Thermal and mechanical properties of the clathrate-II Na₂₄Si₁₃₆

Thermal expansion, lattice dynamics, heat capacity, compressibility, and pressure stability of the intermetallic clathrate Na₂₄Si₁₃₆ have been investigated by a combination of first-principles calculations and experimentation. Direct comparison of the properties of Na₂₄Si₁₃₆ with those of the low-density elemental modification Si₁₃₆ provide insight into the effects of filling the silicon clathrate framework cages with Na on these properties. Calculations of the phonon dispersion only yield sensible results if the Na atoms in the large cages of the structure are displaced from the cage centers, but the exact nature of off-centering is difficult to elucidate conclusively. Pronounced peaks in the calculated phonon density of states for Na₂₄Si₁₃₆, absent for Si₁₃₆, reflect the presence of low-energy vibrational modes associated with the guest atoms, in agreement with prior inelastic neutron-scattering experiments and reflected in marked temperature dependence of the guest atom atomic displacement parameters determined by single-crystal x-ray diffraction. The bulk modulus is only weakly influenced by filling the Si framework cages with Na, whereas the phase stability under pressure is significantly enhanced. The room-temperature linear coefficient of thermal expansion (CTE) is nearly a factor of 3 greater for Na₂₄Si₁₃₆ compared to Si₁₃₆. Negative thermal expansion (NTE), observed in Si₁₃₆ below 100 K, is noticeably absent in Na₂₄Si₁₃₆. In contrast to Si₁₃₆, the thermal expansion behavior in Na₂₄Si₁₃₆ is relatively well described by the conventional Grüneisen-Debye model in the temperature range of 10-700 K. First-principles calculations in the quasiharmonic approximation correctly predict an increase in high-temperature CTE with Na loading, although the increase is less than observed in experiment. The calculations also fail to capture the absence of NTE in Na₂₄Si₁₃₆, perhaps due to anharmonic effects and/or inadequateness of the ordered structural model.