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Cite This: J. Phys. Chem. A 2018, 122, 4092−4098
Melting of the Au20 Gold Cluster: Does Charge Matter? Mathias Rapacioli,*,† Nathalie Tarrat,*,‡ and Fernand Spiegelman† †
Laboratoire de Chimie et Physique Quantiques LCPQ/IRSAMC, UMR5626, Université de Toulouse (UPS) and CNRS, 118 Route de Narbonne, F-31062 Toulouse, France ‡ CEMES, Université de Toulouse, CNRS, 29 rue Jeanne Marvig, 31055 Toulouse, France S Supporting Information *
ABSTRACT: We investigate the dependence upon the charge of the heat capacities of magic gold cluster Au20 obtained from density functional-based tight binding theory within parallel tempering molecular dynamics and the multiple histogram method. The melting temperatures, determined from the heat capacity curves, are found to be 1102 K for neutral Au20 and only 866 and 826 K for Au+20 and Au−20. Both the canonical and the microcanonical aspects of the transition are discussed. A convex intruder, associated with a negative heat capacity, is evidenced in the microcanonical entropy in the case of the neutral cluster but is absent in the melting processes of the ions. The present work shows that a single charge quantitatively affects the thermal properties of the gold 20mer.
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INTRODUCTION The investigation of the thermodynamical behavior of small metal clusters has strongly developed since the early investigations by Pawlow1 and Buffat and Borel2 on finite-sized gold particles, showing that small particles have a lower melting point than bulk materials. After the pioneering nanocalorimetry experiment of Schmidt et al.,3 several experimental techniques4−6 have made possible the determination of the heat capacity curves of metal clusters as a function of temperature down to selected sizes as small as 10 atoms, in particular in the region of the finite-size equivalent of the solid−liquid transition. Unlike in the bulk, the phase transition in finite systems is not abrupt but extends over a finite temperature interval, as formalized by Berry7 and documented in several textbooks.8−10 Both the melting temperatures and the latent heat have been shown to depend strongly on size.3 Alkali metal clusters have been specially documented;3,5 however, caloric curves of other metal particles, such as tin,11 aluminum12−14, and gallium,4,10,15,16 have also been experimentally obtained. Both structural and electronic factors are known to be determinants in the description of the size effects of metal clusters via the completion of either geometric shells or electronic shells. The influence of electronic effects, and in particular charge, on the melting of metal clusters was investigated on sodium,17 aluminum12,18, and gallium.19,20 The case of Al36, a cluster characterized as doubly magic (which simultaneously exhibits closed electronic and geometrical shells) and its neighboring sizes were discussed in detail by Aguado and Lopez18 from density functional theory (DFT) simulations. They showed that the canonical heat capacity peaks of the solid-to-liquid transition in the cation and in the anion were broader than in the neutral species and shifted to lower temperatures. They also reported the occurrence of a negative branch of the microcanonical heat © 2018 American Chemical Society
capacity for Al36, corresponding to a convex intruder in the entropy or equivalently a backbending in the microcanonical temperature curve as a function of the energy. Gold clusters and nanoparticles have been the object of numerous studies due to their remarkable properties in several application fields such as catalysis,21 nanoelectronics,22 nanoluminescence23, and medicine.24 The structural and static properties of AuN clusters and nanoparticles have been widely investigated, and some of the works provide an explicit comparison between the data from calculated structures (mostly in the DFT framework) and experimental data.25−44 However, although the early experiment of Buffat and Borel2 was concerned with gold nanoparticles, no experimental size-selected determination of the caloric curves in the cluster regime range (N less than 50) has been published so far to our knowledge. A number of theoretical investigations45−52 usually achieved with many-body potentials or thermodynamical models have examined thermodynamically induced structural conversion and the melting of large nanoparticles. Simulations of the thermal properties were also carried out in the cluster regime size (from one to a few tens). Several works were concerned with the simulations of specific thermal behaviors related to 2D−3D transitions in small clusters53,54 or to thermodynamical aspects of vibrational heating.55 The caloric curves of neutral gold clusters in the small and medium size range were quite systematically investigated by Soulé du Bas and co-workers.56,57 In their comparison between Au19 and Au20 in particular, they showed that strong differences could be induced by a single missing atom, which is a manifestation of finite size effects where each atom and defect Received: December 20, 2017 Revised: March 23, 2018 Published: March 23, 2018 4092
DOI: 10.1021/acs.jpca.7b12522 J. Phys. Chem. A 2018, 122, 4092−4098
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The Journal of Physical Chemistry A
trajectory was 7.5 ns using a time step of 1.5 fs to integrate the classical equations of motion. Exchanges were attempted using the Metropolis energy criterion every 1.2 ps. We used a Nosé−Hoover chain of five thermostats with a frequency of 80 cm−1 to achieve an exploration of the canonical ensemble. Evaporation did not occur as a crucial problem here since there is quite a large difference between the solid−liquid and the liquid−gas transition temperatures (1337 K72 vs 3243 K,73 respectively, for the bulk). Nevertheless, in order to avoid any problems in the solid−liquid region of the heat capacity, we enclosed the clusters within a large, rigid spherical potential centered at the cluster center of mass defined by V(r) = a(r−r0)8 where a = 0.08 hartree Å−1 and r0 = 20 Å for r > r0. Several approaches exist to determine the heat capacity curve. In the present study, as in the work of Krishnamurty et. al.,57 we applied the multiple histogram method developed by Labastie and Whetten.74 This approach reduces the statistical noise and allows for the extrapolation of heat capacities to temperatures not explicitly simulated. In addition to the canonical heat capacity, those simulations provide the microcanonical entropy S(E) (within a constant), the microcanonical temperature 1/T* = ∂S/∂E, and the microcanonical heat capacity C(E) = −(∂S/∂E)2/(∂2S/∂E2) (the star labels the microcanonical temperature vs the canonical one). To avoid the finite difference calculation of the second derivative, the first derivative of the entropy (1/T*) was fitted by a polynomial function, which was easily derivable. Finally, the isomer population analysis was done after periodic quenching. For each temperature, 1024 quenches (conjugate gradient optimization), regularly distributed along the trajectory, were achieved, allowing the isomer basins (catching areas) to be assigned. The identification of the quenched structures was done by combining an energy threshold (the difference in energy less than 10−4 hartree) together with an ordered-distances criterion (the interatomic distance-based similarity function introduced by Joswig et al.75 with a similarity threshold of 0.95). Note that a number of extra quasi-degenerate isomers, similar but not identical, were obtained with respect to our previous work.69
counts. In the cases of Au16 and Au17, specific features of the heat capacities were also found58 to be associated with the isomerization of the cage global minima of those clusters. The shape of neutral Au20 is a highly symmetric Td pyramid that can be viewed as a part of the fcc lattice. The shapes of ionized clusters Au+20 and Au−20 are also essentially pyramidal, with very small Jahn−Teller deformations from Td symmetry. Those geometries are now well established from theory,35,59 as well as from experiment, namely infrared spectroscopy,35 for the neutral species and trapped ion electron diffraction59 or ion mobility41 techniques for the cation and the anion. Direct atomic imaging insight was given by Wang and Palmer.60 The electronic structure of neutral Au20 can be qualitatively modeled as a closed-shell system with a 1s21p61d102s2 superorbital configuration for the outer delocalized electrons in the simple spherical Jellium model (even though 5d electron bonding and atomic 6s−5d hybridization can certainly not be neglected). This electronic shell closing together with the symmetric fcc tetrahedral packing of Au20 supports a particular large stability and its assignment as a magic cluster.61 In recent works, we have adapted and benchmarked DFTB parameters62,63 for gold materials from clusters up to the bulk.64 We also checked the convergence of the cohesive energies of larger nanoparticles of a few hundred atoms to the bulk values, as well as the structural, elastic, and energetical properties of the bulk itself. As a key advantage, DFTB proved able to yield differential and selective results for charged clusters; namely Au+N and Au−N, in addition to the neutrals, providing fairly consistent ionization potentials and electron affinities. In a subsequent work, we combined DFTB with a global search algorithm based on the parallel tempering molecular dynamics (PTMD) scheme65 completed with periodic quenching to obtain the lowest-energy isomers of Au(0,+,−) and Au(0,+,−) . One conclusion 20 55 of the work for Au20 was that the isomerization energy gap is related to the specific charge state of the cluster, namely the gap is large for the neutral and much smaller for the anion and the cation. The present work is dedicated to the investigation of the influence of the charge state of the cluster on the heat capacity curves in the temperature region about the solid−liquid transition temperatures in the bulk. Apart from this fundamental aspect, the consideration of ions is also of interest because they can be more easily size-selected experimentally via mass spectrometry setups. The nature of the phase changes in the interval 100−1700 K and its dependence upon charge are analyzed from the caloric curves and the temperature-evolution of the isomer populations. Finally, while the present simulations are achieved in the canonical ensemble, we also discuss microcanonical aspects.
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RESULTS AND DISCUSSION The heat capacity curves as a function of the (canonical) temperature are plotted in Figure 1. The first noticeable feature is
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COMPUTATIONAL DETAILS The potential energy of the neutral, cationic, and anionic clusters were determined using the second-order version of DFTB66−68 (self-consistent charge, SCC) and the parameters for gold introduced in our previous publication64,69 (derived from the parameters of references 62 and 63). Note here that the SCC scheme is relevant since it provides a self-consistent account of electrostatics, which is especially important at the surface of charged metal clusters. The potential energy surface (PES) at various temperatures was explored using classical molecular dynamics with a parallel tempering scheme,65 as previously implemented70 in the deMonNano code.71 This technique strongly enhances the ergodicity in the simulations. For each case, we used a temperature range of 50 to 2000 K with 60 replicates using an exponential distribution of temperatures. The length of each
Figure 1. Canonical heat capacity per atom for Au20, Au+20, and Au−20. Note that the classical kinetic contribution is not included. 4093
DOI: 10.1021/acs.jpca.7b12522 J. Phys. Chem. A 2018, 122, 4092−4098
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In the ions, these correspond to various Jahn−Teller distortions, generating several almost isoenergetic isomers. The above deformations affect only the T = 0 K limit of the heat capacities; however, they do not create any visible associated feature on the canonical heat capacity curves. Note that the T = 0 K limit cannot be properly described in the present work due to the neglect of the quantum effects. Let us, however, mention that several groups of quasi-degenerate isomers with neighboring topologies can be found below 1 eV. For instance, isomers 3−12 of the neutral are distinct, but they differ by only one or two bonds or a small distortion. (See Supporting Information Figures S1 and S2.) Such sets of isomers may be thought of as defining meta-basins of the PES. The main differences between the neutral, the cations, and the anions, however, concern the first non-pyramidal isomers. In Au20, the set of isomers 3−12 lies in the range of 0.64−0.79 eV above the lowest one, and the isomerization energy of the next one (13) is 0.97 eV, not considering the barriers. In the cation, the isomerization energy to isomer 4 is 0.34 eV. In the anion, the isomerization energy to isomer 5 is 0.27 eV. Both are significantly smaller than the corresponding gap for the neutral. A continuous progression above the first gap is observed. Again, this decrease of the structural excitation energies above the pyramid meta-basin for ions with respect to the neutral is in reasonable correspondence with the DFT/TPSS values of Letchken et al.59 Partial insight into the atomistic aspect of melting can be gained in the analysis of the isomer basin populations (Figure 3). Note that such analysis provides only energy information about the PES basins but no actual picture about the mechanisms since the barriers between the basins are not considered (except for determining the basins′ catching areas); moreover, the PTMD scheme breaks the true dynamics, preventing the following of the kinetic aspects. In the neutral case, it appears that the decrease of the population of the pyramidal isomer starts at around 800 K. The decrease is quite regular, not showing any steps, and depletion is fully achieved at around 1300 K. Only a single meta-basin (isomers in the range of 0.64−0.76 eV, essentially isomer 5) contributes with a visible population (however minor, 2 eV) than the distance between their maxima (the Tm = 1102 K distribution is weakly asymmetric). E′1 and E′2 correspond to the boundaries of the convex intruder in the microcanonical entropy, as illustrated in Figure 5. This feature appears more clearly in the bottom panel of Figure 5, which reports the difference between the tangents at E′1, E′2, and S(E). We have checked that the convex intruder characteristics and the above microcanonical details remain stable when achieving the multiple histogram treatment from the DFTB-PTMD simulation run with a finer grid (60 temperatures within the 800−1300 K transition region). No entropy convex intruder could be observed in the case of Au+20 and Au−20, again similar to the results of Aguado and Lopez18 for Al+,− 36 . The microcanonical heat capacities of both ions are found to be unimodal, broader, and shifted toward slightly smaller energies with respect to the maximum of the negative heat capacity of the neutral. No significant premelting feature is observed in any of the three cases. One can only notice that the rise in the microcanonical heat capacity with increasing temperature starts earlier for the anion. Finally, taking the example of the neutral case, one can see that the PE(T) distributions at the transition (Figure 6) extend over temperatures between ∼700 and ∼1300 K, framing the melting range extension of the canonical heat capacity discussed above.
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b12522. Structures of isomers 3 to 14 of Au20 (PDF)
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. ORCID
Mathias Rapacioli: 0000-0003-2394-6694 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was granted access to the HPC resources of CALMIP (grants p1303, p0059, and p18009) and IDRIS (grant i2015087375). It was supported by a CNRS-Inphyniti Grant 4096
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(ATHENA project), the CNRS-GDR EMIE, and the NEXT grants ANR-10-LABX-0037 in the framework of the Programme des Investissements d’Avenir (CIM3 and EXTAS projects). We thank J.-M. L'Hermite for helpfull discussions and careful reading of the manuscript.
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DOI: 10.1021/acs.jpca.7b12522 J. Phys. Chem. A 2018, 122, 4092−4098
