Dendritic forms of lithium

Concepts: metallic systems, pairwise cutoff, cfgscale

Lithium metal is widely used as an electrode material for rechargeable batteries, due to its high electropositivity and low weight. During battery operation, lithium at the negative electrode gives up electrons to become $\mathrm{Li^{+}}$, which dissolves into the electrolyte. This process is reversed during the recharge cycle, but the replated lithium adds unevenly to the electrode surface, and as the battery is charged and discharged, dendrites grow from the surface, which can cause short-circuiting in the battery and lead to explosions.

Dendritic growth can be limited by encasing the lithium in large heterogeneous structures, such as intercalating solids or gels. To evaluate such structures and make them lighter and safer, it is useful to have a model that can describe both bulk and large-scale dendritic forms of lithium. We demonstrate the use of eFF for this task.

Bulk lithium in eFF can be described as a periodic structure with $\mathrm{Li^{+}}$ ions assuming face-centered cubic (fcc) lattice positions, and valence electrons assuming octahedral interstitial positions in between the ions. Each interstitial site is occupied by a single spin, which may be up or down. We randomly choose the spins, the only constraint being that the total spin of all the valence electrons should be zero. The energy may be lowered slightly by swapping spins in an annealing procedure, but the improvement is slight ($\sim$2 kcal/mol/atom), suggesting that Pauli interactions between valence electrons in lithium are small.

This model accurately reproduces the lattice constant (4.42 $\mathrm{\AA}$ eFF versus 4.40 $\mathrm{\AA}$ experiment) and the bulk modulus (12.2 GPa eFF versus 13.0 GPa experiment) of lithium. The cohesive energy however is too high (60.3 kcal/mol/atom eFF versus 37.7 kcal/mol/atom experiment). Figure 2.13 shows the eFF equation of state of lithium.

Figure 2.13: Bonding in periodic lithium solid.

For small lithium clusters, ab initio generalized valence bond studies (McAdon and Goddard, J. Phys. Chem. 91, 2607 (1987)) show that electrons concentrate at interstitial positions. The study shows that as the number of atoms increases, there is a progression from linear and zig-zag chains ($N < 8$), to planar and three-dimensional clusters containing tetrahedral packings ( $8 \leq N \leq 20$), to bulk solids. Figure 2.14 shows several eFF $\mathrm{Li_{32}}$ structures obtained by minimizing different initial conditions, with a similar positioning of interstitial electrons as found in the GVB calculations.

Figure 2.14: Nonperiodic lithium, representative local minima with energies per atom given relative to periodic structure.

With eFF, we can study bonding in very large scale hetereogeneous lithium structures. We begin by generating a bulk lithium structure inside a box $\mathrm{(50 bohr)}^{3}$, using the script li_solid_cfg [nx] [ny] [nz]:

  li_solid_cfg 6 6 6 > li_bulk.cfg

To speed up the computation, we apply a pairwise cutoff, which limits the number of interactions that must be considered. With such a scheme, the computational cost scales linearly with the number of particles, making calculations with as many hundreds of thousands of electrons practical. Electrostatic interactions are long range, so the cutoff is large - around 20 bohr. Using such a cutoff, cohesive energy of lithium is affected by $\sim$ 2 kcal/mol/atom, which can be decreased to 0.06 kcal/mol/atom by using an even larger 50 bohr cutoff.

To apply the cutoff, we set taper_cutoff = 20 bohr. The program automatically accelerates the computation by breaking space down into cells on the order of the cutoff length, and only considering interactions between nearest neighbor cells. Periodic boundary conditions for systems larger than $(2 * \mathrm{taper\_cutoff})^3$ become simple too, since each particle only interacts with any other particle at most once, so that a minimum image convention can be applied. To use this scheme, we set the parameter periodic = minimage_xyz. 1D and 2D periodic bounds can be applied too, for example, with the keywords minimage_y or minimage_xz.

We minimize the bulk lithium system:

li_bulk.cfg:
  @params
  x_bound = 0.000000 50.112000
  y_bound = 0.000000 50.112000
  z_bound = 0.000000 50.112000
  taper_cutoff = 20
  periodic = minimage_xyz
  calc = minimize
  @nuclei
  ...

Next, we rescale the box so that it is double in length along every dimension. To do this, we use the script cfgscale [scale_x] [scale_y] [scale_z], which scales the distances between nuclei and electrons uniformly while keeping electron sizes fixed:

  cfgscale li_bulk.cfg.restart 2.0 2.0 2.0 > li_rescale.cfg

With the scaled box in hand, we reoptimize the system, and observe the formation of a dendritic lithium structure:

li_rescale.cfg:
  @params
  x_bound = 0.000000 100.224000
  y_bound = 0.000000 100.224000
  z_bound = 0.000000 100.224000
  taper_cutoff = 20
  periodic = minimage_xyz
  calc = minimize
  num_steps = 100000
  print_every = 100
  @nuclei
  ...

Figure 2.15: Expanding the lithium bulk structure twice in every dimension and reoptimizing causes a dendritic lithium structure to form, shown here as a stereo view.

Figure 2.16: Electrons in dendritic lithium, showing interstitial electrons occupying spaces in between linear chains, two-dimensional meshes, and three-dimensional bulk-like structures. Electrons are scaled to 0.2 times their actual size.

Lower energy structures can be created using a simulated annealing procedure analogous to the one performed in the previous section to create optimized hydrogen boxes.