A. S. Barnard, P. Zapol and L. A. Curtiss- Modeling of Stability and Phase Transformations in Quasi-Zero Dimensional Nanocarbon Systems

8/3/2019 A. S. Barnard, P. Zapol and L. A. Curtiss- Modeling of Stability and Phase Transformations in Quasi-Zero Dimensiona

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MODELING OF STABILITY AND PHASE

TRANSFORMATIONS IN QUASI-ZERO DIMENSIONAL

NANOCARBON SYSTEMS

A. S. Barnarda,, P. Zapola,b,, and L. A. Curtissb,

a Center for Nanoscale Materials and Materials Science and Chemistry Divisions

Argonne National Laboratory,

9700 South Cass Ave,

Argonne, IL, 60439

b Applied Physics,

School of Applied Science

RMIT University GPO BOX 2476V Melbourne,

VIC, 3001, Australia.

Abstract

The transformation of nanodiamonds into carbon-onions (and vice versa) has been observed

experimentally and has been modelled computationally at various levels of sophistication. Also,

several analytical theories have been derived (by a number of research groups) to describe the

size, temperature and pressure dependence of this phase transition. However, in most cases a pure

carbon-onion or nanodiamond is not the final product. More often than not an intermediary is

formed, known as a bucky-diamond, with a diamond-like core encased in an onion-like shell. This

has prompted a number of studies investigating the relative stability of nanodiamonds, bucky-

diamonds, carbon-onions and fullerenes, in various size regimes. Presented here is a review outlining

results of numerous computational and theoretical studies examining the phase stability of carbon

nanoparticles, to clarify the complicated relationship between fullerenic and diamond structures at

the nanoscale.

Keywords: diamond, nanoparticles, theory and simulation, phase transitions

PACS numbers:

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I. INTRODUCTION

For decades it was known that graphite was the stable form of carbon under ambient

temperatures and pressures at the macro-scale, and that diamond was merely metastable.

With the discovery of buckminsterfullerene in 19801 and the one-dimensional analogue car-

bon nanotubes in 19912, it became clear that at the nanoscale, carbon structures differ from

their macroscopic counterparts. Fullerenes and carbon nanotubes share many structural

characteristics with graphite (including sp2 hybridization and six membered rings), and the

relative phase stability of graphite and fullerenes can be attributed to size dependence. The

closing of the shells cause the curved surfaces of fullerenes to gain stability, due to the

absence of the dangling bonds.

Diamond has also been observed at the nanoscale, with a number of forms of nanodiamond

being synthesized using various techniques. Both polycrystalline diamond clusters (ranging

in size from 1 60m in diameter) comprising of nanometer-sized diamond grains from 1

50 nm in diameter, and free nano-size diamond particles (also known as ultradispersed

diamond (UDD) or detonation diamond3) have also been successfully synthesized during

high energy exothermic detonations. Nanocrystalline diamond films have also been grown

using chemical vapor deposition (CVD) methods, with varying plasma chemistries46.

With the variety of quasi-zero dimensional (0-D) nano-carbon phases available, it is not

surprising that the attention of many nanotechnologists is focussed around carbon-based

nanostructures7,8. The existence of multiple carbon phases does however raise questions

as to how they are related to one another, and which phase is energetically preferred in

various size regimes. Experimentally, it has been found that upon annealing 2 5 nm

nanodiamond particles transforms into onion-like carbon (OLC)911. The transition occurs

from the surface inward, with the transformation temperature being dependent on particle

size, with the particles exhibiting preferential graphitization (and exfoliation) of the diamond

(111) surfaces over other lower index surfaces. Further, the results of Chen et al12 show that

while 5 nm nanodiamonds have been found to be unstable with respect to the transition

into OLC, nanodiamonds with a diameter of approximately 2.5 nm still remain after an

annealing; below 1073 K, nanodiamonds with a diameter of 5 nm are stable, implying that

the equilibrium size of diamond is temperature dependent.

In parallel to these discoveries and investigations, the reverse phenomena the transfor-

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mation of OLC into nanodiamond was also observed by Banhart et al13, showing that OLC

undergoes electron irradiation induced self-compression at the center, and decay into dia-

mond nuclei1417. In general, although it has been shown that nanodiamond may transform

into OLC (and visa versa), in most cases a pure carbon onion or nanodiamond is not the

final product. More often than not, an intermediary is formed with a diamond-like core and

an onion-like outer shell. These core-shell nanoparticles are known as buckydiamonds18.

In this review, the application of theory and quantum mechanical based simulations 19 to

the phase stability of nanodiamond will be summarized, including a comparison of several

theoretical models that have been put forward by various authors to describe the relative

phase stability of diamond and other forms of carbon at the nanoscale. A critical issue in the

stability of nanodiamond, buckydiamond and OLC is the role hydrogen absorbed on the

surfaces. In the presence of hydrogen, such as in conventional CVD synthesis of diamond

thin films, nanometer sized diamond clusters dominate since the atomic H in the plasma

etches the sp2 content of the films. The relationship between the various forms of clean

carbon nanoparticles and those in the presence of hydrogen will be discussed, along with

the mechanisms responsible for stabilization.

II. THEORETICAL BACKGROUND

The methods discussed in this review to describe the energy and geometry of carbon

nanostructures are based on the quantum mechanical Schrodinger equation using the Hamil-

tonian operator in the Coulomb approximation20. At the most fundamental level this equa-

tion describes the energy of the total, coupled system of electrons and nuclei (or ions),

which are separated using the Born-Oppenheimer (BO) approximation. This assumes that

the wavefunction t(r, R) for the total system (where r represents the coordinates of the

electrons and R the coordinates of the ions) may be approximated as the product of the

electronic wavefunction e(r, R) which describes the electronic energy for fixed positions of

the ions, and a wavefunction n(R) which describes the energy of the ions.

The wavefunction e(r, R) and the electronic energy Ee(R) are determined by solving

the electronic Schrodinger equation for fixed values of the position of the ions R. The sum,

ET(R) of the electronic energy Ee(R), and the nuclear-nuclear repulsion energy En(R), then

provides the potential (or field) which determines the energy and motion of the ions. A

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numerical approach to the description of the ionic motion may be accomplished by use of a

Monte Carlo (MC) or Molecular Dynamic (MD) simulation21.

A. Wavefunction Based Methods

The most commonly used methods are based on one-electron functions (orbitals) which

are then used to form determinantal wavefunctions which satisfy the Pauli Exclusion

Principle20. The most prevalent method is HartreeFock (HF) theory which (in its sim-

plest form) assumes that the wavefunction is a single determinant constructed from a set

of single electron functions referred to as spin orbitals i(ri, si) which are a product of a

function of the spatial coordinates of electron i, i(ri), and a spin up function (s) or spin

down function (s)20,22, such that:

HF = N 1

2 |1(r1, s1)2(r2, s2).....N(rN, sN)| (1)

If the system is a singlet state then the spatial orbitals are usually assumed to be doubly

occupied, i.e. i = i + 1 for i odd. This means that each space orbital contains two

electrons of opposite spin.

These orbitals may be determined by finding the form of HF i.e. of {i(ri)}, which

gives the lowest possible energy by the use of the quantum mechanical variational principle,

leading to a set of coupled single particle Schrodinger equations called the HartreeFock

equations. These equations are coupled, but may be solved self consistently by an iterative

process which is referred to as the Self Consistent Field (SCF) method. This may be done

numerically for atoms and diatomic molecules but the problem must be tackled in a different

way for more complicated systems20,22.

For large complexes, polymers and solids the usual method to solve the HF equations is

to expand the one electron spatial functions i as a finite, linear combination of a set (a

basis set) of known functions {(r); = 1,.....,M} called basis functions20,22:

i(r) =M

=1

Ci(r) (2)

These linear coefficients Ci are then determined by a set of equations derived from the

Hartree-Fock equations by use of the linear variational method which must also be solved self

consistently. This is called the method of Linear Combination of Atomic Orbitals (LCAO)

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SCF method and if sufficient basis functions are used then the Hartree-Fock result may

be approached to any degree of accuracy20,22. Basis sets in common use are, atom centered

Slater Type Orbitals (STOs), atom centered Gaussian Type Orbitals (GTOs), a combination

of atom and non-atom centered (bond) GTOs and plane waves functions (PWF)20,22,23. If

the system is periodic then symmetry can be used to simplify the construction of the orbitals.

The HF method has been generalized to describe open shells and excited states so the above

form using paired space orbitals is normally referred to as the Restricted HartreeFock

(RHF) method.

The HF method is an independent electron model in the sense that the electrons are

described by individual one electron orbitals or wavefunctions and each electron moves

in the averaged out field due to all the other electrons. Thus, it does not describe the

detailed, correlated motion of the electrons. Hence, the energy given by the HF method,

EHF is not exact and the difference between the exact, non-relativistic energy Ee and EHF

is known as the electron correlation energy, Ecorr. There also exist more accurate methods

which attempt to estimate Ecorr and that use the HF method as a starting point20,2224,

e.g. the Configuration Interaction (CI) method20, the nth order Mollet-Plesset Perturbation

(MPn) theory, the Coupled Cluster (CC) methods CCSD, CCSD(T), CCSDT and the QCCI

methods24. However, none of these methods have as yet been used to describe the type of

systems described in this review, nor periodic systems.

The implementation of the above methods can be performed entirely from first principles

(ab-initio) where all the integrals involved are calculated exactly or semiempirically where

some of the integrals are ignored, some integrals are approximated and the values of some

integrals are adjusted to get agreement with experiment for some properties. Examples of

these semiempirical methods are the MNDO24,25, AM124,25 and PM324,26 methods.

Clearly the ab initio methods are slower and require more computer storage than semi

empirical methods but are more rigorous and, more accurate than the semiempirical meth-

ods (if taken to a high enough order of approximation). Semiempirical methods are, of

course useful for systems containing many atoms and/or many electrons. The approximate

representation between the relative CPU time versus relative error for various quantum me-

chanical methods for electronic structure calculation may be seen in, for example, reference27.

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B. Density Functional Theory

These methods are based on the Hohenberg and Kohn theorem28,29 which states that

for a given type of Hamiltonian operator the energy is a unique functional of the density.

Thus, the electronic energy is a unique functional of the electronic density (r), and we may

write the electronic energy Ee[] of a system of interacting ions and electrons as a unique

functional F[] as below,

Ee[] = F[] +

(r)vext(r)dr + Eion({RI}) (3)

where vext(r) is an external potential of the electron gas (the electrostatic potential of the

nuclei), Eion({RI}) is the ion-ion interaction energy of the ions at fixed positions RI, and

F[] is the universal functional of independent ofvext(r). Thus F[] is the same functional

for electrons in atoms, molecules, polymers or solids. The exact form of this functional F[]

is unknown, and approximations are made such as the Local Density Approximation (LDA),

the Generalized Gradient Approximation (GGA) and, so called, hybrid methods28.

In practice the method of Kohn and Sham28,30 is usually used in which they showed

that the many body problem of an interacting electron gas in the presence of an external

potential may be mapped onto an auxiliary system of non-interacting particles which enables

an orbital description to be used. In this case the orbitals are used to expand the electron

density not the electronic wavefunction.

If the electron density is written in terms of N doubly occupied single-particle orbitals

{i(r) : i = 1, ......., N} as:

(r) = 2

|i(r)|2 (4)

then

Ee{

i(r)}, {R} = 2

i(r)(1

22)

i(r)dr + (5)

vext(r)(r)dr +12

(r)(r)|r r|1drdr

+Exc[] +12

ZIZJ|RI RJ|

1

where the successive terms on the right hand side of equation 5 represent respectively; the

kinetic energy of the electrons, the nuclear-electron coulomb attraction energy, the electron-

electron coulomb repulsion energy, the exchange-correlation energy and the nuclear-nuclear

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coulomb repulsion energy. It is the exchange-correlation energy Exc[] which is the most

difficult term to estimate and it is this term that is approximated by such methods as LDA

and GGA.

The use of the variational method then leads to a set of coupled one-electron equations

called the KohnSham (KS) equations, which determine these orbitals (the KohnSham

orbitals)28,30. The KS equations are then generally solved using basis functions in an anal-

ogous way to the solution of the Hartree-Fock equations28. However, instead of neglect-

ing electron correlation energy Ecorr and using the exact exchange energy as does the HF

method, the KS method approximates the exchange energy but also includes an estimate

of Ecorr. Also in contrast to wavefunction based methods used to estimate Ecorr the KS

method retains an orbital description.

C. Ab-initio Molecular Dynamics

Car and Parrinello31,32 have developed a method based on DFT to enable accurate ab

initio quantum mechanical calculation of the interatomic potentials, and forces and carry

out a MC or MD simulation in order to determine the motion of the ions from first principles.

The method involves some approximations and uses a combination of pseudopotentials66 to

treat the inner shell electrons and DFT with a plane wave basis set to treat the valence elec-

trons. The technique couples the electronic and nuclear motions by introducing an iterative

optimization technique where the plane wave coefficients describing the electrons and the

positions of the ions are treated as fictitious particles interacting via a classical Lagrangian.

Integration of these equations of motion leads to the configuration of minimum energy. Both

electronic and nuclear degrees of freedom are relaxed simultaneously. If the mass of the fic-

titious electrons is adjusted correctly then the correct dynamics is obtained for the nuclear

motion even though the electronic motion is not correctly described. However, the correct

BO energy surface for the electronic ground state is traversed within the approximations of

the method used to describe the electronic state of the system. This method is called first

principles MD, ab initio MD or Car-Parrinello MD (CPMD).

Other implementations of this type of method have been made for example in the

CASTEP33 and Vienna ab initio simulation package (VASP)34 codes. Ab initio MD is a

very valuable method as it allows the treatment of effects which other methods do not read-

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ily treat, such as non-zero temperature, bond making and bond breaking. If conventional

electronic structure methods are used they only describe the structure at zero temperature.

On the other hand MD methods based on the use of empirical potential functions (rather

than on quantum methods) find it hard to describe bond making and breaking. The main

limitations of ab initio MD techniques is that they are very time consuming and are, at

present, largely restricted to systems containing approximately < 103 electrons.

A slightly simpler method is the so-called Tight Binding MD (TBMD) method which

uses the quantum mechanical tight binding Hamiltonian introduced by Slater and Koster35.

Tight binding potentials and TBMD for example, have been shown to be highly applicable

to carbon nanostructures36. This method is simpler than the ab initio MD methods but

makes more approximations in the treatment of the electrons and needs parameterization.

It is, however, based firmly on quantum mechanics and so can be systematically refined. It

is applicable to much larger systems than the ab initio MD method, and at present it can

handle approximately 103104 atoms37,38.

III. CLEAN NANOPARTICLES

In this section, the structural and thermodynamic stability of clean quasi-zero dimen-

sional nanocarbon particles will be discussed, referring to particles in the absence of hydrogen

with C-terminated surfaces. The surface of these particles contain a large number of dan-

gling bonds, making reconstructions of various types of great importance both structurally

and energetically. The effect of H-termination on the stability of such structures will be

addressed in a later section.

We will see, because of the small molar volume of diamond compared to that of graphite,

for a sufficiently small carbon cluster diamond can be more stable than graphite. It should

be noted that in analyzing the stability between the clusters of diamond and graphite the

number of carbon atoms in each cluster should be the same, and thus, the radii of each

cluster are different from each other. The radius of the diamond cluster is smaller than that

of the graphite cluster for the same number of atoms. It is also important to keep in mind

that when a cluster gets smaller, the surface effects become dominant over the bulk effects.

Section III A outlines a number of analytical models used to describe nanocarbon phase

stability (that typically require computations to be performed in obtaining various parame-

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ters), whereas section III B outlines results generating using purely computational methods.

A. Phase Stability

In 1987 Amlof and Luthi39 performed large-scale ab initio restricted HartreeFock (RHF)

calculations of the phase stability of planar graphene and cubic diamond clusters. The

dependence of the total energy on the number of carbon atoms was obtained by fitting the

calculated cluster energies to the equation,

Etotal = NCEC + NDB EDB (6)

where NC is the total number of carbon atoms, NDB is the total number of dangling bonds,

EC is the energy per carbon atom and EDB is the energy per dangling bond. The surface

energy contribution decreases for larger clusters and the cohesive energy approaches the

macroscopic limit (bulk-diamond or bulk-graphite) as NC . Neglecting the contribution

from the cluster zero-point energy, they defined the cohesive energy as the difference between

the energy per carbon atom Etotal/NC and the energy of the free carbon atom39.

By fitting the energies from calculations on small graphene structures they predicted

the cohesive energy for an infinite graphene sheet to be -5.421 eV. The contribution due

to the attractive interactions between the graphitic layers was assumed to be less than

0.217 eV; however, even with the interlayer attraction the results were considerably higher

than the experimental value for the cohesive energy of graphite of -7.371 eV at T = 0 K.

Similarly, in the case of diamond the calculated cohesive energies of the three small sp3

bonded carbon clusters obtained by extrapolation of -5.386 eV was also considerably higher

than the experimental value for bulk diamond of -7.346 eV at T = 0 K. The differences in

the extrapolated cohesive energies of diamond and graphite were attributed to the electron

correlation effects which are not included in the RHF method, but these were important

steps in the right direction39.

Later, Shaw and Johnson40,41 used a diffusion-limited model that assumed that the dif-

ference in energy between the cluster and bulk (E) was proportional to the number of

surface atoms. They demonstrated that in the detonation regime carbon clustering is a slow

reaction, and that carbon particles were built up from random collisions. The model did

not distinguish between the various forms of carbon present during the detonation process,

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however they were able to estimate a particle size from 104 to 105 atoms (50 A), with un-

certainties primarily due to size dependence of the energies of the various carbon clusters.

Using the assumption that E, was proportional to the number of surface atoms, they found

that E n1/3 (where n is the total number of atoms)40,41.

Building on the foundation of this work, a number of theoretical models have been pro-

posed by various authors to compare the relative phase stability of diamond and graphite

particles at the nanoscale.

1. Nanodiamond versus Graphite

In 1996, Gamarnik42 reported on the boundary of the stability regions of diamond and

graphite nanoparticles at low pressure. Beginning with the assumption that an energetic

preference for the nucleation of nanodiamond over nanographite exists, the model asserted

that the stabilization of nanodiamond is a direct result of the small particle diameters. The

model, which also discriminated between atoms located at the particle edges and corners,

began with a statement of the energy of a phase at low pressure, in terms of the Helmholtz

free energy F,

F = Eat T S (7)

where S is the entropy, T is the temperature and Eat is the atomization energy. Note that at

T = 0, the Eat = F. The Eat is defined in terms of the lattice energy E (which is dependent

on the particle size), the kinetic energy Ek, and the ionization potential of bond charges per

atom I, as:

Eat = E+ Ek + I. (8)

The lattice energy was determined by summation of the Born-Lande pair interaction

potentials of all the charges of the crystals43,44,57. The kinetic energy for the bond electrons

in diamond and graphite were derived by relating the Coulomb force FC acting on a bond

electron with the bond electron centripetal force to the neighboring atom Fcp:

FC = Fcp (9)

The centripetal force acting on a bond is due to the interaction of FC with all of the

nearest neighbors in the crystal lattice, rather that with one atom. The kinetic energy of

the diamond bond electrons (per atom) was therefore defined as:

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Ek = QE(1)k , (10)

where,

E(1)k =

2ro(r ro)

r2 I1 (11)

is the kinetic energy of one bond electron, is the bond charge per atom, Q is the atom

charge in the diamond lattice, r is the bond length, ro is the carbon orbital radius and I1 is

the first ionization potential. A similar approach was used to calculated the kinetic energy

of the graphite bond electron charges42.

The diamond and graphite charge lattices were represented by excessive negative covalent

bond charges and positive ions. In the case of graphite, conduction atoms were added (also

represented by excessive negative charges) between the neighboring atoms between the nets,

to secure the stable graphite structure. The attraction between the conduction electrons and

the hexagonal nets compensates the repulsive forces acting between the nets. Therefore, by

assuming that each inner atom relinquishes one electron charge value to establish a bond,

the covalent bond charge value per atom Q0 and the conduction electron charge value 0 was

established. In the case of diamond this is distributed in four directions, and three directions

in the case of graphite, and the bond charge of an inner or surface atom was determined by

considering the number of nearest neighbors n42.

Experimental values of bond charges were then used for the calculation of the size de-

pendencies of Eat, illustrating that Eat was dependent on the size of the nanodiamond and

nanographite particles, as well as the temperature. The model predicted that nanodiamond

was stable for particles below 15 nm at T = 0, and graphite above42. The investigation

was conducted at various temperatures, finding that increasing temperatures decreases the

range of stability of nanodiamond, as shown in Figure 1.

At around the same time, Hwang et al45 outlined a thermodynamic chemical potentialmodel in terms of the surface energy of diamond and graphite clusters that was applicable

in non-equilibrium conditions such as during CVD. Low pressure CVD synthesis of diamond

involves carbon being transferred between gas and solid phases, so that the deposition and

etching was analyzed in terms of the activity of carbon. The model surmises that the

chemical potential of carbon represented the main criterion for the phase transition, and

that the carbon atoms transfer from a higher activity phase to a lower activity phase45.

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FIG. 1: Plot of the phase transition sizes for the nanodiamond-to-graphite transformation as a

function of temperature, as calculated by Gamarnik42.

They asserted that when the activity of carbon in the gas phase (agasC ) is lower than

that of graphite (agraC ) both graphite and diamond phases are etched. When the activity is

between agraC and the activity of diamond (adiaC ) graphite is deposited and diamond is etched.

When it is higher than adiaC both graphite and diamond are deposited. It was also logically

established that agraC is less than a

diaC , and that since diamond was chosen as the reference

state, adiaC is unity at all temperatures45.

Using these activities, the chemical potential of carbon in the gas phase is expressed as:

gasC = sC + RT ln a

gasC (12)

where sC is the chemical potential in the standard state. Thus, the chemical potentials of

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graphite and diamond were expressed, as:

graC = sC + RT ln a

graC (13)

diaC = sC + RT ln a

diaC (14)

respectively. The driving forces for precipitation of graphite and diamond from the gas phase

are, respectively:

gasgraC = graC

gasC = RT ln(a

gasC /a

graC ) (15)

gasdiaC = diaC

gasC = RT ln(a

gasC /a

diaC ) (16)

The authors found that the driving forces for precipitation of graphite and diamond

(depending on the sign) would promote either precipitation or etching. The stability between

diamond and graphite in the carbon phase diagram was considered via the difference in

the driving force between diamond and graphite, showing that the relative magnitude of

the driving force was reversed between diamond and graphite when the pressure was high

enough. From the point of view of metastable diamond formation, this approach predicts

that precipitation of diamond and graphite (from the gas phase) are kinetically equivalent45.

Further, the phase stability was shown differ from the known phase diagram of carbon, due

to the a capillary effect on the nuclei. The capillary pressures were expressed as a function

of the surface energy, and the radius of the curvature of the graphite and diamond clusters

(which were assumed to be spherical) as given by the LaplaceYoung equation. When the

pressure inside the clusters becomes sufficiently high, the chemical potential of the graphite

cluster is higher than the diamond cluster, thereby decreasing the size of the clusters. The

pressure at equilibrium was shown to correspond to the phase boundary between graphite

and diamond in the phase diagram of carbon. Thus, the chemical potentials per mole for

carbon in graphite and diamond clusters (including the capillary effect) were described by:

c,graC = graC + V

gra

2grargra

(17)

c,diaC = diaC + V

dia

2dia

rdia

(18)

where V is the molar volume. By equating these terms, and solving for the number of atoms

when c,diaC =c,graC , the phase transition size was found.

The authors calculated the term diagra to be 1.1201102J/atom, and by inserting

the reported values46 of 3.7 J/m2 and 3.1 J/m2 for the surface energies for diamond and

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graphite (respectively), the number of atoms was estimated to be 104 (at 1200 K). The

surface energy of diamond was then taken as 3.6, 3.5, 3.4 and 3.3 J/m2, resulting in values

of 177, 279, 413 and 584 atoms, respectively45.

This study also went on to examine the nucleation on a substrate; proposing that the

free energy of the contacted particle is decreased when a solid carbon particle forms on

the substrate, and that when the interface energy between the diamond and the substrate

is smaller than that between the graphite and the substrate the stability of diamond is

favored45.

As an extension of this work Hwang and his colleagues47 also detailed a charged clus-

ter model to describe the stability of small charged diamond and graphite clusters. Since

graphite is conducting, the authors assumed the charge to be uniformly distributed over the

conducting bulk and the interface of a graphite cluster will be non-polarizable; whereas in

the case diamond, the charge on the diamond cluster will be localized at the interface, and

the interface will be polarizable. The model assumes that the surface energy of the dia-

mond cluster may be decreased by the presence of charges while that of the graphite cluster

cannot, so that the effects of electrocapillarity (arising when an external field is applied to

a polarizable interface48) will favor diamond formation. The possibility of charge-induced

nucleation of diamond in the gas phase was a key point of the charged cluster model in their

paper47.

Although the charged cluster model (and available experimental data) was not sufficient

to make any quantitative prediction on the sign dependence of the stability between dia-

mond and graphite clusters, based on the concept of sign dependence in nucleation (and the

assumption that a negative charge stabilizes the diamond cluster) the model was used to

describe the electric field at a cluster surface, and the effects of charge on the surface energy.

The model predicts that if the presence of charges increases the intrinsic electric field of this

surface the surface energy will decrease, but the surface polarizibility would increase surface

(and vice versa). Given an uncharged diamond cluster with a negative surface, the accumu-

lation of the negative charges on the surface serves to decrease the surface energy, while the

accumulation of positive charges increases the surface energy, so the presence of negative

charges would be critical to the stability of a diamond cluster. The diamond clusters would

be stable when charged, and transform to a graphitic cluster when the charge is removed 47.

An extensive and complete treatise of this model, outlining the theory and application

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to appropriate systems in given in reference49.

In the models outlined above, the pressure dependence of the nanodiamond/graphite

phase stability was not been explicitly addressed. In an entirely different approach, Jiang et

al50 considered the size dependence of the diamond to graphite transition, as a function of

pressure and temperature. In their model the phase transition between nanodiamond and

nanographite was described, based on the extrapolation of the equilibrium phase bound-

ary (in the bulk diamond to bulk graphite pressure-temperature phase diagram) to the

nanoscale. The extrapolated phase boundary was calculated with the Laplace-Young equa-

tion, with input of experimental values for the bulk phases. The thermodynamic functions

for the diamond to graphite transition were estimated on the basis of the Clausius-Clapeyron

equation. According to the phase diagram of carbon the transition pressure function P(T)

for the diamond to graphite transition is given as,

P = 1700 + 2.06T (19)

where P is the external pressure (in MPa) and T the temperature (in Kelvin). This term

is determined by the terminating points of the graphite to diamond boundary in the phase

diagram at (1700 MPa, 0 K) and (12000 MPa, 5000 K), respectively50.

The contribution of the size-dependent internal pressure was estimated by assuming

spherical, quasi-isotropic nanocrystals, with additional curvature-induced pressure (Pa)

given by the Laplace-Young equation,

Pa = 4/d (20)

where d is the nanoparticle diameter (approximately several nanometers), and is the surface

free energy. To account for Pa, the external pressure P required for the phase transition was

decreased by the same amount; so that P + Pa gave the size dependent phase boundary. In

the case of low-temperature and low-pressure synthesis of diamond, P 0, and,

d =4

1700 + 2.06T. (21)

The equilibrium size of diamond was then calculated with a mean value for the surface

energy of different faces, D = 3.7 J/m2 for diamond. For graphite, a value of G = 3.1

J/m2 was used. These were equated at equilibrium (neglecting effects of surface tension),

with the results predicting the transition size of nanodiamond decreases from approximately

8nm at 0 K to 3nm at 1500 K; as shown in by the solid line in Figure 250.

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FIG. 2: The size-temperature transition diagram of carbon at zero pressure, where the solid line

shows the model prediction, and the segment line denotes the model prediction with surface stress.

The theoretical and experimental results are also plotted in the figure. The symbol denotes the

theoretical estimation based on the surface energy difference between diamond and graphite52. The

symbol gives the theoretical calculation in terms of the charge lattice model of Gamarnik42. The

symbol is based on the experimental observation at 1073 K that nanodiamonds with d = 5 nm are

transformed into nanographite12. The symbol shows an experimental result where nanodiamonds

of 2 nm in size transform to onion-like carbon at 1300 K10. Reprinted from Diamond and Related

Materials, 11, 234, D. Zhao, M. Zhao and Q. Jiang, Diamond. Relat. Mater., c(2002) with

permission from Elsevier.

The model also predicts that since D > G the phase transition would accelerate on the

surface of the nanodiamond, as (and therefore Pa) decreases during surface graphitization.

However, in the reverse case (the graphite to diamond transition), does not decrease and

Pa increases on the surface of graphite. This relates to the experimental observation (under

ambient pressure) of nanodiamonds graphitizing at the surface50.

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Then, by assuming a critical nucleus of100 atoms for nanodiamond, the authors showed

that the critical diameter of the nucleus is located within the stability field of diamond,

thereby promoting the nucleation and growth rate of diamond at lower temperature. How-

ever, as the equilibrium temperature increases the size of the nanodiamond decreases, and

once formed the graphite nanoparticles gain stability due the change in surface energy50.

Finally, by applying the Clausius-Clapeyron equation,

H =dP

dT(VG VD)T (22)

where H is the transition enthalpy in J/mol, dP/dT = 2.06 J/cm3 K (from above) and VG

and VD are the molar volumes of graphite and diamond having sizes of 5.298 and 3.417 in

cm3/mol, respectively, they obtained H = 3.87T, and a transition entropy of S = H/T =

3.87 J/molK.

This model was later extended to include the effects of surface stress on the internal pres-

sure of the nanoparticle51. The value of for a quasi-isotropic nanocrystal was determined

by,

=

3D0sl0

8

12

(23)

where sl0 is the bulk solid-liquid interface energy, is the compressibility and D is the

diameter of a particle where almost all atoms are located on the surface. sl0 was determinedby using the Gibbs-Thomson equation,

sl0 =2hSvibHm(T)

3VmR(24)

where R is the ideal gas constant, h is the atomic diameter, Hm the temperature-dependent

melting enthalpy of crystals, Svib the vibrational part of the overall melting entropy Sm,

and Vm the molar volume of crystals. The melting enthalpy was described in terms of the

Helmholtz function and the temperaturedependent, solid-liquid Gibbs free energy difference

gm(T),

H(T) = gm(T) T dgm(T)dT (25)

For diamond and graphite, the corresponding gm(T) function is expressed as,

gm(T) =Hm(Tm T)T

T2m, (26)

where Hm is the melting enthalpy at the melting temperature Tm. By letting dgm(T)/dT=0,

the isentropic temperature (between the under-cooled liquid and the crystal) Tk=Tm/2 was

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obtained; below which the specific heat difference between a glass and the crystal Cp

approaches zero. As Cp in responsible for the temperature dependence of Hm(T),

H(0 < T < Tk) Hm(Tk) (27)

and,

Hm(T) = Hm

T

Tm

2. (28)

In this case, Hm(T) = Hm(Tk) = Hm/4, and sl0 was written as,

sl0 =hSvibHm(T)

6VmR(29)

The authors then plotted the transition interface between nanodiamond (111) and

nanographite (0001), to find that the transition size of nanodiamond now decreases from

11 nm at 0 K to 4 nm at 1500 K. Furthermore, the transition point at 1300 K was found

to provide a transition size between diamond and the onion-like carbon; as the transformed

product of the onion-like carbon has a lower Gibbs free energy than the graphite, indicating

that the transition size between the nanodiamond and nanographite should be larger than 2

nm. The authors concluded by pointing out that when the surface energy is assumed to be

equivalent to surface stress, the equilibrium transition line is lower than the predicted line,

but approaches the theoretical estimation based on the fact that the value of the averaged

surface stress of 4.75 J/m2 (which is proportional to the equilibrium transition size), is larger

than the averaged value of the surface energies of 3.4 J/m251 .

The results of Jiang et al50,51 clearly show that the effects of surface stress is very im-

portant in the description of the phase stability of nanocarbon, since the transition size of

nanodiamond without surface stress (above) gave 8 nm at 0 K to 3 nm at 1500 K; some

13 nm smaller than when surface stress was included51.

2. Nanodiamonds versus Fullerenes

The variety of models outlined above firmly establish the range of the cross-over for phase

stability of graphite and nanodiamond for the case of large nanocarbon particles (411

nm). The question of the phase stability of smaller (< 2.5nm) was investigated by Barnard

et al53, by extending a familiar thermodynamic model54 to include fullerenes. By treating

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only dehydrogenated nanodiamonds, defined as being nanodiamond structures consisting of

almost entirely of sp3 bonded atoms (as opposed to buckydiamond), a direct comparison

of nanodiamonds with fullerenes was achieved.

The model used to describe the phase stability was based on the heat of formation (as

a function of particle size); a technique used previously by Winter and Ree54. The heat of

formation of graphite (Hof(G)) and diamond (Hof(D)) clusters was expressed in terms of

the bond energies ECC and ECH, and dangling bond energy EDB , such that:

Hof(G)

NC= 3

2EGCC +

NHNC

(EGCH 12

EGCC + Hof(H)) +

+Hof(C) +12

EvdwCC (30)

Hof(D)NC

= 2EDCC +NDBNC

(EDDB 12

EDCC +

+Hof(DB)) + Hof(C) (31)

where, NC is the number of carbon atoms, NDB is the number of dangling bonds on the

surface of the particle, NH is the number of terminating hydrogen atoms, Hof(C) is the

standard heat of formation of carbon at 298.15K and Hof(H) is the standard heat of

formation of hydrogen. The term EvdwCC is the van der Waals attraction between graphite

sheets. In the case of single shell fullerenes EvdwCC =0, and the closed shells ensure that NH/NC

term is also zero, leaving,Hof(F)

NC=

3

2EFCC + H

of(C) (32)

A term for the strain energy (EFstrain) that vanishes in the graphene limit was then included,

to account for the size dependence, so that the heat of formation for a fullerene was given

by,Hof(F)

NC

=3

2

EFCC + Hof(C) +

EFstrain

R2

. (33)

Although EFstrain may be determined in a number of ways, the authors proceeded by as-

suming that if fullerenes may be approximated as homogeneous and isotropic elastic spheres,

the bending energy per unit area (A) of a suitable elastic sheet is given in terms of the sheet

thickness (h) was expressed as,

EstrainA

=

2

+h/2h/2

dzz2

R2=

h3

24R2(34)

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where is the bending modulus of the sheet, and R is the radius of curvature. The strain

energy per carbon atom was therefore,

EstrainNC

=Ah3

NC24R2. (35)

A spherical model was also assumed, where R is equal to the mean radius so that A =

4R2, and NC = 4R2. Therefore, an expression for the strain energy per carbon atom

was obtained, which was linearly dependent on the inverse square of the curvature of the

structure:Estrain

NC=h3

24

1R2

= EFstrain1

R2(36)

Before determining the heat of formation for fullerenes, Barnard et al53 first obtained the

cohesive energy and the strain energy for fullerenic carbon by fitting a linear expression to

the energy per ion for all fullerenes from C20 to C80 (calculated using DFT) versus 1/R2. The

EFstrain and EFCC obtained from the slope and intercept were 5.19eV and 7.81eV respectively.

Using these results equation (33) was plotted as a function of the number of carbon atoms53.

The cohesive energy and dangling bond energy of nanodiamond was determined by calcu-

lating the total energy of a set of stable nanodiamond structures (of various morphologies)

with DFT GGA55, and using the linear fit to the total energy per ion versus number of dan-

gling bonds per ion. The cohesive energy and dangling bond energy of EDCC = 3.855 eV and

EDDB = 1.619 eV were obtained from the intercept and slope of this plot, respectively. By

inserting these bond energies into equation (31), the intersection of the heat of formation of

nanodiamond and fullerenes was determined to be at 1130 atoms, which is approximately

equivalent to a cubic nanodiamond crystals of 1.9 nm in diameter53.

This work established that nanodiamond is not necessarily the stable form of carbon at the

nanoscale. Instead, fullerenes are a more stable form of carbon for small clusters, resulting

in a window of stability for nanodiamond, outside of which nanodiamond is metastable(just as diamond is macroscopically)53. It is important to note that the upper limit of

fullerene stability (the fullerene-to-nanodiamond transitions size) falls above the upper limit

of quantum mechanical based simulations. In the size range of all recent ab initio studies of

nanodiamond (typically less than 103 atoms using current computer technology), fullerenes

are the globally stable phase of carbon.

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3. Phase stability of intermediaries

The studies described above highlight how various approaches may be used to compare

the stability of graphitic/fullerenic with diamond at the nanoscale. However, in most cases

a pure nanodiamond is not formed, but rather buckydiamond18 with a diamond-like core

and a full or partial fullerenic shell. Barnard et al56 also addressed the thermodynamic

stability of multi-shell carbon nanostructure by using the model detailed at the end of the

last section for comparing the phase stability of nanodiamonds and fullerenes, and applying

it to buckydiamonds and OLC. For this purpose they considered particles which exhibited

surface delamination during structural relaxations (as will be described in more detail in

the next section) resulting in sp2 hybridized atoms on the surface, and ignored those that

consist of entirely sp3 hybridized atoms (characterized as stable nanodiamonds)56.

The OLC was treated as nested fullerenes by adding the van der Waals attraction (EvdwCC )

to equation (33), to produce equation (37). The value of EvdwCC = 0.056 eV used in this

study was calculated by Guo57 using a graphite force field. The buckydiamonds have been

treated in the same manner as nanodiamonds by applying equation (31), although obviously

NDB /NC is different for nanodiamonds and buckydiamonds of similar diameter56.

Hof(F)

NC=

3

2EF

CC+ Ho

f(C) +

EFstrain

R2+

1

2Evdw

CC(37)

The heat of formation as a function of particle size for buckydiamond and OLC was

calculated, and plotted along with the nanodiamond and fullerene results, as shown in Figure

3. The authors then pointed out three main points established via this comparison. Firstly,

the sp2 bonded fullerenic structures are most stable below about 900 atoms. Although the

OLC and fullerene results converge below 250 atoms, they diverge very slowly beyond this

value, confirming that the cohesive energy and strain energy of these structures dominate the

formation energy and the contribution from van der Waals inter-layer attraction is small.

In fact, the OLC and fullerene results are indistinguishable (within uncertainties) below

approximately 2000 atoms. Secondly, the atomic heat of formation of buckydiamond is

more akin to the OLC than the nanodiamonds, the latter being the least stable below

1100 atoms, but most stable beyond 1600 atoms. Finally, in the region from 500 to

1850 atoms the simulations predict that a coexistence region has been formed. Within

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FIG. 3: Enthalpy of formation of carbon nanoparticles, indicating the relative subregions of co-

existence of buckydiamond with other phases. Uncertainties in the form of error bars have been

included (shown only for bucky-diamond here, for the purposes of clarity), indicating the quality

of fit of the model to the calculated values. Reproduced with permission from A. S. Barnard, S.

P. Russo and I. K. Snook, Phys. Rev. B, 68, 073406 c(2003) by the American Physical Society.

this region, buckydiamond is coexistent with the other carbon nanoparticles56.

This broad region was then further broken into three subregions, marked as A, B and

C in Figure 3. Within subregion A from 500 to 900 atoms (1.41.7nm), the heat of

formation of buckydiamond is indistinguishable from that of fullerenes (within uncertain-

ties), although carbon-onions represent the most stable form of nanocarbon, and nanodi-

amond the least stable. In subregion B, between 900 and 1350 atoms (1.72.0nm),

buckydiamond and carbon-onions coexist (within uncertainties). In subregion C, bucky

diamond was found to coexist with nanodiamond (within uncertainties) between 1350 and

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1850 atoms (2.02.2nm). Barnard et al56 also point out that the intersection where the

buckydiamond structure becomes more favorable than carbon-onions, is very close to the

intersection for nanodiamonds and fullerenes at 1130 atoms. This, they suggest, indicates

that at approximately 1130 atoms a sp3 bonded core becomes more favorable than a sp2

bonded core, irrespective of surface structure.

In general, these results show that the existence of buckydiamonds provides a smooth

transition from fullerenes to nanodiamond, rather than an abrupt change in phase stability.

The shallow nature of the crossings of the heats of formation do raise questions as to

whether an actual crossing-point may be accurately distinguished. However, such ambiguity

in the assigning of crossing points only supports the hypothesis of a coexistence region,

further highlighting that the relationship between the various types of carbon bonding at the

nanoscale is more complicated than previously thought. Still, understanding the relationship

between the sp3 and sp2 portions of a hybrid structure will be especially critical when

considering the interface between the phases, and the transition mechanisms.

As opposed to the models describing relative stability nanocarbon particles, compar-

atively fewer theories have been developed to describe the mechanisms occurring at the

interface between the sp2 and sp3 phases. The issue of the phase transition between nanodi-

amond and OLC structures was addressed explicitly by Zaiser and Banhart14, who presented

a thermodynamical quasi-equilibrium theory to explain this irradiation-induced transforma-

tion of OLC to nanodiamond. In this model it was theoretically shown that irradiation of

OLC leads to the destabilization of graphitic structures with respect to low-pressure growth

of diamond, due to the large difference in the cross subsections for irradiation-induced dis-

placements of carbon atoms in diamond and graphite. A non-equilibrium phase diagram was

calculated showing the stability of graphite and diamond (as a function of the displacement

rate of atoms), and the results related to the experimentally observed results. The initial

premise of the approach used by Zaiser and Banhart was that the issue of nucleation was

excluded in favor of considering the phase transformation as the motion of a phase boundary

separating the two (solid) allotropes. Considering a sufficiently large cluster of N carbon

atoms, consisting of ND sp3 bonded (diamond) atoms and (1-ND) sp

2 bonded (graphitic)

atoms, separated by a sharp interface. Irradiation of the cluster allows atoms at the interface

to oscillate between diamond and graphite bonding via two mechanisms14.

The first proposed mechanism was via thermally activated processes resulting in thermal

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jump rates GDth and DGth , such that:

GDthDGth

= exp[G], (38)

where G is the Gibbs free energy difference between diamond and graphite, and =(kBT)

1, with kB as Boltzmanns constant and T the temperature. The authors then intro-

duced the total thermally activated jump rate th across the interface, giving:

GDth = pGDth,

DGth = p

DGth, (39)

in terms of the probabilities,

pGD =1

1 + exp[G]

, pDG =exp[G]

1 + exp[G]

. (40)

The second mechanism was via ballistic processes, taking place first by displacing either

sp3 or sp2 bonded atoms due to collisions by irradiating particles, followed by the annihilation

of the displaced atoms on vacant lattice sites of the opposite phase. In describing this

process the authors assumed that only single interstitials and vacancies are produced by the

irradiation, that the exchange of atoms between diamond and graphite occurs only at the

interface, and the probability (either pGD or pDG) for such an exchange was governed by

the Gibbs free energies of the respective phases. The ballistic jump rates were given in

terms of the irradiation flux , by:

GDirr = pGDG,

DGirr = p

DGD, (41)

where G and D are the total displacement cross subsections for sp2 and sp3 atoms, respec-

tively.

The probability PND of finding ND sp3 bonded atoms, in terms of the number Ni of atoms

at the interface and the total jump rates (GD

tot = GD

th + GD

irr , DG

tot = DG

th + DG

irr ),was described by:

1

Ni

PNDt

= GDtot PND1 + DGtot PND+1 [

DGtot +

GDtot ]PND , (42)

for which the solution (under steady-state conditions where there is no net flux of matter),

required that the condition GDtot PND1 = DGtot PND was fulfilled. Thus, PND was found to

be:

PND = Cexp[(ND)] (43)

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where,

(ND) = (0) +

ND10

lnDGtotGDtot

, (44)

was stochastic potential, and C is a normalization constant.

In the absence of irradiation corresponds to G, and for the irradiated system a

non-equilibrium effective free energy Geff was defined to govern the phase stability such

that Geff = kBT[(N) (0)]/N. By making the appropriate substitutions, Zaiser and

Banhart found that,

Geff = G kBT ln

1 + G/th1 + D/th

. (45)

The authors point out that Geff reduces to G when = 0, and at high temperatures

when th G, D, otherwise ballistic jumps predominant as th G, D. However,ifG/D > 1, the non-equilibrium Geff of the sp

3 atoms is reduced by kBT ln[G/D] with

respect to G14.

The impact of irradiation on a crystalline structure was included as the threshold displace-

ment energy Ed (the minimum energy required to permanently displace an atom), which is

weakly anisotropic in nanodiamond and strongly anisotropic in graphite. The ratio G/D

was found to decrease with increasing electron energy, and a ratio of G/D > 1 was found

to be necessary for an irradiation-induced phase transition14.

It was assumed that in equation (45) that th =0exp[Gi] was governed by an Arrhenius

law; where the attempt frequency 0 was taken 1012s1, and the characteristic free activation

energy (required for crossing the interface) Gi was taken as 4 eV. With experimental values

of G(T) (at zero pressure and the atomic volumes) , Geff was determined for different

displacement cross-subsection ratios, and it was found that diamond was more stable than

graphite provided that Gef f < 0. Using this criterion non-equilibrium phase diagrams

at zero pressure were constructed, showing that the phase transition takes place within an

intermediate temperature range, the extension of which depends on the irradiation intensity

as well as on the ratio of the total displacement cross subsections14.

Therefore, the mechanism responsible for the OLC phase transition at the interface was

attributed to ballistic displacements causing interstitials (predominantly from sp2 lattice

sites); as probabilities of relaxing into sp3 or sp2 bonding configurations are of the same

order of magnitude, this causes a net flux of atoms from the sp2 to the sp3 phase. If

T exceeds the critical temperature, thermal jumps predominate so that the sp2 bonding

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remains the stable, but if the irradiation energy is such that only sp2 bonded atoms are

displaced the phase stability may be changed even at very low temperatures14.

The rate of transformation was also considered, expressed in terms of the velocity v of

the motion of the phase interface. It was found that (except in the vicinity of the upper

critical temperature), the transformation velocity was proportional to the displacement rate,

irrespective of the direction of the transformation.

Later, Butenko et al58 derived a kinetic theory to determine a temperature dependent

isotropic rate of migration of the interface between the diamond and graphitic regions of

spherical nanocarbon particles. They employed the reducing sphere model that assumed

the phase conversion begins simultaneously at all interface surface points, and that (under

isothermal conditions) the interface moves with a constant isotropic rate inside the nanopar-

ticle. In a later study, the reaction rate of the nanodiamond to buckydiamond (or OLC)

transition was treated as a migration rate of the interface between the exfoliated fullerenic

shells and the diamond cores. Again, a reducing sphere model was used to obtain the

rates from the changes in densities. Estimated kinetic parameters in an Arrhenius expres-

sion (such as the activation energy) then allowed for the quantitative calculations of the

diamond graphitization rates in and around the critical temperature range58.

In general however, the issue of kinetic processes at the surface and interfaces of carbon

nanoparticles is a topic of further investigation by many in the field. Challenges still exist,

including dependence on the phase transition mechanisms with the morphology of nanodi-

amond particles, as highlighted by a number of computational studies of the crystalline (or

structural) stability.

B. Crystalline Stability

As mentioned in the introduction, the transformation of nanodiamonds into OLC struc-

tures and the reverse phenomena (the transformation of OLC structures into nanodiamond)

have been observed experimentally, and a number of groups have simulated these transition

using quantum mechanical based methods. These simulations maybe loosely grouped into

two categories, the first (sp3 to sp2) examining the delamination of nanodiamond surfaces,

and the second (sp2 to sp3) examining the coalescence and stability of the diamond-like

cores.

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1. sp3 to sp2 transitions

In 1998 Winter and Ree54 used the AMl59 and PM326 parameterizations of the semi

empirical MNDO25 method to optimize the geometry of diamond clusters. For the nanodi-

amond clusters, generated with {111} morphology, NC = N(4N2 1)/3 and NDB = 4N

2,

where N is the number of layers along the c-axis, NC is the number of carbon atoms and

NDB is the number of dangling surface bonds54.

The results of these relaxations showed varying degree of structural change in the C10,

C35, C84, C165 and C286 octahedral nanodiamonds, terminated in all directions with (111)(1

1):1db surfaces (one dangling-bond surfaces). In the small particles the surface atoms of the

relaxed diamond clusters altered the positions so as to increase the 2p -2p overlap and

allow for the formation of double bonds, thereby reducing the number of dangling bonds

and causing an elongation of the bonds connecting the surface and core (interior) atoms.

In addition to this, the surface atoms were found to form 6-membered rings, with C C

distances consistent with sp2 bonds. The optimized geometry of the C84 nanodiamond

exhibited a separation of the 74 surface atoms from the 10 core atoms, with the separation

distance increasing to more than 3 A. This is a consequence of the flattening of the surface

corrugation in order to form bonds from the dangling orbitals. A similar result was

observed in the optimized C165 nanodiamond, with the outer 130 atoms separating from the

inner 35 atom core54.

Examining the diamondtographite phase transition from a different aspect, the prefer-

ential exfoliation of the diamond (111) surface over the (110) and (100) surfaces considered

by Kuznetsov et al10. The authors used a two-layer cluster model for both the (111) and

the (110) surfaces, and physical arguments (rather than direct calculations) to excluding

possibility of exfoliation of the (100) surface.

The calculations were carried out using the semiempirical modified neglect of diatomic

overlap (MNDO) method25, within RHF theory (including only the valence electrons). The

study began with an estimation of the inter-layer binding energy for the pairs of (111) and

(110) planes. These binding energies were found to be 2.83 eV for the (111) surface and

3.93 eV for the (110) surface, respectively, indicating that the inter-layer binding energy

for (110) is significantly greater than that for a corresponding sized (111) cluster. This

was followed by an examination the effect of surface relaxation on the inter-layer binding

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energy, by applying a geometry optimization to the upper (111) and (110) surface of each

cluster; both of which resulted in surface flattening (with no -bonded Pandey chain 2 1

reconstruction on the (111) cluster). The relaxation energies of the (111) and (110) surfaces

of 0.20 eV/surface atom and 0.27 eV/surface (respectively) indicated that relaxation does

not significantly affect the binding energies10.

To study surface graphitization, Kuznetsov and colleagues calculated the total energy

of a graphite single-layer cluster, and defined the graphitization energy of a cluster as the

sum of the energies of the graphite and bottom layers less the total energy of the initial

diamond cluster. The resulting graphitization energies for the (111) and (110) surfaces of

0.003 eV/surface atom and 0.24 eV/surface atom clearly demonstrated that graphitization

of a (111) surface is preferred over that of a (110) surface. The positive graphitization

energies were attributable to the small size of the clusters10.

Inter-layer binding energies were also calculated for larger two-layer clusters with 19 inter-

layer bonds. In this case, the exfoliated top layers consist of 12 six-fold 3 4 rings, with

binding energies per bond of 2.67 eV for the (111) surface and 3.59 and 3.66 eV for the

(110) surface parallel and perpendicular to the surface carbon chains, respectively. These

binding energies although lower than the experimental energy of the CC bond in diamond,

but higher than the experimental graphitization activation energy of 1.953 eV60. The

authors proposed that the graphitization mechanism involves the mutual interaction of the

exfoliated graphitic sheets and the dehydrogenated diamond surface below; and that the

difference between inter-layer binding energy along the (111) and (110) directions arises due

to different packing of the inter-layer bonds in the volume between the parallel planes.

Although all sp3 CC bonds have the same energy in each diamond lattice directions, the

(mainly repulsive) inter-bond interactions depend on the orientations of the bonds with

respect to each other, since the average distances between bonds is relatively small (0.25

nm) and the each bond is occupied by two electrons. The authors also showed that the

attractive interaction between the dangling bonds on the (111) surface is higher than that

for the (110) surface (for the same reason), while the dependence of the inter-layer binding

energy for the two (110) C48H16 clusters on the orientation of the six-fold rings (3 4 or

4 3) was attributed to the different sets of interacting sp3 bonds.

Using the same approach, the authors suggest that the surface graphitization is initiated

by a significant thermal displacement of a single carbon atom at temperatures close to

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the Debye temperature, and the surface delamination of the (111) surface to form curved

graphitic sheets proceeds by a zipper-like migration mechanism. A mechanism for the self-

assembling formation of a mosaic surface structure on larger (micron size and up) diamond

particles was also presented, where the initial diamond surface blisters under tensile stress,

forming fullerenic bubbles, that appear as a partial fullerene-like cage structure that is

anchored to the remaining stable diamond surfaces around its edges10.

The numerous studies outlined above demonstrate the instability of the (111) diamond

surfaces, and a complicated relationship between sp2 and sp3 bonding in nanodiamond

crystals. To further investigate the importance of surface structure on the nano-morphology

of diamond nanoparticles and the relationship between stability and shape (as opposed to

stability and phase), Barnard et al55,6163 undertook a broad ab initio study of nanodiamond

structures of octahedral, cuboctahedral and cubic morphologies, up to approximately 2 nm

in diameter. The calculations were performed with VASP64, using DFT GGA and the

exchange-correlation functional of Perdew and Wang (PW91)65.

To simplify the calculation as much as possible, and reduce the size of the calculations, the

values of orbitals describing the inner shell electrons may be frozen at their atomic values.

This is called the frozen core method, where that effect of the inner shells is represented by

a potential called a pseudopotential66. Various forms of these pseudopotentials are available

in the literature for various atoms. In the case of the work of Barnard and colleagues, ultra-

soft, gradient corrected Vanderbilt-type pseudopotentials (US-PP)67 were used, as supplied

by Kresse and Hafner68.

The three cubic nanodiamonds were included in the study (the C28, C54 and C259 struc-

tures) with (100)(1 1):2db (two dangling bond) and (110)(1 1):1db (one dangling bond)

surfaces. In each case, the initial step of the relaxation process involved the reconstruction

of the (100) surfaces to the (2 1) structure70. Upon relaxation the smallest cubic nanodi-

amond decayed to a tetrahedral amorphous structure, even though the most energetically

favorable structure for a 28 atoms carbon cluster is known to be the C28 fullerene. The

energy per atom (E/NC) for this cluster was found to be -6.351eV, which is considerably

higher than the E/NC of -6.928eV for a relaxed C28 fullerene, even when the relaxation was

extended to a convergence of 106 eV. Clearly the fullerenic structure is energetically pre-

ferred, but the energy barrier for the conversion to a fullerene is too high for the conversion

to occur. The relaxation is therefore trapped in a deep local minima, and would require

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simulated annealing to allow this cluster to convert to the (energetically preferred) structure

of a C28 fullerene63.

The two larger cubic nanodiamonds were found to have surface reconstructions and relax-

ations comparable to bulk diamond, although a slight shearing of the lattice was however

observed in the case of the C54 crystal63. The relaxed C259 nanodiamond is shown in Figure

4(a).

Three octahedral nanodiamonds were also investigated, the bulk-terminated C35, C84 and

C165 octahedral structures. In agreement with the previous results of Winter et al54, the

C35 crystal adopted of a more rounded appearance, but retained the diamond structure even

though the bond lengths altered significantly. The C84 octahedral nanodiamond also adopted

a rounded appearance as a result of the 74 surface atoms separating from the ten atom inner

core, forming an octahedral OLC structure. The shell-core separation distance was found to

be approximately 2.25 A (smaller than that observed by Winter and Ree54). Similarly, the

relaxation of the C165 atom octahedral carbon nanocrystal showed the same transformation

in which the 130 surface atoms separate from the 35 atom core cluster, also forming an

octahedral buckydiamond (again with the shell-core separation distance of approximately

2.25 A)61. The C165 relaxed structures are shown in Figure 4(b).

Finally, three structures were considered with cuboctahedral morphology, which was simi-

lar to that of a truncated octahedron, but with a surface area comprising 40% (111)(11):1db

surface area and 60% (100)(11):2db. The smallest cuboctahedral nanodiamond considered,

the C29 structure, was found to transform into the C@C28 endo-fullerene carbon atom upon

relaxation. The larger cuboctahedron with 142 atoms also exhibited a structural transition.

The (100) surfaces initially reconstructed to the (21) structure, followed by a reorientation

of the surface dimers to form the curved fullerenic cages on the (111) surfaces, as shown

in Figure 4(c). The same effect was however, less pronounced in the C323 structure, where

only the central region of the (111) surface and the edges at the intersection of teh (111)

facets formed a fullerenic cages.

Therefore, the ab initio relaxations performed by Barnard et al55,61 not only demonstrate

the preferential exfoliation of the (111) surfaces over lower index surfaces and the transfor-

mation of the octahedral nanodiamonds into octahedral OLC structures, but also that in

the absence of (111) surfaces nanodiamond structures may be stable. These findings were

supported a semi-quantitative study of the hybridization of the C C within the clusters62.

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FIG. 4: The relaxed (a) C259 cubic nanodiamond, (b) C165 octahedral nanodiamond, and (c) C142

cuboctahedral nanodiamond of Barnard et al62.

The technique uses fractional values of the electron charge density in regions localized at the

center of bonds in the structure, to provided an easy way of visualizing sp2 surface bonds in

isolation from sp3 content, thereby highlighting chemical differences between the core and

surface of their relaxed nanodiamond structures. A computational details and a complete

examination of this visualization method are given elsewhere69.

Around the same time it was shown that similar surface delamination and changes in

structure are also observed in relaxed spherical diamond nanoparticles. Raty et al18 pre-

sented ab initio calculations on the effect of quantum confinement and surface reconstruc-

tions in nanodiamond, performed with GGA and time dependent LDA (using a pseudopo-

tential, plane-wave approach) and semiempirical tight binding. The GGA calculations were

performed for the 1.4 nm cluster sizes, and the tight-binding calculations for the 23 nm

clusters sizes18.

Beginning with (ideal) bulk-diamond terminated spherical particles, the spontaneous low

temperature reconstruction dehydrogenated clusters of 1.4, 2, and 3.0 nm clusters resulted

in graphitization of the first atomic layer of the (111) facets. The smaller clustered (studied

with GGA) exhibited surface delamination, followed by the formation of 5-membered rings

linking the delaminated graphene fragments with the remaining core atoms, producing a

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FIG. 5: Two bucky diamond clusters: C147 and C275 ( 1.2 and 1.4 nm in diameter, respectively).

These carbon clusters have a diamond core (yellow) and a fullerene-like reconstructed surface

(red). Reproduced with permission from J. Y. Raty, G. Galli, C. Bostedt, T. W. Buuren and L. J.

Terminello, Phys. Rev. Lett., 90, 37402 c(2003) by the American Physical Society.

curved surface the same as the cages of Barnard et al55,61,62 and the bubbles of Kuznetsov

et al10. Two examples are shown in Figure 5. The TB simulations performed on the larger

2 and 3 nm clusters (705 and 2425 atoms, respectively) also produced the same surface

reconstructions. The barrier between the ideal (bulk-diamond) surface structure and the

reconstructed surface on a diamond nanocrystal was found to be size dependent (increasing

as the size of the nanoparticle is increased). In the larger nanoclusters the barrier was of the

order of several tens of electron volts. These results are in full agreement with the previous

work of the other researchers outlined above18.

Raty et al also compared their results with the experimental x-ray emission and absorp-

tion spectra (of nanodiamonds from 1.4 to 4 nm in diameter), in an attempt to explain

the presence of pre-edge features. The authors speculated that the above mentioned surface

reconstruction of nanodiamond was responsible for the features observed in absorption spec-

tra, corresponding to the existence of 5-membered rings (usually observed in the absorption

spectra of fullerenes). Based on these results and comparisons, the existence of a new family

of carbon clusters, buckydiamonds, was defined as carbon nanoparticles with a diamond

core (of a few nm) and a partial or complete fullerenic outer shell18.

Also observing these fullerenic bubbles forming on surfaces of nanodiamond structures

during simulations were Fugaciu, Hermann and Seifert71, in their study of the thermal sta-

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bility of spherical diamond nanoparticles. They performed molecular dynamics calculations

based on approximate Kohn-Sham equations, using electronic states with a LCAO to de-

scribe the KS orbitals. The interatomic forces were described using DFTB, and the equations

of motion were integrated numerically using the Verlet algorithm, in time steps of 1.2 fs.

The total simulation time was between 20 to 50 ps, and once the cluster was considered to

be qualitatively stable and it was cooled down to zero temperature within about 2 ps 71.

The simulations were carried out at temperatures in the range of 1400 K to 2800 K, which

is comparable to experimental conditions such as laser evaporation. In addition, irradiation

effects were taken into account by the random introduction of additional stochastic external

momenta, with a maximum kinetic energy of 40 eV. This simulated irradiation (applied

in intervals of 30 to 50 time steps), also introduced elements of the MC method, thereby

improving the chance of the system to exit weak local minima. The authors also reported

that the irradiation accelerated the nanocarbon transformation process and affected the

primary characteristics of the final structures. Using this procedure they analyzed the

structural evolution of 50 diamond nanoparticles, ranging from 64 to 275 atoms.

The authors reported that the nanodiamond particles of approximately 1.1 nm size (120

atoms) underwent fragmentation and formed fullerenic cages, but that the nanodiamond

particles >1.2 nm (150 atoms or more), transformed to form two-shell OLC structures.

Examples of their results are shown in Figure 6. The diamond particles consisting of 159

atoms were shown to transform spontaneously, without simulated irradiation, within a total

simulation time of about 10 ps; while the diamond particles with >191 required the applied

radiation (or extended simulation times) in order to completely transform. The number

of atoms per shell fluctuated. In one example, the inner core contained only a C2 carbon

dimer. The structure of the shells also fluctuated, consisting of pentagons and hexagons,

along with polygons with four and seven vertices. The bond length distribution within the

individual shells was found to be similar to the C C bonds in fullerenes, ranging between

1.38 and 1.50A71.

In most of the OLC clusters the two concentric shells were found to be connected by

cross-links (atoms belonging to the inner and outer shells connected by a sp3 bonds) with

a bond lengths of 1.540.6 A. Particles containing such bonds were termed spiraloidal

particles. It was also shown that simulated high irradiation intensity reduces the number of

sp3 cross-links, and increases the distance between the inner and outer shells. In all cases,

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FIG. 6: Transformation of a diamond particle into a concentric-shell fullerene at 1600 K under

simulated irradiation. (a) Initial state: a 191-atom diamond. (b) Final state after 40 ps: a two-

shell nucleus of a concentric-shell fullerene (green inner shell, yellow outer shell). (c) Example of

a concentric-shell fullerene with cross links between the two concentric shells (green: inner shell,

yellow: outer shell, orange: cross links with diamond-like coordinated atoms). Reproduced with

permission from F. Fugaciu, H. Hermann, and G. Seifert, Phys. Rev. B, 60, 10711 c(1999) by

the American Physical Society.

the cross-link atoms had not been nearest-neighbors in the initial structure, highlighting

that the cross-links appearing in the final structures are not residues of the initial state but

have developed during the transformation at late stages of the formation of the concentric

shells71.

Using the knowledge that the temperature and intensity of simulated irradiation are pa-

rameters which are suitable to control the transformation process, this study was extended

to consider precursors with a graphitic atomic order72. Initial structures with spherical

shape, and rod-like particles consisting of two elongated parallel graphene sheets were an-

alyzed. The spherical particles consisted of two or three circular graphene sheets. In the

case of three layer cluster, the smallest structure was found to evaporate in the first period

of the transformation process, followed by the two remaining layers connecting at one or

more sites at their boundaries resulting in closed or open cages. The rod-like particles often

transformed to elongated closed structures, somewhat similar to short single wall nanotubes.

Certain particles formed carbon nanotips or fullerenic cones, that the authors suggested to

be of use for tips of scanning tunnelling and atomic force microscopes. Overall no correla-

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tion was found between geometry and temperature, with the exception of the observation

that larger particles require higher temperatures to transform into a stable state within the

simulation used during the study72.

The initial part of this work71 was later repeated by Lee et al73, who also examined the

heat induced transformation of spherical nanodiamond particles into elongated fullerenes

using tight-binding molecular dynamics. Using the 275 atom spherical cluster examined

previously by other authors18,71, they confirmed the delamination of the {111} surfaces at

low temperature (and thus, the formation of buckydiamonds) and the transformation of

the cluster into a fullerenic structure (with cross-linked atoms between the outer shell and

the inner core) at a temperature of 2500 K73.

Upon cooling (2500 to 2000 K) the cluster was found to completely transform to an

elongated, tube-like fullerene via three mechanisms. These include a flow out mechanism

where inner atoms exit the core region via holes and defects in the outer shell, a direct

absorption mechanism where atoms within the shell are adsorbed into the shell structure,

and a push-out mechanism where inner atom replace the surface atoms that are pushed

into the vacuum73.

Therefore in summary, it has been shown by the studies described in this section, that the

(111) surfaces of diamond nanoparticles is structurally unstable and delaminated partially or

completely from the core. New family of carbon clusters, denoted as buckydiamonds, has

previously been established to describe dehydrogenated carbon nanoparticles comprising of

a diamond core and a fullerenic reconstructed surface18. It has also been shown that upon

annealing to high temperature, these structures may decay further into complete fullerenes

or OLC structures73.

However, another approach to the crystalline stability of dehydrogenated nanodiamonds

is to examine the stability and the coalescence of the cores, rather than the instability of

the surfaces.

2. sp2 to sp3 transitions

As part of a study on the graphitization of diamond surfaces and the diamond/graphite

interface using tight-binding computer simulations, Jungnickel et al74 examined the equilib-

rium structure of a tetrahedrally bonded icosahedral structure of 300 atoms. The icosahedral

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structure was approximately spherical and had a high degree of symmetry. The structure

was found to be stable in a conjugate gradient relaxation, and although the energy was 0.30

eV/atom higher than a C300 two-shell carbon onion (a C60 fullerene surrounding by an icosa-

hedral C240 fullerene, with average radii of 3.6 and 7.1 A respectively), the cluster could not

be induced to form the corresponding (lower energy) carbon-onion, even at temperatures of

12002700 K. The authors attributed the preservation of the sp3 bonded diamond structure

to the (high) energy barrier represented by the significant restructuring required to form the

corresponding sp2 nanoparticle74.

In a later attempt to model the sp2 to sp3 nanocarbon phase transition, Astala et al75 re-

ported results of atomic simulations using the DFTB method, to model the collision-induced

nucleation and growth of nanodiamond inside the 300 atom OLC structure mentioned above.

The group began by simulating the release of atoms due to knock-on displacements in outer

shells and their transport to the core, by a sequence of random atom additions with zero ini-

tial velocities. The evolution of the structures was found to be dependent upon the number

of interstitial atoms during the ad hoc injection process. The inner C60 core was seeded with

a carefully selected C10 cluster. The irradiation was simulated via the random int

Documents

A. S. Barnard, P. Zapol and L. A. Curtiss- Modeling of Stability and Phase Transformations in Quasi-Zero Dimensional Nanocarbon Systems