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8

A Molecular Dynamics Study of Self-Diffusion in

Stoichiometric B2-NiAl crystals

Marcin Mazdziarz, Jerzy Rojek, and Szymon Nosewicz

Institute of Fundamental Technological Research Polish Academy of Sciences,

Pawinskiego 5B, 02-106 Warsaw, Poland

E-mail: mmazdz@ippt.pan.pl

Abstract. Self-diffusion parameters in stoichiometric B2-NiAl solid state crystals

were estimated by molecular statics/dynamics simulations with the study of required

simulation time to stabilise diffusivity results. An extrapolation procedure to improve

the diffusion simulation results was proposed. Calculations of volume diffusivity for

the B2 type NiAl in the 1224 K to 1699 K temperature range were performed using

the embedded atom model potential. The results obtained here are in much better

agreement with the experimental results than the theoretical estimates obtained with

other methods.

PACS numbers: 02.70.Ns, 66.30.h, 66.30.Fq

Keywords: NiAl, nickel-aluminium, diffusivity, molecular dynamics, molecular statics,

embedded-atom method, sintering

1. Introduction

For numerical material modelling, identification of model parameters for the material

is one of the key tasks which commonly causes many difficulties. A straightforward

approach to parameter determination relies on direct measurement of the wanted

parameters. In some cases, however it is difficult or very expensive to design experiments

that allow direct measurements of the required parameters. Alternatively, parameters of

the model can be identified by numerical simulations at a lower scale. This methodology

used in multiscale modelling is an appealing option, notably when experimental

measurements are not feasible or experimental data are not accessed [1].

This work presents the determination of the diffusion parameters in stoichiometric

B2-NiAl crystals using the molecular dynamic simulations. The nickel-aluminium (NiAl)

type intermetallics are modern materials with low density and advantageous mechanical

properties. These are often used as matrix materials in composites manufactured using

sintering technologies [2]. There is a growing demand for modelling support of design

and optimisation of sintering processes. Sintering is a diffusion driven process, and

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 2

the knowledge of diffusivity of investigated materials is usually necessary to define

parameters of sintering models [3].

Diffusivity can be estimated using both atomic and atomistic modelling. These

approaches can be used to model sintering directly, however, this approach has some

limitations. Direct simulations of particle sintering are limited only to nanoscale,

whereas in general, average particle sizes in NiAl powder can be three orders of

magnitude bigger, see e.g.[4, 5]. Therefore, atomistic models have been aimed to

simulate diffusion in order to evaluate the diffusion properties of NiAl material which

can be used in sintering models [3]. Generally, the diffusivity of NiAl material was

derived from the so called, static and dynamic simulations. Using molecular statics

(MS) approach and modified analytic embedded-atom method (MAEAM) potential

the activation, formation and migration energies of Ni self-diffusion in intermetallic

compound NiAl have been calculated for five diffusion mechanisms in [6]. However,

diffusivity was not analysed during this study. The results show that Ni diffusion

is predominated by the triple-defect diffusion mechanism since it needs essentially

the lowest migration and activation energy among the five diffusion mechanisms

considered. Point defect energetics and the activation energy in NiAl were determined

by molecular statics in [7, 8, 9, 10]. The effective formation energies for a homogeneous

thermodynamically stable ordered compound B2-NixAl1−x were determined by a

combination of the ab initio electron theory with a generalized grand canonical statistical

approach in [11, 12, 13]. By means of molecular statics simulations and the embedded

atom model (EAM) potentials for the assumed six-jump cycle (6JC) mechanism in

the temperature spectrum from 800 to 1500K [14] have found the pre-exponential

factor, D0=1.3 x 10−5(m2s−1) and the activation energy, Q=3.12(eV/atom) (EM=2.44

and EF=0.68). By means of first-principles density functional theory (DFT) calculations

[15] five postulated diffusion mechanisms have been analysed for Ni in NiAl in the

temperature range from 1200 to 1500K and derived the activation energies Q=2.99-

4.15 (eV/atom) and the pre-exponential factor D0=0.46-1.49 x 10−5(m2s−1). Even for a

simpler monatomic fcc Fe system, the DFT calculated self-diffusionD0 is underestimated

by two orders of magnitude, see [16]. A universal tendency can be seen, that the

activation energy Q determined from molecular statics and first-principles are generally

in good agreement with experimental data but pre-exponential factor D0 is significantly

undervalued, see also discussion in Sec.2. It is worth noting that such calculations do

not straightforwardly provide the thermal properties of a material . The harmonic

approximation accuracy even taking into account, the influence of temperature on

other properties of a material, is far from having reached a satisfactory level [17].

Anharmonic effects can be very significant but there are very few analytic calculations

taking into account higher-order terms using perturbation theory, whereas numerical

molecular dynamics calculations can account for anharmonicity to all orders. Some

effects associated with thermal expansion at constant pressure can be described by the

quasiharmonic approximation but anharmonic effects, which explicitly depend on the

magnitude of the atomic vibrational displacements, are present even at fixed volume;

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 3

see [18]. It seems, therefore, that molecular dynamics is a reliable method that allow

one to take into account all anharmonic effects [19]. The experimentally specified

Debye temperature of nominally stoichiometric NiAl varies from 470K to 560K [20]. In

connection to our research, the relevance of the Debye temperature can be twofold. First,

it can provide an estimation of the temperature above which all phonon states become

occupied. This would mean that below Debye temperature the classical considerations

are not likely to apply. Second, the Debye temperature is widely adopted as an

estimation of the temperature under which the underestimation of anharmonicity in

the QHA, QHAD is not very important, even for systems with reduced symmetry (e.g.

with defects). The stochastic Monte Carlo method is another static approach to study

diffusion. The Monte Carlo simulations represent a broad spectrum of computational

algorithms that rely on repeated random sampling to simulate the time evolution of

some processes which appear in nature [21]. It is a fairly straightforward and helpful

technique that can output a sequence of configurations and the times at which transitions

happen between these configurations [22]. Atomistic diffusion in faced-centered cubic

NiAl binary alloys, Ni containing 0.05, 0.1, 0.15, 0.2 and 0.25 atomic fraction of Al, was

examined by Kinetic Monte Carlo (KMC) method by [23]. Fundamental data obtained

from ab initio computer calculations were used as an input to these simulations. The

derived activation energies QNi were in good agreement with experimental data but

pre-exponential factor D0,Ni is unfortunately again highly undervalued.

The most natural dynamic technique to calculate diffusivity is to use molecular

dynamic . An overview of recent molecular dynamics simulations in equiatomic Ni-Al

systems can be found in [24]. In [25], diffusion and interdiffusion in binary metallic

melts, Al-Ni and Zr-Ni, was analysed by molecular dynamics computer simulations

and the mode coupling theory of the glass transition. But the authors claimed

that their ”results not to be quantitative predictions for real Al-Ni or Zi-Ni melts,

but rather for model systems that allow us to understand the relevant mass transport

mechanisms in such melts qualitatively and semiquantitatively”. Molecular dynamics

study of self-diffusion in liquid Ni50Al50 alloy was carried out in [26, 27, 28] and MD

simulation of 2D diffusion in (110) B2-NiAl film in [29]. The first molecular dynamics

simulation of diffusion mechanisms in ordered stoichiometric Ni3Al with the Finnis–

Sinclair interatomic potential was performed by [30], but the author noted that ”results

are at least qualitatively, and in some respect quantitatively, in agreement”. In a

comparative study of embedded-atom methods applied to the reactivity in the NiAl

system [31], the self-diffusion of the liquid Ni-Al mixture was also studied, unfortunately

without giving details of methodology. The consistency of their simulation results with

experiment is rather poor. The authors even stated that ”self-diffusion in a solid state is

much slower and diffusion coefficients are much smaller by several orders of magnitude

than in the liquid state. As a result, MD timescales are generally too short for an

extensive study of these phenomena”, what we refute in this work.

The initial estimation of the volume, surface, and grain-boundary diffusivity for

the B2 type NiAl as the pivotal mechanism of sintering was performed in our previous

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 4

work [32]. That paper presented the methodology , the assumption for the model and

initial results for three temperatures - from 1573 to 1673K. The estimated values were

supposed to be applied to discrete element modelling of the sintering process of NiAl

powder, however such small range of temperatures can be insufficient and can lead to

inconsistency of material data in the context of the connection between atomistic and

microscopic scales. In response to such challenges, the current work concentrates on

improved and extended investigations of volume diffusion determination in NiAl material

in the much wider temperature range - from 1224 to 1699K. Determined materials

parameters will be characterized to better accuracy and agreement with experimental

results. Furthermore, the present paper discusses the inspection of the necessary

simulation time required to ensure the stabilization of diffusivity results. The studied

problem seems to be the one of the major issue of proper determination of diffusivity

results and have a considerable impact on obtained final results. Therefore, it has been

proposed the new procedure to improve the quality of diffusion simulation results, what

is important especially in the analysis of diffusion results in lower temperatures.

2. The mechanism of atomistic diffusion in NiAl

Diffusion is material transport induced by the motion of atoms. A schematic illustration

of volume self-diffusion in the monatomic structure of an atom, from its initial position

into a vacant lattice site, is depicted in Fig.1. The migration energy, EM , has to be

applied to the atom in order to overcome inter-atomic bonds and to move to the new

position.

Figure 1. Schematic representation of the volume self-diffusion process.

Similarly to other solids, self-diffusion is the primary diffusion process in

intermetallics . In binary compounds two tracer self-diffusion coefficients – one for

A atoms and one for B atoms, – are relevant [33]. In the NiAl binary system, we can

identify Ni self-diffusion and Al self-diffusion, where the first one is highly dominative

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 5

in the stoichiometric composition [34, 35, 36]. Interdiffusion, or chemical diffusion,

is different from the tracer diffusion, because it is present in a chemical composition

gradient and not in a homogeneous solid [33].

For the B2 type NiAl, see Fig.3a), various hop mechanisms based on experimental

and theoretical considerations have been suggested [37]. In addition to the dominant

nearest-neighbour (NN) hops in monoatomic structures, next-nearest neighbour (NNN)

hops are also feasible in B2-NiAl due to its open crystal structure. An intriguing property

of B2-NiAl is that NN Al hops from the Al sublattice to the Ni sublattice cannot emerge,

as the end state is predicted to be not mechanically stable. Other suggested mechanisms

include two simultaneous pair-atom (SPA) hop mechanisms, the six-jump cycle (6JC),

two anti-structure bridge (ASB) mechanisms and the triple-defect (TD) sequence.

It is well-known that thermal defects occur at finite temperatures apart from

the constitutional defects. Modelling concepts are usually based on the idea of a

noninteracting point defects gas as suggested by Wagner and Schottky. Thermal defects

in an ordered binary alloy of a fixed composition must occur in a balanced manner in

order to retain the alloy stoichiometry. Hence, since the alloy composition is fixed, they

can occur merely in the composition-conserving combination and any point defects

cannot be a thermal defect alone. There are four basic composition-conserving defects

composed of two types of point defects: Exchange antisite defect (X), Divacancy defect

(D), Triple Ni or simply triple defect (TN) and Triple Al defect (TA), see [38]. It is

worth mentioning that the formation energies of single point defects depend on the

choice of the reference states and they are not insignificant, as opposed to the formation

energies of composition-conserving defects, which do not depend on a specific choice of

the reference state and can be directly compared. Thereby, the first case requires a grand

canonical ensemble, the second a canonical ensemble. In the B2-type intermetallics the

thermal disorder is typical of a triple defect type and the triple defect (TN) formed

by one antisite Ni atom and two Ni vacancies is presumed to be the dominant thermal

defect in NiAl, see [38, 35].

The temperature dependence of diffusivity typically follows an Arrhenius equation

[33] and typically is written as:

D = D0 exp

(

−Q

kBT

)

= D0 exp

(

−EF + EM

kBT

)

, (1)

or in the logarithmic form

log (D) = log (D0)−

(

Q

kB

)(

1

T

)

, (2)

where D is the diffusivity or diffusion coefficient; D0 is the pre-exponential factor; Q is

the activation energy; T is the temperature; kB is the Boltzmann’s constant; EF is the

formation energy and EM is the migration energy.

The thermal equilibrium concentration of point defects is given by the relation:

ND

N= exp

(

−EF

kBT

)

, (3)

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 6

where ND is the number of defects and N is the number of potential defect sites.

Simple Arrhenius behaviour should not however be deemed to be universal. At T >

1500K an upward deviation from this equation can be noted, among others, in B2-NiAl

[14, 38] and bcc-Fe [39].

The activation energy Q for the stoichiometric composition for triple Ni defect (TN)

in the Arrhenius Eq.1 leads to:

Q =

(

ETNi

3

)

+ EM , (4)

where ETNi/3= EF , see [34].

Presently, there exist no directly measured Al tracer diffusion data in NiAl, Al

diffusion cannot be determined experimentally because of the absence of a suitable

isotope, but Ni diffusion in single-crystal NiAl has been determined [40]. For Ni in B2-

NiAl self-diffusion prefactor D0=2.71-3.45 x 10−5(m2s−1) and activation energy Q=2.97-

3.01(eV/atom). It is visible that the spread of results is quite marginal as opposed to

results of self-diffusion in pure Al and in pure Ni. The diffusion prefactor D0 for Al

varies from three orders of magnitude up to eight orders for Ni in different studies [41].

3. Computational methodology

The molecular simulations in this study were made with the use of the Large-scale

Atomic/Molecular Massively Parallel Simulator (LAMMPS) [42] and visualized in the

Open Visualization Tool (OVITO) [43] and Wolfram Mathematica [44].

3.1. Molecular potentials

Two embedded-atom method (EAM) potentials: EAM2002[45] and EAM2009[46] have

been validated for their possible reproduction of NiAl diffusion in the temperature

spectrum from 1224 to 1699K. Using molecular statics approach [47, 48, 49, 50, 51]

the following parameters of NiAl were probed: lattice constant, cohesive energy, bulk

modulus, elastic constants, surface energy, defect energies, migration energies as well as

melting point temperature, see Tab.1. For MS simulations the sample was assumed to

be cube and composed of 103 elementary B2-NiAl cells, see Fig.3a) (∼ 2000 atoms). The

size of the sample allows to fulfil the assumed by Wagner and Schottky condition of non-

interaction of point defects. Energy minimization with periodic boundary conditions

(PBC) applied to all faces of the sample was carried out with a nonlinear conjugate

gradient algorithm [42]. The assumption was, that convergence was reached when the

relative change in the energy and forces, between two successive iterations was less than

10−13. Following the energy minimization procedure an external pressure was applied

to the simulation box to see the volume change effect.

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 7

3.2. Diffusivity simulations

Periodic boundary conditions (PBC) were applied in all simulations. After molecular

statics relaxation, molecular dynamics equilibration for the time period of 1 ns (0.5x106

MD steps, one step = 2 fs) was done. The data gathering simulation covered a time

interval of minimum 1000 ns (500x106 MD steps). Finding the needed simulation time

warranting the stabilization of the results was the first step in the simulation. At a given

temperature and pressure, NPT (constant number of atoms, pressure and temperature)

Nose-Hoover style barostat was used [42, 47]. The diffusivity was calculated as the

average:

Davg. ≡1

2dlimt→∞

⟨

[r(t0 + t)− r(t0)]2⟩

t, (5)

or the instantaneous one:

Dinst. ≡1

2dlimt→∞

∂⟨

[r(t0 + t)− r(t0)]2⟩

∂t, (6)

where, d is the dimensionality (d=2 for surface and grain-boundary diffusion, d=3

for volume); t is the time and⟨

[r(t0 + t)− r(t0)]2⟩

is the ensemble average MSD.

Thermodynamic informations were calculated and stored at intervals of 2 ps (1000 MD

steps).

3.2.1. Volume diffusivity Thermal equilibrium point defect concentration is a function

of temperature and the formation energy EF , see Eq.3. Defect concentration for triple

Ni and Al defect energy found by the molecular potential applied in the MD studies,

see Tab.1, in the temperature interval from 1224 to 1699K is depicted in Fig.2.

TNi

TAl

1300 1400 1500 1600 1700

10-6

10-5

10-4

10-3

Temperature [K]

Defectconcentration

(per

site)

Triple defect concentration

Figure 2. Triple defect concentration for Ni and Al.

Hence, for the cubic computational sample, Fig.3b), using above Eq.3 we can

calculate that one triple Ni defect (TN) for temperature 1224K is in the 223 basic

cells (21296 atoms), for 1244K 213 basic cells (18522 atoms), for 1265K 203 basic cells

(16000 atoms), for 1288K 193 basic cells (13718 atoms), for 1313K 183 basic cells (11664

atoms), for 1341K 173 basic cells (9826 atoms), for 1372K 163 basic cells (8192 atoms),

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 8

for 1406K 153 basic cells (6750 atoms), for 1445K 143 basic cells (5488 atoms), for

1489K 133 basic cells (4394 atoms), for 1540K 123 basic cells (3456 atoms), for 1599K

113 basic cells (2662 atoms) and for 1699K 103 basic cells (2000 atoms), respectively. It

is worth emphasising that the triple defect concentration for Al is about 100 times lower

than for Ni, see Fig.2, what would mean the use of about 100 times larger computational

samples to calculate Al self-diffusion.

3.3. Diffusivity determination from simulation results

In order to answer the question of what value of diffusion D should be read from the

simulation, two procedures were proposed and tested:

I Direct: we take the value of the diffusivity D, Eqs.5,6, from the end of the MD

simulation.

II Extrapolation: we approximate the results of the diffusivity D, Eqs.5,6, versus

simulation time t with a function with a limit, e.g. D(t)=a×exp(b/t), and take its

value for t → ∞.

4. Results

Both considered EAM interatomic potentials , see Tab.1, similarly reproduce the

parameters of B2-NiAl, nevertheless, EAM2002 potential predicts the melting point at

1520K since the experimental value is 1911K. Because of our interest in the temperature

spectrum from 1224 to 1699K for subsequent molecular dynamics computations the

EAM2009 potential, predicting a more reasonable melting temperature of 1780K, was

used.

Figure 3. Atomistic models: a) NiAl basic cell, b) simulation box for volume diffusion.

There is no general rule for how long a simulation has to be run before we reach

the long-time asymptotic behaviour predicted by the Einstein-Smoluchowski relation,

Eqs.5,6. As far as we know the required simulation duration ensuring the stabilization

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 9

Table 1. Material parameters of NiAl from two EAM potentials - Molecular statics

simulations. DFT and experimental data taken from [45, 46]. Migration energy∗

calculated for nearest-neighbour (NN) hop mechanism.

EAM2002 EAM2009 Experiment/DFT

Lattice constant NiAl [A] 2.86 2.832 2.88

Cohesive energy [eV] -4.47 -4.51 -4.50

Bulk modulus B [GPa] 160 159 158

Elastic constant C11 [GPa] 200 191 199

Elastic constant C12 [GPa] 140 143 137

Elastic constant C44 [GPa] 120 121 116

Surface energy [Jm−2] 1.52 2.07 ∼2.2

Exchange energy EX [eV] 2.765 2.075 2.65-3.15

Divacancy energy ED [eV] 2.396 2.569 2.18-3.07

Triple Ni defect energy ETNi [eV] 2.281 2.807 1.58-2.83

Triple Al defect energy ETAl [eV] 5.276 4.406 5.44-6.46

Migration energy∗ ENiM [eV] 2.324 2.49 2.58

Migration energy∗ EAlM [eV] 1.476 1.76 ∼1.6

Melting point [K] 1520 1780 1911

of the slope of the mean-squared displacement (MSD) results was not duly discussed

in the literature. For the molecular dynamics analysis of self-diffusion in bcc Fe [39],

the simulation time was ∼ 41 ns, for hcp and bcc Zr [52] ∼ 20-30 ns and for ordered

stoichiometric Ni3Al in [30] was ∼ 30 ns. For the molecular dynamics simulation of

carbon diffusion in α-iron, the simulation time was ∼50 ns. Similarly, as we can observe

in this work, diffusivity decreases with increasing time and moves towards stabilisation,

see [53].

The conditions where molecular dynamics simulations can be used to calculate

highly converged Arrhenius plots for substitutional alloys with various vacancy

concentrations between 1% and 5% were analysed in [54]. It was found that highly

converged results can be obtained when using an elevated temperature range (i.e.

Tsimulation/Tmelting=0.87-0.98) and an extended simulation time longer than 300 ns. The

higher temperature in the simulations, as well as the higher concentration of defects,

resulted in more convergent results and lower statistical error.

Our observations suggest that the needed simulation time that allows the

stabilization of the MSD slope is substantially longer than that used by mentioned

authors. It can be further observed in Figs.4-7, that the needed simulation time

depends on temperature and as it grows, the time decreases. We also observe, that

instantaneous diffusivity, Eq.6, stabilizes faster than the average diffusivity, Eq.5, and

will be used further as a reference value. It is also observed that in conducted calculations

the simulation time at least 1000 ns is needed. For temperature 1244K the simulation

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 10

time had to be extended to 1800 ns and to 2500 ns for 1224K, respectively.

MSD

0 200000 400000 600000 800000 1×106

0

5

10

15

20

Time [ps]

MSD

[Å^2]

a) Mean Squared Displacement - 1699.864K

Avg.

Inst.

0 200000 400000 600000 800000 1×106

0

1.×10-6

2.×10-6

3.×10-6

4.×10-6

5.×10-6

6.×10-6

Time [ps]

Diffusivity

[Å^2/ps]

a) Diffusivity - 1699.864K

MSD

0 200000 400000 600000 800000 1×106

0

1

2

3

4

Time [ps]

MSD

[Å^2]

b) Mean Squared Displacement - 1599.52K

Avg.

Inst.

0 200000 400000 600000 800000 1×106

0

1.×10-6

2.×10-6

3.×10-6

4.×10-6

Time [ps]

Diffusivity

[Å^2/ps]

b) Diffusivity - 1599.52K

MSD

0 200000 400000 600000 800000 1×106

0.0

0.5

1.0

1.5

2.0

Time [ps]

MSD

[Å^2]

c) Mean Squared Displacement - 1��������

Avg.

Inst.

0 200000 400000 600000 800000 1×106

0

1.×10-6

2.×10-6

3.×10-6

4.×10-6

Time [ps]

Diffusivity

[Å^2/ps]

c) Diffusivity - 1540.295K

MSD

0 200000 400000 600000 800000 1×106

0.0

0.2

0.4

0.6

0.8

1.0

Time [ps]

MSD

[Å^2]

d) Mean Squared Displacement - ��� ���

Avg.

Inst.

0 200000 400000 600000 800000 1×106

0

5.×10-7

1.×10-6

1.5×10-6

2.×10-6

2.5×10-6

3.×10-6

3.5×10-6

Time [ps]

Diffusivity

[Å^2/ps]

d) Diffusivity - ���������

Figure 4. Mean Squared Displacement and Average and Instantaneous Diffusivity

for different temperature (1699-1489K).

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 11

MSD

0 200000 400000 600000 800000 1×106

0.0

0.1

0.2

0.3

0.4

Time [ps]

MSD

[Å^2]

e) Mean Squared Displacement - ������ !

Avg.

Inst.

0 200000 400000 600000 800000 1×106

0

5.×10-7

1.×10-6

1.5×10-6

2.×10-6

2.5×10-6

3.×10-6

Time [ps]

Diffusivity

[Å^2/ps]

e) Diffusivity - 1445.47K

MSD

0 200000 400000 600000 800000 1×106

0.20

0.25

0.30

0.35

Time [ps]

MSD

[Å^2]

f) Mean Squared Displacement - "#$%&'()*

Avg.

Inst.

0 200000 400000 600000 800000 1×106

0

5.×10-7

1.×10-6

1.5×10-6

2.×10-6

2.5×10-6

3.×10-6

Time [ps]

Diffusivity

[Å^2/ps]

f) Diffusivity - +,-./0234

MSD

0 200000 400000 600000 800000 1×106

0.20

0.21

0.22

0.23

0.24

0.25

0.26

Time [ps]

MSD

[Å^2]

g) Mean Squared Displacement - 56789:;<=

Avg.

Inst.

0 200000 400000 600000 800000 1×106

0

5.×10-7

1.×10-6

1.5×10-6

2.×10-6

2.5×10-6

3.×10-6

Time [ps]

Diffusivity

[Å^2/ps]

g) Diffusivity - 1372.287K

MSD

0 200000 400000 600000 800000 1×106

0.19

0.20

0.21

0.22

0.23

Time [ps]

MSD

[Å^2]

h) Mean Squared Displacement - 1341.452K

Avg.

Inst.

0 200000 400000 600000 800000 1×106

0

5.×10-7

1.×10-6

1.5×10-6

2.×10-6

2.5×10-6

3.×10-6

Time [ps]

Diffusivity

[Å^2/ps]

>) Diffusivity - 1341.452K

Figure 5. Mean Squared Displacement and Average and Instantaneous Diffusivity

for different temperature (1445-1341K).

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 12

MSD

0 200000 400000 600000 800000 1×106

?@ABC

DEFGH

0.180

IJKLM

0.190

NOPQR

Time [ps]

MSD

[Å^2]

i) Mean Squared Displacement - STUVWXYZ[

Avg.

Inst.

0 200000 400000 600000 800000 1×106

0

5.×10-7

1.×10-6

1.5×10-6

2.×10-6

2.5×10-6

3.×10-6

Time [ps]

Diffusivity

[Å^2/ps]

i) Diffusivity - \]^_`abcd

MSD

0 200000 400000 600000 800000 1×106

efgij

klmno

0.180

Time [ps]

MSD

[Å^2]

p) Mean Squared Displacement - 1288.341K

Avg.

Inst.

0 200000 400000 600000 800000 1×106

0

5.×10-7

1.×10-6

1.5×10-6

2.×10-6

2.5×10-6

3.×10-6

Time [ps]

Diffusivity

[Å^2/ps]

q) Diffusivity - 1288.341K

MSD

0 200000 400000 600000 800000 1×106

rstuv

wxyz{

0.180

Time [ps]

MSD

[Å^2]

|) Mean Squared Displacement - }~�������

Avg.

Inst.

0 200000 400000 600000 800000 1×106

0

5.×10-7

1.×10-6

1.5×10-6

2.×10-6

2.5×10-6

Time [ps]

Diffusivity

[Å^2/ps]

�) Diffusivity - ���������

Figure 6. Mean Squared Displacement and Average and Instantaneous Diffusivity

for different temperature (1313-1265K).

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 13

MSD

0 500000 1.0×106 1.5×106

0.162

0.164

0.166

0.168

�����

�����

Time [ps]

MSD

[Å^2]

l) Mean Squared Displacement - 1244.0217K

Avg.

Inst.

0 500000 1.0×106 1.5×106

0

5.×10-7

1.×10-6

1.5×10-6

2.×10-6

2.5×10-6

Time [ps]

Diffusivity

[Å^2/ps]

l) Diffusivity - 1244.0217K

MSD

0 500000 1.0×106 1.5×106 2.0×106 2.5×106

0.162

0.164

0.166

0.168

�����

� ¡¢£

¤¥¦§¨

Time [ps]

MSD

[Å^2]

m) Mean Squared Displacement - 1224.443K

Avg.

Inst.

0 500000 1.0×106 1.5×106 2.0×106 2.5×106

0

5.×10-7

1.×10-6

1.5×10-6

2.×10-6

2.5×10-6

Time [ps]

Diffusivity

[Å^2/ps]

m) Diffusivity - 1224.443K

Figure 7. Mean Squared Displacement and Average and Instantaneous Diffusivity

for different temperature (1244-1224K).

The volume diffusivity for different temperatures from our MD computations,

using Direct(I) and Extrapolation(II) procedure of diffusivity determination from MD

simulation results, is summarized in Tab.2 and Fig.8. Our results fit better to the

averaged experimental data, [40], than these from DFT calculations for triple defect

mechanism (TN) [15]. The conformity of our simulation results to experimental results

from 1700K to ∼1400K is excellent. The use of the Extrapolation procedure(II)

improves diffusion simulation results especially for lower temperatures. At lower

temperatures, the agreement is worse, but better still than the results of the DFT

calculations.

Thereafter, using volume diffusivity results for different temperatures from our

molecular dynamics simulations, Tab.2, and the Arrhenius equation, Eq.2, Ni in B2-NiAl

self-diffusion prefactor D0 and activation energy Q for different temperature intervals

was determined, see Tab.3 and Fig.9. The numerical results have been confronted with

experimental values taken from [40]. For the temperature interval 1699-1489K the

calculated D0 and Q perfectly match the experimental results for Direct(I) procedure of

diffusivity determination from MD simulation results. The Extrapolation(II) procedure

extends this interval into the range 1699-1372K. As we can see, the new approach

improves agreement with experimental data, however the inconsistency problem is still

visible in lower temperatures. Moreover, our NiAl diffusion Tsimulation/Tmelting=0.69-

0.95 is much wider than that suggested by the authors in [54], the low temperature

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 14

Table 2. Volume diffusivity for different temperatures from our molecular dynamics

computations (MDI, Direct procedure I and MDII, Extrapolation procedure II), DFT

[15] calculations and from the experiments [40].

Temperature[K] MDI-D[m2/s] MDII-D[m2/s] Exp.-D[m2/s] DFT-D[m2/s]

1224 1.1688E-16 4.5937E-17 1.4659E-17 1.7081E-18

1244 1.6368E-16 7.001E-17 2.2895E-17 2.6801E-18

1265 2.9969E-16 1.2856E-16 3.6542E-17 4.2985E-18

1288 2.9300E-16 1.3336E-16 5.9743E-17 7.0636E-18

1313 3.2353E-16 1.8037E-16 1.0030E-16 1.1923E-17

1341 3.7173E-16 2.1841E-16 1.7347E-16 2.0738E-17

1372 4.2486E-16 2.4826E-16 3.1014E-16 3.7299E-17

1406 6.3452E-16 4.1901E-16 5.7570E-16 6.9678E-17

1445 6.3920E-16 1.0412E-15 1.1153E-15 1.3590E-16

1489 1.8723E-15 2.255E-15 2.2690E-15 2.7852E-16

1540 3.8250E-15 3.0116E-15 4.8868E-15 6.0459E-16

1599 7.1946E-15 7.7964E-15 1.1251E-14 1.4039E-15

1699 3.5237E-14 3.8241E-15 4.0480E-14 5.1172E-15

MD

MD*

Exp.

DFT

1300 1400 1500 1600 1700

0

1.×10-14

2.×10-14

3.×10-14

4.×10-14

Temperature [K]

Diffusivity

[m^2/s]

a) Diffusivity

MD

MD*

Exp.

DFT

1300140015001600

10-18

10-17

10-16

10-15

10-14

1/Temperature[K]

Diffusivity

[m^2/s]

b) Diffusivity

Figure 8. Volume diffusivity for different temperatures: a) normal scale, b) Arrhenius

plot.

results are very noisy and probably have little physical meaning for diffusion, simply

the diffusion lengths are too short.

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 15

Table 3. Ni in B2-NiAl self-diffusion prefactor D0 and activation energy Q

for different temperature intervals from our molecular dynamics simulations (MDI,

Direct procedure I and MDII, Extrapolation procedure II). Experimental D0=2.71-

3.45 x 10−5(m2s−1) and Q=2.97-3.01(eV/atom), see [40].

MDI MDII

Temp. interval[K] D0[m2/s] Q[eV/atom] D0[m

2/s] Q[eV/atom]

1699-1599 3.51317E-3 3.70975 3.90457E-3 3.71324

1699-1540 9.33994E-5 3.18766 1.82291E-3 3.6036

1699-1489 2.76108E-5 3.01506 3.40504E-5 3.03987

1699-1445 9.66945E-5 3.19004 1.44477E-5 2.92018

1699-1406 1.42095E-5 2.92579 2.60354E-5 3.00134

1699-1372 3.61679E-6 2.73949 2.57373E-5 2.99977

1699-1341 7.74201E-7 2.53194 9.39717E-6 2.86412

1699-1313 1.86944E-7 2.34261 2.97039E-6 2.71067

1699-1288 5.10362E-8 2.17135 1.17749E-6 2.58861

1699-1265 1.38062E-8 2.00049 3.94811E-7 2.4458

1699-1244 8.46639E-9 1.93715 2.5614E-7 2.38976

1699-1224 6.42842E-9 1.90179 2.05872E-7 2.36171

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

D0 [

m2/s]

a)

MDI

MDII

Exp.

0

1

2

3

4

Q

b)

MDI

MDII

Exp.

Figure 9. Self-diffusion prefactor D0 a) and activation energy Q b) for different

temperature intervals.

5. Conclusions

The volume diffusivity in the 1224K to 1699K temperature range, has been estimated

in the studies reported in this paper by direct molecular statics/dynamics simulations

applying the embedded atom model potential.

We can conclude that:

• The diffusivity in solid state B2-NiAl intermetallic can be successfully quantified

from direct MS/MD simulations.

• The instantaneous diffusivity stabilizes faster than the average one.

• The simulation times needed to achieve stabilised asymptotic diffusivity must be

A Molecular Dynamics Study of Self-Diffusion in Stoichiometric B2-NiAl crystals 16

in the order of microseconds.

• The application of the extrapolation procedure improves diffusion simulation

results.

Some findings in this paper, especially relating to the simulated by MD diffusivity

of stoichiometric solid state B2-NiAl intermetallic in the 1224K to 1699K temperature

range and the study of required simulation time ensuring the stabilization of diffusivity

results, are the first to be reported and are hopeful that will be confirmed by further

research studies. The estimated values will be used in the discrete element modelling of

a sintering process of NiAl powder [3]. The methodology developed in this study will

be included in future multiscale sintering modelling.

ACKNOWLEDGMENTS

This work was supported by the National Science Centre (NCN – Poland) Research

Project: DEC-2013/11/B/ST8/03287. Additional assistance was granted through the

computing cluster GRAFEN at Biocentrum Ochota and the Interdisciplinary Centre for

Mathematical and Computational Modelling of Warsaw University (ICM UW).

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