The length of production runs was at least 2 ns. Capacitance was computed using Equation 5.1, where V = ∆Ψ is the voltage drop applied across the cell, and Q is the average absolute value of the charge stored on a single electrode. From the production run we also computed local properties of interest using an in-house software package developed for this study,such as the degree of confinement and the charge compensation per carbon , in order to understand the mechanisms of charge storage and gain physical insights into differences in capacitances between the materials. The definitions of these local properties are provided in the Supporting Information. The FAU_1 EDLC initially had an excess ionic charge of 25 e in the cathode, and an equal in magnitude, opposite in sign, excess charge of anions in the anode. After equilibration,the magnitude of excess ionic charge in each electrode is about 16 e. This represents a decrease of about 36% in charge stored in the electrode, since the electrode’s net charge always balances the excess charge of ions inside the electrode. This discharging shows that in the FAU_1 EDLC, the Poisson potential is not the potential we should apply to obtain the charge equivalent to the constant-charge simulation. To assess the actual potential we need to apply to obtain an average atom charge of ±0.01 e, we try several potential differences and follow the evolution of the total electrode charge as a function of time . We found that this “0.01 e–equivalent” voltage is approximately 1.6 V. A longer simulation would be needed to determine the exact voltage to higher precision. Moreover, we were surprised to see that even when the change in equilibrium charge is small, equilibration at constant potential from a “nearby” constant charge configuration still takes several nanoseconds. We believe this is due to two factors: First, the driving force for ions to diffuse in and out of the electrodes is proportional to how far away the EDLC is from equilibrium charge, leading to exponential convergence at constant potential. Second, the configuration of ions in the electrode is not only a function of the total charge in the electrode, but also depends sensitively on the distribution of this charge among the atoms of the electrode. Rearrangement of ions can take on the order of nanoseconds due to diffusive limitations in liquid that are exacerbated by the bulky size of the ions and the small pores of the ZTCs,vertical cannabis growing systems which at their narrowest points have diameters similar to the ion sizes.
The same type of investigation was done for a number of other ZTCs and gives similar results. After switching to a constant-potential simulation using the Poisson potential drop, the resulting equilibrium electrode charges differed from the initial constant charges by 50– 200% . Following this extensive study, we thus conclude that the approach of doing a constant-charge equilibration associated with a Poisson potential calculation is not suitable for the current work. For the constant-charge equilibration to still be interesting, we would need a better way to estimate the potential difference to apply in the constant-potential simulation. In an attempt to improve the determination of the potential difference corresponding to a particular amount of charge stored, we turned to the calculation of the three-dimensional electric potential field as described in Wang et al.We used the software package developed by the authors for constant potential simulations in a previous work,which implicitly computes the electrostatic potential at the position of each electrode atom in order to determine fluctuating charges. We adapted the code to the case of a constant-charge simulation in which the electrostatic potential of each electrode atom fluctuates. The electrostatic potential averaged over all the atoms in each electrode during a constant charge simulation was computed for 5 ZTC and 2 CDC materials and results are shown in Table 5.1. For almost all materials, the potential drop computed from the averaged local electrostatic potential appears to have no correlation with the Poisson potential one. Formaterials which had extremely low or negative capacitances, such as the ZTCs 221_2_6 and SAO, the average local potential drop was approximately 2 V. For FAU_1, where 1.06 V was too low to store a charge of ±0.01 e per atom, the local potential method computed a potential drop of 2.3 V for the ±0.01 e constant-charge simulation. This is too high compared to the “0.01 e–equivalent” potential of 1.6 V determined in Figure 5.3. It seems therefore that while the local electrostatic potential method does not, unlike the Poisson potential method, yield physically unrealistic values such as near-zero and negative potentials drops across the cell, the potential drops computed by the local potential method are still not accurate enough to be useful for significantly decreasing the time needed to converge the electrode charge in a constant-potential simulation. Our initial motivation for calculating the Poisson potential or averaged local potential from a constant-charge simulation was to reduce the time needed for charges on the electrodes to converge in a constant-potential simulation. However, we found that these proxy potentials were not representative of what the applied potential should be to obtain the same amount of stored charge, and the simulation times required for the charge convergence following the constant-charge equilibration were still more than a few nanoseconds. Moving forward, we thus opted to compute constant-potential properties such as capacitance and electrolyte configuration by skipping the constant-charge equilibration step, and directly applying a potential difference of 1 V to the EDLC.
The results of these simulations are detailed in the remainder of this article. In the constant-potential production runs, we excluded materials for which the EDLC simulation cell had more than 12 000 atoms. These larger cells could not be studied due to computational limitations affecting the memory-intensive constant potential method.This suggests that while particular geometric descriptors might be a useful indicator of capacitance within particular families of materials, a clear relationship between capacitance and, for example, pore size is not the rule, but rather the exception for materials which are otherwise geometrically similar. ZTCs, due to their well-defined templated structures, exhibit a diversity of topologies, pore geometries, and local curvatures, which are not well captured by traditional geometric descriptors, but are known to influence charge storage.Thus the insights we can glean from local interfacial properties in ZTCs might be better translated to microporous carbon materials in general. The bottom row of Figure 5.6 plots capacitance versus quantities related to the electrolyteelectrode interfacial configuration, which are computed for an ion in relation to the electrode atoms within its coordination shell: The charge separation is the average distance between the counterion and the carbons within its coordination shell. The degree of confinement is defined as the fraction of the maximum solid angle around a counterion which is occupied by carbon atoms within the coordination shell cutoff .And finally, the charge compensation per carbon , a quantity introduced in this work, is defined as the magnitude of the average charge per electrode atom in the coordination shell. A high CCpC indicates strong and localized charges in the electrode, as opposed to a weak or diffuse charge response. For all quantities, the angle brackets hi denote averaging over all counterions in an electrode. Of particular interest with regard to classical theories of capacitance, a positive correlation can be observed between the capacitance and A/h dsepi , reminiscent of Equation 5.2. This suggests that we can view capacitance in the ZTCs as arising from an “ideal” contribution from a reference electrode with the same A/h dsepi , and a “non-ideal” contribution responsible for the deviations from classical double layer theory, arising from the microporosity. One measure of how micropores influence charge storage is the DoC. Here, we note that we are plotting in Figure 5.6 the average degree of confinement, h DoCi , which obscures differences in the range and distribution of confinement values within a material. We do not observe a strong correlation with capacitance when h DoCi is below 0.25,vertical cannabis grows and when h DoCi is above 0.25 the capacitance seems to be slightly negatively correlated with confinement. This finding adds nuance to the conclusions from previous studies that more confinement is generally a positive influence on charge storage efficiency.We discuss confinement effects further in a later section, where we examine charge storage mechanisms in individual pores.
Finally, the local descriptor which appears to have the best correlation with capacitance is h CCpCi , for which we observe a positive and nearly linear relationship with even less scatter than for A/h dsepi . Capacitance and h CCpCi both aggregate information about the charge stored by the electrode atoms, however their strong correlation is not trivial because only about 30–45% of the electrode atoms are within the coordination shell of a counter-ion at a given time step. These coordination shell carbons have a slightly larger-than-proportional share of charge, carrying between 35 and 50% of the net charge in the electrode . Perhaps surprisingly, the capacitance does not correlate with the total charge compensation within the coordination shell . The observation that per-carbon charge compensation correlates so strongly with the capacitance indicates that localized charge distributions within the electrode store charge more efficiently than disperse charge distributions, as they use less electrode “real estate” to counterbalance an ionic charge. One complication with comparing materials using local properties is that they are computed with a definition of coordination shell that uses a cut-off radius, rcut around the ion. rcut radius was chosen following the literature as the first minimum in the ion-carbon RDF. However, we found in our materials that the first minima were not all at the same location in all materials, and some of them did not have a clear “minimum” at all. Therefore, we opted to use the same rcut of 6.3 Å for all materials, as this was the location of most of the RDF first minima and also was consistent with the literature. Further work is needed to determine how to better define a coordination shell and compute local interfacial properties. However, since we were able to observe quite a strong correlation of capacitance with h CCpCi with the existing coordination shell definition, we leave this complication for a future study. Having investigated geometric descriptors and local interfacial properties of EDLCs, averaged over the entire electrode, we find that almost all of them other than h CCpCi lack a clear correlation with capacitance or, in the case of A/h dsepi , are correlated but exhibit significant scatter. In the following sections we turn our attention to the relationship between pore geometry, local electrolyte properties, and charge storage within individual pores of selected materials. We then move toward a more general framework for rationalizing differences in capacitance among ZTC materials. Due to the structural diversity of ZTC frameworks, we believe insights drawn from ZTCs are also relevant general design rules for porous carbon EDLC electrodes. We begin our examination of individual materials by considering BEA and BEA_beta, which are templated on different polymorphs of the same zeolite as shown in Figure 5.7a. Naturally occuring zeolite beta consists of a mixture of polymorphs A and B, both of which contain layers of the same tertiary building unit which are rotated ±90◦ with respect to each other. In polymorph A , the layers are stacked in a chiral fashion, while in polymorph B , the rotation of the layers alternates. As a result the pore size distributions of BEA and BEA_beta differ, with slightly larger pore sizes for BEA_beta as shown in Figure 5.7b. The capacitances of these ZTCs differ widely, with 34.0 F g−1 gravimetric and 2.33 µF cm−2 areal capacitances computed for BEA . The ions within the pores also have different degrees of confinement, possibly arising from the slight differences in the most probable pore sizes. As seen in Figure 5.7c, the anions in the anode of BEA_beta have a single peak in their DoC histogram around 0.33, while the anions in the BEA have on average higher DoCs, with one peak at 0.35 and another at 0.42. We might suppose from this that BEA should have the higher charge storage efficiency, since Merlet et al. showed that highly confined ions are able to store more charge in super capacitors,however, in this case the opposite is true: h CCpCi DoC is higher in BEA_beta than in BEA for all DoC values . In the cathode, as well, the average charge compensation is lower for BEA than for BEA_beta . One noteworthy feature in the charge compensation distribution of the BEA anode is a minima in h CCpCi DoC at 0.43 DoC , the location of the higher peak in the DoC histogram.