While there have been countless experimental and computational studies on the Lithium-Carbon system, there is a lack of available interatomic potentials for classical simulations with Lithium-Lithium and Lithium-Carbon interactions. Alternatively, Carbon-Carbon interactions are well-described by such potentials as the Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO) potential developed by Stuart or the Tersoff Potential [85, 90, 92, 91]. These potentials are highly accurate for graphite and closely reproduce many experimental aspects of the system. Similar potentials were desired for Li-C and Li-Li and thus various options were explored to accomplish this.
Existing potentials could correctly model bulk lithium-metal behavior such as ReaxFF developed by van Duin et al., but they were not well suited for applications to LIB studies [36]. Based on the goals of this study, a fully (QM) or partial (QM-CL) first-principles approach would not be practical. Within the MD realm, a charge transfer model would provide high levels of accuracy but are highly computationally expensive and again not practical for the duration of simulations needed here. Therefore, a simpler approach was desired for modeling the interatomic interactions.
An existing set of empirical pair potentials were found that were developed for use in finding stable assemblages in Li-C systems by Particle Swarm Optimization and Differential Evolution [21]. Additionally, the authors set out to study advanced anode configurations for LIBs, specifically carbon nanotubes [20]. These potentials were developed by Chakraborti et al. using a least-squares fitting procedure with DFT of the C-Li+ dimer [21]. Equations 3.1 and 3.2 represent the pair potential energy as a function of separation distance, r, for Li-C and Li-Li, respectively. Full details of the associated constants are presented in Ref.
[21] and [20]. The Li-C potential has an attractive and repulsive regime while the Li-Li potential is a strictly repulsive potential and behaves similar to a Coulombic interaction;
it therefore cannot model bulk metallic lithium. Notably these are not charge transfer models and again are restricted to pair interactions.
The original authors provided system energies for several small Li-C clusters via MD which were verified by performing energy minimizations using LAMMPS with a Tersoff Carbon-Carbon Potential [73, 90].
To compare to both experiments and simulations, mass transport characteristics of lithium needed to be established for this system of interatomic potentials. In doing so, single-ion diffusivity tests in bulk graphite were performed to compare to existing QM-CL data. Ohba et al. performed QM-CL simulations of single lithium-ions in bulk graphite to probe the dependence of diffusivity rates on interlayer spacing [67]. The authors provided thorough mean-squared displacement (MSD) data, which is related to particle diffusivity and shown in Equation 3.3, as a means of comparison.
M SD ≡ D
(x (t) − x0)2 E
(3.3)
This work used AIREBO for C-C interactions and the two potentials developed by Chakraborti et al. to simulate single particle diffusion. A single lithium atom was inserted into a graphite crystal containing 15,680 atoms and run dynamically using a microcanonical (NVE) ensemble at T = 423 K for 250 ps. Lithium’s light weight necessitated the use of a t = 0.0001 ps timestep to fully capture atomic trajectories which is 1/10 of the typical timestep used for metallic systems. MSD data was calculated in 12 ps lengths, with a new calculation window beginning every 0.1 ps to collect an increased number of data sets while maintaining statistical independence. The inset of Figure 3.2 shows MSD data from
Ohba and this work, over 3 ps. Based on differences in C-C potentials used, the interlayer spacing of this work (6.8391 ˚A) corresponds to an approximately 2.5% expansion of the lattice in Ohba’s work, which is the case presented in Figure 3.2. In both works, the single-ion exhibited largely ballistic motsingle-ion rather than site hops, which in graphite equates to approximately 5.85 ˚A2. The close correlation between the data sets confirmed the accuracy of the Li-C potential for diffusion studies and hence further verification of the Li-Li system was needed before continuing.
Figure 3.2: Mean squared displacement for single Li atom simulations for the potentials developed here and the original authors, compared to QM-CL simulations by Ohba et al.
[67]. Inset focuses on early time of trends shown in main panel.
Many groups have shown the strong dependence of diffusivity on Li concentration and the next goal was to verify this dependence in bulk graphite via MD [70, 108, 88]. To test the Li-Li potential, several systems were created with varying Li concentrations ranging from x = 0.25 to x = 1.0 in LixC6. Like previously mentioned, expansion of the lattice in the basal direction should be noted for all Li concentrations, with a maximum expansion of 10% at LiC6.
Each system tested was equilibrated using an isobaric-isothermal (NPT) ensemble at T = 300 K and zero pressure to allow for relaxation in the basal direction such that each diffusion study takes place at the equilibrium lattice constant for that loading. Upon
relaxing the compounds, average interlayer spacing was computed to verify the accuracy of the Li-Li potential at various loadings. For all concentrations of Li the system showed compression in the basal direction, which is not seen experimentally, peaking at 17% for Li0.5C6. Figure 3.3 compares Chakraborti’s potentials with experiments by Dahn and first principles by Persson et al., showing compression across all concentrations versus expansion everywhere for the contrary [24, 70]. This compression corresponds to an overpowering Li-C attraction across several planes of Li-C atoms, resisted only by the very weak vdW forces.
The cutoff distance of 10.2 ˚A (chosen to match cutoff radius for AIREBO potential by Stuart) allows for lithium atoms in a gallery to pair with carbon atoms in the three nearest planes, across an empty gallery in Stage II cases which occurs when x ≤ 0.50. Lithium bonds with the six nearest-neighbor carbon atoms in the stable configuration thus the high levels of attraction as the Li concentration increases.
0 0.2 0.4 0.6 0.8 1
Figure 3.3: Basal direction lattice spacing (c) as a function of Lithium concentration at T = 300 K from Chakraborti et al., this work, experiment by Dahn, and first-principles by Persson et al. [21, 24, 70]. The x = 0 values corresponds to bulk graphite for each work.
The goal of this work was to study grain boundary intercalation and subsequent dif-fusion, thus it was decided that although there was good agreement with single Li data, the potentials needed modification to fit the needs of this dissertation. These modifica-tions understandably alter single Li results significantly but should provide valuable insight
into the behavior of mass transport in a covalently bonded grain boundary; something not previously studied with application to energy storage materials.