The Phonon Entropy of Transition Metals and Alloys

A previously unknown correlation was found between the vibrational entropy of the alloy and the difference in electronegativity of the solute and host atoms. The respective contributions of phonons and magnetism to the forward and reverse transformation entropy are evaluated from neutron scattering experiments and scanning calorimetry measurements.

Introduction

Alloy Thermodynamics
Vibrational Entropy
Expected Trends
Overview

Changes in the mass of the constituent nuclei, for example in the case of a mass defect, also induce changes in vibrational entropy. Recent theoretical calculations have also predicted important effects of the vibrational entropy on the solubility limit in dilute alloys.

Theoretical Background

Hamiltonians and Densities

Condensed-Matter Hamiltonian
Born-Oppenheimer Approximation
Density Functional Theory
Kohn-Sham Equations

The first HK theorem states that the external potential ˆVextis is uniquely determined by the ground state density of the interacting many-body system. Kohn and Sham's approach to determining the ground state density of the many-body problem is to introduce an auxiliary Hamiltonian for the independent particles, the Kohn-Sham Hamiltonian.

Lattice Dynamics

Born–von K´ arm´ an Model
Electron-Phonon Interaction

2.23, and leads to the introduction of the "dynamic matrix", the Fourier transform of the interatomic force constant tensor. To calculate the phonon density of states (DOS) of the crystal, we need to diagonalize the dynamical matrix at a large number of points in the first BZ.

First-Principles Phonon Calculations

Frozen Phonons
Hellmann-Feynman Forces
Linear Response
Molecular Dynamics

The direct method for phonon simulations has its foundation in the calculation of the Hellmann-Feynman forces. The first-principles calculation for FCC Al used a 2×2×2 iteration of the cubic unit cell (32 atoms).

Figure 2.1: Phonon DOS of FCC Al (a), antiferromagnetic BCC Cr (b) and ferromagnetic BCC Fe (c), calculated from ﬁrst-principles with the direct method, using VASP [33] and Phonon [30] (Al, Cr), or Phon [34] (Fe)

Thermal Properties

Harmonic Oscillators
Electronic Entropy

In the case of a one-dimensional oscillator, the anharmonicity of the potential can be expressed. Using density functional theory, nV(E) can be calculated for different volumes of the crystal and the change in Sel with thermal expansion can be evaluated.

Inelastic Neutron Scattering

Introduction
Scattering Cross-Section

General Case
Phonon Scattering

Scattering Function
Data Analysis

Background
Multiphonon Scattering
Instrument Resolution
Neutron-Weighting

The first thing to note is the relative intensity of the elastic scattering (central peak) and inelastic scattering (side shoulders). Resolution is the full width at half maximum (FWHM) of the line shape of the instrument under the specified operating conditions.

Figure 3.1: Neutron scattering geometry.

Eﬀects of Alloying on Phonons in Vanadium

Introduction

Previous measurements of phonon DOS in vanadium-based alloys have shown a strong sensitivity of phonons to impurities in vanadium. A clear trend is observed for the solutes in the thed series: impurities to the left of the vanadium cause a softening of the phonon modes, while those to the right cause a strengthening that gradually increases with their number of d-electrons. For heavy impurities such as Pd and Pt, one might expect a general softening of the phonon modes in the alloy, but this is not observed and instead there is a large general strengthening of the phonon DOS.

Experimental Details

Sample Preparation and Characterization
Elastic Moduli
Inelastic Neutron Scattering

A comparison of the lattice parameter in the V-6.25%X alloys with the metallic radii Rm of the pure elements in BCC coordination determined by Teatum and Gschneider [74] reveals a good agreement. As seen in this figure, the contraction of Rm along the d series partially explains the trend observed in the lattice parameter of the random solid solutions. A similar trend is seen in the variation of the bulk modulus B, where Ti induces softening, while elements to the right of vanadium induce a stiffening of the alloy's bulk modulus.

Table 4.1: Samples prepared and relevant physical properties.

Data Analysis

Density of States
Neutron-Weighting

The background emission from the empty Al pan was measured for the same time. Because of this, care must be taken to account for the total scattering cross section for impurities, σscat=σcoh+σinc. However, for the low substitutional impurity concentrations considered here, the overall bias compared to the true phonon DOS is expected to be small.

Phonon DOS

Pure Vanadium
Trend Across the 3 d -Series
Trend Down the Ti Column
Trend Down the V Column
Trend Down the Ni Column
Concentration Dependence in V-Co

Nevertheless, the overall effect of Ta solutions on the phonon DOS appears to be very weak. The shift of the cutoff is about 2.5 meV in the case of Ni and almost 4 meV in the case of Pt. However, these modes affect only a small part of the total DOS for impurity concentrations of 6.25%.

Figure 4.5: Neutron-weighted Phonon DOS for impurities across the 3d-series measured on LRMECS

Born–von K´ arm´ an Inversions

Lattice Dynamics Model and Fitting Procedure
Minimization Results

Some of the guesses for the force constants were captured in local minima in the parameter space during the fitting procedure. In effect, as the size of the impurity replacing the host vanadium atom increases, the X-V bond is set. In the case of Ni impurities, the impurity pDOS does not show any resonance mode and is similar to the pDOS of the surrounding V atoms.

Figure 4.10: Phonon DOS for V-Co alloys of diﬀerent composition. The top curves are from measurements on the LRMECS time-of-ﬂight spectrometer, E i = 50 meV

V experiment V-X de-weighted

Eﬀect of Neutron-Weighting
Vibrational Entropy of Mixing
Concentration Dependence in V-Co
Connection with Elastic Constants
Chemical Trends

Eﬀect of Metallic Radius
Eﬀect of Electronegativity
Comparison with Trends for BCC Transition Metals

Summary

The trend in vibrational entropy is reflected in the plot of the average phonon frequency of figure 4.19. The effect of the impurity metal radius on the average lattice parameter was seen. On the other hand, we show that the vibrational entropy of alloy exhibits a strong correlation with the electronegativity of the solutes.

Table 4.4: Vibrational entropy of alloying and mean phonon energy.

First-Principles Simulations of V-X Alloys

Introduction

Simulation Parameters

Wien2k
VASP

Wien2k calculations were performed with atomic sphere radii of iRMT = 2.2 atomic units (a.e.) for all atoms in the unit cells considered. Additional calculations on relaxed supercells were performed with an increased RMTKmax= 8.0, which did not affect the calculated properties. In the case of magnetic impurities Fe, Co and Ni, spin-polarized calculations were performed on relaxed 2 × 2 × 2 supercells, indicating the absence of magnetization on the impurity atom, and subsequent simulations were performed with non-spin cells. polarized model.

Geometry Relaxation

Procedure
Results

For Co impurities, this ratio decreases to about three, and in the case of Ni, the relaxation mainly affects L2NN. For the other column impurities Ni, Pd, and Pt, the relaxation in the 1NN shell is substantial, at 2.6% and 2.3%. The strain is also positive for 2NN bonds in the case of Pd and Pt impurities (about 0.4% increase).

Figure 5.1: Typical energy-volume curve for the V 15 X 1 supercells calculated with Wien2k.

Electronic Structure

Another possibility is that, in the image of the rigid band, this shift in energy upon introduction of impurities with extrapolated electrons can be seen as a simple filling of the d-band. However, some of these properties are related to the ordered nature of the supercell used in the simulation, and the electronic DOS for a random solid solution is expected to be more washed out. Another important feature of the electronic structure of the alloy is the electronic structure at the Fermi level, n(EF).

Figure 5.5: Impurity-host bond-length for 1NN, 2NN and 3NN bonds in Wien2k 3x3x3 supercells.

Electronic Topological Transition

The Fermi surface is calculated with Wien2k for bcc V, with different band-filling values. This topological transition corresponds to a large decrease in the area (and number of states) of the Fermi surface. This can lead to a significant change in the wave vectors extending from the surface Fermi sheets, which can consequently affect the electron-phonon coupling.

Figure 5.10: Topological transition in Fermi surface of V-Cr alloys. The Fermi surface is calculated with Wien2k for bcc V, with diﬀerent values of band-ﬁlling

Superconductivity and Electron-Phonon Coupling

As mentioned in the previous section, due to the negative slope inn(E) at the Fermi level, the introduction of impurities raises or lowers n(EF) when the solutes have fewer or more d-electrons than vanadium, respectively. The decrease in n(EF) occurs approximately linearly with the number of d-electrons in the impurity, Nd. 5.13 we compare the relative change in the calculated n(EF) with the same experimental quantity, derived from low-temperature measurements of the electronic specific heat.

Figure 5.12: Electronic density at the Fermi level calculated with Wien2k.

Electronic Entropy

We extracted values of λ from these critical temperatures for the alloys whose phonon DOS we measured. Using the electronic densities of states obtained from FP-LAPW simulations for the relaxed 2×2×2 supercells, we have the difference in electronic entropy between the pure vanadium and the alloys V15X1, or electronic entropy of alloy at temperature T, ΔSelal (T) calculated ). Comparing these results with the vibrational entropy of alloy ΔSvibal (295 K) from figure 4.18, one can see that ΔSelal is much smaller than ΔSvibal for the alloys investigated, with the exception of V-6.25%Ni.

Figure 5.15: Electron-phonon coupling strength for V-6.25%X.

Charge Transfer

5.17, the charge transfer to each atom type is weighted by the multiplicity of the atom in the supercell. We see a good correlation between the charge transfer and the electronegativity difference between impurity and host at both scales. It thus appears that charge transfer effects can explain the experimental trend in the phonons and the vibrational entropy.

Deformation Potential

To investigate the change in interatomic potentials upon introduction of impurities into V, we turned to calculations of the deformation potential in a breathing mode of the 1NN shell around the impurity. This deformation mode is not a proper frozen phonon, but it allows us to probe the longitudinal stiffness of the 1NN X-V bond. This is in good agreement with the observed shifts in the frequency of the longitudinal modes, in particular the cutoff frequency.

Figure 5.20: Calculated potential for 1NN atom displacements in a breathing mode. The energy is the total supercell energy calculated with VASP

Summary

The potential is increasingly strong for Cr, Ni, and Pt impurities, while Ti causes a small softening for impurity shifts. The trend in Bader charge transfer behaves linearly with the electronegativity difference between the impurity and the host. This charge transfer provides an explanation for phonon stiffening, as more ionic bonds should be stronger and induce higher energy phonon modes.

Phonons in Vanadium Alloys at High Temperatures

Introduction

High-Temperature Phonon DOS

Pure Vanadium
Vanadium Alloys
Vanadium Alloys
Vibrational Entropy of Alloying

The temperature dependence of the phonon DOS for V-6.25%Pt is very similar to that observed in V-7%Co, with a gradual temperature softening. We recall that these impurities also had similar effects on the phonon DOS of V at room temperature, both causing a large stiffening of the vibrations (see Chapter 4). The results of both studies are in good agreement and attribute the anomalous temperature of the phonon DOS to the anharmonicity of the phonons.

Figure 6.1: Phonon DOS of vanadium at 293, 873, 1273 and 1673 K. Temperatures are as labeled

High-Temperature Properties of BCC Transition Metals

Phonon DOS
Elastic Constants

This intrinsic stiffening appears to dominate over the full temperature range of stability of the BCC phase in group 4 metals, but it is suppressed at very high temperatures in the BCC group 5 metals V and Nb. 6.11, the addition of Mo in Nb shifts the position of the local minimum inC44 to lower temperatures. The significant modifications of the geometry of the Fermi surface during this transition have also been invoked to explain the behavior in the C44 elastic constant [112] .

Figure 6.8: Phonon DOS of BCC Cr at elevated temperatures. Adapted from [142].

Eﬀects of Electron-Phonon Interaction

Theoretical Predictions

Using this result, Allen and Hui deduce the change in heat capacity of the electrons and phonons, due to the electron-phonon coupling. The quasiharmonic heat capacity due to thermal expansion was obtained from the corresponding entropy in Fig. Our DSC result for the total heat capacity of V agrees with the recommended value of Maglic [85].

Figure 6.13: Heat capacity components for pure vanadium. Values are at constant pressure.

Summary

The behavior of phonons with temperature in V- and V-rich alloys is comparable to general trends observed in BCC transition metals, suggesting the importance of the electronic structure in determining the temperature dependence of the lattice vibrations. Careful analyzes of the entropy and heat capacity reveal the necessity to take into account a large number of contributions, many of them having similar magnitudes at elevated temperatures. However, the intrinsic anharmonic components of the entropy and heat capacity, whether originating from phonon–phonon or electron–phonon interactions, are much smaller.

Table 6.2: Average phonon frequencies and anharmonic heat capacity for V and V-7%Co.

Introduction

Kaufman and Cohen [151] showed that the transformation between FCC austenite and BCC martensite in Fe71Ni29 is strongly hysteretic. The γ → α transformation only occurs at a large undercooling below the temperature T0 where the free energies of both phases are equal, while the reverse α → γ transformation requires a similarly large superheating above T0. The transformation therefore takes place out of equilibrium, and it is caused by the large elastic forces of the α−γ transformation.

Experimental Details

Sample Preparation
Calorimetry
Neutron Scattering

We investigate the different contributions to the entropy of transformation in the direct γ → α and reverse α →γ transformations in Fe71Ni29 by inelastic neutron scattering and calorimetry. The amount of γ−α martensitic transformation in Fe71Ni29 was determined by measuring X-ray diffraction patterns of samples cooled to different temperatures. Neutron diﬀraction patterns were obtained from the elastic part of the neutron scattering data and used to monitor martensitic transformation in situ.

Figure 7.1: Diﬀraction patterns from Fe 71 Ni 29 at room temperature for transformation stages corresponding to decreasing temperatures

Analysis of the Phonon Density of States

Data Reduction
Neutron-Weight Correction

Neutron weighting is therefore expected to have at most a moderate effect on the overall shape of the phonon DOS. This technique allows the measurement of the partial phonon DOS (pDOS) associated with the 57Fe resonance nuclei, but is insensitive to the motion of the Ni atoms. The DOS for the intermediate transformation states was then corrected for the neutron weighting of the martensite phase by subtracting the increase in neutron weighting Δgnwα,T =gα,Tnw −gα,T, and.

Results and Discussion

Phonons

These low-energy modes are attributed to the instability of the Ni atoms in the BCC phase. Because Franzet al. performed their measurements of the γ phase at 720 K, there is additional softening in their Fe-pDOS compared to the DOS we measured at 295 K. The authors report no Fe-pDOS for the γ phase at lower temperatures, although the measurements of Kaufmann et al. Cohen [151]. On the other hand, it is much lower than Fe-pDOS above 30 meV. The unweighted total DOS is smaller than its neutron-weighted counterpart at low energies, due to the overcrowding of high-amplitude Ni vibrations at low energies.

Figure 7.5: Neutron-weighted phonon DOS for γ (white circles) and α (black diamonds) Fe 71 Ni 29 at room temperature, measured by inelastic neutron scattering

Analysis of Time-of-Flight Coherent Phonon Scattering

Calculation of S ( Q, E ) for FCC and BCC Fe-Ni

When measuring a polycrystalline sample, one obtains only the average of Scoh(Q, E) over all orientations of the crystallites relative to the neutron beam. We now apply the general considerations from the previous section to the case of the Fe71Ni29 solid solution. However, at the time of writing this thesis, a full fitting procedure represents a heavy computation, due to the cost of computing S(Q, E) based on a Born-von K'arm'an model.

Conclusion