. Kanno, M. Yonemura, T. Kamiyama, Y. Kato, S. Hama, K. Kawamoto and A. Mitsui, Nat. Mater., 2011, 10, 682–686 CrossRef CAS PubMed.
  4. A. Sakuda, A. Hayashi and M. Tatsumisago, Chem. Mater., 2010, 22, 949–956 CrossRef CAS.
  5. T. B. Reddy, D. Linden and D. H. o. b. Linden, Linden's handbook of batteries, McGraw-Hill, New York and London, 4th edn, 2011 Search PubMed.
  6. J. Janek and W. G. Zeier, Nat. Energy, 2016, 1, 16141 EP CrossRef.
  7. L.-O. Hagman, P. Kierkegaard, P. Karvonen, A. I. Virtanen and J. Paasivirta, Acta Chem. Scand., 1968, 22, 1822–1832 CrossRef CAS.
  8. K. Arbi, J. M. Rojo and J. Sanz, J. Eur. Ceram. Soc., 2007, 27, 4215–4218 CrossRef CAS.
  9. V. Thangadurai and W. Weppner, J. Am. Ceram. Soc., 2005, 88, 411–418 CrossRef CAS.
  10. V. Thangadurai and W. Weppner, Adv. Funct. Mater., 2005, 15, 107–112 CrossRef CAS.
  11. Y. Kato, S. Hori, T. Saito, K. Suzuki, M. Hirayama, A. Mitsui, M. Yonemura, H. Iba and R. Kanno, Nat. Energy, 2016, 1, 652 Search PubMed.
  12. X. Han, Y. Gong, K. K. Fu, X. He, G. T. Hitz, J. Dai, A. Pearse, B. Liu, H. Wang, G. Rubloff, Y. Mo, V. Thangadurai, E. D. Wachsman and L. Hu, Nat. Mater., 2017, 16, 572–579 CrossRef CAS PubMed.
  13. A. A. Talin, D. Ruzmetov, A. Kolmakov, K. McKelvey, N. Ware, F. El Gabaly, B. Dunn and H. S. White, ACS Appl. Mater. Interfaces, 2016, 8, 32385–32391 CrossRef CAS PubMed.
  14. S. K. Martha, J. Nanda, Y. Kim, R. R. Unocic, S. Pannala and N. J. Dudney, J. Mater. Chem. A, 2013, 1, 5587 RSC.
  15. S. Larfaillou, D. Guy-Bouyssou, F. Le Cras and S. Franger, J. Power Sources, 2016, 319, 139–146 CrossRef CAS.
  16. D. Danilov, R. A. H. Niessen and P. H. L. Notten, J. Electrochem. Soc., 2011, 158, A215 CrossRef CAS.
  17. K. Smith and C.-Y. Wang, J. Power Sources, 2006, 161, 628–639 CrossRef CAS.
  18. M. Doyle, T. F. Fuller and J. Newman, Electrochim. Acta, 1994, 39, 2073–2081 CrossRef CAS.
  19. S. Zugmann, M. Fleischmann, M. Amereller, R. M. Gschwind, H. D. Wiemhöfer and H. J. Gores, Electrochim. Acta, 2011, 56, 3926–3933 CrossRef CAS.
  20. S. D. Fabre, D. Guy-Bouyssou, P. Bouillon, F. Le Cras and C. Delacourt, J. Electrochem. Soc., 2012, 159, A104 CrossRef CAS.
  21. T. F. Fuller, M. Doyle and J. Newman, J. Electrochem. Soc., 1994, 141, 982 CrossRef CAS.
  22. K. Becker-Steinberger, S. Funken, M. Landstorfer and K. Urban, ECS Trans., 2010, 285–296 CAS.
  23. D. Meng, B. Zheng, G. Lin and M. L. Sushko, Commun. Comput. Phys., 2014, 16, 1298–1322 CrossRef.
  24. M. Landstorfer, S. Funken and T. Jacob, Phys. Chem. Chem. Phys., 2011, 13, 12817–12825 RSC.
  25. K. E. Thomas-Alyea, J. Electrochem. Soc., 2018, 165, A1523–A1528 CrossRef CAS.
  26. J. S. Newman and K. E. Thomas-Alyea, Electrochemical Systems, Wiley, Hoboken, NJ and Chichester, 3rd edn, 2004 Search PubMed.
  27. J. B. Bates, N. J. Dudney, G. R. Gruzalski, R. A. Zuhr, A. Choudhury, C. F. Luck and J. D. Robertson, Solid State Ionics, 1992, 53–56, 647–654 CrossRef CAS.
  28. S. Sicolo and K. Albe, J. Power Sources, 2016, 331, 382–390 CrossRef CAS.
  29. Y. Deng, C. Eames, J.-N. Chotard, F. Lalère, V. Seznec, S. Emge, O. Pecher, C. P. Grey, C. Masquelier and M. S. Islam, J. Am. Chem. Soc., 2015, 137, 9136–9145 CrossRef CAS PubMed.
  30. J. Evans, C. A. Vincent and P. G. Bruce, Polymer, 1987, 28, 2324–2328 CrossRef CAS.
  31. J. W. Patterson, J. Electrochem. Soc., 1971, 118, 1033 CrossRef CAS.
  32. F. Han, A. S. Westover, J. Yue, X. Fan, F. Wang, M. Chi, D. N. Leonard, N. J. Dudney, H. Wang and C. Wang, Nat. Energy, 2019, 4, 187–196 CrossRef CAS.
  33. M. Ménétrier, I. Saadoune, S. Levasseur and C. Delmas, J. Mater. Chem., 1999, 9, 1135–1140 RSC.
  34. C. R. Birkl, E. McTurk, M. R. Roberts, P. G. Bruce and D. A. Howey, J. Electrochem. Soc., 2015, 162, A2271–A2280 CrossRef CAS.
  35. D. Wang, Y. Bao and J. Shi, Energies, 2017, 10, 1284 CrossRef.
  36. Y.-H. Chiang, W.-Y. Sean and J.-C. Ke, J. Power Sources, 2011, 196, 3921–3932 CrossRef CAS.
  37. C. Ho, I. D. Raistrick and R. A. Huggins, J. Electrochem. Soc., 1980, 127, 343 CrossRef CAS.
  38. M. Schönleber, Kramers–Kronig Validity Test Lin-KK for Impedance Spectra, 2015, http://www.iam.kit.edu/wet/english/Lin-KK.php Search PubMed.
  39. B. A. Boukamp, J. Electrochem. Soc., 1995, 142, 1885 CrossRef CAS.
  40. M. Schönleber, D. Klotz and E. Ivers-Tiffée, Electrochim. Acta, 2014, 131, 20–27 CrossRef.
  41. M. Schönleber and E. Ivers-Tiffée, Electrochem. Commun., 2015, 58, 15–19 CrossRef.
  42. H. Schichlein, A. C. Müller, M. Voigts, A. Krügel and E. Ivers-Tiffée, J. Appl. Electrochem., 2002, 32, 875–882 CrossRef CAS.
  43. D. Klotz, D. S. Ellis, H. Dotan and A. Rothschild, Phys. Chem. Chem. Phys., 2016, 18, 23438–23457 RSC.
  44. T. H. Wan, M. Saccoccio, C. Chen and F. Ciucci, Electrochim. Acta, 2015, 184, 483–499 CrossRef CAS.
  45. A. Leonide, V. Sonn, A. Weber and E. Ivers-Tiffée, J. Electrochem. Soc., 2008, 155, B36 CrossRef CAS.
  46. J. Illig, M. Ender, T. Chrobak, J. P. Schmidt, D. Klotz and E. Ivers-Tiffée, J. Electrochem. Soc., 2012, 159, A952–A960 CrossRef CAS.
  47. Y. Matsuda, N. Kuwata, T. Okawa, A. Dorai, O. Kamishima and J. Kawamura, Solid State Ionics, 2019, 335, 7–14 CrossRef CAS.
  48. Y. Iriyama, T. Kako, C. Yada, T. Abe and Z. Ogumi, Solid State Ionics, 2005, 176, 2371–2376 CrossRef CAS.
  49. I. J. Ong and J. Newman, J. Electrochem. Soc., 1999, 146, 4360 CrossRef CAS.
  50. X. Yu, J. B. Bates, G. E. Jellison Jr. and F. X. Hart, J. Electrochem. Soc., 1997, 144, 524 CrossRef CAS.
  51. H. Zhang and P. S. Grant, Sens. Actuators, B, 2013, 176, 52–57 CrossRef CAS.
  52. W. Weppner and R. A. Huggins, J. Electrochem. Soc., 1977, 124, 1569 CrossRef CAS.
  53. M. Ecker, T. K. D. Tran, P. Dechent, S. Käbitz, A. Warnecke and D. U. Sauer, J. Electrochem. Soc., 2015, 162, A1836–A1848 CrossRef CAS.
  54. A. V. Churikov, A. V. Ivanishchev, I. A. Ivanishcheva, V. O. Sycheva, N. R. Khasanova and E. V. Antipov, Electrochim. Acta, 2010, 55, 2939–2950 CrossRef CAS.
  55. N. Ding, J. Xu, Y. X. Yao, G. Wegner, X. Fang, C. H. Chen and I. Lieberwirth, Solid State Ionics, 2009, 180, 222–225 CrossRef CAS.
  56. Y. Li, H. Chen, K. Lim, H. D. Deng, J. Lim, D. Fraggedakis, P. M. Attia, S. C. Lee, N. Jin, J. Moškon, Z. Guan, W. E. Gent, J. Hong, Y.-S. Yu, M. Gaberšćek, M. S. Islam, M. Z. Bazant and W. C. Chueh, Nat. Mater., 2018, 17, 915–922 CrossRef CAS PubMed.
  57. K. M. Diederichsen, E. J. McShane and B. D. McCloskey, ACS Energy Lett., 2017, 2, 2563–2575 CrossRef CAS.
  58. N. Kazemi, D. L. Danilov, L. Haverkate, N. J. Dudney, S. Unnikrishnan and P. H. Notten, Solid State Ionics, 2019, 334, 111–116 CrossRef CAS.
  59. A. Nyman, T. G. Zavalis, R. Elger, M. Behm and G. Lindbergh, J. Electrochem. Soc., 2010, 157, A1236 CrossRef CAS.
  60. P. Nelson, I. Bloom, K. Amine and G. Henriksen, J. Power Sources, 2002, 110, 437–444 CrossRef CAS.
  61. W. Lu, A. Jansen, D. Dees, P. Nelson, N. R. Veselka and G. Henriksen, J. Power Sources, 2011, 196, 1537–1540 CrossRef CAS.
  62. F. Zheng, M. Kotobuki, S. Song, M. O. Lai and L. Lu, J. Power Sources, 2018, 389, 198–213 CrossRef CAS.
  63. S. K. Murthy, A. K. Sharma, C. Choo and E. Birgersson, J. Electrochem. Soc., 2018, 165(9), A1746–A1752 CrossRef CAS.


Electronic supplementary information (ESI) available. See DOI: 10.1039/c9cp03886h

This journal is © the Owner Societies 2019