Anisotropic thermal expansion in flexible materials

A definition of the Grüneisen parameters for anisotropic materials is derived based on the response of phonon frequencies to uniaxial stress perturbations. This Grüneisen model relates the thermal expansion in a given direction (${\alpha }_{ii}$) to one element of the elastic compliance tensor, which corresponds to the Young's modulus in that direction ($Y_{ii}$). The model is tested through ab initio prediction of thermal expansion in zinc, graphite, and calcite using density functional perturbation theory, indicating that it could lead to increased accuracy for structurally complex systems. The direct dependence of ${\alpha }_{ii}$ on $Y_{ii}$ suggests that materials which are flexible along their principal axes but rigid in other directions will generally display both positive and negative thermal expansion.

Figure 1

Crystal structures of materials with highly anisotropic thermal expansion and elastic properties used herein to test anisotropic Grüneisen models [34]. From left to right: zinc, graphite, and calcite. The $c$ axes are aligned vertically.

Figure 2

A comparison of the uniaxial strain and stress models for a simplified system with positive thermal expansion and positive Poisson ratio. Bonds colored in blue are lengthened by the perturbation, leading to a decrease in vibrational frequency and a positive contribution to the bulk Grüneisen parameter, while those colored in red are contracted by the Poisson effect, increasing their frequency and therefore reducing the Grüneisen parameter.

Figure 3

Linear thermal expansion in zinc and graphite along the $a$ (orange lines) and $c$ (blue lines) axes, as predicted by stress (solid lines) and strain (dash-dotted lines) models. Experimental data [24, 42] are shown as squares.

Figure 4

Directional Young's moduli (in GPa, left) and Poisson ratios (right) of calcite. The Young's modulus in a given direction is shown as a green surface. The surface corresponding to the maximum Poisson ratio is shown in blue, and the surface corresponding to the minimum Poisson ratio is shown in green. Visualization generated with elate [61, 62].

Figure 5

Linear thermal expansion in calcite along the $a$ (orange lines) and $c$ (blue lines) axes, as predicted by stress (solid lines) and strain (dash-dotted lines) models. Experimental data are shown as squares [28] and circles [44].

Figure 6

Phonon band structure of calcite, with bands colored according to their axial mode Grüneisen parameters calculated using stress and strain perturbations. Phonons with energies greater than 450 ${\mathrm{cm}}^{-1}$ do not contribute significantly to thermal expansion and are not shown here. The density of states $\rho$, weighted by the Grüneisen parameters as ${\sum }_{\mathbf{k}}{\rho }_{\mathbf{k}}\left(\omega \right){\gamma }_{n,\mathbf{k}}\left(\omega \right)$, is shown as a histogram at the right of each plot, with positive values in blue and negative values in red. Special points in and paths through the Brillouin zone were selected following Ref. [63].

Figure 7

A comparison of indicatrixes of directional thermal expansion and Young's moduli in several extremely anisotropic materials [1, 3, 28, 29, 44, 68]. Directional Young's moduli are shown as orange curves, while blue and red curves correspond to positive and negative linear coefficients of thermal expansion, respectively. The magnitudes of these indicatrixes are normalized for ease of comparison.