Negative Entropy of Mixing for Vanadium-Platinum Solutions O. Delaire, T. Swan–Wood, and B. Fultz California Institute of Technology, W. M. Keck Laboratory, mail 138-78, Pasadena, California 91125, USA (Received 9 April 2004; published 29 October 2004) 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 Ka ´rma ´n 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 mix- ing in the point approximation S mix cf k B clnc 1  cln1  c (1) is always positive (c is concentration, 0 <c< 1). Its contribution to the free energy F  E  TS therefore promotes solubility at higher temperatures, and, in equi- librium with a second phase of lower entropy, the solu- bility 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 [1–5]. There is now a widespread interest in better understanding the effects of vibrational entropy on phase diagrams, an active topic of ab initio investiga- tions. Here we show, to our knowledge for the first time, a negative entropy of mixing at rather low solute concen- trations, caused by the entropy of atomic vibrations. The thermal stability of structurally ordered states and disor- dered 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 depen- dence of solubility in bcc V [6]. Owing to its incoherent neutron cross section and cubic crystal structure, vana- dium is the ideal element for measuring a phonon density- of-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 thermo- dynamics [8–11]. Alloys were prepared by arc melting under a high- purity argon atmosphere. The ingots of V-Ni, V-Pd, and V-Pt were subsequently cold rolled to the desired thick- nesses and annealed in vacuum at 950  C for 1 h and air cooled in their quartz ampoules. X-ray diffractometry showed the samples to be bcc solid solutions. The longitudinal (c L ) and transverse (c T ) sound veloc- ities were measured by transit times of ultrasonic pulses. The shear and bulk moduli G  c 2 T and B  c 2 L  4 3 G were obtained from the acoustic velocities and the alloy densities. The results in Table I show only a slight stiffen- ing in bulk modulus in the alloysV-Pt and V-Ni compared with pure V , and no measurable effect in V-Pd. On the other hand, the solutes cause a substantial stiffening of G. Compared to pure V , the relative stiffenings of G for {V- Ni, V-Pd, V-Pt} are, sequentially, {6%, 16%, 30%}. Inelastic neutron scattering spectra were measured us- ing the IPNS-LRMECS time-of-flight chopper spec- trometer at the Argonne National Laboratory. The incident energy was 55 meV and the detector coverage allowed for momentum transfers between 0.5 and 9  A 1 at zero energy transfer. The FWHM energy resolution was 1:1 meV at 40 meV positive energy transfer, increasing to TABLE I. Measured and calculated elastic moduli. B meas. B calc. G meas. G calc. (GPa) (GPa) (GPa) (GPa) V 149  7 163 46  0:7 35:2 V a 155.2 42.55 V b 157.12 43.768 V-Ni 163  8 174 48:6  0:9 40:8 V-Pd 150  6 159 53:4  0:9 40:9 V-Pt 167  7 170 59  1 47:1 a Reference [12]. b Reference [13]. VOLUME 93, NUMBER 18 PHYSICAL REVIEW LETTERS week ending 29 OCTOBER 2004 185704-1 0031-9007= 04=93(18)=185704(4)$22.50  2004 The American Physical Society 185704-1