Nanomaterials: Design and Simulation

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 ...

Author: Perla Balbuena | Jorge M. Seminario

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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: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-444-52826-1 ISBN-10: 0-444-52826-1 ISSN: 1380-7323 For information on all Elsevier publications visit our website at books.elsevier.com Printed and bound in Italy 07 08 09 10 11 10 9 8 7 6 5 4 3 2 1

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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

vii

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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

4


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 (Pt 3 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


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

6


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.


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.

8

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.


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 Ptn 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 Pt 4 with symmetry D2h are similar for both basis sets. The bond distances and angles of Pt 5 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 Ptn 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 Pt2 and Pt3 . The bond length values for different multiplicities are shown in Table 1

10


Table 2 Structure and energies of Pt4 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 Pt4

Table 3 Bond lengths (Å) of Pt5 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


11

Figure 7 Optimized Structure of Pt5

Table 4 Energies (Ha) of Pt5 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.

12


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


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


(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


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 Pt7 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 Pt3 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 Pt7 Hm clusters Multiplicity Pt 7 H18 Pt 7 H18

3 1 5 1 1 3 1 3 1 3 5 1 5

Pt 7 H12 Pt 7 H12 Pt 7 H10 Pt 7 H9 Pt 7 H6

Pt 7 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. Pt 7 H18 , c. and d. Pt 7 H12 , e. Pt 7 H10 , f. Pt 7 H9 , and g. and h. Pt7 H6


17

Table 17 Geometry of Pt7 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 Pt13 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


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 Pt13 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


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


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


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


Molecule

Tetrahedral symmetry

Silane

Si2H6

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


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


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


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.


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

30


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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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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G. Rossi and R. Ferrando

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.

Structural Properties of Pure and Binary Nanoclusters

37

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.

38


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


39

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

40


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


41

∗ ∗ 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

42


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


43

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

44


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.


45

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

46


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


47

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.

48


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.


49

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,

50

G. Rossi and R. Ferrando Δ–ΔM 1

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.


51

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

52


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

mkB T Es 3N−6 kB T Z=V (14) Is exp − 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.


53

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

54

G. Rossi and R. Ferrando T = 400 K

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)


55

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


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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A. Aguado and J. M. López

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

Computer Simulation of the Solid–Liquid Phase

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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 Na7 – 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 (Na8 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 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


67

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.)

Fy (a.u.)

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

50

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

68


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


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

70

A. Aguado and J. M. López 1/N 0.005

15

0.001

0.015

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 = 4r 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


71

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 Na55 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.

72


Here we will study the melting-like transition of NaN 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


73

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


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


75

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


77

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

78


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


79

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 Na55 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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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

minutes

Micelle formation. Templating nanostructures.

millisec

Chemical reactions and bond formation

time

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Molecular simulations (MD, MC)

picosec

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distance Å

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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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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.



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: (2) C = n1 − x2 x nm − n1 − x 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 (3) − − wall z = 2 ss sf sf 2 × 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 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 understand the relationship between microporosity and surface heterogeneity of porous solids.

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A recent work by Calleja et al. [49] used GCMC simulations to study the adsorption energy heterogeneity in activated carbons following the same idea. They used density profiles and energy distributions for several pore widths, proposing an alternative way of computing energy distribution for individual isotherms. They discussed the implications of obtaining an odd or even number of adsorbed layers in the isotherm shapes, as the basis of further analysis in terms of potential distribution. The local energetic heterogeneities were explained from energy distributions and potential distributions calculated from GCMC. They also calculated the total energy distribution in the activated carbon by considering geometrical aspects, i.e. using the PSD obtained from geometrical considerations to weight the individual “energy distributions” in order to obtain the total energy distribution. This can be considered a useful tool to discriminate geometrical effects in local energetic heterogeneities in calculations that consider a model for local isotherms. The energetic information extracted from GCMC simulations is as straightforward as to count the amount adsorbed. In fact, there is a direct relationship between the amount adsorbed at a given pressure and the energy associated to this quantity. The energetic distribution inside the pore changes as the pore begins to adsorb molecules, even in the case of homogeneous surfaces, due to the different solid–fluid and fluid– fluid interactions felt by the new molecules entering the pore. Figure 5 shows as an illustration, the density and energy profiles of nitrogen adsorbed in a slit-like micropore of H = 1125 nm, at two different relative pressures, p/p0 = 25 × 10−5 and 3.0×10−5 , just before and after the capillary condensation takes place. These results were obtained from the simulations discussed in the previous subsection (Figures 1 and 4). The density profiles show that only two layers of molecules can be formed inside the pore, one adsorbing at each wall, due to the pore width and in consistency with the snapshots depicted in Figure 1. The energy profiles provide the total energy of the adsorbed molecules as well as the contributions coming from the solid–fluid and fluid–fluid interactions. As expected, the energy of adsorption is mainly a consequence of the solid–fluid interactions, due to the proximity of the two walls. The situation is quite different in the case of wider pores, where one surface does not affect the adsorption on the opposing surface, as shown in Figure 6 for a pore of H = 50 nm. In this case several layers are formed in each surface, and there is an energy associated with each layer. Molecules adsorbing on the second and subsequent layers in each surface feel an attraction different than the ones adsorbed on the first layer. This information can be used to investigate the adsorption behavior due to energetic heterogeneities. Of special interest is the case where the surfaces show defects or chemically heterogeneous sites, which can also be investigated by MS. Let us summarize the main ideas behind these simulations. In a similar manner as done for the local amount adsorbed, the energetic heterogeneity of a given adsorbent can be probed by various adsorbates by calculating the energy-dependent local adsorption isotherm p T U, where U is the adsorption energy, p is the adsorbate pressure and T is the absolute temperature [50, 51]. Usually the energetic heterogeneity is characterized, by similarity with the size heterogeneity, by the distribution function of the adsorption energy, fU. This distribution function should be unique for a given adsorbent/adsorbate system, providing a “fingerprint” to recognize the system. It can


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Figure 5 Top figures: total density profiles of nitrogen adsorbed in a slit-like pore of H = 1125 nm from GCMC simulations. Bottom figures: corresponding energy profiles. The full lines stand for the total energy, while the long-dashed lines represent the solid–fluid energy and the dotted lines the fluid–fluid energy

be evaluated by inverting the integral equation for gas adsorption on heterogeneous solids: (9) t p T = l p T UfUdU

where t p T denotes the overall adsorption isotherm measured experimentally (or by simulations) and is the range of the adsorption energy. l p T U represents the local isotherms (individual isotherms) at a given energy value. l p T U can be obtained by MS (using information as that depicted in Figures 5 and 6), or by more classical models, such as the Fowler–Guggenheim method (see [48] for details). The solution of eq. (9) is an ill-posed problem. Equation (9) is a Fredholm integral equation of the first kind; there are several solutions compatible with the experimental adsorption isotherms. Solving eq. (9) relies on the use of mathematical tools to provide unique and reliable solutions. The most standard techniques used nowadays are regularization methods, as

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Figure 6 Top figures: total density profiles of nitrogen adsorbed in a slit-like pore of H = 50 nm from GCMC simulations. Bottom figures: corresponding energy profiles. The full lines stand for the total energy, while the long, dashed lines represent the solid–fluid energy and the dotted lines the fluid–fluid energy

will be explained in detail in the next section. fU depends on the degree of pore filling, on the pore geometry and pore widths present in the material.

3. Pore Size Characterization 3.1. The Mathematical Problem Related to the Adsorption Integral Equation The pore size distribution, fD, of an adsorbent material is obtained through inversion of the adsorption integral equation (AIE), defined as:

Dmax

ae P =

As P DfDdD Dmin

(10)


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where, the fD is the PSD, D is a measure of the pore width, ae P is the experimental adsorption isotherm and As P D, the so-called kernel of the integral equation, is the set of single-pore adsorption isotherm for each pore size D as a function of pressure. This expression is formally identical to eq. (9), used to obtain the adsorption energy distribution function, fU. Hence, the mathematical procedure applied to reliably obtain fD, can be used to obtain fU, provided an adequate kernel can be constructed and there is reliable experimental data available. Since the PSD is not directly obtained from experiments, one deals with several uncertainties in the validity/reliability of this function. Some of them are physical, related to the very nature of the system and the model representing it, and some of them are mathematical, depending on the way in which the AIE is solved, and the robustness of the solution. Among the first group one may need to check several questions: • Is the model used to generate the adsorption isotherms accurate enough regarding the adsorbent and the solid? • What is the influence of the connectivity among the pores, ignored in the pore independent model, in the global adsorption behavior? • How the tortuosity of the material, also ignored in the model, influences the global behavior? • What is the appropriate range of pressures to cover from very low coverage to filling of the adsorbent? • Is the obtained PSD sound regarding the adsorption behavior? There are also several questions regarding the mathematical procedure. In fact, the inversion of the AIE (eq. (10)) is a key issue in obtaining reliable PSDs. The robust procedure to invert the integral turns out to be at least as important as the physical considerations we mentioned. The problem arises from the nature of the equation since solving this integral equation is an “ill-posed” problem, i.e., there are several PSD functions compatible with the experimental adsorption isotherm. As for fU, the inversion of the integral can be performed by proposing an analytical function, as a reasonable representation of the PSD or by direct numerical inversion. The analytical functions are usually based on physical arguments and their parameters are fitted to the adsorption data. However, the use of analytical functions implies a priori the existence of a given function fitting to the data, instead of obtaining the function as solution to the equation, without any assumption about it. In contrast, the numerical inversion of the integral shows the advantage of being more flexible, since the PSD is not constrained a priori to any functional form. An elegant and powerful way to solve this integral equation is by using regularization methods (see, for instance, references [13, 17, 20, 29]). These methods also need additional information on the system, which is used to select one of the PSDs among all possible satisfying the inversion equation from the knowledge of the experimental adsorption isotherms and some additional requirements, some of which will be further reviewed in this chapter. A possible way of checking the reliability of the PSD obtained from the regularization procedure is to apply the standard procedures to materials in which the synthesis process has been mimicked by simulations [13, 33–35], or constructed by simulations from the structural experimental information [36–40]. In both cases the location of the atoms and the available volume are precisely known. These simulations, only attainable nowadays

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by the increasing power of available computers, have greatly helped in understanding the overall process of adsorption in complex materials. The problem this approach poses is that the adsorption integral equation is not solved in this case; the overall adsorption is directly obtained from the simulations. The PSD is calculated from the geometric measurement of the available volume. An alternative approach is to apply the methodology of inverting the AIE to an experimental material in which the pores have regular shape and they are unconnected. In this case, once the individual adsorption isotherms are obtained the problem is just to find the PSD by inverting eq. (10), since the pore independent model holds for this particular material. Recent and outstanding contributions in the field of reliably inverting the AIE come from Neimark and co-authors [14–17] and Seaton and co-authors [23–25]. Both groups have been working on the combination of DFT and MS in conjunction with several regularization procedures to obtain PSD of some given materials, providing valuable tools to interpret the experimental results. Although considerable advances have been performed in the field, there are still several open questions one faces when trying to invert the adsorption integral even with the best mathematical tools – some of them include: • • • • • •

What regularization procedure to use, among the several available ones? What is the best set of individual adsorption isotherms to construct the kernel? What is the adequate pressure range to be considered in these isotherms? How sensitive is the PSD to the regularization parameter? How sensitive is the PSD to errors in the experimental data? How reliable is the PSD obtained from adsorption of nitrogen at 77 K to predict the overall adsorption of other compounds and mixtures at different conditions?

3.2. Individual Adsorption Isotherms in Silica-Based Materials Aiming at answering most of these questions a protocol for obtaining reliable PSDs has been recently proposed [27–29], following the work previously done by Neimark et al. and Seaton et al. on searching for accurate methods to obtain the individual adsorption isotherms and to invert the AIE. The main contribution of our work comes from the way in which the regularization procedure is applied; with several tests to be performed before the final PSD is proposed. In addition, the methodology is systematically applied to materials different in morphology, from a stack of unconnected cylinders (MCMs) [15, 52] to a collection of cylinders in the mesoporous range connected by cylinders in the microporous range (SBA-15, Santa Barbara-x [53]) to meso-microporous materials with some silica deposited in the inner walls of the mesopores (plugged hexagonal templated silica, PHTS, [54]). SBA-15 and PHTS materials were characterized by Ryoo et al. and Kruk et al., using different experimental techniques [55–57]. The availability of accurate data on the characterization of these materials by independent methods provides an excellent framework in which to compare results from any other characterization method, including the combination of MS/DFT with regularization procedures. We will outline here the different steps taken to ensure a reliable and robust PSD from the application of MS and a regularization procedure to these kind of materials.


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In order to invert eq. (10), the first step is to generate the individual adsorption isotherms. Neimark et al. [15] and Ravikovitch and Neimark [17] used DFT to characterize MCM-41 and SBA materials, respectively, assuming cylindrical geometry in both cases. A regularization procedure was used to invert the adsorption integral. The PSD and pore wall thickness obtained for MCM-41 were in excellent agreement with results obtained from XRD scattering. For the SBA-15 material they just determined the distribution of the main channels and the amount of intrawall porosity, without characterizing the microporous region. The structural parameters obtained were in agreement with previously described geometrical considerations and XRD data. However, the global adsorption isotherm obtained by fitting the experimental data with the DFT approach exhibited pronounced layering, while the experimental adsorption isotherm was smooth. This is probably a limitation of the underlying DFT approach [13]. We present here results on the characterization of these materials but with individual isotherms generated by GCMC simulation, a different regularization procedure and additional mathematical tests to assess the robustness of the obtained PSD. For this purpose nitrogen–nitrogen interactions were modeled as single LJ spheres, with nitrogen parameters ff = 03615 nm and ff /k = 1015 K k being Boltzmann’s constant. Those fluid–fluid parameters were chosen by Ravikovitch et al. [58] to fit bulk properties of the adsorbate, including liquid–gas surface tension and reference adsorption isotherms on non-porous substrates. The hexagonal mesoporous silicas were modeled as a collection of individual pores, assumed to be infinite cylinders with silica walls. The silica–gas interactions on such pores were modeled as the LJ interactions with an integrated smooth cylindrical layer of oxygen atoms. The structureless potential of the solid–fluid interaction used in this work is given by [59]:

63 r 9 r −10 9 r 2 F − − 1 1 − 2− 32 sf R 2 2 R

r r −4 3 r 2 3 −3 (11) 2− F − − 1 1 − sf R 2 2 R

wall r R =

2

s sf sf2

−2

where the product s sf = 225369 K Å s , the effective surface number density of the oxygen atoms in the pore wall, being 0153 Å−2 sf /k, the LJ energy parameter between the solid and the fluid, being 147.3 K, and sf , the combined molecular size solid–fluid parameter, = 0317 nm. Those parameters were selected for comparative purposes with the previous work done in the system SBA-15/nitrogen by Ravikovitch et al. [17]. F is the hypergeometric functions [60]. The wall potential was calculated at a given distance r (in the radial direction) when the radius of the pore is R. The individual adsorption isotherms were obtained by GCMC simulations, considering a wide range of pore widths (07 < D < 200 nm). Details on the simulations were provided in [29]. To compare with experimental data, the excess pore fluid density was calculated as: exc =

N − bulk V

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p/p0 Figure 7 Individual nitrogen adsorption isotherm at 77 K for silica-based materials obtained by GCMC. D = 30 nm. The figure on the left is a close-up of the low pressure region, while the figure on the right shows the complete adsorption isotherm. The inner pictures illustrate the information lost about the filling process if the isotherm does not explore the low pressures region

where N is the mean number of particles inside the pore, bulk is the bulk density value at the same conditions, calculated from a molecular-based equation of state [61, 62], and V is the volume of the simulated pore. Figure 7 shows, as an example, the individual adsorption obtained in a cylindrical pore of diameter D = 30 nm. The figure on the left shows the very low relative pressure region. Note that a complete first layer of molecules is already adsorbed at a relative pressure as low as 10−6 , pointing out the need for high resolution experimental adsorption isotherms to precisely know the surface area and the available volume. This is of clear relevance in the case of SBA-15 materials, where there is a microporosity region, besides the mesoporous region. Once the individual adsorption isotherms are obtained for a wide range of pore sizes and at different relative pressures, the next problem to deal with is how to invert the adsorption integral equation.

3.3. Inverting the Integral: The Regularization Procedure Several authors have addressed the calculation of the PSD of porous materials from adsorption data using a variety of different available approaches [13, 17, 24, 25, 27, 29, 56, 57, 63]. Special attention deserves in this context – for its clarity, the work by Davies et al. [25]; they presented a method to determine whether a PSD based on a specified model


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pore geometry can be fitted to a set of experimental data. They used this procedure, in conjunction with regularization techniques that stabilized the calculations, to determine statistically significant PSDs. The procedure was applied to the adsorption of methane at 308 K onto BPL-activated carbons. The regularization procedure they used was to include an additional term taking into account the smoothness of the distribution, as done by some other authors [64–66]. Although these and some other works have helped in understanding the internal structure of several adsorbent materials, unfortunately no many details are given on how these methods are used, with few exceptions; neither do why a regularization procedure is chosen versus another one. This poses a problem for a beginner to the field. In fact, most of the relevant details to check the reliability of the PSD are dispersed in mathematical books and/or papers of relevance to other scientific fields, but not applied to adsorption problems. Among the different possibilities, we have chosen to use Tikhonov’s regularization method through a Singular Value Decomposition (SVD) as we consider it to be most adequate for PSD analysis purposes, for several reasons: (1) it is simple to implement, (2) it is very fast, as it is a direct (as opposed to iterative) method, and (3) it is one of the best mathematically founded methods for that purpose. This method has been also used by Neimark and co-workers. Hence we outline here the mathematical details of the adaptable procedure of deconvolution over the adsorption integral equation followed in this work. The deconvolution procedure implies a grid-size evaluation, i.e. to select the number of pores and relative pressures to be included in the analysis, in addition to the adequate choice of the regularization parameter. In order to obtain the PSD, fD, the adsorption integral equation (10), should be solved. This problem is tantamount to that of solving a Fredholm integral equation of the first kind. As it is well known, this is an ill-posed problem, in the sense that the mapping ae → f , given by eq. (10), is undefined because either the mapping is not continuous or the image f is not unique. From a practical point of view, the lack of continuity implies that f is highly sensitive to arbitrarily small perturbations in the experimental data ae . This poses a first problem of reproducibility of any solution to eq. (10), and thus the problem, as stated before, has physically (and mathematically) no sense. Standard approaches [67] to this problem rely on solving the related problem of finding the solution f that minimizes the functional: 2 2 Dmax D max 2 Jf ≡ a P − A P DfDdD + RD DfDdD (13) s e Dmin Dmin where denotes an appropriately defined norm and is the so-called “regularization parameter”. The additional term renders the former problem a well-defined one for each pair and RD D, and corresponds to the Tikhonov’s regularization method. The criteria selected for uniquely choosing the regularization parameter () is the so-called L-curve. This states that the optimum value of will occur at the corner of the curve defined by the log-log plot of: 2 2 Dmax D max ae P − As P DfDdD RD DfDdD (14) Dmin Dmin

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Additionally, the SVD method requires the experimental data and the kernel to fulfill a mutual suitability criterion given by the Discrete Picard Condition (DPC) [68, 69]. Although this formulation of the problem is mathematically sound, we still face many ambiguities from a physical point of view. First of all, we have the very stringent condition of fD to be a non-negative function (feasibility condition). Second, we aim at finding a systematic and robust procedure to determine a feasible solution. Not only does this mean the obvious idea of robustness against errors in the experimental data and molecular simulations, which the regularization procedure already provides. We shall also require robustness against the very selection of the experimental data and the kernel to be used. In addition, this selection may affect the usual robustness against errors in the data. Thus, this raises the additional ambiguity of choosing a set of input data, as different sets show different robustness. Hence, a complete systematic procedure shall give a prescription for choosing a sound set of experimental data and kernel to be used. Common mathematical procedures, as the one we use, are not guaranteed to neither satisfy nor completely answer these requirements. A way of taking into account these conditions is by performing a systematic procedure in which the different aspects are explicitly considered. A possible protocol includes [27, 29]: • • • •

to chose the kernel size: number of pressure points and pore sizes to be simulated to invert the integral equation with the Tikhonov regularization method through SVD to check for the fulfillment of the DPC to chose the regularization parameter best fulfilling the L-curve criteria and the physical condition that fD should be a non-negative function.

Once these four criteria are applied, the performance of the method for the global adsorption integral and the obtained PSD can be checked. Depending on the agreement with the experimental data the whole cycle is repeated, choosing a new kernel size and imposing the fulfillment of the mathematical requirements in the order just explained. Only the kernels fulfilling the conditions will pass to the next check: for instance, if a kernel does not fulfill the DPC condition, this kernel is not further investigated for fulfilling the L-curve criteria. The robustness of the proposed PSD should be checked versus errors in the experimental data. The mathematical details for each step are explained in detail in [29] and references therein.

3.4. Application to Hexagonal Silica-Based Materials We first applied the methodology to a member of the MCM-41 family. As mentioned, these materials are composed of silica cylinders with well-defined geometry and unconnected pores. Their PSD should be a narrow distribution centered at the main cylinder size diameter, already known from other experimental techniques. For the SBA-15 two ranges of pore sizes should be expected, one corresponding to the mesoporous region and a second region representing the complementary microporosity present in the material [55–57]. Note that in this second case the method will provide these two regions if the very low pressure regions are explored and the connectivity among the pores allows diffusion from one part of the material to the other.


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The experimental nitrogen adsorption isotherm of the selected MCM-41 material was taken from reference [15], while experimental data concerning the SBA-15 was obtained in our laboratory [29]. In order to perform a fair comparison between experimental and simulated results, it is important to note that the experimental adsorption isotherm represents the amount of fluid adsorbed per unit of mass of the solid, while the simulated adsorption isotherm represents the amount of adsorbed fluid per unit of void volume. These two concepts of volume are related by the true density and the porosity of the material. Ignoring this relationship may lead to erroneous PSDs. We have used the value of the true density as 22 g/cm3 for pure silica. The porosity has been calculated by a trial-and-error procedure, checking that ℘ fDdD = 1, where ℘ represents the porosity and fD is expressed in (length)−1 units. As a first step to invert eq. (10), one needs to construct the kernel, that is, to select how many experimental pressures and individual pore isotherms will be considered. It is not clear, a priori, what the influence of these numbers will be neither in the PSD obtained, nor in the global adsorption isotherm generated by fitting to the experimental one. In order to generate the kernel we have calculated a collection of individual adsorption isotherms, As P D, using the GCMC method, in a diameter range of 075 ≤ D ≤ 200 nm, with 55 different pore diameters. This range covers from very narrow pores, well belonging to the micropores regime, to wide mesopores. For the case of the MCM41 material 60 pressure points were selected from those reported in [15], as the starting point to construct the kernel. The pressure points for the SBA-15 kernel were chosen at the experimental relative pressure data. With the calculated set of individual adsorption isotherms, and based on the analysis of the DPC, the next step is to determine how many pores are needed to fit a given set of experimental data by the regularization procedure. For this example we first selected five kernels for AM-5 (a member of the MCM-41 family), the experimental number of relative pressures was fixed to m = 60 (interpolated along the experimental data of Neimark et al. [15]), while different sets of pore sizes were considered, n = 16, 22, 28, 30 and 34. In all the cases the minimum and maximum pore diameter values were 0.75 and 5.40 nm, respectively. As a first choice, the selected pores included in the analysis were equally distributed between these two values. All the studied cases fulfilled the DPC (not shown here). Since, as mentioned in the prescribed protocol, this is the first criterion to be fulfilled, all the selected kernels can be used to check the rest of the criteria. We have checked next the performance of the method for the global adsorption isotherm and the obtained PSD. The agreement with the experimental adsorption data (not shown here) could be improved in the region between 0.2 and 0.5 p/p0 . For this reason we included a new kernel in the analysis with m = 54, in which the pressure range near the inflexion point in the adsorption curve was better refined. The selected number of pores for the kernel was n = 30, since no improvement was found when increasing the number of pores from 30 to 34. The second test would be the application of the L-curve criterion to the kernels fulfilling DPC and the selection of the best feas . It is observed in Figure 8 that the kernel best fulfilling the L-curve criterion (m = 54 n = 30), is precisely the one defining better the inflexion point in the adsorption curve. Although, in principle, we have a feasible, well-sounded solution (54 × 30), it is well known that there are associated errors to the experimental data; then it should be of

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Figure 8 Left figure: The L-curve for m = 60 and n = 16, 22, 28, 30 and 34, with symbols corresponding to feas , (∗), (x), (), () and () respectively; m = 54 n = 30 is represented by a full line, feas for the best solution is denoted by •. Right figure: Graphical representation of the average relative error (over 50 perturbations) of the obtained PSD fD versus the index n. Symbols: () different kernels fulfilling DPC, (•) the single value for m = 54

relevance to know the effect that these errors would have in the obtained PSD, showing the robustness of the procedure. A third criterion to be applied over the six studied kernels is the influence of the errors in the experimental data in the obtained PSD. This is applied based on the effect of Gaussian random errors in the (experimental) data, ae∗ → ae + e. We have generated the error, e, after choosing its variance. This is done through a scaling factor r such that ae∗ = ae 1 + r z, where z is a Gaussian random number. The value of r is chosen such that on average at least 99% of the perturbed values correspond to relative errors not greater than the experimental ones. In the particular cases we are considering here, we have chosen r = 002, and the average values over 50 realizations for the six kernels. Results are summarized in Figure 8 (right figure). As it can be observed in the figure, the 54 × 30 kernel is the most robust among the six studied here. Figure 9 shows the comparison between the experimental and the calculated adsorption isotherm with two selected kernels. Note that although both of them give an overall excellent representation of the adsorption behavior, the 54 × 30 kernel provides a better description of the abrupt changes in the curve. The robustness of the 54×30 kernel is also observed in Figure 9(a), where the PSD for 60 × 34 and 54 × 30 are plotted with their corresponding perturbation e. The dotted lines correspond to a particular case where a perturbation to a has been added, while the continuous lines correspond to the unperturbed cases. Note that although the shape of the PSD function is not exactly the same, being less defined in the 60×34 case, both kernels provide exactly the same value for the peak of the distribution, both of them have captured the relevant pore size of the material. Finally, we compare in Figure 10 results obtained with this methodology (for the case of 54 × 30) with previous results obtained for the same material by Neimark et al. [15]. The agreement obtained is excellent. Using the described methodology the mean pore for AM-5 is found around 3.2 nm, in excellent agreement with the value reported previously [15]. Note that there are two important differences in the way the PSD was

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obtained here and in [15]. Neimark et al. used DFT instead of GCMC to generate the individual adsorption isotherms and they inverted the adsorption integral following a different mathematical procedure. From the previous detailed discussion on the different aspects to be considered when calculating PSD we can conclude that the methodology used is accurate to characterize the adsorption in MCM-41 materials. This proves that the mathematical procedure and the molecular model are accurate for obtaining sound PSDs for materials with unconnected pores, the next step is to use the same methodology to characterize porous

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materials in which pores have a well-defined geometry but there are some connections among them, as it is the case of SBA-15, chosen for its particular structure as well as for its potential applications in several fields of interest. The characterization of the SBA-15 and PHTS materials was performed following a similar procedure to the MCM-41 material described above, including the assumption of the independent pore model for the inversion of the integral equation [29]. Hence, we omit the details here. Figure 11 shows results from the application of part of the protocol to the chosen kernels fulfilling the DPC condition to a SBA-15 material. It is observed that as n increases, the shape of the PSD is better defined, decreasing the noise for wider pores. The fitting to the experimental adsorption isotherm is excellent, as shown in Figure 11(b). The calculated GCMC PSD for SBA-15 shows a sound shape in all cases, with a narrow distribution of mesopores around 7 nm, in accordance with results obtained by other characterization techniques. The left extreme of the PSD can be attributed to the presence of nano- and micropores, as already discussed by other authors [56, 57]. This assesses the validity of the independent pore model for these kind of materials. The next step would be, then, to apply the protocol to a silica-based material but with disordered structure, such as Vycor or CPG [33, 34]. Note that no pore blocking or any other form of diffusion control has been taken into account up to now. The previous discussion dealt with inverting the adsorption integral equation ignoring connectivity and pore blocking effects. Some advances have been performed in recent years to further advance in this sense. López-Ramón et al. [70] have used percolation theory along with GCMC simulations to interpret the isotherms of several adsorbates (methane, perfluoromethane and sulfur hexafluoride) in a microporous carbon. They combined the partial PSDs obtained for each adsorbate, obtaining a much more complex PSD. The comparison of PSDs with two adsorbates (CF4 and SF6 ), with substantial overlapping, showed molecular sieving. These were the two PSDs

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they analyzed in detail with the use of percolation theory, being able to extract some conclusions on the connectivity of the pores.

3.5. Pore Wall Thickness Heterogeneity An important feature inherent to the majority of the local isotherm models refers to the assumption of infinitely thick pore walls, with the corresponding associated potentials similar to those presented in eqs (3) and (11) for slit-like and cylindrical pores, respectively. Actually, most carbons have walls with only a few – typically one to three – planes of carbon atoms in a hexagonal, close-packed structure; hence, a model considering infinite thick pore walls will overestimate the solid–fluid interactions, overpredicting the amount adsorbed. In addition, the carbon framework enveloping the pores is highly disordered, as a result of which the wall thickness may be expected to be random and non-uniform. In fact, results from the atomistic simulations using reconstruction methods are in agreement with those expectations [38–40]. The question is how to deal with this issue if only local isotherms are used. In a series of papers Bhatia [71] and Nguyen and Bhatia [72–74] have presented a method to simultaneously probe the heterogeneity of pore size and pore wall thickness focused on carbons. For this purpose they used a modified DFT formulation that permits a random distribution of wall thickness to be considered. To allow for such possibility a probability distribution for the number of graphene layers in the pore walls was permitted. They used DFT to calculate the local density profiles at the single pore level with a confinement potential considering different (finite) number of layers [72]. The structural heterogeneity of carbons was calculated by simultaneously solving the adsorption integral equation for fD and a similar equation for the pore wall thickness probability distribution, fitting to the experimental adsorption isotherm with the use of Tikhonov regularization method combined with a genetic algorithm. They also used the L-curve criteria to find the regularization parameter. Results obtained from this methodology were in close agreement to those found by other experimental techniques, usually X-ray diffraction. Although Bhatia and Nguyen have used it just for carbons, the same methodology can be applied to other disordered materials. Note that the same procedure can be applied if instead of considering DFT the individual adsorption isotherms are generated by MS.

4. Conclusions and Future Directions The goal of this chapter was to highlight the advantages of using simple molecular simulation tools, such as standard GCMC simulations performed in individual pores, to precisely characterize adsorbent materials. For this purpose, we have illustrated the calculation of the most meaningful quantities needed to reliably characterize these materials: the surface area, the energetic heterogeneities, the pore size distribution and the pore wall thickness distribution. The precise knowledge of these properties will allow the optimal use of these materials, as well as the possibility of tailoring them to some particular applications.

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Although MS is able to provide nowadays a modeled adsorbent material with the same morphological and topological features as experimental disordered materials, these simulations are still very extensive, and they need the availability of high precision experimental complementary information, such as small-angle neutron scattering. A main advantage of using GCMC versus DFT to generate the adsorption isotherms is the fact that the simulations provide additional insights, such as the location of each atom, the local inhomogeneities, etc. In addition, it is straightforward to modify a GCMC code to take into account different adsorbates (with several molecular architectures) as well as different adsorbents. It is also of interest to mention that in most of the cases, not all possible information is extracted from the simulated isotherms. We encourage to use the complementary information provided by GCMC simulations in all possible ways. For instance, GCMC provided independent and consistent ways of obtaining the monolayer capacity value and the C parameter, including the calculation of energetic effects. In some cases one is concerned with developing new methods when actually there are different ways of using the available ones. We have also addressed one of the bottleneck problems in the characterization of materials from adsorption measurements, the inversion of the adsorption integral equation to obtain reliable and robust PSDs. The mathematical issues addressed here can also be applied to obtain the adsorption energy distribution function. We have shown that both physical and mathematical requirements are needed and should be systematically used to obtain reliable PSD. Regarding this issue, more work is still needed in this area, for instance: • most of the measurements/calculations have been performed at the same temperature; adsorption isotherms at different temperatures for the same adsorbate/adsorbent should be combined to obtain the PSD • the available volume would be better captured if adsorption isotherms of different adsorbates on the same adsorbent are combined • the energetic heterogeneities of the surfaces would be better characterized if one of the probe adsorbates is a polar fluid, which will have specific interactions with the polar sites on the surfaces. Finally, one of the best ways to systematically check the assumptions behind these simple models is to apply the methodology usually employed to characterize experimental materials to simulated-modeled materials, taking advantage of the structural information at the atomic level provided by these methods.

Acknowledgements I am indebted to my collaborators C. Herdes, F. Medina and M. A. Santos for their continuous contributions to this work. I also acknowledge the contributions from F. Salvador and M.J. Sánchez-Montero. I am deeply grateful to C. Herdes for his help with some of the figures presented here, and to C. Rey-Castro for a critical reading of the manuscript. Fruitful discussions with K.E. Gubbins, A.M. Morales-Cas and M. Jaroniec are also acknowledged. Financial support for this work has been provided by the Spanish government under projects CTQ2004-05985-C02-01 and CTQ2005-00296/PPQ, and by the Catalan government (2005SGR-00288).


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39. J. Pikunic, R. J. M. Pellenq, K. T. Thomson, J. N. Rouzaud, P. Levitz, K. E. Gubbins, Stud. Surf. Sci. Catal. 132 (2001) 647. 40. J. Pikunic, C. Clinard, N. Cohaut, K. E. Gubbins, J. M. Guet, R. J. M. Pellenq, I. Rannou, J. N. Rouzaud, Langmuir 19 (2003) 8565. 41. S. Brunauer, P. H. Emmett, E. Teller, J. Am. Chem. Soc. 60 (1938) 309. 42. Seri-Levy, D. Avnir, Langmuir 9 (1993) 2523. 43. M. J. Sánchez-Montero, C. Herdes, F. Salvador, L. F. Vega, Appl. Surf. Sci. 252 (2005) 519. 44. F. Salvador, C. Sánchez-Jiménez, M. J. Sánchez-Montero, A. Salvador, Stud. Surf. Sci. Catal. 144 (2002) 379. 45. W. A. Steele, Surf. Sci. 36 (1973) 317. W. A. Steele, The Interaction of Gases with Solid Surfaces, Pergamon, Oxford, 1974; p56. 46. NIST Chemistry WebBook, Saturation properties of nitrogen at 77K, NIST Standard Reference Database Number 69, March 2003 Release, http://webbook.nist.gov/cgi 47. Coasne, K. E. Gubbins, R. J. M. Pellenq, Part. Part. Syst. Charact. 21 (2004) 149. 48. M. Heuchel, M. Jaroniec, Langmuir 11 (1995) 4532. 49. G. Calleja, B. Coto, A. M. Morales-Cas, Appl. Surf. Sci. 252 (2006) 4345. 50. M. Jaroniec, R. Madey, Physical Adsorption on Heterogeneous Solids, Elsevier, Amsterdam, 1988. 51. W. Rudzinski, D. H. Everett. Adsorption of Gases on Heterogeneous Surfaces; Academic Press, New York, 1991. 52. T. Kresge, M. E. Leonowicz, W. J. Roth, J. C. Vartuli, J. S. Beck, Nature 359 (1992) 710. 53. Zhao, Q. Huo, J. Feng, B. F. Chmelka, G. D. Stucky, J. Am. Chem. Soc. 120 (1998) 6024. 54. P. Van Der Voort, P. I. Ravikovitch, K. P. De Jong, M. Benjelloun, E. Van Bavel, A. H. Janssen, A. V. Neimark, B. M. Weckhuysen, E. F. Vansant, J. Phys. Chem. B. 106 (2002) 5873. 55. R. Ryoo, C. H. Ko, M. Kruk, V. Antochshuk, M. Jaroniec, J. Phys. Chem. B 104 (2000) 11465. 56. M. Kruk, M. Jaroniec, C. H. Ko, R. Ryoo, Chem Mater. 12 (2000) 1961. 57. M. Kruk, M. Jaroniec, C. H. Ko, R. Ryoo, J. Phys. Chem. B 107 (2003) 2205. 58. P. I. Ravikovitch, A. Vishnyakov, A. V. Neimark, Phys. Rev. E. 64 (2001) 011602. 59. G. J. Tjatjopoulos, D. L. Feke, J. A. Mann Jr., J. Phys. Chem. 92 (1998) 4006. 60. W. A. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flanery, Numerical recipes in Fortran, 2nd edition, Cambridge University Press, U.S.A., 1992, Chap 6, p263. 61. F. J. Blas, L. F. Vega, Mol. Phys. 92 (1997) 135. 62. J. C. Pàmies, L. F. Vega, Ind. Eng. Chem. Res. 40 (2001) 2532. 63. P. Koalczyk, P. A. Gauden, A. P. Terzyk, D. D. Do, G. Rychlicki, Annales Universitatis Mariae Curie-Sklodowska, Sectio AA, vol. LVII, 2, 46 (2002). 64. J. D. Wilson, J. Mater. Sci. 27 (1992) 3911. 65. M. Szombathely, P. Brauer, M. Jaroniec, J. Comput. Chem. 13 (1992) 17. 66. P. H. Merz, J. Comput. Phys. 38 (1980) 64. 67. C. W. Groetsch, The theory of Tikhonov regularization for Fredholm Equations of the first kind; Pitman Publishing, London, 1984. 68. P. C. Hansen, SIAM Review 24 (1992) 561. 69. P. C. Hansen, BIT Numerical Mathematics 30 (1990) 658. 70. M. V. López-Ramon, J. Jiagello, T. Bandosz, N. A. Seaton, Langmuir 13 (1997) 8. 71. S. K. Bhatia, Langmuir 18 (2002) 6845. 72. T. X. Nguyen, S. K. Bhatia, Langmuir 20 (2004) 3532. 73. T. X. Nguyen, S. K. Bhatia, Carbon 43 (2005) 775. 74. T. X. Nguyen, S. K. Bhatia, J. Phys. Chem. B 108 (2004) 14032.


Chapter 6

Hydrogen Adsorption in Corannulene-based Materials Yingchun Zhanga , Lawrence G. Scanlonb , and Perla B. Balbuenaa a

Department of Chemical Engineering, Texas A&M University College Station, TX 77843 b Air Force Research Laboratory, Electrochemistry & Thermal Sciences Branch WrightPatterson AFB, OH 45433

1. Introduction Hydrogen has been proposed as an ideal energy carrier for future, energy-demanding applications such as power generation, transportation, residential and commercial industries. Hydrogen is expected to replace the fossil fuel energy system, because of considerations of health, environmental safety, and renewable resources. Hydrogenrelated issues have been extensively studied in recent years, including mass production, delivery and storage, conversion and end-use energy applications. Among these issues, hydrogen storage is the key factor for hydrogen uses in transportation applications. So far, an ideal energy hydrogen carrier able to reach the criteria set by U.S. Department of Energy (DOE) of storing 6.5 mass % of hydrogen (density of 625 kg/m3 ) has not been designed yet. The main problems come from the H2 flammable character and from economical considerations. Even though some new materials, such as metal hydrides and the use of high pressurized tanks and liquid hydrogen, have been tested, these materials have several shortcomings such as low capacity, safety problems, or impractical release temperatures. On the other hand, with the advent of nanotechnology, nanotubes, nanofibers, and activated carbon materials have been of great interest experimentally and theoretically [1–8] for the purpose of hydrogen storage. The mechanism of hydrogen adsorption in carbon-based materials is mainly physisorption via van der Waals interactions, especially at ambient temperatures. Nanotubes offer some promise over other carbon materials for hydrogen adsorption. The improved hydrogen adsorption in nanotubes is attributed to the curvature of nanotubes [9], which might increase the interactions between the adsorbent and the hydrogen molecules. However, as a result of 127

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weak interactions between H2 and pure carbon, these materials do not show sufficient storage capacity for commercial use at ambient temperatures and pressures. Modifications of C-based materials which can enhance the interactions between H2 and the absorbents and thus enhance the physisorption of H2 are needed for improvement of hydrogen storage. Since Chen et al. [10] reported that alkali-metal-doped carbon nanotubes exhibit remarkable hydrogen uptake, a great deal of experimental and theoretical work has been done to investigate the hydrogen adsorption in doped carbon materials [11–17]. These studies showed that charge transfer from the alkali metal to these carbon materials polarize hydrogen molecules. As a result, a charge-induced dipole moment enhances the adsorption of hydrogen with doping alkali metal and carbon materials at ambient conditions. Dubot et al. [17] claimed that adsorbed lithium allowed the anchoring of molecular hydrogen on single-wall carbon nanotubes (SWNTs) with a binding energy in a chemisorption regime. Other than the charge-induced dipole moment, doping also increases the separation between molecules generating extra space where hydrogen can be stored [16]. Boron nitride nanotubes have also been studied as an alternative for room-temperature hydrogen storage [18, 19], yielding enhanced binding energies of −173 ∼ −288 kcal/mol, due to heteropolar bonding in boron nitride, and thus larger storage capacity, 18 ∼ 26 wt% under ∼ 10 MPa than that achieved in multiple-wall carbon nanotubes (MWNTs). In this study, we focus on hydrogen adsorption in corannulene-based materials. Similar to carbon nanotubes, corannulene is a bowl-shaped molecule [20] with higher electron density in the peripheric-carbon atoms than in inner-carbon atoms, as illustrated in Figure 1, which shows the calculated Mulliken charge distribution. We have recently shown that this bowl-shaped structure can cause the involvement of more carbon atoms interacting with hydrogen molecules [21], and that the charge distribution in corannulene may create an induced dipole moment in hydrogen [21]. The dipole-induced dipole interaction between corannulene and hydrogen molecules can further enhance physisorption of hydrogen in corannulene-based systems. Here we review our previous work on corannulene and Li-doped corannulene systems and introduce a few new systems with potential as adsorbents of molecular hydrogen.

Figure 1 Charge distribution on the atoms of the corannulene molecule. The color code ranges from red (negative) to green (positive)

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2. Methodology Theoretical studies have been playing a very important role in the investigation of hydrogen adsorption at molecular or atomic levels. These studies explain the mechanism of hydrogen adsorption and the details of the processes taking place, such as identification of hydrogen-binding sites on different materials, adsorption isotherms, and density profiles. Other than developing an understanding of the elementary steps in the adsorption process, theoretical studies can also predict limits in the hydrogen storage capacity and guide the direction of experiments by studying ideal and complex systems. In this section, we describe the methods used and the simulation procedures.

2.1. Ab Initio/DFT Calculations Previous theoretical studies on corannulene indicated that a hybrid DFT method combined with double- plus polarization basis sets would well reproduce the structural parameters of corannulene [22, 23] and protonation and lithium cation binding on corannulene [20, 24]. Here we use B3LYP/6-31G(d,p) for geometry optimization of corannulene and lithium atom–doped corannulene complexes. The optimized geometries are then followed by frequency calculations at the same level to make sure that they are local minima. Details of charge distribution and geometry configurations, as well as the electronic energies of different molecular systems are analyzed. From these studies, we are able to predict the molecular properties of these systems as potential candidates for hydrogen storage applications. Gaussian 03 [25] is used for ab initio/DFT calculations. The Cerius2 3.0 [26] and GaussView 03W [27] programs are used for visualization of the results. In order to find the favorable binding sites for hydrogen adsorption and to assess the interaction strength between the hydrogen molecule and corannulene, as well as with lithium-doped corannulene complexes, we perform potential energy surface scans at B3LYP/6-311G(d,p) with a hydrogen molecule approaching each molecular system from various directions. During the potential energy surface scan, the geometries of the hydrogen molecule, corannulene, and Li–corannulene complexes are fixed at the optimized geometry obtained at the level of B3LYP/6-31G(d,p). Only the distance between hydrogen molecule and a specific site of corannulene or Li–corannulene complexes is changed. Hydrogen adsorption in C-based materials belongs to the physisorption regime, i.e., the interactions are driven by weak van der Waals interactions, and given that DFT methods have some limitations to describe such weak van der Waals interactions, we perform single point calculations using second order Moller Plesset perturbation theory (MP2) to account for weak forces that may be responsible for physisorption. Such single-point calculations are performed at the geometry of the minima of the potential energy surface scan calculated with B3LYP/6-31G(d,p). Based on the energy obtained from MP2/6-31G(d,p), the interaction energy between hydrogen molecule and adsorbent molecules can be calculated precisely. This interaction energy is one of the important criteria to evaluate the potential performance as hydrogen storage materials. Our final aim is to predict the hydrogen storage capacity of corannulene and lithiumatoms-doped corannulene complexes at ambient conditions. To achieve this goal, we

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use classical molecular dynamics (MD) simulations to investigate hydrogen adsorption on the proposed adsorbent systems, which will be described in details in the following sections. In order to set the basic unit cell for the (MD) simulations, we need to establish possible assembly arrangements of these molecules. We use DFT calculations to fully or partially optimize dimers of corannulene molecules, and dimers of the most favorable lithium-doped corannulene complexes at specific lithium-doping concentrations. Full optimizations at the level of B3LYP/6-311G(d,p) are performed for dimers of corannulene molecules in different configurations, and partial optimizations at the same level of theory are performed for dimers of lithium-doped corannulene molecular complexes. During the partial optimizations, each of these complexes is treated as a rigid molecule and the two molecules are only allowed to move in one direction until the dimers reach their most stable states. The results of these partial optimizations provide an estimate of the separation between molecules which are used as input in the molecular dynamics simulations. These results yield intermolecular distance (IMD) and interlayer distance (ILD). The details of the definitions of IMD and ILD will be explained later.

2.2. Classical Molecular Dynamics (MD) Simulations 2.2.1. Simulation details MD simulations are carried out in the NVT ensemble with the Evans thermostat. A total simulation time of 800 ps at either 273 K or 300 K under different pressures is used for hydrogen adsorption, with equilibration runs of 300 ps and production runs of 500 ps. Hydrogen uptake capacities are evaluated as the average value obtained during the production period. Similarly, the corresponding pressure in the gas phase is determined by counting the average number of hydrogen molecules in the gas phase located above the adsorbent molecules. Periodic boundary conditions (PBC) in all three spatial directions are used. The cutoff radius, beyond which intermolecular interactions of the van der Waals potentials are set to zero, is chosen as 11.0 Å, which corresponds to one half of the minimum simulation cell length. The MD simulations are performed with the DL_POLY program, version 2.14 [28]. Initially N hydrogen molecules are located in the gas phase above the adsorbent phase. As simulations start, H2 molecules move from the gas phase to the adsorbent phase until they reach an equilibrium state, which is monitored by counting Ng , the number of H2 molecules in the gas phase as a function of time. The pressure in the gas phase is evaluated after the system reaches the equilibrium state. Hydrogen uptake is calculated by the following equations: N a = N − Ng w =

(1) Na mH2

Na mH2 + Nadsorbent madsorbent

× 100wt%

(2)

where Na is the number of adsorbed hydrogen molecules. For a given N , the molecules in the gas phase Ng are estimated by counting those molecules that are located at distances greater than 4 Å from the surface of the adsorbent. Ng is used to calculate the pressure, based on the ideal gas approximation, taking the average value obtained


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during the last 500 ps simulation period. We find that Na reaches an equilibrium value at about 100 ps. Longer simulation times of up to 1200 ps do not change the number of H2 adsorbed.

2.2.2. Arrangement of adsorbent molecules Corannulene systems. To investigate possible structural arrangements of corannulene dimers, we carried out DFT calculations at the level of B3LYP/6-311G(d,p). Calculations yielded sandwich and T-shaped conformers of the corannulene dimer, shown in the next section. The T-shaped conformer is the global minimum, and the sandwich is a local minimum, higher in 0.08 eV with respect to the T-shaped conformer [21]. These structures somehow resemble those found for the benzene dimer [29], although an additional dimer in parallel-displaced arrangement was not found for corannulene. Thus, two alternative structures were tested as corannulene adsorbent materials for the MD simulations, corresponding to the two dimer configurations, one where the molecules form T-shaped assemblies, as in the corannulene crystal [30], and another with a simple stack of corannulene molecules arranged as in the sandwich configuration. The crystalline structure of corannulene is monoclinic with space group of P21 /c (a = 13.260 Å, b = 11.859 Å, c = 16.520 Å, and = 120.69) at 20 C [30]. In this structure, corannulene molecules are arranged as T-shaped conformers. To represent such structure, we use an MD simulation cell containing 32 corannulene molecules, located at the bottom of the cell of dimensions 26520 Å × 23718 Å × c Å, and an angle = 120.69 . The value of the c parameter for each simulation is estimated on the basis of the number of H2 molecules, Ng , at a temperature T needed to obtain a given pressure P according to the ideal gas law [21]. In the second structural configuration tested for the adsorbent material, the simulation cell contains two layers of corannulene, with 16 molecules per layer, distributed at the bottom of the simulation cell. In this cell, corannulene molecules are arranged as sandwich conformers and also located at the bottom of the simulation cell. The cell dimensions are dependent on the arrangement of adsorbent molecules, decided by two parameters: the ILD and the IMD. The interlayer distance is defined as the distance between two molecules, one located on top of each other overlapping, and the intermolecular distance is that between the centers of two parallel molecules. Lithium-doped corannulene complexes. From DFT calculations, we are able to investigate the energetically favorable configuration of lithium atoms–doped complexes at each specific lithium concentration. Here, we take the most stable configuration at each lithium-doping concentration as the adsorbent for the investigation of hydrogen adsorption. We assume that lithium-atom-doped corannulene arranges in a simple stack configuration as discussed in previous section for the case of undoped corannulene systems. Each simulation cell contains 16 lithium-atom-doped corannulene molecules, with 8 molecules in one layer and distributed at the bottom of the simulation cell. In the MD simulations, the dynamics of the corannulene molecules is not included, thus the adsorbent molecules are kept fixed in their initial positions, and their distribution in each layer is determined by the IMD and ILD parameters obtained from DFT calculations described in the next section.

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2.2.3. Estimation of the IMD and ILD parameters We recently reported DFT calculations at the level of B3LYP/6-311G(d,p) for the characterization of corannulene dimer and six lithium-atoms-doped (at the concave side) of a corannulene dimmer [31]. To investigate available space and collective effects on H2 adsorption, we assumed that lithium-doped corannulene molecules are arranged in a simple stack. Thus, we can adjust the ILD and IMD variables, performing analyses similar to those reported by other researchers using CNT bundles and GNF [16, 32–34]. To have a reference, we computed the optimum ILD and IMD distances for a corannulene dimer and a six-lithium-atoms-doped (at the concave side) corannulene dimer using partial optimization calculations which provide estimates of the IMD (11.4 Å) and the ILD (6.5 Å) parameters for six-lithium-atoms-doped (at the concave side) corannulene dimer, and of the ILD (4.8 Å) for corannulene dimer [31]. To determine the optimum ILD value, only the z co-ordinates of the dimer were allowed to change, whereas only the x co-ordinates of the dimer are allowed to change to investigate the optimum IMD value. In our MD simulations, the IMD is fixed to 11.0 Å for all the adsorbent systems, and additional values of ILD of 8.0 and 10.0 Å are used for a simple stack of two different lithium-doped corannulene systems, and additional values of ILD of 6.0 and 8.0 Å for a simple stack of corannulene molecules. Such systems were chosen to investigate the potential H2 uptake capacity, assuming that substitution of H with bulky alkyl function groups to the rim carbons, for example, t-butyl, isopropyl [35], or bridging of lithium-doped corannulene molecules might increase the ILD. 2.2.4. Force fields used in the classical MD simulations The pair interaction between hydrogen molecules is represented by a simple spherical 12-6 Lennard–Jones (LJ) potential, with = 297 meV = 296 Å [36], which is able to reproduce the vapor–liquid phase behavior of hydrogen gas over a broad range of conditions [37]. Figure 2 shows the potential energy between two hydrogen molecules modeled as single site LJ spheres. We have also tested a two-site model for H–H intermolecular interactions [38]; however, the model fails to reproduce the correct minima found in ab initio calculations for the potential energy of H2 [39]. For the interaction potential between H2 and the carbon atoms of corannulene, we use a one-site 12-6 LJ potential, with parameters initially derived following the procedure described by Cheng et al. [40], to introduce curvature effects of the corannulene

20

V (meV)

15 10 5 0 –5

2

2.5

3

3.5

4

4.5

5

5.5

r (Å)

Figure 2 The potential energy (V ) of two hydrogen molecules modeled as spheres interacting via a Lennard–Jones potential with parameters: = 297 meV = 296 Å


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molecule [9]. The curvature in the corannulene molecule could be compared with that of a cap of a 10 Å diameter carbon nanotube. Therefore, taking a value of 5 Å for the corannulene radius, the parameters C–H = 335 meV C–H = 278 Å are calculated for a curvature-dependent 2-site model. The interaction is then refitted to that of a single-site model, yielding C–H = 252 meV C–H = 295 Å. However, additional interaction needs to be included because of the polarity of the corannulene molecule bearing a dipole moment of 2.07 D determined experimentally. Although H2 does not have a permanent dipole moment, the permanent dipole moment of corannulene creates an induced dipole moment on H2 , enhancing the adsorption of hydrogen molecules on the corannulene system. The general Debye formula for the mean potential energy due to induction by permanent dipoles is [41]: ij = −

i 2j + j 2i

40 2 r 6

(3)

where i and j refer to hydrogen and corannulene, is the molecular polarizability, and as the dipole moment. Polarizabilities are usually reported in units of volume using the relation, =

40

(4)

Values of are 81 × 10−25 cm3 for hydrogen [42] and 255 × 10−23 cm3 for corannulene [43], To calculate the mean potential energy, we first estimate the value of the induced dipole moment of hydrogen in the electric field created by corannulene. The field strength E around corannulene molecules is evaluated using the equation E=

(5)

Equation (3) thus can be simplified to ij = −

223604 meV r6

(6)

This mean potential energy due to dipole and induced-dipole interactions is added to the LJ potential for C–H interactions. As a result, the new 12-6 LJ parameters C–H = 402 meV and C–H = 280 Å are used in the reported MD simulations. Figure 3 shows the fitting results for the C–H interaction, to obtain the LJ parameters. To obtain the interaction between Li and one-site hydrogen molecule, the 12-6 LJ parameters for the Li–H pair are derived using published van der Waals parameters for Li atom in the form: 6 12 xij xij Evdw = Dij −2 (7) + x x where DLiLi = 0025 kcal/mol, and xLiLi = 2451 Å [44]. The parameters are then refitted to the 12-6 LJ form, with LiLi = 119 meV and LiLi = 218 Å. The Lorentz–Berthelot mixing rules are applied to get the cross parameters Li–H = 187 meV and Li–H = 257 Å. For

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a

c

15

V (meV)

b 10 5 0 2

3

2.5

3.5

4

4.5

5

5.5

–5

r (Å)

Figure 3 Fitting results of LJ parameters for the C–H interaction between the C atoms of corannulene and H2 modeled as a single 12-6 LJ sphere. (a) Potential energy of LJ interaction between C and one-site hydrogen molecule, C–H = 252 meV C–H = 295 Å; (b) Sum of dipoleinduced interaction and LJ interaction between C and one-site hydrogen molecule; (c) Fitting of potential energy of LJ interaction between C and one-site hydrogen molecule, C–H = 402 meV C–H = 280 Å

the dipole-induced interaction, we take the dipole moment of Li3 –C20 H10 –Li2 complex of 3.856 De calculated at the level of B3LYP/6-311G(d,p) and lithium polarizability of 243 × 10−23 cm3 [45]. An average pair interaction is calculated as: ij = −

775974 meV r6

(8)

With the addition of the dipole-induced interaction between Li and H, the non-bonded interaction parameters for Li–H used are Li–H = 3906 meV and Li–H = 200 Å in the 12-6 LJ form. The fitting process and results are shown in Figure 4. 100

a

75

V (meV)

50 c

25

b

0 1

2

3

4

5

6

– 25 – 50

r (Å)

Figure 4 Fitting results of LJ parameters for Li–H interaction. (a) Potential energy of LJ interaction between Li and one-site hydrogen molecule, Li–H = 187 meV Li–H = 295 Å; (b) Sum of dipole-induced interaction and potential energy of LJ interaction between Li and one-site hydrogen molecule; (c) Fitting of potential energy of LJ interaction between Li and one-site hydrogen molecule, Li–H = 3906 meV Li–H = 200 Å


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3. Characterization of Corannulene 3.1. Corannulene Monomer and Dimer Corannulene is a bowl-shape molecule with C5v symmetry. Figure 5 illustrates an isodensity surface showing the atomic distribution. Among the various types of non-bonded interactions between molecules, − interactions have been realized as the key roles in molecular recognition, crystal packing, and self-assembly. Zhao and Truhlar [46] recently used multicoefficient extrapolated DFT to study aromatic − interactions of benzene dimer and obtained three conformers of benzene dimer: sandwich (S), T-shaped (T), and parallel-displaced (PD). For corannulene, because of its molecular structure it is not surprising that − interactions also play important roles in molecular self-assembly. Here, we use DFT to study the geometry configurations of corannulene dimer at the level of B3LYP/6-311G(d,p). The optimizations yield sandwich and T-shaped conformers of corannulene dimer, as shown in Figure 6. The T-shaped conformer is the global minimum, and the sandwich is a local minimum, higher in 0.08 eV with respect to the T-shaped conformer. These structures somehow resemble those found for the benzene dimer by Zhao and Truhlar [46], although we did not find an additional dimer in parallel-displaced arrangement for the corannulene dimer.

Figure 5 Isodensity surface of corannulene calculated at the level of B3LYP/6-311G(d,p). The isodensity corresponds to 0.001 e/Å3

(a)

(b)

Figure 6 Conformations of corannulene dimer according to B3LYP/6-311G(d,p): (a) sandwich; (b) T-shaped

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Another point that is worth to be mentioned is the separation in sandwich corannulene dimers. Our previous studies revealed the importance of the spacing between the corannulene layers, which if large enough, allows hydrogen molecules to come under the influence of both corannulene rings and thereby double the binding energy of the hydrogen molecule to the adsorbents as described in Table 1 of our previous publication [21]. In relation to this point, it is interesting to compare the distribution of electronic density for the dimers of corannulene as given in Figure 7. For the H2 adsorption application that we are investigating, it is important to elucidate the spatial regions where the H2 molecules are most likely to be able to adsorb. Figure 7 shows that the T-shaped dimer structure found in crystalline corannulene is very similar to our calculated dimer, although the angle between the two molecular planes is smaller in the crystalline structure than in the single dimer. The sandwich structure, on the other hand, provides an overlap of the electronic densities of the two corannulene molecules. The overlap of the electronic densities results in an enhanced H2 adsorption energy between corannulene dimers, [21] compared to the H2 adsorption energy on a single corannulene molecule.

3.2. AM1 Studies of Corannulene Clusters Having established the S and T comformations of corannulene dimer, we investigate the possible conformations of clusters having more than two corannulene molecules. To reduce computational costs, we choose a semi-empirical method, AM1 (Austin Model 1) [47], instead of DFT calculations, since the idea is to obtain a reasonable initial structure to be used in classical MD simulations. Optimization of 4 corannulene molecules arranged in sandwich conformation failed to converge. However, 4, 8, and 12 corannulene molecules arranged in T-shaped conformation converged successfully. Figure 8 shows the structures of these 4, 8, and 12 corannulene molecular systems, with every pair of corannulene molecules arranged in T-shaped conformation. In the structure of (C20 H10 4 every two neighbor corannulene molecules are well assembled in T-shape, as we observed in our previous DFT study of the corannulene dimer. In the (C20 H10 8 and (C20 H10 12 systems, deformed T-shaped conformers are observed. Instead of having two neighbor corannulene molecules perpendicular to each other, every two neighbor corannulene molecules are inclined one to the other in certain angle.

Figure 7 Isodensity surface (0.001 e/Å3 ) of corannulene dimers: Left – according to the crystalline structure; Center – sandwich; Right – T-shaped dimer


(a)

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(b)

(c)

Figure 8 Optimized structures of n corannulene molecules using AM1. (a) n = 4; (b) n = 8; (c) n = 12

4. Doping of Lithium Atoms on Corannulene Doping of lithium atoms on carbon materials has proven to be an effective way for improvement of hydrogen adsorption. In this section, doping sites of lithium atoms on corannulene are studied. Recently, Kang [48] used DFT to study Li – aromatic sandwich compounds, R–nLi–R, where R is benzene, naphthalene, pyrene. It was found that Li atom was preferentially adsorbed over the six-member ring, instead of over individual or pair of C atoms. For a single Li ion, it was found that the Li cation was bounded on the convex side over a six-member ring of corannulene at the level of B3LYP/6311G(d,p)//B3LYP/6-31G(d,p) [20]. We doped a single Li atom and a single Li ion on the concave/convex side over a six-member ring of corannulene at the level of B3LYP/6-31G(d,p). The results indicate that complexation of Li ions at the convex side is more stable than at the concave side, which is in agreement with the results of Frash et al. [20] However, doping Li atoms on corannulene, we found that complexation at the concave side is more stable (by 1.2 Kcal/mol) than at the convex side [31]. The Li atom

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is more stable over the six-member ring than over the five-member ring, which is due to the larger number of carbon atoms in the first coordination sphere in the first case. Doping of multiple Li atoms is more complicated since these lithium atoms can be doped either at concave side or at convex side. In this study, the doping ratio of lithium atoms to carbon atoms of corannulene is between 1:3 and 1:4. We investigated doping of five and six lithium atoms on corannulene molecule [31]. The lithium doping concentration is in agreement with results reported by other groups [16, 48]. We reported doping of such five lithium atoms on corannulene including (1) five lithium atoms on the concave side; (2) five lithium atoms on the convex side; (3) three lithium atoms on the concave side and two on the convex side; and (4) two lithium atoms on the concave side and three on the convex side [31]. Geometry optimizations of these different complexes were studied at the level of B3LYP/6-31G(d,p); the calculation was followed by frequency calculations at the level of B3LYP/6-31G(d,p). In the initial configuration, all five Li atoms were located at the same side (either convex or concave) of the corannulene molecule, and the optimized structure resulted always with the five Li atoms attached to the concave side, with all five Li atoms doped at the concave side over the six-member rings. Thus, when the optimization was started locating the Li atoms over the convex side, there was an inversion of curvature of the corannulene molecule in agreement with the tendency for a single Li atom. Frequency calculations indicated that both complexes are local minima. Similarly, when the initial configuration contains two Li atoms on one side and three on the other side, the energetically favorable optimized conformations contain more Li atoms on the concave side – which is, three Li atoms on the concave side and the other two Li atoms on the convex side. In this conformation, the five Li atoms are more separated from each other and so the repulsion between them decreases, therefore this conformation has lower energy than that with five Li atoms all attached at the concave side. This complex is about −4 kcal/mol lower than the complex with all five lithium atoms doped on the concave side. Here, we name these two complexes as Li5 –C20 H10 and Li3 –C20 H10 –Li2 respectively. The subscript of Li before C20 H10 indicates the number of lithium atoms doped on the concave side, and that after C20 H10 indicates the number of lithium atoms doped on the convex side. This notation is used all through this chapter in other doping complexes. Overall, it was found that the total charge on Li atoms in Li3 –C20 H10 –Li2 is higher than those in Li5 –C20 H10 . Thus, C atoms of corannulene are more negatively charged. Due to the fact that in the Li3 –C20 H10 –Li2 complex there is a higher charge transfer from Li atoms to corannulene and the Li atoms are located at shorter distances from the six-member ring, the net attractive interaction between Li atoms and corannulene is stronger than in Li5 –C20 H10 . The binding energy per Li atom is −2391 kcal/mol-Li in Li3 –C20 H10 –Li2 and −2311 kcal/mol-Li in Li5 –C20 H10 . For doping of six lithium atoms on corannulene, in which, in each case, the concave side is doped more with lithium atoms than on the convex side, we studied cases including (1) six lithium atoms all in the concave side; (2) five lithium atoms are on the concave side with one lithium atom on the convex side; (3) four lithium atoms on the concave side and two on the convex side; and (4) three each on the concave and convex side. These four complexes were optimized at the level of B3LYP/6-31G(d,p), followed by the frequency calculations. Optimizations show that among these four


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different complexes the complex with six lithium atoms doped at the concave side is the most stable complex. Figure 9 shows the two most stable conformations found in our calculations of six and five Li-doped complexes, Li6 –C20 H10 and Li3 –C20 H10 –Li2 , respectively. With all six Li atoms doped on the same side, forming the complex Li6 –C20 H10 , the first five Li atoms are doped over six-member rings on the concave side, and the sixth lithium atom in the center over these five Li atoms. The Li6 –C20 H10 complex has v symmetry. The binding energy per mole lithium is −2503 kcal/mol-Li for Li6 –C20 H10 . These binding energies are comparable in magnitude to those of the complexes of five Li atoms. The relatively large values of the binding energies might indicate stable lithium adsorption on corannulene, with the ratio of Li:C in the range between 1:4 and 1:3. The shape of the electronic density surface shown in Figure 9 suggests that the Li6 –C20 H10 molecule may offer an extended surface for hydrogen adsorption.

5. Interaction of Hydrogen with Corannulene-based Materials Ab initio/DFT calculations were performed for the adsorption of hydrogen on corannulene and lithium-doped corannulene complexes [31]. Studies of adsorption of hydrogen molecule on lithium-doped corannulene complexes are only performed on the energetically favorable doping complex of each doping concentration, i.e., on the complex Li6 –C20 H10 for six-lithium-atoms-doped corannulene, and the complex Li3 –C20 H10 –Li2 for five-lithium-atoms-doped corannulene. In this section, we perform three different studies. In recent work we investigated the potential energy surface of the systems when hydrogen molecule approaches corannulene or lithium-atoms-doped corannulene complexes, using DFT calculations [31]. We also studied the interaction of the hydrogen molecule with the adsorbent molecule using MP2/6-31G(d,p), to account for weak VDW interactions in these compounds [21, 3]. For hydrogen in corannulene system, two orientations of hydrogen are used (end on or linear vs sideways or parallel) and

Figure 9 Isodensity surfaces (0.000 4 e/Å3 ) calculated from optimized (B3LYP/6-31G(d,p)) conformations of corannulene complexed with Li atoms doped at different positions. Top: Li6 –C20 H10 , Bottom: Li3 –C20 H10 –Li2

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full geometry optimizations are performed. For hydrogen in two-lithium-atoms-doped corannulene complexes, only single-point calculations are performed, due to the expensive computational cost. Here, we briefly summarize these results, and then we describe the adsorption of 18 hydrogen molecules on Li3 –C20 H10 –Li2 using DFT.

5.1. Potential Surface Scan Potential surface scans of hydrogen molecule approaching Li6 –C20 H10 and Li3 –C20 H10 –Li2 complexes were investigated to obtain an estimate of the potential of these materials as H2 adsorbent as compared to pure corannulene [31]. The molecular structures of corannulene, Li6 –C20 H10 complex, and Li3 –C20 H10 –Li2 complex are the optimized results at the level of B3LYP/6-31G(d,p) and the whole molecular structures are kept fixed during the potential energy surface scan. When the hydrogen molecule approaches to corannulene, an attractive interaction between hydrogen and the corannulene molecule is not detected at this level of theory [31]. The most repulsive interaction appears when the hydrogen molecule approaches from the convex side to the six-member ring of corannulene, and the least repulsive when approached from the concave side to the center, five-member ring. However, when a hydrogen molecule approaches Li6 –C20 H10 or Li3 –C20 H10 –Li2 complexes, a clear and relatively strong attractive interaction between H2 and the lithium-atom-doped complex is observed. The strongest attraction appears when the center of hydrogen molecule is 2.47 Å from corresponding lithium atom and the attraction energies were −047 and −206 kcal/mol with complex Li6 –C20 H10 and Li3 –C20 H10 –Li2 , respectively [31]. The attractive interaction between hydrogen molecule and the Li6 –C20 H10 or Li3 –C20 H10 –Li2 complexes confirms that doping of lithium enhances the interaction between the hydrogen molecule and the adsorbent molecule. The enhanced interaction between the hydrogen molecule and the Li-doped corannulene complexes together with the higher available space due to the increase of the interlayer separation caused by lithium doping are the reasons that hydrogen uptake capacity is higher in Li-doped corannulene systems, as will be discussed in the coming sections.

5.2. Single-point Calculations of Binding Energy with MP2 Theory Scanlon et al [21] calculated the binding energies of hydrogen adsorption via physisorption on corannulene. Geometry optimization of a single hydrogen molecule on the concave side and convex side of corannulene with MP2(full)/6-31G(d) yielded binding energies of −094 and −083 kcal/mol for hydrogen molecule on the concave side and the convex side, respectively. Single-point calculation at higher level with MP2(full)/6 − 311 + + G 3df 2p at the optimized geometries with MP2(full)/g-31G(d) yielded −281 and −138 kcal/mol, respectively. Okamoto [9] calculated the potential energy curve of corannulene-H2 by using MP2 with the 6-31G(d) basis set for corannulene and the 6-311G(d,2p) basis set for H2 molecule and obtained 3.05 Kcal/mol of binding energy with H2 molecule adsorbed on the concave side of the corannulene. Geometry optimization of hydrogen molecule on the concave side of corannulene at lower level of B3LYP/6-311G(d,p) does not yield a negative binding energy as obtained with MP2 calculations.


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We studied the interaction of the hydrogen molecule with corannulene using MP2/631G(d,p). Figure 10 shows the top view and the side view of the optimized structures of H2 –C20 H10 systems, one with parallel hydrogen adsorbed (Figure 10(a)), and the other with head-on hydrogen adsorbed (Figure 10(b)). At the same level of theory, adsorption of hydrogen on different locations, such as on the six-member ring C atoms, yields higher energy than adsorption of hydrogen on the five-member ring, both for head-on and parallel orientation. Considering the length of the H–H bond of 0.74 Å, in the head-on orientation, the distance from the center of mass of hydrogen molecule to the center of the fivemember ring is 3.01 Å, which equals to the sum of 2.64 Å and half of H–H bond of 0.74 Å. This distance is the same as the distance from the center of mass of hydrogen molecule in the parallel orientation to the center of the five-member ring, as shown in Figure 10(a). But one hydrogen atom in the head-on orientation is much closer to the center of the five-member ring and the distance is 2.64 Å. In the parallel orientation, both hydrogen atoms have zero charge; while in the head-on orientation, the hydrogen close to the five-member ring bears a charge of +0014 and the other hydrogen on the top is negatively charged with −0015e. The binding energies of the hydrogen molecule to corannulene are −113 and −124 kcal/mol for parallel orientation and head-on orientation, respectively. Adsorption of hydrogen in head-on orientation is more stable. This is in agreement with Scanlon et al studies [21], where single or multiple hydrogen molecules are all adsorbed to corannulene in head-on orientation. In order to account for weak van der Waals forces that are responsible for the hydrogen molecule/lithium-doped corannulene complex interaction, we used second

3.01 Å

(a)

2.64 Å

(b)

Figure 10 Adsorption of a hydrogen molecule on corannulene, MP2/6-31G(d,p) (a) parallel orientation; (b) head-on orientation

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order Möller Plesset perturbation theory to perform single-point calculations of the hydrogen molecule interacting with these two complexes individually with the basis set of 6-31G(d,p). Calculations of the single point energy yield the binding energies of −183 and −282 kcal/mol for H2 – Li6 –C20 H10 ) system and H2 – Li3 –C20 H10 –Li2 system, respectively [31]. Compared to the binding energies of the hydrogen molecule adsorbed on the center, five-member ring of corannulene, the binding energies increase considerably, especially in the H2 – Li3 –C20 H10 –Li2 system, where binding energy is more than double of that in the H2 – C20 H10 system. The interaction is indeed stronger between a hydrogen and lithium atom than between a hydrogen and a carbon atom according to the value of binding energies. As a result, this enhanced binding energy between the hydrogen molecule and lithium-atom-doped complexes results in higher hydrogen uptake capacity, as we discuss in a later section.

5.3. Adsorption of Multiple Hydrogen Molecules on Li3 –C20 H10 –Li2 From our preliminary surface energy scan we found that the binding strength of the hydrogen molecule to the lithium atom was larger in Li3 –C20 H10 –Li2 than in Li6 –C20 H10 . In this section we report adsorption of multiple hydrogen molecules. One of the goals is to get an estimate of the maximum number of hydrogen molecules that can be stably adsorbed per molecule of adsorbent. We choose the Li3 –C20 H10 –Li2 complex and a total of 18 hydrogen molecules, with 10 molecules on the concave side and 8 molecules on the convex side. Due to the computational cost, we use the DFT method at the level of B3LYP/6-31G(d,p) to study the interaction of multiple hydrogen molecules on one adsorbent molecule. The calculation yields a total of −1130 kcal/mol for the interaction of 18 hydrogen molecules and the Li3 –C20 H10 –Li2 complex. By careful examination of the optimized geometry of the system, as shown in Figure 11, we find that 16 of the 18 hydrogen molecules are adsorbed on the first shell around the Li3 –C20 H10 –Li2 complex; among these 16 hydrogen molecules, 8 are on the concave side, and the other 8 on the convex side. If we calculate the binding energy based on 16 adsorbed molecules, the value is −071 kcal/mol-H2 . On the two lithium atoms on the convex side, 6 of 8 hydrogen molecules are surrounding the lithium atoms. The distance between the hydrogen molecules and the Li3 –C20 H10 –Li2 complex is in the range of 2.14–2.42 Å, which is consistent with our study of the potential energy surface scan of the hydrogen molecule on the Li3 –C20 H10 –Li2 complex. On the concave side, it also has 6 hydrogen molecules adsorbed on the three lithium atoms. Most of the hydrogen molecules adsorbed on lithium atoms. For each lithium atom, up to 3 hydrogen molecules can be adsorbed. Also, we notice that hydrogen molecules are in head-on orientation adsorbed to the complex.

6. Modification of Corannulene 6.1. Adsorbent Design In MD simulations, which will be discussed later, we use expanded interlayer distances other than the optimum separation between corannulene molecules in the dimer


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Figure 11 Optimization geometry of 18 hydrogen molecules with Li3 –C20 H10 –Li2 complex at B3LYP/6-31G(d,p)

or the equivalent distance in lithium-doped corannulene dimers. That the corannulene molecule may be modified to achieve such large separation, so the adsorbent system can provide more available space for hydrogen adsorption justifies the use of a larger value of ILD. Here, the objective is to search for suitable substituents to chemically modify the molecule of corannulene. The new material can be expected to have more space available when it is assembled and used as an adsorbent for hydrogen adsorption. Aylon et al. reported two different complex sandwich dimers of corannulene derivatives, tert-butylcorannulene and isopropylcorannulene, where one H was substituted by tert-butyl group and isopropyl group, respectively [35]. Besides, the dimers of monosubstituted corannulene exhibited supramolecular stereochemistry, a meso dimer and a d l dimer. Seiders et al. successfully synthesized different alkyl derivatives of corannulene, for example methylcorannulene, dimethylcorannulene, tetramethylcorannulenes, acecorannulene, C5h symmetric pentamethylcorannulene, and decamethylcorannulene, by substituting a H with a methyl group [49, 50]. Sygula et al. used addition of various alkyllithium reagents to obtain 1-alkyl-1,2-dihydrocorannulenes. X-ray diffraction studies showed the exo-pseudoaxial conformation in the solid state of different derivatives [51]. Recently, these authors reported a clean and quick synthesis of discorannulenosemibullvalene dimethyl dicarboxylate with the exo-endo conformer of the semibullvalene as the most stable conformer [52]. Other groups also reported the substitutions of different functional groups to H attached to the rim carbon atoms of

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corannulene [53, 54]. With the bulky functional groups and self-assembly of these derivatives, more available space in these materials can be expected. The increased free space of these materials justifies the use of higher values of ILD in MD simulations for the adsorption of hydrogen as discussed later in this chapter. Other than substitution of H with bulky functional groups, bridging two corannulene molecules can also change the available space of the materials. Linkers have been successfully used in Metal Oraganic Frameworks (MOFs) to increase the internal surface and porosity of MOFs. Shabtai et al. reported that the use of an octamethylene chain could join two corannulene molecules to form 1,8-dicorannulenyloctane [55]. This is very interesting as it provides another way of modification to corannulene molecules, by bridging two molecules with certain linkers in a certain way to obtain more porous materials. These new materials potentially with more available space can increase the adsorption of hydrogen. Based on this idea, we modeled 1,5-biscorannulenecyclophane (C52 H20 ) that was synthesized at the University of Cincinnati [56]. The molecular structure of 1,5-biscorannulenecyclophane is shown in Figure 12. The molecule has a jaw shape with the mirror plane of the symmetry element v .

6.2. Characterization of 1,5-biscorannulenecyclophane Figure 13 shows another view of the molecular structure, from the convex side of the molecule, optimized at the level of B3LYP/6-31G. From this view, only half of the atoms on the front can be seen, labeled from C1 through H36. The other half of atoms on the back cannot be seen. But we can label these atoms from C1 through H36 , corresponding to symmetric positions with respect to the mirror plane. Table 1 shows the distance between two symmetric carbon atoms separated by the mirror plane. These values of distance somehow are similar to the ILD values that we used in our molecular simulations. The bridging of two corannulene molecules leads to an open-jaw shape, with pairs of atoms in the bottom closer to each other. Compared to the optimum ILD of 4.8 Å between two sandwich corannulene molecules optimized at the level of B3LYP/6-311G(d,p), only three pairs of the carbon rim (Cr ) atoms, Cr –Cr distance are smaller. These three pairs of Cr –Cr are 3.88 Å, 4.26 Å, 4.44 Å for C3–C3 , C2–C2 , C4–C4 , respectively. Other pairs of Cr –Cr , bridge C atom pairs Cb –Cb , and hub C atom pairs Ch –Ch are all larger than the calculated optimum ILD value of the sandwich corannulene dimer. The distance ranges from 5.23 to 9.57 Å, depending on the positions of each pair in the molecule. Taking the average distance between the atoms constituting

Figure 12 Molecular structure of 1,5-biscorannulenecyclophane


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Figure 13 Labeled 1,5-biscorannulenecyclophane, optimized with B3LYP/6-31G

these pairs, the value is about 7.22 Å. Such distance is between ILD values of 6 and 8 Å for the simple stack corannulene configurations. At this point, we could expect that the adsorption of hydrogen molecules on 1,5-biscorannulenecyclophane would show clear improvement compared with adsorption of hydrogen molecules in the system of a simple stack of corannulene molecules with an ILD of 4.8 Å. The distances between C23–C26 and C23 –C26 are 12.85 Å, and the distance between the pairs H35-H36 and H35 -H36 is 14.74 Å. These distances are pointed out because they are much larger than the diameter of corannulene molecule, which can also increase the space between molecules. Other than the potentially higher available space for hydrogen molecules to get adsorbed, the dipole moment of 1,5-biscorannulenecyclophane is also larger than that of the corannulene molecule (2.07 D experimentally, 2.15 D theoretically). DFT calculations with B3LYP yield 3.45 D at the level of 3-21G, and 2.69 D at the level of 6-31G. The higher dipole moment of 1,5-biscorannulenecyclophane can create a stronger dipole-induced interaction between hydrogen and 1,5-biscorannulenecyclophane molecule, thus increasing the hydrogen uptake capacity. Charge distribution on 1,5-biscorannulenecyclophane is also shown in Table 1.

6.3. Intermolecular Interactions The adsorption of hydrogen in different adsorbent systems is attributed to Van der Waals and dipole–induced interactions between hydrogen molecules and adsorbent molecules. The above reported 12-6 LJ interaction parameters between a C atom and a single-site model of hydrogen molecule, C–H = 252 meV, and C–H = 295 Å are used in classical MD simulations. Similarly, additional interactions need to be included because although H2 does not have a permanent dipole moment, the 1,5-biscorannulenecyclophane molecule is polar with theoretical of 2.69 D calculated at the B3LYP/6-31G level, as we mentioned in previous section. This permanent dipole moment of 1,5-biscorannulenecyclophane creates an induced dipole moment on H2 , enhancing the adsorption of hydrogen molecules on the 1,5-biscorannulenecyclophane system. Following the same procedure described

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Table 1 Selective structural parameters (Å) and Mulliken charge distribution (q, in e) of 1,5-biscorannulenecyclophane* at B3LYP/6-31G Cr –Cr

Cr –Cr

Cb –Cb

Ch –Ch

C1–C1 C2–C2 C3–C3 C4–C4 C5–C5

604 426 388 444 535

C6–C6 C7–C7 C8–C8 C9–C9 C10–C10

772 876 957 921 732

C11–C11 C12–C12 C13–C13 C14–C14 C15–C15

605 523 724 938 863

C16–C16 C17–C17 C18–C18 C19–C19 C20–C20

7.65 7.25 8.21 9.25 8.89

Atoms

q

Atoms

q

Atoms

q

Atoms

q

H28,H28 H29,H29 H30,H30 H31,H31 H32,H32 H33,H33 H34,H34 H35,H35 H36,H36

0.127 0.145 0.138 0.129 0.130 0.131 0.145 0.165 0.160

C1,C1 C2,C2 C3,C3 C4,C4 C5,C5 C6,C6 C7,C7 C8,C8 C9,C9

0178 −0120 −0146 −0123 0170 −0122 −0148 −0147 −0144

C10,C10 C11,C11 C12,C12 C13,C13 C14,C14 C15,C15 C16,C16 C17,C17 C18,C18

−0130 0149 0109 0140 0121 0106 −0084 −0062 −0079

C19,C19 C20,C20 C21,C21 C22,C22 C23,C23 C24,C24 C25,C25 C26,C26 H27,H27

−0078 −0052 −0225 −0232 −0023 −0211 −0230 −0024 0141

*The atom numbers correspond to Figure 13.

for corannulene, we can obtain the induced-dipole interaction between hydrogen and 1,5-biscorannulenecyclophane. Equation (3) thus can be simplified to ij = −

372798 meV r6

(9)

This average potential energy (curve c) due to dipole-induced dipole interactions is added to the LJ potential for C–H2 interactions. The overall potential of the sum of dipole-induced interaction and the LJ interaction between C and the one-site hydrogen molecule is plotted in Figure 14(b). Finally, the resulting potential energy is refitted yielding a new set of 12-6 interaction parameters that include the effects of curvature of the adsorbent molecule and the dipole-induced dipole interactions, as in Figure 14(c). As a result, the 12-6 LJ parameters C–H = 615 meV and C–H = 275 Å are used in the reported MD simulations for this system. Figure 14 shows the fitting results of 12-6 LJ parameters for the C–H2 interaction.

6.4. Partial Optimizations of 1,5-biscorannulenecyclophane Dimer To prepare the input to perform MD simulations of adsorption of hydrogen molecules on the 1,5-biscorannulenecyclophane system, we need information about the assembly of 1,5-biscorannulenecyclophane molecules. Unfortunately, there is not such experimental data available yet. Here we use a simple stack of 1,5-biscorannulenecyclophane molecules. By performing the partial optimization of a 1,5-biscorannulenecyclophane dimer we can get some insights about the possible arrangement of molecules, which is then adopted in the MD simulations of hydrogen adsorption.

Hydrogen Adsorption in Corannulene-based Materials a

50

V (meV)

40 30

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b c

20 10 0 – 10 2

3

4

5

6

r (Å)

Figure 14 Fitting results of 12-6 LJ parameters for the C–H2 interaction of the 1,5-biscorannulenecyclophane adsorbent. (a) Potential energy of LJ interaction C and onesite hydrogen molecule, C–H = 252 meV C–H = 295 Å; (b) Sum of dipole-induced interaction and potential energy of LJ interaction C and one-site hydrogen molecule; (c) Fitting of the potential energy of LJ interaction between C and the one-site hydrogen molecule model, C–H = 615 meV C–H = 275 Å

Figure 15(a–d) shows the partial optimization results of the 1,5-biscorannulenecyclophane dimer at the level of B3LYP/3-21G. On these partial optimizations, only the x-coordinates, in (a) and (b), or the y-coordinates, in (c) and (d) of the dimer are allowed to change. The difference between (a) and (b) is that, in (a) one jaw opens up and the other down, while in (b) both jaws open up. The energies are −3 964646 02 Hartrees and −3 964644 71 Hartrees for (a) and (b), respectively. Energetically, the dimer in (a) is more favorable and the distance between the center of mass of two molecules is 13.54 Å. In (c) and (d), only the y-coordinates are allowed to change. In (c) two molecules are totally overlapped along the Y -axis, while in (d) two molecules are jaw-to-jaw. The optimized structures of these two dimers have electronic energies of −3 964649 42 Hartrees and −3 964647 41 Hartrees. Conformation (c) is more stable than conformation (d) and the distance between the centers of mass of these two molecules in (c) is 8.24 Å. For the purpose of the MD simulations of adsorption of hydrogen on 1,5biscorannulenecyclophane system, we build a simulation cell with periodic boundary conditions (PBC) in the three spatial directions. The molecular assembly depends on our partial optimization results. Along the X-axis, the molecules arrange repeatedly as shown in (a). Along the Y -axis, the molecules arrange totally overlapped, as shown in (c). For the simulation cell, the arrangement in the Z-axis needs to be determined. Figure 16 shows the arrangement of 1,5-biscorannulenecyclophane in the YZ plane. Along the Z-axis, the molecules are arranged with the jaw of the molecule up and down alternating. Based on our previous ab initio calculations of the corannulene dimer, which yielded an ILD value of 4.8 Å at the level of B3LYP/6-311G(d,p), the separation between the centers of mass of 1,5-biscorannulenecyclophane dimer along the Z-axis is 14.0 Å, and at this value of separation, the distance between two center five-member rings from two neighbor molecules is 4.8 Å, as shown in Figure 16. This procedure allowed us to build our MD simulation cell using such distribution of adsorbent molecules. A tetragonal cell with 2708 × 1648 × 56 Å3 containing 16 1,5-biscorannulenecyclophane molecules is built. These 16 molecules are distributed in

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Y X (a)

Y X

(b)

Y

Y Z

Z

(c)

(d)

Figure 15 Partial optimization of the 1,5-biscorannulenecyclophane dimer with B3LYP/3-21G. (a) The x-coordinates are allowed to change, one jaw up and the other down, (b) the x-coordinates are allowed to change, both jaws up, (c) the y-coordinates are allowed to change, both jaws up and two molecules overlapped along the y-axis, (d) the y-coordinates are allowed to change, two jaws open to each other

4.8 Å

4.8 Å

Y

Z

Figure 16 Arrangement of 1,5-biscorannulenecyclophane in the YZ plane of the MD simulation cell


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two layers along the Y -axis, with 8 molecules in each layer. When these molecules are used as adsorbent for molecular hydrogen, the dimension of the b parameter is changed according to the predicted desired pressure in the gas phase. In other words, the cell has the dimensions of 2708 Å × b Å × 56 Å, with sixteen 1,5-biscorannulenecyclophane molecules located at the bottom of the cell. Initially, hydrogen molecules are distributed above the adsorbent layers randomly. The simulation procedure is same as reported for hydrogen adsorption in corannulene [21] and Li-doped corannulene complexes [31].

7. MD Simulations of Hydrogen Adsorption 7.1. Hydrogen Adsorption on Corannulene Density functional theory calculations of corannulene dimers yield T-shaped and sandwich conformers. In this section, MD simulations of hydrogen adsorption on corannulene systems are investigated in these two arrangements. In T-shaped conformers, the adsorbent molecules are distributed as in crystalline corannulene; and in sandwich conformers, the adsorbent molecules are distributed as in simple stack, where the arrangement of molecules depends on the values of the parameters ILD and IMD. 7.1.1. Hydrogen adsorption on crystalline corannulene Crystalline structure of corannulene: The first study of hydrogen adsorption focuses on crystalline corannulene systems at different temperatures and pressures. Figure 17 shows the crystalline cell of corannulene, containing 32 molecules with every 2 molecules arranged in a T-shaped fashion. For adsorption of hydrogen molecules, these 32 corannulene molecules are located at the bottom of a simulation cell with dimensions 26520 Å × 23718 Å × c Å, and an angle = 12069 . Hydrogen molecules are all randomly distributed in the gas phase above these corannulene adsorbents. The parameter c used is decided on the basis of the gas phase pressure and the temperature of interest. Temperatures of 273 K and 300 K, and pressures from 50 bar and up to 250 bar are studied. As simulation starts, hydrogen molecules in the gas phase begin to diffuse

Figure 17 Crystalline structure of corannulene, containing 32 molecules [30]

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H2 uptake (wt%)

2

1.5 1

0.5 0 0

50 72 bar 298 K

100

150

200

250

Pressure (bar)

Figure 18 MD results of H2 adsorption on crystalline corannulene: (a) Solid line: at 273 K, (b) Dotted line: at 300 K

into the adsorbent phase. The simulation process during the production period, the last 500 ps, is recorded in the trajectory file. The evaluation of hydrogen uptake and gas phase pressure is based on average values from the whole trajectory file. Hydrogen adsorption: Figure 18 shows the hydrogen uptake on crystalline corannulene at different temperatures as a function of the gas phase pressure [21]. Over all, the predicted H2 adsorption shows a linear relation with pressure in the range of pressure studied, and the amount of H2 adsorbed is higher at the lower temperature 273 K than at 300 K. According to the projection of the hydrogen uptake capacity with pressure, Figure 18 predicts about 0.79 wt% at 72 bar at 273 K and 0.68 wt% at 300 K. This prediction is in very good agreement with 0.8 wt% of H2 adsorption on crystalline corannulene found experimentally at 72 bar at 298 K [21]. This agreement between the experimental data and simulation data validates the parameters for H–H and C–H interactions used in our MD simulations. According to the projection, at 250 bar, H2 adsorption reaches 1.95 wt% at 273 K and 1.69 wt% at 300 K. 7.1.2. Hydrogen adsorption on corannulene assembled in stack arrays Arrangement of corannulene adsorbent molecules: The second assembly of corannulene studied is in a simple stack, in a tetragonal simulation cell. For IMD, a value of 11.0 Å is used in all our MD simulations. Regarding ILD, we used the optimized value of 4.8 Å obtained for the corannulene dimer from full optimization at the level of B3LYP/6-311G(d,p), but also the values of 6.0 and 8.0 Å, to investigate the effect of the available space on the capacity of hydrogen storage. The arrangement of 32 molecules are evenly distributed in 2 layers, each layer of 16 molecules is in the same XY plane. The cell parameters are 384 Å × 22 Å × c Å for ILD = 48 Å 24 Å × 44 Å × c Å for ILD = 6 Å 32 Å × 44 Å × c Å for ILD = 8 Å, where the c parameter depends on the temperature (273 K or 300 K) and estimated pressure (up to 250 bar) of the system. The distribution of hydrogen molecules in the simulation cell at the beginning of the simulation and the evaluation of hydrogen uptake and equilibrium gas phase pressure are


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the same as that described in the previous section of hydrogen adsorption in crystalline corannulene systems. To illustrate the distribution of corannulene molecules and the adsorption of hydrogen molecules on these systems, two systems of final configurations of hydrogen adsorption on corannulene adsorbents are shown in Figure 19. Figure 19(a) shows a snapshot of two different views of adsorbent layers with hydrogen molecules adsorbed, corresponding to ILD = 48 Å at 300 K and 139 bar. A hydrogen uptake of 1.04 wt% is predicted for this system. Figure 19(b) shows a snapshot of two different views of adsorbent layers with hydrogen molecules adsorbed, corresponding to ILD = 80 Å at 300 K and 92 bar. A hydrogen uptake of 2.79 wt% is obtained in this system. An extended discussion regarding the distribution of hydrogen molecules distributed in the adsorbent layers will be provided next. Hydrogen adsorption: Figure 20 shows the predicted hydrogen uptake on corannulene systems assembled in a stack configuration, at various ILDs and 300 K, as a function of pressure. Similar to hydrogen adsorption in crystalline corannulene, we again observe the linear adsorption behavior of hydrogen uptake vs pressure (in the range up to 250 bar), and also an enhanced hydrogen uptake capacity at a lower temperature is found (not shown). As the ILD separation increases from 4.8 Å to 6.0 and 8.0 Å, the hydrogen uptake increases significantly.

(a)

(b) Figure 19 Final configurations of systems of adsorbent layers with hydrogen adsorbed. (a) ILD = 48 Å, 300 K and 139 bar; (b) ILD = 80 Å, 300 K and 92 bar

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Figure 20 Hydrogen uptake capacity at different temperatures and ILD values. The IMD is kept at 11.0 Å. (a) ILD = 48 Å, 273 K; (b) ILD = 48 Å, 300 K; (c) ILD = 60 Å, 273 K; (d) ILD = 60 Å, 300 K; (e) ILD = 80 Å, 273 K; (f) ILD = 80 Å, 300 K

Figure 20 also suggests that desorption of hydrogen from the adsorbent is easier at a higher value of ILD as T increases. So if we can increase the ILD value of corannulene by substituting H with bulky functional alkyl groups, we might be able to get larger amounts of hydrogen adsorbed and more accessible conditions for these adsorbed hydrogen molecules to get desorbed. The predicted H2 adsorption, at the optimized value of ILD of 4.8 Å and pressure of 100 bar, is about 0.75 wt% at 300 K and 0.83 wt% at 273 K, respectively [21]. Guay et al. [57] obtained about 0.75% for SWNT and 0.40% for GNF at a value of VDW gap around 4.8 Å with simulations performed at 293 K and 100 bar. Based on the effect of temperature and pressure on the H2 uptake observed above, our MD results indicate that corannulene may have some advantages over SWNT or GNF, which may be due to the dipole-induced interaction between H2 and corannulene, as we discussed in the force field section. The distribution of hydrogen molecules in the adsorbent layers is very interesting. From the two snapshots of hydrogen adsorption shown in Figure 19, it is observed that at an ILD of 4.8 Å, almost none of the adsorbed H2 molecules are anchored in the regions between two overlapped corannulene molecules, instead they are located at the region in the middle of four corannulene molecules (4-fold interstice) or between two parallel corannulene molecules (2-fold interstice), Figure 19(a). As the ILD increases to 6 and 8 Å, the space between two overlap corannulene molecules increases and the H2 molecules start to fill the space between two overlapped corannulene molecules, as shown in Figure 19(b) for the case of ILD = 8.0 Å. More adsorption sites for hydrogen molecules and an enhanced binding energy between hydrogen molecules and adsorbents are caused by the overlap of the electron densities of the two corannulene molecules [21]. Thus the increased H2 uptake at higher ILD values. Similar results had been found when H2 is adsorbed on the Li-pillared graphene sheet system and Li-pillared SWNT as a function of interlayer distance and intertube distance, respectively [16].


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7.2. Hydrogen Adsorption on Lithium-doped Complexes 7.2.1. Advantages of Lithium doping In pure carbonaceous materials, it is difficult to reach hydrogen storage capacities in the range of the DOE target of 6.5 wt% at room temperature. For the corannulene system, it would require increasing the ILD to 8.0 Å at 300 bar and 300 K, assuming all the adsorbed hydrogen molecules get desorbed at moderate conditions. On the other hand, doping of alkali metal has been investigated and shows enhanced hydrogen storage capacities [10, 12, 13, 15, 16, 58–60]. Generally, the role of alkali metal has been attributed to three factors. First, doping of lithium atoms on carbon materials leads to charge transfer from the metal atom to carbon, which creates an enhanced dipoleinduced interaction with hydrogen molecules. Second, the VDW Li–H2 interactions are stronger than those of C–H2 . Third, the doping of lithium atoms provides more space in the adsorbent materials for hydrogen molecules to get adsorbed. According to our DFT-calculated potential energy surface scan and ab initio calculations of single-point interaction energies using MP2, we found enhanced interactions between the hydrogen molecule and lithium-doped corannulene complexes. Besides, an increased ILD value, as suggested by the partial optimization of Li6 –C20 H10 2 , was obtained as the result of doping of lithium atoms, which could provide more space for hydrogen to get adsorbed in the adsorbent layers. Based on these factors, higher hydrogen uptake capacity can be expected in the systems of lithium-atoms-doped corannulene complexes under the same conditions of temperature and pressure, compared to undoped corannulene systems. In this section, we summarize our findings regarding the hydrogen uptake capacities in two different lithium-atoms-doped complexes, Li6 –C20 H10 and Li3 –C20 H10 –Li2 . We also discuss the effect of lithium doping as well as the effect of doping concentration. 7.2.2. Hydrogen adsorption in Li6 –C20 H10 systems Arrangement of Li6 –C20 H10 adsorbent molecules: The arrangement of Li6 –C20 H10 complex in the simulation cell is in a simple stack, which depends on the ILD and IMD parameters. An IMD value of 11.0 Å is used in all MD simulations [31]. For ILD, we obtained an optimum value of 6.5 Å from partial optimization of Li6 –C20 H10 dimer. Higher values of ILD at 8.0 and 10.0 Å are also tested for hydrogen adsorption. In each simulation cell, there are a total of 16 Li6 –C20 H10 molecules distributed in two layers, with 8 molecules in each layer. The cell parameters are 28 Å × 22 Å × c Å for ILD value of 6.5 Å, 32 Å × 22 Å × c Å for ILD value of 8.0 Å, and 40 Å × 22 Å × c Å for ILD value of 10.0 Å, where parameter c depends on desired temperature (273 K or 300 K) and pressure (up to 250 bar) of the system. The distribution of hydrogen molecules in the simulation cell at the beginning of the simulation and the evaluation of hydrogen uptake and equilibrium gas phase pressure are the same as described before as hydrogen adsorption in undoped corannulene systems [31]. In order to see the different behavior of hydrogen adsorption on lithium atoms doped system and the distribution of hydrogen molecules on these systems, snashots of hydrogen adsorption on Li6 –C20 H10 adsorbents are shown in Figure 21. In such systems, Li6 –C20 H10 adsorbents are arranged as in simple stack. The arrangement is decided by

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(a)

(b)

Figure 21 Snapshots of Li6 –C20 H10 systems of adsorbent layers with hydrogen adsorbed. (a) ILD = 6.5 Å, 300 K and 75 bar; (b) ILD = 100 Å, 300 K and 57 bar

IMD and ILD. As mentioned before, IMD value of 11.0 Å and different values of ILD are used in MD simulations. ILD values include the optimum separation value of 6.5 Å, as well as values of 8.0 and 10.0 Å. Figure 21(a) shows final configuration of two views of adsorbent layers with hydrogen molecules adsorbed, corresponding to the ILD = 65 Å at 300 K and 75 bar. A hydrogen uptake of 2.36 wt% is predicted for this system. Figure 21(b) shows a snapshot including two different views of adsorbent layers with hydrogen molecules adsorbed, corresponding to the condition of ILD = 100 Å at 300 K and 57 bar. A hydrogen uptake of 2.39 wt% is obtained in this system.


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Figure 22 Calculated hydrogen uptake capacity of Li6 –C20 H10 at 300 K. (a) ILD = 65 Å; (b) ILD = 8 Å; (c) ILD = 10 Å

Hydrogen adsorption: Figure 22 shows the predicted H2 uptake at 300 K and various ILD for Li6 –C20 H10 as a function of H2 pressure. Overall, our MD simulations predict (1) at all the conditions studied, the H2 uptake is higher than 2 wt%; (2) the H2 uptake at a given ILD and temperature is almost linear with the increase of pressure in the pressure region studied; (3) adsorption decreases as temperature increases, and H2 uptake is lower at 300 K than in 273 K while keeping other conditions the same according to the adsorption isotherm; (4) increasing ILD while keeping the same T and P, significantly increases the H2 uptake, especially changing ILD from 8.0 to 10.0 Å [31]. According to the projection, the H2 uptake would reach the DOE target of 6.5 wt% at 300 K under 315 bar with an ILD of 8 Å. Compared to the undoped corannulene in the simple stack at the calculated ILD value of 4.8 Å (or in the T-shaped configuration), for H2 uptake to reach DOE target of 6.5 wt% at 300 K, the pressure needed in the gas phase is higher than 900 bar, assuming that the linearity can extend to 900 bar. The dropping of pressure from more than 900 bar to 315 bar indicates the great potential of enhancement of hydrogen uptake upon the doping of lithium atoms. On the other hand, these simulation results indicate that at pressures lower than 250 bar, there is not enough H2 uptake to reach DOE target at the optimum equilibrium ILD of 6.5 Å and temperatures above 273 K. This is due to the limited space available around the Li dopants. In order to enhance the H2 uptake at low pressures, one possible way is to increase the ILD of lithium–atoms-doped complex, by substituting hydrogen atoms attached to rim carbon atoms with bulky alkyl functional groups. With the substitution, we can expect that more space will available for hydrogen to get adsorbed. In our simulations, we assume that ILDs of 8.0 and 10.0 Å can be obtained by such substitution reactions. As we discussed in Section 2 about the evaluation of the hydrogen uptake capacity and the equilibrium gas phase pressure, we stated that 300 ps is long enough for the system to reach the equilibrium state, and 500 ps enough for calculating hydrogen adsorption. This

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# H2 adsorbed

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Figure 23 Number of hydrogen adsorbed in Li6 –C20 H10 adsorbent layers for ILD = 65 Å at 273 K. (a) P = 69 bar with H2 uptake of 2.75 wt%; (b) P = 119 bar with H2 uptake of 3.63 wt%; (c) P = 178 bar with H2 uptake of 4.42 wt%; (d) P = 223 bar with H2 uptake of 5.08 wt%

is illustrated by Figure 23, which shows the number of hydrogen molecules adsorbed in Li6 –C20 H10 adsorbent layers during the molecular dynamics simulations for the systems of ILD = 65 Å at 273 K and various pressures, with a total simulation time of 1200 ps. From Figure 23, it is clear that the number of hydrogen molecules adsorbed in the adsorbent layers fluctuate around the average value at the given conditions at about 150—200 ps in the equilibration period. This indicates that the systems of hydrogen adsorption on Li6 –C20 H10 adsorbent reach the equilibrium state before 200 ps, and 300 ps equilibrium time is long enough for our MD simulations. In the production period, from 300 to 1200 ps, the number of adsorbed hydrogen molecules is about the average. Longer simulation time does not increase the number of hydrogen molecules adsorbed. 7.2.3. Hydrogen adsorption in Li3 –C20 H10 –Li2 systems Arrangement of Li3 –C20 H10 –Li2 adsorbent molecules: The arrangement of Li3 –C20 H10 –Li2 complex in the simulation cell is in a simple stack, dependent on the ILD and IMD parameters. An IMD value of 11.0 Å is used in all MD simulations, and ILD values of 6.5, 8.0 and 10.0 Å are tested for hydrogen adsorption. In each simulation cell, there is a total of 16 Li3 –C20 H10 –Li2 molecules distributed in two layers, with 8 molecules in each layer. The cell parameters are 28 Å × 22 Å × Åc Å for ILD value of 6.5 Å, 32 Å × 22 Å × c Å for ILD value of 8.0 Å, and 40 Å × 22 Å × c Å for ILD value of 10.0 Å, where the parameter c depends on desired temperature (273 K or 300 K) and pressure (up to 250 bar) of the system. The distribution of hydrogen molecules in the simulation cell at the beginning of the simulation and the evaluation of hydrogen uptake and equilibrium gas phase pressure are the same as described for the previous systems. Figure 24(a) shows snapshots corresponding to ILD = 65 Å at 300 K and 131 bar. A hydrogen uptake of 2.87 wt% is predicted for this system. Figure 24(b) shows snapshots corresponding to ILD of 10.0 Å at 300 K and 127 bar. A hydrogen uptake of 3.35 wt% is obtained in this system. Both left side views of Figure 24(a) and


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(b) Figure 24 Final configurations of Li3 –C20 H10 –Li2 systems of adsorbent layers with hydrogen adsorbed. (a) ILD = 65 Å, 300 K and 131 bar; (b) ILD = 100 Å, 300 K and 127 bar

(b) show that hydrogen molecules can be adsorbed in the region between two overlapped Li3 –C20 H10 –Li2 complexes. As the ILD increases, more hydrogen molecules are adsorbed in these regions around lithium atoms and the ratio of hydrogen molecules adsorbed in these regions to hydrogen molecules adsorbed in 2-fold or 4-fold interstice increases. 7.2.4. Effect of lithium doping According to the projection of the MD simulation results at 300 K, the DOE target of 6.5 wt% can be met at 385 bar for ILD = 65 Å, at 335 bar for ILD = 80 Å, and at 251 bar for ILD = 100 Å in Li3 –C20 H10 –Li2 adsorbent systems [31], whereas at 273 K, this target can be met at lower pressures, 344 bar, 295 bar and 219 bar, respectively. These pressures are higher than those corresponding pressures at the same value of ILD

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Table 2 Comparison of H2 uptake at 100 bar on Li-doped corannulene systems. The number in parenthesis is the ratio of the H2 uptake to the number of Li atoms per corannulene molecule ILD (Å)

6.5 8.0 10.0

273 K

300 K

Li6 –C20 H10

Li3 –C20 H10 –Li2

Li6 –C20 H10

Li3 –C20 H10 –Li2

3.26 (0.54) 3.56 (0.59) 4.19 (0.70)

2.63 (0.53) 2.81 (0.56) 3.38 (0.68)

2.78 (0.46) 3.03 (0.51) 3.49 (0.58)

2.17 (0.43) 2.24 (0.45) 2.65 (0.53)

and temperature in Li6 –C20 H10 adsorbent systems, in which the concentration of doped lithium atoms is higher. In order to analyze the role of the concentration of lithium atoms doped on the H2 physisorption, we analyze the H2 uptake behavior on these two different lithium-atomsdoped corannulene systems. Since the relationship of H2 uptake capacity is almost linear to the pressure in the region studied, it is fair to take only one pressure, for example, 100 bar, to explain the effect of lithium concentration on H2 uptake. The data in other pressure ranges should follow similar trend, though the absolute values will not be the same. Table 2 illustrates that the Li6 –C20 H10 adsorbent system has higher H2 uptake capacity than Li3 –C20 H10 –Li2 and the difference at 100 bar is about 0.61 to 0.84 wt%, depending on the temperature and the ILD values. This difference is most likely caused by the lower lithium concentration in the Li3 –C20 H10 –Li2 adsorbent system. Table 2 also shows that the ratio of the H2 uptake to the number of Li atoms per corannulene molecule for each system at the same ILD, temperature and pressure, yield very close values for these two systems, a consequence of the linear dependence of H2 uptake on lithium doping concentration. For example, at ILD of 6.5 Å, the ratios of the H2 uptake to uptake of the number of lithium atoms are 0.54 in Li6 –C20 H10 system and 0.53 in Li3 –C20 H10 –Li2 system at 273 K and 100 bar. The ratios decrease to 0.46 in Li6 –C20 H10 system and 0.43 in Li3 –C20 H10 –Li2 system at 300 K and 100 bar. Similar behavior of the dependence of H2 uptake on lithium atom concentration was found in Li-pillared graphene sheets [16].

7.3. Hydrogen Adsorption on 1,5-biscorannulenecyclophane 7.3.1. Hydrogen adsorption Figure 25 shows a snapshot of hydrogen adsorption on 1,5-biscorannulenecyclophane adsorbent layers, at 273 K and 127 bar illustrating the distribution of H2 molecules in the adsorbent phase. Overall, the hydrogen adsorption behavior on 1,5-biscorannulenecyclophane is similar to what we have observed on corannulene and lithium-doped corannulene complexes. Summarizing, (1) at a given temperature of either 273 K or 300 K, hydrogen uptake shows almost linear response to the increase of the gas phase pressure; and (2) adsorption is lower at higher temperatures. According to the predictions, the hydrogen uptake at 100 bar is about 2.63 wt% at 273 K and 2.26 wt% at 300 K. If the linearity could be


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Figure 25 Snapshot of hydrogen adsorption on 1,5-biscorannulenecyclophane adsorbent layers at 273 K and 127 bar

extended to higher pressures, the DOE target of 6.5 wt% can be reached at 294 bar at 273 K, and 309 bar at 300 K. The calculated isotherms are shown in Figure 26. Distribution of hydrogen on 1,5-biscorannulenecyclophane adsorbent: Table 3 shows the ratio of hydrogen adsorbed in the open jaw of 1,5-biscorannulenecyclophane at different conditions. About 60% of adsorbed hydrogen at different conditions of temperatures and pressures are distributed in the open jaw region of 1,5biscorannulenecyclophane. Two reasons might contribute to this situation. One is that the jaw region has more available volume for hydrogen adsorption, and the other is that the region inside the jaw is more attractive to hydrogen molecules due to the curvature of 1,5-biscorannulenecyclophane molecule and its higher dipole moment. The spacing between the convex surfaces of two jaw structures is 10 Å. Thus, there is a possibility that cooperative interaction between these two convex surfaces is contributing to hydrogen storage. The closed spacing of about 7 Å appears to be more effective as sites for hydrogen storage. Comparison of hydrogen adsorption: Here we describe the effect of the molecular modification to corannulene on the hydrogen adsorption. Figure 27 shows the comparison of hydrogen uptake on 1,5-biscorannulenecyclophane system with that of the corannulene system at 273 K and 300 K. All solid curves are adsorption isotherms at 273 K, and dotted curves are adsorption isotherms at 300 K. Figure 27 shows the comparison with the optimum ILD of corannulene (4.8 Å). It is very clear that hydrogen

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Figure 26 Hydrogen adsorption isotherms on 1,5-biscorannulenecyclophane adsorbent

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Table 3 Ratio of hydrogen adsorbed in the open jaw region of 1,5-biscorannulenecyclophane at different conditions Conditions 273 K, 273 K, 273 K, 273 K, 300 K, 300 K, 300 K, 300 K,

Total number of H2 adsorbed

Number of H2 in open jaw region

168 145 128 98 146 125 113 81

96 86 75 58 88 75 69 50

127 bar 105 bar 90 bar 62 bar 125 bar 107 bar 92 bar 67 bar

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Figure 27 Comparison of hydrogen uptake on 1,5-biscorannulenecyclophane with hydrogen uptake on undoped corannulene at the optimum value of ILD of 4.8 Å, at 273 and 300 K. (1) 273 K, H2 uptake on 1,5-biscorannulenecyclophane. (2) 300 K, H2 uptake on 1,5biscorannulenecyclophane. (3) 273 K, H2 uptake on C20 H10 with ILD = 4.8 Å. (4) 300 K, H2 uptake on C20 H10 with ILD = 48 Å.

uptake capacities on 1,5-biscorannulenecyclophane system are much higher than those on corannulene with an optimum ILD of 4.8 Å at the same conditions of temperature and pressure. According to the prediction, the hydrogen uptake capacities for corannulene with ILD of 4.8 Å are 0.83 wt% at 273 K and 0.75 wt% at 300 K. The hydrogen adsorption on 1,5-biscorannulenecyclophane at 100 bar is three times or more than that in corannulene (with ILD of 4.8 Å). The differences are 1.80 and 1.51 wt% at 273 K and 300 K, respectively. The reason that 1,5-biscorannulenecyclophane has significant higher hydrogen adsorption capacity is that the modification leads to a large volume inside the molecule. The linker (−C ≡ C − CH = CH − C ≡ C−) connects two pairs of rim carbons of corannulene at C1–C1 and C5–C5 , respectively. This leads to the jaw shape of molecule. The distance between two symmetry carbon atoms of the V plane is listed in Table 1. Most of the distances between these carbons are larger than 7 Å, except for those pairs


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located at the bottom of the jaw, which average at 4.99 Å. Even this value is larger than the optimum value of separation of corannulene dimer. The largest separation between two carbons is about 9.57 Å. According to the shape of the molecule of 1,5-biscorannulenecyclophane, we can estimate the average separation of two corannulene molecules of 1,5-biscorannulenecyclophane by averaging the distance of all five pairs of Cb–Cb, which is 7.3 Å. This value is about 2.5 Å larger than the optimum separation distance of corannulene dimer. We can expect that the hydrogen adsorption capability on 1,5biscorannulenecyclophane might be between hydrogen adsorption on corannulene with ILDs between 6 and 8 Å at the same pressure and temperature, since the volume of the jaw of 1,5-biscorannulenecyclophane is open to hydrogen adsorption. Figure 28 shows the comparison of hydrogen adsorption on 1,5-biscorannulenecyclophane with those of that on corannulene with ILD of 6.0 and 8.0 Å both at 273 K and 300 K. It is clear that in the range of pressure studied at both temperatures, hydrogen adsorption capabilities of 1,5-biscorannulenecyclophane are indeed in between hydrogen adsorption capabilities of corannulene with ILD of 6.0 and 8.0 Å. For example, at a pressure of 100 bar, hydrogen adsorption capabilities are (1) 2.63 wt% on 1,5-biscorannulenecyclophane, 2.10 wt% for corannulene with ILD of 6.0 Å, and 3.41 wt% for corannulene with ILD of 8.0 Å, at 273 K; and (2) 2.26 wt% on 1,5biscorannulenecyclophane, 1.72 wt% for corannulene with ILD of 6.0 Å, and 2.95 wt% for corannulene with ILD of 8.0 Å, at 300 K. On the other hand, to reach DOE target of 6.5 wt%, the pressures needed at 273 K are 294 bar for 1,5-biscorannulenecyclophane, 361 bar for corannulene with ILD of 6.0 Å, and 228 bar for corannulene with ILD of 8.0 Å; and at 300 K, the pressures are

6 1 2 3 4 5 6

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Figure 28 Comparison of hydrogen uptake on 1,5-biscorannulenecyclophane with hydrogen uptake on undoped corannulene: (1) 273 K, H2 uptake on 1,5-biscorannulenecyclophane; (2) 300 K, H2 uptake on 1,5-biscorannulenecyclophane; (3) 273 K, H2 uptake on C20 H10 with ILD = 60 Å; (4) 300 K, H2 uptake on C20 H10 with ILD = 60 Å; (5) 273 K, H2 uptake on C20 H10 with ILD = 80 Å; (6) 300 K, H2 uptake on C20 H10 with ILD = 80 Å

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309 bar for 1,5-biscorannulenecyclophane, 408 bar for corannulene with ILD of 6.0 Å, and 253 bar for corannulene with ILD of 8.0 Å.

8. Conclusions In this chapter, the molecular structure of corannulene and doping of lithium atoms on the corannulene molecule are characterized. Three different corannulene-based materials are studied as adsorbents for molecular hydrogen at ambient temperature with pressure up to 250 bar. These simulation results suggest that corannulene-based materials might be options for hydrogen adsorption and the summary is as follows. • It is shown that corannulene has potential advantages as hydrogen storage material over planar graphite and even carbon nanotubes. Such advantages are derived from the specific geometric and electronic characteristics of corannulene, which enhances the interaction with hydrogen molecules. On one hand, the curvature of corannulene can involve more carbon atoms to interact with hydrogen molecules when hydrogen molecules come close to corannulene. On the other hand, further enhanced interactions are due to the permanent dipole moment of corannulene, which can induce dipole–dipole interactions with hydrogen and also to the electronic density distribution between corannulene molecules that double the binding energy per hydrogen molecule through cooperative interaction by corannulene rings due to their close proximity relative to one another. • Lithium atoms are more stably doped over the six-member rings than over the fivemember ring of corannulene with the concave side more favorable for doping than the convex side of corannulene. DFT studies show that the optimized lithium-doped corannulene complexes have more lithium atoms on the concave side than on the convex side. A ratio of lithium to carbon atoms between 1:3 and 1:4 is stable for lithium-doped complexes. • The doping of lithium atoms enhances the hydrogen adsorption due to three factors. Lithium-doped corannulene complexes have higher dipole moment because of charge transfer from lithium to carbon atoms of corannulene, and thus enhance the induced dipole–dipole interaction with hydrogen molecules. Also, the interaction between the lithium atom and hydrogen is stronger than the interaction between carbon atom and hydrogen. Besides, the doping of lithium atoms provides more space in the doping complexes for hydrogen adsorption. • Other factors such as pore volume, temperature and pressure are also important for hydrogen adsorption. Under the pressure range studied, which is up to 250 bar, a linear relationship between hydrogen uptake and pressure is observed. Modification of corannulene such as substitution of hydrogen atom with bulky alkyl functional groups on corannulene can increase the pore volume of corannulene-based adsorbents. The increased available space can enhance the hydrogen adsorption capacity. • 1,5-biscorannulenecyclophane is a modification of corannulene by bridging two corannulene molecules with organic linkers. The modified molecule has v symmetry. The average distance between the symmetric carbon pair is about 7.3 Å. 1,5-biscorannulenecyclophane has large intramolecular volume which is available for hydrogen adsorption. MD simulations of hydrogen adsorption predict enhanced


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hydrogen adsorption capacities compared to the hydrogen adsorption on corannulene at the similar conditions of temperature and pressure. Hydrogen adsorption on 1,5biscorannulenecyclophane has a capacity that is similar to the system of corannulene with ILD values between 6.0 and 8.0 Å. According to the prediction, DOE target of 6.5 wt% can be reached in the modified system at 294 bar at 273 K, and 309 bar at 300 K. About 60% of the hydrogen molecules are adsorbed in the intramolecular region.

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Chapter 7

Toward Nanomaterials: Structural, Energetic and Reactivity Aspects of Single-walled Carbon Nanotubes T. C. Dinadayalane and Jerzy Leszczynski∗ Computational Center for Molecular Structure and Interactions, Department of Chemistry, Jackson State University, 1400 JR Lynch Street, PO Box 17910, Jackson, Mississippi 39217, USA

1. Introduction The structures, classification and properties of single-walled carbon nanotubes (SWNTs) are outlined in this chapter. Previous studies concerning the electronic structures of armchair SWNTs, functionalization via addition and cycloaddition reactions to the sidewalls of perfect and Stone–Wales defect nanotubes are summarized. The significance of the structural aspects in explaining the reactivities of carbon nanotubes are pointed out. We have critically analyzed the structures of perfect and Stone–Wales defect nanotubes of (5, 5) armchair SWNTs. The relative stabilities of defect nanotubes, which contain a single Stone–Wales defect considered at different positions and at two different modes of orientations, with respect to the corresponding defect-free structures were investigated. We examined the structure–reactivity correlation for the Stone–Wales defect SWNTs. The obtained results indicate that pyramidalization angles will be useful for understanding cycloaddition reactions of dienes to the different bonds of the SW defect region of SWNT. Finally, a brief outline of the potential applications of carbon nanotubes and the challenges that remain in this area are presented. Nanomaterials are growing to an immensely advanced level both in scientific knowledge and in commercial applications. Carbon nanotubes (CNTs) have caused fascination in the field of nanomaterials for more than a decade. They have generated much experimental and theoretical interest in recent years due to their peculiar properties such as ∗

Corresponding author

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remarkably high tensile strength, electronic properties ranging from metallic to semiconducting, high current-carrying capacity and high thermal conducting nature [1–9]. They are classified into multi-walled and single-walled carbon nanotubes. Multi-walled carbon nanotubes (MWNTs), formed by concentric shells of apparently seamless cylinders of graphene, were first discovered in 1991 by Iijima and coworkers [10], while the SWNT was first reported in 1993 [11, 12]. MWNTs are practically concentric SWNTs and they may have SWNTs of different chiralities. Recent reports on the large-scale production of SWNTs further encourage both experimentalists and theoreticians to study the properties of these compounds in much detail [13–16]. The SWNTs can be viewed as a hollow cylinder obtained by wrapping a two-dimensional (2D) graphite layer in h is superimposed on its origin (Figure 1). such a way that the end of the roll-up vector C The chiral vector or the roll-up vector is simply specified by a pair of integers (n, m) that h to the two unit vectors a 1 and a 2 . The chiral vector can be described by eq. (1): relates C h = n a1 + ma2 C

(1)

Roll-up

graphene sheet

SWNT (a)

Armchair Zig-Zig

r Ch cto e V ma2 iral Ch θ Chiral Angle 1

a2 a1

na1

(b)

Figure 1 (a) Schematic diagram of rolling up a graphene sheet to SWNT. (b) Illustration of how h = na1 + ma2 and chiral to wrap the graphene sheet to form a carbon nanotube; roll-up vector C angle for an (n, m) SWNT, where a 1 and a 2 are the primitive lattice vectors of a graphene sheet, and n, m are integers ((a) reprinted from [22a] with permission. © copyright (2000) American Chemical Society; (b) reprinted from [22b], © copyright (2001), with permission from Elsevier)

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(a)

1

3 2

3 2

(b)

1

(c) (5, 5)

N

(d)

(10, 0)

(e)

Figure 2 The representative structures for the types (a) armchair, (b) zig-zag and (c) chiral defect-free SWNT with their ends capped by fullerene fragments and open-ended nanotubes of terminal carbons bonded with hydrogens in (d) (5, 5) armchair, (e) (10, 0) zig-zag SWNTs

Single-walled carbon nanotubes are classified into three types: (a) armchair nanotube if m = n, (b) zig-zag nanotube if m = 0 and (c) chiral nanotube if n > m and m = 0. Armchair, zig-zag and chiral carbon nanotubes are in general designated by (n, n), (n, 0) and (n, m) respectively. The representative structures for these three different types of SWNTs are depicted in Figure 2. The armchair (n, n) and zig-zag (n, 0) nanotubes are achiral. The amount of ‘twist’ in the nanotube is decided by the ‘chiral angle’ (), which h and the (n, 0) zig-zag direction. The is defined as the angle between the roll-up vector C value of the chiral angle is 0 and 30 for the zig-zag and armchair carbon nanotubes, respectively. All achiral-type nanotubes have a chiral angle with 0 < < 30 . One can measure the diameter (dt ) and the chiral angle () of the nanotube if one knows (n, m). For measuring the diameter of the nanotube, one should also know the nearest-neighbor carbon atom distance, aCC . Carbon nanotubes with a diameter 6–17 Å are well known, but the smallest diameter tubes (∼4 and 3 Å) have been isolated only recently [17–19]. The mathematical formula for the diameter of an ideal SWNT (n, m), i.e., for an ideal nanotube-cylinder, is given in eq. (2). dt = aCC 3n2 + nm + m2 1/2 /

(2)

where aCC is the nearest-neighbor carbon atom distance, and the value 1.421 Å is generally used. Chiral angle is: = tan−1 31/2 m/2n + m

(3)

Electronic properties of SWNTs depend indeed on the diameter and wrapping angle [20–23]. Electronic band structure calculations indicate that an SWNT can be classified as metallic or semiconducting based on the (n, m) indices [8, 9, 24–28]. For a given (n, m) SWNT, the tube will be metallic when 2n + m/3 is an integer; otherwise semiconducting in nature. Alternatively, if n − m/3 is an integer, then the SWNT is metallic. Thus, all armchair nanotubes are metallic. Theoretical studies have shown that the electronic and magnetic properties of the metallic tubules are diameter dependent [8]. Yakobson et al. have reported that the mechanical properties of SWNTs are controlled by the chirality of the nanotubes [29, 30]. Hence, the physical properties of carbon nanotubes subtly depend on the chiral vector (n, m). Cyclic voltammetery and spectroscopic techniques such as infrared (IR), Raman and X-ray photoelectron spectroscopy (XPS) are used in the characterization of nanotubes

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and their derivatives [28, 31–44]. Microscopy is one of the most important tools for facilitating the characterization of structure in nanotube research [45, 46]. Atomic force microscopy (AFM), scanning electron microscopy (SEM), scanning probe microscopy (SPM), transmission electron microscopy (TEM), high resolution TEM (HRTEM), and scanning tunneling microscopy (STM) are very useful in structural characterization and understanding the electronic properties of nanotube materials [3, 4, 20–22, 45– 51]. Standard spectroscopic techniques are used to observe the electronic transitions between the energy bands of SWNTs [28, 52, 53]. Band gaps of SWNTs are inversely proportional to their diameters; hence, structural information concerning SWNTs can be obtained from band gap transition energies [22, 23]. Theoretical calculations are often used like other instrumental techniques to confirm the experimental results. Recently, vibrational frequencies and 13 C NMR chemical shifts have been reported for finite-length SWNTs based on density functional theory calculations [54–56]. A carbon nanotube is similar to a fullerene but is cylindrical in shape. Both ends are capped with half of a fullerene molecule. SWNT possesses two regions, namely a sidewall and the end cap of the tube with different physical and chemical properties (Figure 2). The properties of fullerenes and carbon nanotubes are different than graphite due to the existence of curvature [2, 8, 9]. Understanding the structural aspects of SWNTs is critical for the functionalization process, which offers the unique opportunity to obtain desired physical, electronic and chemical properties of materials. The functionalization of SWNTs has attracted tremendous attention since functionalized SWNTs are soluble in organic solvents [44, 57–59]. In the successful sidewall functionalization of the carbon nanotubes, the conjugated double-bond network of the cylindrical surface is interrupted. The functionalization of CNTs must amend their structures, thus perturbing the electronic, optical and physical properties. The chemistry of small fullerenes and SWNTs has been reviewed recently [1–4]. Fullerenes and carbon nanotubes show differences in their reactivities despite both of them possess nonplanar sp2 carbons in their structural network. The curvature effects and the different degrees of local strain between the two structures can be attributed to the substantial differences in their reactivities. Additive chemistry of fullerenes is well established, whereas that of SWNTs is less explored [60, 61]. Previous studies have reported that addition reactions involving fullerenes can be explained by the pyramidalization of the carbon atoms, whereas for SWNTs the -orbtial misalignment between adjacent pairs of conjugated carbon atoms is more important than pyramidalization to ascribe the reactivity [53, 62, 63]. A pyramidalization angle (P ), which is a gauge of local curvature, was established to measure the deviation of sp2 -hybridized carbon from the plane passing through the three adjacent atoms connected to it. The pyramidalization angle for the planar sp2 -hybridized carbon is 0 . It is 19 47 for a tetrahedral sp3 -hybridized carbon as shown in Figure 3. The pyramidalization angles for the carbon atoms in the curved polycyclic systems such as fullerenes and nanotubes lie between 0 and 19 47 . Occurrence of defects such as vacancies, Stone–Wales (SW) defects, pentagons, heptagons and dopants is unavoidable in nanotubes either during the growth of the nanotubes or due to stress applied [64–74]. The pre-existing defects at the specified location or defects introduced by stress play a key role in tailoring the physicochemical properties of graphene sheets and carbon nanotubes [73–85]. The defects, which cause dramatic effects on the tube electronic structures, may be created when positioning the


θσπ = 90°

θσπ = 109.47°

θP = 0°

θP = 19.47°

(a)

(b)

171

θσπ

0° < θP < 19.47° (c)

Figure 3 The pyramidalization angle (P = – 90, in degrees) for (a) the planar sp2 -hybridized carbon atom (e.g., in benzene), (b) the tetrahedral sp3 -hybridized carbon atom (e.g., methane), (c) the sp2 -hybridized carbon atom in curved polycyclic systems (e.g., fullerenes and carbon nanotubes)

carbon nanotubes with either STM or AFM tips due to the mechanical exploitation involved [86, 87]. Manipulation of nanotubes for making devices is likely to produce defects. The influence of defects on the electronic and adsorption properties of carbon nanotubes have been recently shown experimentally using STM [75–81]. In early 2006, it has been reported that SWNTs underwent superplastic deformation at high temperatures resulting in a tube that is 280% longer and 15 times narrower before breaking [88]. Defects such as kinks and point defects actively involved in this superplastic deformation process, which is not possible at low temperatures [88]. Introduction of defects to the sidewalls of carbon nanotubes leads to a new structural design. The process of producing different shapes of nanotubes by interrupting the growth has been fascinating in the recent years [89, 90]. Introducing defects including heptagons and pentagons in the carbon nanotubes offers the opportunity to design various shapes such as parallel, X-shaped, and branched T - and Y -shaped nanotube structures [89–94]. These exciting shapes have been realized experimentally [91–93], and some of them were studied theoretically using molecular dynamics simulations [90]. These types of defect structures can be created in a controlled way under certain circumstances [95, 96]. The bent or damaged SWNT was reported experimentally in the treatment of selective opening of SWNTs to fill the metals [97]. Among the possible defects, the Stone–Wales defect is considered to be the most important and the lowest energy defect in carbon nanotubes that will affect the mechanical, chemical and electronic properties of SWNTs [82–85, 98–100]. The Stone–Wales transformation is a reversible diatomic interchange or can be viewed as the rotation of a C–C bond in the hexagonal network of the carbon nanotube by 90 as shown in Scheme 1

5 7

7 5

Scheme 1

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[101]. The Stone–Wales defect nanotube structure contains two pentagons and two heptagons in duo. The 90 bond rotation in the nanotube requires energy of 4–5 eV, thus likely to occur in as-grown nanotubes [80]. The Stone–Wales defect formation has been well recognized in fullerene science as an elegant way to swap the positions of hexagons and pentagons by a 90 rotation of a C–C bond [102–105]. Recently, the Stone–Wales defect in carbon nanotubes has been given significant attention since it plays a key role in nanotube plastic deformation under tension and in modeling the physicochemical properties of SWNTs [85, 106–111]. Nardelli et al. reported that Stone–Wales defects were formed beyond an applied strain of 5–6%. They also added that this defect transformation results in an elongation of the tube along the axis containing the pentagons and in a shrinking along the perpendicular direction [72]. The Stone–Wales defect formation was deemed to be a mechanism for releasing the strain in stressed SWNTs. Molecular mechanics and molecular dynamics simulations have shown that the Stone– Wales defect formation can happen at high temperature and migrate along the tube, and the defect aggregation becomes energetically favored at high strains [72, 98, 112]. The gliding and separation of 5–7–7–5 defects would lead to plastic deformation of carbon nanotubes [106]. A recent experimental characterization of Stone–Wales defects in the carbon nanotubes further stimulates other studies concerning the defects in SWNTs [75]. Fractures in SWNTs were studied by Schatz and coworkers paying attention to the Stone–Wales defect formation at the middle of the nanotube and the aggregation of the SW defects. In that study, various empirical potentials in modeling the fractures in nanotubes were assessed by comparison with the results obtained using low level quantum mechanical calculations [113]. In the later part of this chapter, we have focused on the formation of SW defects in the armchair (5, 5) nanotubes since SW defects play critical role in modeling the physicochemical properties of SWNTs. The forthcoming parts of this chapter are divided into four sections. We have outlined in the first section the geometric and electronic structures of armchair nanotubes of different diameters and results of studies reported on the functionalization, particularly cycloaddition reactions to the sidewalls of the armchair defect-free and the Stone–Wales defect nanotubes. The second section reveals the computational methodologies that we have employed in modeling the Stone–Wales defects in the (5, 5) armchair SWNT. Our results on the formation of the single Stone–Wales defect in (5, 5) armchair SWNTs of different lengths are presented in the third section, where we have also addressed correlation of structures, i.e., local strain with reactivity/exothermicity. We spotlight some of the applications and the challenges existing in the field of carbon nanotubes in the fourth section followed by a summary and outlook.

2. Geometric and Electronic Structures of Armchair Single-Walled Carbon Nanotubes and Sidewall Functionalization Electronic structures play a crucial role in determining the chemical reactivity of molecules. Computing the geometries of defect and defect-free carbon nanotubes at high level quantum chemical calculations are critical for understanding their physical and chemical properties. Many computational studies for carbon nanotubes were limited to the low level calculations since these systems are computationally prohibitive


173

for high level ab initio or DFT calculations. Recently, Budyka et al. have calculated the bond lengths and diameters of armchair SWNTs ranging from (4, 4) to (15, 15) using the semiempirical and density functional theory levels [114]. They have obtained bond lengths of two different kinds of bonds, namely axial (CCa ) and circumferential (CCc ) bonds at the PBEPBE/3-21G(d), B3LYP/3-21G(d) and semiempirical PM3 levels. The computed C–C bond length for the graphene sheet at the B3LYP/3-21G(d) level is 0.003 Å longer than the experimental value of 1.421 Å. Bond lengths of armchair SWNTs range from 1.42 to 1.44 Å. Figure 4 clearly shows that the PM3 method underestimates while PBEPBE/3-21G(d) overestimates the bond lengths compared to the B3LYP/3-21G(d) level. Furthermore, the C–C bond length of the SWNT gradually decreases as the nanotube diameter increases. Good linear correlation was obtained between the diameter of the nanotube computed at the PM3 method and the nanotube index of the armchair (n, n) SWNT as depicted in Figure 5. Budyka et al. have also reported that the heat of formation per carbon atom decreases as the nanotube diameter increases [114]. Haddon and coworkers have reported that pyramidalization and -orbital misalignment angles decrease as the diameter of the armchair SWNT increases, which is evidenced from the data presented in Table 1 [3]. A decrease in the value of pyramidalization and -orbital misalignment angles while going from the (5, 5) to (10, 10) nanotube indicates that there is less strain in large diameter SWNT. Nakamura and coworkers have demonstrated that single-walled armchair nanotubes can be classified into three types, namely Kekule (i and iv), incomplete Clar (ii and v) and complete Clar (iii and vi) networks based on the bonding pattern as shown in Figure 6 [115]. These three types of structural networks emerge periodically by addition of a one-by-one layer of 10 carbon atoms in the (5, 5) nanotube, 12 carbon atoms in the (6, 6) nanotube, and so on. Figure 6 also shows the local aromaticity evaluated through the NICS values using GIAO-SCF/6-31G(d)//HF/6-31G(d) level of calculations. It gives an idea of how the local aromaticity varies as the length of the nanotube increases. The

1.445

CC bond length, Å

1.440 1 2

1.435

3

1.430

4 1.425

5 6

1.420 2

4

6

8

10

12 14 16

∞

n

Figure 4 C–C bond length vs the nanotube index (n for two different kinds of bonds, CCa (2, 4, 6) and CCc (1, 3, 5), calculated at the PBEPBE/3-21G(d) (1, 2), B3LYP/3-21G(d) (3, 4) and PM3 (5, 6) levels. The values for a graphene sheet are shown as limiting values (reprinted from [114], © Copyright (2005), with permission from Elsevier)

174


20

dt Å

16

12

8

4

4

6

8

10

12

14

n

Figure 5 PM3-calculated nanotube diameter (dt ) vs the nanotube index (n) for armchair SWNTs (n, n) (reprinted from [114], © Copyright (2005), with permission from Elsevier) Table 1 Diameter, pyramidalization angle (P ) and -orbital misalignment angle ( ) for the armchair (n, n) nanotubes.a armchair tube (n, n) (5, 5) (6, 6) (7, 7) (8, 8) (9, 9) (10, 10)

diameter (d, in Å)

P , in deg.

6 76 8 11 9 47 10 82 12 17 13 52

5.97 4.99 4.27 3.74 3.33 3.00

, in deg. 0, 0, 0, 0, 0, 0,

21.3 17.6 15.0 13.1 11.6 10.4

(a) The values were taken from [3] with permission, © Copyright (2002) American Chemical Society.

B3LYP/6-31G(d) optimized bond lengths reported for the (5, 5) armchair nanotubes are collected in Table 2. The minimum and maximum bond length values reported are 1.361 and 1.468 Å, respectively. Bond lengths obtained for the bond ‘a’ in all cases are shorter compared to other bond distances since they bear hydrogen atoms. Significantly long bond lengths (1.468 and 1.465 Å) were obtained for bond ‘c’ in the case of complete Clar structural networks C60 H20 (iii) and C90 H20 (vi). Figure 7 shows that energies of the HOMO, LUMO and HOMO-LUMO gaps are oscillating while the length of the nanotubes are increased step by step. The functionalization of SWNTs has generated immense research interest recently since it is an important route in tailoring the structural and electronic properties of carbon nanotubes [71, 116–118]. We have recently reported that the linear hydrocarbons inside SWNTs are stabilized through dispersion interactions [119]. Recent research concerning the modification of fullerenes and carbon nanotubes has generated further interest toward basic research of nanostructured materials [119–125].


(i) C40H20 (5, 5), C48H24 (6, 6)

A

175 NICS (5, 5) (6, 6) –8.62 –9.09

(ii) C50H20 (5, 5), C60H24 (6, 6)

A –3.90 –4.55 B –12.30 –13.09 (iii) C60H20 (5, 5), C72H24 (6, 6)

A B

–9.09 –10.59 0.25 –0.64

A B C

–8.64 –8.67 1.29

(iv) C70H20 (5, 5), C84H24 (6, 6)

–9.84 –9.50 1.08

(v) C80H20 (5, 5), C96H24 (6, 6)

A –4.06 –5.36 B –11.46 –12.30 C –0.77 –1.26 (vi) C90H20 (5, 5), C108H24 (6, 6)

A B C D

–7.01 –9.58 0.04 –1.55 3.35 1.43 –8.54 –11.58

Figure 6 Different types of structural networks such as Kekule (i, iv), incomplete Clar (ii, v) and complete Clar (iii, vi) and color-coded NICS maps with the NICS values obtained at GIAO-SCF/631G(d)//HF/6-31G(d) for finite-length (n, n) CNTs (n = 5 and 6). Hydrogen atoms are omitted for clarity. Chemical bonds are schematically represented by using single-bond (solid single line; bond length > 1 43 Å), double-bond (solid double line; bond length < 1 38 Å), single-bond halfway to double-bond (solid dashed line; 1 43 Å > bond length > 1 38 Å), and Clar structures (i.e., ideal benzene). NICS coding: red, aromatic < −4 5; blue, nonaromatic > −4 5 (reprinted from [115] with permission, © Copyright (2003) American Chemical Society) Table 2 The B3LYP/6-31G(d) optimized bond lengths (Å) for structures of finite-length (5, 5) SWNTs.a

C40 H20 C50 H20 C60 H20 C70 H20 C80 H20 C90 H20

a

b

c

d

e

f

g

h

i

1 366 1 361 1 384 1 372 1 366 1 381

1 435 1 445 1 418 1 432 1 441 1 421

1 452 1 422 1 468 1 451 1 435 1 465

1 415 1 426 1 416 1 416 1 422 1 417

1 444 1 428 1 438 1 441 1 435

1 441 1 434 1 428 1 438

1 409 1 413 1 424

1 437 1 418

1.446

(a) The values were reprinted from [115] with permission, © Copyright (2003) American Chemical Society.


–1.5

6.0

–2.0

5.5

–2.5

5.0

–3.0

4.5

–3.5

4.0

–6.0

3.5

–6.5

3.0

–7.0 –7.5

HOMO (eV)

6.5

LUMO (eV)

HOMO/LUMO gap (eV)

176

–8.0 40

70

100

130

160

190

carbon number (= 10j)

Figure 7 HOMO, LUMO energy values and band gap oscillation of finite-length (5, 5) SWNTs (C10j H20 ) obtained at the PM3 level (reprinted from [115] with permission, © Copyright (2003) American Chemical Society)

The functionalization of carbon nanotubes makes them good candidates for applications to chemical sensors, super-electric batteries, hydrogen storage and precise drug delivery [46, 116, 126–129]. Covalent functionalization in nanotubes has been investigated at different positions such as sidewalls, edge and the defect region depending on the requirement toward specific purpose [130–145]. Covalent and noncovalent functionalizations have been proven to be effective strategies for improving the solubility and processibility of carbon nanotubes [45, 143–148]. Niyogi et al. have reviewed the functionalization to the edge and the sidewall of carbon nanotube exploiting a wide range of chemical processes such as nitrene addition, hydrogenation, fluorination, alkylation, arylation and 1,3-dipolar cycloaddition [3]. Functionalized SWNTs are likely to be precursors for many of the applications that are anticipated for carbon nanotubes. Theoretical studies have demonstrated plausible ways such as 1,3-dipolar cycloaddition, Diels-Alder and other cycloadditions and ozonization for obtaining functionalized SWNTs [148–159]. Chen and coworkers have reported that caution must be exercised in choosing the methods for using the ONIOM approach for [2 + 1] cycloaddition reactions involving SWNTs. They have revealed that the addition of O, CH2 , SiH2 and NH species leads to an opening of sidewalls of the (5, 5) armchair nanotube structure rather than forming 3-membered rings [149]. Opening the sidewall upon [2 + 1] cycloaddition has also been confirmed by a theoretical study of Chu and Su [159]. In addition reactions, the small diameter nanotubes were predicted to have higher reactivity, and this can be attributed to the increased steric strain. The high reactivity of metallic nanotubes compared to semiconducting nanotubes was related to the large differences in their HOMO-LUMO gap [160]. In the rapidly growing field of carbon nanotubes, computational chemistry plays an integral part along with other experimental techniques in designing and proposing new carbon nanotube materials. One piece of evidence is


177

the recent fruitful synthesis of the Diels-Alder cycloadduct of o-quinodimethane with SWNT [161]. The viability of this reaction was theoretically predicted by Lu et al. two years previous to realization by experiment [154]. Computational studies highlight that the reactivity of SWNT depends on the curvature of sidewalls [148–155]. The 1,3-dipolar cycloadditions of a set of 1,3-dipolar molecules to the sidewall of (n, n) armchair nanotubes of various diameters were systematically investigated by Lu et al. using the ONIOM approach [148]. The reactivity decreases when the diameter of the armchair nanotube increases as depicted in Figure 8. The study stresses the importance of understanding the structural details, particularly the pyramidalization angle in order to ascribe the 1,3-dipolar cycloaddition reactivity of carbon nanotubes [148]. The influence of Stone–Wales defects on the reactivities of SWNTs has been the subject of theoretical interest [154–157]. Lu et al. have pointed out that the Stone– Wales defect sites are not always more reactive than defect-free carbon nanotubes as demonstrated in Figure 9 [155]. A theoretical study of Bettinger supports the results of Lu et al. [156]. In the addition reactions, the pyramidalization angle that gauges the local curvature has been used to explain the reactivity, thus indicating the importance of understanding the structural features toward the rational design of carbon nanotube materials [148–157, 162]. Previous studies have considered an impact of the Stone–Wales defect at the middle of the nanotube on the electronic properties and the influence of this topological defect on the reactivity and adsorption of atoms or small molecules [106, 155–157, 163]. Two distinct C–C bonds such as axial and circumferential bonds in the carbon nanotube can be rotated 90 independently to generate Stone–Wales defects with different orientations in the nanotube. However, most of the studies have not considered the two possibilities. Chen et al. have proposed that SWNTs can be cut by simply grinding the tubes in soft organic material such as cyclodextrins [164]. Thus, it may be possible to obtain the SWNT with the Stone–Wales defect at the middle or edge or near the edge of the nanotube. Investigations of a wide range of reactions to the Stone–Wales defect sites of two possible modes at different locations in the nanotubes will offer opportunities to design and realize new nanomaterials.

3. Computational Methods The (5, 5) armchair perfect nanotube structures of three different lengths (1) C60 H20 , (2) C80 H20 and (3) C100 H20 were fully optimized using the Hartree-Fock method employing the 4-31G and the 6-31G(d) basis sets. The geometries were further refined at the hybrid density functional theory, B3LYP, with two different basis sets, 3-21G and 6-31G(d). A single Stone–Wales defect was made in the (5, 5) armchair nanotube at different positions and two possible modes of orientation. All of the structures of the nanotubes with a single Stone–Wales defect were also optimized at all of the above levels. In order to assess the performance of the SW defect formation energies at the lower levels, the energies were calculated at the MP2/6-31G(d) level using the B3LYP/6-31G(d) geometries for all structures possessing 60 and 80 carbon atoms. All calculations were performed using the Gaussian 03 program package [165]. The pyramidalization angle at the selected carbon atoms for all of the structures was calculated using Haddon’s -orbital axis vector (POAV) angle [166].

178


–18

(10, 10) –20

(9, 9)

Er (kcal/mol)

–22

(8, 8)

–24

(7, 7)

–26

(6, 6)

–28 –30

(5, 5)

–32 6

7

8

9

10

11

12

13

14

Diameter (Angstrom) (a) 8

(10, 10)

7

Ea (kcal/mol)

(9, 9) 6

(8, 8) (7, 7)

5

4

(6, 6)

3

(5, 5) 2 6

7

8

9

10

11

12

13

14

Diameter (Angstrom) (b)

Figure 8 The variation of (a) activation energy and (b) exothermicity in 1,3-dipolar cycloaddition reactions of ozone with armchair (n, n) SWNTs of different diameters (reprinted from [148] with permission, © Copyright (2003) American Chemical Society)

4. Results and Discussion: Stone–Wales Defect Formation in Armchair (5, 5) Nanotubes We created the Stone–Wales defect at different positions in (5, 5) armchair SWNTs considering the rotation of two different kinds of bonds such as axial and circumferential. We intend to examine the effect of nanotube length on the formation energy of the Stone– Wales defect. The variation of SW defect formation energies of nanotubes possessing SW defects at different locations such as at the edge, near the edge, and at the center

1.35 1.46

2

1.413

3

1.41

1

2

1

2.05

1.39 2

(a) Perfect tube 1.49

(b) Defective tube

(c) OPS (RE = –74.6 kcal/mol)

1.45 1

1.44

2

1.35 2 1

1.35 2 1

1.41 2.05

1.39

2.03 (d) OSW_I (RE = –62.4 kcal/mol)


3 1 1.43

(e) OSW_II (RE = –73.2 kcal/mol)

1.39

(f) OSW_III (RE = –74.0 kcal/mol)

179

Figure 9 (a) 7-layer defect-free (5, 5) tube model (C70 H20 ; (b) 7-layer defective (5, 5) tube model containing a Stone–Wales defect (colored yellow); (c) optimized O adduct of a defect-free (5, 5) tube model; (d) optimized O adduct at the Stone–Wales defect of defective (5, 5) tube model; (e) optimized O adduct at a 6–6 ring fusion near the Stone–Wales defect of defective (5, 5) tube model; (f) optimized O adduct on a 6–6 ring fusion far away from the Stone–Wales defect of a defective (5, 5) tube model. Bond lengths are in Å (reprinted from [155] with permission, © Copyright (2005) American Chemical Society)

180


of SWNTs was investigated. Previous studies have shown that Haddon’s POAV angle or pyramidalization angle could be used to explain the reactivity of the particular site in fullerenes and nanotubes [3, 148–155, 167, 168]. Hence, we have calculated the pyramidalization angle at the carbons of the perfect nanotube and the Stone–Wales defect region comprised of 16 carbon atoms. We attempt to explain the reactivity of the specific sites of the nanotube using the local curvature measured by the pyramidalization angle. A (5, 5) armchair SWNT containing 60 carbon atoms was built initially; then this structure has been extended by layer-by-layer to obtain two more structures. The ends of all of the nanotube structures were capped with hydrogen atoms. As reported in earlier studies [54, 115], the perfect armchair SWNTs 1–3 can be viewed as complete Clar, incomplete Clar and Kekule structures, respectively. Schemes 2–4 depict the Stone– Wales defect nanotubes generated from structures 1–3, respectively. In the nomenclature, the Stone–Wales defect structures generated by 90 rotation about circumferntial C–C bond are designated as 1a, 2a, 2b, 3a, etc. The SW defect nanotubes created by 90 rotation about an axial C–C bond are specified as 1a , 2a , 2b , 3a , etc. For the defect-free nanotubes, carbon atom numbering is given starting from edge to the center of the nanotube by means of layer-by-layer. In all of the SW defect nanotubes, the numbering is given starting from 7-membered ring then to 5-membered followed by

5 1

6 2

7 3 4

Rotation about circumferential bond

Rotation about axial bond

1, D5d

9

8 1

6 5 4

12

10

7

11 10

11

2 3 16

14 15

1a, Cs

13 1 7

12 13

9

14 2

15 316

6

8

1a′, C2

54


181

9 5 1

6 2

7 3

8 4


Rotation about axial bond 2, D5d

8 7 6 3

5 4

1 2

16

13 12 14 11 10 2 15 1

9 10

8

54

6 5 4

2a′, C2

9 10

1

14 16

9 11

2

3

15

3 16 6

12 13

2a, Cs

7

7

98

11 14 15

10 1

8 7

12 13

65

2b, Cs

4

11

12

2 3

13 14 1615

2b′, C1

Scheme 3

another 7-membered ring, and finally ending up with the other 5-membered ring. Thus, C1–C2 bond is shared by two 7-membered rings in all of the SW defect structures. The principal bond lengths and pyramidalization angles (P ) for the perfect nanotubes obtained from the B3LYP/6-31G(d) optimized geometries are given in Table 3. The bond lengths reported by Matsuo et al. for the defect-free nanotubes 1–3 have been reproduced in our study [115]. The bond lengths for structures 1–3 range from 1.38 to 1.47 Å, and the edge C–C bond lengths are about 1.36–1.38 Å. The pyramidalization angles are more than 5 for all of the carbon atoms except those located at the edge of the perfect nanotubes. The highest pyramidalization angles are observed for C3, C6 and C10 atoms in structures 1–3, respectively.

182


9 5 1

10 11 6 7 2

3 8 4


Rotation about axial bond

3, D5d 8 7 6 5 4

3 16

1 2

12 13 14 11 15 2 10 1 9 3 16 7 8 6 4 5

9 10 11 14 12 15 13

3a, Cs 8 7 6 5 4

3 16

1 2

3a′, C2

9 10 14 15

9

11 12 13

8

10 1

11 12

7

3

6 5

3b, Cs

8 7 6 5 4

3 16

13 14

2

15 16

4

3b′, C1

9 1

12

10 11 10

11

2 14 15

12 13

9 8

13 14 2 15

1

3 16

7 6 5

3c, Cs

4

3c′, C1

Scheme 4

Reliable structural information concerning nanotubes often helps to understand the chemical reactivities at different sites of carbon nanotubes. Thus, the important bond lengths and pyramidalization angles (P ) at the defect region of the Stone–Wales defect SWNTs of B3LYP/6-31G(d) optimized geometries are provided in Tables 4 and 5. Many of the values are not provided in Table 5 since the structures possess the symmetry. Structures 1a and 1a possessing SW defects have been obtained by a 90 rotation of the C5–C6 (circumferential bond) and the C6–C7 (axial bond) bonds, respectively, of pristine structure 1. Similarly, four (2a, 2b, 2a and 2b ) and six structures (3a, 3b, 3c, 3a , 3b and 3c ) with a single Stone–Wales defect in each structure have been generated from the corresponding defect-free nanotubes 2 and 3. In all of the defect structures, the C1–C2 bond that is shared by two 7-membered rings of the SW defect has received


183

Table 3 Important bond lengths (in Å) obtained at the B3LYP/6-31G(d) level and the pyramidalization angles (P in degrees) for the selected carbon atoms of defect-free (5, 5) armchair SWNTs of different lengths. (See Schemes 2–4 for structures, atom numbering and nomenclature.) Parameter r1−2 r2−3 r3–4 r3–6 r5–6 r6–7 r7–8 r7–9a r9–10 r10–11 P 2 P 3 P 6 P 7 P 10

1

2

1 384 1 418 1 468 1 415 1 427 1 441

1 366 1 440 1 435 1 422 1 441 1 428 1 413 1 438

3 0 6 4 4 9

2 3 5 7 5 9 5 7

3 1 373 1 431 1 453 1 416 1 439 1 435 1 415 1 430 1 436 1 416 2 6 6 1 5 6 5 5 6 3

(a) For structure 3, the parameter is for C7–C10.

Table 4 Important bond lengths (in Å) obtained at the B3LYP/6-31G(d) level and the pyramidalization angles (P in degrees) for the carbon atoms in the Stone–Wales defect region of the SW defect nanotubes generated by a 90 rotation of the circumferential bond type of pristine (5, 5) SWNTs. (See Schemes 2–4 for structures, atom numbering and nomenclature.) Parameter

1a, Cs

2a, Cs

2b, Cs

3a, Cs

3b, Cs

3c, Cs

r1–2 r2–3 r3–4 r4–5 r5–6 r6–7 r7–1 r7–8 r8–9 r15–16 r16–3 P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 16

1 367 1 455 1 398 1 459 1 447 1 425 1 453 1 414 1 446 1 361 1 463 0 3 0 3 7 4 4 8 2 7 4 9 8 6 6 1 1 0

1 345 1 460 1 404 1 464 1 461 1 429 1 454 1 407 1 472 1 438 1 416 0 3 1 7 9 1 4 8 2 9 5 0 8 0 6 1 7 8

1 367 1 451 1 436 1 442 1 446 1 410 1 452 1 418 1 413 1 394 1 427 0 7 1 7 8 0 5 1 3 5 4 8 9 8 8 3 1 6

1 348 1 449 1 408 1 464 1 444 1 434 1 452 1 397 1 431 1 434 1 419 0 3 1 0 9 1 4 8 4 0 5 3 10 2 8 2 7 3

1 341 1 454 1 425 1 468 1 456 1 426 1 465 1 396 1 449 1 480 1 408 1 3 0 0 7 5 4 9 2 9 5 1 9 7 9 1 6 0

1 358 1 456 1 455 1 437 1 455 1 408 1 458 1 413 1 433 1 418 1 405 0 8 3 0 8 0 5 4 3 2 4 7 9 4 8 8 1 8

184


Table 5 Important bond lengths (in Å) obtained at the B3LYP/6-31G(d) level and the pyramidalization angles (P in degrees) for the carbon atoms in the Stone–Wales defect region of the SW defect nanotubes generated by a 90 rotation of the axial bond type of pristine (5, 5) SWNTs. (See Schemes 2–4 for structures, atom numbering and nomenclature.) Parameter

1a , C2

2a , C2

2b , C1

3a , C2

3b , C1

3c , C1

r1–2 r2–3 r3–4 r4–5 r5–6 r6–7 r7–1 r7–8 r8–9 r9–10 r10–1 r10–11 r11–12 r12–13 r13–14 r14–2 r14–15 r15–16 r16–3 P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 12 P 13 P 14 P 15 P 16

1 445 1 481 1 387 1 462 1 358 1 453 1 401 1 471 1 408 1 422

1 393 1 441 1 432 1 429 1 464 1 412 1 475 1 423 1 413 1 426

1 367 1 486 1 423 1 458 1 461 1 434 1 470 1 400 1 426 1 402

9 0 4 2 3 0 1 4 3 0 6 0 5 6 6 7

6 7 6 1 2 8 6 1 6 5 5 4 7 2 9 2

1 379 1 461 1 449 1 360 1 458 1 404 1 485 1 409 1 420 1 406 1 471 1 429 1 459 1 457 1 427 1 478 1 407 1 437 1 402 7 1 7 4 4 6 3 5 1 5 2 9 6 5 8 8 7 1 5 2 6 0 5 4 2 4 6 3 8 1 5 2

1 381 1 463 1 414 1 465 1 434 1 426 1 478 1 404 1 412 1 409 1 458 1 439 1 433 1 460 1 415 1 472 1 421 1 418 1 432 7 3 6 4 5 6 6 6 6 0 2 5 6 2 9 7 7 4 6 0 5 8 4 7 3 0 6 7 8 7 7 0

1 399 1 487 1 399 1 456 1 361 1 448 1 431 1 430 1 421 1 418 1 482 1 416 1 459 1 443 1 432 1 446 1 434 1 406 1 413 7 7 7 1 5 8 2 8 1 4 3 6 5 7 5 6 8 0 5 5 2 9 5 3 5 8 6 4 7 6 8 9

6 9 6 8 2 4 4 9 6 0 5 0 6 9 9 3

special attention in many computational studies concerning the reactivities of SW defect CNTs [152, 154–156]. In general, the length of the C1–C2 bond of SW defect structures is significantly shorter compared to the corresponding bond in pristine structures. The C1–C2 bond length is less than 1.4 Å in all of the defect structures. However, it is 1.44 Å in the case of 1a , which is surprisingly longer. The C1–C2 bond distance is


185

notably shorter for SW defect structures yielded by rotation about the circumferential bond compared to rotation about the axial bond (e.g., 1a vs 1a ). As observed in pristine structures, the length of C–C bonds at the edges adjusted by hydrogen atoms in SW defect nanotubes is substantially smaller compared to other bond distances. These edge C–C bond distances are about 1.36 Å except for 2b (1.39 Å) and 3c (1.42 Å). Table 4 reveals that C3–C4 and C7–C8 bond lengths are generally shorter compared to other bond distances in SW defect structures formed by rotation of the circumferential bond type. Many of the bond lengths in the SW defect region are more than 1.4 Å, and some of them are longer than 1.45 Å. A quick look at Table 5 indicates that long bond distances of about 1.5 Å are observed in the case of SW defect nanotubes obtained by a 90 rotation of the axial C–C bond, and those long bonds are shared by 5- and 7-membered rings. As mentioned earlier, the pyramidalization angles have been used to explain the reactivity of the specific site of the nanotube sidewalls. Recent computational studies have pointed out that carbon atoms with high pyramidalization angles exhibit high reactivity [152, 155]. The data presented in Tables 3–5 reveal that at least three or four carbon atoms in the SW defect region have higher pyramidalization angles compared to the highest value observed in the corresponding pristine structures. The values of the pyramidalization angles make evident that the local curvature of the carbon atoms in the 5–7, 5–6–7 and 5–7–7 ring fusions is substantially high. Therefore, the C–C bonds that are in the 5–7, 5–6 and 7–7 ring fusions are expected to be highly reactive compared to the defect-free structures. The C1–C2 bond, which is shared by two 7-membered rings of the 5–7–7–5 defect, has been the subject of theoretical research [152, 155, 156, 163]. In the case of SW defect structures formed by rotation of the circumferential bond, the atoms of the C1–C2 bond have very low pyramidalization angles ranging from 0.3 to 1.3 . This indicates the planar nature of the C1–C2 bond. As pointed out in earlier studies [155, 156], the near planarity of the C1–C2 bond of the SW defect structure explains the lower cycloaddition reactivity across this bond compared to the corresponding bond of the defect-free nanotube. Substantially large pyramidalization angles have been obtained for C1 and C2 atoms of the SW defect structures, which are obtained by the rotation of the axial bond. Due to the high local curvature at the C1 and C2 atoms in the above case, the addition reactions across the C1–C2 bond will be more feasible compared to the corresponding bond of the defect-free nanotubes. Many of the available theoretical studies discuss the cycloaddition reactivities of the sidewalls of CNTs based on the reaction energies [148–156]. Obtaining the right transition state and verifying it for such large systems are challenging as well as computationally demanding. Until recently, there is no reported study on cycloaddition reactions considering many possible C–C bonds in the Stone–Wales defect region of the nanotube. Bettinger has recently investigated the cycloaddition reactions of methylene across various C–C bonds of the SW defect region for 3a and 3b structures [156]. The reaction energies reported by Bettinger along with the relevant bond distances and the pyramidalization angles obtained in this study are given in Table 6. Here, our prime aim is to obtain the correlation between the reaction energies and the pyramidalization angles.

186


Table 6 Reaction energies (Er in kcal/mol) reported at PBE/6-31G(d)//PBE/3-21G for the addition of methylene across various bonds of the Stone–Wales defect region of structures 3a and 3b are given with the relevant bond lengths (in Å) obtained at the B3LYP/6-31G(d) level and the pyramidalization angles (P in degrees). 3b

3a Ci–Cj C1–C2 C5–C6 C6–C7 C7–C1 C7–C8 C8–C9

Er a −62 5 −67 6 −101 1 −61 2 −84 0 −97 8

ri–j

P i

P j

Ci–Cj

1 348 1 444 1 434 1 452 1 397 1 431

0 3 4 0 5 3 10 2 10 2 8 2

1 0 5 3 10 2 0 3 8 2 8 2

C1–C2 C2–C3 C3–C4 C4–C5 C5–C6 C6–C7 C7–C1 C7–C8 C8–C9 C15–C16

Er a −88 5 −82 2 −91 8 −109 6 −65 4 −74 5 −113 2 −82 4 −69 6 −85 6

ri–j

P i

P j

1 381 1 463 1 414 1 465 1 434 1 426 1 478 1 404 1 412 1 418

7 3 6 4 5 6 6 6 6 0 2 5 6 2 6 2 9 7 8 7

6 4 5 6 6 6 6 0 2 5 6 2 7 3 9 7 7 4 7 0

(a) Taken from [156].

The exothermicity of the addition of methylene across the C6–C7 and C8–C9 bonds of 3a is higher (or comparable) with respect to the defect-free nanotube [156]. The cycloaddition reaction across the C1–C2 bond of 3a is the second lowest exothermic process. The nearly planar nature of the C1–C2 bond could be attributed to low exothermicity. In contrast to 3a, the addition of methylene across the C1–C2 bond of 3b is highly exothermic compared to the pristine case, and this can be explained by the large local strain at these carbon sites as evidenced by the high pyramidalization angles. Note that the bond length of C1–C2 of 3b is about 0.03 Å longer than that in 3a. Lu et al. have reported that the C–C bond of the 7–7 ring fusion is chemically less reactive than the corresponding bond in the perfect nanotube [155]. In contrast, our investigation clearly indicates that the cycloaddition reaction across the C–C bond of the 7–7 ring fusion need not always be less reactive compared to the pristine case and depends on the orientation of the SW defect in the carbon nanotubes. In case of 3b , the methylene addition across the C7–C1 bond shows the highest exothermic characteristics followed by the reaction across the C4–C5 bond. It is worth mentioning that nine out of ten C–C bonds considered in the SW defect region of 3b encountered more exothermicity compared to the C–C bond of pristine in the addition of methylene. The pyramidalization angle gives an inkling of the reactivity of the different bonds of the SW defect region in the cycloaddition reactions. Our critical analysis reveals that the pyramidalization angles are useful for understanding the reactivity of different sites of SWNTs. The Stone–Wales defect formation energy (ESW ) is calculated as the energy of the SW defect structure (Edefect−CNT minus the corresponding defect-free nanotube (Epristine . The Stone–Wales defect formation energies computed at various levels of theory are given in Table 7. Choosing a reliable and economical level of theory is critical to systematically model defects in SWNTs. A quick look at Figure 10 indicates that the SW defect formation energies computed at the B3LYP/3-21G level are in agreement with those obtained at the B3LYP/6-31G(d) level. Thus, the B3LYP functional in conjunction


187

Table 7 The Stone–Wales defect formation energies (in kcal/mol) obtained at various levels of theory for (5, 5) armchair SWNTs Structure

HF

1a (Cs 1a (C2 2a (Cs 2b (Cs 2a (C2 2b (C1 3a (Cs 3b (Cs 3c (Cs 3a (C2 3b (C1 3c (C1

MP2/6-31G(d)a

B3LYP

4-31G

6-31G(d)

3-21G

6-31G(d)

77 0 39 3 61 8 84 1 79 0 65 1 89 0 69 5 81 5 70 9 78 0 70 8

64 0 33 8 56 9 78 3 73 5 59 1 82 9 64 4 75 6 64 6 72 3 65 4

67 5 36 9 54 4 64 4 62 0 56 2 71 7 60 3 66 7 61 2 62 3 56 2

62 7 33 5 52 1 61 6 58 2 52 3 69 6 58 4 64 2 57 6 58 4 52 0

60 5 35 9 52 4 57 1 56 2 54 4 – – – – – –

(a) Single-point calculations were done using B3LYP/6-31G(d) geometries.

with the 3-21G basis set may be a better choice to model the defect formation reliably and economically for larger nanotubes. Significant overestimation of the SW defect formation energy at the HF method compared to B3LYP or MP2 indicates the importance of the inclusion of electron correlation in computing reliable energetics. The 4-31G basis set was used in some of the previous computational studies involving SWNTs [76]. Our results show that the HF/4-31G level is the worse choice among the levels considered. Consequently, it is better to avoid the 4-31G basis set for the reliable energetics of the carbon nanotubes. Figure 10 shows that the SW defect formation energies computed at the B3LYP/6-31G(d) level are in good agreement with those obtained at the MP2/631G(d) level. Hence, the B3LYP/6-31G(d) results are used in the discussion related to the SW defect formation energies and the stability ordering of the structures.

HF/4-31G HF/6-31G(d) B3LYP/3-21G B3LYP/6-31G(d) MP2/6-31G(d)//B3LYP/6-31G(d)

ΔESW (kcal/mol)

110 100 90 80 70 60 50 40 30

1a

1a′

2a

2b

2a′

2b′

3a

3b

3c

3a′

3b′

3c′

Figure 10 The variation of Stone–Wales defect formation energies (ESW , in kcal/mol) at different levels of theory for (5, 5) armchair SWNTs

188


Significantly large difference in energy is seen between 1a and 1a structures. The former structure is about 30 kcal/mol more stable than the latter. For system 2, the order of stability is 2a ≥ 2b > 2a > 2b. The stability order of the SW defect structures of system 3 is 3c > 3a ≥ 3b ≈ 3b > 3c > 3a. Our results indicate that SW defect SWNTs by means of axial bond rotation are competitive to those resulting from a 90 rotation of the circumferential C–C bond. Structure 3c is about 18 kcal/mol more stable than 3a. The SW defect is located at the edge of the nanotube for the former structure, while it is positioned at the middle of the nanotube in the latter structure. The energetics of the SW defect nanotubes clearly indicate that the orientation of the SW defect should also be taken into account in modeling nanotube defects for a complete understanding of Stone–Wales defects in SWNTs. Jeng et al. have recently predicted, using molecular dynamics simulations, that the Stone–Wales defects are spontaneously formed in strained nanotubes. They also pointed out the proclivity toward SW transformation as the temperature increases [74]. Zhao et al. demonstrated that SW defect formation energies and activation energies for SW transformations decrease almost linearly with strain as depicted in Figure 11 [73]. The SW defect formation energies obtained in our study also indicate the requirement of a small amount of strain for the SW transformation; otherwise, the SW defect may easily occur at high temperatures. The frontier molecular orbital energies (HOMO and LUMO) and the HOMO-LUMO energy gaps obtained at HF and B3LYP with the 6-31G(d) basis set for all of the structures under the study are given in Table 8. The HOMO-LUMO energy gap values at B3LYP/6-31G(d) are smaller compared to the HF/6-31G(d) level. The discussion on the HOMO, LUMO and HOMO-LUMO gaps is based on the HF/6-31G(d) results. The frontier orbital energies and band gaps oscillate when the length of the armchair perfect nanotube increases as mentioned in earlier studies [115, 116, 169, 170]. This oscillatory behavior can be explained in terms of the periodic changes of the bonding characteristics of the structures. Even though both 1a and 1a structures exhibit higher band gap values compared to pristine structure 1, the latter is envisaged to be kinetically more stable than the former. In contrast to system 1, structures 2a and 2b have lower band gap values compared to other structures in the series including the perfect nanotube 2. The band gaps of the Stone–Wales defect structures of system 3 are less or almost the same compared to defect-free structure 3. 6

10

Energy (eV)

Energy (eV)

4

E(form)

2 0 –2 –4 –6

(a)

0

5

10

Strain (%)

6 4 2 0

15

(b)

E(act)

8

0

10

5

15

Strain (%)

Figure 11 (a) Stone–Wales defect formation energy and (b) activation energy for the formation of Stone–Wales defects as a function of uniaxial strain. Circles and squares correspond to (5,5) SWNT and a graphene sheet, respectively (reprinted with permission from [73], © Copyright (2002) by the American Physical Society; http://link.aps.org/abstract/PRB/V65/e144105)


189

Table 8 Frontier molecular orbital (HOMO and LUMO) energies and band gap values (EHL ) obtained at the HF/6-31G(d) and B3LYP/6-31G(d) levels for the (5, 5) armchair SWNTs of different lengths with Stone–Wales defects and also for defect-free nanotubes. All values are in eV Structure

1 (D5d ) 1a (Cs ) 1a (C2 ) 2 (D5d ) 2a (Cs ) 2b (Cs ) 2a (C2 ) 2b (C1 ) 3 (D5d ) 3a (Cs ) 3b (Cs ) 3c (Cs ) 3a (C2 ) 3b (C1 ) 3c (C1 )

HF/6-31G(d)

B3LYP/6-31G(d)

HOMO

LUMO

EHL

HOMO

LUMO

EHL

−5 28 −5 34 −5 74 −5 49 −5 67 −5 59 −5 19 −5 59 −5 30 −5 11 −5 39 −5 42 −5 39 −5 28 −5 22

0 31 0 63 0 27 0 57 0 78 0 52 0 40 0 18 0 15 −0 07 0 09 0 16 0 06 −0 01 −0 06

5 59 5 98 6 01 6 06 6 44 6 11 5 59 5 78 5 45 5 04 5 48 5 57 5 45 5 27 5 15

−4.17 −3 87 −4 41 −4 30 −4 38 −4 36 −4 05 −4 25 −4 33 −4 11 −4 33 −4 22 −4 35 −4 28 −4 33

−2 61 −2 85 −2 69 −2 54 −2 42 −2 55 −2 64 −2 93 −2 62 −2 78 −2 78 −2 75 −2 90 −2 81 −2 82

1 56 1 02 1 72 1 76 1 96 1 81 1 41 1 32 1 70 1 33 1 56 1 46 1 46 1 47 1 51

5. Applications of Carbon Nanotubes Nanoelectromechanical systems (NEMS) of very small sizes – from hundreds to a few nanometers – are emerging with high technological applications and are rapidly making the world more advanced. They can be fabricated or integrated with electronic, optical and biological systems to produce devices with advanced functions. Fundamental research on these types of systems is significantly engrossed by numerous laboratories worldwide. Carbon nanotubes form the basis for revolutionary novel multifunctional materials that may have promising applications in diverse areas [1–9]. Many applications of carbon nanotubes such as polymer nanocomposite materials, quantum wires, nanoscale test tubes, synthetic membranes, selective adsorbents, biocompatible devices, hydrogen storage, other hydrocarbon fuel storage and nanotube-based sensors have been proposed [3–9, 171–189]. Figure 12 shows a sketch of the utility of carbon nanotubes as sensors. Carbon nanotubes can be used to dissipate heat from tiny computer chips. Nanotubes are thought to have potential applications as catalyst supports in heterogeneous catalysis, catalyst carriers, field emission panel display, actuators, gas storage media, chemical sensing, drug delivery and molecular wires for the next generation electronic devices [126–129, 188–191]. Some of the applications such as sensors, highperformance composite materials, nanotube-based transistors and field emitters have been already realized [16, 22, 176, 192–197]. Figure 13 illustrates a simple model of quantum transport through nanotubes coupled to leads. NASA is committed to developing lightweight materials for space applications from carbon nanotubes due to their high stiffness and strength. Flat panel displays,

190

T. C. Dinadayalane and J. Leszczynski Carbon nanotube

Amplification

Demodulation and Evaluation

Figure 12 A carbon nanotube-based sensor

L

R

Figure 13 Schematic of a typical device consisting of a nanotube coupled to a left (L) and right (R) lead (reprinted with permission from [100b], © Copyright (2001) by the American Physical Society; http://link.aps.org/abstract/PRB/V63/e155412)

gas-discharge tubes in telecom networks, electron guns for electron microscopes, AFM tips and microwave amplifiers are some examples of potential applications of carbon nanotubes as field-emitting devices [46, 177, 196–202]. Samsung has already released its flat panel field emission display product utilizing carbon nanotubes [200, 201]. Some of the applications of carbon nanotubes are depicted in Figures 14 and 15. In order to effectively apply nanotubes to nanocatalytic and sensor applications, a chemical alteration is required for the deposition of catalysts and the other species onto the surface of nanotubes. The derivatization of nanotubes with metal-containing molecular coordination complexes makes the area of metalloorganic chemistry applicable to SWNTs. These nanotube derivatives show potential applications in catalysis including homogeneous catalysis and molecular electronics [118]. Carbon nanotubes have already been established as invaluable as tips for scanning probe microscopy and field-emission devices [199, 200]. Outstanding physical and chemical properties of carbon nanotubes will undoubtedly revolutionize a number of important industries, from biology and medicine to aerospace and electronics. In the direction toward innovative applications of carbon nanotubes, research on the use of CNTs as ferroelectric devices, nano-fluidic devices and their possibility to use in bio-surgical instruments has already been started. Single-walled carbon nanotubes show promise for biomedical applications due to their biocompatibility and high strength [127–129, 185–187]. Some biomedical applications of SWNTs are given here. Baughman et al. have demonstrated that a CNT artificial muscle shows a dramatic increase in work density output and force generation over available technologies along with the ability to operate at low voltage [128]. Carbon nanotube

Pt/Lr Tip

VGCFs

Protruded central nanocore

Silver paste

(a)

(b)

Figure 14 Applications of carbon nanotubes in (a) an AFM tip and (b) a flat panel display from Samsung

source electrode

drain electrode

nanotube molecule Gate oxide

oxide

SiO2

Gate (Al or Ti)

Source (Ti)

silicon wafer

CNT Drain (Ti)

P++ Si

(a)

(b)

Gate Nanotube Source

Drain

Silicon Dioxide Silicon Water Gate Oxide

(c)

(d)

Figure 15 Some applications of carbon nanotubes: (a) a simple nanotube transistor setup, (b) a cross-sectional diagram of an IBM carbon nanotube field effect transistor (CNFET), (c) a three-dimensional model of CNFET created by IBM, and (d) logic circuits with field-effect transistors based on single carbon nanotube Reprinted with permission from [176], © Copyright (2001) AAAS)

192


composites are used for the replacement of bone and teeth, and they have advantage of high mechanical properties compared to skeletal tissues. Chen et al. have shown that carbon nanotubes can be used in specific molecular recognition processes with protein receptors and can resist nonspecific protein binding [202]. A very important biomedical application of SWNTs is the drug delivery system. Carbon nanotubes could be implanted at sites where a drug is needed without trauma and slowly release a drug over time [129]. SWNTs could serve as nanopipettes for the supply of extremely small volumes of liquid or gas into living cells or onto surfaces and could also serve as a medium for the implantation of diagnostic devices. They show promising applications in biology including sensing [203, 204], imaging [205] and scaffolding for cell growth [206]. The low toxicity of carbon nanotubes makes the functionalized nanotubes innovative carriers to deliver peptides, proteins and nucleic acids into cells [187, 207–212]. Bianco and coworkers have reported the successful delivery of antibiotics such as fluorescein and amphotericin B to different types of mammalian cells by selective transport through the membrane [129]. Scientists are anxious to use SWNTs as drug and gene delivery systems since they can retain in organs such as the lungs, the heart and the liver. Very recently, Singh et al. have reported the blood clearance and urenal excretion of functionalized SWNTs that have been used for drug delivery [213]. This will be a good basis for trials in the direction of biomedical applications of SWNTs, particularly as drug delivery systems, in the near future. Carbon nanotubes are now well recognized as the strongest material known [214–216]. They are building blocks for numerous nanomaterials. However, many challenges exist in the area of carbon nanotubes and their utility in several applications. Controlling the diameter and chirality of the nanotube remains challenging in the production of carbon nanotubes [14]. One of the challenges of making very long nanotubes has almost been resolved after the preparation of long super-bundles of SWNTs [50]. The determination and eventual control of chiral indices are of vital importance to make carbon nanotubes useful in the envisaged nanoelectronic device applications. Analytical techniques such as Raman spectroscopy [39, 40], fluorescent spectroscopy [41, 42], scanning probe microscopy [20, 21] and electron diffraction [217–220] have been able to provide an average measurement over a large number of nanotubes. So far, no technique is available to determine the atomic structure of individual carbon nanotubes accurately and rapidly. Very recently, Liu et al. have reported a direct method called the ‘nanobeam electron diffraction pattern’ that facilitates accurately determining the chiral indices (n, m) of a carbon nanotube [221]. Precise spatial alignment, uniform emission properties and low processing temperature are the existing challenges to applying carbon nanotubes to field emission. Another current challenge is the inability to create complex nanotube arrays reliably and rapidly on an industrial scale.

Summary and Outlook This chapter reviews theoretical and experimental studies on the chemistry of SWNTs. It covers classification of carbon nanotubes, sidewall functionalization of SWNTs and detailed theoretical studies of Stone–Wales defects in the (5, 5) armchair nanotube. Carbon nanotubes are at the forefront of nanotechnology and have huge potential for a wide range of applications, thus necessitating the importance of understanding the


193

structures and reactions of SWNTs delineated in this chapter. As an important part of the chapter, we have systematically examined the structures and relative stabilities of the Stone–Wales defect at different positions and orientations of the (5, 5) armchair SWNT using the HF and MP2 methods and the B3LYP functional. We have evaluated the local curvature using the pyramidalization angle at the carbon sites of the Stone–Wales defect region and explained the reactivity based on the pyramidalization angle. The C–C bond shared by two 7-membered rings in the SW defect of SWNT need not always be less reactive than the corresponding bond in the pristine structure. The pyramidalization angles explain the reactivity of different bonds of the SW defect nanotubes. The present study reveals that the C–C bonds that are in the 5–7, 5–6 and 7–7 ring fusions of SW defect region are expected to be highly reactive compared to defect-free structures due to the high local strain at these sites. The SWNT need not always possess an SW defect at the middle portion. The band gaps of the defect-free and defected nanotubes oscillate as the tube length increases. Finally, we have briefly presented the applications of carbon nanotubes and pointed out challenges that remain in the field of CNTs.

Acknowledgements This research was supported by ONR grant # N00014-03-1-0116 and a grant from the US Army Engineer Research and Development Center, grant # W912HZ-05-C-0051. The Mississippi Center for Supercomputing Research (MCSR) and AHPCRC are acknowledged for computational facilities.

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Chapter 8

Thermal Stability of Carbon Nanosystems: Molecular-Dynamics Simulations Sakir ¸ Erkoç, Osman Bar Is¸ MalcIo˘glu and Emre Ta¸scI Department of Physics, Middle east Technical University, Ankara 06531, Turkey

1. Introduction It is now apparent that nanoscience is a critical field of research. This is not at all surprising, because no other topic promised applicability this vast before. Ironically, nanoscience is so huge that, in order to grasp a nanostructure, you need not only master multiple areas of physics, but also chemistry, biology, medicine, and engineering. In fact, nanotechnology may be called the ultimate multidisciplinary topic in this view. Materials at this scale, being at the interface of two worlds, exhibit some very interesting properties. For small enough systems, quantum approaches give very detailed information of what is going on. However, as the dimensionality increases, the number of atoms increases, and the direct quantum approaches become too complicated to solve. For large enough systems, there are statistical methods and formulas originating from experiments, which elegantly give information on what to expect. But being statistical in nature, these approaches do not work tidily enough for smaller systems; fluctuations become too sharp. The problem is that the systems we are interested in are situated just in-between: too big for unmodified quantum approaches, too small for statistical approaches. It is this property that makes this topic so critical, and our everyday life is filled with structures and mechanisms at this scale (the protein mechanisms, for example) and we know very little about them. Fortunately, by modifying atomistic quantum calculation methods, or by introducing new procedures (quantum, empirical or both), one can obtain some theoretical tools to tackle with the problem. Apart from miniaturization, deeper understanding of the mechanisms at this scale will have countless benefits in medicine and biology. A scientist working in this area has much to gain looking at organic chemistry and biology. 201

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One unavoidable conclusion in doing so is the importance of carbon atom. This importance stems from the uniqueness of its bonding characteristic. The valence electrons of the carbon are arranged such that three different bonding schemes are possible; this is called “hybridization” in organic chemistry. The possible hybrids of carbon atoms are named as sp1 sp2 , and sp3 (name indicates the participation level of p orbital to the sp hybrid orbital), which can have two, three, or four stable bonded neighbors correspondingly. The consequences of this property is rather astonishing: for example in sp3 hybridized form, the form of carbon is the diamond, one of the hardest materials known; whereas, in sp2 hybridized form carbon is in the form of graphite, which is quite useful as a writing tool, but neither too shiny nor too hard. It might not be as shiny as diamond, but sp2 hybridized carbon is, perhaps, the most interesting hybrid in the view of nanostructures. This hybrid has unique structural and electronic properties. The exact details of the electronic structure are beyond the scope of this chapter, but the reader is encouraged to read corresponding chapters, for example, in Saito et al. [1]. In this chapter, some example calculations regarding the stability of various carbon nanostructures under heat treatment will be presented, in the hope to introduce a method for such calculations and present some examples to carbon nanostructures that may be of interest in future applications. The question of thermal stability is an important problem; it predicts the stability of the structure of interest, its weak points, and the behavior under (thermal) stress. All these information may be utilized in predicting the usability of the structure. It is more convenient to introduce the individual structures studied in five sub categories.

1.1. Fullerene and some Related Nanocage Structures Chronologically, the first nanostructure to be introduced is the famous ball-shaped Buckminsterfullerene (or fullerene, in short). Using a widely accepted naming scheme, Cn , this structure is abbreviated as C60 . The experimental observation and production of this carbon nanoball in 1985 [2] opened the door for other carbon nanostructures. This marks the beginning of nanoscience as a field. One interesting point is that the non-planar carbon macromolecules were not unknown to society before the introduction of fullerene itself. In fact, the basic building block of C60 , namely corannulene, was encountered and extensively studied in the 1970s [3, 4]. Even a structure in the shape of a football was proposed [5, 6], which was called as the Buckminsterfullerene after 1990. The football shape of this nanostructure is due to symmetrically distributed pentagons. This kind of encapsulating nanostructures is named as “nanocage” due to apparent reasons. Using this nomenclature, many different nanostructures can be grouped together. First of all, there are many ball-shaped carbon nanostructures, which all belong to this category. A further derivation of this nanostructure is the carbon nano-onion, which may be described as concentric carbon nanoballs put together. However, the concentric arrangement of spherical fullerenes possesses extreme robustness, which led to the hypothesis that the quasi-spherical onion-like graphitic particles are the most stable form of carbon particles [7]. In the corresponding theoretical studies presented in this chapter, heat energy is pumped into C20 @C60 and C20 @C60 @C240 composite systems as well as the individual components C20 and C60 in order to get some information which may shed some light

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on to the formation of these fullerenes. C20 is the smallest possible carbon nanoball with the imposed restrictions and C60 is the most stable isolated nanocage structure of carbon at room temperature – thus these geometries are of special interest. Particular attention has been paid to the stability of individual carbon nanoballs against heat treatment. On the other hand, silicon surfaces, due to their inherent scientific interest and technical importance, have been popular substrates for studying the interactions of fullerens with semiconductor devices [8–26]. In the experimental works, the interaction of C60 molecule with silicon surfaces has focused on identifying the C60 bonding sites and their strength as well as the decomposition of these molecules at elevated temperatures [8–23]. In this chapter, two calculations regarding the decomposition of C60 on silicon surfaces, Si(100) and Si(111), by the method of annealing is presented. Nanotoroid is another interesting nanocage structure. This nanostructure also encloses a specific volume, but this time the volume is toroidal [27–36]. The nanotoroidal geometry can be briefly described as a plane formed of hexagons being mapped on the surface of a torus. However, to reduce the torsion on the positively curved outer wall and negatively curved inner wall, pentagons and heptagons must be introduced respectively in place of hexagons [31]. The necessity for these pairs gets smaller as the radius increases, where at some point hexagon-only nanotori are possible. Since many different geometrical approaches are possible in placing the pentagon–heptagon pairs, various methods for obtaining nanotorus geometry have been suggested [37–40]. The common point is that they all can be derived from nanotube geometry point deformations [29]. They also possess the cage structure of fullerenes. Since the nanotori constitute rather a wide concept, it is impossible to cover all the related aspects in this chapter. Instead, two calculations are presented as an introduction to the subject. First one is a comparative study on the structural stabilities under heat treatment of one model for C120 nanotoroid and two models for C240 nanotoroids. The reference nanocage structures used in the comparison are C120 and C240 carbon nanoballs, since the number of carbon atoms is same in both structures. Second work examines the thermal stability of various Fonseca-type nanotori, which consist of nanotube regions, joined by “knee regions”. The nanotori constructed following this procedure have fivefold symmetry and are composed of armchair- and zigzag-type nanotubes. This particular calculation investigates the thermal stability of C170 C250 C360 C520 , and C750 nanotori.

1.2. C60 Chains The structure C60 can be used in a photovoltaic device. After observation of photoinduced electron transfer from a conducting polymer to C60 [41], many novel devices have been suggested regarding the matter [42]. However, C60 structure is inherently non-periodic. They only tend to form molecular films with weak Van der Waals interactions, which is not very welcome either in the processing or in the actual use. Thus a variety of ways of forming stronger films have been suggested [43]. Another interesting property of C60 is its magnetic behavior [44, 45]. Magnetic effects can be observed either in a C60 lattice [46] or by attaching various materials to an isolated fullerene [47]. It has been observed in the former structure that this magnetic behavior is localized in the hexagonal

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planes, and depends on the distance between C60 [48]. Again, creating aligned structures is of importance in this case, for proper utilization. In this chapter, various chain structures that are composed of C60 and interconnecting benzorods have been described. The benzorods are attached to selected hexagonal sites in C60 nanoballs, in order to form a linearly aligned structure. Benzorod is a recently proposed form for a carbon nanorod structure [49], which is described later in this section in more detail. In the structures considered in the corresponding study, chemically more active pentagons are left free which may prove to be more convenient in photovoltaic applications, especially when a bond with a polymer is required. Also, hexagonal planes are in alignment, thus it may have interesting magnetic behavior. Thermal stability of the structures considered is of importance.

1.3. Nanotubes Fullerene marks the beginning of nanoscience, but probably the most famous novel structure introduced in the field is the carbon nanotube, which have more technological applications than other carbon nanostructures. The geometry of the carbon nanotubes is most easily described as graphite folded to form a cylindrical tube. This structure possesses some unique structural and electronic properties. The first experimental observation and systematic study on this structure is in 1991 [50]. Iijima encountered this interesting carbon precipitation while investigating the deposits on one of the electrodes after an arc discharge experiment utilizing carbon. The reported properties made many scientists to become interested in investigating this structure both experimentally and theoretically [51–55]. The most obvious property expected from these superfine structures of carbon is durability and strength. Carbon nanotubes satisfy the expectations even more than anticipated – they are roughly up to 100 times stronger in tensile strength when compared to steel in the same nanometer scale [56, 57] and they are quite flexible. The carbon nanotubes are chemically inert in nature and have good thermal conductivity [58]. Most of the tubules that have been discussed in the literature have closed caps; however, open-ended tubules with dangling bonds have also been reported [59, 60]. The growth mechanism for cylindrical fullerene tubules has still been debated whether the tubules are always capped [61, 62] or open during the growth process and that carbon atoms are added at the open ends of the tubules [59, 63, 64]. For the growth of carbon nanotubes by the arc discharge method, it has been proposed that the tubules grow at their open ends [63, 65]. Finite and infinite systems are very different, and the question of how large a relatively small system must be to yield results that resemble the behavior of the infinite system faithfully lacks a unique answer. In most cases, one is interested in a system which is infinite in at least one direction and the problem arises of deducing from the necessarily finite simulation system the properties of the infinite system. The normal procedure is to invoke the periodic boundary condition (PBC), which overcomes the problem of having the particles at the edge of the simulation box interacting only with particles to one side of themselves. There are two consequences of this periodicity. The first is that an atom that leaves the simulation region through a particular bounding face immediately reenters the region through the opposite face. The second is that atoms lying within a


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cut-off distance of a boundary interact with atoms in an adjacent copy of the system or, equivalently, with atoms near the opposite boundary, which is the well-known minimum image (MI) approximation [66, 67]. In the context of this chapter, the effect of the PBC on the simulation of structural and energetics of carbon nanotubes has been investigated by an example calculation [51]. A further classification of carbon nanotubes uses the number of individual layers in the cylindrical walls. When there are two or more layers, the carbon nanotube is put in the category of multi-wall carbon nanotubes (MWCNTs). MWCNTs can be produced with lesser yield with respect to the single-wall carbon nanotubes (SWCNTs). SWCNTs were first made by the electric-arc discharge method through the introduction of catalyst species along with the evaporated carbon [50]. Later it has been reported that well-aligned MWCNTs, in addition to the arc-discharge method, can be grown by chemical-vapor deposition catalyzed by iron nanoparticles embedded in mesoporous silica [68]. In the case of MWCNTs the spacing between cylinders increases with decreasing diameter of the graphene cylinders, which is due to the increasing curvature of the graphene sheets. The stability and physical properties of nanotubes could be affected by curvature in nanotubes. The example simulation on this matter considers the smallest possible double- and triple-wall nanotubes. The considered multi-wall models can also be put into the category of carbon nanorods. SWCNTs attract more attention than their multi-wall counterparts. This is due to theoretical predictions on their electronic behavior, which shows a strong dependence on the tube size [69] as well as the geometry of the tube [70]. Individual SWCNTs may be considered as quantum wires [71] and their electrical properties vary strongly from tube to tube [72]. SWCNTs are most conveniently identified according to their chiral vector, Ck , which can be expressed by the real space unit vectors a1 and a2 of the hexagonal lattice forming the graphene sheet [73]: Ck = na1 + ma2 ≡ n m = Cn m. The general notation used to define an SWCNT is C(n m), where n and m are integers, (m ≤ n). When m = 0 the tube is called “zigzag” model; when m = n the tube is called “armchair” model; and all the other combinations (0 < m < n) form the chiral models. When the sum (n + m) becomes a multiple of three, the carbon nanotube is conducting, otherwise it is semiconducting in various levels. No electronic device can be realized without junctions, so it is important to study junctions between nanotubes. Multi-terminal nanotube junction devices have been known to work as nanoscale transistors [74–79]. Although such devices are quite important in nanotechnology, junction formation mechanisms are still not well known. There are a number of suggestions and experimentation regarding the matter. Examples include ion bombardment [80], chemical functionalization [81], soldering by selectively depositing carbon [82], low energy carbon ion bombardment [83], and electron beam welding [84]. In the example presented, we investigate the possibility of formation of a junction under pressure. One advantage of this scheme is that there are no contaminating elements, thus structural geometry away from the close vicinity of junction may maintain the original form. The system consists of two identical C(10, 0) SWCNTs, rotated 90 with respect to each other, so that their axis resides on x and y directions. Then, two stiff grapheme layers were introduced above and below the two-tube system. The purpose of these layers was to create an effective pressure without directly interfering with the tube atoms, thus increasing correspondence to a possible experimental method.

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1.4. Nanocapsules In some cases the carbon nanocapsule may be inferred as the derivation of fullerene geometry. In this view, the carbon nanotube consists of the tubular part plus the corresponding half spherical carbon nanoball cap. There is always a corresponding half spherical cap in the geometry of fullerenes. This structure is quite popular among the people working in the area, and a number of applications using its unusual properties are possible [85]. In fact most of the reported nanotube structures in the literature have at least one closed cap. One of the interesting and maybe immediate applications is as a memory device [86]. Another interesting application is in the field of controlled medicine delivery [87]. In this chapter, six models for fullerene-terminated nanotubes are discussed. Using a semi-spherical fullerene as the termination cap restricts the radius strictly, but different chiralities for the tubular part are possible, and they are treated separately. Four of the models contain a C20 nanoball encaged. C20 is the smallest nanoball possible, and have interesting aromatic behavior due to all-pentagon exterior, thus it may be utilized in a shuttle device. In such a device, distinctive locations for the C20 nanoball are the middle of the nanotube and a location close to either of the caps. One of the recent additions to carbon nanotube-related structure literature is the bamboo-shaped carbon nanotubes [88–92]. In all of the cases, reported structure consists of (periodically) compartmented MWCNTs with geometrical aberrations in the vicinity of intersecting layers. The aberrations resemble the bamboo plant. Experimental work suggests that the intersecting layers are either precipitation of the catalyst material or a graphitic interlayer. Although there is no report of a single-wall bamboo-like carbon nanotube structure, in the light of abovementioned growth mechanism it is theoretically possible. Recently, the structural and electronic properties of a coronene-based singlewall carbon nanobamboo structure have been investigated theoretically [93]. In the second example of this section, a structure based on the abovementioned geometry is investigated. The investigated structure is periodically compartmented with coronene spacers. The middle section of the coronene is planar and can be called “graphitic”. Such a compartment spacer is reported in some of the observations in their multi-wall counterparts. The number of involved pentagons in the formation of this structure is higher, and highly bended hexagons are present, each having an adverse effect in the energy of formation. However, the structure is highly symmetric and may balance the stress in the vicinity. As an insight on the formation mechanism, influences such as present coronene in the environment, or a catalysor forcing a planar termination of a single-wall nanotube may work in favor. The tubular section in the structure is restricted to have twelve aligned hexagons in the circumference. This corresponds to C(12, 0) zigzag nanotube. Since twelve is a multiple of three, the tubular section is theoretically conducting, and resultant structure may have diode-like behavior along the axis due to the bamboo regions.

1.5. Nanorods Unlike carbon nanotubes, the nanorod is a non-hollow nanostructure. The number of structures this name covers is overwhelming, as the non-sp2 carbon is also a common acquaintance in nanorods. Four examples will be covered in this section, and they are


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not limited to fullerene-related structures. Structural properties are the forte of these materials. Applications for a nanorod device include advanced mesoscopic electronic and optical devices, inter-connects, probe microscopy tips, as well as superstrong and tough composites [73, 94–98]. The first example in this section covers a comparative study on how a transition from carbon nanotube to carbon nanorod may occur. It is known that electron beam can induce such a transition [99]. Transmission electron microscopy study revealed typical nanorods of 20 nm in diameter and with amorphous structure [99]. In experimental observations, each layer in MWCNTs seems to have the same chirality. Thus the interlayer separation is considerably large. It is not clear that if this is due to formation mechanism, or it is a general restriction. In the second example presented in this section, nanorod structures formed using tightly spaced concentric carbon nanotubes is investigated. Since chirality condition is not applied, inter-wall separation is as low as ∼15 Å. It turns out that these structures are not stable even at very low temperatures. No more restricted to sp2 hybridization, we may turn our attention to other forms of carbon. Use of atomically sharp diamond structures as a tip is demonstrated both theoretically and experimentally [100, 101]. In addition to this, these structures may be used in single point diamond tools, which may be utilized in ultra-precision turning of soft materials such as computer disks and laser mirrors [102]. Thermal transport in these structures, again, may be of interest [103, 104] and may be used in different electronic devices [105]. In this example the stability under heat treatment of diamond nanorods of different diameters and different crystal orientations are investigated. Benzorod is a recently proposed form for a carbon nanorod structure [49]. It is composed of horizontally aligned and dehydrogenated benzene rings stacked together. The “n” in the notation nC6 for benzorods stands for the number of dehydrogenated benzene rings present. Benzene ring is a highly aromatic molecule. In the case of the benzorod, however, interior rings have four nearest neighbors, which causes a huge deviation from the benzene molecular electronic configuration. Since hexagonal orientation of carbon atoms is common in fullerene-related structures and in graphene sheet, benzorod structure may prove to be an important building block in preparing aligned nanostructures. Due to possible field-emitting properties, there may be some number of applications for an aligned array of benzorods. One such application may be a field-emitting display.

2. The PEFs and the MD Simulation The reader should have noticed that although example carbon nanostructure calculations presented in this chapter belong to different categories, the method of heat treatment is mostly common. Once the inter-atomic forces are established, a classical treatment for atoms is possible as long as you are only interested in structural properties. By reducing the atoms to idealized point particles, huge amount of computational time is gained, thus larger systems can be simulated. In molecular-dynamics (MD) method, the geometry is left to realign itself under the influence of the interatomic forces. When the structure can no longer attain such an orientation, or the resultant geometry is significantly different than the initial geometry, it is called “distorted”, and this often marks the end of the particular calculation. Solving the equations of motion in a computer requires a discrete time unit. One MD time step is taken as 10−16 s in carbon systems.

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The canonical ensemble molecular-dynamics NVT [106] is applied in the course of the simulations, where the particle number N , the volume V and the temperature T are fixed. The total energy is not a conserved quantity for constant temperature in canonical ensemble. However, the average kinetic energy is a constant of motion due to its coupling with the temperature. The initial velocities of the particles are determined from the Maxwell distribution at the given temperature. In order to maintain the given temperature, rescaling is taken into account at every MD step. The equations of motion of the particles are solved by considering the Verlet algorithm [106]. For infinitely long tubes we have applied PBC [66, 67] along the tube axis. One of the most suitable empirical, many-body, potential energy function (PEF) options for the pure carbon systems is the Tersoff PEF for carbon [107]. This PEF describes the structural properties and energetics of carbon relatively accurately, including diamond crystal as well as the properties of the individual basal planes of graphite. Furthermore, in its revised form, fullerene-related structures, such as carbon nanotube and buckyball, can also be simulated, and relatively accurate structural properties and energetics are obtained [108]. There is no explicit “bond” information in the Tersoff PEF and thus carbon atoms may rearrange under external influence without bond conservation restriction. The total interaction energy () of a system of particles is taken to be the sum of total two-body (2 ) and total three-body (3 ) contributions [109]. = 2 + 3 Total two-body and three-body energies are expressed, respectively, as 2 = A

N

1 Uij

3 = −B

N

2 Uij

1+

N

n −1/2n Wijk

k=ij

i<j

i<j

n

Here Uij and Wijk represent the two- and three-body interactions, respectively: 1

Uij = fc rij exp− 1 rij

2

Uij = fc rij exp− 2 rij

Wijk = fc rik g ijk

where g ijk = 1 + fc r =

⎧ ⎨

c2 c2 − d2 d2 + h − cos ijk 2

1 − 21 sin 2 r − R/D ⎩ 0 1 2

for r < R − D for R − D < r < R + D for r > R + D

The parameters of the Tersoff PEF for carbon are [109]: A = 1 3936 eV B = 34674 eV 1 = 3487 9 Å−1 2 = 2211 9 Å−1 = 1572 1 × 10−7 n = 0.727 51, c = 380 49 d = 4348 4 h = −0570 58 R = 195 Å, and D = 015 Å. For CSi binary systems a different PEF and method of simulation are required. The PEF used in this case contains again two- and three-body interactions: = 2 + 3 =

N i<j

Uij ri rj +

N i<j3 eV). In these metal oxide photocatalysts, the bottoms of the conduction bands, which consist of the empty transition metal d-orbitals or the hybridized s, p orbitals of typical metal, are located at a potential slightly more negative than 0 V vs NHE at pH 0, whereas the tops of the conduction bands consisting of O2p orbitals are at a potential more positive than 3 eV [11]. This situation makes the band gap energy of the photocatalyst too large to absorb visible light. From the viewpoint of solar energy utilization, it is indispensable to develop a photocatalyst that efficiently works under visible light ( > 400 nm). Some non-oxide materials, such as CdS and CdSe, have been examined as visible-light-driven photocatalysts for overall water splitting; however, successful photocatalytic systems have yet to be achieved due to inherent instability of the materials [12, 13]. Until very recently, reproducible photocatalytic systems for visible-light-driven overall water splitting had not been accomplished, although there are several reports that claimed to demonstrate the decomposition of water under visible light. The difficulty resulted from the lack of materials that meet the requirements mentioned below: • band edge positions suitable for overall water splitting • band gap energy smaller than 3 eV • enough stability during the photocatalytic reaction. Various approaches have been made in an attempt to overcome this problem [14]. One of them is to introduce atomic orbitals with a potential energy higher than O2p orbitals (e.g., N2p and S3p) into metal oxides, thereby making a new valence band instead of O2p orbitals. The authors’ group have studied such materials, in particular (oxy)nitrides and oxysulfides, as photocatalysts for overall water splitting under visible light. Note that this type of materials is different from the material doped with nitrogen or sulfur. Figure 2 shows the schematic band structures of metal oxide and (oxy)nitride. As mentioned above, the tops of the valence band (HOMO) for metal oxides are composed of O2p orbitals. When N atoms are partially or fully substituted for O atoms in a metal oxide, the HOMO of the material must be shifted to higher level compared to corresponding metal oxide, without affecting the bottoms of the conduction bands (LUMO) level. For example, band structure of an oxynitride, BaTaO2 N, is shown in Figure 3, which was obtained by density functional theory (DFT) calculation. The HOMO consists of hybridized N2p and O2p orbitals, whereas the LUMO is composed of empty Ta5d orbitals. The position of HOMO for oxynitrides locates at higher potential energy than that for corresponding oxide due to the contribution of N2p orbitals, making the band gap energy small enough to respond visible light ( 3 eV

Eg < 3 eV O2p

O2p N2p (+ Md) O2p (+ Md)

O2p (+ Md)

Metal oxide

Metal oxynitride

M: Metal cations with d0 or d10-electronic configuration

Figure 2 Schematic band structures of metal oxide and metal (oxy)nitride

Structure of BaTaO2N

LUMO Ba

Ta5d

O Ta HOMO N

N2p+O2p (Hybrid)

Figure 3 Band structure of BaTaO2 N

2. (Oxy)Nitrides and Oxysulfides with d0 -Electronic Configuration 2.1. Ti4+ , Nb5+ , and Ta5+ -Based (Oxy)nitrides [15–21] The (oxy)nitride can be obtained by nitriding a corresponding metal oxide powder. Figure 4 shows the UV–visible diffuse reflectance spectra of some (oxy)nitrides containing transition-metal cations of Ti4+ Nb5+ , and Ta5+ with d0 -electronic configuration. As can be seen, it is clear that the introduction of N3− anions into metal oxides results in

Nano-Particulate Photocatalysts for Overall Water Splitting under Visible Light

305

BaNbO2N

Abs./a.u.

TaON

LaTiO2N

Ta2O5 SrTaO2N

300

400

500 600 Wavelength/nm

700

800

Figure 4 UV–visible diffuse reflectance specra of some oxynitrides

sufficient absorption in visible light region. These (oxy)nitrides possess absorption bands at 500–750 nm, corresponding to the band gap energies of 1.7–2.5 eV [22]. As expected from the result of the DFT calculation mentioned above, these (oxy)nitrides have band edge positions suitable for overall water splitting, which was revealed by UV photoelectron spectroscopy (UPS) and photoelectrochemical (PEC) analysis. Band structures of Ta2 O5 , TaON, and Ta3 N5 determined by UPS and PEC analysis are schematically depicted in Figure 5. It is clear that the tops of the valence bands are shifted to higher

Energy – C.B.

Ta5d

Ta5d

Ta5d

C.B. H+/H2

Eg E g: 3.9 eV

Egg:: 2.4 2.4 eV E

Egg:: 2.1 2.1 eV E O2/H2O V.B. N2p

N2p N2p ++ O2p O2p

V.B. O2p Ta2O5

TaON

Ta3N5

+

Figure 5 Schematic band structures of Ta2 O5 , TaON, and Ta3 N5

306

K. Maeda and K. Domen

potential energies on the order Ta2 O5 < TaON < Ta3 N5 , whereas the positions of the bottoms of the conduction bands are almost same with each other [20]. The photocatalytic activities of some oxynitrides for water reduction or oxidation were investigated in the presence of methanol or silver nitrate as sacrificial reagents. The reactions using sacrificial reagents are not “overall” water splitting, but are often carried out as test reactions for overall water splitting. The equations of water reduction and oxidation in the presence of methanol and silver nitrate as sacrificial reagents are as follows: CH3 OH + H2 O → 3H2 + CO2 (Water reduction) 4AgNO3 + 2H2 O → O2 + 4Ag + 4HNO3 (Water oxidation) These (oxy)nitrides exhibits relatively high photocatalytic activity for water oxidation to form molecular oxygen under visible light irradiation ( > 420 nm). For example, a typical time course of O2 evolution on TaON under visible light irradiation ( > 420 nm) is shown in Figure 6 [15]. No reaction took place in the dark. Upon irradiation with visible light, O2 evolution started and the initial rate was 660 mol h−1 , which corresponds to a quantum efficiency of 34%. The rate of O2 evolution decreased with reaction time due to the decrease in Ag+ concentration and because the surface of TaON became covered with metallic Ag particles that obstructed photon absorption into the TaON. The total amount of evolved O2 (500 mol) corresponds to the amount that can be evolved by the stoichiometric reduction of Ag+ Ag+ 2000 mol O2 500 mol. It was confirmed by X-ray diffraction (XRD), X-ray photoelectron spectroscopy (XPS) and inductive coupling plasma (ICP) analyses that after the reaction for 5.5 h, all Ag+ in the solution was deposited on TaON as metallic Ag. No noticeable difference in the XRD patterns of the catalyst before and after the reaction was observed except for the presence of Ag metal. A small amount of N2 evolution was detected in the early stage of the reaction (first 1–2 h), which is attributed to the oxidation of N3− in the TaON into 800

Amount of evolved gases/μmol

λ > 420 nm 600

Evac. Evac. AgNO33 0.01M AgNO 0.01M added added

O2 O2

400 Q.E. = 34% 200 N2 N 2 0 0

2

4

6 8 Reaction time/h

10

12

Figure 6 Photocatalytic O2 evolution on TaON under visible light (> 420 nm)


307

N2 according to the following equation: 2N3− + 6h+ → N2 . However, the amount of N2 from assumed photogenerated holes corresponded to less than 1% of the catalyst, and further degradation of TaON did not occur. Actually, in the second run after another AgNO3 2000 mol addition, no N2 evolution was observed. The slower rate of O2 evolution was due to Ag metal deposition on TaON. Other (oxy)nitrides, such as Ta3 N5 and LaTiO2 N, also show the relatively high quantum efficiencies (∼10%) for the water photooxidation [16, 18, 19]. In contrast to efficient water oxidation, the photocatalytic activity for H2 evolution via water reduction was about one order of magnitude lower than that for O2 evolution. However, the photocatalytic performance for water reduction is stable. Figure 7 shows the time course of H2 evolution on TaON modified with Pt nanoparticles as H2 evolution promoter under visible light ( > 420 nm [15]. After evacuation of the reaction system, H2 evolution proceeds without any decrease in activity, indicating that TaON also functions as a photocatalyst for photoreduction of H+ into H2 under visible light irradiation. Although a very small level of N2 evolution was observed in the early stage of the reaction, N2 did not evolve during subsequent reactions. These results clearly indicate that TaON functions as a stable photocatalyst for water reduction and oxidation under visible light. The photocatalytic activity of TaON for water reduction increases markedly by photodeposition of Ru nanoparticles as H2 evolution sites [21]. It is important to confirm whether the reactions proceed by irradiation with photon energy corresponding to band gap energy of the catalyst. Figure 8 shows the dependence of the rates of H2 and O2 evolution using Ta3 N5 powder on the wavelength of the incident light [16]. The H2 and O2 evolution rates both decrease as the cutoff wavelength is increased. The longest wavelength available for the reactions is 600 nm, corresponding to the absorption edge of Ta3 N5 . This result clearly indicates that the reactions on Ta3 N5 progressed photocatalytically via the band gap transition from the valence band formed by N2p atomic orbitals to the conduction band formed by the empty Ta5d atomic orbitals.

Amount of evolved gases/μmol

120

λ > 420 nm

100 80

H2

60 Q.E. = 0.2%

40 20

N2 0 0

50

100

Reaction time/h

Figure 7 Photocatalytic H2 evolution on Pt-loaded TaON under visible light (> 420 nm)

308

K. Maeda and K. Domen 6

40

30

O2

20

3 H2

10

0

300

400 500 wavelength/nm

600

rate of evolved gases/µmol h–1

rate of evolved gases/µmol h–1

Absorption spectrum

0 700

Figure 8 Dependence of the rates of H2 and O2 evolution over Ta3 N5 upon the wavelength of the incident light

Although these (oxy)nitrides are confirmed to be stable for water reduction and oxidation, the overall water splitting has yet to be achieved. The main reason seems to be the low activity of H2 evolution on (oxy)nitrides. As mentioned above, these (oxy)nitrides are typically synthesized by nitriding a corresponding oxide precursor under a flow of NH3 at 1073–1173 K [15–19]. This situation causes (oxy)nitrides to be defective, which would hinder prompt electron migration from the bulk to the surface reaction site. Improvement of the preparation method to reduce the defects in the d0 -type (oxy)nitride photocatalysts is therefore required.

2.2. Oxysulfides, Ln2 Ti2 S2 O5 (Ln: Lanthanoid) [23–25] An oxysulfide, Sm2 Ti2 S2 O5 , is a stable photocatalyst that has a sufficient absorption in visible light region ( 440 nm

200 IrO2/Sm2Ti2S2O5 150

100

Sm2Ti2S2O5

50

CdS

0 0

10

20

30

Time/h

Figure 9 Photocatalytic O2 evolution on Sm2 Ti2 S2 O5 under visible light (> 440 nm)

O2 evolution up to 1.1%. It was confirmed by XRD and XPS that the decomposition of Sm2 Ti2 S2 O5 in the bulk and the surface, in particular oxidation of S2− species, does not occur during the water photooxidation, although the valence band partially consists of S3p. Sm2 Ti2 S2 O5 has the Ruddlesden–Popper-type layered perovskite structure in which the layers are composed of S–(TiO2 –O–(TiO2 –S double octahedra. The formation of TiSO5 octahedra results in the hybridization of S3p and O2p orbitals forming the valence band. It is likely that the hybridized S3p and O2p orbitals are more stable than pure S3p orbitals. Water reduction to form H2 on Sm2 Ti2 S2 O5 also proceeds steadily, when modified with Pt nanoparticles. In this case, a mixture of S2− and SO2− 3 is used as a sacrificial electron donor. On the basis of these results, it is concluded that an oxysulfide, Sm2 Ti2 S2 O5 , has functionality as a stable photocatalyst for water reduction or oxidation under visible light irradiation. It has also been demonstrated that Ln2 Ti2 S2 O5 (Ln: Pr, Nd, Gd, Tb, Dy, Ho, and Er) with the same layered perovskite structure as Sm2 Ti2 S2 O5 can evolve H2 or O2 from aqueous solutions containing sacrificial electron donor or acceptor under visible light irradiation without noticeable degradation, respectively. The photocatalytic activities of the Ln2 Ti2 S2 O5 are dependent strongly on the kind of lanthanoids (Ln) which greatly affect the electronic band structure of Ln2 Ti2 S2 O5 .

3. (Oxy)nitrides with d10 -Electronic Configuration 3.1. Difference in the Electronic Band Structure between d0 - and d10 -Based Photocatalyst As mentioned above, non-oxide photocatalysts, such as (oxy)nitrides and oxysulfides, which have a potential for reduction of H+ into H2 or oxidation of H2 O into O2 under

310


visible light, are composed of transition-metal cations of Ti4+ Nb5+ , or Ta5+ with d0 -electronic configuration [15–21, 23–25]. On the other hand, from the viewpoint of electronic band structure, d10 -based semiconducting materials have an advantage as a photocatalyst, compared to materials with d0 -electronic configuration [8–10]. The tops of the valence band for transition metal oxides with d0 -electronic configuration consists of O2p orbitals, whereas the bottoms of the conduction band is composed of empty d orbitals of transition metals. In typical metal oxides with d10 -electronic configuration, however, the bottoms of the conduction band consists of hybridized s, p orbitals of typical metals, although the valence band is formed essentially by O2p orbitals. The hybridized s, p orbitals possess large dispersion, which makes the mobility of photogenerated electrons in the conduction band large, thus leading to high photocatalytic activity [8–10]. This stimulates us to examine (oxy)nitrides or oxysulfides with d10 -electronic configuration as a photocatalyst for overall water splitting.

3.2. -Ge3 N4 [28] A typical metal nitride with d10 electronic configuration, -Ge3 N4 , is synthesized from GeO2 powder by nitridation under a flow of NH3 at 1123–1173 K for 10–15 h. The photocatalytic activity of the as-synthesized -Ge3 N4 for overall water splitting is negligible. However, when modified with RuO2 nanoparticles, the material becomes photocatalytically active for the stoichiometric evolution of H2 and O2 from pure water, as shown in Figure 10. This is the first example of photocatalysis of water using a non-oxide photocatalyst. DFT calculation revealed that the top of the valence band for -Ge3 N4 is formed by N2p orbitals, whereas the bottom of the conduction band consists of hybridized 4s, 4p orbitals of Ge. This result indicates that the photoexcitation under

Amount of evolved gases/mmol

1.2

λ > 200 nm 1.0 H2

0.8 0.6 0.4

O2

0.2 N2 0 0

1

2

3

4

5

Reaction time/h

Figure 10 Overall water splitting on RuO2 -loaded Ge3 N4 under UV light (> 200 nm)


311

irradiation occurs from the N2p orbitals to the Ge4s, 4p hybridized orbitals. Water oxidation to produce O2 on conventional metal oxide photocatalysts takes place as a result of contributions from photogenerated holes in the valence band consisting of O2p orbitals. It is noteworthy that the N2p orbitals in the valence band can generate photogenerated holes and are also able to contribute to the photocatalytic water oxidation. Unfortunately, however, the photocatalyst only works under UV irradiation because of its large band gap energy (ca. 3.8 eV).

3.3. (Ga1–x Znx )(N1–x Ox ) Solid Solution [29–32] Further research on d10 -typical metal (oxy)nitrides revealed that a solid solution of GaN and ZnO, represented as “(Ga1–x Znx N1–x Ox )” hereafter, achieves functionality as a photocatalyst for the stoichiometric decomposition of H2 O into H2 and O2 under visible light irradiation. The (Ga1–x Znx N1–x Ox ) is of interest not only as a novel photocatalyst but also as a new type of material. Both GaN and ZnO are important III–V and II–VI semiconductors, which have been studied extensively for application in light functional materials such as light-emitting diodes and laser diodes [33–35]. The (Ga1–x Znx N1–x Ox ) solid solution possesses a wurtzite crystal structure similar to GaN and ZnO, and is typically synthesized by nitriding a mixture of Ga2 O3 and ZnO. Figure 11 shows the XRD patterns of samples with different compositions. All samples exhibit single-phase diffraction patterns indicative of the wurtzite structure similar to GaN and ZnO. The position of the d(100) diffraction peak was successively shifted to lower angles (2 ) with increasing zinc and oxygen concentrations, indicating that the obtained samples were not physical

Shift

ZnO

Intensity/a.u.

Zn 13.3 atom%

Zn 6.4 atom%

Zn 3.4 atom%

GaN

20

30

40

50

60

70

80

31

32

33

2θ/Degree

Figure 11 Powder X-ray diffraction patterns of Ga1–x Znx N1–x Ox solid solution

312


mixtures of GaN and ZnO phases but rather solid solutions of GaN and ZnO. The same tendency was confirmed by Rietveld analysis using the computer program RIETAN2000 [36]. TEM images and an electron diffraction pattern of the solid solution are shown in Figure 12. The figure clearly reveals that the sample consisted of primary, well-crystallized, submicrometer-order particles with a wurtzite structure, as indicated by the lattice fringe and electron diffraction patterns. Figure 13 shows the UV–visible diffuse reflectance spectra of several samples. Both GaN and ZnO do not absorb visible light due to their large band gap energies of above

200 nm

10 nm

Figure 12 TEM images and electron diffraction pattern of Ga1–x Znx N1–x Ox solid solution (x = 013)

Zn 13.3 atom%

Abs./a.u.

Zn 6.4 atom%

GaN Zn 3.4 atom%

ZnO

300

350

400

450

500

550

600

Wavelength/nm

Figure 13 UV-visible diffuse reflectance spectra of Ga1–x Znx N1–x Ox solid solution


313

Amount of evolved gases/mmol

3 eV, but the Ga1–x Znx N1–x Ox solid solutions have steep absorption edges in visible light region and band gap energies of 2.58–2.75 eV. The absorption edge shifts to longer wavelengths with increasing Zn and O concentrations (x) in the Ga1–x Znx N1–x Ox . DFT calculation indicates that the origin of the visible light response of the solid solution is the contribution of Zn3d atomic orbitals to the valence band formation, where the bonding between Zn and N atoms forms as a result of the formation of solid solution. The as-prepared Ga1–x Znx N1–x Ox alone exhibits little photocatalytic activity for water decomposition even under UV irradiation. However, the modification of Ga1–x Znx N1–x Ox with RuO2 nanoparticles results in clearly observable H2 and O2 evolution. This modification involves the deposition of RuO2 nanoparticles on the surface of the Ga1−x Znx N1–x Ox as H2 evolution sites. The photocatalytic activity increased remarkably with increasing RuO2 content to a maximum at about 5 wt%, with the activity dropping gradually at higher RuO2 contents. It was elucidated by scanning electron microscopy, XPS and X-ray absorption spectroscopy that the enhancement of photocatalytic activity by RuO2 loading is dependent on the formation of crystalline RuO2 nanoparticles with optimal size and coverage. The photocatalytic performance of the RuO2 -loaded Ga1–x Znx N1–x Ox is also dependent strongly on the pH of the aqueous solution. The activity increased as the pH decreases from pH 7, passing through a maximum at pH 3, and then decreasing. In general, (oxy)nitride materials have an inherent instability in basic media, but are stable in acidic media. As a result, the highest activity was obtained at pH 3. Below pH 3, the surface of the catalyst was not completely stable. A typical time course of H2 and O2 evolution on RuO2 (5wt%)-loaded Ga1–x Znx N1–x Ox at pH 3 under visible light (>400 nm) is shown in Figure 14. H2 and O2 evolved steadily and stoichiometrically with no N2 evolution. After evacuation of the gas phase of the reaction system, stoichiometric H2 and O2 evolution is again confirmed, indicative of its good stability. It is known that O2 evolution occurs over ZnO when employed as a photoanode

Evac. →

0.30

Evac. →

> 400 nm H2

0.25 0.20 0.15

O2 0.10 0.05 N2 0 0

5

10 Reaction time/h

15

Figure 14 Overall water splitting on RuO2 -loaded Ga1–x Znx N1–x Ox solid solution (x = 013) under visible light (> 400 nm)

314


for water oxidation in a photoelectrochemical cell, and that the ZnO degrades as a result [37]. It was confirmed by 18 O-isotopic experiment that the O2 evolution on the present catalyst derives from water oxidation. No differences could be identified in the XRD patterns of the samples before and after the reaction. These results indicate that Ga1–x Znx N1–x Ox functions as a stable visible-light-driven photocatalyst for the overall water splitting.

4. Summary and Outlook Various types of non-oxide photocatalysts for overall water splitting have been presented. Until recently, although a material that meets the requirements for visible-light-driven overall water splitting has been limited, (oxy)nitrides and oxysulfides actually function as stable photocatalysts for water reduction or oxidation under visible light irradiation. Among them, the Ga1–x Znx N1–x Ox solid solution achieves functionality as a visiblelight-driven photocatalyst for overall water splitting. Although the quantum efficiency of the material is not high enough to use practical application, this is the first example of achieving overall water splitting by a particulate photocatalyst with a band gap in the visible light region, which opens the possibility of new non-oxide-type photocatalysts for energy conversion.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Khaselev, O.; Turner, J. A. Science 1998, 280, 425. Grätzel, M. Nature 2001, 414, 338. Cortright, R. D.; Davda, R. R.; Dumeslc, J. A. Nature 2002, 418, 964. Kahn, S. U. M.; Al-Shahry, M.; Ingler, W. B., Jr. Science 2002, 297, 2243. Domen, K.; Kondo, J. N.; Hara, M.; Takata, T. Bull. Chem. Soc. Jpn. 2000, 73, 1307, and reference therein. Kudo, A. Catal. Surv. Asia 2003, 7, 31, and reference therein. Kato, H.; Asakura, K.; Kudo, A. J. Am. Chem. Soc. 2003, 125, 3082. Sato, J.; Saito, N.; Nishiyama, H.; Inoue, Y. J. Phys. Chem. B 2001, 105, 6061. Ikarashi, K.; Sato, J.; Kobayashi, H.; Saito, N.; Nishiyama, H.; Inoue, Y. J. Phys. Chem. B 2002, 106, 9048. Sato, J.; Kobayashi, H.; Ikarashi, K.; Saito, N.; Nishiyama, H.; Inoue, Y. J. Phys. Chem. B 2004, 108, 4369. Scaife, D. E. Solar Energy 1980, 25, 41. Williams, R. J. Chem. Phys. 1960, 32, 1505. Ellis, A. B.; Kaiser, S. W.; Bolts, J. M.; Wrighton, M. S. J. Am. Chem. Soc. 1977, 99, 2839. Kudo, A.; Kato, H.; Tsuji, I. Chem. Lett. 2004, 33, 1534. Hitoki, G.; Takata, T.; Kondo, J. N.; Hara, M.; Kobayashi, H.; Domen, K. Chem. Commun. 2002, 1698. Hitoki, G.; Ishikawa, A.; Takata, T.; Kondo, J. N.; Hara, M.; Domen, K. Chem. Lett. 2002, 31, 736. Yamasita, D.; Takata, T.; Hara, M.; Kondo, J. N.; Domen, K. Solid State Ionics 2004, 172, 591. Kasahara, A.; Nukumizu, K.; Hitoki, G.; Takata, T.; Kondo, J. N.; Hara, M.; Kobayashi, H.; Domen, K. J. Phys. Chem. A 2002, 106, 6750.


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19. Kasahara, A.; Nukumizu, K.; Takata, T.; Kondo, J. N.; Hara, M.; Kobayashi, H.; Domen, K. J. Phys. Chem. B 2003, 107, 791. 20. Chun, W.-A.; Ishikawa, A.; Fujisawa, H.; Takata, T.; Kondo, J. N.; Hara, M.; Kawai, M.; Matsumoto, Y.; Domen, K. J. Phys. Chem. B 2003, 107, 1798. 21. Hara, M.; Nunoshige, J.; Takata, T.; Kondo, J. N.; Domen, K. Chem. Commun. 2003, 3000. 22. Band gap energy (Eg ) of the material can be roughly estimated from the following equation: Eg eV = 1240/abs (nm), where abs is the wavelength of the absorption edge of the material. 23. Ishikawa, A.; Takata, T.; Kondo, J. N.; Hara, M.; Kobayashi, H.; Domen, K. J. Am. Chem. Soc. 2002, 124, 13547. 24. Ishikawa, A.; Yamada, Y.; Takata, T.; Kondo, J. N.; Hara, M.; Kobayashi, H.; Domen, K. Chem. Mater. 2003, 15, 4442. 25. Ishikawa, A.; Takata, T.; Matsumura, T.; Kondo, J. N.; Hara, M.; Kobayashi, H.; Domen, K. J. Phys. Chem. B 2004, 108, 2637. 26. Harriman, A.; Thomas, J. M.; Millward, G. R. New J. Chem. 1987, 11, 757. 27. Harriman, A.; Pickering, I. J.; Thomas, J. M.; Christensen, P. A. J. Chem. Soc., Faraday Trans. 1 1988, 84, 2795. 28. Sato, J.; Saito, N.; Yamada, Y.; Maeda, K.; Takata, T.; Kondo, J. N.; Hara, M.; Kobayashi, H.; Domen, K.; Inoue, Y. J. Am. Chem. Soc. 2005, 127, 4150. 29. Maeda, K.; Takata, T.; Hara, M.; Saito, N.; Inoue, Y.; Kobayashi, H.; Domen, K. J. Am. Chem. Soc. 2005, 127, 8286. 30. Maeda, K.; Teramura, K.; Takata, T.; Hara, M.; Saito, N.; Toda, K.; Inoue, Y.; Kobayashi, H.; Domen, K. J. Phys. Chem. B 2005, 109, 20504. 31. Teramura, K.; Maeda, K.; Saito, T.; Takata, T.; Saito, N.; Inoue, Y.; Domen, K. J. Phys. Chem. B 2005, 109, 21915. 32. Yashima, M.; Maeda, K.; Teramura, K.; Takata, T.; Domen, K Chem. Phys. Lett. 2005, 416, 225. 33. Nakamura, S.; Mukai, T.; Senoh, M. Appl. Phys. Lett. 1994, 64, 1687. 34. Nakamura, S. Science 1998, 281, 956. 35. Tsukazaki, A.; Ohtomo, A.; Onuma, T.; Ohtani, M.; Makino, T.; Sumiya, M.; Ohtani, K.; Chichibu, S. F.; Fuke, S.; Segawa, Y.; Ohno, H.; Koinuma, H.; Kawasaki, M. Nat. Mater. 2005, 4, 42. 36. Izumi, F.; Ikeda, T. Mater. Sci. Forum 2000, 321–324, 198. 37. Gerischer, H. J. Electrochem. Soc. 1966, 113, 1174.


Index

Ab initio, 279, 284, 292 Adiabatic (Born–Oppenheimer) state, 277, 278, 284–8 Adiabatic ET, 276–8, 286–93 Adsorption, 6, 109, 112, 114, 127, 142, 149, 153, 158 Adsorption isotherms, 101–104, 105–106, 109, 111, 113, 114–16, 119, 121–2, 123–4, 129, 159, 251, 254–7, 258–62 Alizarin, 277–83, 286–93 Alkali clusters, 62–3, 65, 66, 81 Armchair, 167–9, 172–4, 176–8, 180, 183, 187–90, 192, 193, 203, 205, 216, 217, 227, 228, 230, 239 Array, 85, 150, 219, 220, 223 Axilrod–Teller potential, 209

Catechol, 278–83, 292, 293 Chain, 88–9, 103, 144, 203, 204, 214, 222, 259 Chemical shifts, 170 Chiral, 168–9, 192, 205, 206, 207, 216, 220, 227, 228, 237, 240, 241 Chiral angle, 168, 169, 237, 241 Chiral vector, 168, 169, 205, 227, 228 Chromophore–semiconductor coupling, 276, 277, 278, 283, 288, 290, 293 Classical path, 285, 287 Coherent nuclear vibrations, 278, 290 Computer simulations, 35, 59 Conduction band, 228, 275, 276, 278, 282, 283, 284, 287–93, 302, 303, 306, 307, 310 Corannulene, 127–8, 129–30, 131, 135, 136, 137, 139, 142, 149, 150 Core-shell, 35, 37, 47–51, 52, 55–6, 82 Curvature, 73, 86, 102, 127, 132, 133, 146, 159, 162, 170, 177, 178, 180, 185, 193, 205, 210, 211, 218, 228, 229, 240 Cycloaddition reactions, 167, 172, 176, 177, 178, 185, 186

B3LYP, 129, 130, 131, 132, 134, 135, 137, 138, 139, 140, 142–3, 144, 145, 146, 147, 148, 150, 173, 175, 178, 181–4, 186–9, 193 Back transfer, 275, 276, 285, 292 Band gap, 62, 170, 176, 188, 189, 193, 228, 229, 275, 302, 303, 305, 307, 308, 311, 312, 313 Basin-hopping, 46–7, 51 Basis set, 3–6, 9–12, 15, 19–20, 61, 72, 129, 140, 142, 178, 187, 188, 230, 280, 285–6 Benzorod, 204, 207, 214, 219–20, 221, 222, 223 Bi-isonicotinic acid, 277, 279, 286–93 Buckyball, 208, 214, 216

Decahedron, 210 Density functional theory, 1–3, 37, 59, 60, 87, 102, 149, 173, 178, 279, 303 Density of states, 277, 278, 287, 289 Diffusion, 64, 68, 70, 76–80, 118, 122, 212–13, 250–1, 252, 256, 258, 263–4 Dispersion, 86, 174, 246, 310 Donor–acceptor coupling, 277, 278, 290, 293 Drug delivery, 176, 189, 192

Calculations, 2, 129, 140 Canonical ensemble, 88, 101, 208, 286 Capsule, 217, 218 Carbon nanotubes, 128, 133, 162, 167–78, 182, 186, 187, 189–93, 204–208, 211, 216–17, 218, 220–21, 227, 230, 231, 234–41, 254–64, 275 Catalysis, 1, 3, 28, 35, 189, 190, 246, 294

Electron acceptor, 277, 278, 287, 290–3 Electron density, 38, 60–2, 72, 81, 128, 280, 282, 284, 286 Electron donor, 8, 276, 277, 278, 287, 290–3, 309 Electron relaxation, 276, 278, 283 317

318 Electron transfer, 1–2, 3, 5, 275 Electronic structure, 172, 228, 279 Embedded-atom, 65 Energetic heterogeneities, 104, 105, 110, 123, 124 Exothermicity, 172, 177, 186 Experiments, 246–65 Fermi’s Golden rule, 277 Field emission, 189, 190, 192, 220, 234–6 Force field, 132, 209, 215 Formation energy, 41, 178, 186–8 Fourier transform, 288 Fullerene, 169, 170–2, 174, 180, 202, 203, 204, 206, 207, 208, 209, 216, 227, 238, 275 Functionalization, 167, 170, 172, 174, 176, 192, 205 Gaussian, 2, 4, 5, 102, 120, 129, 178, 280, 282, 289, 292 Genetic algorithm, 45, 47, 123 Global optimization, 36–7, 39, 43–4, 46, 49–52, 80 Grand Canonical Monte Carlo simulations, 93, 103, 251, 255 Grätzel cell, 275, 276, 293 Ground state, 11, 50, 60, 87, 210, 275, 276, 278, 280, 282, 285, 286 Harmonic superposition approximation, 52, 53 Hartree-Fock, 60, 178, 279, 280 Heat treatment, 202, 203, 207, 210, 211, 212, 216, 217, 218, 219, 220, 221, 223 Heterogeneous environment, 261, 278 HF, 173, 175, 187–9, 193 HOMO, 2, 49, 174, 176, 188, 189, 282–3, 292, 303, 304 Homotops, 43–4 Hybrid, 3, 5, 6, 87, 129, 170–1, 178, 202, 207, 214, 228, 229, 303, 304, 309, 310–11 Hydrogen storage, 127–9, 150, 153, 159, 162, 174, 189 Icosahedron, 48, 49, 50, 72 Iijima, 168, 204, 227, 257 Individual adsorption isotherms, 102–104, 105, 106, 114–16, 119, 121, 123 Iodide/tri-iodide, 275 Isomers, 43

Index Junction, 4, 205, 215, 216, 230, 231–4 Kinetic energy functionals, 60, 61, 65, 71, 81 Kohn–Sham orbitals, 284–7 Lennard–Jones potential, 36, 44, 51–2, 105–106, 132, 209, 241, 263 Linear response TDDFT, 280 Localization, 43, 230, 277, 286, 288, 289, 290 LUMO, 2, 49, 174, 176, 188, 189, 282, 283, 292, 303, 304 Magic sizes, 42, 49 Marcus theory, 276, 277 Maxwell distribution, 208 MCM-41, 102, 115, 118, 119, 121, 122, 252 Melting simulations, 37, 49, 52–4, 59 Metal–molecule interface, 3, 6, 7 Microcanonical ensemble, 286 Microscopy, 102, 169, 170, 190, 192, 207, 313 Modeling, 85, 86, 227 Molecular bridge, 277, 290, 293 Molecular dynamics, 36, 52, 59, 87, 103, 130, 201, 207, 249, 252 Molecular dynamics simulations, 52, 64, 66, 130, 156, 171, 172, 188, 201, 236, 239, 240, 252, 293 Molecular mechanics, 172, 209, 215 Monolayer capacity, 104–109, 124 Monte Carlo, 63, 87, 88, 90, 92–3, 95, 101, 103, 104, 251, 255 MP2, 129, 139, 140, 141, 153, 178, 187, 193 Multi-wall, 167, 205, 206, 220, 221, 227, 237 Multiscale modeling, 85, 93 n–pi* transition, 278 Nano-onion, 202, 211, 212 Nanoalloys (or Binary clusters), 35–7, 40–1, 43, 47, 49, 52, 53, 55 Nanoball, 202, 203, 204, 206, 209, 210, 214 Nanobamboo, 206, 218, 222 Nanocage, 202, 203, 209, 210, 212 Nanocapsule, 206, 216, 217 Nanoclusters, 35–7, 52 Nanomaterials, 167,178,192, 214 Nanorod, 204, 205, 206, 207, 211, 218, 219, 220, 221, 222 Nanoscience, 201, 202, 204 Nanostructure, 85, 87, 174, 201, 202, 203, 204, 206, 207, 209, 222

Index Nanotechnology, 127, 192, 201, 205, 245 Nanotoroid, 203, 210 Nanotube, 127, 167, 204–208, 211, 215–18, 220, 221, 227 Nonadiabatic coupling, 285, 286, 293 Nonadiabatic electron transfer, 277, 278, 285–93 Nuclear trajectory, 285, 286 Organic/inorganic interface, 275, 277, 278, 279, 289, 292, 293, 294 Oxygen adsorption, 2, 6 Pair potential, 209 Pentagons, 170–2, 202–204, 206, 210, 211, 214, 216, 218, 230, 231, 232, 235, 239 Periodic boundary, 2, 4, 61, 88, 95, 105, 130, 147, 204, 215, 218, 228, 264, 285 Perylene, 277, 278, 290, 293 Phase transitions, 59, 73, 246, 248–9, 260 Photoexcitation, 276, 278, 282, 283, 286, 287, 289, 290, 292, 293, 310 Photovoltaic, 203–204, 214, 275, 276 pi–pi* transition, 278 Platinum, 6, 8, 9, 11, 17 POAV, 178, 180 Poly-icosehdron, 87 Pore size distributions, 101, 102, 112, 121–2, 253, 260 Pore wall distributions, 101 Potential energy, 37, 39, 43–6, 64–5, 78, 80, 86, 96, 108, 129, 132–4, 139–40, 142, 146–7, 153, 208, 212, 234, 238, 241–2, 285, 303 Potential energy surfaces, 86, 285 Pyramidalization angle, 167, 170, 171, 173, 174, 178, 180–6, 193 Quantum back reaction, 285 Quantum transport, 189, 229

319 Simulated annealing, 44, 46, 67–8, 80, 238 Simulation, 35, 59, 66, 87, 101, 105, 130, 132, 149, 201, 207, 209, 215, 227, 275, 285 Simulation cell, 130, 131, 147–50, 153, 156, 285, 286 Single-wall, 128, 167–8, 172–3, 190, 205, 206, 210, 212, 218, 221, 228, 236–7, 255, 257, 261 Slater Determinant, 284 Solar cell, 275, 277, 279, 289, 293 Spectra, 63–4, 72, 78, 80, 229, 258, 277–83, 292 Spectroscopy, 36, 80, 169, 192, 249, 250, 277, 305, 306, 313 Stability, 41, 50, 54, 69, 201–23 Stone–Wales defect, 167, 170, 171, 172, 178–80, 182–9, 192, 193 Structure, 35, 40, 47, 51, 80, 172, 202, 209, 228, 279 Surface characterization, 105 SWNT, 128, 152, 167–78, 180, 182–90, 192, 193, 227–30, 237, 242, 257, 263, 264 Synthesis of quantum dots, 97 Tersoff potential, 208, 238 Thermal fluctuations, 276, 278, 285, 287–90, 293 Thermal properties, 59, 60, 63, 66, 72, 81 Three-body potentia, 208, 209 Tight-binding, 38, 44, 63, 65, 228, 229, 231, 232, 234, 239 Time-dependent density functional theory, 279, 280 Time-dependent Kohn–Sham theory, 284, 287 Toroid, 203, 210, 211, 222 Transition state, 185, 276, 277, 278, 290, 291 Two-body potential, 38, 96, 208, 209

Reaction coordinate, 277, 287, 290 Reactivity, 167, 170, 172, 176–8, 180, 185, 186, 193 Red-shift, 279, 281, 282, 292 Regularization procedures, 104, 113–15, 116–19

Valence band, 228, 276, 302, 303, 305, 307–11, 313 Vanderbilt pseudopotentials, 286 Vibrational frequencies, 3, 62, 170, 293

SBA-15, 114–16, 118, 119, 122 Sensors, 1, 174, 189, 190 Silicon clusters, 21, 23, 25

Yakobson, 169, 238

Water, 245, 249, 250, 253, 301, 302

Zig-zag, 168, 169


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