Modeling galvanostatic charge–discharge of nanoporous supercapacitors

Molecular modeling has been considered indispensable in studying the energy storage of supercapacitors at the atomistic level. The constant potential method (CPM) allows the electric potential to be kept uniform in the electrode, which is essential for a realistic description of the charge repartition and dynamics process in supercapacitors. However, previous CPM studies have been limited to the potentiostatic mode. Although widely adopted in experiments, the galvanostatic mode has rarely been investigated in CPM simulations because of a lack of effective methods. Here we develop a modeling approach to simulating the galvanostatic charge–discharge process of supercapacitors under constant potential. We show that, for nanoporous electrodes, this modeling approach can capture experimentally consistent dynamics in supercapacitors. It can also delineate, at the molecular scale, the hysteresis in ion adsorption–desorption dynamics during charging and discharging. This approach thus enables the further accurate modeling of the physics and electrochemistry in supercapacitor dynamics.


Part 1. Stable cycle of charging-discharging process
We modeled the galvanostatic charging-discharging process of the supercapacitor for 30 cycles.After a few cycles, it will reach stability.Supplementary Fig. 1 shows that the number density of cations near the positive electrode, obtained from GCD-CPM simulations for an open electrode system with a period of 100 ps.In this work, all the data shown in the main text is in a stable cycle by averaging the data of the last ten cycles.Hence, the time mentioned in this work refers to the time relative to the beginning of the averaged stable cycle.

Part 3. Comparison of EDL structures in open electrode systems
As mentioned in the main text, the number densities of cations near the positive electrode by GCD-CPM and GCD-CCM are almost identical.For a more intuitive comparison, Supplementary Fig. 3 shows the number density of cations near the positive electrode at two moments (0 and 50 ps).Similarly, Supplementary Fig. 4 shows the number densities of cations near the negative electrode, anions near the positive electrode, and anions near the negative electrode.Supplementary Fig. 5 shows orientations of cations adsorbed on the negative electrode.As shown in Supplementary Fig. 5a, η is defined as the angle between the normal of the electrode surface and the normal of the cation plane, α is defined as the angle between the normal of the electrode surface and the vector pointing from the imidazole ring to ethyl, and β is defined as the angle between the normal of the electrode surface and vector pointing from imidazole ring to methyl.The first cation layer refers to the cations at the region between the electrode surface and the first valley of cation number density, as Supplementary Fig. 4   Like the nanoporous electrode system, the charge from the electrolyte side changes slowly than that from the electrode side for open electrode systems.Here we explore the effect of hysteresis of ion adsorption-desorption on the GCD curves. lagging is used to describe how the charge from the electrolyte lags behind that on the electrode, and it is defined as where  is the total surface charge density of one electrode and   is the charge density through the EDL near this electrode. is the distance between the positive and negative electrodes.
For the open electrode systems, the potential difference between the positive and negative electrodes can be derived from charge distribution 1 , as where  0 is vacuum permittivity.We then divide the surface charge  into two parts: the lagging charge σ lagging and the balanced charge  −  lagging .Then  is divided consequently into where  lagging is produced only by  lagging , and  rest is produced by  −  lagging .The degree of charge lagging decreases as P increases (Supplementary Fig. 18a), since electrolyte ions require a longer time to respond to the electrode polarization.The asymmetry and the negative values of the GCD curve are caused by such a lagging charge, as shown in Supplementary Fig. 18b.
are also small for systems with periods of 1000 ps and 3000 ps, but with a period of 200 ps, the temperature of the in-pore region is about 10 K higher than that of the reservoir (Supplementary Fig. 20d).Therefore, these new results suggest that the temperature gradient is very small for the open electrode system; however, there would be a temperature gradient for nanoporous electrode systems in some conditions.

Supplementary Figure 1 |b 2 .
Input current and evolution of cation number density.a, Input current of galvanostatic charge-discharge.b, Time-evolution cation number density near positive electrodes, obtained from GCD-CPM simulations.Only the first 15 cycles are shown.a Part Potential and charge on open electrodes Supplementary Figure 2 | Charge and potential on electrode atoms in open electrode systems.a, Gray contour indicates probability distributions of negative electrode atom charges vary with time obtained from GCD-CPM simulations, and the dashed line is the average.b, Gray contour indicates the probability distributions of negative electrode atom potential vary with time obtained from GCD-CCM simulations.The red dashed line is the average.The solid cyan line is the potential of the negative electrode obtained from GCD-CPM simulations.ba

Supplementary Figure 3 |Supplementary Figure 4 |Supplementary Figure 5 |
shows.One can find that the ion responses obtained by GCD-CPM and GCD-CCM are almost the same.Comparison of the number density of cation near positive electrodes in open electrode systems.a, Number density at time 0 and 50 ps.b-c, Statistical error of time-evolution number density of cation obtained from GCD-CPM (b), and GCD-CCM (c).Ion number density in open electrode systems.a-c, Number density of cation near the negative electrode (a), anion near the positive electrode (b), and anion near the negative electrode (c) as a function of distance from electrode and time.The data was obtained from GCD-CPM simulations.d-f, Number density of cation near the negative electrode (d), anion near the positive electrode (e), and anion near the negative electrode(f) as a function of distance from electrode and time.The data was obtained from GCD-CCM simulations.g-i, Number density of cation near the negative electrode (g), anion near the positive electrode (h), and anion near the negative electrode (i) at time 0 and 50 ps.Orientation of the first cation layer adsorbed at the negative electrode of open electrode systems.a, Schematics for cation orientation.b-d, Probability distribution of η (b) α (c) and β (d) with time.The data was obtained from GCD-CPM simulations.e-g, Probability distribution of η (e) α (f) and β (g) with time.The data was obtained from GCD-CCM simulations.g-i, Probability distribution of η (h), α (i), and β (j) at 0 ps and 50 ps.

6 |Part 5 .Supplementary Figure 7 |Supplementary Figure 8 |
Charge and potential on electrode atoms in nanoporous electrode systems.a, Gray contour indicates probability distributions of negative electrode atom charges vary with time obtained from GCD-CPM simulations, and the dashed line is the average.b, Gray contour indicates the distributions of negative electrode atom potential vary with time obtained from GCD-CCM simulations, and the red dashed line is the average.The solid cyan line is the potential of the negative electrode obtained from GCD-CPM simulations.Evolution of in-pore charge density and effective diffusion Evolution of in-pore charge density along the pore axis of the positive nanoporous electrode and corresponding fitting results.a-c, Evolution of in-pore charge density along the pore axis with the electric current period of 200 ps (a), 1000 ps (b), and 3000 ps (c) obtained from GCD-CPM simulations.d-f, The fitting in-pore charge density with the electric current period of 200 ps (d), 1000 ps (e), and 3000 ps (f), using GCD-CPM simulation data.g-i, Evolution of in-pore charge density along the pore axis with electric current periods of 200 ps (g), 1000 ps (h), and 3000 ps (i), obtained from GCD-CCM simulations.j-l, The fitting in-pore charge density with electric current periods of 200 ps (j), 1000 ps (k), and 3000 ps (l), using GCD-CCM simulation data.Error bars indicate one standard deviation of 4 independent simulations.Evolution of in-pore charge density along the pore axis of the negative nanoporous electrode and corresponding fitting results.a-c, Evolution of in-pore charge density along the pore axis with the electric current period of 200 ps (a), 1000 ps (b), and 3000 ps (c) obtained from GCD-CPM simulations.d-f, The fitting in-pore charge density with the electric current period of 200 ps (d), 1000 ps (e), and 3000 ps (f), using GCD-CPM simulation data.g-i, Evolution of in-pore charge density along the pore axis of the electric current period of 200 ps (g), 1000 ps (h), and 3000 ps (i), obtained from GCD-CCM simulations.j-l, The fitting in-pore charge density with the electric current period of 200 ps (j), 1000 ps (k), and 3000 ps (l), using GCD-CCM simulation data.Error bars indicate one standard deviation of 4 independent simulations.

Figure 20 |
Effect of temperature gradient on molecular simulations of GCD-CPM.a-b, Sketch of open electrode system (a) and nanoporous electrode system (b) in which only half of the nanoporous electrode system is shown.c-d, Local temperature of the open electrode system (c), and nanoporous electrode system (d).The current period is 100 psfor the open electrode system and 200 ps for the nanoporous electrode system.The inset is the temperature difference between the region of EDL (in-pore) and the reservoir region.e, Local temperature of nanoporous electrode system where using thermostats with two groups.The current period is 200ps.f, Effect of temperature difference on the GCD curve in the nanoporous electrode system with a current period of 200 ps.