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Vibrational Entropy Investigation in High Entropy Alloys

The lattice dynamics for NiCo, NiFe, NiFeCo, NiFeCoCr, and NiFeCoCrMn medium to high entropy alloy have been investigated using the DFT calculation. The phonon dispersions along three different symmetry directions are calculated by the weighted dynamical matrix (WDM) approach and compared with the supercell approach and inelastic neutron scattering. We could correctly predict the trend of increasing of the vibrational entropy by adding the alloys and the highest vibrational entropy in NiFeCoCrMn high entropy alloy by WDM approach. The averaged first nearest neighbor (1NN) force constants between various pairs of atoms in these intermetallic are obtained from the WDM approach. The results are discussed based on the analysis of these data.


Introduction
Alloys attracted a lot of attention because of their unique ability to change the physical properties of materials [1][2][3][4] . When there are several components (5 or more) participating in the alloying process, the resulting single-phase solid-solution alloy might be called high entropy alloy (HEA) if the high configurational entropy which is found responsible for the stability of the single solid solution phase over competing intermetallic and elemental phases. However, there are several other contributions in the Gibbs free energy, which determines phase stabilities. The aim of the new concept in the development of new alloy systems with novel mechanical and functional properties 5,6 . Therefore, the role of entropy contributions investigation is vital. Many types of single-phase HEAs have been developed with different crystal structures such as face-centered cubic (FCC) 7 , body-centered cubic (BCC), and hexagonal close-packed (HCP) 8 .
In this work, we employ ab initio-based simulations to investigate the vibrational entropy which is one of the entropy contributions (such as vibrational, electronic, magnetic as well as configurational) for NiCo, NiFe, NiFeCo, NiFeCoCr, and NiFeCoCrMn. NiCo, NiFe, and NiFeCo.
All of the alloys are made of transition metals, and they ave strong force-constant disorder but minimal mass disorder. The contributing elements are isoelectronic and selected from the same row of the periodic table (either 3d or 4d) and have a similar atomic size. We made a comparison between WDM, Supercell approach, and experimental results on the phonon dispersion. Then, we predicted the maximum vibrational entropy for NiFeCo, NiFeCoCrMn.

Computational Details
The density functional theory is used for performing the calculations 9,10 with a plane-wave basis set, as implemented in the Quantum-Espresso 11 . We used the Perdew-Burke-Ernzerhof generalized gradient approximation exchange-correlation functional 12,13 and Optimized Norm-Conserving Vanderbilt

Weighted Dynamical Matrix (WDM) Approach
To find the phonon modes, one needs to construct the dynamical matrix. To do so, The deviation of the i th atom in the l th unit cell of the Supercell from equilibrium position is introduced. Also, the second-order force constants for the − ′ ′ atom pair denoted as Φ (0 , ′ ′ ) (with α and β indices for coordinates) are calculated. Then, the dynamical matrix D( ) at the wavevector is constructed as follows: where is the mass of the i th atom. Phonon frequencies ( , ) and mode eigenvectors ( , ) at , where is the band index, are obtained by solving the eigenvalue equation: To calculate the phonon modes for these alloy samples, first, the Hellmann-Feynman forces of the parent structures (Cu and Au), for the atomic displacement of 0.01 Å from their equilibrium positions, are calculated by DFT calculations. Then from forces ⃗ and ⃗ the force constants Φ and Φ are calculated. The weighted dynamical matrix is constructed as follows: See more details on the WDM approach in references [23][24][25] .

Results and Discussion
We used the WDM approach to calculate the vibrational properties for NiCo, NiFe, and NiFeCo, which are shown in Figure 1  respectively.
The first nearest-neighbor force constants, which shown in Table 1, are an order of magnitude larger than those of the further neighbors, so we used them to comparison our results and the behavior of force constant versus bond distance follows the expected trend

Conclusions
We have investigated the lattice dynamics NiCo, NiFe, NiFeCo, NiFeCoCr, and NiFeCoCrMn medium to high entropy alloys using first-principle density functional theory. The agreement between WDM, supercell approach, and neutron scattering experiments is good, but our results overestimate frequencies for NiFe. That might be due to the lack of exchangecorrelation function and averaging of force-constant. This is important from the point of view of the feasibility of using ab initio ordered alloy force constants to study the disordered high entropy alloys and correctly predict the highest vibrational entropy of them, which are Ni5Fe5Co90 and