Thermodynamics of metastable phase nucleation

at the nanoscale

C.X.Wang,G.W.Yang

*

State Key Laboratory of Optoelectronic Materials and Technologies,School of Physics Science

and Engineering,Zhongshan University,Guangzhou 510275,China

Received 12 April 2005;accepted 2 June 2005

Abstract

Chemical and physical routes under conditions of moderate not extreme temperatures and pressures are generally

used to synthesize nanocrystals and nanostructures with metastable phases.However,the corresponding bulk

materials with the same metastable structures are prepared under conditions of high temperatures or high pressures.

The size effect of nanocrystals and nanostructures may be responsible for the formation of these metastable phases at

the nanometer size.To date,there has not been a clear and detailed understanding of the effects causing the formation

of the metastable structures from the viewpoint of thermodynamics.There is no a clear insight into which chemical

and physical origins leading to the tendency of the metastable phases emerging at the nanoscale.We have proposed

universal thermodynamic approach on nanoscale to elucidate the formation of the metastable phases taking place in

the microphase growth.In this review,we ﬁrst introduce the fundamental concepts and methods of the thermodynamic

approach on nanoscale (so-called nanothermodynamics).Note that our nanothermodynamics,by taking into account

the size-dependence of the surface tension of nanocrystals,differs from the thermodynamics of small systems

proposed by Hill [T.L.Hill,J.Chem.Phys.36 (1962) 3182;T.L.Hill,Proc.Natl.Acad.Sci.U.S.A.93 (1996) 14328;

T.L.Hill,R.V.Chamberlin,Proc.Natl.Acad.Sci.U.S.A.95 (1998) 12779;T.L.Hill,J.Chem.Phys.34 (1961) 1974;

T.L.Hill,J.Chem.Phys.35 (1961) 303;T.L.Hill,Nano Lett.1 (2001) 273;T.L.Hill,R.V.Chamberlin,Nano Lett.2

(2002) 609;T.L.Hill,Nano Lett.1 (2001) 159].Our thermodynamic theory emphasizes the size effect of the surface

tension of nanocrystals on the stable and metastable equilibriumstates during the microphase growth.Then,taking the

syntheses of diamond and cubic boron nitride (c-BN) nanocrystals as examples,we summarize the applications of the

nanothermodynamics to elucidate the nucleation of diamond and related materials nanocrystals in various moderate

environments.Firstly,we studied diamond nucleation upon chemical vapor deposition (CVD),and found out that the

capillary effect of the nanosized curvature of diamond critical nuclei could drive the metastable phase region of the

nucleation of CVD diamond into a new stable phase region in the carbon thermodynamic equilibrium diagram.

Consequently,the diamond nucleation is preferable to the graphite phase formation in the competing growth of

diamond and graphite upon CVD.Similarly,c-BN nucleation upon CVD has been investigated.Secondly,we

investigated the c-BN nucleation taking place in the high-pressure and high-temperature supercritical-ﬂuids systems

under conditions of the low-threshold-pressures (<3.0 GPa) and low-temperatures (<1500 K),and predicted the

threshold pressure of the formation of c-BN in the high-pressure and high-temperature supercritical-ﬂuids system.

Thirdly,to gain a clear insight into the diamond nucleation upon the hydrothermal synthesis and the reduction of

carbide (HSRC),we have performed the thermodynamic approach on nanoscale,in which the diamond nucleation is

preferable to the graphite phase formation in the competing growth between diamond and graphite upon HSRC.We

theoretically predicted that the pressure of 400 MPa should be the threshold pressure for the diamond synthesis by

HSRC in the metastable phase region of diamond in the carbon phase diagram.More importantly,these theoretical

results above are consistent with the experimental data.Additionally,the developed nanothermodynamics was used to

study the theory of nucleation and growth of diamond nanowires inside nanotubes.Accordingly,the thermodynamic

Materials Science and Engineering R 49 (2005) 157–202

* Corresponding author.Tel.:+86 20 8411 3692;fax:+86 20 8411 3692.

E-mail address:stsygw@zsu.edu.cn (G.W.Yang).

0927-796X/$ –see front matter#2005 Elsevier B.V.All rights reserved.

doi:10.1016/j.mser.2005.06.002

approach on the nanometer size seems to provide insight into the metastable phase generation in microphase growth

fromthe viewpoint of thermodynamics.Therefore,we expect the nanothermodynamic analysis to be a general method

to understand the metastable phase formations on nanoscale.

#2005 Elsevier B.V.All rights reserved.

Keywords:Metastable phase;Nucleation;Thermodynamics;Phase diagram;Gibbs free energy;Nanometer size

1.Introduction

Generally,the nucleation process by which clusters of a new phase from a parent phase is a

universal phenomenon in both nature and technology,for example,the basic processes of gas

condensation,liquid evaporation,and crystal growth.There is interest in developing quantitative

accurate theoretical tools such as thermodynamics and kinetics to address the nucleation since the

classical nucleation theory (CNT) was built by the collective pioneer work of Volmer and Weber [1],

Farkas [2],Becker and Do

¨

ring [3],Volmer [4],as well as later developed models by Zeldovich [5],

Frenkel [6],Turnbull and Fisher [7],and Turnbull [8,9].It is well known that CNT ever successfully

predicted the critical supersaturation in gas condensation.However,CNT has come in for scrutiny due

to the improvements in experimental techniques that may now measure the actual nucleation rates.

Oxtoby found that CNT is not accurate for the description of the temperature dependence of the

nucleation rate in some experimental cases [10].The discrepancy has stimulated the development of

the new theoretical tools such as the density-functional theory and computer stimulation [10–12].

Thermodynamically,the fundamental understanding of the nucleation process is still lacking in some

systems.In detail,many denser structures with metastable phases can be realized from their parent

phases under high-pressure and high-temperature conditions,and these metastable structural states

have unique properties without any change in the material composition compared with the stable

states.If sufﬁciently large energetic barriers depress the metastable structural states to transformto the

more energetically favorable structure,the high-pressure phases can be kept in the metastable states

under conditions of the ambient pressure and temperature.The best-known examples are diamond and

cubic boron nitride (c-BNor cubic BN),which are metastable structures compared with their graphite

and hexagonal boron nitride (h-BN) partners.However,many chemical and physical routes under

moderate pressure and temperature are generally used to synthesize these high-pressure phases with

metastable structural states in the corresponding thermodynamic equilibriumdiagramin recent years

[13–28].To our best knowledge,the nucleation and phase transition mechanism involved in the

formation of the metastable phases is still lacking.

On the other hand,in some materials processes,the metastable phases ﬁrst nucleate in the

strongly unstable phase region of the metastable structural states from their parent phases,and then,

after an appreciable time,they are transformed to the stable phase [29].These cases are similar to the

well-known Ostwald stage rule [30].Fortunately,the nucleation and the limited growth of the

metastable phase under the conditions of moderate pressure and temperature have been qualitatively

explained by the capillary effect of the small particles by Garvie [31] and Ishihara [32].They pointed

out that the capillary pressure built up in the nuclei could be so large that the high-pressure phase tends

to become more stable than the low-pressure phase.Since then,the viewpoint has been most frequently

quoted.Nevertheless,the full understanding of the metastable phase nucleation in the strongly

unstable phase regions of the metastable state (MPNUR) in the thermodynamic equilibrium phase

diagramis still qualitative.For this issue,in a series of publications by our group,we have established

the universal quantitative thermodynamic approach at the nanometer scale based on the Laplace–

Young equation and the thermodynamic equilibrium phase diagram to quantitatively elucidate

158 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

MPNURin the microphase growth,as the nucleation usually takes place at the nanometer size,i.e.,the

microphase growth.

This review is devoted to the systematic introduction of the fundamental concepts including

physical and chemical aspects and applications of the quantitative thermodynamic approach at the

nanometer scale in the formation of the metastable phases.We ﬁrst introduce the fundamental outline

of two kinds of nanothermodynamic approaches based on the ﬂuctuations of the temperature and the

Tsallis’ entropy in small systems due to the beginning of the nucleation reactions at the nanometer

scale in Section 2.Then,our nanothermodynamic approach,the thermodynamics of MPNUR on

nanoscale,is introduced in Section 3.Our nanothermodynamics,by taking into account the size-

dependence of the Gibbs free energy of nanocrystals,deﬁnitely differs from the thermodynamics of

small systems proposed by Hill [33–40],as our thermodynamic theory emphasizes the nanosize effect

of the surface tension of nanocrystals on the stable and metastable equilibrium states during the

microphase growth.In this approach,free of any adjustable parameters,the quantitative thermo-

dynamic description of MPNUNis obtained by the appropriate extrapolation of the phase equilibrium

(P,T) line of the generally accepted thermodynamic equilibrium phase diagram and the securable

macroscopically thermodynamic data.Afterward,taking the diamond and c-BN nucleation in the

unstable phase regions of the structure states as examples,we summarize the applications of the

proposed nanothermodynamics to elucidate the nucleation of diamond and cubic boron nitride in

various moderate environments in Sections 4 and 5 [41–48].Finally,the conclusion remarks are given

in Section 6.

2.Nanothermodynamics

2.1.Fundamental concepts

With the advancement of techniques of creating and characterizing materials,a huge of ‘‘small’’

sizes grains (micrometers),nanosystems,molecular magnets,and atomic clusters,has been formed

and displays a variety of interesting physical and chemical properties.Lee and Mori reported the

reversible diffusion phase change in the nanometer-sized alloy particles [49].Nanda et al.found that

the surface energy of the free Ag nanoparticles is signiﬁcantly higher than the bulk values by the

unique method [50].Shibata et al.observed the size-dependent spontaneous alloying of the Au–Ag

nanoparticles [51].The size-dependence of the surface ferromagnetismof Pd nanoparticles was found

only on the (1 0 0) facets [52].Mamin et al.detected the statistical polarization in a small ensemble of

the electron spin centers by magnetic resonance force spectroscopy [53].Dick et al.found size-

dependent melting of the silica-encapsulated gold nanoparticles [54].Masumara et al.measured an

unexpected decrease in the strength of various materials,when the sizes of micrometer change into the

nanometer scale [55].Lopez et al.reported the size-dependent optical properties of VO

2

nanoparticle

arrays [56].Similarly,Katz et al.found the size-dependent tunneling and optical spectroscopy of CdSe

quantum rods [57].Lau et al.found out the size-dependent magnetism of iron clusters [58].Voisin

et al.reported the size-dependent electron–electron interactions in metal nanoparticles [59].There-

fore,all these experimental cases clearly show that the size-dependence of properties is one of

distinguishing features of nanomaterials.Naturally,it is important and timely to develop the new

theoretical tools to address these experimental ﬁndings.On the other hand,the rapid progress in the

synthesis and processing of materials with the structures at the nanometer size has created a demand

for greater scientiﬁc understanding of the thermodynamics on nanoscale (thermodynamics of small

systems).The issue of application of the thermodynamics on nanoscale has been continuously

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 159

attracted,since the nucleation reaction was discovered in the early 1930s [60].Especially,the famous

talk was given by Feynman with the title ‘‘There’s Plenty of Room at the Bottom’’ on December 29,

1959 at the annual meeting of the American Physical Society [61],in which nanotechnology was for

the ﬁrst time formally recognized as a viable ﬁeld of research.For instance,a good example is the

renowned publication of two books with the title ‘‘Thermodynamics of Small Systems’’ by Hill in early

1960s [62,63],and recently,the thermodynamics of small systems is renamed ‘‘nanothermody-

namics’’ [38].

Traditionally,thermodynamics of large systems composing many particles has been well

developed [64–66].Classical thermodynamics describes the most likely macroscopic behavior of

large systems with the change of macroscopic parameters.The really large systems of astrophysical

objects as well as small systems containing a relatively small number of constituents (at the nanometer

scale) are excluded.Therefore,there is a great deal of interest and activity in the present day to extend

the macroscopic thermodynamics and statistical mechanics to the nanometer scale consisting of

countable particles belowthe thermodynamic limit due to the recent developments in nanoscience and

nanotechnology.To generalize the thermodynamics on scale,we need to well understand the unique

properties of nanosystems.It is well known that one of the characteristic features of nanosystems is

their high surface-to-volume ratio.As results of surface effects becoming increasingly important with

decreasing size,and then,the Gibbs free energy relatively increases for some thermodynamic

equilibrium systems.Therefore,the behavior of such nanoscopic clusters differs signiﬁcantly from

the usual thermodynamic limit [67].On the other hand,it is clearly known that when the systemsize

decreases,one has to consider the ﬂuctuations.Based on the nucleation reactions,the ﬁrst con-

siderations are on the temperature ﬂuctuations [60].The quantitative measurements of temperature

ﬂuctuations were realized by superconducting magnetometers [68].Interestingly,it is well explained

in the following statement by the US National Initiative on Nanotechnology [69] that the ﬂuctuations

play an important role:‘‘There are also many different types of time scales,ranging from 10

15

s to

several seconds,so consideration must be given to the fact that the particles are actually undergoing

ﬂuctuations in time and to the fact that there are uneven size distributions.To provide reliable results,

researchers must also consider the relative accuracy appropriate for the space and time scales that are

required;however,the cost of accuracy can be high.The temporal scale goes linearly in the number of

particles N,the spatial scale goes as O(N log N),yet the accuracy scale can go as high as N

7

to N!with a

signiﬁcant prefactor.’’ Therefore,these valuable hints motivate researchers to pursue the thermo-

dynamic description at the nanometer size for the nucleation of the metastable phase.Up to date,there

are two kinds of fundamental approaches to open out the thermodynamics on nanoscale based on the

microscopic and macroscopic viewpoints,respectively.One would go back to the fundamental

theorem of the macroscopic thermodynamics and establish the new formalism of the nanothermo-

dynamics by introducing the new function(s) presenting the ﬂuctuations or the surface effect of

nanosystems [33–40,62,63,70–82].Another one could directly modify the equations of the macro-

scopic thermodynamics and establish the new model of the thermodynamics on nanoscale by

incorporating the Laplace–Young or Gibbs–Thomson relation presenting the density ﬂuctuation of

nanosystems in the corresponding thermodynamic expressions [41–48,83–87].The fundamental

outlines of these approaches will be given in the following section.

2.2.Fundamental approaches

2.2.1.Nanothermodynamics—Hill’s theory

In the early 1960s,Hill [62,63] addressed the subject of the thermodynamics of small systems due

to his interest in thermodynamics of polymers and macromolecules.In order to clarify the relationship

160 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

between the macroscopic thermodynamics and the nanothermodynamics of Hill,ﬁrst of all,let us go

back to the fundamental theorem and recapitulate the foundations of the thermodynamics of

macroscopic systems.

In the case of the equilibrium thermodynamics of a macroscopic system,the fundamental

equation for the internal energy,U,in the absence of an external ﬁeld is expressed as

UðS;V;NÞ ¼ TS PV þmN (2.1)

where S is the entropy (an extensive state function),and it is a function of the extensivevariables (U,V,

N) in one-component system,T the absolute temperature,P the pressure,V the volume,m the

chemical potential,and N is the number of particles.The differential form of Eq.(2.1) may be

represented as

dU ¼ S dT þT dS V dP PdV þN dmþmdN (2.2)

On the other hand,the relationships among U,S,V,N,T,P,and m can be expressed as

m ¼

@U

@N

S;V

;(2.3)

T ¼

@U

@S

N;V

;(2.4)

P ¼

@U

@V

S;N

:(2.5)

Eq.(2.2) will change into the following form by employing one of the above three equations (2.3)–

(2.5)

SdT V dP þN dm ¼ 0 (2.6)

This is the celebrated Gibbs–Duhemrelation,and implies that the changes in the intensive quantities

(m,T,P) are not independent.However,the usual choice (T,P) is made in the literature,deﬁning an

equation of the state for the system.In particular,the Gibbs–Duhem relation implies that

@m

@P

T

¼

V

N

;(2.7)

and

@m

@T

P

¼

S

N

:(2.8)

It is well known that three other functions,besides the internal energy,U,are very useful in

applications to the speciﬁc physical situations.The enthalpy is

HðS;P;NÞ ¼ UðS;V;NÞ þPV (2.9)

The Helmholtz free energy

FðT;V;NÞ ¼ UðS;V;NÞ TS (2.10)

and the Gibbs free energy

GðT;P;NÞ ¼ UðS;V;NÞ TS þPV (2.11)

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 161

According to the dependence relationships of these functions (U,H,F,G) and their continuity

properties in their appropriate variables,four thermodynamic equations called the Maxwell relations

can be yielded as

@T

@V

S

¼

@P

@S

V

;(2.12)

@T

@P

S

¼

@V

@S

P

;(2.13)

@S

@V

T

¼

@P

@T

V

;(2.14)

and

@S

@P

T

¼

@V

@T

P

:(2.15)

However,the state function,U,is not extensive in the variable,N,in a one-component nanosystemand

hence the chemical potential,m,will depend on the number of particles,N,in it.As a result,the other

thermodynamic equations will be invalid including the Maxwell relations in the nanosystems,because

the nanosystemis sensitive to the environment it is placed in,as will be described presently.Hill [62]

approaches the nanothermodynamics based on restating Eq.(2.1) and reﬂecting this feature of

nanometer size by introducing a new function,W(T,P,m),called ‘‘subdivision energy’’ deﬁned as

W ¼ U TS þPV mN (2.16)

Naturally,the differential form of the so-called ‘‘subdivision energy’’ can be expressed as

dW ¼ dU S dT T dS þV dP þPdV N dmmdN (2.17)

By substituting the ﬁrst law of thermodynamics in the differential form

dQ ¼ T dS ¼ dU þPdV mdN (2.18)

For Eq.(2.17),one can obtain the result

dW ¼ S dT N dmþV dP (2.19)

In the macroscopic systems,Eq.(2.16) would be identically zero,while Eq.(2.19) is the Gibbs–Duhem

relationship.These are the ﬁrst step of Hill’s theory,and the rest of the development follows the

traditional path.From the above derivations,one can see that Hill’s theory is a generalized thermo-

dynamicmodel dealingwithnanosystems,as it starts withonlythe ﬁrst lawof thermodynamics relatedto

three independent variables U,V,and N,and does not employ other thermodynamic relations.We well

knowthat the renowned ﬁrst lawof thermodynamics is context independent and another representative

formof the principle of conservation of energy based only on the physical considerations changing of

heat and work in any quasi-static process.On the other hand,another important point of Hill’s theory is

the sensitivity of nanosystems to its environment.For example,the nanosystemincluding the number of

particles,N,in a volume,V,immersed in a heat bath at the temperature,T,is different from the same

systemcontactingwitha reservoir at the same temperature.Thus,Hill introduces the subdivisionenergy,

W,by taking into account of the importance of the ﬂuctuations in nanosystems,as above.

162 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

It is worth noting that Chamberlin et al.have extended Hill’s idea by considering the independent

thermal ﬂuctuations inside bulk materials.In detail,they adapt Hill’s theory to obtain a mean-ﬁeld

model for the energies and size distribution of clusters in condensed matter.Importantly,the model

provides a common physical basis for many empirical properties,including non-Debye relaxation,

non-Arrhenius activation,and non-classical critical scaling [88–97].

2.2.2.Nanothermodynamics—based on Tsallis’ generalization of ordinary Boltzmann–Gibbs

thermostatistics

The thermodynamic theory is on the basis of the Tsallis’ generalization of the ordinary

Boltzmann–Gibbs thermostatistics [98–102] by relaxing the additive properties of the thermodynamic

quantities (the entropy,in particular) to include non-extensivity of nanosystems [103].As described by

Rajagopal et al.[104],the nanothermodynamics differs fromHill’s approach by considering that each

of the nanosystems ﬂuctuates around the temperature of the reservoir,while nanosystems are coupled

to the reservoir.This means that the Boltzmann–Gibbs distribution has to be averaged over the

temperature ﬂuctuations induced by the reservoir.It has been suggested recently that ‘q-exponential’

(x

2

-distributed) distributions

e

b

q

uðxÞ

q

¼

Z

1

0

e

buðxÞ

f ðbÞ db ¼ ½1 þðq 1Þb

0

uðxÞ

1=ðq1Þ

ðq >1;b0Þ;(2.20)

which formthe basis of Tsallis’ non-extensive thermostatistical formalism[98] may be viewed as the

mixtures of the Gibbs distributions characterized by a ﬂuctuating inverse temperature.b

1

q

is a ﬁtting

parameter analogous to the temperature [105–107] and u(x) is the one-particle energy function taken to

be a quadratic or a nearly quadratic function of the velocity variable.The ‘q-exponential’ distribution

is a universal distribution that occurs in many common circumstances such as if bis the sumof squares

of n Gaussian random variables,with

n ¼

2

q 1

(2.21)

Furthermore,the essential point made by Beck (see [105] for the details) is that,if the probability

density,f(b),rules the temperature ﬂuctuations,it has the following form [105]:

f ðbÞ ¼

1

G

1

q1

1

ðq 1Þb

0

1=ðq1Þ

b

ð1=ðq1ÞÞ1

exp

b

ðq 1Þb

0

(2.22)

The constant b

0

is the average of the ﬂuctuating b,and it can be expressed by

q 1 ¼ b

2

0

Z

1

0

ðb b

0

Þ

2

f ðbÞ db (2.23)

When the ﬂuctuation is zero,we recover the usual Boltzmann–Gibbs distribution with q = 1 in the

above expressions.Apoint of interest is that the associated entropy is the non-additive Tsallis entropy

[108],given by

S

q

¼

1

P

i

p

q

i

q 1

(2.24)

when q = 1 goes over to the usual additive Gibbs entropy.We remark that a dynamic reasoning behind

the ﬂuctuation may be thought of as arising fromsome kinds of the Brownian dispersion caused by the

interaction of the heat bath on the nanosystem[109,110].There is a thermodynamics that goes with the

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 163

Tsallis entropy [108].Accordingly,one has here an alternate way to describe the nanothermody-

namics.

The above two models reﬂect the different properties of the real nanosystems.However,these

approaches fail toquantitativelyanalyze the roleof theinteractions betweenthe adjacent systems.Onthe

other hand,theytake intoaccount that the thermodynamic limit does not apply tothe thermodynamics of

MPNUR.Naturally,these theoretical tools may be expected to extend to the thermodynamics of

MPNURbased on their fundamental theorem.Apparently,this mission is really complicated due to the

lack of the accurate potentials for most substances,and difﬁculties in the identiﬁcation and even in the

deﬁnition of physically consistent clusters.Fortunately,in recent years,another fundamental approach

based on incorporating the Laplace–Young or Gibbs–Thomson relation presenting capillary effects of

nanosystems and the generally accepted thermodynamic equilibrium phase diagram into the classical

nucleation theory has been used to describe the thermodynamics of MPNUR [41–48].The detailed

derivation of this thermodynamic approach will be shown in next section.

3.Thermodynamics of metastable phase nucleation on nanoscale

3.1.Classical nucleation thermodynamics

Before starting the analysis of the thermodynamics of MPNUR,we need to look back on some

fundamental concepts of the nucleation theory involved in our model [41–48].Actually,the nucleation

refers to the kinetic processes that initiate the ﬁrst-order phase transitions in non-equilibriumsystems,

and the nucleation of a new phase is largely determined by the nucleation work W.The quantity is

equal to spending the Gibbs free energy having or at least resembling the properties of the new phase

appearance in the parent phase of a density ﬂuctuation and staying in the labile thermodynamic

equilibriumtogether with the parent phase.With a randomacquisition even of a single molecule of a

newphase,the ﬂuctuation may result in the spontaneous formation of the critical nucleation of the new

phase.For this reason,W is the energy barrier (critical energy of cluster formation,DG

*

) of the

nucleation.Therefore,the nucleation work plays an important role in the formation of a new phase.

However,it is well known that the initially homogeneous system is also heterogeneous in character-

ized by the non-uniformdensity and pressure.Therefore,the determination of DG

*

is a hard problem.

Namely,the case above makes it impossible to derive the nucleation work only fromthe method of the

thermodynamics of uniformly dense phases.In CNT,the critical nucleus is regarded as a liquid drop

with a sharp interface (a dividing surface) that separates the newand parent bulk phases.Matter within

the dividing surface is treated as a part of a bulk phase whose chemical potential is the same as that of

the parent phase.In the absence of knowledge of the properties of the microscopic clusters including

the surface tension,the bulk thermodynamic properties with several approximations are used to

evaluate the nucleation work in the discussions below.

In 1878 [111],Gibbs published his monumental work with the title ‘‘On the Equilibrium of

Heterogeneous Substances,’’ and his other publications have a special place in thermodynamics of the

phase’s mixture and equilibrium.Concretely,Gibbs extended the science of thermodynamics in a

general formto heterogeneous systems with and without chemical reactions.Especially,he introduced

the method of the dividing surface (DS) and used it to derive an exact formula for DG

*

in the nucleation

of a newphase in the bulk parent phase.In detail,with the aid of an arbitrarily chosen spherical DS,he

divided the heterogeneous system consisting of the density ﬂuctuation and the parent phase into two

homogeneous subsystems,which are corresponding to the microscopical and macroscopical sub-

systems,respectively.The macroscopically large subsystemequals the parent phase with the uniform

164 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

density and pressure before the ﬂuctuation formed.The microscopically small subsystem is an

imaginary particle (nucleus) replaced a reference new phase with the uniform density and pressure,

and surrounded by the large subsystem.The imaginary particle substitutes for the real nucleus of the

new phase,which is created by the density ﬂuctuation.As Gibbs [111] described the difference

between the imaginary particle (globule by Gibbs deﬁnition) and the density ﬂuctuation by the

following statement:‘‘For example,in applying our formulas to a microscopic globule of water in

steam,by the density or pressure of the interior mass we should understand not the actual density or

pressure at the center of the globule,but the density of liquid water (in large quantities) which has the

temperature and potential of the steam.’’ Furthermore,very recently,Kashchiev detailedly expatiated

the difference between the imaginary particle and the real density ﬂuctuation by the following

statement [112]:‘‘(i) the nucleus size depends on the choice of the DS and may therefore be very

different from the characteristic size of the density ﬂuctuation;(ii) the surface layer of the nucleus is

represented by the mathematical DS and is thus with zero thickness,whereas that of the density

ﬂuctuation is diffuse and can extend over scores of molecular diameters;(iii) the pressure and

molecular density of the nucleus are uniform,and those of the ﬂuctuation are not and might even be

hard to deﬁne when ‘at its center the matter cannot be regarded as having any phase of matter in mass

[113]’;(iv) the uniformpressure and density of the nucleus are equal to those of a reference bulk new

phase rather than to those at the center of the ﬂuctuation.’’ Therefore,based on these approaches

above,Gibbs showed that the reversible work W (free energy of nucleation),required to form the

critical nucleus of a new phase,is

DG

¼ Ag

T

VðP

l

P

v

Þ;(3.1)

where A and Vare the area and volume of the speciﬁc surface energy of a specially chosen DS,P

l

the

pressure of the newbulk reference phase at the same chemical potential as the parent phase,and P

v

is

the pressure of the parent phase far from the nucleus.g

T

is the ‘‘surface of tension,’’ called by Gibbs

[111],of the speciﬁc surface energy of a specially chosen DS,and it is called as the surface tension at

the present day.

In the Gibbs’ analysis,he found out that the classical Laplace–Young equation is valid in his DS,

and governs the pressure of droplets across a curved interface.For a spherical droplet with the critical

nucleus radius r

*

,the Laplace–Young equation reads

P

l

P

v

¼

2g

T

r

:(3.2)

Further,for the spherical critical nucleus,Gibbs showed that with Eq.(3.2),Eq.(3.1) becomes

DG

¼

16p

3

g

3

T

ðP

l

P

v

Þ

2

(3.3)

However,the quantity g

T

could not be obtained by experiment,because it describes such a surface—an

imaginary physical object,i.e.,the nucleus characterized by the surface of tension.Therefore,the

dependent relationships of g

T

and pressure,temperature,and composition of the parent phase,

respectively,are not uncovered.This limits the application of the nucleation theory to various cases

of interest.On the other hand,in order to describe the thermodynamic characterization of various

practical cases,we have to approximate g

T

by a real physical quantity.Clearly,in nearly all nucleation

papers that followed Gibbs’ equation,e.g.,in Refs.[1–12],one used the real interface energy g

0

between the bulk parent and new phases at the phase equilibrium,i.e.,at their coexistence,to replace

the surface tension g

T

of the imaginary DS.However,to apply this famous formula of Gibbs,one has to

know the exact interface energy that is related to the radius of droplets and the droplet reference

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 165

pressure.Unfortunately,there are tremendous challenges because there is no simple way to extract the

interface energy fromthe force measurements in theory.A given interface energy is a function of the

many coordinates of the nanoparticles.Lacking the knowledge of the exact interface energy,the ﬁrst

approximation is to use the experimental interface energy of a ﬂat interface,i.e.,g

0

= g.Actually,the

surface structure of droplets is different fromthat in the bulk,so that,strictly speaking,the boundary

surface never coincides with the equimolecular surface.Nevertheless,they are usually close to each

other and are often taken equally as the physical surface of the droplet.By assuming g

0

= g,one can

obtain the ﬁrst form of the nucleation work

DG

¼

16p

3

g

3

ðP

l

P

v

Þ

2

:(3.4)

In principle,by adopting the approximation,the validity of the Gibbs’s expression on the basis of the

Laplace–Young equation should be limited to the sufﬁciently large nuclei.Interestingly,the

applications of the nucleation theorem [33,63,114–117] in the analysis of experimental data in

various cases of nucleation implied that the Laplace–Young equation could predict well the size of the

nuclei built up of less than a few tens of molecules [83–87,115–121].Hwang et al.compared the

theoretical chemical potential of diamond with that of graphite upon chemical vapor deposition (CVD)

by employing the Laplace–Young equation for the stability of the nuclei,and indicated that the

chemical potential of carbon between diamond and graphite was shown to be reversed when the size of

the carbon cluster is sufﬁciently small [83].Experimentally,Gao and Bando used the Laplace–Young

equation to study the thermal expansion of Ga in carbon nanotubes [86,87].Additionally,the Laplace–

Young equation at the nanometer scale has been extensively developed to study the formation of

quantumdots [84,86,87].For instance,Tolbert and Alivisatos discussed the elevation of pressure in the

solid–solid structural transformation as the crystallite size decreases in the high-pressure systemusing

the Laplace–Young equation [84].Accordingly,it seems to be recognized that the Laplace–Young

equation could be used to predict well the size of nuclei built up of less than a few tens of molecules.

On the other hand,based on the thermodynamic identity,we have

m

l

ðP

l

Þ m

l

ðP

v

Þ ¼

Z

P

l

P

v

V

m

dP;(3.5)

where V

m

is the molar volume of a new phase and m

l

(P

l

) and m

l

(P

v

) are the chemical potential of

matter in the newphase at the pressures P

l

and P

v

.When the critical droplet and the metastable vapor

locate the condition of the unstable equilibrium,one can obtain

m

v

ðP

v

Þ ¼ m

l

ðP

l

Þ (3.6)

Furthermore,if we approximate P

l

by assuming that the droplet is incompressible,and assume that V

m

is a constant.With Eq.(3.5),Eq.(3.6) becomes

P

l

P

v

¼

m

v

ðP

v

Þ m

l

ðP

l

Þ

V

m

¼

Dm

V

m

(3.7)

Eq.(3.7) turns into Eq.(3.4),one can obtain the second form of the nucleation work

DG

¼

16p

3

g

3

V

2

m

ðDmÞ

2

(3.8)

As Obeidat et al.stated [122],the form of the nucleation work is most useful if the chemical

potential difference between a new phase and its parent phase can be obtained.However,the actual

166 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

performance is quite complicated due to the lack of the accurate potentials for most substances.In

order to obtain the Dm,we have to adopt some necessary approximations.Generally,if we assume the

supersaturated and saturated vapors are the ideal gases and the droplet is an incompressible liquid,the

difference of the chemical potential Dm is more commonly derived from the approximate system.In

detail,under the above assumptions,we have

P

l

¼ P

ev

(3.9)

where P

ev

is the equilibrium-vapor pressure.With Eq.(3.9),Eq.(3.5) becomes

m

l

ðP

v

Þ ¼ m

v

ðP

ev

Þ þV

m

ðP

v

P

ev

Þ (3.10)

When the newbulk phase and the parent phase are in the state of thermodynamic equilibrium,one can

obtain

m

l

ðP

ev

Þ ¼ m

v

ðP

ev

Þ (3.11)

With Eqs.(3.10) and (3.11),Eq.(3.7) becomes

Dm ¼ m

v

ðP

v

Þ m

v

ðP

ev

Þ V

m

ðP

v

P

ev

Þ (3.12)

Under the ideal vapor condition,we can easily obtain

m

v

ðP

v

Þ m

v

ðP

ev

Þ ¼ kT ln

P

v

P

ev

(3.13)

where k is the Boltzmann constant,T the absolute temperature,and P

v

is the actual pressure.With

Eq.(3.13),Eq.(3.12) becomes

Dm ¼ kT ln

P

v

P

ev

V

m

ðP

v

P

ev

Þ (3.14)

In Eq.(3.14),compared with the ﬁrst term on the right,the second term on the right is almost

extremely small,and it is customary to neglect it.Therefore,Eq.(3.8) will become the third formof the

nucleation work

DG

¼

16p

3

g

3

V

2

m

kT ln

P

v

P

ev

(3.15)

Applying the ﬁrst two forms of the nucleation work requires the knowledge of the droplet reference

pressure or chemical potential.Usually,this information is unavailable,and the experimental results

are,instead,compared with the rates predicted using the third form,because the supersaturation ratio

is readily determined from the experimental data.Naturally,the size of the critical nucleation,the

critical energy,the phase transition probability,and the nucleation rate would be obtained by the

determined nucleation work.

In summary,fromthe point of CNTabove,one can see that there is an important approximation,

i.e.,assuring g

T

= g

0

= g.Namely,the surface tension (g

T

) of a specially chosen DS,the real interface

energy (g

0

) between the bulk parent and new phases at the phase equilibrium,and the experimental

interface energy of a ﬂat interface (g) are approximated to equal [1–12].Furthermore,CNT indicates

that the Laplace–Young equation seems to be capable of predicting well the size of nuclei built up of

less than a fewtens of molecules [83–87,115–121].However,on the other hand,it is well known that

the CNT describes that a stable new phase forms from a metastable parent phase.Therefore,it is not

directly applied to MPNUR.

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 167

3.2.Application of Laplace–Young equation for the stability of nanophases

As mentioned above,the beginning of the nucleation reactions involves the nanometer size,and it

is useful to brieﬂy review the phase stability at the nanoscale based on the Laplace–Young equation

before starting the description of the thermodynamics of MPNUR due to the established model being

strongly related to the equation [41–48].

The following discussions of a series of representative nanosystems with the abnormally physical

properties described by the Laplace–Young equation in the recent years may give us insight into the

activity of the Laplace law in the nanosystems.In the past few decades,lots of researchers have

reported that the nanometer-sized particles usually showanomalies in the phase stability and the phase

transformation,and resulting in the metastable and unique crystal phases quite different from the

corresponding bulk materials [29,123–137].With regard to the stability of nanocrystals,the best-

known example is that,compared with the corresponding bulk materials,the melting points of

nanocrystals decrease in a wide variety of materials ranging from metals to semiconductors and

insulators [29,135–137].A sample of the typical data that can be obtained and the magnitude of the

effect for the experiments performed on CdS nanocrystals are presented in Fig.1 [137].Furthermore,

there are many excellent approaches for theoretical studies of the melting phenomenon in small

particles,such as the classical thermodynamic method based on the Laplace–Young equation [138],

which predicts a melting point temperature depression can be expressed as [139]

DT ¼ T

bulk

m

T

m

ðrÞ/a

2g

r

;(3.16)

where T

bulk

m

and T

m

(r) are the bulk melting point temperature and the melting point temperature of the

corresponding nanocrystals related to size,respectively,and a is a parameter related to the bulk

melting point temperature,the bulk latent heat of fusion,and the solid phase density.The 2g/r is the

Laplace–Young equation deﬁned by Eq.(3.2).Importantly,Zhang et al.reported the melting behavior

168 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.1.Melting temperature as a function of the size of CdS nanocrystals.The solid line is a ﬁt to a model that describes the

decrease in the melting temperature in terms of the difference in the surface energy between the solid and liquid phases (after

[137]).

of 0.1–10-nm-thickness discontinuous indium ﬁlms by an ultrasensitive thin-ﬁlm scanning calori-

metry technique,and the experimental case is in excellent agreement with their quantitative theoretical

calculation in terms of the above theoretical model [140].Namely,the melting behavior in

nanosystems can be characterized by the Laplace–Young equation resulting from the macroscopic

theory relating the surface tension to the additional pressure,although the melting behavior of

nanosystems is associated with the vibrational instability of crystal resulting fromthe difference of the

amplitude of vibration between the microscopic surface atoms and bulk atoms [141].

In the other cases,Tolbert and Alivisatos [85] developed a general rule for the effect of size on the

abnormallyﬁrst-order solid–solidphasetransitions,comparabletothewell-known1/r dependenceof the

melting temperature on the basis of the Laplace–Young equation.Interestingly,their rule reasonably

explains the high-pressure structural phase transition of semiconductor nanocrystals fromthe point of

view of the kinetics.Furthermore,Jiang et al.calculated the static hysteretic loop widths of the solid–

solidphase transitionof the CdSnanocrystals fromthe considerationof the thermodynamics onthe basis

of the Laplace–Young equation,and their results are reasonable in comparison with theoretical and

experimental results [142].Recently,Jiang et al.[143,144] proposed a thermodynamic approach in the

light of the Laplace–Young equation to analyze the abnormal phase stability between the nanometer-

scaled diamond and graphite,and obtained that the relative stability of diamond increases with the size

and temperature depression.Interestingly,their theoretical results are in agreement with other calcula-

tions interms of the charge lattice model [145] and experimental cases [146,147],respectively,as shown

in Fig.2.Fromthe aforementioned typical cases,one can see that the Laplace–Young equation resulting

fromthe macroscopic theorycanpredict some unusual properties of nanosystems,althoughthese unique

properties result from the ﬂuctuations and the surface effect of the microscopic systems.

In the following section,we will quantitatively describe the MPNUR in the light of the Laplace–

Young equation,CNT,and the thermodynamic equilibrium phase diagram.The brief description of

this topic given here sufﬁces to provoke interest in this open topic.In the section,we hope to present a

purely ‘‘macro’’ viewof the subject matter without directly invoking the microscopic underpinning of

MPNUR arising out of the ﬂuctuations of the nanosystems.

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 169

Fig.2.The size–temperature phase transition diagramof carbon at the zero pressure where the solid line and the segment line

showthe model prediction by adopting the different surface energy.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

graphite.The symbol ‘’ gives the theoretical calculation in terms of the charge lattice model [145].The symbol ‘^’ is

based on the experimental observation at 1073 Kthat nanodiamonds with d = 5 nmare transformed into nanographite [146].

The symbol ‘~’ shows an experimental result where the nanodiamonds of 2 nmin size transformto the onion-like carbon at

1300 K [147] (after [143]).

3.3.Thermodynamics of metastable phase nucleation in unstable region of thermodynamic

equilibrium phase diagram

MPNUR,such as CVDdiamond,CVDc-BN,HSRCdiamond,and c-BNnucleation,seems to be

impossible from the standpoint of thermodynamics,because the nucleation happens in the strongly

unstable region of the metastable structural state in terms of the thermodynamic equilibrium phase

diagram,and violates the second law of thermodynamics.For this issue,as Hwang and Yoon stated

regarding CVD diamond [148],‘‘Something must be wrong either in interpreting the experimental

observation or in applying thermodynamics.’’ In fact,in the early 1965s [31],Garvie pointed out that

MPNURlikely arises out of the capillary pressure built up in the nuclei.Namely,the nanosize-induced

additional pressure could be so large that the high-pressure metastable phase tends to become more

stable than the low-pressure stable phase,as shown in Fig.3.Note that,in the following description,a

phase is metastable or stable if it is stable or metastable without the effect of the nanosize-induced

additional pressure.In our theoretical approach,it also is emphasized that the nanosize-induced

additional pressure [41–48] is reasonably taken into account in the below analysis.

Generally,the Gibbs free energy is an adaptable measure of the energy of a state in phase

transformations among competing phases.At the given thermodynamic condition,both stable and

metastable phases can coexist,but only one of the two phases is stable,with the minimal free energy,

and the other must be metastable and may transform into the stable state.Thermodynamically,the

phase transformation is promoted by the difference of the free energies.The Gibbs free energy of a

phase can be expressed as a function of the pressure–temperature condition,and determined by a

general coordinate or reactive coordinate.According to CNT [149],the Gibbs free energy difference

arises fromthe formation of spherical clusters in the low-pressure gas phase is expressed as a function

of radius r,pressure P,and temperature T

DGðr;P;TÞ ¼

V

s

V

m

Dg þðA

ne

g

ne

þA

sn

g

sn

A

se

g

se

Þ;(3.17)

where V

s

and V

m

are the volume of the spherical clusters with the metastable structural phase and its

molar volume,Dg the Gibbs free energy of molar volume depending on the pressure Pand temperature

170 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.3.Asketch map of MPNURmechanism.Aregion is the metastable structural state of Mphase,and Bregion is the new

stable state of Mphase by the nanosize-induced additional pressure driving.The inset shown in the sketch map displays the

spherical nuclei nucleated on the hetero-substrate.

Tin the phase transition,A

ne

and g

ne

the interface area and the energy between the spherical clusters of

the metastable structural phase and the environment gas phase,A

sn

and g

sn

the interface area and the

energy between the spherical clusters of the metastable phase and the hetero-substrate,and A

se

and g

se

are the interface area and the energy between the hetero-substrate and the environment gas phase.The

formation of the spherical clusters with the metastable structural phase produces two interfaces,i.e.,

the interface A

ne

between the spherical clusters and the environment gas phase and the interface A

sn

between the spherical clusters and the hetero-substrate,and makes the original interface A

se

(be equal

to A

sn

) between the hetero-substrate and the environment gas phase vanish.According to the geometry,

the volume V

s

of the spherical clusters of the metastable structural state,the interface area A

ne

between

the spherical clusters of the metastable state and the environment gas phase,and the interface area A

sn

between the spherical clusters and the hetero-substrate are expressed as

V

s

¼

pr

3

ð2 þmÞð1 mÞ

2

3

;(3.18)

A

ne

¼ 2pr

2

ð1 mÞ;(3.19)

and

A

sn

¼ pr

2

ð1 m

2

Þ;(3.20)

where r is the curvature radius of spherical clusters of the metastable structural phase and mis given by

m ¼ cos u ¼

g

se

g

sn

g

ne

;(3.21)

where u is the contact angle between the spherical clusters of the metastable structural state and the

hetero-substrate,as shown in the inset of Fig.3.Here,g

ne

is assumed to be approximately equal to the

surface tension value of the metastable structural phase (g),g

se

for the interface energy between the

hetero-substrate and the environment gas phase is taken to be equal to the surface tension value of the

hetero-substrate,and the interface between the spherical clusters and the hetero-substrate is assumed

to be incoherent interface;therefore

g

sn

¼

g

ne

þg

se

2

(3.22)

Thus,we can obtain

DGðr;P;TÞ ¼

4

3

pr

3

Dg

V

m

þ4pr

2

g

ð2 þmÞð1 mÞ

2

4

(3.23)

where the factor of

f ðuÞ ¼

ð2 þmÞð1 mÞ

2

4

(3.24)

is called as the heterogeneous factor,and its value is in the range of 0–1.Especially,when the clusters

nucleated on the homo-substrate,its value is 1.

According to thermodynamics,we have

@DgðT;PÞ

@P

T

¼ DV (3.25)

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 171

Then,the difference of the Gibbs free energy per mole can be deﬁned by

DgðT;PÞ DgðT;P

0

Þ ¼

Z

P

P

0

DV dP DVðP P

0

Þ ¼ DV DP (3.26)

where DV is the mole volume difference between the metastable and the stable phase.When the

conditions are near the equilibrium line,one can approximately have Dg(T,P

0

) = 0.Thus,Eq.(3.26)

would be deﬁned as

DgðT;PÞ ¼ DV DP:(3.27)

On the other hand,due to the nanosize-induced additional pressure DP

n

,the clusters enduring pressure

will increase by the same amount [41–48].Under the assumptions of spherical and isotropic clusters,

the nanosize-induced additional pressure is denoted by the Laplace–Young equation,i.e.

DP

n

¼

2g

r

(3.28)

Furthermore,as mentioned above,the nanosize-induced additional pressure can drive the metastable

phase regions into the stable phase region near the boundary line of the high-pressure phase in the

equilibriumphase diagram.Therefore,one can obtain the size-dependent equilibriumphase boundary

line between the metastable and the stable phases,and it can be approximately deﬁned as

P ¼ P

b

2g

r

(3.29)

where P

b

is the equilibrium phase boundary equation between the metastable and the stable phases.

From Fig.3,one can see that the equilibrium phase boundary between the metastable and the stable

phases can be expressed by

P

b

¼ k

0

T þb

0

(3.30)

where k

0

and b

0

are the slope and intercept in the P coordinate axis of the equilibriumphase boundary

line between the metastable and the stable phases.With Eq.(3.30),Eq.(3.29) can be deﬁned as

P ¼ k

0

T þb

0

2g

r

(3.31)

Therefore,DP would change into

DP ¼ P k

0

T b

0

þ

2g

r

:(3.32)

With Eq.(3.32),Eq.(3.27) can be denoted by

DgðT;PÞ ¼ DV P k

0

T b

0

þ

2g

r

:(3.33)

With Eq.(3.32),Eq.(3.23) can be expressed as

DGðr;P;TÞ ¼

4

3

pr

3

DV P k

0

T b

0

þ

2g

r

V

m

þ4pr

2

g

0

@

1

A

f ðuÞ:(3.34)

172 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

When @DG(r)/@r = 0,the critical size of the high-pressure phase is obtained as

r

¼

2g

2

3

þ

V

m

DV

k

0

T þb

0

P

:(3.35)

Substituting Eq.(3.35) into Eq.(3.34),the critical energy of the high-pressure phase nuclei is given by

DGðr

;P;TÞ ¼

4

3

p

2g

2

3

þ

V

m

DV

k

0

T þb

0

P

!

3

DV

V

m

P k

0

T b

0

þ

k

0

T þb

0

P

2

3

þ

V

m

DV

!

þ4p

2g

2

3

þ

V

m

DV

k

0

T þb

0

P

!

2

1

A

f ðuÞ (3.36)

On the other hand,it is well known that the phase transition is determined by the probability.We have

studied the nanosize effect on the probability of the phase transformation based on the thermodynamic

equilibriumphase diagram.The probability of the phase transformation fromthe metastable phase to

the stable phase is related not only to the Gibbs free energy difference Dg(T,P),but also to an

activation energy (E

a

Dg(T,P)),which is necessary for the phase transition,as shown in Fig.4.

When the two phases are at the equilibriumcondition,i.e.,Dg(T,P) = 0,E

a

is the maximumpotential

energy for both sides with respect to the general coordinate.The general expression of the probability f

of the phase transformation from the initial states to ﬁnal states is [150]

f ¼ exp

E

a

DgðT;PÞ

RT

exp

E

a

RT

;(3.37)

where Ris the gas constant and Dg(T,P) is deﬁned by Eq.(3.32).Accordingly,we have established the

thermodynamic approach at the nanometer size to quantitatively describe the nucleation and the phase

transition of the metastable phase in the strongly unstable phase region of the metastable structural

state in the thermodynamic equilibriumphase diagram.In fact,the developed approach is a useful and

effective theoretical tool to address MPNUR,although it looks a little bit simple in thermodynamic.

Importantly,the validity of our thermodynamic theory has been substantively checked by use in the

nucleation of diamond and c-BN.

In the following section,using the proposed thermodynamic model above,we will consider

diamond and c-BN as examples to elucidate their nucleation and phase transition under various

unstable phase regions of these structural states in their phase diagrams.

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 173

Fig.4.The schematic diagram of Gibbs free energy vs.coordinate (after [43]).

4.Thermodynamic descriptions of diamond nucleation in the unstable phase

regions of the structural state

The diamond lattice is composed of two interpenetrating face-centered cubic lattices,one

displaced 1/4 of a lattice constant in each direction from the other.Each site is tetrahedrally

coordinated with four other sites in the other sublattice,as shown in Fig.5.The structure is

responsible for the very strongly carbon–carbon bonds,resulting in its several unique properties

including extreme high hardness,very high thermal conductivity,large band gap,and chemical

inertness,etc.[151–155].Therefore,metastable diamond is viewed as an ideal material for many

applications [156–170] due to its particular properties mentioned above.On the other hand,these

unique properties have therefore led to considerable efforts to create diamond since the ﬁrst report of

diamonds synthesized through a high-pressure and high-temperature process (HPHT) [171].Up to

date,the syntheses of diamonds include HPHT [171],CVD[162],shock-wave method [172],pulsed-

laser-induced liquid–solid interface reaction [173–176],hydrothermal synthesis and the reduction of

carbide (HSRC) [177–184],and so on.Interestingly,the diamond nucleation upon CVDand HSRCare

in the strongly unstable phase regions of the metastable structural states on the basis of the general

accepted thermodynamic equilibriumphase diagramof carbon [185].However,these cases seemto be

paradoxes fromthe thermodynamic point of view,because they contradict the fundamental principle

of the chemical thermodynamics.Why can diamonds form in the strongly unstable regions of the

structural state?For this issue,ﬁrst of all,we will review this case in the next section.

4.1.CVD diamond

4.1.1.Historical aspects of CVD diamond

Before starting the analysis of the nucleation of CVDdiamonds using our thermodynamic model,

let us look back the historical aspects of CVD diamonds.In 1961 [186],Eversole found the ﬁrst

method of CVDdiamonds under the lowpressure by employing the developed cyclic process.In 1967,

Angus et al.extended Eversole’s work and deposited diamonds on virgin,natural diamond powders

from the methane gas at 1050 8C and 0.3 Torr [187].In 1976 [188],Eversole’s work was further

174 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.5.Diamond can be viewed as two interpenetrating face-centered cubic lattices shifted along the body diagonal by (1/4,

1/4,1/4)a,where a is the dimension of the cubic (mineralogical) unit cell.

expanded by Deryaguin et al.who performed the careful physical chemistry experiments and

published a set of very beautiful photos of diamond crystals grown from the vapor phase under

activated low pressures.However,the application of the cyclic pyrolysis method above is unrealistic

due to a very slowdiamond deposition rate (1 nm/h) or requirement of a diamond seed substrate.In

1982 [189],Matsumoto et al.overcame the bottlenecks of the growth velocity and substrate,and led to

the application of CVD diamonds becoming realistic.They grew the diamond ﬁlms onto the non-

diamond substrate and obtained a high growth velocity by employing hot ﬁlaments (2000 8C) to

directly activate the hydrogen and hydrocarbon,which are passed through the hot ﬁlament.Before

long,the new activated diamond technology has been widely applied and became an important

research project through the whole world [190],and various activating methods for CVD diamonds

such as dc-plasma,rf-plasma,microwave plasma,electron cyclotron resonance-microwave plasma

CVD (ECR-MPCVD),and their modiﬁcations have been developed [191].

However,the important innovation of the synthesis of diamond under low-pressure conditions

was generally acknowledged until the middle of 1980s due to diamond being the metastable phase

under the low-pressure on the basis of the thermodynamic equilibriumphase diagramof carbon.In the

early days of the synthesis of diamond under the lowpressure,fewpeople accepted the case,and even

it had been joked as ‘‘alchemy’’ [192],because it was regarded as ‘‘thermodynamic paradox,’’ and

maybe ‘‘violating the second law of thermodynamics’’ [148,193–196].Up to date,most of explana-

tions for the formation of CVDdiamonds are that the atomic hydrogen plays an important role by the

hydrogen activation process [197–201].Namely,their hypothesis is based on the preferential etching

of graphite over diamond by the atomic hydrogen [197,198].Then,Yarbrough [196] indicated that the

atomic hydrogen hypothesis is contrary to the thermodynamic concept.Nevertheless,the atomic

hydrogen hypothesis seems to be most frequently quoted [148].Furthermore,several models attempt

to explain the formation of CVD diamonds using the exclusively kinetics [202,203].However,as

Piekarczyk described [194],‘‘However,a chemical process cannot proceed if it is thermodynamically

impossible,even though it is kinetically favored.Kinetics should be exercised within thermodynamics

and never go against it.’’

In thermodynamic aspects,several qualitative models have been proposed so far,respectively,

i.e.,Sommer’s quasi-equilibriummodel [204],Yarbrough’s surface reaction model [205],Bar-Yam’s

defects-induced stabilization model [206],Hwang’s charged cluster model [207,208],and Wang’s

chemical pump model [190,193,209–211],and so on.However,none of these models clearly gives a

completely satisfying insight into the formation of CVDdiamonds,each model just tended to focus on

one aspect of the complicated process of the CVD diamond nucleation [212].

4.1.2.Nanothermodynamic analyses of CVD diamond nucleation

Generally,CVD diamond is usually a typical quasi-equilibriumprocess [41],and the pressure is

in the range of 10

2

–10

5

Pa and temperature is in the range of 1000–1300 K[212].In the carbon phase

diagram shown in Fig.6 [185],the general thermodynamic region of the diamond nucleation upon

CVDis shown as Gregion,which belongs to the strongly unstable or metastable region of the diamond

structural state,i.e.,the stable region of graphite structural state.It is well known that the graphite

nucleation would be prior to diamond nucleation in the G region from the point of view of

thermodynamics.Therefore,the diamond nucleation would not happen unless the graphite nucleation

is restrained or stopped.For the issue,the most popular explanation is that the atomic hydrogen plays

an important role.Atomic hydrogen is an essential factor in CVD diamonds due to its higher etching

rate for the graphite phase and less etching rate for the diamond phase.Unfortunately,some

researchers have reported that diamond ﬁlms are grown upon CVDwith a hydrogen-free environment

[213–215].Further,Gruen [216] concluded that CVD diamonds do not require the reactant gas

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 175

mixtures consisting primarily of hydrogen,and the microstructure of diamond ﬁlms can change

continuously from micro- to nanocrystalline when hydrogen is successively replaced by a noble gas

such as argon.Moreover,they pointed out that a chief function of the atomic hydrogen is to reduce the

secondary nucleation rates.Therefore,these experiments clearly indicate that,besides enhancing the

growth of the diamond nuclei,the atomic hydrogen may have little function for the diamond primary

nucleation [216].Our question is:would the diamond nucleation be really in the strongly unstable

region of the diamond structural phase upon CVD?

The nucleation of CVD diamonds should happen in the D region of Fig.6 on the basis of the

nanosize-induced additional pressure.When we assume the surface tension of diamond is 3.7 J/m

2

[217],the dependence relationships of the nanosize-induced additional pressure based on the Laplace–

Young equation and the size of diamond clusters can be obtained,as shown in the inset of Fig.6.From

the inset,one can see that the additional pressure increases with the crystal particle’s size decreasing.

Notably,when the radius is less than 4 nm,the additional pressure goes up to above 2.0 GPa,which is

above the phase equilibrium line shown as the D region,i.e.,the diamond stable region,in Fig.6.In

other words,the nanosize-induced additional pressure could drive the metastable region (Gregion) of

the diamond nucleation into the newstable region (Dregion) in the thermodynamic equilibriumphase

diagramof carbon.These deductions are supported by the experimental cases fromthe CVDdiamonds

on non-diamond substrates [218–220].For instance,Lee et al.reported that the size of the nuclei of

CVD diamonds on Si substrates is in the range of 2–6 nm[219].Consequently,the nanosize-induced

additional pressure of 1–3 nmradius of the diamond nuclei would be enough to drive the Gregion into

the D region in Fig.6.Therefore,the nucleation of CVD diamonds should happen in the D region in

Fig.6 based on our nanothermodynamic approach.In the next section,we will take into account the

CVD diamond nucleation on silicon substrates as an example to present a quantitative description

based on the aforementioned thermodynamic model of MPNUR [41,221].

According to Eq.(3.34),g = 3.7 J/m

2

[217],V

m

= 3.417 10

6

m

3

mol

1

[45],DV = 1.77

10

6

m

3

mol

1

[45],k

0

= 2.01 10

6

[45],and b

0

= 2.02 10

9

Pa [45],one can obtain the relation-

ship curves between the size of the diamond critical nuclei and the pressure at the temperature of

1300 K upon the CVD diamond case,and it is displayed in Fig.7,in which the inset shows the

dependent relations of the pressure and the critical radius at the given various temperatures.Clearly,

we can see that the radii of the critical nuclei are less than 5 nm in a broad range of the pressure

176 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.6.Carbon thermodynamic equilibrium phase diagram.The G region means a metastable phase region of the diamond

nucleation upon CVD;the D region means a new stable phase region of the CVD diamond nucleation with respect to the

effect of the nanosize-induced additional pressure.The inset shows the relationship between the nanosize-induced additional

pressure and the nuclei size (after [41]).

and temperature.The diamond nucleation upon CVD seems to be in the stable phase region of

diamond due to the driving of the nanosize-induced additional pressure of the diamond nuclei.In

addition,from this ﬁgure,one also can see the very weak dependence of the pressure on the critical

radius.In other words,at the given temperatures,the critical radii are hardly changed with the external

pressure change,because the external pressure is quite small compared with the nanosize-induced

additional pressure.

Based on Eq.(3.35),the value of the surface tension of silicon (1.24 J/m

2

) [45],and the given

parameters above,we display the relationship curves between the pressure and the critical energy of

CVD diamonds at the temperature of 1200 K,as shown in Fig.8,in which the inset displays the

dependent relations of the pressure on the critical energy at the given various temperature.Obviously,

one can see in Fig.8 that the critical energy of the diamond nuclei slowly increases with the pressure

increasing at a given temperature,and approximately remains as unchanged.The case results fromthe

too little external pressure compared with the nanosize-induced additional pressure.These results

indicate that the critical energy of the diamond nucleation upon CVDis quite low(10

16

J),suggesting

that the heterogeneous nucleation of CVD diamonds does not require high forming energy.

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 177

Fig.7.The relationship curves between the radii of the critical nuclei and the pressure at the temperature of 1300 K upon

CVD systems.The inset shows the dependence relation of the pressure and the critical radii at given various temperatures.

One can see that,in faith,the radii of the critical nuclei of diamond upon CVD for a broad range of pressures and

temperatures are less than 5 nm.Namely,the nucleation of CVDdiamond could occur in the stable phase region of diamond

(after [221]).

Fig.8.The relationship curves between the pressure and the critical energy of the nucleation upon CVD diamond at the

temperature of 1200 K,and the inset showing the curves at various given temperatures (after [221]).

Apparently,the lowforming energy of the heterogeneous nucleation of CVDdiamonds implies that it

is not difﬁcult to nucleate diamond,and the diamond nucleation could happen in the diamond stable

region (Gregion) as shown in Fig.6.Based on the analyses above,the diamond nucleation upon CVD

would happen in the diamond stable region in the carbon phase diagram from the point of view of

thermodynamics.In fact,when the size of the crystalline particles is in the nanometer scale,the

additional pressure induced by the curvature of the nanometer-sized particles is so high as to exceed

the equilibrium pressure between diamond and graphite,i.e.,going up to break through the phase

equilibrium boundary line,which means that the additional pressure could drive the thermodynamic

phase region of the diamond nucleation fromthe metastable to the stable as shown in Fig.6.Therefore,

the diamond nucleation upon CVD is not actually the ‘‘thermodynamic paradox,’’ and ‘‘violating the

second lawof thermodynamics.’’ These results indicate that the presence of the atomic hydrogen is not

a vital factor to grow diamonds upon CVD from the viewpoint of the nanothermodynamics above.

However,why did most experimental studies all showthat the atomic hydrogen plays a very important

role in CVD diamonds?In fact,it is recognized experimental evidence that the atomic hydrogen

etching the graphite phase (more etching rate to the graphite phase and less etching rate to the diamond

phase) and helping the sp

3

hybridization of carbon atoms [218].Naturally,the diamond nucleation

could be enhanced in the low-pressure gas,only when the graphite phase forming is restrained or

stopped by the atomic hydrogen or other factors.Thus,the presence of the atomic hydrogen could

increase the rates of the diamond growth.Accordingly,the effect of the atomic hydrogen on the

diamond growth is much more larger than that on the diamond nucleation upon CVD [222].In other

words,the inﬂuence of the atomic hydrogen on the diamond nucleation would be small fromthe point

of view of the experimental investigations involved in how to enhance the diamond nucleation upon

CVD [162].

In conclusion,aiming at a clear insight into the nucleation of CVD diamonds,we studied the

diamond nucleation from the point of the view of a nanoscaled thermodynamics.Notably,these

theoretical results show that the diamond nucleation would happen in the stable phase region of

diamond in the thermodynamic equilibrium phase diagram of carbon,due to the nanosized effect

induced by the curvature–surface tension of the diamond nuclei.In other words,at the nanometer size,

the diamond nucleation is prior to the graphite nucleation in competing growth of diamond and

graphite upon CVD.

4.2.Diamond formation in the hydrothermal synthesis and reduction of carbide systems

Recently,the important progress of the diamond syntheses has been made by HSRC [177–184].

By using diamond seeds,Syzmanski et al.[177] synthesized diamonds by the hydrothermal synthesis

in the different supercritical-ﬂuid systems in 1995,and Gogotsi et al.almost simultaneously prepared

diamonds by using the containing-carbon-element water solution in the hydrothermal synthesis

[178,179].Following Syzmanski and Gogotsi,a few groups have synthesized diamonds by the

hydrothermal synthesis without diamond seeds,in which the non-diamond carbon and various

carbides with chlorine or supercritical-ﬂuid water solution containing hydrogen were used as raw

materials [179–181].Furthermore,without chlorine and supercritical-ﬂuid water solution containing

hydrogen,Lou et al.synthesized diamonds through the reduction of carbon dioxide and reduction of

magnesium carbonate with metallic supercritical-ﬂuid sodium [183,184].However,compared with

the rapid experimental progress of the diamond synthesis by HSRC,the thermodynamic nucleation of

diamond upon HSRC supercritical-ﬂuid systems still has much less theoretical understanding,so far.

For instance,the phase region that diamonds are synthesized by HSRC is in the range of 713–1273 K

and 0.1–200 MPa [179,181–184].Then,the phase region is located below the boundary line between

178 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

diamond and graphite,i.e.,so-called Bundy’s line (B line),in the carbon thermodynamic equilibrium

phase diagram as shown in Fig.9 [45].Note that the carbon phase diagram proposed by Bundy has

been generally accepted,so far.In other words,the diamond nucleation would not be expected to take

place in the phase regions created by HSRCin the carbon phase diagram,because the diamond phase is

metastable and the graphite phase is stable in the phase region mentioned above.Why would the

results of the diamond synthesis in HSRC systems not be consistent with the prediction of the carbon

thermodynamic equilibriumphase diagram?The convincing understanding for this issue has not been

reached yet.To our best knowledge,few studies concerning the thermodynamic nucleation of

diamonds upon HSRC are found in the literature.

To gain a better understanding to the diamond nucleation upon HSRCsupercritical-ﬂuid systems

fromthe point of view of thermodynamics,we proposed the nanothermodynamic analysis to address

the seed-free diamond nucleation upon HSRC based on the thermodynamics of MNPUR mentioned

above.It is noticed that the size of the diamond critical nuclei is limited in the range of several

nanometers upon CVD[219,220],and the supercritical-ﬂuid systems are suggested to have the liquid-

like densities but gas-like properties [223].It is therefore a convincing suggestion that the size of the

diamond critical nuclei should be limited within several nanometers in the HSRC supercritical-ﬂuid

systems [180].According to the established thermodynamic model,we ﬁrst calculated the size and the

forming energy of the critical nucleation of diamond upon HSRC,respectively,in which all data are

from the securable literatures about the diamond synthesis in the HSRC supercritical-ﬂuid systems.

More importantly,our theoretical results are consistent with the experiment data and other calculations

from ﬁrst-principles [180,224,225].

In detail,on the basis of Eq.(3.34) and the aforementioned thermodynamic parameters,we show

the relationship curves between the size of the critical nucleation and the pressure at various

temperatures in Fig.10.Then,it is noticed that the data points of the symbols (~,$,!,&,

and ^) derived from Refs.[183,184,181,179,182],respectively.Clearly,one can see that the size of

the diamond nuclei increases with the pressure increasing at a given temperature,and decreases with

the temperature increasing at the certain pressure in Fig.10.In addition,we can see that the sizes of the

critical nucleation are close to a constant at the pressures below 400 MPa under the condition of the

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 179

Fig.9.Carbon thermodynamic equilibriumphase diagrambased on pressure and temperature.Gregion means a metastable

phase region of diamond nucleation;D region means a new stable phase region of the diamond nucleation by the

hydrothermal synthesis or reduction of carbide under the nanosize-induced interior pressure conditions.The inset shows the

enlarged G and D regions.The data point of the symbols (~,$,!,&,and ^) of the G region derived from Refs.

[183,184,181,179,182],respectively (after [45]).

certain temperature.However,the sizes of the critical nuclei increase quickly when the pressures

exceed 400 MPa.These results indicate that 400 MPa seems to be a pressure threshold for the diamond

synthesis by HSRC,and the corresponding size of the diamond critical nuclei is about 5 nm.Our

theoretical results are not only in excellent agreement with Kraft et al.experimental results,but also

good consistent with Badziag and Winter’s calculations from the ﬁrst-principle [181,224,225].The

ﬁrst-principle calculations suggested that when the size of carbon clusters is in the range of 3–5 nm,

the diamond phase should be thermodynamically more stable than graphite phase [224,225].

In terms of Eq.(3.35) and the given the value of f(u),we display the relationship curves of the

nucleation energy of the diamond critical nuclei and the pressure at the conditions of the various

temperatures and the heterogeneous factor equaling to 0.5,and as the evidence shown in Fig.11.

Similarly,one can see that the forming energy of the critical nuclei increases with the pressure

increasing at the given temperature and the heterogeneous factor,and the values of the nucleation

energy of the critical nuclei decrease with the temperature increasing at a given pressure in Fig.11.In

addition,we can see that the values of the nucleation energy are close to a constant at the pressures

below 400 MPa under the certain temperature.However,the values of the nucleation energy of the

critical nuclei greatly increase when the pressures exceed 400 MPa.Importantly,these results show

180 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.10.The relationship curves between the size of the critical nucleation and the pressure at various temperatures.The data

point of the symbols (~,$,!,&,and ^) derived from Refs.[183,184,181,179,182],respectively (after [45]).

Fig.11.The relationship curves of the critical energy and the pressure at various temperatures (the heterogenous factor is

0.5) under considering the nanosize-induced interior pressure condition.The inset shows the relationship curves of the

critical energy and the heterogeneous factor at the given pressure and temperature.The data point of the symbols (~,$,!,

&,and ^) derived from Refs.[183,184,181,179,182],respectively (after [45]).

that the diamond nucleation upon HSRC need not be the relatively high nucleation energy when the

pressure is less than 400 MPa,or the size of the critical nuclei is less than 5 nm.Apparently,the low

nucleation energy of diamond in HSRCsuggests that it is not difﬁcult for the diamond nucleation to take

place upon HSRC.Therefore,the diamond nucleation of HSRC seems to happen in the stable phase

region of diamond,i.e.,the D region as shown in Fig.9,based on our thermodynamic theory.As a

comparison,we give the relationship curves of the nucleation energy of diamond and the heterogeneous

factor at the given pressure and temperature in inset of Fig.11.From the inset,one can see that the

nucleation energy increases with the heterogeneous factor increasing.Eventually,fromFigs.10 and 11,

we can predict that 400 MPa should be the threshold pressure for the diamond synthesis by HSRCin the

metastable phase region of diamond in the carbon phase diagram.The diamond synthesis would thus

hardlytake place inthe thermodynamic metastable phase regionof diamondinthe carbonphase diagram

when the pressure of HSRCexceeds 400 MPa.In fact,all pressures carried out in the diamond synthesis

by HSRC are less than 400 MPa in the present literatures.

More recently,it has been reported that various kinds of precursors containing carbon such as

SiC,CO

2

,MgCO

3

,etc.,are used to synthesize diamonds in hydrogen or hydrogen-free systems in the

thermodynamic metastable region of diamond of the carbon phase diagram by HSRC [182–184].

From these experiments mentioned above,we can deduce that hydrogen is not essential for the

diamond nucleation upon HSRC.Similar evidence has existed in CVD diamonds.On the other hand,

the nucleation kinetics of the diamond synthesis fromthe SiC reduction in HSRC suggested that Si is

extracted from SiC to result in the residual carbon structures forming carbon atoms in the sp

3

hybridization by kinetic regime.In addition,the diamond formation fromCO

2

in HSRCcould deduce

that the reductant ﬁrst combines with oxygen,and then carbon atoms form sp

3

hybrid bonds by

complicated chemistry and physics processes.Therefore,it should be noted that the diamond

nucleation upon HSRC is the relatively complicated chemical and physical process just like

CVD.The detailed kinetics has not been fully understood yet.

Following Gleiter [226,227],many chemical and physical routes under the conditions of the

moderate temperatures and pressures are generally used to synthesize nanocrystals with metastable

structures.However,the corresponding bulk materials with the same metastable phases are prepared

under the conditions of the high temperatures or high pressures.Deﬁnitely,the nanosized effect of nano-

crystals should be responsible for the formation of these metastable structures at the nanometer scale.

The above theoretical results display that the diamond nucleation upon HSRC would happen in

the stable phase region of diamond in the thermodynamic equilibrium phase diagram of carbon.

Furthermore,the threshold pressure of 400 MPa is predicted for the diamond synthesis in the

metastable phase region of diamond by HSRC.

4.3.Diamond nanowires growth inside nanotubes

One-dimensional nanostructures such as wires,rods,belts,and tubes have become the focus

of intensive research owing to their unique applications in mesoscopis physics and fabrication of

nanoscale devices.For instance,they not only provide a good system to study the electrical and

thermal transport in one-dimensional conﬁnement,but also are expected to play an important role in

both interconnection and functional units in fabricating electronic,optoelectronic,and magnetic

storage devices with nanoscaled dimension [228].Recently,one-dimensional nanostructures of

diamond have received intensively increasing interesting in theoretical [229–234],even though

any successful syntheses of diamond nanowires (DNWs) have not been found in the literature yet.For

example,diamond nanorods are expected to be an important and viable target structure for synthesis,

due to stronger than fullerene nanotubes [231].In this section,we therefore propose a thermodynamic

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 181

nucleation and kinetic growth approach at the nanoscale under the consideration of the effect of the

surface tension induced by the nanosized curvature conditions.

Based on the nanothermodynamic nucleation [41–48],we herein theoretically perform the

formation of DNWs inside nanotubes upon CVD.This theoretical model is formulated based on the

assumptions:(i) the nanoscale nuclei are perfectly spherical without the structural deformation

comparison with the bulk one;(ii) the nanoscaled nuclei are mutually non-interaction.The schematic

illustration of a DNWgrown in a nanotube upon CVD is shown in Fig.12.When the reactant gases

CH

4

and H

2

ﬂow along the nanotube,the carbon clusters are condensed on the inner wall in the

nanotube by a series of the surface reactions and diffusions.Sequentially,the diamond nucleation will

occur inside the nanotube by the phase transition.Now,we discuss the nucleation of carbon clusters

with diamond structure inside nanotubes.Thermodynamically,the phase transformation is promoted

by the difference of the free energies.The Gibbs free energy of a phase can be expressed as a function

of the pressure and temperature,and determined by a general coordinate or reactive coordinate [174].

The Gibbs free energy difference of a cluster can be expressed as

DG ¼ ðs

sc

s

sv

ÞS

1

þs

cv

S

2

þDg

v

V (4.1)

where s

sc

,s

sv

,and s

cv

are the substrate–nucleus,the substrate–vapor,and the nucleus–vapor interface

energy,S

1

and S

2

the corresponding interface areas (as shown in Fig.12(b)),V the volume of the

182 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.12.Schematic illustration of a DNWnucleation and grown inside a SiNT.(a) Adiamond nucleus on the inner wall of a

nanotube.(b) The cross-section of the case (a),S

1

and S

2

are the areas of the substrate–nucleus and nucleus–vapor interfaces,

respectively.(c) A diamond nucleus on the surface of a Si substrate.(d) A DNW grown inside a nanotube (after [47]).

diamond clusters,and Dg

v

is the Gibbs free energy difference per unit volume,which can be expressed

by [47]

Dg

v

¼

RT

V

m

lnðP=P

e

Þ

(4.2)

where P and T are the pressure and temperature upon CVD,and ln(P/P

e

) = 0.8 [46].P

e

is the

equilibrium-vapor pressure of diamond,R the gas constant,and V

m

is mole volume of diamond.

Further,considering the effect of the surface tension induced by the nanoscaled curvatures of the

diamond nuclei and nanotubes upon CVD,applying the Laplace–Young equation and the Kelvin

equation,Dg

v

can be expressed as:

Dg

v

¼

1

2

RT

V

m

ln

P

P

e

þs

cv

1

r

þ

1

r

0

(4.3)

where r and r

0

are the radii of the nanotube and the diamond cluster,respectively.Therefore,

substituting these relations,i.e.,Eqs.(4.2) and (4.3),into Eq.(4.1),we attain the Gibbs free energy of

the formation of diamond clusters inside nanotubes.Note that one can see that the values of S

1

,S

2

,and

V in Eq.(4.1) can be determined when the radii of the nanotube and the spherical nuclei are given.

However,it is not easy in our case to obtain the analytical expressions of S

1

,S

2

,and V.Accordingly,it

is not easy to deduce the analytical expression of the critical radius and the forming energy of an

atomic cluster with diamond structure inside a nanotube.Therefore,in the case [48],we calculated

the critical radius and the forming energy of a diamond cluster inside the nanotube by a numerical

method.

It is well known that the thermodynamic nucleation just provides the probability for the formation

of DNWs inside nanotubes upon CVD.On the other hand,the kinetic growth will play a key role in the

achievement of the probability,when the thermodynamics operates.We therefore develop a growth

kinetic approach to the growth of DNWs inside nanotubes upon CVD,based on the growth kinetics of

one-dimensional nanostructures inside nanotubes [46],which originates fromWilson–Frenkel growth

law [235,236].Generally,the growth velocity V

s

of the crystalline nucleus can be expressed as [47]:

V

s

¼ hn exp

E

a

RT

1 exp

jDgj

RT

(4.4)

where h,n,and E

a

are the lattice constant of the crystalline nucleus in the growth direction,the thermal

vibration frequency,the mole adsorption energy of adatoms attached at the surface sites,the R,and T

are deﬁned by Eq.(4.3).The Dg is the Gibbs free energy difference per mole.According to Eq.(4.3),

Dg can be denoted by

Dg ¼

1

2

RT ln

P

P

e

þ

s

cv

M

R

0

r

þ

s

cv

M

rr

(4.5)

where M,R

0

,r,and r are the mole mass of diamond,the curvature radius of the diamond nucleus,the

radius of the nanotube,and the density of diamond,respectively.

In order to validate the models above,based on the sufﬁcient securable thermodynamic

parameters,we take DNWs growth in silicon nanotubes (SiNTs) upon CVD as an example to check

its operation.From Fig.12(d),one can see that R

0

= r/cos u,and cos u ¼

s

sv

s

sc

s

cv

,in which u is the

contact angle between the diamond nucleus and the wall of a SiNT.When the growth direction of

DNWs is assured along the (1 0 0) direction,h,n,and E

a

are 0.218 nm,2.5 10

13

Hz,and

2.4 10

5

J/mol,respectively [237–239].According to Eqs.(4.1) and (4.3),one can obtain the

comparison curves of the free energy of the diamond nucleation between inside a SiNTand on a ﬂat Si

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 183

substrate under various given temperature conditions in Fig.13.Meanwhile,the dependence of the

critical radius of the diamond nucleus inside a SiNT on the radius of the SiNT under various given

temperature conditions is shown Fig.14.Clearly,one can see,comparing the nucleation barrier of the

diamond nuclei inside SiNTs with that on a ﬂat Si substrate,the former is much less than the latter,

fromFig.13.In other words,the diamond nucleation inside SiNTs would be preferable to that on the

ﬂat Si substrates,due to the effect of the surface tension induced by the nanosized curvatures of the

nanotubes and the critical nuclei.On the other hand,it can be found fromFig.13 that with increasing

the substrate temperature,the nucleation barrier and the critical radius of the diamond nucleation (the

R corresponding to peak value in Fig.13) will increase.The result indicates that the decrease of the

substrate temperature (in the limited range) is favorable for the nucleation of diamond.Importantly,

these results are consistent with the experiment cases of the diamond nucleation on a ﬂat Si substrate

by CVD[240–242].Furthermore,we can see that the radius of the diamond critical nucleus increases

with the radius of SiNTs increasing fromFig.14.Apparently,these results indicate that the diamond

nucleation is relatively easy inside SiNTs with fewer radii.Similarly,one can see from Fig.14 that

with increasing of the substrate temperature,the critical radius of the diamond nucleation will

184 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.13.The comparison of the free energy of the diamond nucleation between in a SiNT and on a ﬂat Si substrate under

various given temperature conditions,and the radius of a SiNT,r = 5 nm (after [47]).

Fig.14.The dependence of the critical radius of the diamond nucleus inside a SiNTon the radius of the SiNTat various given

temperatures (after [47]).

increase,suggesting that the diamond nuclei are relative stable at the lowsubstrate temperature (in the

limited range).Note that these theoretical results are in agreement with the experiment cases of

diamond nucleation on a ﬂat Si substrate [240–242].

According to Eqs.(4.4) and (4.5),we can attain the relationship curve between the growth

velocity of DNWs and the radius of SiNTs in Fig.15.Clearly,fromFig.15,one can see that the growth

velocity of DNWs inside the SiNTs increases with the radius of the SiNT decreasing at given

deposition temperatures.Deﬁnitely,when r <10 nm,the increasing of the growth velocity goes to

much high with the radius of the SiNTcontinually decreasing.When r >10 nm,and the decreasing of

the growth rate becomes apparent.In fact,the growth of DNWs seems not correlative with the size of

SiNTs when the radius of SiNTs goes to too large.In addition,the growth rate of CVDdiamond ﬁlms

based on the Wilson–Frenkel equation is also shown in Fig.15.Clearly,the calculated value of the

growth rate of CVD diamond ﬁlms on a ﬂat Si substrate is in good agreement with the experiment

cases [243–245].Further,it can be seen that the growth rate of DNWs inside SiNTs is close to the

growth rate of CVDdiamond ﬁlms when the radius of SiNTs is more than 100 nm.In other words,the

growth rate of DNWs inside SiNTs is nearly the same as one of CVD diamond ﬁlms on a ﬂat Si

substrate when the radius of the SiNTs is large enough.

In summary,based on a thermodynamic nucleation on nanoscale,we found out that the diamond

nucleation inside SiNTs would be energetically preferable to that on the ﬂat surface of silicon wafers,

due to the nanosized effect induced by the curvature of the nanotubes and the critical nuclei upon

CVD.Meanwhile,in kinetic,the growth rate of DNWs inside SiNTs would go to much high once the

diamond nuclei forming inside SiNTs.Therefore,considering the fabrication of the near-perfect one-

dimensional nanoscaled device consisted of DNWs and SiNTs,we expected SiNTs to be a template to

grow the DNWs by CVD.

5.Cubic boron nitride nucleation in the unstable regions of the structural state

Cubic BN,a pure artiﬁcial III–Vcompound and structural properties similar to cubic diamond,as

shown in Fig.16,has attracted great interest due to its outstanding physical and chemical natures such

as the second to diamond hardness,high thermal stability,and chemical inert [246],since it was for the

ﬁrst time synthesized successfully using the high-pressure and high-temperatures method with a help

of a suitable catalyst (which is similar to the synthesis of diamond) in 1957 by Wentorf [247].In the

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 185

Fig.15.The relationship curve between the growth velocity of DNWand the radius of SiNTs,and the comparison with the

case of CVD diamond ﬁlms on the Si substrate (after [47]).

past several decades,many methods have been developed to create c-BNexcept HPHT,such as CVD

[248–252],pulsed-laser deposition (PLD) [253–259],physical vapor deposition (PVD) [260–264],

hydrothermal synthesis [265–270],pulsed-laser induced liquid–solid interface reaction [271,272],and

direct current arc discharge method [273],and so on.Most experimental preparations take place above

the unstable phase region of c-BN in the thermodynamic equilibriumphase diagramof boron nitride.

However,to date,thermodynamics of the nucleation of c-BN remains much less understood and

largely relies on trial and error.In this section,we will take the c-BN nucleation upon CVD and

supercritical-ﬂuid systems as examples to quantitatively describe the nucleation thermodynamics in

the light of the MPNUR model.

5.1.Nucleation of CVD cubic boron nitride

Cubic BN ﬁlms are typically grown as a thermodynamic metastable phase by the means of the

low-pressure CVD over the last 20 years [193,274].Although much signiﬁcant progress has been

made in the intensive research and developed in the past few years,the preparation of the single

crystalline c-BN ﬁlms still remains a great challenge for physicists and materials scientists [275].

Actually,the nucleation of CVD c-BN is a complicated chemical and physical process.In order to

provide a qualitative description of the c-BNnucleation,six different models have ever been proposed

so far,respectively,i.e.,the compressive stress model [276–278],the dynamic stress model [279–282],

the preferential sputter model [283,284],the subplantation model [285–288],the cylindrical thermal

spike model [289–291],and the nanoarches model [292].However,none of those models could clearly

give a completely satisfying picture of the c-BNformation,and each model just tended to focus on one

aspect of the complicated process of the nucleation of CVD c-BN [275].There have been several

excellent review papers surveying current theories of the c-BN nucleation [274,275,292],in which

they pointed out that the c-BN nucleation occurs by a mechanism that the structural changes are

accomplished by the high compressive stress of several GPa and the growth of a layered structure

consisting of an amorphous (a-BN) interface.After the interlayers are grown on a substrate,the

hydrostatic component of the compress stress is sufﬁcient to place the growth conditions inside the c-

BNstable zone (C zone of Fig.17) in the range from500 to 1300 K,using the Corrigan–Bundy’s line

[293] to deﬁne the phase boundary [277],as shown in Fig.17.Moreover,a lot of literature employs the

compress stress model to explain the nucleation of c-BN [276–278,294,295].However,few studies

186 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.16.Sketch map of c-BN of a diamond-like material with a zinc-blende crystal structure.

have involved in the nucleation thermodynamics at the nanometer size.In this review,we thus focus on

the thermodynamic description of the nucleation of CVD c-BN in terms of the model of MPNUR.

It is well known that CVD could be considered to be close to the thermodynamic equilibrium

process,the typical thermodynamic parameters of CVD c-BN are that the pressure is a few Torr and

the temperature is about 500–1300 K[278,296–298].In the boron nitride thermodynamic equilibrium

phase diagram(Fig.17),the general thermodynamic region of the c-BNgrown upon CVDis shown as

the H region,which belongs to the metastable region of c-BN phase,i.e.,the stable region of h-BN

phase.Under the assumption of spherical,isotropic c-BN nanocrystals,the size-induced additional

pressure would drive the metastable phase into the stable region of c-BN phase (C region of Fig.17).

According to the Laplace–Young equation (the surface tension g = 4.72 J/m

2

for c-BN[299]),we can

obtain the dependence of the radius of the c-BNnuclei on the additional pressure,as shown in the inset

of Fig.17.Distinctly,one can see that the additional pressure increases with the crystal particle’s size

decreasing.Notably,in the size range below several nanometers,the additional pressure goes up to

above several gigaPascal,which is above the C–B line shown as the C region [293],i.e.,the stable

region of c-BN phase in Fig.17.

In principle,Eq.(3.35) should give the relationship curves of the forming energy of the nuclei and

the temperature.However,we could not obtain the curve due to the uncertain surface energy of the

interlayers.In order to ﬁnd out the relationship curve of the forming energy and the temperature under

the conditions of the given additional pressure,we supposed that c-BN directly nucleates on Si

substrates (the value of the surface energy of silicon for 1.24 J/m

2

[299]).Thus,we attain the forming

energy curves of the c-BNnuclei with the size effect,as shown in Fig.18 [299].Clearly,we can see in

Fig.18 that the forming energy of the c-BNnuclei decreases with the temperature increasing at a given

pressure,and increases with the pressure increasing.Therefore,these results showthat the nucleation

of CVDc-BNdoes not need the relatively high forming energy.Apparently,the lowforming energy of

the c-BN nucleation upon CVD implies that it is not difﬁcult to the c-BN nucleation,and the c-BN

nucleation seems to happen in the c-BN stable phase region (C region) as shown in Fig.17.

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 187

Fig.17.Boron nitride thermodynamic equilibriumphase diagramand the relationship curve between the nanosize-induced

additional pressure and the nuclei size (as an inset).Hregion means a metastable phase region of the nucleation upon CVDc-

BN;C region means a new stable phase region of the c-BN nucleation with respect to the effect of the nanosize-induced

additional pressure (after [299]).

On the other hand,according to Eq.(3.34),one can calculate the relationship curves of the

forming energy and the heterogeneous factor of f(u) under the conditions of the given additional

pressure and temperature,as shown in Fig.19 [299].It is noticed that the different heterogeneous

factors f(u) represent the different interlayers between the c-BNnuclei and the substrate.FromFig.19,

we can ﬁnd that the forming energy increased with f(u) increased.The result indicates that the low

interfacial energy between the nuclei and the interlayer is advantageous to the nucleation of c-BN.

Thus,the c-BN nucleation would be preferred on those substrates that have lower lattice mismatch

with c-BN,which would be one of the physical origins of the interlayers grown for CVD c-BN.

Actually,a characteristic nucleation sequence a-BN!textured h-BN (c-axis parallel to the sub-

strate)!c-BNhas been established to precede the c-BN-growth,observed by Kester et al.[300,301],

and their experimental results indicated that the nucleation of c-BNon the surface of the interlayer not

only relies on the combined effect of a 2:3 lattice matching between h-BNand c-BN,but also concerns

in h-BN density on the surface of interlayers,i.e.,so-called ‘‘h-BN densify.’’ Naturally,‘‘h-BN

densify’’ would result in the surface energy of the interlayer increasing,and then,the heterogeneous

factor (f(u)) would be decreased from our calculations.

Since the phase transition is determined quantitatively by the probability of the h-BN molecules

crossing a potential barrier of intermediate phase [174].According to Eq.(3.36) and the securable

188 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.18.Under the condition of the direct nucleation of c-BN on Si substrates,the relationship curve of the forming energy

and the temperature in various nanosize-induced additional pressures (after [299]).

Fig.19.The relationship curves between the forming energy and the heterogeneous factor under conditions of the nanosize-

induced additional pressures and various temperatures (after [299]).

thermodynamics parameters (see Ref.[299]),one can obtain the probability curves of the h-BN–c-BN

transition in the new stable region of the temperature–pressure phase diagram of boron nitride when

r = 4.0 nm as shown in Fig.20 [299].It can be seen in Fig.20 that the probability of the phase

transition fromh-BNto c-BNis about 1.0 10

10

to 1.0 10

9

in the newstable region (Cregion in

Fig.17).Actually,the probabilities of the h-BN to c-BN transition,1.0 10

10

to 1.0 10

9

,are

really lowin the Cregion.Thus,the result implies that the c-BNnucleation density should be very low

during CVD,although it could happen.In fact,the deduction is in agreement with the experiment

evidence.(Attempts to grow c-BN by simple chemical process alone failed so far [302,303].)

Therefore,in order to enhance the probabilities of the h-BN to c-BN transition and promote the

c-BN nucleation density,many ways,e.g.,electron cyclotron resonance,inductively coupled plasma

(ICP),and radio-frequency (rf),etc.,are employed to assist CVD [274].

Fromthe discussions above,one can see that the c-BN nucleation seems to happen in the stable

region of c-BN phase in the thermodynamic equilibriumphase diagramof boron nitride based on the

thermodynamics of MPNUR,i.e.,the c-BN nucleation would be prior to the h-BN nucleation in

competing growth of c-BN and h-BN upon CVD.

5.2.c-BN nucleation in high-pressure and high-temperature supercritical-ﬂuid systems

Recently,the important progress of the c-BN synthesis by HPHT,which was developed by

Solozhenko and Singh et al.,seems to break through the general accepted thermodynamic equilibrium

phase diagramof boron nitride proposed by Corrigan and Bundy.Solozhenko and Singh et al.for the

ﬁrst time synthesized c-BNunder the supercritical conditions using non-conventional catalysts such as

the volatile hydrazine NH

2

NH

2

and MgB

2

(so-called HPHT supercritical-ﬂuid systems) in the phase

region that the pressure is in the range of 1.8–3.8 GPa and the temperature is in the range of 1200–

1600 K by HPHT [304–309].Importantly,these phase regions are located below the C–B line of the

BN thermodynamic equilibriumphase diagram,in which the c-BN phase is metastable and the h-BN

phase is stable.In other words,the c-BN synthesis would not be expected upon HPHT in these phase

regions (shown in Fig.21).For this issue,according to the experiments and the theoretical calculations

based on a series of hypotheses,Solozhenko and coworkers suggested that the C–B line should move

down in the boron nitride thermodynamic equilibrium phase diagram developed by Corrigan and

Bundy.Further,they proposed a new BN thermodynamic equilibriumphase diagramto substitute for

the Corrigan–Bundy equilibrium phase diagram [304,310–312].Would the Corrigan–Bundy equili-

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 189

Fig.20.The probability of the h-BNto c-BNtransition with respect to the effect of the nanosize-induced additional pressure

in the new stable phase region (C region) of the c-BN nucleation upon CVD (above the C–B line) (after [299]).

brium phase diagram be really outdated for the c-BN synthesis by HPHT?In fact,it is an essential

issue involved in the fundamental thermodynamics in the HPHT supercritical-ﬂuid systems of the

c-BNsynthesis.To our best knowledge,there are hardly any thermodynamic descriptions concerned in

the HPHT supercritical-ﬂuid systems,as the full understanding of the complicated interactivities

during the non-conventional catalyst/solvent synthesis has not been obtained in detail yet.Therefore,

the developed thermodynamic approach at the nanometer scale is used to address the formation of

c-BN upon the HPHT supercritical-ﬂuid systems.

In fact,the nucleation and growth of c-BNare complicated in a ﬂux of another material (so-called

‘‘catalyst’’) upon the HPHT supercritical-ﬂuid systems.Generally,c-BN is considered to be the

spontaneous-crystallization in the so-called conventional ‘‘solvents–catalysts system’’ [313].Actu-

ally,no matter what the conventional solvents–catalysts systemor the supercritical-ﬂuid systems,the

nucleation and growth of c-BN must meet simultaneously the conditions as follows:(i) the raw

material and catalyst molten in the system,(ii) the high-supersaturation raw material in the solvent

catalysts,and (iii) the ambient pressure (including the nanosize-induced additional pressure) and

temperature of the spontaneous-crystallization of c-BN clusters above the C–B line (or the sponta-

neous-crystallization of c-BNclusters is unstable).However,the results of Solozhenko and Singh et al.

[305–309] showed that the nucleation and growth of c-BN in a wide temperature and pressure ranges

in the supercritical-ﬂuid systems (as shown in Fig.21,the H region).Then,the case would be

impossible in the conventional solvents–catalysts system.Thus,it is reasonable that the structure of the

resulting solutions and the mechanismof the BNcrystallization fromthese solutions are expected to be

heavily dependent on the ﬂuid phase composition.Naturally,the results are incompatible with the

general accepted phase diagram proposed by Corrigan and Bundy.

On the other hand,since c-BN was synthesized by HPHT,many theoretical models have

discussed the nucleation and growth of c-BN upon HPHT.For example,the solid–solid transition

model thinks that h-BN melt in the catalyst solvent under the certain conditions of the pressure and

temperature,and then,the formation of the new solvent (BN-rich).Subsequently,the temperature

190 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.21.Boron nitride thermodynamic equilibriumphase diagramand the relationship curve between the nanosize-induced

interior pressure and the nuclei size (as an inset).H region means a metastable phase region of c-BN nucleation;C region

means a newstable phase region of the c-BNnucleation with respect to the effect of nanosize-induced interior pressure (the

data of the square symbols of the H region derived from Ref.[311]) (after [42]).

increasing leads to the formation of a second compound that is more BN-rich compared with that

mentioned above.Finally,the new BN-rich compound formed under the high pressures and

temperatures becomes instable and decomposes into c-BN and another products (unknown structure

and composition yet) by the fast solid–solid transition process [314].However,very recently,

Solozhenko [315] reported that the formation of any crystalline intermediate phases was not observed

fromthe beginning to the ending of the c-BNcrystallization in NH

4

F–BNsystem,and only h-BNand a

melt coexist in the systemby in situ measuring using X-ray diffraction with synchrotron radiation.In

addition,as aforementioned above,the supercritical-ﬂuid may have liquid-like densities but gas-like

properties [223].These cases made us revise the solid–solid transition model and allowed us to draw

some conclusions about the mechanismof the nucleation and growth of c-BNin the supercritical-ﬂuid

systems based on the thermodynamic model of MPNUR.

According to Eq.(3.34) and the given macroscopic thermodynamic parameters [42],one can

deduce the relationship curves of the pressure and the critical radius of the c-BNnuclei at various given

temperatures,as shown in Fig.22.Clearly,we can see that the critical radius ranges from2.8 to 4.8 nm.

Importantly,these results are in agreement with the calculation values by the chemical potential

method [42].Furthermore,Fig.22 displays that the critical radius increases with the pressure

increasing at a given temperature,and increases with the temperature decreasing at a given pressure.

Actually,these results just indicate that the nanosize-induced additional pressure plays an important

role on the c-BN nucleation upon supercritical-ﬂuid systems.

In the light of Eq.(3.35),the dependence of the forming energy of the c-BNnuclei on the pressure

at various given temperatures can be obtained,as shown in Fig.23.Obviously,one can see that the

forming energy of the c-BNnuclei increases,when the total pressure (external and additional pressure)

is close to the C–B line.Thus,the result is in agreement with CNT.Namely,on the C–B line,c-BN

cannot nucleate.Importantly,these results show that the heterogeneous nucleation of c-BN does not

need the high forming energy in the supercritical-ﬂuid systems.Apparently,the lowforming energy of

the heterogeneous nucleation of c-BN implied that it is not difﬁcult to the c-BN nucleation in

the HPHT supercritical-ﬂuid systems,and the c-BN nucleation would happen in the stable phase

region (C region) of c-BN as shown in Fig.21.

According to Eq.(3.36),we can obtain the probability curves of the h-BN to c-BN transition in

the newstable region of the temperature–pressure phase diagramof boron nitride when r = 1.6 nmas

shown in Fig.24 [44].The f

c

constant curves display a ‘V’-shape:one side approaches the Corrigan–

Bundy line and the other stands nearly vertical.Additionally,fromFig.24,we can see that the values

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 191

Fig.22.The dependence of the critical radius on the pressure at various given temperatures upon the supercritical-ﬂuid

systems (after [44]).

of the phase transition probability fromh-BNto c-BNare in the range of 1.0 10

8

to 1.0 10

7

in

the new stable phase region (C region in Fig.21).Actually,the probabilities of the h-BN to c-BN

transition,1.0 10

8

to 1.0 10

7

,are really low in the C region.Thus,these results show that the

c-BN nucleation density should be low in the HPHT supercritical-ﬂuids system,although it could

happen.In other words,the c-BN nucleation would not be favored in the HPHT supercritical-ﬂuid

systems.In fact,these deductions are in agreement with the experiment evidence [316].For instance,

recently,some attempts to repeat the Solozhenko’s experiment in the HPHT supercritical-ﬂuid

systems did not meet with success by Gonna et al.[316].On the other hand,based on the above

thermodynamic model,the dependence of the probability of the h-BN to c-BN transition on the

pressure under the conditions of T = 1500 Kand r = 1.6 nmis shown in Fig.25 [44].It is clearly seen

that the shape of these curves in Fig.25 is similar to that of the Arrhenius line,i.e.,the probability of

the h-BNto c-BNphase transition is in agreement with the Arrhenius rule.Meanwhile,we can see that

the f

c

increases fast when the pressure is in the range 1.8–2.4 GPa,and then,it goes to saturation with

further increasing of the pressure.

In addition,we calculate the probability of the phase transition fromh-BN to c-BN in the HPHT

supercritical-ﬂuid systems without the consideration of the nanosize-induced additional pressure (i.e.,

192 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.23.The relationship curves between the external pressure and the forming energy upon the supercritical-ﬂuid systems

(after [44]).

Fig.24.The probability of the phase transition from h-BN to c-BN upon the HPHT supercritical-ﬂuids system under the

consideration of the nanosize-induced additional pressure conditions (i.e.,the phase transition probability of the C region in

Fig.21) (after [44]).

the phase transition probability of the Hregion in Fig.21),as shown in Fig.26 [44].Interestingly,from

Fig.26,one can see that the values of the phase transition probability are in the range of 10

10

to 10

9

in the H region.Furthermore,one can see that the area under the C–B line in the H region is not a

superposition with the f

c

constant curves.Namely,the c-BN nucleation would be hardly expected in

the area under the C–Bline in the Hregion based on the deductions above.These results imply that the

experimental synthesis of c-BN in the H region seems impossible.Accordingly,the reasonable phase

region of the c-BNnucleation upon the HPHTsupercritical-ﬂuid systems should be the Cregion rather

than the H region.

On the other hand,the dependence of the probability on the external pressure without taking into

account the nanosize-induced additional pressure is shown in Fig.27,when the temperatures are in the

range of 1300–1500 K [43].Actually,one can see that,under the condition of the pressure below

3.5 GPa,the probability of the phase transformation is close to zero in the temperatures range from

1300 to 1500 K.Additionally,we can see that the threshold pressure is 3.5–4.0 GPa in the

temperatures range from 1300 to 1500 K.Distinctly,the result is incompatible with Solozhenko

et al.’s experimental data [317].However,it is in excellent agreement with the Corrigan–Bundy

C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 193

Fig.25.The dependence relationship of the phase transition probability of h-BN to c-BN on the pressure upon HPHT

supercritical-ﬂuids system under certain temperature (T = 1500 K) and the radius of the nucleation (r = 1.6 nm) conditions

(after [44]).

Fig.26.The probability of the phase transition fromh-BN to c-BN upon the HPHT supercritical-ﬂuids system without the

consideration of the nanosize-induced interior pressure conditions (i.e.,the phase transition probability of the H region in

Fig.21) (after [44]).

equilibriumphase diagram.In order to clarify the puzzle,based on the nanothermodynamic nucleation

proposed by us,we obtain the threshold pressure in the temperatures range above again,and

these results are shown in Fig.28(a and b) [43].One can see clearly that the threshold pressures

are 2.3–3.0 GPa (r = 2.8 nm) and 2.4–3.0 GPa (r = 3.2 nm),respectively.These results indicate that

the threshold pressure decreases with the size of the critical nuclei of c-BN decreasing.These results

are in excellent agreement with the experimental data [317].

In conclusion,based on the nanothermodynamic analysis,our approach provided a clear physical

and chemical insight into the c-BN nucleation in the supercritical-ﬂuid systems.These theoretical

results indicate that the c-BNnucleation would actually occur in the stable phase region of c-BNin the

boron–nitride phase diagram developed by Corrigan and Bundy.

6.Summary

Thermodynamics of the metastable phases nucleation in the strongly unstable regions of the

metastable structural states in the thermodynamic equilibrium (P,T) phase diagram is reviewed and

194 C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202

Fig.27.The relationship curves of the nucleation probability of c-BN vs.the pressure in the given temperatures without

taking into account the nanosize-induced interior pressure conditions (after [43]).

Fig.28.The relationship curves of the nucleation probability of c-BNvs.the pressure in the given temperatures under taking

into account the nanosize-induced interior pressure conditions:(a) the size of the critical nuclei is 2.8 nmand (b) the size is

3.2 nm (after [43]).

assessed.It is well known that the beginning of the nucleation reactions involves the nanometer scale;

thus,it is important to consider the hierarchy of the phase stability and the operation of the kinetic

limitations at the nanometer size.With regard to the relative thermodynamic stability,the well-known

Laplace–Young formula (simple capillary theory) provides a building block for the comparison.Thus,

the equation indicates the relative increase in the Gibbs free energy,for gas–liquid or liquid–solid

equilibrium,due to a ﬁne size scale.Recently,we developed a quantitative thermodynamic model at

the nanometer scale based on the Laplace–Young formula and the thermodynamic equilibriumphase

diagram to describe the thermodynamic phenomenon of the metastable phase nucleation in the

strongly unstable regions of the metastable structural states.In this approach,free of any adjustable

parameters,the quantitative nanothermodynamic descriptions of MPNUN are attained by the

appropriate extrapolation of the phase equilibrium(P,T) line of the generally accepted thermodynamic

equilibrium phase diagram and the macroscopic thermodynamic data.The established nanothermo-

dynamic theory seems to open up a newavenue to understand MPNUR.However,we point out that the

developed nanothermodynamic approach is universal and not only applicable to MPNUR.Very

recently,we have extended the nanothermodynamic theory to address the nucleation of the metastable

phase in the stable region of the metastable structural states in the corresponding thermodynamic

equilibrium phase diagram.For instance,we have elucidated the nucleation of diamond and c-BN

nanocrystals upon the pulsed-laser ablation in liquid (i.e.,PLIIR) [318,319],in which the diamond

nucleation takes place in the stable phase region of diamond in the carbon phase diagram.We also

extended the nanothermodynamic approach to study the nucleation and growth of one-dimensional

structures on the basis of the vapor–liquid–solid mechanism(VLS).For example,using the established

nanothermodynamic analysis,we have not only theoretically predicted the thermodynamic and kinetic

size limit of nanowires upon the catalyst assistant CVD [320],but also proposed the nucleation

thermodynamic criteria and diffusion kinetic criteria for the issue of catalyst nanoparticles on

nanowires tip or substrate [321].Therefore,these new achievements of the nanothermodynamic

theory deﬁnitely indicate that the theoretical tool could be expected to be a universal approach to

elucidate the nucleation and growth of materials at the nanometer size.

Acknowledgments

The National Science Foundation of China under Grants Nos.50072022,90306006,and

10474140,the Distinguished Creative Group Project of the National Natural Science Foundation

of China,and the Natural Science Foundation of Guangdong province under Grant No.036596

supported this work.The authors are grateful to Dr.J.B.Wang,Dr.Q.X.Liu,and Dr.C.Y.Zhang,who

ever worked in G.W.Yang’s group and made the important contributions to the research ﬁeld covered

by this review.Additionally,the authors are greatly grateful to Professor N.S.Xu for the strong support

and stimulation discussions.

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