**4.1 Thermal and mechanical properties**

We can classify our results on thermal and mechanical properties of Au in to seven different categaries (i) the *P*-*V* diagram has been analyzed to determine the bulk modulus under zero pressure, (ii) the specific heat has been determined by using the changes of the enthalpy with temperature, (iii) the radial distribution function has been obtained in solid and liquid phases for the estimation of structural properties, (iv) the *P*-*T* graph, which is plotted by using the variation in melting temperatures with increasing pressure acted on the system, have been examined. (v) the pressure dependence of *V/Vo* has been obtained, (vi) elastic constants and pressure derivatives of elastic constants and bulk modulus has been investigated.

The change on the atomic volume with the gradually increasing pressure, which acts on the system at 300K temperature, is given in Fig.2 for Au. The bulk modulus calculated from the *P-V* diagram shown in Fig.2 is obtained as *B*=174.3 GPa for Au. The calculated bulk modulus is in good agreement with their experimental values (see Table 1) within an error of ~3.4% for Au.

Fig. 2. P-V diagrams for Au.

The variations of enthalpy with temperatures under zero pressure for solid Au is given in Fig.3, and this graph is used to compute specific heats under the constant pressure. The calculated values of specific heats over 0-300K are found to be *C*p= 28.2 J/molK for Au.

For the calculation of glass formation and crystallization, firstly, we run 20 000 time steps to make the system into equilibrium state, then the liquid phase is cooled to 100K at the rate of 1.5833x1013 K/s and 1.5833x1012 K/s , respectively to examine the formation process of

We can classify our results on thermal and mechanical properties of Au in to seven different categaries (i) the *P*-*V* diagram has been analyzed to determine the bulk modulus under zero pressure, (ii) the specific heat has been determined by using the changes of the enthalpy with temperature, (iii) the radial distribution function has been obtained in solid and liquid phases for the estimation of structural properties, (iv) the *P*-*T* graph, which is plotted by using the variation in melting temperatures with increasing pressure acted on the system, have been examined. (v) the pressure dependence of *V/Vo* has been obtained, (vi) elastic constants and pressure derivatives of elastic constants and bulk modulus has been

The change on the atomic volume with the gradually increasing pressure, which acts on the system at 300K temperature, is given in Fig.2 for Au. The bulk modulus calculated from the *P-V* diagram shown in Fig.2 is obtained as *B*=174.3 GPa for Au. The calculated bulk modulus is in good agreement with their experimental values (see Table 1) within an error

> **Au**, T=300K Bm= 173.4 GPa

0 2 4 6 8 10 12 14 16

The variations of enthalpy with temperatures under zero pressure for solid Au is given in Fig.3, and this graph is used to compute specific heats under the constant pressure. The calculated values of specific heats over 0-300K are found to be *C*p= 28.2 J/molK for Au.

**P(GPa)**

amorphization and crystallization.

**4.1 Thermal and mechanical properties** 

**4. Results and discussion** 

investigated.

of ~3.4% for Au.

16.0

Fig. 2. P-V diagrams for Au.

16.4

16.8

**V**(**A**

**3**)

17.2

17.6

Considering the experimental data in Table 1, it can be seen that the specific heat is calculated with an error of 9.8 % for Au.

Fig. 3. Variation of the enthalpy with temperature for Au.

There are several methods for determining the melting temperature of a crystal. MD simulations are performed on system at various temperatures, and the cohesive energy is plotted as a function of temperature in one of these methods, as we did here. At the melting point, a discontinuity occurs in the cohesive energy. The other way of determining the melting temperature is to plot caloric curve which is the change of the total energy of crystal versus kinetic energy [36]. Indeed, the melting temperature of metal is obtained as the temperature at which the Gibbs free energy of the solid and liquid phases become equal. The entropy is required to compute the free energy, but it can not be directly calculated from MD simulations. For this reason, some other approaches are required [3]. Another way of determining the melting temperature is to simulate the solid-liquid interface [14]. In this way, the temperature for which the interface velocity goes to zero is determined as the melting temperature and it is reproduced more correctly than the way of caloric curve. Karimi et al [14] estimated the melting temperature for Ni as 1630±50K within an error of - 5.6%, using the solid-liquid interface technique.

In the present work, the variations of cohesive energy with temperature for different pressures acted on the system are given in Fig. 4 for Au. We have computed the melting temperatures under zero pressure as 1100±20K for Au. When these values are compared with the experimental ones of 1337K given in Table 1, the error for Au becomes 21%.

The radial distribution function (RDF) is used to investigate the structural properties of the solid and liquid phases. The plot of radial distribution functions acquired in solid and liquid phases for Au is given in Fig. 5. First peak location of radial distribution curves represents the distance of the nearest neighbor atoms, *r*0. The second peak location denotes the distances of next nearest neighbors, *a*0. These distances are found to be 2.907Å and 4.144Å, respectively for Au. By comparing with experimental data given in Table1, the calculated

A Molecular Dynamics Study on Au 209

Au p=0GPa

Fig. 5. The radial distribution curves in solid and liquid phases forAu.

0 4 8 12 16 20 P(GPa)

*fcc*

700K

300K

900K

1200K

1700K

1900K

liquid

02468 r(A<sup>0</sup> )

Fig. 6. P-T diagrams for Au.

900

1400

1900

2400

T(K)

0

2

4

6

8

10

g(r)

12

14

16

18

20

2900

A

3400

error on *a*0 and *r*0 are 0.8% and 1.5% for Au. So, the present errors can be omitted since the parameters of the potential energy function were fitted to the crystal properties in static case. Since the peak locations shown in Fig. 5 satisfy the certain peak locations at 2 , 3 , 4 , 5 , etc. times *r*0 in an ideal FCC unit cell, the metal of Au has an FCC unit cell under zero pressure.

The *P*-*T* diagrams plotted by using the melting temperatures under different pressures are given in Fig. 6 for Au. The binding energies of the metals can be reduced by increasing temperature. At high temperatures near the melting point, it is generally expected that the Gibbs free energy is lowered by phase transition like martensitic types from one structure to another one which has lower energy at higher temperatures, like a BCC lattice.

Fig. 4. The cohesive energy as a function of temperature at different pressure for Au. The symbols , , , , , , + reppresents the pressure values of 0.0, 0.5, 1.0, 1.5, 2.5, 5.0, 7.5 GPa, respectively.

We calculated *V/Vo* as a function of pressure (0-45 kbar) for Au and added experimental points [37] for comparing with MD results. The plot of *V/Vo* versus pressure for Au is given in Fig. 7. Here *V*o is the volume under the zero pressure. MD results are in very good agreement with the experimental data at pressures below 25GPa.

We also calculated elastic constans and pressure derivatives of the elastic constants and bulk modulus at 0 K and in P=0 GPa pressure. The results are summarized in Table 2. Obtained results are in good agreement with available other theoretical results.


Table 2. Second order elastic constants and pressure derivatives of elastic constants and bulk modulus (P=0 GPa).

error on *a*0 and *r*0 are 0.8% and 1.5% for Au. So, the present errors can be omitted since the parameters of the potential energy function were fitted to the crystal properties in static case. Since the peak locations shown in Fig. 5 satisfy the certain peak locations at 2 , 3 , 4 , 5 , etc. times *r*0 in an ideal FCC unit cell, the metal of Au has an FCC unit cell under

The *P*-*T* diagrams plotted by using the melting temperatures under different pressures are given in Fig. 6 for Au. The binding energies of the metals can be reduced by increasing temperature. At high temperatures near the melting point, it is generally expected that the Gibbs free energy is lowered by phase transition like martensitic types from one structure to

Fig. 4. The cohesive energy as a function of temperature at different pressure for Au. The symbols , , , , , , + reppresents the pressure values of 0.0, 0.5, 1.0, 1.5, 2.5, 5.0, 7.5

We calculated *V/Vo* as a function of pressure (0-45 kbar) for Au and added experimental points [37] for comparing with MD results. The plot of *V/Vo* versus pressure for Au is given in Fig. 7. Here *V*o is the volume under the zero pressure. MD results are in very good

We also calculated elastic constans and pressure derivatives of the elastic constants and bulk modulus at 0 K and in P=0 GPa pressure. The results are summarized in Table 2. Obtained

This study 195.43 163.67 44.56 6.99 3.98 2.01 4.02 [38] 192.9 162.8 41.5 5.72 4.96 1.52 4.66 [39] 192.2 162.8 42.0 7.01 6.14 1.79 6.43 Table 2. Second order elastic constants and pressure derivatives of elastic constants and bulk

(GPa) <sup>11</sup> *<sup>T</sup> C P* <sup>12</sup> *<sup>T</sup> C P* <sup>44</sup> *<sup>T</sup> C P <sup>T</sup> B P*

agreement with the experimental data at pressures below 25GPa.

results are in good agreement with available other theoretical results.

*C44* 

another one which has lower energy at higher temperatures, like a BCC lattice.

zero pressure.

GPa, respectively.

*C11*  (GPa)

modulus (P=0 GPa).

*C12*  (GPa)

Fig. 5. The radial distribution curves in solid and liquid phases forAu.

Fig. 6. P-T diagrams for Au.

A Molecular Dynamics Study on Au 211

ΔT/Δt=1.5833x1013 K/s

Fig. 8. Average volume of Au during heating and cooling at a rate of (a) 1.5833x1013 K/s and

T(K)

cooling

T= 1100K (melting)

(b)

100 400 700 1000 1300

stma

heating

*T�T/�t=1,5833×1013* 

T c =350 K

broad peaks shows that the structure has melted. The sample was heated to 1500K and then cooled back to 1200 K, leading to the same structure as for heating, indicating a stable liquid state. Cooling to 500K, from RDF we still see the structure of a liquid, in fact a supercooled liquid. However, after cooling to 300K, we see that the second peak of RDF is split. This splitting of the second peak is a well-known characteristic feature in the RDF of a

(b) 1.5833x1012 K/s.

15

17

19

V(nm3)

21

23

25

metallic glass.

Fig. 7. Variation of pressure as a function of V/Vo for Au. Experimental points are taken from Ref.[37].

#### **4.2 Glass formation and crystallization**

Traditionally,the heating and cooling processes are applied to examine the formation process of amorphization and crystallization. The Fig.8(a) and (b) show the variation of volume at the rate of 1.5833x1013 K/s and 1.5833x1012 K/s, respectively. The sudden jump in volume in the temperature range of 1000 to 1100K for the heating process is due to the melting of the Au. In contrast to heating, cooling curves show a continuous change in volume.

The slope of the volume versus temperature curve in Fig.8(a) at the rate of 1.5833x1013 K/s decreases below 500K. This is a sign of glass formation. Since the glass is a frozen liquid, the change in configurational entropy vanishes. Thus, the derivative of entropy with respect to pressure is the derivative of volume with respect to temperature[40]. The Fig. 8(b) at rate of 1.5833x1012 K/s shows a sharp change in the volume as the temperature is lowered below 300K. At 350 K system shows that the cooled Au has crystallized.

Different methods are suggested to determine the glass transition temperature (*T*g) which is observed widely in amorphous materials. According to one of these definitions, which is known as Wendt-Abraham ratio [41], to determine *T*g in MD simulations, the *g*min/*g*max ratios of RDF curves at different temperatures are calculated [39]. Here, *g*min is the first minimum value and *g*max is first maximum value of RDF curve. In such a plot, two lines in different slopes occur, and glass transition temperature is taken as intersection point of these lines. The graph of *g*min/*g*max ratios versus temperature obtained in this study is given in Fig. 9. The *T*g is obtained from this figure to be 500K.

The RDF curves of the model structure during the heating and cooling processes at different temperature are given in Fig10. The RDF shows an fcc crystal structure as the sample is heated from 0 to 500 K. But, at 1200 K (above the melting temperature) the emergence of

**Au**

15 30 45

P(kbar)

Fig. 7. Variation of pressure as a function of V/Vo for Au. Experimental points are taken

Traditionally,the heating and cooling processes are applied to examine the formation process of amorphization and crystallization. The Fig.8(a) and (b) show the variation of volume at the rate of 1.5833x1013 K/s and 1.5833x1012 K/s, respectively. The sudden jump in volume in the temperature range of 1000 to 1100K for the heating process is due to the melting of the Au. In contrast to heating, cooling curves show a continuous change in

The slope of the volume versus temperature curve in Fig.8(a) at the rate of 1.5833x1013 K/s decreases below 500K. This is a sign of glass formation. Since the glass is a frozen liquid, the change in configurational entropy vanishes. Thus, the derivative of entropy with respect to pressure is the derivative of volume with respect to temperature[40]. The Fig. 8(b) at rate of 1.5833x1012 K/s shows a sharp change in the volume as the temperature is lowered below

Different methods are suggested to determine the glass transition temperature (*T*g) which is observed widely in amorphous materials. According to one of these definitions, which is known as Wendt-Abraham ratio [41], to determine *T*g in MD simulations, the *g*min/*g*max ratios of RDF curves at different temperatures are calculated [39]. Here, *g*min is the first minimum value and *g*max is first maximum value of RDF curve. In such a plot, two lines in different slopes occur, and glass transition temperature is taken as intersection point of these lines. The graph of *g*min/*g*max ratios versus temperature obtained in this study is given in Fig.

The RDF curves of the model structure during the heating and cooling processes at different temperature are given in Fig10. The RDF shows an fcc crystal structure as the sample is heated from 0 to 500 K. But, at 1200 K (above the melting temperature) the emergence of

300K. At 350 K system shows that the cooled Au has crystallized.

9. The *T*g is obtained from this figure to be 500K.

0.97

**4.2 Glass formation and crystallization** 

0.98

V/Vo

from Ref.[37].

volume.

0.99

1.00

Fig. 8. Average volume of Au during heating and cooling at a rate of (a) 1.5833x1013 K/s and (b) 1.5833x1012 K/s.

broad peaks shows that the structure has melted. The sample was heated to 1500K and then cooled back to 1200 K, leading to the same structure as for heating, indicating a stable liquid state. Cooling to 500K, from RDF we still see the structure of a liquid, in fact a supercooled liquid. However, after cooling to 300K, we see that the second peak of RDF is split. This splitting of the second peak is a well-known characteristic feature in the RDF of a metallic glass.

A Molecular Dynamics Study on Au 213

Fig. 10. Radial distribution function (RDF) of Au during the heating and cooling processes at

It has been found that the present version of EAM with a recently developed potential function, which makes it more flexible owing to the parameter *n*, represents quite well the interactions between the atoms to simulate the studied mono atomic systems. Since the parameterization technique of our potential is based on the bulk properties of metals at 0K, it can describe the temperature-dependent behaviors of our crystals particularly, qualitatively. As a whole, present model well describes the many physical properties ,and our results are in reasonable agreement with the corresponding experimental findings, and

provide another measure of the quantitative limitations of the EAM for bulk metals.

[1] T. Çağn, G. Dereli, M. Uludoğan, and M. Tomak, Phys. Rev. B, 59,5, (1999)3468. [2] P. Haasen, Physical Metallurgy, 2 nd ed., Cambridge Univ. Press., UK, 1992.

and Solids, Kluwer Academic Publishing, USA, 1990, pp. 1-28.

Chapman& Hall, T. J. Press (Padstow), UK, 1992.

[9] M. Hasegawa, K. Ohno, J. Phys. Condens. Matter, 9 (1997) 3361. [10] G.D. Barrera, R.H. Tendler, Comput. Phys. Commun.,105 (1997) 159. [11] S.A. Ostanin, V.Y. Trubitsin, Comput. Mater. Sci., 17 (2000) 174.

[6] M.C. Moody, J.R. Ray, J. Chem. Phys. 84 (3) (1986) 1795.

[7] J. Ihm, Rep. Prog. Phys. 51 (1988) 105.

(13) (1992) 7036.

[3] D. A. Porter, K.E. Easterling, Phase Transformation in Metals and Alloys, 1, 2nd ed.,

[4] J.M. Haile, Molecular Dynamics Simulation, Elementary Methods, Wiley, Canada, 1992. [5] C.R.A Catlow., in: C.R.A Catlow. et al. (Eds.), Computer Modelling of Fluids Polymers

[8] W.C. Kerr, A.M. Hawthorne, R.J Gooding, A.R Bishop., J. A Krumhansl, Phys. Rev. B 45

rate of 1.5833x1013 K/s (a) at 0K (b)at 500 K , and (c) at 1200K.

**5. Conclusion** 

**6. References** 

Fig. 9. Determination of glassy transition temperature.

0 200 400 600 800 1000

Tg= 500K

**T(K)**

0.0

0.1

**gmin/gmax**

0.2

0.3

A u

Fig. 9. Determination of glassy transition temperature.

Fig. 10. Radial distribution function (RDF) of Au during the heating and cooling processes at rate of 1.5833x1013 K/s (a) at 0K (b)at 500 K , and (c) at 1200K.

#### **5. Conclusion**

It has been found that the present version of EAM with a recently developed potential function, which makes it more flexible owing to the parameter *n*, represents quite well the interactions between the atoms to simulate the studied mono atomic systems. Since the parameterization technique of our potential is based on the bulk properties of metals at 0K, it can describe the temperature-dependent behaviors of our crystals particularly, qualitatively. As a whole, present model well describes the many physical properties ,and our results are in reasonable agreement with the corresponding experimental findings, and provide another measure of the quantitative limitations of the EAM for bulk metals.

#### **6. References**



**11** 

*Singapore* 

Eldin Wee Chuan Lim *National University of Singapore* 

**Gelation of Magnetic Nanoparticles** 

The study of magnetic nanoparticles and ferrofluids has gained considerable interests among research workers in recent years. The potential range of application that these novel magnetic nanomaterials can offer is gradually being recognized and continues to be explored. As described in a recent review article (Pamme, 2006), magnetic particles have already been used for such diverse applications as the fabrication of ferrofluidic pumps, solid supports for bioassays, fast DNA hybridization, giant magnetoresistive sensors and superconducting quantum interference devices (SQUID). At a more fundamental level, one of the most important and widely investigated aspects of magnetic nanoparticles and ferrofluids is the formation of self-organized microstructures under the influence of an externally applied magnetic field. A suspension of magnetic nanoparticles in a fluid medium can generally be considered as a single magnetic domain with macroscopic properties that are dependent on the properties of individual nanoparticles as well as the interactions between them (Rosensweig, 1985). In the presence of an external magnetic field, the magnetic domain will be oriented in the direction of the field and may approach saturation magnetization. When the external magnetic field is removed, the domain will revert to a randomly oriented state which exhibits no macroscale magnetism. Although it is well-established that the magnetization of a magnetic fluid or ferrofluid is related to the arrangement of the suspended magnetic nanoparticles, which in turn arises due to the effects of interactions between various types of forces present such as Brownian and dipoledipole interactions for example, current understanding of the kinetics, dynamics and resulting microstructure of the nanoparticle aggregation process is far from complete.

In the research literature, a variety of experimental, theoretical and computational approaches have been applied towards studies of the aggregation and microstructure formation process of magnetic nanoparticles and ferrofluids. In particular, the computational techniques that have been used for such investigations include Monte Carlo simulations (Davis et al., 1999; Richardi et al., 2008), Brownian dynamics (Meriguet et al., 2004, 2005; Yamada and Enomoto, 2008), lattice-Boltzmann method (Xuan et al., 2005), molecular dynamics simulations (Huang et al., 2005), combination of analytical density functional theory and molecular dynamics (Kantorovich et al., 2008), stochastic dynamics (Duncan and Camp, 2006) and analytical methods (Furlani, 2006; Furlani and Ng, 2008; Nandy et al., 2008). Further, several recent studies have also reported comparisons between experimental and theoretical or computational results. For example, the chain formation process of magnetic particles in an external magnetic field and under

**1. Introduction** 

