CHINESE JOURNAL OF PHYSICS
VOL. 51, NO. 2
April 2013
Effect of Crystal Defects on the Melting Temperature of Ni and Al
Fayyaz Hussain,1 Sardar Sikandar Hayat,1, 2, ∗ Zulfiqar Ali Shah,2 and S. A. Ahmad1
1
Department of Physics, The Islamia University of Bahawalpur, Bahawalpur 63120, Pakistan
2
Department of Physics and Astronomy,
Hazara University, Mansehra 21300, Pakistan
(Received September 24, 2011; Revised March 12, 2012)
The molecular dynamics simulation technique has been applied to study the effects of
temperature on Ni and Al with point and planar defects. For this purpose a well established
program ‘dyn86’ containing the DYNAMO subroutine has been used. Semi-empirical potentials based on the embedded atom method (EAM) have been employed to calculate the
lattice parameter and energy per atom in order to determine the melting point. The effects
of point defects, including self, substitution, and interstitial, on the melting point have been
investigated. The twin formation energy of some low index (111), (112), (113), and (114)
twin-interfaces and their effects on the melting point of Ni and Al are also studied. It is
observed that the presence of defects (point or planar) lowers the melting point of metals,
as compared to the simulated melting point of a defect-free crystal.
DOI: 10.6122/CJP.51.347
PACS numbers: 07.05.Tp, 61.50.Ks, 64.70.dj
I. INTRODUCTION
The presence of any type of defect is very important in pure metals, because it may
alter the thermal properties of materials [1–4]. The study of the nature and concentration of defects in metals, which control many diffusion processes, that formed in thermal
equilibrium at high temperatures, is needed. Differential thermal expansion measurements
have been performed in some metals [5], which showed that vacant lattice sites are the
thermal equilibrium defects which are dominant near the melting temperature (Tm ) [6].
The melting temperature Tm is an important parameter of a material, because it indicates
its instability against heating.
Studies of equilibrium defects at high temperatures are generally preferred for quenching experiments [7]. With the development of the positron annihilation techniques, the
study of defect formation in high-melting metals became feasible for the first time [7, 8].
The increase of lattice defects in crystalline compounds causes a decrease in their melting
temperature; therefore, these compounds undergo solid-state amorphization [9].
An ion or electron beam causes various kinds of atomic defects in target materials. The primary defects are self-interstitial atoms (SIA) and vacancies, if the bombarding
energy is in the intermediate (keV) energy range. Some SIA and vacancies can diminish immediately by recombination. Otherwise, these primitive point defects remain, and
∗
Electronic address: [email protected]
http://PSROC.phys.ntu.edu.tw/cjp
347
c 2013 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
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EFFECT OF CRYSTAL DEFECTS ON THE MELTING . . .
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preferably form extended defects [10]. Fractures of bi-materials usually occur at or near
the interface due to the effect of defect and stress connection. The role that individual
interfaces play in the mechanical behavior of a material is a hard phenomenon to observe
experimentally. To simplify this analysis, many experiments have been conducted on a crystal, which motivates the study of deformation and incompatibilities across a single grain
boundary [11]. The resulting defect structures will depend strongly on the target temperature. At high temperatures under self-irradiation, SIA can form different kinds of extended
defects. Linear defects of chain structure, loops, planar defects, or three-dimensional defect
agglomeration can be produced [10].
Heterogeneous nucleation of the liquid phase at extrinsic defects, such as grain boundaries, voids, or the crystal surface, turned out to be the most rapid—and hence the most
dominant—mechanism of melting, entirely in accordance with the experimental observations. The investigations [12, 13] demonstrate in particular that the introduction of any of
these defects nucleates melting at any temperature above the equilibrium Tm [12, 14].
In this work, attention has been focused on the melting temperature and the role of
crystal defects (point or planar) on the melting of Ni and Al by means of MD simulations
using the EAM potentials. The octahedral and the dumbbell sites for interstitial atoms are
pointed out in Figs. 1 (a) and 1 (b), respectively. The dumbbell interstitial site carries two
atoms attached at the single lattice site which form an atom pair in the <100> direction;
it has been reported on by Jesson et al. [15]. We have calculated the lattice parameter and
energy per atom at various temperatures and deduced the melting temperature of a defectfree crystal. Interstitial atoms are introduced at octahedral and dumbbell sites and low
index: (111), (112), (113), and (114), twin-boundaries are generated, and finally melting
checked in the presence of these defects.
FIG. 1: (a) Interstitial atom at an octahedral site. (b) Interstitial atom at a dumbbell site.
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FAYYAZ HUSSAIN, SARDAR SIKANDAR HAYAT, ET AL.
349
II. SIMULATION TECHNIQUE
We used the ‘dyn86’ code contained within a main routine called dynamo. The
details of the MD technique can be found in the literature [16, 17]. According to our
problem, we introduced routines for the generation of an fcc lattice and twin-boundaries
inside the lattice. The salient features of the technique as applied in our case are given
here. Nordsieck’s algorithm [18] with time step of 10−15 s was used in order to solve the
classical equations of motion for the atoms interacting by the EAM potential [16, 17]. To
study point defects, a model crystallite was generated with 1440 moveable atoms, while
in order to study a twin-interface, a model crystallite was generated with 1536 moveable
atoms. This crystallite has a rectangular block of atoms. Periodic boundary conditions
are applied along the parallel direction, while fixed boundary conditions are applied along
the perpendicular directions, and the cut-off distance for the potential was kept between
the 3rd and 4th nearest neighbor. A preliminary simulation was carried out under the
condition of a constant number of atoms, pressure, and temperature (NPT ensemble), with
the computational cell comprising 1536 atoms arranged on an fcc lattice. Now the crystallite
was allowed to evolve till the cell edges and volume became constant using a constant
number of atoms, volume, and temperature (NVT ensemble). The lattice constants thus
obtained are used to generate an fcc crystal at various temperatures. We have checked
the convergence of the lattice parameter and energy per atom with respect to system size.
Use of periodic boundary conditions diminishes the effect of the size of a crystal on the
calculations.
The energy of a perfect crystal (Ep ) is first calculated using a constant number of
atoms, volume, and energy (NVE ensemble), and then a twin-interface is introduced at the
middle of the computational cell, as described above. The energy of the crystal is again
calculated using the NVE ensemble. The crystal is allowed to relax to its minimum energy
by the conjugate gradient method [19]. Then the energy of a twin-interface Eγ is calculated
as Eγ = Ep+γ − Ep , where Ep+γ is the energy of a relaxed crystalline in the presence of a
twin. For the case of an Al interstitial in Ni crystal, we introduced two species with respect
to the two potentials of Al and Ni. Similarly, for a Au interstitial in Al crystal also, the
potentials of both Au and Al are provided.
III. RESULTS AND DISCUSSION
The internal energy per atom E(T ) is calculated as a function of the temperature in
the range 300–2500 K for Ni and 300–1200 K for Al in steps of 100 K; near the melting point
the step must be used ±5 K using the NPT simulation. The results are plotted in Figs. 3
and 4, which exhibit a sudden jump between 2085 and 2095 K for Ni and between 895 and
900 K for Al. The NPT simulation is also used to calculate the lattice parameter α(T ) at
various temperatures. The computational cell is generated at a particular temperature and
then, keeping the pressure and temperature constant, the system is allowed to evolve till
the cell edges and the volume become constant. The lattice parameter is calculated from
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EFFECT OF CRYSTAL DEFECTS ON THE MELTING . . .
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FIG. 2: Lattice parameter α (Å) and the energy per atom E (eV) for (a) Ni and (b) Al at various
temperatures.
α(T ) = (4Ω)1/3 , where Ω is the calculated average atomic volume at each temperature.
Plots of the calculated lattice parameters as a function of temperature are also given in
Figs. 2 (a) and 2 (b) for Ni and and Al, respectively. Transitions can again be noted
at the same temperatures as those observed for the energy per atom curves. The lattice
parameter and energy per atom are calculated using the NPT ensemble in the temperature
range 300–2100 K for Ni and 300–1200 K for Al. The transitions of the curves for the lattice
parameter and energy per atom show that these metals have melting temperatures equal
to Tm = 2085 ± 10 for Ni and Tm = 895 ± 5 K for Al. These results are also suggested by
recent calculations for Ni and Al [20, 21]. The value of the melting point obtained for Ni
in the present work is 20% higher as compared to the experimental value of 1726 K [22].
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FAYYAZ HUSSAIN, SARDAR SIKANDAR HAYAT, ET AL.
351
The value of 895 K as Tm of Al agrees reasonably well with the experimental value of 933
K [23], as compared to the value reported by Mitev et al. [24].
FIG. 3: Lattice parameter α (Å) for (a) Ni having Al as a self-interstitial at the octahedral position,
and (b) Al having Au as a self-interstitial at a dumbbell site at various temperatures (SIOS – self
interstitial at octahedral site, SIDS – self interstitial at dumbbell site and AIOS – Au interstitial at
octahedral site).
The presence of defects may change the properties of metals considerably. We have
studied the effect of an interstitial atom on the melting temperature of Ni and Al. When an
interstitial atom is introduced inside the crystal then the coordination of nearest-neighbor
atoms is changed, which disturbs the location of atoms in the model crystal. As the atomic
density is different in the neighborhood of the interstitial/substitutional atom compared
to a normal crystal, the potential energy of the crystal is increased. It seems reasonable
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EFFECT OF CRYSTAL DEFECTS ON THE MELTING . . .
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FIG. 4: (Color online) The relaxed structure of a twin-interface projected on the (110) plane showing
two adjacent layers of (a) (111), (b) (112), (c) (113), and (d) (114) twin-boundary for Ni. Horizontal
mid-lines represent twin-boundaries.
that the defects can physically disorder the structure of a crystal; due to this the lattice
parameter also increases, which results in a decrease of the melting temperature as compared
to the simulated melting point of a defect-free crystal. In the presence of a substitutional
atom inside the crystal the melting point of the crystal is also decreased, because of its
own potential energy which affects the nearest neighbor atoms. The arrangement of atoms
inside the crystal is disturbed, which decreases the melting point. When the interstitial
is introduced, it finds no space to adjust inside the crystal, so it disturbs the remaining
crystal. The results of self interstitial, an Al interstitial having an octahedral position inside
the Ni crystal, and a Au interstitial at octahedral and dumbbell sites inside the Al crystal
are plotted in Figs. 3 (a) and 3 (b). From these results it is observed that in the presence
of a self interstitial the melting point of Ni crystal is decreased by 30 K, as compared
to the simulated normal melting point of the crystal, and for Al a self interstitial at an
octahedral site it is decreased by 95 K. In the presence of a self interstitial at a dumbbell
site, the melting point of an Al crystal is decreased by 25 K, and in the presence of an
Al interstitial inside a Ni crystal the melting point of the Ni crystal is decreased by 45 K,
and in the presence of a Au interstitial inside an Al crystal the melting point of the Al
crystal is decreased by 65 K. The results are summarized in Table I. From these results it
is observed that if the size of the substitutional interstitial is large then the decrease of the
melting point of the crystal is also large, as compared to a small size of the substitutional
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FAYYAZ HUSSAIN, SARDAR SIKANDAR HAYAT, ET AL.
353
interstitial atom and a self interstitial atom (same size).
TABLE I: Variation of the melting point of a crystal having an interstitial.
Metal
Interstitial
Decrease of Tm (K)
Ni
Ni
30
Ni
Al
45
Al
Au
65
Al
Al (OS)
95
Al
Al (DS)
25
To study the effect of twin-boundaries on the melting temperature of Ni and Al
we generated a model fcc crystal consisting of a three dimensional rectangular block of
atoms. The twin-boundary is located as close as possible to the centre of the model. After
generating a model twined crystal, atoms are allowed to relax from their initial approximate
positions to their equilibrium configuration using the conjugate gradient method. The size
independence is achieved by the use of periodic boundaries in the calculation of the twin
formation energy. The crystallite developed for the (111) twin boundary having 1440 atoms
is a rectangular block of atoms with the number of planes 30(112), 48(111), and 6(110).
The (111) faces are kept fixed, while the other faces are under periodic boundary conditions.
The twin-boundary is generated on the central plane of the model. The relaxed structure
of the (111) twin-interface is shown in Fig. 4 (a) for Ni. The computational cell having
1408 atoms for a (113) twin boundary consisted of 44(332), 88(113), and 8(110) planes.
Fixed boundary conditions are applied on the faces parallel to the (113) twin-interface, and
cyclic periodic boundary conditions are imposed on the faces perpendicular to the (113)
twin-interface. The twin boundary is generated at the middle of the (113) plane. The
fully relaxed structure of the (113) twin for Ni is given in Fig. 4 (b). The model crystal
with 1536 atoms used to create a (112) twin-boundary is a rectangular block of atoms with
24(111), 48(112), and 8(110) planes. The (112) faces are simulated under rigid boundary
conditions, whereas the other four faces have periodic boundary conditions. The twinboundary is generated at the center of the model. The fully relaxed structure of the (112)
twin for Ni is shown in Fig. 4 (c). Similarly, for the creation of the (114) twin-boundary, the
crystallite has 1536 atoms. It has 48(221), 144(114), and 8(110) planes. Periodic boundary
conditions are applied perpendicular to the (114) direction, while along the (114) direction
fixed boundary conditions are applied. Fig. 4 (d) represents the fully relaxed structure of
the (114) twin-boundary of Ni.
To calculate the twin formation energy at various temperatures, the NVE ensemble
is used. The (111), (112), (113), and (114) twin-interfaces are simulated in the temperature
range 300–2000 K for Ni and 300–800 K for Al, in steps of 100 K. The twin formation
energies of these twin-interfaces against temperature are plotted in Figs. 5 (a) and 5 (b),
respectively. It is observed that the (111) twin-interface, with high planar atomic density
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EFFECT OF CRYSTAL DEFECTS ON THE MELTING . . .
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and greater interplanar spacing, has a low twin formation energy. The (114) twin-interface,
with low planar atomic density and smaller interplanar spacing, has a high twin formation
energy. The present results of twin-interfaces are satisfactory, in the sense that the atomic
relaxations which occur are all consistent with what might be anticipated using the hard
sphere model. The relative energies which have been deduced are acceptable. The relaxed
structures of all the twin-boundaries projected on the (110) plane are shown in Fig. 4. Our
calculated twin formation energies for Ni are 14.04, 873.61, 1540.71, and 3217.71 mJ/m2
at 300 K for the (111), (113), (112), and (114) twin, respectively. Similarly, the twin
formation energies at 300 K for Al are 60.64, 147.96, 375.57, and 765.07 mJ/m2 for the
same twins, respectively. The present values of the twin formation energy are lower than
the earlier simulated results, which were deduced by Faridi et al. using Ackland potentials
for fcc metals [25], and the relaxations near the twin boundaries are comparable with
recent calculations [26]. We also studied the effect of these low index twin-interfaces on
the melting point of Ni and Al. The results for the lattice parameter of a twinned-crystal
against temperature for Ni and Al are plotted in Figs. 6 (a) and 6 (b), respectively.
The energy per atom and interplanar spaces are increased with a rise of temperature. With large distances among the atoms and high energy, the generation of twins and
relaxations of the crystal are easy. Therefore, the twin formation energy is decreased with
an increase of temperature. The interfaces with high atomic density have large interplanar
spacing and vice-versa. Atomic relaxation of a plane is proportional to the interplanar spacing and inversely proportional to the atomic density of that plane. Therefore, interfaces
with low atomic density have high twin formation energy and vice-versa. The maximum
disturbance of the atoms is at the interfaces, while the other part of the crystal is perfect.
Therefore, the interlayer and plane registry relaxations (change of interplanar spacing and
registry of all the atoms of the plane) decrease moving away from the interfaces and become
negligible after more than ten planes (approximately).
From these results it is observed that the lattice parameter of twinned crystallite
is increased by an increase of temperature; atoms at the grain-boundary cores have a
potential energy higher than that of bulk atoms. The interface atoms can become physically
disordered, and thus have their own melting transition. Indirect experimental evidence and
theoretical consideration [27, 28] supports this behavior. The atomic density for a normal
plane is different as compared to the density in the region of the twin-interfaces. With an
increase of temperature, the number of self interstitials and vacancies increases around the
twin-interface, so that the diffusion of atoms is likely to begin at the twin-interfaces. At
higher temperatures, in the vicinity of a twin-interface the atoms are completely disordered.
This disordering and diffusion mechanism leads to premelting near the twin-interfaces,
which enhances the melting process; as a result, a crystal having twin-interfaces melts at a
lower temperature as compared to the defect-free crystal.
It is observed that with an increase of temperature, the twin formation energy is
decreased, and compact planes have low twin formation energy such as the (111) plane, and
open planes have high twin formation energy such as for the (114) plane. Our calculations
estimated for Ni, Tm (111) ∼
= 2060 K ±10 K, Tm (112) ∼
= 1980 K ±10 K, Tm (113) ∼
= 2050
∼
∼
K ±10 K, and Tm (114) = 2000 K ±10 K. Similarly, for Al, Tm (111) = 820 K ±10 K,
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FAYYAZ HUSSAIN, SARDAR SIKANDAR HAYAT, ET AL.
355
FIG. 5: A plot of the twin formation energy (mJ/m2 ) versus temperature (K) for the (111), (113),
(112), and (114) twin-boundaries for (a) Ni and (b) Al. All the plot lines show that with an increase
of the temperature the twin formation energy is reduced, except for the (111) twin.
Tm (112) ∼
= 800 K ±10 K, Tm (113) ∼
= 800 K ±10 K, and Tm (114) ∼
= 700 K ±10 K. It is also
observed that in the present study there is a minimum effect on the melting temperature
in the presence of a (111) twin-interface and a maximum effect in the presence of a (112)
twin-interface for Ni, while for Al it is in the presence of a (114) twin-interface.
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EFFECT OF CRYSTAL DEFECTS ON THE MELTING . . .
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FIG. 6: Lattice parameter α (Å) in the presence of the (111), (112), (113), and (114) twin-interfaces
for (a) Ni and (b) Al at various temperatures.
IV. SUMMARY
In this study, the melting temperature was computed in two different ways, i.e., by
looking at a variation in the lattice parameter and the energy per atom. Our calculations
estimate Tm ∼
= 2085 ± 10 for a defect-free Ni crystal and Tm ∼
= 895 ± 5 K for a defect-free
Al crystal. This value for Al is closer to the experimental value of 933 K [23], as compared
to the value recently simulated by Mitev et al. [24]. The value of Tm for Ni lies within
20% deviation from the experimental results [22]. The twin formation energies and the
change in the melting temperature in the presence of defects are comparable with recent
findings [20, 27].
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FAYYAZ HUSSAIN, SARDAR SIKANDAR HAYAT, ET AL.
357
The interplanar spacing is 0.5773α, 0.3015α, 0.2357α, and 0.2041α for the (111),
(113), (114), and (112) planes, respectively. It is concluded that:
1. The transition temperature decreased with any type of defect.
2. The decrease in the melting temperature of metals having substitutional interstitial
depends on the size of the interstitial.
3. The (111) twin-interface has a low twin formation energy and shows a smaller decrease
in the melting temperature, because of the high planar atomic density and larger
interplanar spacing as compared to others.
4. The (112) twin-interface has a higher twin formation energy and shows a greater
decrease in the melting temperature, because of the lower planar atomic density and
smaller interplanar spacing as compared to others.
5. The (113) twin-interface has a higher twin formation energy than that of the (111)
twin-interface and a lower from the (112) twin-interface, because its planar atomic
density and interplanar spacing lies between the (111) and (112) planar atomic density
and the interplanar spacing.
6. The (114) twin-interface has a high twin formation energy and shows a greater decrease in the melting temperature, because of the coalescence of three planes at the
interface, giving rise to a nearly reflection situation with a slightly higher energy than
that of a perfect reflection of a twin.
Acknowledgements
This work is supported by the Higher Education Commission (HEC) of Pakistan.
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