Like mentioned above, there was a lot of excitement soon after the discovery of graphene with thoughts that it could have a higher theoretical storage capacity than graphite. How-ever, it has since been proven that pristine graphene falls well short of the charge capacity of graphite. When evaluating this hypothesis, Lee and Persson published extensive ab-sorption energy data for lithium concentrations on graphene ranging from pure graphite to Li6C6 as computed by first-principles. It was believed that this data would provide an appropriate baseline for potential modifications as it accentuates both the Li-C attraction and the Li-Li repulsion with Li atoms straddling either side of the graphene layer. How-ever, graphene is a two-dimensional material and therefore this study alone will not probe any change to the interlayer spacing with increases in Li concentration. Therefore, exper-imental expansion data is also considered to provide additional parameters for modifying the existing empirical potentials.
Using conjugate-gradient energy minimizations, Lee’s calculations were repeated for two configurations of Li on graphene, Li1.5C6 and Li6C6. The total number of atoms in the system were 810 and 1296 across 108 fundamental units (FU), which is equivalent to those constructed by previous work. Although a range of energies were reported at each loading, values of 0.8 and 3.1 eV as calculated by Lee et al. using VASP were used.
Equation 3.4 represents the absorption energy calculation performed, where E (x) is the computed absorption energy per FU of LixC6, E (0) is the energy of pure graphene, x is Li concentration, and ELi is the energy of a bulk metallic Li atom, taken as -1.67 eV/FU [48].
Ea(x) = E (x) − E (0) − xELi (3.4)
From the onset, this modification was not intended to be a full potential development,
Table 3.1: Possible scaling factor combinations based on a least-squares fit to absorption energy data from Lee and Persson and energy minimizations performed by the authors [50].
δ1 δ2 Ea(1.5) Ea(6) Error 0.1014 0.0628 0.3397 3.2047 0.2229 0.0885 0.050 0.6314 3.8105 0.5333 0.1142 0.0885 0.1036 3.3768 0.5616 0.1014 0.0500 0.2840 2.4245 0.7225 0.1142 0.0757 0.047 2.5989 0.8169 0.1014 0.0757 0.3954 3.9832 0.9438
thus a simple scaling approach of the existing potentials was taken, as noted in Equations 3.5 & 3.6.
ULiC∗ = δ1ULiC (3.5)
ULiLi∗ = δ2ULiLi (3.6)
Scaling factors ranging from 0 to 1 were applied to both the Li-C and Li-Li potential and implemented in LAMMPS prior to performing energy minimizations. The absorption en-ergy for the configuration was recorded along with the error calculated by a least-squares fit to Lee’s data for each scaling combination. Scaling factors were only considered acceptable if they produced positive absorption energies at both the low and high Li concentrations, consistent with previous studies. This produced numerous combinations of acceptable scales, thus further refinement was necessary. Deeming an error less than 1.0 as a suitable starting point, several combinations of scaling factors were returned as viable options based on the initial criteria, as seen in Table 3.1.
Upon investigation of Table 3.1 it is obvious that scaling factors of δ1 = 0.1014 and δ2 = 0.0628 lead to the greatest correlation to Lee and Persson’s findings. With the first of two modification criteria satisfied, dynamic simulations were performed on a bulk LiC6
system to quantify changes in interlayer spacing. Like the original potentials, the new selections resulted in compression of the graphite lattice, which again is non-physical.
Aside from contradicting experiment, compression of the graphite lattice has been shown to greatly stifle intraplanar diffusion of lithium [67]. With the driving force of potential redevelopment being diffusion studies, this suppression would severely increase simulation times and perhaps alter the dynamics of the system from a ballistic nature to more site hops or no motion at all for lower Li concentrations.
The obvious issue with both this choice and the original potentials is that the relative magnitude of Li-C and Li-Li potentials are quite skewed. The overwhelming nature of the Li-C attraction fails to allow the Li-Li strictly repulsive potential to become effective and increase the layer spacing of the system. When the potentials are scaled back such as those selected from the authors’ optimization, there is less system-wide compression however again the Li-Li repulsions are not strong enough to overcome attractive forces.
At this point, it is understood that the best scaling parameters for the desired applica-tion must deviate from the published ab initio data to correctly capture crystallographic expansion. Like previously stated, experimental data shows that fully-liathiated graphite expands 10% in the basal direction. Provided that this effect could be captured while maintaining positive absorption energies, albeit it with greater errors than before, the scaling factors would be considered acceptable.
Various scaling factors were chosen and applied to a LiC6 system running dynamically with an isobaric-isothermal ensemble at T = 300 K. The basal plane lattice constant was measured and compared to the pure graphite lattice constant to verify expansion. One such combination that resulted in an approximate 10% expansion at LiC6 was δ1= 0.0885 and δ2 = 0.90. For these factors, the absorption energies were 4.33 and 41.0 eV, respectively.
While the Li1.5C6 energies remained reasonable, the Li6C6 grew quite rapidly leading to the large deviations from published data. However, it was decided that these factors accomplished the initial goals of this modification with adequate precision and were selected as the final values. Figure 3.4 shows Chakraborti’s original potentials alongside the scaled potentials developed in this work.
0 1 2 3 4 5
Figure 3.4: Potential energy as a function of particle separation distance for both Lithium-Carbon and Lithium-Lithium potentials. Shown are the unmodified potentials from Chakraborti et al. as well as the two potentials modified here. Although not shown, the cutoff radius for each potential is 10.2 ˚A.