A One-Dimensional Reduction Method for Calculating Thermal Expansion in Solids: Application to Orthorhombic Systems

Abstract

Anisotropic thermal expansion plays a critical role in the performance and reliability of functional materials, yet its theoretical description remains limited. Here, a computational framework that reduces the calculation of thermal expansion in solids to an effective one-dimensional problem is presented and applied to orthorhombic lattice. Using this method, a comprehensive set of thermodynamic and mechanical properties is determined.

Figures (11)

• Figure 1: Schematic representation of the unit cell of the orthorhombic Mo$_2$C phase, where Mo atoms at ($8d$) sites are shown in blue, and C atoms at ($4c$) sites are shown in red
• Figure 2: Graphs of the change in the total energy $\Delta E = E_{tot}(V_0,\delta_i) - E_{tot}(V_0,0)$, obtained for different distortion matrices D1 - D9 as functions of the applied deformations ($\delta$) of the crystal lattice of $\alpha$-Mo$_2$C, calculated for the ground state. Dashed lines are polynomial approximations.
• Figure 3: Scheme for determining thermal expansion paths of orthorhombic $\alpha-$Mo$_2$C. The lattice parameters are plotted along the XYZ axes. The point V$_0$ = (a$_0$, b$_0$, c$_0$) corresponds to the optimized equilibrium volume at T = 0 K obtained from first-principles calculations. The red arrow indicates the isotropic thermal expansion/contraction direction, lying in the red plane and forming an angle $\theta$ with the OZ axis. The angles $\theta$ and $\phi$ correspond to latitude and longitude, respectively. The blue arrow represents the predicted thermal expansion direction. Multiple candidate expansion paths within the same blue plane are shown by colored arrows, and their projections onto the XOY plane are displayed below.
• Figure 4: Projection of the $\beta = 60^\circ$ plane onto the XOY plane. The intersection point of all paths (30, 20, 10, 0, -10, -20, ..., -60) corresponds to the projection of $V_0$ with optimized lattice parameters ($a_0$, $b_0$, $c_0$) of orthorhombic $\alpha$-Mo$_2$C. The red dashed line (0) denotes the pseudo-isotropic thermal expansion path, where only the ${a/b}$ ratio is constant. The blue solid line (-30) represents the projection of the predicted thermal expansion path. Projections of experimental and calculated lattice parameters reported in previous studies are included for comparison.
• Figure 5: Total energies $E_{\mathrm{tot}}(V)$ of orthorhombic $\alpha$-Mo$_2$C calculated from first principles as functions of volume along the candidate thermal expansion paths (30°, 20°, 10°, 0°, -10°, -20°, …, -60°) within the $\beta$-plane.