Lithium-sulfur (Li-S) batteries are technologically attractive due to their high energy density (2600 Whkg-1), large theoretical capacity (1675 mA h g-1), low cost, and environmental friendliness 1. The optimizations of cathode materials 7–9, multifunctional separators 10,11, electrolyte chemistries 12, and anode structures13 enabled remarkable progress in this area, but the challenges with simultaneous attainment of long life cycle, high areal capacity, power density, and discharge rate2–4 compatible with standards for broadly accessible electric vehicles (EV) and other sustainability-critical technologies remain. An analysis of the latest breakthroughs indicate that sulfur host materials, such as porous carbon14, conductive polymers15, graphene16, metal-organic frameworks (MOCs)17,18, metal oxides/sulfides8–10, and their hybrids 8,19–22 all play critical roles in the implementation of Li-S batteries. Battery components structured as reticulated nanofiber composites (RNCs) based on interconnected nanowires 23, nanotubes 24, and nanofibers 25 can be intuitively selected as promising candidates. However, their structural design is nontrivial because the requirements of mechanical, electrical, and chemical properties of Li-S cathodes are stringent and mutually exclusive. Materials for Li-S cathodes must (1) accommodate large volumes of sulfur, (2) have high electrical conductivity, and (3) withstand large volumetric changes during multiple charge-discharge cycles. They also must have high chemical affinity to lithium polysulfides (LiPs) to prevent their cross-electrode transfer21 and related anode poisoning. Also, sulfur is an insulator, which makes it particularly hard to deliver charge uniformly across the electrode and independently of local stresses to avoid ‘dead volumes’ and low Coulombic efficiency.
Theoretical studies point to dependences between (1) stiffness of truss networks and (2) conductance through wire networks on their respective connectivity patterns.26–30 However, it remains unclear whether these relations are transferable to practical systems, such as sulfur cathodes. We also note that these and similar topological relations are scaleless. The pathways for embedding these relationships in the Euclidian space and rendering them material- and device-specific are uncertain. Bridging these knowledge gaps, our study suggests topometric parameters describing materials characteristics in topological and physical spaces simultaneously will enable a simple methodology for the design of RNCs for high performance Li-S batteries.
As a model system we used aramid nanofibers (ANFs) obtained by spin-coating (see Methods).31 Three-dimensional (3D) nanoscale networks, self-assembled from aramid fibrils with a diameter of 25 nm, display characteristic open pores 500-1000 nm in size (Fig. 1), which enables ANFs to accommodate other nanoscale components (Fig. 1A and B).25 Nanoparticles (NPs) of zeolitic imidazolate framework-67 (ZIF-67) were self-assembled on the surface of ANFs to produce an RNC composite that can be denoted as ZIF@ANF (Fig. 2A). When ANF scaffolds with and without ZIF were immersed into a 20.0 mM solution of Co(NO3)2·6H2O in methanol, Co2+ adsorb onto the nanofibers, forming coordination bonds with the amide groups 32. The density of ZIF-67 NPs can be varied by immersion time (figs S1-S3) that should not exceed four hours to avoid blocking the pores between nanofibers (fig S3). Single-step carbonization under N2 atmosphere in presence of melamine33–35 resulted in the formation of Co NPs from Co2+ coordinated into ZIF-67 (Fig. 2B and fig S3-S5). These NPs concomitantly spur the growth of carbon nanotubes (CNTs) 32,35 that have outer and inner diameters of ∼15 nm and ~8 nm, respectively (Fig. 2A-F). RNCs produced from carbonized pristine ANFs and those carrying CNTs will be referred to as CANF and CANN, respectively. From the C1s part of the X-ray photoelectron spectroscopy (XPS) spectra (Fig. 2G),36 we conclude that N atoms were successfully doped into all carbonaceous segments. Three types of observed N1s XPS peaks correspond to pyridinic (398.1 eV), graphitic (400.8 eV), and oxidic (404.9 eV) nitrogen atoms (Fig. 2H) derived from the decomposition of ANF 34, ZIF-67 37, and melamine 32, respectively.
Cobalt NPs are naturally located in the junctions between CANF and CNTs. The latter have diameters of 20 nm and can be clearly identified in transmission electron microscopy (TEM) images by the characteristic lattice spacing of 0.20 nm (Fig. 2E-F) as well as by their signatures in X-ray diffraction (XRD), Raman scattering (fig S6)38, and XPS, ∼780 eV, fig S7) data. After the exposure to air, 25.3 wt% Co (fig S8) undergoes partial oxidation as indicated by the appearance of Co2p1/2 (796.1 eV) and Co2p3/2 (780.4 eV) in the Co2p XPS peaks (Fig. 2I) 38, thus creating polar sites capable of interacting with charged molecules such as LiPs.
CANN can be particularly promising for Li-S cathodes because it is expected to have high conductivity, strength, polarity, and porosity. For example, Brunauer-Emmett-Teller (BET) analysis shows that CANN exhibits a high specific surface area of 653 m2 g-1 and BET pore sizes of ~5 nm (fig S9). Prior studies14–16,23,40–42 indicated, however, that in addition to the these parameters, the connectivity patterns43 in composites are critical for the performance of cathodes, anodes, and membranes. The engineering of their complex reticulated structure is potentially possible by solving systems of differential equations describing charge transport (Kirchhoff’s law), ion diffusion (Fick's laws), and stress distribution (Timoshenko relationships). However, there are fundamental concerns about the realism of such calculations for complex nanoscale architectures as shown in Fig. 1. The relevance of such calculations performed for a small number of filaments15–22 compared to a practical material exceptionally complex and large system and undergoing multiple charge-discharge cycles can also be questioned. The materials design problem here is that failure points in RNCs can form stochastically as collective phenomena over a large ensemble of fibrils and can be difficult to model on a small number of fibrils.
Table 1. Graph theoretical and topometric descriptors for reticulated composites used in the study.
Parameter
(see Methods)
|
Cathode Material
|
ANF
|
CANF
|
CANN
|
rGO
|
CF
|
Average Nodal Degree, k̄
|
2.87
|
2.96
|
2.98
|
2.70
|
2.77
|
Volumetric Nodal Density, kV, 1/m3
|
7.3*1020
|
1.3*1021
|
2.1*1021
|
1.7*1016
|
9.4*1019
|
Average Betweenness Centrality, β̄
|
0.0197
|
0.0168
|
0.0127
|
0.019
|
0.017
|
Ohm Centrality,
βΩ, Ω
|
2.2*1014
|
3310
|
561
|
1900
|
1270
|
Average Clustering Coefficient, C̄
|
0.059
|
0.058
|
0.053
|
0.082
|
0.078
|
Freundlich Nodal Density, k̄F nm/eV
|
|
1480
|
3430
|
1330
|
2460
|
Assortativity Coefficient, AC
|
0.044
|
0.036
|
0.020
|
0.059
|
0.051
|
Graph theory (GT) provides a mathematical framework to describe the complex architectures of nanostructures 25,44. The filaments in RNCs can be described as K2 graphs, i.e. bipartite complete graphs, with junctions between them being as nodes 45. GT processing of microscopy images can provide a rapid account of their organization based on nodal degree (k), betweenness centrality (β), and clustering coefficient (C), assortativity coefficient (AC) (Table 1). Besides ANF, CANF, and CANN, we obtained these parameters for RNCs from reduced graphene oxide (rGO) and Co-embedded carbon framework (CF). Li-S based rGO and CF composites for use as cathodes provide comparative examples for ANF-based RNCs.
GT parameters describe patterns of structural organization rather than periodic translation in space, quantifiable by crystallography. GT can also capture the patterns in particle interactions taking place during the self-assembly. For example, differences in the synthetic protocols of RNCs translates into the differences in AC (Fig. 1G, Table 1). Integration of GT descriptors with the theory of charge, stress, and mass transport makes it possible to create simplified design criteria for such structurally complex materials. An important measure of materials performance is uniformity of stress transport, which is essential for the elimination of the potentially rare but critical ‘weak points’ and propagation of catastrophic failures during frequent charging cycles. Maps of k, β, and C that are essential for mechanical properties 25,46, can be immediately used for assessment (Fig. 1B-G and fig S10-12) of areas most sensitive to disruption. For example, they can be identified by the high values of β because the loss of filaments in these points can result in catastrophic disruption of both stress and charge transport through the network. The comparison of β-maps for different RNCs indicates that CANN composites display more uniform connectivity patterns across all GT parameters, which is needed for deformation-resilient materials.
Mechanical robustness of the materials is also determined by the Maxwell isostaticity threshold,47,30 which can be expressed in GT terms as k̄ = 3 29. ANF, CANF, and CANN display k̄ values between 2.87 and 2.98, while for rGO and CF scaffolds, k̄ ~ 2.7. RNCs with lower k̄ are expected to be less resilient to deformations.
The robustness of charge transport pathways for RNCs can be evaluated using computational simulations based on Kirchhoff’s law with specific thresholds for electrical breakdown. Such models lead to k̄ > 2.5, the threshold required for resilient charge transport 28. All RNCs considered here exceed this requirement. Additional topological requirement emerges from need to have multiple strain-resilient charge transport pathways in the nanoscale system undergoing large cyclic deformations. Integration of percolation theory and GT parameters show that a system of conductive filaments acquires robustness for β̄ < 0.02 48. This requirement is satisfied for all materials considered in this study with CANN displaying the lowest β̄ = 0.0127.
The GT parameters above and other topological descriptors49 do not reflect materials properties of the filaments and remain scaleless and unitless. The physics and chemistry of the filaments are certainly relevant, which necessitates the introduction of ‘hybrid’ topometric descriptors that combine parameters from topological and Euclidian (physical) spaces. They can be obtained by using weighted graphs and calculating weights corresponding to average lengths of the edge-filaments, l̄ (Table 1 and Fig. 1). Being measured in Ohms, a topometric quantity that accounts for materials characteristics would be Ohm centrality βΩ = β*l̄/Aσ, where A and σ are the cross-sectional area and the conductivity of the corresponding filaments, respectively. Areas with high values of βΩ are problematic because the breakdown of long and thin edge filaments with high resistivity will be most disruptive. Low values of average β̄Ω identify RNCs with efficient and resilient charge transport as needed in the sulfur cathodes. Among the materials that we consider here, CANN displays the lowest value of 561 for β̄Ω, while that of ANF is unacceptably high, and β̄Ω for CANF, rGO, and CF display intermediate values of 1000-3000.
The usefulness of topometric characteristics can also be demonstrated for volumetric nodal density, kV = 1/l3. The high values of kV for cathode materials correspond to the high local density of junctions embedded in the Euclidian space. In the context of Li-S cathodes, the node-junctions contain Co NPs and defect sites of carbon lattice. Both of them facilitate redox kinetics50 and provide chemical environment increasing affinity to LiPs 51,52. Furthermore, high values of kV counteract the poor conductivity of sulfur and facilitate in-and-out electron transport from potentially inaccessible parts of the cathode. CANN and CANF have the highest average k̄V = 1/l̄3 values.
In the context of this work, it was important to evaluate whether battery performance correlates with the topometric parameters and represents a performance advancement compared to other cathodes. Li-S cells with sulfur-infused cathodes, that is, CANN@S, CF@S, rGO@S, and CANF@S exhibit typical two-plateau voltage profiles (Fig. 3A) and two-stage electron transport curves in cyclic voltammetry (CV) (fig S13).53 Among the tested RNCs, CANN cathodes display the lowest potential gap between the discharge and charge curves, sharper peaks, and smaller electrochemical polarization in CV, suggesting more facile redox kinetics in CANN than in other networks 54. Simultaneously, the galvanostatic cycling (Fig. 3B) showed that CANN@S cathodes deliver an initial capacity of 1351 mAhg-1, which considerably exceeds those of rGO-based sulfur hosts made by us (1192 mAhg-1) and other groups (800 mAhg-1 - 1200 mAhg-1 Table S3). The capacity of CANN-based cells also exceeds that of based on CF (1296 mAhg-1) and CANF (1158 mAhg-1). The shape of charge/discharge curves is virtually unchanged from the 1st to the 100th cycle for CANN-based cell (fig S14). Capacity retention of 1205 mAhg-1 and a Coulombic efficiency (CE) of ~99% for 100 cycles of CANN cathodes imply excellent sulfur utilization and exceed those used for other RNCs: 970 mAhg-1 and 97% for CF; 832 mAhg-1 and 93% for rGO; 783 mAhg-1 and 92% for CANF.
Clear dependence of the key battery performance parameters on the topometric characteristics of electrodes were observed. For example, CE shows a distinct linear dependence with β̄Ω, which can be directly related to the stress-resilience of the conductive networks (Fig. 3F-G). To confirm this dependence, we acquired SEM images of CANN@S before and after 100 cycles. No significant changes in the organization of the material after cycling were observed (fig S15). Both k̄V and C describe the local organization of the filaments around nodes determining the charge transport to sulfur, which makes both of significant to accessibility of sulfur and reduction of the ‘dead’ volumes in cathodes.
Encouraged by the performance of CANN@S cathodes and potential significance of the topometric parameters for other battery performance characteristics, we carried out additional charge-discharge cycle and rate tests (Fig. 3C-H). Even at a high current density up to 10C, cells with CANN@S deliver a capacity as high as 600 mAhg-1; the capacity reversibly recovers to 1040 mAhg-1 when the current density is reduced to 0.5C, indicating the fast charge transport to sulfur. Despite the voltage plateaus on discharge curve drops at the high current density, the sloping shape remains almost unchanged (Fig. 3D), indicating rapid sulfur redox conversion commensurate with high k̄V . This conclusion is further supported by the Nyquist plots (Fig. 3F), where the semicircle in the high-frequency region is much smaller for CANN than for CF, rGO, and CANF electrodes due to the lower charge transfer resistance. Perhaps not surprisingly, CF-, rGO-, and CANF-based cells display reduced capacities of 420 mAhg-1, 302 mAhg-1, and 275 mAhg-1, respectively at the high current density of 10C (Fig. 3C). CANN cells also show cycling life exceeding 2500 cycles with a negligible capacity decay of 0.011% per cycle and a CE consistently above 98% at a current rate of 1.0C (Fig. 3H). The rate capability, cycle life, and capacity retention of Li-S batteries with CANN as sulfur hosts exceed those for metal organic framework (MOF)-derived or carbon-based sulfur hosts electrodes (fig S16 and Table S2-3).
To quantitatively describe the LiPs node-based chemisorption, better understand the chemical foundations for topometric correlations, and sulfur hosts to better cathodes, the interactions between LiPs and different forms of nanocarbons were evaluated by density functional theory (DFT) (figS17-18). In contrast to weak binding of LiPs to rGO, their binding to CANN and CF is much stronger due to interactions with C-Co-N structural segments (fig S16 and Table S1). The intense red color of LiPs helps to visualize this important difference in properties of RNCs. After 24 hours of rGO and CANN immersed in Li2S4 solution, the vial with CANN exhibits a much lighter color than that with rGO (fig S19), which was further confirmed by XPS (fig S20). The ability of RNCs to entrap LiPs at the node-junctions can be quantified as a topometric relationship. Nanoscale carbons obey the Freundlich isotherm when adsorbing charged species such as LiPs 55,56. Using the data from DFT and SEM, we calculated Freundlich nodal density, k̄F (see Methods) – a cumulative metric combining topological, geometrical, and chemical characteristics of the material (fig S18). The decay rate and initial capacity for all tested cathodes was observed to increase as k̄F decreased (Fig. 3G). This dependence originates from the fact that (1) high volumetric density of node-junctions, (2) high binding energy, and (3) high accessibility C-Co-N sites are all needed to reduce cross-electrode transport of LiPs while allowing Li+ ions to pass through efficiently.
Building on these design parameters, we continued optimization of cathodes. Cathodes with high C, k̄V, k̄F, and β̄Ω, should be able to accommodate high areal sulfur loadings. We tested CANN@S cells with 7.68, 11.52, 15.36, and 19.20 mg cm-2 while maintaining an E/S ratio of 8:1 (Methods). The cells with the sulfur loading of 15.36 mg cm-2 showed a reversible areal capacity up to 17.0 mAh cm-2 for over 50 charge/discharge cycles (Fig. 4A), which outperforms most state-of-the-art sulfur hosts including those specifically designed for high sulfur loadings (Fig. 4B and Table S4). Further increase of sulfur loading to 19.20 mg cm-2 results in limited capacity improvement. At a high current density of 5C, the CANN cells with 15.36 mg cm-2 sulfur content achieve reversible capacity as high as 9.9 mAhcm-2 (642.8 mAhg-1) (Fig. 4C-D). When the charge-discharge rate is switched back to 0.5C, a quick rise of capacity to 15.8 mAhcm-2 (1023.3 mAhg-1) is observed. CANN@S cathodes in the cycling performance tests (Fig. 4E) also displayed a reversible areal capacity as high as 10.2 mAhcm-2 (662.3 mAhg-1) at 0.2 C with a capacity decay rate as low as 0.20 % over 200 cycles.
In summary, the structural design of materials needed for high-performance batteries can be drastically simplified using topometric parameters for charge, stress, and mass transport. The simultaneous optimization of complex connectivity patterns combined with physicochemical characteristics, resulted in Li-S battery cells with performance exceeding state-of-the-art Li-S batteries (see SI), high-performance Li+ ion batteries 57,58, and government battery targets for EVs 6. The GT design framework can be potentially generalized to other RNCs and corresponding composites based on nanotubes, nanofibers, nanowires, and nanoparticles being designed for batteries with different chemistries, structural supercapacitors, water purification membranes, implantable devices, and lightweight thermal insulation, where mutually restrictive property requirements necessitate nanoscale architecture of high complex.