金属Mo和W纳米晶表面能及形成焓的计算
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http://www.paper.edu.cn The calculation of surface energies and formation enthalpies of metallic Mo and W nanocrystals D. Xie *, M. P. Wang, L. F. Cao, Y. L. Jia School of Materials Science and Engineering, Central South University, Changsha 410083, China * Author to whom correspondence should be addressed. Tel.: +86-731-8830264. Fax: +86-731-8876692. E-mail: patriot_csu@163.com
Abstract A simple but valid equation for the higher surface energies of metallic nanocrystals is obtained, the formation enthalpies of nanocrystals are calculated by using the equation of surface energies, and the structure variation induced formation enthalpies are also discussed. The calculated surface energy values of Ag and Au nanocrystals are in excellent agreement with the corresponding newest experimental values, and the predicted formation enthalpy values are consonant with the corresponding experimental values of Mo and W nanocrystals. Keywords：Metallic nanocrystals, Surface energy, Formation enthalpy
Since Kim et al. [1] reported the first experimental data on the formation enthalpies of Mo and W by measuring the oxidation enthalpies of nanocrystals, several authors began to investigate the important thermodynamic parameter theoretically [2-4]. Despite the success, most of the models established in the literatures under different theoretical origins, avoided to discuss the structure variations of nanocrystals which could directly influence the formation enthalpies ( for example, the structure of Mo and W nanocrystals can be changed from BCC structure to FCC structure with crystals size decreasing [1] ), and can not determine the surface induced and structural 1http://www.paper.edu.cn induced formation enthalpies of nanocrystals. Recently, Qi made efforts to present a thermodynamic method to calculate the surface contribution to formation enthalpies of nanocrystals, and tried to calculate the structural induced formation enthalpies indirectly by considering the structure variations of nanocrystals [5], but unfortunately the calculations presented by the author seems wrong which we will discuss later. In this letter, a simple but valid equation for higher surface energies of metallic nanocrystals is obtained to calculate the formation enthalpies of Mo and W nanocrystals, and the formation enthalpy induced by structure variation are also discussed. According to the discussions on the surface and structural effects of nanocrystals by Kim et al. [1], the formation enthalpy of nanocrystal ΔH should be written as follows
surfsvHHH∆=∆+∆ (1) where ΔHsurf denotes the formation enthalpy induced by surface, ΔHsv denotes the formation enthalpy induced by structure variation, whose absolute value could also be regarded as the value of solid structure transition energy. ΔHsv =0 when the structure of the nanocrystals has no difference from that of the corresponding bulk. It is know that the surface induced formation enthalpy of a nanocrystal can be written as ΔHsurf = Sγn, where S is the surface area, γn is the surface energy per unit area at 0 K [6]. Though the γn value of free nanocrystal is supposed to be size and structure independence and equal to the corresponding bulk value by a few of authors [5], there are lots of literatures to confirm both experimentally and theoretically that the value of γn is much higher than that of the corresponding bulk and varies with the changed structure [7-10]. For instance, the size independence surface energies of γn have been experimentally determined by Nanda et al. as 7.2 J/m2 for free FCC Ag
2http://www.paper.edu.cn nanoparticles [7] and 9.0 J/m2 for free FCC Au nanoparticles [11] by studying the size dependence evaporation of nanoparticles relating to Kelvin effect in latest 2 years, and it is apparently that the experimental values of nanoparticles are significantly higher than the corresponding bulk values of 1.25 J/m2 and 1.5 J/m2 [10] respectively. Based on our previous work in theoretical studying the thermal stability of nanocrystals [4,12-14], the following two equations are obtained by a continuous model [14] and a discontinuous model [4] on the cohesive energy 2nnb1323
6aEEnCµγ=− (2)
nbb13232
36114EEE
nCk
µµα
π
⎛⎞⎛⎜⎟⎜⎜⎟⎜⎜⎟⎜⎝⎠⎝=−=−⎞⎟⎟⎟⎠
(3)
Where En and Eb denote the cohesive energy of nanocrystal and corresponding bulk respectively, α and n denote the surface-to-volume atomic ratio and the atomic
number of nanocrystal respectively, a denotes lattice parameter, μ denotes the shape factor which defined as the ratio of surface areas between random shape and cube of nanocrystals in identical volume [4,12,15], k is the ratio between atomic radius and lattice parameter, and C is the atomic number of one structure cell. For FCC, BCC and HCP structures, k are 24, 34 and 1/2, C are 4, 2 and 2 respectively [16]. By combining Eq. (2) and Eq. (3), a simplified formula for surface energy of nanocrystal can be given
