J. Phys. Chem. C 2010, 114, 4929–4933
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Vacancy Formation Energy in Metallic Nanoparticles under High Temperature and High Pressure G. Ouyang,*,† W. G. Zhu,† G. W. Yang,‡ and Z. M. Zhu§ School of Electrical & Electronic Engineering, Nanyang Technological UniVersity, Singapore 639798, Singapore, State Key Laboratory of Optoelectronic Materials and Technologies, Institute of Optoelectronic and Functional Composite Materials, Nanotechnology Research Center, School of Physics & Engineering, Zhongshan (Sun Yat-sen) UniVersity, Guangzhou 510275, Guangdong, People’s Republic of China, and Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control, Ministry of Education, and Department of Physics, Hunan Normal UniVersity, Changsha 410081, Hunan, People’s Republic of China ReceiVed: January 20, 2010; ReVised Manuscript ReceiVed: February 8, 2010
We present an analytic model to address the vacancy formation energy (VFE) of vacancies in metallic nanoparticles under the conditions of high temperature and high pressure based on the size-dependent cohesive energy consideration. It is found that the VFE decreases with the size of metallic nanoparticles and the VFEs on the surface and at the interface are smaller than those in the core interior. The temperature and pressure can effectively influence the VFE. The atomic energy state in the surface shell in the bond imperfection seems to be the physical origin of the anomalous properties of nanoparticles. Theoretical predictions for the size dependency of the VFE are in agreement with the available results from the simulations. Introduction Point defects, vacancies, or cavities in solid specimens have attracted intensive attention due to their fundamental significance in novel mechanical, optical, and electrical properties.1–7 For instance, some physical quantities of low-dimensional structures such as the melting temperature, the mechanical modulus, and the surface energy can be properly tuning the density and size of the nanosized pores.1,5,8,9 Recently, on the basis of the nanothermodynamics and the continuum mechanics method, Ouyang et al.9 have demonstrated that the surface energy of nanocavities in lattice matrix with negative curvature increases with the inverse size of nanocavities. As one of the fundamental physical quantities, the vacancy formation energy (VFE) plays a crucial role in the understanding the atomic diffusion and other physical properties such as transport and thermal properties, which is considered as the energy to break all the atomic bonds of the specific site to its surroundings. However, theoretical consideration of the VFE under the condition of applied pressure and temperature is still not clear, and very few quantitatively theoretical analyses involved in the issue have been reported in the open literature.1,7 Theoretically, there have been many attempts to investigate the VFE based on the thermodynamics theory,10 the semiempirical method,11–14 the ab initio calculations,15 etc. In 1975, Tiwari and Patil16 proposed a correlation between the VFE (EV) and the cohesive energy by combining the approaches established by Enderby et al.,17 Mnkherjee,18 and McLachlan et al.19 This correlation is EV ) KECmn/z, where K is equal to 0.08557, z is the coordination number (CN), and m and n are constants determined by the potential parameters.16 In fact, in the 1950s, by assuming the isotropic of bulk materials, Brook20 proposed a semiempirical model to calculate the VFE of bulk materials, * Corresponding author,
[email protected]. † Nanyang Technological University. ‡ Zhongshan (Sun Yat-sen) University. § Hunan Normal University.
giving the relation, EV ) 4π r02γ0[1/(1 + γ0/(2Gr0))], whereG, γ0, and r0 are the shear modulus, surface energy per unit area of the surrounding crystal, and the radius of the atom, respectively. Further, by considering the change of the atomic radius induced by the surface stress, Qi et al.13 replaced the r0 in Brook’s model with rp, extended it to the metallic nanoparticles, and found out that EV is dependent on the lattice strain. Experimentally, the void nucleation has been observed at the grain boundaries,21–25 interfaces, and the edges of metallic interconnections,26 owing to vacancy supersaturation or thermomechanical stresses. However, the VFE considerations are basically limited by the usual conditions as aforementioned. Therefore, there are some issues about vacancies and the VFE as why vacancies are easily produced at sites surrounding grain boundaries, surfaces, or interfaces and how about the VFE of nanostructures under the coupling effects of high temperature and high pressure. It is well-known that the cohesive energy is one of the important physical quantities of nanomaterials, which determines many physical and chemical performances. Thus, it is essential to correlate the detectable quantities to the cohesive energy, and further connect with the bonding identities such as the order, nature, atomic distance, and bond strength involved. In this contribution, by considering the response of the energy and the length of all the bonds or an average of their representatives to the external stimuli, we focus on the correlation among the VFE of metallic nanoparticles under the applied temperature, pressure and the solid size from the perspective of the bond identities based on the local bond average (LBA) consideration. Importantly, these theoretical results provide an atomistic understanding of the external temperature and pressure, and the size effect induced the variation of the VFE of specimens. Principle. According to the LBA correlation mechanism,27–29 the statistic information from a large number of atomic bonds represents the true situation of experimental measurements and theoretical models. The relative change of a physical quantity
10.1021/jp100583n 2010 American Chemical Society Published on Web 03/01/2010
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J. Phys. Chem. C, Vol. 114, No. 11, 2010
Ouyang et al. coupling stimuli of temperature and pressure in terms of shell-core configuration is given by
∑ NiziEis + zbEB(N - ∑ Ni)
EC(D, p, T) )
i
(2)
i
with Figure 1. Schematic illustration of a nanoparticle with two vacancies (A and B) showing in the shell and the core interior, respectively. The dashed circle means the boundary between the surface shell and the interior. The atomic bonds will be shorter and stronger in the shell due to CN imperfection.
such as bond length and cohesive energy under the external stimuli approach can be connected with the bulk counterpart, which focuses merely on the performance of the local representative atomic bonds disregarding the number of atomic bonds in the given specimen. Therefore, we demonstrate that the pressure, temperature, and solid size dependence of VFE variation could be described from the perfective of bond identities and LBA method that causes additional energy induced by the compression energy and thermal energy. Physically, a vacancy is characterized by its formation energy, which is equal to the energy that can form a vacancy in a material.30 Importantly, the VFE is proportional to the atomic cohesive energy.16 This proportionality can be inferred from a thermal stability viewpoint. Actually, vacancies in lattice matrix can lead to a significant decrease of the instability point in comparison with an ideal case and hence vacancy concentrations can be a sign of melting point.31 Therefore the VFE is a function of the bond energy of the solid.32 In general, for metals it has been reported that the VFE is only one-third of cohesive energy in the bulk case.33 Gladkikh and Kryshtal34 calculated the size-dependent VFE of nanomaterials using the relation: EV ) ϑTm, where ϑ and Tm denote the constant determined by the specific material and the melting temperature, respectively. It is noted that the Tm is proportional to the EC. Hence by transferring the proportionality between EV and EC to the nanoscale and under the condition of pressure and temperature, we could obtain the relation
EV(D, p, T) EC(D, p, T) ) EV,B EC,B
(1)
where D, p, and T are the diameter of a nanoparticle, the applied pressure, and the temperature, respectively. The denominators in the relation represent the bulk case. Conceptually, the cohesive energy of a solid is defined as the energy required to break all the atoms of the solid into isolated atomic species. For an isolated atom, EC ) zEb, where Eb is the single bond energy in bulk. According to our previous considerations,35 the temperature, pressure, and solid sizedependent cohesive energy of nanoparticles can be linked from the perspective of superposition principle of energy. The applied stimuli such as temperature, pressure, and their joint effect can be taken as the perturbations of the Hamiltonian or the cohesive energy of the system. The schematic illustration of the core-shell structural model of a nanoparticle with the diameter D is shown in Figure 1. The total cohesive energy of a nanoparticle can be decomposed as the energy of the core interior and the surface energy.35 Thus, the cohesive energy of a spherical nanoparticle under the
∑ ∆El
EB ) Eb +
Eis ) Ei +
and
l)p,T
∑ ∆Eil
l)p,T
where ∑l)p,T∆El denote the energy perturbation induced by the external stimuli. In detail
∆ET ) -
∫0T η1(t) dt
∆EiT ) -
∫0T η1i(t) dt
and
Ei ) ci-mEb Here η1(t) is the heat capacity of single atomic layer. zb is the CNs of bulk. Eis is the single bond energy in surface shell. ∆ER(R ) p,T) are the energy gain due to the p and T, respectively. i is the ith atomic layer, which may be conventionally counted up to three from the outmost atomic layer to the center of the solid.35 ci is the CN dependence of the bondcontraction coefficient.28 The index m is an adjustable parameter introduced to describe the change of single bond energy. It is noted that the single bond energy in the surface layers of nanoparticles is stronger than those in the core interior due to bond order loss of an atom.35 In addition
∆Ep ) -
∫VV p(V) dV 0
where V0 is the volume of unit cell at zero pressure and zero temperature. It is generally known that the external pressure would enhance the unit cell distortion energy. As a first order approximation, we assume the same elastic energy of the individual bond in the surface shell to that of those in the core interior, ∆Eip ≈ ∆Ep. Notably, the distortion energy, ∆Ep, can be obtained by the integral area based on the p-V relationship.27,36 Thus, we have
EC(D, p, T) B
EC (∞, 0, 0)
)
∑ γi[(zibci-m(1 + ∑ i
(1 +
∑ ∆l)] + (1 + ∑ ∆l)
l)p,T
with
γi ) 2τcih/D and
∆il)) -
l′)p,T
l)p,T
(3)
Vacancy Formation Energy in Metallic Nanoparticles
∆p )
∆Ep , Eb
∆ip )
∆Eip , Ei
∆T )
∆ET , Eb
J. Phys. Chem. C, Vol. 114, No. 11, 2010 4931
∆EiT Ei
∆iT )
where γi is the surface-to-volume ratio. h is the diameter of an atom. Here τ ) 1, 2, and 3 corresponds to a thin plate, a cylindrical rod, and a spherical dot, respectively. According to the relation between the VFE and the cohesive energy, EV(D,p,T) function is shown as EV(D, p, T) B
EC (∞, 0, 0)
)
EC(D, p, T) ECB(∞, 0, 0)
)
(1 +
∑ γ [(z c i
-m
ib i
(1 +
i
∑∆
il))
-
l′)p,T
∑ ∆ )] + (1 + ∑ ∆ ) l
l
l)p,T
(4)
l)p,T
As is well-known, the surface effect is crucial to the nanoscale materials due to the ratio of surface-to-volume increases and the shorter and stronger surface bonds. The CN imperfection of atoms at the surface layer would largely influence their physical and chemical properties.37,38 The average CN varies with the dimension owing to the change of curvature of nanoparticles. Therefore, the VFE in the surface layer (SVFE) is so different compared with that of the core bulk. In terms of the considerations mentioned above, we could obtain the SVFE of nanoparticles as
Figure 2. Size-dependent VFE of an Au nanoparticle. The solid line denotes the consequences of eq 2 and the symbols 330 and ]10 are calculation results of Au. The symbol 0 is the simulation data from ref 44. The necessary parameters in the calculations are h ) 0.2884 nm and EC ) 3.81 eV/atom for Au.45,46
n
∑
ziEb,i EVshell(D) 〈Ecohshell〉 1 i ) ) EV(∞) zbEb n* zbEb
(5)
where n* is the total surface atomic layer of nanoparticles. Therefore, combining eqs 4 and 5, the VFE and SVFE of nanoparticles under the joint effect of pressure and temperature can be analyzed. Results and Discussion Taking a Au nanoparticle with face-centered cubic structure as an example, we first calculate the size-dependent VFE as shown in Figure 2. It is shown that the VFE of a nanoparticle decreases when the size is reduced. The VFE increases smoothly until reaching the bulk value when the diameter of the nanoparticle is larger than 20 nm. This finding indicates that 20 nm could be a threshold of VFE of a spherical nanoparticle for its size dependence. Theoretical predictions agree well with the simulation results. Furthermore, in order to predict the VFE of nanoparticles under the condition of external stimuli, we calculate the VFE of Au by taking into account the coupling effect of pressure, temperature, and solid size, respectively. Results shown in parts a and b of Figure 3 indicate clearly the ratio between the VFE of nanoparticles and that of the bulk decreases with the increase of solid size. Meanwhile, the elevated temperature could lead to the VFE reduction, while the applied pressure does oppositely. On increasing temperature, the bond lengths increase and the bond strength decreases. The higher temperature results in the VFE decrease. Mott et al. proposed that the formation enthalpy to linearly decrease with increasing temperature.39 Thus, the heat energy induced by applied temperature will lead to relaxation of an atom, rendering a certain portion of the cohesive energy. In addition, an obvious increase in VFE is observed for the 30 nm Au nanoparticles and that of the bulk counterpart shown in Figure 3b. Due to the
Figure 3. Dependence of the VFE on the diameter and the T (a) and the diameter and the p (b). The inset in (b) is the pressure-dependent distortion energy of Au nanoparticles and that of the bulk.
competition between the solid size effect and the applied pressure-induced compression dominates the VFE of nanoparticles. The VFE is enhanced by increasing the pressure, which implies that the vacancies formation could be effectively tuned by external stimuli. Importantly, at a fixed pressure, the VFE of nanoparticles is smaller than that of the bulk. Strikingly, these predicted changes are consistent with the molecular dynamics simulations conducted by Fuks and co-workers.40 The validity of eq 4 supports the aforementioned approaches to the VFE of nanoparticles. Stiffening occurs due to the decrease of atomic
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Ouyang et al. pressure and high temperature. Introducing the joint effect of CN imperfection to the shell of nanoparticles and energy gain under the conditions of applied stimuli has led to a new energy state that revolves the cohesive energy, the VFE and SVFE of nanoparticles. This approach allows us to discriminate the contributions from surface effect with bond order loss and the external stimuli of nanoparticles. The agreement between theoretical predictions and available evidence suggests that the energy states in the surface shell of nanoparticles are different from those of the core interior. The developed approach not only provides the physical origin of VFE estimation but also indicates an effective way to tune the related physical and chemical properties.
Figure 4. Size-dependent ratio between the average surface shell VFE, the first atomic layer VFE, and the core interior.
distances and work hardening energy storage under applied pressure whereas the thermal energy and the shortage of neighboring atoms in the surface shell cause weakening of bond strength. In this way it seems reasonable to predict the VFE of nanoparticles under the conditions of applied stimuli. On the other hand, on the basis of eq 5, we calculated the SVFE of nanoparticles as shown in Figure 4. Clearly, the VFE both in surface shell and in core interior decrease when the solid size increases, while the SVFE is much smaller than that of the bulk due to CN imperfection at the surface shell. Moreover, the first atomic layer has the smallest VFE. That is why the vacancies will be shown first at the surface shell or the corner and edge of nanomaterials. Thus, from our results we can anticipate that the vacancies in surface layer are more easily formed when the solid size is diminished. Wen et al.41 also indicated that the VFE in the surface layer is smaller than that of the core interior in terms of atomistic simulations. So, the concentration of vacancies in surface layer will be larger in comparison with the situation of the bulk, which is in agreement with the experimental measurements.42 In addition, according to the relationship among the VFE, cohesive energy and the melting temperature, i.e., EV ∝ Ecoh ∝ Tm, the Tm of nanoparticles has also shown the similar behavior for its inverse linear dependence.43 The surface melting phenomena of nanomaterials plays the dominant role for their solid-liquid transition. In fact, the variation of the average cohesive energy with solid size responsible for the phase transition behavior and other physical-chemical effects such as catalytic reactivity, crystal growth, atomic diffusion, and so on. Therefore, the compressed bond under external pressure will enhance the VFE, while the thermal stimulus and the CN imperfection will lower the EV. Cansequently the atomic binding energy can be enhanced by either surface-bond contraction or additional compression. More strikingly, these factors can strengthen each other on the electronic structures and relevant properties of nanoparticles. The VFE depends on a sum of the binding energies over the entire solid. In detail, the binding energy, or interatomic potential, depends on atomic distance and charge quantity of the atoms. Thus, the energy gained by pressure, temperature, and nanosized effect can tune the VFE of materials. It implies that it is possible to devise new sources for electronic or optical electronic desired wavelength by controlling the physical size or external stimuli. Conclusion In summary, we have developed an analytic model to pursue the atomistic mechanism of the VFE of nanoparticles under high
Acknowledgment. Financial support from Singapore Millennium Foundation, National Natural Science Foundation of China (NSFC) (Nos. 10804030, U0734004, and 10974260), Key Project of Chinese Ministry of Education (No. 209088), and Scientific Research Fund of Hunan Provincial Education Department (No. 08B052) is acknowledged. References and Notes (1) Kraftmakher, Y. Phys. Rep. 1998, 299, 79. (2) Look, D. C.; Hemsky, J. W. Phys. ReV. Lett. 1999, 82, 2552. (3) Zhang, S. B.; Wei, S. H.; Zunger, A. Phys. ReV. 2001, B 63, 075205. (4) Zeng, H. B.; Cai, W. P.; Liu, P. S.; Xu, X. X.; Zhou, H. J.; Klingshirn, C.; Kalt, H. ACS Nano 2008, 2, 1661. (5) Ding, Y.; Sun, C. Q.; Zhou, Y. C. J. Appl. Phys. 2008, 103, 084317. (6) Zeng, H. B.; Xu, X. J.; Bando, Y.; Gautam, U. K.; Zhai, T. Y.; Fang, X. S.; Liu, B. D.; Golberg, D. AdV. Funct. Mater. 2009, 19, 3165. (7) Ouyang, G.; Wang, C. X.; Yang, G. W. Chem. ReV. 2009, 109, 4221. (8) Ouyang, G.; Yang, G. W.; Sun, C. Q.; Zhu, W. G. Small 2008, 4, 1359. (9) Ouyang, G.; Tan, X.; Cai, M. Q.; Yang, G. W. Appl. Phys. Lett. 2006, 89, 183104. (10) Yang, C. C.; Li, S. Phys. ReV. 2007, B 75, 165413. (11) Daw, M. S.; Baskes, M. I. Phys. ReV. 1984, B 29, 6443. (12) Oh, D. J.; Johnson, R. A. J. Mater. Res. 1988, 3, 451. (13) Qi, W. H.; Wang, M. P. Physica B 2003, 334, 432. (14) Qi, W. H.; Wang, M. P. J. Mater. Sci. 2004, 39, 2529. (15) Sahli, B.; Vollenweider, K.; Fichtner, W. Phys. ReV. 2009, B 80, 075208. (16) Tiwari, G. P.; Patil, R. V. Scr. Metall. 1975, 9, 833. (17) Enderby, J. E.; March, N. H. In Phase stability in metals and alloys; Rudman, P. S., Stringer, J., Jaffee, R. I., Eds.; McGraw Hill: New York, 1970; p 479. (18) Mnkherjee, K. Philos. Mag. 1965, 12, 915. (19) McLaohlan, D., Jr.; Chamberlain, L. L. Acta Metall. 1964, 12, 571. (20) Brooks, H. Impurities and Imperfection; American Society for Metals: Cleveland, OH, 1955. (21) Kolluri, K.; Gungor, M. R.; Maroudas, D. Appl. Phys. Lett. 2007, 90, 221907. (22) Nucci, J. A.; Shachamdiamand, Y.; Sanchez, J. E. Appl. Phys. Lett. 1995, 66, 3585. (23) Nucci, J. A.; Keller, R. R.; Sanchez, J. E.; Shachamdiamand, Y. Appl. Phys. Lett. 1996, 69, 4017. (24) Nucci, J. A.; Keller, R. R.; Field, D. P.; Shachamdiamand, Y. Appl. Phys. Lett. 1997, 70, 1242. (25) Sekiguchi, A.; Koike, J.; Kamiya, S.; Saka, M.; Maruyama, K. Appl. Phys. Lett. 2001, 79, 1264. (26) Guedj, C.; Barnes, J. P.; Papon, A. M. Appl. Phys. Lett. 2006, 89, 203108. (27) Ouyang, G.; Sun, C. Q.; Zhu, W. G. J. Phys. Chem. 2008, B 112, 5027. (28) Sun, C. Q. Prog. Solid State Chem. 2007, 35, 1. (29) Sun, C. Q. Prog. Mater Sci. 2009, 54, 179. (30) Shandiz, M. A. J. Phys.: Condens. Matter 2008, 20, 325237. (31) Zhukov, V. P. SoV. Phys.sSolid State 1985, 27, 723. (32) Mukherjee, K. Phys. Lett. 1964, 8, 17. (33) Zhang, C. J.; Alavi, A. J. Am. Chem. Soc. 2005, 127, 9808. (34) Gladkikh, N. T.; Kryshtal, O. P. Funct. Mater. 1999, 6, 823. (35) Ouyang, G.; Zhu, W. G.; Sun, C. Q.; Zhu, Z. M.; Liao, S. Z. Phys. Chem. Chem. Phys. 2010, 12, 1543. (36) Ouyang, G.; Sun, C. Q.; Zhu, W. G. J. Phys. Chem. 2009, C 113, 9516.
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