Vibrational entropy of ordered and disordered Ni3Al

An Unexpected Decrease in Vibrational Entropy of Multicomponent Rutile Oxides

High-entropy oxides (HEOs), featuring infinite chemical composition and exceptional physicochemical properties, are attracting much attention. The configurational entropy caused by a component disorder of HEOs is popularly believed to be the main driving force for thermal stability, while the role of vibrational entropy in the thermodynamic landscape has been neglected. In this study, we systematically investigated the vibrational entropy of multicomponent rutile oxides (including Fe0.5Ta0.5O2, Fe0.333Ti0.333Ta0.333O2, Fe0.25Ti0.25Ta0.25Sn0.25O2, and Fe0.21Ti0.21Ta0.21Sn0.21Ge0.16O2) by precise heat capacity measurements. It is found that vibrational entropy gradually decreases with increasing component disorder, beyond what one could expect from an equilibrium thermodynamics perspective. Moreover, all multicomponent rutile oxides exhibit a positive excess vibrational entropy at 298.15 K. Upon examinations of configuration disorder, size mismatch, phase transition, and polyhedral distortions, we demonstrate that the excess vibrational entropy plays a pivotal role in lowering the crystallization temperature of multicomponent rutile oxides. These findings represent the first experimental confirmation of the role of lattice vibrations in the thermodynamic landscape of rutile HEOs. In particular, vibrational entropy could serve as a novel descriptor to guide the predictive design of multicomponent oxide materials.

Vibrational entropy of disordering in omphacite

The cations of an ordered omphacite from the Tauern window were gradually disordered in piston cylinder experiments at temperatures between 850 and 1150 °C. The samples were examined by X-ray powder diffraction and then investigated using low-temperature calorimetry and IR spectroscopy. The low-temperature heat capacity data were used to obtain the vibrational entropies, and the line broadening of the IR spectra served as a tool to investigate the disordering enthalpy. These data were then used to calculate the configurational entropy as a function of temperature. The vibrational entropy does not change during the cation ordering phase transition from space group C2/c to P2/n at 865 °C but increases with a further temperature increase due to the reduction of short-range order.

Supplementary Information

The online version contains supplementary material available at 10.1007/s00269-023-01260-7.

Entropy Landscaping of High-Entropy Carbides

The entropy landscape of high-entropy carbides can be used to understand and predict their structure, properties, and stability. Using first principles calculations, the individual and temperature-dependent contributions of vibrational, electronic, and configurational entropies are analyzed, and compare them qualitatively to the enthalpies of mixing. As an experimental complement, high-entropy carbide thin films are synthesized with high power impulse magnetron sputtering to assess structure and properties. All compositions can be stabilized in the single-phase state despite finite positive, and in some cases substantial, enthalpies of mixing. Density functional theory calculations reveal that configurational entropy dominates the free energy landscape and compensates for the enthalpic penalty, whereas the vibrational and electronic entropies offer negligible contributions. The calculations predict that in many compositions, the single-phase state becomes stable at extremely high temperatures (>3000 K). Consequently, rapid quenching rates are needed to preserve solubility at room temperature and facilitate physical characterization. Physical vapor deposition provides this experimental validation opportunity. The computation/experimental data set generated in this work identifies "valence electron concentration" as an effective descriptor to predict structural and thermodynamic properties of multicomponent carbides and educate new formulation selections.

Lattice Vibrations Change the Solid Solubility of an Alloy at High Temperatures

We develop a method to accurately and efficiently determine the vibrational free energy as a function of temperature and volume for substitutional alloys from first principles. Taking Ti_{1-x}Al_{x}N alloy as a model system, we calculate the isostructural phase diagram by finding the global minimum of the free energy corresponding to the true equilibrium state of the system. We demonstrate that the vibrational contribution including anharmonicity and temperature dependence of the mixing enthalpy have a decisive impact on the calculated phase diagram of a Ti_{1-x}Al_{x}N alloy, lowering the maximum temperature for the miscibility gap from 6560 to 2860 K. Our local chemical composition measurements on thermally aged Ti_{0.5}Al_{0.5}N alloys agree with the calculated phase diagram.

The vibrational and configurational entropy of α-brass

The heat capacities of two samples of a fcc Cu-Zn alloy with the composition CuZn15 and CuZn34 were measured from T = 5 K to 573 K using relaxation and differential scanning calorimetry. Below ∼90 K, they are characterised by negative excess heat capacities deviating from ideal mixing by up to -0.20 and -0.44 J · mol-1 · K-1 for CuZn15 and CuZn34, respectively. The excess heat capacities produce excess vibrational entropies, which are less negative compared to the excess entropy available from the literature. Since the literature entropy data contain both, the configurational and the vibrational part of the entropy, the difference is attributed to the excess configurational entropy. The thermodynamics of different short-range ordered samples was also investigated. The extent of the short-range order had no influence on the heat capacity below T = 300 K. Above T = 300 K, where the ordering changed during the measurement, the heat capacity depended strongly on the thermal history of the samples. From these data, the heat and entropy of ordering was calculated. The results on the vibrational entropy of this study were also used to test a relationship for estimating the excess vibrational entropy of mixing.

Positive vibrational entropy of chemical ordering in FeV

Inelastic neutron scattering and nuclear resonant inelastic x-ray scattering were used to measure phonon spectra of FeV as a B2 ordered compound and as a bcc solid solution. The two data sets were combined to give an accurate phonon density of states, and the phonon partial densities of states for V and Fe atoms. Contrary to the behavior of ordering alloys studied to date, the phonons in the B2 ordered phase are softer than in the solid solution. Ordering increases the vibrational entropy by +0.22±0.03 kB/atom, which stabilizes the ordered phase to higher temperatures. First-principles calculations show that the number of electronic states at the Fermi level increases upon ordering, enhancing the screening between ions, and reducing the interatomic force constants. The effect of screening is larger at the V atomic sites than at the Fe atomic sites.

Structural, vibrational and thermodynamic properties of Ag(n)Cu(34-n) nanoparticles

We report results of a systematic study of structural, vibrational and thermodynamical properties of 34-atom bimetallic nanoparticles from the Ag(n)Cu(34-n) family using model interaction potentials as derived from the embedded atom method and invoking the harmonic approximation of lattice dynamics. Systematic trends in the bond length and dynamical properties can be explained largely from arguments based on local coordination and elemental environment. Thus an increase in the number of silver atoms in a given neighborhood introduces a monotonic increase in bond length, while an increase of the copper content does the reverse. Moreover, for the bond lengths of the lowest-coordinated (six and eight) copper atoms with their nearest neighbors (Cu atoms), we find that the nanoparticles divide into two groups with the average bond length either close to (∼2.58 Å) or smaller than (∼2.48 Å) that in bulk copper, accompanied by characteristic features in their vibrational density of states. For the entire set of nanoparticles, we find vibrational modes above the bulk bands of copper/silver. We trace a blue shift in the high-frequency end of the spectrum that occurs as the number of copper atoms increases in the nanoparticles, leading to shrinkage of the bond lengths from those in the bulk. The vibrational densities of states at the low-frequency end of the spectrum scale linearly with frequency as for single-element nanoparticles, with a more pronounced effect for these nanoalloys. The Debye temperature is found to be about one-third of that of the bulk for pure copper and silver nanoparticles, with a non-linear increase as copper atoms increase in the nanoalloy.

Charge redistribution and phonon entropy of vanadium alloys

The effects of alloying on the lattice dynamics of vanadium were investigated using inelastic neutron scattering. Phonon densities of states were obtained for bcc solid solutions of V with 3d, 4d, and 5d transition metal solutes, from which vibrational entropies of alloying were obtained. A good correlation is found between the vibrational entropy of alloying and the electronegativity of transition metal solutes across the 3d row and down columns of the periodic table. First-principles calculations on supercells matching the experimental compositions predicted a systematic charge redistribution in the nearest-neighbor shell around the solute atoms, also following the Pauling and Watson electronegativity scales. The systematic stiffening of the phonons is interpreted in terms of the modified screening properties of the electron density around the solutes.

Negative entropy of mixing for vanadium-platinum solutions

The phonon densities of states for pure vanadium and the solid solutions V-6.25% Ni, Pd, Pt were determined from inelastic neutron scattering measurements. The solute atoms caused a large stiffening of the phonons, resulting in large, negative vibrational entropies of mixing. For V-6.25%Pt, the negative vibrational entropy of mixing exceeds the conventional positive chemical entropy of mixing. This negative total entropy of mixing should extend to lower concentrations of Pt, and the effect on the bcc solvus line is discussed. The experimental data were inverted to obtain interatomic force constants by using a Born-von Kármán model with an iterative optimization algorithm. The stiffening of bonds responsible for the decrease of entropy was found to occur mainly in first-nearest-neighbor solute-host bonds, and correlates in part with the solute metallic radius.

Entropically favored ordering: the metallurgy of Al(2)Cu revisited

The famous sequence of precipitates which form upon heat treating Al-Cu is part of nearly every metallurgical textbook. Numerous precipitation (and other) experiments have led to a long-standing belief that the energetic ground state of Al(2)Cu is the theta phase. Modern first-principles calculations at T = 0 K surprisingly predict the energy of the observed Al(2)Cu-theta phase to be higher than that of its metastable counterpart, theta('). We show that vibrational entropy reverses this energetic preference at T approximately 150--200 degrees C, resolves the apparent discrepancy between theory and experiment, and hence plays a critical (but previously unsuspected) role in the precipitation sequence.

Large harmonic softening of the phonon density of states of uranium

Phonon density-of-states curves were obtained from inelastic neutron scattering spectra from the three crystalline phases of uranium at temperatures from 50 to 1213 K. The alpha-phase showed an unusually large thermal softening of phonon frequencies. Analysis of the vibrational power spectrum showed that this phonon softening originates with the softening of a harmonic solid, as opposed to vibrations in anharmonic potentials. It follows that thermal excitations of electronic states are more significant thermodynamically than are the classical volume effects. For the alpha-beta and beta-gamma phase transitions, vibrational and electronic entropies were comparable.

Large vibrational effects upon calculated phase boundaries in Al-Sc

The fcc portion of the Al-Sc phase diagram is calculated from first principles including contributions to alloy free energies associated with ionic vibrations. It is found that vibrational entropy accounts for a 27-fold increase in the calculated solubility limits for Sc in fcc Al at high temperatures, bringing calculated and measured values into very good agreement. The present work gives a clear example demonstrating a large effect of vibrational entropy upon calculated phase boundaries in substitutional alloys.