3D characterisation of S/C electrode microstructure
Figure 1 (a) shows the internal structure of the Li-S battery. The morphological properties of the S/C electrode (Fig. 1(b-d)) in Li-S batteries are composed of multiple phases and vary over multiple length scales. Figure 2 (a) shows the reconstructed S/C pristine electrode and the detailed orthoslice, both of which show the S/C cathode consisting of three phases: elemental sulfur, CBD and electrolyte (which fills the pores), segmented based on the greyscale attenuation factor. Figure 2 (b-d) shows the volume of Sulfur (yellow), pores filled with electrolyte (pink), and CBD (grey) present in the S/C electrode and highlights the complex and heterogeneous 3D nature of the phase distribution in the electrode. Figure 2 (e-f) shows the coloured spatial distribution of sulfur particle size and pore sizes, respectively. Each colour represents different size particles (e) and pores (f), highlighting the extent of non-uniformity. As experimentally observed by various researchers, the heterogeneity of the particles’ morphology affects the rate of sulfur dissolution, i.e., dissolution of solid sulfur (S8(solid)) into the electrolyte [4, 29]. Figure 2 (g) displays the effect of microstructure on the distribution of voltage across the electrode in the through-plane direction at any given instance; the results here are independent of the electrochemical reactions and emphasises the non-uniform voltage distribution in the electrodes due to electronic resistance of the microstructure only, highlighting the importance of considering electrode heterogeneity while modelling the electrode structure. It can be inferred from the shape factor (sphericity) distribution shown in Fig. 2 (h) that sulfur particles significantly deviate from being perfectly spherical, with an average shape factor of 0.81. However, as shown by Cao et al. who conducted a CFD-DNS analysis of differently shaped particles with an identical surface area, dissolution rates can vary significantly with varying shape factor, emphasising the need to include particle shape analysis in Li-S battery modelling and ideally to include realistic sulfur particle geometries from X-ray CT data in the modelling studies [30]. Previous Li-S battery modelling studies used the Bruggeman correlation to represent the effective transport parameters in the C/S electrode [3, 10, 11, 13, 14, 16, 18, 27, 29, 31, 32]. However, the derivation of the Bruggeman correlation is based on a perfect sphericity assumption; the current results emphasise the use of an image-based modelling approach to examine the suitability of using the empirical approaches [33]. Figure 2 (i) displays microscopic sulfur particle size distribution using equivalent diameter; 90% of the sulfur particles were smaller than 12 µm in diameter and the average particle diameter observed was ca 7.5 µm. Figure 2 (j) shows the changes in volume fraction of individual phases across the electrode thickness. Collectively, Fig. 2 highlights the microstructural heterogeneities present in the C/S electrodes which should be considered while evaluating current electrodes, battery modelling studies, and, importantly, when designing new electrodes.
Sampling of Subvolumes
Figure 3 illustrates the schematic of a Li-S battery domain and reaction pathway during the discharge process. To analyse the effect of heterogeneity on the electrochemical performance, representative elementary volume sampling is not viable due to enormous computational expense (more information can be found in Supplementary Information), therefore, a multiple subsampling approach was adopted. Four subvolumes, cut through the entire thickness of the electrode, were selected from the various points (Fig. 4). Table 1 shows the phase volume fraction present in each sub-volume, highlighting the variation of CBD, sulfur and electrolyte volume present. Electrolyte to sulfur ratio (E/S ratio) is the critical parameter that defines the sulfur dissolution process [34], the initial discharge capacity, and the cell durability [11, 29]. It should be noted that calculated E/S ratios are based entirely on the local microstructure and assume zero excess electrolyte – in reality, Li-S cells will be flooded with electrolyte and likely have much higher E/S ratios for the whole cell, and so the reported values here are all much lower than cell-level E/S ratios. Nevertheless, relative local E/S ratios are a key and useful parameter in determining the local performance of the cell based on the heterogenous microstructure. As shown in Table 1, Subvolume 1 and Subvolume 4 have the highest E/S ratio whereas Subvolume 2 has the highest sulfur content and lowest E/S ratio. The effect of E/S ratio, sulfur content and CBD content on the cell performance is discussed later.
Table 1
Phase volume fraction present in each Subvolume and compared to the total electrode volume from the X-ray CT data set
|
Sulfur
|
Electrolyte
|
CBD
|
Specific surface area of CBD (m− 1)
|
Effective local E/S ratio
(µL mg− 1)
|
S/SSA
(ug um)
|
1C Current density (A m− 2)
|
Subvolume 1
|
27.50
|
12.40
|
60.10
|
8.51 x105
|
0.70
|
0.03
|
22.12
|
Subvolume 2
|
31.10
|
4.70
|
64.20
|
7.76 x105
|
0.56
|
0.04
|
25.01
|
Subvolume 3
|
20.70
|
5.35
|
73.95
|
1.05 x106
|
0.61
|
0.02
|
16.65
|
Subvolume 4
|
30.16
|
14.70
|
55.14
|
7.47 x105
|
0.72
|
0.04
|
24.26
|
Full electrode
|
27.96
|
39.29
|
32.75
|
1.23 x105
|
1.17
|
4.92
|
23.59
|
Effect of localised microstructure on the discharge performance
Figure 5 demonstrates the discharge curve comparison between the microstructurally resolved 1D model and image-based 3D models at different subvolumes. The 1D model depicted in Fig. 5(a) was developed using the effective properties obtained from the actual electrode microstructure and electrochemical properties adapted from the Kumaresan’s model [7]. The simulated discharge curve was compared with the measured experimental data provided by Soni et al. [35], where two different volumes of electrolyte were added inside the cell. The deviation between the predicted and realistic curves implies the weakness of the 1D model, which overpredicts the capacity of the studied Li-S cell. Subsequent 1D models (Fig. 5(b), dashed lines) of the subvolumes use the same approach, but with the volume averaged parameters from each individual subvolume.
As expected, the higher activation overpotential at larger global discharge current results in a lower cell voltage. At the upper voltage plateau, the reduction of \({S}_{8 \left(dissolved\right)}\) to \({S}_{8}^{2-}\) and shorter chain polysulfides is prominent [7, 27, 36]. From the high voltage plateau, the voltage decreases continuously until it reaches the minimum voltage where the domain is saturated with Li+, \({S}_{4}^{2-}\) and \({S}_{2}^{2-}\) and the formation of short-chained polysulfides begins, promoting the start of the nucleation of solid polysulfides. The reduction of \({S}_{2}^{2-}\) to \({S}^{2-}\) and the formulation of non-soluble short-chained solid polysulfides, i.e., Li2S2 and Li2S, results in the lower-voltage plateau; however, with further accumulation of solid-polysulfides and subsequent loss in the electrochemically active surface area, the cell reaches 100% depth of discharge (DoD). This is the well-documented discharge performance captured in the effective property-based 1D model (Fig. 6a).
The discharge performance of image-based models at different subvolumes was analysed at global C-rates of C/30, C/20 and C/10 (Fig. 5(b)); the global C-rate C is calculated based on the total sulfur mass loading of the pristine full electrode, which is equal to 23.59 Ah m-2. Overall, the simulated performance varies with the localised microstructure of the subvolumes, and different capacities are obtained for each subvolume, despite being normalised to the local sufur content in the subvolume. Subvolume 1, which has inherent porosity of 40% and an effective local E/S ratio of 0.70 µL mg-1, shows better performance compared to the other subvolumes, and the closest match to the volume averaged 1D model for this subvolume (inherent porosity here refers to the porosity of the electrode without any sulfur impregnates and any polysulfide precipitation, i.e., 1 minus the CBD volume fraction). Subvolume 3 performs worst and has the lowest inherent porosity at 26%; it is also the only subvolume to display differing normalised capacity at different global C-rates. Moderate and similar performance was observed at Subvolume 2 and 4; however, the inherent porosities and E/S ratios differ significantly. There is also a larger discrepancy between the 3D and 1D model capacities for Subvolume 2 than Subvolume 4. As conventionally considered, lower E/S ratio will increase the energy density of the battery, but at the expense of the electrochemical performance (rate capability and sulfur utilisation) [11, 29], and this is often the approach taken in lower order continuum models; however, the fact that Subvolumes 2 and 4 (relatively low and high E/S ratio, respectively) have a similar electrochemical performance highlights the role of microstructure in the actual performance of the battery. In the case of Subvolume 4, for example, the performance is lower than might be expected from simply considering the E/S ratio and this is contrary to what might be expected from the higher inherent porosity and microstructural properties (tortuosity etc) of the CBD in this subvolume, which should allow for favourable sulfur dissolution, distribution in the electrolyte phase and electrochemical reaction during discharge. Conversely, Subvolume 2, which has a very low E/S ratio, performs almost as well as Subvolume 4 despite having a lower inherent porosity than it. The only volume averaged parameter that is similar for Subvolumes 2 and 4 is the ratio of sulfur to specific surface area (S/SSA); however, this parameter does not explain the high performance for Subvolume 1, which has a moderate value of S/SSA. Additionally, despite the low E/S ratio, increasing the C-rate does not cause a reduction in normalised capacity for Subvolume 2 as it does for Subvolume 3 which has almost double the E/S ratio. It should be noted that global C-rates (of C/30, C/20 and C/10) are applied to all subvolumes based on the total sulfur loading of the entire electrode volume (1C current density of 23.59 A m-2), and therefore that the effective local C-rate of each subvolume will be different depending on its sulfur volume fraction. (According to the actual sulfur capacity, the corresponding 1C current density of each subvolume 1–4 is 22.12 A m-2, 25.01 A m-2, 16.65 A m-2 and 24.26 A m-2, respectively.) Therefore, for example, Subvolume 3 will experience local C-rates higher than the applied global C-rate, and Subvolumes 2 and 4 (which have similar and the highest S volume fractions) would experience the lowest local C-rates (see Fig. 7). This could be the cause of the deviation in normalised capacities with C-rate in Subvolume 3; the overpotential at the global rate of C/10 (2.36 A m-2 which would be higher effective C-rate for this volume) is too great to obtain the ‘full’ capacity, whereas in all other subvolumes, the same capacity is achieved at all C-rates. This does not fully explain the trend in normalised capacity and electrochemical performance, however, as Subvolume 1 significantly outperforms the others. It also does not apply to the volume averaged 1D simulations for Subvolume 3, which achieve the same capacity at all C-rates. In fact, none of the trends observed in the modelled electrochemical performance of the subvolumes can be explained by trends in the subvolume-averaged parameters outlined in Table 1. This is a finding that can only be reached via using real microstructures in combination with a 3D electrochemical model and is a shortcoming of lower ordered modelling approaches, which homogenise the distribution of phases in 1D. Because the 3D model is applied to the actual microstructure, it is inherently set up to include local variations in concentration of soluble species, reaction gradients in 3D, localised current densities and overpotential variations, and reduction in active surface area caused by deposition of lower-order (low conductivity) polysulphide species. Because of this, the outputs of the model can provide insights that volume-averaged parameters simply cannot – even when the parameters are averaged over subvolume domains within the electrode.
Effect of microstructure on DoD and temporal distribution of dissolved sulfur
Figure 6(a-c) highlights the effect of microstructure at increasing C-rates on DoD. With an increase in the discharge rate, the impact of heterogeneity in the electrode microstructure on the performance becomes more prominent [Figure 6(c)]. At the lowest discharge rate (C/30) Subvolumes 1, 2 and 4 display similar performance when plotted against DoD; however, Subvolume 3, shows a lower voltage performance. This could be due to a lower sulfur loading at Subvolume 3 causing higher activation overpotential that results in a lower local voltage. Figure 6(d-f) shows the simulated spatio-temporal distribution of dissolved sulfur, normalised with the maximum sulfur solubility in DOL:DME at 20%, 60% and 95% DoD. The normalised concentration of dissolved sulfur is relatively uniform at all the subvolumes when plotted at 20% DoD and discharge current equivalent to a global rate of C/10 [Figure 6(d)]. At 60% DoD, the non-uniform concentration of dissolved sulfur in all the subvolumes shows areas of entire utilisation of dissolved sulfur (blue) and areas showing isolated pockets of dissolved sulfur, not utilised in the sulfur reduction process (red) [Figure 6(e)]. The total dissolved sulfur concentration for Subvolumes 1–4 starts from 17.15, 16.35, 15.05, and 15.9 mol mm-3 at 20% DoD, drops to 4.3, 2.32, 10.92 and 0.79 mol mm-3 at 80% DoD, respectively. The concentration of dissolved sulfur is unchanged from 60–95% DoD, highlighting the presence of unutilised sulfur. However, it is important to note that, in this study, isolated pockets of dissolved sulfur present in the subvolumes could be artefacts generated due to the artificial boundaries imposed by the subvolume approach and could be connected to the neighbouring pockets, albeit with extremely long and tortuous paths resulting in higher resistance to diffusion of Li+ ions. Quantitatively, Subvolume 3 shows a lower sulfur utilisation, the concentration of unreacted sulfur remaining in the electrolyte is 10.92 mol mm-3, leading to a reduction in the performance.
Evolution Of Local Electrode Current Density
As mentioned previously, owing to the existing heterogeneous microstructures in each sub volume, unlike in the homogeneous structure, the local current density will differ within the electrode at various locations, even though the applied global C-rates are the same. Figure 7 presents the magnitude evolution of local electrode current density within the four representative sub-volumes applying the same global C-rate C/10 (2.38 A m-2) during the discharging process. It is obvious to observe that, at 50%, 80% and 100% DoD, the local current density is varied with location and above zero, and the maximum current densities are over 5 A m-2. For Subvolumes 1, 2 and 4, from 50–80% DoD, the area where the current densities are generally higher (red) are expanding, and the lower current density areas (blue) are reducing and, as illustrated numerically, the volume-averaged current density is increasing. Subsequently, from 80–100% DoD, the trend is reversed, with a decreasing volume-averaged current density. However, for Subvolume 3, it is noted that it has the highest average current density at the evaluated DoD states compared to other sub-volumes. Additionally, the volume-averaged current densities are increasing with DoD, unlike others. This also has implications for the inferior discharging behaviour demonstrated by Subvolume 3 in previous sections; the applied current density is too great for this local microstructure, which prevents Subvolume 3 obtaining the full capacity. For Subvolumes 1, 2 and 4, comparable current densities are applied, and the slightly lower current density which applies to Subvolume 1 yields the best capacity performance.
Evolution of solid polysulfide deposition
Since the deposition of solid polysulfides, i.e., Li2S2 and Li2S, over an electrochemically active surface area affects the performance, it is important to understand the interplay between discharge rates, DoD and electrode microstructure. Here, electrochemically active surface area refers to the interface between the electrolyte and CBD, and, as previously stated, we are assuming the CBD to be solid and non-porous. With this modelling approach, it is not possible to model the morphology or thickness of the deposits, but we are able to predict their location, and the percentage of the available surface area covered with solid polysulfides; therefore, the effect on the subsequent performance once non-conductive deposits have formed on a portion of the CBD is accounted for in this model. In a real system, the solid polysulfide deposits would likely be more 3D in nature, and therefore we are likely to be overestimating the fraction of CBD surface that is rendered inactive via the assumed 2D film deposits in our model; nevertheless, it is still valuable to study the effect the microstructure has on the heterogeneity of deposits in the electrode. Figure 8 depicts the extent of deposition of polysulfides on an electrochemically active surface area at each subvolume, obtained at 50%, 80% and 100% DoD and at the global discharge rate of C/10.
At 50% DoD, the interface between CBD and electrolyte is comparatively free from the deposition of solid polysulfides; however, some deposition was observed at Subvolume 3 towards the separator, equating to ~ 5% electrochemically active surface area covered with solid polysulfides. With further increase in DoD, electrochemically active surface areas are covered under precipitation with more extensive deposition observed towards the separator. The amount of electrochemically active surface area covered by polysulfide deposition was ~ 20% for Subvolumes 1 and 2, ~ 35% for Subvolume 3 and 30% for Subvolume 4. This is in accordance with previously developed 1D models, although they of course cannot spatially resolve this deposition behaviour in 3D [7, 10, 15, 37].
At 100% DoD, most of the interface between CBD and electrolyte (i.e., for subvolumes 1 and 2, approximately 70%, for Subvolume 3, ~ 90% and for Subvolume 4 ~ 80%) is covered under polysulfides, reducing the active area. However, as expected, the active areas near unutilised sulfur are free to form solid deposits. It is interesting to note that while the normalised discharge behaviour showed Subvolumes 2 and 4 to behave similarly (and Subvolume 1 to out-perform all), when considering the polysulfide deposition behaviour, Subvolumes 1 and 2 have similar levels of coverage to each other at all DoD, and the lowest amount of coverage compared to the other two subvolumes. This is despite the different inherent porosities, sulfur volume fractions, effective local E/S ratios and S/SSA ratios. This, again, highlights the effect the microstructure, rather than the subvolume-averaged parameters, has on the local performance of the electrode and the strength of the 3D modelling approach taken here.
Despite the presence of a global deposition gradient, the image-based models have shown that the deposition of solid polysulfides is heterogeneous. Precipitation of solid polysulfides results in the expansion of the battery electrode that leads to mechanical stresses [8, 32, 38]. Inhomogeneous precipitation will, in turn, lead to an inhomogeneous volume expansion and contraction of the electrode, resulting in inhomogeneous material stresses and degradation. However, this study aims to first discern the advantage of image-based modelling to understand the impact of heterogeneities present in the electrode on the battery performance, thus the morphology of the solid polysulfide precipitation is assumed in the form of film-type precipitation. However, the precipitation could be film type, particulate or finger type, further altering the electrode morphology and the balance between film and finger-type precipitation would alter the transport properties and mean pore sizes [26], which will be a focus of future modelling work that allows for morphological changes to the phases in the image volume.