◦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
Pavel Anisimovich Dolin
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
Pavel Anisimovich Dolin
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
Figure 3.2a shows the calculated formation energies for the Laves phases and several
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
Figure 3.4a shows the calculated formation energies for the Laves phases and several
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
Figure 3.6a shows the calculated formation energies for the Laves phases and several
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
Discussion Chapter 4
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
Discussion Chapter 4
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
Zr Hf V Nb Ta Cr Mo W
Ti
Zr
Hf
V
Nb
Ta
Cr
Mo
W
Figure 6.2: DFT calculations of alloys with elements from group 4 and group 6
23
Appendix Chapter 6
Cr Mo
V
Nb
Ta
W
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 6.3: DFT calculations of alloys with elements from group 5 and group 6
24
Appendix Chapter 6
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
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
Zr Hf V Nb Ta Cr Mo W
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
Zr Hf V Nb Ta Cr Mo W
Ti
Zr
Hf
V
Nb
Ta
Cr
Mo
W
BCC
HCP
Laves C14
Laves C15
Laves C36
Figure 6.6: Volumes of alloys with elements from group 4 and group 6
27
Appendix Chapter 6
Cr Mo
V
Nb
Ta
W
BCC
Laves C14
Laves C15
Laves C36
Zr Hf V Nb Ta Cr Mo W
Ti
Zr
Hf
V
Nb
Ta
Cr
Mo
W
Figure 6.7: Volumes of alloys with elements from group 5 and group 6
28
Zr Hf V Nb Ta Cr Mo W
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
Bibliography
[1] G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Phys.
Rev. B 47 (Jan, 1993) 558–561.
[2] G. Kresse and J. Furthmuller, Efficient iterative schemes for ab initio total-energy
calculations using a plane-wave basis set, Phys. Rev. B 54 (Oct, 1996)
11169–11186.
[3] G. Kresse and J. Furthmller, Efficiency of ab-initio total energy calculations for
metals and semiconductors using a plane-wave basis set, Computational Materials
Science 6 (1996), no. 1 15 – 50.
[4] G. Kresse and J. Hafner, Ab initio molecular-dynamics simulation of the
liquid-metal–amorphous-semiconductor transition in germanium, Phys. Rev. B 49
(May, 1994) 14251–14269.
[5] J. C. Thomas and A. V. d. Ven, Finite-temperature properties of strongly
anharmonic and mechanically unstable crystal phases from first principles, Phys.
Rev. B 88 (Dec, 2013) 214111.
[6] B. Puchala and A. Van der Ven, Thermodynamics of the zr-o system from
first-principles calculations, Phys. Rev. B 88 (Sep, 2013) 094108.
30
[7] A. V. der Ven, J. Thomas, Q. Xu, and J. Bhattacharya, Linking the electronic
structure of solids to their thermodynamic and kinetic properties, Mathematics and
Computers in Simulation 80 (2010), no. 7 1393 – 1410. Multiscale modeling of
moving interfaces in materials.
[8] B. Predel, Cr - W (Chromium - Tungsten), pp. 241–241. Springer Berlin
Heidelberg, Berlin, Heidelberg, 2012.
[9] B. Predel, Nb-V (Niobium-Vanadium), pp. 1–3. Springer Berlin Heidelberg, Berlin,
Heidelberg, 1997.
[10] B. Predel, Nb-W (Niobium-Tungsten), pp. 1–3. Springer Berlin Heidelberg, Berlin,
Heidelberg, 1997.
[11] A. Dinsdale, Sgte data for pure elements, Calphad 15 (1991), no. 4 317 – 425.
[12] B. Predel, Hf-Zr (Hafnium-Zirconium), pp. 1–4. Springer Berlin Heidelberg,
Berlin, Heidelberg, 1996.
[13] P. Franke and H. J. Seifert, eds., Binary System Mo-Ti, pp. 47–47. Springer Berlin
Heidelberg, Berlin, Heidelberg, 2012.
[14] P. Franke and H. J. Seifert, eds., Binary System Ti-W, pp. 63–63. Springer Berlin
Heidelberg, Berlin, Heidelberg, 2012.
[15] B. Predel, Nb-Zr (Niobium-Zirconium), pp. 1–2. Springer Berlin Heidelberg,
Berlin, Heidelberg, 1997.
[16] B. Predel, Ta-Zr (Tantalum-Zirconium), pp. 1–1. Springer Berlin Heidelberg,
Berlin, Heidelberg, 1998.
31
[17] B. Predel, Hf-Ta (Hafnium-Tantalum), pp. 1–2. Springer Berlin Heidelberg, Berlin,
Heidelberg, 1996.
32