A tight binding molecular dynamics calculation has been conducted to study the size and coordination dependence of bond length and bond energy of Pd atomic clusters of 1.2–5.4 nm in diameter. It has been found that the bond contraction associated with bond energy increases in the outermost layer about 0.24 nm in a radial way, yet in the core interior the bond length and the bond energy remain their corresponding bulk values. This surface bond contraction is independent of the particle size.
KeywordsLattice parameters Atomic simulation Bond energy Nanoparticles
It is reported that the lattice parameter (LP) of nanoparticles depends on particle size [1–12], and several theoretical models have also been established to find the relation between LP and the particle size [13–16]. For non-spherical particles, the shape effect on LP can be approximately predicted by the shape factor , where the shape factor is a modified parameter to describe the shape effect. However, the previous work in theories or experiments only gave the mean values of LP, and the difference between the interior core and the exterior surface is not considered. It is known that the surface atoms of nanoparticles have large dangling bonds, but low coordination number (CN) than the bulk, which may cause the surface to be different from the bulk. Also, the LP in the surface may be different from the bulk. To understand the structure of nanoparticles, it is needed to study the structure rigorously.
Back to 1995, Lamber et al.  reported that the LPs of small Pd particles of 1.4–5 nm decrease with decreasing particle size due to the surface effect, which clarify the contradictions of LP of Pd particle reported in literatures [5–7]. Silva et al. simulated the LP of Pd particles by molecular dynamics simulation method, where the simulated results are consistent with the experiments of Lamber et al. . Also, the experiments are well consistent with results of Continuous Media (CM) model , where the CM model regarded the nanoparticles as ideal spherical crystals generated from bulk, and then approaching to thermodynamic equilibrium to form a nanoparticle. However, in the experiments, the simulation or the theory, only the mean values of LP are obtained, and we do not know the different lattice variation between the surface and the interior core.
In this paper, we will reconsider the LP of Pd nanoparticles in detail, and discover the different structure of surface and core. Also, the bond energy and the CN will be discussed.
Cleri and Rosato Potential parameters for Pd 
The spherical Pd nanoparticles were generated from the ideal Pd crystal. The number of atoms was 38, 68, 92, 164, 298, 370, 490, 682, 1,048, 1,830, 2,598, 3,396, and 4,874, where the particle size ranges from 1.2 to 5.4 nm. The free boundary condition was applied, and the time step was chosen as 2 fs. To obtain the most stable structure, the annealing method was used presently. Since the melting temperature of nanoparticles depends on particle size, different annealing temperature was chosen to avoid the phase transition. For 38, 68, 92, 164, 298, 370, and 490, respectively, the simulation started from 300 K, and the initial 30,000 steps was to relax the structure at 300 K. The following 70,000 step decreases the temperature from 300 K to 0 K. For 682, 1,048, 1,830, 2,598, 3,396 and 4,874, respectively, the simulation started from 500 K, and the initial 30,000 steps was to relax the structure at 500 K. The following 120,000 step decreases the temperature from 500 K to 0 K.
Results and Discussion
Comparing Eq. 2and Eq. 3, one can find that a term was added in Eq. 3, where this term is just the increased energy due to the surface relaxation. According to thermodynamics, every system approaches to the configuration with low energy, thus the free energy decreases after relaxation, and the cohesive energy increases. For unsupported particles, the relaxation factorδ is smaller than 1, where the value can be obtained by fitting simulation or experimental values. The present fitted value is δ = 0.42. Apparently, the results considered the relaxation effect are more close to the simulation values.
Based on the discussion above, the structure of a small Pd particle can be classified into three regions from center, i.e., the core, the subsurface, and the surface. The thickness of the surface is about 0.24 nm, and the subsurface is about 0.16 nm, where the values are independent of the particle size. In the core, the bond length and the bond energy are almost the same as the corresponding bulk values; in surface, the bond length contracts but the bond energy increases; and in the subsurface, the bond energy keeps the bulk value but the bond length contracts. The subsurface can be regarded as a transitional region from the core to the surface. The three shell model has also been found in copper particles by Meyer and Entel . It should be mentioned that in our previous BE model , we assumed that the atoms in a nanoparticle can be classified as the exterior and the interior atoms, where the exterior atoms is only the first layer of nanoparticles. According to the present simulation results, this assumption is reasonable and applicable.
The tight binding molecular dynamics simulation method has been used to study the structure of small Pd particles. The simulated mean LP decreases with decreasing particle size, which is well consistent with the experimental values. It is found that the structure of an unsupported Pd particle can be divided into three regions, i.e., the core, the subsurface, and the surface. The thickness of the surface is about 0.24 nm, and the subsurface is about 0.16 nm, where both the values are independent of the particle size. Furthermore, the bond energy increases and the bond length decreases with the decrease in CN.
This work was supported by China Postdoctoral Science Foundation (No. 20070420185) and Postdoctoral Science Foundation of Central South University.