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Relative Stability of Diamond and Graphite As Seen Through Bonds and Hybridizations Cite This: Phys PCCP View Article Online PAPER View Journal | View Issue Relative stability of diamond and graphite as seen through bonds and hybridizations Cite this: Phys. Chem. Chem. Phys., 2019, 21, 10961 ab c abc Ilya V. Popov, Arno L. Go¨rne, Andrei L. Tchougre´eff * and Richard Dronskowski *cd The relative stability of the two most important forms of elemental carbon, diamond and graphite, is readdressed from a newly developed perspective as derived from historically well-known roots. Unlike other theoretical studies mostly relying on numerical methods, we consider an analytical model to gain fundamental insight into the reasons for the quasi-degeneracy of diamond and graphite despite their extremely different covalent bonding patterns. We derive the allotropes’ relative energies and provide a Received 12th December 2018, qualitative picture predicting a quasi-degenerate electronic ground state for graphite (graphene) and Accepted 26th March 2019 diamond at zero temperature. Our approach also gives numerical estimates of the energy difference and DOI: 10.1039/c8cp07592a interatomic separations in good agreement with experimental data and recent results of hybrid DFT Creative Commons Attribution-NonCommercial 3.0 Unported Licence. modeling, although obtained with a much smaller numerical but highly transparent effort. An attempt to rsc.li/pccp extend this treatment to the lowest energy allotropes of silicon proves to be successful as well. Introduction Carbon allotropes are a hot topic with a plethora of newly proposed ones, predicted to have fascinating physical proper- ties. To navigate this large zoo of allotropes, we recommend a This article is licensed under a comprehensive and timely database,1 a survey2 and two recent articles dealing with the bonding of these carbon allotropes3 8 9 and sp-mixing triangle maps from first principles.4 The experi- Fig. 1 Crystal structures of diamond, graphite and lonsdaleite, the hexagonal form of diamond.10 Open Access Article. Published on 20 May 2019. Downloaded 7/19/2020 9:56:53 PM. mentally available allotropes are by far less numerous but of interest in different areas: graphene and fullerenes are popular materials in material sciences and electrochemistry,5,6 and modified nanotubes have been found to be efficient catalytic local arrangements of C–C bonds but differing by the hexagonal systems.7 Despite the impressive diversity of novel carbons, the ABAB stacking as compared to the cubic ABCABC one (Fig. 1). well-known diamond and graphite (Fig. 1, as well as hexagonal The most intriguing fact about diamond and graphite is diamond dubbed lonsdaleite) form the centerpiece of the entire their relative stability. Experimentally, graphite is thermodynami- carbon story. These two allotropes have been known to humans cally more stable at 1 atm and 298 K, but only barely (ca. 2 kJ molÀ1, for thousands of years, being the only crystalline forms present that is, they are degenerate within a chemical accuracy of in nature rather than prepared artificially. Also, they are the 1kcalmolÀ1). Any attempt to find a rigorous explanation of this most studied carbons and prototypical for other allotropes quasi-degeneracy stumbles upon the problem that diamond and variously combining sp3- and sp2-hybridized C atoms.2,11–13 graphite are just too different from the point of view of, say, Remarkably, diamond itself can exist in two forms, the usual topology, which is accepted as a basis of allotrope classification in cubic diamond and hexagonal lonsdaleite10 featuring the same ref. 1. Indeed, diamond and graphite adopt different space-group symmetry, different unit cells, different band structures and a A.N. Frumkin Institute of Physical Chemistry and Electrochemistry of the Russian extremely different physical properties, so why should they have Academy of Science, Moscow, Russia (almost) equal energies? We recall that solid-state chemistry b Independent University of Moscow, Moscow, Russia would portray diamond and graphite as species featuring c Chair of Solid State and Quantum Chemistry, RWTH Aachen University, different hybridization patterns (sp3 vs. sp2), different localiza- 52056 Aachen, Germany. E-mail: andrei.tchougreeff@ac.rwth-aachen.de, [email protected] tion, and different covalent bonding types. Thus, it might look d Ju¨lich–Aachen Research Alliance, JARA-HPC, RWTH Aachen University, easier to explain if graphite were significantly more stable since 52056 Aachen, Germany the delocalized p-system might also decrease graphite’s total This journal is © the Owner Societies 2019 Phys. Chem. Chem. Phys., 2019, 21, 10961--10969 | 10961 View Article Online Paper PCCP energy as habitually evoked in organic chemistry. Vice versa,if above diamond. This suggests that, within chemical accuracy, diamond were significantly more stable, one might argue in diamond and graphene (and graphite, too) are almost degen- terms of the larger number of strong covalent bonds per atom. erate states of carbon. Thus, we use graphene in our considera- A recent, comprehensive DFT study of the relative stability of tions for reasons of simplicity. diamond and graphite14 provides valuable numerical informa- tion on the exact allotropic energy difference under different Model conditions, but in our contribution, we try to understand how diamond and graphite can come so close on the energy scale in Now, we turn to constructing an analytical model of the electronic terms of an analytical model. structuresforbothcarbonallotropes, relying upon fundamental chemical concepts such as bonding, hybridization etc. Like in the Experimental situation Molecular Mechanics or Force Field22–27 approaches, we represent the energy of each allotrope as a sum of energies of the bonds First, we specify the quantity to be reproduced. Under standard formed by differently hybridized orbitals, complemented by the conditions, the difference in Gibbs energies DG1 between energies of atoms in the respective hybridization states and by the diamond and graphite falls in the range of 2.87–2.90 kJ molÀ1, energy of repulsion of the atomic ‘‘cores’’, that is, the atoms bereft graphite being lower,15 but this quantity is contaminated by the of their valence electrons.† Instead of aprioriinferring the form of lattice contribution. Since the crystal structures and elasticity the bond energies, we use expressions stemming from a general modules and, thus, the phonon spectra are rather different, one quantum-mechanical theory of the molecular electronic structure, 28–30 cannot a priori exclude changes in the order of electronic the formal background being the group-function formalism. 3 energies. We therefore extrapolated the difference in Gibbs The diamond energy per atom is that of the sp hybridized energies to zero temperature; the standard experimental data carbon atom supplied by the energies of two s-bonds formed (NIST) of entropy and heat capacities for graphite (see Appendix A)16 by these hybrids (four s-bonds counted 50% per atom) and 3 Creative Commons Attribution-NonCommercial 3.0 Unported Licence. are fit to the Debye–Sommerfeld model, Cp = gT + aT . For diamond, the structure-dependent repulsion energy. Likewise, graphene 2 this yields g = 0 (nonmetallic and no other degrees of freedom contains 3/2 s-bonds formed by sp hybrids per atom, but their yielding a linear term) but graphite features so called flexural energy must be augmented by that of a delocalized p-electron vibrations (Fig. 2),17 whose dispersion law produces a T2 con- per atom. Thus, we write tribution to the low-temperature heat capacity, as predicted 3 3 ðDÞ EDiamond ¼ E0ðsp Þþ2Esðsp ÞþE quite long ago19 and documented experimentally.20 Even with rep (1) accounting for the flexural modes, graphite is still more stable 3 3 E ¼ E ðsp2Þþ E ðsp2Þþ E þ EðGÞ than diamond at 0 K, but the difference in Gibbs energies is Graphite 0 2 s 2 p rep even smaller (1.39–1.55 kJ molÀ1) than at room temperature. This article is licensed under a and call E0(h) the hybridization-dependent one-center energies, Taking into account the ZPEs (as estimated in ref. 14) of either that is, the sums of core attraction and mutual repulsion of the allotrope yields a difference of electronic energies in the range À1 valence electrons in the atoms. Es(h) are the energies of s- of 0.14–0.78 kJ mol in favor of graphite. 3 2 bonds formed by the sp and sp hybrid orbitals, Ep is the Open Access Article. Published on 20 May 2019. Downloaded 7/19/2020 9:56:53 PM. Another uncertainty is the weak interaction between the gra- energy of the p-electrons in graphene recalculated per number phene layers forming graphite. To avoid this overwhelmingly of nearest neighbors of each atom, so the multipliers equal 3/2, complex problem (in particular when relying on DFT approaches), (A) and Erep represents the core–core repulsions of the carbon we notice that adding the recently21 determined exfoliation energy atoms specific for each allotrope due to structural differences. of graphite (about 3 kJ per mole of carbon atoms) puts the Clearly, the carbon atoms in the allotropes have different electronic energy of an individual graphene sheet ca. 2.5–3 kJ molÀ1 hybridizations. Normally, the one-center energies E0(h) strongly depend on hybridization,31–33 but the allotropes of the group IV elements represent a remarkable exception.31 For them, the populations of all atomic hybrid orbitals are equal to unity, and the one-center terms are equal in graphene and diamond (see proof in Appendix B), so we can exclude them from now on. (D) (G) The repulsion energies Erep and Erep come from the general formula for the repulsion of the atomic charge distribution if one takes into account the differences between interactions of the core point charges and of the electron distributions with a spatial extent.30,32 Hence, repulsive interactions between elec- troneutral atoms acquire a quasi-Yukawa form Y(d), that is, dÀ1 † For the carbon atom, the effective core charge arrives at +4 because there is an electronic configuration of 1s22s22p2 from which the 2s22p2 valence electrons are Fig.
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