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Distinct point defect behaviours in body-centered cubic medium-entropy alloy NbZrTi induced by severe lattice distortion

March 2022
Acta Materialia 229(3):117806

March 2022
229(3):117806

DOI:10.1016/j.actamat.2022.117806

Authors:

Tan Shi

University of Michigan

Zhengxiong Su

Xi'an Jiaotong University

Jing Li

Xi'an Jiaotong University

C.G. Liu

Yantai University

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The point defect properties of body-centered cubic medium-entropy alloy NbZrTi were studied by first-principles calculations. Due to severe lattice distortion, a significant portion of conventional vacancy and interstitial structures are unstable and require large structural relaxation, indicating an irregular energy landscape with large site-to-site variations. The average vacancy and interstitial formation energy are 0.95 eV± 0.34 eV and 1.92 eV ± 0.39 eV, respectively, much lower than that of Nb (2.77 eV and 4.38 eV). The vacancy migration energy exhibits a wide distribution extending to 0 eV, resulting in preferential vacancy migration through low barrier sites. The interstitial diffusion is slower than that of pure Nb due to the reduction of long <111> diffusion induced by the site-to-site variations in stable interstitial orientations. Ti atoms diffuse much faster than Nb and Zr atoms due to the preferential interstitial binding with Ti. The effect of atomic composition and short-range order on elemental and total interstitial diffusion was also investigated. The obtained first-principles results are important for the development of interatomic potentials for radiation damage studies. When irradiated with 3-MeV Fe ions at 675 ∘C to a peak dose of ∼100 dpa, NbZrTi reduced the void formation at high temperature compared to Nb owing to its higher equilibrium vacancy concentration and closer mobility between vacancies and interstitial atoms.

(a) Valence charge density of NbZrTi and the corresponding atom types in the (1$\overline{1}$0) plane. The partial charge density from Ef - 5~eV to Ef was selected, where Ef is the Fermi energy. (b) Phonon band spectrum of NbZrTi. (c) Spatial distribution of atomic positions of NbZrTi at a temperature of 300 K, 600 K and 900 K. Atom positions at each timestep are projected in the (001) plane with four atoms aligned in the same column.

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(a) Distribution of vacancy migration energies in NbZrTi. (b) Correlation between the forward and backward migration energy. (c) Vacancy migration energies for paths to all the 1NN sites for three different Nb vacancy sites. (d) Representative vacancy migration barriers. The corresponding transition structures can be found in Fig.~S11 of the Supplementary Materials

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(a) Interstitial formation energies for different final interstitial configurations (blue error bar: standard deviation; gray error bar: maximum and minimum). (b) Occurrence probabilities of different types of final interstitial configurations. (c) Occurrence probabilities of different elements in the final interstitial configurations.

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(a) Mean square displacement of interstitial migration in NbZrTi and Nb at 1200 K. (b) Representative trajectories of interstitial pairs in NbZrTi and Nb with the same diffusion time scale. The interstitial trajectories are unwrapped from the simulation box with periodic boundaries and are enclosed here in boxes of the same length scale.

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(a) Cross-sectional STEM-BF image of Nb irradiated at 675 °C. Depth profiles of the dpa and Fe ion concentration predicted by the SRIM code are shown. (b-c) BF TEM images of void formation in the irradiated regions of pure Nb at two different depths acquired at under-focused condition. (d) Cross-sectional STEM-BF image of NbZrTi irradiated at 675 °C along with the depth profiles of dpa and Fe ion concentration. (e-f) No formation of voids and visible defect microstructures in the irradiated NbZrTi.

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Distinct point defect behaviours in body-centered cubic medium-entropy alloy

NbZrTi induced by severe lattice distortion

Tan Shia, Zhengxiong Sua, Jing Lia, Chenguang Liuc, Jinxue Yanga, Xinfu Hed, Di Yuna, Qing Penge,∗,

Chenyang Lua,b,∗

aSchool of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, 710049, China

bState Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, 710049, China

cCollege of Nuclear Equipment and Nuclear Engineering, Yantai University, Yantai, 264005, China

dChina Institute of Atomic Energy, Beijing, 102413, China

eState Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

Abstract

The point defect properties of body-centered cubic medium-entropy alloy NbZrTi were studied by ﬁrst-principles

calculations. Due to severe lattice distortion, a signiﬁcant portion of conventional vacancy and interstitial structures

are unstable and require large structural relaxation, indicating an irregular energy landscape with large site-to-site

variations. The average vacancy and interstitial formation energy are 0.95 eV±0.34 eV and 1.92 eV ±0.39 eV,

respectively, much lower than that of Nb (2.77 eV and 4.38 eV). The vacancy migration energy exhibits a wide

distribution extending to 0 eV, resulting in preferential vacancy migration through low barrier sites. The interstitial

diﬀusion is slower than that of pure Nb due to the reduction of long <111>diﬀusion induced by the site-to-site

variations in stable interstitial orientations. Ti atoms diﬀuse much faster than Nb and Zr atoms due to the preferential

interstitial binding with Ti. The eﬀect of atomic composition and short-range order on elemental and total interstitial

diﬀusion was also investigated. The obtained ﬁrst-principles results are important for the development of interatomic

potentials for radiation damage studies. When irradiated with 3-MeV Fe ions at 675 °C to a peak dose of ∼100 dpa,

NbZrTi reduced the void formation at high temperature compared to Nb owing to its higher equilibrium vacancy

concentration and closer mobility between vacancies and interstitial atoms.

Keywords: Medium-entropy alloys, Point defect properties, First-principles calculations, Ion irradiation, Radiation

resistance

1. Introduction

Medium-entropy alloys (MEAs) and high-entropy al-

loys (HEAs) are important groups of multi-principal el-

ement alloys (MPEAs), a class of materials with two

or more principal alloying elements. The entropy-

based deﬁnitions originate from the contribution of

high mixing entropy to the formation of solid-solution

phase [1]. MPEAs have demonstrated excellent me-

chanical properties [2, 3] and damage resistance [4]

owing to their unique features of lattice distortion,

sluggish diﬀusion, high-entropy eﬀect and cocktail ef-

fect [1]. Among MPEAs, MEA NbZrTi and NbZrTi-

based refractory HEAs have exhibited great mechani-

∗Corresponding authors.

Email addresses: pengqing@imech.ac.cn (Qing Peng),

chenylu@xjtu.edu.cn (Chenyang Lu)

cal performance and have shown great promise for ap-

plications as high-temperature structural materials [3,

5, 6]. This group of materials has been recently

considered for potential employment in advanced nu-

clear reactors [7]. The advanced reactors pose se-

vere challenges to the material performance, as mate-

rials need to suﬀer from high operating temperature (up

to 1050 °C), high irradiation damage (up to 150 dis-

placement per atom (dpa)), harsh corrosive environ-

ment, and also ideally a longer service time [8, 9].

Therefore, there is a continuous demand for ideal ma-

terial candidates that can meet these extreme require-

ments. As conventional refractory alloys, NbZrTi-based

refractory MPEAs have excellent mechanical strength

at high temperature [3]. In addition, various types

of NbZrTi-based alloys have also shown good room-

temperature ductility [7], such as NbZrTi [5], NbZr-

Preprint submitted to Acta Materialia March 5, 2022

TiHf [10], NbZrTiHfTa [11, 12], NbZrTiMo0.3V0.3[13],

NbZrTiTaHfMo0.75 [14], (NbZrTiHf)0.95Al0.05 [15], etc.

The body-centered cubic (BCC) structure also has an

intrinsically better resistance to radiation-induced void

swelling [16]. The concept of MPEAs oﬀers a large

compositional space for the material property tuning,

but the actual application of a material is usually a

multifaceted problem where diﬀerent properties need

to be balanced at the stage of alloy design. While the

aspects of alloy production and mechanical properties

of various alloy compositions have been studied previ-

ously [3], the point defect properties have rarely been

explored. As point defect energetics have a great im-

pact on the mechanical properties [17], corrosion be-

haviours [18] and radiation resistance [19], they are

studied in this work with NbZrTi as a representative ma-

terial to show the distinct point defect properties of this

group of materials.

Point defect behaviours have been systematically

studied in face-centered cubic (FCC) structures, in-

cluding Ni-based binary and ternary MPEAs [19,

20], Co-free HEA NiFeMnCr [21] and CoCrNi-based

MPEAs [22], and have also been studied in BCC HEA

VTaCrW [23]. These studies have all shown that point

defect formation and migration energies exhibit a cer-

tain degree of energy ﬂuctuation due to the diﬀerence in

the local chemical environment, and the interstitial dif-

fusion is reduced due to lattice distortion and sluggish

diﬀusion eﬀect. The closer mobility between vacancies

and interstitials is believed to promote the point defect

recombination during irradiation. However, the defect

energy seems to be also alloy dependent. For example,

there is no consistent trend for the relative magnitude of

vacancy formation energy compared to the constituent

elements. In addition, due to the more severe lattice dis-

tortion of BCC MPEAs, the energy distribution is much

wider [23], indicating signiﬁcantly diﬀerent point defect

properties from their widely studied FCC counterparts.

Recently, there have been controversies regarding

the currently established theory on the radiation resis-

tance of MPEAs. FCC MPEAs, which are more ex-

tensively studied, have shown great radiation resistance

including reduced void swelling [4, 24], delayed de-

fect microstructure growth and less degree of radiation-

induced segregation [25]. It is believed to be attributed

to the enhanced point defect recombination, slower en-

ergy dissipation, extended incubation period of dislo-

cation loop growth and slower diﬀusion of defect clus-

ters [4, 20, 25, 26]. However, Parkin et al. found similar

saturation dose for defect clusters between BCC NbTa-

TiV and V under heavy ion irradiation [27]. If the de-

fect recombination rate is higher and the defect cluster

growth is delayed, the threshold dose should be higher

in MPEAs. The same work also showed that NbTaTiV

with a large lattice distortion did not lead to a greater re-

duction in defect generation compared to its FCC coun-

terparts, whereas the reduction in defect mobility is usu-

ally believed to be associated with the magnitude of

lattice distortion. In addition, by the same method of

molecular dynamics (MD) simulations of collision cas-

cades induced by energetic particles [28], discrepancies

exist on whether enhanced point defect recombination

during the initial displacement cascade is a major con-

tributor to the radiation resistance of MPEAs, which

may be only caused by defect evolution on a longer time

scale [29, 30]. Recently, there has been great progress

in the development of interatomic potentials for defect

and radiation damage studies for BCC MPEAs, includ-

ing MoNbTaVW [31], TiVTa [32, 33] and VTaW [33].

Thus, for NbZrTi-based MPEAs and also other BCC

MPEAs, it is crucial to ﬁrst understand their unique

point defect behaviours by ﬁrst-principles method so

that accurate interatomic potentials can be constructed

and their impact on radiation resistance can be accu-

rately characterized.

In this work, we present a ﬁrst-principles investiga-

tion on the energy properties and migration behaviours

of point defects in NbZrTi. First, the magnitude of

lattice distortion of NbZrTi was characterized, show-

ing more severe lattice distortion compared to FCC and

some BCC MPEAs. Next, vacancy formation energy

and migration energy were systematically calculated,

revealing an unusual energy landscape induced by the

large lattice distortion and chemical complexity. The

interstitial formation energy and migration behaviours

were then studied, along with the impact of atomic com-

position and short-range order (SRO) on interstitial mi-

gration. The obtained ﬁrst-principles results were then

used to benchmark an interatomic potential based on the

embedding-atom method (EAM) potential. Ion irradia-

tion experiment was performed at 675 °C with 3-MeV

Fe ions at a peak dose of ∼100 dpa to study the diﬀer-

ence in radiation damage behaviours between pure Nb

and NbZrTi, showing a direct impact of point defect en-

ergetics on the material irradiation resistance.

2. Methodology

2.1. First-principles calculations

The ﬁrst-principles calculations were performed with

the Vienna ab initio simulation package (VASP) us-

ing the projector-augmented wave method [34]. The

generalized gradient approximation of Perdew-Burke-

Ernzerhof (PBE) formalism was used to describe the

exchange-correlation functionals [35]. Semi-core elec-

trons were included in all calculations to describe

short interatomic distances for interstitial conﬁgura-

tions, where 13, 12 and 12 electrons were considered

for Nb, Zr and Ti, respectively. The studied system con-

sists of 128 atoms with 42 Nb atoms, 43 Zr atoms, and

43 Ti atoms. The alloy structure was constructed by

the special quasi-random structure (SQS) method [36]

using the Alloy Theoretic Automated Toolkit [37, 38]

to ensure the equal probability of appearance of each

type of atom at the ﬁrst nearest neighbour (1NN) and

second nearest neighbour (2NN) sites. The Brillouin

zone was sampled with a Monkhorst-Pack mesh of 3

×3×3. This choice of k-point grid was validated by

vacancy and interstitial formation energies of pure Nb,

Zr and Ti (see Fig. S1 of the Supplementary Materi-

als for the comparison of vacancy and interstitial for-

mation energies with diﬀerent k-point meshes) [39–41].

The cutoﬀenergy for the plane-wave basis was set to

400 eV, and the convergence criteria for ionic relax-

ation and self-convergence loop were set to 10−4eV and

0.01 eV/Å, respectively. The spin polarization was ini-

tially included by using three diﬀerent initial magnetic

moments, but no magnetism was manifested after en-

ergy minimization. Thus, spin polarization was not in-

cluded for all the following calculations of NbZrTi.

First, the system volume and atomic coordinates of

the perfect lattice structure was determined based on

the conjugate gradient optimization. The phonon cal-

culation was performed to verify the structural dynami-

cal stability. Second-order force constants of the super-

cell were calculated with a ﬁnite atomic displacement

of 0.01 Å to each atom of the supercell for a total 768

sets of displacement. We maintained a k-point mesh of

3×3×3 and used an additional support grid for the

calculation of augmentation charges. The energy con-

vergence criteria was set to 10−7eV. The band unfolding

method from Ref. [42, 43] was used to decompose the

phonon modes onto the Brillouin zone of the primitive

BCC cell based on the crystallographic symmetry using

the PHONOPY package [44].

Next, diﬀerent vacancy and interstitial structures

were constructed and relaxed with a ﬁxed system vol-

ume. For the calculation of vacancy formation ener-

gies, 15 vacancies of each type of atom were randomly

selected while it was ensured that the average number

of each element is approximately equal at neighbour-

ing sites. The initial structures for the calculation of

interstitial formation energies include in total 18 <110>

dumbbells, 18 <111>dumbbells, 9 <100>dumbbells

and 9 tetrahedral sites with equal number of Nb, Zr or Ti

atoms being added as the interstitial atoms. The choice

of initial interstitial structures is based on the relative

stability of diﬀerent interstitial structures in pure Nb.

Our simulations show that the ﬁnal interstitial structures

have weak dependence on the initial structures. In ad-

dition to the random generation of interstitial sites, dif-

ferent <110>and <111>dumbbell orientations at the

same interstitial site was also simulated to study the im-

pact of local chemical environment. The point defect

formation energy was calculated as:

def =Edef −Eper ±µx,(1)

where Eper is the energy of the perfect structure, Edef

is the system energy with a vacancy/interstitial, and µx

is the chemical potential of the removed atom (+) or

added atom (-). The chemical potential of each ele-

ment was determined by the Widom-type substitution

technique [45] as adopted by previous defect studies of

MPEAs [20, 21]. By substituting one element to an-

other and calculating the energy diﬀerence of the two

systems, the chemical potential of each element can be

determined. 20 substitutions were computed in total and

the chemical potentials for Nb, Zr and Ti were deter-

mined to be -10.18 eV ±0.03 eV, -8.39 eV ±0.04 eV

and -7.76 eV ±0.02 eV, respectively. The Bader charge

analysis was performed to determine the charge and vol-

ume of the removed atom [46].

The vacancy migration energies were computed

using the climbing-image nudged elastic band (CI-

NEB) method with a force convergence tolerance of

0.03 eV/Å [47]. In total, 60 migration paths were com-

puted in which the ﬁrst 30 migration paths used 3 or

5 intermediate images to characterize the shape of the

migration barriers and the rest of the migration paths

used 1 intermediate image to increase the statistics of

the average migration energy. As there is a large varia-

tion among migration barriers of diﬀerent sites, one in-

termediate image has suﬃcient accuracy to evaluate the

average migration energy.

The interstitial migration dynamics behaviours were

studied by the ab initio molecular dynamics (AIMD)

method. In dynamics simulations, a single k-point was

employed with an energy cutoﬀof 400 eV for the plane-

wave basis. A systematic comparison of point defect

formation energies was performed between a k-point

mesh of 1 ×1×1 and 3 ×3×3, showing similar

average formation energies and correct order for diﬀer-

ent types of interstitial pairs (see Table S1 and S2 of

the Supplementary Materials for the comparison of va-

cancy and interstitial formation energies). The system

was controlled by the NVT ensemble using the Nose-

Hoover thermostat. The interstitial migration of diﬀer-

ent structures was compared at a temperature of 1200 K.

The results were averaged over four diﬀerent simula-

tions in which the interstitial atom was introduced at a

random site. For each simulation, after an initial equili-

bration time of 3 ps, a total simulation time of 32 ps was

computed with a timestep of 2 fs. The mean square dis-

placement (MSD) was analyzed by the VASP analysis

program VASPKIT [48]:

MS D(m)=1

i=1

Nt−m

Nt−m−1

k=0

(ri(k+m)−ri(k))2,(2)

where mis the length of the time window, ri(k) is the

position vector of atom iat a timestep of k,Ntis the

total number of timesteps, and Nais the total number

of atoms. The MSD of the ﬁrst half of the simulation

time was selected to avoid large statistical variations at

longer time due to limited integration timesteps [19, 49].

The interstitial trajectory was analyzed via the Wigner-

Seitz defect analysis from the Ovito program [50]. The

distribution of atomic positions of the perfect lattice

was also studied with similar settings using the AIMD

method at 300 K, 600 K and 900 K. The NVT ensem-

ble was employed with a timestep of 3 fs for a total of

15 ps. The atom positions after 1000 timesteps were

used to ensure the equilibration of the system.

Lattice Monte Carlo (MC) simulations were per-

formed to obtain the SRO structure of NbZrTi [51]. The

positions of two random atoms of diﬀerent species were

exchanged with an acceptance probability based on the

Metropolis-Hastings sampling [52]. The applied tem-

perature was 500 K. The supercell volume was opti-

mized every 50 swaps. A k-point mesh of 1 ×1×1 was

used during the MC process. The Warren-Cowley SRO

parameter was used to describe the extent of chemical

ordering [51, 53]:

αν

i j =1−

pν

i j

,(3)

where pν

i j is the probability of ﬁnding element jaround

element iin the νth neighbouring shell, cjis the atomic

concentration of element j. A random solid solution

corresponds to a SRO parameter of zero. Positive and

negative values correspond to a tendency of deceasing

and increasing the number of i,jpairs, respectively. As

shown in Fig. 1, with the increase of MC steps, there is

a general trend of energy decrease and a clear tendency

of elemental ordering. Although it is diﬃcult to obtain

a full convergence with limited MC steps, the variations

of system energy and SRO parameters become more

gentle around 1500 steps. Thus, the SRO structure at

1500 MC steps was used for the following study of in-

terstitial migration.

(b)

(a)

Figure 1: (a) Evolution of accepted total energy relative to the initial

system energy as a function of the Monte Carlo steps. (b) Evolution

of Warren-Cowley SRO parameters of diﬀerent atom pairs in the ﬁrst

nearest neighbour shell for the accepted Monte Carlo steps.

For comparison among BCC and FCC structures,

equiatomic NiCoCr and NiCoFeCrMn with a FCC

structure and equiatomic VTaTi and VTaW with a BCC

structure were also studied by ﬁrst-principles calcula-

tions with similar simulation settings. NiCoCr, VTaTi

and VTaW were used to compare the extent of lattice

distortion and NiCoFeCrMn was used to study the pro-

cess of vacancy structural relaxation. For FCC struc-

tures, a supercell of 108 atoms was constructed with the

SQS method. A Monkhorst-Pack grid of 2 ×2×2 was

used to sample the Brillouin zone with an energy cut-

oﬀof 500 eV for the plane-wave basis. Spin polariza-

tion was considered for all calculations to account for

the magnetic properties of the constituent elements. For

each alloy system, three diﬀerent initial paramagnetic

spin moments were considered, and the relaxed struc-

ture with the lowest system energy was selected.

2.2. Irradiation experiment and characterization

The NbZrTi with a nominal composition of

Nb33Zr34 Ti33 (at.%) was prepared by arc melting and

casting by mixing pure metals (purity >99.9 wt.%)

in a vacuum induction furnace. The as-cast alloys

were homogenized at 1200 °C for 2 h in an Ar atmo-

sphere to achieve a uniform elemental distribution. The

as-homogenized bulk sample presented a single BCC

structure, as evidenced by the X-ray diﬀraction pattern

(see Fig. S2 of the Supplementary Materials). For com-

parison, Nb with a purity >99.99 wt.% was also pre-

pared for the irradiation experiment. Before irradiation,

all samples were mechanically polished using colloidal

silica and then vibratory polished to remove the stress

layer, resulting in a mirror surface with a roughness of

less than 3 nm. Subsequently, the samples were irra-

diated at 675 °C with 3-MeV Fe ions at a ﬂuence of

1.02 ×1017 ions/cm2. The depth proﬁle of displacement

damage was calculated by the SRIM code [54], assum-

ing a displacement threshold energy of 60 eV, 40 eV and

30 eV for Nb, Zr and Ti, respectively [55]. The peak

dose was calculated to be 99.8 dpa and 129 dpa for Nb

and NbZrTi, respectively, with the Kinchin–Pease op-

tion in the quick calculation mode [56].

After irradiation, a focused ion beam (FIB) lift-out

technique was utilized to prepare the cross-sectional

transmission electron microscope (TEM) samples. The

FIB-induced damage was completely removed from the

TEM samples using “ﬂashing polishing” technique to

avoid artifacts [57]. The characterization of radiation-

induced defect microstructures was carried out on a FEI

Talos-F200X operated at 200 kV.

3. Results

3.1. Local lattice distortion

Local lattice distortion is one of the important prop-

erties of MPEAs, which has a signiﬁcant impact on

their mechanical properties, electrical and thermal con-

ductivity, and damage tolerance [4, 58, 59]. Based

on ﬁrst-principles calculations, the lattice constant for

NbZrTi is 3.381 Å, which is in good agreement with

experimental results of 3.36 – 3.41 Å [5]. The BCC

HEAs tend to have a larger lattice distortion than FCC

NiCoFeCrMn-based alloys due to the larger atomic size

mismatch [60, 61]. It is shown from Fig. 2(b) and (c)

that after structural relaxation, the broadening in the

atomic distance is much wider in NbZrTi than the FCC

MEA NiCoCr. Severe lattice distortion can be seen

by the overlap between the 1NN and 2NN in the ra-

dial distribution function, which is also seen in other

refractory MEAs and HEAs including VTaCrW, HfN-

bZr, HfNbTiZr and HfNbTaTiZr [23, 61].

The extent of lattice distortion of NbZrTi is quantiﬁed

here by several commonly used parameters [60, 61, 63],

(a)

(b)

(c)

Figure 2: Radial distribution function of the BCC NbZrTi structure

constructed by the special quasi-random structure method (a) before

and (b) after structural relaxation. The distributions from diﬀerent

atom pairs are overlapped in (a). (c) Radial distribution function of

the FCC equiatomic NiCoCr alloy after structural relaxation.

Table 1: Quantiﬁcation of the local lattice distortion in NbZrTi. The

atomic radii are taken from Ref. [62].

Atomic size Local atomic distortion Average atomic

mismatch δ(%) parameter α2displacement ∆d(Å)

5.03 0.023 0.171

as listed in Table 1:

δ=v

ci(1 −ri/r)2,r=

ciri,(4)

α2=

j≥i

cicjri+rj−2r

2r,(5)

∆d=1

NaX

irrelaxed,i−runrelaxed,i,(6)

where riand ciare the atomic radius and atomic percent-

age of element i,runrelaxed and rrelaxed are atomic position

vectors in unrelaxed and relaxed structures, Nais the to-

tal number of atoms, and α2adopts the parameter repre-

sentation from Ref. [60] to reﬂect the pairwise relation-

ship between diﬀerent atom types. δis frequently used

to estimate the lattice distortion due to the atomic size

diﬀerence [63, 64], and α2has shown strong correlation

with diﬀerent alloy properties including phase, intrinsic

elastic strain energy and excess entropy [60]. However,

these two parameters are calculated directly from the

elemental radii, whereas ∆dis determined by the ﬁrst-

principles method, which should provide a more accu-

rate measure of the lattice distortion. The parameter of

∆dwas compared among supercells of diﬀerent sizes,

showing that a supercell of 128 atoms is suﬃcient to

represent the lattice distortion (see Fig. S3 of the Sup-

plementary Materials for the comparison of supercell

sizes of 54 atoms, 128 atoms and 250 atoms). The ∆d

parameters for Nb, Zr and Ti are 0.14 Å, 0.15 Å and

0.22 Å, respectively, where Ti has the largest displace-

ment from its ideal lattice site.

By examining the 1NN bond length of diﬀerent atom

pairs, it is shown that Zr atoms tend to have larger

bond length while Nb-containing bond length tends to

be smaller. Ti atom, as the lightest element in NbZrTi,

tends to accommodate these diﬀerences by atomic dis-

placements. By analyzing the projected density of states

(DOS) of diﬀerent elements, it is also shown that Ti

has a large DOS near Fermi level, which generally sug-

gests its inferior lattice instability compared to Nb and

Zr [23, 65]. Thus, large relaxation can further help sta-

bilize Ti atoms in the BCC structure. The supporting

evidence related to bond length and DOS analysis can

be found in Table S3, Fig. S4 and Fig. S5 of the Sup-

plementary Materials. In addition, there is a tendency

of forming larger atomic displacement when there are

fewer Nb atoms in the 1NN. The d-orbitals of a BCC

structure can be classiﬁed into the d-egand d-t2gcom-

ponent. The d-t2gcontribution points along the <111>

direction (direction of 1NN) and helps stabilize the BCC

structure, whereas the d-egcontribution points along the

<100>direction (direction of 2NN) [66]. The valence

charge density of NbZrTi is presented in Figure 3(a)) in

the (110) plane to visualize both 1NN and 2NN chemi-

cal bonding. Clearly, Nb atoms have a strong d-t2gcom-

ponent along the <111>direction, which is manifested

by the higher charge density between nearest neigh-

bours. In contrast, the 1NN bonding is relatively weaker

for Ti and Zr atoms. Thus, when fewer Nb atoms are

present in the surroundings, especially with 0 or 1 Nb

atom in the 1NN, the lack of stabilizing element leads

to larger local lattice distortion.

Although local lattice distortion has proved to be es-

sential in stabilizing BCC structures [67, 68], atomic

displacements from chemically-induced lattice distor-

tion could be intermixed with atomic displacements in-

duced by the tendency of structural transformation [68].

Thus, before performing systematic defect calculations,

the phonon dispersion of the perfect lattice was calcu-

lated, as shown in Fig. 3(b). Below zero frequency, there

is little imaginary component (less than 0.05 THz−1

in the spectral function), showing the stability of the

studied structure. Signiﬁcant phonon broadening can

be observed due to the ﬂuctuations of mass and force

constant in the MPEAs and the extent of broadening

is also consistent with other refractory MPEAs [43].

As the structural instability is usually associated with

the transition to HCP and ωphases, three HCP and ω

phase SQS structures were constructed using 96 atoms

and 81 atoms, respectively. If the cell shape was not

ﬁxed, all six conﬁgurations relaxed to BCC structure

after energy minimization (see Fig. S6 of the Supple-

mentary Materials for the structural transformation). If

the cell shape was ﬁxed, the energy of HCP phase was

33 meV/atom higher than the BCC phase, whereas the

ωphase still transformed to the BCC phase, showing

the stability of the BCC structure. In addition to the

stability study at 0 K, the atom trajectories from vibra-

tion were also examined at 300 K, 600 K and 900 K

(see Fig. 3(c)). If there is a tendency of phase trans-

formation, the atom positions will exhibit a large devia-

tion from the ideal BCC lattice sites and show a consis-

tent shift along a speciﬁc direction for all atoms, as evi-

denced in the case of TiZrHf [68]. For NbZrTi, most of

the atoms vibrate around BCC lattice sites with a circu-

lar distribution proﬁle within the projection plane. Only

few atoms (∼3 out of 128 atoms) show a deviation from

the ideal lattice site at 300 K, and this localized insta-

bility is greatly reduced at 600 K and 900 K. Based on

these analyses, we show that the constructed SQS struc-

ture is stable with little dynamical instability.

Based on the parameter of ∆d, the extent of local

lattice distortion in NbZrTi is larger than BCC NbTiV

and VTaCrW, and similar to AlNbTiV and HEAs from

the HfNbTaTiZr-based alloy system [23, 61]. Although

point defect properties depend on the physical and elec-

[001]

[110]

1.0

0.0

0.2

0.4

0.6

0.8

e/Å3

(a)

2 NN

1 NN

(b)

Figure 3: (a) Valence charge density of NbZrTi and the corresponding atom types in the (110) plane. The partial charge density from Ef- 5 eV

to Efwas selected, where Efis the Fermi energy. (b) Phonon band spectrum of NbZrTi. (c) Spatial distribution of atomic positions of NbZrTi at

a temperature of 300 K, 600 K and 900 K. Atom positions at each timestep are projected in the (001) plane with four atoms aligned in the same

column.

tronic properties of the constituent elements, the local

lattice distortion has a great impact on the structural sta-

bility and energy landscape, which also inﬂuences the

point defect behaviours.

3.2. Vacancy

Based on ﬁrst-principles calculations, the vacancy

formation energies (Ev

f) of Nb, Zr and Ti are

0.97 eV ±0.39 eV, 0.95 eV ±0.36 eV and

0.92 eV ±0.29 eV, respectively, as shown in Fig. 4(a).

There is a wide range of energies for each element and

the average energy is approximately equal. The vacancy

formation energy is much smaller than those of its con-

stituent elements (2.77 eV from BCC Nb, and 2.00 eV

and 2.01 eV from Zr and Ti of HCP structure).

Interestingly, in 24% of the relaxed vacancy struc-

tures, the initial vacancy site was ﬁlled by a neighbour-

ing atom after energy minimization. For one instance,

the new vacancy site was subsequently ﬁlled by its own

neighbouring atom again. To our best knowledge, this

type of instability of vacancy site was not reported in

other MEAs or HEAs. There is no clear diﬀerence in

the local chemical environment around these unstable

vacancy sites when the number of each element in the

1NN and 2NN shell is assessed. In addition, these sites

are distributed randomly in the studied structure with

no clear spatial aggregation. For comparison, vacancy

relaxation was also performed in FCC HEA NiCoFe-

CrMn and no such instability was identiﬁed. By com-

paring the atomic force of each atom in the unrelaxed

vacancy structures (see Fig. 4(b) and (c)), it is found that

the atomic forces have similar distributions and magni-

tudes between NbZrTi and NiCoFeCrMn. One might

expect that the extreme values of the atomic force in the

unrelaxed structure, corresponding to the neighbouring

atoms of the vacancy site, is higher in NbZrTi due to the

larger lattice distortion. However, it is shown here that

the diﬀerence in the initial atomic force is not signiﬁ-

cant between NbZrTi and NiCoFeCrMn, although two

diﬀerent structures are compared. In addition, the un-

stable atom that ﬁlls into the vacancy site does not nec-

essarily experience a large initial atomic force, which

could be quite low in some cases (see Fig. 4(b)). The

spontaneous replacement can occur when the path to the

(a)

(d) (e)

NbZrTi

NiCoFeCrMn

NbZrTi

(b) (c)

Figure 4: (a) Vacancy formation energy of diﬀerent types of atoms in NbZrTi (blue error bar: standard deviation; gray error bar: maximum and

minimum). (b) Atomic force of each atom in NbZrTi in the unrelaxed structure of Zr vacancy, where the unstable atom refers to the neighbouring

atom that ﬁlls into the vacancy site during the structural relaxation. In total, 15 Zr vacancy structures were simulated. Results for Nb and Ti

vacancies are also included in Fig. S7 of the Supplementary Materials. (c) Atomic force of each atom in NiCoFeCrMn in the unrelaxed vacancy

structure. (d-e) Average displacement per atom in NbZrTi and NiCoFeCrMn after removal of a vacancy.

vacancy site has no energy barrier regardless of the mag-

nitude of the atomic force. Therefore, it is the unusual

local potential landscape that leads to the spontaneous

replacement of the vacancy site by the neighbouring

atoms. The large lattice distortion changed the poten-

tial energy landscape signiﬁcantly from a lattice-based

ﬂuctuation, completely suppressing vacancy migration

barriers between certain neighbouring sites, while the

slope of the potential of these unusual local sites were

not greatly enhanced.

By comparing the atomic displacement during the va-

cancy relaxation using Eq. (6), it is shown from Fig. 4(d)

and (e) that NbZrTi experienced much larger structural

adjustment. The average atomic displacement is ∼6×

larger than that of NiCoFeCrMn. The overall larger de-

gree of structural adjustment may lead to the lower Ev

of NbZrTi compared to that of its constituent elements.

One clear evidence is that for unstable vacancy struc-

tures with spontaneous exchange with a neighbouring

atom, the average Ev

fis 0.69 eV ±0.29 eV, which is

lower than the overall average Ev

fof 0.95 eV ±0.34 eV.

The removal of an atom can trigger a large extent of

system relaxation, which lowers the total system en-

ergy and Ev

f. However, there is no strong correlation

between individual Ev

fand the total relaxed distance of

the system because the contribution of atomic displace-

ment to the lowering of the system energy also depends

on the atom locations relative to the vacancy site. The

relationships between Ev

fand the number of each ele-

ment in the 1NN shell, the number of each element in

the 1NN and 2NN shell, Bader volume of the removed

atom and Bader charge of the removed atom all show

weak correlations (see Fig. S8 - Fig. S10 of the Sup-

plementary Materials for the dependence of Ev

fon these

parameters). Further exploration is needed to determine

whether a single parameter or combined parameters can

be used to estimate the individual vacancy formation en-

ergy. Here, we show that due to the large local lattice

distortion and local chemical complexity, the structural

relaxation is a complicated process and depends heavily

on its local environment.

In previous studies of MPEAs, no signiﬁcant de-

crease of Ev

fwas observed. The average Ev

fof NiFeM-

nCr HEA is 1.96 eV [21], which is comparable to its

constituent elements of Ni [69, 70], Fe [69, 71], Cr [69]

and Mn [71]. In the Ni–containing MPEAs, the aver-

age Ev

fis equal or slightly higher than pure Ni [20]. In

BCC equiatomic VTaCrW, the Ev

falso has a large en-

(a) (b)

Figure 5: (a) Distribution of vacancy migration energies in NbZrTi. (b) Correlation between the forward and backward migration energy. (c)

Vacancy migration energies for paths to all the 1NN sites for three diﬀerent Nb vacancy sites. (d) Representative vacancy migration barriers. The

corresponding transition structures can be found in Fig. S11 of the Supplementary Materials.

ergy spread, and the average Ev

fis slightly lower than

pure W, but higher than pure Ta [23]. Thus, there seems

no clear trend on the relative magnitude of vacancy for-

mation energy compared to pure elements based on pre-

vious studies, and such low Ev

fwas not observed.

First, lower vacancy formation energy in NbZrTi

leads to a higher equilibrium vacancy concentra-

tion [55]:

Cv=exp Sv

k!exp −Ev

kT !,(7)

where Sv

fand Ev

fare vacancy formation entropy and en-

ergy, respectively. For example, for a temperature of

1000 K, the exp (−Ev

f/kT ) term is ∼109higher than that

of pure Nb. Here, energy calculations were performed

at 0 K and we assume that Sv

fdoes not decrease sig-

niﬁcantly relative to its constituent elements [72]. Fur-

ther evaluation of Sv

fis necessary by taking into account

diﬀerent entropy components [73–76]. Recent studies

have shown that for a perfect lattice, the change of vi-

bration entropy is of the same magnitude as that of con-

ﬁgurational entropy among diﬀerent BCC MPEAs [43],

and entropy can be important in predicting phase sta-

bility of binary random solid solutions [77]. From the

perspective of Cv, the equivalent average Ev

fshould be

actually lower than the arithmetic mean due to the wide

distribution of Ev

fand the tendency of vacancies to form

near low-energy neighborhoods [73]. In addition to the

impact on Cv, lower vacancy formation energy also in-

dicates that less energy is required to produce a vacancy,

which means that vacancy defects can be more easily

formed.

These two diﬀerences have implications for materials

used for nuclear applications. First, radiation-induced

swelling is a major concern for long-term mechanical

integrity of structural materials. High equilibrium va-

cancy concentration promotes the thermal emission of

vacancies by void, reducing the extent of swelling at

high temperature [55]. Second, it is believed that dur-

ing the initial stage of irradiation when the displacement

cascade takes place, the reduced heat dissipation from

MPEAs can result in enhanced defect recombination,

leading to a lower number of surviving defects after the

thermal spike period [26, 29, 78]. If the defect forma-

tion energies are much lower, the same imparted energy

from the incident radiation can create a higher number

of defects during the thermal spike period, which com-

petes with the aforementioned mechanism of enhanced

defect recombination.

Figure 5(a) shows the distributions of vacancy migra-

tion energies (Ev

m) in NbZrTi. All three elements have

similar wide energy distributions with the lower end ex-

tending to nearly 0 eV. This is consistent with previ-

ous calculations of vacancy formation energy with no

migration barrier. There is a weak correlation between

the forward and backward migration energy, showing a

Pearson’s correlation coeﬃcient of 0.22 (see Fig. 5(b)).

We need to emphasize that there is a distinct diﬀerence

from FCC MPEAs, which usually have stronger positive

linear correlations between the forward and backward

migration barriers [21, 23].

Figure 5(c) lists the migration barriers to all eight

neighbouring sites at three diﬀerent Nb vacancy sites.

At the same vacancy site, depending on the migration

direction, the energy barrier varies substantially. For in-

stance, the barrier can be 1.4 eV in one direction and

0.1 eV in a diﬀerent direction. At site 3, several mi-

gration paths were unstable, where the ﬁnal states were

spontaneously relaxed to the initial state. Representa-

tive migration paths were presented in Fig. 5(d): the

path 1 and 2 are conventional barrier proﬁles with good

symmetry; the path 3 and 4 have large diﬀerences be-

tween forward and backward migration energies; the

path 5 and 6 exhibit irregular barrier shapes. All these

results indicate that due to the large lattice distortion,

the energy landscape could diﬀer signiﬁcantly at a very

local scale. The site-to-site variation is more promi-

nent compared to FCC MPEAs and even some BCC

MPEAs [23]. For future material modeling at a larger

scale, such as MD and kinetic Monte Carlo (KMC) sim-

ulations, these characteristics need to be taken into ac-

count to reﬂect the distinct defect behaviours of NbZrTi.

Due to the wide distribution of the vacancy migration

energy, using the arithmetic mean to represent the av-

erage migration energy could potentially bias the actual

migration behaviours. For each vacancy site, assum-

ing only the vacancy jumps through the 1NN, there are

eight possible migration pathways. Since the jump fre-

quency scales with Ev

m/kT , the vacancy will preferen-

tially migrate through sites of low Ev

m. A simple KMC

algorithm was used to estimate the equivalent Ev

m. For

each migration event, the number of each type of atom

in the 1NN shell was randomly generated and Ev

mof

the corresponding atom type was randomly assigned to

these sites based on the ﬁrst-principles results. This is

supported by the weak correlation among vacancy mi-

gration energies at the same site (see Fig. 5(c)). The mi-

gration event and step time were then determined based

on the Monte Carlo method [79]. Because the corre-

lation between the forward and backward Ev

mis weak

(see Fig. 5(b)), we assume that each jump is indepen-

dent from its previous jump. In total, 106jumps were

computed. Here, we assume that the Debye frequency

and vacancy migration entropy do not vary, and we only

focus on the impact of the variation of Ev

m. The KMC-

equivalent mean was then determined by equating the

total time and total number of jumps of each element.

As shown in Table 2, the arithmetic mean overesti-

mates the average Ev

mof NbZrTi. This is expected be-

cause it is highly likely that the migration takes place

quickly through the low Ev

msites among all eight neigh-

bouring sites. Since the method of arithmetic mean

gives equal weight to large Ev

mvalues, it could give

inaccurate Ev

morder among diﬀerent elements, which

aﬀects the analysis related to elemental diﬀusion and

segregation. The KMC method is less sensitive to ex-

treme values from the limited data set, as each jump

event is determined collectively from eight diﬀerent

mvalues. Based on the KMC method, the average

mis approximately ∼0.35 eV, which is smaller than

0.45 eV from pure Nb. For pure Zr, the Ev

mvalues are

0.54 eV and 0.63 eV for the basal and non-basal migra-

tion, respectively [80]. For pure Ti, the basal and non-

basal Ev

menergies are 0.40 eV and 0.53 eV [81]. Al-

though Zr and Ti have a diﬀerent crystal structure, we

show here that the Ev

mof NbZrTi is smaller than all of

its constituent elements under ambient conditions. We

note that in NbZrTi, there is a small diﬀerence in Ev

among Nb, Zr and Ti. Due to the large energy spread of

m, more statistics are needed to fully characterize the

relative order of Ev

mamong diﬀerent elements.

3.3. Interstitial

Diﬀerent initial structures including <111>dumb-

bells, <110>dumbbells, <100>dumbbells and tetra-

hedral sites were simulated, as described in Section 2.1.

Among all the conﬁgurations, 61% of the interstitials

moved to a diﬀerent lattice site by rotating their direc-

tions and replacing other neighbouring atoms. The in-

terstitial instability originates from the irregular energy

landscape induced by the large lattice distortion. In the

work of Zhao et al. [23], unstable interstitial conﬁgura-

tions have also been observed in VTaCrW but were not

counted in the calculation of Ei

fdue to the involvement

of antisite exchange. In this work, those conﬁgurations

were included in the calculation of Ei

fusing Eq. (1). We

have them considered because of two factors: (1) they

are representative events in NbZrTi, and (2) they have

stable ﬁnal states based on energy minimization.

Table 2: Average vacancy migration energy based on the KMC-equivalent method and the arithmetic mean. The jump percentage of each type of

atom is normalized by the total number of jumps from the KMC method.

Atom type KMC-equivalent mean at 1000 K (eV) Jump percentage (%) Arithmetic mean (eV) Jump percentage (%)

Nb 0.352 33.3 0.526 ±0.506 4.8

Zr 0.348 34.5 0.378 ±0.298 20.0

Ti 0.354 32.2 0.498 ±0.388 6.5

(a) (b) (c)

Figure 6: (a) Interstitial formation energies for diﬀerent ﬁnal interstitial conﬁgurations (blue error bar: standard deviation; gray error bar: maximum

and minimum). (b) Occurrence probabilities of diﬀerent types of ﬁnal interstitial conﬁgurations. (c) Occurrence probabilities of diﬀerent elements

in the ﬁnal interstitial conﬁgurations.

As shown in Fig. 6(a), in the ﬁnal conﬁgurations,

Ti-Ti dumbbell has the lowest formation energy, fol-

lowed by Ti-Nb and Ti-Zr. Other types are not listed

individually due to their limited number of occurrences.

The average Ei

fis 1.92 eV ±0.39 eV, much lower than

4.38 eV of the most stable <111>dumbbell of pure

Nb. For pure Zr and Ti with HCP structure under

ambient conditions, according to diﬀerent studies, the

fvalues for the most stable interstitial conﬁguration

are in the range of 2.70 eV – 3.03 eV and 2.27 eV –

2.42 eV, respectively, which are also larger than that of

NbZrTi [39, 40, 80, 82]. The occurrence probabilities

of interstitial conﬁgurations with low Ei

fhave a higher

tendency to form (see Fig. 6(b) and (c)). The occurrence

probability here corresponds to static calculations with

the same number of each type of atom being added in

the initial structures. For dynamic simulations, such as

cascade collision simulations for radiation damage stud-

ies, we still expect a higher occurrence probability of

Ti-containing interstitials and a lower probability of Zr-

containing interstitials based on Ei

f. However, we note

that the defect formation dynamics is not reﬂected from

static calculations.

The preferred dumbbell directions are diﬀerent for

diﬀerent types of atoms, as presented in Table 3. For

example, the Nb-Ti interstitial can form both <110>

Table 3: Preferred directions for diﬀerent types of dumbbell intersti-

tials.

Dumbbell Preferred Occurrence Second-preferred Occurrence

type direction probability direction probability

Nb-Ti <110>9/17 <111>7/17

Ti-Ti <110>13/14 <100>1/14

Zr-Ti <110>9/12 <113>2/12

Nb-Nb <111>5/6<012>1/6

Zr-Nb <111>1/2<110>1/2

Zr-Zr <013>1/1

Table 4: Stable ﬁnal interstitial conﬁgurations at the same initial lat-

tice site with diﬀerent initial orientations for a Nb-Nb dumbbell. The

corresponding ﬁnal structures are provided in Fig. S12 of the Sup-

plementary Materials (Lc: lattice parameter, Dinitial−ﬁnal : distance be-

tween the initial and ﬁnal interstitial site).

Initial dumbbell Final dumbbell Dinitial−ﬁnal Ei

f(eV)

direction conﬁguration

[111], [110], [101], [011] Ti-Ti [010] √3Lc1.82

[111], [011] Ti-Ti [011] √3Lc1.79

[111], [110], [101] Ti-Ti [011] 2√3Lc1.31

[111] Nb-Ti [111] 1.5√3Lc1.61

and <111>dumbbells, but Ti-Ti and Nb-Nb pair have

a strong preference for <110>dumbbell and <111>

dumbbell, respectively. Due to the large distortion of

the lattice, unusual dumbbell directions can be formed.

For the same initial interstitial site with diﬀerent dumb-

bell directions, the ﬁnal conﬁgurations can be diﬀerent,

as illustrated in Table 4. The location of the ﬁnal inter-

stitial site can be far away from the initial site and the Ei

of the stable interstitials can also be diﬀerent. For nu-

clear materials, the interstitial structures and formation

energies have a great impact on the dislocation loop nu-

cleation, which further impacts the irradiation harden-

ing and defect microstructure evolution [55, 83–86].

The interstitial migration behaviours were studied by

AIMD simulations at a temperature of 1200 K. NbZrTi

has a lower MSD than pure Nb due to the more tor-

tuous migration pathways (see Fig. 7). The diﬀusion

coeﬃcients for Nb and NbZrTi are 1.02 ×10−10 m2/s

and 3.88 ×10−11 m2/s, respectively, according to the

Einstein relation [87]. The interstitial atom in pure

Nb can form long one-dimensional (1D) diﬀusion path

along <111>direction, whereas the interstitial atom in

NbZrTi diﬀuses in a three-dimensional (3D) manner

with more frequent changes of directions. The 3D mi-

gration behaviour leads to a larger cross section of in-

teractions with defects of low mobility, such as vacancy

and vacancy clusters, and enhances the defect recombi-

nation [88]. The jump rate for NbZrTi is actually 9.6%

higher than that of Nb, but due to the 3D spatial migra-

tion behaviour, the diﬀusion coeﬃcient is only 38% of

Nb at 1200 K.

The correlation factor fcis used to examine the angu-

lar correlation between consecutive jumps [19, 88]:

fc=1+cos θ

1−cos θ,(8)

where θis the angle between two consecutive jumps and

fcis calculated by averaging cosθover all trajectories.

Nb and NbZrTi have a fcof 0.19 and 0.20, respectively,

showing negligible diﬀerence. A value between 0 and

1 means that there is a tendency for the next jump to

return back to the initial location of the previous jump.

Although the average angular correlation between con-

secutive jumps is similar, two mechanisms contribute to

the diﬀerence in diﬀusion coeﬃcients between Nb and

NbZrTi. First, interstitial atoms in NbZrTi have their

own preferred dumbbell directions and have a higher

probability to rotate their orientations (see Table 3).

Therefore, it is more diﬃcult to maintain the same 1D

<111>direction. Second, certain lattice sites can act as

“trap sites” for interstitial atoms and make it diﬃcult for

interstitials to escape their surroundings. Based on the

AIMD trajectories, there are six lattice sites in NbZrTi

that an interstitial spends longer time at or returns to

for more number of times compared to the maximum

case from pure Nb. In addition to the stability of these

sites, we also conjecture that as a large fraction of in-

terstitial structures are unstable at their initial sites, the

distance between stable interstitial structures is farther

than the regular atomic spacing from conventional ma-

terials. Once an interstitial is trapped at a stable site,

it is more diﬃcult to escape due to the longer distance

from the neighbouring stable site. However, this needs

further conﬁrmation by simulations at a larger scale.

In NbZrTi, Ti diﬀuses much faster than Nb and Zr,

as shown in Fig. 8. By examining the time fraction and

number fraction of each type of interstitial, we ﬁnd that

Ti-Ti dumbbell has the highest occurrence probability,

followed by Ti-Nb and Ti-Zr. In contrast, Nb-Zr and

Zr-Zr dumbbells have the lowest probability to form.

This is in good agreement with the order of Ei

f(see

Fig. 6), where Ti-Ti dumbbell has the lowest Ei

fand

Zr-containing interstitials tend to have higher Ei

f. For

each type of interstitials, the time fraction and number

fraction have similar values because diﬀerent interstitial

pairs have similar distributions of dwelling time at each

site (see Fig. S13 of the Supplementary Materials for the

dwelling time distribution of diﬀerent interstitial pairs).

According to the model for radiation-induced segrega-

tion [89], assuming the vacancy diﬀusion behaviours are

identical for diﬀerent alloy elements, a strong binding

of interstitials to a certain type of atom leads to a higher

tendency of its segregation near defect sinks, such as

surface or grain boundary [55]. Here, Ti atom is prefer-

entially associated with the interstitial, which can be ex-

plained by its positive binding energy, i.e., the energy of

converting one type of interstitial to the preferred type

of interstitial [55]. Thus, Ti atom diﬀuses faster through

the interstitial diﬀusion mechanism. The overall segre-

gation behaviour is then determined collectively by the

interstitial and vacancy diﬀusion mechanism [90–92].

Attempts have been made to tune the interstitial dif-

fusion behaviours through changing the elemental com-

position and local atomic bonding. The ﬁrst attempt

is to reduce Ti composition to reduce the diﬀusion of

Ti. When the composition changes from equiatomic

NbZrTi to Nb0.4Zr0.4Ti0.2, as shown in Fig. 9(a) and (b),

the total diﬀusion coeﬃcient is reduced due to the de-

crease of Ti content and the corresponding decrease of

Nb and Ti diﬀusion coeﬃcients. The Ti-Ti dumbbell be-

comes less easily formed due to the reduced Ti compo-

sition. Accordingly, the proportions of Ti-Nb and Ti-Zr

dumbbells become higher (see Fig. 9(c)). As the prefer-

ential Ti diﬀusion is promoted by the preferential inter-

stitial binding of Ti-Ti pair, Ti diﬀusion coeﬃcient be-

comes lower. Although the fraction of Ti-Nb dumbbell

becomes higher, the Ti atom in Ti-Nb dumbbell is more

(a) (b)

NbZrTi

Figure 7: (a) Mean square displacement of interstitial migration in NbZrTi and Nb at 1200 K. (b) Representative trajectories of interstitial pairs in

NbZrTi and Nb with the same diﬀusion time scale. The interstitial trajectories are unwrapped from the simulation box with periodic boundaries

and are enclosed here in boxes of the same length scale.

(a)

(b)

Figure 8: (a) Mean square displacement of diﬀerent elements in

NbZrTi at 1200 K. (b) Time fraction and number fraction of inter-

stitial sites of diﬀerent dumbbell types.

active and the Nb atom is relatively immobile, which

leads to the lower Nb diﬀusion coeﬃcient.

The second attempt is to study the impact of SRO

structure while keeping the same equiatomic composi-

tion. The SRO structure was created by a combined

Metropolis MC and density functional theory (DFT)

scheme, as described in Section 2.1. The overall dif-

fusion coeﬃcient is reduced due to the decrease of Ti

and Nb diﬀusion coeﬃcients (see Fig. 9(d) and (e)).

Fig. 1(b) shows that the SRO structure has a strong ten-

dency of Nb-Nb clustering and a moderate tendency of

Zr-Ti clustering. Interestingly, the interstitial prefers to

migrate in the Ti- and Zr-enriched region as revealed by

the diﬀusion trajectories, which leads to the lower diﬀu-

sion coeﬃcient of Nb. This can also be reﬂected by the

decrease of Ti-Nb dumbbell fraction and the increase of

Ti-Zr dumbbell fraction (see Fig. 9(f)). As the intersti-

tial is mostly conﬁned in the Ti- and Zr-enriched region,

the long-range diﬀusion becomes less favoured, which

results in the decrease of elemental and total diﬀusion

coeﬃcients. The ordering tendencies among diﬀerent

atom pairs all have an impact on the interstitial migra-

tion behaviour. In contrast to these two structural tuning

methods, another attempt by simply reducing the per-

centage of the fast diﬀusion species of Ti atom in the

1NN shell shows no improvement for the reduction of

Ti and total diﬀusion coeﬃcients (see Section 2 of Sup-

plementary Materials for the comparison).

3.4. Ion irradiation experiment

To study the impact of point defect energy on

radiation-induced void swelling, ion irradiation exper-

iment was performed at 675 °C with 3-MeV Fe ions at a

ﬂuence of 1.02 ×1017 ions/cm2, corresponding to a peak

dose of 99.8 dpa and 129 dpa in Nb and NbZrTi, respec-

tively. Void swelling is known to be pronounced at in-

termediate homologous temperature (T/Tm) because the

void growth is limited by the low defect mobility at low

temperature and is limited by the high vacancy super-

saturation concentration at high temperature [55]. As

explained in Section 3.2, the vacancy formation energy

SRO NbZrTi

(MC+DFT)

Nb0.4Zr0.4Ti0.2

(a) (b) (c)

(d) (e) (f)

Figure 9: (a-b) Comparison of MSD of interstitial migration between NbZrTi and Nb0.4Zr0.4Ti0.2. (c) Number fraction of diﬀerent dumbbell types

in Nb0.4Zr0.4Ti0.2. (d-e) Comparison of MSD between SQS-based NbZrTi and NbZrTi with a SRO structure based on the MC +DFT method.

(f) Number fraction of diﬀerent dumbbell types in NbZrTi with a SRO structure based on the MC +DFT method. The dotted lines in (c) and (f)

represent the results from the SQS-based NbZrTi structure.

0 500 1000 1500 2000 2500 3000 3500 4000

120 dpa

Depth (nm)

dpa

0.0

1.5

3.0

4.5

Fe (at.%)

20 nm

(e) (f)

(b) (c)

20 nm

20 nm 20 nm

void void

400-520 nm

950-1070 nm

400-520 nm

1100-1220 nm

(a) (d)

0 500 1000 1500 2000 2500 3000 3500 4000

120

Fe (at.%)

dpa

Depth (nm)

dpa

0.0

1.5

3.0

1 µm1 µm

Figure 10: (a) Cross-sectional STEM-BF image of Nb irradiated at 675 °C. Depth proﬁles of the dpa and Fe ion concentration predicted by the

SRIM code are shown. (b-c) BF TEM images of void formation in the irradiated regions of pure Nb at two diﬀerent depths acquired at under-focused

condition. (d) Cross-sectional STEM-BF image of NbZrTi irradiated at 675 °C along with the depth proﬁles of dpa and Fe ion concentration. (e-f)

No formation of voids and visible defect microstructures in the irradiated NbZrTi.

of NbZrTi is much lower than that of pure Nb, the equi-

librium vacancy concentration should be signiﬁcantly

higher, limiting the void formation at high temperature.

According to previous experiments [93–97], voids can

be formed in Nb by heavy ion irradiation from 650 °C

to 1120 °C, corresponding to a T/Tmof 0.34 to 0.51.

With 3.3-MeV 58Ni+ions, the maximum void swelling

occurred at a temperature of 1010 °C, corresponding to

a T/Tmof 0.47 [93]. Based on these previous eﬀorts,

an irradiation temperature of 675 °C was selected be-

cause in addition to the expected void formation in Nb at

this temperature, a T/Tmof 0.48 in NbZrTi [98] is also

in the normal void swelling temperature window [55].

Furthermore, this temperature falls into the operating

temperature range of fourth-generation nuclear reactors

including gas-cooled fast reactor and molten salt reac-

tor [8, 9].

As shown in Fig. 10(a), in the irradiated region of

pure Nb, voids and dislocation loops were formed by

agglomeration of vacancies and interstitial atoms. Dis-

location loops were also formed outside the SRIM-

predicted ion range by diﬀusion of radiation-induced

defects at high temperature. A great number of voids

were observed throughout the irradiation area with an

average diameter of 5.5 nm ±1.7 nm (see Fig. 10(b) and

(c)). In contrast, no voids and other radiation-induced

defect microstructures were observed in NbZrTi, as

shown in Fig. 10(d) – (f). The dislocation lines at a

depth range of 1500 nm – 2500 nm correspond to pre-

irradiation defects. The energy dispersive spectroscopy

(EDS) mapping also showed that the spatial distribu-

tions of Nb, Zr and Ti are uniform in the irradiated

and unirradiated region, and the atomic composition re-

mains equiatomic (see Fig. S14 and Table S4 of the Sup-

plementary materials for the EDS mapping results). As

expected, the higher equilibrium vacancy concentration

limits the growth of voids, where the vacancy inﬂux to

voids are counterbalanced by the thermal vacancy emis-

sion [55].

In addition, the enhanced recombination of point

defects contribute to the observation of reduced de-

fects. As reported for FCC Ni-based MPEAs and BCC

VTaCrW [20, 23], the recombination probability of

point defects should be higher than pure metals due to

the closer mobility between the vacancy and interstitial

atom. When the point defect recombination is higher,

there are less surviving defects that can further agglom-

erate into large defect microstructures, such as voids and

dislocation loops. This has been considered as one of

the important mechanisms for the enhanced radiation

resistance of HEAs [20, 26]. The same mechanism ap-

plies to NbZrTi because of the lower average vacancy

migration energy (see Table. 2) and lower interstitial dif-

fusion coeﬃcient (see Fig. 7), i.e, higher vacancy mobil-

ity and lower interstitial mobility. However, with lower

vacancy and interstitial formation energy, more vacan-

cies and interstitial atoms are created during the ther-

mal spike period. The relevant importance of these two

competing mechanisms needs to be further evaluated.

4. Discussions

Based on ﬁrst-principles calculations, NbZrTi ex-

hibits distinctive point defect properties from pure

Nb. The average vacancy formation energy is

0.95 eV ±0.34 eV compared to 2.77 eV in Nb. The av-

erage interstitial formation energy is 1.92 eV ±0.34 eV,

whereas the interstitial formation energies in Nb are

4.38 eV – 5.12 eV for diﬀerent interstitial conﬁgura-

tions. In NbZrTi, the interstitial formation energy has a

strong dependence on the dumbbell atom types, which

can be clearly seen from Fig. 6(a). The dependence on

the dumbbell orientation is insigniﬁcant relative to the

site-to-site variation. For example, for a ﬁnal conﬁgura-

tion of <110>dumbbell and <111>dumbbell, the aver-

age energies are 1.83 ±0.37 eV and 2.07 ±0.49 eV, re-

spectively. The diﬀerence could also be attributed to the

preferential formation of <110>dumbbell of the most

stable Ti-Ti pair. According to the KMC-equivalent

method computed at 1000 K (see Table 2), the aver-

age vacancy migration energy is ∼0.35 eV in NbZrTi,

which is lower than 0.45 eV from pure Nb. Based on

AIMD simulations, the interstitial diﬀusion coeﬃcient

in NbZrTi is 38% of that in Nb at 1200 K. In summary,

NbZrTi has much lower vacancy and interstitial forma-

tion energies, higher vacancy mobility and lower inter-

stitial mobility.

The instability of defect structures was observed in

both vacancy and interstitial conﬁgurations. In the va-

cancy structures, it was manifested by the spontaneous

replacement of the vacancy site by a neighbouring atom,

whereas in the interstitial structure, it was manifested by

the large relaxation distance of the interstitial atom away

from the initial interstitial site (see Table 4). This can

be attributed to the severe lattice distortion of NbZrTi

that leads to an irregular energy landscape (see Fig.5(d))

and the resulting suppression of migration barriers at

local sites. The lower vacancy formation and inter-

stitial formation energies can also be explained by the

large degree of relaxation resulting from such an irreg-

ular energy landscape. The BCC VTaCrW, which has

only interstitial instability, has a local lattice distortion

parameter ∆dof 0.087 Å [23], which is smaller than

0.171 Å from NbZrTi. It is also reported that BCC

MEA VTaTi and VTaW have lower interstitial forma-

tion energies due to the large local relaxation based on

classical molecular static simulations [33]. For com-

parison, the ∆dvalues of VTaTi and VTaW were cal-

culated by ﬁrst-principles calculations and were found

to be 0.132 Å and 0.061 Å, respectively, which are

also smaller than that of NbZrTi. By comparing dif-

ferent BCC MPEAs, there seems to be a trend that with

the increase of lattice distortion, the structural instabil-

ity extends from interstitial structures to vacancy struc-

tures and leads to the decrease of the corresponding de-

fect formation energy. NbZrTi-based MPEAs, such as

NbZrTiTaHf and NbZrTiHf, all have similar or even

larger ∆dvalues, ranging from 0.16 Å– 0.20 Å [61].

Thus, it is worth exploring further whether these alloys

have similar low point defect formation energies and

large extent of defect structural relaxation.

As explained previously, it was proposed that the

point defect recombination in MPEAs is higher than

pure elements under irradiation due to the closer mo-

bility between vacancies and interstitials [20]. This was

largely attributed to the slower diﬀusion of interstitial

atoms and the overlap between vacancy and interstitial

migration energy distributions. Here, due to the ex-

tremely large distribution of vacancy migration energy

and also the skewed distribution proﬁle towards low en-

ergy (see Fig. 5(a)), the vacancy migration can occur

much more easily because the probability of migration

through a low migration barrier is high. Therefore, the

higher vacancy mobility can further enhance the point

defect recombination under irradiation. It is noted that

higher mobility of vacancy can also promote the cluster-

ing of vacancies by increasing the vacancy arrival rate at

void embryos [55]. Considering also the other distinct

point defect properties, the overall impact on irradiation

resistance could diﬀer signiﬁcantly from FCC MPEAs

and needs to be further evaluated with MD simulations

and irradiation experiments.

Regarding the diﬀusion capabilities of diﬀerent ele-

ments, Nb, Zr and Ti have similar vacancy migration

energies (see Table 2), and the relative energy diﬀer-

ence cannot be clearly distinguished. On the other hand,

interstitial diﬀusion through Ti is much faster than Zr

and Ti (see Fig. 8) due to the preferential binding of in-

terstitials with Ti atom. The reduction of Ti content in

NbZrTi and the creation of low-energy SRO structure

based on the lattice MC method are both eﬀective in de-

creasing the Ti and total diﬀusion coeﬃcients. A slower

diﬀusion coeﬃcient is usually associated with more

trapping around certain lattice sites or within certain lo-

cal regions, leading to a lower probability of long-range

diﬀusion. When there is a strong short-range clustering

tendency of atom pairs in NbZrTi, the diﬀusion of in-

terstitials is preferred in regions where Zr and Ti are en-

riched, which limits the long-range interstitial diﬀusion.

The existence of SRO structure has been conﬁrmed ex-

perimentally and has shown great impact on the me-

chanical and radiation properties of MPEAs [99–101].

In this work, a SQS model with no SRO was used to cal-

culate the defect properties, but the actual local ordering

tendency in BCC MPEAs could complicate the defect

properties, which needs additional investigation [102].

The obtained ﬁrst-principles results are useful for the

validation of interatomic potentials. A systematic com-

parison of point defect properties of NbZrTi, including

vacancy formation energy, vacancy migration energy

and interstitial formation energy, was demonstrated with

an EAM potential with potential parameters taken from

Ref. [103, 104] (see Section 3 of Supplementary Mate-

rials for the detailed analysis). Although a quantitative

agreement was not fully reached, several conclusions or

implications can be drawn from such comparison. First,

accurate defect energies from pure metal elements need

to be guaranteed. For instance, the underestimation of

interstitial formation energy of pure Nb leads to the

underestimation of Nb-containing interstitial formation

energies in NbZrTi. Second, the EAM potential with a

relatively simple formalism is able to capture some im-

portant characteristics of NbZrTi, including lower va-

cancy and interstitial formation energy, unstable initial

vacancy and interstitial structures, as well as wide en-

ergy distribution in vacancy migration energy. Third,

the currently used EAM potential underestimates the

lattice distortion and overestimates the defect formation

energies. This seems aligned with our conjecture that

the decrease of defect formation energies is correlated

with the increase of lattice distortion. A good agreement

on lattice distortion relies on an accurate description of

cross interactions among principal elements.

According to the ion irradiation experiment at

675 °C, we showed that the concept of MPEAs can

greatly modify the point defect properties and lead to a

diﬀerent response to the irradiation temperature. Com-

pared to void swelling in Nb at both an absolute tem-

perature of 675 °C and a homologous temperature of

0.48 [93], negligible radiation-induced defects were ob-

served in NbZrTi under the same condition. The re-

duction in swelling at high temperature is aligned with

the high equilibrium vacancy concentration originating

from the low vacancy formation energy. The increase of

the point defect recombination also plays an important

role in the reduction of radiation-induced defects. The

characteristics of radiation-induced defects at a range

of temperatures and simulations of defect cluster be-

haviours need to be studied in the future to fully un-

derstand the irradiation response of NbZrTi.

5. Conclusions

First-principles calculations were performed to assess

the vacancy and interstitial formation energies and their

migration behaviours in body-centered cubic medium-

entropy alloy NbZrTi. Due to the irregular energy land-

scape induced by the large local lattice distortion, un-

usual structural instability was observed in both va-

cancy and interstitial structures. Compared to pure Nb,

NbZrTi exhibits much lower vacancy and interstitial for-

mation energies, higher vacancy mobility and lower in-

terstitial mobility. Because of the wide energy distri-

bution and the preferential vacancy migration through

low-energy sites, the equivalent vacancy migration en-

ergy is lower than pure Nb. The slower interstitial dif-

fusion and the chemically-biased diﬀusion among con-

stituent elements are highly related to the stability of

diﬀerent interstitial conﬁgurations. The atomic compo-

sition and chemical short-range order can inﬂuence the

elemental and total interstitial diﬀusion behaviour. The

obtained ﬁrst-principle results are useful for the bench-

mark of classical interatomic potential for defect stud-

ies. The ion irradiation experiment at 675 °C with 3-

MeV Fe ions at a total dose of ∼100 dpa showed re-

duced void swelling in NbZrTi, which is consistent with

the high equilibrium vacancy concentration induced by

the low vacancy formation energy and enhanced point

defect recombination induced by the closer mobility be-

tween vacancies and interstitials.

Acknowledgement

This work is supported by the National Key Re-

search and Development Program of China under Grant

No. 2019YFA0209900, the National Natural Science

Foundation of China under Grant No. 12075179 and

No. 12105219, the China Postdoctoral Science Founda-

tion under Grant No. 2021M702583, the Nuclear Mate-

rial Technology Innovation Center Project under Grant

No. ICNM 2020 ZH05, the Continuous Basic Scien-

tiﬁc Research Project under Grant No. WDJC-2019-

10, and the Innovative Scientiﬁc Program of CNNC. Q.

P. would like to acknowledge the support provided by

LiYing Program of the Institute of Mechanics, Chinese

Academy of Sciences through Grant No. E1Z1011001.

In addition, we would like to thank Instrument Anal-

ysis Center of Xi’an Jiaotong University for the TEM

characterization and Lanzhou Heavy Ion Accelerator

National Laboratory for the support on heavy ion ex-

periment.

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Microstructures and hardening effects of refractory high entropy alloys irradiated by Proton & He ion dual beam

Article

Feb 2024
MATER CHARACT

Influence of magnetic properties on elemental vacancy migration energy in Fe49.5Mn29.4Co10.1Cr10.1C0.9 high-entropy alloy

Article

Full-text available

Apr 2024

Randomly mixing ferromagnetic (FM) and antiferromagnetic (AFM) elements in high-entropy alloys (HEAs) can create fluctuating local magnetic moments that influence the energetics of point defects. In this study, we employed first-principles calculations to investigate the influence of magnetic properties on vacancy migration energy in Fe49.5Mn29.4Co10.1Cr10.1C0.9, alongside equiatomic NiCoFeCrMn alloy. By examining structures with paramagnetism, ferromagnetism, and no spin polarization, our study reveals significant impacts of magnetic interactions on vacancy migration barriers, potentially altering the sequence of elemental migration energies if overlooked. In Fe49.5Mn29.4Co10.1Cr10.1C0.9, the order of vacancy migration barriers is Co > Fe > Mn > Cr across all magnetic states, suggesting the dominant roles of atomic properties and inherent chemical bonding. Conversely, the NiCoFeCrMn HEA exhibits a pronounced magnetic state-dependent elemental migration energy order, indicating that magnetic interactions significantly influence vacancy migration behavior in this alloy. In addition, while FM elements generally exhibit higher migration barriers, AFM elements display lower barriers in the investigated Cantor alloys, with notable variations between the studied compositions. These findings underscore the critical role of magnetism in accurate migration energy calculations, which is important for studying chemically biased diffusion and radiation-induced segregation in HEAs.

Effects of Nitrogen Doping on Microstructures and Irradiation Resistance of Ti–Zr–Nb–V–Mo Refractory High-Entropy Alloy

Article

Apr 2024
Acta Metall Sin

Interstitial strengthening with nitrogen (N) is one of the effective ways to improve the mechanical properties of HEAs, but the effects of N on the microstructures and mechanical properties of the irradiated HEAs have not been studied extensively. Here, the microstructures and mechanical properties of N-free and N-doped Ti2ZrNbV0.5Mo0.2 HEAs before and after He irradiation were investigated. The results showed that the solid solution strengthening caused by interstitial N improved the yield strength at room temperature and 1023 K without significantly reducing plasticity. N doping significantly promoted the growth, aggregation and wider spatial distribution of He bubbles by enhancing the mobility of He atoms/He-vacancy complexes, with the average size of He bubbles increasing from 10.4 nm in N-free HEA to 31.0 nm in N-doped HEA. In addition, N-doped HEA showed a much higher irradiation hardness increment and hardening fraction than N-free HEA. Contrary to conventional materials doped with N, the introduction of N into Ti2ZrNbV0.5Mo0.2 HEA had adverse effects on its resistance to He bubble growth and irradiation hardening. The results of this study indicated that N doping may not improve the irradiation resistance of HEAs.

Defects Act in an “Introverted” Manner in FeNiCrCoCu High-Entropy Alloy under Primary Damage

Article

Full-text available

Feb 2024

High-entropy alloys (HEAs) attract much attention as possible radiation-resistant materials due to their several unique properties. In this work, the generation and evolution of the radiation damage response of an FeNiCrCoCu HEA and bulk Ni in the early stages were explored using molecular dynamics (MD). The design, concerned with investigating the irradiation tolerance of the FeNiCrCoCu HEA, encompassed the following: (1) The FeNiCrCoCu HEA structure was obtained through a hybrid method that combined Monte Carlo (MC) and MD vs. the random distribution of atoms. (2) Displacement cascades caused by different primary knock-on atom (PKA) energy levels (500 to 5000 eV) of the FeNiCrCoCu HEA vs. bulk Ni were simulated. There was almost no element segregation in bulk FeNiCrCoCu obtained with the MD/MC method by analyzing the Warren–Cowley short-range order (SRO) parameters. In this case, the atom distribution was similar to the random structure that was selected as a substrate to conduct the damage cascade process. A mass of defects (interstitials and vacancies) was generated primarily by PKA departure. The number of adatoms grew, which slightly roughened the surface, and the defects were distributed deeper as the PKA energy increased for both pure Ni and the FeNiCrCoCu HEA. At the time of thermal spike, one fascinating phenomenon occurred where the number of Frenkel pairs for HEA was more than that for pure Ni. However, we obtained the opposite result, that fewer Frenkel pairs survived in the HEA than in pure Ni in the final state of the damage cascade. The number and size of defect clusters grew with increasing PKA energy levels for both materials. Defects were suppressed in the HEA; that is to say, defects were “cowards”, behaving in an introverted manner according to the anthropomorphic rhetorical method.

First-principles study of behaviors of point defects in Mo-14 Re alloy

Article

Jun 2024

Effect of short-range order on lattice distortion, stacking fault energy, and mechanical performance of Co-Fe-Ni-Ti high-entropy alloy at finite temperature

Article

May 2024

The characterization of short-range order (SRO) and its influence on performance are a widely debated topics in high-entropy alloys (HEAs). In this work, taking the Co-Fe-Ni-Ti alloy without complicated magnetism as a benchmark of 3d HEAs, we investigate the effect of SRO on local lattice distortion (LLD), general stacking fault energy (GSFE), tensile and shear strength of FCC Co30Fe16.66Ni36.67Ti16.67 via the combination of Monte Carlo (MC) and molecular dynamics (MD). This alloy shows the typical SRO of Ti-X (X=Fe,Co) atomic pairs, while the segregation of Ti-Ti atomic pairs. The SRO has a minor inhibition on LLD. Considering the thermal vibration induced atomic displacement, the degree of LLD increases nonlinearly with increasing temperature. Both the severe LLD and SRO are helpful to tune the GSFE at finite temperature. The SRO enhances the degree of deformation twinning and delays the appearance of the HCP phase but increases the number of HCP-type atoms as the energy buffers. For the polycrystalline systems, SRO promotes the precipitation of BCC phase at grain boundaries and the number of HCP-type atoms in the grain and activates the deformation of slip surfaces. Therefore SRO could play a key role for the outstanding strength and plasticity of Co-Fe-Ni-Ti HEAs.

High friction stress and weak grain-refinement strengthening effect induced by severe lattice distortion in ZrTiNb medium entropy alloys

Article

Apr 2024

Radiation-enhanced precipitation and the impact on He bubble formation in V-Ti-based refractory alloys containing interstitial impurities

Article

Apr 2024
J NUCL MATER

V-Ti-based refractory high entropy alloys (RHEAs), e.g., VTiTa and VTiTaNb, have been found with outstanding mechanical properties and irradiation resistance, granting them application potential in nuclear engineering. Nonetheless, similar to conventional V-Ti-based alloys such as V-4Ti-4Cr, these alloys are sensitive to C/N/O interstitial impurities. Under irradiation, such impurities may promote the formation of precipitates which may affect the defect evolution, especially at the appearance of helium. Previous studies have observed different helium behavior in the above alloys, including the heterogeneous distribution of helium bubbles in VTiTa, but the origin was not clear. In this work, detailed microstructural characterization is performed to analyze the irradiation induced precipitates and their impact on helium bubble formation in these alloys. Under helium ion irradiation at 700 °C, the TiN precipitates with NaCl structure are formed in all the three alloys. Nonetheless, the interfaces between precipitates and matrix in VTiTa and VTiTaNb are semi-coherent while those in V-4Ti-4Cr are coherent, due to the various lattice parameters of the alloys. Moreover, severe lattice distortion is found also inside the precipitates in the two RHEAs. Consequently, significant heterogeneous nucleation of He bubbles are observed in these two RHEAs but not in V-4Ti-4Cr. Among the three alloys, the growth of precipitates is the fastest in VTiTa, causing the formation of bubble clusters. Furthermore, the growth direction/shape of precipitates affect the shape of bubbles, leading to the {001} cuboids shape in V-4Ti-4Cr and the truncated {110} dodecahedra shape in V-Ti-Ta and V-Ti-Ta-Nb.

Ultrabroad distribution of multiple anelasticities in O-doped refractory multiprincipal element alloys

Article

Apr 2024
ACTA MATER

The Snoek anelastic internal friction (IF) peak due to stress-induced dipole ordering associated with interstitial atoms was well observed in IF curves of single-principal element alloys (SPEAs) as early as in the 1950s, however, its behavior remains elusive in multi-principal element alloys (MPEAs) to date. Here the temperature-dependent IF spectra of NbTiV0.5Zr refractory MPEAs with varying O contents were presented, and the analysis of these data indicates that for O-doped NbTiV0.5Zr MPEAs, the O addition triggers two anelastic IF peaks (the PO1 and PO2), corresponding to the O-Snoek-type relaxation in random solid solution (RSS) and local chemical ordering (LCO) structures, respectively. The half-peak width of the PO1 is 2–3 times broader than that for SPEAs, associated with the wide O migration energy barrier distribution in MPEAs. These phenomena show a striking contrast to the single and narrow Snoek-type peak observed for SPEAs. Our analysis further reveals an approximate linear correlation between the half-peak width of the Snoek-type peak and the mixing entropy. The introduction of O also shifts grain boundary relaxation peaks (PG) towards higher temperatures. Interestingly, high-temperature anomalous modulus changes were observed for the first time in TiV-based lightweight refractory MPEAs, which is suggested to be originated from the order reduction of the body-centered cubic (bcc) structure at elevated temperatures. Moreover, the 1 at.%O doped NbTiV0.5Zr MPEA with a favorable balance between the peak heights of the PO2 and PG, exhibits outstanding specific yield stress (SYS) among refractory MPEAs while maintaining an elongation as high as 20%. This work provides not only a bridge between the anelastic behavior of SPEAs and MPEAs for upcoming studies and theories concerning bcc MPEAs with interstitial atoms, but contributes to the holistic design of bcc MPEAs with high strength and excellent ductility.

Uncovering origin of grain boundary resistance to irradiation damage in NiCoCr multi-principal element alloys

Article

Feb 2024
INT J PLASTICITY

Modeling refractory high-entropy alloys with efficient machine-learned interatomic potentials: Defects and segregation

Article

Full-text available

Sep 2021

We develop a fast and accurate machine-learned interatomic potential for the Mo-Nb-Ta-V-W quinary system and use it to study segregation and defects in the body-centered-cubic refractory high-entropy alloy MoNbTaVW. In the bulk alloy, we observe clear ordering of mainly Mo-Ta and V-W binaries at low temperatures. In damaged crystals, our simulations reveal clear segregation of vanadium, the smallest atom in the alloy, to compressed interstitial-rich regions such as radiation-induced dislocation loops. Vanadium also dominates the population of single self-interstitial atoms. In contrast, due to its larger size and low surface energy, niobium segregates to spacious regions such as the inner surfaces of voids. When annealing samples with supersaturated concentrations of defects, we find that in complete contrast to W, interstitial atoms in MoNbTaVW cluster to create only small (∼1 nm) experimentally invisible dislocation loops enriched by vanadium. By comparison to W, we explain this by the reduced but three-dimensional migration of interstitials, the immobility of dislocation loops, and the increased mobility of vacancies in the high-entropy alloy, which together promote defect recombination over clustering.

Current development of body-centered cubic high-entropy alloys for nuclear applications

Article

Full-text available

May 2021

High-entropy alloys greatly expand the alloy design range and offer new possibilities for improving material performance. Based on the worldwide research efforts in the last decade, the excellent mechanical properties and promising radiation and corrosion resistance of this group of materials have been demonstrated. High-entropy alloys with body-centered cubic (BCC) structures, especially refractory high-entropy alloys, are considered as promising materials for high-temperature applications in advanced nuclear reactors. However, the extreme reactor conditions including high temperature, high radiation damage, high stress, and complex corrosive environment require a comprehensive evaluation of the material properties for their actual service in nuclear reactors. This review summarizes the current progress on BCC high-entropy alloys from the aspects of neutron economy and activation, mechanical properties, high-temperature stability, radiation resistance, as well as corrosion resistance. Although the current development of BCC high-entropy alloys for nuclear applications is still at an early stage as the large design space of this group of alloys has not been fully explored, the current research findings provide a good basis for the understanding and prediction of material behaviors with different compositions and microstructures. Further in-depth understanding of the degradation mechanisms and characterization of material properties in response to conditions close to reactor environment are necessary. A critical down-selection of potential candidates is also crucial for further comprehensive evaluation and engineering validation.

Direct observation of chemical short-range order in a medium-entropy alloy

Article

Full-text available

Apr 2021
NATURE

Complex concentrated solutions of multiple principal elements are being widely investigated as high- or medium-entropy alloys (HEAs or MEAs)1–11, often assuming that these materials have the high configurational entropy of an ideal solution. However, enthalpic interactions among constituent elements are also expected at normal temperatures, resulting in various degrees of local chemical order12–22. Of the local chemical orders that can develop, chemical short-range order (CSRO) is arguably the most difficult to decipher and firm evidence of CSRO in these materials has been missing thus far16,22. Here we discover that, using an appropriate zone axis, micro/nanobeam diffraction, together with atomic-resolution imaging and chemical mapping via transmission electron microscopy, can explicitly reveal CSRO in a face-centred-cubic VCoNi concentrated solution. Our complementary suite of tools provides concrete information about the degree/extent of CSRO, atomic packing configuration and preferential occupancy of neighbouring lattice planes/sites by chemical species. Modelling of the CSRO order parameters and pair correlations over the nearest atomic shells indicates that the CSRO originates from the nearest-neighbour preference towards unlike (V−Co and V−Ni) pairs and avoidance of V−V pairs. Our findings offer a way of identifying CSRO in concentrated solution alloys. We also use atomic strain mapping to demonstrate the dislocation interactions enhanced by the CSROs, clarifying the effects of these CSROs on plasticity mechanisms and mechanical properties upon deformation.

Introduction to Computational Materials Science: Fundamentals to Applications

Book

Mar 2013

Richard LeSar

Emphasising essential methods and universal principles, this textbook provides everything students need to understand the basics of simulating materials behaviour. All the key topics are covered from electronic structure methods to microstructural evolution, appendices provide crucial background material, and a wealth of practical resources are available online to complete the teaching package. Modelling is examined at a broad range of scales, from the atomic to the mesoscale, providing students with a solid foundation for future study and research. Detailed, accessible explanations of the fundamental equations underpinning materials modelling are presented, including a full chapter summarising essential mathematical background. Extensive appendices, including essential background on classical and quantum mechanics, electrostatics, statistical thermodynamics and linear elasticity, provide the background necessary to fully engage with the fundamentals of computational modelling. Exercises, worked examples, computer codes and discussions of practical implementations methods are all provided online giving students the hands-on experience they need.

A systematic investigation on quaternary NbTiZr-based refractory high entropy alloys using empirical parameters and first principles calculations

Article

Oct 2021

Defect accumulation and evolution in refractory multi-principal element alloys

Article

Aug 2021
ACTA MATER

Refractory multiple principal elemental alloys (MPEAs) hold great promise for structural materials in future nuclear energy systems. Compared to the extensively studied face-centered cubic (FCC) MPEAs, the irradiation resistance of body-centered cubic (BCC) refractory MPEAs is relatively less known. In this work, we study defect accumulation and evolution in two BCC VTaTi and VTaW MPEAs comparatively through atomistic simulations. For this purpose, we have parameterized the Embedded Atom Method (EAM) potential parameters for V metals. Combined with available potential parameters for other elements, we construct average atom models for the considered alloys to elucidate the effects of chemical complexities in BCC MPEAs. Our results based on Frenkel pair accumulation simulations suggest that the major influence of chemical fluctuations in BCC MPEAs is on the clustering behavior of defects, which leads to discrete point defects or small defect clusters, in contrast to the large defect clusters observed in the average atom model. We further show that the diffusion of interstitial clusters exhibits different modes due to chemical complexity. While interstitials show either three-dimensional or one-dimensional diffusion in the average atom model, the mean free path of interstitials in the random alloy is strongly suppressed. These results provide fundamental insights into the irradiation response of BCC MPEAs and pinpoint the critical role of chemical complexity on defect evolution, which lay the basis for the future development of irradiation-resistant structural materials based on BCC MPEAs.

Finnis–Sinclair-type potential for atomistic simulation of defects behaviour in V-Ti-Ta ternary system

Article

Aug 2021
J NUCL MATER

Vanadium (V)-based alloys are potential candidates for structural materials of fusion reactors because of their excellent properties; recently, the V–Ti–Ta alloy has received considerable attention. The irradiation-resistant performance is significant for the development and application of fusion reactor materials. In this study, combining with our previously developed potentials of V and Ta, we developed the V-Ti-Ta interatomic potential based on the Finnis–Sinclair (FS) formalism, which is an important step towards an atomic-scale investigation of V-based alloys. The potential parameters were determined by fitting to a large database of experimental data and density functional theory (DFT) calculations. The formation energies of vacancy and self-interstitial atoms, as well as the pressure-volume relation obtained from the present Ti potential, were well reproduced. The typical defect energies of solute atoms Ti and Ta in the V matrix and Ti and V in the Ta matrix, such as the formation energies of substitutional solute atoms, binding energies between solute atoms and point defects, agree well with DFT results. Thus, the developed potentials are expected to be suitable for atomistic simulations of point defects evolution in the V-Ti-Ta ternary system.

VASPKIT: A user-friendly interface facilitating high-throughput computing and analysis using VASP code

Article

May 2021
COMPUT PHYS COMMUN

We present the VASPKIT, a command-line program that aims at providing a powerful and user-friendly interface to perform high-throughput analysis of a variety of material properties from the raw data produced by the VASP code. It consists of mainly the pre- and post-processing modules. The former module is designed to prepare and manipulate input files such as the necessary input files generation, symmetry analysis, supercell transformation, k-path generation for a given crystal structure. The latter module is designed to extract and analyze the raw data about elastic mechanics, electronic structure, charge density, electrostatic potential, linear optical coefficients, wave function plots in real space, and etc. This program can run conveniently in either interactive user interface or command line mode. The command-line options allow the user to perform high-throughput calculations together with bash scripts. This article gives an overview of the program structure and presents illustrative examples for some of its usages. The program can run on Linux, MacOS, and Windows platforms. The executable versions of VASPKIT and the related examples, together with the tutorials, are available in its official website vaspkit.com. Program summary Program title: VASPKIT CPC Library link to program files: https://doi.org/10.17632/v3bvcypg9v.1 Licensing provisions: GPLv3 Programming language: Fortran, Python Nature of problem: This program has the purpose of providing a powerful and user-friendly interface to perform high-throughout calculations together with the widely-used VASP code. Solution method: VASPKIT is able to extract, calculate and even plot the mechanical, electronic, optical and magnetic properties from density functional calculations together with bash and python scripts. It can run in either interactive user interface or command line mode.

Simulations of primary damage in a High Entropy Alloy: Probing enhanced radiation resistance

Article

May 2021
ACTA MATER

High Entropy Alloys (HEA) attract attention as possible radiation resistant materials, a feature observed in some experiments that has been attributed to several unique properties of HEA, in particular to the disorder-induced reduced thermal conductivity and to the peculiar defect properties originating from the chemical complexity. To explore the origin of such behavior we study the early stages (less than 0.1 ns), of radiation damage response of a HEA using molecular dynamics simulations of collision cascades induced by primary knock-on atoms (PKA) with 10, 20 and 40 keV, at room temperature, on an idealized model equiatomic quinary fcc FeNiCrCoCu alloy, the corresponding “Average Atom” (AA) material, and on pure Ni. We include accurate corrections to describe short-range atomic interactions during the cascade. In all cases the average number of defects in the HEA is lower than for pure Ni, which has been previously used to help claiming that HEA is radiation resistant. However, simulated defect evolution during primary damage, including the number of surviving Frenkel Pairs, and the defect cluster size distributions are nearly the same in all cases, within our statistical uncertainty. The number of surviving FP in the alloy is predicted fairly well by analytical models of defect production in pure materials. All of this indicates that the origin of radiation resistance in HEAs as observed in experiments may not be related to a reduction in primary damage due to chemical disorder, but is probably caused by longer-time defect evolution.

Local Ordering Tendency in Body-Centered Cubic (BCC) Multi-Principal Element Alloys

Article

Mar 2021

Shijun Zhao

The recent development of high-entropy alloys (HEAs) has opened a new avenue for alloy design by incorporating multiple principal elements into a simple crystal lattice. In HEAs, short-range chemical ordering may arise due to the different mixing properties of various constituent elements. In this work, we explore the trend of elemental arrangement in five typical body-centered cubic (BCC) multi-principal element alloys (MPEAs), including CrFeV, HfNbZr, TaVW, NbTaV, and TaTiV. Combining the Monte Carlo method and density-functional theory calculations, we calculated the evolution of short-range order (SRO) parameters in these alloys at 500 K. Our results demonstrated that all these MPEAs might develop a certain degree of SRO, depending on the mixing properties of different element pairs. Using a modified quasi-chemical model, we show that the SRO values in these MPEAs can be accurately predicted as long as the mixing enthalpy values between different atomic pairs are known. The presence of SRO has a profound impact on the local structure of MPEAs, which further alters the electronic and elastic properties of MPEAs. Our results provide a theoretical understanding of the basic alloy structure of novel MPEAs, which is an essential step toward tailoring their properties.

Distinct point defect behaviours in body-centered cubic medium-entropy alloy NbZrTi induced by severe lattice distortion

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