Physica B 406 (2011) 1060–1064

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Physica B

0921-45

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/physb

Interaction of small vacancy clusters with (1 1 4) twin-boundary in gold

Fayyaz Hussain a, Sardar Sikandar Hayat b,n, Muhammad Imran a

a Department of Physics, The Islamia University of Bahawalpur, Bahawalpur 63120, Pakistanb Department of Physics and Astronomy, Hazara University, Mansehra 21300, Pakistan

a r t i c l e i n f o

Article history:

Received 26 August 2010

Received in revised form

31 October 2010

Accepted 1 November 2010Available online 5 November 2010

Keywords:

Molecular dynamics

(1 1 4) Interface

Vacancy clusters

Gold

26/$ - see front matter & 2010 Elsevier B.V. A

016/j.physb.2010.11.001

esponding author. Tel.: +92 346 8834889; fax

ail address: sikandariub@yahoo.com (S.S. Hay

a b s t r a c t

The molecular dynamics simulation technique with many-body and semi-empirical potentials is used to

calculate the (1 1 4) twin-boundary in gold at different temperatures. Relaxations are found on both sides

of the interface with the same magnitude and the phenomenon of coalescence is observed near the

interface. The interactions of single-, di- and tri-vacancies with twin-interface at 300 K on mirror and

off-mirror sites are calculated. Off-mirror arrangements are favorable for all vacancy clusters, except for

the single-vacancy cluster, which is less repulsive on the mirror site. Vacancy clusters energetically prefer

to lie at planes closest to the (1 1 4) interface rather than away from it. The effect of temperature on

interaction behavior is also calculated.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Interaction between solute atoms and vacancies in metals hasbeen the subject of studies in the recent years. The energy andstructure of close packed vacancy clusters are discussed in manyearly studies [1–3]. As for the mechanism of atom transport in agehardening alloys, it is observed that the interactions play a verysignificant role. The interaction of crystal defects plays a verycrucial role in many theories of the strength of materials. Theformation of cavities, denuded zones and grain-boundary precipi-tates expresses these effects in a striking manner. Unfortunatelythe elasticity theory cannot be used to investigate the interactionenergy. Therefore, clearly relativistic results of interaction betweendifferent defects can be obtained using a simulation tool.

Point defects provide fast grain-boundary diffusion in grain-boundaries. The study of grain-boundaries shows structural effectsassociated with point defects, like delocalization of vacancies aswell as vacancy instability at certain grain-boundary sites andinterstitials in grain-boundary core [4]. The point defects have asignificant role in controlling characteristics such as plasticity,strength, electrical and thermal conductivities, etc. [5]. The inter-action of point defects with twin-boundaries may contribute tochange in mechanical properties. Bacon and Osetsky [6] havestudied extensively the atomic-scale processes involved indislocation–defect interactions.

The interaction of point defects with other lattice defects likedislocation and grain-boundaries ultimately changes mechanical

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at).

properties of materials. The strength and rupture properties ofmaterials can be manipulated strongly by isolation of alloying orimpurity atoms to grain-boundary or other interface. Atomicstructure of the boundary and character of dislocation may causeinteraction of crystal dislocations with twin-boundaries due to awide range of reactions. The probably major cause of twinningdislocations is dislocation absorption by boundary and the con-sequent transformation into a boundary defect. Restrictions ontwin-boundary motion and intensification by pinning twinningdislocations may be caused by the result of point defect clusters.The moving boundaries can operate as recombination centers orsinks for defects and can provide resources to remove defects fromregions of radiation damage [7]. The analysis of strain field aroundtwin-boundaries shows non-dislocation elastic stresses. The elasticfield stress is determined using microscope in twins and symmetricgrain-boundaries [8,9].

There are four possible distinct tri-vacancies (see Fig. 1) and asingle di-vacancy cluster configuration on the (1 1 1) plane of fcc

crystals [10]. In fcc metallic crystals twinning is a common planardefect and frequently observed [11–13]. Forwood and Clarebrough[14] in 1984 experimentally investigated the

P9 (1 1 4) twin-

boundary in Cu+6 at% Si alloy and rigid-body displacement at theboundary was shown. Interactions that exist between low indextwins and vacancies in fcc metals are attractive in fcc metals[15–17]. Static calculations for interaction energies with low-indextwin-boundaries are available [17] but the field still requires anexplanation in terms of interactions of point defects with twin-boundaries at high temperatures as they can be more reliable(approaches reality) when atomic dynamics is involved. Attentionis focused on the interaction of small vacancy clusters with the(1 1 4) twin-boundary (which is an open interface) in Au.

Fig. 1. Different geometries of tri-vacancy cluster in fcc closed packed structure

projected on the (1 1 1) plane.

F. Hussain et al. / Physica B 406 (2011) 1060–1064 1061

The present work is arranged in the following way; Section 2represents the salient features of the technique. Embedded AtomMethod (EAM) potentials [18,19] with interactions extending up tothe fourth nearest neighbors are used to calculate interactions.Steps of the procedure involved in the simulation work made havebeen explained. In Section 3, twin formation energy of the (1 1 4)twin is calculated at different temperatures. Single-, di- and tri-vacancies are generated and their interaction behavior is observedat the (1 1 4) interface at 300 K along with details of twins. Thegraphical presentation of calculations is also made. Section 4contains the conclusions of this work.

Fig. 2. Comparison between calculated and experimental values of lattice para-

meters at various temperatures in the range of 300–1000 K for Au.

2. Details of calculation

We used the ‘‘Dyn86’’ code contained within a main DYNAMOroutine, which is based on classical molecular dynamics. Theliterature on the MD technique can be found in Refs. [18,19]. Inthis code the Nordsiesk Algorithm [20] is used with a time step of10�15 s. The computational cell used consists of 256 atoms,arranged in the fcc lattice. To achieve the required temperature,preliminary NPT simulations are carried out with periodic bound-ary conditions in all directions of the cubic simulation cell. We havechecked the convergence of the lattice parameter and energy peratom with respect to system size. The use of periodic boundaryconditions diminishes the effect of size of the crystal on thecalculation of interaction energies.

The twin-boundary is generated on the central plane. Firstly avacancy cluster is created on the model at the plane nearest to thetwin-boundary and then proceeds on to next planes away from thetwin. A vacancy cluster that has more than one atoms, never existsin a single plane but we assume that the vacancy cluster lies at theplane closest to the boundary containing at least one vacancy ofthat cluster. Energy of the crystal is minimized by NVE ensembleusing the conjugate gradient method [21]. The energy of vacancyinteraction is calculated in each case. The energy of the crystal inthe presence of a twin is given by

ET ¼ EPþT�EP :

Here EP +T is the energy of the crystal containing the twin-boundaryand EP the energy of the perfect crystal. In the next step the energyEV is calculated, which is associated with a single-vacancy and isgiven by

EV ¼ EPþV�EP :

Here the term EP+ V gives the energy of the crystal containing asingle-vacancy. The energy of di-vacancy E2V is calculated using therelation

E2V ¼ EPþ2V�2EV�EP :

Here the term EP +2V gives the energy of the crystal containing adi-vacancy. Similarly, the energy of tri-vacancy E3V is calculatedusing the relation

E3V ¼ EPþ3V�3EV�EP :

Here the term EP +3V gives the energy of the crystal containing a tri-vacancy. We have to find energies of both twin-boundary and

vacancy. The interaction energy of twin-boundary and single-vacancy is calculated using the relation

E1C ¼ EPþTþV�EPþT�EV :

Here EP+ T +V represents the energy of the crystal containing bothtwin- and single-vacancies. Then combined interaction energy fordi-vacancy can be evaluated from the equation

E2C ¼ EPþTþ2V�EPþT�E2V :

Here EP+T+2V is the energy of the crystal in the presence of di-vacancyand a twin-boundary. Similarly, for tri-vacancy the combined inter-action energy can be calculated as

E3C ¼ EPþTþ3V�EPþT�E3V :

Here EP+T+3V is the energy of the crystal in the presence of tri-vacancyand a twin-boundary.

3. Results and discussion

We used an experimental lattice parameter of 4.08 A to generatean Au fcc crystal at 0 K with 256 atoms. The crystal is generatedusing a preliminary NPT ensemble at different required tempera-tures. This produces the lattice parameter at different tempera-tures, which can be used to develop a slab-like structure with therequired geometry of planes. The lattice constant obtained from thepreliminary MD simulation is plotted in Fig. 2 as a function oftemperature along with experimental corresponding values [22].The lattice parameter increases at an average of 0.0062 A for every100 K increase in temperature. The plot of lattice parameter atdifferent temperatures shows that the calculated values of latticeparameter are in good agreement with experimental values.

The simulated values of lattice parameter at various tempera-tures are on an average higher by 0.013 A when compared to theexperimental values. The overall increase in the calculated latticeconstant is 0.31%. The reason behind it can be the choice of purecrystal for simulation whereas real crystals have a number ofimpurities or defects and also there are possibilities of otherexperimental limitations. However, the trend of increase in bothcalculated and experimental values is the same. Deviation incalculated lattice parameters at various temperatures from experi-mental values is in a good range as compared to the calculatedvalues reported by Kallinteris et al. [23] using tight bindingpotentials.

A rectangular block of 1728 atoms is used for the high-angle(1 1 4) twin-boundary with mutually perpendicular 36(2 2 1),72(1 1 4) and 24(1 1 0)planes. The relaxed twin formation energyat 0 K is 2033.92 mJ/m2. Twin formation energy decreases with

F. Hussain et al. / Physica B 406 (2011) 1060–10641062

increase in temperature, with an average decrease of 53.17 mJ/m2

for 100 K. The twin formation energy for the (1 1 4) twin-interfacein the temperature range 0–1000 K, in steps of 100 K, is shown inFig. 3. The relaxed (1 1 4) twin structure projected on the (1 1 0)plane is presented in Fig. 4.

The (1 1 4) twin-boundary is generated by shuffling the upperhalf of the crystal along the [1 1 4] direction. The atomic density ofthe (1 1 4) plane is one-ninth of that of the close-packed (1 1 1)plane. In the case of the (1 1 4) twin-boundary maximum interlayerand registry relaxations are observed near the interface and morethan 85% contraction of two planes is calculated. Therefore, threeplanes coalesce completely near the boundary and two pairs ofplanes coalesce partially on either side. In the conventionalgeometric structure of the (1 1 4) twin-interface, atoms at theboundary are extremely close to each other and the model mayexplode. To avoid this situation atoms of the boundary aretranslated (shifted) by 1.9305 A along the negative [2 2 0] direc-tion. Due to this near the twin across the boundary, reflection isachieved, which prevents the crystal from exploding. Therefore, thetwin formation energy is low as compared to that of explodingcrystal. These calculations agree with early findings [24].

To calculate the interaction energy in cluster 1.1 at 300 K, asingle-vacancy is introduced at the plane closest (first) to eitherside of the (1 1 4) twin-boundary. The interaction energy for asingle-vacancy with the (1 1 4) twin-boundary at the first plane is1.903549 eV. This process is repeated in the next 2–8 planes. It isobserved that the interaction energy shows repulsive behavior forthe single-vacancy with the (1 1 4) interface at all planes. Similarly,cluster 2.1 is generated near the (1 1 4) twin-boundary by intro-ducing two vacancies. The interaction energy for cluster 2.1 at the

Fig. 3. Plot of twin formation energy (mJ/m2) versus temperature (K) for the (1 1 4)

twin-boundary. It shows that twin formation energy decreases with increase in

temperature.

Fig. 4. Relaxed structure projected on the (1 1 0) plane of the (1 1 4) twin-interface.

Horizontal mid-line shows the twin-boundary.

first plane near the (1 1 4) twin-boundary is 3.728004 eV at 300 K.The di-vacancy is repulsive at all (1–8) planes. Similarly, theinteraction energies at 300 K of the tri-vacancy with differentgeometries of 3.1, 3.2, 3.3 and 3.4 with the (1 1 4) twin-boundaryare 5.49384, 5.09687, 5.29316 and 4.95917 eV, respectively, at thefirst plane. All tri-vacancy configurations are repulsive at all planeswith some variations in magnitude.

A pair of single- and di-vacancies is generated on both sides ofthe twin-interface on- and off-mirror sites. Single- and di-vacanciesare shifted to one interplanar spacing to get off-mirror positionsalong the [2 2 0] direction. Figs. 5 and 6 show that clusters 1.1 and2.1 are repulsive at all planes with on- and off-mirror arrange-ments, respectively. However, di-vacancy is more repulsive ascompared to single-vacancy at all planes on mirror positions,except at the first plane. Single-vacancy is favored at on-mirrorsites as compared to off-mirror sites. The off-mirror site is favorableas compared to the on-mirror site for di-vacancy. Di-vacancy is lessrepulsive at the first plane as compared to all other planes.Similarly, tri-vacancy clusters are also calculated in pair form aton- and off-mirror positions. The interaction energies of tri-vacancy clusters at 300 K on either side of the (1 1 4) twin-boundary on- and off-mirror sites are presented in Figs. 7 and 8,respectively. For both mirror and off-mirror sites tri-vacancyclusters are repulsive. This repulsion is more for mirror sites ascompared to off-mirror sites for all tri-vacancy clusters. Theinteraction energy at the first plane on off-mirror position for

Fig. 5. Interaction energies of two single-vacancy clusters, one on either side of the

(1 1 4) twin at 300 K.

Fig. 6. Interaction energies of two di-vacancy clusters, one on either sides, of the

(1 1 4) twin at 300 K.

Fig. 7. Interaction energies of tri-vacancy cluster at 300 K on either side of the

(1 1 4) twin-boundary on mirror sites.

Fig. 8. Interaction energies of tri-vacancy cluster at 300 K on either side of the

(1 1 4) twin-boundary on off-mirror sites.

Fig. 9. Interaction energies of single-, di- and tri-vacancies as a function of

temperature at the first plane near the (1 1 4) twin-boundary.

F. Hussain et al. / Physica B 406 (2011) 1060–1064 1063

the cluster 3.1 is high as compared to other tri-vacancy clusters andat other planes.

The effect of temperature on interaction energies in single-,di- and tri-vacancies is calculated near the (1 1 4) twin-boundary atthe first plane in the temperature range 300–1000 K, in steps of100 K (see Fig. 9). The clusters 1.1, 2.1 and 3.1 have interactionenergies 1.44348, 1.70407 and 5.49384 eV, respectively, at 300 K.

At the plane closest to the (1 1 4) twin-boundary, all these vacancyclusters represent repulsive behavior and with increase in tem-perature interaction energy increases. The interaction energychange (increase) with increase in temperature for clusters 1.1,2.1 and 3.1 on an average for 100 K is 0.505344, 0.457951 and0.433874 eV, respectively. The interaction energies become4.98089, 4.97645 and 9.57645 eV for single-, di- and tri-vacancies,respectively, at 1000 K.

Compact interfaces relax less as compared to the high-indexinterfaces because of the lack of symmetry near the high-indexinterface [25,26]. The (1 1 4) interface is more open (has less atomicdensity) as compared to the (1 1 1), (1 1 2) and (1 1 3) interfaces.Therefore, comparatively more relaxations are found near the(1 1 4) twin-boundary. The bulk atoms have electric and ionicsymmetry arrangements throughout the bulk; therefore somemodification is required in the charge configuration near theinterface. The force field is changed as compared to bulk atomswith normal coordinates. Some relaxations appeared as the fieldchanged; the interlayer spacing was changed around the twin-interface. During the motion of planes perpendicular to the inter-face, the registry of atomic planes may change due to relaxation ofplanes parallel to the boundary. Proper charge redistributionaround the grain-boundaries is facilitated by these multilayerrelaxations. In the process of relaxation, spacing around thetwin-interface relaxed more as compared to bulk interplanarspacing.

The energy per atom and interplanar spacing increase withincrease in temperature. With large distances between atoms andhigh energy, generation of twins and relaxation in the crystal iseasy. Therefore, twin formation energy decreases with increase intemperature. The interface with high atomic density has largeinterplanar spacing and vice-versa. Atomic relaxation of a plane isproportional to interplanar spacing and inversely proportional toatomic density of that plane. Therefore, an interface with lowatomic density has high twin formation energy and vice-versa. Themaximum disturbance of atoms is at the interface, while the otherpart of the crystal is perfect. Therefore, plane registry and interlayerrelaxations decrease with movement away from the interface andbecome negligible after more than ten planes (approximately).

Atoms in the grain-boundary core have high potential energy ascompared to the bulk atoms and it seems reasonable that interfacecan get thermally disordered before the bulk or has its own meltingtransition. For such behavior there exists an indirect experimentalevidence and theoretical considerations [19,27,28]. Around thetwin-boundary the atomic density is different as compared to thatin the normal plane. The number of self interstitials and vacanciesincrease around the twin plane with increase in temperature; thusdiffusion of atoms is likely to begin in the crystal at twin-boundaries. At a sufficiently high temperature, in the vicinity oftwin plane, the atoms are in complete disorder. This disorderingand diffusion can lead to pre-melting near the twin-boundary,which enhances the effect of the melting phenomenon. Therefore,the bulk crystal melts at lower temperatures than that of thenormal melting point [19].

The present results of interaction energies of small vacancyclusters with the (1 1 4) twin-boundary are satisfactory in thesense that atomic relaxations that occur are all consistent withwhat might be anticipated using the hard sphere model and arecomparable with the results given by Ahmad and Ramzan [17]using static calculations. The values of twin formation energy arecomparable with the simulated results for other interfaces ofaluminum [19], copper [29], palladium and platinum [26] metals.The results presented in this study can be relied upon as they areobtained using a well established suite of computer programs thathave already produced plausible results for other fcc metals[3,19,26,30–32].

F. Hussain et al. / Physica B 406 (2011) 1060–10641064

4. Conclusions

The lattice parameter for Au is calculated and compared withexperimental values [25]. The simulated values of lattice parameteragreed well with experimental values in the temperature range of300–1100 K. Results of MD simulation are closer to experimentalvalues as compared to the calculated values given by Kallinteriset al. [23] using tight binding potentials.

We concluded that:

(i)

Twin formation energy decreases with increase in tempera-ture. With the increase of 100 K in temperature the twinformation energy decreases on an average by 53.17 mJ/m2.

(ii)

(1 1 4) twin-interface has a higher twin formation energy andshows greater interlayer and registry relaxations around thetwin-interface because of lower planar atomic density andsmaller interplanar spacing as compared to (1 1 1), (1 1 3) and(1 1 2) interfaces, which have been studied in Refs. [22,29].There is coalescence of three planes at the (1 1 4) interface.

(iii)

Atomic density near the interface increases due to thecoalescence process of planes and the energies of core (dis-turbed) atoms become high as compared to bulk atoms, whichcreates repulsion of vacancy clusters from the twin-boundary.

(iv)

Energy per atom and interlayer spacing increase with increasein temperature; therefore, at high temperature, interactionenergy (repulsion) is increased for all vacancy clusters.

(v)

Almost all vacancy clusters are favorable at nearest planes andrepulsion increases as they move away from the boundaries ofboth on- and off-mirror sites. Off-mirror sites are favored ascompared to on-mirror sites.

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