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Freezing structures of free silver nanodroplets: A molecular dynamics simulation study
Physics Letters A 373 (2009) 1667–1671
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Physics Letters A www.elsevier.com/locate/pla
Freezing structures of free silver nanodroplets: A molecular dynamics simulation study Ze-An Tian a,b , Rang-Su Liu a,b,∗ , Ping Peng a,b , Zhao-Yang Hou a,b , Hai-Rong Liu a,b , Cai-Xing Zheng a,b , Ke-Jun Dong c , Ai-Bing Yu c a b c
School of Physics and Microelectronic Science, Hunan University, Changsha 410082, China School of Materials Science and Engineering, Hunan University, Changsha 410082, China Centre for Simulation and Modelling of Particulate Systems and School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW, 2052, Australia
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Article history: Received 10 February 2009 Accepted 21 February 2009 Available online 25 February 2009 Communicated by V.M. Agranovich PACS: 61.20.Ja 61.25.Mv 61.46.Df 64.70.Dv 83.10.Tv Keywords: Freezing structures Free sliver nanodroplet Non-magic number Rapid cooling Molecular dynamics simulation
a b s t r a c t Freezing structures of free silver nanodroplets containing different numbers of atoms have been studied by using MD simulation and adopting quantum Sutton–Chen (QSC) potential. It is demonstrated that during most of the solidification processes there are a first order and a continuous phase transitions. By means of the cluster-type index method (CTIM-2) and three dimension graphic techniques, the internal structures of final nanoparticles have been investigated intensively. Besides regular crystalline, decahedral and icosahedral nanoparticles, some very interesting novelty morphologies have also been found. From geometrical views, these novelty structures can be called surface-isomers, because they can be constructed on the base of regular crystalline, decahedral or icosahedral nanoparticles by adding few layers with specific atomic arrangement. These new morphologies have well global (three- or five-fold) symmetry and lower energy than other structures, with same size and one of them is in agreement with the observation in experiment [Phys. Rev. Lett. 92 (2004) 196102]. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Owing to the absence of the translational symmetry, the exact determination of the internal structures of nanoparticles is a challenging problem. Recently, many new methods have been developed to detect the structures of the nanoparticles such as the electron tomography [1,2] and aberration-corrected scanning transmission electron microscopy [3,4]. After many crystal nanoparticles such as the truncated octahedron (TO), the decahedron (Dh), and icosahedron (Ih) have been observed in experiments [4,5], and very recently, the bi-decahedron (Bi-Dh) [6], double icosahedron [7] and the dodecahedron [8] were also discovered in laboratory. However, a lot of experimental samples [4,9] are still waiting for being identified up to now. Almost all of the methods identifying the atomic arrangement in a specific three-dimensional nanoparticle need to combine the qualitative or quantitative analysis with some model structure procedures and realistic image contrast sim-
ulations. Therefore, adopting simulation method to search more candidates for structure modeling is of importance for determining the internal structure of the unidentified samples in experiments. Recently, for free gold nanoparticles, not only the Bi-Dh has been verified but also a more general family of poly-decahedral (pDh) nanoparticles has been found by molecular dynamics (MD) simulation [10]. In this Letter, the freezing structures of free silver nanodroplets have been studied by MD simulation adopting QSC many-body potential [11,12], and the internal microstructures of nanoparticles at final temperature of 273 K have been examined through a new structure analysis method carefully. Highly interesting is that not only the above-mentioned morphologies such as Dh, Bi-Dh, Ih, and pDh have been obtained, but also some new structures found in the simulations. These new nanoparticles contain three to few decades of fivefold symmetry axes (FSAs) and at least one fcc (tetrahedral) domains enclosed by twinned hcp planes. 2. Simulation and analysis methods
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Corresponding author at: School of Physics and Microelectronic Science, Hunan University, Changsha 410082, China. Tel.: +86 731 8822817; fax: +86 731 8822817. E-mail address: [email protected] (R.-S. Liu). 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.02.041
To investigate the freezing behavior of the free silver nanodroplet, the simulation conditions and methods are set as fol-
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Fig. 1. (Color online.) 3D views of the typical Ih, tDh, bcc, hcp and fcc basic clusters (for clarity, all bonds between the centre atom and near neighbors have been omitted).
lows: First of all, put a certain number of atoms into a cubic box, the interaction potential between atoms are calculated with the QSC potential [11,12], which is cut off at 22 a.u. The equation of motion is integrated using the leapfrog algorithm with the time step 2.0 fs. Let the system run 50 000 time steps (100 ps) at the initial temperature (T init ) to obtain an equilibrium liquid state which determined by monitoring the energy change of the system. Without period boundary condition and external pressure (namely P ext = 0), the initial cubic nanodroplet transferred into a sphericallike shape very soon. Thereafter, the system is cooled continuously by coupling to a Gaussian thermostat [13,14] from T init = 1273 K to the final temperature (T f = 273 K) at a cooling rate of 100 K/ns. The numbers of atoms in the simulated nanodroplets are designed as 500, 1000, 1500, 2000, 2500, 5000 and 10 000, respectively. It should be pointed out that these numbers of atoms are far from any magic number of well-known polyhedrons such as Ih, Dh and TO. Many simulations [10,15] and experiments [16] have indicated that lots of different isomers can be resulted from identical thermal processes because of the kinetic effects. Thus a number of cooling processes have been performed for each number of atoms at the identical simulation conditions (namely, for the same number of atoms, same cooling rate, same zero external pressure, and same initial thermal state). The structures of the resulting nanoparticles are analyzed by the cluster-type index method (CTIM) [17,18], in which we have defined the basic cluster as the structure composed of a core atom and its surrounding neighbor atoms. Most basic clusters can be expressed by the CTIM-2 [17] which is extended from the CTIM. In CTIM-2, a set of six integers represents a basic cluster. The first integer equals the total number of near-neighbors of the center atom, and the following five integers represent the numbers of the 1441, 1551, 1661, 1421 and 1422 bond-types respectively, expressed by using Honeycutt and Anderson index method [19], by these bond-types the surrounding atoms are connected with the center one of the basic cluster. Five typical (Ih, tDh, bcc, hcp and fcc) basic clusters are shown in Fig. 1. We can classify all the atoms (except for those on the surface) in a nanoparticle according to the type of the basic cluster they located in. Using this classifying technique, the red atoms in each basic cluster in Fig. 1 are called Ih-, tDh-, bcc-, hcp-, fcc-atom respectively. For convenience of discussion, here we define the atom located at the fivefold vertex (FV) of a 1551 bond-type as a FV-atom; and define the plane/facet in which the atoms are all hcp or fcc atoms as a hcp or fcc plane/facet. In fact, from Fig. 1, it can be clearly seen that a smallest basic cluster of Ih marked as (12 0 12 0 0 0) is composed of an Ih-atom at the center and 12 FV-atoms on the surface. A basic cluster marked as (12 0 2 0 0 10) is a smallest truncated decahedron (tDh), in which the central atom (tDh-atom) and two FV-atoms constitute a single FSA. Using CTIM-2, as shown in Fig. 2, the atoms in a larger perfect Dh/Ih, can be classified into four types as: the surface atoms (on the outmost surface, not displayed in Fig. 2), tDh- and FVatoms (white balls on the middle and the end of single/twelve FSAs respectively), hcp-atoms (dark blue balls on five/thirty twinned planes), and fcc-atoms (light blue balls in five/twenty cells separated by hcp-planes). Therefore, by looking for the Ih-, tDh-, and
Fig. 2. (Color online.) Results of CTIM-2 applied to a perfect decahedral cluster of 309 atoms (a), and to a perfect 309-atom icosahedron (b). Both in (a) and (b), all atoms as the center of a tDh-cluster with a (12 0 2 0 0 10) signature are shown as white balls. These tDh-atoms constitute the single FSA of the decahedron and the twelve FSAs of the icosahedron (shown in the second column). All hcp-atoms are shown (in the third column) as dark blue balls and all fcc-atoms are shown (in the fourth column) as light blue balls.
FV-atoms in a particular nanoparticle, we can single out all atoms on FSAs, to investigate the fivefold symmetry. And with the help of the graphic technique, the distribution of all kinds of atoms (such as fcc-, hcp-, Ih, Dh-atoms), and global characteristic can also be found and displayed in 3D views. 3. Results and discussion 3.1. General trends Several general trends can be obtained from all simulations, although the structural evolutions and the final structures of all nanodroplets at 273 K are different from each other. The solidification characteristics of amorphous and crystalline can be found in the energy curve for most nanodroplets as shown in Fig. 3. From the point A to B, the nanodroplet is in liquid state. A typical first order phase transition (generally corresponding to crystallization) takes place between the temperature B and C, and the following evolvement of the energy curve is a continuous phase transition closely similar to the amorphous-solidification. This indicates that the final structure of the nanoparticle would exhibit both the crystalline and the non-crystalline characteristics. Similar to the experimental results performed by Turnbull and Cech [20] for small metal droplets, from the statistical viewpoint, the solidification temperature T is (that is defined as the temperature at which the slope-break of energy curve takes place, as shown in Fig. 3) is increased with the system size increasing. The averages of T is are in turn as 668, 810, 831, 860, 862, 871, and 880 K corresponding to the system sizes of 500, 1000, 1500, 2000, 2500, 5000 and 10 000 atoms, respectively. However, it is significant that the solidification behavior of particular nanodroplets often deviated far from their average values. For instance, for all simulations containing 2000 atoms, the range of T is is from 795 K to 889 K; the interval is close to 100 K.
Z.-A. Tian et al. / Physics Letters A 373 (2009) 1667–1671
At the final temperature 273 K and without relaxation, the average of the total energy per atom is lowered with the increase of system size. The average energies are −259.8, −265.8, −268.3, −269.4, −270.6, −273.1, and −275.5 kJ/mol at sizes of 500, 1000, 1500, 2000, 2500, 5000 and 10 000 atoms, respectively. Using the equation [21] E = a + bN 1/3 + cN 2/3 + dN 1 to fit the atom energy E vs. nanoparticle size N, one can get a = −288.99 kJ/mol. That is to say, for bulk system, the energy for each atom is −288.99 kJ/mol. This is in close agreement with the simulation result (−2.923 eV per atom i.e. −282.01 kJ/mol) for bulk silver system (see Fig. 3(a) of Ref. [17]) under the cooling rate of 138 K/ns. 3.2. Novelty morphologies Thereafter the nanoparticles discussed in the text are all at the final temperature 273 K and without any relaxation. Freezing nanoparticles adopt many different outlines such as sphere, ellipsoid, peanut, rod and even worm, and their morphology is various. However, all the final nanoparticles resulted from our freezing simulations are always well-defined. In other words, some defects maybe present on the surface but they are not amorphous. There are always one or more FSAs, each of them surrounded by a local (truncated) decahedral arrangement, in all non-crystalline nanoparticles. For the conveniences of description, we further define the length of a FSA is the number of atoms on it. Thus the nanoparticles can be classified according to the number and arrangement of the FSAs in them. In this Letter, several nanoparticles with special symmetry will be reported as follows. Tri-Dh is here defined as such a kind of nanoparticles in which there are at least one three-fold vertex of FSAs and no more than three FSAs meeting together. Fig. 4 shows a simplest tri-Dh containing 1000 atoms just holds three FSAs which do not share one atom, but intersect at a common vertex (see Fig. 4a and d). In other words, any couple of FSAs is co-planar and between them
Fig. 3. A typical energy curve for a solidification process of a nanodroplet. From A to B, the nanodroplet is in liquid state; between B and C, there is a first order phase transition; and from C through D to E, a continuous phase transition takes place. The temperature at B is defined as T is .
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three hcp facets formed; furthermore the angle between any two FSAs is just near 60◦ (the numerical result is 59.969 ± 0.028◦ ). Therefore, a rather canonical tetrahedron region with an unclosed facet is formed between the three hcp planes; and the tetrahedron region is packed by fcc atoms. Figs. 4b–d display the rather perfect global three-fold symmetry; and the TO-like outline can be found from Fig. 4c and d. Furthermore, it holds the lowest energy in all same size ones we studied. Tetra-Dh is another kind of nanoparticles with at least one tetrahedral skeleton constituted of equal-length-FSAs. Fig. 5 displays a tetra-Dh of 1000 atoms and its hexahedral skeletons composed of (two antipodal canonical tetrahedral skeletons) nine FSAs with the length as seven atoms. Among the five vertexes of hexahedral skeletons, there are three local Ih sites (marked by A, B, C in Fig. 5a) and two three-fold vertexes of FSAs (marked by D and E in Fig. 5a). There are several decades twinned hcp-planes across two or three co-planar FSAs (totally 33 FSAs in Fig. 5b). Between these twinned hcp-planes, there are many tetrahedral domains that are filled by fcc atoms; and there are one or two opened facets for most of these tetrahedral domains, except for two perfect ones outlined by two FSA-skeletons ABCD and ABCE (see Fig. 5a). The global three-fold symmetry and local five-fold symmetry of the nanoparticle can be seen in Fig. 5c and d, respectively. An Umb-Dh is a nanoparticle in which the skeleton composed of FSAs is like an umbrella. In an Umb-Dh, no Ih-atoms can be found; but a main FSA and some sub-FSAs are necessary; and all FSAs join together, usually at the end of the main FSA. Fig. 6 displays a perfect Umb-Dh containing 2000 atoms which holds an atom where all six FSAs meet together. The length of the main FSA (umbrella handle) is 12 atoms, that of four sub-FSAs is identical as 9 atoms and only one sub-FSA is 8 atoms. Just because of the five extra FSAs, the nanoparticle is not a Marks decahedron (m-Dh) [21], although obvious reentries can be found (indicated by arrows in Fig. 6d). As shown in Fig. 6b, there are five twinned hcpplanes; each contains the main FSA and one sub-FSA. And there are another five hcp-planes; each crosses two neighboring sub-FSAs. Therefore the nanoparticle is divided into five isolated fcc domains (shown in Fig. 6c) by the 10 hcp-planes. Comparing the Figs. 6d and 6e, it is easy to be found that the atom arrangement on the top and the bottom of the nanoparticle is different. On the top surface shown in Fig. 6d, five (1 1 1) facets share five edges (coffee atoms); while on the bottom one in Fig. 6e, five (1 1 1) facets are separated by five narrow (1 0 0) facets (wine atoms). As the illustration in Fig. 6f, similar to the bimetallic decahedral nanoparticles with surface reconstruction reported in Ref. [22], the five narrow (1 0 0) facets and five sub-FSAs (see again Fig. 6a) are both resulted from the specific atom arrangement in the last two layers, which does not follow the fcc sequence (. . . ABCAB), but arrange as (. . . ABCBA). However, it is different from the single layer of surface reconstruction [22] that in our simulation, the result reveals that there exists another atom layer out of the reconstruction layer. Therefore, on the surface the narrow (1 0 0) facets (wine atoms) contain three rows of atoms. The atoms in the penultimate layer
Fig. 4. (Color online.) A tri-Dh containing 1000 atoms with the length of FSAs as 8, 9, 9. (a) The (yellow) tDh- and (green) FV-atoms on three FSAs, and (dark blue) hcp-atoms in twinned hcp planes. (b), (c) 3D views along the same direction as that in (a). (d) A 3D view along the direction invert to that in (a).
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Fig. 5. (Color online.) A tetra-Dh composed of 1000 atoms with two tetrahedral skeletons which share a triangle ABC; Red, yellow, green, and gray balls represent the Ih, tDh, FV, and other type atoms respectively. (a) Atoms on two tetrahedral skeletons. (b) All atoms on all FSAs. (c), (d) 3D views projected along the D → E and A → C axes shown in (a).
are fcc-atoms except for those (dark blue) hcp-ones. And within the third layer (from the surface), the atoms on five facets are (dark blue) hcp-atoms, and those on five edges are (yellow) tDhatoms. A fractal-Ih has been found in all simulated samples. It is well known that in a shell-closed icosahedron [23], twelve FSAs with identical lengths meet at the center, and on the surface, there are 12 FVs, 30 edges and 20 regular-triangle-facets. As shown in Fig. 7, the fractal-Ih containing 1500 atoms holds surprising high symmetry and hierarchical structure. There is NOT ONLY an Ih-core with 13 atoms at the center of it, BUT ALSO a larger and rather perfect Ih-skeleton with the identical orientation as that of the core-Ih (Fig. 7a and d). In addition to the Ih-core and the Ih-skeleton constituted of 42 FSAs, there are still two local icosahedrons with two outmost FV-atoms as their centers (red atoms in Fig. 7). So from the viewpoint of the Ih-skeleton, this nanoparticle possesses a self-similar hierarchy-structure. This is why the nanoparticle is assigned as a fractal-Ih. As shown in Fig. 7b, up to the 6th shell, there is still a perfect icosahedron containing 561 atoms, and atoms on all edges in 2–5 layers are (dark blue) hcp-ones. However, in the 7th shell, two triangle-facets (wine atoms in Fig. 7c) do not follow the fcc (. . . ABCA) sequence but arrange in (. . . ABCB). Refer to the terms Used in Ref. [24], the two facets can be called anti-Mackey surface terminations (AMSTs). At the joint of the two AMSTs appear an extra (1 0 0) facet which composed of 10 atoms lining in two rows (see the center part of Fig. 7c). And other four triangle-facets which neighbor to the two AMSTs have somewhat distortion. Just because of the arrangement change of atoms in 7th shell, the type of some atoms in 6th shell changed: 4 yellow atoms sited in the diagonal of the green diamond in Fig. 7b, no longer are hcp-atoms but tDh-ones; and anther 20 green atoms are FVs. In other words, four local FSAs constituted of the 24 atoms formed in the 6th shell. Likewise, as shown in Fig. 7f, in the 8th shell, another twenty-six (1 0 0) facets formed between 14 AMSTs and 6 other type facets. And in 7th shell, 26 local FSAs formed (see yellow atoms in Fig. 7e). So far 30 local FSAs formed in the 6th and 7th shell at the edges of an icosahedron. Including 12 main FSAs of an icosahedron, an Ih-skeleton constituted of 42 FSAs formed. Because not all 30 outer FSAs are in an identical shell, there are some faults with it. Another incomplete shell covers the eighth facets so that there is one or two layers out of each fcc domain with the sequence of . . . ABCBA. And similar to the case of the Fig. 6f, some (1 0 0) facets including three rows of atoms can be found on the fractal-Ih surface.
Fig. 6. (Color online.) A Umb-Dh containing 2000 atoms. (a) The umbrella skeleton composed of 6 FSAs. (b), (c) the twinned hcp planes and five fcc domains. (d), (e) 3D views respectively along the B → A and A → B directions shown in (a). (f) The illumination for the specific atom arrangement in the outmost two layers, under which there are several FSAs.
Fig. 7. (Color online.) A fractal-Ih contains 1500 atoms. Three views of (a)–(c) are along an main axis of five-fold symmetry; and three views (d)–(f) are along another main axis of five-fold symmetry. (a), (d) different views of the skeleton composed of 42 FSAs. (b), (c) 3D views contains six and seven shells. (e), (f) 3D views contains seven and eight shells. Color configuration: in all views, red, dark blue, gray balls present Ih-, hcp-, and other type atoms; For clarity, green for FV-atoms, yellow only for hcp-atoms in (b); and all wine ball present atoms on the facets of AMST. (c), (e) 3D views which are both including seven shells but along different directions.
3.3. Discussion From the geometrical viewpoint, for a edge which two neighboring (1 1 1) facets share, if the next layers on the two (1 1 1) facets do not follow the ABC . . . repeat sequence but arranged as ABCB, the atoms on the edge will be tDh-atoms and a FSA formed. Another outer layers cover these facets to form . . . ABCBA sequence as just mentioned above for Umb-Dh (Fig. 6) and fractal-Ih (Fig. 7). In other words, from the “C” layer toward two sides, both are fcc sequence (CBA), so the atoms on the “C” layer are hcp-atoms. This change takes place in surface layers, so the Umb-Dh and fractal-Ih can be called as surface-isomers. About 25% nanoparticles resulted from our simulation are surface-isomers, they have at least one peripheral FSA in the 2nd or 3rd layer under surface. And a new surface-isomer with a m-Dh as its center should be obtained if more cooling processes were performed; in this kind of surfaceisomers, all ten edges at two ends in a canonical m-Dh have been transferred into FSA and the skeleton of FSAs can be regarded as two antipodal Umb-Dhs, like a leftrightarrow ↔.
Z.-A. Tian et al. / Physics Letters A 373 (2009) 1667–1671
Further analysis demonstrates that from the hcp planes shown in Fig. 4a to surface, there are three or four outer layers, and the sequence of closed-packed planes are [. . . (ABC)BAC] for three outer layers and [. . . (ABC)BCAB] for four outer layers. Therefore the tri-Dh is also a surface-isomer. The configuration of atoms outside the three hcp-planes is some similar to a Leary tetrahedron [25] which is global minimum for 98-atom Lennard-Jones cluster. Perhaps similar reason leads to the tetra-Dh has the lowest energy among all 1000-atom silver clusters. As for the tetra-Dh depicted in Fig. 5, it has a hexahedral (two twinned fcc tetrahedra) center on which many incomplete fcc tetrahedra cover, and every five fcc tetrahedra share a FSA (shown in Fig. 5a and b). It looks like but not two antipodal Leary tetrahedrons, because of the three additional local Ihs. However, the fcc tetrahedra exists in most of noncrystalline nanoparticles obtained by our simulations, i.e., for silver the fcc tetrahedron is the primary structural unit; this is in agreement with the earlier simulation result that the tetrahedral structure was lowest in energy for silver [11]. The diversity of nanoparticle structures is ascribed to the formation of metastable structures and growth under the conditions dominated by kinetic factors [21,26]. In our simulations, the cooling rate of 100 K/ns is much faster than that in experiment; at such a cooling rate, nanodroplets are difficult to reach the minimum free-energy state, thus their structures should be any trapped one from some metastable configurations, especially when the nanodroplets are further cooled down after their solidification. However, for the nanoparticles with high symmetrical FSA-skeleton as above mentioned, we think they should be stable enough to be observed in laboratory. In fact, the bimetallic nanoparticles similar to the Umb-Dh have been found in experiment [22], and both the Umb-Dh (Fig. 6) and the fractal-Ih (Fig. 7) have no any disassembly or deformation even heated to 1573 K at heating rate 100 K/ns. To determine whether they are the global minimum, more studies should be performed. Perhaps just because of the isomerous surfaces, the canonical decahedra are difficult to be obtained (the Umb-Dh in Fig. 6 can be regarded as a surface-isomers of a canonical m-Dh). Many freezing nanoparticles have the characteristics of tDh and Ih. However, the total proportion of the crystal-like (including those in which there are a very short peripheral FSA) is close to 30%. The icosahedrallike (containing at least one local icosahedron) is close to 35%. And if take all the Umb-Dh nanoparticles as Dh-nanoparticles, the proportion of crystal-, Ih-, and Dh-nanoparticles is in good agreement with the experiment results performed by Reinhard et al. [26].
Analysis indicates that the CTIM-2 combining 3D views is a good method for discomposing the microstructure of nanoparticles; and that the inner structures of all freezing nanoparticles are different from each other. While the statistic distribution about the crystal-, Ih-, and Dh-like ones is consistent with previous simulations and experiments. The freezing non-crystalline nanoparticles are made up of several (complete and incomplete) fcc tetrahedra, and two neighboring fcc domains sharing a hcp plane. Surface-isomers usually can be obtained. That is, the ABC . . . repeat sequence of (1 1 1) facets usually change into ABCBA in outer layers. This leads to many local FSAs in the third or penultimate layer formed at the edges shared by two neighboring fcc domains. Several unfamiliar structures have been found which not only have well global three/five-fold symmetry but also have lower or lowest energy than others with same size, and our primary calculation reveals that they should be stable enough to be observed in experiment. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 50831003, 50571037, 50771044), and Project supported by Hunan Provincial Natural Science Foundation of China (Grant Nos. 05JJ30086, 06JJ3003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
4. Conclusion In summary, series of MD simulations at an identical cooling rate 100 K/ns have been performed for free silver nanodroplets in the size range of 2.5–10 nm containing non-magic number of atoms. A first order phase transition and a continuous phase transition usually take place in turn in a solidification process. With the size increasing, the T is is increasing, while the atomic average energy in final nanoparticles at 273 K is decreasing.
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Physics Letters A www.elsevier.com/locate/pla
Freezing structures of free silver nanodroplets: A molecular dynamics simulation study Ze-An Tian a,b , Rang-Su Liu a,b,∗ , Ping Peng a,b , Zhao-Yang Hou a,b , Hai-Rong Liu a,b , Cai-Xing Zheng a,b , Ke-Jun Dong c , Ai-Bing Yu c a b c
School of Physics and Microelectronic Science, Hunan University, Changsha 410082, China School of Materials Science and Engineering, Hunan University, Changsha 410082, China Centre for Simulation and Modelling of Particulate Systems and School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW, 2052, Australia
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Article history: Received 10 February 2009 Accepted 21 February 2009 Available online 25 February 2009 Communicated by V.M. Agranovich PACS: 61.20.Ja 61.25.Mv 61.46.Df 64.70.Dv 83.10.Tv Keywords: Freezing structures Free sliver nanodroplet Non-magic number Rapid cooling Molecular dynamics simulation
a b s t r a c t Freezing structures of free silver nanodroplets containing different numbers of atoms have been studied by using MD simulation and adopting quantum Sutton–Chen (QSC) potential. It is demonstrated that during most of the solidification processes there are a first order and a continuous phase transitions. By means of the cluster-type index method (CTIM-2) and three dimension graphic techniques, the internal structures of final nanoparticles have been investigated intensively. Besides regular crystalline, decahedral and icosahedral nanoparticles, some very interesting novelty morphologies have also been found. From geometrical views, these novelty structures can be called surface-isomers, because they can be constructed on the base of regular crystalline, decahedral or icosahedral nanoparticles by adding few layers with specific atomic arrangement. These new morphologies have well global (three- or five-fold) symmetry and lower energy than other structures, with same size and one of them is in agreement with the observation in experiment [Phys. Rev. Lett. 92 (2004) 196102]. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Owing to the absence of the translational symmetry, the exact determination of the internal structures of nanoparticles is a challenging problem. Recently, many new methods have been developed to detect the structures of the nanoparticles such as the electron tomography [1,2] and aberration-corrected scanning transmission electron microscopy [3,4]. After many crystal nanoparticles such as the truncated octahedron (TO), the decahedron (Dh), and icosahedron (Ih) have been observed in experiments [4,5], and very recently, the bi-decahedron (Bi-Dh) [6], double icosahedron [7] and the dodecahedron [8] were also discovered in laboratory. However, a lot of experimental samples [4,9] are still waiting for being identified up to now. Almost all of the methods identifying the atomic arrangement in a specific three-dimensional nanoparticle need to combine the qualitative or quantitative analysis with some model structure procedures and realistic image contrast sim-
ulations. Therefore, adopting simulation method to search more candidates for structure modeling is of importance for determining the internal structure of the unidentified samples in experiments. Recently, for free gold nanoparticles, not only the Bi-Dh has been verified but also a more general family of poly-decahedral (pDh) nanoparticles has been found by molecular dynamics (MD) simulation [10]. In this Letter, the freezing structures of free silver nanodroplets have been studied by MD simulation adopting QSC many-body potential [11,12], and the internal microstructures of nanoparticles at final temperature of 273 K have been examined through a new structure analysis method carefully. Highly interesting is that not only the above-mentioned morphologies such as Dh, Bi-Dh, Ih, and pDh have been obtained, but also some new structures found in the simulations. These new nanoparticles contain three to few decades of fivefold symmetry axes (FSAs) and at least one fcc (tetrahedral) domains enclosed by twinned hcp planes. 2. Simulation and analysis methods
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Corresponding author at: School of Physics and Microelectronic Science, Hunan University, Changsha 410082, China. Tel.: +86 731 8822817; fax: +86 731 8822817. E-mail address: [email protected] (R.-S. Liu). 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.02.041
To investigate the freezing behavior of the free silver nanodroplet, the simulation conditions and methods are set as fol-
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Fig. 1. (Color online.) 3D views of the typical Ih, tDh, bcc, hcp and fcc basic clusters (for clarity, all bonds between the centre atom and near neighbors have been omitted).
lows: First of all, put a certain number of atoms into a cubic box, the interaction potential between atoms are calculated with the QSC potential [11,12], which is cut off at 22 a.u. The equation of motion is integrated using the leapfrog algorithm with the time step 2.0 fs. Let the system run 50 000 time steps (100 ps) at the initial temperature (T init ) to obtain an equilibrium liquid state which determined by monitoring the energy change of the system. Without period boundary condition and external pressure (namely P ext = 0), the initial cubic nanodroplet transferred into a sphericallike shape very soon. Thereafter, the system is cooled continuously by coupling to a Gaussian thermostat [13,14] from T init = 1273 K to the final temperature (T f = 273 K) at a cooling rate of 100 K/ns. The numbers of atoms in the simulated nanodroplets are designed as 500, 1000, 1500, 2000, 2500, 5000 and 10 000, respectively. It should be pointed out that these numbers of atoms are far from any magic number of well-known polyhedrons such as Ih, Dh and TO. Many simulations [10,15] and experiments [16] have indicated that lots of different isomers can be resulted from identical thermal processes because of the kinetic effects. Thus a number of cooling processes have been performed for each number of atoms at the identical simulation conditions (namely, for the same number of atoms, same cooling rate, same zero external pressure, and same initial thermal state). The structures of the resulting nanoparticles are analyzed by the cluster-type index method (CTIM) [17,18], in which we have defined the basic cluster as the structure composed of a core atom and its surrounding neighbor atoms. Most basic clusters can be expressed by the CTIM-2 [17] which is extended from the CTIM. In CTIM-2, a set of six integers represents a basic cluster. The first integer equals the total number of near-neighbors of the center atom, and the following five integers represent the numbers of the 1441, 1551, 1661, 1421 and 1422 bond-types respectively, expressed by using Honeycutt and Anderson index method [19], by these bond-types the surrounding atoms are connected with the center one of the basic cluster. Five typical (Ih, tDh, bcc, hcp and fcc) basic clusters are shown in Fig. 1. We can classify all the atoms (except for those on the surface) in a nanoparticle according to the type of the basic cluster they located in. Using this classifying technique, the red atoms in each basic cluster in Fig. 1 are called Ih-, tDh-, bcc-, hcp-, fcc-atom respectively. For convenience of discussion, here we define the atom located at the fivefold vertex (FV) of a 1551 bond-type as a FV-atom; and define the plane/facet in which the atoms are all hcp or fcc atoms as a hcp or fcc plane/facet. In fact, from Fig. 1, it can be clearly seen that a smallest basic cluster of Ih marked as (12 0 12 0 0 0) is composed of an Ih-atom at the center and 12 FV-atoms on the surface. A basic cluster marked as (12 0 2 0 0 10) is a smallest truncated decahedron (tDh), in which the central atom (tDh-atom) and two FV-atoms constitute a single FSA. Using CTIM-2, as shown in Fig. 2, the atoms in a larger perfect Dh/Ih, can be classified into four types as: the surface atoms (on the outmost surface, not displayed in Fig. 2), tDh- and FVatoms (white balls on the middle and the end of single/twelve FSAs respectively), hcp-atoms (dark blue balls on five/thirty twinned planes), and fcc-atoms (light blue balls in five/twenty cells separated by hcp-planes). Therefore, by looking for the Ih-, tDh-, and
Fig. 2. (Color online.) Results of CTIM-2 applied to a perfect decahedral cluster of 309 atoms (a), and to a perfect 309-atom icosahedron (b). Both in (a) and (b), all atoms as the center of a tDh-cluster with a (12 0 2 0 0 10) signature are shown as white balls. These tDh-atoms constitute the single FSA of the decahedron and the twelve FSAs of the icosahedron (shown in the second column). All hcp-atoms are shown (in the third column) as dark blue balls and all fcc-atoms are shown (in the fourth column) as light blue balls.
FV-atoms in a particular nanoparticle, we can single out all atoms on FSAs, to investigate the fivefold symmetry. And with the help of the graphic technique, the distribution of all kinds of atoms (such as fcc-, hcp-, Ih, Dh-atoms), and global characteristic can also be found and displayed in 3D views. 3. Results and discussion 3.1. General trends Several general trends can be obtained from all simulations, although the structural evolutions and the final structures of all nanodroplets at 273 K are different from each other. The solidification characteristics of amorphous and crystalline can be found in the energy curve for most nanodroplets as shown in Fig. 3. From the point A to B, the nanodroplet is in liquid state. A typical first order phase transition (generally corresponding to crystallization) takes place between the temperature B and C, and the following evolvement of the energy curve is a continuous phase transition closely similar to the amorphous-solidification. This indicates that the final structure of the nanoparticle would exhibit both the crystalline and the non-crystalline characteristics. Similar to the experimental results performed by Turnbull and Cech [20] for small metal droplets, from the statistical viewpoint, the solidification temperature T is (that is defined as the temperature at which the slope-break of energy curve takes place, as shown in Fig. 3) is increased with the system size increasing. The averages of T is are in turn as 668, 810, 831, 860, 862, 871, and 880 K corresponding to the system sizes of 500, 1000, 1500, 2000, 2500, 5000 and 10 000 atoms, respectively. However, it is significant that the solidification behavior of particular nanodroplets often deviated far from their average values. For instance, for all simulations containing 2000 atoms, the range of T is is from 795 K to 889 K; the interval is close to 100 K.
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At the final temperature 273 K and without relaxation, the average of the total energy per atom is lowered with the increase of system size. The average energies are −259.8, −265.8, −268.3, −269.4, −270.6, −273.1, and −275.5 kJ/mol at sizes of 500, 1000, 1500, 2000, 2500, 5000 and 10 000 atoms, respectively. Using the equation [21] E = a + bN 1/3 + cN 2/3 + dN 1 to fit the atom energy E vs. nanoparticle size N, one can get a = −288.99 kJ/mol. That is to say, for bulk system, the energy for each atom is −288.99 kJ/mol. This is in close agreement with the simulation result (−2.923 eV per atom i.e. −282.01 kJ/mol) for bulk silver system (see Fig. 3(a) of Ref. [17]) under the cooling rate of 138 K/ns. 3.2. Novelty morphologies Thereafter the nanoparticles discussed in the text are all at the final temperature 273 K and without any relaxation. Freezing nanoparticles adopt many different outlines such as sphere, ellipsoid, peanut, rod and even worm, and their morphology is various. However, all the final nanoparticles resulted from our freezing simulations are always well-defined. In other words, some defects maybe present on the surface but they are not amorphous. There are always one or more FSAs, each of them surrounded by a local (truncated) decahedral arrangement, in all non-crystalline nanoparticles. For the conveniences of description, we further define the length of a FSA is the number of atoms on it. Thus the nanoparticles can be classified according to the number and arrangement of the FSAs in them. In this Letter, several nanoparticles with special symmetry will be reported as follows. Tri-Dh is here defined as such a kind of nanoparticles in which there are at least one three-fold vertex of FSAs and no more than three FSAs meeting together. Fig. 4 shows a simplest tri-Dh containing 1000 atoms just holds three FSAs which do not share one atom, but intersect at a common vertex (see Fig. 4a and d). In other words, any couple of FSAs is co-planar and between them
Fig. 3. A typical energy curve for a solidification process of a nanodroplet. From A to B, the nanodroplet is in liquid state; between B and C, there is a first order phase transition; and from C through D to E, a continuous phase transition takes place. The temperature at B is defined as T is .
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three hcp facets formed; furthermore the angle between any two FSAs is just near 60◦ (the numerical result is 59.969 ± 0.028◦ ). Therefore, a rather canonical tetrahedron region with an unclosed facet is formed between the three hcp planes; and the tetrahedron region is packed by fcc atoms. Figs. 4b–d display the rather perfect global three-fold symmetry; and the TO-like outline can be found from Fig. 4c and d. Furthermore, it holds the lowest energy in all same size ones we studied. Tetra-Dh is another kind of nanoparticles with at least one tetrahedral skeleton constituted of equal-length-FSAs. Fig. 5 displays a tetra-Dh of 1000 atoms and its hexahedral skeletons composed of (two antipodal canonical tetrahedral skeletons) nine FSAs with the length as seven atoms. Among the five vertexes of hexahedral skeletons, there are three local Ih sites (marked by A, B, C in Fig. 5a) and two three-fold vertexes of FSAs (marked by D and E in Fig. 5a). There are several decades twinned hcp-planes across two or three co-planar FSAs (totally 33 FSAs in Fig. 5b). Between these twinned hcp-planes, there are many tetrahedral domains that are filled by fcc atoms; and there are one or two opened facets for most of these tetrahedral domains, except for two perfect ones outlined by two FSA-skeletons ABCD and ABCE (see Fig. 5a). The global three-fold symmetry and local five-fold symmetry of the nanoparticle can be seen in Fig. 5c and d, respectively. An Umb-Dh is a nanoparticle in which the skeleton composed of FSAs is like an umbrella. In an Umb-Dh, no Ih-atoms can be found; but a main FSA and some sub-FSAs are necessary; and all FSAs join together, usually at the end of the main FSA. Fig. 6 displays a perfect Umb-Dh containing 2000 atoms which holds an atom where all six FSAs meet together. The length of the main FSA (umbrella handle) is 12 atoms, that of four sub-FSAs is identical as 9 atoms and only one sub-FSA is 8 atoms. Just because of the five extra FSAs, the nanoparticle is not a Marks decahedron (m-Dh) [21], although obvious reentries can be found (indicated by arrows in Fig. 6d). As shown in Fig. 6b, there are five twinned hcpplanes; each contains the main FSA and one sub-FSA. And there are another five hcp-planes; each crosses two neighboring sub-FSAs. Therefore the nanoparticle is divided into five isolated fcc domains (shown in Fig. 6c) by the 10 hcp-planes. Comparing the Figs. 6d and 6e, it is easy to be found that the atom arrangement on the top and the bottom of the nanoparticle is different. On the top surface shown in Fig. 6d, five (1 1 1) facets share five edges (coffee atoms); while on the bottom one in Fig. 6e, five (1 1 1) facets are separated by five narrow (1 0 0) facets (wine atoms). As the illustration in Fig. 6f, similar to the bimetallic decahedral nanoparticles with surface reconstruction reported in Ref. [22], the five narrow (1 0 0) facets and five sub-FSAs (see again Fig. 6a) are both resulted from the specific atom arrangement in the last two layers, which does not follow the fcc sequence (. . . ABCAB), but arrange as (. . . ABCBA). However, it is different from the single layer of surface reconstruction [22] that in our simulation, the result reveals that there exists another atom layer out of the reconstruction layer. Therefore, on the surface the narrow (1 0 0) facets (wine atoms) contain three rows of atoms. The atoms in the penultimate layer
Fig. 4. (Color online.) A tri-Dh containing 1000 atoms with the length of FSAs as 8, 9, 9. (a) The (yellow) tDh- and (green) FV-atoms on three FSAs, and (dark blue) hcp-atoms in twinned hcp planes. (b), (c) 3D views along the same direction as that in (a). (d) A 3D view along the direction invert to that in (a).
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Fig. 5. (Color online.) A tetra-Dh composed of 1000 atoms with two tetrahedral skeletons which share a triangle ABC; Red, yellow, green, and gray balls represent the Ih, tDh, FV, and other type atoms respectively. (a) Atoms on two tetrahedral skeletons. (b) All atoms on all FSAs. (c), (d) 3D views projected along the D → E and A → C axes shown in (a).
are fcc-atoms except for those (dark blue) hcp-ones. And within the third layer (from the surface), the atoms on five facets are (dark blue) hcp-atoms, and those on five edges are (yellow) tDhatoms. A fractal-Ih has been found in all simulated samples. It is well known that in a shell-closed icosahedron [23], twelve FSAs with identical lengths meet at the center, and on the surface, there are 12 FVs, 30 edges and 20 regular-triangle-facets. As shown in Fig. 7, the fractal-Ih containing 1500 atoms holds surprising high symmetry and hierarchical structure. There is NOT ONLY an Ih-core with 13 atoms at the center of it, BUT ALSO a larger and rather perfect Ih-skeleton with the identical orientation as that of the core-Ih (Fig. 7a and d). In addition to the Ih-core and the Ih-skeleton constituted of 42 FSAs, there are still two local icosahedrons with two outmost FV-atoms as their centers (red atoms in Fig. 7). So from the viewpoint of the Ih-skeleton, this nanoparticle possesses a self-similar hierarchy-structure. This is why the nanoparticle is assigned as a fractal-Ih. As shown in Fig. 7b, up to the 6th shell, there is still a perfect icosahedron containing 561 atoms, and atoms on all edges in 2–5 layers are (dark blue) hcp-ones. However, in the 7th shell, two triangle-facets (wine atoms in Fig. 7c) do not follow the fcc (. . . ABCA) sequence but arrange in (. . . ABCB). Refer to the terms Used in Ref. [24], the two facets can be called anti-Mackey surface terminations (AMSTs). At the joint of the two AMSTs appear an extra (1 0 0) facet which composed of 10 atoms lining in two rows (see the center part of Fig. 7c). And other four triangle-facets which neighbor to the two AMSTs have somewhat distortion. Just because of the arrangement change of atoms in 7th shell, the type of some atoms in 6th shell changed: 4 yellow atoms sited in the diagonal of the green diamond in Fig. 7b, no longer are hcp-atoms but tDh-ones; and anther 20 green atoms are FVs. In other words, four local FSAs constituted of the 24 atoms formed in the 6th shell. Likewise, as shown in Fig. 7f, in the 8th shell, another twenty-six (1 0 0) facets formed between 14 AMSTs and 6 other type facets. And in 7th shell, 26 local FSAs formed (see yellow atoms in Fig. 7e). So far 30 local FSAs formed in the 6th and 7th shell at the edges of an icosahedron. Including 12 main FSAs of an icosahedron, an Ih-skeleton constituted of 42 FSAs formed. Because not all 30 outer FSAs are in an identical shell, there are some faults with it. Another incomplete shell covers the eighth facets so that there is one or two layers out of each fcc domain with the sequence of . . . ABCBA. And similar to the case of the Fig. 6f, some (1 0 0) facets including three rows of atoms can be found on the fractal-Ih surface.
Fig. 6. (Color online.) A Umb-Dh containing 2000 atoms. (a) The umbrella skeleton composed of 6 FSAs. (b), (c) the twinned hcp planes and five fcc domains. (d), (e) 3D views respectively along the B → A and A → B directions shown in (a). (f) The illumination for the specific atom arrangement in the outmost two layers, under which there are several FSAs.
Fig. 7. (Color online.) A fractal-Ih contains 1500 atoms. Three views of (a)–(c) are along an main axis of five-fold symmetry; and three views (d)–(f) are along another main axis of five-fold symmetry. (a), (d) different views of the skeleton composed of 42 FSAs. (b), (c) 3D views contains six and seven shells. (e), (f) 3D views contains seven and eight shells. Color configuration: in all views, red, dark blue, gray balls present Ih-, hcp-, and other type atoms; For clarity, green for FV-atoms, yellow only for hcp-atoms in (b); and all wine ball present atoms on the facets of AMST. (c), (e) 3D views which are both including seven shells but along different directions.
3.3. Discussion From the geometrical viewpoint, for a edge which two neighboring (1 1 1) facets share, if the next layers on the two (1 1 1) facets do not follow the ABC . . . repeat sequence but arranged as ABCB, the atoms on the edge will be tDh-atoms and a FSA formed. Another outer layers cover these facets to form . . . ABCBA sequence as just mentioned above for Umb-Dh (Fig. 6) and fractal-Ih (Fig. 7). In other words, from the “C” layer toward two sides, both are fcc sequence (CBA), so the atoms on the “C” layer are hcp-atoms. This change takes place in surface layers, so the Umb-Dh and fractal-Ih can be called as surface-isomers. About 25% nanoparticles resulted from our simulation are surface-isomers, they have at least one peripheral FSA in the 2nd or 3rd layer under surface. And a new surface-isomer with a m-Dh as its center should be obtained if more cooling processes were performed; in this kind of surfaceisomers, all ten edges at two ends in a canonical m-Dh have been transferred into FSA and the skeleton of FSAs can be regarded as two antipodal Umb-Dhs, like a leftrightarrow ↔.
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Further analysis demonstrates that from the hcp planes shown in Fig. 4a to surface, there are three or four outer layers, and the sequence of closed-packed planes are [. . . (ABC)BAC] for three outer layers and [. . . (ABC)BCAB] for four outer layers. Therefore the tri-Dh is also a surface-isomer. The configuration of atoms outside the three hcp-planes is some similar to a Leary tetrahedron [25] which is global minimum for 98-atom Lennard-Jones cluster. Perhaps similar reason leads to the tetra-Dh has the lowest energy among all 1000-atom silver clusters. As for the tetra-Dh depicted in Fig. 5, it has a hexahedral (two twinned fcc tetrahedra) center on which many incomplete fcc tetrahedra cover, and every five fcc tetrahedra share a FSA (shown in Fig. 5a and b). It looks like but not two antipodal Leary tetrahedrons, because of the three additional local Ihs. However, the fcc tetrahedra exists in most of noncrystalline nanoparticles obtained by our simulations, i.e., for silver the fcc tetrahedron is the primary structural unit; this is in agreement with the earlier simulation result that the tetrahedral structure was lowest in energy for silver [11]. The diversity of nanoparticle structures is ascribed to the formation of metastable structures and growth under the conditions dominated by kinetic factors [21,26]. In our simulations, the cooling rate of 100 K/ns is much faster than that in experiment; at such a cooling rate, nanodroplets are difficult to reach the minimum free-energy state, thus their structures should be any trapped one from some metastable configurations, especially when the nanodroplets are further cooled down after their solidification. However, for the nanoparticles with high symmetrical FSA-skeleton as above mentioned, we think they should be stable enough to be observed in laboratory. In fact, the bimetallic nanoparticles similar to the Umb-Dh have been found in experiment [22], and both the Umb-Dh (Fig. 6) and the fractal-Ih (Fig. 7) have no any disassembly or deformation even heated to 1573 K at heating rate 100 K/ns. To determine whether they are the global minimum, more studies should be performed. Perhaps just because of the isomerous surfaces, the canonical decahedra are difficult to be obtained (the Umb-Dh in Fig. 6 can be regarded as a surface-isomers of a canonical m-Dh). Many freezing nanoparticles have the characteristics of tDh and Ih. However, the total proportion of the crystal-like (including those in which there are a very short peripheral FSA) is close to 30%. The icosahedrallike (containing at least one local icosahedron) is close to 35%. And if take all the Umb-Dh nanoparticles as Dh-nanoparticles, the proportion of crystal-, Ih-, and Dh-nanoparticles is in good agreement with the experiment results performed by Reinhard et al. [26].
Analysis indicates that the CTIM-2 combining 3D views is a good method for discomposing the microstructure of nanoparticles; and that the inner structures of all freezing nanoparticles are different from each other. While the statistic distribution about the crystal-, Ih-, and Dh-like ones is consistent with previous simulations and experiments. The freezing non-crystalline nanoparticles are made up of several (complete and incomplete) fcc tetrahedra, and two neighboring fcc domains sharing a hcp plane. Surface-isomers usually can be obtained. That is, the ABC . . . repeat sequence of (1 1 1) facets usually change into ABCBA in outer layers. This leads to many local FSAs in the third or penultimate layer formed at the edges shared by two neighboring fcc domains. Several unfamiliar structures have been found which not only have well global three/five-fold symmetry but also have lower or lowest energy than others with same size, and our primary calculation reveals that they should be stable enough to be observed in experiment. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 50831003, 50571037, 50771044), and Project supported by Hunan Provincial Natural Science Foundation of China (Grant Nos. 05JJ30086, 06JJ3003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
4. Conclusion In summary, series of MD simulations at an identical cooling rate 100 K/ns have been performed for free silver nanodroplets in the size range of 2.5–10 nm containing non-magic number of atoms. A first order phase transition and a continuous phase transition usually take place in turn in a solidification process. With the size increasing, the T is is increasing, while the atomic average energy in final nanoparticles at 273 K is decreasing.
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