Modeling of Nanomaterials for Supercapacitors: Beyond Carbon Electrodes

Abstract

Capacitive storage devices allow for fast charge and discharge cycles, making them the perfect complements to batteries for high power applications. Many materials display interesting capacitive properties when they are put in contact with ionic solutions despite their very different structures and (surface) reactivity. Among them, nanocarbons are the most important for practical applications, but many nanomaterials have recently emerged, such as conductive metal-organic frameworks, 2D materials, and a wide variety of metal oxides. These heterogeneous and complex electrode materials are difficult to model with conventional approaches. However, the development of computational methods, the incorporation of machine learning techniques, and the increasing power in high performance computing now allow us to tackle these types of systems. In this Review, we summarize the current efforts in this direction. We show that depending on the nature of the materials and of the charging mechanisms, different methods, or combinations of them, can provide desirable atomic-scale insight on the interactions at play. We mainly focus on two important aspects: (i) the study of ion adsorption in complex nanoporous materials, which require the extension of constant potential molecular dynamics to multicomponent systems, and (ii) the characterization of Faradaic processes in pseudocapacitors, that involves the use of electronic structure-based methods. We also discuss how recently developed simulation methods will allow bridges to be made between double-layer capacitors and pseudocapacitors for future high power electricity storage devices.

Keywords: 2D materials; Double layer; MXene; Machine-learning; Metal oxides; Metal−organic framework; Molecular dynamics; Pseudocapacitors.

Figure 1

Electrode nanomaterials beyond carbon and their applications in supercapacitors are based on different electrochemical charge-storage mechanisms.

Figure 2

Classification of the simulation methods in the field of electrochemical energy storage. The panel for molecular mechanics is reproduced with permission from ref (49). Copyright 2014 AIP Publishing. The panel for machine learning is reprinted with permission under a Creative Commons CC BY License from ref (50). Copyright 2017 The Royal Society of Chemistry. The panel for quantum mechanics is reproduced with permission from ref (51). Copyright 2021 John Wiley and Sons. The panel for hybrid mechanics is reproduced from ref (52). Copyright 2021 Elsevier.

Figure 3

A typical workflow for generating MLFFs. Initial reference data are labeled and prepossessed, which usually come from DFTMD simulations. This provides atomic configurations as well as energies and/or forces labels. Then, before applying any machine learning method (e.g., neural network), atomic configurations need to be encoded using a chosen descriptor such as atom centered symmetry functions (ACSFs), smooth overlap of atomic positions (SOAP) descriptor, the Faber-Christensen-Huang-Lilienfeld (FCHL) descriptor, or a graph convolutional neural network PiNet. More labeled data with unique configurations as compared to the initial reference data are usually needed to produce a well-functioning MLFF and this can be done using so-called active learning schemes, see section 2.4 in ref (116) and references therein.

Figure 4

MD simulations of three c-MOFs in ionic liquids. (a) Schematics of molecular simulations of c-MOF-based supercapacitor. (b) Differential areal capacitance of c-MOFs. (c) Intrinsic relaxation time and ionic conductivity of ILs in the MOF pores at different temperatures obtained by using transmission line model. (d) Gravimetric Ragone plot for c-MOFs-based supercapacitors from 300 to 400 K. Reproduced with permission from ref (27). Copyright 2020 Springer Nature.

Figure 5

QM/MM simulations on the c-MOF in ionic liquids. (a) Schematics of QM/MM simulation system of c-MOF-based supercapacitor. (b) Schematic to show two primary charging mechanisms when the charge density σ < 0. The upper panel shows the counterion insertion (count.in.) mechanism, while the lower panel shows the co-ion removal (co.rem.) (c) Density of excess charge is illustrated with respect to the radial distance (r) from the center of the MOFs, when σ is −4.5 μC cm^–2. (d) Differential capacitance from QM/MM simulations of the Cu₃(HHTP)₂ with different charging mechanisms. (e) σ–ΔU curve along two different charging mechanisms. (f) Potential drop inside the MOFs is shown with respect to the X value using cylindrical capacitor models. The total potential drop (ΔU_model) is decomposed into three terms, where two of them are ΔV₁ and ΔV₂. The potential drop due to effective dielectric screening from the solvent molecules (solvent) is also shown. Adapted with permission under a Creative Commons CC BY License from ref (137). Copyright 2023 American Chemical Society.

Figure 6

MD simulations on 1T-MoS₂ in ionic liquids. (a) MD simulations setup. (b) Differential capacitance of 1T-MoS₂ with interlayer distances of 0.80, 1.05 and 1.30 nm. (c) Time evolution of electrode charge (solid lines) and its fit (dashed lines) of MoS₂ electrodes with various interlayer distances. Reproduced with permission from ref (138). Copyright 2021 John Wiley and Sons. (d) Partial charge and Gaussian width of Mo and S atoms. (e) The number of ions inside the interlayers for the CPM_χ simulations (left) and the CPM simulations (right). (f) Charging mechanism parameter (X) obtained by the two methods. Adapted with permission from ref (84). Copyright 2022 American Chemical Society.

Figure 7

(a) The voltage-dependent population of RuO₂ subsurface protons, presented for different values of EDL capacitance. Reproduced with permission from ref (148). Copyright 2020 American Physical Society. (b) RuO₂ charge storage mechanisms at potentials below and above the PZC. Proton coverage is illustrated at the right-hand side of the panel Adapted with permission from ref (145). Copyright 2016 IOP Publishing Ltd. (c) Possible proton jumps within and between channels of RuO₂. Reproduced with permission from ref (147). Copyright 2013 American Chemical Society. (d) Inhomogeneity in K⁺-distribution during initial GCMC-simulation of birnessite, visualized at the top of the panel. Adding excess charge in a stratified manner leads to the distance-dependent interaction energies and radial distribution functions of the bottom plots. Reproduced with permission from ref (149). Copyright 2021 Springer Nature.

Figure 8

Joint-DFT simulations of MXenes-based pseudocapacitors. (a) Schematics of the simulation on MXenes in the H₂SO₄ electrolyte. (b) Gibbs free energy of Ti₃C₂T_x (T = O, OH) as a function of H coverage x at different electrode potentials, ϕ, relative to SHE. (c) Faradaic charge (blue), EDL charge (black), and total charge (red) stored at different electrode potentials. (d) Differential capacitance of Ti₃C₂T_x in 1 M H₂SO₄. (e) Total electronic DOS and local DOS of the Ti₃C₂T_x electrode with a H coverage of 1/9 (upper) and a H coverage of 2/9 (bottom). (f) Planar charge density change (ΔQ) of the Ti₃C₂T_x electrode after charging from a neutral surface with different H coverages. Reproduced with permission from ref (155). Copyright 2018 American Chemical Society. (g) Side and top view of the simulated model showing intercalated ions and surface termination, used in the work of Ando et al. (h) Comparison of the PDOS curves of hydrated and adsorbed Li atoms. (i) Schematic pictures of the capacitive and pseudocapacitive conditions formed inside the MXene electrodes. Electric-double layer is formed and behaves capacitively (left), and orbital coupling occurs and behaves pseudo capacitive (right). Reproduced with permission from ref (156). Copyright 2020 John Wiley and Sons.

Figure 9

Finite-field DFTMD simulations on TMO-aqueous interfaces. (a) Simulation setup overlaid with corresponding density profiles of ions at surface charge q = 4e. (b) Time evolution of total dipole moment M_z at PZPC when switching electric boundary condition from E̅ = 0 to D̅ = 0. (c) Dynamics of adsorbed water molecules at positively and negatively charged (q = 2e) rutile TiO₂(110)-aqueous interfaces. (d) Helmholtz capacitance at different surface charges. Adapted with permission from ref (99). Copyright 2019 American Chemical Society. (e) The differential Helmholtz capacitance for the SnO₂(110)/NaCl interfaces. (f) A schematic illustration of the coupling between the EDL field and the water orientation/dissociation at the SnO₂(110)/NaCl interface at low pH. Reprinted with permission under a Creative Commons CC BY License from ref (161). Copyright 2021 American Chemical Society.

Figure 10

Graphene oxide simulations using the PiNNwall interface. (a) Schematic illustration of the PiNet-χ framework for predicting response properties. Reprinted with permission under a Creative Commons CC BY License from ref (131). Copyright 2022 IOP Publishing Ltd. (b) A snapshot of the electrode surface with a 10% surface coverage (electrolyte solution is not shown for clarity). (c) Snapshot of the carboxyl-terminated electrode surface with a 10% surface coverage of OH (electrolyte solution is not shown for clarity, and the locations of deprotonated carboxyl groups are highlighted). (d) Helmholtz capacitance for the positive and negative electrodes as a function of the surface coverage. Dashed lines correspond to the positive electrode, while dotted lines correspond to the negative electrode. (e) Electrode charge as a function of the applied potential. V_PZFC is identified when the electrode charge becomes zero. Reprinted with permission under a Creative Commons CC BY License from ref (130). Copyright 2023 American Chemical Society.