Dynamic stabilities of icosahedral-like clusters and their ability to form quasicrystals

Dynamic stabilities of icosahedral like clusters and their ability to form quasicrystals Dynamic stabilities of icosahedral like clusters and their ability to form quasicrystals Xiaogang Liang, Ilyar[.]

Dynamic stabilities of icosahedral-like clusters and their ability to form quasicrystals Xiaogang Liang, Ilyar Hamid, and Haiming Duan Citation: AIP Advances 6, 065017 (2016); doi: 10.1063/1.4954741 View online: http://dx.doi.org/10.1063/1.4954741 View Table of Contents: http://aip.scitation.org/toc/adv/6/6 Published by the American Institute of Physics AIP ADVANCES 6, 065017 (2016) Dynamic stabilities of icosahedral-like clusters and their ability to form quasicrystals Xiaogang Liang, Ilyar Hamid, and Haiming Duana College of Physics Science and Technology Xinjiang University, Urumqi 830046, People’s Republic of China (Received 15 March 2016; accepted 13 June 2016; published online 20 June 2016) The dynamic stabilities of the icosahedral-like clusters containing up to 2200 atoms are investigated for 15 metal elements The clusters originate from five different initial structures (icosahedron, truncated decahedron, octahedron, closed-shell fragment of an HCP structure, and non-closed-shell fragment of an HCP structure) The obtained order of the dynamic stabilities of the icosahedral-like clusters can be assigned to three groups, from stronger to weaker, according to the size ranges involved: (Zr, Al, Ti) > (Cu, Fe, Co, Ni, Mg, Ag) > (Pb, Au, Pd, Pt, Rh, Ir), which correspond to the predicted formation ability of the quasicrystals Thedifferences of the sequences can be explained by analyzing the parameters of the Gupta-type manybody inter-atomic potentials C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4954741] I INTRODUCTION Since their discovery,1 quasicrystals have attracted great interest in many fields, such as material science, industry, and technology.2 Quasicrystals have already been shown to be important in hydrogen storage and as an aeronautical alloy.3,4 The quasicrystal formed in nature is mainly composed of Al-Cu-Fe5 (although the Al-Ni-Fe quasicrystal has also recently beendiscovered6), and other quasicrystals have been synthesized.2 Basically, the Al-based,1,2,5–10 Zr-based,11–15 and Ti-based16–18 quasicrystals are of fundamental concern, although the reason why these are indispensable components in such quasicrystals still remains unclear Hence, the origin of the preferential formation ability of the Al-,Zr-, and Ti-based quasicrystals requires further investigation As is well known, a five-fold symmetry in crystals is not possible; however, at the microscopic level, clusters containing several hundred or fewer atoms prefer adopting icosahedral geometries with five-fold symmetry.19,20 Many studies have investigated the local atomic structures and the corresponding dynamic properties when determining the structural properties of the quasicrystals owing to the lack of translational periodicity.21 Although the naturally or synthesized quasicrystals are all composed of several components2,22,23 and no single-component quasicrystal has been found to date, to avoid the high complexity in the multi-component quasicrystals in theoretical investigations, many studiesinvestigate the structural and dynamic properties of quasicrystals using the one-component model24,25 (an important reason for such achoice arises from there being no proper force fields between the multi-component interactions) For example, Engel et al used a tunable pair potential to investigate the structures and dynamics of a one-component icosahedral quasicrystal.25 Although many investigations have been performed either on the melting behavior or on the structural variations for several metal clusters, there is still a lack of studies for a wide range of sizes and elements In this work, we use molecular dynamics simulations to study the dynamicstabilities of icosahedral-like clusters of 15 Mn (M = Mg, Al, Ti, Fe, Co, Ni, Cu, Zr, Rh, Pd, a Corresponding author: Tel: +86-991-8582404; Fax: +86-991-8582405 E-mail: dhm@xju.edu.cn 2158-3226/2016/6(6)/065017/9 6, 065017-1 © Author(s) 2016 065017-2 Liang, Hamid, and Duan AIP Advances 6, 065017 (2016) Ag, Ir, Pt, Au, and Pb; n = 13 - 2157) clusters.The differences of the size ranges involved in the different icosahedral-like dynamic stabilities are emphasized and the different formation abilities of the quasicrystals are analyzed according to the local structure and dynamics in the quasicrystals II COMPUTATIONAL DETAILS A Initial structures Several different initial structures are considered, as shown in Figs and Fig shows three different closed-shell structures of a cluster with the same number of atoms (cluster size): the icosahedron (Ih) (Fig 1(a)), the truncated decahedron (TD) (Fig 1(b)), and the octahedron (Oh) (Fig 1(c)) The size sequences of all the three geometries can be written as:26 N(k) = 11 10 k − 5k + k −1 3 (1) where k denotes the number of shells in the cluster The clustersizes of the three geometries involved in this study are up to 2100 atoms (13, 55, 147, 309, 561, 923, 1415, and 2057) For the geometries shown in Fig 1, the icosahedron and the truncated decahedron are unable to exist in the crystal form (at the macroscopic level) and the octahedron can exist as a crystal (it can be assigned as a fragment of the face-centered cubic (FCC) crystal structure) In addition to the octahedron, another common crystal structure is hexagonal closed-packed (HCP), and the closed-shell fragment of the HCP structure (called the CHCP structure in this study) is shown in Fig 2(a) The cluster sizes of the CHCP structures are different from those with the geometries listed in Fig 1, and the cluster sizes involvedinthe CHCP structures are up to 2200 atoms (13, 57, 153, 323, 587, 967, 1483, and 2157) To further investigate the dynamic stabilities of the non-closed geometries, the unclosed-shell structures of the HCP geometries with the same sizes as the Ih structures are also considered (termed as the UHCP structures, shown in Fig 2(b)) The initial structures considered above are by no means to represent the lowest-energy structures of the clusters For example, for the icosahedral-like clusters, the structures with reconstructed surfaces27,28 or with central vacancies29,30 may be lower in energy than the perfect icosahedral ones B Inter-atomic potential In this study, the cohesive energies of the clusters are calculated from the Gupta-type manybody potential The Gupta potential is a semi-empirical potential based on the second-moment approximation of the tight-binding model It has been widely applied in studying the geometrical structures31,32 and dynamic behavior33 of metal and alloy clusters The formula of the Gupta FIG Three different closed-shell structures of a cluster with the same number of atoms (cluster size) From the left to the right: the icosahedron (Ih) (a), the truncated decahedron (TD) (b) and the octahedron (Oh) (c) 065017-3 Liang, Hamid, and Duan AIP Advances 6, 065017 (2016) FIG The closed-shell (CHCP) (a) and non-closed-shell (NHCP) (b) structures of the hexagonal closed-packed geometries The latter has the same number of atoms as that of the closed-shell structure shown in Fig TABLE I The Gupta-potential parameters p and q and the product p*q of 15 metal elements, where N denotes the atomic number of the element N Element p q p*q Reference 77 78 79 46 82 45 47 12 26 27 29 13 22 28 40 Ir Pt Au Pd Pb Rh Ag Mg Fe Co Cu Al Ti Ni Zr 16.98 10.612 10.229 10.867 9.576 18.45 10.928 12.82 10.50 11.604 10.96 8.612 8.62 16.999 8.25 2.691 4.004 4.036 3.742 3.648 1.867 3.139 2.257 2.60 2.286 2.278 2.516 2.39 1.189 2.249 45.693 42.490 41.284 40.664 34.933 34.446 34.303 28.935 27.300 26.527 24.967 21.668 20.602 20.212 18.554 31 31 31 31 31 31 31 31 32 31 31 31 31 31 31 potential can be written as:34    ri j ri j B2 exp[−2q( − 1)]) V= ( A exp[−p( − 1)] − r0 r0 i j(,i) j(,i) (2) where r i j is the distance between two atoms i and j, and A, B, p and q are the parameters related to the types of atoms involved The Gupta-potential parameters used in this paper are taken from the investigations by Cleri et al.35 and Stanek et al.36 Table I provides the Gupta-potential parameters of the 15 metal elements involved in the present work The order (from top to bottom) is characterized by the product of two Gupta-potential parameters (p*q) and will be discussed later C Molecular dynamics simulation The canonical molecular dynamics (MD) simulation method is used to investigate the dynamic properties of different metal clusters originating from different initial geometries (as shown in Figs and 2) The temperature is controlled by the Berendsen thermostat37 and the Newtonian equation is integrated with a time step of fs by applying the velocity Verlet algorithm; 1×106 MD steps are propagated ateach temperature point The temperature is controlled from 100 K to a temperature higher than the bulk melting temperature of each metalcluster, and the temperature 065017-4 Liang, Hamid, and Duan AIP Advances 6, 065017 (2016) interval (∆T) is chosen as 20K Generally, several different MD simulations are performed for each cluster so as to obtain the collections within statistical convergence D Physical qualities The energies and the heat capacities of the clusters are calculatedat each temperature The heat capacity of a cluster containing N atoms can be calculated as: Cv = < Et2 >T − < Et >T2 2N KbT (3) whereKb is the Boltzmann constant and Et is the total energy of the cluster T is the temperature and T denotes the ensemble average III RESULTS AND DISCUSSION For each cluster within two specific sizes (13, 55, 147, 309, 561, 923 1415, and 2057 for the initial Ih, TD, Oh, and NHCP geometries and 13, 57, 153, 323, 587, 967, 1483, and 2157 for the initial CHCP geometries) of all elements investigated, the MD simulation is performed from a relatively low temperature (to ensure the cluster is in a solid-like state) to a relatively high temperature (to ensure the cluster is in a liquid-like state) Fig shows the typical features (the energy and heat capacity curves) of the melting properties of a metal cluster originating from each of the geometries listed in Figs and The melting curves of the clusters originating from each of the Ih geometries over the entire size range (13 - 2057) can be obtained,as shown in Fig 3(b) There is a sharp increase in the energy curve at the melting point, which corresponds to the sharp peak in the heat capacity curve However, if the cluster originates from any of the other geometries (not Ih), the situation is quite different with the melting behavior being strongly dependent on the cluster size (size range) and element For each of the clusterswith smaller sizes (for example, all 13- and 55-atom clusters), a structural transition from the initial-like structure (non-Ih-like) to the Ih-like structure occurs before melting for all elements This indicates the inherent nature of the high dynamic stabilities of the icosahedral clusters at small sizes (for some elements, the initial non-Ih structures are unable to be dynamically stable at the initial temperature of 100K) For larger sizes (more than 100 atoms), the structural variations strongly correlate with the metal elements Different structural transition areas emerge for the different elements Fig shows the energy and heat capacity curves of Ti923 and Ti1415 clusters originating from an initial Oh geometry Fig 3(a) showsthe two peaks in the heat capacity curve of Ti923 The first peak at approximately 620K corresponds to a structural transition from the Oh-like to the Ih-like (solid-to-solid FIG Energy and heat capacity curves of the clusters Ti923 (a) and Ti1415 (b) originating from the initial TD structures 065017-5 Liang, Hamid, and Duan AIP Advances 6, 065017 (2016) transition) geometry (It should be noticed that this peak should be in negative physically, since the heatcapacity can be viewed as the derivative of energy to temperature, and the transition from Oh-like to Ih-like corresponds to a decrease in energy In our calculations, theheat capacity is always positive from the formula (3) in this text This discrepancy is reflecting an artifact due to the non-optimal initial configuration.), and for a further increase of the temperature, the Ih-like structure dominates the dynamic stability until the cluster melts at approximately 1040K (the melting point, which corresponds to the second peak in the heat capacity curve) Hence, the dynamic stability of the Ih-like structure becomes dominant before melting There is only one peak corresponding to the melting point of the Ti1415 cluster originating from the initial TD structure, as shown in Fig 3(b), which means that for a larger size, the dynamic stability of the initial TD structure is sustained over the whole temperature range before melting With an increasing cluster size, the dynamic stabilities of the clusters become very different for the different elements For each element, with increasing the cluster size to a certain number, the dynamic stabilityoriginating from each of the non-Ih geometries can be sustained over the whole temperature range before melting This critical size, accompanied by its previous neighbor size (the size where the structural transition occurs) can be termed as the transition size range For example, the transition size range of the Ti clusters, as discussed above, is from 923 to 1415 if originating from the initial TD structures The transition size range also strongly depends on the elements For example, if originating from the initial TD structures, the transition size ranges for Ir, Pt, Au, Pd, Pb, and Rh are all from 147 to 309; those for Fe, Co, and Cu are all from 309 to 561; but larger values can be obtained for Ti and Zr from 923 to 1415 Therefore, the dynamic stabilities of the Ih-like structures dominate the much larger cluster size ranges for Ti and Zr compared with the other elements, like Ir, Pt, and Au As stated above, to obtain the simulation results within statistical convergence, several different MD simulations were performed for each cluster originating from the different initial geometries at different cluster sizes of each element An interesting phenomenon was observed for some elements where a statistical randomness of the dynamic stability of the Ih-like structure occurs at a certain size At this size, either the dynamic stability of the Ih-like structure occurs (solid-to-solid structural transition) before melting or the dynamic stability of the initial non-Ih structure remains over the whole temperature range up to the melting point In sucha case, the cluster size can be termed as the transition size of the dynamic stability of the Ih-like geometry, otherwise one obtains a transition size range (such as 923 and 1415 for Ti clusters originating from theinitial TD structures, as shown in Fig 3) Fig shows two typical melting curves of an Al923 cluster from five different MD simulations originating from the same initial TD structure Three MD simulations provided similar results as those shown in Fig 4(a), where a solid-to-solid structural transition occurs before melting (the only difference is that the solid-to-solid transition temperature varies for different MD processes), and in the other two MD simulations The non-Ih (TD) structure can sustain its dynamic stability FIG Energy curves in the melting process of Al923 at two different MD simulations originating from the same initial TD structure 065017-6 Liang, Hamid, and Duan AIP Advances 6, 065017 (2016) FIG Transition sizes (size ranges) of the stabilities of the Ih-like clusters of 15 metal elements originating from the initial TD ((a)) and Oh ((b)) geometries over the whole temperature range before melting (as shown in Fig 4(b)) Such phenomena, with statistical randomness accompanying the structural changes, also exist for several other metal clusters at different sizes and with different initial structures These sizes are considered as crossover sizes between the Ih-dominated size areas and the non-Ih-dominated size areas It should be noticed that the statistical randomness in transitions originates to a good extent from the finite time of the simulations, which may be insufficient to cause the transition to free-energy minima As stated above, for the clusters of different elements originating from different geometries, the crossover sizes (or size ranges) of the clusters are quite different Fig 5(a) and 5(b) show the crossover sizes (size ranges) of 15 metal clusters originating from the initial TD and Oh structures, respectively The order of the different metal elements is determined according to the value of the product of the two Gupta potential parameters p*q (with decreasing values of p*q from left to right) Fig shows that the crossover sizes (size ranges) of the different metal elements are substantially different and are strongly correlated with the p*q values For some elements, such as Ir, Pt, Au, Pd, Pb, and Rh, the crossover sizes (size ranges) are very small (about 100 - 350 atoms), but for some other elements, such as Al, Ti, and Zr, the crossover sizes (size ranges) are quite large (about 900 - 2100 atoms), and for the other metal elements (Ag, Mg, Fe, Co, Cu, and Ni), the crossover sizes (size ranges) are intermediate The cluster sizes (size ranges) of the dynamic stabilities of the Ih-like clusters of different metal elements originating from the initial closed-shell TD and Oh structures are discussed above, where the cluster sizes (13, 55, 147, 309, 561, 923 1415, and 2057) are the same as those of the 065017-7 Liang, Hamid, and Duan AIP Advances 6, 065017 (2016) FIG Transition sizes (size ranges) of the stabilities of the Ih-like clusters of 15 metal elements originating from the initial NHCP ((a)) and CHCP ((b)) geometries closed-shell Ih structures One may ask if the cluster originates from other closed-shell structures with sizes different from those of the Ih structures, whether the same results (the strong correlation between the transition sizes or size ranges of the dynamic stabilities of the Ih-like geometries and the values of p*q of different metal elements, as listed in Fig 5) can also be obtained The transition sizes (size ranges) of the dynamic stabilities of the Ih-like geometries originating from the closed-shell hexagonal closed-packed structures (the CHCP structures shown in Fig 2(a)) are provided in Fig 6(b) For comparison, the transition sizes (size ranges) of the dynamic stabilities of the Ih-like geometries originating from the non-closed-shell hexagonal closed-packed structures (the NHCP structures which have the same cluster sizes as those of the closed-shell Ih clusters, as shown in Fig 2(b)) are provided in Fig 6(a) In general, Fig shows that one can also get the similar results as that from Fig The transition sizes (size ranges) of the dynamic stabilities of the Ih-like clusters are substantially different for the different elements and also show strong correlations with the p*q values For example, for some metal elements with large p*q values,such as Pt and Au, the crossover size ranges are quite small, and for other metal elements with small p*q values, such as Al, Ti, and Zr, the crossover sizes (size ranges) are quite large As we know, an important feature is the local icosahedral atomic assembly with five-fold symmetry in the quasicrystal The formation ability of the local icosahedral assembly can be strongly correlated with its dynamic stability The local atomic structures and their dynamics of the local icosahedral environmentsin quasicrystals can be described by using the cluster model From Figs and 6, one may deduce that the dynamic stabilities of the icosahedral-like geometries of the 15 065017-8 Liang, Hamid, and Duan AIP Advances 6, 065017 (2016) metal elements can be assignedinto three different groups according to the cluster’s transition sizes (size ranges) involved A larger the transition size (size ranges) results in a higher dynamic stability of the Ih-like structures As a result, the order of the dynamic stabilities of the Ih-like structures of the 15 metal elements can be shown as (Zr, Al, Ti) > (Cu, Fe, Co, Ni Mg, Ag) > (Pb, Au, Pd, Pt, Rh, Ir), and such an ordercorresponds to the predicted formation ability of the quasicrystals of the different metal elements From this observation, one may conclude that the formation abilities of the quasicrystals of some metal elements, such as Zr and Al, are much stronger than those of some other elements, such as Au and Pd, which is consistent with the available experimental observations of quasicrystals The Al-based and Zr-based quasicrystals are foundational amongst the available quasicrystals The reason for the higher formation abilities of Al- or Zr-based quasicrystals can be attributed to the stronger dynamic stabilities of the local Ih-like surroundings, as shown from the above discussions even though the single-component model was used As stated above, for the 15 metal elements, according to the decreasing ordering of the product of the two interaction parameters p and q, the obtained results from Ir to Zr, shown in Figs and 6, generally correspond to the increasing formation abilities of quasicrystals (with the exception of Ni) The first comprehensive in-depth analysis of the parameters (p and q) and their products was performed by Ferrando et al in their investigation of the energetics and stabilities of several statically stable geometries of various metal elements (Cu, Ag, Au, Pd, and Pt) They showed that a smaller p*q value corresponds to a less sticky interaction.26 We have also previously applied such a criterion to study the dynamic stabilities of the octahedral structures and the melting behaviors of different metal clusters.38 As shown in this study, this criterion can also be used for describing the different inter-atomic interactions among clusters of different metal elements IV CONCLUSIONS Applying the molecular dynamics simulation method, based on the Gupta-type many-body inter-atomic potential, the structural transitions are investigated in the melting processes of metal clusters containing up to 2000 atoms of 15 differentmetal elements, and the dynamic stabilities of the icosahedral-like clusters are emphasized Five different structures, the Ih, the TD, the Oh, the CHCP, and the NHCP, are considered as initial structures Strong correlations are observed between the transition sizes (size ranges) of the dynamic stabilities of the Ih-like structures and the metal elements, regardless of any differencein originating structures The origin of the different crossover sizes (size ranges) of different metals can be inferred by analyzing the products of the interaction parameters p and q We propose thatthe formation abilities of the quasicrystals involved can be classified into three groups, as 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