Page 1

© Copyright by International OCSCO World Press. All rights reserved. 2008

VOLUME 31

ISSUE 1

November

2008

Research paper

41

of Achievements in Materials

and Manufacturing Engineering

and Manufacturing Engineering

of Achievements in Materials

Molecular dynamics study

of Cu-Pd ordered alloys

S. Özdemir Kart a,*, A. Erbay a, H. Kılıç a, T. Cagin b, M. Tomak c

a Department of Physics, Pamukkale University, 20020 Denizli, Turkey

b Department of Chemical Engineering, Texas A&M University, TX77845-3122 Texas, USA

c Department of Physics, Middle East Technical University, 06531 Ankara, Turkey

* Corresponding author: E-mail address: ozsev@pau.edu.tr

Received 27.06.2008; published in revised form 01.11.2008

Analysis and modelling

AbstrAct

Purpose: The goal of the paper is to study the molecular dynamics of Cu-Pd ordered alloys.

Design/methodology/approach: The thermal and mechanical properties of Cu, Pd pure metals and their ordered

intermetallic alloys of Cu3Pd(L12) and CuPd3(L12) are studied by using the molecular dynamics simulation. The

melting behavior of the metals considered in this work is studied by utilizing quantum Sutton-Chen (Q-SC) many-body

potential. The effects of temperature and concentration on the physical properties of Cu-Pd system are analyzed.

Findings: A wide range of properties of Cu, Pd pure metals and their Cu3Pd and CuPd3 ordered intermetallics

is presented. It was found that this potential is suitable to give the general characteristics of the melting process

in these systems.

Practical implications: The simulation results such as cohesive energy, density, elastic constants, bulk modulus,

heat capacity, thermal expansion and melting points are in good agreement with the available experimental data

and other theoretical calculations.

Originality/value: To the best our knowledge this work presents, for the first time, a wide range of physical

properties of alloys focusing on Cu-Pd ordered compounds.

Keywords: Cu-based intermetallic alloys; Molecular dynamics simulations; Quantum Sutton-Chen potentials;

Melting properties

1. Introduction

1.?? Introduction

With the rapid development of advanced technologies, the

demands for the design of components and the production of new

materials which are strong, stiff and ductile at high temperature

and the study of the final product properties are increasing. Stable

materials responding these requirements are the intermetallic

compounds. They have structural properties which differ greatly

from constituent metals.

Cu-based intermetallics have motivated the strong interest in

their fundamental properties including interatomic bonding, long

range order, crystalline defects, order-disorder transition and

diffusion [1-14]. Among them, CuPd has receiving considerable

attention due to their promising use as catalysts in many

technologically important areas, including petroleum refining and

automotive emission control, and membrane for hydrogen

production and purification. On the other hand, Cu3Pd and CuPd3

being ordered alloys of Cu-Pd system offer to be investigated due

to a little of theoretical and experimental studies. In fact,

relatively attention has been paid to the temperature dependence

of the thermodynamical and mechanical properties of the ordered

Cu3Pd and CuPd3 intermetallic compounds. In this work this

specific problem is addressed.

Some first principles (ab initio) calculations responsible for the

physical properties of Cu-Pd system have recently been carried out

successfully [1, 2, 11-13]. Because ab initio methods are

computationally expensive, the calculations have been limited to

short time scales and to a few hundred atoms. On the other hand,

Page 2

Research paper

42

Journal of Achievements in Materials and Manufacturing Engineering

S. Özdemir Kart, A. Erbay, H. Kılıç, T. Cagin, M. Tomak

Volume 31 Issue 1 November 2008

there are empirical and more practical approaches that can afford to

investigate many system and trends in physical properties [15-18].

These potentials provide sufficiently accurate and quick description

in metallic systems. Molecular dynamics (MD) simulation can

provide an important insight by allowing one to determine static

and dynamical properties of materials at finite temperatures, when

provided with suitable interatomic potentials.

In this study, we have performed MD simulations using

Sutton-Chen (SC) [18] potential with new potential parameter set,

namely the Quantum Sutton-Chen (Q-SC) potential developed by

Ça?n and co-workers [19] to investigate the physical properties

of Cu and Pd pure metals and ordered Cu-Pd alloys (especially,

Cu3Pd(L12)and CuPd3(L12)). This potential has been used in

various applications, ranging from random alloys, glass

formation, crystallization, surfaces, clusters, nanowires and single

crystal plasticity of pure metals to transport properties of fcc

metals [20-26]. One purpose of this study is to verify the validity

of potential parameters to show their transferability from pure

elements to ordered alloys, from low temperatures to high

temperatures. Another aim is to obtain comprehensive data of the

physical properties of ordered Cu-Pd, which is scientifically and

technologically important material. In this study, we have

calculated the lattice parameter, cohesive energy, density, elastic

constant and bulk modulus at various temperatures and deduced

the melting temperature, heat capacity and thermal coefficient of

the volume expansion by using Q-SC potential parameters. The

results are compared with experimental and theoretical data

available in the literature. To the best our knowledge this work

presents, for the first time, a wide range of physical properties of

alloys focusing on Cu-Pd ordered compounds.

This paper has the following structure: simulation details are

presented in Section 2. Section 3 deals with the results of computer

simulations. The final section gives the conclusion of the results.

2. Simulation details

2.?? simulation?details

Three successive simulations are performed for heating

process of Cu-Pd alloys by using the algorithm based on extended

Hamiltonian formalism [27-31]. First, constant-enthalpy constant-

pressure (HPN) MD simulation is carried out to heat the system

from 0.1 K to 2500 K with the increments of 50 K. This

increment is reduced to 10 K near the melting temperature to get

more accurate values of the melting temperature. The heating

procedure is carried out by scaling velocities with the ratio of the

increment temperature to the specific number of steps depending

on target temperature, (1 K/step). At each temperature, 2000 time

steps are carried out for equilibrium. Then 20000 additional steps

in constant-temperature constant-pressure (TPN) dynamics are

taken to obtain some statistical properties, such as volume,

density and energy of the system. Finally, 50000 steps of

microcanonical ensemble (EVN) dynamics follows by using the

resulting zero strain average matrix to obtain pressure dependent

properties of the system, such as elastic constant.

The simulation box is made up of 2048 particles arranged on

the fcc structure for the pure Cu and Pd, L12 structure for the

Cu3Pd and CuPd3 ordered intermetallic systems. In the case of

Cu3Pd, the Pd atoms occupy the corner cites, while Cu atoms

occupy the face centres of the basis cube; the opposite occurs for

CuPd3. The MD simulation performed in this study uses the

Sutton-Chen (SC) type potential [18]. This potential is recently

reparametrized by Ça?n et al. [19] by including quantum

corrections accounting for zero-point energy, hence called

quantum Sutton-Chen (Q-SC) potential, to improve some physical

properties at high temperatures. The potential parameters are

obtained by fitting to some experimental properties, such as lattice

parameter, cohesive energy, bulk modulus, phonon frequency at

the X point, vacancy formation energy and surface energy. The

total potential energy has the following form:

?

??

?

?

?

i

?

i

?

?

i

?

?

j

?

?

?

?

?

?

?

?

?

??

?

?

?

??

?

?

?

??

?

?

??

?

?

?

?

??

?

??

NN

j

ij

m

ij

ij

iii

ij

n

ij

ij

i

iji tot

r

a

c

r

a

UU

2 / 1

2

1

??

(1)

The first term in Eq. 1 is a two body repulsive interaction

between the atoms i and j, separated by a distance rij. The second

term represents the many-body cohesion term associated with

atom i. The square root term introduces a many-body component

into the energy summation. The popularity of SC potentials is

partly due to the computationally tractable form adopted for the

many-body forces. It is the relatively simple analytic form of the

potential that enables one to calculate the many physical

properties of the materials.

In the Eq. 1, a is a length parameter scaling to the lattice

spacing of the crystal, c is a dimensionless parameter scaling the

attractive terms, ? is an energy parameter determined from

experiment, and

mn,

are integer parameters with

determine the range of the two components of the potential. The

interaction length of potential is taken as two lattice parameters for

the efficiency of the computer simulation time. The temperature

effects in the simulations are considered by giving an additional

length of half the lattice parameter. The parameters for the Cu-Pd

alloys are obtained through the following mixing rules [32]:

?

ji ij

????

,

ij

a

?

mn ?

which

?

2 / 1

2

ji

aa

?

,

(2)

2

ji

ij

mm

m

?

?

,

2

ji

ij

nn

n

?

?

Q-SC potential parameters [19] for pure Cu and Pd metals are

given in Table 1.

Table1.

Q-SC potential parameters for Cu and Pd pure metals

Metal

n m

)(meV

?

c

)(

?

Aa

3.6030 Cu 10 5 5.792184.843

Pd 12 6 3.2864 148.205 3.8813

The fluctuation formula for the calculation of the elastic

constants [33-35] is given as:

?

klij

B

T Nk

??

????

0

?

??

ijkljk iljlik

B

klijijkl

V

PPPP

Tk

V

C

??

??

0

2

(3)

Page 3

43

Analysis and modelling

Molecular dynamics study of Cu-Pd ordered alloys

The first term represents the contribution of the fluctuation of the

microscopic stress tensor

energy contribution, and third term is the so-called Born term. <>

denotes the averaging over time and

volume for the model system. The bulk modulus for the cubic

systems can be obtained from

B

?

and coefficient of thermal volume expansion can be determined

from the differential of the enthalpy and that of volume,

respectively, as follows:

ij P , the second term is the kinetic

00

dethV ?

is the reference

3 / )

12

C

2(

11

C

?

. Specific heat

P

p

T

TH

?

TC

?

?

?

?

?

?

?

?

)(

)(

, (4)

P

p

T

TV

?

TV

T

?

?

?

?

?

?

?

??

)(

)(

1

)(

?

. (5)

3. Results

3.?? results

Table 2 shows some thermodynamical properties, namely

density ? , lattice constanta ,

compressibility ? of CuPd ordered metal alloys calculated from

TPN ensemble average over 20000 time steps at 300 K, together

with available experimental or theoretical data. The results for Cu

and Pd pure metals are in good agreement with the experimental

data [36]. The discrepancy between Q-SC and experimental

values for the first three properties is less than 1%, for

compressibility of Cu 20.5% and for that of Pd 12.5%. We expect

that the results of compressibility will be improved at high

temperatures because Q-SC potential describes well temperature

dependent properties. There is not any experimental data for

Cu3Pd and CuPd3 alloys to compare our results with them. The

values of lattice constants calculated for the ordered alloys are

cohesive energy

c

E and

close to that of other theoretical studies [10, 37]. The increasing

concentration of Pd in Cu-Pd compounds gives rise to decreasing

the cohesive energy of the Cu-Pd intermetallics, while it causes an

increment in lattice parameter and density, as expected.

We calculated the heat capacity as a function of temperature

and concentration by fitting the enthalpy of Cu-Pd alloys to a

quadratic polynomial using the data below the melting

temperature. The quadratic function form may be given as

2

)(

cTbTaT

Here T is the temperature. Heat capacity can be found by

taking the derivative of the polynomial function of Eq. 6

according to Eq. 4. The resulting

H

???

(kJ/mole).

(6)

p

C should not be extrapolated

)(TH

which is fitted to to T=0 K, as it is derived from

simulation results between 100 K and 700 K. The coefficients of

expression in Eq.6 are given in Table 3.

As shown in Table 3, the agreement between the simulated results

and experimental data [38] is very good. For example, the heat

capacities of Cu and Pd at 300 K are reported as 26.359 Jmole-1K-1 and

25.791 Jmole-1K-1, respectively. Deviations from the experimental

values for Cu and Pd are 7.75 % and 0.73 %, respectively.

We have also fitted the volume and temperature curve by the

same type of quadratic polynomial function as done in the heat

capacity to analyze further the volume thermal expansion

behaviour. The function used in the fitting procedure is

2

)(

cT bTaTV

???

(nm3mole-1). (7)

The coefficients in Eq.7 and the values of the thermal volume

expansion calculated from Eq. 5 at 300 K are presented in Table 4.

The value for copper, 7.642x10-5 K-1, is in better agreement with the

value from the experiment [38] than other experimental value

measured by Moruzzi et al. [39]. For palladium, our simulation

result is greater than experimental value of 1.160x10-5 K-1 [39].

Table 2.

Density ? , lattice constanta , cohesive energy

300 K, together with a comparison with available experimental [36] or theoretical data. Stars correspond to other theoretical results [10, 37]

E (kJ/mol)

Exp Q-SC

Exp Q-SC

Cu -336.0 -339.58 3.61

Cu3Pd -339.89 3.694*, 3.705**

CuPd3 -358.87 3.839*, 3.834**

Pd -376.0 -377.91 3.89

* from [37], ** from [10]

c

E and compressibility ? of CuPd ordered metal alloys calculated by using Q-SC potential parameters at

Material

c

)(

?

Aa

? (g/cm3)

Exp Q-SC

8.93

12.00

? (10-11m2/N)

Exp Q-SC

0.73

0.55

3.62

3.70

3.84

3.90

8.93

9.73

11.22

11.96

0.58

0.62

Table 3.

Coefficients of polynomial function used to find the heat capacity of Cu, Pd pure metals and their alloys. Heat capacity values of the metals along

with whenever available experimental data [38] at 300 K

Material

a

4

10?

bx

6

10?

cx

p

C (J/moleK)

Q-SC Exp

Cu

Cu3Pd

CuPd3

Pd

-339.672

-347.680

-366.437

-377.824

249.787

255.138

244.646

246.235

2.300

1.609

2.247

1.945

26.359

26.479

25.813

25.791

24.464

25.980

Page 4

Research paper

44

Journal of Achievements in Materials and Manufacturing Engineering

S. Özdemir Kart, A. Erbay, H. Kılıç, T. Cagin, M. Tomak

Volume 31 Issue 1 November 2008

We calculated the elastic constants of Cu-Pd ordered alloys to

study the mechanical and dynamical properties. The elastic constants,

in particular, provide valuable information on the stability and

stiffness of materials. In this study, elastic constants are calculated by

using the fluctuation expression (3) by taking the average ensemble of

EVN over the 50000 time steps. The elastic constants and bulk

modulus results for Cu and Pd pure metals and their ordered alloys

are listed in Table 5. At each temperature, the density obtained from

TPN ensemble by averaging over the 20000 time step is used to

specify the volume of the EVN ensemble. The detailed calculation

methodology related to the elastic constants can be found in Refs.

[35]. The elastic constants and bulk modulus of Cu and Pd pure

metals predicted from Q-SC parameters are compared with the

available experimental values and the results of previous works using

different potential models at 0 K in Table 5. As shown, the potential

used by Cleri and Rosato [40] for the elastic constants is yielding

better results. The elastic constants and bulk modulus from Q-SC

agree with the available experiment and other calculations except for

elastic constant of C12. The percentage differences of C11, C12 and C44

of Cu for the Q-SC are 9.8%, 2.9% and 11.1% at 0 K, respectively.

The accuracy of the Q-SC elastic constant of C11 for Pd, showing a

deviation of 8% from experiment, is comparable to that of the

embedded atom model (EAM) [41]. The experimental data on the

elastic constants for Cu-Pd alloys are not available for comparison.

Table 4.

Coefficients of polynomial function used to find the thermal

volume expansion of Cu, Pd pure metals and their alloys. Thermal

expansion values of the metals along with whenever available

experimental data [38, 39] at 300 K

Material

4

10?

ax

9

10?

bx

12

10?

cx

5

10?

x

p

?

Q-SC Exp

(K-1)

Cu 116.023 734.596 281.536 7.642

1.6701

4.9502

25.980

Cu3Pd

CuPd3

Pd

1 from [39], 2 from [38]

124.252

139.136

145.387

757.483

749.161

748.575

232.696

200.403

179.893

7.090

6.149

5.802

Table 5.

Comparison of calculated and experimental (Exp) [42] values for

elastic constants (Cij) and bulk modulus (B) at 0 K for pure metals

in the units of GPa. For Cu-Pd alloys, the elastic constants are

predicted from Q-SC potential parameters at 300K. Our results

(Q-SC) for Pd and Cu are compared with the other potential

models: embedded atom model (EAM) [41] and tight binding

second moment approximation (TBSMA) [40, 43]

Metals Model C11

Q-SC

Exp1

TBSMA2

TBSMA3

142.00

Cu3Pd Q-SC 166.71

CuPd3 Q-SC 190.44

Q-SC

Exp1

TBSMA2

EAM4

218.00

1 from [42], 2 from [40], 3 from [43], 4 from [41]

C12

121.30

124.94

125.00

180.00

117.97

132.90

150.25

176.14

178.00

184.00

C44

72.74

81.80

82.00

122.00

67.85

77.08

91.66

71.17

73.00

65.00

B

Cu

159.00

176.20

176.00

127.50

142.03

142.00

142.00

134.22

152.08

172.17

195.00

196.00

195.00

Pd

216.00

234.12

232.00

We are also interested in investigating the temperature and

concentration dependence of elastic constants of Cu-Pd ordered

system to see the effect. The variation of elastic constants of Cu-

Pd in the fcc structure for pure metals and in the L12 structure for

Cu3Pd and CuPd3ordered alloys as a function of temperature are

given in Figure 1a-d, respectively. These quantities decrease

linearly with increasing temperature, as seen in the Figures.

Thermal softening increases with increasing the concentration of

Cu in Cu-Pd system at each temperature.

The melting temperatures of Cu-Pd ordered alloys have been

determined by analyzing the behaviour of density, energy, volume

and pair distribution function as a function of temperature. We

obtain the same melting temperatures from all these physical

properties. Computer simulations are carried out by 10 K

increments around the melting point in order to make better

predictions of the melting point. In this manner, the melting points

of the pure elements and intermetallic systems are predicted for

Q-SC parameters. The results are listed in Table 6, along with the

available experimental [44] and other theoretical data [41, 45, 46].

As shown in the table, the melting points of pure Cu (1370±10 K)

and Pd (1820 ±10 K) metals are in very good agreement with

experimental values. As we go into the alloy, this accuracy

decreases with the maximum deviation of 3.20%. This is due to

the potential parameters of binary metal alloys calculated by using

those of pure metals using combination rules (Eq. 2). Our result

for the melting temperature of Cu are in excellent agreement with

the other calculation based on EAM [35].

Fig. 1. The variation of elastic constants as a function of temperature

for a) Cu, b) Cu3Pd, c) CuPd3 and d) Pd

Page 5

45

Analysis and modelling

Molecular dynamics study of Cu-Pd ordered alloys

Table 6.

Melting points of Cu-Pd ordered alloys and Cu and Pd pure

metals along with experimental [44] and other calculation results

[41, 45, 46], where available

Tm(K)

Q-SC Exp Other calculations

Cu 1370±10 1356 13701,10732±17,13403

Cu3Pd 1450±10 1405

CuPd3 1700±10 1650

Pd 1820±0 1825

1 from [45], 2 from [46], 3 from [41]

Mater.

Devia.

(%)

1.03

3.20

3.03

0.27

1215±202,13903

The temperature dependence of density, volume and energy of

Cu, Pd, Cu3Pd and CuPd3 are shown in Fig. 2(a), (b) and (c)

respectively. The discontinuity in the Figures shows the structural

transformation from solid phase to liquid phase. The melting

temperature is identified by monitoring the jump in the Figures. At

the melting temperatures, we find the density for Pd and Cu to be

10.53±0.06 and 7.64 g/cm3, respectively. These values are consistent

with experimental values which are 10.49 and 8.00, respectively [47].

The way we follow to predict the melting temperatures from

the pair distribution function g(r) is observed in Fig. 3 plotted at

the selected temperatures; 1600, 1810, 1820 and 2000 K for Pd.

The peaks at 1600 K could be one by one related to the different

coordination shells of a perfect fcc structure. With increasing

temperature up to 1810 K the peaks are broadened and lowered,

showing the structure with the peaks at near some of the ideal fcc

position. At the temperature of 1820 K some peaks disappear,

indicating that a diffusion dynamics is thermally activated.

Finally, the crystal order is broken and melting occurs after this

temperature. That is, the metal goes into the liquid state (2000 K).

Fig. 2. (a) Density, (b) energy and (c) volume of Cu, Pd , Cu3Pd

and CuPd3 as a function of temperature

Fig. 3. Pair distribution function g(r) for Pd at various temperatures

4. Conclusion

4.?? conclusions

We have presented a wide range of properties of Cu, Pd pure

metals and their Cu3Pd and CuPd3 ordered intermetallics. We have

found that this potential is suitable to give the general characteristics of

the melting process in these systems. The transferability of the potential

is an important conclusion which can be made from this work.

Although the potential parameters were fitted to solid experimental

properties of the pure system, the Q-SC model describes the thermal

and mechanical properties of the liquid Cu-Pd system. Because the only

experimental data for Cu-Pd metal alloys exits for the melting points,

we can test the transferability from experimental case to alloy case for

melting. That the results for density, lattice constants, cohesive energy,

compressibility, elastic constants, heat capacity and thermal expansion

coefficients of pure metals show satisfactory agreement with available

experimental values leads us to conclude that transferability of the

potential is proved for pure metal cases.

To our knowledge, the temperature dependence of physical

properties for Cu3Pd and CuPd3 ordered alloys are presented for the

first time, in this study. If the potential energy function considered

here is fitted to the solid properties of the intermetallic compounds

of Cu3Pd and CuPd3, the results may be improved further.

Acknowledgements

Acknowledgements

This work is supported by Pamukkale University Scientific

Research Fund through Project No: BAP-2006-FEF-019.

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