First Principles Investigation of 36 Binary Refractory Alloys · 2020. 2. 6. · component alloys made of only refractory elements. Refractory elements in this study involve elements

$Page 1: First Principles Investigation of 36 Binary Refractory Alloys · 2020. 2. 6. · component alloys made of only refractory elements. Refractory elements in this study involve elements$
◦Bachelors of Science

University of CaliforniaSanta Barbara

First Principles Investigation of 36 Binary

Refractory Alloys

A thesis submitted in partial satisfaction

of the requirements for the degree

in

Physics

by

Pavel Anisimovich Dolin

Committee in charge:

Professor Anton Van der VenProfessor Doug Eardley

June 2019

The Dissertation of Pavel Anisimovich Dolin is approved.

Professor Anton Van der Ven

Professor Doug Eardley

March 2019

First Principles Investigation of 36 Binary Refractory Alloys

Copyright c© 2019

by


iii

iv

Acknowledgements

I’m very grateful for the opportunity to be a part of Van Der Ven lab. Over the past

years it has been an amazing journey leading to a publication. A big thank you goes to

Professor Anton Van der Ven for allowing me to be a part of this amazing project and

mentorship. Another big thank you goes to Dr. Anirudh Natarajan for his mentorship,

patience and help. I additionally thank Professor Doug Eardley for giving me an academic

credit for this research.

v

Abstract

First Principles Investigation of 36 Binary Refractory Alloys

by


Multi-principal element refractory alloys offer attractive applications in aerospace,

medical and nuclear industries. In this study I present the report on the phase stability

of 36 binary alloys derived from the nine refractory elements in groups and rows four,

five and six. The study reveals excellent agreement with experimental data with a few

exceptions. The study also predicts a low-temperature Laves-phase in the Nb-V system.

vi

Contents

Abstract vi

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Crystal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Overview of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Methods 32.1 Density Functional Theory (DFT) . . . . . . . . . . . . . . . . . . . . . . 32.2 Cluster Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Scientific Group Thermodata Europe (SGTE) . . . . . . . . . . . . . . . 5

3 Results 83.1 Binary thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.1 Cr-W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Nb-V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.3 Nb-W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.4 Hf-Zr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.5 Mo-Zr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.6 Mo-Ti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Discussion 174.1 Occurring Phases Across 36 Binaries . . . . . . . . . . . . . . . . . . . . 174.2 Computational and Experimental Phase Diagrams . . . . . . . . . . . . . 18

5 Conclusion 21

6 Appendix 226.1 DFT Formation Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2 DFT Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

vii

Chapter 1

Introduction

1.1 Motivation

With a growing demand by aerospace industries for alloys that can withstand high

temperature as well as medical industries that demand alloys that are highly resistant

to corrosion; there has been a renewed interest in understanding the behavior of multi-

component alloys made of only refractory elements. Refractory elements in this study

involve elements from group and row IV, V and VI in the periodic table. The nine

elements in this group show several interesting phenomena ranging from polymorphic

phase-transitions and the formation of topologically close-packed crystal structures such

as the Laves phase.

1.2 Crystal Structures

The elements in groups V and VI: vanadium, niobium, tantalum, chromium, molyb-

denum and tungsten are all found to occur in the body-centered cubic (bcc) crystal

structure at ambient pressures. The elements in group IV (titanium, zirconium and

1

Introduction Chapter 1

hafnium) show a series of polymorphic phase transitions with increasing temperatures.

At ambient temperatures and pressures all three elements are stable in the hexagonal

close-packed crystal structure (hcp). With increasing temperature they undergo a sec-

ond order phase transition to the bcc crystal structure. This structure is thought to be

stabilized through anharmonic vibrational entropy at higher temperatures.

Similar to the elements, the binary alloys comprising elements from the refractory

group also show several interesting phenomena. The high-temperature bcc phase formed

by titanium, zirconium and hafnium is known to be stabilized at lower temperatures

through the addition of certain group V and VI elements. Some of these binary alloys

also form a brittle, hard intermetallic Laves phase. Binary alloys such as Cr-Ti show a

polymorphic phase transition between the three well-known Laves phases C15, C14 and

C36. The Laves phases in these alloys are typically viewed as undesirable since they

degrade the mechanical properties of these alloys.

1.3 Overview of the Study

In this study I explore the thermodynamic properties of binary alloys comprising

elements from groups IV, V and VI in the periodic table. High-throughput first-principles

calculations reveal that the high-temperature phase diagrams of every binary alloy may

be grouped into six classes. The link between the 0K formation energies and the high-

temperature phase diagrams is then demonstrated through rigorous statistical mechanics

techniques. I find that the calculations agree with experimental investigations with the

exception of a single system where we predict a new phase and one family of binary alloys

where we do not predict the existence of a high-temperature miscibility gap.

2

Chapter 2

Methods

2.1 Density Functional Theory (DFT)

The finite temperature phase stability across all binary and unary systems was as-

sessed by combining first-principles electronic structure calculations with statistical me-

chanics techniques. Total energies were calculated with density functional theory (DFT)

as implemented in the Vienna Ab-initio Simulation Package (VASP) (1),(2),(3),(4), using

projector augmented wave (PAW) pseudopotentials. DFT calculations were performed

within the Perdew-Burke-Ernzerhof (PBE) parametrization of the generalized gradient

approximation (GGA). For all calculations an automatic k point mesh was used with 42

k-points per A−1 with an energy cutoff of 480 eV for the plane wave basis set. The pseu-

dopotentials for each element were chosen as shown in table 2.1. All degrees of freedom

were relaxed for every structure with a force convergence criterion of 10−3 and Gaussian

smearing of width 0.1. Total energies were extracted from a final static calculation using

the tetrahedron method with Blochl corrections to ensure accurate relaxed energies.

3

Methods Chapter 2

Elements Ti V Cr Zr Nb Mo Hf Ta WValence electrons 12 13 12 12 13 14 10 11 12

Table 2.1

2.2 Cluster Expansion

Phase stability may be determined from the relative energies of several symmetri-

cally distinct orderings across all crystal structures of interest. The Clusters Approach to

Statistical Mechanics (5),(6),(7), code was used to enumerate all symmetrically distinct

orderings on the bcc and hcp crystal structures. Symmetrically distinct decorations of

each constituent element in the binary alloy were enumerated across the full composi-

tion range for all thirty six binary alloys. Orderings on the bcc crystal structure were

enumerated in cells containing up to 5 atoms for all alloys, while orderings on hcp were

enumerated for alloys containing at least one element from group IV in cells containing up

to 6 atoms. Each ordering may be uniquely identified by assigning occupation variables

σi at each site i in the crystal that takes a value of zero if specie A is at the site and zero

otherwise. Any ordering may be represented by the collection of these site occupation

variables across all N sites in the crystal given by ~σ = {σ1, σ2, · · · , σN}. The formation

energies of each ordering in an A−B alloy is given by:

Ef (~σ) =E(~σ)−NAE

refA −NBE

refB

NA +NB

(2.2.0.1)

where E(~σ) is the total energy calculated from DFT, NA and NB are the number of A

and B atoms in the cell, and ErefA and Eref

B are the reference energies of A and B in their

reference crystal structures. In this study we define the reference crystal of an element

to be the lowest energy structure across the bcc and hcp crystals at 0K.

The calculated formation energies were used to parameterize a cluster expansion.

4

Methods Chapter 2

Within the cluster expansion formalism, the formation energy of an arbitrary ordering

on a parent crystal structure is written as:

Ef (~σ) = Vo +∑α

Vαφα(~σ) (2.2.0.2)

where the sum extends over all clusters of sites α, the φα are products of the occupation

variables σi over all sites in the cluster α, and Vα are the effective cluster interactions.

We started with a pool of clusters including 2,3 and 4-body clusters with a maximum

radius of 9A, 8A and 5.5A respectively. We then used the genetic algorithm with a k-fold

cross-validation score to estimate the best fit for the given alloy and crystal structure.

Grand-canonical Monte-Carlo simulations were coupled with free-energy integration

techniques to estimate phase stability at elevated temperatures. Grand-canonical free

energies are calculated from ensemble averages of composition and internal energy at fixed

chemical potential and composition. Phase diagrams were constructed with the common

tangent construction through the convex hull algorithm. Details of the cluster expansion

can be found in tables 2.2 and 2.3. Grand-canonical Monte Carlo was performed using

metropolis method.

Constant temperature Grand-canonical Monte-Carlo simulations were all performed

with a cell of volume 2 ·153 A3 (matrix with zeros on the diagonal and 15 off diagonal)

which accommodated all of the ground states. Additionally for Nb-W for constant µ

runs we used cell of volume 2 ·163 A3.

2.3 Scientific Group Thermodata Europe (SGTE)

The bcc crystal structure in alloys containing at least one group IV element is sta-

bilized through anharmonic vibrational entropy. We approximate the anharmonic vibra-

5

Methods Chapter 2

System No. Cofig. No. Basis Fun. CV[eV/atom] RMS[eV/atom]Cr-W 53 15 0.005 0.003Nb-V 57 15 0.002 0.002Nb-W 103 15 0.002 0.001Hf-Zr 52 11 0.004 0.004Mo-Ti 83 15 0.007 0.006Mo-Zr 69 10 0.008 0.007

Table 2.2: BCC cluster expansion details

System No. Cofig. No. Basis Fun. CV[eV/atom] RMS[eV/atom]Hf-Zr 62 15 0.000 0.000Mo-Ti 44 4 0.004 0.013Mo-Zr 31 4 0.09 0.08

Table 2.3: HCP cluster expansion details

tional entropy in binary alloys by extrapolating the values for pure elements from the

empirical SGTE database.

We use HCP Gibbs free energies ∆Gαclex(1, 0),∆Gα

clex(0, 0) as our reference for BCC

and HCP. In order to insure that the HCP-BCC difference in elemental Gibbs free

energies from cluster expansion at T = 0 K, that is: ∆Gβclex(1, 0) − ∆Gα

clex(1, 0) and

∆Gβclex(0, 0) − ∆Gα

clex(0, 0) is not contributing to the HCP-BCC time dependent differ-

ence in elemental Gibbs free energies from SGTE database; we first reference BCC Gibbs

free energy ∆Gβclex(x, T ) to the HCP-BCC difference of elemental Gibbs free energies

from cluster expansion and then add linearly interpolated temperature dependent HCP-

BCC difference in SGTE Gibbs free energies, that is: ∆Gβsgte(1, T ) − ∆Gα

sgte(1, T ) and

(∆Gβsgte(0, T ) − ∆Gα

sgte(0, T ). By linearly interpolating HCP-BCC difference in SGTE

Gibbs free energies we create a simple, yet effective way to calculate the temperature

dependant HCP-BCC transition not only for the elements but for the entire composition

6

Methods Chapter 2

range.

∆Gβ(x, T ) = ∆Gβclex(x, T )

+x · ((∆Gβsgte(1, T )−∆Gα

sgte(1, T ))− (∆Gβclex(1, 0)−∆Gα

clex(1, 0)))

+(1− x) · ((∆Gβsgte(0, T )−∆Gα

sgte(0, T ))− (∆Gβclex(0, 0)−∆Gα

clex(0, 0)))

(2.3.0.1)

7

Chapter 3

Results

3.1 Binary thermodynamics

Using DFT we calculated total energies of symmetrically distinct orderings on bcc,

hcp as well as Laves C14, C15 and C36 crystal structures for 36 binary alloys fig. 6.1,

fig. 6.2, fig. 6.3, fig. 6.4. By examining the constructed DFT convex hulls across 36

systems we were able to identify 6 distinct types of thermodynamic features at 0 K.

This observation allowed us to organize 36 alloys into 6 unique groups. For each group

we selected a representative alloy and calculated its phase diagram via cluster expan-

sion, Grand-canonical Monte-Carlo simulations and Gibbs free energy integration. The

representative alloys are Cr-W, Nb-V, Nb-W, Zr-Hf, Zr-Mo and Ti-Mo. Due to a great

similarity in DFT convex hull features within the group, DFT free energies and calculated

phase diagrams of the representatives can serve as prototypical calculations for the entire

group. Thus, DFT free energies and calculated phase diagrams of all 6 representatives:

fig. 3.2,fig. 3.3,fig. 3.4,fig. 3.5 and fig. 3.6 serve as a summary of all occurring thermody-

namic behaviour across 36 binary refractory alloys. Each figure shows DFT free energy,

calculated phase diagram of the representative alloy and a triangular diagram showing

8

Results Chapter 3

group members marked as black. Each square in the triangular diagram represents one

out of 36 alloys. Two shades of gray are used for easier visual identification.

6 prototypical thermodynamic behaviours can be broken into 2 categories according

to the elements involved. fig. 3.1, fig. 3.2, fig. 3.3 represent alloys with only group 5,6

elements. They characterize 3 types of a low temperature behavior: solid-state miscibility

gap in the bcc phase, two phase region in Laves C14 - bcc solid solution, presence of

multiple bcc orderings across the composition range that dissolve into bcc solid solution

at low temperatures. fig. 3.4, fig. 3.5, fig. 3.6 represent alloys with at least one element

from group 4. They characterize another 3 types of a low temperature behavior: hcp

to bcc phase transition of both alloy elements at higher temperatures, hcp to bcc phase

transition of group 4 element with the presence of Laves C15 ordering, hcp to bcc phase

transition of group 4 element without the presence of Laves C15 ordering.

3.1.1 Cr-W

0.00 0.25 0.50 0.75 1.00

x Cr

� 0.1

0.0

0.1

0.2

0.3

0.4

En

erg

y [

eV

/ato

m]

CrW

Figure 3.1: Cr-W: DFT, calculated phase diagram. Alloys with similar thermodynamicbehaviour

Figure 3.1a shows the calculated formation energies for the Laves phases and several

orderings on the bcc crystal structure in the binary Cr-W system. At zero Kelvin the

stable compounds on the global convex hull are found to be only bcc-Cr and bcc-W. The

large positive formation energies of orderings on the bcc crystal structure suggests a solid-

9

Results Chapter 3

state miscibility gap in the bcc phase. As shown in fig. 3.1b, rigorous statistical mechanics

calculations reveal the miscibility gap in the predicted temperature-composition phase

diagram. We calculate the critical temperature of the miscibility gap to be ≈ 1800 ◦C.

The presence of the miscibility gap is consistent with the experimental results, however

the experimental critical temperature of the miscibility gap is measured to be ≈ 1680 ◦C

(8). The discrepancy between the computed and experimental values can be attributed

to the vibrational entropy, which we did not consider in our calculations. Above this

temperature the phase diagram shows a bcc solid solution. An investigation of the DFT

formation energies reveals that Cr-Mo is the only other binary alloy predicted to show a

similar phase diagram. Cr-W and Cr-Mo are listed in fig. 3.1c.

3.1.2 Nb-V

0.00 0.25 0.50 0.75 1.00

xV

� 0.1

0.0

0.1

0.2

0.3

0.4

En

erg

y [

eV

/ato

m]

VNb

Figure 3.2: Nb-V: DFT, calculated phase diagram. Alloys with similar thermodynamicbehaviour


orderings on the bcc crystal structures in the binary Nb-V system. We picked Nb-V as a

group representative since C14-Laves phase in Nb-V have not been confirmed experimen-

tally at low temperatures (9). We note that at zero Kelvin the stable compounds on the

global convex hull are found to be bcc-Nb, the C14-Laves phase and bcc-V. We predict no

metastable orderings on the bcc crystal across the full composition range. The large pos-

10

Results Chapter 3

itive formation energies of orderings on the bcc crystal structure suggests a limited solid

solubility of molybdenum in niobium. The C14-Laves phase is very stable at 0K due to

its large negative formation energy, and is likely to remain stable to high-temperatures

in the equilibrium phase diagram. As shown in fig. 3.2b, rigorous statistical mechan-

ics calculations do reveal these features in the predicted temperature-composition phase

diagram. At low temperatures the bcc solid solution show limited solid-solubility and

exhibit two-phase region with the C14-Laves phase. We predict the critical temperature

of the two phase region to be ≈ 1650 ◦C. Above this temperature the phase diagram

shows a bcc solid solution. An investigation of the DFT formation energies reveals that

3 other binary alloys are predicted to show a similar phase diagram. These systems are

listed in fig. 3.2c.

3.1.3 Nb-W

0.00 0.25 0.50 0.75 1.00

xW

� 0.1

0.0

0.1

0.2

0.3

0.4

En

erg

y [

eV

/ato

m]

Nb W

BCC

WNb

Figure 3.3: Nb-W: DFT, calculated phase diagram. Alloys with similar thermodynamicbehaviour

Figure 3.3a shows the calculated formation energies for the Laves phases and multi-

ple orderings on the bcc crystal structures in the binary Nb-W system. At zero Kelvin

the stable compounds on the global convex hull are found to be bcc-Nb, bcc-W among

with multiple continuous orderings on the bcc crystal structure. The presence of mul-

tiple ordered bcc phases mostly on W-rich compositions indicates a low order-disorder

11

Results Chapter 3

temperature on the bcc crystal structure. The large positive formation energies of Laves

phases suggest their absence in the system at 0 K and higher temperatures. As shown in

fig. 3.3b, rigorous statistical mechanics calculations reveal these features in the predicted

temperature-composition phase diagram. At low temperatures, the bcc solid solutions

show limited solid-solubility on the niobium rich side and exhibit a two-phase region

with the continuous orderings on bcc crystal structure. The critical temperature of the

two-phase region is ≈ −60◦C. The critical temperature of the a solid-state miscibility

gap in the bcc phase on the niobium rich side is at ≈ 40◦C. The critical temperature

of continuous orderings on bcc crystal structure is ≈ 75◦C. Above ≈ 75◦C the phase

diagram shows bcc solid solution. The rapid disorder of continuous orderings on bcc

crystal structure can be attributed the energy penalty of a defect that gets compensated

by the configurational entropy, facilitating the formation of bcc solid solution at lower

temperatures. Low temperature behaviour has not been confirmed experimentally. Data

is only available for temperatures above ≈ 2126◦C (10). The formation of bcc solid so-

lution at higher temperatures is consistent with experimental data. An investigation of

the DFT formation energies reveals that 8 other alloys are predicted to show a similar

phase diagram. In total including Nb-W, 7 out of 9 alloys involve elements from separate

elemental groups, that is group 5 or group 6 elements. Systems are listed in fig. 3.3c.

12

Results Chapter 3

3.1.4 Hf-Zr

BCCLaves

BCC

0.00 0.25 0.50 0.75 1.00

x Hf

� 0.1

0.0

0.1

0.2

0.3

0.4

En

erg

y [

eV

/ato

m]

Zr

Figure 3.4: Hf-Zr: DFT, calculated phase diagram. Alloys with similar thermodynamicbehaviour


orderings on the hcp and bcc crystal structures in the binary Hf-Zr system. At zero

Kelvin the stable compounds on the global convex hull are found to be hcp-Zr, several

orderings on hcp crystal structure and hcp-Hf. Orderings on the bcc crystal structure are

predicted to be metastable. The large positive formation energies of Laves phases imply

their absence not only at 0 K but at high temperatures as well. The formation energies

of all orderings on the hcp crystal structure are ≈ 0 ev/atom. This suggest that all of

them are likely to dissolve into a solid solution at really low temperatures. Additional

evidence of the formation of hcp solid solution at lower temperatures can be see via values

of effective cluster interactions (eci) from the hcp cluster expansion. Eci values are near

zero, implying the alloy behaves nearly like a solid solution at 0 K. As shown in fig. 3.4b,

rigorous statistical mechanics calculations augmented with empirical shifts (11) to the free

energies reveal these features in the predicted temperature-composition phase diagram.

At low temperatures, the hcp solid solutions exists across the entire composition range.

The hcp→bcc transformation of both elements results in hcp solid solution exhibiting a

two-phase region with the bcc solid solution. Linearly interpolated elemental empirical

13

Results Chapter 3

shifts across the composition range are in agreement with experimental observations

(12). An investigation of the DFT formation energies reveals that 2 other binary alloys

are predicted to show a similar phase diagram. All systems within this group involve

elements strictly from group 4. These systems are listed in fig. 3.4c.

3.1.5 Mo-Zr

0.00 0.25 0.50 0.75 1.00

x Mo

� 0.1

0.0

0.1

0.2

0.3

0.4

En

erg

y [

eV

/ato

m]

MoZr

Figure 3.5: Mo-Zr: DFT, calculated phase diagram. Alloys with similar thermodynamicbehaviour

Figure 3.5a shows the calculated formation energies for the Laves phases and sev-

eral orderings on the hcp and bcc crystal structures in the binary Mo-Zr system. At

zero Kelvin the stable compounds on the global convex hull are found to be hcp-Zr, the

C15-Laves phase and bcc-Mo. Orderings on the bcc crystal structure are predicted to be

metastable for molybdenum-rich compositions, while we predict no metastable orderings

on the hcp crystal across the full composition range. The presence of several metastable

ordered bcc phases at Mo-rich compositions indicates a low order-disorder temperature

on the bcc crystal structure. The large positive formation energies of orderings on the

hcp crystal structure suggests a limited solid solubility of molybdenum in hcp-Zr. The

C15-Laves phase is very stable at 0K due to its large negative formation energy, and is

likely to remain stable to high-temperatures in the equilibrium phase diagram. As shown

in fig. 3.5b, rigorous statistical mechanics calculations augmented with empirical shifts

14

Results Chapter 3

(11) to the free energies do indeed reveal these features in the predicted temperature-

composition phase diagram. At low temperatures, the hcp and bcc solid solutions show

limited solid-solubility and exhibit two-phase regions with the C15 Laves phase. The

hcp→bcc transformation of zirconium coupled with the solid solubility of molybdenum

in bcc-Zr results in an eutectoid reaction at ≈ 790◦C. This is in agreement with experi-

mental observations that peg this invariant reaction at ≈ 780◦C. Above this temperature

the phase diagram shows two-phase regions between the bcc solid solutions and the Laves

phase, with the Laves phase being thermodynamically stable up to the melting temper-

ature. An investigation of the DFT formation energies reveals that 6 other binary alloys

are predicted to show a similar phase diagram. These systems are listed in fig. 3.5c.

3.1.6 Mo-Ti

0.00 0.25 0.50 0.75 1.00

x Mo

� 0.1

0.0

0.1

0.2

0.3

0.4

En

erg

y [

eV

/ato

m]

MoTi

Figure 3.6: Mo-Ti: DFT, calculated phase diagram. Alloys with similar thermodynamicbehaviour


orderings on the hcp and bcc crystal structures in the binary Mo-Ti system. At zero

Kelvin the stable compounds on the global convex hull are found to be hcp-Ti, bcc-Mo

and multiple continuous orderings on the bcc crystal structure, mostly at Mo-rich com-

positions. Just as in the case of fig. 3.3a the presence of multiple ordered bcc phases

mostly on Mo-rich compositions indicates a low order-disorder temperature on the bcc

15

Results Chapter 3

crystal structure. The presence of several metastable ordered hcp phases at Mo-rich

compositions indicates a low order-disorder temperature on the hcp crystal structure

as well. The large positive formation energies Laves phases indicate their absence not

only at 0 K but also at higher temperatures. As shown in fig. 3.6b, rigorous statistical

mechanics calculations augmented with empirical shifts (11) to the free energies reveal

these features in the predicted temperature-composition phase diagram. At low tem-

peratures, the hcp and bcc solid solutions show limited solid-solubility. The hcp→bcc

transformation of titanium occurs at ≈ 840◦C, as dictated by the empirical shift. Above

this temperature the phase diagram shows bcc solid solution. An investigation of the

DFT formation energies reveals that 8 other binary alloys are predicted to show a similar

phase diagram. These systems are listed in fig. 3.6c. Some of the alloys in this group,

including our representative alloy, Mo-Ti have spurred a lot of debates in the literature

as some experiments indicate the opening of the solid-state miscibility gap in the bcc

phase at higher temperatures (13). With methods used in this paper we are not able to

reproduce this type of the behaviour.

3.2 Volumes

We used crystalographic data from DFT calculations with orderings on hcp, bcc,

Laves C14, C15 and C36 crystal structures for 36 binary alloys to show that alloys tend to

obey Vegard’s law. Figure 6.5, fig. 6.6, fig. 6.7, fig. 6.7 show volumes of the configurations

per atom as a function of composition. It is evident that both bcc and hcp configurations

follow Vegard’s law. Laves phases in some cases seem to slightly deviate from the trend,

most notably in fig. 6.7 and fig. 6.8.

16

Chapter 4

Discussion

4.1 Occurring Phases Across 36 Binaries

Based on the features of DFT convex hulls we were able to group 36 alloys into 6

categories. Alloys in each category are displayed in fig. 3.1c, fig. 3.2c, fig. 3.3c, fig. 3.4c,

fig. 3.5c, fig. 3.6c. Using this information and prototypical phase diagrams belonging to

the group we are able to summarize the occurrence of bcc, hcp and Laves phases across

36 binaries, fig. 4.1a. Additionally using thermodynamic assessments from Landolt-

Brnstein database we were able to construct a similar diagram, but with phases seen

experimentally, fig. 4.1b.

DFT calculations are phenomenally consistent with experimental results. We make

a prediction that Laves C14 phase is stable at lower temperatures in Nb-V alloy. Exper-

imentally this system has not been investigated at low temperatures (9). 2 alloys Hf-V

and Zr-V are not consistent with experimentally determined phases. The discrepancy is

due to the absence of the vibrational term in our calculation. From DFT calculations

Laves C15 phase is not on the global convex hull. Since we are only considering configu-

rational entropy it is not enough to stabilize the Laves phase at higher temperatures. If

17

Discussion Chapter 4

we considered vibrational entropy in our calculations, we would be able to reproduce the

stability of Laves C15 phase. Other studies have shown that with harmonic vibrations in

is possible to achieve this result computationally and reproduce the experimental result.

Laves

HCP

BCC

(a) Predicted ground states based on first-principles

calculations in the binary alloys

Laves

HCP

BCC

(b) Expected ground states based on experiments

Figure 4.1: Comparison of zero K ground states across experiments and calculations

4.2 Computational and Experimental Phase Diagrams

Just as discussed in the previous section, based on our results fig. 3.1, fig. 3.2, fig. 3.3,

fig. 3.4, fig. 3.5, fig. 3.6 and experimental phase diagrams from Landolt-Brnstein database

we were able to compare the computed phase diagrams and experimental phase diagrams

across 36 alloys, fig. 4.2.

We can see that computational phase diagrams are predominantly in agreement with

experimental ones. The correct predictions are marked as blue. Two exceptions, Hf-V

and Zr-V are not in agreement due to the absence of the vibrational correction in our

18


calculations; as mentioned in the previous section. Those alloys are marked as purple. We

make a prediction that Nb-V will form Laves C14 phase at lower temperature. Current

experimental data (9) is only available for higher temperatures where Laves C14 is absent.

Nb-V is marked as green in fig. 4.2.

The 5 major disagreements are Ti-Mo, Ti-W, Zr-Nb, Zr-Ta, Hf-Ta. In these systems

miscibility gap in the bcc phase is experimentally observed at higher temperatures (13),

(14),(15),(16), (17). With the calculation techniques that we used to construct the phase

diagrams it is not possible to reproduce this behaviour. The presence of the miscibility

gap can be attributed to new physics in the form of contaminations in the experiments

or the coupling of vibrational degrees of freedom with configurations degrees of freedom

for which we did not account in this study.

19


Miscibility gap dissagreement

Vibrationally stabilized Laves phase

Laves phase prediction

Agreement

Figure 4.2: The predicted agreement with experiment and theory

20

Chapter 5

Conclusion

In this study I explored the thermodynamic properties of binary alloys comprising ele-

ments from groups IV, V and VI in the periodic table. High-throughput first-principles

calculations reveal that the high-temperature phase diagrams of every binary alloy may

be grouped into six classes. I find that the calculations agree with experimental investi-

gations with the exception of a Nb-V where I predict a formation of Laves C14 phase and

one family of binary alloys where I do not predict the existence of a high-temperature

miscibility gap. Further experimental and computational investigation is needed to de-

termine the physics behind the formation of high-temperature miscibility gap.

21

Chapter 6

Appendix

6.1 DFT Formation Energies

V Nb

Ti

Zr

Hf

Ta

BCC

BCC related

HCP

HCP related

Laves C14

Laves C15

Laves C36

Zr Hf V Nb Ta Cr Mo W

Ti

Zr

Hf

V

Nb

Ta

Cr

Mo

W

Figure 6.1: DFT calculations of alloys with elements from group 4 and group 5

22

Appendix Chapter 6

Cr Mo

Ti

Zr

Hf

W

BCC

BCC related

HCP

HCP related

Laves C14

Laves C15

Laves C36


Ti

Zr

Hf

V

Nb

Ta

Cr

Mo

W


23

Appendix Chapter 6

Cr Mo

V

Nb

Ta

W

BCC

BCC related

HCP

HCP related

Laves C14

Laves C15

Laves C36


Ti

Zr

Hf

V

Nb

Ta

Cr

Mo

W


24

Appendix Chapter 6

BCC

BCC related

HCP

HCP related

Laves C14

Laves C15

Laves C36


Ti

Zr

Hf

V

Nb

Ta

Cr

Mo

W

G4 G5

R4-R5

R4-R6

R5-R6

G6Ti-Zr V-Nb Cr-Mo

Cr-W

Mo-WNb-TaHf-Zr

Ti-Hf V-Ta

Figure 6.4: DFT calculations of alloys with elements from group 4, group 5, group 6

25

Appendix Chapter 6

6.2 DFT Volumes

V Nb

Ti

Zr

Hf

Ta


Ti

Zr

Hf

V

Nb

Ta

Cr

Mo

W

BCC

HCP

Laves C14

Laves C15

Laves C36

Figure 6.5: Volumes of alloys with elements from group 4 and group 5

26

Appendix Chapter 6

Cr Mo

Ti

Zr

Hf

W


Ti

Zr

Hf

V

Nb

Ta

Cr

Mo

W

BCC

HCP

Laves C14

Laves C15

Laves C36


27

Appendix Chapter 6

Cr Mo

V

Nb

Ta

W

BCC

Laves C14

Laves C15

Laves C36


Ti

Zr

Hf

V

Nb

Ta

Cr

Mo

W


28


Ti

Zr

Hf

V

Nb

Ta

Cr

Mo

W

G4 G5

R4-R5

R4-R6

R5-R6

G6

Ti-Zr V-Nb Cr-Mo

Cr-W

Mo-WNb-TaHf-Zr

Ti-Hf V-Ta

BCC

HCP

Laves C14

Laves C15

Laves C36

Figure 6.8: Volumes of alloys with elements from group 4, group 5, group 6

—– Bibliography —————-

29

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32

First Principles Investigation of 36 Binary Refractory Alloys · 2020. 2. 6. · component alloys made of only refractory elements. Refractory elements in this study involve elements

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