Nucleation theorems, the statistical mechanics of molecular clusters, and a revision of classical nucleation theory

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1 PHYSICAL REVIEW E VOLUE 56, NUBER 5 NOVEBER 997 Nucleaton theorems, the statstcal mechancs of molecular clusters, and a revson of classcal nucleaton theory I. J. Ford Department of Physcs and Astronomy, Unversty College London, Gower Street, London WCE 6BT, Unted Kngdom Receved 2 February 997; revsed manuscrpt receved 5 June 997 The nucleaton theorems relate the temperature and supersaturaton dependence of the rate of nucleaton of droplets from a metastable vapor phase to propertes of the crtcal molecular cluster, the sze that s approxmately equally lkely to grow or decay. They are derved here usng a combnaton of statstcal mechancs and cluster populaton dynamcs, usng an arbtrary model cluster defnton. The theorems are employed to test the valdty of the classcal theory of homogeneous nucleaton and ts nternally consstent form. It s found that the propertes of the crtcal cluster for these models are ncorrect, and t emerges that ths occurs because the classcal theory employs the free energy of a fxed droplet, rather than one free to take any poston n space. Thus a term representng postonal, or mxng, entropy s mssng from the cluster free energy. A revsed model s proposed, based on the capllarty approxmaton but wth such a term ncluded, and t s shown that t s fully consstent wth the nucleaton theorems. The model ncreases classcal rates by factors of approxmately Other nucleaton models should be tested for nternal consstency usng the same methods. Fnally, the nucleaton theorems are used to extract the excess nternal energes of molecular clusters from expermental data for several substances. S063-65X PACS numbers: Qb I. INTRODUCTION The formaton of clouds and fog n the atmosphere s perhaps the most famlar example of the transformaton of a vapor nto lqud droplets. Water vapor n ar that cools below the so-called dew pont becomes thermodynamcally unstable wth respect to the lqud phase and droplets are formed. The thermodynamcs of the phase change are well understood, but the dynamcs are not: no fully successful theory of the rate of droplet formaton has emerged n spte of seventy years of effort. Conceptually, the process of nucleaton s smple enough. Free molecules are contnually colldng wth one another and occasonally becomng bound: dmers and larger clusters of molecules are bult up n ths way. Clusters can also lose molecules by occasonal evaporaton and so ndvdual clusters follow a fluctuatng hstory of growth and decay. When the vapor phase s thermodynamcally stable wth respect to the condensed phase, decay s more lkely than growth, and large clusters tend to fall back to smaller szes, or break apart completely nto free molecules. However, when the bulk condensed phase becomes thermodynamcally stable wth respect to the vapor phase, occasonal growth excursons by ndvdual clusters nto larger sze classes can sometmes lead to contnued growth. Ths s because when the condensed phase s stable, growth s more lkely than decay for large clusters. However, such szes cannot be reached wthout the pror formaton of small clusters except by very unlkely multmolecular collsons and these often reman thermodynamcally unstable wth respect to the free molecules. Decay contnues to wn for small clusters, even when the bulk condensed phase s the thermodynamcally stable state. The phase transton s therefore brought about by nfrequent fluctuatons by ndvdual clusters through the unstable sze range, past the crtcal sze, beyond whch the lkelhood of growth exceeds that of decay, enablng them to grow nto large droplets. Ths s droplet nucleaton, and t ncreases n frequency as the vapor s made more and more metastable wth respect to the bulk condensed phase, snce ths alters the mean growth and decay rates. For the smplest systems, the parameters that control the nucleaton rate are the temperature and the supersaturaton of the vapor, whch s the rato of the vapor pressure to the saturated vapor pressure. If droplet nucleaton takes place on the surface of an exstng partcle, whch s common n the atmosphere, the process s called heterogeneous nucleaton, but the more fundamental homogeneous nucleaton process nvolves only the nteractons of the vapor molecules among themselves. The latter has receved more attenton, both theoretcally and expermentally. The statstcal mechancs of molecular clusterng s perhaps the most natural theoretcal framework for descrbng nucleaton. However, progress has been hampered by uncertantes n how to represent a physcal cluster n statstcal mechancs. The pcture of growth fluctuatons of embryonc droplets that has been sketched above requres a cluster to have a certan stablty, so that t can truly be held to exst as a physcal entty on the tme scale of molecular collsons. However, physcal clusters cannot be absolutely stable, or evaporaton would never take place. ayer clusters do not represent physcal clusters, for example, snce there s no lmtaton on the separaton between molecules: the defnton wll therefore nclude confguratons that are unbound and ephemeral. ost defntons of physcal clusters nvolve the confnement of the molecules to a small spatal volume n the hope that all such confguratons wll be bound and can serve as model physcal clusters as requred n the knetc treatment of nucleaton, and that no bound states are excluded. Ths s a dffcult and perhaps mpossble task, and some mscountng seems to be X/97/565/5655/$ The Amercan Physcal Socety

2 566 I. J. FORD 56 nescapable. The cluster defnton has an effect upon the predctons of the model, as we shall see. The most common defnton used n the lterature requres each molecule of a cluster to le wthn a sphere centered on the center of mass of the group 2 and n further refnements, for a molecule also to le on the surface of the sphere 3 5. Ths has been proposed snce t s possble to enumerate all the confguratons of molecules n a system that satsfy ths cluster defnton. Ths means that the phase space ntegrals can be evaluated and cluster partton functons found. It s assumed that such a constructon wll nclude all the bound states of the component molecules and wll nclude no undesred unbound states. Other cluster defntons exst that would appear to be more natural, such as the Stllnger cluster 6, where each molecule need only le n a partcular regon centered around any other molecule already n the cluster. However, the enumeraton of molecular confguratons satsfyng ths defnton s not easy analytcally, nor can the defnton exclude unbound transtory states. Possbly the best cluster defnton of all would requre all the molecules n the cluster to have a negatve total energy 7,8. It would be the ntermolecular potental that determned whether a partcular molecular confguraton should be vewed as a physcal cluster. However, ths prescrpton would be complcated to realze n practce, and several dffcultes exst 8. In fact the defnton s not at all perfect snce t would fal to account for physcal clusters where ndvdual molecular energes become postve but where crcumstances, such as a collson, prevent the evaporaton of such an unbound molecule. Furthermore, one would need to calculate knetc energes wth respect to some reference frame, and t s unclear whch choce to make. Perhaps the best that can be done s to demand that the molecular potental energy should be less than the equparttoned molecular knetc energy. The uncertantes n the constructon of statstcal mechancal models 2,9, and also the poor state of knowledge about the ntermolecular forces, have made t attractve to study smpler models nstead. The prme example s the classcal theory, whch s based on the dea that clusters can be vewed as tny droplets wth a well-defned surface area and densty, characterzed by the propertes of bulk condensed matter 0. A related approach s to use a phenomenologcal cluster model, where the dffcultes n the status of clusters are set asde by fttng parameters n the model to expermental data 3. Nevertheless, a well-founded theory of nucleaton s more lkely to emerge from statstcal mechancal consderatons. However, n the lght of the above dscusson, t would be best to try to obtan results that were not dependent on any partcular choce of cluster defnton. Ths possblty has been pursued n the present paper. Results regardng the temperature and supersaturaton dependence of the nucleaton rate are derved from the statstcal mechancs and populaton dynamcs of arbtrarly defned clusters. These results are the two nucleaton theorems. The frst nucleaton theorem was orgnally a conjecture based on classcal nucleaton theory 4, whch was then placed on frmer ground usng statstcal mechancs 5 and then thermodynamcs 6. The second nucleaton theorem was derved thermodynamcally by the present author 7. These prevous dervatons studed the probablty of occurrence 5 or the work of formaton 6,7 of the crtcal cluster. The nucleaton rate s proportonal to a Boltzmann factor contanng the crtcal cluster work of formaton, and the remander of the expresson s often called the knetc prefactor. The knetc prefactor from classcal theory was used to complete the dervaton of the theorems. In the present paper, however, the temperature and supersaturaton dependence of the rate of nucleaton s obtaned drectly, wthout the need to estmate the knetc prefactor. Indeed the valdty of the classcal prefactor can be tested usng the more general results obtaned. The dervaton usng statstcal mechancs also ndcates how mcroscopc calculatons based on a grand ensemble for a sngle cluster can be used, together wth the theorems, to predct the senstvty of the nucleaton rate to expermental condtons. The equlbrum statstcal mechancs of a subsaturated vapor are descrbed n Sec. II. The pressure of such a vapor s less than the saturated vapor pressure at whch the vapor and lqud are n thermodynamc equlbrum. olecular confguratons are classfed as collectons of clusters of varous szes, and the grand partton functon of the vapor s represented n terms of the partton functon of a sngle cluster, accordng to some chosen cluster defnton. The equlbrum system s then nterpreted knetcally to obtan rate coeffcents for varous growth and decay processes. The nonequlbrum stuaton, where the vapor s supersaturated and undergong the nucleaton of condensed phase droplets, s dscussed n Sec. III. An expresson for the nucleaton rate s derved n terms of the propertes of equlbrum systems and ths expresson s then used to derve the nucleaton theorems n Sec. IV. These are then appled n Sec. V to test the valdty of the classcal theory of nucleaton and an nternally consstent dervatve of t. Both models fal to satsfy the theorems, and n both cases t s because the work of formaton of a fxed droplet s employed. The postonal, or mxng entropy dscussed recently by Ress et al. 8,9 s mssng, and the models are therefore nconsstent wth the statstcal mechancs. In the lght of ths, we propose a model based on the same capllarty approxmaton, but whch ncludes postonal entropy and s n accord wth the theorems. Fnally, the nucleaton theorems are used n Sec. VI to extract cluster propertes from expermental data, and conclusons follow n Sec. VII. II. STATISTICAL ECHANICS OF SUBSATURATED VAPOR A. Equlbrum populatons The statstcal mechancal treatment of an mperfect vapor s tradtonally developed usng a canoncal ensemble of systems of volume V contanng N molecules that have the ablty to assocate nto bound clusters due to mutual nteractons. The law of mass acton that determnes the cluster sze dstrbuton n terms of cluster partton functons can be derved, but the method s not entrely satsfactory snce there s no external control over the pressure of the cluster mxture, and hence the supersaturaton of the vapor phase. The vapor pressure depends nstead on the nternal parameters V and N, whch need to be chosen to obtan the desred pressure.

3 56 NUCLEATION THEORES, THE STATISTICAL The development n ths paper wll take a slghtly dfferent route by examnng a grand canoncal ensemble nstead. The system of volume V s consdered to be n contact wth a partcle reservor at a chemcal potental so that the number of molecules N n the system fluctuates. These molecules assocate nto clusters as before, and by varyng the chemcal potental, the mean pressure of the mxture of clusters n the system can be controlled. The reservor also acts as a heat bath at a prescrbed temperature T. If H(N) s the Hamltonan of the system when N molecules are present, and k s Boltzmann s constant, the grand partton functon of the system (,T,V) s then proportonal to the ntegral of exp(h(n)n)/kt over the entre phase space of molecular postons and momenta, summed over all N from zero to nfnty:,t,v N0 N N!h 3N j d 3 x j d 3 p j exphnn/kt, where x j and p j are the poston and momentum of the jth molecule, h s Planck s constant, and N! corrects for molecular ndstngushablty. The calculaton of can be greatly smplfed by consderng the system to be occuped by populatons of molecular clusters. The grand partton functon of the system can then be constructed from modfed canoncal partton functons for sngle clusters of molecules an -cluster. The defnton of a molecular cluster wll be left open, but t could, for example, requre that all the molecules le wthn a sphere of a certan volume centered on the center of mass of the whole set of molecules. Alternatvely, t mght nstead be requred that the separaton between molecules wthn the cluster should not exceed a maxmum dstance the Stllnger cluster. One could use any rule: a cluster could be defned as a confguraton of molecules lyng n a sngle plane. A snapshot of the molecules n the system would then be classfed as a collecton of clusters, wth the numbers of clusters present dependng on the cluster defnton chosen. Ths s llustrated n Fg.. Dfferent cluster populatons emerge f dfferent cluster defntons are used. However, as has been stressed above, a nucleaton theory based on a poor cluster defnton s unlkely to be very successful, and one should attempt through the defnton to nclude all physcal clusters and exclude all ephemeral states. There s a second mplcaton of usng an napproprate cluster defnton, whch s the followng. The calculaton of the system grand partton functon n terms of cluster partton functons wll be accurate only f the nteractons between molecules lyng n dfferent clusters are neglgble. The total energy for a partcular confguraton of the molecules n the system can then be separated nto ndependent contrbutons from each cluster. We wsh to wrte H(N) n H c (), where n s the number of clusters present n a gven system confguraton usng the chosen cluster defnton and H c () s the -cluster Hamltonan, whch depends on the postons and momenta of the molecules that make up the cluster. The cluster defnton may be arbtrary, but clearly the level of approxmaton n the evaluaton of wll depend upon t. Ths wll gude the choce of FIG.. A partcular molecular confguraton that contrbutes to the grand partton functon of a subsaturated vapor, llustratng how dfferent cluster defntons can affect the evaluaton of cluster populatons. Cluster A s defned usng a sphercal shell centered on the center of mass of a set of closely assocated molecules. Cluster B s defned nstead usng a crteron that molecules should be colnear. Accordng to ths crteron, only four molecules n case A can be consdered to be a cluster. Cluster C s defned by the requrement that the molecules are a fxed dstance apart. Those molecules that do not satsfy a chosen cluster defnton are consdered to be monomers: accordng to defnton C, therefore, none of the molecules n cases A and B s clustered. cluster defnton. For example, f ntermolecular forces were sotropc, then a defnton that favors sphercal clusters would be a better choce than clusters based on a planar crteron. On the other hand, f the ntermolecular forces were planar n character, then such a defnton mght not be unsutable. These ponts support the dea that a cluster should be defned as a collecton of molecules that are bound energetcally. Ths should mnmze the cluster-cluster contrbutons to the total energy: f a nearby molecule had a strong nteracton wth a cluster, an energy-related defnton would be lkely to nclude the molecule wthn the cluster. Clustercluster nteractons could be taken nto account n the form of a vral seres 20, but here we shall smply gnore any cluster-cluster nteractons n the Hamltonan. We now ntroduce a modfed canoncal partton functon Z for a cluster contanng molecules. Ths s related to the ntegral of exph c ()/kt over the phase space accessble to molecules n the cluster: Z d 3 x!h 3 j d 3 p j exph c /kt. j Clearly Z Q exp(/kt), where Q s the canoncal partton functon. The cluster defnton ntroduces constrants on the phase space avalable to the consttuent molecules, whch can act upon both molecular postons and momenta, and whch s ndcated n Eq. 2 as a prme on the ntegraton symbol. The phase space of molecules that form a cluster s smply a subset of the total phase space of a collecton of free molecules. The subset depends on the cluster defnton, and the dffculty n actually evaluatng the partton functon ntegrals wll of course depend on the defnton chosen. 2

4 568 I. J. FORD 56 Wth a lttle thought, t s evdent that the grand partton functon of the complete system can be expressed n the followng form: n Z n n!, where, as ndcated, the sum s over all possble cluster sze dstrbutons n. Ths grand partton functon then contans, as t should, contrbutons from all possble confguratons of molecules n the system, weghted by the approprate energes f the total Hamltonan separates nto cluster contrbutons. The factor of n! corrects for overcountng due to the ndstngushablty of clusters. any prevous evaluatons of a cluster partton functon have proceeded from ths pont by defnng molecular postons wth respect to the cluster center of mass, and then ntegratng the coordnates of the cluster center of mass over the system volume. Ths makes the modfed partton functon Z proportonal to the system volume V. However, ths proportonalty would result from any crteron that uses the phase space confguraton of the consttuent molecules to determne whether they form a cluster. The fundamental pont s that for any confguraton of molecules that satsfes the defnton, there wll be others that correspond smply to spatally translated copes of the frst. The summaton of these contrbutons to the partton functon ntroduces a proportonalty to V. The center of mass defnton s not exclusve, and n order to develop the statstcal mechancs t s not necessary to ntroduce t. We note n passng that snce Z s dmensonless, t should also be nversely proportonal to a quantty wth the dmensons of volume. We shall return to the nature of ths scalng volume later on. The grand partton functon n Eq. 3 s a sum of contrbutons over a new phase space of all sets n,.e., all possble cluster sze dstrbutons. The molecular poston and momentum coordnates are subsumed nto the cluster partton functon Z. A system contanng no molecules, and a system contanng a sngle cluster fllng the entre system, are among the confguratons taken nto account n Eq. 3. There wll be many possble other arrangements, correspondng to ntermedate molecular denstes. There s no restrcton n the grand ensemble on n, the number of molecules wthn the system. The next step that s tradtonally taken but normally wthn a canoncal and not a grand canoncal ensemble s to note that n Eq. 3 s domnated by a contrbuton from one partcular sze dstrbuton n e. In order to determne ths dstrbuton, we smply extremze the logarthm of the expresson wthn the sum n Eq. 3. We requre n ln Z n 4 n!0, whch leads, usng Strlng s approxmaton, to the followng expresson for the most probable, or equlbrum, sze dstrbuton n e for the gven condtons T and : n e Z. 3 5 Usng ths approxmaton, the grand partton functon becomes Z n e n e! exp n e. The pressure p v exerted by the nonnteractng clusters wthn the system for a sze dstrbuton n e s gven by Dalton s law p v V n e kt, and we see that the vapor pressure s a functon of and T, and that exp(p v V/kT) as requred. The grand potental of the whole system s p v V. The grand canoncal ensemble does not allow the vapor pressure to be fxed exactly, snce fluctuatons n cluster populatons and therefore p v can occur, but these are neglgble for a large system, and so to a very good approxmaton, the vapor pressure n the system s controlled by the external parameters and T, as we requre. Equaton 5 leads to the law of mass acton: n e Q (n e /Q ). Puttng Eq. 5 nto Eq. 7 we fnd p v VkT Z. The sum n Eq. 8 wll only converge f Z decreases sutably as a functon of. It turns out that ths lmts the statstcal mechancal ensemble to the study of vapors at or below the saturated vapor pressure. Systems at a hgher vapor pressure wll not be globally stable: the true equlbrum state wll be the bulk condensed phase. Ths constrant translates nto an upper lmt for the chemcal potental. Snce n e Q exp(/kt) from Eq. 5, we can ntroduce a reference populaton n es, whch s the free molecule, or monomer, populaton for a saturated vapor, and wrte n es Q exp( s /kt). s s the chemcal potental that produces ths reference populaton. We then deduce that s kt, where we have ntroduced the saturaton rato Sn e /n es, whch so far s lmted to values below unty. To a good approxmaton, for condtons well below the crtcal pont, n es ktp vs V, and Sp v /p vs, where p vs s the saturated vapor pressure. Ths confrms that whle does not control the total pressure exactly, t does control the partal pressure of the monomers, and hence provdes a very good external control over the saturaton rato. Furthermore, the approxmaton that the Hamltonan should separate nto contrbutons from ndependent clusters wll also fal as the system becomes denser. No attempt wll be made to descrbe the metastable, or supersaturated vapor usng statstcal mechancs, snce none wll be needed. The sngle cluster modfed partton functon Z can be expressed n terms of the sngle cluster grand potental (): Z exp/ktexpf/kt

5 56 NUCLEATION THEORES, THE STATISTICAL () and F() are the grand potental and Helmholtz free energy, respectvely, of a sngle -cluster n an otherwse empty system of volume V, at a temperature T and chemcal potental. They depend on the cluster defnton. From Eq. 5, the equlbrum populatons for szes and j satsfy n e /n j e exp j/kt. B. Knetc treatment We now ntroduce a knetc nterpretaton of ths equlbrum stuaton. We consder that the equlbrum sze dstrbuton n e s the statonary soluton of the followng set of populaton dynamcs equatons: dn dt j n j W j n j W j, 2 where W j s the coeffcent that determnes the mean rate at whch transtons are made that convert a j-cluster nto an -cluster. Recall that snce the clusters defned n the statstcal mechancs are supposed to model real physcal clusters, the rate coeffcents n Eqs. 2 wll descrbe such processes as molecular capture and molecular evaporaton to and from embryonc droplets. The connecton between rate coeffcents and cluster statstcal mechancs s then gven by W j e n exp j/kt. W 3 j n j e The knetc nterpretaton can be taken a step further by assumng that the only mportant transtons are those that are brought about by the addton or loss of sngle molecules to or from the cluster. The only nonvanshng rate coeffcents are then W for cluster growth, and W for cluster decay. They are related through exp/kt. 4 Transtons such as dmer addton to an -cluster to form an (2)-cluster have been consdered elsewhere 2 and n most cases they alter the nucleaton rate only slghtly, unless for some reason the dmer populaton s unusually large. The rate of combnaton of sngle molecules monomers and -clusters to form an ()-cluster s proportonal to n e n e,so n e. In fact, accordng to knetc theory, the growth rate s the molecular flux tmes the surface area A of the -cluster, assumng all collsons stck. The usual expresson s n e kta V2mkT S /2 s, 5 where s n es kta /V(2mkT) /2, where m s the molecular mass. We shall use the growth coeffcent n Eq. 5 n the followng development, whch brngs wth t a certan temperature and supersaturaton dependence. Other growth regmes could be consdered, for example, where dffuson lmts the absorpton rate. The decay coeffcent, however, s ndependent of supersaturaton. Vasl ev and Ress 22,23 have suggested that Eq. 5 can underestmate the true absorpton rate by a factor of up to 2, f the nomnal surface area A s used. Ths s brought about by an attractve nteracton between the cluster and a free molecule, whch can alter the trajectores of approachng molecules. Ths enhancement wll be neglected here, but would n any case only alter the nucleaton rate by the same relatvely small factor. A more sgnfcant mplcaton of the effect would be to ntroduce a dependence of growth rate upon the carrer gas pressure. The reasonng s that nert gas molecules, whch have been gnored htherto, but whch are necessarly present n nucleaton experments, mght nterfere wth the attracton between free molecule and cluster, and change the absorpton rate. Assumng that the evaporaton rate s not affected by the carrer gas n an equvalent way, the nucleaton rate could then be suppressed. However, ths would seem to be a large effect only for hgh carrer gas pressures, when carrer gas molecules are often lkely to be found wthn the attractve range of clusters, and so we wll neglect t here. The stuaton consdered so far s lmted to saturaton ratos S controlled by the external chemcal potental. At s the vapor just becomes saturated: S and p v p vs. The grand potental of the -cluster for such condtons wll be gven a subscrpt s and wrtten as s F s. The rato of rate coeffcents s then s exp s s /kt. III. NUCLEATION RATE IN A SUPERSATURATED VAPOR 6 7 The statstcal mechancal treatment of a subsaturated vapor n the prevous secton provdes rate coeffcents that can be used to study the populaton dynamcs of clusters for a supersaturated vapor (S). In ths way, the process of nucleaton can be modeled. We rewrte Eq. 2 n the form dn dt n n n n J J, 8 where J n n s the mean cluster current between szes and : the excess of growth transtons over decay transtons between the two szes. The populatons here no longer carry the superscrpt e, whch denoted equlbrum n a subsaturated vapor. They are now a general sze dstrbuton determned by the dynamcal equatons 8. The growth rate s stll gven by Eq. 5, but wth S: we now refer to S as the supersaturaton rather than the saturaton rato. Ths enhanced growth rate s the drvng force that causes the nucleaton of the new phase. Note that we consder nucleaton here to be the result of mean rates of transtons n the populaton dynamcs of clusters, whereas the pcture drawn earler was one of ndvdual

6 5620 I. J. FORD 56 clusters growng and decayng stochastcally, and occasonally beng drven to large, stable szes. The populaton equatons are solved as usual for the followng boundary condtons. Frstly the monomer concentraton s S tmes greater than that whch occurs n the saturated vapor,.e., n Sn es wth S). Secondly, the populaton at a sze s set to zero. It turns out that as long as s large enough, t does not matter whch partcular value s chosen. The condton n 0 prevents the system from relaxng to the global equlbrum state where the condensed phase flls the avalable volume. The cluster populatons are therefore held somewhat artfcally n a state of perpetual nucleaton of the phase transton. If we consder the steady-state soluton, such that dn /dt0 and J s a constant J for all, a process of elmnaton wthn the system of equatons 8 leads to the followng expresson for the nucleaton rate: J n j2 2 j / j. 9 The product j2 j / j can also be wrtten as ( / ) j2 j / j and usng Eqs. 5 and 7 we then get j2 j j exp s s /kt. Substtutng nto the rate expresson then gves 20 J n exp 2 s s /kt Sp vsv kt exp skt/kt exp s kt /kt. 2 Note that the nucleaton rate derved here s the number of clusters per second reachng the maxmum sze () n the volume V. The dmensons are nverse tme. Nucleaton rates are more usually defned as the number of partcles generated per second per unt volume, but we shall consder the total current and not ts densty. The usual procedure s now to represent the sum over as an ntegral between, and to expand the argument of the exponental about the pont where t reaches a maxmum, whch defnes the crtcal sze *: s kt 0. * 22 To see that the crtcal sze s loosely the sze that s equally lkely to grow or to decay, consder Eq. 4 for *. We have * * exp kt, 23 * usng Eqs. 9, 0, 6, and 22. If we ntroduce the cluster work of formaton then to a good approxmaton s kt, 24 J Sp vsv kt exp/ktz * exp*/kt, 25 where Z s the so-called Zeldovch factor gven by Z 2kT 2 s */ A small contrbuton from the dervatve of wth respect to has been neglected. Note that J s proportonal to *,soan enhancement of ths coeffcent accordng to the deas of Vasl ev and Ress 22,23 would enhance the nucleaton rate by the same factor. Why s () the cluster work of formaton? The cluster work of formaton s the change n the grand potental n gong from an empty system to a system contanng an -cluster, for constant external condtons of and T. Snce the grand potental of an empty system s zero, the cluster work of formaton s smply (), and Eq. 24 follows from Eqs. 9, 0, and 6. Equaton 25 takes the form that s often proposed on the followng heurstc grounds. Snce the crtcal cluster s the sze that s equally lkely to evaporate or to grow, the frequency of nucleaton should be proportonal to * n *, the rate of attachment of monomers to crtcal clusters. The populaton of crtcal clusters s equal to exp(*)/kt. Strctly, ths latter result s not vald for a supersaturated vapor, but s an extrapolaton of Eqs. 5 and 0 for

7 56 NUCLEATION THEORES, THE STATISTICAL s. Ths accounts for the last two terms n Eq. 25. However, addtonal factors such as Z and exp()/kt emerge from the more rgorous approach outlned above. Some of the factors n Eq. 25 correspond to the knetc prefactor referred to earler. m * P P. 3 IV. NUCLEATION THEORES A. Frst nucleaton theorem Now we can examne the supersaturaton and temperature dependence of the nucleaton rate and derve the nucleaton theorems. We start wth the exact expresson for J n Eq. 2 rather than the less unweldy but more approxmate verson n Eq. 25: The sze m * s equal to the crtcal sze * under the same approxmatons that were used earler to derve Eq. 25 from Eq. 2, namely, to replace s s n Eq. 29 by * and to expand the exponent to second order about * gven by Eq. 22. Then P()P(*)expZ 2 (*) 2 wth Z gven by Eq. 26. The sums n Eq. 3 can then be replaced by ntegrals over and so expz 2 * 2 J Sp vsv kt exp s/kt exp s s kt /kt, 27 where a factor of /S n both numerator and denomnator has been canceled. Takng the dervatve wth respect to, holdng T constant, we fnd J J T J J P T P P P, where the weghtng functon P() s gven by We therefore fnd that P s exp skt /kt. lnj m *, T where the cluster sze m * s defned as the expectaton value m * expz 2 * 2 *. 32 The dstncton between * and m * wll henceforth be dropped. Equaton 30 s the nucleaton theorem 4 6. It tells us that the supersaturaton dependence of the nucleaton rate s related to the crtcal sze. It has been used prevously to extract * from the slope of nucleaton rate data plotted aganst supersaturaton on a log-log scale at constant temperature. Earler dervatons concentrated on the supersaturaton dependence of the crtcal cluster work of formaton (*). The proof gven here would appear to be more general, and the steps taken n reachng t have been carefully set out. The proof does not rely on a partcular choce of cluster defnton n the statstcal mechancs. Some ndependence of cluster defnton was to be expected, snce the earler dervatons were made usng arguments from thermodynamcs. Some approxmatons have been made, but we beleve they are tenable n most stuatons. A sgnfcant advantage of the present statstcal mechancal knetc dervaton s that Eq. 30 s an exact expresson of the nucleaton theorem, whch takes nto account the knetc prefactor n the rate expresson as well as the exponental term. There s actually a hdden dependence on the cluster defnton n Eq. 30. We have stressed several tmes that the cluster defnton affects the calculaton of the grand potental, and ths means that each cluster defnton wll produce a dfferent crtcal sze, and therefore a dfferent dependence of the model nucleaton rate upon supersaturaton. However, when we come to use Eq. 30 to analyze expermental data, we mplctly make the reasonable assumpton that a perfect defnton exsts for physcal clusters, and that the expermental data are gvng us the crtcal sze for that defnton: the actual crtcal cluster, whch s equally lkely to grow or decay. B. Second nucleaton theorem A second nucleaton theorem has been derved recently 7 usng the methods of small system thermodynamcs 24. It concerns the temperature dependence of the nucleaton rate at constant supersaturaton, and s n a sense the

8 5622 I. J. FORD 56 conjugate to Eq. 30, whch we shall now refer to as the frst nucleaton theorem. We now prove t wthn the statstcal mechancal formalsm. It s a rather lengthy dervaton, wth partcular effort spent n carefully evaluatng small terms that are then neglected. The fnal result appears n Eq. 48, and the reader could proceed drectly to that pont f desred. Takng the dervatve of Eq. 27 wth respect to T, we fnd T J T Sp vs V kt exp s/kt vs Jp p vs T st s kt 2 P s / s s T s /kt 2 P P, 33 where a prme ndcates a partal dervatve wth respect to temperature. By expandng the terms n curly brackets nsde the sum about * we can wrte lnj T * s kt 2 s * p vs p vs T st s s*t s * kt 2 P 2 2 *2 2kT st 2 s * P. 34 No lnear term appears snce wth the use of prevous approxmatons, the expectaton value of (*) weghted by P() s zero. The term nvolvng s has been taken outsde the sum snce accordng to Eq. 5 we can wrte s T p vs p vs 2T A A T, s 35 and f we assume that A (v l ) 2/3 for sphercal clusters, then the fnal term n the last equaton s smply 2v l /(3v l ) and the whole expresson s ndependent of. The remanng expectaton value s best dealt wth by replacng the sums by ntegrals over wth lmts, and usng the approxmate form for P() used n Eq. 32. The last term n Eq. 34 can then be wrtten as d kt 2 2 *2 st 2 s * exp * kT st 2 s, * 36 where The last term n Eq. 34 s therefore 2T 2 s * 2 2 2Z2 0 kt 2 s 2 * 2 s*t 2 s * 2T T ln 2 s * Note the shorthand denotng the evaluaton of the dervatves at the crtcal sze. The partal dervatve wth respect to T n the last term s performed holdng * constant. The dmensonalty of the argument of the logarthm can be taken care of, f wshed, by the nserton of an arbtrary constant. Equaton 34 then becomes 2p vs 2 p vs T st s. 39 lnj T kt 2 2v l 3v l s*t s * kt 2 2 T ln 2 s * 2

9 56 NUCLEATION THEORES, THE STATISTICAL Now, s T s T FT F T s T s T, and snce Q expf()/kt s proportonal to the ntegral of exph c ()/kt over the phase space avalable to the molecules n the cluster as determned by the cluster defnton, where H c () s the -cluster Hamltonan, we can deduce that 40 FT F T lnq kt j j d 3 x j d 3 p j H c exph c /kt d 3 x j d 3 p j exph c /kt Ē, 4 where the ntegrals over the molecular postons and momenta are restrcted by the cluster defnton, and Ē() s the mean energy of an -cluster n a canoncal ensemble at temperature T. The bar over E emphaszes that fluctuatons n the cluster nternal energy are lkely to be substantal for such a small system. Now we need the dervatve of s wth respect to T n Eq. 40. s s the molecular Gbbs free energy of a bulk vapor phase at a pressure p vs and we can use the Gbbs-Duhem relaton s l dtv l dp vs d s 0, 42 where s l and v l are the entropy and volume per molecule n the bulk lqud phase, to deduce that s T s T h dp vs lts l Tv l dt s lh l Tv l p vs e l v l p vs Tp vs, 43 where h l and e l are the enthalpy and nternal energy per molecule n the bulk lqud phase when n equlbrum wth the vapor. Therefore Eq. 39 becomes 2p vs 2 p vs T E x kt E x* * v lp vstp vs 2v l 2 kt 2 kt 2 3v l 2 T ln 2 s *, 44 2 where lnj T E x Ēe l 45 s the excess nternal energy of an -cluster. Ths s the mean energy of the cluster mnus the energy the molecules would have, on average, n the bulk lqud phase at the same temperature and pressure of the vapor. Usng the Clausus-Clapeyron equaton dp vs /dt(h v h l )/(v v v l )T and p vs v v kt, the frst term on the rght-hand sde of Eq. 44 can be shown to be equal to 2Lv l /(v v v l )/(kt 2 ), where Lh v h l s the latent heat per molecule, and h v and v v are the enthalpy and volume per molecule n the bulk saturated vapor. Snce Ē() s the mean energy of a sngle vapor molecule, we can also wrte E x Ēe l h v h l p vs v v v l LkTp vs v l. 46 Equaton 44 then can be wrtten as lnj T L kt 2 2v l v v v l T p vsv l kt * v lp vstp vs 2 kt 2 E x* 2v l kt 2 3v l 2 T ln 2 s * Neglectng terms of order v l /v v, and also the last two logarthmc terms, whch n most crcumstances wll be small compared to the others, Eq. 47 reduces fnally to the second nucleaton theorem: lnj T kt LkTE x* Furthermore, t s possble to wrte LkTe v e l e x, whch defnes the mean excess nternal energy e x of the vapor. The rght-hand sde of Eq. 48 then reduces to e x E x (*)/kt 2. The dervaton of the theorem gven here s more rgorous than the earler treatment 7, and ncludes a number of small terms n Eq. 47. However, n the orgnal verson of ths theorem an addtonal term (LkT)/kT 2 appeared on

10 5624 I. J. FORD 56 the rght-hand sde of Eq. 48. Ths arose because the thermodynamc treatment focused on the temperature dependence of the cluster work of formaton: the exponental factor n the rate expresson. The knetc prefactor from classcal nucleaton theory was assumed, and ths ultmately gave rse to the addtonal term. Ths suggests that the classcal knetc prefactor s ncorrect, and ths wll be explored n the next secton. The use of the classcal prefactor to complete the dervaton of the theorem meant that the excess nternal energes extracted from expermental nucleaton data n Ref. 7 are too small by approxmately 20kT. A reanalyss of the data s gven n Sec. VI. V. TESTS OF ODELS A. Classcal nucleaton theory: The problem The nucleaton theorems can be used to test the nternal consstency of the classcal theory of homogeneous nucleaton. Ths model was derved orgnally from thermodynamc arguments. The man assumpton s that the equlbrum cluster populatons are gven by where, for large 0, e n,cl n e exp cl cl /kt, cl cl A kt cl () s a functon that plays the role of the work of formaton of an -cluster. We comment on what t represents later. The frst term s the surface free energy of a sphercal droplet wth the bulk lqud densty and bulk surface tenson. The surface area A s taken to be equal to A 0 2/3 wth A 0 (36v 2 l ) /3. n e s the monomer populaton, gven approxmately by n e p v V/kT. Equaton 49 s strctly vald for S. Usng Eq. 49 we can go drectly to the knetc dervaton of the nucleaton rate, startng from Eq. 3 and proceedng to Eq. 25 wth () replaced by cl (), or equvalently s () replaced by A. Insertng the classcal expresson nto Eqs. 22, 25, and 26 gves the classcal crtcal sze cl *2A 0 /(3kT ) 3 and classcal nucleaton rate J cl 2 V S2 p 2 vs v l exp A 0 cl * 2/3. 5 3kT /2 m kt 2 We now test the compatblty of the classcal rate wth the frst nucleaton theorem by calculatng the dervatve of lnj cl wth respect to. Wefnd lnj cl T2 A 0 3 kt 2 cl * 2/3 2 cl *. 52 The classcal theory s therefore nconsstent wth the frst nucleaton theorem, gven n Eq. 30. It s well known that the classcal theory and the law of mass acton are ncompatble and the falure to comply wth the frst nucleaton theorem s another reflecton of ths. Both problems can be corrected by multplyng J cl by a factor of /S 25, and there have been several attempts to justfy ths wthn the classcal formalsm. Now we calculate the partal dervatve of lnj cl wth respect to T: lnj cl d T 2 dt 2 L kt T 2 cl * d lnv l dt d kt2t dta 0 cl * 2/3. 53 We must compare ths wth Eq. 47. For the classcal theory 2 s 2 A A 0 4/3. 54 Note that the last term on the rght-hand sde of Eq. 47 only contrbutes, therefore, through the temperature dependence of the surface tenson and lqud densty. It s a small term, as suggested earler, but we wll retan t rather than use the approxmate form of the second nucleaton theorem, gven n Eq. 48. However, we neglect all terms of order v l /v v n Eq. 47, to obtan lnj cl T d 2 dt L kt T d lnv l 2 dt E x cl cl *, kt 2 55 so the excess nternal energy for the crtcal cluster n the model, as determned by the second nucleaton theorem, s E x cl cl * T d dta 0 cl * 2/3 kt d lnv 2 l dt cl *LkT. 56 Now, snce cl *kt (2/3)A 0 cl * 2/3, the frst two terms are proportonal to the surface area of the crtcal droplet. If these were the only terms, the nternal energy of the droplet would be gven by a term proportonal to the volume (e l ) plus a term proportonal to the surface area the excess nternal energy, and ths would be consstent wth the underlyng capllarty approxmaton. However, the last term n Eq. 56 spols ths pcture. Unfortunately, t s not suffcent smply to neglect t n comparson wth the other terms. It s a symptom of a deeper nconsstency wthn classcal theory that needs to be resolved. The falure to comply wth the two nucleaton theorems tell us that the classcal theory s ncomplete. The volatons arse because Eq. 50 does not represent the work of formaton of a cluster correctly. The rght-hand sde of Eq. 50 s n fact the work of formaton of a classcal droplet, whch has a fxed poston n the system. We can dentfy cl () wth the work of formaton of such a droplet and neglect cl () n comparson. What we really need, however, s the free energy of a cluster whch can appear anywhere n the system. A symptom of ths problem s that exp cl ()/kt wth cl () gven by the rght-hand sde of Eq. 50 s not proportonal to V as t should be. The mssng term s the postonal entropy mxng entropy 8 arsng from the translaton of clusters throughout the system volume. Note ths s not the same as ntroducng the translatonal knetc energy of a cluster nto the excess free energy, whch we shall comment on shortly.

11 56 NUCLEATION THEORES, THE STATISTICAL B. Reparng classcal nucleaton theory Let us now return to Eq. 25 and see how we mght be able to derve the classcal rate expresson takng nto account postonal entropy, and therefore repar classcal nucleaton theory. The ngredent that has to be provded by some physcal model s the cluster work of formaton: s kt F s kt. 57 Now, snce s g l, the Gbbs free energy per molecule n the bulk lqud phase when n equlbrum wth the vapor, we can wrte s f l p vs v l, where f l s the molecular Helmholtz free energy n the lqud phase, so that F x p vs v l kt F x kt, 58 where F x () s the excess Helmholtz free energy of the -cluster, defned by F x Ff l. 59 Agan, ths excess quantty s the free energy of the cluster mnus tmes the free energy per molecule n the condensed phase. We need to calculate () for use n Eq. 25. We wrte F s kt, 60 where F() s the Helmholtz free energy of a monomer n the volume V, and s s the common chemcal potental of a vapor and ts condensate at equlbrum at a temperature T. From elementary statstcal mechancs, F()kT ln(v) and s kt ln(/ vs ), where we have approxmated the monomer densty n the saturated vapor by the molecular densty vs. The factor s equal to (2mkT) 3/2 /h 3, where h s Planck s constant. Then exp/kt S vs V v v SV. 6 scaled-down propertes of a sphercal droplet of bulk lqud. However, we shall contnue to pursue ths model n the sprt of tryng to repar classcal theory. The second term n Eq. 62 s the postonal entropy term due to the contrbuton to the partton functon from translated copes of every cluster. It was stated n Sec. II that the cluster free energy should be proportonal to the volume. The so-called scalng volume v c appears n Eq. 62 n order to mantan the correct dmensons. It acts as a means of resolvng and countng translated states n the system. Ths quantum of volume has been dscussed extensvely elsewhere 8,9. It appears n coarse-graned statstcal models where the poston of a mesoscale object s a degree of freedom; droplet models are n ths class, as are models of mcroemulson behavor. Often the sze of the scalng volume can be obtaned ntutvely 8. Otherwse one needs to refer back to the statstcal mechancs of the underlyng system, defned n the full phase space of all the degrees of freedom. The work of formaton based on the capllarty approxmaton s then, to a good approxmaton, cap A 0 2/3 kt lnv/v c kt. 63 Ths form should not be expected to apply for small, and when t s used, t wll be assumed that s large. The crtcal sze s found by solvng cap (*)/0, whch yelds the classcal expresson * cl *, f the scalng volume v c s ndependent of. The nucleaton rate s now obtaned from Eqs. 25, 54, and 6: J cap m 2 /2 Sp vs v l kt * 2/3 V v c exp A 0 cl. 64 3kT We shall now construct a repared classcal theory, startng by nvokng the capllarty approxmaton, so that the excess free energy for a sphercal droplet wth the bulk lqud densty and bulk surface tenson s F x cap A kt lnv/v c. 62 The frst term, the classcal excess free energy, has been crtczed on the grounds that contrbutons from cluster translatonal and rotaton knetc energy do not appear, so that the free energy does not represent all the degrees of freedom avalable. However, t s the excess free energy that s requred, and t seems lkely that there s no strong contrbuton to F x () from molecular knetc energy. The molecules n the cluster probably have a smlar mean knetc energy to that whch they have n the bulk lqud. Ths pont has been debated for thrty years and has shrouded the applcaton of statstcal mechancs to nucleaton n controversy. Perhaps the above argument clarfes the pont: t s largely the molecular potental energy that plays a role n nucleaton and not the knetc energy terms. A second crtcsm of the capllarty approxmaton s that small molecular clusters are most unlkely to possess the FIG. 2. The classcal nucleaton rate predctons dvded by the expermental rates 5,29,3 are shown as open symbols for varous substances and temperatures. The flled symbols show the same rato accordng to a revsed model based on the capllarty approxmaton but satsfyng the nucleaton theorems. The enhancement factor s v v /(Sv l ).

12 5626 I. J. FORD 56 If v c dd depend on, the crtcal sze of the revsed model would not be the same as n the classcal model. To a frst approxmaton, however, we could gnore ths nfluence and smply evaluate v c n Eq. 64 at the classcal crtcal sze. We now can see that the classcal rate expresson, Eq. 5, can be obtaned from a more rgorous statstcal mechancal knetc approach usng the capllarty approxmaton, but only f we take the scalng volume n Eq. 64 to be the molecular volume n the supersaturated vapor, namely, v c v v /SkT/(Sp vs ). However, the fact that the classcal rate does not satsfy the nucleaton theorems warns us aganst ths choce. Recent work has suggested that the scalng volume for a droplet model based on the capllarty approxmaton should be of the order of the molecular volume n the condensed phase, not n the supersaturated vapor 8. Indeed Ress et al. 9 have recently examned the scalng volume for a droplet model based on the capllarty approxmaton and proposed that for szes relevant to nucleaton, t takes the approxmate form v c v l /2. We can examne the propertes of the scalng volume v c by requrng that Eq. 64 should satsfy the nucleaton theorems. Repeatng the steps of Eqs we fnd that f v c s ndependent of both S and T, then both nucleaton theorems are satsfed, wth the excess cluster nternal energy beng gven by Eq. 56 wthout the unwanted fnal term. As forecast, ths extra term was a symptom of a major problem, whch s now seen to be ether the total neglect of postonal entropy n the classcal cluster free energy, or equvalently the use of an ncorrect scalng volume. We conclude that the classcal rate should be corrected by a factor of v v /(Sv c ), where v c s a temperature- and supersaturaton-ndependent volume. Several other nucleaton models not necessarly based on the capllarty approxmaton have suggested that the classcal rate should be corrected by a factor v v /(Sv l ) 2,26,27, together wth addtonal numercal factors and powers of *. We see that ths correcton factor has a sutable form, apart from the mnor temperature dependence of v l, to ensure complance wth the nucleaton theorems. The scalng volume s then ndeed proportonal to the molecular volume n the lqud. We examne the effect of the rate enhancement factor v v /(Sv l ) n Fg. 2 for varous substances over typcal expermental temperature ranges. The open symbols denote the rato of J cl to expermentally measured nucleaton rates. The flled symbols denote the same rato multpled by the approprate factor of v v /(Sv l ), whch takes values between 0 4 and 0 6. Unfortunately, the revsed model s no more successful than the classcal theory n collapsng all the data onto a sngle lne, or even n accountng for the expermental temperature dependence. In several cases the predcted rate has an mproved temperature dependence, but ths s not unversal, as demonstrated by the data for n-butanol. Nevertheless, the revsed theory s better founded than classcal theory, and presumably t s the capllarty approxmaton that leads to the poor agreement. C. Internally consstent classcal theory The nucleaton theorems allow us also to study the socalled nternally consstent classcal theory ICCT 28 for whch the nucleaton rate s J ICCT m 2 /2 2 v l V Sp vs kt expa 2 0 /ktexp A 0 cl 3kT * 2/3. 65 We can derve ths expresson n two ways. The model was orgnally developed by employng Eq. 49 wth the terms n the exponent gven by ICCT ICCT A 0 2/3 kt A 0 kt. 66 Ths s motvated by a desre for an expresson that gves the correct result at n Eq. 49: ths s the nternal consstency that the model s desgned to acheve. The crtcal sze n the model s the same as for classcal theory, cl *, and the supersaturaton dependence of Eq. 65 satsfes the frst nucleaton theorem. However, when we use the second nucleaton theorem to extract the excess nternal energy, the same problem we encountered n the classcal theory appears. We fnd that lnj ICCT T d 2 dt 2 L kt T 2 cl * cl * /3 d lnv l dt kt d 2T dta 0 cl * 2/3, 67 so the excess nternal energy for the ICCT s E ICCT x cl * T dta d 0 cl T d 2T dt 3 * 2/3 kt d lnv 2 l dt cl * cl * /3 LkT d lnv l dt A 0 cl * 2/3 LkT. 68