Edinburgh Research Explorer High Pressure Hydrocarbons Revisited: From van der Waals Compounds to Diamond

: Methane and other hydrocarbons are major components of the mantle regions of icy planets. 1 Several recent computational studies have investigated the high-pressure behaviour of speciﬁc 2 hydrocarbons. To develop a global picture of hydrocarbon stability, to identify relevant decomposition 3 reactions, and probe eventual formation of diamond, a complete study of all hydrocarbons is 4 needed. Using density functional theory calculations we survey here all known C-H crystal structures 5 augmented by targeted crystal structure searches to build hydrocarbon phase diagrams in the ground 6 state and at elevated temperatures. We ﬁnd that an updated pressure-temperature phase diagram for 7 methane is dominated at intermediate pressures by CH 4 :H 2 van der Waals inclusion compounds. 8 We discuss the P-T phase diagram for CH (i.e. polystyrene) to illustrate that diamond formation 9 conditions are strongly composition dependent. Finally, crystal structure searches uncover a new 10 CH 4 (H 2 ) 2 van der Waals compound, the most hydrogen-rich hydrocarbon, stable between 170 and 11 220 GPa. 12

pure H 2 and diamond. 48 Beyond methane, longer hydrocarbons could have facilitated the production of organic molecules 49 on the early Earth through impact-shock events [26][27][28][29]. It is important to understand their response 50 to the extreme pressure and temperature conditions present in such events, in particular the creation 51 of specific reaction products. However, hydrocarbon chemistry is driven by kinetic effects, which also 52 applies to reactions under high pressure: compressed benzene, C 6 H 6 , can polymerise under specific 53 experimental conditions to form carbon nanothreads [30][31][32], which have promising mechanical 54 properties and can be recovered to ambient conditions despite being clearly metastable. While exciting 55 for materials science purposes, this path-dependence and longevity of metastable phases further 56 complicates our understanding of hydrocarbons under pressure.

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Dynamic compression experiments on hydrocarbons are used to probe their properties at 58 combined high-pressure and -temperature conditions [33][34][35][36][37][38][39][40], both to investigate their potential 59 decomposition, but also to better understand a critical ablator material used in inertial confinement 60 fusion experiments. The thermal equation of state of hydrocarbons has therefore been a subject 61 of several ab initio molecular dynamics studies [41][42][43][44][45][46]. Recent shock experiments on polystyrene 62 ([C 8 H 8 ] n ) [37,38] reported its decomposition into diamond, however only above 140 GPa and 4000 K 63 in an apparent disagreement with diamond anvil cell experiments [23,24]. On the other hand, 64 compressed polyethylene did not form diamond up to 200 GPa and 5000 K, instead retaining a 65 crystalline polyethylene structure up to 190 GPa and 3500 K [40]. Kraus et al. [37] compare their 66 polystyrene measurements to the diamond formation conditions found in static methane compression 67 as well as an extrapolation of the diamond formation line (C 4 H 10 + 3 H 2 → 4 C + 8 H 2 ) obtained in 68 first-principles calculations [18]. However, diamond formation in carbon rich samples like [C 8 H 8 ] n 69 could begin at significantly lower pressures than in CH 4 , because the global stoichiometry suggests 70 that decomposition into long hydrocarbons and excess carbon could be a viable reaction pathway.

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There is then also scope to revisit the C-H phase diagram and its stable compounds as obtained 72 from first-principles calculations. This will allow us to predict more precisely the conditions upon 73 which diamond should form from various initial hydrocarbon species. It is also timely to update 74 the methane phase diagram reported by Gao et al., as several potentially stable C-H species have 75 been proposed in the intervening period. These comprise a new stable phase for C 2 H 6 [47] and 76 various van der Waals inclusion compounds of the type (CH 4 ) m (H 2 ) n . A series of 1:1 compounds 77 (stoichiometry CH 6 ) have been reported to be more stable than methane and hydrogen above 28 GPa, 78 before decomposing above 233 GPa [48]. A 2:3 compound (CH 7 ) was found to be stable between 10 79 and 215 GPa [49]. However, none of these recent works consider all of the other reported structures, 80 and it is therefore unclear at present which hydrocarbons are in fact stable. Here, we construct an updated picture of the C-H binary phase diagram by compiling all known 82 C x H 1−x structures present in the literature [18,[47][48][49]51]. We also include results from new structure 83 searching calculations that are motivated by the presence of stoichiometric van der Waals inclusion 84 compounds. A recent experimental study reported the van der Waals compound Ar(H 2 ) 2 to be stable 85 at least up to 358 GPa [52]. Since Ar is of similar size to the CH 4 molecule, it is conceivable that a 1:2 86 inclusion compound (CH 4 )(H 2 ) 2 can form at high pressure; and we find indeed that such a compound is 87 stable. With stoichiometry CH 8 this is not only the most hydrogen-rich hydrocarbon compound known, 88 but also with 20 wt-% releasable hydrogen content one of the most efficient hydrogen storage materials.

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In addition, we find another new inclusion compound with composition 2:1 (CH 5 stoichiometry) 90 that is very close to stability at lower pressures. Overall, the picture that emerges for compressed 91 hydrocarbons is notably more complex than previously thought.   We performed structure searches for various hydrogen-rich hydrocarbons. we anticipate that the stability of C-H compounds will depend heavily on the zero point energy 156 (ZPE), these calculations provide a good first approximation. Figure 2 shows the relative enthalpies Considering the (CH 4 ) m (H 2 ) n inclusion compounds is crucial at higher pressures: the methane 171 phases are unstable above 95 GPa against a decomposition into C 2 H 6 (P2 1 /c) and an appropriate 172 amount of any of the three inclusion compounds CH 4 (H 2 )-P2 1 /c or -P2 1 2 1 2 1 , or (CH 4 ) 2 (H 2 ) 3 . This we reproduce a transition from C 4 H 10 + H 2 to CH 2 + H 2 at 380 GPa, and see no decomposition into 183 diamond and hydrogen up to at least 500 GPa.

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In Figure 3   Eventually, however, the presence of CH 4 molecules becomes unstable. This is less due to volume 208 effects (the p∆V terms in Figure 4 do not show drastic changes) and more due to loss of the internal 209 energy advantage ∆E by the CH 4 -containing phases. A local analysis of all structures that contain 210 methane molecules reveals increasingly distorted CH 4 molecules under pressure, with all four bonds 211 of slightly different lengths and bond angles. We quantify the tetrahedrality of a methane molecule as where θ ijk is the angle between the j th and k th C-H bonds in the i th molecule. Q i = 1 for an 213 ideal tetrahedron, and Q i = 1/2 for a square-planar arrangement. A general trend is for the unit 214 cell-averaged measure Q to decrease monotonously with increasing pressure, see Figure A3. to remain closer to their ideal structure. The molecular distortions correspond to an increase in internal

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Considering now all C x H 1−x structures, we construct a series of relative formation enthalpy diagrams across the full hydrocarbon composition range. Then, the convex hull of ∆H f (x) given as represents all C-H phases that are stable against any decomposition reactions. In Figure 5 we

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The enthalpy values ∆H f (x) in Figure 5a refer to the energy cost associated with full 230 decomposition of a hydrocarbon into pure carbon and hydrogen. In that sense, the most stable 231 hydrocarbons are CH 4 and CH 3 (i.e., ethane, C 2 H 6 ) at low pressures, and CH 2.5 (C 4 H 10 ) and CH 2 at 232 high pressures. For instance, CH 4 is stable against decomposition by 0.21 eV/atom (101 kJ/mole CH 4 ) 233 at 40 GPa, and CH 2 is stable by 0.04 eV/atom (11.6 kJ/mole CH 2 ) at 500 GPa. [52], we found that a straightforward replacement of Ar with a methane molecule led to a very unstable 239 structure (See Figure A2 for details). However, our own structure searches at the same stoichiometry 240 found a new phase of P2 1 /c symmetry. This new inclusion compound does not play a role in the 241 methane decomposition sequence as calculated at the PBE level, because it is too hydrogen-rich.

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However, it is stable in its own right as the most hydrogen-rich hydrocarbon phase with global 243 stoichiometry CH 8 between 170 and 220 GPa, and could be of importance in very hydrogen-rich 244 environments. is very similar to CH 4 (H 2 )-P2 1 /c (see Figure 6) and also (CH 4 ) 2 (H 2 ) 3 ; all three of these compounds 251 differ in the amount of H 2 molecules incorporated between the methane molecules. By comparison, 252 the methane sublattice of the new CH 4 (H 2 ) 2 phase is a distorted version of that found in CH 4 -P2 1 2 1 2 1 253 and CH 4 (H 2 )-P2 1 2 1 2 1 , such that it can accommodate an additional two hydrogen molecules in a new 254 vacancy channel present in the distorted network. is the most stable.
with a large enthalpy-pressure gradient and as such the phase boundary is essentially independent of shown in Figure 7), but CH 4 molecules persist in various hydrogen-rich van der Waals compounds 279 to much higher pressures. Amongst those we find (CH 4 ) 2 (H 2 ) 3 (overall CH 7 ) to be the most stable.

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For carbon-rich phases we find that longer hydrocarbon chains are stable to successively higher 285 pressures. Our data suggests there are no stable hydrocarbon phases on the carbon-rich side of CH 2 286 above 30 GPa so diamond formation should always be energetically favourable above that pressure in 287 samples with more than 33-at% carbon. immediately upon compression and/or heating due to partial decomposition of CH. Diamond 295 appearance in a sample of polystyrene stoichiometry [37] should not coincide with a complete 296 breakdown of hydrocarbons, i.e. the presence of only diamond and hydrogen. In fact, we find that with the final decomposition reaction curve, and not the initial decomposition curve [37].  of (a) CH composition relative to graphane, and (b) CH 2 composition relative to CH 2 -Cmcm. Lower panel: pressure -temperature phase diagrams for (c) CH and (d) CH 2 calculated using the harmonic approximation. Black lines indicate predicted transitions between labelled phases, the blue dashed line indicates the H 2 melt line as calculated by Bonev et. al. [61]. Shaded coloured areas represent the metastability region (within 10 meV/atom) of both graphane and polyethylene.
We can illustrate the same point on a more hydrogen-rich example, CH 2 , or polyethylene. In have a much larger enthalpy-pressure gradient, and the enthalpic drive towards decomposition of CH 2 316 increases strongly above 280 GPa; this should make diamond formation in polyethylene samples much 317 more likely above 300 GPa. The calculated phase diagram of CH 2 , shown in Figure 9(d), highlights the 318 large region of metastability for CH 2 (again defined as being unstable by less than 10 meV/atom) and 319 its rapid destabilisation beyond the decomposition into diamond and pure hydrogen. compounds between 70 and 150 GPa. At least four such compounds, with CH 4 :H 2 stoichiometries of 325 2:1, 1:1, 2:3, and 1:2, are competitive in this pressure range. This suggests that the decomposition of the 326 methane molecule is a very protracted process: some CH 4 molecules will react to form C 2 H 6 , and the 327 H 2 molecule created in this process will form a van der Waals compound with some of the remaining 328 CH 4 molecules. The ratio of "surviving" CH 4 molecules ranges from 50% (in a 2:1 inclusion compound) 329 to 20% (in a 1:2 inclusion compound). Since the different inclusion compounds are so close in free 330 energy, it is possible that a methane sample can form more than one of these compounds simultaneously  Methane's subsequent evolution, beyond the van der Waals compounds, is through hydrocarbon 341 chains of length 2 (C 2 H 6 -ethane), 4 (C 4 H 10 -butane) and then infinite (CH 2 ) -the latter seen in 342 ground state enthalpies only. It seems natural to expect a series of crystalline structures formed of 343 increasingly long hydrocarbon chains, i.e. C 6 H 14 -hexane, C 8 H 18 -octane and so on [18]. In their 344 solid state these would necessarily form increasingly large unit cells and be much harder to access by 345 unbiased structure searching [65]. For this reason and due to the small energy differences between the 346 competing phases, the phase transition between C 4 H 10 and CH 2 (or diamond) in the methane phase 347 diagram should serve as a substitute for a series of cascading hydrocarbon chain formation reactions.

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The hydrogen-rich inclusion compound CH 4 (H 2 ) 2 reported here, its less hydrogen-rich variant 349 CH 4 (H 2 )-P2 1 2 1 2 1 , and the CH 4 -P2 1 2 1 2 1 phase are all structurally similar and all stable or close to calculations suggest that any of the best pure methane phases are far from thermodynamically stable 364 above 100 GPa in contrast to experiment in which methane in its pure form is found in a cubic phase at 365 these pressures [21,22]. The high pressure chemistry of this process should be studied carefully, either 366 highlighting this as failure of DFT, a failure to predict the correct cubic structure of methane, or as 367 restrictions present in experiments.

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Experiments aiming for the formation of diamonds in compressed C-H mixtures have used the 369 decomposition curve of methane [37], similar to that found in Figure 7. A more appropriate phase 370 diagram for the experimental stoichiometry is shown here, in Figure 9c. The thermodynamics of CH 371 phases such as polystyrene imply excess carbon (diamond) production already at 25 GPa at room temperature and above, when longer hydrocarbons become more stable than graphane. Only parts of 373 the carbon atoms present are able to form diamond, and full decomposition occurs at higher pressures.

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Diamond nucleation rates depend heavily on the composition of the sample as well as the pressure 375 and temperature conditions [67]. It is certainly intriguing that shock experiments on a carbon-rich 376 sample such as polystyrene do not produce diamond until much higher pressures and temperatures 377 [37] when there is clear computational evidence for a non-zero amount of excess carbon in the system 378 following the formation of hydrocarbon chains. However, the free energy gain for these reactions is

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In conclusion, we present an internally consistent computational study of all known hydrocarbon 390 compounds, and discuss their stability as function of pressure, temperature, and composition, based 391 on semilocal DFT calculations and the harmonic approximation. In addition, we predict two new 392 methane-hydrogen van der Waals inclusion compounds that are relevant at high-pressure conditions.

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One of these (CH 8 , or CH 4 (H 2 ) 2 ) is the most hydrogen-rich hydrocarbon compound known, and 394 contains 20 wt-% releasable hydrogen, on par with the best known hydrogen storage materials.

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For methane, we present an updated P − T phase diagram, where the various van der Waals 396 compounds appear prominently in an intermediate pressure region, between 60 and 150 GPa. This 397 suggests that, purely on thermodynamic grounds, the decomposition of methane with increased 398 pressure is quite a protracted process with several intermediate stages. 399 We also present phase diagrams for more carbon-rich phases, CH 2 and CH, which have been 400 studied in recent dynamic compression experiments. These phase diagrams disagree with experimental 401 findings regarding diamond formation (similar to methane), but we explore possible explanations 402 around the metastability of the hydrocarbon phases and the lack of enthalpic gains unless full 403 decomposition into diamond and hydrogen is favourable.

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The interiors of icy planets are chemically very diverse, and a more realistic description requires 405 consideration of other molecular ices such as water and ammonia. In addition, the temperatures 406 that correspond to the pressure regime studied here will require studies beyond the harmonic 407 approximation. Nonetheless, this study presents a step forward in our understanding of hydrocarbon 408 diversity and their evolution with pressure, and enables follow-up studies to focus on relevant 409 hydrogen-carbon mixtures.

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The energies of the stable structures for the explored stoichiometries are presented in Figure A2. Figure A2. Convex hulls of low enthalpy structures found in the structure searches performed in this work at 100 and 200 GPa (black circles). Red circles denote enthalpies of hydrocarbon structures from the literature. Notable CH 4 (H 2 ) 2 structures P2 1 /c and the Ar(H 2 ) 2 inspired compound are indicated in green and blue, respectively. Of these, a star symbol indicates thermodynamic stability. 419 We defined the tetrahedrality measure Q in equation 1. Figure A3a shows the unit cell averages 420 Q for various phases that contain CH 4 molecules. At the highest pressure, the Q values range 421 from 0.999 to 0.988. To quantify the energy costs typical for such distortions, we created randomly 422 angularly distorted single methane molecules and calculated their energies in 10 × 10 × 10 Å boxes 423 with a Monkhorst-Pack k-point grid of 3 × 3 × 3. In Figure A3b we show the energy difference from the 424 ground state ∆E against the corresponding tetrahedrality measure Q . There is a strong linear trend 425 of increasing energy with increasing tetrahedral distortion. For example, CH 4 -Cmcm changes Q by 426 0.01 between 60 and 220 GPa, corresponding to a change in internal energy of around 50 meV/CH 4 .

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The equation of state data in figure 3 is fitted to Birch-Murnaghan equations of state of the form hydrocarbon content in the CH phase diagram as function of pressure, as shown in Figure A4. Firstly, Figure A3. Left stoichiometry. All values are normalised per unit of CH 4 . For example, the "Diamond + H 2 " volume V 0 corresponds to V 0 = V 0 (C-dia) + 2V 0 (H 2 ). Errors are associated with 90% confidence limits of fits.
at 40 GPa and T = 0 K, graphane (W hc = 100%) decomposes into 1/6 C 2 H 6 + 2/3 diamond, with 437 W hc = 33%. At the onset of formation of 1/10 C 4 H 10 + 3/5 diamond, W hc increases to 40%. Finally the 438 transition to diamond + hydrogen (W hc = 0%) occurs at 280 GPa. In the low temperature regime W hc 439 has a local minimum between 30 and 160 GPa, which corresponds to a local maximum in diamond