[Paper Review] Evaluation of crystal free energy with lattice dynamics

Tran Doan Huan|arXiv (Cornell University)|Jun 30, 2015

Microstructure and mechanical properties1 references4 citations

TL;DR

This paper presents a DFT-based lattice dynamics approach to compute the temperature-dependent free energy of crystals within the harmonic approximation, enabling thermodynamic stability analysis at finite temperatures. It demonstrates accurate prediction of phase transitions in ZrO₂ and HfO₂, while highlighting limitations in highly dynamic materials like KBH₄ due to anharmonicity and small energy differences.

ABSTRACT

Within the framework of density functional theory (DFT), the total energy of crystal structures is calculated at zero temperature. Herein, we briefly discuss the DFT-based lattice-dynamics approach for computing crystal free energy, the quantity needed in various non-zero-temperature contexts. We illustrate this well-established approach by examining the temperature-dependent thermodynamic stability of several crystalline materials, including ZrO$_2$, HfO$_2$, KBH$_4$, and Zn(BH$_4$)$_2$.

Motivation & Objective

• To enable finite-temperature thermodynamic analysis of crystalline materials using DFT-computed free energy.
• To assess the thermodynamic stability of crystal phases, including phase transitions, via free energy differences.
• To evaluate the reliability of the harmonic lattice dynamics approximation in materials with strong anharmonic effects.
• To investigate the feasibility of predicting synthesis feasibility for complex borohydrides using free energy changes.
• To identify limitations of the harmonic approximation in systems with large atomic displacements or rotational dynamics.

Proposed method

• Compute the static DFT total energy $U_0$ at the relaxed ionic configuration $\{\mathbf{r}_i^0\}$.
• Construct the force constant matrix from second derivatives of the DFT total energy with respect to atomic displacements.
• Diagonalize the dynamical matrix to obtain phonon frequencies $\omega_\nu$ using harmonic approximation.
• Calculate the vibrational free energy $F_{\text{vib}}(T)$ using the quantum harmonic oscillator partition function: $F_{\text{vib}}(T) = k_B T \sum_{\nu=1}^{3N} \ln \left[ 2 \sinh\left( \frac{\hbar \omega_\nu}{2k_B T} \right) \right]$.
• Combine $U_0$ and $F_{\text{vib}}(T)$ to obtain the Helmholtz free energy $F(T) = U_0 + F_{\text{vib}}(T)$, used to compare phase stability.
• Use VASP and phonopy for DFT and phonon calculations, and compute free energy differences to assess phase transitions and reaction spontaneity.

Experimental results

Research questions

• RQ1 How accurately can the DFT-lattice dynamics approach predict phase transition temperatures in zirconia and hafnia?
• RQ2 Why does the harmonic approximation fail to correctly predict the transition temperature in KBH₄, despite its success in other oxides?
• RQ3 To what extent does the small energy difference (~3 meV/atom) between low- and high-temperature phases of KBH₄ affect the reliability of free energy predictions?
• RQ4 Can the free energy change of a solid-state reaction like $2\text{NaBH}_4 + \text{ZnCl}_2 \to \text{Zn(BH}_4)_2 + 2\text{NaCl}$ be used to assess the thermodynamic feasibility of synthesizing Zn(BH₄)₂?
• RQ5 How do anharmonic effects, such as BH₄ group rotation, limit the applicability of the harmonic lattice dynamics approach in complex borohydrides?

Key findings

• The harmonic lattice dynamics approach correctly predicts phase transitions in ZrO₂ and HfO₂ at 1600 K and 2100 K, respectively, showing good agreement with experimental values.
• The predicted transition temperature for KBH₄ is ~550 K, significantly higher than the experimental value of ~197 K, indicating failure of the harmonic approximation in this system.
• The small energy difference (~3 meV/atom) between low- and high-temperature phases of KBH₄ suggests that the system is thermodynamically marginal, making free energy predictions highly sensitive to approximations.
• The free energy change $\Delta F(T)$ for the synthesis of Zn(BH₄)₂ via reaction (12) is negative up to 500 K, indicating thermodynamic feasibility under these conditions.
• The method's accuracy is compromised in highly dynamic materials like KBH₄ due to unaccounted anharmonic effects such as group rotations and large-amplitude motions.
• While the approach works well for oxides with small anharmonicities, it should be applied with caution to complex borohydrides where anharmonicity dominates.