Synthesis, Characterization and Properties of Nanostructures [1 ed.] 9783038132882, 9783908451730

Citation preview

Synthesis, Characterization and Properties of Nanostructures

Synthesis, Characterization and Properties of Nanostructures Computational and Experimental Approach

Special topic volume, invited papers only.

Edited by

Prafulla K. Jha and Arun Pratap

TRANS TECH PUBLICATIONS LTD Switzerland • UK • USA

Copyright © 2009 Trans Tech Publications Ltd, Switzerland

All rights reserved. No part of the contents of this publication may be reproduced or transmitted in any form or by any means without the written permission of the publisher.

Trans Tech Publications Ltd Laubisrutistr. 24 CH-8712 Stafa-Zurich Switzerland http://www.ttp.net

Volume 155 of Solid State Phenomena ISSN 1012-0394 (Pt. B of Diffusion and Defect Data - Solid State Data (ISSN 0377-6883)) Full text available online at http://www.scientific.net

Distributed worldwide by

and in the Americas by

Trans Tech Publications Ltd Laubisrutistr. 24 CH-8712 Stafa-Zurich Switzerland

Trans Tech Publications Inc. PO Box 699, May Street Enfield, NH 03748 USA

Fax: +41 (44) 922 10 33 e-mail: [email protected]

Phone: +1 (603) 632-7377 Fax: +1 (603) 632-5611 e-mail: [email protected]

Printed in the Netherlands

Preface  Editorial on the Special issue on “Synthesis, Characterization and Properties of Nanostructures: Computational and Experimental Approach” Reducing the dimension of matter domains down to the nanometer scale, confines the electronic and vibrational wavefunctions resulting in unique properties and opens a wide range of potential applications in domain as optics, mechanics, electrical, magnetic devices, reactivity and biomedicine. Nanostructures, characterized by at least one dimension in the nanometer range are considered to constitute a bridge between single molecule and bulk counterpart. The challenges for the so called nanotechnologies are to achieve perfect control of nanoscale-related properties, which obviously requires correlating the parameters of the synthesis process with the resulting nanostructures. Nanostructures are also ideal for computer simulation and modeling. In computations related to nanomaterials one deals with a spatial scaling from few nanometer to few micrometer and a time scaling from few femto-second (fs) to 1 second with the limit of accuracy going beyond 1 kcal mol-1. This special issue of solid state phenomena documents novel computational and experimental approaches summoned to resolve questions on growth of nanostructures, its characterization and modeling. Motivated by the increasing need to synthesize and understand the properties of materials at nanoscale this special issue is timely and a step toward improving our knowledge of how nanomaterials are useful for modern technologies. The special issue contains a collection of ten original review/papers including the reviews concerned with experimental approaches, theoretical analysis and numerical models. Seven papers are related to growth and characterization of nanomaterials while three papers deal theoretical approaches for understanding their properties. Another positive point of this special issue is the one common aim of several of these papers to achieve a deeper understanding of the underlying functionality of the properties of nanomaterials. The special issue has been divided in two sections namely the Computational Nanomaterials and Experimental Nanomaterials. The special issue contains three extensive topical review articles. Contributions appear in alphabetical order of their first authors in each section. The paper by Lu et al aims to develop better understanding of the size dependent interface energy of nanomaterials. The classic thermodynamics as a powerful theoretical tool is used to model different bulk interface energies in the computations of the size dependence of coherent energy of atoms within nanocrystals. This exhaustive review may lead to improve insight into the understanding of surface and interface energy of nanomaterials of low dimensional materials in different shapes with different chemical bond natures. Maiti examines the electron transport properties of some molecular wires and an unconventional disordered thin film and persistent current in metallic Rings and Cylinders under the frame work of tight-binding approach in his two of his articles. The model calculations provide a physical insight into the behavior of electron conduction across a bridge system. The characteristic properties of persistent current strongly depend on total number of electrons, chemical potential, randomness and total number of channels. The paper by Acharya reports the microscopic analysis of track etched polymeric membranes produced by bombardment of energetic ions on polymer membranes. The track results due to the energy loss during the formation of loosely bound passage in membrane. The Atomic Force Microscopy (AFM) used for the characterization gives the size and distribution of

the pores. The pore size is observed in nano regime and the pore density was found to depend on the irradiation dose. The ultimate objective of the paper by Ding et al is to prepare a ZnO thin film for the high optical transmittance in the visible region and strong absorption in ultraviolet region. The ZnO film is prepared from sol-gel precursors using electrospray method. The surface images obtained from AFM showed the compact ZnO films composed of wurtzite ZnO nanoparticles. The review article by Dutta and Tyagi aims to better understand the inorganic phosphor materials for solid state white light generation. Various examples are discussed based on oxide, fluoride, nitride, sulfide and phosphate  based host lattices. The important concepts like CIE coordinates and color correlated temperature (CCT) are also discussed. Eu3+-doped NaYF4 and YOF nanocrystallites were seen to be good red emitters. The paper by Mehta et al discuses the study of the dielectric properties of thin films of multiferroic compound Sr[(Mg0.32Co0.02) Nb0.66]O3 (SMCN) prepared by Pulsed Laser Deposition Technique (PLD). The dielectric properties show enhanced ε' values compared to bulk and decrease in ac susceptibility with temperature suggesting promising applications in alternative technologies for the switching devices. Technologically important two step activation energy is also observed in the film of SMCN. The aim of Parekh et al is to synthesize the monodispersed magnetite and cobalt ferrite nanoparticles via non-aqueous route. The characterization techniques used reveals the size distribution less than 5 % and particles of spherical shape. The magnetic moment obtained at room temperature is higher than that obtained earlier using other techniques. The work of Singhal et al reports the electronic structure study of Mn doped ZnO diluted magnetic semiconductor by using x-ray absorption spectroscopy for its possible use for future spin-electronic applications. The changes in the electronic structure are correlated with the observed magnetic properties. They observed that the most of the Mn ions of the ferromagnetic samples are in divalent state. Yadav proposes a simple citrate gel process to prepare magnetoelectric nanocomposite. TEM observation showed that the average particles size is around 40 nm. The variation of the dielectric constant and the dielectric loss with frequency showed dispersion in the low frequency range. The large values of ME coefficients are attributed to low coercivity, large magnetization and small crystallite size of constituent phases.

It is hoped that the present selection of contributions is of interest to the scientists working in the area of nanoscience and that it will contribute to the dynamic and rapidly expanding field of nanoscience and nanotechnology particularly with regards to growth, characterization and modeling. Our heartfelt thanks are due to all the contributors for the submission of their valuable work and the referees of the papers for sparing their valuable time. We especially thank the Editorial Board of Solid State Phenomena for their support in bringing out this special topical volume and all the members of the TRANS TECH PUBLICATIONS LTD office for their efforts throughout the publication process. Prafulla K. Jha and Arun Pratap (Guest Editors)

Table of Contents Preface

CHAPTER I: Computational Nanomaterials Size Dependent Interface Energy of Nanomaterials H.M. Lu Quantum Transport in Bridge Systems S.K. Maiti Persistent Current in Metallic Rings and Cylinders S.K. Maiti

3 71 87

CHAPTER II: Experimental Nanomaterials Microscopic Analysis of Track Etched Polymeric Membranes N.K. Acharya Inorganic Phosphor Materials for Solid State White Light Generation D.P. Dutta and A.K. Tyagi Dielectric and Conductivity Studies of Sr[(Mg0.32Co0.02) Nb0.66]O3 Thin Film P.K. Mehta, B. Bishnoi, R. Kumar, R.J. Choudhary and D.M. Phase The Preparation and Optical Property of ZnO Thin-Film by Electrospray Y.H. Ding, P. Zhang, Y. Jiang, F. Xu, J. Chen and H.L. Zheng Monodispersed Magnetic Fluids: Synthesis and Characterization K. Parekh, R.V. Upadhyay and V.K. Aswal An Electronic Structure Study of Mn Doped ZnO Diluted Magnetic Semiconductor Using X-Ray Absorption and Photoemission Techniques R.K. Singhal, M.S. Dhawan, S.K. Gaur and E. Saitovitch Synthesis and Characterization of Magnetoelectric Nanocomposites K.L. Yadav

107 113 145 151 155 163 173

CHAPTER I Computational Nanomaterials

Solid State Phenomena Vol. 155 (2009) pp 3-70 © (2009) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.155.3

Size Dependent Interface Energy of Nanomaterials H.M. Lu Department of Materials Science and Engineering, National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China [email protected] Keywords: Solid-liquid interface energy; Solid-solid interface energy; Solid-vapor interface energy; Liquid-vapor interface energy; Size and temperature dependences; Thermodynamics

Abstract. The reduction of size of the low dimensional materials leads to a dramatic increase of surface-to-volume ratio. The properties of a solid are essentially controlled by related surface/interface energies. Although such changes are believed to dominate behaviors of nanoscale structures, little experience or intuition for the expected phenomena, especially the size dependent properties and their practical implications, are modeled. In this contribution, the classic thermodynamics as a powerful traditional theoretical tool is used to model different bulk interface energies and the corresponding size dependences where emphasis on the size dependence of interface energy is given, which is induced by size dependence of coherent energy of atoms within nanocrystals. It is found that solid-vapor interface energy, liquid-vapor interface energy, solid-liquid interface energy, and solid-solid interface energy of nanoparticles and thin films fall as their diameters or thickness decrease to several nanometers while the solid-vapor interface energy ratio between different facets is size-independent and is equal to the corresponding bulk ratio. The predictions of the established analytic models without any free parameter, such as size and temperature dependences of these four kinds of interface energies, are in agreement with the experimental or other theoretical results of different kinds of low dimensional materials with different chemical bond natures. Introduction When interface size at least in one dimension is at nanometer size range, the corresponding materials properties could not be readily interpreted based on ″classical″ atomic or solid solution theories, and the regions of space involved were beyond the scope of existent experimental techniques [1]. Science to interpret these phenomena has taken a firm theoretical and experimental hold on the nature of matter at its two extremes: at the molecular, atomic, and subatomic levels, and in the area of bulk materials, including their physical, chemical and electrical properties. Between two extremes lies the nanometer size range, or called as mesoscale. Even with the latest advanced techniques for studying the phases and the region between phases, a great many mysteries remain to be solved. That ″region″ of the physical world represents a bridge not only between chemical and physical phases, but also plays a vital but often unrecognized role in other areas of physics, chemistry, materials science, biology, medicine, engineering, and other disciplines [1]. ″Interfacial″ phenomena may be defined as those related to the interaction between one phase (solid or liquid) and another (solid, liquid, gas or vacuum) in a narrow region where the transition from one phase to another occurs. As will quickly become apparent, the two classes of phenomena are intimately related and often cannot be distinguished [1]. Our understanding of the nature of the interfacial region and the changes or transitions that occur in going from one chemical (or physical) phase to another has historically lagged behind that in many other scientific areas because of the development and implementation of both theoretical and practical concepts. In the late nineteenth and early twentieth centuries, great strides have been made in the theoretical understandings of interactions at interfaces by thermodynamics [1]. Modern

4

Synthesis, Characterization and Properties of Nanostructures

computational and analytical techniques available in the last few years have led to significant advances toward a more complete understanding of the unique nature of interfaces and the interactions that result from their unique nature due to the rapid increase of computation ability/price ratio of computers [2]. The so-called computation materials science considers the interface properties from three different size scales [2]: 1. From atomic scale with the ab initio calculation based on the first principle, which considers many-body interaction behavior of several ten to hundreds molecules together; 2. From nanosize scale with molecular dynamics and Monte Carlo methods, which studies many-body interaction behavior of several thousands to several million molecules; 3. Engineering behavior of large-scale construction problems, or bulk materials using finite element methods where averaging constitutive laws are used to incorporate the microstructure. However, because of the unusual and sometimes complex character of interfaces and associated phenomena, the development of fully satisfying theoretical models has been slow. There exists a great deal of controversy in many areas related to interfaces [1]. Also, the classic thermodynamics has more or less neglected to interpret the interface phenomena in modern science due to the appearance of computer simulation technique. Note that many present controversies in theories and experiments in interface phenomena are not bad, since it represents the fuel for continued fundamental and practical research. However, for the practitioner who needs to apply the fruits of fundamental research, such uncertainty can sometimes complicate attempts to solve practical interfacial problems [1]. Nanoscience and nanotechnology are rapidly developed field in materials science and engineering in recent years. As size of materials drops to nanometer size range, interface/volume ratio increases and thus interface effect on materials properties becomes evident. When the sample size, or grain size, or domain size becomes comparable with the specific physical length scale such as the mean free path, the domain size in ferromagnets or ferroelectrics, the coherence length of phonons, or the correlation length of a collective ground state as in superconductivity, then the corresponding physical phenomenon will be strongly affected [3]. The interface phenomena, which affect the materials behavior, are in nature produced by different energetic states of molecules on the interface in comparison with those within the materials. Since the molecular interactions on the interfaces differ from those on the interior of phases, the excess specific Gibbs free energy or interface energy for molecules of unit interface area exists, which is equal to the difference between the total Gibbs free energy of interface molecules and that within the phases per unit area. The interfaces and corresponding interface energies studied here are sometimes called as other ones. The following is several examples of them. When the interface is composed of the same solids, the interface is also called as grain boundary and the corresponding interface energy is named as grain boundary energy. Usually solid-vapor interface energy and liquid-vapor interface energy are considered to have the same size of solid-vacuum interface energy and that of liquid-vacuum interface energy since their differences are often at least one order smaller when the external pressure is ambient one although it is known that this difference just leads to the absorption of vapor on interfaces. The solid-vapor interface energy is also named as solid-vapor interface energy while the liquid-vapor interface energy is made as liquid-vapor interface energy [4]. Although liquid-vapor interface energy of liquid can mean both solid-vapor interface energy and surface stress where they have the same size and meaning, this identity fails for solids [5]. The former describes the reversible work per unit area to form a new solid surface, whereas the latter denotes the reversible work per unit area due to elastic deformation, which is equal to the derivative of γ with respect to the strain tangential to the surface. All above phenomena change as materials dimension and broken symmetry vary. The dimensionality is dictated by the size in a certain direction of the object compared to an appropriate length scale for the phenomena being discussed. If only one dimension of the sample is small compared to this length parameter, then the object is two-dimensional. Similar definitions can be given for one-dimensional and zero-dimensional.

Solid State Phenomena Vol. 155

5

Although an increasing body of excellent computational materials science, the computer simulations remain many uncertainties in nanosize range. Thus, the classic thermodynamics still remain importance to model the above phenomena since the history of science demonstrate a belief in the universal nature of thermodynamics. The laws of thermodynamics were formulated as phenomenological laws about observable and operationally defined quantities. One of the main points of classical thermodynamics is that the thermodynamic approach is applicable to macroscopic systems only, i.e. to systems containing great number of molecules, atoms, ions, or to black holes and the entire Universe [6]. From this point of view, a single small object, including a nanosized droplet and a nanocrystal, does not satisfy the definition of the macroscopic thermodynamic system. At the same time, an ensemble of many small objects will be a macroscopic system. Such an ensemble may be rather speculative or correspond to a real dispersed system (aerosol, micro-emulsion, composed material etc). However, if the system is monodispersed (a contemporary example is an ensemble of the same, in a well-defined approximation, working elements in micro- or nanoelectronics), the treatment may be reduced to the investigation of a single modeling ensemble element. Introducing adequately defined distribution functions, the polydispersed systems may be also replaced (at least in a first approximation) by a modeling monodispersed one. Thus, extension of thermodynamic methods to very small objects, including nanoparticles seems possible but faces many principal difficulties. It is also noteworthy that, in the context of some modern experimental techniques such as the atomic force microscopy, the thermodynamic method is equally promising as a theoretical and empirical description of the experimentally investigated nanosystems. Indeed in many cases, experimental data on nanoparticle properties are rather scanty and contradictory. Thus if the particle is not entirely rigid, the experimental arrangement can have an effect on the object under investigation. It was demonstrated by Monte Carlo computer experiments that the tip of the atomic force microscopy could have a noticeable effect on a nanodroplet in the space between the tip and the solid substrate [ 7 ]. Hence difficulties of principle inevitably occur not only at the thermodynamic treatment but also at the experimental study of very small objects. However, no comprehensive overview of that field exists. This contribution aims to fill that gap, which is essentially based on the author′s own recent works. This gives a review how the traditional thermodynamics treats these interface amounts where particular emphasis is given on the size dependences of interface energies. From Section 2 to Section 5, the modeling of solid-liquid interface energy γsl, solid-solid interface energy γss, solid-vapor interface energy γsv, and liquid-vapor interface energy γlv are presented. Section 6 summarizes and provides a future prospect. 2. Solid-liquid interface energy 2.1. The bulk solid-liquid interface energy γsl0 The bulk solid-liquid interface free energy γsl0, which is defined as the reversible work required to form or to extend a unit area of interface between a crystal and its coexisting fluid plastically, is one of the fundamental materials properties [8,9,10,11,12,13,14,15]. It plays a key role in many practically important physical processes and phenomena like homogeneous nucleation, crystal growth from the melt, surface melting and roughening transition, etc. Thus, a quantitative knowledge of γsl0 values is necessary. However, direct measurements for γsl0 are not at all easy even for elements in contrast to the case of interface energy of liquid-vapor γlv0 [9,10,12]. Various techniques have been applied to estimate γsl0 from experimental data like maximum supercooling [8,9], dihedral angles (DA) [16,17], contact angles (CA) [17], the depression of melting points of small crystals (DMP) [10,17,18], the shape of the grain-boundary-grooves (GBG) [15], and molecular dynamics simulations (MD) [19,20,21,22,23], etc. Despite of the efforts, the obtained values show a considerable scattering with conflicts. Therefore, a theoretical determination of γsl0 is of vital importance.

6

Synthesis, Characterization and Properties of Nanostructures

Some theoretical attempts have been made to obtain γsl0 [8-10,14,16,24,25,26,27,28]. Based on the nucleation experiments and the classical nucleation theory (CNT), Turnbull proposed an empirical relationship that γsl0 is proportional to its melting enthalpy ∆Hm [9], γCNT = τ∆Hm/(Vs2a)1/3

(2.1)

where Turnbull coefficient τ is considered to be 0.45 for metals (especially closed-packed metals) and 0.34 for nonmetallic elements at about 20% of undercooling below the melting point Tm, Vs is the g-atom volume of the crystal, and a is the Avogadro constant. The γsl values measured by Turnbull are recognized now to be lower than real ones for metals [26-28]. According to the review papers of Eustathopoulos and Kelton, τ = 0.55±0.08 [11] and τ = 0.49±0.08 [13] for metals while τ value increases noticeably for molecules having more asymmetry [14]. Moreover, Eq. 2.1 overlooks some important pieces of physics. Moreover, the existence of τ to be determined also weakens the theoretical meaning of this equation, and leads to it only to be an empirical rule. Although Eq. 2.1 underestimated γsl values, the form of Eq. 2.1 has been rationalized in terms of interfacial bonding models [16,26-28]. The formulation developed by Ewing is [26], b′

γsl = h∆Hm/(4Vs)+(ab′k/Vl) ∫ g l (r ) ln g (r )dr 0

(2.2)

where where h is the atomic diameter in crystals, gl(r) is the liquid radial distribution function, b′ is the cut-off distance beyond which gl(r) shows no significant deviation from unity, k is Boltzmann′s constant, and Vl is the g-atom volume of the liquid. The approach of Miedema and den Broeder results in [27], γsl = 0.211∆Hm/(Vs2a)1/3+0.52×10-7T/Vs2/3

(2.3)

while that of Gránásy and Tegze yields [16,28], γsl = φ(∆Hm+TSm)/[2(Vs2/a)1/3]

(2.4)

where Sm is melting entropy and φ is a geometric factor ranging from 0.29 and 0.63 for cubic or hcp structure, depending on the interface orientation. It is known that the most powerful method available for theoretically estimating γsl is to make direct use of the so-called Gibbs-Thomson equation (known also as the Kelvin equation) [10], which describes the equilibrium between a small solid nucleus of spherical shape and the infinite amount of its liquid, γsl = Dn∆Hm(Tn)(1-Tn/Tm)/(4Vs)

(2.5)

where Dn and Tn are the critical diameter of the solid nucleus and the nucleation temperature. Eq. 2.5 is valid at T < Tm where the equilibrium size of the solid phase has a finite positive value. The physical meaning of Eq. 2.5 is that the growth of the solid phase will be thermodynamically more favorable than its dissolution for solid nucleus with size larger than Dn at Tn. Thus, the γsl value can be determined in terms of Eq. 2.5 as long as the Dn value at Tn is known. Through combining Eq. 2.5 and a model for the size-dependent melting temperature, γsl0 is also read as [14], γsl0 = 2hSvib∆Hm/(3RVs)

(2.6)

where R is the ideal gas constant and Svib is the vibrational part of the overall melting entropy Sm.

Solid State Phenomena Vol. 155

7

Recently, the CNT was re-examined through introducing size-dependence of solid-liquid interface energy where the Gibbs free energy change ∆G(D,T) of the metallic assembly consisting of the nucleus and the rest fluid can be expressed as following [29], 1 − 3h / D 7T S vib Tm − T  7T ∆G ( D, T ) ( D / h) 3 . = − 3 2Tm  Tm + 6T πh ∆H m /(3Vs )  D /(2h) Tm + 6T R

(2.7)

Obviously, Dn can be determined through letting ∂∆G(D,T)/∂D = 0 by Eq. 2.7. Letting ∆T = Tm-Tn denote the amount of undercooling and θ = ∆T/Tm the normalized amount of undercooling with Svib/R ≈ 0.9 for metals in terms of Table 1 of Ref. [29], it is found in terms of Eq. 2.7 that for any Tn and the corresponding Dn the correlation between Dn and θ can be determined as, Dnθ/h ≈ 2.1

(2.8)

For any pure, isotropic, spherical, condensed material, h in Eq. 2.8 can be estimated from the known Vs value and the packing density η of the solid crystals as, h = [6ηVs/(πa)]1/3

(2.9)

Taking Vs1/3 described by Eq. 2.9 and substituting it and Eq. 2.8 into Eq. 2.5, there is, γsl(T) = α∆Hm(T)/(Vs2/a)1/3

(2.10)

with α ≈ [27η/(32π)]1/3. In terms of Helmholtz function, ∆Hm(T) = gm(T)-Tdgm(T)/dT where gm(T) is the temperature-dependent Gibbs free energy difference between the bulk liquid and the corresponding bulk crystal, which has the following form for metals [30,31], gm(T) = 7∆HmT(Tm-T)/[Tm(Tm+6T)].

(2.11)

Thus, ∆Hm(T) function for metals can be read as, ∆Hm(T) = 49∆HmT2/(Tm+6T)2.

(2.12)

Note that gm(T) reaches its maximum when the temperature reduces to the ideal glass transition temperature or Kauzmann temperature Tk ≈ 0.27Tm for metals [30], which is determined by letting dgm(T)/dT = 0. Below Tk, the liquid must transform to glass where the specific heat difference between the crystal and glass approaches zero. Since ∆Hm(T) is induced by the specific heat difference, ∆Hm(T) at T < Tk will remain constant, namely ∆Hm(0 0 [83]. This is the case of free nanoparticles. When the interface atoms of different elements are intermixing, C or the bond strength may increase, such as alloying or compound formation. The alloying and chemical reaction may alternate the atomic valences, which may introduce repulsive stress among the ions. An alternative interpretation is that the element with lower Tm has lower bond strength within the layer than that on the interface, bond expansion of interface atoms is present, which should results in negative f.) Although some data exist for the solid-vacuum surface stress fsv, little is known about the solid-liquid interface stress fsl. It is assumed that fsl(Tm) ≈ fsv(Tm) because the interface stress induced by the elastic strain of a solid remains almost constant when the vacuum is substituted by a liquid on the interface of solid, since a liquid affects little the elastic strain of the solid. It is known that for the most elements fsv is one order larger than γsl0 according to theoretical and experimental results [84,85,86,87]. Although some MD works based on a hard-sphere model show that f has the same magnitude of γsl0 [88], however, the hard-sphere model itself may lead to this result where strain is absent, which co-exists with f [83]. Thus, Eq. 2.21 may be simplified as a first order approximation, γsl(D)/γsl0 ≈ 1-D0/D.

(2.22)

Fig. 2 compares the model prediction of Eq. 2.22 and the computer simulation result of γsl(D)/γsl0 [19]. It is evident that the model prediction is consistent with the computer simulation result: γsl(D) value reduces as D decreases. In Fig. 2, since an unknown fcc crystal of solid-liquid interface energy is cited for comparison [19], the simplified form of Eq. 2.22 is employed. At the same time, γsl(D) values of five organic nanocrystals have also been calculated in terms of Eq. 2.21 and shown in Table 3. The predictions are consistent with the experimental observation with the note that there exists a size distribution of nanocrystals [12]. As the size of crystals decreases, γsl(D) decreases. At D = 4 nm, the decrease in γsl(D) for different substances is different, which is mainly induced by different D0 or D0/D values. The drop of γsl(D) values in comparison with the corresponding bulk values reaches 20~40% where the critical diameter of a nucleus may be near 4 nm. Thus, the energetic resistance for the nucleation procession in liquid may be smaller than what the CNT has estimated [78].

16

Synthesis, Characterization and Properties of Nanostructures

1.0

γsl(D)/γsl0

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

D0/D Fig. 2. γsl(D)/γsl0 as a function of D0/D according to Eq. 2.22. The symbol denotes the computer simulation result of γsl(8h) and γsl(8h)/γsl0 = 0.58 [19]. D0 = 3h in terms of Eq. 2.20a. Table 3. The comparison of γsl(D)/γsl0 values between the model predictions in terms of Eq. 2.21 with D0 = 3h and the corresponding experimental results [12] with κ in unit of GPa-1 ×10-8.

Benzene Naphthalene Chlorobenzene Heptane Trans-decalin Ref

γsl(D)/γsl0 0.67 0.66 0.69 0.68 0.63 0.89 0.80 0.63 0.68 0.60 Eq. 2.21 [12]

κ 87 ≈ 87 67 134 ≈ 87 [64]

Since κ values of crystals are not found, κ values of the corresponding liquid are used, which leads to minor error.

The above agreements between Eq. 2.21 or Eq. 2.22 and experimental results denote that the energetic and structural differences between crystal and liquid decrease with size, which is proportional to surface/volume ratio with a 1/D relationship. The success of prediction for γsl(D) values in return confirms again that ∆Hm determines the sizes of γsl0 values as shown in Eq. 2.13. 2.3. &ucleus-liquid interface energy of elements Combining Eqs. 2.6, 2.7, 2.12 and 2.22, the integrated size- and temperature-dependent interface energy can be read as [89], γ sl ( D, T ) / γ sl 0 (Tm ) = (1 −

3h 7T )( )2 D Tm + 6T

(2.23a)

γ sl ( D, T ) / γ sl 0 (Tm ) = (1 −

3h T 2 )( ) D Tm

(2.23b)

Substituting Eq. 2.23 into Eq. 2.7, Dn can be determined by letting ∆G(D,T)/∂D = 0 [89],

Solid State Phenomena Vol. 155

17

Dnθ / h = 2( B + B 2 − 3Bθ / 2 )

(2.24a)

Dnθ / h = 2( B′ + B′2 − 3B′ / 2 )

(2.24b)

with θ = Tm − T , B = Tm

14Svib 1 − θ 2S 1 − θ , and B′ = vib . 3R 7 − 6θ 3R θ

Substituting Eq. 2.24 into Eq. 2.23 with experimentally determined θ values, γsl(rn,Tn) of the nucleus-liquid interface can be determined quantitatively. Fig. 3 shows a comparison between the predicted γsl(Dn,Tn) values in terms of Eq. 2.23 and experimentally determined γCNT values [9] with good agreement. Moreover, good agreement shown in Fig. 3 indicates that the γCNT value by Eq. 2.1 is not the bulk solid-liquid interfacial energy, but the nucleus-liquid interfacial energy. 300

Ni

γsl(Dn,Tn) (mJ/m2)

Co Ge

200

Cu Au

Pd

Pt

Mn

Fe

Ag Al

100

Sb

Sn Pb Hg

Ga Bi

0 0

100

200

300

γCNT (mJ/m2) Fig. 3. The comparison between γsl(Dn,Tn) in terms of Eq. 2.23 for a variety of elemental systems and γCNT calculated from the CNT [9].

Combining our previous work [90], the model predictions for both Tn and γsl(Dn,Tn) correspond well to the undercooling experimental results although both the expressions of γsl(D,T) and gm(T) functions differ from those in the CNT. The possible reason may be the mutual compensation between γsl(D,T) and gm(T) functions. However, there is about 40% difference in the value of Dn between the CNT (Dn ≈ 8h) and present model (Dn ≈ 11h), which may result from the neglect of the derivative of γsl(D,T) to D. Although we cannot confirm the above difference from experiments due to the experimental difficulties, we believe that a little larger Dn is more reasonable. Moreover, according to Eq. 2.24, the size of Dn is decided by Svib and θ due to the introduction of γsl(D,T) function, and it increases with an increase in Svib or a decrease in θ. Since Svib and θ values for elements with different bond natures are similar, Dn is in fact independent on the elemental types. Thus, Dn ≈ 11h for all concerned elements here is reasonable. It is obvious from Eq. 2.1 that the size of h also affects γCNT value. To accurately estimate the influence of h, h of all elements should be unified to h′ where the elements with different structures have the same coordination number (C) of 12. According to the Goldschmidt premise for lattice contraction [91], (1-h/h′) will be 3%, 4%, and 12% if CN of the atom reduces from 12 to 8, 6, and 4, respectively. The additional correlation between h and h′ for the elements, which CN is not 4, 6 or 8,

18

Synthesis, Characterization and Properties of Nanostructures

can be found in Ref. [34]. Replacing h with h′, Eq. 2.1 can be simplified as, γCNT = τ2∆Hm/Vs

(2.25-a)

with τ2 ≈ 0.11 ± 10% nm except semi-metals Pb, Sn and Ga as shown in Fig. 4. 300

Ni

γsl(Dn,Tn) (mJ/m2)

Pt Pd Ge

Mn

200

Co

Cu Fe Au Ag Al Sb

100

Sn Ga Bi Hg Pb 0 0

1

9

2 3

3

∆Hm/Vs (10 J/m ) Fig. 4. γsl(Dn,Tn) as a function of ∆Hm/Vs for a variety of elemental systems in terms of Eq. 2.25-a (the solid line) where the symbols ■ and ∆ denote the γsl(Dn,Tn) values of metals and semiconductors, respectively.

In a similar way, Eq. 2.6 can also be simplified as, γsl0(Tm) = τ3∆Hm/Vs

(2.25-b)

with υ3 = 2Svibh′/(3R) ≈ 0.18 ± 15% nm except Fe, Al and Ga. Eq. 2.25 can represent γCNT or γsl0(Tm) with a unique τ2 or τ3 rather than two different τ values by Eq. 2.1 due to the different bond natures of the elements. Moreover, the disappearance of h or h′ in Eq. 2.25 implies that the unique parameter in deciding γsl is ∆Hm. Substituting Eq. 2.24 into Eq. 2.23 and plotting, the curve can be linearly regressed as a function of θ, which leads to, γsl(Dn′,Tn′) ≈ (1.78-3.83θ)×10-10×∆Hm/Vs.

(2.26)

where Tn′ is any nucleation temperature and Dn′ is the corresponding diameter of a nucleus. The standard deviations for 1.78 and 3.83 are only 0.01 and 0.16, respectively. Eq. 2.26 indicates that: Eqs. 2.25-a and 2.25-b denote two extreme cases where θ = θn (the maximum degree of undercooling, which is nearly a constant of 0.18±0.02 for the most elements [92]) and θ = 0, respectively. In fact, at any θ value, γsl(Dn′,Tn′) is always proportional to ∆Hm/Vs. As an example, Fig. 5 shows such a relationship at θ = 0.1 where the slope linearly regressed is equal to 1.40 as indicated by Eq. 2.26.

Solid State Phenomena Vol. 155

19

400

γsl(Dn′,Tn′) (mJ/m2)

Ge Pt Mn Pd Cu Fe

300

Au

200

Co

Ni

Ag Sb

Hg

Sn

θ = 0.1

Al

Ga

100

Bi

Pb

0 0

1

9

2 3

3

∆Hm/Vs (10 J/m )

Fig. 5. γsl(Dn′,Tn′) as a function of ∆Hm/Vs with θ = 0.1 for a variety of elemental systems in terms of Eq. 2.26 (the solid line) where the symbols ■ and ∆ denote the γsl(D,T) values of fcc and non-fcc elements, respectively.

This linear relationship of Eq. 2.26 between γsl and ∆Hm can be considered as that ∆Hm is related to bond energy of crystalline atoms while γsl denotes the bond energy difference between surface atoms and interior ones of a crystal. Since γsl is proportional to the solid-vapor surface energy γsv while γsv is also found to be directly proportional to Eb [89], it is thus found that γsl ∝ Eb. The size-dependent Eb has been determined as [93],

 2S  1 1    E ( D) / Eb = 1 − exp − b   2 D / h − 1  3R 2 D h − 1 

(2.27)

1.2

1.2

0.9

0.9

0.6

0.6

0.3

0.3

0.0 0

20

40

60

D/(2h)

80

0.0 100

E(D)/Eb

γsl(D)/γsl0

where Sb = Eb/Tb is the bulk solid-vapor transition entropy of crystals with Tb being the bulk solid-vapor tansition temperature. Fig. 6 shows a comparison between γsl(D,T)/γsl0(T) function in terms of Eq. 2.22 and E(D)/Eb function by Eq. 2.27 (Sb ≈ 12R is assumed except for Sb and Bi [32]) with good agreement.

20

Synthesis, Characterization and Properties of Nanostructures

Fig. 6. Comparison between γsl(D)/γsl0 shown as solid line in terms of Eq. 2.22 and E(D)/Eb shown as dot in terms of Eq. 2.27.

Another linear relationship between γsl(Dn,Tn) or γCNT and Tm with large scatter among different groups of elements [9] has been rejected as the basis for an empirical rule in favor of the correlation with ∆Hm [38]. However, direct calculations for γCNT values of transition fcc metals with hard-sphere systems have shown a strong correlation with Tm [38,94], which can also be extended to γsl0(Tm) for all fcc metals since ∆Hm = TmSm and υ3Sm = 1.59 ± 7% nm J/mol K in terms of 2.25-b. For other elements, υ3Sm values show large scatter in a range of 1.04 to 5.44 nm J/mol K due to the scatter of Sm values. Thus, it is not Tm but ∆Hm that generally characterizes γsl0(Tm) value better. 4. Solid-solid interface energy 3.1. The bulk solid-solid interface energy γss0 For coherent or semi-coherent solid-solid interface, its interface energy can be determined according to some classic dislocation models [95]. While for semi-coherent interface, atomic diameter misfit on the interface must be smaller than 0.15-0.25, γss0 is very strongly dependent on the mis-orientation of the two crystal halves and remains challenge. However, for high angle grain boundary, γss0 is almost a constant. Here, the considered grain boundary energy is the high angle one. Table 4. Comparison of γss0(Tm) between γss0 by Eq. 3.1 and available theoretical or experimental results γ′ss0 for metals [96,97,98,99,100,101,102] and organic crystals [43,44] with γ in unit of mJ/m2. System Cu Ag Au Al Ni Co Nb Ta Sn Pb Bi Pivalic acid Succinonitrile

γss0 584 398 430 302 844 706 804 944 158 96 176 4.8 14.6

γ′ss0 601 392,375 400 300~380 866,757 650 760 900 160,164 111.5±15.6 140.5±14.1 5.2±0.4 15.0±2.0

γ′sl0

2.7±0.2 7.9±0.8

It is well known that a liquid may be regarded as a solid with such a high concentration of dislocation cores that these are in contact everywhere [103]. Based on this consideration, solid-solid interface energy γss0 at Tm is considered to be twice of the solid-liquid interface energy γsl approximately [104], γss0(Tm) ≈ 2γsl0(Tm). Combining with Eq. 2.6, γss0(Tm) ≈ 4hSvibHm/(3VsR).

(3.1)

As shown in Table 4, the γss0(Tm) values based on Eq. 3.1 for eleven elemental crystals (Cu, Ag, Au, Al, Ni, Co, Nb, Ta, Sn, Pb and Bi) and two organic crystals (Pivalic acid and Succinonitrile) correspond to the available theoretical values γ′ss0 [43,44,96,97,98,99,100,101,102]. In addition, the

Solid State Phenomena Vol. 155

21

listed data of γ′ss0 and γ′sl0 for Pivalic acid and Succinonitrile determined by the equilibrated grain boundary groove shapes also confirm the validity of γss0(Tm) ≈ 2γsl0(Tm) with the absolute deviation smaller than 5.5%. Thus, Eq. 3.1 can be used to quantitatively calculate the γss0(Tm) values, at least for metals and organic crystals. 3.2. The size dependence of solid-solid interface energy γss(D)

1.0

γss(D)/γss0

0.9

0.8

0.7

0.6 0.0

Cu

0.2

0.4

D0/D 0.6

0.8

1.0

Fig. 7. γss(D)/γss0 as a function of D0/D according to Eq. 3.4. The solid line and the segment line are obtained by use of negative and positive f, respectively. The symbol denotes the computer simulation results of Cu [105]. D0 = 3h in terms of Eq. 2.20a.

In the deduction of γss(D), the way is similar to the deduction of γsl(D) while Eq. 2.15 and Eq. 2.16 must be modified due to different interface conditions [78]. We assume that f as a first order approximation keeps constant for both solid-liquid and solid-solid interface at Tm, which leads to the same strain on both sides of the interface when the grains are isotropic. However, elastic modulus of grain boundaries should be larger than that of the solid-liquid interface with less strain under the same stress. This is introduced by the fact that A = 3V/(D) for solid-solid interface because two solid-liquid interfaces of particles combine to form one grain boundary with P = 2fA/(3V) = 2f/D

(3.2)

ε = -2κf/(3D).

(3.3)

Both equations indicate that the strain on the grain boundary is only a half of that on the solid-liquid interface and thus ∆A/A = 2ε = -4κf/(3D). Now ∆γss = γss(D)-γss0 = (∆A/A)(f-γss)= -4κf/(3D)(f-γss), or γss(D)/γss0 = [1-D0/(4D)]/[1-γss0D0/(4fD)].

(3.4)

Similar to the simplification of γsl(D), Eq. 3.4 can also be simplified as γss(D)/γss0 = 1-D0/(4D).

(3.5)

Fig. 7 shows an agreement between the predictions of Eq. 3.4 and the computer simulation

22

Synthesis, Characterization and Properties of Nanostructures

results for Cu [105]. It is interesting that the use of a negative interface stress for γss(D) in Eq. 3.4 leads to a full agreement with the computer simulation results, which implies that f < 0. If we compare Eq. 2.22 and Eq. 3.5, it can be found that γsl(D)/γsl0 < γss(D)/γss0, which implies that the stiffer surrounding of particles brings out less decrease of the interface energy as D decreases. For the grain boundaries, even if when D→D0, γss(D)/γss0 ≈ 75% while γsl(D)/γsl0 = 0. Since when D ≈ 2D0, the grains are no more stable and will transform to amorphous solids in terms of the computer simulation results [105], the smallest value of γss(D0)/γss0 could be about 85%. 4. Solid-vapor interface energy 4.1. The bulk solid-vapor interface energy γsv0 The solid-vapor interface energy γsv0, usually defined as the difference of free energy between surface atoms and interior ones, is one of the basic quantities to understand the surface structure and phenomena [106]. Despite of its importance, γsv0 value is difficult to determine experimentally. The most of these experiments are performed at high temperatures where liquid-vapor interface energy of liquid is measured, which are extrapolated to zero Kelvin. This kind of experiments contains uncertainties of unknown magnitude and corresponds to only γsv0 value of an isotropic crystal [107,108,109]. Moreover, many published data determined by the contact angle of metal droplets or from peel tests disagree with each other, which can be induced by the presence of impurities or by mechanical contributions, such as dislocation slip or the transfer of material across the boundary. In addition, there are hardly the experimental data on the more open surfaces except for the classic measurements on Au, Pb and In to our knowledge [110,111]. Therefore, a theoretical determination of γsv0 values especially for open surface is of vital importance. During the last years there have been several attempts to calculate γsv0 values of metals using either ab initio techniques [112,113,114] with tight-binding (TB) parameterizations [115] or semi-empirical methods [116]. γsv0 values, work functions and relaxation for the whole series of bcc and fcc 4d transition metals have firstly been studied [112] using the full-potential (FP) linear muffin-tin orbital (LMTO) method in conjunction with the local-spin density approximation to the exchange-correlation potential [117,118,119]. In the same spirit, γsv0 values and the work functions of the most elemental metals including the light actinides have been carried out by the Green′s function with LMTO method [113,114,120,121]. Recently, the full-charge density (FCD) Green′s function LMTO technique in the atomic-sphere approximation (ASA) with the generalized gradient approximation (GGA) has been utilized to construct a large database that contains γsv0 values of low-index surfaces of 60 elements in the periodic table [119,122,123,124,125]. The results denote a mean deviation of 10% for the 4d transition metals from FP methods [126]. This database in conjunction with the pair-potential model [127] has been further extended to estimate the formation energy of mono-atomic steps on low-index surfaces for an ensemble of the fcc and bcc metals [128]. On the other side, the traditional broken-bond model is again suggested to estimate γsv0 values of the transition metals and the noble metals with different facets [110,129,130]. The simplest approach to get a rough estimation of γsv0 values at T = 0 K is to determine the broken bond number Zhkl for creating a surface area by cutting a crystal along certain crystallographic plane with a Miller index (h k l). Zhkl = ZB-ZS where ZS and ZB are coordination numbers (Cs) of surface atoms and interior ones, respectively. Multiplying this number with the cohesion energy per bond Eb/ZB for the non-spin-polarized atom at 0 K, γsv is determined by [131], γsv0 = (1-k1)Eb/(aAS) where k1 = ZS/ZB and AS denotes the area of the two-dimensional unit cell of solid.

(4.1)

Solid State Phenomena Vol. 155

23

In Eq. 4.1, Eb is independent on crystalline structures as a first order approximation since energy differences between solid structures are several orders of magnitude smaller than Eb for any structure when the bond type remains unaltered. Although the broken-bond rule seems to contradict the basic knowledge about the electronic structure since Eb, in general, does not scale linearly with ZS, the above estimation provides the order of magnitude of γsv0 and shows a possible relationship between γsv0 and atomic binding strength. Despite the absence of verification from experiments, such a rule has been used to give a reasonable description of the γsv0 value of Al [111]. Since the bond strength becomes stronger for an atom with a smaller C, this C-bond-strength relation can be quantified using tight-binding approximation. In the second-moment tight-binding approximation, the width of the local density of states on an atom scales with ZS, leads to an energy gain to be proportional to the square root of ZS due to the lowing of the occupied states [129]. Neglecting the repulsive terms, the energy per nearest neighbor is then proportional to the square root of ZS. By assuming that the total crystalline energy is a sum of contributions of all bonds of an atom, the solid-vapor interface energy is suggested to follows the relation: γsv0 = (1-k11/2)Eb/(aAS).

(4.2)

However, Eq. 4.1 does not consider the variation of bond strength with C, and Eq. 4.2 is also not complete because only attractive forces are taken into account [129]. Namely, the former neglects while the latter overestimates the effect of relaxation on γsv0 [132], which results in that Eq. 4.1 can be an acceptable concept for strongly covalent crystals while Eq. 4.2 is especially suitable for noble metals. Although a direct utilization of Eq. 4.1 or Eq. 4.2 is reasonable, one of them cannot alone give satisfied predictions for γsv0 values in comparison with the experimental and theoretical results [112]. To obtaining a more general formula, we arbitrarily assume that both of Eqs. 4.1 and 4.2 could make up the deficiency each other with the same weight to both formulae. Thus, γsv0 values may be determined by an averaged effect of them without elaborate estimation on the relaxation energy [132], γsv0 = (2-k1-k11/2)Eb/(2aAS).

(4.3)

Eq. 4.3 implies that γsv0 values still depend on the bond-broken rule although they are scaled by both of ZS and the square root of ZS. In Eq. 4.3, ZS can be determined according to the crystalline structure by determining Zhkl in terms of a geometric consideration [133,134]. For fcc or hcp structure, ZB = 12; For a bcc lattice, although ZB = 8 is taken according to the nearest-neighbor definition by some authors (probably the majority), others prefer to regard ZB = 14 since the difference between the nearest neighbor bond length and the next-nearest neighbor bond length is small [135]. Here, the latter is accepted. By assuming that the total energy of a surface atom is the sum of contributions from both of the nearest neighbor and the next-nearest neighbor atoms, Eq. 4.3 should be rewritten for bcc metals after normalization [132], γsv0 = [(2-k1-k11/2)+ϕ(2-k1′-k1′1/2)]Eb/[(2+2ϕ)aAS]

(4.4)

where the superscript comma denotes the next-nearest C on a surface and ϕ shows the total bond strength ratio between the next-nearest neighbor and the nearest neighbor [136].

24

Synthesis, Characterization and Properties of Nanostructures

4.1.1. Solid-vapor interface energy of elementary crystals To roughly estimate the size of ϕ, the Lennard-Jones (LJ) potential is utilized. The potential is expressed as u(r) = -4ϖ[(ξ/r)6-(ξ/r)12] with ε being the bond energy and ξ insuring du(r)/dr(r=h) = 0, i.e. ξ = 2-1/6h where r is the atomic distance. For fcc crystal, h = 21/2a/2 and h′ = a, respectively. Let r = a, ϖ′ ≈ ϖ/4, and ϕ ≈ [(1/4)×6]/12 = 1/8. Thus, the effect of the next-nearest C can be neglected as a first order approximation, which is also applicable to hcp crystals. Namely, Eq. 4.4 can be simplified as Eq. 4.3 for fcc and hcp crystals; For bcc crystal, h = 31/2a/2 and h′ = a, respectively. Let r = a, ϖ′ ≈ 2ϖ/3. Thus, ϕ = [(2/3)×6]/8 = 1/2. Adding this value into Eq. 4.4, γsv0 = [3-k1-k11/2-k1′/2-(k1′/4)1/2]Eb/(3aAS).

(4.5-a)

Note that the bonding of the LJ potential, which is utilized to justify ϕ value in Eq. 4.4, is different from the metallic bond in its nature. For instance in a LJ bonded system, the surface relaxation is outwards whilst in the transition metals it is inwards. However, this difference leads to only a second order error in our case and has been neglected. The effect of next-nearest C also occurs for sc and dc structure crystals because there are twice and thrice as many the second neighbors as the first neighbors, respectively. Similar to the above analysis, for sc crystals: ϖ′ ≈ ϖ/4 and ϕ = [(1/4)×12]/6 = 1/2, which is the same for bcc and thus Eq. 4.5-a can also hold for sc crystals. For dc structure crystals, ϖ′ ≈ ϖ/10 and ϕ ≈ [(1/10)×12]/4 = 3/10. With this ϕ value, Eq. 4.4 is rewritten as, γsv0 = [26-10k1-10k11/2-3k1′-(9k1′)1/2]}Eb/(26aAS).

(4.5-b)

Zhkl can be determined by some known geometrical rules. For any surface of a fcc structure with h ≥ k ≥ l [133,134], Zhkl = 2h+k

for h, k, l being odd

Zhkl = 4h+2k for the rest

(4.6-a) (4.6-b)

In a similar way, Zhkl for any surface of a bcc structure is determined with the consideration of the next-nearest C [133], Zhkl = 2h+(h+k+l) Zhkl = 4h+2(h+k+l)

for h+k+l being even for h+k+l being odd and h-k-l ≥ 0

Zhkl = 2(h+k+l)+2(h+k+l) for h+k+l being odd and –h+k+l > 0

(4.7-a) (4.7-b) (4.7-c)

where the 2nd item of the right-hand side of Eq. 4.7 denotes the broken bond number of the next-nearest neighbors. For sc crystals, Zhkil values of the nearest and the next-nearest atoms are 1 and 4 for (100) surface as well as 2 and 5 for (110) surface, respectively. For dc structure crystals, Zhkl values of the nearest and the next-nearest atoms are 1 and 6 for (110) surface. For several surfaces of a hcp structure, Zhkil is obtained by [134], Zhkil = 4(h+k)+3l Zhkil = 4(h+k)+(8h+4k)/3

for (0001)

(4.8-a) −

for (10 1 0)

(4.8-b)

Solid State Phenomena Vol. 155

25

where the 1st item of the right-hand side of Eq. 4.8 denotes the average number of basal broken bonds while the 2nd item is that of non-basal broken bonds. Table 5. AS, ZS and ZB values for different surfaces and structures where a is lattice constant while ZS and ZB are determined by Eqs. 4.6 to 4.8. Structure Fcc

AS

Surface (111) (100) (110)

2

3 a /4

a2/2

ZS 9 8 6

ZB 12 12 12

ZS = 6, ZS′ = 4 ZS = 4, ZS′ = 4 ZS = 2, ZS′ = 0

ZB = 8, ZB′ = 6 ZB = 8, ZB′ = 6 ZB = 8, ZB′ = 6

9 16/3

12 12

ZS = 3, ZS′ = 6

ZB = 4, ZB′ = 12

ZS = 5, ZS′ = 8 ZS = 4, ZS′ = 7

ZB = 6, ZB′ = 12 ZB = 6, ZB′ = 12

2 2 a /2

Bcc

(110) (100) (111)

2 2 a /2

a2 a2 2 3 a /2 2 8 /3 a 3

Hcp

(0001) −

(10 1 0) Diamond

(110)

SC

(100) (110)

2 2 a /4

a2 2 2a

Table 5 shows some necessary parameters in the Eqs. 4.3 and 4.5, and Tables 6-8 give the predicted γsv0 values for fcc, bcc, hcp, dc and sc structure crystals in terms of these two equations where the related parameters are taken from references [33,137]. As a comparison, the first principle calculations γ′sv0 and two sets of experimental results γ′′sv0 are also shown [107-109]. Note that the experimental results are not orientation-specific but are averaged values of isotropic crystals. Thus, they should be close to those of the most close-packed surface. Table 6. Comparison of surface energies of fcc metals among the predicted values γsv0 of Eq. 4.3, FCD calculations γ′sv0 [109], and experimental results γ′′sv0 [107,108]. Eb in kJ/g-atom [132], a in nm [33], and γ in J/m2, which are the same as those in Tables 7 and 8.

Eb

a

Cu

336

0.366

Ag

284

0.418

Au

368

0.420

Ni

428

0.358

(h k l) (111) (100) (110)

γsv0 1.83 2.17 2.35

γ′sv0 1.95 2.17 2.24

γ′′sv0 1.79, 1.83

(111) (100) (110)

1.20 1.40 1.51

1.17 1.20 1.24

1.25, 1.25

(111) (100) (110) (111) (100) (110)

1.52 1.80 1.94 2.44 2.88 3.11

1.28 1.63 1.70 2.01 2.43 2.37

1.51, 1.50

2.38, 2.45

26

Synthesis, Characterization and Properties of Nanostructures

Pd

Pt

Rh

376

564

554

1.85 2.15 2.35

1.92 2.33 2.23

2.00, 2.05

0.385

(111) (100) (110)

2.54 2.98 3.24

2.30 2.73 2.82

2.49, 2.48

0.402

(111) (100) (110)

2.70 3.15 3.41

2.47 2.80 2.90

2.66, 2.70

0.387

(111) (100) (110) (111) (100) (110)

3.19 3.74 4.06

2.97 3.72 3.61

3.05, 3.00

(111) (100) (110) (111) (100) (110) (111) (100) (110) (111) (100) (110) (111)

0.55 0.64 0.70 1.45 1.68 1.84 0.43 0.50 0.55 0.33 0.39 0.43 1.65

0.32 0.38 0.45 1.20 1.35 1.27 0.57 0.54 0.58 0.43 0.41 0.43 3.10

0.59, 0.60

(111) (100) (110) (111) (100) (110)

0.90 1.03 1.14 1.61 1.85 2.36

0.87 0.73 0.68 1.48 1.47 1.45

Ir

670

0.391

Pb

196

0.511

Al

327

0.405

Ca

178

0.562

Sr

166

0.617

Mn*

282

0.353

Ac

410

0.579

Th

598

0.519

1.14, 1.16

0.50, 0.49

0.42, 0.41

1.54, 1.60

1.50

The symbol *, which has the same meaning in Tables. 7 and 8, denotes that when the low temperature equilibrium crystal structure has a lower symmetry than a close packing phase at high temperature or under a high pressure, the latter is utilized [109]. Table 7. Comparison of surface energies of crystals in bcc, sc and diamond structures among the predicted values γsv0 based on Eq. 4.5, FCD calculations γ′sv0 [109], and experimental results γ′′sv0 [107,108].

Li

Na

E

a

158

0.399

107

0.420

(h k l) (110) (100) (111)

γsv0 0.50 0.58 0.72

γ′sv0 0.56 0.52 0.59

γ′′sv0 0.52, 0.53

(110) (100) (111)

0.29 0.34 0.41

0.25 0.26 0.29

0.26, 0.26

Solid State Phenomena Vol. 155

K

Rb

90.1

82.2

0.530

0.571

27

(110) (100) (111) (110) (100) (111)

0.16 0.18 0.23 0.12 0.15 0.18

0.14 0.14 0.15 0.10 0.11 0.12

0.13, 0.15

0.10 0.12 0.14

0.08 0.09 0.09

0.10, 0.10

0.38, 0.37

0.12, 0.11

Cs

77.6

0.626

(110) (100) (111)

Ba

183

0.503

(110) (100) (111)

0.36 0.41 0.51

0.38 0.35 0.40

Ra

160

0.537

(110) (100) (111)

0.27 0.32 0.40

0.30 0.29 0.32

179

0.458

(110) (100) (111)

0.43 0.50 0.61

0.49 0.46 0.52

0.45, 0.45

Eu

(110) (100) (111) (110) (100) (111) (110) (100) (111) (110) (100) (111) (110) (100) (111) (110) (100) (111) (110) (100) (111) (100) (110)

2.74 3.26 4.04 2.39 2.83 3.50 2.52 2.92 3.62 2.58 2.99 3.72 3.20 3.81 4.62 3.40 4.05 5.01 3.36 3.90 4.84 0.66 0.77

3.26 3.03 3.54 3.51 3.98 4.12 2.43 2.22 2.73 2.69 2.86 3.05 3.45 3.84 3.74 3.08 3.10 3.46 4.01 4.64 4.45 0.61 0.66

2.62, 2.56

0.49, 0.49

V

512

0.302

*

395

0.285

Fe

413

0.286

Cr

Nb

730

0.376

Mo

658

0.317

Ta

782

0.335

W

859

0.358

Sb (sc*)

265

0.336

Bi (sc*)

210

0.326

(100) (110)

0.55 0.64

0.54 0.54

Po (sc*)

144

0.334

(100) (110)

0.38 0.44

0.44 0.37

2.35, 2.30

2.42, 2.48

2.66, 2.70

2.91, 3.00

2.90, 3.15

3.27, 3.68

0.60, 0.54

28

Synthesis, Characterization and Properties of Nanostructures

Si (A4)

446

0.771

(110)

1.06

1.14

Ge (A4)

372

0.810

(110)

0.80

0.88

For both noble and transition metals, the predictions γsv0 agree nicely with the experimental results and FCD calculations as shown in Tables 6-8 although γsv0 for transition metals have slightly larger deviations than those for the noble metals due to the fact that their d-bands are not fully filled and they present peaks at the Fermi level, which can slightly change from one surface orientation to the other and consequently the energy needed to break a bond changes also a little. As shown in these tables, γsv0 values of transition metals increase along an isoelectronic row where a heavier element has a larger γsv0 value. This is because the d-level of a heavier element is higher in energy and the corresponding d-wave function with a stronger bonding is more extended. This is also true for elements in the same row in the periodic table where a heavier element has more d-electrons [110]. An exception is in VA series where γsv0 value of Nb is smaller than that of V possibly due to the rehybridization of Nb where Nb, whose d shell is less than the half-full, rehybridizes in the opposite direction, i.e., depletes their dz2 orbitals based on a charge density difference analysis [138,139,140]. γsv0 values for sp metals except for Be are smaller than those for d-metals due to their bond nature of s- and p-electrons, which are more mobile than the localized d-electrons and therefore less energy is needed to break these bonds. For fcc metals except Ca, Sr and Al, the mean-square root error σ between the predicted and the experimental results for the most close-packed (111) is about 7.5%. For aluminum, the degree of covalent Al-Al bonding increases or the nature of the bonding changes with reduced C [141], which leads to model prediction deviating from the experimental results since our formula neglects the variation of bonding type. However, the reason of deviations for Ca and Sr is not clear. For hcp metals, σ ≈ 10% except Mg, Zn, Cd and Tl. For Cd and Tl, both of the predictions and FCD calculations deviate evidently from the experimental results. In the case of Zn and Cd, c/a ratios (1.86 and 1.89) are larger than the ideal value of 1.633. Thus, the nearest C values will differ from the ideal condition, which should contribute the deviation of our prediction. For sc metals, σ = 2.6% where Sb and Bi with the rhombohedral structure are assumed to have slightly a distorted sc structure [109]. For bcc metals, σ = 10%. The smallest value of σ in all considered structures appears for diamond structure crystals with σ = 1.4%, which implies that the pure coherent bond does not change after a C deduction. Note that the temperature dependence of solid-vapor interface energy is ignored here although the experimental results listed in Tables 6-8 are calculated at 0 K while the most lattice constants cited are measured at room temperature. This temperature effect decreases the prediction accuracy and can be partly responsible for the disagreement with other experimental and theoretical results. Table 8. Comparison of surface energies of hcp metals among the predicted values γsv0 of Eq. 4.3, the FCD calculations γ′sv0 [109], and the experimental results γ′′sv0 [107,108].

Be

E 320

a 0.222

(h k i l) (0001) −

(10 1 0)

γsv0 2.40 2.88

γ′sv0 1.83 2.13

γ′′sv0 1.63, 2.70

Solid State Phenomena Vol. 155

Mg

145

0.320

(0001) −

(10 1 0) Zn

Cd

Tl

130

112

182

0.268 (c/a=1.86) 0.306 (c/a=1.89) 0.371

(0001) −

(10 1 0) (0001) −

(10 1 0) (0001) −

(10 1 0) Sc

376

0.330

(0001) −

(10 1 0) Ti

468

0.295

(0001) −

(10 1 0) Co

424

0.253

(0001) −

(10 1 0) Y

422

0.355

(0001) −

(10 1 0) Zr

603

0.325

(0001) −

(10 1 0) Tc

661

0.274

(0001) −

(10 1 0) Ru

650

0.272

(0001) −

(10 1 0) La*

431

0.387

(0001) −

(10 1 0) Lu

428

0.351

(0001) −

(10 1 0) Hf

621

0.320

(0001) −

(10 1 0) Re

775

0.276

(0001) −

(10 1 0) Os

788

0.275

(0001) −

(10 1 0)

29

0.53 0.65

0.79 0.78

0.79, 0.76

0.66 0.72

0.99

0.99, 0.99

0.44 0.47

0.59

0.76, 0.74

0.49 0.60

0.30 0.35

0.60, 0.58

1.25 1.53

1.83 1.53

1.28

1.96 2.39

2.63 2.52

1.99, 2.10

2.42 2.95

2.78 3.04

2.52, 2.55

1.22 1.49

1.51 1.24

1.13

2.08 2.54

2.26 2.11

1.91, 2.00

3.22 3.93

3.69 3.90

3.15

3.20 3.90

3.93 4.24

3.04, 3.05

1.05 1.28

1.12 0.92

1.02

1.27 1.55

1.60 1.42

1.23

2.22 2.71

2.47 2.31

2.19, 2.15

3.72 4.54

4.21 4.63

3.63, 3.60

3.80 4.64

4.57 5.02

3.44, 3.45

In the FCD calculations, there are often exceptions that the most close-packed surface does not

30

Synthesis, Characterization and Properties of Nanostructures

have the lowest γsv0 values or there exists a weak orientation-dependence [109]. These physically unacceptable results are fully avoided here. Moreover, the anisotropy of γsv0 is perfectly considered. The ratios 1.16 and 1.27 of (100) and (110) to (111) facets for fcc metals agree with theoretical values of 1.15 and 1.22 [110,130] while that of (100) to (110) facets for monovalent sp metals metals is 1.16, which is comparable with 1.14 based on the jellium model [142]. If the experimental results are taken as reference, 60% of γsv0 values of the most close-packed surfaces of 52 elements shown in Tables 6-8 have better agreements with the experimental ones than those of the FCD calculations do while 20% of the FCD calculations are in reverse. Note that local density approximation (LDA) is implied here while GGA is used in FCD. Recently, it has been shown that both methods need to be corrected due to the neglect of surface electron self-interactions where GGA is worse than LDA [138-140]. This is surprising because GGA is generally considered to be the superior method for energetic calculations [109]. For the transition metals and noble metals, the formula works better than for others as the greatest contribution to bonding is from the s-d interaction and the orbitals of the latter localize, which is more like a pair interaction. According to Tables 6-8, the predicted γsv0 values of divalent sp metals have bad correspondence with the experimental results since the many body (e.g. trimer) terms are here critical to understanding the cohesive energy. Thus, the used pair potentials physically may be not fully correct. It is possible that the background of the formula, i.e. the broken-bond model, is not universally applicable although the lattice constants used in Eqs. 4.3 and 4.5 have measuring error of about 2% [107,123]. According to the first principles calculations, the effect of relaxation on the calculated γsv0 value of a particular crystalline facet may vary from 2 to 5% depending on the roughness [139,143]. The semi-empirical results indicate further that the surface relaxation typically affects the anisotropy by less than 2% [116]. Surface relaxations for vicinal surfaces have been studied mainly using semi-empirical methods due to the complexity arisen by the simultaneous relaxation of a large number of layers [110]. Here, the relaxation effect is simply considered by adding Eq. 4.2 into Eq. 4.1. According to Tables 6-8, this measure leads to satisfactory results. As a simple model without free parameter, the above formula supplies a new insight and another way for a general estimation of solid-vapor interface energy of elements. This success is difficult to achieve for present first principles calculation. Moreover, this model supplies a basis of comparison and supplement for further theoretical and experimental considerations on γsv0 of elements. Recently, Lodziana et al. have proposed that solid-vapor interface energy for θ-alumina is negative [144]. Their use of the term ″negative solid-vapor interface energy″ can and has caused confusion in the scientific community [145]. Mathur et al. nicely summarized and clarified definitions of solid-vapor interface energies for single- and multi-component systems [145]: Solid-vapor interface energy for single-component systems is always positive, whereas for multi-component systems it can become negative due to chemical effects, which has been confirmed experimentally [146]. 4.1.2. Solid-vapor interface energy of several ceramics with &aCl structure The alkaline metal oxides (AMO) as one kind of three supporting industries in material domain hold the balance in daily life and industrial manufacture [147]. The transition metal carbides (TMC) and the transition metal nitrides (TMN) have been widely applied as the layers of cutting tools, electrically conducting diffusion barriers in electronic devices, in coatings for solar applications and for corrosion protection [148,149]. All of these applications due to their unique properties (e.g. high hardness and high melting temperatures) are closely related with its surface states. The solid-vapor interface energy as important concept influences the growth rate, catalytic behavior, adsorption, surface segregation and the formation of grain boundaries. However, γ values of these metallic compounds are difficult experimentally to measure due to their ionic character and the hardness, which lead to less reliable experimental data [150] although some computer simulation results can

Solid State Phenomena Vol. 155

31

be found [151,152,153,154]. Although Eqs. 4.3 and 4.5 are deduced for elemental crystals, they have been developed for insulating or metallic compounds in literatures [153,155]. Since the main source of the bonding in these ceramics is the interaction between metal and non-metal atoms while all these ceramics can be treated in the NaCl structure [153], ZB = 6 for the NaCl structure and ZS are 5, 4, and 3 respectively for (100), (110) and (111) surfaces [156]. Moreover, because γsv0 values of compounds are usually reported in unit of ev/atom, γsv0 value in this section is also in unit of ev/atom. Table 9. Comparison of γ100 values in eV/atom of TMC among γ1 of Eq. 4.3, γ2 of Eq. 4.2, γ3 by using FP-LMTO calculations based on LDA, simulation results γ4 using GGA-PW91 and γ5 by LDA [156], theoretical results γ6 [153], and other available results γ7 using pseudopotential plane-wave-based DFT [151,152]. Eb [156] is in eV/atom.

Eb

γ1

γ2

γ3

γ4

γ5

γ6

ScC TiC VC CrC MnC FeC

6.37 7.16 6.94 5.80 5.14 5.67

0.79 0.89 0.86 0.72 0.64 0.71

0.57 0.65 0.62 0.50 0.43 0.47

0.67 0.83 0.77 0.71 0.70 0.71

0.47 0.54 0.37 0.34 0.30 0.34

0.56 0.69 0.55 0.42 0.47 0.55

0.68, 0.86 0.84, 0.98 0.88, 0.92 0.88, 0.84 0.87, 0.85 0.86

CoC NiC YC ZrC NbC

5.69 5.65 6.39 7.93 8.26

0.71 0.70 0.80 0.99 1.03

0.48 0.49 0.43 0.64 0.72

0.72 0.45 0.65 0.86 0.87

0.20 0.21 0.33 0.53 0.49

0.32 0.45 0.44 0.66 0.69

0.83, 0.85 0.75, 0.77 0.62, 0.78 0.82, 0.94 0.90, 0.87

MoC TcC RuC RhC PdC

7.22 6.88 6.73 6.23 5.36

0.90 0.86 0.84 0.77 0.67

0.66 0.66 0.68 0.62 0.52

0.77 0.69 0.69 0.68 0.47

0.38 0.37 0.31 0.38 0.32

0.64 0.61 0.45 0.41 0.48

0.90, 0.77 0.88, 0.77 0.86, 0.77 0.82, 0.75 0.73, 0.65

LaC HfC TaC WC ReC

5.74 8.11 8.56 8.25 7.47

0.71 1.01 1.07 1.03 0.93

0.39 0.68 0.70 0.74 0.73

0.70 0.90 0.88 0.77 0.66

0.25 0.52 0.52 0.47 0.34

0.33 0.70 0.68 0.65 0.44

0.63, 0.81 0.82, 0.97 0.87, 0.89 0.85, 0.76 0.81, 0.73

OsC IrC PtC

7.36 6.84 6.34

0.92 0.85 0.79

0.78 0.75 0.69

0.62 0.59 0.49

0.28 0.31 0.14

0.41 0.35 0.29

0.77, 0.71 0.72 0.64, 0.68

γ7 0.50 0.36

Table 9 gives γ100 values of TMCs in terms of Eqs. 4.2 and 4.3. Other computer simulation and theoretical results for TMCs are also shown for comparison [151-153,156]. It can be found that the agreements among different methods for stable TMCs are better than that for metastable TMCs, such as CoC, NiC and OsC.1 The Fermi levels of both CoC and NiC lie in the upper part of the d region [157], which may lead to the simulation results being much lower than those in terms of Eq. 4.3. While for OsC, the Fermi level contains the most d states that do not hybridize with carbon states [157]. Thus, the simulation results represent different trends compared with the theoretical 1

3d transition metal compounds from ScC to VC are stable [156]; 4d transition metal compounds from YC to TcC are stable [156]; 5d transition metal compounds from LaC to WC can be regarded as stable [156]. TMCs consisting of the groups VIIIB are metastable when they have a NaCl structure.

32

Synthesis, Characterization and Properties of Nanostructures

ones. The theoretical results of Eq. 4.2 are in good agreement with LDA simulation ones while those of Eq. 4.3 have good correspondences with other theoretical results based on modification of cohesive energy of the classic broken-bond rule [153]. It is found that all γsv values calculated from GGA are smaller than those from LDA, which is the same as other similar calculations [154]. Table 10 shows γ100 values of twelve existed stable TMNs with the same methods [151,152].2 γ100 values of the stable TMNs determined by GGA-PW91 with the corresponding potential are nearly half of the results in terms of Eq. 4.3 while are larger than other simulation results [151,152]. LDA results are relatively larger and in agreement with the modified broken bond model better. Since the existing results are little, it is difficult to evaluate the accuracy of different simulation methods. Table 10. Comparison γ100 values in eV/atom of TMNs among γ1 of Eq. 4.3, γ2 of Eq. 4.2, simulation results γ3 using GGA-PW91 and γ4 using LDA [156], and other simulation results γ5 [151,152]. Eb γ1 γ2 γ3 γ4 γ5 ScN TiN VN CrN

6.72 6.69 6.25 5.14

0.84 0.83 0.78 0.65

0.59 0.58 0.55 0.44

0.39 0.38 0.29 0.28

0.54 0.61 0.47 0.36

YN

6.98

0.86

0.61

0.45

0.50

ZrN NbN MoN TcN LaN HfN TaN

7.52 7.50 6.20 5.48 6.27 7.62 7.63

0.94 0.95 0.78 0.69 0.79 0.96 0.96

0.66 0.66 0.55 0.48 0.55 0.67 0.67

0.48 0.41 0.38 0.36 0.34 0.45 0.39

0.73 0.56 0.45 0.38 0.50 0.69 0.55

0.36 0.27

For TMCs and TMNs, the existence of the d-electron of transition metals leads to almost the same lattice constants for the same group when the period number of transition metals increases. At the same time, their Eb increases gradually. Thus, γsv values increases a little as the period number increases. In the other side, as the group number in the same period varies, a maximum of Eb values is reached at VB or IVB group. Thus, the corresponding γsv0 values also approach their maxima. It is known that the variation of the bandwidth of compounds should be proportional to that of γsv0 values [153,154], which has been confirmed for 4d-TMC by analyzing the density of states (DOS) and the Fermi energy using LMTO method [153]. It is found that the bandwidth developed an early maximum for ZrC and NbC around 11 eV, after which it remains fairly constant, and then shifts downwards for AgC only around 5 eV. The same trend is found in the 3d, 4d and 5d carbides [153]. Considering the chemical similarities between TMNs and TMCs, the variations of γsv values in TMN series should follow the same rule. Table 11. Comparison of γsv0 values in eV/atom of AMOs among γ1 of Eq. 4.3, γ2 of Eq. 4.2, simulation results γ3 using GGA-PW91 and γ4 using LDA [156], and other two series of available LDA results γ5 and GGA results γ6 [154]. MgO

2

Eb 5.15

(h k l) (100) (110) (111)

γ1 0.64 1.32 2.04

γ2 0.45 0.95 1.51

γ3 0.24 0.75 2.03

γ4 0.32 1.03 2.30

γ5 0.32 0.98

γ6 0.29 0.87

The calculated results for γ100 values of metastable phases have worse accuracy, partly due to their metastable natures with possibly unsuitable potentials. Thus, the corresponding calculations are not shown.

Solid State Phenomena Vol. 155

CaO

5.50

SrO

5.20

BaO

5.05

(100) (110) (111) (100) (110) (111) (100) (110) (111)

0.65 1.41 2.18 0.65 1.34 2.06 0.62 1.23 2.00

0.48 1.01 1.61 0.45 0.96 1.53 0.44 0.93 1.49

0.24 0.73 1.62 0.23 0.68 1.54 0.21 0.66 1.04

0.31 1.05 2.07 0.28 0.96 1.65 0.27 0.88 1.62

33

0.30 0.92

0.24 0.73

0.28 0.83

0.25 0.73

0.27 0.75

0.20 0.62

Table 11 gives the calculated γsv0 values of AMOs in terms of Eq. 4.3, Eq. 4.2, LDA and GGA-PW91 [156], and other theoretical results [154]. The packing densities of (100), (110) and (111) surfaces of NaCl structure are roughly in the ratio of 1: 0.71: 0.58, which suggests a sequence of γ100 < γ110 < γ111 [158] where the (100) surface is the most stable surface and the (111) one is the most reactive [155]. As expected, the calculated results agree with the above sequence. In our knowledge, γ111 values have not been determined up to now except γ111 value of MgO with γ111 = 2.54 eV/atom [159], which was obtained by the DFT based, full-potential linearized augmented plane-wave method. When compared with γ100 = 0.26 eV/atom, γ111 is one order larger than γ100. This can be induced by its polar nature with instability. From Table 11, comparison between our simulation results [156] and Eq. 4.3 show that the deviation between them increases along the series from MgO to BaO. This may be due to the increase of the crystalline lattice constant with drop of the binding energy induced by the increase of cation repulsion and decrease of anion-anion overlap. For a certain surface, γsv0 values decrease along the sequence going from MgO to BaO where the energy loss of MgO is most obvious. MgO has much broader bandwidth and higher dispersion than the rest of the series. Through the analysis of the bulk electronic density of states (DOS) for the compounds series, the O (2p) band becomes narrower along the series going from MgO to BaO [156]. Only the bandwidth of MgO is prominent broader while that of CaO, SrO, and BaO are roughly the same [154]. These could be the cause of the surface energy decrease along the period numbers of the corresponding metals. The above fact can also been considered by Eq. 4.3. The reverse ratio between the period number of the metals and γ values of AMOs is induced by increase of a due to the rapid increase of s-electron orbital as the period number increases while Eb remains almost constant for the same group of elements. From Tables 9-11, it is quite clear that our simulation results γ3 and γ4 [156] are in good agreement with other available results using pseudopotential plane-wave-based DFT (such as TiC100 [152] and VC100 [151] in Table 9, TiN100 [152] and VN100 [151] in Table 10, and all AMOs [154] in Table 11). But unfortunately, the results from the pseudopotential method used in this paper are much lower than the results from FP-LMTO (linear-muffin-tin-orbitals) method for TMCs calculations [153], seeing Table 9. For TiC, even the largest pseudopotential result (0.68 eV/atom) from LDA [160] is about 18.1% lower than the result (0.83 eV/atom) from LMTO [153]. This is because of the great difference between the two approaches during solving the DFT equations. The method used in Ref. [153] treats the effective one-electron potential without any shape approximation and allows one to solve the electronic-structure problem for bulk as well as for the surface from first principles within a single scheme [161]. LMTO often employs the atomic sphere approximation (ASA) that is clearly inadequate for structures with large interstitial regions or with low coordination symmetry, as in the present surface calculations. Thus, the LMTO approach is generally regarded as efficient but relatively inaccurate method [162].

34

Synthesis, Characterization and Properties of Nanostructures

4.2. The size-dependent solid-vapor interface energy γsv(D) 4.2.1 γsv(D) of nanomaterials with positive curvature The thermodynamic behavior of nanocrystals differs from that of the corresponding bulk materials mainly due to the additional energetic term of γsv(D)A—the product of the surface (or interface) excess free energy and the surface (or interface) area. This term becomes significant to change the thermal stability of the nanocrystals due to the large surface/volume ratio of nanocrystals or A/V ∝ 1/D [163,164,165,166]. When the surfaces of polymorphs of the same material possess different interfacial free energies, a change in phase stability can occur with decreasing D [167]. Despite of the fundamental thermodynamic importance of γsv(D), few reliable experimental or theoretical values are available [63,109]. The effects of size and surrounding of nanocrystals on γsv(D) are hardly studied [78,167,168] However, in mesoscopic size range, the size-dependence of the liquid-vapor interface energy γlv(D) was thermodynamically considered fifty years ago by Tolman and Buff, respectively [106,169]. The final form of the analytical equation is as follows [106], γlv(D)/γlv0 = 1-4δ/D+…

(4.9)

where γlv0 is the corresponding bulk value of γlv(D), δ denotes a vertical distance from the surface of tension to the dividing surface where the superficial density of fluid vanishes. As a first order approximation, although there is no direct experimental evidence to support Eq. 4.9, Eq. 4.9 should also be applicable to predicting γsv(D) since the structural difference between solid and liquid is very small in comparison with that between solid and gas or between liquid and gas. In addition, it is unknown whether D in Eq. 4.9 can be extended from micron size to nanometer size. Hence, a theoretical determination of γsv(D) is meaningful. Although both the expressions of Eqs. 4.1 to 4.5 and the corresponding results are different, all of them indicate that, γsv = υEb/(aAS)

(4.10)

where υ < 1 is a function of C. If the nanocrystals have the same structure of the corresponding bulk, υ is size-independent. Thus, Eq. 4.10 may be extended to nanometer size as [170], γsv(D) = υE(D)/(aAS).

(4.11)

Combining Eq. 4.11 with Eq. 2.27, there is [170], 

1



 2S

1



γ sv ( D) / γ sv = 1 − exp − b .  2 D / h − 1  3R 2 D / h − 1 

(4.12)

Solid State Phenomena Vol. 155

35

2.0

B e (0001)

2

γsv(D) (J/m )

1.5

1.0

Al (110) M g (0001)

0.5

N a (110) 0.0 0.0

0.1

0.2

0.3

0.4

0.5

-1

1/D (nm ) Fig. 8. γsv(D) as a function of 1/D in terms of Eq. 4.12 (solid lines) and Eq. 4.14 (segment lines) for nanocrystals Be, Mg, Na and Al with different facets. The symbols ■, ▲ and ● denote the experimental results for Be and Mg (0001) [171], the theoretical values for Na (110) [172] and Al (110) [173].

In terms of Eq. 4.12, comparisons of γsv(D) of Be, Mg, Na, Al thin films and Au particles with different facets between model predictions and experimental and other theoretical results [171,172,173,174] are shown in Figs. 8 and 9 where the related parameters in equations are listed in Table 12. It is evident that our predictions are in agreement with the experimental values of Be and Mg (0001), other theoretical results for Na (110), and those for three low-index surfaces of Au. The deviations in all comparisons are smaller than 5% except that for Al (110) with a deviation of about 10%.

A u (110)

2

γsv(D) (J/m )

1.8

A u (100) 1.2

A u (111)

0.6 0.0

0.1

0.2

0.3

-1

1/D (nm ) Fig. 9. γsv(D) as a function of 1/D for nanocrystals Au with different facets in terms of Eq. 4.12 (solid lines) and Eq. 4.14 (segment lines). The symbols ■, ▲and ● denote the calculated results for (111), (100) and (110) facets in terms of a modified embedded-atom-method potential [174], respectively.

36

Synthesis, Characterization and Properties of Nanostructures

Table 12. Necessary parameters in equations with D [171-174] and h [32] in nm, Eb [32] in kJ/mol, Tb [32] in K, Sb in J/mol-K, and γsv0 [109,111] in J/m2. Element Be Mg Na Al

Surface (0001) (0001) (110) (110)

Au

(111) (100) (110)

D 1.90 2.80 3.45 4.29 5.72

h 0.222 0.320 0.372 0.286

Eb 292.4 127.4 97.0 293.4

Tb 2745 1363 1156 2793

Sb 106.5 93.5 83.9 105.0

γsv0 1.83 0.79 0.26 1.30

3.80

0.288

334.4

3130

106.8

1.28 1.63 1.70

Note that only the γsv0 value of Al is cited from Ref. [111] since this value in Ref. [109] is doubtful where γsv0 of Al (110) is smaller than that of Al (111), which is physically unacceptable.

As shown in the Figs. 8 and 9, γsv(D) decreases with a decrease in size. This trend is expected since E(D) of the nanocrystals increases as the size decreases. In other words, γsv(D) as an energetic difference between surface atoms and interior atoms decreases as energetic state of interior atoms increases. Considering the mathematical relation of exp (-x) ≈ 1-x when x is small enough, Eq. 4.12 can be rewritten as, γsv(D)/γsv ≈ 1-Sbh/(3RD)

(4.13)

Eq. 4.13 is in agreement with the general consideration that the decrease of the any size-dependent thermodynamic quantity is proportional to 1/D [29]. If γsv(D) function of Eq. 4.13 and γlv(D) function of Eq. 4.9 have the same size dependence, δ = Sbh/(12R) ≈ h when Sb ≈ 12R as seen in Table 12. Namely, the transition zone separating a solid phase and a vapor phase is only one atomic layer, which is an understandable result. This determined δ value is expected since when the atomic distance is larger than h, the bond energy decreases dramatically. Thus, Eq. 4.9 can be rewritten as, γlv(D)/γlv0 ≈ γsv(D)/γsv0 ≈ 1-4h/D.

(4.14)

It is known that the solid-vapor interface energy ratio between different facets is a more important parameter in determining the crystalline shapes. Eq. 4.12 indicates that,

γ sv1 ( D) γ sv1 0 = γ sv2 ( D) γ sv2 0

(4.15)

where the superscripts 1 and 2 denote different facets. Eq. 4.15 implies that although the solid-vapor interface energy is size-dependent, the solid-vapor interface energy ratio between different facets is size-independent and is equal to the corresponding bulk ratio. Eq. 4.15 can also be compared with the theoretical results for Au. For example, γ sv(100 ) (3.8nm) / γ sv(111) (3.8nm) ≈ 1.24 and

γ sv(110 ) (3.8nm) / γ sv(111) (3.8nm) ≈ 1.28 [174], which correspond well to the corresponding bulk ratios of 1.27 and 1.32 [109]. Noted that the structures of Be and Mg, Na, Al and Au belong to hcp, bcc and fcc structures, respectively. Owing to the above agreements shown in Figs. 8 and 9, the model should be applicable for all crystalline structures with different facets. Thus, Eq. 4.12 not only supplies a

Solid State Phenomena Vol. 155

37

simple way to determine γsv(D) values of different facets without any free parameter but also has an evident thermodynamic characteristic.

4.2.2 γsv(D) and Tm(D) of nanocavities with negative curvature In recent years, nanocavities have attracted much more interest because of important implications for microscopic physics, chemistry, biology, and medicines [175,176,177]. In contrast to nanoparticle, which is thought to be a cluster of atoms in a vacuum, nanocavity in condensed matter can usually be considered as a cluster of vacancies [177]. These two metastable condensed matter structures have an antisymmetry relation. For example, the melting temperature Tm of nanoparticles increases with the increasing particle size [178,179,180] while that of nanocavities decreases with increasing of the cavity radius r and finally reaches a plateau at a large cavity size [181,182,183]. Recently, nanocavities have been reported to shrink under thermal activation and the shrinkage slows down when the pore size is reduced to 2 nm, where the cavity surface energy is considered as the driving force not only for the external shrinkage but also for the internal shrinkage [184]. Moreover, cavity melting is thought to be governed by the variations and interplay of surface and interface energies, etc [185]. Since no special assumptions are made in the deductions of Eq. 4.10 and Eq. 4.11, these two equations should also be applicable for the nanocavities. Combining with Eq. 2.27 and considering the negative curvature of the nanocavities, γsv(D) functions of the nanocavities can be determined as [186],



1



 2S

1



γ sv ( D) / γ sv = 1 + exp b .  2 D / h + 1    3R 2 D / h + 1 

(4.16)

Table 13. Several necessary parameters used in the Eq. 4.16 with h in nm, Eb in kJ/mol, Tb in K and Sb in J/mol-K.

Cu Si Ar

h 0.256 0.157 0.176

Eb 300.3 384.2 6.4

Tb 2836 3540 87.3

Sb 105.9 108.5 73.3

In terms of Eq. 4.16, γsv(D)/γsv0 functions of Si and Cu nanocavities (the solid lines) are shown in Fig. 10. As a comparison, the predictions of Eq. 4.9 with negative curvature and δ = h are also shown. It is evident that our model predictions are in good agreement with those of Eq. 4.9. This correspondence in return confirms the validity of the assumption employed in the deduction of Eq. 4.10. Moreover, although Eqs. 4.10 and 4.11 are deduced in terms of the relationship between the bulk surface energy and the broken bond number of surface atoms for transition metals, the agreement for Si as shown in Fig. 10a indicates that this relation should also be applicable to other types of materials.

38

Synthesis, Characterization and Properties of Nanostructures

2.0 (a)

Si

γsv(D)/γsv0

1.6 1.2 2.0

(b)

Cu

1.6 1.2 0

2

4

6

8

10

D/2 (nm) Fig. 10. Comparisons of the D dependence of γsv(D)/γsv0 described by different models predictions in terms of Eq. 4.16 (the dotted lines) and Eq. 4.9 with negative curvature (the solid lines) for (a) Si and (b) Cu.

Usually, the surface energy of nanocavities is considered to consists of chemical part γchem and structural part γstru, namely γsv = γchem+γstru. The former results from the dangling bond energy at the inner surface of the nanocavity while the latter originates from elastic strain energy in the inner skin of one atomic layer thick of the nanocavity. However, γstru is one or two orders smaller than γchem at several nanometer sizes as shown in Fig. 2a of Ref. [187]. Thus, γstru is negligible as a first order approximation. Furthermore, the agreement shown in Fig. 10 also indicates that this neglect does not result in a big error. As shown in Fig. 10, γsv(D) of nanocavities increases with the decreasing size, which is in contrast with the trend of γsv(D) of nanoparticles [170,188]. This difference can be explained as the following: Due to the negative curvature in inner skin of nanocavity, the atomic bonding energy and the elastic strain energy are contributed to the total surface free energy. As shown in Fig. 12 of Ref. [177], the atomic bonds at the surface layer of nanocavities and nanoparticles are schematically illustrated. Clearly, the density of atomic bonding energy increases with the decreasing size at negative curvature surface, while the positive curvature surface has the opposite trend. At the same time, the density of elastic strain energy also increases with the decreasing size. These results indicate that the excess surface free energy in inner skin of nanocavity is different from or contrary to that in the nanocrystal. Moreover, the radius of 2 nm seems to be the threshold as shown in Fig. 10, at which the surface energy becomes stable. Since the intrinsic modulus in the cavity skin becomes larger than that of the bulk and local hardening may take place with the increase of the surface energy [187], the shrinking is faster when the cavity′s radius is larger than the threshold value of 2 nm and it will be slow down for the larger surface energy with smaller size, which is consistent with the observation of Zhu [184].

Solid State Phenomena Vol. 155

39

4

2

γsv0 (J/m )

3

2

1

0 0

1000

2000

3000

4000

Tm0 (K) Fig. 11. γsv0 as a function of Tm0 for transition metals.

Since the surface energy denotes the bond energy difference between surface atoms and interior ones while the melting temperature is directly proportional to the bond strength, an empirical relation between surface energy and melting temperature should also exist. As shown in Fig. 11, the surface energy is plotted against the melting temperature of transition metals where the experimental values of bulk surface energies are taken from Ref. [109] while those of the bulk melting temperature Tm0 are cited from Ref. [32]. In this case, the linear relation between γsv0 and Tm0 can be linearly regressed as, γsv0 ≈ nTm0

(4.17)

where the linearly regressed slope n is equal to 1.06×10-3 J/m2-K with the standard deviation of 5.96×10-3 J/m2-K noted that the correlation coefficient of the fit is 0.955. Also under the assumption that the elemental nanocavities have the same structure of the corresponding bulk, Eq. 4.17 may be extended to nanometer size with the same form, γsv(r) ≈ nTm(r)

(4.18)

Combining Eqs. 4.16-4.18, there is,

 2S  1 1    Tm ( D) / Tm 0 = 1 + exp b   2 D / h + 1  3R 2 D h + 1 

(4.19)

Fig. 12 indicates the comparison of Tm(D)/Tm0 between the model predictions in terms of Eq. 4.19 (the solid line) and available results of MD simulation (♦, ∆, and●) for Ar where ♦, ●, and ∆ denote that the systems consist of 256000, 108000 and 32000 atoms, respectively [185]. Although it seems that the difference between the model predictions and the MD simulation results is large for the system with 32000 atoms, the actual value is smaller than 10% at D/2 > 1 nm. Even at the smallest radius of 0.58 nm, the difference is only 20%. Thus, it can be concluded that Eq. 4.19 can predict the size dependence of the melting temperature of nanocavities with good accuracy. The agreement shown in Fig. 12 also in return confirms the validity of the assumption employed in the deduction of Eqs. 4.17 and 4.18.

40

Synthesis, Characterization and Properties of Nanostructures

It is known that the curvature-induced compressive stress will build up on the surface of the inner wall of a nanocavity when the size of a nanocavity reduces to a nanometer range. This compressive stress will lead to the increase of intrinsic modulus in the cavity skin and a speeding up of the vibration of the surface dangling bonds, where the latter will thus increase the Debye temperature θD of a region surrounding the nanocavity. According to Lindemann′s criterion, θD = c[Tm/(MV2/3)]1/2, where c is a constant, M and V denotes the molar weight and molar volume and can be assumed to be size-independent. Thus, the melting temperature will increase with the increase of Debye temperature since Tm(D) has the same size dependence of θ D2 ( D ) as a first order approximation. On the other hand, the increase of the intrinsic modulus also suggests that Tm(D) should increase with decreasing of the size because both two physical quantities characterize the bonding strength. Since the size dependence of the intrinsic modulus is reported to be directly proportional to that of the surface energy, the same size dependence for γsv(D) and Tm(D) is thus reasonable.

Tm(D)/Tm0

1.3

1.2

1.1

1.0 0

2

4

6

8

D/2 (nm) Fig. 12. Comparison of the D dependence of Tm(D)/Tm0 between the model predictions in terms of Eq. 4.19 (the solid line) and the results of MD simulation (♦, ●, and ∆) for Ar [185].

It has been reported that cavity melting in four stages is governed by a unique mechanism that originated from the variations and interplay the following properties: surface energy, surface tension, solid-liquid interface energy, the curvature of the void surfaces, and the elastic energy [185]. Since all of surface energy, surface tension, solid-liquid interface energy, and the elastic energy are related to the bond energy of surface atoms while the cohesive energy determines the size of bond strength, it is understandable and inevitable that Eqs. 4.17 and 4.18 developed according to the model for the size-dependent cohesive energy can describe the size dependence of the melting temperature of nanocavities.

5. Liquid-vapor interface energy 5.1. The bulk liquid-vapor interface energy γlv0 and temperature coefficient γ′lv0 Temperature dependent γlv0(T) is known to be one of the fundamental and important quantities in the theory and practice of materials processing (e.g., crystal growth, welding, and sintering), and its temperature coefficient γ′lv0(T) = dγlv0(T)/dT governs the well-known Marangoni convection on the surface of melt. There are several characteristics of the liquid surface. First, the liquid surface usually takes an equilibrium configuration with the minimum energy due to the high mobility of

Solid State Phenomena Vol. 155

41

liquid molecules. Second, because the liquid fails with respect to elastic deformation resistance, γlv0(T) (liquid-vapor interface energy) equals surface stress when surface adsorption is not taken into account, which is defined as the reversible work per unit area involved in forming a new surface of a substance plastically [5]. Although early methods of measurement of γlv0(T) are sufficiently precise, there is still uncertainty regarding its absolute values and particularly regarding γ′lv0(T) function mainly due to the effect of impurities, which strongly changes the measured results. Therefore, considerable efforts have recently been directed towards the experimental determinations of γlv0(T) and γ′lv0(T) of metals, and progress has been achieved with the advent of levitation processing and oscillating drop techniques [189,190,191]. However, such an experiment often suffers from the ambiguities in the interpretation of the resulting frequency spectra, it is also unlikely that experimental measurements will ever encompass all possible temperature ranges of interest and for all metals. In contrast to the determination of the γlv0(Tm) value, the γ′lv0(Tm) value is not well known experimentally even for elemental metallic liquids. A recent analysis of existing data shows that this quantity is known with accuracy better than 50% for only 19 metals while for 28 metals, the accuracy is worse; For 18 metals (mainly refractory metals) there are no experimental results [192]. Computer stimulations with Monte Carlo or molecular dynamics methods are considered to be one of the reliable methods [193], with which γlv0 can be calculated either using the mechanical expression for the surface stress or from the viewpoint of the solid-vapor interface energy. Unfortunately, the former approach suffers from rather high fluctuation and statistical uncertainty, while the latter introduces additional complexity into performance. Thus, the demand of developing reliable prediction methods has never declined. Semiempirical predictions based on the correlation between the surface and bulk thermodynamic properties are always active [194,195,196,197]. Stephan firstly links γlv0 to the heat of evaporation Hv′ at T = 0 K [194], γlv0(Tm) = c′Hv′/Vl2/3

(5.1)

with c′ being an unknown constant. Eq. 5.1 seems to apply only to transition metals. Although Eq. 5.1 has existed for more than 100 years, attempts to theoretically determine c′ value are rare. On the other hand, γlv0(T) of pure substances may be evaluated from values of critical temperature Tc by the Eötvos or Guggenheim empirical equations [198], γlv0(T)Vl2/3 = Q(1-T/Tc)

(5.2-a)

Or γlv0(T)/γlv0(Tm) = (1-T/Tc)W

(5.2-b)

where the exponent W is system-dependent, e.g., 4/5 for strongly hydrogen bonded substances or 11/9 for H2, N2 and CO, etc. However, W value for liquid metals has not been determined to the best of our knowledge. Moreover, unlike those of organic fluid, Tc values of liquid metals are only available for alkali metals and mercury [32], which severely restricts the use of Eq. 5.2-a. When γlv0(Tm) and γ′lv0(Tm) values are known, under the assumption that γ′lv0(T) is nearly a constant being equal to γ′lv0(Tm), the γlv0(T) function is also calculated by [199], γlv0(T) = γlv0(Tm)+γ′lv0(Tm)(T-Tm).

(5.3)

However, Eq. 5.3 has not been strictly examined. Thus, both γlv0(T) and γ′lv0(T) functions need to be further considered. As stated in Section 4.1, Eq. 4.4 can be used to calculate γsv0 of elements [132]. Due to the

42

Synthesis, Characterization and Properties of Nanostructures

structural similarity of the liquid and solid at least near Tm, Eq. 4.4 for γsv0 should give suggestions for γlv0 modeling and analytical determination of c′ value in Eq. 5.1. About 60 years ago, noting that fusion has only small effects on volume, cohesive forces, and specific heat of substance, Frenkel reached the conclusion that ″the character of the heat motion in liquid bodies, at least near the crystallization point, remains fundamentally the same as in solid bodies, reducing mainly to small vibrations about certain equilibrium position″ [200]. The very slight change in volume on melting is also thought to imply that the atoms in a liquid are tightly bound to one another like those in a crystalline solid [201]. Thus, the structural and energetic differences between a solid and a liquid are very small in comparison with those between a solid and a gas or between a liquid and a gas. Consequently, Eq. 4.4 can be extended to determine γlv0 with several modifications [202]: (i) Since T ≥ Tm for all the amounts that have been studied, which is much higher than 0 K, Eb at 0 K should be replaced by Hv(T) and AS should be substituted with AL(T), where the subscript L denotes liquid; (ii) The influence of surface entropy S(T) should contribute γlv0 due to the high temperature; (iii) The first coordination number of a liquid is usually determined by integrating the radial distribution function (RDF) up to the first minimum while the distance of the second minimum of RDF is approximately twice that, the effect of the next nearest-neighbors thus may be neglected in terms of the Lennard-Jones potential function, namely, ϕ ≈ 0. Thus, Eq. 4.4 can be rewritten for determining γlv0(T) [202], γlv0(T) = [mHv(T)-TS(T)]/[aAL(T)]

(5.4)

with m = (2-k1-k11/2)/2. Since metallic liquid is closely packed, the packing density of a random close packing (η = 0.637) can be employed for the liquid [34]. As discussed in reference [34], the volume change on melting ∆V/Vs is not solely determined by the difference in packing density η between two phases, and the Goldschmidt premise for lattice contraction should also be consideration. Otherwise, when the ηL of 0.637 is compared with ηfcc of 0.74, ∆V/Vs = ηfcc/ηL-1 ≈ 16%, which is larger than the experimentally observed value of 2-6% [34]. Note that the ηL of 0.637 is the maximum value that the single-component liquid can take, which leads to the fact that the specific volume difference between a solid with bcc structure and the corresponding liquid is only 0.2% in terms of Eq. 2 of Ref [34]. Whether a metal can be undercooled depends on the energetic nucleation barrier. When the volume change on crystallization is small, the corresponding nucleation barrier should be also small, which leads to a large degree of undercooling [203]. If this rule can also be applied to elementary substance, the local order in the metallic liquid is very similar to the bcc-type short-range order (SRO) [ 204 ]. This is the case of liquid Zr [ 205 ]. Correspondingly, this consideration can also be applied on the surface structure of liquid metals with a similar (110) surface of bcc structure to ensure the minimum of the solid-vapor interface energy [195]. Thus, the expression of Abcc is assumed to be applicable also to the liquid as a first-order approximation. As a result, k1 = 3/4 and AL = 81/2h2/3. For any pure, isotropic, condensed material, h = (6ηV/π)1/3. V can be calculated from the atomic weight M and ρ(T) by V = M/[aρ(T)]. Thus, AL(T) can be determined as, AL(T) = λ′{M/[aρL(T)]}2/3

(5.5)

with λ′ = (81/2/3)(6η/π)2/3. ρL(T) is equal to ρL(Tm)+(dρL/dT)(T-Tm) with dρL/dT being the temperature coefficient of liquid density since dρL/dT ≈ dρL(Tm)/dT for liquid metals in the temperature range of Tm-2Tm. This range could be up to 3Tm for Rb and Cs and 4Tm for Li and K [66].

Solid State Phenomena Vol. 155

43

It is known that Hv(T) for most substance is zero at Tc and reaches the maximum at the triple point Tt where Tt is very close to Tm for metals [206]. Recently, an empirical equation Hv(T)/Hv(Tm) = (1-t)it+j has been proposed for liquids having a triple point where t = (T-Tm)/(Tc-Tm), i = 0.44 and j = -0.137 [207]. In terms of the known Tm, Tb and Tc values for alkali metals (mercury is not involved in this work), it is found that the temperature dependence of Hv(T) between Tm and Tb is very small (