The entropy of lithiation of Li x CoO 2 for 0.5Ͻ x ഛ 1.0 was determined from measurements of the temperature dependence of equilibrated voltages of electrochemical cells. Measured changes in the entropy of the lithiation reaction were as large as 9.0 k B / atom, and as large as 4.2 k B / atom within the "O3" layered hexagonal structure of Li x CoO 2 . Three contributions to the entropy of lithiation for the O3 phase were assessed by experiment and calculation. The phonon entropy of lithiation was determined from measurements of inelastic neutron scattering. Phonon entropy can account for much of the negative entropy of lithiation, but its changes with lithium concentration were found to be small. Electronic structure calculations in the local density approximation gave a small electronic entropy of lithiation of the O3 phase. The configurational entropy from lithium-vacancy disorder was large enough to account for most of the compositional trend in the entropy of lithiation of the O3 phase if ordered structures exist at lithium concentrations of x =1/2 and x =5/6. The electrochemical measurements showed the existence of a two-phase region in the composition range between x =5/6 and 0.95. Electronic structure calculations gave evidence that these phases were metallic and insulating, respectively. Changes of the electronic and configurational entropy were found to be of comparable importance for this metal-insulator transition.

No abstract

Phonon densities of states ͑DOS͒ were obtained from inelastic neutron scattering measurements on Ce 0.9 Th 0.1 at temperatures from 10 to 300 K. The ␣ phase showed a significant softening of its phonon DOS when heated from 10 to 140 K. Despite the 17% volume collapse, the phonon DOS showed little change between the ␥ phase at 150 K and the ␣ phase at 140 K. This is supported by analysis of the magnetic spectra showing that most of the transition entropy can be accounted for with the crystal field and changes in the ground-state spin fluctuations. We argue that the anomalous behavior of the phonon DOS originates with the volume dependence of the ground-state spin fluctuations.

Inelastic neutron-scattering spectra were measured on stoichiometric Fe 0.50 Cr 0.50 prepared as a bodycentered-cubic ͑bcc͒ solid solution, and after increasing amounts of chemical unmixing on the bcc lattice induced by annealing the solid solution at 773 K. These spectra were reduced by a conventional procedure to a neutron-weighted vibrational density of states. Mössbauer spectrometry was used to characterize the extent of decomposition after annealing. A neutron-weight correction was performed, using results from the Mössbauer spectra and recent data on inelastic nuclear resonant scattering from 57 Fe-Cr. The vibrational entropy of decomposition was found to be −0.17± 0.01k B / atom, nearly equal to the change in configurational entropy after spinodal decomposition. Effects of vibrational entropy on the thermodynamics of unmixing are analyzed, showing a large effect on the free energy with the formation of Cr-rich zones, and a large effect on the critical temperature for spinodal decomposition for equiatomic Fe 0.50 Cr 0.50 .

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 Ká rmán 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. DOI: 10.1103/PhysRevLett.93.185704 PACS numbers: 64.75.+g, 61.12.-q, 63.70.+h Mixing creates structural disorder, so the entropy of a mixed system is expected to be larger than an unmixed system. The well-known configurational entropy of mixing in the point approximationis always positive (c is concentration, 0 < c < 1). Its contribution to the free energy F E ÿ TS therefore promotes solubility at higher temperatures, and, in equilibrium with a second phase of lower entropy, the solubility limit of a solid solution should increase with temperature. This is typically observed, but exceptions are noteworthy, including unexplained cases where solid solubility decreases over a range of temperature, termed ''retrograde'' solubility. Experimental and theoretical investigations have shown that differences in vibrational entropy play an important role in the relative thermodynamic stabilities of solid phases [1][2][3][4][5]. There is now a widespread interest in better understanding the effects of vibrational entropy on phase diagrams, an active topic of ab initio investigations. Here we show, to our knowledge for the first time, a negative entropy of mixing at rather low solute concentrations, caused by the entropy of atomic vibrations. The thermal stability of structurally ordered states and disordered states is thereby reversed.Although the solubility of Ni in the body-centered cubic (bcc) V-rich phase is qualitatively consistent with the entropy of Eq. (1), the elements below Ni in the periodic table, Pd and especially Pt, have similar phase diagrams but show a much weaker temperature dependence of solubility in bcc V [6]. Owing to its incoherent neutron cross section and cubic crystal structure, vanadium is the ideal element for measuring a phonon densityof-states (DOS) by inelastic neutron scattering [7], as are V-rich bcc solid solutions. From the phonon DOS the vibrational entropy and other phonon thermodynamic functions can be obtained, at least at low temperatures. Previous studies of resonance modes of heavy solutes in vanadium suggest important effects on phonon thermodynamics [8][9][10][11].Alloys were prepared by arc melting under a highpurity ar...