Investigation of the Ionic Liquid Graphene Electric Double Layer in Supercapacitors Using Constant Potential Simulations

In this work, we investigate the effect of the cation structure on the structure and dynamics of the electrode–electrolyte interface using molecular dynamics simulations. A constant potential method is used to capture the behaviour of 1-ethyl-3-methylimidazolium bis (trifluoromethane)sulfonimide ([C2mim][NTf2]) and butyltrimethylammonium bis(trifluoromethane) sulfonimide ([N4,1,1,1][NTf2]) ionic liquids at varying potential differences applied across the supercapacitor. We find that the details of the structure in the electric double layer and the dynamics differ significantly, yet the charge profile and capacitance do not vary greatly. For the systems considered, charging results in the rearrangement and reorientation of ions within ∼1 nm of the electrode rather than the diffusion of ions to/from the bulk region. This occurs on timescales of O(10 ns) for the ionic liquids considered, and depends on the viscosity of the fluid.


Bulk Ionic Liquid
We randomly placed 256 anions and cations for each IL system in a cubic simulation cell using PACKMOL [Martínez L., Andrade R., Birgin E.G. and Martínez J.M., J. Comput. Chem., 30 (2009), 2157-2164. We then equilibrated the samples in the NPT-MD ensemble at 294 K and 1 atm over a period of 10-30 ns. After this step, we applied a simulated annealing procedure to the samples so that the mixing of the ions was ensured. The details of this simulated annealing procedure were reported in the main text.
The system temperature and pressure were controlled using the Nosé-Hoover thermostat and barostat, respectively [Nosé, S., J. Chem. Phys., 81 (1984), 511-519, Hoover W.G., Phys. Rev. A, 31 (1985), [1695][1696][1697] . The Lennard-Jones and Coulombic interactions were cut off at an interatomic distance of 12 Å and long-range Coulombic interactions were calculated using a particle-particle-particle-mesh (PPPM). Figure S1 shows the density evolution of each IL system at 294 K and 1 atm.  Figure S2 shows the normalised mass density distribution for the [C2mim][NTf2] system obtained during the last 2 ns part of the equilibration simulations at 294 K (no applied potential difference). Figure S2. Normalised mass density distribution for (a) the entire ionic liquid, (b) [C2mim] + and [NTf2]ions. Normalisation is with respect to the density in the bulk region (more distant than 30 Å from either electrode). Figure S3 shows the normalised mass density distribution for the [N4,1,1,1][NTf2] IL obtained during the last 2 ns part of the equilibration simulations at 294 K (no applied potential difference). Figure S3. Normalised mass density distribution for (a) the entire ionic liquid, (b) [N4,1,1,1] + and [NTf2]ions. Normalisation is with respect to the density in the bulk region (more distant than 30 Å from either electrode).

Charging/Discharging Dynamics
We fitted the charging/discharging dynamics of the supercapacitor (see Figure 3 of the main text) using Eqns. 1 and 2 of the main text in the charge process and Eqns. 3 and 4 of the main text for the discharge process. The value of was fixed at 0.5 in Eqn. 1 based on fitting to DY = 4 V. The fitted curves and corresponding parameters for the charge process are shown in Figure  S6 and Table 1. Those for the discharge process are shown in Figure S7 and     Figure S8 shows the sum of the partial charges of the atoms found within a specific distance from the electrode surface for the [C2mim][NTf2] system at ∆Ψ = 4 V. Our results indicate that the total charge accumulated within a distance of 5 Å from the negative electrode surface has a positive value. This reflects the presence of [C2mim] + ions. This could also be attributed to the presence of positively charged atoms (e.g. hydrogen atoms of the [C2mim] + ions) found within this distance even if some atoms of the [C2mim] + ions are found outside this 5 Å-thick layer. Figure S8. Sum of the partial charges of the atoms found within a specific distance from the electrode surface for the [C2mim][NTf2] system at ∆Ψ = 0 V and 294 K. Each data point represents a value averaged over 50 ps. NE and PE represent negative electrode and positive electrode, respectively. Figure S9. Sum of the partial charges of the atoms found within a specific distance from the electrode surface for the [C2mim][NTf2] system at ∆Ψ = 1 V and 294 K. Each data point represents a value averaged over 50 ps. Ne and PE represent negative electrode and positive electrode, respectively. Figure S10. Sum of the partial charges of the atoms found within a specific distance from the electrode surface for the [C2mim][NTf2] system at ∆Ψ = 4 V and 294 K. Each data point represents a value averaged over 50 ps. NE and PE represent negative electrode and positive electrode, respectively.

Charge evolution
When we calculated the total charge within the distance of 10 Å and beyond this distance from the negative electrode surface, it seems that the charge neutrality was satisfied.
Interestingly, the total charge calculated within the distance of 5 Å from the positive electrode surface, resulted in highly positive. This can be attributed to the fact that some of the [C2mim] ions were still found in the vicinity of the positive electrode. In other words, the ratio of [C2mim]/[NTf2] on the negative electrode is larger than the ratio of [NTf2]/[C2mim] on the positive electrode for a distance of 5 Å. Figure S11. Sum of the partial charges of the C2mim and NTf2 atoms found within a specific distance from each electrode surface for the [C2mim][NTf2] system at ∆Ψ = 0 V and 294 K. Each data point represents a value averaged over 50 ps. NE and PE represent negative electrode and positive electrode, respectively. ALL stands for the entire supercapacitor system. Figure S12. Sum of the partial charges of the C2mim and NTf2 atoms found within a specific distance from each electrode surface for the [C2mim][NTf2] system at ∆Ψ = 1 V and 294 K. Each data point represents a value averaged over 50 ps. NE and PE represent negative electrode and positive electrode, respectively. ALL stands for the entire supercapacitor system. Figure S13. Sum of the partial charges of the C2mim and NTf2 atoms found within a specific distance from each electrode surface for the [C2mim][NTf2] system at ∆Ψ = 4 V and 294 K. Each data point represents a value averaged over 50 ps. NE and PE represent negative electrode and positive electrode, respectively. ALL stands for the entire supercapacitor system.   Figure S16 shows the reference points used to calculate the angle distribution of ions as a function of distance from the electrode surface.

Angle Distribution Analyses
We calculated the angle between the electrode surface normal and the line that is the cross product, vi ´ wi (                                 , where i and N are the index of atom and total number of atoms for each ion type in the simulation cell, respectively), for ions found within 10 Å of the electrode surface for the [C2mim][NTf2] system at ∆Ψ = 4 V. The results show the entire simulation time for the charging process. Each data point represents a value averaged over 50 ps. NE and PE represent negative electrode and positive electrode, respectively.