Article pubs.acs.org/JPCA
Tetrahelix Conformations and Transformation Pathways in Pt1Pd12 Clusters Rafael Pacheco-Contreras,† Maribel Dessens-Félix,‡ Dora J. Borbón-González,§ L. Oliver Paz-Borbón,∥ Roy L. Johnston,*,⊥ J. Christian Schön,*,¶ and Alvaro Posada-Amarillas*,† †
Departamento de Investigación en Física, Universidad de Sonora, Apdo. Postal 5-088, 83190 Hermosillo, Sonora, México Programa de Doctorado en Ciencias de Materiales, Universidad de Sonora, 83000 Hermosillo, Sonora, México § Departamento de Matemáticas, Universidad de Sonora, 83000 Hermosillo, Sonora, México ∥ Department of Applied Physics and Competence Centre for Catalysis, Chalmers University of Technology, SE-412 96 Göteborg, Sweden ⊥ School of Chemistry, University of Birmingham, Edgbaston, B15 2TT Birmingham, U.K. ¶ Max-Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany ‡
ABSTRACT: The threshold method is used to explore the potential energy surface of the Pt1Pd12 bimetallic cluster, defined by the Gupta semiempirical potential. A set of helical structures, which follow a Bernal tetrahelix pattern, correspond to local minima for the Pt1Pd12 cluster, characterizing the region of the energy landscape where these structures are present. Both right-handed and left-handed chiral forms were discovered in our searches. Energetic and structural details of each of the tetrahelices are reported as well as the corresponding transition probabilities between these structures and with respect to the icosahedron-shaped global minimum structure via a disconnectivity graph analysis.
1. INTRODUCTION The regular tetrahedron is the basic structural element from which it is possible to build more complex structures from interconnected (twinned) groups of tetrahedra, even though Euclidian space cannot be tiled by this Platonic solid.1 A large number of examples are obtained by means of geometrical reasoning, however. In structural studies of inorganic nanoparticles, archetypes of these twinned structures are the icosahedron or the pentagonal bipyramid, which are found even for a small number of atoms in clusters of atoms. Likewise, the Bernal spiral or tetrahelix2−4 is another type of structure that can be described as an assembly of regular twinned tetrahedra. Right- or left-handed spirals with an irrational twist are formed by chains of face-sharing tetrahedra.5 Studies of a variety of systems have shown that the tetrahelix is found as a linear chain that is part of the structural diversity in complex materials.6−13 Also, a very recent experimental study on Au−Ag nanowires14 has demonstrated the formation of the tetrahelix. In that work, a structural model was built in order to explain the different diffraction patterns and the exceptional symmetry of synthesized Au−Ag nanowires. However, a correlation between the model and the nanowire’s energy landscape was not presented in that study. The energy landscape is a key concept to understand the properties of a large number of physical systems, such as atomic clusters, proteins, colloids, liquids, and both glassy and crystalline solids, and to resolve many of the most fascinating problems in chemical physics.15,16 All physical and chemical © 2012 American Chemical Society
phenomena exhibited by atomic clusters are determined by the potential energy surface (PES). In turn, the PES is a mathematical model of the interatomic bonding in such a way that it is a uniquely defined function once we have prescribed the chemical composition and size of the atomic cluster to be studied. Analyzing the PES requires a tremendous computational effort and efficient mathematical optimization algorithms in order to search for and determine the local and global minima of potential energy functions and the barriers separating them. Even for the conventional Morse or LennardJones pair-potentials, the search for minimum energy structures can be a daunting computational task, which depends on the system’s size, composition, and peculiarities of the energy landscape, such as roughness or degree of frustration.17 With the aim of gaining insight into the energetic origin of the tetrahelix in nanoparticles, we have performed comprehensive structural optimizations on the Pt1Pd12 bimetallic atomic cluster, with a fixed composition and number of atoms, using the threshold method (TM)18 to explore a PES defined by the semiempirical Gupta potential.19−21 The feasibility of using the Gupta potential to analyze the structural conformations and dynamics of transition and noble metal clusters has been demonstrated in previous computational studies.22−25 For bimetallic nanoparticles, the PES is far more Received: March 12, 2012 Revised: May 4, 2012 Published: May 4, 2012 5235
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quench to efficiently reach the bottom of the basin. This procedure yielded the helical structures and made it possible to build the disconnectivity graph for the energy region where these structures were found computationally. A word of caution is necessary here in view of inherent limitations of all stochastic search algorithms. The success of finding minimum energy structures depends on a series of parameters designed to facilitate an extensive exploration of the PES, whose values depend on the system’s size, the number of chemical elements involved, the PES function complexity, etc. In this research, the existence of enantiomeric pairs (right- and left-handed isomers of the same helical homotop) not reached in our searches, was verified by generating the enantiomers from initial helical guess structures followed by a local minimization using the L-BFGS-B method. After identifying the tetrahelix structure with the lowest energy, H0, we repeatedly explored the PES to determine possible decay pathways to the icosahedral global minimum structure via neighboring minimum structures. For this set of explorations, the disconnectivity graph was determined by employing a set of thresholds with a spacing between energy lids of 0.02 eV/atom and the highest energy lid at −3.12 eV/ atom, starting from the lowest energy tetrahelix. After overcoming the first energy barrier(s), a set of new local equilibrium structures similar to the helical structures was reached, and the threshold exploration was restarted from these new minima. It is worth mentioning that not all the structures found exhibited (approximately) helical symmetry, but all of them had a lower energy than H0. This procedure was repeated until the global minimum structure was reached. The transition probability depicts the probability of a walker crossing an energy barrier and reaching a different local minimum, that is to say, for the system to change from one modification to another. We are interested in these transition probabilities in order to have a qualitative idea of the structural stability of each of the structures found in the decay pathway as well as in predicting the structural changes taking the helical structure (1-D) to the icosahedron, a 3-D structure, which is the ground state for the Gupta potential. Considering the stochastic nature of the TM and the fact that the number of steps performed by the walkers is finite, there is always a possibility that some minima cannot be reached even if there exists a trajectory under some energy lid. In this way, the probability of occurrence of a structural transition from a given structure to another one is approximately given by the ratio of the number of successful transitions with respect to the total number of isomeric structures.
complex than for monometallic ones because of the existence of homotops.26 Homotops are structures with the same geometry where the two types of atoms are distributed in different ways for a given composition and number of atoms. The TM approach applies to continuous energy landscape systems and has been previously used in the search of new crystalline structures27,28 and in the determination of the barrier landscape of both crystalline solids27−30 and clusters,31,32 with potential application in the energy landscape study of nanoparticles. This method explores the PES by means of random walks, which are constrained to move below an energy lid. When the exploration finishes, the energy lid is displaced upward. In this way, the random walker has a larger configurational space to explore as the energy lid is displaced toward higher energy values. A new exploration is then started after every energy lid displacement. In this article, we discuss the existence of tetrahedral helices in 13-atom Pt1Pd12 systems over a range of energy lid values. We have calculated the transition probabilities associated with energy barriers between similar helical structures and with respect to the lowest energy icosahedral structure. The TM procedure requires a starting configuration. We provided the lowest energy configuration as the starting point in the threshold runs, which had been found to be a 13-atom icosahedron with the Pt atom at the core by means of a genetic algorithm described elsewhere.33
2. METHODOLOGY In this work, we use the TM to determine a number of local minima associated with complex spiral structures known as Bernal spirals or tetrahelices; the energy barriers that separate them as well as the transition probabilities for a particular decay pathway from one of these local minima toward the lowest energy structure. Additionally, we build disconnectivity (tree) graphs18,34 to characterize the PES of the atomic cluster both globally and in a given energy region. The TM has been used to visualize PES landscapes of crystalline solids27 through such disconnectivity graphs and transition maps. In these plots, minima or basins of the PES are represented by a sequence of nodes weighted by the number of states below an energy lid.18 Computer search runs of the threshold algorithm were implemented with the G42 code35 to explore the accessible energy hypersurface associated with the Pt1Pd12 cluster below the energy lids. For each run, the starting structure (starting minimum) corresponds to a Mackay icosahedron with the Pt atom at the center of the structure that had been obtained through a genetic algorithm implemented in the BCGA code.33 Particular details of the PES exploration are the following: a mesh of 30 lids was built in order to carry out each one of the runs; the energy gap between lids was 0.1 eV/atom. The first lid was placed at −3.52 eV/atom, a value slightly above the energy corresponding to the icosahedral structure, whereas the final lid was placed very close to the cluster’s dissociation energy, at −0.6 eV/atom. For each run below every energy lid, a different seed was used, giving each one 2.5 × 105 Monte Carlo steps. Random walks were realized in each run accepting movements limited by the value of the energy lid. Three different seeds were used for each energy lid, 10 random walkers for each seed and 20 stochastic quenches for each walker. For each of the 30 energy lids, periodically, we carried out stochastic quenches of 2.5 × 105 steps each, to find out if the random walk had led to another local minimum. The LBFGS-B algorithm of Zhu et al.36 was used at the end of each
3. RESULTS AND DISCUSSION 3.1. Initial Search for Low Energy Minimum Structures. A detailed analysis of the energy window between −3.6 eV/atom and −0.60 eV/atom yields a number of structures, some of them exhibiting a screw axis of symmetry, both righthanded and left-handed. This type of helical structure has been found experimentally for Au−Ag nanowires14 and in computer simulation studies of liquid nickel.7 This may be an indication of their more general existence on the energy landscape of systems composed of transition and noble metal atoms. Our results also support the idea that they can be found in bimetallic nanoparticles or nanoalloys.37 In this work, we focus the analysis on the helical structures of 13-atom Pt1Pd12 nanoparticles. Relevant energetic information is shown in Table 1. 5236
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Table 1. Energy of the Helical Structures Found in the Global Search (Ebottom), Energy Lid Value at Which the Transition Occurs (Etop), and Energy Barrier Value (ΔE)a
a
structure
Ebottom
Etop
ΔE
H5 H4 H3 H2 H1 H0 ICO
−3.2607 −3.2801 −3.2966 −3.2983 −3.3026 −3.3110 −3.5385
−3.2200 −3.2200 −3.2600 −3.2600 −3.2800 −3.2800 −3.3400
0.0407 0.0601 0.0366 0.0383 0.0226 0.0310 0.1985
Energies are given in eV/atom.
In most cases, both the right- and left-handed enantiomeric helices were found in the search. Figure 1 depicts the right- and left-handed helices found in our PES exploration: H0 is the spiral with the lowest and H5 is the one with the highest energy, respectively. The right-handed structures, H0R and H1R, are homotops; and isomers H1R and H2L are inverted, right-handed and left-handed nonenantiomeric structures, respectively. Isomers H2L to H5L are all left-handed helices, but the Pt atom is present in different strands, so they are homotops. It should be noted that 13 homotops are expected for these Bernal spirals, corresponding to the Pt atom occupying one of the positions in any of the three strands (containing 5, 4 and 4 atoms), with each homotop giving rise to two iso-energetic enantiomers. However, the helix symmetry leads to a number of the atom sites being symmetry equivalent, thereby reducing the number of inequivalent homotops to 7 (each existing as an enantiomeric pair). For completeness, we have generated the missing homotop (H6) and the two missing enantiomers (H1R and H3R) by hand (also shown in Figure 1). The local structure was analyzed using the common neighbor analysis (CNA) technique,38 which has been used in the structural analysis of disordered systems to distinguish between various local structures like fcc, hcp, bcc, and icosahedral environments.25,39,40 This analysis indicates that the helical structures are described by pair-types 211, 322, and 432. Pair abundances (PA) are shown in Table 2. Note that this information is particularly relevant when comparing the results of this study with investigations of larger, disordered extended systems like amorphous or liquid metals because traces of helical structures might be identified through the presence of these pairs. 3.2. Tree Graph Analysis. The disconnectivity graph, which resembles a palm tree,15 is shown in Figure 2. The icosahedral structure is given as a reference ground state corresponding to the lowest energy structure for the Gupta potential. The energy region of the energy landscape shown here contains an abundance of elongated structures some of which correspond to spiral arrangements of atoms, the multitwinned Bernal spirals built from regular tetrahedra. This palm-tree-type landscape permits us to visualize possible transitions between minima after overcoming the respective energy barriers, also suggesting possible decay pathways into the ground state structure. In this figure, the magnitudes of the energy barriers are represented by the red vertical lines. Such hypothetical decay pathways can be studied by considering the helical structures as starting modifications on the PES. Depending on the experimental thermal conditions or after annealing, each one will decay toward the ground state following various routes that facilitate structural transitions. We
Figure 1. Set of 7 right- and left-handed Bernal spirals found in the exploration of the 13-atom Pt1Pd12 energy landscape. Handedness is evident for all the isomeric spiral structures. Local minimizations were carried out on initial helical guess structures to reach the right-handed H1R and H3R structures. Enantiomers H6 were not reached in the TM search but they were found performing local minimizations, swapping the Pt atom to the right position on the same strand of the corresponding right- and left-handed helices H0.
Table 2. CNA Indices and Relative Abundance of Pair-Types Associated to the Helical Structure H0 structure
CNI
RA
H0
211 322 432
0.42424242 0.33333333 0.24242424
are unaware of structural relaxation experiments on Pt−Pd nanoparticles at any concentration and we hope these predictions will inspire experimental work. As an example, we have chosen a transition from the lowest energy helix structure, H0, to the icosahedron (see Figure 3). 5237
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Table 3. Transition Probabilities in the Decay Pathway from H0 to ICO Structuresa Start/End
H0
A1
A2
A3
ICO
H0 A1 A2 A3 ICO
23.95 0.00 0.00 0.00 0.00
0.05 16.53 0.00 0.00 0.00
0.33 0.00 31.48 0.00 0.00
0.33 0.00 0.00 14.77 0.00
24.50 25.59 21.30 24.16 69.77
a
The highest return probability (i.e., the likelihood of ending up in the start minimum after the threshold walk) occurs for the icosahedral global minimum structure and the lowest for structure A3, nearly 70% and 15%, respectively. Note that the rows do not add up to 100% since there is always some likelihood that other minima on the landscape besides H0, A1, A2, A3, and ICO can be reached.
Table 4. Structures Found Close in Energy to the Helical Structure H0a
Figure 2. Analysis of the PES of the Pt1Pd12 nanoparticle. The disconnectivity graph shows the palm-tree type topology of the energy landscape. (Only the most relevant minima are shown.)
structure
Ebottom
Etop
ΔE
H0 A1 A2 A3 ICO
−3.3110 −3.3357 −3.3450 −3.3820 −3.5385
−3.28 −3.30 −3.30 −3.36 −3.34
0.0310 0.0357 0.0450 0.0220 0.1985
a
In each instance, the local minimum energy (Ebottom), the energy where the transition occurs (Etop), and the energy barrier value (ΔE) are shown. Energies are given in eV/atom.
energy landscape as shown here, especially given the growing evidence regarding the existence of chirality in bimetallic nanoparticles43 and their potential technological applications, such as catalytic enantioselectivity.44
4. CONCLUSIONS The threshold method is a powerful global search algorithm that can be applied to study homoatomic or heteroatomic nanoparticles provided we have a function to model interatomic bonding. Through the threshold method, we have predicted the existence of right-handed and left-handed helical structures and the barriers separating them by means of the energy landscape analysis of 13-atom Pt1Pd12 nanoparticles. This study has provided a greater understanding of both global and local energy landscape features around specific energetic regions containing elongated nanostructures described by the semiempirical Gupta potential, as well as their decay pathways. Understanding mechanisms of nanoparticle formation and transformations and detailed knowledge of their energy landscapes will enable insight to be gained into their physical and chemical properties for specific technological applications, of which heterogeneous catalysis is only one example.
Figure 3. Possible structural relaxation trajectory from tetrahelix H0 to the ground state icosahedron for the 13-atom Pt1Pd12 nanoparticle.
For the decay analysis, we do not follow the atomistic (Newtonian) dynamics because of the stochastic nature of our exploration method; instead we selected 3 energy minima (A1−A3) that are neighbors of the H0 structure and calculated the energy barriers following the route H0 → A1... → ICO. Transition rates above 12% were taken into account in the calculation of the return probability to the original basin. In Table 3, we give more details of the transition and return probabilities of the structures given in this decay pathway. It is apparent that, as we move down in energy, the 5-fold symmetry characteristics become evident until we reach the ground state. The calculated energy barrier values and other energy-related data are listed in Table 4. Novel nanoscale devices have already been envisaged41,42 exploiting handedness inversion in helical molecules. The procedure described here can also be used to follow different transition pathways, e.g., looking for reversals of the helix chirality, which would be useful in the design of novel nanowires with the capability of switching between two chirally different catalytic functionalities. This is a complex but feasible study using the TM to analyze structural transitions on the
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (A.P.-A.); r.l.johnston@bham. ac.uk (R.L.J.);
[email protected] (J.C.S.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS A.P.-A. acknowledges CONACYT for financial support through project 24060. R.L.J. acknowledges COST Action MP0903: 5238
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″Nanoalloys as Advanced Materials: From Structure to Properties and Applications″.
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(35) Schön, J. C.; Jansen, M. Angew. Chem., Int. Ed. 1996, 35, 1286− 1304. (36) Zhu, C.; Byrd, R. H.; Nocedal, J. ACM Trans. Math. Software 1997, 23, 550−560. (37) Ferrando, R.; Jellinek, J.; Johnston, R. L. Chem. Rev. 2008, 108, 845−910. (38) Faken, D.; Jónsson, H. Comput. Mater. Sci. 1994, 2, 279−286. (39) Urrutia-Bañuelos, E.; Posada-Amarillas, A.; Garzón, I. L. Phys. Rev. B 2002, 66, 144205−1−144205−5. (40) Urrutia-Bañuelos, E.; Posada-Amarillas, A. Int. J. Mod. Phys. B 2003, 17, 1011−1025. (41) Chakrabarti, D.; Wales, D. J. Soft Matter 2011, 7, 2325−2328. (42) Chakrabarti, D.; Fejer, S. N.; Wales, D. J. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 20164−20167. (43) Elgavi, H.; Krekeler, C.; Berger, R.; Avnir, D. J. Phys. Chem. C 2012, 116, 330−335. (44) Szöllosi, G.; Mastalir, Á .; Király, Z.; Dékány, I. J. Mater. Chem. 2005, 15, 2464.
REFERENCES
(1) Haji-Akbari, A.; Engel, M.; Keys, A. S.; Zheng, X.; Petschek, R. G.; Palffy-Muhoray, P.; Glotzer, S. C. Nature 2009, 462, 773−777. (2) Bernal, J. D. Proc. R. Soc. London Ser. A 1964, 280, 299−322. (3) Steinhardt, P. J.; Nelson, D. R.; Ronchetti, M. Phys. Rev. Lett. 1981, 47, 1297−1300. (4) Nelson, D. R. Phys. Rev. B 1983, 28, 5515−5535. (5) Zheng, C.; Hoffmann, R.; Nelson, D. R. J. Am. Chem. Soc. 1990, 112, 3784−3791. (6) Vázquez-Polo, G.; José-Yacamán, M. Mater. Res. Soc. Symp. Proc. 1990, 174, 163−170. (7) Likhachev, V. A.; Mikhailin, A. I.; Zhigilei, L. V. Philos. Mag. A 1994, 69, 421−436. (8) Sadoc, J. F.; Rivier, N. Eur. Phys. J. B 1999, 12, 309−318. (9) Campbell, A. I.; Anderson, V. J.; van Duijneveldt, J. S.; Bartlett, P. Phys. Rev. Lett. 2005, 94, 208301−1−208301−4. (10) Shevchenko, V. Y.; Samoilovich, M. I.; Talis, A. L.; Madison, A. E. Glass Phys. Chem. 2005, 31, 823−828. (11) Sciortino, F.; Tartaglia, P.; Zaccarelli, E. J. Phys. Chem. B 2005, 109, 21942−21953. (12) Shevchenko, V. Y.; Madison, A. E.; Mackay, A. L. Struct. Chem. 2007, 18, 343−346. (13) Oguz, E. C.; Messina, R.; Löwen, H. Europhys. Lett. 2011, 94, 28005-p1−28005-p5. (14) Velázquez-Salazar, J. J.; Esparza, R.; Mejía-Rosales, S. J.; EstradaSalas, R.; Ponce, A.; Deepak, F. L.; Castro-Guerrero, C.; José-Yacamán, M. ACS Nano 2011, 5, 6272−6278. (15) Wales, D. J. Energy Landscapes with Applications to Clusters, Biomolecules and Glasses; Cambridge University Press: Cambridge, U.K., 2003. (16) Schön, J. C.; Jansen, M. Z. Kristallogr. 2001, 216, 307−325 & 361−383. (17) Doye, J. P. K.; Wales, D. J. J. Phys. B: At. Mol. Opt. Phys. 1996, 29, 4859−4894. (18) Schön, J. C.; Putz, H.; Jansen, M. J. Phys.: Condens. Matter 1996, 8, 143−156. (19) Gupta, R. P. Phys. Rev. B 1981, 23, 6265−6270. (20) Massen, C.; Mortimer-Jones, T. V.; Johnston, R. L. J. Chem. Soc., Dalton Trans. 2002, 23, 4375−4388. (21) Cleri, F.; Rosato, V. Phys. Rev. B 1993, 48, 22−33. (22) Lloyd, L. D.; Johnston, R. L.; Salhi, S.; Wilson, N. J. Mater. Chem. 2004, 14, 1691−1704. (23) Fernández, E. M.; Balbas, L. C.; Pérez, L. A.; Michaelian, K.; Garzón, I. L. Int. J. Mod. Phys. B 2005, 19, 2339−2344. (24) Paz-Borbón, L. O.; Johnston, R. L.; Barcaro, G.; Fortunelli, A. J. Phys. Chem. C 2007, 111, 2936−2941. (25) Posada-Amarillas, A.; Garzón, I. L. Phys. Rev. B 1996, 53, 8363− 8368. (26) Jellinek, J.; Krissinel, E. B. Chem. Phys. Lett. 1996, 258, 283−292. (27) Wevers, M. A. C.; Schön, J. C.; Jansen, M. J. Phys.: Condens. Matter 1999, 11, 6487−6499. (28) Schön, J. C. Z. Anorg. Allg. Chem. 2004, 630, 2354−2366. (29) Putz, H.; Schön, J. C.; Jansen, M. Comput. Mater. Sci. 1998, 11, 309−322. (30) Putz, H.; Schön, J. C.; Jansen, M. Z. Anorg. Allg. Chem. 1999, 625, 1624−1630. (31) Doll, K.; Schön, J. C.; Jansen, M. J. Chem. Phys. 2010, 133, 024107−1−024107−8. (32) Neelamraju, S.; Schön, J. C.; Doll, K.; Jansen, M. Phys. Chem. Chem. Phys. 2012, 14, 1223−1234. (33) Johnston, R. L. Dalton Trans. 2003, 4193−4207. (34) (a) Hoffmann, K. H.; Sibani, P. Phys. Rev. A 1988, 38, 4261− 4270. (b) Becker, O. M.; Karplus, M. J. Chem. Phys. 1997, 106, 1495− 1517. (c) Wales, D. J.; Miller, M. A.; Walsh, T. R. Nature 1998, 394, 758−760. 5239
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