Statistical mechanics of nucleation: a review

SPECIAL ISSUE PAPER

Statistical mechanics of nucleation a review

I J Ford

Department of Physics and Astronomy University College London Gower Street London WC1E 6BT UK

Abstract The principles of statistical mechanics have been used to develop a theory of thenucleation of a phase transition but a number of subtle questions remain and are highlighted in thisreview A central issue is the cluster definition the mathematical scheme which distinguishes amolecular cluster from a collection of separate molecules There is also the question of whetherthermodynamic transition state theory is suitable to describe the process or whether the developmentof kinetic rate equations describing cluster growth and decay is a better approach The classical theoryof nucleation is flawed but appears to provide useful estimates in some cases including waterPhenomenological extensions of the classical theory can improve matters However improvements inthe theory from microscopic considerations are not simple to apply require major computationaleffort and suffer from uncertainties due to lack of knowledge of the fundamental intermolecularinteractions Calculations of the nucleation rate of water droplets are especially difficult since thissubstance is notoriously difficult to model Nevertheless as capabilities improve accurate calculationsshould come within reach which will offer better understanding of the process for practicalapplications such as the transition from dry to wet steam

Keywords statistical mechanics nucleation

NOTATION

A parameter in the fit to experimental dataB parameter in the fit to experimental dataf mean force on the moleculef0 free energy per molecule in the metastable

phaseFethiTHORN free energy of an i-clusterf Langevin forceHethNTHORN system Hamiltoniani cluster size in moleculesi critical sizeI1 I2 I3 principal moments of inertia of clusterIm1 I

m2 I

m3 principal moments of inertia of the

moleculeJ nucleation ratek Boltzmann constantm molecular massM cluster massni population of i-clustersncei constrained equilibrium cluster

populationsnei equilibrium cluster populations in

subsaturated vapour

nsi i-cluster population in saturated vapournssi steady state cluster population in

supersaturated vapourns1 monomer population in saturated vapourpv vapour pressurepsv saturated vapour pressurePij transition probability in population

dynamicsq sticking coefficientq0 parameter in the DillmannndashMeier modelr radial position from the cluster centre of

massre escape radiusS supersaturationt timeT temperatureU0ethiTHORN cluster potential energy at restUm

0 molecular potential energy at restV system volumew molecular probability density in spaceZd Zeldovich factorZi i-cluster canonical partition functionzi modified i-cluster canonical partition

function

ai parameter in the DillmannndashMeiermodel

bi growth rate of an i-clusterThe MS was received on 15 July 2003 and was accepted after revision forpublication on 7 April 2004

883

C12903 IMechE 2004 Proc Instn Mech Engrs Vol 218 Part C J Mechanical Engineering Science

g Langevin friction coefficientgkin cluster decay rate in the Langevin modelgi decay rate of an i-clusterDF change in free energyDW work of cluster formationDWcl classical work of cluster formationyi parameter in the DillmannndashMeier modelki parameter in the DillmannndashMeier model chemical potentials chemical potential in saturated vapourX grand canonical partition functionrl liquid phase densityrv vapour phase densityrsv saturated vapour densitys planar surface tensiont parameter in the DillmannndashMeier modelok cluster vibrational angular frequencyom

k molecular vibrational angular frequency

1 INTRODUCTION

One of the most remarkable properties of matter is itsability to take different physical forms for differentvalues of parameters such as temperature or pressureFor example a vapour will condense by the nucleationof droplets when its pressure is raised above thesupersaturated vapour pressure or when its temperatureis reduced below the dew point Such metamorphosesare fundamentally a consequence of the second law ofthermodynamics a principle that can be characterizedas a desire on the part of physical systems to minimizetheir free energy by any available means This caninclude the repositioning of molecules on a grand scalea phase change Phase transformations are the statisticalconsequences of the ability of large physical systemsand any environment with which they interact toexplore a huge range of microscopic configurationsThis dynamical behaviour underpinning the secondlaw remains only partially understood [1] but it seemsto be universal

A practical situation where a change in phase createsproblems is the behaviour of steam in turbines Thetemperature and pressure gradients are such that waterdroplets might nucleate with undesirable effects on theperformance of the machine including the erosion of theturbine blades On the other hand the steam must beexpanded as much as possible to extract the most workfrom it Some of the drawbacks can be avoided if thephase change can be controlled

The need for an understanding of the transition fromdry to wet steam in turbines has motivated this reviewbut this is just an example of a process which affectsmany aspects of the world The formation of droplets in

a turbine presents practical difficulties but withoutatmospheric clouds which form by a similar process [2]the global climate would be rather different from what itis at present [3] Aerosol formation in the atmosphereand dust in space [4] and in the case of condensedphases the freezing of liquids all proceed by the samebasic mechanism The principal idea is that there exists abottleneck in the transformation which is passedthrough only by fragments or molecular clusters ofthe new phase The bottleneck is narrowest when theclusters reach a so-called critical size and therefore theproperties of these critical clusters are central to thetheory of nucleation [5]

Experimental progress in understanding droplet for-mation in turbines has been significant [6 7] but at firstsight it would appear that progress on the theoreticalside has been less impressive The tool used most oftenin modelling studies [8] remains the classical nucleationtheory (CNT) derived many decades ago and based onthe most primitive ideas of the properties of the criticalcluster [9ndash13] The capillarity approximation isemployed whereby the critical cluster however smallis considered as a scaled-down macroscopic droplet ofthe condensed phase The model is a version oftransition state theory (TST) in which a realization ofthe system is identified which is in unstable thermo-dynamic equilibrium equally able to develop with timeinto the initial or the final phase How is it that thismodel is still employed when developments in micro-scopic modelling have proceeded so rapidly in recentyears Why has no better model been developed

One reason is pragmatism It is very surprising thatthe classical theory successfully predicts nucleation rateswithin a couple of orders of magnitude in the case ofwater perhaps the most important substance to under-stand for a temperature range of about 210ndash260K anda range of supersaturations Evidence for this has beencollected through various studies over the last 20 years[14ndash20] This level of agreement is good in the field ofnucleation theory However in these conditions thecritical cluster consists of only a few tens of moleculesand the capillarity approximation is questionable in theextreme

However if a broader view is taken and data for othermaterials are considered the failure of CNT is moreapparent The nucleation rate seems to have a depen-dence on supersaturation suggested by CNT but thetemperature dependence does not agree Often atemperature range where classical theory is reasonablyconsistent with experiment does not exist

Experimental data gathered during recent years canbe reviewed briefly Some early data on n-nonaneshowed deviations from CNT of four orders ofmagnitude either way [21] A study of several membersof the n-alkane series [22] confirmed the need forcorrection factors of again up to four orders ofmagnitude Early data for a range of alcohols showed

I J FORD884

Proc Instn Mech Engrs Vol 218 Part C J Mechanical Engineering Science C12903 IMechE 2004

deviations from CNT of over five orders of magnitude[23] In an effort to acquire definitive data for a singlesubstance the nucleation of n-pentanol was givenparticular attention by a number of research groups inthe late 1990s and early 2000s For temperatures in therange 260ndash290K most data lie three orders of magnitudeabove the CNT predictions [24] although some measure-ments lie between one and three orders of magnitudebelow [25] Further results for n-pentanol show goodagreement with the CNT at 273K [26] but withdeviations away from this temperature In the light ofthe reasonable success of CNT in explaining water datait is disappointing to find that hydrogen-bonded organicmaterials such as a series of glycols disagree verystrongly with the temperature dependence of thenucleation rate suggested by CNT [27] Similarly theorganic species dibutyl phthalate deviates by up to sixorders of magnitude from the model [28] The situationbecomes even more unsatisfactory for the nucleation oftwo or more species together Strong deviations frombinary CNT are found for the ethanolndashwater system [29]and for a range of n-nonanendashn-alcohol mixtures [30]However it is with metals that the failure of CNT ismost dramatic examples in the literature being caesium[31] mercury [32] and lithium [33] For caesium thecritical supersaturation (that required to drive nuclea-tion at a specified rate) is between two and ten timeslarger than the CNT predictions The difference betweenmeasured and predicted nucleation rates is astronom-ical Although in most materials the deviation is lesssevere than this it is certainly not the case that a smallcorrection to CNT is all that is needed to account for alldata

The usual CNT expression for the rate is

JCNT frac14 q2spm

1=2 rvrl

pvV

kT

6 exp 16ps3

3ethkTTHORN3r2l ethlnSTHORN2

eth1THORN

where q is a sticking or accommodation coefficient m isthe molecular mass k is Boltzmannrsquos constant s is thesurface tension of a planar interface between thecondensate saturated vapour pv is the vapour pressureand rl and rv are the densities of the bulk condensedphase and the vapour respectively in molecules per unitvolume S is the supersaturation defined by S frac14 pv=p

sv

where psv is the saturated vapour pressure at thetemperature T Note that J in this expression is thenumber of particles formed per second and it isproportional to the system volume V There are goodreasons to insert a factor of 1S into the prefactor of theabove expression but this is the often-quoted formCalculations are easy agreement is reasonable and sowhy should anyone bother to develop the theoryfurther

The problem in employing this formula for water isthat the laboratory data for water are taken at lowtemperatures and need to be extrapolated to hightemperatures for application to turbine conditions Asalready noted water seems to be a lucky case wheretheory and experiment actually agree for a particular setof conditions namely at a temperature of about 240Kand for a range of supersaturations Unfortunately adiscrepancy appears and grows as the temperature israised and at 260K the theory overpredicts the rate byan order of magnitude [18] The ratio of measurednucleation rate to CNT prediction has been fitted to asimple function by Wolk and Strey [18]

JexpethT STHORN frac14 JCNT exp Athorn B

T

eth2THORN

where A frac14 2756 and B frac14 6500K This formula fitsdata in the temperature range 220ndash260K but has alsobeen found to account for data at temperatures sometens of kelvins higher (B Wyslouzil personal com-munication 2003)

Nevertheless caution must be exercised in extrapolat-ing to temperatures around 370K for steam turbinesDevelopments in the theory are certainly needed It is achallenge to understand why CNTworks reasonably wellat 240K forwaterwhat terms are cancellingout andwhyHow canmeasured data be extrapolated with confidence

In this review the determination of the properties ofmolecular clusters using the tools of statistical mech-anics is again considered Similar accounts of the widerpicture have been given in various reviews and books[34 35] Starting from very general arguments attemptsare made to discern how a theory such as the CNTmight emerge The subtleties of defining a clusterprecisely in terms of the positions and momenta of itsconstituents are discussed It is noted that the transitionstate theory based on equilibrium cluster properties isequivalent to an approach based on the kinetic theory ofcluster decay at least in certain circumstances Somephenomenological models developed from CNT arereviewed and finally a perspective of future develop-ments in this field is given

By the very nature of a review the discussion is ratherbroad while at the same time some of the technicalitiesare outlined The intention is to give a reasonablycomplete picture of the current state of nucleationtheory at least from the point of view of the presentauthor and to show that significant progress has beenmade in the approximately 80 years of its history whilemore is still clearly necessary

2 TRANSITION STATE THEORY

A theory of nucleation based on statistical ideas isnecessary since direct simulation of the process from a

STATISTICAL MECHANICS OF NUCLEATION 885


microscopic point of view is very laborious and has beenrealized in only a very few example cases [36ndash39] Thetime scales involved make a nucleation event simply tooslow to follow directly for realistic conditions and withcurrent computational resources

The simplest statistical treatment of the nucleationprocess comes from TST within a framework of thetheory of free energy fluctuations According to thermo-dynamics a system with externally fixed temperatureand pressure (or volume) will adjust any internal prop-erties in order to minimize its Gibbs (or Helmholtz) freeenergy [40 41] Such adjustments will include changes inphase Nevertheless statistical fluctuations in systemproperties will naturally take place and it is useful tointerpret these as fluctuations in the free energy of thesystem away from the minimum Considering constant-volume conditions for the moment general arguments[42ndash45] suggest that the probability of a fluctuationDF in Helmholtz free energy is proportional toexpfrac12DF=ethkTTHORN

The physical fluctuations that are important in thenucleation process are those where a molecular clusterof new phase is formed temporarily from the originalphase The change in free energy accompanying theformation of a cluster of i molecules in the new phase iswritten DF frac14 DWethiTHORNFethiTHORN if0 where FethiTHORN is the freeenergy of the i cluster and f0 is the free energy permolecule in the old phase This is also called the lsquoworkof formationrsquo since it corresponds in thermodynamics tothe external work needed to insert an additional clusterreversibly into the system while maintaining the overall

temperature and pressure (or volume) Note that thisinsertion of a new cluster while maintaining suchconstraints is a theoretical operation since it would behard to achieve in practice

For a metastable initial phase the magnitude of thefree-energy fluctuation DF peaks at the so-called lsquocriticalsizersquo i This is illustrated in Fig 1 The perturbed statewith the additional critical cluster is therefore athermodynamic transition state since reversible changesin size of the extra cluster are accompanied byreductions in free energy It is presumed that this meansthat the new cluster has equal probabilities of growingor decaying and that the rate of nucleation is thereforerelated to the rate of formation of these critical clusters

The probability of forming the critical clusteraccording to this picture is proportional toexpfrac12DWethi THORN=ethkTTHORN but the proportionality constant(essentially a time scale) is not yet determined Tointroduce this simple kinetic theory [46] can be used tocalculate the collision rate bi between single molecules(monomers) and an i-cluster modelled as a sphere withbulk condensed phase molecular density rl

bi frac14qpv

eth2pmkTTHORN1=21thorn 1

i

1=2

1thorn i1=3 2

eth36pTHORN1=3

r2=3l

eth3THORN

where m is the molecular mass It is more usual toreplace the expression in square brackets with i2=3 thelimit for large i The simplest TST expression for the

Fig 1 Sketch of the cluster work of formation for a range of (constrained) cluster sizes The maximum in thecurve defines the critical size i and the critical work of formation DWethi THORN

I J FORD886


nucleation rate is then J bini where ni is the meannumber of critical clusters in the system

More sophisticated treatments of thermodynamicscan provide the time scale without recourse to the abovekinetic argument which is reasonably valid for gas-to-particle conversion but not for freezing for exampleAttempts based on linearized non-equilibrium thermo-dynamics [47 48] originate from the work of Onsager[49]

The next step is crucial and it is to relate thepopulation ni to the probability of a fluctuation and toan explicit model of the critical work of formation Theassumption in classical nucleation theory is that

ni frac14 n1 expDWclethi THORN

kT

eth4THORN

where DWclethiTHORN is the classical work of formation givenby

DWclethiTHORN frac14 4pR2s ikT lnS eth5THORN

and DWclethi THORN is the maximum of this function Thecluster is envisaged to be a sphere of radius R given by4pR3rl=3 frac14 i

Unfortunately some uncertainty accompanies thesesteps in the derivation The theory is flawed on severalcounts The work of formation contains a termassociated with the creation of a spherical interface(characterized by the bulk value of surface tension) anda change kT lnS in free energy per molecule due tothe change in phase from supersaturated vapour tocondensate Moreover the factor n1 appearing inequation (4) appears to have been introduced heuristi-cally Might it not instead be the monomer populationin the saturated vapour How does DWclethiTHORN comparewith the actual work of formation DWethiTHORN Is thedroplet really spherical with a sharp interface Have

any important factors been left out Furthermoreshould there be concern that equation (4) is in-consistent when i frac14 1 This has been a controversialissue [50ndash54]

3 KINETIC INTERPRETATION

The expression for the classical rate of nucleation givenin equation (1) emerges from the models in equation (4)and (5) when the thermodynamic transition stateanalysis is embellished by a kinetic interpretation [11]Instead of using the probability of fluctuations awayfrom the equilibrium state (the TST route) a set of rateequations is introduced to describe the dynamics ofcluster populations

The evolution of cluster populations is modelled usingthe following equations

dni

dtfrac14Xj

njPji niXj

Pij eth6THORN

where Pji is the mean rate at which a j cluster convertsinto an i cluster The transition processes are assumed tobe Markovian so that the rate coefficients depend onthe properties of the cluster and its environment but noton their history

The kinetic interpretation can be taken a step furtherby assuming that the only important transitions arethose involving the addition or loss of single moleculesto or from the cluster The scheme of transitionprocesses is illustrated in Fig 2 The only non-zerorate coefficients are then bi frac14 Piithorn1 for cluster growthand gi frac14 Pii1 for cluster decay The rate equations

Fig 2 The scheme of the birth and death of clusters addition of a monomer to a cluster will take it up thechain towards the right while the loss of a monomer will take it in the opposite direction Attachmentand detachment of dimers and larger clusters are usually ignored



reduce to

dni

dtfrac14 bi1ni1 gini bini thorn githorn1nithorn1 eth7THORN

These are the BeckerndashDoring equations [11] and theyprovide a useful reinterpretation of the TST approachThere are two steady state solutions In the first thepopulations satisfy equation (7) and also the detailedbalance condition

bi1nei1 frac14 gin

ei eth8THORN

and the fnei g are taken to represent the cluster popula-tions in an equilibrium vapour at or below the saturatedvapour pressure For true thermal equilibrium themonomer population has to be less than or equal to itsvalue for the saturated vapour ns1 The cluster populationsin a saturated vapour are denoted fnsig Nevertheless asituation can be invented which is known as lsquoconstrainedequilibriumrsquo where the detailed balance condition is stillsatisfied but with n1 gt ns1 Since bi pvS while gi isindependent of vapour pressure equation (8) then definesso-called constrained equilibrium populations

ncei ampSinsi eth9THORN

for a supersaturated vapour It has been assumed thatS frac14 pv=p

svampn1=n

s1

The second steady state solution to the BeckerndashDoring equations is more realistic It is not meant torepresent thermal equilibrium The supersaturation isset to a value greater than unity and a boundarycondition of zero population at some cluster size imax isimposed These conditions will drive a steady productionrate of large clusters from the vapour The approach isfor mathematical convenience only and the same steadystate cluster populations nssi and nucleation rate wouldemerge if imax was sent off to infinity The detailedbalance condition (8) is replaced by J frac14 bi1n

ssi1 gin

ssi

defining the nucleation current J which can then becalculated analytically as shown in another paper in thisvolume [55] To a good approximation

J frac14 Zdbincei eth10THORN

which involves the constrained equilibrium population atthe critical size i the size with the lowest constrainedequilibrium population The condition imax 4 i musthold Zd is the so-called Zeldovich factor [12] givenapproximately by

Zd frac14 1

2pq2ethln ncei THORN

qi2

ifrac14i

1=2

eth11THORN

When equations (4) and (5) are used to specify ncei and theresult is substituted into equations (10) and (11) then theclassical nucleation rate given in equation (1) isrecovered as long as the large i limit of the attachment

rate coefficient in equation (3) is taken It is also possibleto derive an expression for J purely in terms of the ratecoefficients bi and gi This possibility is considered againin section 8

The main assumptions used in this derivation of theclassical theory principally the validity of equations (4)and (5) are now examined In order to explore this it isnecessary to see how the equilibrium cluster populationsnei in subsaturated vapours may be calculated usingstatistical mechanics

4 CLUSTER STATISTICAL MECHANICS

The statistical mechanical treatment of a vapour is bestdeveloped using a grand canonical ensemble [56ndash59] Thesystem is placed in contact with a particle reservoir at achemical potential and temperature T so that thenumber of molecules N in the system can change Thegrand partition function of the system X is then propor-tional to the integral of expf frac12HethNTHORN N=ethkTTHORNg overall possible configurations of molecular positions andmomenta whereHethNTHORN is the Hamiltonian of the system

Evaluating X exactly is exceedingly difficult and it ismuch simpler to proceed by considering the configura-tions of the system as arrangements of molecularclusters The vapour then is viewed as a mixture ofideal gases each composed of clusters of differentsizes The grand partition function of the system canbe reconstructed from partition functions for singlei-clusters In order to do this however a cluster must bedefined in a unique way in terms of molecular positionsand velocities An example of such a definition would beto require the separation between molecules in a clusterto be less than a maximum distance Rc [60] as illustratedin Fig 3

Fig 3 According to the Stillinger definition a molecule isconsidered part of a cluster if it lies closer than adistance Rc from a constituent molecule of the cluster

I J FORD888


At first sight the cluster definition is completelyarbitrary a means of categorizing and resumming allthe elementary configurations of the system Howeveraccurately representing the system as a mixture of idealgases requires the interactions between molecules lyingin different clusters to be minimized The clusterdefinition affects whether a particular interaction isregarded as intercluster or intracluster and so it plays acrucial role This will be discussed further in section 6Residual clusterndashcluster interactions can be taken intoaccount in the form of a virial series [61]

These ideas are now put into practice Define amodified canonical partition function zi for a clustercontaining i molecules This is simply the integral ofexpf frac12HcethiTHORN i=ethkTTHORNg over configurations involvingmolecules in the cluster alone with Hc the Hamiltonianof the i molecules comprising the cluster Using thepicture of a mixture of ideal gases the grand partitionfunction of the complete system can then be expressed as

XampXfnig

Yifrac141

znii

ni

eth12THORN

where the sum is over all possible cluster populationdistributions fnig This is demonstrated rigorously inappendix A of reference [57]

The next step is to determine the distribution fnei gthat dominates the sum in equation (12) The logarithmof the summand is extremized

qqni

Xifrac141

lnz

nii

ni

frac14 0 eth13THORN

which gives the following very simple expression for themost probable or equilibrium cluster size distributionfor the given conditions T and

nei frac14 ziethTTHORN frac14 ZethTTHORN ei=ethkTTHORN eth14THORN

which is equivalent to the so-called law of mass action

nei frac14 Zine1Z1

i

eth15THORN

where ZiethTTHORN frac14 expfrac12FethiTHORN=ethkTTHORN frac14 zi expfrac12 i=ethkTTHORN isthe canonical partition function for an i cluster and FethiTHORNis the i-cluster Helmholtz free energy This relationshipis often used as the starting point in models of clusterpopulations

The vapour pressure is now given by a sum of partialpressures

pvV frac14 kTXifrac141

nei frac14 kTXifrac141

ziethTTHORN eth16THORN

and this series will only converge if the equilibriumpopulations or the zi decrease sufficiently quickly forlarge i This limits the statistical mechanics approach to

the study of saturated or subsaturated vapours How-ever as in the preceding section the equilibriumpopulations nei in equation (14) (for S lt 1) can beextrapolated into constrained equilibrium populationsncei (with S gt 1) for use in equation (10) Saturatedvapour corresponds to a particular chemical potentialms and the saturated vapour cluster populations are nsi The constrained equilibrium populations are obtainedby inserting into equation (14) the chemical potential ofthe supersaturated vapour which is frac14 s thorn kT lnSgiving ncei frac14 Sinsi which is consistent with the resultobtained in equation (9) for dilute vapours

The constrained equilibrium populations are there-fore expressed as

ncei frac14 SiziethsTTHORN

frac14 expfrac12FethiTHORN is ikT lnS

kT

eth17THORN

and the correspondence with CNT can be made closerby introducing the n1 prefactor

ncei frac14 n1 exp frac12FethiTHORN is thorn kT ln n1 ikT lnS

kT

eth18THORN

The next task is to explore how the combinationFethiTHORN is thorn kT ln n1 might be approximated by theCNT result 4pR2s This can be explored most easilyfor solid rather than liquid clusters

5 SOLID CLUSTERS

The partition function of a solid cluster of i moleculescan be factorized into translational rotational andvibrational parts through a transformation to suitablerelative positions and momenta [62ndash64] Starting withthe vibrational degrees of freedom the atom positionsare defined with respect to their mean positions whichare determined by a fixed centre of mass and set oflattice vectors It is assumed that the potential energypart of the Hamiltonian is quadratic in the displace-ments from these mean positions Now the partitionfunction is worked out for configurations of the particlesrestricted to having zero total linear and total angularmomentum about the centre of mass This is writtenZvib corresponding to a product of partition functionsfor 3ji 6 independent oscillators (if there are j atomsper molecule and therefore a total of 3ji degrees offreedom if the molecule is not linear) each of which inthe classical limit takes the form kT=ethhoTHORN where o isthe angular frequency of the specific vibrational modeThe partition function has an additional factor ofexpfrac12U0ethiTHORN=ethkTTHORN where U0ethiTHORN is the system potentialenergy evaluated at the mean atom positions A



different expression applies for low temperatures whenthe oscillator is not classically activated [65]

The rotational and translational degrees of freedomare next The previous restriction of the configurationsto zero angular momentum is lifted The approximate(classical) rotational partition function is

Zrot frac14 p1=28p2kTh2

3=2

ethI1I2I3THORN1=2 eth19THORN

where Ik are the principal moments of inertia about thecentre of mass for atoms fixed in their mean positions

Finally the translational free energy of the centre ofmass of the cluster is given by the partition function

Ztrans frac14 V2pMkT

h2

3=2

eth20THORN

where V is the system volume and M frac14 im is the clustermass The free energy of the cluster (plus kT ln n1 forconvenience) is therefore

FethiTHORN thorn kT ln n1

frac14 kT lnZtransZrotZvib

n1

frac14 U0ethiTHORN kT ln

eth2pMkTTHORN3=2

Srsvh3p1=2

8p2kTh2

3=2

6ethI1I2I3THORN1=2Y3ji6

kfrac141

kT

hok

eth21THORN

where rsv frac14 ns1=V is the molecular density of dilutesaturated vapour

By similar arguments the chemical potential of dilutesaturated vapour is

s frac14 Um0 kT ln

eth2pmkTTHORN3=2

rsvh3p1=2

8p2kTh2

3=2

6ethIm1 Im2 Im3 THORN1=2Y3j6

kfrac141

kT

homk

eth22THORN

where Um0 is the potential energy of a molecule when the

atoms are at their mean positions Imk are the threeprincipal moments of inertia of the molecule about thecentre of mass and om

k are the oscillatory frequencies ofmolecular internal vibrations

The classical work of formation DWclethiTHORN should nowcorrespond to the leading-order contribution to thedifference between equation (21) and i times equation(22) Calculations [54 64 66ndash68] demonstrate that theleading term is indeed roughly proportional to i2=3 thusjustifying the form of the classical model The suitabilityof the capillarity approximation depends on the extentto which the coefficient of this leading term correspondsnumerically to the planar surface free energy Howeverthe full statistical mechanical expression for the work of

formation contains additional terms and these will beimportant for small clusters and need to be taken intoaccount In order to evaluate them the variousstructural and energy properties of cluster are requiredall of which can be obtained from microscopic modelsand they can then be inserted into the formulae for thethermodynamic properties [equations (21) and (22)] theconstrained equilibrium populations [equation (18)] andultimately the nucleation rate [equation (10)] Followingthis route employs the almost universally held assump-tion that equilibrium properties such as free energies canbe used to describe non-equilibrium properties specifi-cally the cluster decay rate This assumption is re-examined in section 8

6 LIQUID DROPLETS AND THE CLUSTER

DEFINITION

Defining what was meant by a solid cluster in theprevious section was not an issue Within the assump-tions of the model (harmonic interactions) it wasimpossible for a molecule to escape from the clusterThe cluster definition in the case of liquids cannot bedismissed so easily Liquids are more volatile thansolids and the molecules in a cluster are much moremobile It is imperative now to characterize a clusterusing the molecular positions and velocities andperhaps some time scale and length scale too

The traditional microscopic method for defining acluster is to impose a geometric constraint on molecularpositions An example is to restrict molecules to a spherecentred on the centre of mass [69ndash71] or to impose amaximum allowable separation between the molecules(the Stillinger criterion [60]) More sophisticated treat-ments exist such as the n=v Stillinger cluster introducedby Reiss and co-workers [72ndash75] Here the confiningvolume v used to define the cluster is considered to be anadditional defining characteristic on the same level asthe cluster size n As time progresses a cluster changesits confining volume as the component molecules followtheir respective trajectories just as they change their sizewith the gain and loss of molecules The association of amolecular configuration with a precise confining volumeis complicated but tractable The dependence of the freeenergy on v determines the statistical dynamics of thiscluster volume A similar attempt to associate a clusterwith a specified volume within a grand canonicalensemble description was developed by Kusaka andco-workers [58 59 76] The simulation box was taken tobe the confining cluster volume Many of these ideasmay be viewed as a treatment of the cluster definition(eg the Stillinger radius) as a variational parameter inthe representation of the partition function Further

I J FORD890


refinement of this point of view using a treatment ofclusterndashcluster interactions to optimize the definingcluster volume has been presented [77] A review ofrecent progress in this area and the application of theseideas to nucleation theory has been provided by Sengeret al [78]

Such geometric constraints can be implemented buttheir realism is questionable Situations better regardedas close encounters between separate clusters would beclassed as snapshots of single larger clusters It shouldbe recalled that the cluster definition is an internalfeature of the model which is to be optimized to reduceclusterndashcluster interactions and to improve the repre-sentation of the grand partition function Misclassifyingconfigurations is fraught with consequences A secondissue to consider is that for convenience the clustersshould behave in a manner that can be modelled withBeckerndashDoring equations This means that the growthand decay rates should be free of memory effects ie thetransition processes should be Markovian The extent towhich this is satisfied (if indeed it is physically realistic)depends on the cluster definition

An intuitively reasonable choice of definition whichaddresses both these points is that a cluster shouldconsist of molecules that are bound energetically Thisshould minimize the clusterndashcluster contributions to thetotal energy According to such a definition if amolecule interacts strongly with a cluster then it islikely to be regarded as part of that cluster Also anenergetically bound cluster is likely to decay by thermalfluctuation in a stochastic Markovian fashion ratherthan by the separation of passing poorly boundfragments

Ideas have been proposed which require the cluster tobe energetically bound (in a loose sense since a cluster isintrinsically prone to decay) The time evolution of amolecular configuration and not simply the set ofinstantaneous positions of its component molecules isof central importance Hill [79] introduced a clusterdefinition requiring the total energies of pairs ofmolecules in a cluster to be negative (in their centre ofmass frame) The intention was to exclude situationswhere a molecule could leave the cluster within the timeit would take for a molecule to travel a typical lineardimension of the cluster Soto and Cordero [80] havedeveloped Hillrsquos criterion and Barrett [81] has per-formed Monte Carlo modelling of molecular configura-tions excluding those which decay in moleculardynamics within a given period A number of otherenergy-based criteria have been developed [82ndash85]

Harris and Ford [86] have recently proposed a newenergy-based cluster definition As with Barrettrsquosapproach it is based on the requirement that amolecular snapshot is energetically bound for a chosensubsequent period Evaporation according to thisscheme requires not only that a molecule should makea positive energy excursion but also that the subsequent

dynamics carry it far away from the cluster Onlyperhaps one in eight molecules manage to avoidrecapture once having satisfied the positive energycriterion [86] A snapshot of a cluster in this casesurrounded by a carrier gas is shown in Fig 4 Thehistory of the energy of a molecule in such a cluster isillustrated in Fig 5 three excursions into positive energyoccur in this sequence but only one leads to actualescape corresponding to the maintenance of positiveenergy for a significant time The instant of escape istaken to be the point at which the energy goes positivefor the third and final time

Cluster definitions implemented in moleculardynamics and Monte Carlo codes allow cluster freeenergies and hence the decay rate to be calculated Anenormous amount of simulation work has been carriedout and calculations can be laborious Many differentschemes and approximations have been made to easethe evaluation of these properties The principalconclusion from various studies using a range of clusterdefinitions and interaction models [49 57 58 70 85 87ndash92] seems to be that the work of formation of liquidcritical clusters does indeed contain a dominantcontribution proportional to i2=3 consistent with theclassical work of formation Hale and collaboratorspresent cluster free energies for water and other speciesin a way that makes this particularly clear indeed it issurprising how dominant the surface term can be foreven very small clusters [70 93 94] Neverthelessfurther contributions exist small clusters are notcharacterized just by scaled-down properties of largeclusters and it is impossible to escape the need todetermine the additional terms

In all the above discussion it has been assumed thatthe intermolecular interactions are accurately specifiedThis is a situation that is rarely possible Althoughfirst-principles quantum-mechanical models could be

Fig 4 Snapshot of a cluster of 50 argon atoms (brightspheres) bathed in a gas of helium (darker spheres)



developed in practice empirical potentials are devel-oped on the basis of physical intuition containingparameters fitted to the measured properties of thesubstance in question [95] Usually these are bulkproperties and it is not always clear that the sameinteractions can be used for the very different circum-stances of a molecular cluster [96] In view of thesensitivity of nucleation rates to the thermodynamicproperties of clusters this is perhaps a crucial failingProgress can be made however by computing knowncluster properties where they are available and byrefining the potentials to suit Extraction of clusterproperties through use of the so-called nucleationtheorems (which relate derivatives of the nucleationrate to properties of the critical cluster [16 56 97ndash99])may provide just this sort of experimental information

A technique that has not yet been described is the useof density functional theory (DFT) to model theproperties of liquid droplets [100ndash104] This powerfulapplication of statistical mechanics promises to avoidlaborious computations at the microscopic level byinstead focusing on the density profile of moleculesmaking up the droplet This profile may be regarded asthe average distribution of molecules about the clustercentre of mass Although this might appear to involvecoarse-graining and approximation it is a well-foundedapproach and in principle can be exact In practical

implementations however a mean field approach istaken such that the constituent molecules move in aneffective potential well created by the other molecules inthe cluster [105] Also the interactions are assumed to becharacterized by strong repulsive cores and weaklyattractive tails

Considerable analytical work has been carried out[104 106] Various developments have been made todescribe droplets consisting of simple atoms as well aschain-like molecules [107] and polar molecules [108]The main drawback in these implementations is thatspecific interactions such as hydrogen bonds are hardto represent Certain details of the modelled systemare neglected [109 110] and the (implicit) clusterdefinition used is geometric and not energetic [111]On the other hand quite sophisticated systems havebeen modelled including multi-component dropletsThe method has been compared with otherapproaches [112 113] and certain features are betterdescribed particularly the approach to spinodalconditions where the nucleation barrier vanishesDFT calculations can be performed relatively easilyand they can provide a qualitative and perhapsquantitative insight into quite complicated clusteringproblems They provide an intermediate-level descrip-tion between the simple capillarity approximation anda microscopic description

Fig 5 The energy of a molecule in the centre-of-mass frame of a cluster occasionally becomes positive butthis does not necessarily lead to escape Only the third positive energy excursion in this sequenceindicated by an arrow results in particle escape A cluster definition must capture this behaviour

I J FORD892


7 PHENOMENOLOGICAL MODELS

The reader might be dismayed by the mathematicalcomplexity of the expressions for the work of formationof solid clusters in section 5 might be alarmed by theprocedural complexity of implementing a cluster defini-tion in section 6 may recoil from the difficulties incalculating free energies from molecular simulationmethods and ultimately may give up in view of theuncertainties in the underlying intermolecular interac-tions Considering these complications it is tempting totry to develop phenomenological models of the work offormation After all if the capillarity approximationworks reasonably well for water then surely a minorcorrection to account for small cluster effects ought tobe sufficient to improve matters

However after perhaps two decades of work nomodel has emerged offering broad success across arange of different substances It is perhaps asking toomuch to expect universal improvement CNT works as afirst approximation for water for a particular tempera-ture range but as discussed earlier it possesses thewrong temperature dependence so that predictionsaway from a temperature in the range 210ndash260K areuncertain A similar picture emerges for other sub-stances although in some cases the failure is even moresevere

The simplest phenomenological adjustment to equa-tion (4) the basis for CNT is to write

ncei frac14 n1 expfrac12DWclethiTHORN DWcleth1THORN

kT

eth23THORN

where DWclethiTHORN is given by equation (5) This amounts toa uniform downward shift of the classical free energybarrier illustrated in Fig 1 At least now the expressionis consistent at i frac14 1 The theory is known as internallyconsistent classical theory (ICCT) [51] Unfortunately itis not clear whether this constant shift in the work offormation should be a realistic representation of all thesubleading terms in the work of formation [52] It is anarbitrary modification and its success in accounting fordata is mixed For example nucleation rates for a rangeof alkanes [114] require further modification by factorsof between one and nine orders of magnitude to comeinto agreement with ICCT

It should be noted at this point that the correctionterm kT lnS which appears in DWcleth1THORN can beregarded as rigorous There are good reasons for thisbased on general considerations involving the nucleationtheorems [56] or more simply the principles of chemicalequilibrium represented by the law of mass action givenin equation (15) The correction converts the prefactorn1 in equation (4) into ns1 and results in the insertion of afactor 1=S in the nucleation rate in equation (1)

The Fisher droplet model is a slightly more sophis-ticated scheme [115] Here the constrained equilibrium

populations are modelled by

ncei frac14 ns1 exp yii2=3 thorn t ln i lnq0kT

psv

thorn i lnS

eth24THORN

The term proportional to yi may be viewed as a surfaceterm while the other two are corrections of a typesuggested by the free energy expressions developed insection 5 The parameters q0 yi and t are chosen toensure that the model correctly reproduces knownproperties of clusters and vapours including virialcoefficients and critical properties the theory is there-fore phenomenological

For large cluster sizes and for temperatures wellbelow the critical temperature the free energy offormation should tend towards that of a macroscopicliquid droplet yi should therefore approach A1s=ethkTTHORNfor large i where A1i

2=3 is the surface area 4pR2 of aspherical droplet containing i monomers at the bulkliquid density so that A1 frac14 eth36pTHORN1=3r2=3

l Dillmann andMeier [116] suggested writing

yi frac14kiA1skT

eth25THORN

with the Ansatz

ki frac14 1thorn a1i 1=3 thorn a2i 2=3 eth26THORN

The coefficients a1 and a2 can be found in terms of virialcoefficients of the vapour The form of this Ansatz wasmotivated by the curvature dependence of the bulksurface tension derived in classical thermodynamics byTolman [117]

The initial success of the DillmannndashMeier theory wasshort lived since modifications had to be made toovercome an internal inconsistency [118] Virial expan-sions for the vapour pressure and density were usedincorrectly to calculate the virial expansion of the ratioof the two Correcting this error [61] changed predictedrates by several orders of magnitude Nevertheless theapproach was very influential [119] In a furtherdevelopment Kalikmanov and van Dongen [120] seta2 frac14 0 (this term can be absorbed into q0 anyway) andchose a1 such that the experimental saturated vapourpressure psv is reproduced by the model The DillmannndashMeier model matched the first two virial coefficientsonly The Kalikmanovndashvan Dongen theory shouldtherefore approximate the (revised) DillmannndashMeiertheory at low temperatures but ought to provide abetter description at higher temperatures

Other phenomenological models have been proposedincluding the Hale scaling relation [121] This modelbased on CNT but employing simplified models for thesurface tension and liquid density based on scalingbehaviour near the critical point together with a rather



more straightforward rate prefactor seems to be quitesuccessful in accounting for data for a wide range ofmaterials even metals [122] and two-component systems[123] Another phenomenological model is the diffuseinterface theory [124] which is based on a treatment ofthe phase interface using simplified density profiles ofthermodynamic properties Another general class ofmodel makes use of conditions at the vapourndashliquidspinodal to constrain the form of the model free energy[125ndash128]

The main drawback in phenomenological models isthe arbitrariness in the form of the proposed work offormation The parameter fitting in practice ensures thatthe populations of small clusters especially the mono-mer and dimer are well represented by the model butthe properties of larger clusters depend on the expres-sion assumed for the model and are therefore uncertainIntuition suggests that phenomenological modelling islikely to have mixed success and that microscopiccalculations of cluster properties do matter There mightbe better value in phenomenological models however ifthey were based securely on the results of microscopiccalculations

8 KINETICS OR THERMODYNAMICS

In section 3 kinetic equations representing the nuclea-tion process were described However the coefficients inthose equations were related to constrained equilibriumpopulations and hence to the equilibrium statisticalmechanics of clusters through the detailed balancecondition (8) Is it necessary to do this Could acompletely kinetic route be followed by evaluating themean growth and decay coefficients bi and gi directly bysome statistical approach The main problem is ofcourse the difficulty in performing such calculationsand this has restricted the development of such models[129ndash131] It is expected that the kinetic coefficients willcorrespond to those derived from the detailed balanceargument for near-equilibrium conditions but circum-stances can be imagined where non-equilibrium effectsare present which cannot be dealt with within standardstatistical mechanics

The framework for evaluating kinetic coefficients ofcluster decay can be briefly introduced [86 132] Amolecular dynamics simulation of a cluster can be set upand evolved according to the interactions until bychance an evaporation event occurs By extensivesimulation statistics for the mean evaporation rates gican be amassed in this way Similarly moleculardynamics simulations where a single molecule iscaptured by a cluster could be performed and thegrowth rates bi determined These will be functions ofcluster energy and so would fit best into a rate equationscheme involving both size and energy [133]

This is a laborious approach but statistical physicsoffers an approximate alternative representation of theescape and capture processes The basic tool is theLangevin equation The radial motion of a molecule inthe cluster is described by [134]

meurorr frac14 f ethrTHORN thornfethr _rr tTHORN eth27THORN

where r is the radial position m is the molecular mass fis the mean (time- and velocity-averaged) force on themolecule at position r and f is the stochastic forceThe latter is usually represented as a velocity-dependent friction term mg _rr where g is the so-calledfriction coefficient plus a fluctuating term ~ff ethr tTHORN withzero mean and correlation in time h ~ff ethr tTHORN~ff ethr t0THORNi frac14eth2gkTmTHORNdetht t0THORN where k is Boltzmannrsquos constant andwhere T has the characteristics of a temperature as willbe seen It is a standard manipulation [135] to convertthe Langevin description in the large friction limit intoa FokkerndashPlanck equation

qwqt

frac14 1

mg qethfwTHORN

qrthorn kT

q2wqr2

eth28THORN

which represents the evolution of wethr tTHORN the prob-ability density that the molecule should lie at the radialposition r It should be noted in passing that FokkerndashPlanck equations have often been used to describe thestochastic dynamics of a cluster in size space [48 136] Incontrast the use of the method described above refers tomotion of a molecule in real coordinate space relative tothe centre of mass of the cluster

The steady state equilibrium solution of equation (28)is

wethrTHORN exp FethrTHORNkT

eth29THORN

where FethrTHORN is called the potential of mean force relatedto the mean force on the molecule f throughf frac14 dF=dr It can now be seen how the parameter Tin the stochastic force plays the role of temperaturesince equation (29) looks like a Boltzmann distributionThis solution is not what is sought however The escapeproblem is characterized by a boundary conditionwethreTHORN frac14 0 where re is the radius at which a particleescapes from the system The resulting escape rate maybe shown to be [134]

gkin frac14 KethreT m gTHORN exp DFkT

eth30THORN

where the prefactor K depends on the parameters listedas well as the shape of the potential of mean force Theprincipal feature of equation (30) is the exponentialdependence on the depth DF of the potential of meanforce Therefore if the potential of mean force FethrTHORN can

I J FORD894


be evaluated by studying a molecular dynamics trajec-tory for example then the kinetic decay rate can bedetermined This could be extended to account for aradially dependent temperature and friction coefficientboth of which might be determined from simulation Asketch of a potential of mean force is given in Fig 6

Kinetically determined cluster growth and decayrates gkini and bkini respectively could provide a newroute for the determination of nucleation rates inde-pendent of assumptions about detailed balance and theuse of equilibrium statistical mechanics in a non-equilibrium situation

9 CONCLUSIONS

This review has been brief and in some ways partisan tothe personal views of the author It attempts to sketch acoherent picture of nucleation theory and how thevarious disparate strands relate to one another One ofthe fascinations of nucleation theory is the variety ofapproaches and the number of difficult issues that areencountered How well does free energy fluctuationtheory represent a real non-equilibrium process Howwell does the idea of detailed balance provide aframework for determining non-equilibrium decaycoefficients from equilibrium statistical mechanicsHow well do various methods perform in the evaluation

of cluster free energies Can cluster growth and decayrates be calculated directly without the need for such aframework How exactly are clusters to be defined

The theoretical problems are many but the issues canbe resolved to various degrees of satisfaction Thereremains of course the fundamental task of choosing anintermolecular potential that adequately represents thephysical interactions Without this all the finer points ofcluster definition and treatments of non-equilibriumprocesses are unimportant Modelling many problems inscience relies on the availability of an accurate micro-scopic description and so nucleation rate calculationsare not unique in this respect Most empirical molecularpotentials are fitted to the properties of bulk and surfaceproperties and in the very best examples attention isgiven to their transferability to more unusual environ-ments such as clusters In view of these capabilities andmodern theoretical developments in nucleation theoryit seems likely that successful microscopic models of thenucleation process based on statistical physics are atlast coming within reach

ACKNOWLEDGEMENTS

The present author is grateful to colleagues SarahHarris and Marco Arosio for providing some of thefigures

Fig 6 A molecule in a cluster may be considered to be confined within it by a potential of mean force FethrTHORN Itcan escape by reaching a radial position re driven by a phenomenological stochastic force



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7 Bakhtar F Ebrahimi M and Bamkole B O On the


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force field J Comput Chemistry 1998 19 535


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1999 111 5938

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Phase transitions in systems small enough to be clusters


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111 Ford I J (unpublished)

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approach J Chem Physics 1995 103 1175

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nucleation J Chem Physics 1995 99 2865

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tional Conference on Nucleation and Atmospheric Aerosols

(Eds N Fukuta and P E Wagner) 1992 pp 11ndash18

(Deepak Publishing Hampton Virginia)

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point Physics 1967 3 255

116 Dillmann A and Meier G E A A refined droplet

approach to the problem of homogeneous nucleation

from the vapor phase J Chem Physics 1991 94 3872

117 Tolman R C J Chem Physics 1949 17 333

118 Ford I J Laaksonen A and Kulmala M Modification

of the DillmannndashMeier theory of homogeneous nuclea-


119 Laaksonen A Ford I J and Kulmala M Revised

parametrization of the DillmannndashMeier theory of homo-

geneous nucleation Phys Rev E 1994 49 5517

120 Kalikmanov V I and van Dongen M E H Semipheno-

menological theory of homogeneous vaporndashliquid nuclea-


121 Hale B N Application of a scaled homogeneous

nucleation-rate formalism to experimental-data at

T 5Tc Phys Rev A 1986 33 4156

122 Martinez D M Ferguson F T Heist R H and Nuth

J A Application of scaled nucleation theory to metallic

vapor condensation J Chem Physics 2001 115 310

123 Hale B N and Wilemski G A scaled nucleation model

for ideal binary systems Chem Physics Lett 1999 305

263

124 Granasy L Diffuse interface theory for homogeneous

vapor condensation J Chem Physics 1996 104

5188

125 McGraw R and Laaksonen A Interfacial curvature free

energy the Kelvin relation and vaporndashliquid nucleation


I J FORD898


126 Talanquer V A new phenomenological approach to

gasndashliquid nucleation based on the scaling properties

of the critical nucleus J Chem Physics 1997 106

9957

127 Koga K and Zeng X C Thermodynamic expansion of

nucleation free-energy barrier and size of critical nucleus

near the vaporndashliquid coexistence J Chem Physics 1999

110 3466

128 Kusaka I On the scaling behavior of the free energetics of


129 Nowakowski B and Ruckenstein E A kinetic approach

to the theory of nucleation in gases J Chem Physics

1991 94 1397

130 Nowakowski B and Ruckenstein E Homogeneous

nucleation in gases a three-dimensional FokkerndashPlanck

equation for evaporation from clusters J Chem Physics

1991 94 8487

131 Wilcox C F and Bauer S H Estimation of homo-

geneous nucleation flux via a kinetic model J Chem

Physics 1991 94 8302

132 Hanggi P Talkner P and Borkovec M Reaction rate

theory fifty years after Kramers Rev Mod Physics 1990

62 251

133 Barrett J C Clement C F and Ford I J Energy

fluctuations in homogeneous nucleation theory for aero-

sols J Physics A 1993 26 529

134 Ford I J and Harris S A Molecular cluster decay

viewed as escape from a potential of mean force J Chem

Physics 2004 120 4428

135 Risken H The FokkerndashPlanck Equation Methods of

Solution and Applications 2nd edition 1989 (Springer-

Verlag Berlin)

136 Shizgal B and Barrett J C Time dependent nucleation




g Langevin friction coefficientgkin cluster decay rate in the Langevin modelgi decay rate of an i-clusterDF change in free energyDW work of cluster formationDWcl classical work of cluster formationyi parameter in the DillmannndashMeier modelki parameter in the DillmannndashMeier model chemical potentials chemical potential in saturated vapourX grand canonical partition functionrl liquid phase densityrv vapour phase densityrsv saturated vapour densitys planar surface tensiont parameter in the DillmannndashMeier modelok cluster vibrational angular frequencyom

k molecular vibrational angular frequency

1 INTRODUCTION

One of the most remarkable properties of matter is itsability to take different physical forms for differentvalues of parameters such as temperature or pressureFor example a vapour will condense by the nucleationof droplets when its pressure is raised above thesupersaturated vapour pressure or when its temperatureis reduced below the dew point Such metamorphosesare fundamentally a consequence of the second law ofthermodynamics a principle that can be characterizedas a desire on the part of physical systems to minimizetheir free energy by any available means This caninclude the repositioning of molecules on a grand scalea phase change Phase transformations are the statisticalconsequences of the ability of large physical systemsand any environment with which they interact toexplore a huge range of microscopic configurationsThis dynamical behaviour underpinning the secondlaw remains only partially understood [1] but it seemsto be universal

A practical situation where a change in phase createsproblems is the behaviour of steam in turbines Thetemperature and pressure gradients are such that waterdroplets might nucleate with undesirable effects on theperformance of the machine including the erosion of theturbine blades On the other hand the steam must beexpanded as much as possible to extract the most workfrom it Some of the drawbacks can be avoided if thephase change can be controlled

The need for an understanding of the transition fromdry to wet steam in turbines has motivated this reviewbut this is just an example of a process which affectsmany aspects of the world The formation of droplets in

a turbine presents practical difficulties but withoutatmospheric clouds which form by a similar process [2]the global climate would be rather different from what itis at present [3] Aerosol formation in the atmosphereand dust in space [4] and in the case of condensedphases the freezing of liquids all proceed by the samebasic mechanism The principal idea is that there exists abottleneck in the transformation which is passedthrough only by fragments or molecular clusters ofthe new phase The bottleneck is narrowest when theclusters reach a so-called critical size and therefore theproperties of these critical clusters are central to thetheory of nucleation [5]

Experimental progress in understanding droplet for-mation in turbines has been significant [6 7] but at firstsight it would appear that progress on the theoreticalside has been less impressive The tool used most oftenin modelling studies [8] remains the classical nucleationtheory (CNT) derived many decades ago and based onthe most primitive ideas of the properties of the criticalcluster [9ndash13] The capillarity approximation isemployed whereby the critical cluster however smallis considered as a scaled-down macroscopic droplet ofthe condensed phase The model is a version oftransition state theory (TST) in which a realization ofthe system is identified which is in unstable thermo-dynamic equilibrium equally able to develop with timeinto the initial or the final phase How is it that thismodel is still employed when developments in micro-scopic modelling have proceeded so rapidly in recentyears Why has no better model been developed

One reason is pragmatism It is very surprising thatthe classical theory successfully predicts nucleation rateswithin a couple of orders of magnitude in the case ofwater perhaps the most important substance to under-stand for a temperature range of about 210ndash260K anda range of supersaturations Evidence for this has beencollected through various studies over the last 20 years[14ndash20] This level of agreement is good in the field ofnucleation theory However in these conditions thecritical cluster consists of only a few tens of moleculesand the capillarity approximation is questionable in theextreme

However if a broader view is taken and data for othermaterials are considered the failure of CNT is moreapparent The nucleation rate seems to have a depen-dence on supersaturation suggested by CNT but thetemperature dependence does not agree Often atemperature range where classical theory is reasonablyconsistent with experiment does not exist

Experimental data gathered during recent years canbe reviewed briefly Some early data on n-nonaneshowed deviations from CNT of four orders ofmagnitude either way [21] A study of several membersof the n-alkane series [22] confirmed the need forcorrection factors of again up to four orders ofmagnitude Early data for a range of alcohols showed

I J FORD884




JCNT frac14 q2spm

1=2 rvrl

pvV

kT

6 exp 16ps3


eth1THORN


sv




T

eth2THORN















bi frac14qpv


i

1=2

1thorn i1=3 2

eth36pTHORN1=3

r2=3l

eth3THORN



I J FORD886






kT

eth4THORN









dni

dtfrac14Xj

njPji niXj

Pij eth6THORN






reduce to

dni



bi1nei1 frac14 gin

ei eth8THORN




svampn1=n

s1


ssi1 gin

ssi




Zd frac14 1


qi2

ifrac14i

1=2

eth11THORN








I J FORD888




XampXfnig

Yifrac141

znii

ni

eth12THORN



qqni

Xifrac141

lnz

nii

ni

frac14 0 eth13THORN




nei frac14 Zine1Z1

i

eth15THORN











kT

eth17THORN



kT

eth18THORN


5 SOLID CLUSTERS







3=2





h2

3=2

eth20THORN




n1


eth2pMkTTHORN3=2

Srsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

hok

eth21THORN



s frac14 Um0 kT ln

eth2pmkTTHORN3=2

rsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

homk

eth22THORN







DEFINITION



I J FORD890


















I J FORD892







kT

eth23THORN






psv

thorn i lnS

eth24THORN





yi frac14kiA1skT

eth25THORN

with the Ansatz















qwqt

frac14 1

mg qethfwTHORN

qrthorn kT

q2wqr2

eth28THORN




eth29THORN



eth30THORN


I J FORD894




9 CONCLUSIONS




ACKNOWLEDGEMENTS





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)







JCNT frac14 q2spm

1=2 rvrl

pvV

kT

6 exp 16ps3


eth1THORN


sv




T

eth2THORN















bi frac14qpv


i

1=2

1thorn i1=3 2

eth36pTHORN1=3

r2=3l

eth3THORN



I J FORD886






kT

eth4THORN









dni

dtfrac14Xj

njPji niXj

Pij eth6THORN






reduce to

dni



bi1nei1 frac14 gin

ei eth8THORN




svampn1=n

s1


ssi1 gin

ssi




Zd frac14 1


qi2

ifrac14i

1=2

eth11THORN








I J FORD888




XampXfnig

Yifrac141

znii

ni

eth12THORN



qqni

Xifrac141

lnz

nii

ni

frac14 0 eth13THORN




nei frac14 Zine1Z1

i

eth15THORN











kT

eth17THORN



kT

eth18THORN


5 SOLID CLUSTERS







3=2





h2

3=2

eth20THORN




n1


eth2pMkTTHORN3=2

Srsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

hok

eth21THORN



s frac14 Um0 kT ln

eth2pmkTTHORN3=2

rsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

homk

eth22THORN







DEFINITION



I J FORD890


















I J FORD892







kT

eth23THORN






psv

thorn i lnS

eth24THORN





yi frac14kiA1skT

eth25THORN

with the Ansatz















qwqt

frac14 1

mg qethfwTHORN

qrthorn kT

q2wqr2

eth28THORN




eth29THORN



eth30THORN


I J FORD894




9 CONCLUSIONS




ACKNOWLEDGEMENTS





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)











bi frac14qpv


i

1=2

1thorn i1=3 2

eth36pTHORN1=3

r2=3l

eth3THORN



I J FORD886






kT

eth4THORN









dni

dtfrac14Xj

njPji niXj

Pij eth6THORN






reduce to

dni



bi1nei1 frac14 gin

ei eth8THORN




svampn1=n

s1


ssi1 gin

ssi




Zd frac14 1


qi2

ifrac14i

1=2

eth11THORN








I J FORD888




XampXfnig

Yifrac141

znii

ni

eth12THORN



qqni

Xifrac141

lnz

nii

ni

frac14 0 eth13THORN




nei frac14 Zine1Z1

i

eth15THORN











kT

eth17THORN



kT

eth18THORN


5 SOLID CLUSTERS







3=2





h2

3=2

eth20THORN




n1


eth2pMkTTHORN3=2

Srsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

hok

eth21THORN



s frac14 Um0 kT ln

eth2pmkTTHORN3=2

rsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

homk

eth22THORN







DEFINITION



I J FORD890


















I J FORD892







kT

eth23THORN






psv

thorn i lnS

eth24THORN





yi frac14kiA1skT

eth25THORN

with the Ansatz















qwqt

frac14 1

mg qethfwTHORN

qrthorn kT

q2wqr2

eth28THORN




eth29THORN



eth30THORN


I J FORD894




9 CONCLUSIONS




ACKNOWLEDGEMENTS





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)









kT

eth4THORN









dni

dtfrac14Xj

njPji niXj

Pij eth6THORN






reduce to

dni



bi1nei1 frac14 gin

ei eth8THORN




svampn1=n

s1


ssi1 gin

ssi




Zd frac14 1


qi2

ifrac14i

1=2

eth11THORN








I J FORD888




XampXfnig

Yifrac141

znii

ni

eth12THORN



qqni

Xifrac141

lnz

nii

ni

frac14 0 eth13THORN




nei frac14 Zine1Z1

i

eth15THORN











kT

eth17THORN



kT

eth18THORN


5 SOLID CLUSTERS







3=2





h2

3=2

eth20THORN




n1


eth2pMkTTHORN3=2

Srsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

hok

eth21THORN



s frac14 Um0 kT ln

eth2pmkTTHORN3=2

rsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

homk

eth22THORN







DEFINITION



I J FORD890


















I J FORD892







kT

eth23THORN






psv

thorn i lnS

eth24THORN





yi frac14kiA1skT

eth25THORN

with the Ansatz















qwqt

frac14 1

mg qethfwTHORN

qrthorn kT

q2wqr2

eth28THORN




eth29THORN



eth30THORN


I J FORD894




9 CONCLUSIONS




ACKNOWLEDGEMENTS





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)





reduce to

dni



bi1nei1 frac14 gin

ei eth8THORN




svampn1=n

s1


ssi1 gin

ssi




Zd frac14 1


qi2

ifrac14i

1=2

eth11THORN








I J FORD888




XampXfnig

Yifrac141

znii

ni

eth12THORN



qqni

Xifrac141

lnz

nii

ni

frac14 0 eth13THORN




nei frac14 Zine1Z1

i

eth15THORN











kT

eth17THORN



kT

eth18THORN


5 SOLID CLUSTERS







3=2





h2

3=2

eth20THORN




n1


eth2pMkTTHORN3=2

Srsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

hok

eth21THORN



s frac14 Um0 kT ln

eth2pmkTTHORN3=2

rsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

homk

eth22THORN







DEFINITION



I J FORD890


















I J FORD892







kT

eth23THORN






psv

thorn i lnS

eth24THORN





yi frac14kiA1skT

eth25THORN

with the Ansatz















qwqt

frac14 1

mg qethfwTHORN

qrthorn kT

q2wqr2

eth28THORN




eth29THORN



eth30THORN


I J FORD894




9 CONCLUSIONS




ACKNOWLEDGEMENTS





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)







XampXfnig

Yifrac141

znii

ni

eth12THORN



qqni

Xifrac141

lnz

nii

ni

frac14 0 eth13THORN




nei frac14 Zine1Z1

i

eth15THORN











kT

eth17THORN



kT

eth18THORN


5 SOLID CLUSTERS







3=2





h2

3=2

eth20THORN




n1


eth2pMkTTHORN3=2

Srsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

hok

eth21THORN



s frac14 Um0 kT ln

eth2pmkTTHORN3=2

rsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

homk

eth22THORN







DEFINITION



I J FORD890


















I J FORD892







kT

eth23THORN






psv

thorn i lnS

eth24THORN





yi frac14kiA1skT

eth25THORN

with the Ansatz















qwqt

frac14 1

mg qethfwTHORN

qrthorn kT

q2wqr2

eth28THORN




eth29THORN



eth30THORN


I J FORD894




9 CONCLUSIONS




ACKNOWLEDGEMENTS





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)








3=2





h2

3=2

eth20THORN




n1


eth2pMkTTHORN3=2

Srsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

hok

eth21THORN



s frac14 Um0 kT ln

eth2pmkTTHORN3=2

rsvh3p1=2

8p2kTh2

3=2


kfrac141

kT

homk

eth22THORN







DEFINITION



I J FORD890


















I J FORD892







kT

eth23THORN






psv

thorn i lnS

eth24THORN





yi frac14kiA1skT

eth25THORN

with the Ansatz















qwqt

frac14 1

mg qethfwTHORN

qrthorn kT

q2wqr2

eth28THORN




eth29THORN



eth30THORN


I J FORD894




9 CONCLUSIONS




ACKNOWLEDGEMENTS





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)





















I J FORD892







kT

eth23THORN






psv

thorn i lnS

eth24THORN





yi frac14kiA1skT

eth25THORN

with the Ansatz















qwqt

frac14 1

mg qethfwTHORN

qrthorn kT

q2wqr2

eth28THORN




eth29THORN



eth30THORN


I J FORD894




9 CONCLUSIONS




ACKNOWLEDGEMENTS





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)










kT

eth23THORN






psv

thorn i lnS

eth24THORN





yi frac14kiA1skT

eth25THORN

with the Ansatz















qwqt

frac14 1

mg qethfwTHORN

qrthorn kT

q2wqr2

eth28THORN




eth29THORN



eth30THORN


I J FORD894




9 CONCLUSIONS




ACKNOWLEDGEMENTS





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)













qwqt

frac14 1

mg qethfwTHORN

qrthorn kT

q2wqr2

eth28THORN




eth29THORN



eth30THORN


I J FORD894




9 CONCLUSIONS




ACKNOWLEDGEMENTS





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)







9 CONCLUSIONS




ACKNOWLEDGEMENTS





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)





REFERENCES











320 553







115ndash124









211(C3) 195ndash210


119 277


236



(Leipzig) 1935 24 719















2000 112 8205



Chemistry B 2001 105 11683



Physics 2002 117 6176




Physics 2002 116 10 772




1989 90 1856




Physics 1996 105 4707




2325














Physics 1996 105 7617





7696


















4093






New Jersey)







I J FORD896


















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)





















York)


(Pergamon London)





Physica A 1998 259 10









1990 93 1273






98 6408




















1999 110 5249





2080















Physics 1968 48 5553






1999 111 4688







5766














2000 112 294


















2002 116 1097





Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)







Physics 2002 116 4317



118 9216



85 2178




Physics 1998 109 9901







112 4193













2000 pp 31ndash34












76 5098




7665






1988 89 7521







Physics 1994 100 5190



1999 111 5938



Physics 1997 107 7989



Chem Physics 1999 111 10 602



Physics 1995 103 3686



7982





(submitted)






























T 5Tc Phys Rev A 1986 33 4156






263



5188




I J FORD898





9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)








9957




110 3466





1991 94 1397




1991 94 8487



Physics 1991 94 8302



62 251






Physics 2004 120 4428



Verlag Berlin)





Statistical mechanics of nucleation: a review

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Transcript of Statistical mechanics of nucleation: a review