Predicting experimentally stable allotropes: Instability of penta-graphene

In recent years, a plethora of theoretical carbon allotropes have been proposed, none of which has been experimentally isolated. We discuss here criteria that should be met for a new phase to be potentially experimentally viable. We take as examples Haeckelites, 2D networks of sp²-carbon–containing pentagons and heptagons, and “penta-graphene,” consisting of a layer of pentagons constructed from a mixture of sp²- and sp³-coordinated carbon atoms. In 2D projection appearing as the “Cairo pattern,” penta-graphene is elegant and aesthetically pleasing. However, we dispute the author’s claims of its potential stability and experimental relevance.One of the joys of carbon research is the huge flexibility of carbon bonding (1–4), resulting in many varied allotropes that have already been experimentally identified. Computational modeling opens the floor to predicting many more, and tools such as graph theory (5) and evolutionary algorithms (6) allow systematic exploration of potential bonding networks. New computationally proposed phases are typically identified as metastable via positive phonon modes, and sometimes via molecular-dynamics (MD) simulations showing lattice coherence at experimental operating temperatures. However, there are common criteria beyond these two tests that link those allotropes that have been experimentally isolated.First, they occupy deep potential wells in the surrounding energetic landscape. Additionally, the surrounding energy wells are all higher in energy, “funneling” toward the stable structural form. Finally, barriers to subsequent conversion to alternative structures are typically high. Buckminsterfullerene, I_h-C₆₀, is a good example. The disconnectivity graph for C₆₀ connecting the 1,812 isomers with pentagonal and hexagonal faces via branches whose height indicates the transformation barrier has a “willow tree pattern,” with a gentle funnel running toward the stable I_h-C₆₀ isomer (7) (Fig. 1A). The relatively high barriers are accessible during high-temperature growth and alternatively can be catalyzed via the presence of impurities or carbon interstitial atoms (8–10).Open in a separate windowFig. 1.(A) “Willow tree” pattern of different C₆₀ isomers, with the lower points of each vertical bar representing the calculated formation enthalpy relative to I_h-C₆₀, and bar heights representing the calculated barrier to transformation. This shows that I_h-C₆₀ is significantly more stable than other isomers and lies at the center of an “energetic funnel.” Adapted from ref. 7, with permission from Macmillan Publishers Ltd.: Nature, copyright 1998. (B) Calculated formation enthalpies of B₄₀, showing the D_2d cage structure is significantly more stable than alternative isomers. Reproduced from ref. 12, with permission from Macmillan Publishers: Nature Chemistry, copyright 2014.In contrast, attempts to experimentally isolate higher-order boron fullerenes have been largely unsuccessful to date. For the proposed fullerene B₈₀, this can be understood because the energy landscape was shown to feature many closely related isomers with similar (and sometimes lower) energies (11). In contrast, calculations for B₄₀, for which there are first experimental indications (12), show a single (D_2d, 1A1) cage isomer, energetically well separated from alternative isomers (Fig. 1B). This behavior is consistent with the rules discussed above.We apply here a similar analysis for experimental viability to other proposed phases, starting with “penta-graphene,” a 2D carbon allotrope proposed by Zhang et al. (4). The structure can be viewed as a series of out-of-plane distorted ethylene units connected via tetrahedral sp³-carbon linkers. The result is a corrugated layer that in projection matches the “Cairo pattern” of distorted pentagons (Fig. 2A).Open in a separate windowFig. 2.A calculated structural transformation route from (A) penta-graphene to (D) graphene; each step is exothermic. Red arrows indicate direction of motion of atoms for 90° rotation of carbon–carbon bonds. Red (blue) lines indicate C–C bonds that are broken (formed). Note that structures A–C were constrained within orthogonal unit cells; this constraint was lifted for step C to D. The final structure, graphene, is 0.761 eV per atom more stable than A. Unit cells are marked with dotted lines; calculated cell dimensions are (A) 5.095 × 5.095 Å, (B) 4.769 × 5.510 Å, (C) 4.888 × 5.318 Å, and (D) 4.883 × 6.476 Å, α = 100.88°.