Effect of twin grain boundary on material thermal expansion

Based on first-principles phonon calculations within the framework of quasiharmonic approximation, we study the effect of twin grain boundary (TGB) on the thermal expansion of insulating materials. We select diamond-structure solids (C, Si, and Ge) and rutile $\left({\mathrm{SnO}}_2\right)$ as representatives of the covalent and ionic bonding materials that are important systems commonly accompanied with TGB. We found distinct effects of TGB on thermal expansion in two types of materials. For diamond-structure solids, the thermal expansion of (111)-oriented twinning structure varies subtly from that of pristine material within the temperature range studied, which is primarily induced by the modification of vibrational phonon mode by the TGB effect. Distinctly, the thermal expansion of $\mathrm{SnO}_2$ twinnings increase substantially by comparison with the pristine case and depend strongly on the twin orientation. The (101) twinning shows much larger thermal expansion than the (301) twinning. Further analysis indicates the physical mechanism can be mainly attributed to the effect of rotational degree of freedom of the $\mathrm{SnO}_6$ octahedron motif and the stress induced by TGB. Our work reveals the physical mechanism underlying the distinct effect of TGB on thermal expansion caused by different chemical bonding, and meanwhile provides an insightful understanding of the relationship between specific structural features and thermal expansion in solid-phase materials.

Figure 1

(a) Illustration of the twin grain boundary. The CTE ($\alpha$) dependent on temperature of (b) C/Si/Ge crystal/twinning, and (c) ${\mathrm{SnO}}_2$ crystal, (101) twinning, and (301) twinning. The solid (dotted or dashed) line marks the crystal (twinning) thermal expansion.

Figure 2

(a)–(c) Cumulative Grüneisen parameters at different frequencies at 300 K. (d) The vibrational modes of some specific phonons at $L$ (0, 0.5, 0.5) point around twin boundaries. (e) Schematic diagram of the vibrational modes. The tree modes in the red dashed box present the vibrational modes corresponding to positive thermal expansion; the two modes in the blue dashed box contribute to the negative thermal expansion.

Figure 3

(a) Calculated phonon dispersion of the ${\mathrm{SnO}}_2$ crystal. (b) The low-frequency (2.16 THz) negative vibrational mode at S ($-0.5$, 0.5, 0) point. Two insets on the bottom and right illustrate the connection and vibration patterns of polyhedrons. The corresponding mean distance between oxygen atoms ${\overline{d}}_{\text{O-O}}$ in the polyhedrons of (c) the crystal, (d) (101) twinning, and (e) (301) twinning. The lower panel in (d): the charge distribution profile, i.e., the averaged Bader charge results along the direction perpendicular to the twining plane of the (101) TGB.

Figure 4

Projected phonon densities of state of (a) crystal, (b) (101) twinning, and (c) (301) twinning. (d) The vibrational mode at the low-frequency peak (6.5 Thz) in the crystal. (e) The vibrational mode at the high-frequency peak ($\sim 23Thz$) in the (101) twinning.