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Physica B ] (]]]]) ]]]]]]Contents lists available at SciVerse ScienceDirectPhysica B0921-45

http://d

n Corr

E-m

Pleasjournal homepage: www.elsevier.com/locate/physbCrystallization analysis and determination of Avrami exponents of CuAlNialloy by molecular dynamics simulationF.A. Celik a,n, S. Kazanc b

a Bitlis Eren University, Faculty of Arts & Sciences, Physics Department, 13000 Bitlis, Turkeyb Firat University, Faculty of Education, Physics Teaching Programme, 23119 Elazig, Turkeya r t i c l e i n f o

Article history:

Received 16 May 2012

Received in revised form

14 September 2012

Accepted 15 October 2012

Keywords:

Crystal growth

Computer simulation

Nanostructures

Recrystallization

JohnsonMehl and Avrami (JMA) model26/$ - see front matter & 2012 Published by

x.doi.org/10.1016/j.physb.2012.10.015

esponding author. Tel.: 90 0434 228 5270;ail address: facelik@beu.edu.tr (F.A. Celik).

e cite this article as: F.A. Celik, S. Kaa b s t r a c t

In this study, local atomic rearrangements of Cu%26.8Al%2.5Ni ternary alloys (3 A) are investigated

during their crystallization processes from amorphous phase using molecular dynamics (MD) simula-

tions. These simulations are based on the SuttonChen type of embedded atom method (SCEAM) that

employs many-body interactions. In order to analyse the structural development obtained from MD

simulation, the simulation techniques are used as bond order parameter, radial distribution function

(RDF). Local atomic bonded pairs and short range order properties in the model alloy have been

analysed using the HoneycuttAndersen (HA) method. The kinetics of the crystallization is described by

Johnson, Mehl and Avrami (JMA) model, which has been analysed with MD method by using the

crystalline bonded pairs. The simulation results show that the structural variation of local atomic

bonded pairs is of great importance to understand the crystallization kinetics from amorphous phase to

crystal phase during the crystallization.

& 2012 Published by Elsevier B.V.6768697071727374757677787980818283848586878889909192931. Introduction

Many studies of rapidly quenched amorphous alloys focus oneasy glass forming ability or the crystallization onset as a measureof kinetic stability. In fact, the initial annealing response has beenused to distinguish between microcrystalline and amorphousstructures in terms of a continuous grain growth or a sharp onsetfor a nucleation and growth [14]. Since amorphous alloys are notin thermodynamic equilibrium, supplying such alloys with suffi-cient energy can promote a phase transformation from anamorphous to a more stable crystalline phase during the anneal-ing process [4,5]. In this process, nucleation plays a key role incrystal growth, crystal formation, high quality single crystals andits study is currently being stimulated by the development of newexperimental and theoretical techniques [69].

According to the classical nucleation theory, the nucleationrate mostly depends on the nucleation energy barrier, diffusioncoefficient and free energy between solid and liquid. It is under-stood from these physical factors that the nucleation and crystalgrowth are effective factors in amorphous structure stability[1012]. On the other hand, the critical number of crystalline-type clusters occurred in the amorphous phase is an effectivefactor in the stability of amorphous structure and crystallizationbehaviours from amorphous phase to crystal phase [13,14].94959697

Elsevier B.V.

fax: 90 0434 228 5271.

zancPhysica B (2012), httpMetallic alloys are especially popular due to their low costsand relatively easy manufacturing. Especially, copper-basedalloys have been preferred in the technological and commercialapplications because of their good shape memory properties, easyand low cost of production [15,16]. During amorphous phase tocrystal phase transformation process, it is quite difficult toexperimentally investigate the local order and nano-cluster char-acteristics determining the structural properties of the system ata considerable level. Considering all of these, it is needed thatsimulation studies should be conducted in order to determine theinternal dynamical properties of such transformations that havean effect on the physical behaviours of the materials [1719].

Crystallization investigations of amorphous alloys are veryimportant to understand the mechanism of phase transforma-tions and produce well-controlled nanostructures. Experimentalthermal analysis (TA) methods such as differential scanningcalorimetry are widely used for crystallization kinetics analysisof amorphous alloys. This method recognizes three mechanisms,i.e. nucleation, growth and impingement of growing new-phaseparticles, and is applicable to both isothermal and non-isothermaltransformations [2022].

Cu-based alloys, such as CuAlNi, are attractive for fundamentalresearch of the thermo-mechanical properties of shape memorymaterials (superelasticity, shape memory effect, twinning inmartensite) because the martensitic transformations in them arefairly well known [23]. It can be said from our early studies that thethermoelastic phase transformation at CuAlNi ternary alloy systemhas been modelled by SuttonChen type of embedded atommethod9899

://dx.doi.org/10.1016/j.physb.2012.10.015

www.elsevier.com/locate/physbwww.elsevier.com/locate/physbdx.doi.org/10.1016/j.physb.2012.10.015dx.doi.org/10.1016/j.physb.2012.10.015dx.doi.org/10.1016/j.physb.2012.10.015mailto:facelik@beu.edu.trdx.doi.org/10.1016/j.physb.2012.10.015dx.doi.org/10.1016/j.physb.2012.10.015dx.doi.org/10.1016/j.physb.2012.10.015dx.doi.org/10.1016/j.physb.2012.10.015

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F.A. Celik, S. Kazanc / Physica B ] (]]]]) ]]]]]]2based on many-body interaction [24,25]. On the other hand, we wantto emphasize that the goal of this study is to investigate the crystal-lization behaviours of amorphous CuAlNi alloy during isothermalannealing by MD method and recrystallization process of ternaryalloy nanoclusters of Cu, Al and Ni in the nano-size range. In thispurpose, first, MD simulation method proposed by Parrinello andRahman was used to study the crystallization process of amorphousin CuAlNi (Cu%26.82Al%2.47Ni atomic percentage) ternary alloysystem during isothermal annealing. The nucleation and growthmechanisms at the microscopic level were investigated by HAanalysis method based on the bonded pairs and bond orientationalorder parameter for the model alloy system by using MD method atdifferent annealing temperatures. Second, we determine the Avramiexponents based on the experimental thermal analyses of the ternarymodel alloy system by using crystalline atomic bonded pairs obtainedfrom MD simulation results.8384858687888990919293949596979899

1001011021031041051061071081091101111121131141151161171181192. Potential energy function

Interatomic potentials are very important for the systemswhich have been modeling in the atomic scale. PEF which areused for modeling of binary alloy systems have been used formodelling ternary alloys systems as well [26]. In the presentstudy a SCEAM potential function is used. The total energy of acrystal with N atoms in the SCEAM methodology is given by [27]

ET Xi

Ei Xi

1

2

Xja i

eijf rij

cieiiffiffiffiffiffiri

p2435 1

where rij is the separation between atom i and j, c is a positivedimensionless parameter, and e is a parameter with the dimen-sions of energy. f(rij) is a pairwise repulsive potential

f rij

Aijrij

n2

between atoms i and j (arising primarily from Pauli repulsionbetween the core electrons), while the metallic bonding iscaptured in a local energy density, ri, associated with the atomi and defined as

ri Xja i

r rij

Xja i

Aijrij

m3

here A is a parameter with the dimensions of length; m and n arepositive integers.

The potential parameters representing the interactionbetween different type atoms can be written from LorentzBerthelet as follows [17];

Aij Aji AiAj

2, nij nji

ninj2

, mij mji mimj

24

eij eji ffiffiffiffiffiffiffieiej

p5120121122123124125126127128129130131132

Table 1Atom occupation fractions in sublattices I and II of B2type ordered phases [30].

Atom Sublattice I Sublattice II

Cu 1(1WB2) (CAlCNi) 1(1WB2) (CAlCNi)Al (1WB2) CAl (1WB2) CAlNi (1WB2) CNi (1WB2) CNi3. Calculations methods

3.1. The simulation technique

In the MD method considering the variation of MD cell inshape and size, the lengths of the MD cell axis are described withthree vectors A(t), B(t) and C(t) as a function of time. More detailedexpositions of MD simulation method can be found from litera-ture [28,29].

The simulations were performed on three dimensional arraysof 303030 unit cells (54,000 atoms). The ideal crystal DO3Please cite this article as: F.A. Celik, S. KazancPhysica B (2012), httpsuper lattice structure of the 3 A model alloys was chosen as thestarting configuration for the MD computer simulation. The initialvelocities were derived from random number generator so as toconfirm a MaxwellBoltzmann distribution at a given tempera-ture initially. The temperature of the system has been controlledby rescaling the atomic velocities at every two integration steps.The equation of motion is numerically solved by using the Gearsfive order predictor-corrector algorithm. Each molecular dynamicstime step corresponds to about 6.64 fs.

Structural analysis is examined from the radial distributionfunctions (RDF), g(r), which is given as

g r VN2

PNi 1 nir

4p r2Dr

* +6

Here r is the radial distance, ni(r) is the coordination number ofatom i separated with r within Dr interval, and bracket denotesthe time average [29].

The atomic configuration depends on the degree W of long-range order. For the B2 type ordered (CsCl type) phase, there aretwo sublattices. In the perfect B2 type order state (WB21),sublattice I is occupied by Cu atoms while sublattice II is occupiedby Al, Ni and the remaining Cu atoms. In the A2 type disorderedstate (WB20) the occupations of the two sublattices are thesame. CCu, CNi and CAl are the atomic fractions of Cu, Ni, Al atoms,respectively, (CCuCNiCAl1); then the fractions of each atomin CuAlNi alloys on sublattices I and II are those listed in Table 1[30].

For the DO3 type (or more accurately L21 type) ordered phase,above mentioned sublattice II is divided further into two sub-lattices, i.e. sublattice 3 and 4. In the perfect L21 type order state(WL21), all Ni atoms lie on sublattice 3 while Al atoms lie onsublattice 4. The ordering state with WL20 corresponds to thatwith WB21. Therefore, we have the atom occupation fractionsfor L21 type ordered CuAlNi alloys as listed in Table 2 [30].

In order to investigate the crystallization process of the modelalloy system from amorphous to crystalline phase, the equili-brated model in the structure of DO3 type super lattice at 500 Ktemperature was, first, heated up to 1100 K at intervals of 100 Kstarting from 500 K. Later, the system was quenched up to 300 Ktemperature at intervals of 100 K (11013 K/s). It was observedthat the system was stable at amorphous phase in all theseholding time and that the transformation to crystalline phasewas not realized. The model alloy system was held at amorphousphase and following this it was heated up to the annealingtemperatures of 400 K and 500 K starting from amorphous phase(300 K).

3.2. The HoneycuttAndersen (HA) method

It is known that the pair analysis technique can be used todescribe the microscopic local structure and short-range order inthe structures [31]. In this technique the local structure isclassified by using a sequence of four integers (ijkl) which havethe following meanings, the first integer (i) is 1 when the atoms inthe root pair are bonded, otherwise it is 2. The second integer (j) isthe number of near-neighbour atoms shared in common by theroot pair. The third integer (k) is the number of nearest-neighbour://dx.doi.org/10.1016/j.physb.2012.10.015

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Table 2Atom occupation fractions in sublattices III and IV of DO3-type ordered phases

[30].

Atom Sublattice III Sublattice IV

Cu 2CCu12WDO3 (CAlCNi) 2CCu12WDO3 (CNiCAl)Al (1WDO3) 2CAl (1WDO3) 2CAlNi (1WDO3) 2CNi (1WDO3) 2CNi

Fig. 1. Black atoms (A and B) represent pair atoms (root pair); white atomsrepresent neighbour atoms with root pair. Straight thick line denotes the bond

between A and B, dotted lines denote near-neighbour bond numbers of root pair,

dashed lines denote bond among neighbours. (a) 1421 bonded pair in fcc cluster

(b) 1441 bonded pair in bcc cluster (c) 1422 bonded pair in hcp cluster (d) 1551

bonded pair in icos cluster.

F.A. Celik, S. Kazanc / Physica B ] (]]]]) ]]]]]] 3bonds among the shared neighbours. The fourth integer (l) isneeded to difference between the cases when the first threeindices are the same but the bond geometries are different.

It is believed that the method is convenient to analyze thegeometric characteristics of an atomic cluster. If any atom systemAB forms a bond, i1, otherwise i2, A and B atoms representroot pair. For example, the 1551 bonded pairs represent the tworoot pair atoms with five common neighbours that have fivebonds forming a pentagon of near-neighbour contact. In this case,some typical bonded pairs are examined to understand the localstructure of systems. The 1551 bonded pairs have fivefoldsymmetry, and the ratio of 1551 bonded pairs gives a measureof the degree of ideal icosahedral (ICOS) order. The 1421 and 1422bonded pairs are characteristic bonded pairs for fcc and hcpcrystal structures, respectively. The 1661 and 1441 bonded pairsare the characteristic bonded pairs for bcc crystal structure [31].The pair fractions shown here are normalised so that the sumover all cases for the nearest neighbours is summed to unity. The1541 and 1431 bonded pairs are characteristic disordered ICOSorder. The 1201, 1331, 1321, 1311 bonded pairs represent therhombus symmetrical features of short-range order. An exampledisplaying four integers in some atomic clusters is given by HAmethod in Fig. 1.Please cite this article as: F.A. Celik, S. KazancPhysica B (2012), http3.3. Measuring the order parameter

The calculation of the bond orientational order parametershas, recently, been used to determine the local structures of theatoms in the MD cell [32,33]. The formalism is basically runningsuch that the nearest neighbours of each atom in the MD cell,which constitute a cluster, is determined from a cut-off distancetaken from the minimum of the RDF between the first and secondpeaks. The spherical coordinates of each bond in the cluster iscalculated from

Qlmr Ylm yr ,fr 7

where Ylm(y,f) are spherical harmonics, and y(r) and f(r) are thepolar and azimuthal angles of the bond measured with respect tosome reference frame. The local structure around particle i isgiven by

Qlm i 1

NbiXNbij 1

Qlmr 8

where the sum runs over all Nb(i) bonds that particle i has with itsneighbours. On the other hand, the global orientational orderparameters can be obtained from the average of Qlmi over all Nparticles, i.e.,

Qlm PN

i 1 Nbi QlmiPNi 1 Nbi

9

The second-order invariants (Ql) is determined from therelations,

Ql 4p

2l1Xl

m lQ lm

2" #1=2 10Since the lowest non-zero Ql is common with the ICOS and

crystal cubic symmetry corresponds to l6, it has been argued byseveral authors that the value of Q6 is very sensitive to any kind ofcrystallization and increases significantly when order appears[34,35].

3.4. Crystallization kinetics

The kinetic study of crystallization was performed by usingJohnsonMehlAvrami (JMA) [36,37] isothermal analyses forvolume fraction x transformed as a function of time t based onthe following equation:

ln 12x k t n 11

Eq. (11) can be written as

ln2ln 12x nlnknlnt 12

where k is effective rate constant, x is the crystallized volumefraction and n is the Avrami exponent. The Avrami plot ofln[ ln(1x)] versus ln(t), yields a straight line with slope andintercept n ln(k). n may be 4, 3, 2 and 1, which are related todifferent crystallization mechanism [20]: n4, volume nucleationand three dimensional growth; n3, volume nucleation and twodimensional growth; n2, volume nucleation and one dimen-sional growth; n1, surface nucleation and one dimensionalgrowth.4. Results and discussion

In this study, the validity of the PEF and its parameters for themodel system were determined before investigating phase trans-formations. To determine the stable lattice structure of the modelsystem, the variation of cohesive energy with volume was://dx.doi.org/10.1016/j.physb.2012.10.015

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F.A. Celik, S. Kazanc / Physica B ] (]]]]) ]]]]]]4calculated for three different lattice structures such as disorderB2, DO3 and disorder A2 lattices. The variation of cohesive energy,which we have obtained from SCEAM approach, with compres-sing volume for these different structures is shown in Fig. 2. Thevalue of cohesive energy for order B2, DO3 and disorder A2structures was determined as 3.41 eV, 3.43 eV and 3.38 eV,respectively. It is well known that the stable structure of a systemcorresponds to the minimum enthalpy. As a result, it can be saidthat the DO3 type super lattice structure for 3 A model system isthe most stable structure for our MD study.

The effect of annealing temperatures on the crystallizationprocess at the 3 A model alloy system was performed as seenfrom Fig. 3. From this figure, the variations (or changes) of thestructural quantities, are observed such as the cohesive energy (E)during the crystallization process. At the annealing temperatureof 400 K, cohesive energy decreases very slowly and a nearlyconstant energy during the crystallization processes, whichmeans that amorphous state is conserved without crystallizingduring the process. Therefore, at the annealing temperature of400 K, there is only some structural relaxation. At the annealingtemperature of 500 K, the energy drops during the crystallizationprocess. This means that there are some crystalline embryos whichgrow in the amorphous matrix, and therefore crystallization has90919293949596979899

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0.8-3.5

-3.4

-3.3

-3.2

-3.1

-3.0

E (

eV)

DO3

B2

A2

V / V0

1.0 1.2 1.4

Fig. 2. The cohesive energy against the variation of volume for order B2, DO3 anddisorder A2 structures of 3 A model alloy system.

1Time (ns)

-3.6

-3.59

-3.58

-3.57

-3.56

E (

eV)

400K

500K

1.2 1.4 1.6 1.8 2

Fig. 3. The variation of potential energy during annealing process for differenttemperatures.

Please cite this article as: F.A. Celik, S. KazancPhysica B (2012), httphappened at higher annealing temperatures because of the rapidgrowth of the crystalline phase.

Fig. 4(ac) shows the RDFs results for different annealing timesduring crystallization at 400 K, 500 K and 600 K annealing tem-peratures. It can be seen that the annealing process starts formamorphous phase (0 ns), which means that a splitting can be seenin the second peak at 300 K, which is a characteristic of themetallic glass as stated in [7,10]. At 400 K annealing temperature,about at the time of 3.6 ns, the crystallization process starts fromamorphous phase due to the splitting of the second peak becom-ing less obvious. Since the peaks of RDF at certain atomicpositions are not very sharp, it can be said that it only starts withthe surface crystallization in the amorphous phase and a stablecrystalline structure does not appear. At 500 K annealing tem-perature and the time of 0.82 ns, the crystallization process startsfrom amorphous phase and the curves obviously show that thetransition occurs from amorphous structure to crystalline struc-ture due to the nucleation of crystalline phase in amorphousphase. After 0.82 ns, the peaks are sharper and this indicates thatthe degree of crystalline order of the system has increased andthat a stable crystalline structure has appeared obviously. Thisimplies that the crystallized phase undergoes coarsening withincreasing annealing time at high annealing temperature. At thetime of 2.1 ns, the peaks of RDF become sharper compared with400 K annealing temperature; the RDF curves show the feature ofmore stable crystal structure due to the rapid growth of thecrystal grains. At annealing temperature of 600 K, we note thatthe model structure is in liquid phase for all annealing times. Thecrystal phase does not occur and the atoms are distributed inmore disordered.

It is also known that it is necessary to investigate the atomicbonded pairs to determine small sized embryos structures [14].These atomic groups are yet embryos at amorphous phase andthese embryos reach the critical number depending on thestructural rearrangements of different types of atomic bondedpairs being within a system because of nucleation. During thetransformation process from amorphous phase to crystal phase,the structural properties of bonded pairs (these pairs will becalled as embryos) have determined from HA method and theireffect on nucleation and growth process has been investigated inthis study.

Fig. 5 shows the variation of the number of crystal-typebonded pairs (total number of 1421, 1422, 1441 and 1661 bondedpairs) during from amorphous phase to crystal phase at 400 K and500 K annealing temperatures for the model system. It can beseen that from beginning of annealing time to t1, the number ofcrystal bonded pairs increases slowly. This time period may beinterpret as the crystal nucleation in the amorphous phase. Fromt1 to t2 the number of crystal bonded pairs increases very fast, andthis time period may be related to the crystal growth during thecrystallization process. After time of t2, the number of crystalbonded pairs goes to constant values and the crystal phase remainstable with the time. In brief, it can be said that the structuralrelaxation in the amorphous phase, the nucleation of crystallineclusters and crystal embryos growth occur during this annealingprocess. The results lead to the increasing of number of crystalbonded pairs forming crystalline grains which occur faster at thehigh annealing temperature. This case is in agreement with thetheories based on the classical crystallization kinetics [9,11].Fig. 5b shows the variation of amorphous-type bonded pairs(total number of 1551, 1541, 1431 bonded pairs) during fromamorphous phase to crystal phase at 400 K and 500 K annealingtemperatures for the model system. The curves clearly show thatthe total number of the crystal bonded pairs increases, while thetotal number of the amorphous bonded pairs decreases corre-spondingly. This result clearly shows that the amorphous phase://dx.doi.org/10.1016/j.physb.2012.10.015

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2r ()

g (r

)

0 ns

400K 500K

0.82 ns

2.1 ns

3.6 ns

3.7 ns

600K

0 ns

0.82 ns

2.1 ns

3.6 ns

3.7 ns

0 ns

0.82 ns

2.1 ns

3.6 ns

3.7 ns

3 4 5 6 7 2r ()

3 4 5 6 7 2r ()

3 4 5 6 7

Fig. 4. The RDF curves at different annealing times and temperatures during the crystallization process.

0.8

24

26

28

30

The

num

ber o

f cry

stal

bon

ded

pairs

(%)

20

30

40

50

60

70

t1

t2

55

56

57

58

59

60

61

The

num

ber o

f am

orph

ous b

onde

d pa

irs (%

)

10

20

30

40

50

60

70

400K

500K

400K

500K

1.2 1.6 2Time (ns)

0.8 1.2 1.6 2Time (ns)

Fig. 5. Variation of various bonded pairs during annealing times (a) crystal bonded pairs (b) amorphous bonded pairs.

Table 3The various bonded pairs at 400 K and 500 K annealing temperatures.

400 KPairs

Percentage(%)

500 KPairs

Percentage(%)

1301 0.67 1301 0.4

1311 5.27 1311 4.25

1321 1.84 1321 0.88

1331 0 1331 0

1211 0.45 1211 0.11

1201 0.36 1201 0.24

1101 0.22 1101 0.13

Total 8.81 Total 6.01

F.A. Celik, S. Kazanc / Physica B ] (]]]]) ]]]]]] 5transforms to crystal phase during crystallization process. On theother hand, the number of disordered structural units, such as1201, 1211, 1301, 1311 and 1321 only amounts to 8.81% and6.01% at 400 K and 500 K annealing temperatures, respectively asseen Table 3. The relative bonded pairs, such as 1431, 1551, 1541,1311 and 1321 which are corresponding to disorderly system canpreserve the structures from amorphous phase to crystal phaseduring annealing process because of under the high quenchingfrom liquid phase. 1331, 1211 and 1301 bonded pairs are notfound or found in a very small amount in crystal phase becausethere are not only internal relations but also structural and rangeorder differences between liquid and crystal phase. On the otherhand, super cooled liquids contain many disorderly structuralPlease cite this article as: F.A. Celik, S. KazancPhysica B (2012), http://dx.doi.org/10.1016/j.physb.2012.10.015

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100101102103104105

00.2

0.3

0.4

0.5

0.6

0.7

0.8

Vol

ume

frac

tion

(x)

400K

500K

Time (ns)1 2 3 4 5

Fig. 7. Crystallized volume fraction x as function of annealing time for model alloysystem.

F.A. Celik, S. Kazanc / Physica B ] (]]]]) ]]]]]]6units not in solids such as clusters 1221, 1331, etc. The existenceof various clusters in liquid manifests the variety of microstruc-tural in liquid or amorphous metal. Under slow annealing condi-tions, various types of clusters in amorphous metal will berearranged in a particular order to form metal crystal duringcrystallization process. As the annealing temperature increases,the relative numbers of crystal-type bonded pairs increaseslightly and disordered bonded pairs (other pairs) tend to beremoved from the system.

Moreover, it is known that the bond orientational orderparameters are also very important parameters to detect thecrystalline order during crystallization process. In particular, sincethe values of Q6 are very sensitive to crystallization [35], thevalues increase significantly indicating the order of crystallinephase. In the process of transition from the amorphous to crystal-line phase, rearrangement of the atoms in the system can beassessed by examining the variation in global bond orientationalorder parameter values. During any phase transition, global bondorientational order parameter values have considerable varia-tions. As previously established, particularly in the crystallizationprocess, a sudden increase in the numerical values of the Q6parameter indicates a transformation into the crystalline phase orordering. In this case, the variation of Q6 value for the modelternary alloy obtained at annealing temperatures of 400 K and500 K starting from amorphous phase with the time interval isshown in Fig. 6. The order parameter Q6 has very weak values inthe amorphous phase and with the time; it has much largervalues in the crystal phase at two annealing temperatures. Asseen in Fig. 6, a sudden change (increase) was observed in thenumerical values of the Q6 parameter in time at the annealingtemperature of 500 K. On the other hand, at 400 K, no suddenly orsharply change of Q6 values was observed in time. This resultshows that the model system of alloys has a better crystallizationand crystal growth process at high annealing temperatures.

One of the most well known statements regarding nucleationand growth kinetics is the JohnsonMehlAvrami (JMA) kinetics.The Avrami exponent n provides the characteristics of nucleationand growth during the crystallization mechanisms at any tem-perature. In this study, the effect of the number of bonded pairs10.33

0.34

0.35

0.36

0.37

0.38

0.39

Q6

500K

400K

2 3 4 5Time (ns)

Fig. 6. Variation of the bond order parameter at 400 K and 500 K temperaturesduring annealing.

Please cite this article as: F.A. Celik, S. KazancPhysica B (2012), httprepresenting the crystal structures determined via MD method onthe nucleation and growth process was investigated. The crystal-lized volume fraction in a given time during crystallization wasdetermined by measuring the percentage of crystalline bondedpairs for system.

With the above HA analysis, all crystalline embryos (or nuclei)in the system can be confirmed, and the number of crystallinebonded pairs can be calculated. Fig. 7 shows the evolutions of thecrystallized volume fraction of crystal phases during the relaxa-tion process of amorphous alloy at 400 K and 500 K annealingtemperatures. The crystallized volume fraction (x) in a given timeduring crystallization is determined by crystalline bonded pairsfor which corresponding results are shown in Fig. 8. Plottingln[ ln(1x)] versus ln(t) using data x%, the isothermal JMAplotted at different annealing temperatures for model alloysystem are obtained. The Avrami exponents n are calculated by106107108109110111112113114115116117118119120121122123124125126127128129130131132

0-1.2

-0.8

-0.4

0

0.4

ln (

-ln

(1-x

))

400K

500K

0.5 1 1.5 2ln (t)

Fig. 8. Isothermal JMA plots of crystallization at different annealing temperaturesfor model system.

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F.A. Celik, S. Kazanc / Physica B ] (]]]]) ]]]]]] 7the slope of curves and our most simulation results are located atthese fitted curves. Under the isothermal condition, the Avramiexponent at 400 K annealing temperature is obtained as 0.48. Thisvalue smaller than 1, which indicates that the crystallization isgoverned by one dimensional growth from surface to the insideand only surface nucleation. At high annealing temperature of500K, the value of Avrami exponent is 2.78. This value is largerthan 2.5 and close to value of 3, which indicates that volumenucleation and two dimensional growths with an increasednucleation rate. At low temperatures, because of random packedatomic configuration and a multi-component system of amor-phous phase, mobility of atoms in such atomic configuration isvery difficult. The formation of new crystalline nuclei needs theatomic rearrangements and mobility. The number of crystallinebonded pairs increases with increasing annealing temperature as8283848586878889909192939495969798

Fig. 9. Snapshots of the atomic configuration at different annealing temperaturesof model system. (a) 400 K and (b) 500 K.

Please cite this article as: F.A. Celik, S. KazancPhysica B (2012), httpexhibited in Fig. 5. At higher temperature, growth of criticalcrystalline embryos leads to change the composition in theirneighborhood and as a result the crystal growth increasesbecause of the easy mobility of atoms as seen in Fig. 3. On theother hand, it is considered that since the model alloy system issubsequently heated to a high annealing temperature, the struc-tural defects such as slipping of atomic planes and stacking faults,in the transformed regions may be removed or alternativelyrearranged at nucleating recrystallized grains in configurationsof lower energy during isothermal annealing processes. With non-crystalline bonded pairs decreasing in the amorphous phaseduring annealing, the crystallized bonded pairs or atomic groupsare particularly efficient at nucleating these grains. From thesimulation results, the necessary time for formation of stablecrystal phase and completion of recrystallization decreases withincreasing temperature as seen in Figs. 5, 6 and 8, which isconsistent with the nucleation theory [9,11,12]. On the otherhand, since form a stable crystal phase from amorphous phase,the most ideal annealing temperature for model alloy system isdetermined as 500 K with MD method.

Fig. 9 shows snapshots of atomic configuration during crystal-lization at 400 K and 500 K annealing temperatures. Fig. 9a showsthe atomic configuration at 400 K and it can be said from thisimage that some crystal nuclei are formed in the amorphousphase nucleate from the amorphous phase. Fig. 9b shows theatomic configurations of the final crystallization at the annealingtemperature of 500 K. The ordered crystalline structure is almostcompletely formed, namely the crystal embryos have grownforming stable crystal phase after growth and the grain coarsen-ing can be clearly observed. As it is obviously seen in the figuresthat some areas possess disordered structures in this atomicconfiguration, which are also in agreement with Fig. 5.991001011021031041051061071081091101111121131141151161171181191201211221235. Conclusion

The EAM potential for the CuAlNi alloy system was performedto study the crystallization process of the amorphous alloy duringisothermal annealing. The transition from amorphous structure tocrystal structure of the system at different annealing tempera-tures is a continuous process, associated with the sharp variationsin the RDF, bond order parameter, the number of bonded pairs byusing HA method. The simulation results also show that thehigher the annealing temperature, the faster the crystallizationprocess. These simulation results are consistent with experimen-tal findings and nucleation theory. On the other hand, the Avramiexponent of about 3 implied that the crystallization process of thebulk amorphous alloy is a three dimensional growth and volumenucleation at high annealing temperature. At low annealingtemperature, the value of Avrami exponent is smaller than 1.This value is consistent with a surface nucleation mechanismaccording to which the crystallization occurs mainly from theglass surface and crystal growth proceeds into the interior. Wecalculate Avrami exponents at different annealing temperaturesby using crystallized volume fraction of CuAlNi model alloysystem estimates from present MD simulation.124125126127128129130131132Acknowledgement

This work is supported by the Research Foundation of FiratUniversity contact grant-number: FUBAP 1496. In addition to, theauthors wish to thank Professor Jamal Beraktar from Martin-Luther-University for supplying the researches.://dx.doi.org/10.1016/j.physb.2012.10.015

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Crystallization analysis and determination of Avrami exponents of CuAlNi alloy by molecular dynamics simulationIntroductionPotential energy functionCalculations methodsThe simulation techniqueThe Honeycutt-Andersen (HA) methodMeasuring the order parameterCrystallization kinetics

Results and discussionConclusionAcknowledgementReferences