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Frustrated supermolecules: the high-pressure phases of crystalline methane

Marcin Kirsz, Miguel Martinez-Canales, Ayobami D. Daramola, John S. Loveday, Ciprian G. Pruteanu, Graeme J Ackland

Abstract

Methane is the simplest hydrocarbon, yet it exhibits an extraordinarily complicated series of crystal phases. Notably, the non-plastic phases have large unit cells with nearly, but not quite cubic symmetry. Furthermore, although non-polar molecules interact very weakly, their reorganisation across phase transitions is very sluggish. Here, we demonstrate that these complex structures can be understood as simple packing of near-spherical supermolecular clusters of methane molecules: the departure from cubic symmetry arising from the non-spherical nature of the molecules. We use molecular dynamics based on density functional theory calculations to simulate the finite-temperature crystal structures of methane, finding that the complex Phase A is based around a 13-molecule regular icosahedron, with 8 additional molecules forming the 21-molecule unit cell. Similarly, Phase B is based on a body-centred cubic bcc packing of 17-molecule Z16 polyhedra, with the remaining 12 molecules per cell in tetrahedral interstices. We demonstrate that the favored intermolecular separation depends sensitively on molecular orientation, leading to hindered rotation and suppressed entropy. The structures are determined by a trade-off between efficient packing and entropy.

Frustrated supermolecules: the high-pressure phases of crystalline methane

Abstract

Methane is the simplest hydrocarbon, yet it exhibits an extraordinarily complicated series of crystal phases. Notably, the non-plastic phases have large unit cells with nearly, but not quite cubic symmetry. Furthermore, although non-polar molecules interact very weakly, their reorganisation across phase transitions is very sluggish. Here, we demonstrate that these complex structures can be understood as simple packing of near-spherical supermolecular clusters of methane molecules: the departure from cubic symmetry arising from the non-spherical nature of the molecules. We use molecular dynamics based on density functional theory calculations to simulate the finite-temperature crystal structures of methane, finding that the complex Phase A is based around a 13-molecule regular icosahedron, with 8 additional molecules forming the 21-molecule unit cell. Similarly, Phase B is based on a body-centred cubic bcc packing of 17-molecule Z16 polyhedra, with the remaining 12 molecules per cell in tetrahedral interstices. We demonstrate that the favored intermolecular separation depends sensitively on molecular orientation, leading to hindered rotation and suppressed entropy. The structures are determined by a trade-off between efficient packing and entropy.
Paper Structure (19 sections, 1 equation, 8 figures, 3 tables)

This paper contains 19 sections, 1 equation, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Molecular Dynamics
  4. X-ray and Neutron Diffraction Patterns
  5. Rotational Free Energy
  6. Crystal Structure and Symmetry Analysis
  7. Results
  8. Phase I
  9. Phase A
  10. Phase A Simulations
  11. Phase A Structure
  12. Phase A Simulated Diffraction Patterns
  13. Phase B and HP
  14. Phase B and HP Structure Molecular Orientations
  15. Bond Enthalpies, Energies and Frustration
  16. ...and 4 more sections

Figures (8)

  • Figure 1: (Left) Mean squared displacement (MSD) of CH$_4$ in phases I, A, B and HP. For each phase, C- and H-only MSDs are shown. The high mobility of the H atoms compared to the nearly stationary C atoms is characteristic of plastic solid phases. (Right) Partial radial distribution functions for C--C, H--H, and C--H pairs.
  • Figure 2: Top left and bottom left: site--site RDFs and ADFs for C--C pairs in CH$_4$ Phase I ($Fm\overline{3}m$, single Wyckoff site a). RDF and ADF curves computed separately for each symmetrically equivalent carbon (all on the a site) lie on top of one another, coinciding with the corresponding ensemble-averaged distributions and confirming the expected $fcc$ positional order. Top right: Mollweide projection of the ensemble-averaged orientational PDF, showing pronounced preferred orientations and avoidance of eight distinct directions; the corresponding rotational entropy is $S^\mathrm{rot} = 0.78$ per molecule, indicating that the CH$_4$ molecules are not free rotors. Bottom right: the same PDF, smoothed with a Gaussian kernel with $\mathrm{FWHM} = 10^\circ$ and labelled with the cubic crystallographic directions. C--H bonds avoid the cube body-diagonal $\langle 111 \rangle$ directions, so C--H--C contacts with the 12 nearest neighbours are formed and broken along preferred, off-diagonal orientational pathways. This behaviour is consistent with CH$_4$ molecules acting as hindered rotors on the $fcc$ lattice sites, rather than freely sampling all orientations.
  • Figure 3: C--C radial and angular distribution functions showing the five environments (Wyckoff sites) of molecules in methane Phase A simulations. The rotating molecule on the 1b site has the largest near-neighbour separations. Molecule in the 1b site is hindered-rotor like; one 6f site, labelled f1, has notably higher entropy than the remaining f-sites. Note that PDF (right) colour scale is different than in Fig. \ref{['fig:phaseI']}. None of the molecules can be considered as "static".
  • Figure 4: Phase A at 8 GPa and 300 K, based on 50 uncorrelated frames from BOMD. (Left) Neutron diffraction simulation (all H replaced by D) for run with a triclinic cell. (Middle) Measured neutron diffraction pattern by Maynard-Casely et al.maynard2010. Red circles mark diamond peaks coming from the anvils of the pressure cell. (Right) Simulated neutron diffraction for run with a rhombohedral cell.
  • Figure 5: Phase B (top row) and HP (bottom row). The symmetry breaking in B-HP relates $\text{2a} \rightarrow \text{a2}$; $\text{8c} \rightarrow \text{a1, b7}$; $\text{24g1} \rightarrow \text{b1, b2, b6, b9}$; $\text{24g2} \rightarrow \text{b3, b4, b5, b8}$. Phase B molecules on the c and g1 sites show some reorientational behavior, other sites are largely fixed.
  • ...and 3 more figures