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 (socalled nanothermodynamics).Note that our nanothermodynamics,by taking into account
the sizedependence 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 (cBN) 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,cBN nucleation upon CVD has been investigated.Secondly,we
investigated the cBN nucleation taking place in the highpressure and hightemperature supercriticalﬂuids systems
under conditions of the lowthresholdpressures (<3.0 GPa) and lowtemperatures (<1500 K),and predicted the
threshold pressure of the formation of cBN in the highpressure and hightemperature supercriticalﬂuids 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.
Email address:stsygw@zsu.edu.cn (G.W.Yang).
0927796X/$ –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 densityfunctional 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 highpressure and hightemperature 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 highpressure phases can be kept in the metastable states
under conditions of the ambient pressure and temperature.The bestknown examples are diamond and
cubic boron nitride (cBNor cubic BN),which are metastable structures compared with their graphite
and hexagonal boron nitride (hBN) partners.However,many chemical and physical routes under
moderate pressure and temperature are generally used to synthesize these highpressure 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
wellknown 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 highpressure phase tends
to become more stable than the lowpressure 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 cBN 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 nanometersized 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 sizedependent spontaneous alloying of the Au–Ag
nanoparticles [51].The sizedependence 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 silicaencapsulated 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 sizedependent optical properties of VO
2
nanoparticle
arrays [56].Similarly,Katz et al.found the sizedependent tunneling and optical spectroscopy of CdSe
quantum rods [57].Lau et al.found out the sizedependent magnetism of iron clusters [58].Voisin
et al.reported the sizedependent electron–electron interactions in metal nanoparticles [59].There
fore,all these experimental cases clearly show that the sizedependence 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 surfacetovolume 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 onecomponent 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 onecomponent 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 socalled ‘‘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 quasistatic 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 meanﬁeld
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 nonDebye relaxation,
nonArrhenius activation,and nonclassical 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 nonextensivity 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 ‘qexponential’
(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’ nonextensive 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 oneparticle energy function taken to
be a quadratic or a nearly quadratic function of the velocity variable.The ‘qexponential’ 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 nonadditive 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 ﬁrstorder phase transitions in nonequilibriumsystems,
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 nonuniformdensity 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 highpressure 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 equilibriumvapor 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 nanometersized 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 0.1–10nmthickness discontinuous indium ﬁlms by an ultrasensitive thinﬁlm 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
abnormallyﬁrstorder solid–solidphasetransitions,comparabletothewellknown1/r dependenceof the
melting temperature on the basis of the Laplace–Young equation.Interestingly,their rule reasonably
explains the highpressure 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 onionlike 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,CVDcBN,HSRCdiamond,and cBNnucleation,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 nanosizeinduced
additional pressure could be so large that the highpressure metastable phase tends to become more
stable than the lowpressure 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 nanosizeinduced
additional pressure.In our theoretical approach,it also is emphasized that the nanosizeinduced
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 lowpressure 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 nanosizeinduced additional pressure driving.The inset shown in the sketch map displays the
spherical nuclei nucleated on the heterosubstrate.
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 heterosubstrate,and A
se
and g
se
are the interface area and the energy between the heterosubstrate 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 heterosubstrate,and makes the original interface A
se
(be equal
to A
sn
) between the heterosubstrate 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 heterosubstrate 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
heterosubstrate,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
heterosubstrate and the environment gas phase is taken to be equal to the surface tension value of the
heterosubstrate,and the interface between the spherical clusters and the heterosubstrate 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 homosubstrate,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 nanosizeinduced 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 nanosizeinduced additional pressure is denoted by the Laplace–Young equation,i.e.
DP
n
¼
2g
r
(3.28)
Furthermore,as mentioned above,the nanosizeinduced additional pressure can drive the metastable
phase regions into the stable phase region near the boundary line of the highpressure phase in the
equilibriumphase diagram.Therefore,one can obtain the sizedependent 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 highpressure 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 highpressure 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 cBN.
In the following section,using the proposed thermodynamic model above,we will consider
diamond and cBN 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 facecentered 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 highpressure and hightemperature process (HPHT) [171].Up to
date,the syntheses of diamonds include HPHT [171],CVD[162],shockwave method [172],pulsed
laserinduced 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 facecentered 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 dcplasma,rfplasma,microwave plasma,electron cyclotron resonancemicrowave plasma
CVD (ECRMPCVD),and their modiﬁcations have been developed [191].
However,the important innovation of the synthesis of diamond under lowpressure conditions
was generally acknowledged until the middle of 1980s due to diamond being the metastable phase
under the lowpressure 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 quasiequilibriummodel [204],Yarbrough’s surface reaction model [205],BarYam’s
defectsinduced 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 quasiequilibriumprocess [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 hydrogenfree 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
nanosizeinduced additional pressure.When we assume the surface tension of diamond is 3.7 J/m
2
[217],the dependence relationships of the nanosizeinduced 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 nanosizeinduced 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 nondiamond 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 nanosizeinduced
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 nanosizeinduced additional pressure.The inset shows the relationship between the nanosizeinduced 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 nanosizeinduced 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 nanosizeinduced
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 nanosizeinduced 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 nanometersized 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 lowpressure 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 supercriticalﬂuid systems in 1995,and Gogotsi et al.almost simultaneously prepared
diamonds by using the containingcarbonelement 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 nondiamond carbon and various
carbides with chlorine or supercriticalﬂuid water solution containing hydrogen were used as raw
materials [179–181].Furthermore,without chlorine and supercriticalﬂuid water solution containing
hydrogen,Lou et al.synthesized diamonds through the reduction of carbon dioxide and reduction of
magnesium carbonate with metallic supercriticalﬂuid sodium [183,184].However,compared with
the rapid experimental progress of the diamond synthesis by HSRC,the thermodynamic nucleation of
diamond upon HSRC supercriticalﬂuid 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.,socalled 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 HSRCsupercriticalﬂuid systems
fromthe point of view of thermodynamics,we proposed the nanothermodynamic analysis to address
the seedfree 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 supercriticalﬂuid systems are suggested to have the liquid
like densities but gaslike 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 supercriticalﬂuid
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 supercriticalﬂuid systems.
More importantly,our theoretical results are consistent with the experiment data and other calculations
from ﬁrstprinciples [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 (~,$,!,&,
and ^) derived from Refs.[183,184,181,179,182],respectively.Clearly,one can see that the size of
the diamond nuclei increases with the pressure increasing at a given temperature,and decreases with
the temperature increasing at the certain pressure in Fig.10.In addition,we can see that the sizes of the
critical nucleation are close to a constant at the pressures below 400 MPa under the condition of the
C.X.Wang,G.W.Yang/Materials Science and Engineering R 49 (2005) 157–202 179
Fig.9.Carbon thermodynamic equilibriumphase diagrambased on pressure and temperature.Gregion means a metastable
phase region of diamond nucleation;D region means a new stable phase region of the diamond nucleation by the
hydrothermal synthesis or reduction of carbide under the nanosizeinduced interior pressure conditions.The inset shows the
enlarged G and D regions.The data point of the symbols (~,$,!,&,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 ﬁrstprinciple [181,224,225].The
ﬁrstprinciple 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 (~,$,!,&,and ^) derived from Refs.[183,184,181,179,182],respectively (after [45]).
Fig.11.The relationship curves of the critical energy and the pressure at various temperatures (the heterogenous factor is
0.5) under considering the nanosizeinduced interior pressure condition.The inset shows the relationship curves of the
critical energy and the heterogeneous factor at the given pressure and temperature.The data point of the symbols (~,$,!,
&,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 hydrogenfree 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
Onedimensional 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 onedimensional 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,onedimensional 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 noninteraction.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 crosssection 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
equilibriumvapor 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
onedimensional 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 nearperfect 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 highpressure and hightemperatures 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 cBNexcept HPHT,such as CVD
[248–252],pulsedlaser deposition (PLD) [253–259],physical vapor deposition (PVD) [260–264],
hydrothermal synthesis [265–270],pulsedlaser 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 cBN in the thermodynamic equilibriumphase diagramof boron nitride.
However,to date,thermodynamics of the nucleation of cBN remains much less understood and
largely relies on trial and error.In this section,we will take the cBN nucleation upon CVD and
supercriticalﬂuid 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
lowpressure 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 cBN ﬁlms still remains a great challenge for physicists and materials scientists [275].
Actually,the nucleation of CVD cBN is a complicated chemical and physical process.In order to
provide a qualitative description of the cBNnucleation,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 cBNformation,and each model just tended to focus on one
aspect of the complicated process of the nucleation of CVD cBN [275].There have been several
excellent review papers surveying current theories of the cBN nucleation [274,275,292],in which
they pointed out that the cBN 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 (aBN) 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 cBN [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 cBN of a diamondlike material with a zincblende 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 cBN 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 cBN 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 cBNgrown upon CVDis shown as
the H region,which belongs to the metastable region of cBN phase,i.e.,the stable region of hBN
phase.Under the assumption of spherical,isotropic cBN nanocrystals,the sizeinduced additional
pressure would drive the metastable phase into the stable region of cBN phase (C region of Fig.17).
According to the Laplace–Young equation (the surface tension g = 4.72 J/m
2
for cBN[299]),we can
obtain the dependence of the radius of the cBNnuclei 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 cBN 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 cBN 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 cBNnuclei with the size effect,as shown in Fig.18 [299].Clearly,we can see in
Fig.18 that the forming energy of the cBNnuclei decreases with the temperature increasing at a given
pressure,and increases with the pressure increasing.Therefore,these results showthat the nucleation
of CVDcBNdoes not need the relatively high forming energy.Apparently,the lowforming energy of
the cBN nucleation upon CVD implies that it is not difﬁcult to the cBN nucleation,and the cBN
nucleation seems to happen in the cBN 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 nanosizeinduced
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 cBN nucleation with respect to the effect of the nanosizeinduced
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 cBNnuclei 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 cBN.
Thus,the cBN nucleation would be preferred on those substrates that have lower lattice mismatch
with cBN,which would be one of the physical origins of the interlayers grown for CVD cBN.
Actually,a characteristic nucleation sequence aBN!textured hBN (caxis parallel to the sub
strate)!cBNhas been established to precede the cBNgrowth,observed by Kester et al.[300,301],
and their experimental results indicated that the nucleation of cBNon the surface of the interlayer not
only relies on the combined effect of a 2:3 lattice matching between hBNand cBN,but also concerns
in hBN density on the surface of interlayers,i.e.,socalled ‘‘hBN densify.’’ Naturally,‘‘hBN
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 hBN 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 cBN on Si substrates,the relationship curve of the forming energy
and the temperature in various nanosizeinduced 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 hBN–cBN
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 fromhBNto cBNis about 1.0 10
10
to 1.0 10
9
in the newstable region (Cregion in
Fig.17).Actually,the probabilities of the hBN to cBN transition,1.0 10
10
to 1.0 10
9
,are
really lowin the Cregion.Thus,the result implies that the cBNnucleation 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 cBN by simple chemical process alone failed so far [302,303].)
Therefore,in order to enhance the probabilities of the hBN to cBN transition and promote the
cBN nucleation density,many ways,e.g.,electron cyclotron resonance,inductively coupled plasma
(ICP),and radiofrequency (rf),etc.,are employed to assist CVD [274].
Fromthe discussions above,one can see that the cBN nucleation seems to happen in the stable
region of cBN phase in the thermodynamic equilibriumphase diagramof boron nitride based on the
thermodynamics of MPNUR,i.e.,the cBN nucleation would be prior to the hBN nucleation in
competing growth of cBN and hBN upon CVD.
5.2.cBN nucleation in highpressure and hightemperature supercriticalﬂuid systems
Recently,the important progress of the cBN 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 cBNunder the supercritical conditions using nonconventional catalysts such as
the volatile hydrazine NH
2
NH
2
and MgB
2
(socalled HPHT supercriticalﬂuid 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 cBN phase is metastable and the hBN
phase is stable.In other words,the cBN 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 hBNto cBNtransition with respect to the effect of the nanosizeinduced additional pressure
in the new stable phase region (C region) of the cBN nucleation upon CVD (above the C–B line) (after [299]).
brium phase diagram be really outdated for the cBN synthesis by HPHT?In fact,it is an essential
issue involved in the fundamental thermodynamics in the HPHT supercriticalﬂuid systems of the
cBNsynthesis.To our best knowledge,there are hardly any thermodynamic descriptions concerned in
the HPHT supercriticalﬂuid systems,as the full understanding of the complicated interactivities
during the nonconventional 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
cBN upon the HPHT supercriticalﬂuid systems.
In fact,the nucleation and growth of cBNare complicated in a ﬂux of another material (socalled
‘‘catalyst’’) upon the HPHT supercriticalﬂuid systems.Generally,cBN is considered to be the
spontaneouscrystallization in the socalled conventional ‘‘solvents–catalysts system’’ [313].Actu
ally,no matter what the conventional solvents–catalysts systemor the supercriticalﬂuid systems,the
nucleation and growth of cBN must meet simultaneously the conditions as follows:(i) the raw
material and catalyst molten in the system,(ii) the highsupersaturation raw material in the solvent
catalysts,and (iii) the ambient pressure (including the nanosizeinduced additional pressure) and
temperature of the spontaneouscrystallization of cBN clusters above the C–B line (or the sponta
neouscrystallization of cBNclusters is unstable).However,the results of Solozhenko and Singh et al.
[305–309] showed that the nucleation and growth of cBN in a wide temperature and pressure ranges
in the supercriticalﬂuid 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 cBN was synthesized by HPHT,many theoretical models have
discussed the nucleation and growth of cBN upon HPHT.For example,the solid–solid transition
model thinks that hBN melt in the catalyst solvent under the certain conditions of the pressure and
temperature,and then,the formation of the new solvent (BNrich).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 nanosizeinduced
interior pressure and the nuclei size (as an inset).H region means a metastable phase region of cBN nucleation;C region
means a newstable phase region of the cBNnucleation with respect to the effect of nanosizeinduced 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 BNrich compared with that
mentioned above.Finally,the new BNrich compound formed under the high pressures and
temperatures becomes instable and decomposes into cBN 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 cBNcrystallization in NH
4
F–BNsystem,and only hBNand a
melt coexist in the systemby in situ measuring using Xray diffraction with synchrotron radiation.In
addition,as aforementioned above,the supercriticalﬂuid may have liquidlike densities but gaslike
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 cBNin the supercriticalﬂuid
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 cBNnuclei 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 nanosizeinduced additional pressure plays an important
role on the cBN nucleation upon supercriticalﬂuid systems.
In the light of Eq.(3.35),the dependence of the forming energy of the cBNnuclei 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 cBNnuclei 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,cBN
cannot nucleate.Importantly,these results show that the heterogeneous nucleation of cBN does not
need the high forming energy in the supercriticalﬂuid systems.Apparently,the lowforming energy of
the heterogeneous nucleation of cBN implied that it is not difﬁcult to the cBN nucleation in
the HPHT supercriticalﬂuid systems,and the cBN nucleation would happen in the stable phase
region (C region) of cBN as shown in Fig.21.
According to Eq.(3.36),we can obtain the probability curves of the hBN to cBN 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 supercriticalﬂuid
systems (after [44]).
of the phase transition probability fromhBNto cBNare 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 hBN to cBN
transition,1.0 10
8
to 1.0 10
7
,are really low in the C region.Thus,these results show that the
cBN nucleation density should be low in the HPHT supercriticalﬂuids system,although it could
happen.In other words,the cBN nucleation would not be favored in the HPHT supercriticalﬂuid
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 supercriticalﬂuid
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 hBN to cBN 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 hBNto cBNphase 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 fromhBN to cBN in the HPHT
supercriticalﬂuid systems without the consideration of the nanosizeinduced 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 supercriticalﬂuid systems
(after [44]).
Fig.24.The probability of the phase transition from hBN to cBN upon the HPHT supercriticalﬂuids system under the
consideration of the nanosizeinduced 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 cBN 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 cBN in the H region seems impossible.Accordingly,the reasonable phase
region of the cBNnucleation upon the HPHTsupercriticalﬂuid 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 nanosizeinduced 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 hBN to cBN on the pressure upon HPHT
supercriticalﬂuids 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 fromhBN to cBN upon the HPHT supercriticalﬂuids system without the
consideration of the nanosizeinduced 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 cBN 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 cBN nucleation in the supercriticalﬂuid systems.These theoretical
results indicate that the cBNnucleation would actually occur in the stable phase region of cBNin 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 cBN vs.the pressure in the given temperatures without
taking into account the nanosizeinduced interior pressure conditions (after [43]).
Fig.28.The relationship curves of the nucleation probability of cBNvs.the pressure in the given temperatures under taking
into account the nanosizeinduced 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 wellknown
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 cBN
nanocrystals upon the pulsedlaser 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 onedimensional
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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