[Original] The effect of graphene interplanar spacing and carbon nanotube diameter on the capacitance of electric double layer capacitors

Abstract: As a new type of energy storage device, electric double-layer capacitors have great potential in the fields of equipment energy storage, electric vehicles and power grids due to their advantages of high power density, long service life, clean and environmental protection. Nevertheless, its low energy density has hindered its application. The energy density can be increased by increasing its capacitance. Therefore, molecular dynamics simulation (MD) is used to study the influence of graphene interplanar spacing (slit aperture) and carbon nanotube diameter (circular hole diameter) on area specific capacitance. This indirectly reflects the influence of graphene interplanar spacing and carbon nanotube diameter on energy density. By analyzing the distribution law of K+ and H2O, it is found that in the slit hole, when K+ is distributed in a single layer (the interplanar spacing is less than 0.5 nm), the capacitance increases as the interplanar spacing decreases; K+ is distributed in two layers (interplanar spacing). In the case of 0.5～0.803 nm), the opposite is true. In a circular hole, the capacitance varies with the diameter oscillating, and due to the curvature, its area is much larger than that of a slit hole.

Keywords: supercapacitor; graphene; carbon nanotube; interplanar spacing; diameter
Electric double-layer capacitors belong to a category of supercapacitors and have broad application prospects in wind power generation, electric vehicles, renewable energy and other fields. However, its energy density is very low (usually no more than 20 W·h/kg, and the energy density of lithium batteries currently used in electric vehicles is about 100 to 150 W·h/kg), which greatly limits its application. In order to break this limitation and enable it to be better used in fields such as portable electronic products and new energy vehicles, it is necessary to increase its energy density. According to the energy density formula E=CV2/2 of the electric double layer capacitor, it can be seen that the energy density can be increased by increasing the capacitance. The capacitance of a capacitor is related to the electrode material, electrode spacing, shape, and electrolyte. Chmiola et al. studied the abnormal increase in capacitance when the electrolyte is an organic solution and the slit aperture is reduced from 1 nm to 0.6 nm, breaking the view that solvated ions larger than the electrode aperture cannot contribute to capacitance. Huang et al. [5] studied the capacitance of round holes in the size of mesopores and micropores, and proposed an EDCC/EWCC model to explain the variation of capacitance with size. Feng et al. studied the relationship between the capacitance of the room temperature ionic liquid slit-hole supercapacitor and the interplanar spacing, and found that the capacitance changes when the spacing is 0.67～1.8 nm, and the abnormal increase of capacitance occurs in the interval of 0.7～1 nm. Qiu et al. studied the ionic structure and capacitance of negatively charged graphene nanochannels in NaCl aqueous solution, and concluded that when the nanopores can only accommodate ions of the opposite electrical property to the wall, the nanopores will reach the maximum capacitance and energy density. . Nicolas et al. considered the complexity of pore size dispersion and the complexity of two different solvents, and established a model to understand the capacitance of microporous carbon materials. They also observed that the surface normalized capacitance decreases with the spacing when most pores are less than 1 nm. Small but increasing result. Feng et al. studied the capacitance change of the slit holes with an interplanar spacing of 0.936～1.47 nm, and studied the distribution of K+ and H2O in the electrolyte, and explained the change of capacitance with the pore size from the perspective of configuration, and based on this The classic slit hole capacitance formula is modified to Cs=εrε0A/deff, which quantitatively explains the change of capacitance with the aperture, but this work still has certain deficiencies. First of all, according to the classic electric double layer (EDL) theory, the capacitance should decrease with the decrease of the interplanar spacing. The research of Feng et al. showed that when the aperture width is 0.936 ~ 1.47 nm, the capacitance increases abnormally, so it is smaller. Whether there will be new changes in the size has not been studied. Secondly, when the size is 0.936～1.47 nm, the experimental data points do not completely fall on the curve fitted by the formula (R2=0.926), and when the slit hole is replaced with a round hole (EWCC model), the fitted R2 is 0.921 , Which shows that the curvature should be considered in the quantitative calculation of capacitance. In order to determine the change rule of Cs and the influence of curvature on the capacitance under small size, this paper uses the probe averaging method (PA) to study the relationship between the capacitance and spacing of the plate and the hole under the smaller size, and uses MD to study the capacitor The configuration of K+ and H2O is used to explore the microscopic mechanism of capacitance change. In order to ensure the control variables, all factors other than the size are the same as Feng et al., that is, graphene plates, K+ and H2O are selected as simulated substances, and the radius of K+ (0.138 nm) is taken into account. Select the surface spacing to be 0.4～1.203 nm. In addition, it is ensured that the diameter of the circular hole corresponds to the surface spacing of the slit hole to be equal, and while the capacitance of the circular hole changes with the diameter, the two are compared to finally obtain the effect of curvature.
1 Simulation method
First, establish two graphene planes with a charge of -0.55 C/m2, adjust the distance between the two planes to 0.936 nm, and define the interplanar distance as the distance between the center planes of the two graphene layers. Secondly, 8 and 210 K+ and H2O were filled between the two plates. Under different surface spacing, the H2O concentration was kept constant and the system remained electrically neutral, as shown in Figure 1. Under the canonical ensemble (NVT), a 1.3 ns equilibrium dynamics simulation was performed with a time step of 1 fs. The Nose-Hoover thermostat is used to control the temperature to 300 K, the force field parameters of sp2 type carbon are used for carbon atoms, and the force field parameters of Lee etc. are used for H2O and K+. To ensure accuracy, the last 300 ps data of equilibrium kinetics was taken for research, and the data was recorded every 10 ps for result analysis. The round hole model is shown in Figure 2.
 
Figure 1 The model when the pitch is 0.936 nm, where the pink, red, white, and cyan balls represent K, O, H, C respectively
 
Figure 2 The model when the diameter is 0.936 nm, where the pink, red, white, and cyan balls represent K, O, H, C respectively
2 Slit holes and round holes

2.1 Slit hole
In order to explore the microscopic mechanism of capacitance change, the distribution of K+ and H2O under different sizes was studied. Prior to this, simulations with interplanar spacings of 0.803 nm, 0.936 nm, 1.08 nm and 1.203 nm have been carried out. It is found that the particle distribution, K+ hydration, interaction energy and capacitance changes are consistent with Feng et al., verifying the model Accuracy. On the basis of the accuracy of the model, further study the distribution of particles when the interplanar distance decreases. The distribution of particles is shown in Figure 3. It can be seen from Figure 3(a) that when the interplanar spacing is 0.6 nm, K+ and H2O are distributed on both sides of the polar plates in a double layer. As the interplanar spacing decreases, the double-layer distribution of K+ gradually transitions to a single-layer distribution, and H2O changes to a single-layer distribution, as shown in Figure 3(b). When the surface spacing is further reduced, the distribution of K+ and H2O in the center of the plate becomes more concentrated, as shown in Figure 3(c). When the interplanar spacing is reduced to 0.4 nm, the distribution of K+ and H2O keeps the single-layer distribution and no longer changes, as shown in Figure 3(d). It can be seen from the above analysis that when the interplanar spacing is less than 0.6 nm, due to the size limitation, K+ and H2O can only be distributed between the plates in a single layer.
 
Figure 3 Distribution of K+ and H2O [(1Å=0.1 nm) Figure (b) The distribution of K+ in the 0.1～0.2 nm section is caused by periodicity. To explain the distribution of K+ and H2O, study K+ and wall, H2O and wall, K+ Four interaction energies between coordination water with K+ hydration water and K+ hydration water and K+ hydration water. The interaction energy corresponding to each interval and the hydration number of K+ are shown in Figure 4 and Table 1. When the spacing is 0.6 nm, the energy of K+ hydration at the center, the energy between the coordination water of K+ hydration water and K+ hydration water is lower, and the number of hydration is higher, but K+ is double-layered, which indicates that hydration is at this time The role played by K+ and H2O is very small, and the role between K+ and H2O and the wall is negligible. Therefore, only the electrostatic repulsion other than the four interactions plays a dominant role, that is, according to the classic EDL theory, the solvent and the Regarding ions as point charges, the main consideration is electrostatic repulsion. When the spacing is 0.55 nm, the energy of K+ hydration at the center, the interaction energy between K+ and H2O and the wall surface, and the energy between K+ hydrated water and K+ hydrated water are lower than those near the wall This further explains the reason why K+ forms a single-layer arrangement from the perspective of energy.
 
Figure 4 The relationship between the hydration number of K+ in the slit hole and the surface spacing
Table 1 The interaction energy corresponding to each interval in the slit hole

Finally, this paper uses the probe averaging method (PA) to calculate the relationship between the capacitance of the parallel plate capacitors with pitches of 0.6 nm, 0.55 nm, 0.5 nm and 0.4 nm. PA is a method of dividing the space into a uniform three-dimensional grid, using the Ewald summation formula to find the potential on each grid point, and then averaging in the (X, Y) direction to obtain the potential distribution curve in the Z direction . Finally, the area specific capacitance is calculated according to Cint=Q/(Vele-Velepze)A, where A is defined as the electrode surface area, Vele is the electrode potential, and Velepze is the potential of the zero-charge electrode. This method is suitable for both the traditional fixed charge method (FCM) and the recently developed constant potential method (CPM). It is more accurate than the traditional numerical solution of the one-dimensional Poisson equation without supplementary assumptions. The calculation results in this paper are shown in Figure 5. When the distance is 0.5～0.803 nm, the capacitance decreases with the decrease of the distance. It shows the opposite change law from 0.803～1.203 nm, which is the law described by Cs=εrε0A/deff Different; when the spacing is 0.4-0.5 nm, the capacitance increases as the spacing decreases, and the trend becomes consistent with Cs=εrε0A/deff.

 
Figure 5 When the surface spacing is 0.4～1.203 nm, the corresponding capacitance-spacing relationship curve comprehensively analyzes the distribution structure of K+ and the change of capacitance, and it is found that there is a certain correspondence between the two. In the range of 0.5～0.803 nm, the K+ distribution transitions from double layer to single layer, and the capacitance decreases as the distance decreases; until the distance becomes 0.5 nm, K+ becomes a single layer distribution, and the capacitance becomes monotonous. The performance becomes increasing as the spacing decreases. This enlightens this article to explain the change of capacitance from the distribution structure of K+. When K+ is distributed in the center of the capacitor in a single layer, Feng et al. equivalently regards the capacitance of the capacitor as two capacitors in parallel, and the plane formed by K+ is used as the common anode. , The walls are used as two cathodes respectively. The formula deduced based on this hypothesis does illustrate the law of capacitance change when a single layer K+ is distributed. But when the K+ distribution is a double layer, this assumption will be problematic, because the capacitance formed between the two layers of K+ will be ignored according to this assumption. At this time, a reasonable assumption should be that two capacitors are connected in series and then connected in parallel with another capacitor. In order to prove the rationality of the hypothesis, make the change curve of the electric potential of the capacitor, as shown in Figure 6. It can be seen that there is a potential difference between the two K+ layers, which indicates that a capacitance can be formed between the two K+ layers. Careful observation revealed that, apart from the interplanar spacing of 0.6 nm and 0.803 nm, the curve is divided into 3 segments due to the double-layer distribution of K+. At 0.55 nm and 0.5 nm, the potential also has a similar 3-segment curve, although K+ is not Double layer, but due to the asymmetry of the K+ distribution, the curve is still divided into 3 segments. As for the quantitative explanation of 3 capacitors in series and parallel connection, it is still to be studied.
 
Fig. 6 The change curve of capacitor potential when the interplanar distance is 0.4～0.803 nm
2.2 round hole
Starting from Cs=εrε0A/deff, this article has discussed the changing law of capacitance at smaller sizes, and found that this formula is only applicable when K+ forms a single layer. But even when K+ forms a single layer, the value of capacitance is not completely consistent with that described by Cs=εrε0A/deff (R2=0.926). It can be seen that it is not enough to only consider the interaction between the particles when calculating the capacitance quantitatively. In fact, in addition to the interaction, the influence of curvature on capacitance cannot be ignored, and in actual production, capacitors are not all flat plates. Therefore, studying the influence of curvature on capacitance is not only helpful to accurately reveal the law of capacitance change, but also in practical applications. Has a broader meaning. In order to solve the problem that the above formula is not accurate and the capacitor is not a flat plate, this paper studies the change of the capacitance of a circular hole with the diameter of the circular hole, and also studies the particle distribution and interaction energy to explore the microscopic mechanism of the capacitance change with the diameter. As shown in Figure 7, when the diameter of the hole is 1.199 nm, K+ is distributed in two layers in the center of the hole, but the peak difference of the two layers is obvious. When the combined diameter is 1.086 nm, K+ is distributed in the center of the hole in a single layer As a result, it can be judged that when the diameter is 1.199 nm, the distribution of K+ is transitioning from double layer to single layer, so the two peaks are quite different. The distribution of H2O constitutes 3 uniform peaks. The distribution of H2O still has three peaks, but due to the increased space constraint, the space for particles to move freely decreases, so the peak value is lower than the original. When the diameter is 0.936 nm, the distribution of K+ has an obvious peak and a peak that is being formed. Under the further constraints of size, H2O changed from 3 layers to 2 layers. When the diameter is 0.797 nm, the peak on the left in the K+ distribution graph is reduced and split into two, and the peak on the right is completely formed. H2O is still double-layer, but the peak value drops again, transitioning from double-layer to single-layer. When the diameter of the circular hole is 0.608 nm, K+ changes from 3 peaks to 2 peaks, and the distribution moves closer to the center. H2O is distributed in the center in a single layer, which is caused by the diameter of H2O being approximately 0.4 nm. As the size decreases, H2O will only form a single-layer distribution, which has been directly reflected in the figure, and will not be repeated here. When the diameter is reduced to 0.55 nm, K+ forms a very obvious peak and a less obvious peak, which is the double peak of K+ is becoming a single peak. Since the radius of K+ is 0.138 nm, and considering that K+ may form close packing, the diameter of the circular hole containing the double-layer K+ should be about 0.519 nm. The experimental results in this paper also show that when the diameter is reduced to 0.519 nm, K+ can still form a double layer. But when the diameter is reduced below 0.519 nm, K+ can only exist in the center of the capacitor.
 
Figure 7 The distribution of K+ and H2O when the diameter of the circular hole is 0.4519 ~ 1.199 nm. The asymmetry of the curve in the figure is due to the fact that the K+ number is too small to explain the formation of particles with different diameters, and to study the interaction energy of particles and K+ hydration. As shown in Table 2 and Figure 8, when the diameter is 1.199 nm, the energy of K+ hydration at the center is much lower than that near the wall. This explains why K+ is distributed in a single layer, and the peak of the hydration number is at the center. Occurs, which shows that hydration is dominant at this size. When the diameter is 0.936 nm, the energy of K+ hydration at the center is less than that near the wall, but K+ is not distributed in a single layer, but in a state of transition from a single layer to a double layer. The hydration situation is also different from before. The peak value of the hydration number is only at 0.608 nm<x<0.8 nm, 0.3 nm<y<0.5 nm