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Dependence of self-diffusion coefficient, surface energy, on size, temperature, and Debye temperature on size for aluminum nanoclusters
Fluid Phase Equilibria 335 (2012) 26–31
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Dependence of self-diffusion coefficient, surface energy, on size, temperature, and Debye temperature on size for aluminum nanoclusters Farid Taherkhani a,b,∗ , Hamed Akbarzadeh c,∗∗ , Hadi Abroshan d , Alessandro Fortunelli b a
Department of Chemistry, Razi University, Kermanshah, Iran CNR-IPCF, Istituto per i Processi Chimico-Fisici del CNR, Pisa, Italy c Department of Chemistry, Sabzevar Tarbiat Moallem University, Sabzevar, Iran d Department of Chemistry, Carnegie Mellon University, Pittsburgh, PA, USA b
a r t i c l e
i n f o
Article history: Received 27 May 2012 Received in revised form 29 July 2012 Accepted 13 August 2012 Available online 20 August 2012 Keywords: Surface energy Self-diffusion coefficient Size effect Molecular dynamics simulation
a b s t r a c t Molecular dynamics simulations are performed to investigate the surface energy and self-diffusion coefficient in aluminum nanoclusters (AlN ) as a function of temperature, T (T = 300–1100 K), and size (N = 108–4000 atoms, with N the number of atoms in the cluster), with the self-diffusion coefficient compared with the bulk limit. Debye temperature, cohesive energy, and average coordination number of AlN are also explored as a function of the cluster size. The surface energy decreases as a function of size as well as temperature, and becomes very small at sizes larger than N ≥ 2000, while the average coordination number and the Debye temperature increase as a function of nanocluster size. The self-diffusion coefficient decreases with increasing size as N−2/3 , while increasing as a function of temperature, and exhibits values substantially larger than in the bulk, quantitatively confirming the much greater structural freedom encountered of nanoscale systems. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Nanoclusters have peculiar properties because of their finite size. For such systems, the constraint of translational invariance on a lattice does not apply, and nanoclusters can present noncrystalline structures containing e.g. five-fold symmetry axes, icosahedra and decahedra being the best known [1]. Metal particles in the nanometer regime containing tens to millions of atoms, and have attracted a great deal of interests in both science and technology because of a number of exciting potential applications in catalysis as well as in electronic and optical nanodevices [2–4]. These systems may consist of identical atoms, or be composed of two or more different species and can be studied in a number of media, such as molecular beams, the gas phase, colloidal suspensions, or isolated in inert matrices or on surfaces [4–9]. Depending on the medium, nanometer-size metal particles can be produced experimentally in several ways, such as atomic deposition, mechanical milling, chemical methods, and gas-aggregation techniques [3]. In recent decades, cluster-assembled materials, i.e.
∗ Corresponding author at: Department of Chemistry, Razi University, Kermanshah, Iran. Tel.: +98 8314274569; fax: +98 8314274559. ∗∗ Corresponding author. E-mail addresses: [email protected] (F. Taherkhani), [email protected] (H. Akbarzadeh). 0378-3812/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fluid.2012.08.011
solids intimately composed of nanometer-size clusters, have also drawn a great deal of attention because of their unique thermomechanical, electrical, and magnetic properties [9,10]. Interest in clusters arises because they constitute a new type of material with distinct properties from those of individual atoms and bulk matter. An important subject in cluster science is therefore the size-dependent evolution of their properties such as their structure [8,11–13]. In fact, both the geometric shape and the energetic stability of the clusters may drastically change with size. In particular, clusters containing no more than a few hundred particles (with diameters of 1–3 nm) are expected to have strongly size-dependent properties [13], such as geometric and electronic structure, binding energy, and melting temperature. Larger clusters, with many thousands of atoms and diameters in the range of 10 nm and more, exhibit a more smoothly varying behavior, which tends to the bulk limit as size increases [1]. Knowledge of the shape and precise atomistic structure of the particles is thus very important in the determination of their stability and associated properties. Al nanoclusters (AlN , with N the number of atoms) are very interesting under this viewpoint and have been the subject of previous investigations. At the experimental level, Martin et al. showed that, above size N ≥ 250 magic numbers were due to geometric shell closings associated with face-centered-cubic octahedral [19]. At the theoretical level, for N = 13 the global minimum has been predicted to be icosahedral [14–17] while for N = 55 the most stable structure depends on the chosen method (ab initio or classical
F. Taherkhani et al. / Fluid Phase Equilibria 335 (2012) 26–31
molecular dynamics simulation): icosahedral [15], decahedral [17], cubooctahedral [18], and disorder structures [14] were found as the most stable structures. The choice of the theoretical method, e.g. the chosen interatomic potential, is important for determining the structure of Al nanoclusters. The Gupta potential favors structures based on the Mackay icosahedron, whereas glue potentials tend to stabilize polytetrahedral structures. For the latter, it is generally preferred to have five tetrahedra around a nearest-neighbor contact, but beyond a certain size, edges surrounded by six tetrahedra must also be present [20]. Despite this previous work, there are not many studies addressing a thorough energetic and thermal analysis of nanoparticles in a wide size interval. The melting of Al clusters in the size range 49 ≤ N ≤ 62 has been studied by Calvo et al. via atomistic simulations using glue potentials to find the melting temperatures, the latent heats of fusion and the entropies of fusion as a function of size, and the heat capacities as a function of temperature [20]. The size dependence of the melting point of aluminum nanoparticles containing 55–1000 atoms was studied as a bistability between the solid and liquid phases by Alavi et al. [21]. The surface and interface energies are crucial quantities of nanostructures because, as the surface-to-volume ratio increases with decreasing size, these quantities will greatly affect the physical properties of nanostructures, as in the case of nanostructures with negative curvature for which a clear and detailed understanding of surface energy is still lacking [22]. Given the above described context, the aim of work is to investigate the self-diffusion coefficient, the surface energy, the average coordination number, the cohesive energy, and the Debye temperature in aluminum nanoclusters as a function of size and temperature in a wide interval of both variables. Because of the appreciable dependence on size of such physical properties in nanosystems, a thorough exploration of size effects seems interesting and instructive. It will be shown for example that the sharp variation of the self-diffusion coefficient with temperature found in the bulk limit at the melting point becomes appreciably dispersed in a wider temperature range for small clusters, and that the self-diffusion coefficient in the smaller particles presents values significantly higher than in larger ones, quantitatively confirming the much greater structural freedom encountered in nanoscale systems. In Section 2 we give details of the computational approach. In Section 3 we present results on the energetics and in Section 4 on the self-diffusion coefficients obtained through our approach. Conclusions are summarized in Section 5.
2. Molecular dynamics simulation Molecular dynamics (MD) simulations of bulk fcc aluminum are carried out using the NVT ensemble and periodic boundary conditions. In the case of Al nanoclusters, simulations are carried out in a NVE ensemble without periodic boundary conditions. Temperature is controlled by Nose–Hoover thermostat [23] and the equations of motion are integrated using the Verlet leapfrog algorithm [24] with a time step of 0.001 ps. The systems were equilibrated during 400 ps, then the averages of structures were computed over the following 500 ps. The DL-POLY-2.20 package was used to conduct the MD simulations [25]. In order to obtain starting structures, a block of fcc Al was constructed from a fcc unit cell, and then replicated in three dimensions. The Sutton–Chen (SC) empirical potential was chosen to describe interactions in these systems. This potential has a many-body character, and its parameters are optimized to describe the lattice parameter, the cohesive energy, the bulk modulus, the elastic constants, the phonon dispersion, the vacancy formation energy, and the surface energy, leading to an accurate description of many properties of metals and alloys [26–29]. In this
27
study, the parameters ε, a, n, m, c of the SC force field were set to: 0.033147 (eV), 4.05 (Å), 7.0, 6.0, and 16.399, respectively [21]. fcc initial arrangements are locally optimized. After optimization the fcc structure is not kept and the structure tends to be become disordered. An analysis of our result regarding the effect of size on the shape of aluminum nanocluster in the range N = 256–2048 atoms is shown in Appendix (A) (in Supplementary Material). Our results show that as the cluster size is increased, it is the nanocluster structure approaches a spherical shape. The shape of the N = 864 particle as a function of temperature at 300–800 K is shown in Appendix B (in Supplementary Material). In Appendix B, it is shown that the melting point of the N = 864 AlN particle occurs at 580 K when using a Sutton–Chen type potential, whereas it occurs at 630 K using the Streitz–Mintmire potential [21]. Increasing the temperature, the structure of the solid nanoparticle expands and eventually the ordered solid configuration is completely lost at the melting point. 3. Energetics 3.1. Dependence of surface energy on nanocluster size The surface energy can be calculated conventionally by semiempirical methods based on a broken bond rule [22,30] as being determined by the number of bonds that have to be broken in order to create a unit area of surface. The relation is specified by [22] hkl =
Whkl 2Ahkl
(1)
where and W are the surface energy and the reversible work involved in creating a new surface, respectively, and the subscripts h, k, and l represent the Miller indices specifying the orientation of a plane. Therefore, the surface energy is the complement to the binding energy of the surface atoms. To calculate the surface energy per atom, we use the definition [31]: =
Ecluster − Ebulk
(2)
4Rc2
where Ecluster and Ebulk are the potential energies per atom of cluster and bulk, respectively, and Rc is the cluster radius at a certain temperature which is defined as:
Rc = Rg
5 + Ri 3
(3)
where Rg and Ri are gyration and atomic radius, respectively. For ˚ Rg can be obtained using the following aluminum, Ri equals 1.25 A. equation Rg2 =
1 (Rj − Rcm )2 N
(4)
j
Rj − Rcm in Eq. (4) is the distance of atom j from the cluster center of mass [26–31]. We have calculated the cluster radius (Rc ) and the surface energy (per unit surface area and atom) at 300 K for different nanocluster size (N = 256–4000 atoms), and the results are shown in Fig. 1. Spherical symmetry is applied for calculating the cluster radius (Rc ). In general, the binding energy Eb of a cluster of size N can be expressed in the following form: Eb = a1 N + b1 N 2/3 + c1 N 1/3 + d1
(5)
The first term in Eq. (5) is related to bulk energy while other terms are related to surface, edges, and vertices, respectively [1]. On the basis of Eq. (5), the binding energy per atom of cluster EbN , namely Ecluster in Eq. (2), can be written as EbN = Ecluster = a1 + b1 N −1/3 + c1 N −2/3 +
d1 N
(6)
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Fig. 2. (a) Radial distribution function for surface atom in platinum nanocluster for different size 256 [a] and 500 [b] atoms. (b) Radial distribution function for surface atom in aluminum nanocluster for different size 256 [a] and 500 [b] atoms.
Fig. 1. (a) Surface energy as function number of particle aluminum nanoparticle at T = 300 K. (b) Surface energy is fitted as a power law (N−1 ) to the size of system with slope m = 0.7688 and R-square 0.9977 linearly.
On the basis of Eq. (6) energy per atom for bulk (namely Ebulk in Eq. (2)) is a1 . On the basis of Eq. (5) the expression 4Rc2 is proportional to b1 N2/3 , by substituting the energy per atom for the cluster Ecluster in Eq. (6), the energy per atom for bulk Ebulk and 4Rc2 in Eq. (2) we have = =
Ecluster − Ebulk 4Rc2
≈
(a1 + b1 N −1/3 + c1 N −2/3 + d1 /N) − a1
b 1 c 1 1 b
N
+
b
bN 2/3
N −4/3 +
d 1
b
N 1/3
(7)
On the basis of Eq. (7) the surface energy can be fitted linearly as a N−1 . The result of this good fitting including slope and Rsquare values is shown in Fig. 1b. Surface atoms have less binding energy, compared to bulk atoms, hence with decreasing number of atoms, the potential energy of the cluster is increased. On the basis of Eq. (2) the reduction of the potential energy with increasing size is due to the less important contribution of surface atoms to the total energy which leads to a decrease of the surface energy
(). Based on the previous literature [32] surface energy for platinum nanocluster increase as a function of size however surface energy decreases as a function of size for aluminum nanocluster. The nature of chemical bond for surface atom and bulk atom in platinum nanocluster is completely different to each other. Atoms in bulk are stable than atom in surface for platinum nanocluster. Increase of cluster size due to increase of coordination number in inside of platinum nanocluster and diminishing of surface effect, amount of energy from transferring to surface becomes difficult. As a result surface energy increases with number of atom in platinum nanocluster. From computational point of view radial distribution function [32] g(r) = (1/nN) ı(r − r ) for surface atom in i,j i j platinum nanocluster is calculated for 256 and 500 particles at 300 K. Result of radial distribution for surface atom of platinum nanocluster as a function of r (distance) is shown in Fig. 2a. Based on Fig. 2a with increase number of atom the value of area, which is obtained from taking integration of radial distribution of surface atom (Fig. 2a), increases. As a result, increase of the number of atom in platinum nanocluster increases the surface energy then atom in platinum nanocluster prefer go to inside of nanocluster not remained in surface, for big size of cluster. For aluminum nanocluster, atom in surface is stable than atom in bulk. When atom ads to aluminum nanocluster, it prefers to choose the surface not bulk. Our result for radial distribution function for surface atom of aluminum nanocluster is shown in Fig. 2b. Based on Fig. 2b increase
F. Taherkhani et al. / Fluid Phase Equilibria 335 (2012) 26–31
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of number of atom the value of area, which is obtained from integration of radial distribution function graph, decreases. As a result with increase of number of atom surface energy due to decrease number of atom in surface, decreases. 3.1.1. Average coordination number for AlN versus size Nanoparticles have a very high surface area to volume ratio. If we define the coordination number as the number of bond per atom, it is obvious that the number of bond for surface atoms is smaller than for bulk atoms. As a result, there are different coordination numbers in a nanosystem: surface coordination number (Cs ) and bulk coordination number (Cb ). The average coordination number is defined as 1 C¯ = N
M
Cbi +
N1
i=1
Csi
(8)
i=1
where M, N1 , and N are number of atom in bulk and surface, and the total number of atom in the system, respectively. The average coordination number is related to the cohesive energy using following equation [33] C¯ n Ecn d = =1− Ecb D C¯ b
(9)
where C¯ b = 12 for the fcc structure. In Eq. (9) C¯ n , C¯ b , Ecn , Ecb , d, D are average coordination number for aluminum nanoclusters, average coordination number for bulk, cohesive energy for nanoclusters, cohesive energy for bulk, diameter of atom and diameter of particle, respectively [34]. It is clear that for fcc structure d = (3/2)1/3 a, where a is lattice constant [34]. Results regarding the average coordination number at 300 K for AlN are shown in Appendix C. This figure illustrates that increase of nanocluster size leads to an increase of the average coordination number of AlN , and approaching the coordination number in the bulk. Also the ratio of surface to bulk decrease with increasing size of AlN . As a result average coordination number for AlN increases as a function of size and finally it approaches to 12 for face center cubic structure in the bulk limit. 3.1.2. Debye temperature of AlN as a function of size The Debye temperature ( D ) is a significant quantity determining thermal transport parameters, thermal vibration of atoms, and phase transitions [35,36]. Using Lindemann theory, Reisland [37,38] derived the expressions of Debye temperature ( D ) and atomic mean square displacement (u2 ) for a bulk crystal as
D = Clin
u2 =
Tm
1/2 (10)
2/3
MVm
92 T 2 mkB D
,
=
h 2
(11)
where Clin is the Lindeman constant and its value for metal and non-metal materials is 137 and 200, respectively [37]. Here Tm , M, Vm , h, T, m, and kB are melting temperature, molar mass, molar volume, Planck constant, temperature, mass of an individual atom, and the Boltzmann constant. The relationship between the Debye temperature and the cohesive energy is [37,38] Dc = Db
E 1/2 cn
Ecb
(12)
The melting point of bulk Al in this study is obtained at 875 K (Appendix D in Supplementary Material shows the jump of heat capacity at the melting point), while the molar volume is obtained to be 9.997 × 10−6 M3 /mol which is in good agreement with Refs.
Fig. 3. Debye temperature versus size of AlN from 108 to 4000 particles.
[37,39]. For aluminum the value of atomic mean square displacement (MSD) u2 = (1/N) i (xi − x¯ )2 (where xi and x¯ , N are atomic coordination in nanocluster, average atomic coordination and total number of atom, respectively) is calculated to be 2.27 × 10−22 m2 as 1.25 percentage of nearest-neighbor distance. For small size of cluster because of different mobility of surface atom compare to the bulk atomic mean square displacement is obtained as our previous definition for MSD. However value of MSD is very big for small size of nanocluster and it can be extended for small size of nanocluster [37]. MSD for atoms in surface is completely different from bulk, the MSD which is reported in this work for small size of nanocluster is, average of MSD in surface and bulk atom. Using Eq. (11) and the calculated MSD, the Debye temperature to be 460 K which is in good agreement with the experimental result (394 K) for aluminum in bulk [40]. The calculated Debye temperature for AlN with 108–4000 particles as a function of cluster size is shown in Fig. 3. On the basis of Eq. (12) the Debye temperature is proportional to the cohesive energy. The coordination number for surface atom is less than in the bulk; therefore, the cohesive energy for a nanocluster is less than in the bulk. As a result the Debye temperature increases with increasing the size of the nanocluster. It is worthwhile to notice that our simulation result via Sutton–Chen type potential regarding the melting point of AlN is close to experimental value which is obtained via differential scanning calorimetry [41]. Based on experimental data melting point for aluminum nanoparticle sizes with 10 nanometer is 920 K, which is close to the our simulation data 873.38. Our result for melting point as a function of inverse size of particle is fitted as a linear function (R-square = 0.82) and it is shown in Fig. 4. On the basis of Fig. 4 our simulation result regarding melting point shows good agreement with experimental result [41] for small size of aluminum nanocluster via Sutton–Chen type potential. 3.2. Temperature effect on surface energy Molecular dynamics simulations have been finally been performed for different sizes of AlN as a function of temperature from 300 to 1200 K. At 1200 K the solid structure is destroyed completely. The surface energy as a function of temperature is shown in Fig. 4 for N = 256. As Fig. 5 indicates the surface energy decreases as a function of temperature. Increase of temperature leads to an increase of the kinetic energy of system. On the other hand, the cluster radius increases with temperature before melting point [32–42].
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F. Taherkhani et al. / Fluid Phase Equilibria 335 (2012) 26–31
Fig. 4. Melting point of AlN which is fitted linearly inverse of particle diameter with R-square = 0.82.
Fig. 6. Diffusion coefficient of AlN versus number of particles at 300 K.
phase transition. However the jump in self-diffusion coefficient and its large value around 875 K due to increase of atomic distance in liquid phases can be explained for solid to liquid phase transition. Comparison of our melting point result 875 K with the experimental one (933 K) shows 6% relative error [43]. Difference melting temperature between our simulation result and experiment value is 58 K. Results of the self-diffusion coefficient in an Al nanocluster (256 atoms) versus temperature are shown in Appendix F Supplementary Material. Similar to the bulk state, the self-diffusion coefficient for AlN increases versus temperature, and close to the solid–liquid phase transition there is a significant change in the self-diffusion coefficient. However, this change is dispersed on a much wider temperature range. The self-diffusion coefficient as a function of the size of the particles was calculated via MD simulations in the NVE ensemble, and the corresponding results are shown at a temperature of 300 K in Fig. 6. As apparent in the figure, the self-diffusion coefficient decreases as a function cluster size. Two points are worth noting. First, the steep increase for small particles (below 1000 atoms). Second, the fact that the self-diffusion coefficient of aluminum atoms
Fig. 5. Surface energy versus temperature for AlN with 256 particles.
As a result, binding energy between atoms of AlN decreases; therefore, a smaller amount of energy is needed to bring Al atoms from the bulk to the surface of the nanocluster. 4. Self-diffusion coefficients The self-diffusion coefficient (D2 ) can be calculated from the positions or velocities of atoms as: D2 ≡
1 [r(t0 + t) − r(t0 )]2 lim t 2d2 t→∞
(13)
where d2 and r are the dimensionality of the system and the vector position of the cluster atoms at time t, respectively [28]. The self-diffusion coefficient is proportional to the atomic mean square displacement (MSD), but it is necessary to run the simulations long enough to get a reliable result. Results of self-diffusion coefficient for Al bulk as a function of temperature are shown in Appendix E Supplementary Material. The self-diffusion coefficient of aluminum atoms naturally increases as a function of temperature, as – at higher temperatures – the inter-atomic distances increase and aluminum atoms move with a larger mean free path. The jump at 875 K in the self-diffusion coefficient corresponds to the solid–liquid
Fig. 7. Diffusion coefficient is fitted as a power law (N−2/3 ) to the size of system with slope m = 6.026e−008 and R-square 0.9836.
F. Taherkhani et al. / Fluid Phase Equilibria 335 (2012) 26–31
in the nano regime is much greater than its bulk limit (Appendix E Supplementary Material). Surface atoms can move more freely than those in the inner part of cluster, explaining why the self-diffusion coefficient in nanoclusters is larger than in bulk state. Since increase in the cluster size will result in a reduction of the ratio of surface to inner atoms, this effect will be less important for bigger cluster. Indeed, in Fig. 6 we interpolate the self-diffusion coefficient as an inverse function of N2/3 finding a good fit. The result of self diffusion coefficient fitting with inverse function of N2/3 is presented in Fig. 7. Our results indicates that the order of magnitude of selfdiffusion coefficient for aluminum nano particle is about 10−9 m2 /s (for 256–4000 particles at 300 K) which is bigger by three orders of magnitude with respect to the bulk limit (10−12 m2 /s).
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.fluid.2012.08.011. References [1] [2] [3] [4] [5] [6] [7]
5. Conclusions Molecular dynamics simulations are used to investigate the effect of the size dependence on the surface energy, the selfdiffusion coefficient, the cohesive energy, the Debye temperature, and the average coordination number of Al nanoclusters. The effect of temperature on surface energy and self-diffusion coefficient is also examined. The calculations are conducted for particles in a wide size interval: from 108 to 4000 atoms. The surface energy decreases as a function of size and temperature for aluminum nanoparticles, and practically vanishes for large sizes testifying the transition to bulk-like particles. Increase of temperature leads to an increase in the kinetic energy of system. Consequently, the binding energy of Al nanocluster decreases, as well as the energy needed to bring an Al atom from the bulk to the surface. The self diffusion coefficient increases as a function of temperature for both nanoclusters and bulk, but with order-of-magnitude larger values in the nano regime with respect to the bulk. There is also a significant change for self diffusion coefficient during the solid–liquid phase transition. Increase of atomic distance among atoms leads to an increase of the mean free path, hence there is a significant change in the diffusion coefficient at the phase transition point, which is however extremely sharp in the bulk and much more dispersed for the nanoclusters. The Debye temperature, which in the bulk is in relatively good agreement with experimental value [40], as well as the cohesive energy and the average coordination number increase with the size of AlN . Acknowledgment Financial support from Razi University and the SEPON Project within the ERC Advanced Grants is gratefully acknowledged.
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Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid
Dependence of self-diffusion coefficient, surface energy, on size, temperature, and Debye temperature on size for aluminum nanoclusters Farid Taherkhani a,b,∗ , Hamed Akbarzadeh c,∗∗ , Hadi Abroshan d , Alessandro Fortunelli b a
Department of Chemistry, Razi University, Kermanshah, Iran CNR-IPCF, Istituto per i Processi Chimico-Fisici del CNR, Pisa, Italy c Department of Chemistry, Sabzevar Tarbiat Moallem University, Sabzevar, Iran d Department of Chemistry, Carnegie Mellon University, Pittsburgh, PA, USA b
a r t i c l e
i n f o
Article history: Received 27 May 2012 Received in revised form 29 July 2012 Accepted 13 August 2012 Available online 20 August 2012 Keywords: Surface energy Self-diffusion coefficient Size effect Molecular dynamics simulation
a b s t r a c t Molecular dynamics simulations are performed to investigate the surface energy and self-diffusion coefficient in aluminum nanoclusters (AlN ) as a function of temperature, T (T = 300–1100 K), and size (N = 108–4000 atoms, with N the number of atoms in the cluster), with the self-diffusion coefficient compared with the bulk limit. Debye temperature, cohesive energy, and average coordination number of AlN are also explored as a function of the cluster size. The surface energy decreases as a function of size as well as temperature, and becomes very small at sizes larger than N ≥ 2000, while the average coordination number and the Debye temperature increase as a function of nanocluster size. The self-diffusion coefficient decreases with increasing size as N−2/3 , while increasing as a function of temperature, and exhibits values substantially larger than in the bulk, quantitatively confirming the much greater structural freedom encountered of nanoscale systems. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Nanoclusters have peculiar properties because of their finite size. For such systems, the constraint of translational invariance on a lattice does not apply, and nanoclusters can present noncrystalline structures containing e.g. five-fold symmetry axes, icosahedra and decahedra being the best known [1]. Metal particles in the nanometer regime containing tens to millions of atoms, and have attracted a great deal of interests in both science and technology because of a number of exciting potential applications in catalysis as well as in electronic and optical nanodevices [2–4]. These systems may consist of identical atoms, or be composed of two or more different species and can be studied in a number of media, such as molecular beams, the gas phase, colloidal suspensions, or isolated in inert matrices or on surfaces [4–9]. Depending on the medium, nanometer-size metal particles can be produced experimentally in several ways, such as atomic deposition, mechanical milling, chemical methods, and gas-aggregation techniques [3]. In recent decades, cluster-assembled materials, i.e.
∗ Corresponding author at: Department of Chemistry, Razi University, Kermanshah, Iran. Tel.: +98 8314274569; fax: +98 8314274559. ∗∗ Corresponding author. E-mail addresses: [email protected] (F. Taherkhani), [email protected] (H. Akbarzadeh). 0378-3812/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fluid.2012.08.011
solids intimately composed of nanometer-size clusters, have also drawn a great deal of attention because of their unique thermomechanical, electrical, and magnetic properties [9,10]. Interest in clusters arises because they constitute a new type of material with distinct properties from those of individual atoms and bulk matter. An important subject in cluster science is therefore the size-dependent evolution of their properties such as their structure [8,11–13]. In fact, both the geometric shape and the energetic stability of the clusters may drastically change with size. In particular, clusters containing no more than a few hundred particles (with diameters of 1–3 nm) are expected to have strongly size-dependent properties [13], such as geometric and electronic structure, binding energy, and melting temperature. Larger clusters, with many thousands of atoms and diameters in the range of 10 nm and more, exhibit a more smoothly varying behavior, which tends to the bulk limit as size increases [1]. Knowledge of the shape and precise atomistic structure of the particles is thus very important in the determination of their stability and associated properties. Al nanoclusters (AlN , with N the number of atoms) are very interesting under this viewpoint and have been the subject of previous investigations. At the experimental level, Martin et al. showed that, above size N ≥ 250 magic numbers were due to geometric shell closings associated with face-centered-cubic octahedral [19]. At the theoretical level, for N = 13 the global minimum has been predicted to be icosahedral [14–17] while for N = 55 the most stable structure depends on the chosen method (ab initio or classical
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molecular dynamics simulation): icosahedral [15], decahedral [17], cubooctahedral [18], and disorder structures [14] were found as the most stable structures. The choice of the theoretical method, e.g. the chosen interatomic potential, is important for determining the structure of Al nanoclusters. The Gupta potential favors structures based on the Mackay icosahedron, whereas glue potentials tend to stabilize polytetrahedral structures. For the latter, it is generally preferred to have five tetrahedra around a nearest-neighbor contact, but beyond a certain size, edges surrounded by six tetrahedra must also be present [20]. Despite this previous work, there are not many studies addressing a thorough energetic and thermal analysis of nanoparticles in a wide size interval. The melting of Al clusters in the size range 49 ≤ N ≤ 62 has been studied by Calvo et al. via atomistic simulations using glue potentials to find the melting temperatures, the latent heats of fusion and the entropies of fusion as a function of size, and the heat capacities as a function of temperature [20]. The size dependence of the melting point of aluminum nanoparticles containing 55–1000 atoms was studied as a bistability between the solid and liquid phases by Alavi et al. [21]. The surface and interface energies are crucial quantities of nanostructures because, as the surface-to-volume ratio increases with decreasing size, these quantities will greatly affect the physical properties of nanostructures, as in the case of nanostructures with negative curvature for which a clear and detailed understanding of surface energy is still lacking [22]. Given the above described context, the aim of work is to investigate the self-diffusion coefficient, the surface energy, the average coordination number, the cohesive energy, and the Debye temperature in aluminum nanoclusters as a function of size and temperature in a wide interval of both variables. Because of the appreciable dependence on size of such physical properties in nanosystems, a thorough exploration of size effects seems interesting and instructive. It will be shown for example that the sharp variation of the self-diffusion coefficient with temperature found in the bulk limit at the melting point becomes appreciably dispersed in a wider temperature range for small clusters, and that the self-diffusion coefficient in the smaller particles presents values significantly higher than in larger ones, quantitatively confirming the much greater structural freedom encountered in nanoscale systems. In Section 2 we give details of the computational approach. In Section 3 we present results on the energetics and in Section 4 on the self-diffusion coefficients obtained through our approach. Conclusions are summarized in Section 5.
2. Molecular dynamics simulation Molecular dynamics (MD) simulations of bulk fcc aluminum are carried out using the NVT ensemble and periodic boundary conditions. In the case of Al nanoclusters, simulations are carried out in a NVE ensemble without periodic boundary conditions. Temperature is controlled by Nose–Hoover thermostat [23] and the equations of motion are integrated using the Verlet leapfrog algorithm [24] with a time step of 0.001 ps. The systems were equilibrated during 400 ps, then the averages of structures were computed over the following 500 ps. The DL-POLY-2.20 package was used to conduct the MD simulations [25]. In order to obtain starting structures, a block of fcc Al was constructed from a fcc unit cell, and then replicated in three dimensions. The Sutton–Chen (SC) empirical potential was chosen to describe interactions in these systems. This potential has a many-body character, and its parameters are optimized to describe the lattice parameter, the cohesive energy, the bulk modulus, the elastic constants, the phonon dispersion, the vacancy formation energy, and the surface energy, leading to an accurate description of many properties of metals and alloys [26–29]. In this
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study, the parameters ε, a, n, m, c of the SC force field were set to: 0.033147 (eV), 4.05 (Å), 7.0, 6.0, and 16.399, respectively [21]. fcc initial arrangements are locally optimized. After optimization the fcc structure is not kept and the structure tends to be become disordered. An analysis of our result regarding the effect of size on the shape of aluminum nanocluster in the range N = 256–2048 atoms is shown in Appendix (A) (in Supplementary Material). Our results show that as the cluster size is increased, it is the nanocluster structure approaches a spherical shape. The shape of the N = 864 particle as a function of temperature at 300–800 K is shown in Appendix B (in Supplementary Material). In Appendix B, it is shown that the melting point of the N = 864 AlN particle occurs at 580 K when using a Sutton–Chen type potential, whereas it occurs at 630 K using the Streitz–Mintmire potential [21]. Increasing the temperature, the structure of the solid nanoparticle expands and eventually the ordered solid configuration is completely lost at the melting point. 3. Energetics 3.1. Dependence of surface energy on nanocluster size The surface energy can be calculated conventionally by semiempirical methods based on a broken bond rule [22,30] as being determined by the number of bonds that have to be broken in order to create a unit area of surface. The relation is specified by [22] hkl =
Whkl 2Ahkl
(1)
where and W are the surface energy and the reversible work involved in creating a new surface, respectively, and the subscripts h, k, and l represent the Miller indices specifying the orientation of a plane. Therefore, the surface energy is the complement to the binding energy of the surface atoms. To calculate the surface energy per atom, we use the definition [31]: =
Ecluster − Ebulk
(2)
4Rc2
where Ecluster and Ebulk are the potential energies per atom of cluster and bulk, respectively, and Rc is the cluster radius at a certain temperature which is defined as:
Rc = Rg
5 + Ri 3
(3)
where Rg and Ri are gyration and atomic radius, respectively. For ˚ Rg can be obtained using the following aluminum, Ri equals 1.25 A. equation Rg2 =
1 (Rj − Rcm )2 N
(4)
j
Rj − Rcm in Eq. (4) is the distance of atom j from the cluster center of mass [26–31]. We have calculated the cluster radius (Rc ) and the surface energy (per unit surface area and atom) at 300 K for different nanocluster size (N = 256–4000 atoms), and the results are shown in Fig. 1. Spherical symmetry is applied for calculating the cluster radius (Rc ). In general, the binding energy Eb of a cluster of size N can be expressed in the following form: Eb = a1 N + b1 N 2/3 + c1 N 1/3 + d1
(5)
The first term in Eq. (5) is related to bulk energy while other terms are related to surface, edges, and vertices, respectively [1]. On the basis of Eq. (5), the binding energy per atom of cluster EbN , namely Ecluster in Eq. (2), can be written as EbN = Ecluster = a1 + b1 N −1/3 + c1 N −2/3 +
d1 N
(6)
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Fig. 2. (a) Radial distribution function for surface atom in platinum nanocluster for different size 256 [a] and 500 [b] atoms. (b) Radial distribution function for surface atom in aluminum nanocluster for different size 256 [a] and 500 [b] atoms.
Fig. 1. (a) Surface energy as function number of particle aluminum nanoparticle at T = 300 K. (b) Surface energy is fitted as a power law (N−1 ) to the size of system with slope m = 0.7688 and R-square 0.9977 linearly.
On the basis of Eq. (6) energy per atom for bulk (namely Ebulk in Eq. (2)) is a1 . On the basis of Eq. (5) the expression 4Rc2 is proportional to b1 N2/3 , by substituting the energy per atom for the cluster Ecluster in Eq. (6), the energy per atom for bulk Ebulk and 4Rc2 in Eq. (2) we have = =
Ecluster − Ebulk 4Rc2
≈
(a1 + b1 N −1/3 + c1 N −2/3 + d1 /N) − a1
b 1 c 1 1 b
N
+
b
bN 2/3
N −4/3 +
d 1
b
N 1/3
(7)
On the basis of Eq. (7) the surface energy can be fitted linearly as a N−1 . The result of this good fitting including slope and Rsquare values is shown in Fig. 1b. Surface atoms have less binding energy, compared to bulk atoms, hence with decreasing number of atoms, the potential energy of the cluster is increased. On the basis of Eq. (2) the reduction of the potential energy with increasing size is due to the less important contribution of surface atoms to the total energy which leads to a decrease of the surface energy
(). Based on the previous literature [32] surface energy for platinum nanocluster increase as a function of size however surface energy decreases as a function of size for aluminum nanocluster. The nature of chemical bond for surface atom and bulk atom in platinum nanocluster is completely different to each other. Atoms in bulk are stable than atom in surface for platinum nanocluster. Increase of cluster size due to increase of coordination number in inside of platinum nanocluster and diminishing of surface effect, amount of energy from transferring to surface becomes difficult. As a result surface energy increases with number of atom in platinum nanocluster. From computational point of view radial distribution function [32] g(r) = (1/nN) ı(r − r ) for surface atom in i,j i j platinum nanocluster is calculated for 256 and 500 particles at 300 K. Result of radial distribution for surface atom of platinum nanocluster as a function of r (distance) is shown in Fig. 2a. Based on Fig. 2a with increase number of atom the value of area, which is obtained from taking integration of radial distribution of surface atom (Fig. 2a), increases. As a result, increase of the number of atom in platinum nanocluster increases the surface energy then atom in platinum nanocluster prefer go to inside of nanocluster not remained in surface, for big size of cluster. For aluminum nanocluster, atom in surface is stable than atom in bulk. When atom ads to aluminum nanocluster, it prefers to choose the surface not bulk. Our result for radial distribution function for surface atom of aluminum nanocluster is shown in Fig. 2b. Based on Fig. 2b increase
F. Taherkhani et al. / Fluid Phase Equilibria 335 (2012) 26–31
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of number of atom the value of area, which is obtained from integration of radial distribution function graph, decreases. As a result with increase of number of atom surface energy due to decrease number of atom in surface, decreases. 3.1.1. Average coordination number for AlN versus size Nanoparticles have a very high surface area to volume ratio. If we define the coordination number as the number of bond per atom, it is obvious that the number of bond for surface atoms is smaller than for bulk atoms. As a result, there are different coordination numbers in a nanosystem: surface coordination number (Cs ) and bulk coordination number (Cb ). The average coordination number is defined as 1 C¯ = N
M
Cbi +
N1
i=1
Csi
(8)
i=1
where M, N1 , and N are number of atom in bulk and surface, and the total number of atom in the system, respectively. The average coordination number is related to the cohesive energy using following equation [33] C¯ n Ecn d = =1− Ecb D C¯ b
(9)
where C¯ b = 12 for the fcc structure. In Eq. (9) C¯ n , C¯ b , Ecn , Ecb , d, D are average coordination number for aluminum nanoclusters, average coordination number for bulk, cohesive energy for nanoclusters, cohesive energy for bulk, diameter of atom and diameter of particle, respectively [34]. It is clear that for fcc structure d = (3/2)1/3 a, where a is lattice constant [34]. Results regarding the average coordination number at 300 K for AlN are shown in Appendix C. This figure illustrates that increase of nanocluster size leads to an increase of the average coordination number of AlN , and approaching the coordination number in the bulk. Also the ratio of surface to bulk decrease with increasing size of AlN . As a result average coordination number for AlN increases as a function of size and finally it approaches to 12 for face center cubic structure in the bulk limit. 3.1.2. Debye temperature of AlN as a function of size The Debye temperature ( D ) is a significant quantity determining thermal transport parameters, thermal vibration of atoms, and phase transitions [35,36]. Using Lindemann theory, Reisland [37,38] derived the expressions of Debye temperature ( D ) and atomic mean square displacement (u2 ) for a bulk crystal as
D = Clin
u2 =
Tm
1/2 (10)
2/3
MVm
92 T 2 mkB D
,
=
h 2
(11)
where Clin is the Lindeman constant and its value for metal and non-metal materials is 137 and 200, respectively [37]. Here Tm , M, Vm , h, T, m, and kB are melting temperature, molar mass, molar volume, Planck constant, temperature, mass of an individual atom, and the Boltzmann constant. The relationship between the Debye temperature and the cohesive energy is [37,38] Dc = Db
E 1/2 cn
Ecb
(12)
The melting point of bulk Al in this study is obtained at 875 K (Appendix D in Supplementary Material shows the jump of heat capacity at the melting point), while the molar volume is obtained to be 9.997 × 10−6 M3 /mol which is in good agreement with Refs.
Fig. 3. Debye temperature versus size of AlN from 108 to 4000 particles.
[37,39]. For aluminum the value of atomic mean square displacement (MSD) u2 = (1/N) i (xi − x¯ )2 (where xi and x¯ , N are atomic coordination in nanocluster, average atomic coordination and total number of atom, respectively) is calculated to be 2.27 × 10−22 m2 as 1.25 percentage of nearest-neighbor distance. For small size of cluster because of different mobility of surface atom compare to the bulk atomic mean square displacement is obtained as our previous definition for MSD. However value of MSD is very big for small size of nanocluster and it can be extended for small size of nanocluster [37]. MSD for atoms in surface is completely different from bulk, the MSD which is reported in this work for small size of nanocluster is, average of MSD in surface and bulk atom. Using Eq. (11) and the calculated MSD, the Debye temperature to be 460 K which is in good agreement with the experimental result (394 K) for aluminum in bulk [40]. The calculated Debye temperature for AlN with 108–4000 particles as a function of cluster size is shown in Fig. 3. On the basis of Eq. (12) the Debye temperature is proportional to the cohesive energy. The coordination number for surface atom is less than in the bulk; therefore, the cohesive energy for a nanocluster is less than in the bulk. As a result the Debye temperature increases with increasing the size of the nanocluster. It is worthwhile to notice that our simulation result via Sutton–Chen type potential regarding the melting point of AlN is close to experimental value which is obtained via differential scanning calorimetry [41]. Based on experimental data melting point for aluminum nanoparticle sizes with 10 nanometer is 920 K, which is close to the our simulation data 873.38. Our result for melting point as a function of inverse size of particle is fitted as a linear function (R-square = 0.82) and it is shown in Fig. 4. On the basis of Fig. 4 our simulation result regarding melting point shows good agreement with experimental result [41] for small size of aluminum nanocluster via Sutton–Chen type potential. 3.2. Temperature effect on surface energy Molecular dynamics simulations have been finally been performed for different sizes of AlN as a function of temperature from 300 to 1200 K. At 1200 K the solid structure is destroyed completely. The surface energy as a function of temperature is shown in Fig. 4 for N = 256. As Fig. 5 indicates the surface energy decreases as a function of temperature. Increase of temperature leads to an increase of the kinetic energy of system. On the other hand, the cluster radius increases with temperature before melting point [32–42].
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F. Taherkhani et al. / Fluid Phase Equilibria 335 (2012) 26–31
Fig. 4. Melting point of AlN which is fitted linearly inverse of particle diameter with R-square = 0.82.
Fig. 6. Diffusion coefficient of AlN versus number of particles at 300 K.
phase transition. However the jump in self-diffusion coefficient and its large value around 875 K due to increase of atomic distance in liquid phases can be explained for solid to liquid phase transition. Comparison of our melting point result 875 K with the experimental one (933 K) shows 6% relative error [43]. Difference melting temperature between our simulation result and experiment value is 58 K. Results of the self-diffusion coefficient in an Al nanocluster (256 atoms) versus temperature are shown in Appendix F Supplementary Material. Similar to the bulk state, the self-diffusion coefficient for AlN increases versus temperature, and close to the solid–liquid phase transition there is a significant change in the self-diffusion coefficient. However, this change is dispersed on a much wider temperature range. The self-diffusion coefficient as a function of the size of the particles was calculated via MD simulations in the NVE ensemble, and the corresponding results are shown at a temperature of 300 K in Fig. 6. As apparent in the figure, the self-diffusion coefficient decreases as a function cluster size. Two points are worth noting. First, the steep increase for small particles (below 1000 atoms). Second, the fact that the self-diffusion coefficient of aluminum atoms
Fig. 5. Surface energy versus temperature for AlN with 256 particles.
As a result, binding energy between atoms of AlN decreases; therefore, a smaller amount of energy is needed to bring Al atoms from the bulk to the surface of the nanocluster. 4. Self-diffusion coefficients The self-diffusion coefficient (D2 ) can be calculated from the positions or velocities of atoms as: D2 ≡
1 [r(t0 + t) − r(t0 )]2 lim t 2d2 t→∞
(13)
where d2 and r are the dimensionality of the system and the vector position of the cluster atoms at time t, respectively [28]. The self-diffusion coefficient is proportional to the atomic mean square displacement (MSD), but it is necessary to run the simulations long enough to get a reliable result. Results of self-diffusion coefficient for Al bulk as a function of temperature are shown in Appendix E Supplementary Material. The self-diffusion coefficient of aluminum atoms naturally increases as a function of temperature, as – at higher temperatures – the inter-atomic distances increase and aluminum atoms move with a larger mean free path. The jump at 875 K in the self-diffusion coefficient corresponds to the solid–liquid
Fig. 7. Diffusion coefficient is fitted as a power law (N−2/3 ) to the size of system with slope m = 6.026e−008 and R-square 0.9836.
F. Taherkhani et al. / Fluid Phase Equilibria 335 (2012) 26–31
in the nano regime is much greater than its bulk limit (Appendix E Supplementary Material). Surface atoms can move more freely than those in the inner part of cluster, explaining why the self-diffusion coefficient in nanoclusters is larger than in bulk state. Since increase in the cluster size will result in a reduction of the ratio of surface to inner atoms, this effect will be less important for bigger cluster. Indeed, in Fig. 6 we interpolate the self-diffusion coefficient as an inverse function of N2/3 finding a good fit. The result of self diffusion coefficient fitting with inverse function of N2/3 is presented in Fig. 7. Our results indicates that the order of magnitude of selfdiffusion coefficient for aluminum nano particle is about 10−9 m2 /s (for 256–4000 particles at 300 K) which is bigger by three orders of magnitude with respect to the bulk limit (10−12 m2 /s).
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.fluid.2012.08.011. References [1] [2] [3] [4] [5] [6] [7]
5. Conclusions Molecular dynamics simulations are used to investigate the effect of the size dependence on the surface energy, the selfdiffusion coefficient, the cohesive energy, the Debye temperature, and the average coordination number of Al nanoclusters. The effect of temperature on surface energy and self-diffusion coefficient is also examined. The calculations are conducted for particles in a wide size interval: from 108 to 4000 atoms. The surface energy decreases as a function of size and temperature for aluminum nanoparticles, and practically vanishes for large sizes testifying the transition to bulk-like particles. Increase of temperature leads to an increase in the kinetic energy of system. Consequently, the binding energy of Al nanocluster decreases, as well as the energy needed to bring an Al atom from the bulk to the surface. The self diffusion coefficient increases as a function of temperature for both nanoclusters and bulk, but with order-of-magnitude larger values in the nano regime with respect to the bulk. There is also a significant change for self diffusion coefficient during the solid–liquid phase transition. Increase of atomic distance among atoms leads to an increase of the mean free path, hence there is a significant change in the diffusion coefficient at the phase transition point, which is however extremely sharp in the bulk and much more dispersed for the nanoclusters. The Debye temperature, which in the bulk is in relatively good agreement with experimental value [40], as well as the cohesive energy and the average coordination number increase with the size of AlN . Acknowledgment Financial support from Razi University and the SEPON Project within the ERC Advanced Grants is gratefully acknowledged.
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