- Email: [email protected]
Cluster size dependence of surface energy of Ni nanoclusters: A molecular dynamics study
Chemical Physics Letters 558 (2013) 57–61
Contents lists available at SciVerse ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Cluster size dependence of surface energy of Ni nanoclusters: A molecular dynamics study Hamed Akbarzadeh a,⇑, Farid Taherkhani b a b
Department of Chemistry, Hakim Sabzevari University, Sabzevar, Iran Department of Chemistry, Razi University, Kermanshah, Iran
a r t i c l e
i n f o
Article history: Received 12 November 2012 In final form 19 December 2012 Available online 3 January 2013
a b s t r a c t In this Letter we have presented a method for applying pressure on nanoparticles in the computer simulations, using an ideal gas as the pressure medium. Simulations have been performed under different isothermal conditions (200–400 K) and for pressures up to 60 GPa for different sizes of Ni nanoclusters (N = 336, 484, 736, 1004, 1956 Ni atoms). We have noticed that the surface energy depends on temperature, pressure, and cluster size and its dependence on the cluster size is significant. At constant temperature and pressure, surface energy decreases with cluster size. Also, at constant temperature and N, surface energy increases with pressure. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent decades, consolidated solids composed of nanometer size clusters (nanophase materials) have drawn a great deal of attention because of their unique thermomechanical, electrical, and magnetic properties [1,2]. In contrast to conventional polycrystalline solids, nanophase materials have a large fraction of atoms in the interfacial regions, which have a dramatic effect on the structure and physical properties of these materials [3–7]. One of the most commonly used parameters for the description of the energetic situation on the surface of a solid is the surface energy. The surface energy is analogous to the surface tension of a liquid. Surface energy describes the interaction between cohesive and adhesive forces which, in turn, is dictated if wetting occurs. The surface energy may be defined as the excess energy of the surface of a material compared to its bulk [8,9]. Although the surface tension of liquids has been understood and measured since the time of Young and Laplace, the surface energy of solids has eluded understanding and evaded measurement, despite its postulated importance to catalysis, crystal growth, colloidal behavior, sintering and fracture [10]. Its surface energy must depend on its structure. Information about low-energy structures of Ni nanoparticles can be found in [11,12]. In non-crystalline structures the difference from the bulk energy is due not only to surface contributions but also to internal strain. Grunwald and Dellago [13] used an ideal gas flow through a surface to apply pressure on CdSe nanoparticles for their computer simulations. ⇑ Corresponding author. E-mail address: [email protected] (H. Akbarzadeh). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.12.039
We have recently presented an approach for applying pressure on the nanosystems with molecular dynamics simulation [14]. This method is especially appropriate for finite systems for which no periodic boundary condition is applicable. The purpose of the present work is to calculate the surface energy for Ni nanoclusters, relative to the solid. Molecular dynamics, MD, simulation technique was employed to perform the relevant calculation. Simulations have been performed under different isothermal conditions (200–400 K) and pressures up to 600 kbar for different Ni nanoclusters sizes. 2. Molecular dynamics simulation We have used molecular dynamics simulation to calculate the surface energy of Ni nanoclusters. In this Letter, the procedure used is similar to that introduced in Ref. [14]. We will briefly explain this method. At the beginning of the simulation, we consider spherical clusters. In our simulations, the pressure medium consists of particles that do not interact with each other (ideal gas), but do interact with the particles of the crystal via a soft sphere potential of the form
r12 UðrÞ ¼ e r
ð1Þ
where r denotes the distance between two particles, r is the interaction range and e is the interaction strength. We have used argon gas as pressure medium in these simulations. In our simulations of the Ni nanocrystals, however, if one takes merely the interaction potential of Eq. (1) among each Ar atom and the atoms of nanocluster, the pressure never converges. To overcome this problem, we may assume that a small fraction of argon atoms interact with the atoms of the nanocluster as follows, instead of Eq. (1),
58
H. Akbarzadeh, F. Taherkhani / Chemical Physics Letters 558 (2013) 57–61
12 r
V LJ ¼ 4e
r
r6
ð2Þ
r
The Lennard–Jones potential parameters for a nickel–argon pair are: e = 8.642 kJ/mol, r = 2.84 Å [15,16]. The fraction of Ar atoms that interact through Eq. (2) may be found by try and error, in such a way that it leads to pressure convergence. We have used the quantum Sutton–Chen potential for the Ni–Ni interactions [17,18]. The MD simulations are carried out in a NpT ensemble with the periodic boundary conditions, for the system including the nanocluster and argon gas. Temperature is controlled by a Nose–Hoover thermostat [19] and the equations of motion are integrated using the Verlet Leapfrog algorithm [15] with a time step of 0.001 ps. The system was equilibrated for 500 ps (500 000 time steps), the averages were computed over the following 1 ns (1 000 000 time steps). We have used the DL-POLY-2.20 program [20]. The results for the bulk Ni were obtained from specific MD simulations for different isotherms and isobars. Simulations have been performed under different isothermal conditions (200–400 K) and for pressures up to 60 GPa for different sizes of Ni nanoclusters (N = 336, 484, 736, 1004, 1956 Ni atoms). The bulk calculations were done with the NPT ensemble at ambient pressure and with the periodic boundary conditions, applying a Nose–Hoover thermostat–barostat with the relaxation times for the temperature and pressure of 0.1 and 1.0 fs, respectively. The Verlet leapfrog scheme was used to integrate the equations of motion. In these calculations for bulk system, a time step of 1 fs was used and trajectories were carried out for a total of 800 ps, with the first 500 ps used to equilibrate the system. 3. Surface energy Surface energy is defined as the energy needed in bringing a molecule to the surface for forming and maintaining the surface
area in equilibrium, which is represented by the surface free energy per unit surface area in thermodynamics (the reversible work per unit area to form a new surface of a substance) [21,22]. In this Letter, we have calculated the surface energy of a cluster (compared to the bulk) from [23]
c ¼ ðP E; cluster PE; Bulk Þ=ð4pR2c Þ
ð3Þ
where PE, cluster and PE, Bulk are the molecular potential energies of the cluster and bulk Ni, respectively, at given temperature and pressure, and Rc is the cluster radius. Firstly, we have calculated the potential energy of the bulk nickel using molecular dynamics simulation with the QSC potential. Secondly, we have calculated the potential energy of the nickel nanoclusters with different sizes.
4. Results and discussion Eq. (3) may be used to calculate the surface energy of a cluster, compared to its corresponding bulk solid. The calculated potential energy of cluster at different pressures and temperatures is summarized for some given cluster sizes in Table 1. Then, the calculated PE, cluster, along with PE, bulk, can be used to obtain c, via Eq. (3). In Figure 1, we have plotted surface energy (c) as a function of cluster size for the Ni nanoclusters for different isotherms and isobars. This figure shows a decrease in surface energy when cluster size increases. As cluster size decreases, a larger fraction of atoms are on the surface of cluster. Since surface atoms have less binding energy, compared to the bulk atoms, therefore with decrease in number of particles, magnitude of potential energy per atom of cluster is expected to decrease (less negative). Therefore, a de-
Table 1 The calculated potential energy from MD simulations for given sizes of Ni nanoclusters (in eV) at given temperatures and pressures, compared to the bulk. N
P (kbar)
PE,
336 336 336 336 336 336 336 484 484 484 484 484 484 484 736 736 736 736 736 736 736 1004 1004 1004 1004 1004 1004 1004 1956 1956 1956 1956 1956 1956 1956
1 100 200 300 400 500 600 1 100 200 300 400 500 600 1 100 200 300 400 500 600 1 100 200 300 400 500 600 1 100 200 300 400 500 600
3.897 3.866 3.820 3.768 3.715 3.669 3.636 5.712 5.668 5.602 5.525 5.448 5.381 5.334 8.834 8.766 8.665 8.549 8.430 8.327 8.255 12.189 12.097 11.959 11.799 11.636 11.495 11.397 24.232 24.056 23.787 23.475 23.156 22.880 22.688
cluster
(T = 200)
PE,
cluster
3.888 3.858 3.814 3.763 3.713 3.669 3.635 5.700 5.656 5.592 5.518 5.444 5.381 5.332 8.813 8.747 8.649 8.537 8.423 8.326 8.251 12.160 12.070 11.936 11.782 11.626 11.493 11.391 24.179 24.008 23.747 23.445 23.140 22.880 22.680
(T = 250)
PE,
cluster
3.881 3.852 3.809 3.760 3.711 3.668 3.634 5.687 5.646 5.583 5.512 5.440 5.377 5.328 8.793 8.731 8.635 8.526 8.417 8.321 8.245 12.131 12.046 11.916 11.766 11.617 11.485 11.381 24.125 23.964 23.710 23.417 23.124 22.865 22.663
(T = 300)
PE,
cluster
3.873 3.845 3.803 3.756 3.709 3.666 3.631 5.675 5.636 5.575 5.505 5.437 5.375 5.324 8.775 8.715 8.622 8.516 8.411 8.317 8.239 12.103 12.022 11.894 11.749 11.606 11.476 11.369 24.073 23.919 23.670 23.386 23.106 22.851 22.643
(T = 350)
PE,
cluster
3.865 3.840 3.799 3.753 3.707 3.665 3.629 5.664 5.627 5.567 5.501 5.434 5.373 5.321 8.754 8.698 8.607 8.505 8.404 8.310 8.231 12.076 12.001 11.876 11.737 11.597 11.468 11.360 24.016 23.873 23.631 23.358 23.085 22.833 22.621
(T = 400)
H. Akbarzadeh, F. Taherkhani / Chemical Physics Letters 558 (2013) 57–61
59
Figure 1. Size dependence of the surface energy for different isotherms and isobars of given Ni nanoclusters.
crease in surface energy with increase of cluster size should be expected which is in accordance with Figure 1. Figure 2 shows pressure dependence of the surface energy for some Ni nanoclusters at specified temperatures. As this figure shows, an increase in surface energy at the pressure can be observed. At P = 0, atoms in a solid are in their equilibrium positions, i.e. at the minimum potential energy. Applying pressure on the solid will increase its potential energy. Owing to the fact that c increases with P (Figure 2), means that the increase in PE is more significant for surface atoms than those in the bulk. In the other words, compression of a solid will influence more on the surface atoms. In Figure 3, we have plotted surface energy (c) as a function of temperature (T) at constant pressure for some given Ni nanoclusters. As this figure shows, the surface energy decreases with temperature. As temperature increases, the binding energy of Ni
atoms decreases. The decrease in binding energy is more significant for those atoms that have binding with more atoms, than those on the surface. Therefore, by raising temperature up, less energy needed in bringing an atom to the surface. Hence, decrease of the surface energy with T is expected, which is in accordance with Figure 3.
5. Conclusion In this Letter and previous work [14], we have presented a method for applying pressure on nanoparticles in the computer simulations, using an ideal gas as the pressure medium. This method is especially suitable for the finite systems for which the periodic boundary condition does not apply. In Table 1, the calculated molecular potential energy for the Ni nanoclusters is
60
H. Akbarzadeh, F. Taherkhani / Chemical Physics Letters 558 (2013) 57–61
Figure 2. Pressure dependence of surface energy for some Ni nanoclusters at given temperatures (The lines are drawn simply to guide the eye).
Figure 3. Temperature dependence of surface energy for given Ni nanoclusters at specified pressure. (The lines are drawn simply to guide the eye.)
H. Akbarzadeh, F. Taherkhani / Chemical Physics Letters 558 (2013) 57–61
listed for different sizes, obtained by molecular dynamics simulation for pressures up to 600 kbar at different temperatures. We have investigated the effects of pressure, temperature and cluster size on the surface energy of Ni nanoclusters. A decrease in surface energy with increasing cluster size and temperature can be observed (Figs. 1 and 3). Also, an increase in surface energy with pressure can be seen (Fig. 2). On the basis of Figure 1, although the surface energy of a solid depends on T and P, its dependence on the size is very remarkable. It means that nanomaterials are expected to be much more reactive, compared to corresponding materials in the bulk state, especially for the small clusters (N < 800). References [1] [2] [3] [4] [5] [6]
H. Gleiter, Nanostruct. Mater. 1 (1992) 1. R.W. Siegel, Mater. Sci. Eng. A 168 (1993) 189. R.W. Siegel, J. Phys. Chem. Solids 55 (1994) 1097. E.A. Stern et al., Phys. Rev. Lett. 75 (1995) 3874. R.K. Kalia et al., Phys. Rev. Lett. 78 (1997) 689. T. Campbell, R.K. Kalia, A. Nakano, P. Vashishta, Phys. Rev. Lett. 82 (1999) 4866.
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
[23]
61
Z. EI-Bayyari, J. Mol. Struct. THEOCHEM 716 (2005) 165–174. D.J. Burnett, A.R. Garcia, J. Nantechnol. Online 388 (2010) 88. S.W. Ip, J.M. Toguri, J. Mater. Sci. 29 (1994) 688. K. Kendall, N.McN. Alford, J.D. Birchall, Nature 325 (1986) 794–796. G. Rossi, L. Anghinolfi, R. Ferrando, F. Nita, G. Barcaro, A. Fortunelli, Phys. Chem. Chem. Phys. 12 (2010) 8564. C. Mottet, J. Goniakowski, F. Baletto, R. Ferrando, G. Tréglia, Phase Transitions 77 (2004) 101. M. Grunwald, C. Dellago, Mol. Phys. 104 (2006) 3709. H. Akbarzadeh, H. Abroshan, F. Taherkhani, G.A. Parsafar, Solid State Commun. 151 (2011) 965. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquid, Clarendon, Oxford, 1997. Y.Y. Cheng, C.C. Lee, Nucl. Instrum. Methods Phys. Res. B 267 (2009) 1428. A.P. Sutton, J. Chen, Philos. Mag. Lett. 61 (1990) 139. Y. Qi, T. Cagin, Y. Kimura, W.A. Goddard III, Phys. Rev. B 59 (1999) 3527. S. Nose, J. Phys.: Condens. Matter 2 (1990) 115. W. Smith, I.T. Todorov, Mol. Simul. 32 (2006) 935. C.A. Miller, P. Neogi, Interfacial Phenomena: Equilibrium and Dynamic Effects, Marcel Dekker, Inc., New York, 1985. J. Lykema, G.J. Fleer, J.M. Kleijn, F.A.M. Leermakers, W. Norde, T. Van Vliet, Fundamentals of Interface and Colloid Science, Liquid–Fluid Interfaces, vol. 3, Academic Press, 2000. L. Wang, Y. Zhang, X. Bian, Y. Chen, Phys. Lett. A 310 (2003) 197.
Contents lists available at SciVerse ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Cluster size dependence of surface energy of Ni nanoclusters: A molecular dynamics study Hamed Akbarzadeh a,⇑, Farid Taherkhani b a b
Department of Chemistry, Hakim Sabzevari University, Sabzevar, Iran Department of Chemistry, Razi University, Kermanshah, Iran
a r t i c l e
i n f o
Article history: Received 12 November 2012 In final form 19 December 2012 Available online 3 January 2013
a b s t r a c t In this Letter we have presented a method for applying pressure on nanoparticles in the computer simulations, using an ideal gas as the pressure medium. Simulations have been performed under different isothermal conditions (200–400 K) and for pressures up to 60 GPa for different sizes of Ni nanoclusters (N = 336, 484, 736, 1004, 1956 Ni atoms). We have noticed that the surface energy depends on temperature, pressure, and cluster size and its dependence on the cluster size is significant. At constant temperature and pressure, surface energy decreases with cluster size. Also, at constant temperature and N, surface energy increases with pressure. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent decades, consolidated solids composed of nanometer size clusters (nanophase materials) have drawn a great deal of attention because of their unique thermomechanical, electrical, and magnetic properties [1,2]. In contrast to conventional polycrystalline solids, nanophase materials have a large fraction of atoms in the interfacial regions, which have a dramatic effect on the structure and physical properties of these materials [3–7]. One of the most commonly used parameters for the description of the energetic situation on the surface of a solid is the surface energy. The surface energy is analogous to the surface tension of a liquid. Surface energy describes the interaction between cohesive and adhesive forces which, in turn, is dictated if wetting occurs. The surface energy may be defined as the excess energy of the surface of a material compared to its bulk [8,9]. Although the surface tension of liquids has been understood and measured since the time of Young and Laplace, the surface energy of solids has eluded understanding and evaded measurement, despite its postulated importance to catalysis, crystal growth, colloidal behavior, sintering and fracture [10]. Its surface energy must depend on its structure. Information about low-energy structures of Ni nanoparticles can be found in [11,12]. In non-crystalline structures the difference from the bulk energy is due not only to surface contributions but also to internal strain. Grunwald and Dellago [13] used an ideal gas flow through a surface to apply pressure on CdSe nanoparticles for their computer simulations. ⇑ Corresponding author. E-mail address: [email protected] (H. Akbarzadeh). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.12.039
We have recently presented an approach for applying pressure on the nanosystems with molecular dynamics simulation [14]. This method is especially appropriate for finite systems for which no periodic boundary condition is applicable. The purpose of the present work is to calculate the surface energy for Ni nanoclusters, relative to the solid. Molecular dynamics, MD, simulation technique was employed to perform the relevant calculation. Simulations have been performed under different isothermal conditions (200–400 K) and pressures up to 600 kbar for different Ni nanoclusters sizes. 2. Molecular dynamics simulation We have used molecular dynamics simulation to calculate the surface energy of Ni nanoclusters. In this Letter, the procedure used is similar to that introduced in Ref. [14]. We will briefly explain this method. At the beginning of the simulation, we consider spherical clusters. In our simulations, the pressure medium consists of particles that do not interact with each other (ideal gas), but do interact with the particles of the crystal via a soft sphere potential of the form
r12 UðrÞ ¼ e r
ð1Þ
where r denotes the distance between two particles, r is the interaction range and e is the interaction strength. We have used argon gas as pressure medium in these simulations. In our simulations of the Ni nanocrystals, however, if one takes merely the interaction potential of Eq. (1) among each Ar atom and the atoms of nanocluster, the pressure never converges. To overcome this problem, we may assume that a small fraction of argon atoms interact with the atoms of the nanocluster as follows, instead of Eq. (1),
58
H. Akbarzadeh, F. Taherkhani / Chemical Physics Letters 558 (2013) 57–61
12 r
V LJ ¼ 4e
r
r6
ð2Þ
r
The Lennard–Jones potential parameters for a nickel–argon pair are: e = 8.642 kJ/mol, r = 2.84 Å [15,16]. The fraction of Ar atoms that interact through Eq. (2) may be found by try and error, in such a way that it leads to pressure convergence. We have used the quantum Sutton–Chen potential for the Ni–Ni interactions [17,18]. The MD simulations are carried out in a NpT ensemble with the periodic boundary conditions, for the system including the nanocluster and argon gas. Temperature is controlled by a Nose–Hoover thermostat [19] and the equations of motion are integrated using the Verlet Leapfrog algorithm [15] with a time step of 0.001 ps. The system was equilibrated for 500 ps (500 000 time steps), the averages were computed over the following 1 ns (1 000 000 time steps). We have used the DL-POLY-2.20 program [20]. The results for the bulk Ni were obtained from specific MD simulations for different isotherms and isobars. Simulations have been performed under different isothermal conditions (200–400 K) and for pressures up to 60 GPa for different sizes of Ni nanoclusters (N = 336, 484, 736, 1004, 1956 Ni atoms). The bulk calculations were done with the NPT ensemble at ambient pressure and with the periodic boundary conditions, applying a Nose–Hoover thermostat–barostat with the relaxation times for the temperature and pressure of 0.1 and 1.0 fs, respectively. The Verlet leapfrog scheme was used to integrate the equations of motion. In these calculations for bulk system, a time step of 1 fs was used and trajectories were carried out for a total of 800 ps, with the first 500 ps used to equilibrate the system. 3. Surface energy Surface energy is defined as the energy needed in bringing a molecule to the surface for forming and maintaining the surface
area in equilibrium, which is represented by the surface free energy per unit surface area in thermodynamics (the reversible work per unit area to form a new surface of a substance) [21,22]. In this Letter, we have calculated the surface energy of a cluster (compared to the bulk) from [23]
c ¼ ðP E; cluster PE; Bulk Þ=ð4pR2c Þ
ð3Þ
where PE, cluster and PE, Bulk are the molecular potential energies of the cluster and bulk Ni, respectively, at given temperature and pressure, and Rc is the cluster radius. Firstly, we have calculated the potential energy of the bulk nickel using molecular dynamics simulation with the QSC potential. Secondly, we have calculated the potential energy of the nickel nanoclusters with different sizes.
4. Results and discussion Eq. (3) may be used to calculate the surface energy of a cluster, compared to its corresponding bulk solid. The calculated potential energy of cluster at different pressures and temperatures is summarized for some given cluster sizes in Table 1. Then, the calculated PE, cluster, along with PE, bulk, can be used to obtain c, via Eq. (3). In Figure 1, we have plotted surface energy (c) as a function of cluster size for the Ni nanoclusters for different isotherms and isobars. This figure shows a decrease in surface energy when cluster size increases. As cluster size decreases, a larger fraction of atoms are on the surface of cluster. Since surface atoms have less binding energy, compared to the bulk atoms, therefore with decrease in number of particles, magnitude of potential energy per atom of cluster is expected to decrease (less negative). Therefore, a de-
Table 1 The calculated potential energy from MD simulations for given sizes of Ni nanoclusters (in eV) at given temperatures and pressures, compared to the bulk. N
P (kbar)
PE,
336 336 336 336 336 336 336 484 484 484 484 484 484 484 736 736 736 736 736 736 736 1004 1004 1004 1004 1004 1004 1004 1956 1956 1956 1956 1956 1956 1956
1 100 200 300 400 500 600 1 100 200 300 400 500 600 1 100 200 300 400 500 600 1 100 200 300 400 500 600 1 100 200 300 400 500 600
3.897 3.866 3.820 3.768 3.715 3.669 3.636 5.712 5.668 5.602 5.525 5.448 5.381 5.334 8.834 8.766 8.665 8.549 8.430 8.327 8.255 12.189 12.097 11.959 11.799 11.636 11.495 11.397 24.232 24.056 23.787 23.475 23.156 22.880 22.688
cluster
(T = 200)
PE,
cluster
3.888 3.858 3.814 3.763 3.713 3.669 3.635 5.700 5.656 5.592 5.518 5.444 5.381 5.332 8.813 8.747 8.649 8.537 8.423 8.326 8.251 12.160 12.070 11.936 11.782 11.626 11.493 11.391 24.179 24.008 23.747 23.445 23.140 22.880 22.680
(T = 250)
PE,
cluster
3.881 3.852 3.809 3.760 3.711 3.668 3.634 5.687 5.646 5.583 5.512 5.440 5.377 5.328 8.793 8.731 8.635 8.526 8.417 8.321 8.245 12.131 12.046 11.916 11.766 11.617 11.485 11.381 24.125 23.964 23.710 23.417 23.124 22.865 22.663
(T = 300)
PE,
cluster
3.873 3.845 3.803 3.756 3.709 3.666 3.631 5.675 5.636 5.575 5.505 5.437 5.375 5.324 8.775 8.715 8.622 8.516 8.411 8.317 8.239 12.103 12.022 11.894 11.749 11.606 11.476 11.369 24.073 23.919 23.670 23.386 23.106 22.851 22.643
(T = 350)
PE,
cluster
3.865 3.840 3.799 3.753 3.707 3.665 3.629 5.664 5.627 5.567 5.501 5.434 5.373 5.321 8.754 8.698 8.607 8.505 8.404 8.310 8.231 12.076 12.001 11.876 11.737 11.597 11.468 11.360 24.016 23.873 23.631 23.358 23.085 22.833 22.621
(T = 400)
H. Akbarzadeh, F. Taherkhani / Chemical Physics Letters 558 (2013) 57–61
59
Figure 1. Size dependence of the surface energy for different isotherms and isobars of given Ni nanoclusters.
crease in surface energy with increase of cluster size should be expected which is in accordance with Figure 1. Figure 2 shows pressure dependence of the surface energy for some Ni nanoclusters at specified temperatures. As this figure shows, an increase in surface energy at the pressure can be observed. At P = 0, atoms in a solid are in their equilibrium positions, i.e. at the minimum potential energy. Applying pressure on the solid will increase its potential energy. Owing to the fact that c increases with P (Figure 2), means that the increase in PE is more significant for surface atoms than those in the bulk. In the other words, compression of a solid will influence more on the surface atoms. In Figure 3, we have plotted surface energy (c) as a function of temperature (T) at constant pressure for some given Ni nanoclusters. As this figure shows, the surface energy decreases with temperature. As temperature increases, the binding energy of Ni
atoms decreases. The decrease in binding energy is more significant for those atoms that have binding with more atoms, than those on the surface. Therefore, by raising temperature up, less energy needed in bringing an atom to the surface. Hence, decrease of the surface energy with T is expected, which is in accordance with Figure 3.
5. Conclusion In this Letter and previous work [14], we have presented a method for applying pressure on nanoparticles in the computer simulations, using an ideal gas as the pressure medium. This method is especially suitable for the finite systems for which the periodic boundary condition does not apply. In Table 1, the calculated molecular potential energy for the Ni nanoclusters is
60
H. Akbarzadeh, F. Taherkhani / Chemical Physics Letters 558 (2013) 57–61
Figure 2. Pressure dependence of surface energy for some Ni nanoclusters at given temperatures (The lines are drawn simply to guide the eye).
Figure 3. Temperature dependence of surface energy for given Ni nanoclusters at specified pressure. (The lines are drawn simply to guide the eye.)
H. Akbarzadeh, F. Taherkhani / Chemical Physics Letters 558 (2013) 57–61
listed for different sizes, obtained by molecular dynamics simulation for pressures up to 600 kbar at different temperatures. We have investigated the effects of pressure, temperature and cluster size on the surface energy of Ni nanoclusters. A decrease in surface energy with increasing cluster size and temperature can be observed (Figs. 1 and 3). Also, an increase in surface energy with pressure can be seen (Fig. 2). On the basis of Figure 1, although the surface energy of a solid depends on T and P, its dependence on the size is very remarkable. It means that nanomaterials are expected to be much more reactive, compared to corresponding materials in the bulk state, especially for the small clusters (N < 800). References [1] [2] [3] [4] [5] [6]
H. Gleiter, Nanostruct. Mater. 1 (1992) 1. R.W. Siegel, Mater. Sci. Eng. A 168 (1993) 189. R.W. Siegel, J. Phys. Chem. Solids 55 (1994) 1097. E.A. Stern et al., Phys. Rev. Lett. 75 (1995) 3874. R.K. Kalia et al., Phys. Rev. Lett. 78 (1997) 689. T. Campbell, R.K. Kalia, A. Nakano, P. Vashishta, Phys. Rev. Lett. 82 (1999) 4866.
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
[23]
61
Z. EI-Bayyari, J. Mol. Struct. THEOCHEM 716 (2005) 165–174. D.J. Burnett, A.R. Garcia, J. Nantechnol. Online 388 (2010) 88. S.W. Ip, J.M. Toguri, J. Mater. Sci. 29 (1994) 688. K. Kendall, N.McN. Alford, J.D. Birchall, Nature 325 (1986) 794–796. G. Rossi, L. Anghinolfi, R. Ferrando, F. Nita, G. Barcaro, A. Fortunelli, Phys. Chem. Chem. Phys. 12 (2010) 8564. C. Mottet, J. Goniakowski, F. Baletto, R. Ferrando, G. Tréglia, Phase Transitions 77 (2004) 101. M. Grunwald, C. Dellago, Mol. Phys. 104 (2006) 3709. H. Akbarzadeh, H. Abroshan, F. Taherkhani, G.A. Parsafar, Solid State Commun. 151 (2011) 965. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquid, Clarendon, Oxford, 1997. Y.Y. Cheng, C.C. Lee, Nucl. Instrum. Methods Phys. Res. B 267 (2009) 1428. A.P. Sutton, J. Chen, Philos. Mag. Lett. 61 (1990) 139. Y. Qi, T. Cagin, Y. Kimura, W.A. Goddard III, Phys. Rev. B 59 (1999) 3527. S. Nose, J. Phys.: Condens. Matter 2 (1990) 115. W. Smith, I.T. Todorov, Mol. Simul. 32 (2006) 935. C.A. Miller, P. Neogi, Interfacial Phenomena: Equilibrium and Dynamic Effects, Marcel Dekker, Inc., New York, 1985. J. Lykema, G.J. Fleer, J.M. Kleijn, F.A.M. Leermakers, W. Norde, T. Van Vliet, Fundamentals of Interface and Colloid Science, Liquid–Fluid Interfaces, vol. 3, Academic Press, 2000. L. Wang, Y. Zhang, X. Bian, Y. Chen, Phys. Lett. A 310 (2003) 197.