High Temperature Entropy Research Articles

Investigation of Planckian behavior in a high-conductivity oxide: PdCrO2

The layered delafossite metal PdCrO[Formula: see text] is a natural heterostructure of highly conductive Pd layers Kondo coupled to localized spins in the adjacent Mott insulating CrO[Formula: see text] layers. At high temperatures, T, it has a T-linear resistivity which is not seen in the isostructural but nonmagnetic PdCoO[Formula: see text]. The strength of the Kondo coupling is known, as-grown crystals are extremely high purity and the Fermi surface is both very simple and experimentally known. It is therefore an ideal material platform in which to investigate "Planckian metal" physics. We do this by means of controlled introduction of point disorder, measurement of the thermal conductivity and Lorenz ratio, and studying the sources of its high-temperature entropy. The T-linear resistivity is seen to be due mainly to elastic scattering and to arise from a sum of several scattering mechanisms. Remarkably, this sum leads to a scattering rate within 10[Formula: see text] of the Planckian value of k[Formula: see text]T/[Formula: see text].

Predicting the vibrational contribution to thermal and EOS properties of [formula omitted] ([formula omitted]) intermetallic compounds

Two computational approaches to the vibrational part of thermal properties and equation-of-state parameters of the Ni3Sn(hP8), Ni3Sn(cP4), Ni3In(hP8) and Ni3In(cP4) intermetallic phases, of interest in the study of lead-free soldering materials, have been critically compared. The first approach, referred to as the “phonon method” (PHM) combines the zero-kelvin energy versus volume values obtained ab initio using the VASP, with an account of the vibrational frequencies provided by the PHONOPY code. The second method, the “quasi-harmonic Debye approach” (QDA), implemented in the GIBBS Program, introduces a volume-dependent Debye temperature θD(V). Extensive comparisons between the predictions of the PHM and the QDA are reported, which show significant discrepancies. In particular, it is found that the QDA systematically underestimates the entropy of the current phases. The discrepancy is explained in terms of the difference between θD(V), estimated by the GIBBS Program from the elastic properties, and the Debye temperature θD(0), which accounts for the high-temperature entropy, to be determined by the logarithmic average of all phonon frequencies. A discussion is also reported of the possible effect upon the discrepancies of varying the only parameter kept constant in the QDA calculations, viz., the Poisson ratio.

Thermodynamic picture of vitrification of water through complex specific heat and entropy: A journey through "no man's land".

We investigate thermodynamic properties of supercooled water across the "no man's land" onto the formation of amorphous ice. The calculations are aided by very long computer simulations, often more than 50 μs long, with the TIP4P/2005 model potential. Density fluctuations that arise from the proximity to a putative liquid-liquid (LL) transition at 228 K, cast a long shadow on the properties of water, both above and below the LL transition. We carry out the calculations of the quantum mechanical static and frequency-dependent specific heats by combining seminal studies of Lebowitz, Percus, and Verlet and Grest and Nagel with the harmonic approximation for the density of states. The obtained values are in quantitative agreement with all available experimental and numerical results of specific heats for both supercooled water and ice. We calculate the entropy at all the state points by integrating the specific heat. We find that the quantum corrected-contributions of intermolecular vibrational entropy dominate the excess entropy of amorphous phases over the crystal over a wide range of temperatures. Interestingly, the vibrational entropy lowers the Kauzmann temperature, TK, to 130 K, just below the experimental glass-to-liquid water transition temperature, Tg, of 136 K and the calculated Tg of 135 K in our previous study. A straightforward extrapolation of high temperature entropy from 250 K to below however would give a much higher value of TK ∼ 190 K. The calculation of Lindemann ratios shows the melting of amorphous ice ∼135 K. The amorphous state exhibits an extremely short correlation length for the distance dependence of orientational correlation.

Temperature-dependent stable phase domains of Zr–C–N (Zr–ZrC–ZrC0.96–Zr(C1−xNx)−ZrN, x = 0.1–0.9) systems from ab initio calculations and experimental results

Despite the importance of ZrC, ZrN, and their intermediate solid solutions, the temperature-dependent phase stability of the Zr–C–N system has not previously been determined precisely. This work reports these phase stabilities determined experimentally and using ab initio calculations within the quasi-harmonic approximation. Special quasi-random structure models and supercell models with random substitution were used to mimic the random atomic distribution of C and N in the sublattice of Zr(C1−xNx) solid solutions. The results indicated that volume and high-temperature entropy changes are important to determine the free energy of the solid solutions and that moderate temperatures (∼1500–2000 K) were needed to synthesize Zr(C1−xNx) solid solutions. The calculated stable phase domains were then successfully produced. Our results clarify the synthesis and stable phase prediction of the Zr–C–N system under certain external conditions, and they will help broaden the applicability of ZrC, ZrN, and their solid solution phases.

Synthetic Control to Achieve Lanthanide(III)/Pyrimidine-4,6-dicarboxylate Compounds by Preventing Oxalate Formation: Structural, Magnetic, and Luminescent Properties

Control over the synthetic conditions in many metal/diazinedicarboxylato systems is crucial to prevent oxalate formation, since dicarboxylato ligands easily undergo degradation in the presence of metal salts. We report here an efficient route to obtain oxalato-free compounds for the lanthanide/pyrimidine-4,6-dicarboxylato (pmdc) system on the basis of the reaction temperature and nonacidic pH or oxygen free atmosphere. Two different crystal architectures have been obtained: {[Ln(μ-pmdc)(1.5)(H(2)O)(3)]·xH(2)O}(n) (1-Ln) and {[Ln(2)(μ(4)-pmdc)(2)(μ-pmdc)(H(2)O)(2)]·H(2)O}(n) (2-Ln) with Ln(III) = La-Yb, except Pm. Both crystal structures are built from distorted two-dimensional honeycomb networks based on the recurrent double chelating mode established by the pmdc. In compounds 1-Ln, the tricapped trigonal prismatic coordination environment of the lanthanides is completed by three water molecules, precluding a further increase in the dimensionality. Crystallization water molecules are arranged in the interlamellar space, giving rise to highly flexible supramolecular clusters that are responsible for the modulation found in compound 1-Gd. Two of the coordinated water molecules are replaced by nonchelating carboxylate oxygen atoms of pmdc ligands in compounds 2-Ln, joining the metal-organic layers together and thus providing a compact three-dimensional network. The crystal structure of the compounds is governed by the competition between two opposing factors: the ionic size and the reaction temperature. The lanthanide contraction rejects the sterically hindered coordination geometries whereas high-temperature entropy driven desolvation pathway favors the release of solvent molecules leading to more compact frameworks. The characteristic luminescence of the Nd, Eu, and Tb centers is improved when moving from 1-Ln to 2-Ln compounds as a consequence of the decrease of the O-H oscillators. The magnetic properties of the compounds are dominated by the spin-orbit coupling and the ligand field perturbation, the exchange coupling being almost negligible.

Improved model for the transit entropy of monatomic liquids

In the original formulation of vibration-transit (V-T) theory for monatomic liquid dynamics, the transit contribution to entropy was taken to be a universal constant, calibrated to the constant-volume entropy of melting. This model suffers two deficiencies: (a) it does not account for experimental entropy differences of +/-2% among elemental liquids and (b) it implies a value of zero for the transit contribution to internal energy. The purpose of this paper is to correct these deficiencies. To this end, the V-T equation for entropy is fitted to an overall accuracy of +/-0.1% to the available experimental high-temperature entropy data for elemental liquids. The theory contains two nuclear motion contributions: (a) the dominant vibrational contribution S_{vib}(T/theta_{0}) , where T is temperature and theta_{0} is the vibrational characteristic temperature, and (b) the transit contribution S_{tr}(T/theta_{tr}) , where theta_{tr} is a scaling temperature for each liquid. The appearance of a common functional form of S_{tr} for all the liquids studied is a property of the experimental data, when analyzed via the V-T formula. The resulting S_{tr} implies the correct transit contribution to internal energy. The theoretical entropy of melting is derived in a single formula applying to normal and anomalous melting alike. An ab initio calculation of theta_{0} , based on density-functional theory, is reported for liquid Na and Cu. Comparison of these calculations with the above analysis of experimental entropy data provides verification of V-T theory. In view of the present results, techniques currently being applied in ab initio simulations of liquid properties can be employed to advantage in the further testing and development of V-T theory.

Calculation of debye temperature for crystalline structures—a case study on Ti, Zr, and Hf

The methods to calculate the Debye temperature from elastic moduli have been reviewed. The approximation approach due to Moruzzi et al. was critically examined by considering experimental elastic constant data for all the cubic elements. It was found that many cubic elements are exceptions with regard to the assumed constant scaling factor for the expression of the average sound velocity in terms of the bulk modulus, and consequently the Debye temperature of a cubic element must be calculated from the knowledge of all the elastic constants of the system. On the other hand, a fairly constant scaling factor has been found to exist for the hexagonal elements. Through the study of experimental data, some empirical relationships have been observed between the high temperature entropy–Debye temperature θ D (0) and the low temperature limit of the Debye temperature θ D (−3). For those structures that are dynamically unstable at low temperatures, we proposed a way to obtain their θ D (0) from the calculated isotropic bulk moduli. The methods have been applied to calculate the Debye temperatures of hcp, bcc, and fcc Ti, Zr, and Hf from their elastic moduli derived from ab initio calculations. The calculated results agree very well with the experimental data.

Thermal expansion of Ce3Bi4Pt3 at ambient and high pressures.

${\mathrm{Ce}}_{3}$${\mathrm{Bi}}_{4}$${\mathrm{Pt}}_{3}$ is a cerium compound that exhibits a hybridization gap and concomitant semiconducting behavior in the coherent ground state. Using neutron powder diffraction, we have determined the variation of the lattice constant with temperature at two pressures (P=${10}^{\mathrm{\ensuremath{-}}3}$ and 17.7 kbar), and determined the bulk modulus and atomic mean-square displacements for ${\mathrm{Ce}}_{3}$${\mathrm{Bi}}_{4}$${\mathrm{Pt}}_{3}$ and its normal analog, ${\mathrm{La}}_{3}$${\mathrm{Bi}}_{4}$${\mathrm{Pt}}_{3}$. The thermal expansion \ensuremath{\Delta}\ensuremath{\beta} in ${\mathrm{Ce}}_{3}$${\mathrm{Bi}}_{4}$${\mathrm{Pt}}_{3}$ exhibits a maximum at ${\mathit{T}}_{\mathrm{max}}$=50 K at ambient pressure. Below 50 K, the bulk modulus \ensuremath{\Delta}B is proportional to T\ensuremath{\Delta}\ensuremath{\beta}. We apply a Gr\"uneisen analysis, under the assumption that the free energy exhibits T/${\mathit{T}}_{0}$(V) scaling. The Gr\"uneisen parameter deduced from the ratio \ensuremath{\Delta}B/T\ensuremath{\Delta}\ensuremath{\beta} is consistent with the value \ensuremath{\Omega}=36 deduced from the pressure variation of ${\mathit{T}}_{\mathrm{max}}$. The analysis allows us to predict the temperature dependence of the 4f specific heat (\ensuremath{\Delta}C\ensuremath{\propto}\ensuremath{\Delta}\ensuremath{\beta}/\ensuremath{\Omega}), which thus has a maximum at 50 K and a high-temperature entropy nearly equal the expected value R ln6. We argue that the maximum at 50 K reflects an (indirect) hybridization gap of order 100 K. We show further that the lattice constant anomaly \ensuremath{\Delta}${\mathit{a}}_{0}$ is proportional to the effective moment T\ensuremath{\chi}; this suggests that a relationship known to be valid for antiferromagnets, namely \ensuremath{\partial}(T\ensuremath{\chi})/\ensuremath{\partial}T\ensuremath{\propto}\ensuremath{\Delta}C where \ensuremath{\Delta}C is the specific heat, may be valid for ${\mathrm{Ce}}_{3}$${\mathrm{Bi}}_{4}$${\mathrm{Pt}}_{3}$. Finally, we show that the temperature dependence of the mean-square atomic displacements is nearly identical for ${\mathrm{Ce}}_{3}$${\mathrm{Bi}}_{4}$${\mathrm{Pt}}_{3}$ and ${\mathrm{La}}_{3}$${\mathrm{Bi}}_{4}$${\mathrm{Pt}}_{3}$ and can be fitted by a simple Debye-Waller model; hence the expansion anomaly does not affect the average lattice dynamics.

A generic model for the ionic contribution to the equation of state

Abstract We describe an extended standard model that yields the thermodynamics of the ionic contribution for general materials, from the low temperature solid region, through melting, to the ideal gas limit. We use the Debye model for the solid. Melting is determined by the Lindemann formula with standard rules of thumb used to determine density and energy discontinuities. The model interpolates through the liquid regime to the ideal gas assuming that the specific heat drops monotonically from about 3R at melting, to 9R/4 at five times melting, and continuing to 3R/2 at high temperatures. The area under the specific heat curve is constrained in the model to reproduce the correct high temperature entropy. Thus, for a compound the extra contribution from the entropy of mixing forces into the model, in a crude way, the extra specific heat due to dissociation.

An analysis of the high-temperature entropy of transition metals

The high-temperature entropy at constant pressure, obtained from heat capacity data, is analyzed for Mo, Nb, Pd, Rh, Ru, Sc, Ti, V, W, Y, and Zr. The vibrational contribution to the entropy defines an entropy Debye temperature, θ S(T). Due to anharmonicity, θ S(T) decreases with the temperature T, typically by 10–20% between room temperature and the melting point. An unusual temperature dependence of θ S(T) may indicate inadequacies in the underlying data. On this ground, the C p data recommended by Hultgren et al. (1973) for W and Ru seem doubtful. A comparison is made between θ S and the elastic-limit Debye temperature θ D for the metals with a cubic lattice structure. θ S of Nb is found to be almost independent of T, a feature related to the fact that the elastic constant c 44 increases with T.

Vertex models for the antiferroelectric transition in squaric acid

We consider two-dimensional 4- and 12-vertex models on a square lattice for the phase transition in the layered hydrogen-bonded antiferroelectric crystals H2C4O4 and its deuterated isomorph. An exact solution for the 4-vertex model, which strictly obeys the ’’ice-rules’’, leads to a vanishing free-energy. The 12-vertex model, which takes into account the single ionized configurations, gives a second order phase transition in the framework of the Bethe approximation. The high-temperature entropy of the 12-vertex model is calculated exactly and agrees with the value predicted by the Bethe approximation. An exact calculation of the partition function for a modified 12-vertex model, with a more degenerate ground state, leads to a well behaved thermodynamic potential. Also, we add a mean-field interlayer interaction to the 12-vertex model for predicting the correct low-temperature ordering in squaric acid. Some thermodynamic quantities are explicitly calculated for this model.

Calorimetric Study of Glassy State. IX. Thermodynamic Properties of Stannous Chloride Dihydrate and Dideuterate Crystals

Abstract The heat capacity of SnCl2·2H2O and SnCl2·2D2O crystals, substances with layer structure, was measured from 13 to 300 K. Anomalies due to ordering of the hydrogen (deuterium) position were observed at TC(H2O)=217.94 K and at Tc(D2O)=234.47±0.05 K. The total entropy change of the transition was estimated to be 4.6 J K−1 mol−1 for both compounds. An exact calculation of the high temperature entropy based on the two-dimensional lattice model gave 3.13 J K−1 mol−1. The critical indices α± of the divergent heat capacity anomalies are as follows: α+(H2O)=0.53, α−H2O)=0.48, α+(D2O)=0.57, α−(D2O)=0.54. Another heat capacity anomaly was found at 140–160 K where the heat capacity values depended on the thermal history of the specimen. The enthalpy relaxation observed at the temperature region was analysed by use of the first-order rate equation, giving the activation enthalpies ΔHa(H2O)=49.9 kJ mol−1 and ΔHa(D2O)=49.8 kJ mol−1. The concept of long-range-ordered glassy state was introduced for description of the non-equilibrium state of the present substances below 140 K. A general consideration of the relation between molecular relaxation and phase change in terms of the Deborah number is presented.

High-Temperature Entropy and Specific Heat of Interacting Electrons in a Solid

The virial expansion for a fermion system is employed to derive an expression for the high-temperature entropy of electrons which interact according to the Hubbard Hamiltonian. The results are applied to an itinerant electron ferromagnet at temperatures well above the Curie temperature. In contrast with the results of the Stoner theory, a nonvanishing interaction contribution to the entropy and specific heat is obtained above ${T}_{C}$.

Estimation of High Temperature Entropy of Solids by Entropy Correspondence Principle

Estimation of High Temperature Entropy of Solids by Entropy Correspondence Principle