1. Introduction
The development of new high energy density materials (HEDM) is important for a range of applications including explosives and chemical energy storage [
1]. Desired properties for such materials include high energy density and stability in common environments as well as under stimuli. Also desired are low cost of inputs and of synthesis as well as low environmental impact of both inputs and the products of decomposition. New HEDMs are actively researched experimentally as well as computationally [
2,
3,
4,
5,
6,
7,
8,
9]. Most effective molecular HEDM contain nitrogen- and oxygen-containing heterocycles and/or NO
x moieties [
10]. The decomposition of such molecules usually leads to the release of CO
x, NO
x and cyanogen molecules which are toxic or environmentally damaging [
2,
3,
4,
5,
6,
7,
8,
9].
HEDMs based solely on nitrogen are attractive due to the potential for both high energy density and environmental friendliness. Nitrogen gas, N
2, that is part of the atmosphere and is much used in research laboratories, is inert because the N≡N triple bond is one of the most stable chemical bonds. Alpha, beta, and gamma crystalline phases of nitrogen (molecular crystals) are well-studied. Nitrogen exhibits a uniquely large difference in energy between a single bond and a triple bond as compared to other common elements such as carbon. A large amount of energy, therefore, is released under the transformation from the single-bonded nitrogen to the molecular state. To put this statement into perspective, the average bond energy of an N–N single bond (38 kcal/mol) is considerably less than of that of a double bond (100 kcal/mol), which is around 2/5 of that of a triple bond (226 kcal/mol) [
11]. In comparison, the average bond energy of a C–C single bond is much higher (80 kcal/mol) compared to an N–N single bond (38 kcal/mol), while the average bond energy of C≡C (200 kcal/mol) is considerably smaller than that of a N≡N (226 kcal/mol) [
11]. This is to say, on decomposing from a single bonded to a triple bonded form, nitrogen would release much more energy as compared to other elements.
Singly-bonded allotropes of nitrogen are known. For example, cg-N (cubic gauche nitrogen) is a covalently bound crystalline material that had been computationally predicted [
12,
13,
14,
15] before being synthesized [
16]. cg-N possesses an extremely high energy density of 6.7 kcal/g [
16], more than six times that of trinitrotoluene, TNT (1.1 kcal/g). However, it was not stable at room temperature and pressures below 25 GPa [
16]. The crystal structure of cg-N is shown in
Figure 1. The discovery of cg-N is a proof simultaneously of the power of
ab initio materials design (usually at the DFT, or Density Functional Theory, level [
17]) and of the possibility of synthesis of nitrogen-only HEDM. However, no singly bonded allotrope of nitrogen stable at normal conditions have yet been made.
Polynitrogen compounds consisting of N
n molecular units involving single and double N–N bonds, and corresponding
molecular solids stable at ambient temperatures and pressure, are expected to be easier achievable than covalent crystals. Indeed, many molecules (including all common fuels) are synthesized in a metastable state (positive heat of formation), and the vdW forces by which molecules are held together in a molecular solid are always attractive at long range. However, intermolecular interactions between N
n moieties could potentially result in decomposition [
18]. As of today, no molecular N
n solid has been produced, although isolated N
n molecules have been detected with
n = 3–5 [
19,
20,
21,
22,
23]. Recently, Hirshberg, Gerber and Krylov [
24] predicted, by
ab initio simulations, the existence of N
8, as a (meta)stable molecular crystal, at ambient pressures. The crystal structure is shown in
Figure 2. It is a vdW-bound crystal consisting of two kinds of N
8 isomers, dubbed EEE and EZE in [
24]. The NBO (natural bond orbital) analysis of N
8 allowed identifying multiple single bonds, which are responsible for the high degree of metastability [
24]:
Figure 1.
The crystal structures (left to right) of α, β, γ, and cg-nitrogen. Unit cells are shown. Visualization here and elsewhere by VESTA [
25].
Figure 1.
The crystal structures (left to right) of α, β, γ, and cg-nitrogen. Unit cells are shown. Visualization here and elsewhere by VESTA [
25].
Figure 2.
The unit cell of N8 as computed by (top to bottom): DFT-D and DFTB-D with the parameter sets 3ob-2-1, matsci-0-3, and pbc-0-3. Projections along (left to right): a*, b*, c* axes. (N.B. As visualization completes molecular units beyond the unit cell, their number might appear to be different in some cases).
Figure 2.
The unit cell of N8 as computed by (top to bottom): DFT-D and DFTB-D with the parameter sets 3ob-2-1, matsci-0-3, and pbc-0-3. Projections along (left to right): a*, b*, c* axes. (N.B. As visualization completes molecular units beyond the unit cell, their number might appear to be different in some cases).
As a result, this material would release 260 kcal/mol (or 32.5 kcal per mol of N) when decomposing into N
2 gas [
24]. This is about 4.6 kcal/g, and more than four times the gravimetric energy density of TNT and about a third smaller than that of cg-N. The lowest computed barrier to decomposition of the N
8 isomers was 22 kcal/mol (0.95 eV) and the cohesive energy of the crystal 9.8 kcal/mol (0.42 eV). Clearly, this is an extremely promising HEDM.
Importantly, these metastable phases of nitrogen have been discovered computationally using DFT, and while cg-N has since been synthesized, N8 has not. DFT, while well suited to the study of bulk materials (as unit cells can be considered, which typically involve small length scales and small numbers of atoms), suffers from significant computational cost, which makes it impractical to study interfaces of such materials or the dynamics of their decomposition or reactions (e.g., of energy release). This would involve length and time scales which may be too computationally expensive. Indeed, the scaling of DFT is near-cubic with the system size (number of explicitly treated electrons or pairs thereof), which limits routinely doable calculations to simulation cells containing 101–102 atoms.
Density Functional Tight Binding (DFTB) is an approximate DFT-based method that is about three orders of magnitude faster compared to DFT. It can be considered as a parametrization/tabulation of the most expensive parts of DFT [
26,
27,
28,
29]. It achieves DFT-like accuracy for systems for which it is well parametrized. Several parameter sets have been benchmarked for DFTB with which DFTB provides
ab initio accuracy for several classes of materials, including organic molecules, selected inorganic solids, and interfaces [
29,
30,
31,
32,
33,
34]. While several parameterizations of N–N interactions for DFTB are available, the suitability of these parametrizations for the modeling of nitrogen allotropes has, however, not been established.
The main purpose of this work is therefore to establish the applicability and accuracy of DFTB and its different parameterizations to the modeling of different phases of nitrogen covalent and molecular crystals including highly metastable phases like cg-N or the N8. Dispersion-corrected DFTB (DFTB-D) calculations are compared against dispersion-corrected DFT (DFT-D) calculations using atom centered basis sets. To ensure reliability of DFT-D calculations, the basis and dispersion parameters are tuned here to reproduce the structures and energetics of different phases of nitrogen.
Another purpose of this work is, therefore, a DFT-D setup based on localized basis functions that is tuned for accurate description simultaneously of several nitrogen allotropes including covalently and vdW-bound crystals and including high-energy phases. The use of localized basis functions is advantageous for speed of calculations as well as for parallelizability and scalability, as they allow implementing order-N approaches.
The paper is organized as follows:
Section 2 details the methodologies used in this work,
Section 3 presents the results of simulations, and
Section 4 concludes.