18~ T H E O R E T I CA L A N D C O M P U TAT I O NA L C H E M I S T RY
Nanomaterials: Design and Simulation
T H E O R ET I CA L A N D C O M P UTAT I O NA L C H E M I ST RY
S E R I E S E D IT O R S
Professor P. Politzer Department of Chemistry University of New Orleans New Orleans, LA 70148, U.S.A.
Professor Z.B. Maksi´c Rudjer Boškovi´c Institute P.O. Box 1016, 10001 Zagreb, Croatia
VOLUME 1 Quantitative Treatments of Solute/Solvent Interactions P. Politzer and J.S. Murray (Editors)
VOLUME 10 Valence Bond Theory D.L. Cooper (Editor)
VOLUME 2 Modern Density Functional Theory: A Tool for Chemistry J.M. Seminario and P. Politzer (Editors)
VOLUME 3 Molecular Electrostatic Potentials: Concepts and Applications J.S. Murray and K. Sen (Editors)
VOLUME 4 Recent Developments and Applications of Modern Density Functional Theory J.M. Seminario (Editor)
VOLUME 5 Theoretical Organic Chemistry C. Párkányi (Editor)
VOLUME 6 Pauling’s Legacy: Modern Modelling of the Chemical Bond Z.B. Maksic andW.J. Orville-Thomas (Editors) VOLUME 7 Molecular Dynamics: From Classical to Quantum Methods P.B. Balbuena and J.M. Seminario (Editors)
VOLUME 8 Computational Molecular Biology J. Leszczynski (Editor) VOLUME 9 Theoretical Biochemistry: Processes and Properties of Biological Systems L.A. Eriksson (Editor)
VOLUME 11 Relativistic Electronic Structure Theory, Part 1. Fundamentals P. Schwerdtfeger (Editor) VOLUME 12 Energetic Materials, Part 1. Decomposition, Crystal and Molecular Properties P. Politzer and J.S. Murray (Editors) VOLUME 13 Energetic Materials, Part 2. Detonation, Combustion P. Politzer and J.S. Murray (Editors) VOLUME 14 Relativistic Electronic Structure Theory, Part 2. Applications P. Schwerdtfeger (Editor) VOLUME 15 Computational Materials Science J. Leszczynski (Editor) VOLUME 16 Computational Photochemistry M. Olivucci (Editor) VOLUME 17 Molecular and Nano Electronics: Analysis, Design and Simulation J.M. Seminario (Editor) VOLUME 18 Nanomaterials: Design and Simulation P.B. Balbuena and J.M. Seminario (Editors)
18~ T H E O R ET I CA L A N D C O M P UTAT I O NA L C H E M I ST RY
Nanomaterials: Design and Simulation
Edited by Perla B. Balbuena Jorge M. Seminario Department of Chemical Engineering and Department of Electrical and Computer Engineering Texas A&M University College Station, Texas, USA.
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2007 Copyright © 2007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email:
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Contents
Preface
vii
1 Electrical Characteristics of Bulk-Molecule Interfaces
1
Perla B. Balbuena, Lina R. Saenz, Carolina Herrera, and Jorge M. Seminario
2 Structural Properties of Pure and Binary Nanoclusters Investigated by Computer Simulations
35
Giulia Rossi and Riccardo Ferrando
3 Computer Simulation of the Solid–Liquid Phase Transition in Alkali Metal Nanoparticles
59
Andrés Aguado and José M. López
4 Multiscale Modeling of the Synthesis of Quantum Nanodots and their Arrays
85
Narayan Adhikari, Xihong Peng, Azar Alizadeh, Saroj Nayak, and Sanat K. Kumar
5 Structural Characterization of Nano- and Mesoporous Materials by Molecular Simulations
101
Lourdes F. Vega
6 Hydrogen Adsorption in Corannulene-based Materials
127
Yingchun Zhang, Lawrence G. Scanlon, and Perla B. Balbuena
7 Toward Nanomaterials: Structural, Energetic and Reactivity Aspects of Single-walled Carbon Nanotubes
167
T. C. Dinadayalane and Jerzy Leszczynski
8 Thermal Stability of Carbon Nanosystems: Molecular-Dynamics Simulations
201
S¸ akir Erkoç, Osman BarIs¸ MalcIog˘ lu and Emre Ta¸scI
9 Modeling and Simulations of Carbon Nanotubes
227
Alper Buldum
10 Nano-Confined Water
245
Alberto Striolo v
vi
Contents
11 Ab Initio Simulations of Photoinduced Molecule–Semiconductor Electron Transfer
275
Walter R. Duncan, William Stier and Oleg V. Prezhdo
12 Nano-Particulate Photocatalysts for Overall Water Splitting under Visible Light
301
Kazuhiko Maeda and Kazunari Domen
Index
317
Preface
Since the Richard Feynman dictum “there is plenty of room at the bottom” in 1959, several approaches have been developed to design materials using a bottom-up approach, i.e., designing nanostructured or molecularly structured materials. This novel avenue has revolutionized practically all fields of science and engineering, providing an additional design variable, the feature size of the nanostructures, which can be tailored to provide new materials with very special characteristics. A particularly important role towards a rational design of nanostructures is played by atomistic modeling, including a variety of methods from ab initio electronic structure techniques, ab initio molecular dynamics, to classical molecular dynamics, also being complemented by coarse-graining and continuum methods. Such rainbow of computational tools offers the great possibility of exploring nanoscopic details and using such information for the prediction of physical and chemical properties in some cases impossible to be obtained experimentally, and in others providing an invaluable new instrument for guiding and interpreting experiments. This volume covers several aspects of the simulation and design of nanomaterials analyzed by a selected group of active researchers in the field. The editors thank all the contributors for their kind collaboration, effort, and patience to make this book a reality. The editors would like to recognize the effort and dedication of Ms Mery Diaz, who helped putting together this camera-ready volume. The editors also acknowledge the US Army Research Office and the US Department of Energy for their sustained interest in several aspects of the nanomaterials field. Perla B. Balbuena and Jorge M. Seminario Texas A&M University
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Nanomaterials: Design and Simulation P. B. Balbuena & J. M. Seminario (Editors) © 2007 Elsevier B.V. All rights reserved.
Chapter 1
Electrical Characteristics of Bulk-Molecule Interfaces a
Perla B. Balbuena, a Lina R. Saenz, a Carolina Herrera, and ab Jorge M. Seminario a
Department of Chemical Engineering and Department of Electrical and Computer Engineering, Texas A&M University, 3122 TAMU, College Station, TX 77842 USA
b
1. Introduction Electron transfer reactions between molecules and surfaces of nanosized clusters and of bulk materials are of paramount importance for the development of new materials for catalysis, sensors, and power source devices. For example, the oxygen reduction reaction is one of the surface electrode reactions of low temperature fuel cells, characterized by its slow kinetics even when taking place on Pt, the best catalyst currently known for this reaction [1]. Improving the performance of such catalytic process may significantly contribute to an enhanced fuel cell performance, which is strongly needed in order to make fuel cells a commercial reality [2]. Thus, achieving such goal involves first to develop a thorough understanding of the Oxygen Reduction Reaction mechanism, which is still debated in spite of vast research efforts [3]. According to the generally accepted knowledge, the oxygen reduction reaction on Pt surfaces in acid medium may proceed via a direct four-electron pathway that reduces O2 to H2 O via a sequence of four electron and proton transfers, or via a series pathway where H2 O2 is produced as an intermediate, which is then reduced to water [3]. Recently Balbuena et al. have reported ab initio molecular dynamics [4] and density functional theory [5] results which suggest that a parallel (direct and series) mechanism may be in place at the fuel cell operational conditions, with the direct as the dominant pathway. The electrolyte used in proton-exchange membrane fuel cells is a polyelectrolyte membrane, which conducts protons from the anode, where a fuel such as H2 is oxidized, to the cathode where O2 is reduced [2, 6]. In such environment, the production of H2 O2 is highly undesired because of its potential for generating radicals that degrade the polymer membrane [7], one of the most expensive materials of the fuel cell, along with the catalyst itself. 1
2
P. B. Balbuena et al.
Several reaction intermediates (OOH, OH, O, H2 O2 ) and H2 O (the reaction product) of the oxygen reduction reaction are produced by combined electron and proton transfer which take place sequentially or simultaneously with the O2 dissociation on the catalytic surface [4, 8]. In previous work [5, 9–11] we investigated the adsorption of intermediate species of the oxygen reduction reaction on bimetallic clusters using density functional theory in small clusters, and we identified the variation of the adsorption strength produced by the presence of one or more foreign atoms (such as Cr, Co, Ni) located in a neighbor position to a Pt active site. Moreover, we analyzed [12] the effect of the metal on the discrete molecular orbitals of O2 . One important feature is given by the broadening of the peaks corresponding to the energy states of the molecule. The energy of the HOMO orbital of the O2 molecule is −654 eV, lying just below the Fermi level of Pt (−593 eV), and that of the LUMO orbital is −511 eV. In the molecule–metal system, we found that those states are shifted revealing the transfer of electrons from the metal to the antibonding ∗ states of O2 [12]. On the other hand, in a totally different application, which uses the same techniques shown here, electron transfer reactions at molecule–bulk interfaces are examined because of their vital importance for the development of molecular electronics circuits able to continue the scaling-down of integrated circuits after present CMOS technology reaches its physical limits [13–22]. In this chapter, we illustrate with a few examples details of electron transfer at the metal–molecule and semiconductor–molecule interfaces as calculated from first principles. First, we analyze the effect on the local density of states of Pt and Pt-alloys of the main intermediate species that adsorb on the catalytic surface (H2 O2 , OOH, O, OH, and H2 O) during the oxygen reduction reaction and for comparative purposes similar examinations are done for the adsorption of H atoms on Pt clusters. Second, a semiconductor–hydrogen interface is analyzed with the same methods. In all cases, the molecule-cluster system is embedded in a continuum bulk, thus adding carefully the non-local effects while keeping high accuracy for the local effects.
2. Computational Methods Three different types of first principles methods were used in our approach to understand electron dynamics on metal surfaces. The single molecule or cluster calculations, which given the advance on hardware and software are able to contain all atoms needed to consider local effects, i.e., most of the chemistry. The second group of methods are the ab initio methods for extended systems, which are very similar to the single molecule calculations except that they allow us to calculate a single cell with periodic boundary conditions and therefore providing the effects of the continuum or bulk material. And finally, a Green’s function approach that allows us to consider the effects of the bulk on the single molecule, providing a precise interpretation of any physics and chemistry taking place at the reaction sites.
2.1. Calculation of Single Molecules and Clusters Except indicated otherwise, density functional theory (DFT) as implemented in Gaussian 2003 [23] is used to study the optimized structures, binding energies, Mulliken charge
Electrical Characteristics of Bulk-Molecule Interfaces
3
distribution, and vibrational frequencies of molecule and clusters. All calculations were performed using the B3PW91 hybrid functional, which uses a combination of the Becke3 (B3) [24] exchange functionals and the Perdew–Wang (PW91) [25, 26] correlation functionals. The combined functional B3PW91 is used with the quasi-relativistic pseudopotential LANL2DZ (Los Alamos National Laboratory, doble-) to describe the 1s to 4f core electrons for Pt, Ni, Co, and Cr using effective core pseudopotentials [27–29]. For hydrogen, first and second row atoms we use the standard basis set 6-31G(d) also named 6-31G*; this combination of functional, basis set, and pseudopotential used to describe the cluster-molecule complexes in this study has been found to provide excellent results in several related applications [9, 11, 12]. In particular, when tested against results from precise experiments of molecules containing only first and second row atoms, results are of chemical accuracy [30, 31]. Validations on systems containing higher atoms from third row or higher are difficult to make because the scarcity of experimental data with chemical accuracy. Following the structural optimization by calculating the derivatives of the energy with respect to the Cartesian coordinates of all atoms, which assures that the atomistic system is in equilibrium, i.e., forces are approximately zero in all atoms, a second derivative calculation is performed to determine the existence of a true local minimum and to find zero-point energies. The second derivative is needed because the fact that the forces are zero does not guarantee a local minimum for such geometrical structure. If negative eigenvalues are found in the Hessian matrix, the geometry is modified in order to get away from those negative eigenvalues whose eigenvectors determine dissociative states. The self-consistency of the non-interactive wave function was performed with a convergence threshold on the density matrix of 10−6 and 10−8 for the root-mean-square and maximum density matrix error between iterations, respectively. These settings provide correct energies of at least five decimal figures, three for the atom lengths and one for the bond angles within the level of theory.
2.2. Electron Transfer at Interfaces Bulk-Molecule Using a combination of the Green’s function theory and density functional theory we study the electron transport characteristics of different Metal–molecule interfaces [32–34]. In its present version, our approach is able to handle two different bulk materials interfaced to the molecule, although for most practical applications in catalysis only one bulk material would be enough. The case with two interfaces is very important to determine the electrical characteristics of single molecules approached by two different electrodes [21]. In general, an interface, [Bulk 1 -Cluster 1 ]-[M]-[Cluster 2 -Bulk 2 ], consists of an extended cluster that is composed of a molecule or central cluster M, a cluster of Bulk 1 atoms from the bulk interfacing the molecule and a cluster of Bulk 2 atoms interfacing the molecule on another site 2; both clusters are followed by a semi-infinite bulk (Bulk 2 ) on the left, and a semi-infinite bulk material (Bulk 2 ), as shown in Figure 1. All the chemistry takes place at site M and the two bulk materials do not react with each other. Partial DOS of the bulk materials represent the s p, and d band contributions to the bulk or catalyst materials, which are attached to discrete clusters of the bulk materials atoms, representing the interface to the system M. The system M is a molecule or cluster containing a finite number of atoms where the electron transfer reactions are
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Figure 1 A molecule embedded in a Bulk2 material (usually vacuum, air, or solvent) is interacting with a Bulk1 material (usually a catalyst). An extended molecule, M, is defined as the reactive species (H2 O2 in this case) augmented with the nearest atoms from the bulk materials (Pt3 in this example)
calculated. To do this and to keep the chemistry of reactant and products as close to reality, calculations of M are actually performed on an extended molecule or cluster Cluster1 -[M]-Cluster2 thus M has the local effect of the contacts. A Green’s function approach takes the discrete characteristics of the extended molecule and includes the nonlocal effect of the contacts. Preparing a calculation of electron transport through a discrete chemical system interconnected to two large contacts requires extreme care as performing the experiment on a single molecule. This is why these calculations need to be performed with rigorous ab initio methods, avoiding any empirical or phenomenological theory. Basically, three different types of calculations are performed in our approach: the ab initio DFT calculations for the extended cluster or molecule, the ab initio DFT calculations with periodic boundary conditions for the bulk materials, and the Green’s function calculation for the electron transport through the junction. Crystal-2003 [35] is used to study the bulk materials using the linear combination of atomic orbital approximation with periodic boundary conditions. Crystal-2003 is a suite of programs that can calculate electronic structure, total energy, and wave functions, including its band structure, density of states (DOS), electron charge distribution, electron momentum distribution, Compton profile, Mulliken charges, electrostatic potentials, and X-ray structure factors [36] using a DFT with a linear combination of atomic orbitals with periodic boundary conditions [37]. The basis set expands the Bloch functions built using, s p, and d Gaussian functions. In order to include the chemistry of the junction in the calculations, full geometry optimizations and second derivative evaluation are performed for the extended molecule. The extended cluster or molecule [Bulk 1 ]-[M]-[Bulk 2 ], calculation optimizes the geometry to make sure that all forces in all atoms are zero, then the second derivative of the energy is calculated to guarantee that the Hessian matrix has no negative eigenvalues confirming that the optimized structure of the extended cluster is a local minimum [38]. If the optimized structure is not a local minimum, the structure of the extended cluster is adjusted and re-optimized until a local minimum is reached. Chemically speaking all these are very important steps in order to consider the important local effects at the
Electrical Characteristics of Bulk-Molecule Interfaces
5
interface. Then, a series of single point calculations of the extended cluster or molecule under the effect of different applied electric fields are performed; and the Hamiltonian and overlap matrices obtained for each external electrical field are used in the Green’s function approach. All the calculations of the extended clusters and molecules were performed with the Gaussian 2003 program [23]. The partial DOS, s p deg , and dt2g bands for the bulk catalysts are obtained and used to construct the self-energy matrices for Green’s function approach. For the sake of consistency, both the discrete and continuous systems are calculated using the same method and basis sets. The extended molecule and the bulk calculations are both first principles quantum mechanics calculations at the B3PW91/LANL2DZ level of theory, which corresponds to a Kohn–Sham (KS) Hamiltonian [39, 40] with the Becke-3 hybrid exchange functional [41] and the generalized-gradient approximation (GGA) Perdew–Wang 91 correlation functional [42, 26]. The basis set used is the LANL2DZ, which also includes effective core potentials for heavy atoms [27, 28, 43]. The Green’s function approach requires as input: (a) the Hamiltonian and overlap matrices of the extended cluster or molecule under different electric field biases and (b) the partial DOS of the bulk materials. From the Green’s function matrix, the DOS at the interfaces are calculated [32]. In the electron transfer calculation, the matrix representation of Green’s function is used. The retarded Green’s function matrix, GR , satisfies the following matrix equation [32, 44]: EI − Hinterface GR = I
(1)
where Hinterface is the interface Hamiltonian matrix, I is the unit matrix, and E is the energy of the injecting electron. Since the interface is an open system, the interface Hamiltonian matrix should count the whole interface system including the cluster or molecule, the contacts, and the semi-infinite bulk. The contributions from the quasiinfinite bulk are modeled by self-energy 1 and 2 terms for the Bulk 1 and the Bulk 2 materials, respectively [45, 46]. In this calculation, the construction of self-energy follows the procedure introduced in [32]. The contribution to the interface Hamiltonian KS from the cluster and bulk is evaluated from the Kohn–Sham Hamiltonian HExtended and overlap matrix SExtended of the extended cluster (cluster + bulk) as follows: A transformation, KS H = S−1 Extended HExtended
(2)
is performed because the atomic basis sets are not orthogonal [47]. And then, H is rearranged into sub-matrices [48]: ⎤ H11 H1C H12 H = ⎣ HC1 HCC HC2 ⎦ H21 H2C H22 ⎡
(3)
The sub-matrices H11 , H22 , HCC , H12 (H21 ), H1C (HC1 ), and H2C (HC2 ) correspond to matrix elements of the left bulk and right bulk materials, the cluster, the coupling between the two bulk materials, the coupling between the cluster and left bulk, and
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P. B. Balbuena et al.
the coupling between the cluster and right bulk, respectively. Thus, the corresponding Green’s function is expressed as, ⎤−1 0 − 1 g1−1 GE V = ESV − HV −1 = ⎣ − 1+ ESM V − HM V − 2+ ⎦ g2−1 0 − 2 ⎡
and the interface Hamiltonian can then be written as Hinterface = HCC + 1 + 2
(4)
where i E = i+ gi i for j = 1 2. As opposed to an isolated cluster or molecule, which has real eigenvalues, an interface has complex eigenvalues since the interface Hamiltonian matrix is not Hermitian as a result of the imaginary nature of the selfenergy [49]: = 0 − + i /2
(5)
With complex eigenvalues, the originally discrete electronic states of the isolated cluster become broadened peaks with width , and their positions shift from 0 to 0 − . These broadened peaks are described by a continuous function, the DOS: DOSE =
1 2 2 + 2 E −
0
+ /2
(6)
3. Adsorption of OOH, OH, O, H2 O2 and H2 O on a Platinum Surface A number of oxygenated species result upon oxygen reduction in acid medium catalyzed by platinum. Among them, the radical OOH results from the interaction of a physisorbed or chemisorbed O2 – the strength of the interaction depending on pH, solvent, degree of coverage of the surface – with a hydrated proton; this radical has a short life on the surface and rapidly becomes dissociated into adsorbed O and OH, which in turn become involved in new electron and proton transfer reactions producing water molecules. As an alternative pathway, the adsorbed OOH radical before dissociation may combine with a proton yielding weakly adsorbed H2 O2 . This molecule may easily desorb from the surface and decompose, providing highly reactive radical species that may cause the degradation of the electrolyte membrane. Thus, it is important to characterize the differences among the metal–molecule interface for each of these species. We have chosen four typical cases to analyze the effect of the adsorbates on the local density of states of bulk Pt. In all cases, the local interaction involves one or more O atoms interacting with Pt top sites. The optimized geometries are shown in Figure 2, and Figure 3 depicts possible schemes of the interface molecule–bulk metal. The adsorption geometries shown in Figure 2 were obtained from DFT optimizations using the hybrid B3PW91 functional together with the LANL2DZ pseudopotential and basis set for Pt atoms and the 6-31G* basis set for H and O atoms.
Electrical Characteristics of Bulk-Molecule Interfaces
7
Figure 2 DFT optimized structures for adsorption of intermediate species of the oxygen reduction reaction on Pt3 . Top left: H2 O2 adsorbed through one of its oxygen atoms on top of one of the Pt atoms; Top right: two water molecules each adsorbed via O–Pt interaction; Bottom left: O + OH; Bottom right: adsorption of radical OOH
Figure 3 Most likely metal–molecule interfaces for the geometries of Figure 2. Eventually, our procedure can provide a specific bulk DOS for any shape of the material
The calculated local density of states is shown in Figure 4 for the cases displayed in Figure 3. The DOS corresponding to Pt (blue curve) is characterized by a broad band of electronic states located approximately at the Fermi level of the metal (∼593 eV). Note that the presence of the adsorbates strongly modifies the local electronic characteristics.
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P. B. Balbuena et al. 3
Pt3 O + OH 2H2O H2O2 OOH H2O2
2.5
2
DOS
* * * * * *
1.5
1
0.5
–10
–9
–8
–7
–6
–5
–4
–3
–2
–1
0
Energy (ev)
Figure 4 Local density of states (arbitrary units) of the systems shown in Figure 3. The experimental Fermi level of Pt is located at −593 eV. The various curves describe the local effect of the adsorbate on the DOS of the bulk metal. Such local DOS is found very sensitive to the position of the adsorbate site
Platinum is an electron donor for the O atom that becomes negatively charged, especially in the cases of adsorbed OOH, O, and OH. Thus, in these cases (yellow and purple cases), even though there is a depletion of states (compared to pure Pt) in the range −5 to −7 eV, the DOS in that range shows a relatively high population which is attributed to the electronic states of the adsorbates. In contrast, adsorption of H2 O and H2 O2 are much weaker, and those adsorbates do not contribute new states at energies between −7 and −5 eV. This example clearly illustrates the analysis that can be performed to investigate details of the electronic states at the interface of the reactants, intermediates, and products with the catalytic sites. In the next sections we describe details of the DFT calculations carried out to analyze the metal–cluster and metal–semiconductor systems.
4. Toward a Platinum Testbed A challenging problem is to find a testbed to perform reactions simulating events on a platinum surface. The initial step for this is to study systematically the characteristics of small clusters resembling one of the typical surfaces where reactions are going to take place. For this preliminary work we have chosen the (111) surface of platinum. The goal is to obtain a cluster of platinum atoms showing a geometry compatible with a (111) surface that is in static equilibrium and that corresponds to a local minimum. Then molecules of interest are attached in order to study the effects on the electron characteristics of the interface. We focus mainly in these studies on the DOS at the interfaces. The structure and energies of platinum, hydrogenated platinum clusters, and oxygenated platinum clusters have been performed using the B3PW91 functional.
Electrical Characteristics of Bulk-Molecule Interfaces
9
4.1. Platinum Clusters The LANL2DZ and SDD basis sets were employed to study the platinum clusters [50–53]. The geometries and energies of the Pt n n = 1 2 3 clusters are shown in Table 1 and Figure 5. The lowest energy corresponds to the triplet state using the SDD basis set for platinum monomer and dimer. As shown in Table 1, the bond lengths obtained with the two basis sets are different. We have found recently that the SDD basis set does not provide better results than the LANLDZ. For the Pt trimer there is not a marked difference between the singlet and triplet states. Using the LANDZ basis set the singlet and the triplets correspond to D3h and C2v point groups. The bond lengths, angles, and energies of Pt 4 are shown in Table 2. The geometry is displayed in Figure 6. The SDD basis set provides lower energy values than LANL2DZ basis set. The minimum energy corresponds to triplet state. The bond distances of Pt4 with symmetry D2h are similar for both basis sets. The bond distances and angles of Pt5 for different multiplicities are shown in Table 3 and Figure 7. The bond length values yielded by the two basis sets are very similar. Table 4 displays the total energies for the lowest states of each multiplicity. The lowest energy corresponds unambiguously to the quintet state.
Table 1 Structure and Energies for the Pt n n = 1 2 3 cluster by using the B3PW91 functional with the LANL2DZ and SDD basis sets Molecule
Pt
Pt2
Pt 3
m
3 1 3 5 7 1 3 7
Bond lengths (Å)
Energy (Ha)
LANL
SDD
LANL
SDD
– 2.662 2.356 2.395 2.394 2.499 1–2 2.558 1–3 2.478 2–3 2.558 2.545
– 2.672 2.528 2.533 2.388 2.499
−119118 51 −238256 87 −238335 96 −238275 72 −238114 65 −357560 49
−119343 86 −238711 01 −238767 48 −238733 40 −238572 53 −358233 54
2.527
−357560 23
−358232 70
2.537
−357484 07
−358157 41
Figure 5 Optimized structures of Pt 2 and Pt 3 . The bond length values for different multiplicities are shown in Table 1
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Table 2 Structure and energies of Pt 4 by using LANL2DZ and SDD basis set using the B3PW91 functional m
Basis set
Bond length and angle optimized ( )
(Å)
1 3 5
7
LANL SDD LANL SDD LANL SDD LANL SDD
Energy (Ha)
R1
R2
2.557 2.559 2.577 2.551 2.556 2.552 2.529 2.526
2.495 2.494 2.471 2.702 2.542 2.616 2.731 2.720
121.6 121.7 122.7 116.0 120.4 118.3 114.6 114.8
−476756 90 −477657 01 −477657 01 −477660 73 −476763 96 −477658 51 −476735 42 −477633 84
R1
θ
R2
Figure 6 Structure optimized of Pt 4
Table 3 Bond lengths (Å) of Pt 5 for different multiplicities using the B3PW91 functional with the LANL2DZ and SDD basis sets m=1
1–2 1–3 1–4 1–5 2–3 3–4 4–5
m=3
m=5
m=7
LANL
SDD
LANL
LANL
SDD
LANL
SDD
2.475 2.797 2.797 2.475 2.520 2.520 2.520
2.504 2.660 2.661 2.504 2.484 2.768 2.484
2.531 2.925 2.638 2.535 2.487 2.695 2.488
2.493 3.143 2.612 2.549 2.513 2.540 2.489
2.504 2.636 2.636 2.504 2.568 2.652 2.568
2.510 2.776 2.776 2.510 2.491 2.673 2.491
2.501 2.648 2.648 2.501 2.584 2.514 2.584
Electrical Characteristics of Bulk-Molecule Interfaces
11
Figure 7 Optimized Structure of Pt 5
Table 4 Energies (Ha) of Pt 5 using the B3PW91 functional with the LANL2DZ and SDD basis sets for several multiplicity states. The triplet state of Pt5 using SDD did not optimize Multiplicity 1 3 5 7
LANL
SDD
−595956 31 −595980 17 −595994 58 −595986 35
−597094 59 – −597113 28 −597102 10
4.2. Hydrogenated Platinum Clusters The LANL2DZ and the SDD basis sets were used to study Pt n Hm clusters (n = 3 7 13) for their singlet, triplet, and quintet states. Also several other theoretical and experimental studies have been already performed [52, 54–56]. The geometrical parameters for PtHm (m = 1, 2, 3, 4) are shown in Table 5. We can observe in Table 5 that the difference in predicted geometries between the two basis sets is larger for the smaller systems and it improves for the larger systems. Tables 6–12 display the geometries and energies of PtHm (m = 5 6 11) clusters. All distances are reported in Angstroms (Å). Table 6 shows that the lowest state corresponds to a quartet (three unpaired electrons), which is roughly at 68 kcal/mol below the lowest doublet state (one unpaired electron). Table 7 shows the results for PtH6 . The ground state is definitely a triplet, which is several eV below the nearest lowest quintet state. In the case of PtH7 the doublet state is the ground state. The sextet state is far above. Table 9 shows the results for the singlet and triplet of PtH8 . The singlet is the ground state. For PtH9 the doublet is the ground state followed by the sextet (with five unpaired electrons) and then the quartet (three unpaired electrons). The singlet of PtH10 becomes the ground state followed by the triplet and then the quintet (Table 11). PtH11 yield the quartet as ground state 50 kcal/mol below the doublet. Looking carefully at Figure 8, it can be determined that PtH11 has dissociated into the triplet PtH10 leaving one free hydrogen atom.
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Table 5 Geometries and Energies to PtHm (m = 1, 2, 3, 4) by using LANL2DZ and SDD basis sets for different multiplicities. PtH3 with multiplicity 2 did not optimize by using SDD basis set Bond length (Å)
PtH
PtH2
PtH3 PtH4
m
LANL
SDD
2 4 6 1 3 5 2 4 6 3 5
1.536 1.722 3.182 1.656 1.683 2.920 1.874 1.666 2.827 1.667 1.676
1.722 1.701 3.549 1.663 1.672 1.673 – 1.654 4.154 1.661 1.664
Angle ( ) LANL
1800 1800 1800 1200 1200 1200 900 900
Energy (Ha)
SDD
LANL
SDD
1800 1800 1800 1200 1200 1200 900 900
−119742 63 −119643 12 −119428 70 −120112 64 −120267 20 −120062 12 −120612 28 −120845 02 −120592 35 −121418 51 −121242 55
119959 57 −119874 78 −119671 10 −120490 78 −120497 05 −120327 43 – −121081 44 −120849 39 −121658 62 −121482 77
Table 6 Geometries and energies of PtH5 for doublet and quartet states Bond lengths and angles R1 R2 1 2 Energy (Ha)
2
4
1560 2792 900 1200 −121886 31
1653 1665 900 1200 −121994 74
Table 7 Geometries and energies of PtH6 for triplet and quintet states. All bond lengths are the same and the angle between hydrogen–platinum–hydrogen is 90 Bond lengths
3
5
Pt–H Energy (Ha)
1664 −122574 39
1506 −122112 52
Table 8 Geometries and energies of PtH7 for doublet and sextet states Bond lengths 1–2, 1–3, 1–5, 1–6, 1–7 1–4 1–8 Energy (Ha)
2
6
164 164 167 −123230 13
201 166 166 −122902 90
Electrical Characteristics of Bulk-Molecule Interfaces
13
Table 9 Geometries and energies of PtH8 for singlet and triplet states. The value of platinum–hydrogen bond length is the same Bond lengths
1
3
Pt–H Energy (Ha)
165 −123852 47
228 −123346 08
Table 10 Geometries and energies of PtH9 for doublet, quartet, and sextet states Bond lengths and angles 1–2, 1–3, 1–4 1–5, 1–6, 1–7, 1–8, 1–9, 1–10 Energy (Ha)
2
4
6
184 164 −124298 34
207 228 −123993 43
495 164 −124127 34
Table 11 Geometries and energies of PtH10 for singlet, triplet, and quintet Bond lengths and angles 1–2, 1–3, 1–4, 1–5, 1–6, 1–7, 1–9, 1–10 1–8, 1–11 Energy (Ha)
1 165 258 −124768 39
3
5
173 179 −124745 65
188 174 −124597 76
Table 12 Geometries and energies of PtH11 for doublet and quartet states Bond lengths and angles
2
4
1–2 1–3, 1–5 1–4 1–6, 1–7, 1–8 1–9, 1–12 1–10, 1–11
186 192 345 186 170 192
162 208 517 162 205 208
Energy (Ha)
−125284 32
−125366 80
The lowest energy of PtH5 PtH6 , and PtH7 correspond to multiplicities 4, 1, and 6, respectively. The lowest energy singlet states of PtH8 PtH9 and PtH10 correspond to D4h C3v , and D2h , point groups, respectively. Notice that the hydrogen atoms tend to join or to move away from the molecule due their short distance between themselves in the PtHm clusters for m = 6 7 11. The PtH12 molecule with symmetry Oh was run with different bond distances but none of them optimized. This was very important to simulate a single atom connected to the 12 nearest neighbors as it takes place in the face-centered cubic structure of platinum. Table 13 shows the bond lengths that were tested with their corresponding energies.
14
P. B. Balbuena et al.
(a)
(b)
(c) 3
6 3 2
5 4
1
1 7 2
4 5
(d)
(e) 8 2
5
7
8
1
7
1
7 9
4
5
(h) 4
5
9
6
8
1
3
(g)
3
3
4 6
6
(f)
6
4
2
5
10
2
(i)
10 6
11 1 7
8
9 2
( j)
3
(k)
(l)
Figure 8 Optimized structures of a. PtH, b. PtH2 , c. PtH3 , d. PtH4 , e. PtH5 , f. PtH6 , g. PtH7 , h. PtH8 , i. PtH9 , j. PtH10 , k. PtH11 , l. PtH12 Table 13 Trial and final bond lengths and their corresponding energies. None optimized correctly Bond length (Å)
Energy (Ha)
1.7 1.8 1.9 2.0 2.5 3.0
−125330 −125545 −125545 −125530 −125489 −125502
Final bond length (Å) 1.913 1.789 1.959 1.827 2.045 2.051
Table 14 summarizes the bond lengths, multiplicities, and energies for the lowest states of PtHn clusters. Their corresponding optimized structures are shown in the Figure 8. Once the chemistry of a single platinum atom interacting with a group of hydrogen atoms is understood, we decided to increase the number of platinum atoms in our
Electrical Characteristics of Bulk-Molecule Interfaces
15
Table 14 Symmetries, multiplicities, and energies of the lowest states of the PtHn clusters using the B3PW91 with the LANL2DZ and SDD basis sets Molecule
Symmetry
Multiplicity
PtH
Dh
2
PtH2
D3h
3
PtH3
D3h
4
PtH4
D4h
3
PtH5 PtH6 PtH7 PtH8 PtH9 PtH10 PtH11 PtH12
D3h Oh D5h D4h C3h D2h C2v Oh
4 3 2 1 2 1 4
Less energy (Ha/particle)
−119742 6 −119959 6 −120267 2 −120497 1 −120845 0 −121081 4 −121418 5 −121658 6 −121994 7 −122574 4 −123230 1 −123852 5 −124298 3 −124768 4 −125366 8 did not optimize
Basis set LANL SDD LANL SDD LANL SDD LANL SDD LANL LANL LANL LANL LANL LANL LANL LANL
clusters toward forming a stable testbed representing a surface of platinum. The process was not straightforward because finding a small cluster of platinum atoms resembling a (111) surface is not trivial because these structures are very unstable. In addition, the optimization of these structures is not straightforward and sometimes impossible to converge as is the case of a planar cluster of platinum atoms. This compelled us to insert hydrogen atoms to the cluster in order to improve their static stabilization. The energies and the number of imaginary frequencies of Pt 3 Hn clusters with n = 12 22, and 30 are shown in the Table 15. For all these cases, the number of imaginary frequencies is too large to be of practical interest. Imaginary frequencies correspond to negative eigenvalues in the Hessian matrix and relate to unstable geometrical conformations. The Pt 7 Hm clusters were developed in order to find an appropriated geometry that can yield zero imaginary frequencies. The number of hydrogen atoms m was varied in order to reach the goal of zero imaginary frequencies. The results are shown in Table 16, and Table 15 Energies and number of imaginary frequencies for Pt 3 Hm clusters for different multiplicities Molecule Pt3 H12 Pt3 H22
Pt3 H30
Multiplicity 1 3 1 3 5 3
Energy (Ha) −364713 27 −364625 96 −370496 22 −370324 38 −370324 37 −375009 81
# of imaginary frequencies 4 13 3 11 10 9
16
P. B. Balbuena et al. Table 16 Energies and number of imaginary frequencies for Pt 7 Hm clusters Multiplicity Pt 7 H18 Pt7 H18
3 1 5 1 1 3 1 3 1 3 5 1 5
Pt7 H12 Pt7 H12 Pt7 H10 Pt7 H9 Pt7 H6
Pt7 H6
Energy (Ha)
# Negative frequencies
−845048 61 −845093 02 −845025 61 −841550 51 −841660 17 −841508 87 −840455 10 −839813 35 −837967 63 −837990 21 −838045 09 −838101 45 −838096 69
28 15 20 11 7 17 9 10 4 6 5 0 0
Figure 9 displays the optimized geometries. It was fortunate to find out that the three hydrogen atoms strategically located on the cluster make the seven platinum clusters stable and still with a structure that resembles the (111) surface of platinum. This structure corresponds to a singlet state (or spin zero structure) and have a quintet excited state at only 3 kcal/mol above. Figures show that steric effects are the main reasons for unstable structures. The lowest energy and stable structure corresponds to Figure 9h. The advantage of having actually two totally different electronic structures for practically the same geometric conformation it is a great advantage for the testbed. We can say that this is the smallest testbed that we can use. The corresponding geometries are shown in Table 17. It can be observed that the singlet state has larger Pt–Pt distances, however but shorter Pt–H distances are.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 9 Structures of Pt7 Hm clusters: a. and b. Pt7 H18 , c. and d. Pt7 H12 , e. Pt 7 H10 , f. Pt 7 H9 , and g. and h. Pt 7 H6
Electrical Characteristics of Bulk-Molecule Interfaces
17
Table 17 Geometry of Pt 7 H6 with zero imaginary frequencies for singlet and quintet state Bond Length (Å) Pt1 –Pt2 Pt2 –Pt3 Pt1 –Pt3 Pt1 —H Pt2 —H
1 2684 2683 2686 1526 1525 − 83810145
Energy (Ha)
5 2693 2649 2679 1664 1543 − 838096 69
Pt 13 Hm clusters were also developed, and are shown in Figure 10. The planar structure of Pt 13 H12 has 23 imaginary frequencies, while Pt 13 H6 (singlet and triplet) presented zero imaginary frequencies with a non planar structure in the singlet state and almost planar in the triplet. However, the singlet state has the lowest energy (−1 553543 29 Ha). Even with the distorted surface, the singlet state still resembles the shape distribution of platinum atoms on a (111) surface and may be used to represent not quite perfect surfaces. Other trial should be performed to eliminate the hydrogen atom that has migrated inside the surface. Fortunately, this migration did not take place with the triplet structure.
4.3. DOS of Reacting Molecules on Platinum Substrates Figure 11 shows the local DOS plots for the stable structure of Pt 7 H6 at different sites in the molecule shown in Figure 12. This is our smallest testbed and certainly still work is needed to get larger ones that allow us performing a larger set of reactions. This is probably useful only for one atom reacting with the central platinum atom. The DOS on the Pt site shows the sharpest features. At the PtH site these features have been blurred and practically have disappeared at the hydrogen site. These sites are explicitly shown in Figure 12. A stable cluster containing 13 platinum atoms has been obtained by adding six hydrogen atoms to compensate for any dissociative behavior as it was done with the
(a)
(b)
(c)
Figure 10 Pt 13 Hm clusters: a. m = 12, and b. and c. m = 6 singlet and triplet, respectively
18
P. B. Balbuena et al. 1.5 Pt Pt–H H
DOS
1
0.5
0 –10
–9
–8
–7
–6 –5 –4 Energy (ev)
–3
–2
–1
0
Figure 11 Partial DOS plots on top of the central Pt (blue), one of the PtH corners (red), and one of the hydrogen atoms (orange) in the Pt7 H6 cluster. DOS units are states/eV atom
8
8 1
3
(a)
(b)
(c)
Figure 12 Different sites where the local DOS have been calculated and shown in Figure 11 using the Pt7 H6
seven atoms cluster. The intercalated symmetric distribution of in and out hydrogen atoms to the surface seems to be the proper and easy way to stabilize difficult surfaces such as the platinum one. The advantage of having this larger cluster is that the effect of the presence of hydrogen atoms at the reactive site is practically zero. The DOS at three different sites in the stable triplet structure of Pt 13 H6 is shown the Figure 13. The sites are explicitly shown in Figure 14. Notice that the shape of this Pt 13 cluster allows performing chemical reactions within the central seven atoms with little of not interference from the borders or from the hydrogen atoms. On the other hand, there is a similarity between the curves obtained with the Pt7 cluster shown in Figure 11. Work is in progress to include a second layer of platinum atoms and in general to increase the size of the cluster.
5. Silicon Testbeds A similar problem related to electron transport and distribution at interfaces corresponds to the field of electronics. As electronic devices approach molecular sizes, the atomistic study of interfaces becomes of paramount importance. What it used to be called microelectronics, it is right now nanoelectronics as the feature size of present electronic
Electrical Characteristics of Bulk-Molecule Interfaces
19
3 H Pt H–Pt
2.5
DOS
2 1.5 1 0.5 0 –10
–9
–8
–7
–6 –5 –4 Energy (ev)
–3
–2
–1
0
Figure 13 Local DOS at three different sites of the Pt 13 H6 (triplet). These sites are explicitly indicated in Figure 14. DOS units are states/eV atom
(a)
(b)
(c)
Figure 14 Specific sites (blue) of the triplet state of Pt 13 H6 to obtain the local DOS shown in Figure 13
devices approaches the ∼30 nm range. This range is practically just one order of magnitude away from most of the small molecule sizes, which is about 1–3 nanometers (nm). Therefore, regardless of whatever is going to be the next generation of electronic devices, the study of interfaces by ab initio techniques will be important and needed. In this section, we extend our analysis to the development of a testbed for silicon, which is by far the most used material in the electronic industry and a key element for the development of ultra fast computers. We start our systematic approach calculating ionization potentials, electron affinities, and atomization energies of several small silicon-based molecules. We calculated them using the B3PW91 functional and the 6-31G**, cc-pVTZ, and the cc-pV5Z basis sets and compare them with the experimental data available in the literature [57]. Probably of practical importance for our purposes only the ionization potentials for the neutral and the first two ions are needed; however, for the sake of completeness, all ionization energies are shown in Table 18. The errors are close to chemical accuracy for the neutral atom and the positive ion. Table 19 shows the electron affinities for the atom and the dimer of silicon. The results are also in good agreement with experiment. The calculation of electron affinities is very important to determine electron transport properties, which cannot be obtained precisely using empirical or semiempirical methods. Table 20 shows the atomization energies, these calculations show that the errors with respect to the experimental measurements are smaller when the cc-pVTZ or cc-pV5Z basis set are used. In addition, it is not
20
P. B. Balbuena et al.
Table 18 Errors of the ionization potentials obtained using the B3PW91 functional with the cc-pVTZ (M1), 6-31G** (M2) and cc-pV5Z (M3) basis sets Atom/Cation Si (m = 3) Si+1 m = 2 Si+2 m = 1 Si+3 m = 2 Si+4 m = 1 Si+5 m = 2 Si+6 m = 3 Si+7 m = 4 Si+8 m = 3 Si+9 m = 2 Si+10 m = 1 Si+11 m = 2 Si+12 m = 1 Si+13 m = 2
Si(I) Si(II) Si(III) Si(IV) Si(V) Si(VI) Si(VII) Si(VIII) Si(IX) Si(X) Si(XI) Si(XII) Si(XIII) Si(XIV)
Energy (Ha)
M1 (eV)
M2 (eV)
M3 (eV)
Exp (eV)
−289342 96 −289039 93 −288434 81 −287212 29 −285544 00 −279393 90 −271797 93 −262648 05 −251378 28 −238381 29 −223609 19 −206321 96 −187305 70 −97880 75
009 012 −023 025 059 143 248 313 255 060 −595 −596 −425 −971
009 016 −017 014 231 512 648 428 037 −596 −661 −722 −446 −1005
009 015 −023 026 021 043 076 142 170 194 −471 −282 −437 −955
815 1635 3349 4514 16677 20527 24650 30354 35112 40137 47636 52342 2 43763 2 67318
Table 19 Errors of the electron affinities obtained using the B3PW91 with the cc-pVTZ (M1), 6-31G** (M2) and cc-pV5Z (M3) basis sets EA (eV) E(Si)−E(Si−1 E(Si2 −E(Si−1 2
M1
M2
M3
Experimental
−020 −030
−022 −027
000 −019
139 218
Table 20 Atomization energies obtained using the B3PW91 with the cc-pVTZ (M1), 6-31G** (M2) and cc-pV5Z (M3) basis sets. Experimental and calculated atomization energies at 298 K
O–Si F–Si N–Si H–Si S–Si Se–Si
M1
M2
M3
Experimental
788 594 444 313 612 548
771 580 426 311 597 583
802 605 455 315 619 552
829 ± 014 573 ± 002 487 ± 016 ≤310 6.46 5.68
necessary to increase the size of the basis set in order to improve its performance because of the accuracy already offered by the cc-pVTZ basis set, which has a lower cost of computational resources than the cc-pV5Z basis set. Table 21 shows the lowest energies for several Sin (n = 2, 3, 4, 5) clusters and ions. The lowest energies for the Si, Si2 , and Si3 corresponded to triplet multiplicities and the
Electrical Characteristics of Bulk-Molecule Interfaces
21
Table 21 Energies for neutral and ionic Si clusters of Si1 –Si5 obtained with the B3PW91/cc-pVTZ levels of theory Energy (Ha) Cluster Si Si+ Si− Si2 Si+ 2 Si− 2
Si3 Si+ 3 Si− 3
Si4 Si+ 4 Si− 4
Si5 Si+ 5 Si− 5
m
6-31G**
cc-pVTZ
1 3 2 4 2 4 1 3 2 4 2 4 1 3 5 2 4 6 2 4 6 1 3 5 2 4 6 2 4 6 1 3 5 2 4 6 2 4 6
−289271 26 −289318 86 −289015 92 −288832 38 −289309 14 −289361 85 −578722 87 −578753 17 −578438 95 −578461 26 −578823 31 −578774 20 −868211 06 −868216 03 −868139 16 −867913 80 −867912 27 −867840 70 −868298 03 −868244 05 −868167 28 −1 157695 26 −1 157667 30 −1 157582 59 −1 157401 41 −1 157351 60 −1 157247 59 −1 157775 17 −1 157711 87 −1 157662 36 −1 447148 63 −1 447132 16 −1 447072 99 −1 446849 71 −1 446836 44 −1 446767 32 −1 447225 44 −1 447164 79 −1 447085 94
−289296 56 −289342 96 −289039 93 −288855 21 −289336 50 −289386 54 −578775 35 −578807 08 −578492 93 −578516 38 −578875 85 −578826 13 −868292 61 −868298 28 −868220 43 −867998 46 −867996 40 −867924 28 −868378 46 −868324 61 −868248 29 −1 157802 68 −1 157776 11 −1 157691 86 −1 157513 48 −1 157464 18 −1 157362 80 −1 157881 91 −1 157819 26 −1 157759 36 −1 447291 01 −1 447271 58 −1 447215 93 −1 446996 22 −1 446979 25 −1 446907 22 −1 447363 02 −1 447299 90 −1 447219 04
lowest for Si4 and Si5 corresponded to singlets. These structures were obtained based on previous research [58–61]. The optimized structures and bond lengths of the neutral Si clusters are shown in Figure 15 and Table 22. Our experience shows that small silicon clusters always yield triangular shapes.
22
P. B. Balbuena et al.
Figure 15 Lowest energy structures for n-atoms Si clusters for n = 2 3 4 and 5, optimized using the B3PW91/cc.pVTZ level of theory. Bond lengths are described in Table 22.
Table 22 Characteristic bond lengths of small Si clusters calculated with the B3PW91/cc-pVTZ levels of theory Molecule
Symmetry
Si2 m = 3
D3h
Si3 m = 3
C2v
Si4
D2h
Si5
C1
Bond lengths (Å) 1–2 1–3 1–2 2–4 2–3 2–5 1–5
2.264 2.287 2.288 2.407 2.317 2.312 2.312
This is also a typical characteristic at boundaries making also difficult to have good surfaces. Fortunately, the solution in the case of silicon is simply using hydrogen atoms to fill in the dangling bonds. In order to obtain the most stable structure of the Si5 cluster, different configurations were optimized. The structure with the lowest energy is the squashed trigonal bipyramid shown in Figure 16, and the energies are shown in Table 23. As mentioned, Si clusters are highly energetic and react immediately with other molecules. To avoid this instability of Si clusters, hydrogen atoms were added to the silicon structures. These hydrogen atoms were added to the structure producing two different conformations: the tetrahedral and the silanes [62, 63]. Several of these structures are shown in Figures 17 and 18. Tables 24 and 25 show the energies and bond lengths of these structures. Our goal is to develop structures resembling a Si surface. These surfaces can be used later on to connect to a metallic surface such as gold or simply to attach molecules to develop further the field of molecular electronics. Recently it has been determined that highly doped silicon could be a strong alternative to metal contacts to attach molecules.
Electrical Characteristics of Bulk-Molecule Interfaces D3h
C1
4 5
C1/D3h
Cs
3
5
1
3
1
23
2
3
4
5 2
2
4
5
2
3 4
1
1
Figure 16 Different configurations for Si5 clusters obtained by using the B3PW91/cc-pVTZ levels of theory
Table 23 Energies of Si5 cluster with the B3PW91/cc-pVTZ levels of theory Structure Pointed trigonal bipyramid Low symmetry structure Squashed trigonal bipyramid Squashed trigonal bipyramid Tetragonal pyramid
Point group
Energy
D3h C1 C1 D3h CS
−1447258 97 −1447267 24 −1447291 00 −1447290 99 −1447256 66
In order to study the interaction of a Si–H cluster with other molecules, the size of the Si–H cluster was increased up to 26 Si atoms and 32 hydrogen atoms and then optimized (Figure 19). The optimized lengths of the Si–H cluster with tetrahedral symmetry are shown in Table 27. Partial and full optimizations were performed to find out the most stable structures. Results are shown in Table 26. Therefore, optimizations of the silicon clusters using the hydrogen terminal atoms seem to finish successfully. Preliminary optimizations seem to indicate that a larger size of cluster than the one needed for platinum is needed to keep the nature of the surface realistic. In the case of the platinum clusters, a single layer of platinum atoms was obtained and stabilized by a small number of hydrogen atoms. These results show, as expected, that the most stable structure for the cluster is the one obtained when a full optimization is performed. Once a cluster of a reasonable size was obtained, the next step was to obtain the local DOS on few sites at the tip and at the center of the (111) surface. Actually, three different sites were tested at the tip and base side (Figure 20): at the H site on top of the central Si atom, at the SiH region of the central region, and at the central Si on the surface when the H atom at that site has been removed. This resembles the approach followed in the platinum surfaces in the last section. Finally, the common organic molecule benzene was attached to the Si cluster to obtain the density of states on the organic molecule. The benzene was optimized previously with B3PW91/6-31G* level of theory and attached to the already optimized silicon cluster. Then, the DOS was also obtained at different distances from the tip and from the
24
P. B. Balbuena et al.
Molecule
Silane
Si2H6
Tetrahedral symmetry 7
3 4
1
8
2 6
5 7
Si3H8
6 5
2
11 1 4
Si4H6
8 10
3
13
9 6
5 14
6
11
4
4
2
9
12
7 13
1
7
3
14 7 8
12
5
13
5
6
11
1 2
9
12 10
8
6
7 4
11
3
10 2
Si5H12
8 5
1
11
16 8
3
14
16
10
17 9
13
12
14
9
4 17
15
15
1
3 10
2
Si10H22
25 26 29
10
28
7 30
5 8
17 15
31
6
11
14
24 23
12 13
15
20 27
21 22
19
11
1
10 9
4
3 2
Si10H16
Figure 17 Optimized geometries of small Si clusters with hydrogen atoms obtained by using the B3PW91/cc-pVTZ levels of theory
base of the cluster. This also allowed us to obtain an equilibrium structure. The results are shown in Figures 22 and 23. The most stable structure was found when benzene is attached to the Si cluster at ∼1.9 Å. The importance of this arrangement is because practically covers the two extremes of possible attachments to organic molecules. In the case of Figure 21a the structure corresponds to the attachment to a smooth surface; however, in the case of
Electrical Characteristics of Bulk-Molecule Interfaces
25
Table 24 Small silanes and silicon clusters energies and geometries using the B3PW91/cc-pVTZ level of theory. Blank entries mean that such structures do not exist as silanes or with tetrahedral symmetry Energy (Ha) Molecule
Silanes
Clusters with tetrahedral symmetry
−291864 29 −582543 88 −873224 93 −1 163906 24 −1 454587 59 – −2 907994 50
SiH4 Si2 H6 Si3 H8 Si4 H10 Si5 H12 Si10 H16 Si10 H22
– – – −1 163907 42 −1 454591 35 −2 904453 34 –
Table 25 Bond lengths for the smallest Si cluster with the B3PW91/ccpVTZ level of theory Bond lengths (Å) Si–Si Si–H
2.352 1.492
Angles ( ) Si–Si–Si H–Si–H
109.7 107.9
Figure 18 Smallest Si cluster with terminal hydrogen atoms, fully optimized structure obtained using the B3PW91/cc-pVTZ level of theory, no imaginary frequencies were found
Figure 21b the structure corresponds to the attachment to a tip or point structure. Figures 22 and 23 show the local DOS at the benzene molecule when it is connected to the surface and to the tip, respectively. For each case, the tip and the surface, an additional plot showing the DOS in smaller scale corresponding to the values of the DOS in the neighborhood of the Fermi level. In most situations, values around the Fermi level are the most important as they determine the amount of electrons available for conduction at one end as well as the amount of free levels to receive electrons at
26
P. B. Balbuena et al.
Figure 19 Si cluster with terminal hydrogen atoms, fully optimized structure obtained using the B3PW91/cc-pVTZ level of theory, no imaginary frequencies were found
Table 26 Energy obtained from a partial and full optimization for singlet state with the B3PW91/ccpVTZ levels of theory. The symmetry for all these cases was Td # Of cells 1 1 5 5
Energy (Ha)
(full optimization) (partial optimization) (full optimization) (partial optimization)
−2 904453 34 −2 904453 05 −7 545939 28 −7 545937 69
Table 27 Bond lengths and angles of a Si cluster (Figure 19) with the B3PW91/cc-pVTZ levels of theory Bond lengths (Å) 54–51 56–28 15–3 51–13 30–17 35–31 8–7
1.497 1.493 1.496 2.352 2.356 2.352 2.358
Angles ( ) 54–51–13 57–30–58 28–51–30 51–28–16 8–3–15 29–16–27 16–27–17
109.8 107.7 109.1 110.3 108.6 109.6 110.3
Electrical Characteristics of Bulk-Molecule Interfaces
1
27
1 H
0.8
SiH
0.8
Si
0.6
(
)
0.6
0.4
0.4
0.2
0.2
0 –10
–8
–6 –4 Energy (eV)
–2
0
0 –10
H SiH Si
–8
–6 –4 Energy (eV)
(a)
–2
0
(b)
Figure 20 Local DOS on the (a) tip and (b) center plane of an Si26 H32 cluster. For each case the DOS on the central H, SiH, and Si are calculated. DOS units are states/eV atom
(a)
(b)
Figure 21 Structure of the benzene attached to the (a) center plane and (b) tip of a Si26 H32 cluster. For each case the DOS at different distances of attachment were calculated and shown in Figures 22 and 23
28
P. B. Balbuena et al. 20 a c 15
b d
e
f
g i
h j
10
5
0 –10
–8
–6
–4 Energy (eV)
–2
0
(A) 1 a c e g i
b d f h j
0.5
0 –10
–8
–6
–4
–2
0
Energy (eV)
(B)
Figure 22 Local DOS at the benzene attached to the surface side Si26 H32 cluster. For each case the DOS are calculated at different distances from the benzene to the cluster. (a) 1.7 Å, (b) 1.9 Å, (c) 2.1 Å, (d) 3.0 Å, (e) 4.0 Å, (f) 5.0 Å, (g) 6.0 Å. The local DOS at the (h) H, (i) Si–H, (j) central Si sites are also shown. Dashed region in (A) has been amplified in (B). DOS units are states/eV atom
the other end. There is plenty of work already developed for the field of molecular electronics and our approach actually consists in adapting such procedures for catalysis and electro-catalysis problems. This structure will allow us to study equally the insertion of molecules on the surface or the injection of dopand atoms into the silicon surface using ab initio techniques that do not require the use of experimental information, opening the possibility of obtaining chemically accurate information about the chemistry of silicon.
Electrical Characteristics of Bulk-Molecule Interfaces
29
20 a c e g i
15
b d f h j
10
5
0 –10
–8
–6 –4 Energy (eV) (A)
–2
0
1
0.8
0.6
a
b
c
d
e
f
g
h
i
j
0.4
0.2
0 –10
–8
–6
–4
–2
0
Energy (eV) (B)
Figure 23 Local DOS at the benzene attached to the tip side of the Si26 H32 cluster. For each case the DOS are calculated at different distances from the benzene to the tip. (a) 1.7 Å, (b) 1.9 Å, (c) 2.1 Å, (d) 3.0 Å, (e) 4.0 Å, (f) 5.0 Å, (g) 6.0 Å. The local DOS at the (h) H, (i) Si–H, (j) central Si sites are also shown. Dashed region in (A) has been amplified in (B). DOS units are states/eV atom
Acknowledgements We acknowledge our group members that have contributed in related topics reviewed in this paper, among them: Mery Diaz Campos, Luis Agapito, Yuefei Ma, Sergio Calvo, and Juan Sotelo. PB and JS acknowledge the financial support from the Department of Energy/Basic Energy Sciences, (DE-FG02-05ER15729). JS also
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acknowledges the support from the US Army Research Office (ARO) and the U.S. Defense Threat Reduction Agency (DTRA) and PB acknowledges the use of computational facilities at the National Energy Research Scientific Computing Center (NERSC).
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Nanomaterials: Design and Simulation P. B. Balbuena & J. M. Seminario (Editors) © 2007 Elsevier B.V. All rights reserved.
Chapter 2
Structural Properties of Pure and Binary Nanoclusters Investigated by Computer Simulations Giulia Rossi and Riccardo Ferrando Dipartimento di Fisica, Università degli studi di Genova Via Dodecaneso 33, 16146 Genova, Italia
1. Introduction 1.1. Why Studying Nanoclusters? Clusters are aggregates of atoms or molecules, with an average diameter ranging from a few nanometers up to thousands of nanometers. The properties of these finite systems have attracted the interests of researchers coming from the chemical, physical and biological areas. Their characteristics are linked to the high value of their surface/volume ratio, and therefore the structure of clusters plays a fundamental role in determining the whole spectrum of their physical properties. Clusters have a wide range of interesting applications. Semiconductor and metal clusters can modify the optical properties of polymers [1]; supported pure and binary metallic clusters, as Pd, Au, and PdPt clusters, are considered of special interest in catalysis [2–6]. In the biological area, metal nanoparticles have proved to be a promising tool for nanomedicine application: DNA detection, drug delivering and tissue repairing [7]. In the following we will deal with the structure of pure and binary clusters, focusing on the chemical elements we have studied in the more recent years. We have to point out that the physical properties of pure clusters are directly related to their structure, but binary clusters’ characteristics depend also on their chemical ordering. The different atomic radii of the species in a binary cluster, for example, can lead to favour mixed or core-shell structures, as well as a crystalline or non-crystalline ordering. Given a fixed structure, two clusters with a different chemical ordering could exhibit different electronic or magnetic properties. 35
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1.2. Experimental Techniques for the Production of Free and Supported Nanoclusters On the experimental side, production and analysis of clusters in the free gas phase can be achieved by molecular beam techniques [8]. Clusters over the whole range of elements can be produced by vapourization of an atomic or molecular source. In seeded supersonic nozzle and inert-gas aggregation sources, a piece of bulk material (with a sufficiently high vapour pressure) is heated up; in laser evaporation and ion sputtering sources, vapour is obtained by hitting a target that can be an alloy rod if binary clusters are desired [9–11]. When the starting metallic target is an alloy, the stoichiometry of the clusters quite well reproduces the initial one. The vapourization stage is followed by cooling and condensation. Generally these goals are achieved by a supersonic expansion or by the interaction with a cold buffer gas, like He. The first is quite common for rare gas clusters, because their vapour pressures are high. In the latter case the hot vapour is mixed with a cold inert-gas flow, which acts as a collisional thermostat: this method is successfully applied to metallic clusters. Gas pressure is one of the factors determining the size of the clusters produced. Cluster sizes are then characterized by mass spectroscopy. If the metallic clusters are deposited on surfaces or embedded in inert-gas matrices, they can be studied by various spectroscopic methods: electron spin resonance, infrared and ultraviolet techniques. And of course, they can be observed by microscopes [12, 13]. The surface chemical ordering of binary clusters can be investigated by ion spectroscopy techniques, like in [10, 14], or Raman-scattering experiments [15].
1.3. An Overview of the Computational Techniques Used in Cluster Field Today a wide range of computational instruments are available for the study of nanoclusters. As regards the dynamics of their formation, Molecular Dynamics (MD) simulations have been successfully applied to the characterization of the growth of rare gas, alkalimetal and noble and quasi-noble metal clusters. Besides growth simulations, structural and thermodynamics transitions have been also investigated by MD simulations. During the last decades, another important computational effort has been devoted to the global optimization of cluster structures: that is, finding the lowest energy geometrical configuration of a cluster with fixed size and chemical composition [16]. This goal is a key step of the theoretical study of cluster structures, and in the following we will offer a deeper description of the algorithms used to locate the minimum energy structures. The computational study of clusters is possible when a model of the interatomic interactions is available. Apart from ab initio and first-principle approaches (all-electron calculations for very small clusters, or density functional–based calculations for larger ones), several semi-empirical potentials have been developed to study metallic and raregas clusters. These potentials are usually fitted to experimental properties of the bulk material. Moreover, simple interaction models like Lennard–Jones and Morse potential can be used, particularly when comparing the efficiency of several algorithms upon the same benchmark system.
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The article is structured as follows. Section 2 presents some common energetic models applied to the study of pure and binary clusters; Section 3 shows some of the more common structural motifs for pure clusters, together with some useful index for their energetic characterization. Section 4 is devoted to the description of the global optimization procedure and its computational implementation. In this section we present the structural analysis of the global minima of both bimetallic and binary rare-gas clusters. Section 5 is devoted to the dynamical behaviour of binary metallic clusters. Growth sequences of core-shell and three-shell clusters and the melting of singleimpurity doped clusters are studied.
2. Energetic Models 2.1. Volume and Surface Contributions Clusters are finite systems, and the energy contribution of surface atoms has to be taken into account when evaluating their energetic configuration. The following is a rough description of nanoclusters’ binding energy: Eb = aN + bN 2/3 + cN 1/3 + d
(1)
The first term is the volume contribution: this is the only term considered when dealing with bulk systems. If the cluster has a crystalline structure, a = coh , the cohesion energy of the material. The second, third and fourth terms correspond to the surface contributions. Respectively, they represent the energy contribution of surface, edges and vertex atoms. This basic description of the energetics of a cluster is useful to understand how crucial the geometrical ordering of the cluster can be. The smaller the cluster’s size, the more important and shape-dependent these surface contributions become. The first question we have to face when approaching the computational prediction of the energy of a given cluster configuration is: what is the accuracy we need? Ab initio and first-principles calculations, like Density Functional Theory (DFT) calculations, are the most accurate. In metal clusters, quantum effects (such as the shell-closure effect) can be important in determining the energy of a cluster, and these techniques are appropriate. As regards nanoalloys, DFT calculations can take into account charge transfer effects that are really important if the difference between the electronegativities of the two elements in the clusters is large. DFT-based first-principles studies of nanoalloy clusters have been performed recently by Peng Yang and coworkers [17] and by several other groups [18]. Of course, these approaches are computationally more expensive, and they can be applied only to small size clusters (less than one hundred atoms). If the desired accuracy is not so high, semi-empirical potentials are often used to describe cluster’s energy. These methods, based on the fit of a potential energy function to experimental data, can be easily applied to large size clusters, and are especially efficient in global optimization procedures, when a large number of structural configurations have to be explored in order to locate the minimum energy one.
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The tight-binding approach is based on the one-electron, self-consistent field approximation. The cohesion between atoms is due to the band energy that is dominated by the contribution of the valence electrons. The orbitals included in the calculations depend on the material considered. As regards the repulsive term, the functional forms used in the framework of the tight-binding model often contain a two-body repulsive potential, so that the energy can be expressed as: E = Ebond + Erep = Ebond +
1 u r 2 ij r ij
(2)
where rij is the distance between the atoms i and j. The bonding term in the tight-binding approach usually shows a dependence on the square root of atoms’ coordination. That is, the energy of each atom does not depend linearly on the number of its neighbours. A large number of neighbours implies a weaker effective attraction per bond. This effect in metals is named bond order–bond length correlation, and results in the contraction of surface bonds and in the relaxation of volume bonds. In order to show an example of potential function based on the tight-binding approach, here we show the functional form of the Rosato–Guillopé–Legrand (RGL) potential (formally identical to Gupta potential) [19, 20]:
ERGL =
i
Ei = A
e
−p
r
ij r0
−1
−2q rij r0 − 2 e
i
i
−1
(3)
j=i
Function parameters, A p q, are fitted to bulk experimental values, such as the cohesive energy, the lattice parameter and the elastic constants. The model can be applied to binary systems too, but the parameters are interaction-dependent, so that pure (A–A and B–B) and mixed (A–B) bonding are treated with different sets of parameters. The heteroatomic interactions can be fitted to the experimental value of the solubility energy of an impurity A into a B bulk, and vice versa. Another example of potential derived in the tight-binding scheme is the Embedded Atom (EA). In this case, the electron density perceived by atom i is derived as the superposition of the electron densities of all the other individual atoms j: i =
f rij
(4)
j=i
The total energy in EA models is thus expressed by: EEA =
i
Fi i +
1 u r 2 ij r ij
(5)
The term Fi i is the embedding energy required to place atom i in an electron density i , while the second term is the standard two-body repulsive contribution. Several functional forms have been proposed for F : it can be obtained by first-principles
Structural Properties of Pure and Binary Nanoclusters
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calculations, or more practically by total semi-empirical approach. Sutton–Chen potential [21], for example, uses the following expressions for i and Fi i : i =
j=i
a rij
m
√ Fi = c i
The whole energy of the systems is thus expressed by: 1 a n √ − c i E= 2 j=i rij
(6) (7)
(8)
In Sutton–Chen potential [22, 23], c is a dimensionless parameter, is a parameter with dimensions of energy, a is the lattice constant, and m and n are positive integers with n > m. Parameters n m and c assume different values depending on the material described. The Sutton–Chen potential provides a reasonable description of various bulk properties, with an approximate many-body representation of the delocalized metallic bonding. However, in analogy with Gupta potential, it does not include any directional terms, which are likely to be important for transition metals with partially occupied d shells.
2.2. Model Potentials and their Computational Applications The most simple way to explain why ab initio and semi-empirical methods have different range of application is probably to show how much computational time is required to get a local minimization by the two approaches. Starting from a random point of the configurational space, (that is, from a cluster characterized by a given spatial distribution and chemical composition), it is possible to look for the nearest local minimum of the Potential Energy Surface (PES) of the cluster. Let us consider first that the energy is evaluated by means of a semi-empirical approach, and let us suppose that the cluster to be minimized is a binary metallic cluster in the size range 10 < N < 100. The computational time required by a standard processor to locally minimize the structure is a small fraction of a second. In the DFT case, the time required is enormously larger: one can estimate [24] that a 1.9 GHz processor could perform the local minimization in one hundred hours or more, depending on the distance between the starting configuration and its nearest local minimum. There is a significant gap to be filled: first-principles approaches are more accurate, so that their application is advisable for the evaluation of the potential energy of small size clusters. On the contrary, semi-empirical approaches are more suitable for large size clusters and can be used during the simulation of the dynamical behaviour of such systems (for example in MD simulations), but they are less reliable. In order to get information about the cluster structures with a low potential energy, a common strategy consists in coupling semi-empirical models to ab initio methods: semi-empirical potentials are implemented in global optimization algorithms that explore
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large portions of the configuration space of the cluster, and when a set of candidate minima is available, they are locally minimized by ab initio codes. But the simulation of the dynamical processes is cumbersome and in many cases impossible for ab initio models, and only semi-empirical approaches can be applied.
3. Structures 3.1. Useful Energy Indexes, Structures and Magical Sizes As shown before, the binding energy of a cluster is strongly related to its surface/volume ratio, and consequently to its geometrical structure. The simplest idea to identify possible structural configurations in clusters is just the search for the geometrical ordering that presents a low surface/volume ratio, which is a quasi-spherical shape, and a surface ordering that maximizes the number of bonds among surface atoms, like closely packed facets. Of course, the surface stability has to be balanced by volume strain that would destabilize a perfectly spherical configuration. In order to formalize these considerations, it is useful to define some energetic quantities explicitly related to the structural characteristics of the cluster. We consider specifically the case of binary clusters. The first one is the excess energy Eexc , whose value in a cluster formed by m atoms of the A species and n atoms of the B species is given by: Eexc Am Bn = EAm Bn − mAcoh − nBcoh
(9)
Here Acoh and Bcoh are the cohesive energies per atom of the bulk metals A and B. This quantity is a sort of comparison between the energetic configuration of two separate pieces of bulk matter, Am and Bn , and the one of the binary cluster. The excess energy of a cluster is a positive quantity, and it takes into account energy losses and gains with respect to what happens in bulk matters. These losses and gains are due to three factors: (a) the presence of surface atoms, normally neglected in the bulk matter, (b) the possibility that the geometrical ordering of the cluster does not correspond to the typical one in the bulk systems, and (c) the energetic contribution coming from the combination of two different chemical species. In binary metallic systems like the ones we will consider in following, (a) corresponds usually to an energy loss, while the geometrical ordering, often different from the fcc typical of several bulk metals, constitutes a gain with respect to the crystalline structure. Moreover, when considering binary clusters the cohesion between the two atomic species has to be compared to that of the pure interactions. This last theme will be discussed in more detail in Section 4. However, since metals with larger cohesive energy tend to present pure clusters with higher excess energy, Eexc may be a biased index. For this reason, it could be preferable to use Eexc defined as follows: ∗ Am Bn = E Am Bn − m Eexc
EBN EAN −n N N
(10)
namely subtracting to the total energy the energy per atom of the pure clusters of the ∗ is unbiased, being zero same size instead of the bulk cohesive energy. In this way Eexc
Structural Properties of Pure and Binary Nanoclusters
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∗ ∗ for pure clusters. A negative value of Eexc means that mixing is favourable. Eexc is the analogous of the formation energy of bulk alloys, adapted to the case of nanoclusters. In order to compare the energetic stability of different-sized clusters, usually the value of the excess energy is divided by a factor of N 2/3 that roughly scales as the number of surface atoms in a quasi-spherical cluster. The index is so defined as:
= Eexc N 2/3
(11)
The second index of energetic stability we would like to describe is 2 . It is the second difference in energy, defined as: 2 Am Bn = EAm+1 Bn−1 + EAm−1 Bn+1 − 2EAm Bn
(12)
The stability of the cluster formed by m atoms of the A species and n atoms of the B species is compared to the one of the two clusters with close composition. This index is useful when comparing clusters in a sequence with fixed size and variable chemical composition: high 2 values correspond to stable structures. Both and 2 are suitable indices to deal with pure or binary clusters. As stated before, the definition of the excess energy of a cluster is based on the comparison between the energetic configuration of atoms in the bulk matter and that in the cluster configuration. Another possible approach, suitable to the study of binary clusters, consists in comparing the energetic configuration of the binary cluster to that of same cluster composed by the pure constituents. This index is called mixing energy Emix : Emix Am Bn = EAm Bn − EAm ⊂ Am An − EBm ⊂ Bm Bn
(13)
The expression EAm ⊂ Am An is the energy of the m atoms of species A when put inside the cluster obtained by substitution of the B atoms of the starting cluster with A atoms. The same procedure is applied to B atoms to get EBn ⊂ Bm Bn .
3.2. An Example of Application of Index The parameter has been widely used when dealing with pure metallic clusters. It has been useful, for example, in comparing the energetic trends of several structural motifs as increasing the size of the clusters. Let us consider three different (and quite common) structural orderings: the bulk-like fcc, the icosahedral (Ih) and the decahedral (Dh) ordering (see Figure 1). Metals like Ag, Pd, Pt, Cu, Ni, Au have an fcc arrangement in the bulk. These clusters often exhibit small squared (100) facets resulting from the truncation of their vertices, and large compact (111) facets. Decahedral clusters are made up of two pentagonal pyramids sharing a common basis. Each pyramid contains five tetrahedra. Icosahedra are composed of 20 regular tetrahedra sharing a common vertex (at the centre of the structure). In both structures, the tetrahedra are slightly distorted, and this distortion causes the inner strain that is typical of Dh and Ih structures. In icosahedra, the surface is a composition of triangular closely packed facets, while the volume atoms are arranged
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Figure 1 From left to right, an fcc, a decahedral, and an icosahedral cluster are shown. In the bottom row, the size of atoms is reduced to better appreciate the symmetry of the different structures
according to a shell structure. In cluster science, every structural type is characterized by some geometrical magic sizes, that is, sizes corresponding to the completion of a perfect structure. At small sizes, for example, 38 is magic size for the fcc truncated octahedron, while 55 and 75 are magic sizes for Ih and Dh structures, respectively. For more details about structures, see [25]. Once collected the magic sizes of different structural motifs in a given size range, it is possible to wonder whether they correspond to stable energetic configurations. In Figure 2, the qualitative behaviour of for magic Ih, Dh and fcc structures is reported. Calculations about transition metal clusters are reported in [26]. Δ
Ih Dh
fcc Transition Ih → Dh
Transition Dh → fcc
Size N
Figure 2 Here the qualitative behaviour of is shown as a function of the size of metal clusters. The red line refers to icosahedral structures, the dashed line to decahedral structures and the dotted line to crystalline, fcc structures. Two structural transition regions are highlighted
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At small sizes, icosahedra are the most favourable structures: their surface/volume ratio is low and the strain, which is a volume contribution, is not very important due to the small number of atoms involved. Increasing size, it is possible to locate the first region of structural crossover. As strain increases in icosahedra, decahedra become more favourable. And at large sizes, the fcc bulk arrangement is more and more likely to be found. Even if the transition sizes are strongly material-dependent, the structural trend identified by the evaluation of for magic structures has been proved experimentally, for example for Cu clusters [27, 28]. Of course, these considerations cannot be simply transferred to binary clusters. As we will show later in detail, metal binary clusters can exhibit structural motifs different from the Ih Dh and fcc ordering.
4. The Global Optimization of Pure and Binary Metallic Clusters 4.1. A Description of the More Common Global Optimization Techniques The global optimization of the potential energy function of a cluster is the search for the point of the configuration space corresponding to the lowest value of the potential energy of the cluster. This is a very hard task: the number of local minima on the PES of a pure cluster scales exponentially with cluster’s size. Because of that, these optimizations are classified as NP-hard problems, which means that there is no polynomial function that can describe the increasing of the number of local minima on the PES as a function of the size of the system. Global optimization algorithms in cluster science should be able to locate the structural conformation that minimizes the potential energy of a cluster, whose size and chemical composition are known. A further observation is important. A binary cluster of size N has a larger number of local minima than a homogeneous cluster of the same size. This can be explained introducing the concepts of isomers and homotops. With the term isomer we identify clusters with the same size and chemical composition, which differ only for the spatial arrangement of their atoms. The binary clusters in the first and second row of Figure 3 are isomers. With the term homotops we refer to two clusters of the same size, chemical composition and structure, differing only in two or more atom-labels. The clusters in the first and third row of Figure 3 are homotops. One cannot distinguish between the homotops of a pure cluster that have all the same potential energy. On the contrary, homotops of a binary cluster can be different: starting from a given binary cluster structure, it is possible to get a different energy homotop just by exchanging the positions of two atoms of different species. In terms of the computational effort devoted to the localization of the global minimum structure, we can roughly say that on the PES of a pure cluster every isomer corresponds to one local minimum energy, while on the PES of a binary cluster every isomer corresponds to several local minima, each one related to a homotop with different energy. In order to get the structure with the lowest potential energy, the number of local minima to be explored is much higher in binary clusters than in pure clusters. Different approaches have been proposed in literature to solve the global optimization problem. As underlined by Wales [16], a first distinction regards to biased and unbiased methods. Biased methods are based on some previous knowledge about the
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Figure 3 In the top and middle row, two isomers of the Ag11 Ni27 cluster. Size and composition do not change, but the spatial arrangement of atoms is different. Clusters in the top and bottom row are homotops: the structure is the same, but the chemical order is different
global minimum of the cluster. For example, one could bias the search for the global minimum of a Lennard Jones cluster by favouring structures with a compact shape [29], or one could bias the search for the global minimum of a AgNi binary metallic cluster, introducing a penalization for the configuration that does not respect the Agshell –Nicore ordering that is suggested by experiments [10]. It has to be observed that the best global optimization strategy is often system-dependent, and the knowledge of the global minima of a family of systems can be often successfully exploited when looking to systems with similar characteristics. In the following we are going to illustrate some of the more common unbiased optimization algorithm. 4.1.1. Simulated annealing Simulated annealing consists in equilibrating the cluster at high temperature, and gradually reducing it in order to get the coincidence between the global free energy minimum and the global potential energy minimum of the system [16]. Simulated annealing, first proposed in [30], has been applied to a variety of cluster systems, like Lennard–Jones clusters [31] and tight-binding clusters (calcium, gold and silicon clusters in [32–34]). Today, it is widely recognized that simulated annealing has two basic defects. First, it is too time-consuming. Secondly, it is possible that during cooling the cluster remains trapped in a funnel separated by high barriers from the global minimum one. As the temperature decreases, the probability that the structure overcomes the energy barrier becomes more and more low, so that global minimum cannot be found in a reasonable scale of time.
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4.1.2. Genetic algorithms Genetic algorithm is based on the principle of natural evolution. The code deals with a population of individuals. Each individual is a cluster, represented by a string containing its genetic material, i.e. the coordinates of its atoms. In Figure 4, a graphical representation of the fundamental steps of the genetic evolution is shown. Couples of clusters from the starting population are mated through the recombination of some of their string digits, giving birth to two offspring clusters. Moreover, with a certain probability the individuals go through mutations of their genetic code. Every time a new population is built, through mating and mutation stages, the energies of all the individuals are evaluated and compared, in order to select the clusters that better fit the request to have a low potential energy. Individuals with a good energetic configuration are chosen for mating, and bad energetic configuration clusters are not transmitted to the next generation. Missing clusters can be replaced by random individuals so that the number of individuals in the population is kept constant. This procedure allows us to select the best genetic material, and in principle it can be applied to a variety of systems and optimization problems just by changing the way of selecting the survivors through generations. An important remark has to be made: in cluster energy optimization, this architecture gets a substantial improvement if it is coupled to a procedure of local minimization. When a new cluster is born or an old one is mutated, a local minimization procedure can
Figure 4 In a genetic code, a cluster of N atoms is represented by a string of digits that codifies the spatial positions of the atoms. During the evolution of the starting population of clusters, the genetic material is recombined through mating and mutations, so as to favour individuals with a low potential energy. One of the possible mating strategies consists of exchanging some digits between the parent clusters, in order to obtain two offspring clusters. Mutation can also be performed in several ways. In the mutation process illustrated in the picture, some digits of a cluster string are replaced with random values
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be performed before its energy is compared to the others. Maintaining the evolutionary metaphor, one could say that a Lamarck evolution is better than a Darwin evolution in cluster optimization, because the local minimization can be interpreted as a learning process that modifies the genetic code of the individual, and is thus transmitted to the offspring. Genetic optimization has revealed to be a quite effective instrument in the global optimization of clusters, both pure and binary. Its application to molecular geometry optimization was proposed by [35], and already at that stage they demonstrated that genetic algorithm is faster and more effective than simulated annealing is. Gold clusters have been optimized by Garzon and coworkers [36, 37], binary gold–copper and nickel– aluminum clusters have been investigated by the group of Johnston [38, 39], together with several other binary compositions in [40]. 4.1.3. Basin-hopping algorithm The basin-hopping algorithm can be simply described as the coupling of a Monte Carlo–Metropolis procedure to local minimization. The unbiased basin-hopping algorithm starts from a random configuration, and explore the PES through moves that modify the geometry of the cluster. Every time a move is attempted, the new configuration is compared to the starting one by a Metropolis procedure only after a local minimization is performed. This corresponds to an effective replacement of the actual PES with a modified surface that is the step-function ideally obtained associating each point of the configuration space to the energy of its nearest local minimum. In Figure 5 an example of a basin-hopping walk is shown.
Figure 5 In the basin-hopping algorithm, the Potential Energy Surface is transformed in a step function. In the transformed PES, each point of the configuration space is associated to the energy of its nearest local minimum. In the figure, points from A to F show a possible path explored by the algorithm. Starting from the random configuration A, the structure of the cluster is locally minimized to get the local minimum B. The algorithm then applies a move that gives the cluster C, and a further local minimization reaches the point D. Move from B to D is accepted because D has an energy lower than B. The next random structural modification applied to D gives the E cluster, which corresponds to the F global minimum through local minimization
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The basin-hopping algorithm has revealed to be very efficient in locating the global minimum structure of single-funnel PES, and its results can be considered as effective as the ones coming from genetic algorithms. Nevertheless, several improvements are possible when dealing with multiple-funnel PES. During basin-hopping optimization, temperature is the only parameter that influences the move acceptance probability. If several funnels are present on the PES, the optimization goal can be reached quickly if the starting configuration lies in the global minimum funnel, or if the barriers separating the starting funnel from the global minimum one are low. And, what if this is not the case? A way to improve basin hopping effectiveness consists in performing parallel walks on the PES. Each walker, or trajectory, interacts with the others so that the spreading of the walkers through all the different funnels of the PES is maintained through the optimization run. In order to define what an interaction between walkers is, one can use an order parameter. Two walkers interact if there is overlapping between their order parameter values. Energy itself can be considered an order parameter, but better results can be achieved if the parameter distinguishes between structural motifs (for example, Common Neighbor Analysis signatures can be helpful). We tested such an algorithm on the bimetallic system, Ag32 Cu6 , and on the LJ38 cluster, and as a result we found that the standard basin-hopping results are improved by a factor of 3 and 2 respectively [41].
4.2. The Structures of Binary Metal Clusters Here we shall deal with the determination of the structural properties of binary metal clusters (nanoalloy clusters in the following), studied by means of both genetic and basin-hopping algorithms. We focus on nanoalloys of transition and noble metals, treating a series of systems which have been theoretically investigated in the last years [18, 38, 40, 42, 43], such as AgCu, AgNi, AgPd, AgAu, AuCu, AgCo and PdPt. Determining the structure of a nanoalloy cluster implies both the determination of its most favourable geometry together with its preferential chemical ordering. Possible structures of nanoalloy clusters belong to several structural motifs, like fcc bulk-like clusters, icosahedra, decahedra, polytetrahedra. To this series of structures, quite common in homogeneous clusters too, one can add polyicosahedral structures. Polyicosahedra are constructed by packing up together elementary icosahedra of 13 atoms (see Figure 6). As for chemical ordering, nanoalloy clusters can be intermixed, core-shell or even multishell. Presently, we are not aware of systems presenting complete segregation of the two elements separated by a planar interface. The most favourable structure is determined by the interplay of several factors, corresponding to different parameters. Some of these parameters may be deduced from quantities measured on the corresponding bulk systems. Other parameters, on the other hand, take into account specific surface or nanosize effects. In the following, we try to list the main factors governing the structure of nanoalloy clusters. a) Tendency of the two metals to separate or mix in the bulk phase. This is an important parameter; however, the tendency to mixing is enhanced in nanoalloys with respect to pure systems, so that elements that are immiscible in the bulk may mix at the nanoscale.
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Ag27Cu7
Ag32Cu6
Ag32Cu13
1
2
3
Figure 6 Some bimetallic polyicosahedral clusters. Silver atoms are grey, copper atoms are red. Starting from the left: the fivefold pancake (symmetry group D5h ), composed of seven interpenetrating Ih13 , the sixfold pancake (symmetry group D6h ), with its peculiar six copper atom ring, and the anti-Mackay icosahedron (icosahedral symmetry group)
b) Difference of surface energy between the two elements. The metal with the lowest surface energy may prefer to occupy surface sites, thus favouring a core-shell chemical ordering. c) Mismatch between atomic sizes. For example, transition and noble metals present a strong bond order-bond length correlation, their surfaces have a clear tendency to contract. In the case of nanoalloys, surface contraction favours inner positions for smaller atoms and surface positions for large atoms, because large atoms are more free to contract in this way. Size mismatch is thus another factor leading toward core-shell arrangements.
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d) Ability of the elements to accommodate atomic strain. Elements characterized by sticky atomic interactions do not accommodate the strain easily, so that they are not well suited for strained geometries, like those of icosahedra, polytetrahedra, polyicosahedra and (weakly) decahedra. e) Electronegativity difference between the elements. A difference of electronegativity usually favours intermixing. f) Quantum size effects. These effects play an important role especially for small sizes, at which electronic shell closing may favour specific sizes and structures. Nice examples of the interplay of these factors are found in the systems that we are going to analyse below. These systems are studied first within a semi-empirical Gupta potential model, by performing global optimization in the full configuration space. Then, the most representative structures are locally optimized within a densityfunctional approach. Two sizes are considered, 34 and 38 atoms, for each possible composition [18, 40, 42, 43]. AgCu. In bulk systems, Ag and Cu present a very weak tendency to mix. Indeed, their bulk phase diagram shows an extended miscibility gap. Ag has a lower surface energy than Cu. Moreover, Ag atoms are about 13% larger than Cu atoms, and the electronegativity difference between the two elements is negligible. All these features indicate that phase separation into core-shell structures should occur, with Ag atoms placed in the external shell. This is indeed the case, as demonstrated by the global optimization results. Silver atoms are always placed at surface sites. A rather striking result of the optimization is that binary cluster structures are almost always different from those of the pure elemental clusters. At size 38, for example, the Gupta potential model favours the fcc truncated octahedral structure for both pure Ag and pure Cu. Binary clusters of the same size, on the contrary, adopt structures which are either (possibly distorted) fragments of the icosahedron of 55 atoms or polyicosahedra. Among these clusters, the polyicosahedral family is of remarkable stability, especially at those compositions where perfect core-shell structures are formed. In fact, polyicosahedra are compact structures, with a large number of nearest-neighbour bonds for a given size. However, pure polyicosahedra have strongly compressed inner atoms. In AgCu clusters, the tendency to surface segregation of Ag brings the small Cu atoms to the interior of the cluster. Because of their size, these Cu atoms are not compressed in the interior of polyicosahedral structures. Several polyicosahedral clusters of remarkable stability have been singled out by the global optimization within the Gupta potential model. The most important (see again Figure 6) are Ag27 Cu7 D5h symmetry group, called fivefold pancake), Ag32 Cu6 D6h symmetry group, sixfold pancake), Ag30 Cu8 (Cs symmetry group), and Ag32 Cu13 (anti-Mackay icosahedron, icosahedral symmetry group). Among these clusters, Ag27 Cu7 is the most interesting. This composition presents clearly the lowest excess energy at size 34 (see Figure 7), with a global minimum which is very well separated from higher isomers. As we shall see below, this large separation leads to a high melting temperature. Moreover, Ag27 Cu7 presents a strong electronic stability, associated to a HOMO-LUMO gap of about 0.8 eV, as found by means of DFT calculations. This large gap can be understood from the fact that 34 is a magic size of the spherical jellium model. The high-symmetry structure of Ag27 Cu7 , which is common to several other AgCu clusters at this size, allows electronic shell closing,
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Size 38
Size 34
0.5
0 Δ21 0.5 0 –0.5 0
10
20
30
0
10
20
30 N1
Figure 7 and 2 as a function of the number of silver atoms, N1 , for AgCu (red) and AgNi (blue). Open symbols refer to the results of global optimization within semi-empirical potential model, while full symbols are the results of DFT calculations. At size N = 34 the magic composition (27, 7) is singled out. Reprinted from [42]
which causes a high gap and enhances the energetic stability of the cluster. On the contrary, the structures based on the sixfold pancake seem to be less favourable at the DFT level than in semi-empirical calculations. AgNi. This system behaves in analogy with AgCu, due to the even larger size mismatch and even stronger tendency of Ag to segregate at the surface. Since Ni is much more cohesive than Ag, core-shell polyicosahedral clusters with compact Ni cores are even more favourable. Because of that, sixfold pancakes are less favoured here than in AgCu, already at the level of semi-empirical modelling. Also in this system, the most remarkable cluster is Ag27 Ni7 . At variance with the corresponding AgCu cluster, Ag27 Ni7 has a magnetic ground state. The DFT calculations give a gap of 0.46 eV. Another structure of great stability is the anti-Mackay icosahedron Ag32 Ni13 . This structure becomes favourable when size mismatch is relevant. AuCu. This system is again characterized by a relevant size mismatch, since the size of Au atoms is very close to the size of Ag atoms. The main difference between AuCu and the previous systems is that Au and Cu mix in the bulk, forming a series of ordered phases. Moreover, the difference in surface energy of the two elements is somewhat smaller. For these reasons, polyicosahedral clusters are still formed, but the core-shell arrangement is less clearly favoured. This is especially evident in DFT calculations, while the semi-empirical modelling still favours perfect core-shell clusters. In fact, compared to AgNi and AgCu, in AuCu there are more discrepancies between DFT and semi-empirical results. The latter predicts fcc structures in the Cu-rich part of the sequence at size 38. This prediction has not yet been controlled by ab initio methods.
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AgCo, AuNi and AuCo. Again, these systems present a relevant size mismatch, no tendency to bulk mixing, clear surface segregation of either Ag or Au atoms. Therefore, the same kind of behaviour of AgCu and AgNi could be expected. To investigate this issue, several polyicosahedral structures have been optimized by DFT methods. The results have fully confirmed the expectations in the case of AuNi clusters, where the Au-rich perfect core-shell structures present the lowest excess energies. On the contrary, AgCo and AuCo clusters favour intermediate compositions, indicating a tendency of these metals to mix at the nanoscale. AgPd and AgAu. These systems are characterized by a small (or absent) size mismatch, and by bulk mixing, with the formation of a series of solid solutions. From these properties, we may expect that clusters with intermixed compositions would be the most favourable ones. This is confirmed by the results of the global optimization within the Gupta potential model. The global minima are polyicosahedral clusters on the Ag-rich side. These polyicosahedra are based on the sixfold pancake structure, with Ag occupying preferentially surface positions (this is due to the lower surface energy of silver). Compared to the fivefold pancake, the sixfold pancake is better suited to systems with smaller size mismatch for geometrical reasons. On the Pd or Au-rich side, the global minima are fcc structures. PdPt. These metals are again with a very small size mismatch. None of them is able to accommodate easily the strain, so that PdPt clusters are preferentially fcc. Fcc structures have a lower number of nearest-neighbour bonds than polyicosahedra, but they are essentially without strain. According to the Gupta potential model, Pd atoms occupy preferentially surface sites.
4.3. The Structures of Noble Gas Clusters Lennard–Jones potential has been first applied to the study of noble gas clusters, and a huge database of global minimum structures has been built [44–46 and references therein]. There are some exceptions, but on the whole the icosahedral ordering is the most common through the minima sequence up to N ∼ 1000. Despite the simple functional form of Lennard–Jones potential, there are several cluster sizes for which the global optimization task has revealed to be quite hard to accomplish. This is the case of N = 38, 75, 98, 102–104: all these global minima are not icosahedral, but the correspondent PES are multiple funnel PES, and the global minimum funnel has found to be in strict competition with the icosahedral one. This is why today this model represents the usual benchmark of any new global optimization algorithm. One expects that, as for bimetallic clusters, binary Lennard–Jones systems can be more complicated than homogeneous systems. Experimentally, there is some evidence that mixed rare-gas clusters can exhibit anomalous enrichment effects and radial segregation [47–49]. A computational analysis of these systems has been proposed by Florent Calvo and Ersin Yurtsever, who have studied through a modified basin-hopping algorithm mixed rare-gas systems like Ar–Xe and Kr–Xe at size 38. Their basin-hopping procedure consists in the parallel evolution of clusters of different composition. Beyond the usual move-and-check strategy of the Metropolis algorithm, each trajectory can exchange its structural motif with its adjacent trajectory, provided that the transmutation of two atoms is performed. As concerns Ar–Xe clusters, the authors find that
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this mixed system favours the polytetrahedral arrangement, and that at highly mixed or Xe-rich compositions global minima exhibit a core-shell arrangement, with the small Ar atoms in the cluster cores. The authors attribute this behaviour to the size mismatch between Ar and Xe, which implies a release of strain in polytetrahedral clusters. In fact, homogeneous polytetrahedral clusters have compressed internal atoms. Substituting internal atoms with atoms of smaller size causes a strain release. A sort of phase separation is observed in Kr–Xe too, but in this case the preferred structures are based on the fcc ordering, and there is no substantial mismatch between the two elements. Xe atoms in the core part of the cluster ensure a high number of mixed bonds, favoured by the mixed interaction parameterization. As a size-mismatch effect in Lennard–Jones clusters, we can also recall the stabilization of the four-atom Xe ring capped by an Ar atom in ArXe clusters, studied in [50]. Another computational study for heterogeneous Lennard–Jones clusters has been proposed by Doye and Meyer [51]. They have studied how the equilibrium distance of the mixed interaction can condition the size of high stability clusters, that is magic clusters. From a computational point of view, they perform global optimization runs during which cluster composition is allowed to change, so that both the geometrical and chemical ordering are optimized at the same time. Interestingly, they find that increasing the gap between the two equilibrium distances involved, polyicosahedral structures become more and more favourable, replacing the polytetrahedral minima that are typical of homogeneous Lennard–Jones clusters. Even if the metallic bond order– bond length correlation is absent, the formation of polyicosahedral clusters is possible as a size-mismatch effect.
5. Melting and Growth of Binary Metallic Clusters 5.1. Melting of Core-Shell Nanoclusters Core-shell nanoclusters can present interesting melting behaviour. For example, differential melting could take place: the external shell may melt at considerably lower temperature than the inner core. This has been shown by molecular dynamics simulations for clusters containing hundreds of atoms in CuNi [52] and AgCo [53]. Also in smaller 55-atom LiNa clusters, DFT-based MD simulations have shown a premelting of the external Na layer [54]. Besides MD simulations, melting and structural transformations of nanoalloy clusters have been studied also within the harmonic superposition approximation (HSA) [25, 42]. This approach was developed for, and applied to, homogeneous clusters, for example Lennard–Jones clusters [55]. Within this approach, a huge set of local minima is collected, and each minimum is weighted by its harmonic entropy. In this way, the cluster free energy F is evaluated by F = −kB T ln Z, where the partition function Z reads:
2 2 3/2 Es 3N−6 kB T mkB T (14) Is exp − Z=V 4 kB T i=1 si s where m is the mass of the cluster, Is and Es are the average momentum of inertia and the energy of minimum s. The frequencies si are those of the normal modes of minimum s and V is the volume of the box where the cluster is enclosed.
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δE 1
PGM
0 1
0
C 1.5
1
200
400
600
800
1000
T
Figure 8 Melting of pure clusters and of core-shell polyicosahedra. The top panel reports the caloric curves obtained by MD simulations ( E = E − EGM − 3N − 1kB T EGM is the global minimum energy), with energies in eV and the temperature T in Kelvin. The middle panel reports the occupation probability of the global minimum. The bottom panel reports the vibrational specific heat per degree of freedom, in units of the Boltzmann constant. Stars, diamonds and triangles refer to pure clusters (Ag38 Cu38 Ni38 respectively) while circles and squares refer to the five-fold pancakes (Ag27 Cu7 and Ag27 Ni7 ). The latter clusters melt at considerably higher temperatures than pure clusters in the same size range. Reprinted from [42]
The results of the harmonic superposition approximation are in good agreement with those of the MD simulations of the melting of highly symmetric, magic nanoalloy clusters [42]; however, there are indications that the harmonic approximation is not accurate in general. In the case of the fivefold pancake modelled by Gupta potentials, the agreement between the estimate of the melting temperature by means of the HSA and by the full MD simulation of the caloric curve is quite good (see Figure 8). Both Ag27 Ni7 and Ag27 Cu7 show a remarkable thermodynamic stability, which is associated to high melting temperatures. The analysis of the set of local minima collected for the HSA analysis shows that the global minima of these high-symmetry perfect coreshell clusters are very well separated from higher isomers. For this reason, melting temperatures are high.
5.2. The Role of Impurities in Metal Cluster Melting The melting of icosahedral Ag clusters, doped by a single impurity, has been recently studied by Mottet et al. [56] by means of MD simulations within a Gupta-like potential
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T = 525 K
T = 600 K
Figure 9 Snapshots from MD simulations of the melting of icosahedra of 55 atoms. Top row: pure silver. Middle row: silver with a Ni impurity. Bottom row: silver with a Pd impurity. One can observe that the cluster with the Ni impurity preserves its structure up to T = 600 K. Reprinted from [56]
model (Figure 9). Cu, Ni, Pd and Au impurities were considered. In the case of Cu and Ni impurities, the most favourable site is at the cluster centre. In fact, Cu and Ni atoms are considerably smaller than Ag atoms, so that an incosahedral cluster with these central impurities can contract and partially release the strain of the icosahedral structure. In fact (see for example [25]) radial (intershell) interatomic distances in icosahedra are contracted with respect to the optimal value of bulk crystal, while intrashell distances are expanded. A small central impurity can thus allow the cluster to relax toward a configuration with better interatomic distances, so that the resulting structure is of increased stability (even a central vacancy can increase the stability, see [57]). This can lead to an upward shift of the melting temperature, even in clusters containing hundreds of atoms. For a Ni impurity, the upward shift is of 70, 50, 30, and 20 K for clusters of 55, 147, 309 and 561 atoms respectively. On the other hand, Pd or Au impurities present a much less significant size mismatch, and do not prefer to be located at the central site. For these impurities, the shift of the melting temperature is negligible (Figure 10). The enhanced stability of the clusters with Cu and Ni impurities can be rationalized also by the inspection of the solution energy of the impurity Eimp , which follows from eq. (10) with m = N − 1 and m = 1. In our case Eimp reads: Eimp = EAgN −1 X1 −
N −1 1 EAgN − EXN N N
(15)
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where X = Cu, Ni, Au, Pd; EAgN is the energy of the (icosahedral) global minimum of the pure AgN cluster and EXN is the energy of the global minimum of the cluster XN . Even though the solution energy of Cu and Ni impurities in bulk Ag is strongly positive, the solution energy in icosahedra is clearly negative, more negative than for Au or Pd impurities whose solution energy is already negative in the bulk. Moreover, the atomic stress on the central site is greatly reduced in the case of Ni and Cu impurities, while it is slightly increased for Au and Pd impurities (see [56]).
5.3. Growth of Core-Shell and Three-Shell Nanoparticles The formation process of nanoalloy clusters by the addition of single atoms has been studied by MD simulations within the Gupta potential model for a series of binary systems (Ag–Cu, Ag–Ni, Ag–Pd) in [58, 59]. The simulations were started from a seed, namely from a pure cluster of element A given size and structure. Above this seed, atoms of B metal are deposited one by one. Both the cases of direct deposition (metal B has tendency to surface segregation with respect to A) and inverse deposition (metal B has the tendency to incorporate inside A) were treated (Figure 11).
5.0 55 atoms
ΔE (eV)
4.0 3.0 2.0 1.0 0.0 15
ΔE (eV)
147 atoms 10
5
0 450
550
650 Temperature (K)
750
Figure 10 Caloric curves for Ag icosahedra with a single impurity atom. The quantity plotted is E = E − EGM − 3N − 1kB T , where E is the total cluster energy, EGM is the minimum energy at 0 K. Crosses refer to pure Ag, solid circles to AgNi, squares to AgCu, diamonds to AgPd, and stars to AgAu. Reprinted from [56]
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Figure 11 Growth sequence for the deposition of Cu above an Ag-truncated octahedron of 201 atoms at 500 K. Single atoms are deposited each 7 ns. Snapshots are taken (from left to right) at the beginning of the simulation and after the deposition of 67, 108, 122, and 201 atoms. In the bottom row we show the cluster surface, in the top row a cross section of the cluster to show its internal arrangement. Ag and Cu atoms are represented in light and dark grey, respectively. Reprinted from [59]
In the case of direct deposition [58], the initial core was either a Cu or Pd cluster of size close to 200 atoms. Both icosahedral and fcc seeds were considered, and Ag atoms were deposited at different temperatures and deposition fluxes. The simulations showed the possibility of growing well-defined (although strained) Ag shells of monoatomic thickness in a wide range of temperatures and fluxes. In the case of an fcc Cu seed, the external Ag shell induced a significant rearrangement of the Cu core to reduce the strain. Inverse deposition was simulated starting from either icosahedral or fcc Ag cores, depositing Cu, Ni and Pd atoms above them [59]. Depending on the seed structure and on temperature, either core-shell stable structures or three-shell onion-like metastable structures were formed. The three-shell onion-like structures, made of an external Ag layer of monatomic thickness, an intermediate Cu, Ni or Pd shell, and an internal Ag core, were obtained in the case of inverse deposition above fcc clusters. On the contrary, deposition on Ag icosahedral clusters was followed by fast incorporation of the incoming atoms and by the formation of a central core in the cases of Cu and Ni. Pd deposition above Ag icosahedral cores led to a structural transformation of the cluster into a decahedral particle. The formation of three-shell onion-like clusters was rationalized noting that single Cu or Ni impurities inside fcc Ag clusters preferentially occupy subsurface sites, where they can achieve a better strain relaxation. Therefore, the first deposited atoms stop just one layer below the surface, where they trigger the formation of an intermediate metastable shell. In the case of icosahedral Ag clusters, the most favourable site for a single impurity is the one at the centre of the cluster, and this triggers the formation of simple core-shell structures. The preferential subsurface position for single impurities and small aggregates has been also found in the equilibrium MC simulation of Ag–Co clusters [60].
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6. Conclusions In summary, we have shown that the computational methods and resources that are now available allow detailed studies of the properties of nanoparticles. Computational studies are of great help in the determination of the best structures depending on size and composition, of the evolution of the structures with temperature, and of the growth shapes that are formed in non-equilibrium conditions. These studies have a crucial role in the interpretation of experiments, and also in suggesting new possible fascinating structures and properties of nanoaggregates.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
25. 26. 27. 28. 29. 30. 31. 32. 33.
S. Schelm et al., Appl. Phys. Lett. 82 (2003) 4346 G. Prévot et al., J. Phys. Chem. B 106 (2002) 12191 K. Judai et al., J. Am. Chem. Soc. 126 (2004) 2732 A. S. Worz et al., J. Am. Chem. Soc. 125 (2003) 7964 K. Koszinowski et al., J. Am. Chem. Soc. 125 (2003) 3676 N. Toshima, Pure Appl. Chem. 72 (2000) 317 P. Alivisatos, Scientific American (sept. 2001) C. Binns, Surf. Sci. Rep. 44 (2001) 1 W. A. de Heer, Rev. Mod. Phys. 65 (1993) 611 M. Gaudry et al., Phys. Rev. B 67 (2003) 155409 W. Bouwen et al., Rev. Sci. Instr. 71 (2000) 54 L. Bardotti et al., Phys. Rev. B 62 (2000) 2835 K. Koga et al., Surf. Sci. 529 (2003) 23 J. L. Rousset et al., Phys. Rev. B 58 (1998) 2150 H. Portales et al., Phys. Rev. B 65 (2002) 165422 D. J. Wales, Energy Landscapes, Cambridge University Press, Cambridge, 2003 P. Yang et al., J. Mol. Struct. 755 (2005) 75 R. Ferrando et al., Phys. Rev. B 72 (2005) 085449 V. Rosato et al., Phil. Mag. A 59 (1989) 321 R. P. Gupta, Phys. Rev. B 23 (1981) 6265 http://www-doye.ch.cam.ac.uk/jon/structures/SC/potential.html A. P. Sutton et al., Phil. Mag. Lett. 61 (1990) 139 J. P. K. Doye et al., New J. Chem. 22, 733-744 (1998) The estimate is courtesy of Giovanni Barcaro, who has performed the local minimization of a Ag27 Cu13 cluster, starting from the minimum located by the semi-empirical model. The cluster has been minimized without any simmetry consideration, and with the DF module of the NWChem package that uses Gaussian-type orbitals for the solution of the Kohn-Sham equations F. Baletto et al., Rev. Mod. Phys. 77 (2005) 371 F. Baletto et al., J. Chem. Phys. 116 (2002) 3856 D. Reinhard et al., Phys. Rev. Lett. 79 (1997) 1459 D. Reinhard et al., Phys. Rev. B 58 (1998) 4917 M. Locatelli et al., Comput. Optim. Appl. 21 (2002) 55 S. Kirkpatrick et al., Science 220 (1983) 671 J. Lee et al., Phys. Rev. Lett. 91 (2003) 080201 X. Dong et al., Phys. Rev. B 70 (2004) 205409 I. L. Garzon et al., Phys. Rev. B 54 (1996) 11796
58 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
G. Rossi and R. Ferrando M. R. Lemes et al., Phys. Rev. B 56 (1997) 9279 D. M. Deaven et al., Phys. Rev. Lett. 75 (1995) 288 I. L. Garzon et al., Phys. Rev. Lett. 81 (1998) 1600 J. M. Soler et al., Phys. Rev. B 61, (2000) 5771 S. Darby et al., J. Chem. Phys. 116 (2002) 1536 M. S. Bailey et al., Eur. Phys. J. B 25 (2003) 41 A. Rapallo et al., J. Chem. Phys. 122 (2005) 194308 G. Rossi et al., Chem. Phys. Lett. 423 (2006) 1 G. Rossi et al., Phys. Rev. Lett. 93 (2004) 105503 G. Rossi et al., J. Chem. Phys. 122 (2005) 194309 Cambridge Cluster Database, http:// brian.ch.cam.ac.uk /CCD.html H. Leary et al., Phys. Rev. E 60 (1999) R6320 D. Wales et al., J. Phys. Chem. A 101 (1997) 5111 E. Fort et al., J. Chem. Phys. 110 (1999) 2579 M. Tchaplyguine et al., J. Chem. Phys. 120 (2004) 345 M. Tchaplyguine et al., Phys. Rev. A 69 (2004) 031201 S. M. Cleary et al., Chem. Phys. Lett. 418 (2006) 79 J. P. K. Doye et al., Phys. Rev. Lett. 95 (2005) 063401 S.-P. Huang et al., J. Phys. Chem. B 106 (2002) 7225 T. Van Hoof et al., Phys. Rev. B 72 (2005) 115434 A. Aguado et al., Phys. Rev. B 71 (2005) 075415 J. P. K. Doye et al., Phys. Rev. Lett. 86 (2001) 3570 C. Mottet et al., Phys. Rev. Lett. 95 (2005) 035501 C. Mottet et al., Surf. Sci. 383 (1997) L719 F. Baletto et al., Phys. Rev. B66 (2002) 155420 F. Baletto et al., Phys. Rev. Lett. 90 (2003) 135504 T. Van Hoof et al., Eur. Phys. J. D 29 (2004) 33
Nanomaterials: Design and Simulation P. B. Balbuena & J. M. Seminario (Editors) © 2007 Elsevier B.V. All rights reserved.
Chapter 3
Computer Simulation of the Solid–Liquid Phase Transition in Alkali Metal Nanoparticles Andrés Aguado and José M. López Department of Theoretical Physics, University of Valladolid, 47011 Valladolid, Spain
First-principles simulations of the structural, electronic and thermal properties of clusters constitute a subject of current intensive interest in computational physics and chemistry. However, obtaining many of those properties (search for the minimum-energy isomer as a function of cluster size, determination of the melting point, etc.) becomes computationally prohibitive already at small cluster sizes for the traditional (ab initio) methods of quantum chemistry. In this chapter, we demonstrate that the orbital-free version of Density Functional Theory (DFT) is a useful and accurate alternative to orbital-based methods for some of these problems. The computational expense of the orbital-free method scales linearly with the number of atoms N forming the cluster, which contrasts with the scaling laws (proportional to N 3 ) of traditional ab initio methods, and allows to considerably expand both the range of tractable sizes and the length of molecular dynamics (MD) simulations, resulting in improved ergodicity. Armed with this equipment, we analyze the melting behavior of unsupported alkali metal nanoparticles. The irregular size dependence of melting points, latent heats and entropies of fusion observed in calorimetric experiments on Na clusters is properly reproduced. The anomalously high melting point of some clusters is related to a higher-than-average compactness degree, which is only possible for specific cluster sizes, and results in an enhanced stability of the cluster surface shell. The evolution of the entropy of melting in Na135 –Na147 clusters is rationalized in terms of a surface premelting mechanism. The dynamical instabilities leading to melting are analyzed in detail, providing a tentative explanation for the local melting point maximum at Na+ 142 . We also provide a brief review of previous simulations of melting of alkali nanoparticles by us and other groups. 59
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1. Introduction There is considerable interest nowadays in obtaining an understanding of the physical and chemical properties of small atomic clusters, both because of fundamental and practical reasons. At the applied level, the continuous request for further miniaturization of electronic and/or optical devices is pushing the size of their several components to the nanoscale limit. Also, atomic clusters can serve as building blocks in the construction of new, nano-assembled materials, which may possess novel properties. At the fundamental level, it is very interesting to analyze how the bulk-like properties emerge from those of large clusters of increasing size, which could enhance our present understanding of solid-state physics. Computer modeling of materials, based on well-established quantum-mechanical theories (essentially, Hartree-Fock, Density Functional Theory and their variants), has already become an indispensable tool in the physical interpretation and understanding of many properties of small molecules and macroscopic solid-state systems. In order to extend the range of applicability of computer modeling to complex problems, it is necessary to devise first-principles electronic structure methods whose computational efficiency scales linearly with system size. This is a very important practical issue as traditional methods from quantum chemistry cannot yet routinely afford the efficient simulation of, for example, thermal properties of large clusters (containing from several tens to several thousands of particles) due to their inherent poor scaling (the computational expense of typical implementations of the Hartree-Fock method, for example, increases with the number of particles N as N 3 ). In this chapter, we show some simulation results obtained in our group about the structural and thermal properties of alkali nanoparticles of medium and large size. We employ the Orbital-Free Density Functional Theory (OFDFT) first-principles method [1], whose efficiency scales linearly with the number of particles and that is ideally suited to the study of systems with metallic bonding, combined with classical MD to follow the time evolution of clusters at finite temperature. The OFDFT method is based on the Hohenberg–Kohn theorem [2] establishing that knowledge of the electron density alone suffices to obtain all ground state (GS) properties of an electronic system. Combined with the Born–Oppenheimer approximation, it allows all the atomic properties of a system of electrons and nuclei to be obtained. A very brief description of this model is given in Section 2, together with references to previous works that provide a full description. After that, we offer in Section 3 our most recent simulation results on the melting-like transition in Na clusters, together with a short location summary of previous works on the structural and thermal properties of alkali nanoparticles. Finally, Section 4 closes the chapter with some concluding remarks.
2. Theory The details of our implementation of the OFDFT scheme have been described at length in previous work [1], so we just present briefly the main technical issues. Within this formalism, the GS energy of a system composed of N ionic cores and Ne electrons is written as a functional of the electron density n(r) containing the following terms: the kinetic energy functional Ts n for a noninteracting electron gas with the same density as
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that of the interacting system of interest, the classical electron–electron repulsion energy (Hartree term), the interaction energy between the electrons and any external potential, the exchange-correlation energy functional, and the classical Coulomb repulsion between positive ions. In practice, we take n(r) to be the electron density due just to valence electrons, and thus the external potential always contains the contribution provided by the instantaneous configuration of ions (here, each ion contains a nucleus and the corresponding set of core electrons). Three key approximations in the energy functional involve the electronic kinetic and exchange-correlation components, and the electron–ion interaction. The local density approximation is used for exchange and correlation [3, 4]. The ionic field acting on the electrons is represented by an evanescent-core local pseudopotential developed by Fiolhais et al. [5]. The parameters entering this pseudopotential as given in the original work are optimal only for bulk systems under linear response theory conditions, and thus may not be fully appropriate for cluster studies [6]. Thus, in the main results shown in this chapter we will employ cluster-adapted pseudopotential parameters, obtained by requiring the OFDFT to reproduce Kohn–Sham (KSDFT) results for the interatomic forces and relative energies of selected cluster configurations, as explained elsewhere [7, 8]. OFDFT reproduces KSDFT interatomic forces for sodium clusters in a wide size range to within 5%, and energy differences between isomers to within 2%. The results are of a slightly lower quality for heavier alkali metals like Cs, and the error in the forces can grow up to 10% for alkali nanoalloys (which is nevertheless an acceptable error, given the much lower computational complexity which results in better statistical sampling of phase space along the MD simulations). In some previous results on the melting of alkali clusters obtained by our group [9–12], we directly employed the original pseudopotential parameters [5], which constitutes an obvious source of error in those older calculations. Regarding the kinetic energy functional, in our initial set of calculations [9–12] we employed the gradient expansion carried to second order [13], which is the sum of a Thomas–Fermi term (TTF n) and a fraction (1/9TvW n) of the von Weizsacker term. By comparing the results of OFDFT and KSDFT calculations, we have recently verified that this is not a very good kinetic energy functional for alkali clusters. Thus, in our most recent set of results [7, 8] we use either the combination (TTF + 04TvW ) or a nonlocal kinetic energy functional based on the average density approximation [1, 14–16], which is fully derived from first-principles arguments and therefore contains no empirical parameters. The results obtained from these two functionals are quite comparable for the systems considered here, although the first-principles functional is obviously to be preferred on theoretical grounds. Assuming now that our computer experiments are performed under periodic boundary conditions, we expand the valence electron density in a basis set formed by the plane waves periodic in the superlattice, and regard the coefficients of that expansion as generalized coordinates of a set of fictitious classical particles each of mass m, which allows us to write down the Lagrangian for the whole system of electrons and ions following Car and Parrinello [17]. For each fixed atomic configuration, the optimal electron density is obtained from the molecular dynamics of the electrons alone, reducing their velocities at every time step (electron annealing). Once the optimal electron density is known, Hellmann–Feynman theorem allows the forces on ions to be obtained, which results in an ab initio Born–Oppenheimer molecular dynamics procedure for simulations
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at finite temperature. By finding the correct GS density for each atomic configuration (Born–Oppenheimer MD), we can adopt the natural time step of the ionic system for integration of the ionic Lagrangian equation. This is determined by the highest vibrational frequency of the atomic system, which is usually much lower than typical frequencies of oscillation for the electron coefficients, due to the difference between electron and ion masses. Integration of both electronic and atomic equations of motion is performed by employing Verlet algorithm [18]. Technical details, such as the choice of appropriate time steps or electronic fictitious masses, are different in general for each specific problem. The explicit functional of the density we employ for the electronic energy is much superior in computational speed and memory requirements to the conventional KSDFT orbital approach, allowing the treatment of larger systems for longer simulation times. First, we need to solve M Lagrangian equations, where M is the number of plane waves, instead of Ne M orbital coefficient equations found with the KS functional; secondly, orthogonality constraints are avoided. Thus, the scaling of computer time with system size is approximately linear. Memory requirements also scale linearly, as opposed to quadratic scaling in KS approaches. Linear scaling is thus obtained in a natural way, without invoking the existence of a band gap in the spectrum of quasiparticles, as it does in the KS scheme [19]. Therefore, the orbital-free technique is particularly appealing for metallic systems, which do not have a finite band gap. A related advantage for extended metal systems is that Brillouin-zone sampling, which is necessary to obtain accurate KS results for metals, is not needed in OF calculations as it only affects the wavefunction, not the density itself. At a practical level, the fundamental difference between KS and OF realizations of DFT is that an explicit approximate expression for the functional dependence of the electronic kinetic energy on the electron density is needed in OFDFT. This accounts for the computational superiority of OFDFT, discussed in the previous paragraph, but at the same time results in comparatively less accurate results. Nevertheless, finding a very accurate expression for Ts n would not immediately imply that OF calculations with fully predictive power are possible for metals. The reason is that the flexibility provided by the auxiliary orbitals can be additionally exploited in KSDFT by defining different pseudopotentials for atomic orbitals of each angular momentum. Thus, the interaction of core electrons with s- and p-like valence electrons, for example, is described by different channels of a nonlocal pseudopotential [20]. In OFDFT, only a representation of the electron density is available, and thus employment of local pseudopotentials is mandatory.
3. Melting-like Transition in Unsupported Homogeneous Alkali Clusters 3.1. Background The melting point Tm of a finite atomic system is expected to decrease from its corresponding bulk limit as the number of atoms N is reduced, because of the increased surface-to-volume ratio. This is indeed the observed behavior at the mesoscale level. At the nanoscale, however, clusters formed by a few hundred or less atoms show important
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deviations from such a classical law. Jarrold and coworkers [21] have demonstrated that small gallium and tin clusters melt at temperatures higher than Tmbulk . Regarding alkali clusters, which are the main subject of this chapter, calorimetry experiments by Haberland and coworkers [22–26] show that the size dependence of Tm is not monotonous for Na+ N clusters in the size range N≈50–350. In the first set of calorimetric experiments [22], where only a reduced size range was covered, maxima in the Tm N curve were associated to joint electronic and structural enhanced stability. For example, a local maximum in Tm was observed for Na+ 142 and the conjecture was made that the + close proximity of both geometrical (Na+ 147 ) and electronic (Na 139 ) shell closings produced that maximum. However, later experiments in a wider size range [26] confirmed that Tm -maxima are not correlated in general either with known electronic or geometrical shell closings. A very recent conjoint analysis of calorimetry experiments and photoelectron spectra [27] has demonstrated that maxima in the latent heat and entropy of melting are indeed correlated with geometrical shell closings, which demonstrate that electronic effects do not play an important role in the melting of sodium clusters with more than 50 atoms. For clusters with less than 50 atoms there is some ambiguity in defining a melting temperature, as the melting transition proceeds in a gradual way. However, measurements of the temperature dependence of the photoabsorption cross sections for Na + N (N = 4–16) have been reported [28, 29]. Although the spectra do not show evidence of a sharp melting transition, they show a characteristic temperature evolution, where the different peaks appreciated at low temperature gradually disappear (merge into a single broad peak) upon increasing the temperature over a certain critical value. The detailed analysis of calorimetric experiments has also allowed to demonstrate that the microcanonical specific heat may be negative at melting for some cluster sizes [25]. Also, the relative importance of energetic and entropic effects on the melting transition of Na clusters has been elucidated [30]. There have been several attempts in the past to analyze the melting-like transition in alkali clusters from computer simulation, some of which we briefly review now in roughly chronological order. Röthlisberger and Andreoni [31] first employed ab initio KS-MD simulations in an analysis of the melting transition of small NaN clusters (N = 8–20) and found that the cluster structure is still rigid at 240 K. However, due to the computational complexity of the calculations at that time, simulation lengths of 3–6 ps were used, which are nowadays recognized as clearly insufficient to obtain meaningful statistical averages. Bulgac and Kusnezov [32] considered the thermal properties of Na 7 – Na9 clusters in an isothermal statistical ensemble, with the atomic interactions modeled by a many-body parameterized potential. Their results suggest that melting appears at approximately 100 K, whilst evaporation sets in at about 800 K. The clusters are found to adopt elongated cigar-like shapes just before the evaporation threshold temperature. Poteau et al. [33] used Monte Carlo (MC) simulations of small Na clusters with a distance-dependent tight-binding hamiltonian to describe the electronic system. They found Tm = 200 K for Na8 and Tm = 300 K for Na20 , with a two-step melting transition in the case of Na20 . Blaise et al. [34] performed extended Thomas–Fermi calculations on finite temperature properties of Na clusters of large size, observing that the averaged radial ion densities loose their structure as the temperature is increased, but did not include a detailed analysis of the variation of Tm with cluster size. Quadrupolar-shape deformations in the liquid clusters were identified and analyzed. Bonacic–Koutecky et al. [35] reported ab initio KS-MD simulations of melting in small Li clusters. They
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analyzed in detail the different dynamical behaviors of several isomers of Li+ 9 , and also concluded that even at 500 K these small clusters cannot be considered to be in a fully developed liquid state, as they remain for relatively long times on the potential energy surface (PES) basin corresponding to the GS isomer. Rytkönen et al. [36] reported a KSMD simulation of melting in medium-size Na clusters with up to 55 atoms, employing, however, a high heating rate of 5 K/ps. The main conclusion from their work is that a cluster of this size can already show a well-defined (more or less sharp) melting transition at temperatures which, at least for Na55 , are within the range of experimental values. Aguado et al. [9, 10] reported the first orbital-free molecular dynamics simulations trying to explain the size variation of Tm for sodium clusters in a wide size range. Specifically, NaN clusters with N = 8, 20, 55, 92 and 142 were analyzed, well within the size range considered in the calorimetric experiments. The melting points obtained were in very good agreement with experiment for Na92 and Na142 , but for Na55 the experimental melting temperature is substantially higher than the OFMD result. Important premelting effects were observed for all sizes except Na55 , and identified as surface melting for the two largest clusters. For the smallest-size clusters (Na 8 and Na20 ), the premelting phenomena are associated with isomerization transitions involving just specific subsets of atoms, which occur at temperatures lower than that corresponding to homogeneous melting [9]. For Na8 , the calculated melting point agrees well with the temperature at which the experimental photoelectron spectra starts to undergo significant variations [28], and the melting transition proceeds over a wide temperature interval, which also agrees with the experimental observation that melting proceeds in a gradual manner. Low-temperature structures of Na clusters are usually icosahedral-like, and thus present a well-defined distribution of atoms in concentric shells, that is, the radial density of atoms is very low or zero in the intershell regions. An interesting question then is, to what degree the way the melting transition proceeds is determined by this radial order. Aguado et al. [11] have analyzed this problem by performing OFMD simulations of the melting-like transition in Na clusters showing no apparent order (amorphous clusters). Results from this study show that the melting transition is not abrupt in amorphous Na clusters, but develops in an almost continuous way upon increasing the temperature, which means there are no appreciable free energy barriers against diffusion of atoms, just kinetic impediment at low temperatures. Therefore, the transition proceeds without any significant latent heat of fusion. However, this is true only for those clusters which show both orientational and radial disorder: an apparently disordered isomer of Na142 showed a well-defined first-order-like melting transition. A structural analysis found a distribution of atoms in radial shells, even though those shell have orientational disorder. This cluster cannot be considered to be fully amorphous. The melting-like transition in icosahedral clusters of K, Rb and Cs of several sizes has been studied by Aguado [12]. Qualitative features of the melting transition were found to be similar to those of Na clusters, which means a separate surface premelting stage was observed in all cases. The homogeneous melting temperature decreases with increasing atomic number as in the bulk limit, but the percentage value of this reduction in Tm is larger than for bulk materials. The height of the specific heat peaks decreases and their width increases with increasing atomic number, and the premelting effects are more important the heavier the alkali atom. These trends were rationalized in terms
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of physical features of the local pseudopotentials employed to describe electron–ion interactions. The preponderance of surface premelting effects in alkali clusters was also identified in MD and MC simulations performed at that time by Calvo and Spiegelmann [37], who were able to investigate a larger number of different sizes by employing a phenomenological many-body interatomic potential. They observed that premelting and melting peaks in the specific heat of sodium clusters approach each other as the cluster size increases. All our calculations described in the previous paragraphs were obtained with the methodology described in Section 2, but with the (TTF + 1/9TvW kinetic energy functional and bulk-adapted local pseudopotentials. Thus, while those results are certainly useful to analyze qualitative trends, there is no reason for them to reproduce experimental observations with quantitative accuracy. We will present in the next section our most recent set of results on the melting of Na clusters, obtained with an improved OFDFT methodology. For example, Aguado and López [38] have very recently revisited the melting of Cs55 employing cluster-adapted pseudopotentials obtained from a forcematching procedure, as described above. These more accurate results show that Cs55 melts homogeneously, without a separate surface premelting stage. Previous calculations with bulk-adapted pseudopotentials may therefore have magnified the importance of surface premelting. Also, Vichare et al. [39] found that specific heat peaks for Na8 and Na20 are wider than found in our initial work. We are presently revisiting also this problem with the more accurate energy functional and longer simulation lengths in order to assess the accuracy of our previous simulations. Lee et al. [40] reported computer simulations of melting in clusters of several metals modeled by a simple potential. The virtue of this work is that the simplicity of the potential employed allows breaking the potential energy of interaction into atomic components. In this way, they were able to interpret a rich variety of behavior (premelting versus homogeneous melting, sublimation and tendency to amorphization) in terms of a single parameter measuring the relative stability of surface and interior atoms. However, although such a correlation is very appealing because of its conceptual simplicity, Calvo et al. [41] have provided counterexamples which demonstrate that it is by no means universal. In particular, it may be a consequence of nonergodic behavior in the MD simulations. Calvo and Spiegelmann [42] considered the possible influence of a finite electronic temperature on the melting of Na clusters through exchange-MC simulations. While this effect was found to be small, this work also demonstrated that the improved ergodicity in exchange-MC as compared to usual MC tends to suppress the premelting peaks in the specific heat, which in fact are not observed in the experiments. Reyes-Nava et al. [43] were able to reproduce a negative specific heat in microcanonical MD simulations of the melting transition of some sodium clusters, and provided an interpretation in terms of the widths of kinetic energy distributions. Calvo and Spiegelmann [44] made the observation that premelting and melting peaks are much closer in tight-binding than in embedded-atom simulations, suggesting that an accurate description of atomic interactions and, in particular, an explicit description of electronic degrees of freedom may be important in quantifying the relevance of premelting effects to the melting phenomenon in Na clusters. Manninen et al. [45] have recently performed MD simulations of melting in NaN with N = 40–355 and reproduced a nonmonotonic size evolution of Tm , in qualitative agreement with experiment. However, due to computational limitations,
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interatomic forces in all clusters with N <93 are calculated employing a phenomenological potential, and the Tm N curve is not in good agreement with the experimental one for N >142. The reasons for this discrepancy may be: (a) purely electronic effects (not well described by any phenomenological potential) have a strong influence; (b) interatomic forces provided by that potential might not be accurate enough so that correct GS isomers (for example, those predicted by an ab initio methodology) are not reproduced; and (c) even if the potential correctly reproduces the ab initio energetic ordering of isomers, it is very difficult to locate the GS isomer when so many atoms are involved; etc. Agreement with experiment is much better for smaller clusters. Specifically, Manninen + et al. [46] reproduce the experimental result Tm Na+ 55 > Tm Na 93 . Finally, Lee et al. [47] and Chacko et al. [48] have made an intensive computational effort to study the melting transition in Na clusters with up to 142 atoms with KSDFT calculations. Where direct comparison to experimental melting points is possible, agreement is very good, which leads the authors to conclude that a quantum-mechanical description of metallic bonding is crucial to attain quantitatively accurate results. Irregular variations of Tm are also observed in the small size regime N = 8–50.
3.2. OFMD Simulations of Melting in Na Clusters We are going to describe here the main results of our most recent simulations of the melting phenomenon in alkali clusters [7, 8]. In our opinion they represent an important contribution to cluster research, as their results reproduce many of the experimental observations with surprising fidelity and identify some systematics in the evolution of thermal properties with size and structure. Taken in conjunction with all previous experimental and theoretical efforts summarized in the previous subsection, the present results demonstrate that the melting process in sodium clusters finally begins to be understood on a sound physical basis. 3.2.1. Irregular variation of the melting point in a broad size range As a first step toward obtaining an understanding of the calorimetric experiments, we consider here the melting-like transition in unsupported NaN clusters with N = 55, 92, 147, 181, 189, 215, 249, 271, 281 and 299, as modeled by orbital-free isokinetic molecular dynamics simulations. These specific sizes are chosen because they are close to local minima or maxima in the experimental Tm N curve of Na+ N cluster ions [23, 26]. We will find that the OFDFT simulations are able to reproduce the irregular size dependence of the melting temperatures Tm observed in the calorimetry experiments at a quantitative level. We will also find that structural effects alone can explain all broad features of experimental observations. Specifically, maxima in Tm N correlate with a high surface stability and with structural features such as a high compactness degree. The two parameters entering the Fiolhais pseudopotential (named and R in [5]) are obtained by fitting to KSDFT calculations. Specifically, we have performed KSDFT static calculations on 20 representative cluster geometries obtained by preliminary MD simulations (including different sizes and structures – icosahedra, decahedra, disordered, etc. – at different temperatures). KSDFT calculations are performed with the SIESTA code [49], employing nonlocal Troullier-Martins pseudopotentials [50] and the LDA
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approximation [3, 4] to the exchange-correlation functional EXC n (this is the same EXC n employed in the OFDFT calculations). We request that OFDFT calculations, performed with our local pseudopotential, reproduce SIESTA results for the interatomic forces and energy differences between cluster geometries. For the optimal pseudopotential thus obtained ( = 3799 R = 0445), the average deviation between OF and KS forces is smaller than 5%. The energy differences are reproduced with an average error smaller than 1%. Figure 1 provides an explicit demonstration that OF forces are of KS accuracy. This level of agreement is uniform with respect to cluster size (within the considered size range) and temperature, and is the same for structures other than those included in the fitting set. Once we have our energy model defined, we have located candidate GS isomers of NaN clusters of sizes defined above. Each cluster is placed in a unit cell of a cubic superlattice of length L (L = 25 a.u. for Na55 and 48 a.u. for Na299 ). The energy cutoff in the plane wave expansion was of 20 Ryd. Minimum-energy isomers were located by simulated annealing, performed at a cooling rate of 0.2 K/ps. As starting geometries, we have employed high-T liquid-like (disordered) and solid-like structures with icosahedral, decahedral and cuboctahedral symmetries. The initial temperature T of the solid-like structures was the highest one which does not lead to melting of the cluster. For all sizes, lowest-energy structures were found by annealing the icosahedral structures. The final structures (excepting Na55 Na147 and Na299 ), although resembling incomplete Mackay icosahedra, are distorted rounded geometries which try to attain an optimal packing and be as spherical as possible. Two examples of these structures are shown in Figure 2. A subsequent heating–cooling run of these structures did not locate lower-energy isomers. For clusters of this size, it is quite unlikely that we find
0.006
Fz (a.u.)
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Fx (a.u.)
0.004 0.002 0 –0.002 0.004 0.002 0 –0.002 –0.004 –0.006 –0.008 0.004 0.002 0 –0.002 –0.004 –0.006
0
10
20
30
40
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60
Atom number
Figure 1 Comparison between SIESTA (full symbols) and OFDFT (continuous line) interatomic forces, for a Na55 atomic configuration extracted from a run at 150 K, which was not directly employed in the pseudopotential fitting. Reprinted with permission from [Phys. Rev. Lett. 94, 233401 (2005)]. Copyright (2005) American Physical Society
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Figure 2 OFDFT GS isomer of Na181 and Na215 found by simulated annealing. Atoms at the surface are represented by golden spheres and those at the interior by blue spheres. Reprinted with permission from [Phys. Rev. Lett. 94, 233401 (2005)]. Copyright (2005) American Physical Society
the absolute minimum-energy isomer, irrespective of the sampling method. We have chosen simulated annealing because it provides a reasonable efficiency/cost ratio, and is sufficiently inexpensive that it allows for the employment of first-principles forces at all stages of optimization. For each cluster size, we have performed isokinetic Born Oppenheimer MD runs, in which the average kinetic energy is kept constant by velocity rescaling and the atomic forces are evaluated from Hellmann–Feynman theorem. The time step is 3 fs, and the total simulation length for each size is 5 ns (for those temperatures close to the transition region, the simulations were 300 ps long, while shorter runs of 150 ps were employed at other temperatures). We employ multiple histogram techniques [51] in order to extract smooth caloric and specific heat curves. Statistical sampling is necessarily less complete than that achieved in [51] due to the computational expense of firstprinciples calculations, but we have checked that it is adequate to extract accurate caloric curves. A representative example of these curves is given in Figure 3, while melting temperatures Tm (read from the specific heat maxima) and latent heats q (estimated from the step height between liquid and solid branches of the caloric curve) are given in Table 1 and Figure 4. Both values are in quantitative agreement with the experimental determinations [26]. Although not shown explicitly, this means that the entropies of melting also agree with experiment. This level of agreement has not been achieved in previous simulations over a broad size range. The specific heat curves are similar for all sizes, and contain a single peak, which is slightly wider on the low-temperature side. Analysis of the diffusion constants of different sodium atoms indicates that melting initiates at surface atoms, providing an explanation for the peak asymmetry. Surface and homogeneous melting temperatures are so close that merge into a single specific heat peak. The absence of premelting signatures in the experimental melting curves (note that this does not mean that surface premelting does not appear – see next section) is thus reproduced by the OFDFT calculations. In the following, we try to identify some energetic and/or structural trends that help to rationalize the oscillations in Tm N . If, as we observe, melting nucleates at the cluster surface, it is sensible to think that the stability of surface atoms plays a role in
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69 16
–6
14
–6.02
12
latent heat
10
–6.04
Cv /NkB
Total Energy (eV/atom)
T*
8
6 –6.06 4
160 180 200 220 240 260 280 160 180 200 220 240 260 280
T (K)
T (K)
Figure 3 Caloric curve (left) and specific heat per particle in units of kB (right) of Na299 . The latent heat q is obtained from the caloric curve as shown. T ∗ signals the beginning of melting. Reprinted with permission from [Phys. Rev. Lett. 94, 233401 (2005)]. Copyright (2005) American Physical Society
Table 1 Size variation of melting temperatures Tm , latent heats q, and several energetic and structural properties described in the text Size
Tm (K)
q (eV/atom)
(eV)
dsb (Å)
dss (Å)
d T ∗
55 92 147 181 189 215 247 271 281 299
280 206 256 252 217 249 225 245 222 251
0.011 0.004 0.014 0.013 0.006 0.011 0.006 0.007 0.008 0.013
0122 −0182 0057 0037 −0095 0012 −0203 0009 −0067 0014
3.552 3.612 3.604 3.603 3.640 3.612 3.645 3.638 3.650 3.615
3.683 3.670 3.708 3.709 3.698 3.716 3.708 3.710 3.704 3.720
0.009 0.008 0.005 0.006 0.002 0.004 0.003 0.002 0.001 0.002
determining Tm . This idea has already been put forward by Lee et al. [40] within the context of parameterized potential models. As a measure of surface stability, we employ i , with the average surface evaporation energy, defined as < Eevap N >= 1/Ns Eevap i = −EN + E i N − 1 + ENa the energy Ns the number of surface atoms and Eevap required to remove atom i from the cluster surface. This is a zero temperature calculation, and E i N − 1 is obtained from local structural relaxation to the nearest minimum after evaporation of atom i. As a measure of the stability of a surface atom, relative to an
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A. Aguado and J. M. López 1/N 0.005
15
0.015
0.001
V/N (Å3)
14.5 14 13.5 13 Experiment OFDFT
Tm (K)
280 260 240 220 200
50
100
150
200
250
300
N
Figure 4 Size variation of the volume per atom (top) and melting temperature (bottom) in sodium clusters. The dashed line in the upper panel is the best linear fit to the data. Reprinted with permission from [Phys. Rev. Lett. 94, 233401 (2005)]. Copyright (2005) American Physical Society
average atom in the cluster, we show in Table 1 the quantity N = < Eevap N > −Eb N , where NEb N = −EN − NENa is the average binding energy per atom.
N provides a measure of relative surface stability against evaporation. Although a corresponding measure involving diffusion instead of evaporation would be more appropriate in a discussion of melting, N provides at least a rough idea of the relative strength of surface bonds. N takes positive values precisely for the same sizes where a maximum in Tm is observed; moreover, the larger the value of N , the higher the melting temperature. This suggests that melting temperatures of sodium clusters are directly related to relative surface stability. It also agrees with the experimental observation that entropic effects play only a secondary role in determining melting temperatures, as compared to energetic effects [30]. Trying to find a structural origin for the enhanced surface stability of selected cluster sizes, we have evaluated the average volume per atom, defined as V/N = 4 r 3 /3N , with r the gyration radius of the cluster. Let us note that this quantity decreases with decreasing cluster size as the average interatomic distance contracts. Figure 4 shows the variation of V/N with cluster size, together with the best linear fit to the data. It is immediately appreciated that clusters with an “anomalously” high melting temperature are more compact. The enhanced surface stability is thus shown to be induced by a higher compactness degree, which can only be obtained for some particular sizes. In order to proceed further, we have divided each cluster into surface and “bulk” atoms (as shown in Figure 2), and evaluated the average distance between bulk atoms (dbb ), between surface atoms (dss ) and between surface and bulk atoms (dsb ). We have
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found that dsb shows a clear correlation with V/N (see Table 1): the higher the melting temperature of a given cluster, the shorter its dsb value. Table 1 also shows that clusters with shorter dsb have larger interatomic distances between surface atoms. As surface tension is driven by the tendency of dss to decrease (which contributes to the internal cluster pressure), those clusters with higher Tm have also larger surface tensions. As the cluster is heated, we observe that dsb distances expand, while dss slightly contract (which leads to the expected decrease in surface tension as temperature increases). At the very initial stages of the melting-like transition (identified from the initial increase in the specific heat curve, see Figure 3), Table 1 shows that d = dss T ∗ − dsb T ∗ is very close to zero. This result is independent of size and thus serves to identify a systematic behavior in the melting transition of Na clusters: all clusters show a reduction of d upon heating and all melt when d is close to zero, but different sizes approach this critical stage at different temperatures. Our calculations thus predict a simple systematics for the melting behavior of small sodium clusters in a broad size range. It is a well-known fact that surface metal atoms tend to undergo an inwards relaxation (bond-length contraction) in order to compensate for the reduced electronic density at the surface. At the same time, there is a strong energy penalty for dangling atoms, which would possess a very low coordination. Satisfaction of these rules leads to GS isomers with smooth surfaces (without any surface steps or dangling atoms; see Figure 2) which try to optimize ionic packing. The allocation of all surface atoms into a single, rounded, surface shell induces oscillations in dss and dsb as a function of size. When dss is relatively large, dsb may be more efficiently reduced, resulting in an additional stabilization of the cluster surface which inhibits melting. For example, the “anomalous” very high melting temperature of Na 55 is simply due to its higher compactness relative to other sizes. García–González et al. [52] have shown that kinetic energy functionals with a higher degree of nonlocality than the one employed here would be needed in order to reproduce electronic shell closing effects. Our results thus show that electronic effects can only play a secondary role, and are not needed in an explanation of the main Tm oscillations. We have calculated the energy eigenvalue gap at the Ne = 58 electronic shell closing for the Na55 cluster, for both solid and liquid configurations at temperatures close to Tm , using SIESTA. These values are averages over 10 solid-like and 10 liquid-like solid independent configurations extracted from the MD runs. The results (Egap = 023 eV liquid and Egap = 020 eV) show that both solid and liquid phases are similarly “magic” from the electronic point of view, which explains why electronic shell effects do not affect the melting transition significantly. A recent analysis of calorimetric and photoelectron experiments by Haberland et al. [27] also supports the idea that the size variation of Tm is controlled by geometric effects, providing further, independent support for our conclusion.
3.2.2. Variation of the melting point in a narrow size range As explained in Subsection 3.1, calorimetric experiments find a local maximum in the melting temperature for Na + 142 . This cluster size is very close and bracketed by an electronic and a geometric shell closing, which led to the initial assumption that a combination of electronic and structural features might explain the local maximum at N = 142.
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Here we will study the melting-like transition of Na N clusters with N = 135−147. Our goal is twofold: on the one hand, we would like to know if the orbital-free methodology also succeeds in reproducing the subtle size dependence of Tm in such a narrow size range. If it does, then we would like to find a physical interpretation of this observation. In this subsection, all details regarding the employed OF energy functional and the MD strategy are exactly the same as those employed in the previous subsection. The photoelectron spectra measured by Haberland et al. [27] corroborate that icosahedral symmetry dominates the structure of cold NaN clusters in the size range covered by the present study. Therefore, we assume that neutral NaN clusters, with N = 135–147, have icosahedral symmetry. Each cluster is placed in a unit cell of a cubic superlattice of length L = 33 a.u. The set of plane waves periodic in that superlattice, up to an energy cutoff of 20 Ryd, is used as a basis set to expand the valence electron density. Forces on atoms are calculated through Hellman–Feynmann theorem only after the GS density has been found. For all sizes within this range, it is found that the least bound atom is at a surface vertex position. Therefore, in order to locate the GS structure of Na141 , for example, we consider all geometrically inequivalent possibilities of removing 6 vertex atoms from a perfect 3-shell Na147 icosahedron, and find the optimal set of atomic coordinates by using a conjugate gradients routine. Some examples of GS structures obtained this way are shown in Figure 5. The structural trends may be summarized in the following terms: for N = 142–145, the surface vacancies created by the removal of vertex-like atoms tend to be as separated as possible – that is, they repel each other; for N = 137–140 – it is the vertex atoms that repel each other. Nevertheless, the energy differences between isomers obtained this way are of the order of 0.1 meV/atom, which means they can be considered as degenerate isomers in a practical sense. That is, isomers with different distributions of vacancies in vertex positions will be presented in the experimental beams. The thermal properties of the different isomers are not expected to be very different, due to surface premelting effects discussed below. Thus, from now on we only consider the GS isomers predicted by OFDFT. The total simulation length for each size in this study is 5 ns. Multiple histogram techniques [51] are again employed in order to extract smooth caloric and specific heat
Figure 5 OFDFT GS icosahedral isomers of Na138 and Na144 . Atoms at surface vertex positions are represented by red spheres and the rest of atoms by blue spheres. Reprinted with permission from [J. Phys. Chem. B 109, 13043 (2005)]. Copyright (2005) American Chemical Society
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243 K
– 6.24 Na147 Na141 Na137
Energy (eV/atom)
– 6.25
– 6.26
q – 6.27
269 K
– 6.28
255 K
– 6.29 200
220
240
260
280
300
T (K)
Figure 6 Representative sample of caloric curves obtained in this study. The arrows signal the corresponding melting temperatures, at which the derivatives of the caloric curves (the specific heat per particle) attain a maximum value. The latent heat per particle q is obtained from the caloric curves by measuring the energy difference between liquid and solid branches at T = Tm , as shown for Na137 . Reprinted with permission from [J. Phys. Chem. B 109, 13043 (2005)]. Copyright (2005) American Chemical Society
curves from the isokinetic runs performed at a discrete set of temperatures. A representative example of caloric curves is given in Figure 6, while melting temperatures Tm (read from the specific heat maxima), latent heats per particle q (estimated from the step height between liquid and solid branches of the caloric curve at T = Tm ) and entropies of melting per particle s (obtained from Clapeyron’s relation Tm = q/ s) are given in Figure 7. There is a striking agreement with experimental observations [27], namely: (1) q and s increase on average in the size range N = 135–147; and (2) a maximum in Tm is observed for N = 141 in OFDFT and for N = 142 in the experiment. The global trends are quantitatively reproduced and this is what we are interested in. The agreement is not expected to be perfect because of the approximations in the T n and EXC n energy functionals, possible inaccuracies in statistical sampling at the transition region, and the fact that experimental clusters are charged while simulated clusters are neutral. Another point of agreement is that there are no appreciable discontinuous signatures of premelting transitions in the caloric curves. For the smallest sizes, where premelting effects (see below) are most important, Figure 6 shows that there is a slight curvature of the caloric curve at temperatures lower than Tm , but not a discontinuous change as expected from a first-order phase transition. Our results for Na142 are also in very reasonable agreement with a recent KS simulation by Chacko et al. [48], which reports a melting point of 290 K and a latent heat of 14.9 meV/atom, while the experimental values are between OF and KS results. The average size evolution of q = Eliquid Tm – Esolid Tm /N is easiest to rationalize. In fact, if we instead consider the quantity q0 = Eliquid Tm – Esolid 0 K/N (not explicitly shown), it already displays the same average behavior. This means that the
A. Aguado and J. M. López
q (meV/atom)
74
16 14 12 10 8 6 270
Tm (K)
260 250 240 230 220
Δs/kB
0.7 0.6 0.5 0.4 135
136
137
138
139
140
141
142
143
144
145
146
147
Number of Atoms
Figure 7 Size variation of the latent heat per atom (top), melting temperature (middle) and entropy of melting per atom (bottom) of NaN (N = 135–147) clusters. OFDFT results are indicated by full circles, and error bars are experimental results [27]. Reprinted with permission from [J. Phys. Chem. B 109, 13043 (2005)]. Copyright (2005) American Chemical Society
binding energy decrease upon removal of an atom is smaller in the liquid than in the solid phase, that is, it costs less energy to evaporate an atom in a liquid cluster within the size range considered. This is expected as all atoms are tightly bound (highly coordinated) in the geometrically compact icosahedra considered here. This might not be the case if the solid phase contains, for example, floater atoms, that is, atoms outside a complete atomic shell, with a low coordination number. Oscillations of q about this average monotonous behavior, which are observed both in the experimental and OF results, are small in magnitude and more difficult to rationalize. We have evaluated the evaporation energies at zero temperature, Eevap N = E1 + EN –1 – EN , and observed that local deviations in q from its average size dependence are correlated with Eevap N (see Figure 8). We stress here that the energy differences between different icosahedral isomers are so small that they do not affect the results of Figure 8. No matter which allocation of vertex atoms is chosen, it costs more energy to evaporate an atom from Na141 than from Na142 , for example. Thus, deviations from strictly monotonous behavior in qN can be safely traced back to the size dependence of Eevap in (that is, relative stability of) the solid phase, but no simple explanation can be found for the size dependence of Eevap itself. The entropy per atom of a liquid cluster can be safely assumed to be approximately size independent [27], at least within a narrow size range as that considered here. Therefore, the size evolution of s must be explained in terms of the entropy per particle of the solid-phase clusters, for temperatures close to Tm . Our simulations predict (see below) that premelting effects are negligible for Na147 . However, important surface premelting
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1.29
3.57
Energy (eV/atom)
3.55 1.28 3.54
distance (Å)
3.56
1.285
1.275
1.27
Ebind/N
dss – 0.06
Eevap – 0.07
dsb
136 138 140 142 144 146
136 138 140 142 144 146
Number of atoms
Number of atoms
3.53
Figure 8 Size dependence of Ebind /N and Eevap (left) and of dss and dsb (right). In both sides, one of the quantities has been displaced along the vertical axis to help visualization. Reprinted with permission from [J. Phys. Chem. B 109, 13043 (2005)]. Copyright (2005) American Chemical Society
effects are observed for N = 135–146, which are associated with the diffusive motion of atomic vacancies at the surface. Specifically, two different premelting mechanisms have been observed: first, isomerizations where an atomic vacancy at a surface vertex site moves to a different surface vertex site. This mechanism, which is dominant in the size range N = 142–146, is illustrated in Figure 9: three surface atoms conjointly move along an icosahedral edge so that the vacancy can jump between two different vertex sites directly. A similar mechanism has been observed by Shimizu et al. [53] in simulations of the rapid alloying process in 2-dimensional bimetallic clusters, and named edge-running mechanism. Here we show that the same mechanism is operative
Figure 9 Snapshots taken from an OFDFT run on Na145 , showing the edge running premelting mechanism. The surface vacancy can jump directly between two different vertex sites due to the cooperative displacement of a whole edge of atoms (marked in red). Other two atoms are shown in green in order to better appreciate the mechanism. Reprinted with permission from [J. Phys. Chem. B 109, 13043 (2005)]. Copyright (2005) American Chemical Society
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in realistic, 3-dimensional clusters. The second is isomerizations where a surface atom moves from an edge to a hollow vertex position, so that the vacancy can explore also the edge sites at the surface. This mechanism is dominant for N = 135–141. We have never observed, within the length of the simulations, the vacancy to occupy a face position at the surface, which means that there are 20 surface atoms that do not displace during the premelting stage. From all the surface atoms, face-like atoms have the largest coordination, so creation of a vacancy at a face position costs more energy. If we take into account the fact that face, edge and vertex-like atoms at the surface shell have different radial distances (that is, the surface shell is formed by three different radial subshells), excitation of an edge atom to a hollow vertex site can be viewed as temporary thermal generation of a floater atom [54], which is later neutralized by a neighboring surface vacancy. These premelting effects produce an increase in the entropy per atom of the solid phase before the cluster melts, and provide an explanation for the size evolution of s. They also support the entropy model advanced by Haberland et al. [27] (which is indeed based on assuming mobility of surface atoms at temperatures lower than Tm ), at least for those sizes immediately smaller than a geometrical shell closing. The premelting effects start approximately 20 K below Tm for Na146 , while for Na135 start at temperatures as low as 100 K, more than 50% below Tm ! Figure 7 shows that present simulations reproduce the experimental observation that q and s are highly correlated in their size dependence. s is larger the higher the structural order of the solid phase just before melting, and therefore attains a maximum at N = 147. The fact that q is maximum at exactly the same size provides direct experimental evidence that electronic shell closing effects are at most of secondary importance to the cluster melting phenomenon. Otherwise, we would expect q to be maximum at a different size, closer to the electronic shell closing at N = 138. As the number of electrons does not change upon melting, this observation is not so strange. In any case, this is one important reason why present OFDFT studies, which do not reproduce electronic shell closing effects [1], may give accurate predictions about the melting of unsupported Na clusters. It should be stressed here that, although the premelting effects observed will result in surface diffusion in the long-time term, they are not characteristic of a surface melting stage (namely, a liquid layer on top of a solid core shell). The structural surface disorder is never high, as only a well-defined number of icosahedral isomers are visited during the premelting stage, which should be best described as a highly concerted isomerization stage. In fact, this is the reason why premelting does not lead to any marked feature in the caloric curve: the isomers visited have a substantially lower energy than those corresponding to a fully disordered surface. It has been pointed out by Haberland et al. [27] that the size evolutions of q and s show a high correlation with geometric shell closings, while the melting temperatures themselves show just a partial correlation. Also, Clapeyron’s equation tells us that knowledge of q and s suffices to determine Tm . Therefore, q and s are apparently more basic quantities than Tm to explain the irregular size dependence of the meltinglike transition. However, the intensive variable (in this case T is the one directly controlled in the experiments and present OFDFT simulations. Moreover, Clapeyron’s relation describes just the thermodynamics of melting, but the melting-like transition is a dynamic phenomenon, which must be initiated by a well-defined structural instability
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in the solid-phase cluster. This instability acts as a seed that triggers the development of the liquid phase. Thus, Clapeyron’s equation alone does not provide any physical insight as to why the maximum in Tm is located at N = 141. Let us elaborate this point in detail: With decreasing size starting from N = 147, the decrease of s is initially more marked than that of q, which results in an increase of the melting temperature Tm . In the size range N = 135–141, however, the rate of change of q is larger than that of s, which results in a Tm -maximum at N = 141. Why do the rates of change of q and s evolve with size in such a way that their ratio is a maximum at N = 141? It is important to notice that the value of s is not very sensitive to the precise location of the melting point, as the entropy of the solid phase increases for temperatures considerably lower than Tm . On the contrary, as the liquid and solid branches of the caloric curves have different slopes, q is much affected by a change in melting point. That is, if Tm was reduced by, for example, 10 K, satisfaction of Clapeyron’s equation would imply that the change in q would be much larger than the change in s. s is therefore a more stable quantity than q. In what follows, we suggest that identification of the structural instability mechanism may provide a more satisfying explanation for the size dependence of Tm than direct interpretation of the size evolution of q. The viewpoint adopted here is that the structural instability occurs at a given critical temperature, which is therefore the basic variable, while the latent heat q is considered a consequence of the melting transition. Irrespective of the premelting effects discussed above, the structural instability mechanism which triggers the melting transition is observed to be the same in the size range N = 141–147, and involves the conjoint thermal excitation of surface atoms. Just below Tm , the surface of the cluster is very fluxional, with surface atoms undergoing very large amplitude vibrations. The displacements of atoms are highly concerted, both in the tangential and radial directions: when the instantaneous radial position of a surface atom is displaced inwards, those nearest atoms in the radial shell immediately below expand in the tangential direction, and vice versa. Also, a set of neighboring atoms at the surface shell has instantaneously short interatomic distances only when their radial positions increase, and vice versa. At T = Tm , a critical stage is achieved when the distance between two neighboring surface atoms is temporarily so large that a neighboring third surface atom may move across the space left by the bond expansion. The net result is that the identity of vertex, edge and face-like atoms at the surface is interchanged. The whole process involves the concerted rearrangement of the positions of many surface atoms (as opposed to the surface vacancy migration outlined above), and propagates to involve the whole cluster surface very fast, which results in a disordered surface structure. Once the surface disorder is initiated, it acts as a catalyst for homogeneous melting, which occurs at the same temperature. Shimizu et al. [53] have also observed that surface disorder may lead to radial diffusion in 2-dimensional metal clusters. Our OFDFT simulations thus do not predict a separate surface melting stage, although surface diffusion is present at T < Tm for N < 147 through the premelting mechanisms discussed above. Figure 8 shows that evaporation energies are always larger than the corresponding binding energies per atom, implying that surface atoms have a higher stability than an average atom, in energetic terms. This may be the reason why homogeneous melting sets in as soon as the surface melts. It also shows that the relative stability of surface atoms increases from N = 147 to N = 135. This can be traced back to the radial distance
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dsb separating the surface atomic shell from the shell immediately below. This distance decreases with N by close to 1% within the size range considered here, while the average distance between surface atoms, dss , changes by less than 0.1%, which stabilizes surface atoms. Although barriers against evaporation are not necessarily correlated with barriers against diffusion, the results of Figure 8 are at least indicative that it costs more energy to break bonds at the cluster surface the smaller the size of the cluster. Thus, a higher temperature is needed to activate the structural instability when passing from N = 147 to N = 141. Figure 8 thus shows that there exists a correlation between the melting points in the size range N = 141–147 and the distribution of potential energy into core and surface regions. As mentioned in Section 1, Calvo et al. [41] have demonstrated such correlation may not be universal. Our OFDFT results predict that the liquid phase is directly accessed from the icosahedral isomer, which has the lowest energy at zero temperature. In contrast, a typical MC simulation would include contributions to the thermodynamic functions from other sets of solid-like isomers (decahedral, bcc-like, etc.), each with its appropriate statistical weight, which would make the melting point to depend on the global properties of the PES. It is still an open question which of these scenarios is more appropriate to describe the experiments of Haberland’s group, because there is no detailed information about the relative populations of isomers in the experimental beams as a function of temperature. Haberland and coworkers [27] report, however, that measured photoelectron spectra in this size range are best fitted by icosahedral structures, suggesting a very low population for isomers with other symmetries, at least at those temperatures where the spectra were measured. We have checked that the distribution of potential energy into core and surface regions is qualitatively the same for a family of decahedral isomers with N = 135–147, which would enlarge the limit of validity of the correlation for the case of Na clusters. In any case, this correlation, which considers just the local information about the PES provided by the icosahedral basin, is valid within our MD framework, which samples only the thermal instabilities of the icosahedral isomer. Were the structural instability mechanism which is a precursor of melting be the same in the whole range N = 135–147, we would expect an approximately monotonous increase of Tm with decreasing size, as the energy cost of breaking surface bonds continues to increase. OFDFT simulations predict, however, that the dynamical melting mechanism is different in the size range N = 135–140, providing a tentative explanation for the maximum in Tm . When the number of surface vacancies is considerable, wide amplitude concerted vibrations at the surface lead to the temporary creation of large “voids”, which can be filled with an atom from the atomic shell immediately below. This process does not leave a vacancy in the interior of the cluster because a surface atom (specifically, we always observe it to be a face-like atom, which has the shortest radial distance) moves to the inner part almost at the same time. This means that radial diffusion is activated at a lower temperature than surface diffusion for these sizes, that is, interlayer mixing is easier than intralayer mixing. The process is again highly concerted as the two atoms involved (one leaving and the other entering the surface shell) are always far apart. Homogeneous melting sets in directly in these clusters, apart from the premelting effects leading to the increase of the solidphase entropy, discussed above. It is intuitively clear that interlayer mixing will be less impeded the larger the number of surface vacancies, which explains the trends observed in the size evolution of Tm . The picture that emerges from this study is that the decrease
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in Tm induces the large size dependence of the latent heat in this size range, that is, the energy of the liquid phase which is accessed at T = Tm is lowered relative to that of the solid phase on account of the difference in heat capacities of the two phases (see Figure 6). A realistic MD simulation of melting in clusters requires both accurate interatomic forces and statistical sampling. Improved ergodicity is usually obtained by employing phenomenological parameterized descriptions of atomic interactions, at the cost of employing much less accurate interatomic forces. On the opposite side, KSDFT simulations provide accurate forces but it is presently difficult to routinely achieve ergodicity for clusters of this size. (Although Chacko et al. [48] have very recently demonstrated that reasonably converged KSDFT simulations of melting of Na142 are feasible, such calculations are by no means routine. Present OFDFT simulations, on the contrary, may run on a single Pentium processor routinely. Moreover, the computational expense of the calculations does not increase much when dealing with clusters of about 300 atoms [7], as expected from a code which scales almost linearly with size. In this work, we have been able to simulate the melting-like transition for 13 different clusters with about 140 particles, which enlarges by more than one order of magnitude the total simulation length covered by Chacko et al.). We would like to stress here that present OFDFT results try to meet both accuracy criteria as close as possible. It has already been demonstrated that OFDFT forces closely match KSDFT forces for Na clusters of these sizes. Now, the added computational simplicity of OFDFT simulations [1] can be exploited to simulate larger systems for longer times in a routine way, as compared to KSDFT. Present simulations are between 300—and 400 ps long at the transition region, which is a considerable time for a first-principles method, especially if we are considering 13 different sizes. It is true that much longer simulations are needed in order to observe some phenomena. For example, the possible dynamic coexistence of liquid and solid phases at a given temperature has to be studied with parameterized potentials. Also, quantities such as diffusion constants or rms bond-length fluctuations are possibly not fully converged by present OFDFT simulations for those temperatures where the premelting effects are important (although they are well converged for fully solid or fully liquid phases). These structural quantities are converged only when the surface vacancies have explored all possible surface sites, which may take a long time at the premelting stage. The caloric curve, on the other side, is not so sensitive to this complete exploration because the premelting mechanism samples many isomers which differ only in the permutation of atoms, and thus have the same energy. We have checked that the caloric curves (the main results presented here) obtained by reducing by 30% the simulation times are the same within round-off errors. Full ergodicity is thus just a sufficient (as opposed to necessary) condition to obtain sufficiently accurate melting points. Present OFDFT simulations thus provide accurate forces, which is essential for a realistic simulation that can be meaningfully compared to experiment, and make an important effort in obtaining a reasonable statistical sampling. In summary, we have reported first-principles MD simulations that reproduce and explain for the first time the main trends observed in the calorimetric experiments [27] in the size range N = 135–147. The size evolutions of the latent heat and entropy of melting are found to be maximal for N = 147, which means they are correlated with geometric shell closings. It has been stressed this observation implies that electronic shell closing
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effects can only be of secondary importance to the melting process, at least within the size range considered in this work, which makes the OFDFT suitable to address the melting problem in sodium clusters. Important premelting effects, associated with the diffusion of atomic vacancies at the surface, are observed at temperatures below Tm for all sizes except Na147 , which explains the size evolution of the entropy of melting. In order to find an explanation for the maximum of Tm , observed for Na141 , the thermally induced structural instabilities leading to melting have been analyzed. It has been found that two different structural instabilities trigger melting in the size ranges N = 135–141 and N = 141–147. The size dependence of the activation energies for these mechanisms explains the size dependence of Tm . The fact that OFDFT reproduces experimental results with almost quantitative accuracy is probably somewhat fortuitous due to the several approximate functionals employed and possible inaccuracies in statistical sampling, but no much better agreement should be expected either from phenomenological potentials or from presently affordable KSDFT simulations. It is the authors’ opinion that the emphasis of this work should be put instead on the correct reproduction of experimental trends. 3.2.3. Structure of sodium clusters The most basic property of a cluster is its preferred structure at given temperature conditions (that with the minimum possible free energy), as it determines any other property, either thermal or electronic. All our results point to the prevalence of icosahedral symmetry for Na clusters in the size range considered. Icosahedral symmetry is also revealed by low-temperature photoelectron spectroscopy [28]. Nevertheless, although photoelectron spectra are useful to distinguish between icosahedral and, say, fcc or bcc crystalline fragments, they are probably not sensitive enough to cluster structure as to distinguish between different isomers based on the same underlying icosahedral symmetry. From a theoretical perspective, what is needed is an unbiased method for locating the global minimum on the potential energy surface. Very recently, Noya et al. [55] have reported such a global optimization of the structure of Na clusters of up to 380 atoms, performed by using the basin hopping method [56]. This method requires a very large number of energy evaluations, so that it is not computationally feasible nowadays to employ it in conjunction with an ab initio energy model. Thus, atomic interactions are modeled with a parameterized Murrell–Mottram potential. This global minimization also supports the view that Na clusters in this size range are icosahedral. For those sizes intermediate between two geometric shell closings, the surface shell of these icosahedra are twisted with respect to the cluster core so that only {111}-like faces are exposed at the surface. A plot of the energy of these clusters as a function of size shows enhanced stabilities for specific sizes which are in very good correspondence with the local maxima in the melting points. It is very satisfactory that the orbital-free structures found by simulated annealing are very similar (and, for some specific sizes, exactly the same) to the isomers found by Noya et al. This provides some support for the accuracy of the Murrell–Mottram potential and also for the cooling rate employed in our simulated annealing simulations. Nevertheless, one should keep in mind that some unsolved issues remain. For example, the photoelectron spectra of Na299 [27] shows no features of icosahedral symmetry, so
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it is probable that our calculations have not located the correct GS isomer for this size, nor have the basin hopping optimizations. If this is true, our good reproduction of Tm for this size may have been fortuitous.
4. Summary and Conclusions In this contribution we have demonstrated that the orbital-free MD method allows to perform reliable large-scale simulations of alkali clusters. The computational advantages (mostly linear scaling) and disadvantages (use of local pseudopotentials as well as an approximate expression for the electronic kinetic energy functional) of such a methodology have been highlighted. The method has then been applied to the efficient simulation of a specific problem, namely, that of the thermal properties and the meltinglike transition of Na clusters of varying sizes, which nowadays would be very difficult to obtain routinely from traditional (orbital-based) ab initio methods. Let us now put the orbital-free MD method in a more general perspective. With the presently developed approximations for the electronic kinetic energy functional and local pseudopotentials, the orbital-free technique provides reliable results (competing in accuracy with KS methods indeed) for simple bulk metals such as the alkalis, alkalineearths, some trivalent elements like aluminum, and their alloys. At the cluster level, it has been tested just on the alkalis, also with satisfactory results. Further progress in our understanding of the Ts n functional, and/or in the methods for generating local pseudopotentials, is needed in order to extend the applicability of the orbital-free method to other metals such as Ga or Ge, for which only qualitatively accurate results can be obtained at present, or to clusters of more complex metals such as aluminum, for which a non-monotonous size dependence in the melting point has also recently been identified [57]. In general terms, it can be concluded that presently developed Ts n functionals work better the closer is the simulated system to a nearly free electron metal. Strongly inhomogeneous electron densities, such as those of ionic (MgO, for example) or van der Waals crystals (Ne, Ar, etc.) are much more difficult to represent with the available Ts n functionals, partly because all these functionals have been generated by trying to reproduce known exact limits, most of which are related with small-amplitude departures from a homogeneous electron density. We have good prospects for future work on metallic systems employing the orbitalfree methodology. On the one hand, we plan to check in depth the possibility of extending the range of applicability of OFMD to more complicated metals. On the other hand, we plan to study other problems which cannot yet be routinely addressed by orbital-based ab initio methods, such as the thermal properties of nanoalloys. Good initial progress has indeed been already made in our group on this subject [38, 58–62]. The effect on a single substitutional alkali impurity in a Na 55 cluster has been considered in [58–60]. It was found that substitution of a single atom, even though does not break the icosahedral symmetry of the GS isomer, is able to introduce qualitative changes in the melting behaviour, and that selective doping can be useful in order to tune the melting point of nanoparticles. The structural patterns and melting behavior in alkali binary and ternary nanoalloys of several compositions have been analyzed in [38, 61, 62]. It has been found that the preferred structural ordering is based on a polyicosahedral packing. This is driven by size mismatch and surface tension effects, which force the smaller atomic
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species to remain at the cluster interior (core) whilst the larger species segregate to the cluster surface. For some specific compositions, a perfect core-shell nanoparticle may form [38]. They have an enhanced energetic and thermal stabilities, and thus present higher melting points. On the contrary, when some mixing is present (as is the case in Li–Na mixtures [61], or the core-shell structure is not perfect (see [62]), the melting point is lowered and substantial premelting effects are observed.
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Nanomaterials: Design and Simulation P. B. Balbuena & J. M. Seminario (Editors) © 2007 Elsevier B.V. All rights reserved.
Chapter 4
Multiscale Modeling of the Synthesis of Quantum Nanodots and their Arrays Narayan Adhikari,a Xihong Peng,b Azar Alizadeh,c Saroj Nayak,b and Sanat K. Kumara a
Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180 b Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic Institute, Troy, NY 12180 c GE Global Research Center, Niskayuna, NY.
1. Introduction It is by now well appreciated that reducing the sizes of crystals to the nanometer range strongly affects their optical and electrical properties. This effect has been attributed to changes in the density of states of electrons as a function of the “dot” size – i.e., a quantum size effect. [1] Nanostructures are also of great interest from a practical viewpoint due to increasing miniaturization, e.g. in computer chips. The convergence of these two very diverse facts have made the synthesis of monodisperse nanocrystals with control of their shape and size a well-studied research topic. Popular methods, such as molecular beam epitaxy (MBE) and metal-organic chemical vapor decomposition (MOCVD), can create such nanocrystals but their utility is limited by the fact that the resulting materials are attached to a substrate or embedded in a matrix. In contrast, in colloidal methods, surfactants are dynamically adsorbed onto the surfaces of the growing crystals, thereby controlling their size and presumably their polydispersity. [2–17] The surfactant adsorption has to be carefully balanced, i.e., it should be strong enough to prevent the aggregation of nanocrystals, whereas the surfactants have to have sufficient mobility to allow for the addition of metal atoms to the growing nanocrystal. This second factor is critical toward obtaining particles with narrow size distributions. Since experimentalists have empirically found several combinations of surfactants and nanocrystal structures which can help to achieve this delicate balance, there has been unprecedented growth of research in this field. 85
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While there are many theoretical works in this area, [18–24] probably the best developed molecular concepts are from Gelbart and coworkers who exploited the analogy between nanocrystal dispersions and the dispersion of oil in water (i.e., microemulsions). [20] The main idea is that the surfactant will bond to the nanocrystal surface, and hence the surface-to-volume ratio of the particles will be determined by the amount of surfactant relative to nanocrystal. Since the particle diameter uniquely determines its surface-to-volume ratio (or its curvature), the nanocrystal size will be essentially fixed by the nanocrystal/surfactant composition. The role of surfactant and metal concentration on the sizes of gold nanocrystals could thus be predicted. [20] Further, Gelbart et al. postulated that the nanoparticles assembled into a variety of higher order structures if they interacted with a long range repulsion in addition to a short range attraction. [23] While this work provides interesting conjectures on the mechanism of the surfactantbased synthesis of nanodot arrays, the molecular understanding of these issues remains poor at this time. Our work has focused on bridging this gap, specifically on obtaining a quantitative understanding of the experiments, and the molecular factors which govern them. The relevant phenomena in the surfactant-mediated nanocrystal growth and assembly occur over disparate length scales, not easily connected within a single calculation. At the subatomic level, quantum mechanical simulations are necessary to delineate the potential energy surfaces which govern intermolecular interactions. Atomistic simulations, then, are appropriate to model the assembly of a single nanodot, but cannot simulate large enough systems to capture their mesoscale assembly into arrays. Mesoscopic models thus are required at this larger length scale. Previously, we have presented the first results of a multiscale simulation method designed to understand the surfactant-mediated synthesis of cobalt nanodots and their assembly into two-dimensional hexatic arrays in a unified manner. [25–27] In this paper, we summarize these previous results, which explicitly account for the chemical details of the molecules of interest. More importantly, here we focus on the generality of these conclusions. To establish this point we employ a generic lattice-based model, which does not incorporate the chemical details of the molecules of interest. Thus, while this model sacrifices chemical fidelity, it is able to clearly delineate the principles underlying the synthesis of these interesting materials at a variety of different length (and hence time) scales.
2. Modeling Methodology 2.1. Retaining the Chemical Reality of the Molecules Figure 1 schematically illustrates the underlying philosophy in our multiscale simulation strategy. We make explicit contact with the experiments of Wang, [5] who had synthesized cobalt nanodots by dissolving cobalt carbonate in toluene in the presence of a surfactant: sodium bis(2-ethylhexyl) sulfosuccinate (termed Na(AOT) or “surfactant”). The cobalt salt was reduced to cobalt on raising temperature. The cobalt had limited solubility in the solvent and attempted to precipitate out: this “precipitation” was controlled and templated by the presence of the surfactant micelles. The resulting solution of dots was then cast on a surface, the dots ordered into hexatic arrays under these conditions.
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Supramolecular ordering of nanostructures
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Figure 1 Representation of the length and time scales, and various physical phenomena to be simulated.
2.1.1. Quantum mechanics simulations To understand these experiments we began by enumerating the interaction potentials between the surfactant, toluene (solvent) and cobalt (metal). As discussed in [25] we used all electron spin polarized density functional theory (DFT) [28] to delineate both the ground state and the distance-dependent interaction potential between any two moieties. Since it is difficult to handle the whole surfactant molecule in this calculation, we divide it into two different parts: (a) the hydrophilic “head”, which contains Na, S, all the oxygens and (b) two hydrophobic “tails”. Each tail is a string of (CH2 ) groups: for simplicity these are modeled as a string of T groups, where each T fragment is equivalent to two catenated (CH2 ) groups. We generate several configurations of a pair of moieties (chosen from head, cobalt and toluene) at a pre-specified separation. Each structure is optimized to get the local minimum energy configuration. The difference between this energy and the sum of the energies of the two moieties in their respective ground states gives the binding energy. 2.1.2. Atomistic simulations These potentials are used as input for Monte Carlo simulations so as to understand the assembly of individual nanodots. We discretize the space into a simple cubic lattice with a lattice spacing of a = 225 Å. We make this simplification since off-lattice Molecular Dynamics (MD) simulations for the micellization of Na(AOT) in hexane takes ∼1 year CPU time. Additional motivation for this simplification comes from our previous work which suggests that lattice models with such discretized potentials quantitatively
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reproduce the thermodynamics of off-lattice systems so long as molecular sizes are larger than the spacing of the underlying lattice. [29] The surfactant, Na(AOT), is modeled as T3 H2 T4 : here each T was equivalent to two methylene units and H is a head fragment. All interactions involving a H or a Co are truncated at 4.5 Å. All interactions with the T groups and the toluene–toluene interaction are modeled as hard cores. (A T group has a hard core diameter of 4.67 Å while that of toluene is 4.30 Å. For T–H and H–toluene interactions they are 3.5 Å and 3.4 Å, respectively). While we sacrifice quantitative accuracy in making some of these computationally expedient assumptions, we expect our results to be qualitatively, and even semi-quantitatively, accurate. We use the Grand Canonical Ensemble Metropolis Monte Carlo method with periodic boundary conditions in all the three directions at a temperature of T ∼500 K. A few methodological issues need to be stressed here. In a typical simulation on a 403 lattice the number of Co atoms (= 384, volume fraction = 0.006) and that of toluene molecules (= 3400, volume fraction = 0.72) are kept fixed, while the number of surfactant molecules are allowed to vary. Thus, these simulations are conducted in the Grand Canonical Ensemble for the surfactant molecules alone. The only elementary moves implemented on the cobalt and toluene molecules are random displacements. In this move, a randomly chosen particle is moved to a randomly chosen lattice point with a maximum displacement selected to ensure that roughly 50% of these trial moves were accepted. The trial move is accepted or rejected according to the Metropolis criterion. At low temperatures the system becomes extremely sluggish, and in particular the probability of moving a cobalt atom from a cluster becomes prohibitively small. To improve the success rate of this move, which we believe to be crucial for size equilibration, we use the aggregation volume-biased Monte Carlo (AVBMC) elementary move. [30] The AVBMC method has previously been successfully used to facilitate another difficult calculation, namely Gibbs ensemble simulations of gas–solid coexistence. The use of this move allowed for the transfer of atoms from the solid to the gas phase, which is typically a very low probability event. In our simulations we employed the AVBMC moves only for the cobalt particles because they aggregate and form clusters. For these AVBMC moves the “inside” volume is a sphere of radius 4. For the surfactant molecule, we employed reptation, and chain insertion/removal. In the reptation move, we chose a surfactant molecule randomly, and reptated the chain by one bond length in a direction that was also chosen at random. Note that the bond lengths could sample any one of the 26 nearest neighbors on the lattice, and hence the molecules have a somewhat variable bond length. This trial move was accepted following the Metropolis criterion. In the chain insertion/removal, the temperature and the chain chemical potential were specified as input parameters to the simulation. The surfactant molecules were created or annihilated at random: the acceptance criteria for these trial moves were derived from standard Grand Canonical Ensemble Monte Carlo rules. The probability of insertion/removal was augmented by the use of the configurational-bias sampling method. [31, 32] The first monomer of the surfactant was placed randomly on an unoccupied lattice site. The position for the next catenated monomer was chosen randomly from the unoccupied 26 neighbors of the first lattice site. This process was repeated till the chain was fully grown. The Rosenbluth weight for each growth step was calculated as the ratio of the number of unoccupied sites to the total number of possible sites (=26). The product of Rosenbluth
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weights of these sequential steps gives the Rosenbluth weight of the chain. During the removal of the chain, the reverse occurred. Namely, a randomly selected chain was unzipped from one randomly selected end to the other. The addition/removal step was accepted following the Metropolis method, after accounting for the bias introduced through the use of the CBMC move. For each simulation we specified the temperature and chain chemical potential, and obtained histograms of system energy and number of surfactant molecules. We then performed a series of simulations where the surfactant chemical potentials were varied systematically: these different simulations were “connected” by the histogram reweighting technique, which then allows us to delineate system behavior over a broad range of phase space. Typical simulations involved 500 million MC moves: 12.5% random moves for solvent (acceptance ∼36%), 12.5% AVBMC moves for Co (acceptance ∼10%), 45% reptation moves for the Na(AOT) (acceptance ∼1%) and 30% insertion/deletion events (acceptance ∼0001%). The clustering of metal atoms is determined by the following algorithm. Two metal particles are considered to be in the same “cluster” if they are within the interaction range of the cobalt–cobalt potential (i.e., 0.45 nm). Similarly, two surfactant molecules are considered to be in the same cluster if at least one head from each cluster is in the interaction range of another head. One cobalt particle and a surfactant particle are also considered in the same cluster if at least one head of the surfactant is within the interaction range of the cobalt. We find that, as in the experiments, the metal atoms are in the center of the micelle. These are sequentially surrounded by head, tail and then solvent. Our results stress the validity of the experimentally based conjectures that (a) the surfactant templates the growth of the metal dots and (b) the uniformity of nanodot size is attributed to the slow exchange of metal atoms between different micelles, thus permitting the “equilibration” of their size. Figure 2 presents simulation results for the sizes of spherical cobalt nanoparticles for a fixed concentration of solvent and cobalt but with different concentrations of the surfactant. As mentioned above, the surfactant concentration is varied by changing the value of the chain chemical potential. The temperature for all the three cases are the same, ∼500 K. It can be seen from the figures that, as the surfactant concentration decreases, the size of the nanoparticles increases. To understand the effect of the surfactant concentration on the size distribution of the metal clusters, in Figure 3 we consider the distribution of radii of gyration of the cobalt nanoparticles for these different surfactant concentrations. It can be seen from the figure that, as the concentration of surfactant decreases, the radius of gyration of the cobalt nanoclusters increases. Further, if the concentration of surfactant is significantly less than the concentration of cobalt, then the polydispersity increases. In Figure 3, we even see double peaks for the lowest surfactant concentration. This means the nanodots are not monodisperse, a highly undesirable result. Thus, we conclude that monodisperse cobalt nanoclusters are only obtained if the ratio of the concentration of surfactant to the concentration of metal particles is ∼1. We have also examined the role of the metal concentration in this context. Figures 4 and 5 show the distribution of nanodot sizes for two different cobalt concentrations, which are different roughly by a factor of 4, as well as instantaneous system snapshots in both cases. Since the surfactant concentration is ∼001 we are in a situation where the metal-to-surfactant concentration is ∼1. It is clear that the size of the metal particles
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Figure 2 Snapshots of micelles formed for different values of surfactant chemical potential. The surfactant concentration is lowest top left (volume fraction ∼0003 3), higher top right (0.006 6) and highest at the bottom (0.019 9). Red is the cobalt, yellow the toluene and blue-green the surfactant.
increases with metal concentration. However, their polydispersities remain about the same. These results, in conjunction with our previous work, reiterate the notion that the relevant quantity is the metal-to-surfactant ratio, so long as it is not very different from unity. 2.1.3. Supramolecular ordering of Cobalt nanodots To understand the molecular basis for the supramolecular assembly of these nanodots into two-dimensional arrays we construct the pair distribution function for the centers of mass of a pair of nanoclusters decorated by surfactant, gr, where r is the spatial distance between the centers of mass of the dots. We employ the gr values to deduce an effective distance-dependent interaction, ur, between two nanodots. To construct this ur, which precisely reproduces these gr, we perform inverse Monte Carlo
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P(Rg)
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Figure 3 Distributions of radii of gyration of the micelles as a function of the mean concentration of surfactant. Note that the surfactant concentration was varied by changing its chemical potential, as noted in the text.
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Figure 4 Distribution of the radii of gyration of the micelles as a function of the concentration of metal. The concentration of surfactant is essentially constant in both cases as discussed in the text.
simulations. [33–39] In this approach we model each quantum dot (i.e., metal with decorating surfactant) by a single sphere. Since each nanodot on average is ∼ 45 nm in diameter it occupies a volume of ∼ 173 lattice sites. This coarse graining step thus reduces the computational effort by at least a factor of 176 ∼107 . Initially, the interaction
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Figure 5 Snapshots of system for two different cobalt concentrations. Left picture is for a volume fraction of 0.005 7, the right for 0.023.
potential between pairs of coarse-grained spheres is set to: ur = − ln gr. We now perform a Monte Carlo simulation with this ur, and update the ur following: gr ur = urold + ln (1) gtarget r where gtarget r is the target value of the radial distribution function between the nanodots, gr is the corresponding function obtained using the current estimate of ur and f = 01. The resulting converged ur is somewhat attractive (ur ≈ −15) close to contact, and then goes to zero with increasing r. Since this exclusion-driven attraction is relatively weak, we conjecture that the nanodots should show little sign of binding irreversibly. Note that this form of the potential does not possess a long range repulsion between particles, postulated by Gelbart to be the origin of long range assembly of dots. [23] We perform two-dimensional Monte Carlo simulations on these coarse grained spheres using ur so as to understand the experimentally observed ordering of these nanodots when they are cast on substrates. As expected from past work, the relatively short range of the attractive potential between nanodots prevents them from undergoing a liquid–gas phase separation with increasing concentration. Rather, they form an orientally ordered hexatic phase with increasing concentration, and into hexatic crystals at even higher filling fractions. These results are very reminiscent of the ordering of the cobalt nanodots in the experiments, and strongly suggest that a combination of long-range repulsion and short-range attraction is unnecessary in this context. Rather, the physics of twodimensional assembly of nanodots into ordered arrays is equivalent to the behavior of spheres with short-range attractions. In brief, then, these results present clear evidence that we can go from details of intermolecular interactions (from quantum mechanics simulations), to the assembly of single dots, to the macroscopic ordering of these dots into arrays in a single multiscale simulation protocol. At each scale we have compared our findings to known experimental results and find nearly quantitative agreement. While these results are for only one
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system, and much more critical comparison to experiment is crucial to validate this scheme, we are encouraged by the success of this multiscale simulation tool.
2.2. Generic Models To establish the robustness of our multiscale modeling tool and also to verify the generic validity of the conclusions discussed above, next we employ a model which does not incorporate the reality of the molecules of interest. Space is discretized into a cubic lattice with coordination number, z = 26. The incompressible system simulated has three components: solvent (or empty lattice sites), “metal” particles and surfactant. The surfactant architecture is H4 T4 . In the notation we adopt, T denotes a “solvent soluble” tail moiety, while H represents a “solvent-phobic” head group. Each solvent molecule, metal particle and a surfactant moiety (either a single H or T) occupies one lattice site. In our calculation the H moiety and the “metal” are assumed to be identical, while a T group is identical to a solvent molecule. A single moiety is placed at a lattice site at most, and only an unfavorable interaction between a H (or metal) moiety and a T (or solvent) group is included when they are immediate lattice neighbors (i.e., corresponding to z = 26 neighbors). There is therefore one energy scale: the energetic dislike between the H and T moieties, 2. We define the reduced temperature as T ∗ = kB T/, and most simulations utilized T ∗ = 3. The “metal” atoms undergo a gas–liquid transition, but not a liquid–solid transition due to the symmetry of the lattice. To calculate the critical micelle concentration, cmc , from the simulations we followed the procedure suggested by Floriano et al. [40] These workers have calculated the osmotic pressure of a surfactant solution as a function of volume fraction surfactant. The “critical micelle concentration” (cmc) was defined as the volume fraction of surfactant at which the ideal line of unit slope (at low volume fraction) intersected a linear fit to data at higher concentrations. At a fixed temperature and concentration of metal particles, a number of Grand Canonical Monte Carlo simulations were performed over a range of surfactant chemical potentials. The osmotic pressure vs. surfactant concentration behavior was calculated using the histogram reweighting technique. Figure 6 shows that the resulting cmc decreases with increasing concentration of metal particles at a fixed temperature. This suggests that the presence of the metal particles does not qualitatively affect the micellization of the surfactant, but only affects it quantitatively. We then examined the spatial distribution of “metal”, and H and T moieties in a given micelle. Two metal particles (or a metal and a H group etc.) are considered to be in the same cluster if they are within the interaction range (within 26 neighbors) of each other. As expected, the metal atoms are in the center of the micelle. These are sequentially surrounded by H groups, T groups and then the solvent. Although the particles have very different sizes at the start of the simulations, after sufficient equilibration we find that the size distribution of metal clusters is narrow with a standard deviation which is ≈5% of the mean (see inset to Figure 7). These results echo experimental findings which find that size equilibration is slow, but that it always yields spherical, nearly monodisperse spheres. The number of metal particles inside the micelles depends upon temperature, concentration of surfactant and
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concentration of the metal particles. Figure 7 shows the ratio of the number of metal particles outside the micelles to the number of metal particles inside the micelles, K, as a function of temperature. From Figure 7, it is seen that significant micellar templating of the nanoparticles only occurs when the temperature is below the lattice-gas critical temperature of the metal particle (shown as an arrow). This follows since, for
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higher temperatures, there is a relatively high probability for metal atoms to exist in the outside solution: this situation is not optimal for the design of monodisperse nanocrystals (Figure 7). Note also that the partition coefficient decreases very strongly below the critical temperature, again emphasizing that this templating becomes exponentially stronger with decreasing temperature. However, in these cases, we expect size equilibration to become progressively slower, basically due to the increased barrier for molecules to migrate across clusters, a fundamental process that is required for size equilibration.
2.3. Supramolecular Ordering of Nanocrystals In previous work, we have studied the supramolecular ordering of nanodots for models which have been designed to closely mimic experiment. [25] We shall show here that similar results can be obtained by the generic model. Figure 8 shows the pair distribution function for the centers of mass of the clusters. It can be seen that there is just one peak at a distance comparable to the diameter of the clusters. From this pair distribution function, one can calculate the effective interaction potential between a pair of clusters. To calculate this effective potential we employed the method of reverse MC simulations. Again, we model each dot as a sphere, and we consider a system size of 603 containing 8100 spheres of unit diameter. The density of spheres corresponds to that obtained from the MC simulations discussed above where both the surfactant and the metal atoms were fully accounted for. Note that the system is now modeled off-lattice with periodic boundary conditions in all the three directions. The starting potential between two spheres is given by ur = − ln gr. This form of the potential is used in a standard Metropolis Monte Carlo scheme to generate the pair distribution function
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between spheres. An update to the intermolecular potential in then obtained following eq. (1). We find that this updating procedure is remarkably stable for an f = 01, and that the effective potential between dots converges within 10–20 steps of this updating procedure. We use the resulting converged potential (inset to Figure 8) to explore the two-dimensional supramolecular ordering of the nanodots. We performed several MC simulations to study the distribution of the nanodots using the effective two-body mean field potential in two dimensions. Figure 9 presents the distribution of nanodots in two dimensions obtained on a square of 202 . The area fraction of the circles are ∼70% in this case. Clearly, the nanodots order into a hexatic phase. Figure 9 also presents the pair distribution function of the two-dimensional nanodots. It is clear that there is longrange spatial ordering of the nanodots if the area fraction is greater than ∼ 70%. It is apparent that this ordering allows us to go directly from a “gas” phase of nanodots to a hexatic array because the range of attractions is relatively short. Previous works, first by Frenkel and his coworkers [41] and then by several others, have reiterated the notion that in the case of hard spheres one sees this sort of behavior when the width of the attractive well is smaller than ∼25% of that of the hard core. While this quantitative result is expected to be strongly dependent on the absolute form of the interdot potential, we can draw a few generic conclusions based on this result. First, if one chose to reduce dot size, then, the position of the potential energy minimum relative to the hard core size (or, alternatively, the well width relative to the hard core in a square well model) will increase. We thus postulate that it should become progressively harder to order smaller dots into hexatic arrays, with, ultimately, the system going over to a disordered liquid phase. This result is highly undesirable if one chooses to create addressable arrays of dots. A second point to note is that these results only hold if the dots themselves are nonmagnetic. It is now well appreciated that larger dots have magnetic moments: in these cases we expect that the diamagnetic interactions between
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dots will tend to order them into lines (and closed loops) even at low densities. Under these conditions, the ordering of dots into higher order structures probably becomes more difficult. Simulating this new class of phenomena is an aspect we are currently focusing our efforts on.
3. Conclusions Our results presented here – for two models, one fully chemically realistic and one generic – confirm the robustness of the multiscale simulation tool in describing the experimental protocol for the synthesis of quantum dots and their ordering into twodimensional arrays. As expected from experiments we find that sizes of the nanodots are controlled by the surfactant-to-metal concentration, as long as this ratio is in the vicinity of unity. If the surfactant concentration becomes much lower we find that the polydispersity of the dots increases significantly, which is an undesirable conclusion. Similarly, we show that the ordering of dots into hexatic arrays is only expected for large, nonmagnetic dots where the behavior of these materials are expected to be akin to large colloidal particles, with weak short-ranged attractions. In the opposing limit, especially of small colloids, we expect the formation of a disordered liquid phase, which is undesirable for applications. While our results appear to be in good agreement with experiment, we stress that more systems have to be examined so as to conclusively demonstrate the quantitative accuracy of this tool. Looking out into the future, it is clear that the extension of these concepts to quantitatively understand other synthesis strategies, e.g., the kinetically driven shape control of nanoparticles will become more important. These longer-ranged questions constitute one major focus of the work that is in progress in our groups.
Acknowledgements The authors acknowledge the financial support from the National Science Foundation, first through an NER grant and more recently from a GOALI grant. We thank Shekhar Garde, Thanasis Panagiotopoulos and Igal Szleifer for helpful comments.
References 1. Bimberg, D., Quantum Dot Heterostructures. Wiley: 1999. 2. Yang, H. T.; Shen, C. M.; Wang, Y. G.; Su, Y. K.; Yang, T. Z.; Gao, H. J., Stable cobalt nanoparticles passivated with oleic acid aind triphenylphosphine. Nanotechnology 2004, 15 (1), 70–74. 3. Yang, Z. H.; Zhang, W. X.; Wang, Q.; Song, X. M.; Qian, Y. T., Synthesis of porous and hollow microspheres of nanocrystalline Mn2O3. Chemical Physics Letters 2006, 418 (1–3), 46–49. 4. Yi, D. K.; Lee, S. S.; Papaefthymiou, G. C.; Ying, J. Y., Nanoparticle architectures templated by SiO2/Fe2O3 nanocomposites. Chemistry of Materials 2006, 18 (3), 614–619. 5. Yin, J. S.; Wang, Z. L., Preparation of self-assembled cobalt nanocrystal arrays. Nanostructured Materials 1999, 11 (7), 845–852.
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6. Yin, Y.; Alivisatos, A. P., Colloidal nanocrystal synthesis and the organic-inorganic interface. Nature 2005, 437 (7059), 664–670. 7. Zhang, H.; Wang, D. Y.; Mohwald, H., Ligand-selective aqueous synthesis of one-dimensional CdTe nanostructures. Angewandte Chemie-International Edition 2006, 45 (5), 748–751. 8. Hosokawa, Y.; Maki, S.; Nagata, T., Gold nanoparticles stabilized by tripod thioether oligomers: Synthesis and molecular dynamics studies. Bulletin of the Chemical Society of Japan 2005, 78 (10), 1773–1782. 9. Manna, L.; Scher, E. C.; Alivisatos, A. P., Shape control of colloidal semiconductor nanocrystals. Journal of Cluster Science 2002, 13 (4), 521–532. 10. Manna, L.; Scher, E. C.; Alivisatos, A. P., Synthesis of soluble and processable rod-, arrow-, teardrop-, and tetrapod-shaped CdSe nanocrystals. Journal of the American Chemical Society 2000, 122 (51), 12700–12706. 11. Pan, B. F.; He, R.; Gao, F.; Cui, D. X.; Zhang, Y. F., Study on growth kinetics of CdSe nanocrystals in oleic acid/dodecylamine. Journal of Crystal Growth 2006, 286 (2), 318–323. 12. Peng, X. G.; Manna, L.; Yang, W. D.; Wickham, J.; Scher, E.; Kadavanich, A.; Alivisatos, A. P., Shape control of CdSe nanocrystals. Nature 2000, 404 (6773), 59–61. 13. Querner, C.; Reiss, P.; Sadki, S.; Zagorska, M.; Pron, A., Size and ligand effects on the electrochemical and spectroelectrochemical responses of CdSe nanocrystals. Physical Chemistry Chemical Physics 2005, 7 (17), 3204–3209. 14. Scher, E. C.; Manna, L.; Alivisatos, A. P., Shape control and applications of nanocrystals. Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences 2003, 361 (1803), 241–255. 15. Wang, F.; Xu, G. Y.; Zhang, Z. Q.; Xin, X., Synthesis of monodisperse CdS nanospheres in an inverse microemulsion system formed with a dendritic polyether copolymer. European Journal of Inorganic Chemistry 2006, (1), 109–114. 16. Wang, Q. A.; Pan, D. C.; Jiang, S. C.; Ji, X. L.; An, L. J.; Jiang, B. Z., Luminescent CdSe and CdSe/CdS core-shell nanocrystals synthesized via a combination of solvothermal and two-phase thermal routes. Journal of Luminescence 2006, 118 (1), 91–98. 17. Xu, S.; Kumar, S.; Nann, T., Rapid synthesis of high-quality InP nanocrystals. Journal of the American Chemical Society 2006, 128 (4), 1054–1055. 18. Barnard, A. S.; Curtiss, L. A., Computational nano-morphology: Modeling shape as well as size. Reviews on Advanced Materials Science 2005, 10 (2), 105–109. 19. Jorge, M.; Auerbach, S. M.; Monson, P. A., Modeling spontaneous formation of precursor nanoparticles in clear-solution zeolite synthesis. Journal of the American Chemical Society 2005, 127 (41), 14388–14400. 20. Leff, D. V.; Ohara, P. C.; Heath, J. R.; Gelbart, W. M., Thermodynamic Control of Gold Nanocrystal Size – Experiment and Theory. Abstracts of Papers of the American Chemical Society 1995, 209, 20-PHYS. 21. Nishio, K.; Morishita, T.; Shinoda, W.; Mikami, M., Molecular dynamics simulation of icosahedral Si quantum dot formation from liquid droplets. Physical Review B 2005, 72 (24). 22. Takaki, T.; Hasebe, T.; Tomita, Y., Two-dimensional phase-field simulation of self-assembled quantum dot formation. Journal of Crystal Growth 2006, 287 (2), 495–499. 23. Gelbart, W. M.; Sear, R. P.; Heath, J. R.; Chaney, S., Array formation in nano-colloids: Theory and experiment in 2D. Faraday Discussions 1999, (112), 299–307. 24. Sear, R. P.; Gelbart, W. M., Microphase separation versus the vapor-liquid transition in systems of spherical particles. Journal of Chemical Physics 1999, 110 (9), 4582–4588. 25. Adhikari, N. P.; Peng, X. H.; Alizadeh, A.; Ganti, S.; Nayak, S. K.; Kumar, S. K., Multiscale modeling of the surfactant mediated synthesis and supramolecular assembly of cobalt nanodots. Physical Review Letters 2004, 93 (18). 26. Adhikari, N. P.; S., G.; A., A.; Nayak, S. K.; Kumar, S. K., Surfactant Mediated Synthesis of Quantum Dots and Their Arrays. Physical Revie E 2006.
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27. Adhikari, N. P.; S., G.; A., A.; Nayak, S. K.; Kumar, S. K., Surfactant Mediated Synthesis of Quantum Dots and Their Arrays. Journal of Computational Mechanics 2006. 28. Perdew, J. P.; Wang, Y., Accurate and Simple Analytic Representation of the Electron-Gas Correlation-Energy. Physical Review B 1992, 45 (23), 13244–13249. 29. Panagiotopoulos, A. Z.; Kumar, S. K., Large lattice discretization effects on the phase coexistence of ionic fluids. Physical Review Letters 1999, 83 (15), 2981–2984. 30. Chen, B.; Siepmann, J. I.; Klein, M. L., Direct Gibbs ensemble Monte Carlo simulations for solid-vapor phase equilibria: Applications to Lennard-Jonesium and carbon dioxide. Journal of Physical Chemistry B 2001, 105 (40), 9840–9848. 31. Escobedo, F. A.; Depablo, J. J., Extended Continuum Configurational Bias Monte-Carlo Methods for Simulation of Flexible Molecules. Journal of Chemical Physics 1995, 102 (6), 2636–2652. 32. Siepmann, J. I.; Frenkel, D., Configurational Bias Monte-Carlo – a New Sampling Scheme for Flexible Chains. Molecular Physics 1992, 75 (1), 59–70. 33. Soper, A. K., Maximum-Entropy Methods in Neutron-Scattering – Application to the Structure Factor Problem in Disordered Materials. Institute of Physics Conference Series 1989, (97), 711–720. 34. Garde, S.; Ashbaugh, H. S., Temperature dependence of hydrophobic hydration and entropy convergence in an isotropic model of water. Journal of Chemical Physics 2001, 115 (2), 977–982. 35. Ashbaugh, H. S.; Patel, H. A.; Kumar, S. K.; Garde, S., Mesoscale model of polymer melt structure: Self-consistent mapping of molecular correlations to coarse-grained potentials. Journal of Chemical Physics 2005, 122 (10). 36. Sun, Q.; Faller, R., Crossover from unentangled to entangled dynamics in a systematically coarse-grained polystyrene melt. Macromolecules 2006, 39 (2), 812–820. 37. Sun, Q.; Faller, R., Systematic coarse-graining of atomistic models for simulation of polymeric systems. Computers & Chemical Engineering 2005, 29 (11–12), 2380–2385. 38. Soper, A. K., Partial structure factors from disordered materials diffraction data: An approach using empirical potential structure refinement. Physical Review B 2005, 72 (10). 39. Milano, G.; Muller-Plathe, F., Mapping atomistic simulations to mesoscopic models: A systematic coarse-graining procedure for vinyl polymer chains. Journal of Physical Chemistry B 2005, 109 (39), 18609–18619. 40. Floriano, M. A.; Caponetti, E.; Panagiotopoulos, A. Z., Micellization in model surfactant systems. Langmuir 1999, 15 (9), 3143–3151. 41. Hagen, M. H. J.; Meijer, E. J.; Mooij, G.; Frenkel, D.; Lekkerkerker, H. N. W., Does C-60 Have a Liquid-Phase. Nature 1993, 365 (6445), 425–426.
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Nanomaterials: Design and Simulation P. B. Balbuena & J. M. Seminario (Editors) © 2007 Elsevier B.V. All rights reserved.
Chapter 5
Structural Characterization of Nano- and Mesoporous Materials by Molecular Simulations Lourdes F. Vega Institut de Ciència de Materials de Barcelona. Consejo Superior de Investigaciones Científicas (ICMAB-CSIC). Campus de la U.A.B. 08193 Bellaterra. Barcelona. Spain.
The increasing speed of modern computers along with new simulation algorithms and more refined, high quality available experimental data on structural properties of porous materials have contributed to a better understanding of the microscopic behavior of gases confined in these materials, and hence to a more reliable interpretation of the experimental results. The purpose of this chapter is to highlight some of the recent applications of molecular simulations to the characterization of adsorbent materials. We will focus here on the information that can be extracted from adsorption isotherms simulated in modeled individual pores, as well as some mathematical procedures to obtain reliable Pore Size Distributions. The information extracted refers to the most meaningful quantities to reliably characterize micro-, nano- and mesoporous materials, including the surface area, the pore size heterogeneity, the pore wall distribution and the energy distribution function. This valuable information can be extracted from standard Monte Carlo simulations in the Grand Canonical ensemble, which can be done now with high accuracy in a desktop PC.
1. Introduction The precise knowledge of the internal structure is essential for the optimal use of porous materials. These materials serve as adsorbents and catalytic supports for several industrial applications, ranging from chemical to pharmaceutical, among others. Some of most important characteristics of internal structure include the surface area, the pore size distribution, as well as chemical and other surface heterogeneities. The pore structure 101
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characterization is not done in a direct manner; it requires both an effective experimental probe of the porous solid and an appropriate theoretical or numerical model to interpret the experimental data. The principal experimental technique used to probe the structure of the porous material is gas adsorption porosimetry [1]; indeed, although various experimental techniques have been proposed as alternatives, such as immersion calorimetry [2], positron annihilation [3], transmission electron microscopy [4], small-angle X-ray scattering [5], and neutron scattering [6], gas adsorption porosimetry continues to be the most used technique. Due to its scientific and technical relevance, the development of consistent methods for the interpretation of adsorption isotherms has been the subject of several research efforts for more than 60 years. The most extended methods developed from the classical approach are those of Brunauer, Emmet and Teller (BET), the Gibbs–Kelvin method (GK), the Dubinin and Radushkevich (DR) equation, and their modifications, based on phenomenological assumptions with well-known limitations. For instance, the GK method assumes subcritical adsorbate homogeneity and incompressibility, gas phase ideality and independence of the interfacial tension of the liquid of its curvature; the BET model was derived assuming identity to bulk liquid of all adsorbed layers beyond the first and absence of lateral interactions in adsorbed layers; the DR method assumes volume filling of pores and Gaussian distribution of micropores, etc. Although their limitations are well known, some of these methods are still in use for the characterization of adsorbent materials. For a detailed discussion on the subject the reader is referred to [1]. As a result of the known limitations of the most extended techniques, the development of reliable methods for the accurate characterization of porous materials remains an on-top and motivating problem at the present time [1, 2], especially for materials with a wide range of pore sizes and shapes and for heterogeneous surface materials. Statistical mechanics offers an attractive alternative approach to characterize porous materials. It provides two methods to obtain the individual adsorption isotherms: the density functional theory (DFT) and molecular simulations (MS). The individual adsorption isotherms calculated by these techniques are used in conjunction with some mathematical procedures to propose the Pore Size Distribution (PSD) of adsorbent materials. DFT and MS have been applied, with great success, to a wide variety of materials, making these methods standard, reliable tools for the interpretation of adsorption data nowadays. In particular, different versions of DFT have been used to characterize materials, most of them focused on carbons [7–12]. Other materials also characterized by DFT include controlled pore glasses [13, 14], MCM-41 and MCM-48 [14–16], and other hexagonal mesoporous materials [17]. Until very recently DFT was preferred versus MS, essentially because it requires considerable less computational efforts. However, a main limitation of DFT in this context is that it is difficult to discern the uncertainties coming from the approximations made in the theory to those coming from the inversion of the adsorption integral equation and/or the model used. Also, the extension of the theory to non-spherical fluids, polar fluids and non-homogeneous surfaces is not straightforward. Due to these limitations MS is becoming increasingly used in recent times. Although more expensive from the computational point of view, the high speed of available computers makes simulations a feasible tool for the generation of the individual adsorption isotherms as well as for more complicated adsorption studies, including
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atomistic models of disordered materials. In addition, MS offers several advantages over the DFT approach, such as: a) the statistical mechanical equations are solved exactly for the prescribed model of the pore geometry and intermolecular interactions, b) its versatility permits the incorporation of surface structure and heterogeneity, a variety of pore geometries and irregularities, and c) it is straightforward to implement for different adsorbates, including chain, polar and associating molecules. The most appropriate MS technique applied to adsorption in porous materials is the Grand Canonical Monte Carlo simulation method (GCMC) [18]. The GCMC simulation method involves the determination of the properties of a system at a constant volume V (the pore in the case of confined fluids) in equilibrium with an infinite fictitious reservoir of molecules. Equilibrium thermodynamics imposes the condition of equality of chemical potential and temperature between the confined and the free phases. Since the chemical potential is the same in both phases, the number of molecules inside the pore varies, depending on the thermodynamic conditions, being its average one of the most meaningful quantities to be extracted from these simulations. GCMC has been used by several authors for the characterization of porous materials, mainly focused on carbons [19–25], although it has also been applied to other materials, including hexagonal mesoporous silicas [26–30] and aluminum methyl phosphonates [31, 32], among others. In addition to obtaining individual adsorption isotherms, MS has been used in recent years to reproduce the pore structure of disorder microstructures by atomistic models. The global adsorption behavior of the experimental material is interpreted with the help of simulations of gas adsorption in the “modeled” material. There are two approaches to reproduce this internal structure by MS: (1) to mimic the experimental procedure used in the laboratory to fabricate the material, obtaining an amorphous structure that is then statistically analyzed to get the desired structural information, and (2) to use adsorbent structural data (e.g. small-angle neutron scattering) to construct a model disordered porous structure statistically consistent with the experimental measurements. Gelb and Gubbins presented the first type of amorphous material belonging to the first group, modeled by the use of quench Molecular Dynamics (MD). The simulations provided model porous silica glasses topologically similar to controlled pore glasses (CPGs) or Vycor glasses [33, 34]. These authors were able to tailor the pore structure of the model material (pore sizes and connectivities) by changing the quench time. In a very recent paper Schumacher et al. [35] used a kinetic Monte Carlo (kMC) scheme to generate realistic atomic models for periodic mesoporous silicas. By using simplified potential models in kMC they were able to simulate the reaction path of the hydrothermal synthesis and calcinations of the silica material. The simulated materials were characterized by adsorption of different gases simulated by GCMC. An advantage of the availability of such model materials is that their geometrical PSD can be obtained by sampling the pore volume accessible to probe molecules of different radii, instead of by inversion of an adsorption integral equation. Belonging to the second approach are the works presented in references [36–40], in which the MC simulation method is used to reconstruct the disordered microstructure of realistic models of materials. Levitz and co-workers used
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off-lattice reconstruction methods [36, 37] to build models of porous glasses, while Gubbins and co-workers used a similar technique, known as reverse Monte Carlo, to match the structure of disordered microporous carbons to data obtained from small-angle X-ray or neutron scattering [38–40]. Details on the abovementioned approaches to reconstruct amorphous and regular materials, their implementations and applications can be found in the original articles. We will focus here on the information (or lack of information) obtained from the use of individual adsorption isotherms generated by simulations and used to interpret the adsorption data of experimental materials. In order to obtain the PSD of an adsorbent from individual adsorption isotherms the independent pore model is assumed in most of the cases [8]. This model states that the global adsorption behavior of the material is due to the contribution of the individual pores with different diameters integrating the material, ignoring the effects derived from the connectivity among them. This assumption is also known as the bundle of straight pores (BSP) model [24]. The PSD function is then obtained by inverting the adsorption integral equation, constructing a kernel with the individual adsorption isotherms for pores with different sizes but with same geometry, and a regularization procedure. In fact, the information needed to precisely characterize a material refers to both, the pore size heterogeneity (pores of different sizes present in the adsorbent, usually known as the PSD) and the energetic heterogeneity (due to the inner nature of the adsorption process and/or to chemical heterogeneities exposed to the surface). Most of the recent studies have focused on solving the PSD, since it is directly related to the adsorbent capacity, while few studies have been devoted to the energetic heterogeneities, of special relevance for catalytic applications. We will present here some recent results concerning the pore size and the energetic heterogeneity characterization by using GCMC simulations as the primary tool, combined with a regularization procedure to obtain the PSD in a reliable manner.
2. Surface Area and Energetic Heterogeneity The most widely used method to obtain the surface area is still the BET model, due to Brunauer, Emmer and Teller [41]. The theory is based on kinetic arguments and it assumes a reversible adsorption/desorption process with multilayer formation. The BET model owes its widespread use to the fact that it is straightforward to apply and a linear transformation of the original equation, as proposed by the authors, provides two of the most meaningful textural quantities, the monolayer capacity and the C parameter, giving fairly acceptable results for several solids. In spite of its success, BET is also one of the most criticized theories since its inception [42]. The main criticism of the BET model comes from two of the basic (inaccurate) assumptions made when developing it: (1) all adsorbing sites are energetically homogeneous and (2) only vertical interactions between adsorbed molecules and the adsorbing surface take place, hence ignoring any lateral interaction. There have been several attempts to correct one or the two assumptions in order to improve the BET model; however, there is not general consensus on an accurate method to substitute it for surface area calculations [1]. In this context, not only does MS serve to generate the kernel to obtain the PSD, or to construct the structure of a
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material, as previously explained; simulations can also be used as an additional tool to calculate the surface area and the energetic heterogeneities, providing additional insights to interpret results obtained from these classical methods.
2.1. The BET Model Revisited from MC Simulations. Surface Characterization Most of the published works devoted to characterize adsorbent materials by Statistical Mechanic tools have used DFT or MS just to generate the individual adsorption isotherms needed to construct the kernel for obtaining the PSD. The purpose of this section is to show that there is additional information in the individual adsorption isotherms generated by simulations that can be extracted to further interpret adsorption data. In this sense, Sánchez-Montero et al. [43] have recently published results on the application of GCMC simulations to interpret the BET model. For this purpose, the BET model with different linear transformations was applied to the simulated isotherms, in the same manner as done to the experimental adsorption isotherms [44]. The linearizations of the BET equation can be used to extract the monolayer capacity (nm ) and an energetic parameter related to the heat of adsorption, C. The goal was not to improve the BET model, but to use it, as it is, and to understand the implications of applying it from a molecular perspective. The original equation proposed by Brunauer, Emmett and Teller is [41]: (1) n nm = Cx 1 − x 1 − x + Cx where n represents the moles of gas adsorbed at the equilibrium pressure p nm is the number of moles of gas adsorbed when the surface is covered by a complete monomolecular layer (the monolayer capacity); x is the relative pressure (p/p0 ), p0 is the vapor pressure of the adsorbate and C is an energetic parameter. The monolayer capacity provides the surface area, assuming a given value of the nitrogen diameter. The C parameter can be obtained directly from eq. (1) once the monolayer capacity is known: C = n1 − x2 x nm − n1 − x (2) The model can be directly applied to obtain the monolayer capacity and the C parameter from direct linearizations of eq. (1). We illustrate here this application to nitrogen adsorption isotherms obtained with GCMC simulations for a crude model of activated carbons. The adsorbent is modeled as a collection of independent slit-like pores, with periodic boundary conditions and minimum image convention in the xy plane. Both walls are taken to be the basal plane of a graphite surface, made up of Lennard–Jones (LJ) atoms of diameter ss = 0340 nm, arranged hexagonally in each plane [45]. H is defined as the perpendicular distance separating the graphite planes from the center to center of carbon atoms. The interaction of the nitrogen molecules with each wall is calculated by the structureless 10-4-3 potential due to Steele [45]. 4 10 4 2 sf sf sf wall z = 2 ss sf sf 2 × (3) − − 5 z z 3 z + 061 3
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where = 035 nm is the distance between carbon planes, ss = 114 nm−3 is the solid density, ss = 280 K is the Lennard–Jones solid energy parameter, and the unlike sf and sf solid–fluid interaction parameters are calculated following the standard Lorentz– Berthelot combining rules. For a given slit-like pore of width H, the external potential exerted to any molecule inside the pore is given by: sf z = wall z + wall H − z
(4)
Individual adsorption isotherms of nitrogen, modeled as a spherical LJ fluid with parameter values ff = 03575 nm and ff /k = 9445 K, k being the Boltzmann’s constant, were obtained using standard GCMC simulations at different pore widths, ranging from micropores (0.8 nm) to mesopores (5 nm). Figure 1 shows the nitrogen adsorption isotherm obtained at 77 K in a modelactivated carbon of width H = 1125 nm. This pore size is chosen to be representative of micropores but with enough volume available to present capillary condensation. This capillary condensation is clearly observed in the snapshots depicted in the figure. Equation (1) can be transformed into a linear form in different ways [43]: x 1 C −1 = + x n1 − x nm C nm C
(5)
x 1 1 x = + 2 n1 − x n m C nm 1 − x
(6)
The first expression was provided by Brunauer et al. as a direct way of obtaining the monolayer capacity value and the C parameter from the slope and the intercept
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Figure 1 Nitrogen adsorption isotherm obtained from GCMC simulations for a slit-like pore of width H = 1125 nm. Symbols represent the simulation results, while the dashed line is a guide to the eyes. The figure on the right represents the molecules distribution inside the pore at reduced pressures: p/p0 = 10−5 , 2.5×10−5 and 3.0×10−5 , from bottom to top, respectively
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of the line represented by this equation. Brunauer et al. observed that the linearity of the BET representation was reduced to a small range of pressures (p/p0 = 005 − 03). Later studies have shown that the range is much more reduced for different materials and/or different adsorbates. We have applied eqs (5) and (6) to the data presented in Figure 1, obtaining a monolayer capacity value of nm = 6570 molecules/nm2 . Results for the linearization are depicted in Figure 2. The value obtained when applying the same BET linearization to the experimental adsorption isotherm of nitrogen in an activated carbon is nm = 609 molecules/nm2 [43]. Although slightly overestimated, there is a good agreement between the two values, especially considering that the simulated pore is into the microporous region, where both walls exert a great influence in the adsorption behavior. The monolayer capacity can also be directly obtained from simulations performed in a slit-like pore wide enough so that one wall does not influence the adsorption at the opposing wall. This was accomplished in our case for a pore width of 5nm. If one represents the amount of nitrogen adsorbed as a function of the reduced pressure a change in the isotherm shape is observed once the first layer is filled. Simulations allow to count the number of molecules in this layer, the resulting number being nm = 611 molecules/nm2 , in close agreement with the number obtained from the experimental data and from the narrower pore. There is still a third independent way of checking the monolayer capacity value using results from the same simulations. Figure 3 shows the total energy of the molecules inside the pore as a function of the pressure for H = 50 nm. Note that there is a continuous increase in energy as the number of molecules adsorbed at the surface increases, until there is a change in the slope of the curve, occurring when the first layer is filled. The point at which this change takes place can be identified as the value of the monolayer capacity nm = 647 ± 003 molecules/nm2 . Again, this value is in agreement, within the error uncertainties, with the value obtained by counting the molecules and also from the linearization of the BET equation. An advantage of obtaining this value from molecular simulations is that we can actually see the distribution of the molecules at this particular point, as shown in the snapshots. Note that although the adsorbed molecules begin to create the second layer, there is still
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Figure 2 Two linear regressions of the BET model to the simulation data presented in Figure 1. The plot in the left corresponds to eq. (5), while the one on the right represents eq. (6). Lines are guides to the eyes
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Figure 3 Total potential energy versus total number of molecules in a simulated pore of width H = 50 nm. The arrow indicates the change in slope of this line, occurring at a relative pressure p/p0 = 001. Symbols represent the simulation results, while the dashed line is a guide to the eyes. The figure on the right are two views of the pore, the first one is a top view of one of the walls, while the second one is a lateral view of the pore at the pressure where the energy line changes its slope
some free volume available in the first layer; this volume will be occupied at higher pressures. In summary, the same adsorption isotherm, obtained by simulations of simple LJ spheres in a model slit-like pore can serve as three independent ways of calculating the monolayer capacity in more complicated adsorbent materials with slit-like geometry. As mentioned already, the BET model also provides what is known as the C parameter. In the original linearization of the model C comes out to be a number. The magnitude of its value accounts, somehow, for the adsorption energy magnitude. C can be directly obtained from the BET model, once the monolayer capacity is known (eq. (2)). In addition, this parameter can also be extracted from independent measurements of heats of adsorption, which show a clear non-constant value with pressure (see references [2, 44] and references therein). Following Rouquerol et al. [2], the C parameter is obtained from the expression:
l Cii ≈ exp E1 − EL RT = exp u˙ exc T − uT /RT
(7)
where ulT = −34419 kJ mol is the molar internal energy of bulk liquid nitrogen [46], at T = 77k, and: exc exc
u˙ exc n TVA T = U
(8)
is the differential surface excess internal energy. The C parameter obtained from eq. (7) can thus be compared to that obtained by eq. (2), results for the pore H = 1125 nm are presented in Figure 4. Note that although the values of the C parameter are very different, depending on the approach used to calculate it, the trend of the curve is very similar in both cases. The shape of this curve can be justified by a competition between energetic and entropic effects [1, 43]: the first part of the adsorption process is dominated by the energy gain inside the pore; once the pore has some molecules adsorbed, there is
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Figure 4 Variation of the C parameter versus surface coverage calculated from eq. (2) (left axis) and obtained from eq. (7) (right axis), in a simulated pore H = 1125 nm. Lines are guide to the eyes
an entropy lost by getting more molecules inside the pore. Although no quantitative agreement with experimental data is obtained, as expected, the shape of the C curve is in qualitative agreement with that obtained by experimental data and by independent heats of adsorption measurements. It should be mentioned, however, that the agreement of the BET model is not as good as the results provided here for simulated materials with different topologies and morphologies. Recently, Coasne et al. [47] showed that the BET model deviates significantly for materials in which the degree of confinement is increased, i.e. regular cylindrical pores, as well as for materials with morphological disorder. In spite of this, what is clear from the results just shown here is that simple molecular simulations provide reliable independent ways of calculating the monolayer capacity, without any a priori assumption on the mechanism of adsorption, but as a consequence of the obtained results, as opposed to the empirical methods [2].
2.2. Energetic Heterogeneity from Adsorption Data As stated, simulations provide additional information related to the energy of adsorption that can be used to characterize energy surface heterogeneities. Besides its practical importance, published results in this area are not as abundant as those related to the pore size heterogeneity, or as atomistic models of materials. In order to provide some illustrations of the information extracted by simple simulations in this field it is worth mentioning some of the representative works in the area. Heuchel and Jaroniec [48] used GCMC adsorption isotherms to explore the possibility of extracting information about surface and structural heterogeneity of microporous solids from experimental data. These authors focused on evaluating the adsorption energy distribution function as a function of the degree of pore filling and pore width, pointing out that additional simulation studies are needed to
