Heterogeneous multicomponent nucleation theorems for the analysis of nanoclusters

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1 THE JOURNAL OF CHEMICAL PHYSICS 16, Heteroeneous mutcomponent nuceaton theorems for the anayss of nanocusters Hanna Vehkamäk, a Ann Määttänen, b Antt Laur, and Markku Kumaa Department of Physca Scences, P.O. Box 64, Unversty of Hesnk, Hesnk, Fnand Pau Wnker, Aron Vrtaa, and Pau E. Waner Insttut für Expermentaphysk, Unverstät Wen, Botzmannasse 5, A-1090 Venna, Austra Receved 19 December 006; accepted 15 March 007; pubshed onne 7 May 007 In ths paper we present a new form of the nuceaton theorems appcabe to eroeneous nuceaton. These eroeneous nuceaton theorems aow, for the frst tme, drect determnaton of propertes of nanocusters formed on pre-exstn partces from measured eroeneous nuceaton probabtes. The theorems can be used to anayze the sze frst theorem and the eneretcs second theorem of eroeneous custers ndependent of any specfc nuceaton mode. We appy the frst theorem to the study of sma water and n-propano custers formed at the ace of 8 nm sver partces. Accordn to the experments the sze of the two-component crtca custers s found to be beow 90 moecues, and ony ess than 0 moecues for pure water, ess than 300 moecues for pure n-propano. These vaues are drastcay smaer than the ones predcted by the cassca nuceaton theory, whch ceary ndcates that the nuceatn custers are too sma to be quanttatvey descrbed usn a macroscopc theory. 007 Amercan Insttute of Physcs. DOI: / I. INTRODUCTION Frst-order phase transtons are cruca n many branches of physcs and chemstry. The formaton of a new phase can occur n a oeneous parent phase or eroeneousy around some nuceaton seeds such as mpurtes or partce aces. 1 In the case of as-to-partce transton, condensaton rowth, evaporaton, and eroeneous chemstry are processes that defne the fate of the newborn qud or sod custers after nuceaton has occurred. Nanopartces have receved ntensve attenton n many branches of technooy, and eroeneous nuceaton s an mportant part of ther formaton processes. Atmospherc nanopartces can affect human heath, and when they row to arer szes they aso reduce vsbty and pay a roe n determnn the Earth s radaton budet and thus cmate chane. 3 6 Recenty, eroeneous nuceaton was suested to be mportant n atmospherc nanopartce formaton. 7 We nvestate the sze of sma custers formed at the ace of a preexstn aeroso partce. The number of moecues n a crtca custer actn as a startn pont of phase transton can be obtaned usn the frst nuceaton theorem, whch we derve here for eroeneous nuceaton. We aso derve the second eroeneous nuceaton theorem, whch ves the bndn enery of the crtca custer, athouh expermenta data for the appcaton of ths theorem does not yet exst. Genera forms for the nuceaton theorems have been presented earer, 8 but the ack of a specfc form appcabe to anayss of eroeneous nuceaton probabty data has been hampern the use of these powerfu anaytca toos. s the excess number of moecues of component n the, crtca custer. N s the number of moecues n the buk, qud phase of the custer, N,, and N,,so, are the numbers of moecues on the as-qud and qud-sod aces of, the custer, N,so,A,so s the number of moecues on a asa Eectronc ma: hanna.vehkamak@hesnk.f b Aso at: Fnnsh Meteorooca Insttute, P.O. Box 503, Hesnk, Fnand In Sec. II we revew the enera form for the frst theorem, wth temperature constant and varyn as-phase actvtes, n the case of eroeneous nuceaton, and n Sec. III we derve the enera form for the second theorem studyn a case where ony temperature vares, but as-phase actvtes are constant. In Secs. IV and V we show that the cassca eroeneous nuceaton theory, and especay the eometrc factors used n t, obey the form of the theorems. Secton VI shows how the theorems are expressed n terms of the measurabe nuceaton probabty, and Sec. VII dscusses estmaton of the roe of the knetc pre-factor. Secton VIII descrbes the expermenta resuts and how they are anayzed, and Sec. IX contans the resuts of the data anayss. In Sec. X we fnay ve concusons. II. GENERAL FORMALISM: FIRST THEOREM Kashchev 8 has shown that for the sotherma case the nuceaton theorem for eroeneous as-qud mutcomponent nuceaton s G, = N,, 1 T,,j where G s the formaton free enery of the eroeneous crtca custer,, s the as-phase chemca potenta of component, T s the temperature, and N,, = N, + N,, + N,so,, N,so,A,so V,, /007/1617/174707/1/$ , Amercan Insttute of Physcs Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see

2 Vehkamaek et a. J. Chem. Phys. 16, sod ace whch has the same area A,so as the qud-sod nterface of the custer, V s the voume of the custer, and, s the number densty of component n the as phase. For as-qud ace and the qud phase we have used the subscrpt to expcty ndcate that we are dean wth the eroeneous custer, snce aso the oeneous custer has a as-qud nterface and qud phase core. In most cases we have omtted the superscrpt referrn to the crtca custer n the nterest of smpfyn the notatons. A the quanttes we dea wth are those of the crtca custer uness otherwse stated. In practce, experments provde nuceaton rate, or nuceaton probabty, as a functon of as-phase actvtes A,. The nuceaton rate s proportona to exp G /, and the as-phase actvty s connected to the as-phase chemca potenta, by, =,,pure p,pure sat + n A,, 3 whch s vad for an dea mxture of dea ases. The chemca potenta of saturated pure vapor,,,pure p,pure sat depends ony on temperature, and k s the Botzmann constant. Usn Eq. 3 the nuceaton theorem 1 can be wrtten as G = N n A,. 4,T,A,j For one-component systems N, ven by takes a smper form N, =N, V, snce the dvdn aces between the phases can be chosen to be the equmoar aces, 9 and thus the numbers of ace moecues are zero. The oeneous case s ready obtaned as a speca case of the eroeneous theorem by settn the numbers of moecues on the as-sod and qud-sod nterfaces as zero. III. GENERAL FORMALISM: SECOND THEOREM The formaton free enery of a eroeneous crtca custer s G = P P V +, 5 where P and P are the pressure n the qud custer and the as-phase pressure, respectvey, and the effectve ace enery 8 s =,so +,,,so,a,so, 6 where,so s the contrbuton of the qud-sod ace,,, s the contrbuton of the as-qud ace, and,so,a,so s a contrbuton of a as-sod ace whch has the same area A,so as the qud-sod ace. The temperature dervatve of the formaton free enery s G T P P = V T + TV + P P V T + V V T. 7 The enera expresson for the formaton free enery of a not necessary crtca custer s 9 11 G = P P V +,,so, N,so,, N,, + +,,, N, +,so,,,,,, N,so,A,so. 8 The crtca custer satsfes the condton G /V N,,N,so,,N,so,,N,,,,T =0, where the dervatve s taken wth respect to the ocaton of the dvdn ace, but keepn the actua physca custer unchaned. 1 Ths eads to the enerazed Lapace equaton P P + =0, 9 V whch s vad for the crtca custer, and any choce of the dvdn ace, 10 and thus Eq. 7 reduces to G P P = V T T TV For each ace phase we use the Gbbs adsorpton equaton, whch n a nonsotherma case reads d A = S = dt d, N,, 11 where S stands for entropy, and for the buk qud and as we have the Gbbs-Duhem equatons and V V where V dp = S dt + dp = S dt + d,, N d, N,, 1 13 s the as-phase voume and N, =V, s the number of as moecues. Assumn that keepn the voume constant aso keeps the ace area constant n other words the shape of the custer s unchaned Eq. 10 can then be wrtten as G = V T S, + N, T V where S,so,A,so has area A,so. S S,so, + N, T + S,, +S,so,A,so, N,,so,so T + N,, +,,,, T,,,so N,so,A,so, 14 T s the entropy of a as-sod ace whch Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see

3 Heteroeneous mutcomponent nuceaton J. Chem. Phys. 16, The crtca custer s n a metastabe equbrum wth the vapor, and thus the chemca potentas are equa throuhout the system,, =,, =,,so =, =,.,so 15 If the as-phase chemca potentas are kept constant whe takn the dervatve wth respect to temperature, the second nuceaton theorem s smpy G T = S S,so, S,, + V V S + S,so,A,so. 16 In Eq. 16, the combnaton of the neatve terms ve the tota entropy of the eroeneous crtca custer, and the postve terms the entropy that the space whch the custer occupes voume V and the ace area A,so when fed wth the as phase, and thus the equaton can be wrtten G = S T. 17, In practca appcatons t s however more convenent to keep the as-phase actvtes, rather than the chemca potentas, constant and Eq. 3 toether wth the Gbbs-Duhem equaton and Causus-Capeyron equaton aow then the temperature dervatve of the as-phase chemca potenta to be expressed as 13 d, dt A,,, where h pure, and T h, pure T, T e, pure T, 18 are the enthapy and enery, respectvey, per moecue n a pure buk qud. In dervn the frst equaty of resut 18, the parta moecuar voume n the qud has been assumed nebe compared to the parta moecuar voume n the as. Usn 15 and 18 n Eq. 14 resuts n G T = 1 A, T V V TS,, TS + N + N,,,,,,,,, TS + TS,so,A,so, + N,,, + TS,so, N,so,A,so,so +,,. N,,so,,,so For the dervatve of the nuceaton rate or the nuceaton probabty we need aan the dervatve of G / whch, usn the crtca custer formaton enery ven by Eq. 5, can be wrtten as G = 1 T A, T G + G T A, = 1 G G T A, = V V P V + TS + N +,so + TS,so +,so,a,so + TS,so,A,so N,,so,so +,,,,, + P V +,, + TS,,, N,so,A,so,,,so 1 + TS. + N + N,,,,,,,,,, 19 0 Wth the hep of the tota enery of a ace phase U = + TS +, N,, and the eneres of qud and as phases U = P V U = P V + TS +, N,, + TS +, N,, Equaton 0 can be wrtten as 1 G = 1 T A, U U,so,A,so + V V, + U,, + U,so, N,,, N,, N, + V V U,, N,so,, N,so,A,so. 3 In Eq. 3 the frst three terms ve the tota enery of the Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see

4 Vehkamaek et a. J. Chem. Phys. 16, v, v,j =. n A, /A, n A,j /A,j 6 FIG. 1. Geometry of a custer cap-shaped part of a sphere wth radus r formn on a spherca seed partce wth radus R p. s the contact ane. eroeneous custer; the next two terms represent the enery that the space occuped by the custer woud have f fed wth as; the ast fve terms ve the enery the custer moecues woud have n pure buk quds, and the enery the moecues n the as-fed custer voume and on the as-sod ace woud have n pure buk quds. Thus a smpe form for the second nuceaton theorem reads G T A, = pure,u, 4 where pure, refers to the dfference compared to pure quds, and the frst refers to the dfference between the custer and the same space occuped by as phase. For onecomponent system the theorem 3 can be smpfed, snce by usn equmoar aces as the dvdn aces the numbers of moecues on the aces can be set to zero. The oeneous case s aan obtaned as a speca case of the eroeneous theorem by settn the eneres and numbers of moecues reated to the as-sod and qud-sod nterfaces as zero. For comparson wth earer forms of second oeneous nuceaton theorem, see remarks after Eq. 55. IV. CLASSICAL FORMALISM: FIRST THEOREM In the cassca eroeneous nuceaton theory the nuceatn custer s treated as a cap-shaped embryo wth radus r formn on a spherca seed partce wth radus R p, and the contact ane between the custer and the underyn ace s denoted by. Fure 1 shows the eometry of the stuaton. Ths eometry s a speca case of a more enera stuaton overned by the resuts n the precedn sectons. We aso want to show that the we-known eometrc factors 14 arsn n ths speca case can expcty be manpuated so that we arrve n the eroeneous nuceaton theorems. In the same as-phase actvtes and temperature, the radus of the eroeneous custer s the same as that of the oeneous one, and t s ven by the Kevn equaton r v,, =, 5 n A, /A, where v, and A, are, respectvey, the qud phase parta moecuar voume actvty of component,, s the asqud ace tenson, k s the Botzmann constant and T s the temperature. The crtca custer composton can be soved from equaton For the oeneous crtca custer the formaton enery can be wrtten as G = 4r,. 7 3 The frst oeneous nuceaton theorem 10 ves the excess compared to the custer voume fed wth vapor number of component moecues n the crtca custer as N, = G n A,T,A,j = 4r, 3 r r n A, + 1,, n A,. 8 The formaton enery for a eroeneous crtca custer can be expressed wth the hep of the oeneous formaton enery as 14 G = f G G. 9 The eometrc factor f G can be expressed as 14 f G = Xm + X m X m where = 1+X Xm, + X 3 3 X m X m 1, 30 X = R p r, 31 and the contact parameter s m=cos. Youn s equaton 15 reates the contact parameter to the ace tensons between as and sod,so, qud and sod,so, and as and qud, as m = cos =,so,so. 3, The number of moecues n the qud phase of the eroeneous crtca custer s connected to the oeneous case by N,, = f N N, where f N s the rato of eroeneous and oeneous custer voumes, V and V, assumn the same qud densty n both cases 14,16 f N = V V = Xm X 3 3 X m + X m 1 Xm For a panar pre-exstn ace X and f G = f N, but for a spherca condensaton nuceus f G f N. Usn the equaty of chemca potentas 15, as-phase chemca potenta 3 and =A, Gbbs adsorpton Eq. 11 at constant temperature eads to equatons Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see

5 Heteroeneous mutcomponent nuceaton J. Chem. Phys. 16, ,so A,so A,so A,,, n A, = N,so,so, 34 = N n A,so,A,so, 35,,so,so, =4r = N n A, n A,,, 36, where the ace area of the oeneous custer s A, =4r. 37 The as-qud ace area n the eroeneous case s A mx 1, =r The qud-sod nterface area s A,so =R 1 X m p, 39 and thus the dervatve of the factor f G wth respect to contact parameter m can be expressed as 4r f G 3 m =R X m p 1 = A,so. 40 Youn s Eq. 3 ves m = 1 n A,,,so,so m, n A, n A,, and Eq X r = R p r = X r. 4 The dependence of the eroeneous custer formaton free enery on the as-phase actvty A, s ven by G G = f G G n A,T,A n A,,j = f G N = f G N + X + G X r f G X f G X r X r + f G m n A, m n A, r + m f G n A,, m f G 4r, r X 3 r + 1 n A,,, 1 f G n A,, m,,so,so n A, n A, + 4r 3 m f G m X f X G, n A, f G,so,so, 43 m n A, where n the ast stae we have added and subtracted term 4r /3X/f G /X, / n A, to be abe to dentfy the ast form of Eq. 8. Usn Eqs. 8, 34 36, and 40, Eq. 43 can be transformed to read G n A,T,A,j =f G X f G XN 3m 1 f G m X f G, XN,, + N,,,so N,so,A,so = f N N 3m 1 f G m X f G, XN,,, + N,,so N,so,A,so, 44 where we have used the foown reaton between the eometrca factors: f G X f G f N. 45 X= The excess number of moecues n the oeneous custer,, conssts of buk qud N and ace phase N,, contrbutons N, = N, + N,, V,. Equaton 44 can then be wrtten as G, n A,T,A,j = f N N X +f N 3m 1 f G m X f G, N,, + N,,,so N,so,A,so f N V,. Usn ace areas 37 and 38 we et f N 3m 1 f G m X f G X= 1 1 mx + = A, A,, and the frst eroeneous nuceaton theorem takes the enera form 4 Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see

6 Vehkamaek et a. J. Chem. Phys. 16, G, n A,T,A,j = f N N + A, A,, N,so,A,so, = N N,,, f N V,, + N,so, + N,, + N,,,so N,so,A,so V, = N, 49 where we have used reatons A, /A,, f N N =N,, and f N V =V, N,, =N,,,,. Ths resut concudes that the cassca Fetcher 14 theory for eroeneous nuceaton, and the frequenty used eometrca factors nvoved, are consstent wth the frst nuceaton theorem. U = G T A, = 4r, 3 r r T + 1,, T 1 T. 50 Usn equaty of chemca potentas 15 and formua 18 for the temperature dervatve of the chemca potenta n the Gbbs adsorpton Eq. 11 n a nonsotherma case wth =A thus eads to equaton A TA, = S N,, T A, = T 1 TS +, N,, N, = T 1 U, A N,, 51 V. CLASSICAL FORMALISM: SECOND THEOREM The second oeneous nuceaton theorem 13 reates the temperature dervatve of the formaton free enery to the excess enery of the crtca custer compared to the same moecues n pure buk quds where we have used Eq. 1 for the ace phase enery. Youn s Eq. 3 ves s m T = 1,,so,so m,. 5 T T The temperature dervatve of the formaton free enery G T G = f G A, T U = f G U = f G + X G + G m f G m X f G f G X r X r T + f G m m T X f G r r X T + m f G,, m T 1 f G, m f G 4r, r X 3 r T + 1,, X, T f G m,so,so T T 1 T + 4r, 3 X f G X + 4r 3,so,so T, 53 where we have added and subtracted terms 4r /3X/f G /X1/,, /T 1/T to be abe to dentfy the ast form of Eq. 50. Usn Eqs. 50, 45, 40, 37, and 51 for oeneous as-qud and eroeneous as-sod and qud-sod aces, Eq. 53 can be wrtten as G =f G X f G T A, X U + 3m 1 f G U = f N 1 3 m f G m X f G 1 U,so,A,so U = f N U,so,A,so U,, A,so,so,, 3 m X X U,,,, N,,, N,so,A,so +, f G XA,, T + A, 3, N,so,A,so A,,, X f G X + A,so,so,so T, + A,, X f G 3 X N,, U,so + A,so,so +,, N,so m f G m X f G X 1 m,a,so 1 A,so,so +,so, U,so, N,so. 54 Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see

7 Heteroeneous mutcomponent nuceaton J. Chem. Phys. 16, The combnaton of terms proportona to A,so equas zero accordn to Youn s Eq. 3. The excess enery of the oeneous custer conssts of buk qud U and ace phase U,,, contrbutons U = U + U,,,, N,, V V + V V U,,, N, N. 55 In the prevous versons 13,17,18 of the second oeneous nuceaton theory, the terms proportona to V /V have been omtted as nebe, or because the reference state used has been an empty system rather than the custer voume fed wth as. These terms represent the enery of the moecues that the custer space woud have f fed wth as, and the enery those moecues n pure buk quds, and are ndeed sma, snce the custer occupes a tny porton of the tota voume of the nuceatn as, but we have ncuded these terms here to be consstent wth our enera formasm. Usn resut 48 and reatons A,,, f N N =N,, f N V A, /A, U,, =U,, f N U =U we et the resut /A,, N,, =N,,,, =V, and G T N U = f A, U,, N U = f U,so,A,so = 1 U + V V,, N +,, N,,,, N +, + U,, + U,so,, N + V V U + V V U,so,A,so V V, N,so,A,so V V U + V + V U,so,, N,, + f N 3m 1 f G m X f G U,,so +, N,so,A,so, N, U U,so,A,so A +, A,, N,so,, N U,,, X, N,so,, N,,, N,,,, N,so,, N,so,A,so, 56 whch s equa to formua 3 snce we study the oeneous and eroeneous nuceaton n the same vapor, and, thus N /V =N, /V =,. Thus, we have expcty shown that the Fetcher 14 theory s aso consstent wth the second nuceaton theorem. we et n J =nn 1 P 1 nt, 58 VI. THEOREMS IN TERMS OF THE NUCLEATION PROBABILITY So far, the forms of nuceaton theorems presented n the terature have ony nked the behavor of the nuceaton rate to the crtca custer propertes. In eroeneous nuceaton experments the quantty of prme nterest s, however, the nuceaton probabty, whch tes the fracton of pre-exstn partces that have a nuceated custer rown on the ace. We want to nk ths drecty observabe quantty to the propertes of the custers. From the defnton of the nuceaton probabty P n a tme perod t see, for exampe, Ref. 19 P =1 exp J t, 57 where J s the nuceaton rate per pre-exstn partce per unt tme unts 1/s. To obtan the nuceaton probabty as functons of the as phase actvtes as n F. the expermentast count the number of pre-exstn partces whch have actvated as nuceaton centers and started to row after a certan fxed tme. To et the probabty curve as a functon of the vapor actvty, the atter s chaned, but the tme perod after whch the probabty measured s kept constant. Thus, we can keep t constant when anayzn ths knd of expermenta data. We take the dfferenta of Eq. 58 wth respect to n A, and T usn the fact that the nuceaton rate s connected to the formaton enery by J =K exp G /, where K s a knetc pre-factor. We can express the eroeneous nuceaton theorems as Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see

8 Vehkamaek et a. J. Chem. Phys. 16, than zero but ess than one, and ths s the rane of expermenta resuts we can use to obtan crtca custer propertes. VII. KINETIC PRE-FACTOR In the mutcomponent nuceaton the contrbuton of the pre-factor K cannot be exacty cacuated, 18 but the cassca form of the knetc pre-factor K can be used to estmate the effect of the knetcs on the nuceaton theorems. Accordn to the cassca theory the pre-factor for nuceaton rate per pre-exstn partce, unts 1/s reads K =4R p R av c ads s,tot Z. 61 The tota number of moecues adsorbed on the preexstn partce ace s c ads s,tot =c ads s,1 +c ads s,, and the expressons for c ads s, 1/m are cacuated usn a steady state between ncomn and outon moecue fuxes 0 FIG.. The expermenta nuceaton probabtes markers, ft functons for monodsperse 8 nm seed partces sod nes and nterated probabtes for the actua expermenta partce sze dstrbuton dash-dotted nes. The top fure shows the water-rch cases, and the bottom fure the n-propano rch cases. Dfferent markers are used for dfferent as-phase fractons of n-propano ven n the eends. The x-axs shows the as-phase actvty of water top fure or n-propano bottom fure. nn 1 na, nn 1 1 PT,A,j 1 P T A, F n A,T,A,j = n J = N n A,,T,A,j + F TA, = n K, n A,T,A,j n J T A, 59 = pure, U n K +, 60 T A, whch nk expermentay accessbe nuceaton probabty P to quanttatve propertes of crtca custers rown on partce aces. We have defned Fnn 1/1 P to shorten the notaton ater n ths paper. Note that F ony has meannfu vaues when the nuceaton probabty s reater c ads p, s, = m exp F des,, 6 where m s the mass of a moecue, p, s the pressure n the nuceatn vapor, s the vbraton frequency of a moecue on the ace, and F des, s the desorpton enery for component. The averae rowth rate R av s defned as R av = 1 1 sn + cos. 63 s the drecton ane of the crtca custer rowth vector obtaned as the eenvector assocated wth the neatve eenvaue of the matrx product R W, where the rowth matrx contann the coson rates s R = Matrx W s formed from the second dervatves of the formaton free enery = G n W 1 G n 1 n G n 1 n G n W 11 W 1, W 1 W 65 wth the dervatves performed wth respect to the tota numbers n the eroeneous custer. 1 The tota numbers of moecues consst of buk qud contrbutons pus ace excess correctons for both as-qud and qud-sod nterfaces. The drect vapor deposton approach takes nto account ony the vapor monomers codn drecty wth the crtca custer, whereas the ace dffuson approach 0 consders ony the monomers that have coded and adhered to the ace of the pre-exstn partce, after whch they dffuse to the custer. The ace dffuson approach s used n ths mode, and t ves 7 8 orders of mantude hher coson rates than the drect vapor deposton mode. The coson rates 1/s are ven as the product of the number Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see

9 Heteroeneous mutcomponent nuceaton J. Chem. Phys. 16, of adsorbed moecues n poston to jon the erm, and the frequency exp F sd, / wth whch they jump to jon t =R p sn Dc ads s, exp F sd,, 66 where D s the mean jump dstance of a moecue, and F sd,, s the ace dffuson enery. The enth of the crcuar contact ne between the as, qud and sod s cacuated as R p sn, 14 wth the ane ven by cos =X m/. The Zedovch factor Z appearn n the formua 61 s ven by Z = W 11 +W 1 tan + W tan tan det W. 67 Detaed descrpton of the knetcs of mutcomponent eroeneous nuceaton s presented by Määttänen et a., 1 and the parameters used for water-n-propano mxture are the same as used by Kumaa et a. 3 In a cassca cacuatons, uness otherwse stated, we have used the mcroscopc contact ane 4 = x x, where x s the moe fracton of n-propano n the qud. VIII. EXPERIMENTAL DATA 68 No expermenta data so far s avaabe for the appcaton of the second eroeneous nuceaton theorem; here we demonstrate the power of the frst theorem n a practca appcaton. In recent experments Waner et a. 4 have obtaned nuceaton probabtes for unary and bnary eroeneous nuceaton of water and n-propano vapors on sver nanopartces. Neary monodspersed popuatons of A partces wth a eometrc mean partce dameter of 8 nm and eometrc standard devaton were used as seed partces. Ths eometrc standard devaton s somewhat smaer as compared to the vaue reported prevousy. 4 The smaer eometrc standard devaton of the seed partces has actuay been determned n the present study by more accurate accountn for the transfer functon of the eectrostatc aeroso cassfer 5 used n the experments. The experments were conducted wth severa as-phase actvty fractons of n-propano X = A, /A,1 + A,, 69 X =0 pure water, 0.18, 0.5, 0.406, 0.541, 0.75, 0.819, and 1 pure n-propano. Athouh the A partces are qute narrowy dstrbuted, the nfuence of the fnte wdth of the partce sze dstrbuton on the measured nuceaton probabtes must be taken nto account. For extractn the nfuence of poydspersty, we consder nuceaton probabty functons P for strcty monodspersed partces, whch, for a constant actvty fracton X can be approxmatey expressed by the formua TABLE I. Lambda vaues that ft Eq. 70 to the expermenta data for dfferent as-phase actvty fractons X. The tabe aso shows the mass fractons of n-propano X M, n the qud used to enerate the nuceatn vapor. X X M, P = 1 tanh 1A + + 1, 70 where A = A, +A,1 s used as a representatve vaue correspondn to the as phase actvtes n the bnary vapor mxture. The parameter 1 s reated to the sope of the nuceaton probabty P when potted as a functon of mean as-phase actvty A. Ths sope s assumed to be ndependent of the partce sze over the narrow rane of partce szes consdered n the experments. The rato / 1 s the onset saturaton rato correspondn to nuceaton probabty P=0.5. Ths onset saturaton rato has been obtaned as a functon of the partce dameter from addtona nuceaton measurements performed for A partces wth dameters shty above and beow 8 nm. Interaton of the nuceaton probabty functons P for strcty monodspersed partces over the actua expermenta partce sze dstrbuton yeds an nterated nuceaton probabty functon, whch can be drecty compared to the expermenta resuts. The expermenta data ponts can be ftted by approprate choce of 1. Each expermenta as-phase actvty fracton X s thus assocated wth a 1, par sted n Tabe I as a resut of the fttn. It shoud be noted that ths procedure s not dependent on any specfc theoretca mode. Fure shows the expermenta data and the ftted nuceaton probabty functons P for monodsperse 8 nm seed partces sod nes. Interaton of the nuceaton probabty functons P over the actua expermenta partce sze dstrbuton resuts n nterated nuceaton probabty functons dashed nes n ood areement wth experments. The numbers of moecues n the crtca custer N, are cacuated from the expermentay determned nuceaton probabty functon P for strcty monodspersed partces, Eq. 70. From the ft functons to the expermenta data, we can determne the dervatve F/A X whch accordn to Strey, Vsanen, and Waner 6,7 can be used to cacuate the dervatves needed n the frst nuceaton theorem 59 as F n A,1T,A, = A,1 F A X 1+ A, A, A,1F A,1 F A, /A,1, 71 A, A,1 F A, /A,1 Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see

10 Vehkamaek et a. J. Chem. Phys. 16, FIG. 3. Gas-phase actvtes that ve a constant nuceaton probabty P. The owest curve s reated to the owest probabty. The crces mark the vaues obtaned for expermenta as-phase actvty rato from Eq. 70, the dashed nes the resuts of the nterpoaton, and the crosses, whch accuratey overap the P=0.5 crces, are the measured expermenta onset actvtes for P=0.5. F n A,T,A,1 = A, wth F defned as F nn 1/1 P. F A X 1+A, /A,1 A, /A,1 A, A,1 F, 7 73 We thus need the dervatve A, /A,1 F for the onset curve wth constant F n other words constant nuceaton probabty P. Gas-phase actvty A, as a functon of A,1 for any constant P can be soved from Eq. 70 anaytcay for the expermenta as-phase actvty fractons. To obtan the dervatves we need A, A,1 aso for actvty fractons shty off the expermenta vaues, and we have used tabe ook-up near nterpoaton scheme 8 to obtan these ntermedate vaues. Fure 3 shows the onset curves for nuceaton probabtes 0.1, 0.5, and 0.9. IX. RESULTS The sub-pots of F. 4, each representn a constant as-phase actvty fracton, show the numbers of moecues n the crtca custer extracted from the expermenta data usn the frst nuceaton theorem 59. The moecuar numbers are wth a coupe of exceptons at the eft end of the x axs where P approaches zero smooth monotonous functons of as-phase actvtes, as s physcay reasonabe. We aso show the cassca theory predctons for comparson. The cassca theory ves orders of mantude arer crtca custer than the anayss of the expermenta data. If the macroscopc contact anes are used nstead of the mcroscopc ones, the cassca custers contan n some cases even more moecues. We have mted the y-axes and thus eft out some cassca vaues to better show the expermenta resuts. It has been shown earer, 4 that the cassca theory predcts we the onset condtons for eroeneous nuceaton but the present anayss shows that the transton from nuceaton probabty vaue 0 to vaue 1 s n reaty not as steep as the cassca mode predcton. We checked the correctness and consstency of our data anayss by checkn that at the onecomponent mt, the two-component theorems aso n practce ve the resuts obtaned by one-component anayss; ths check aso assured us that the ft based on Eq. 70 behaves we at the one-component mts. Usn cassca formuas the dervatve of the knetc pre-factor n K wth respect to n A, can be estmated numercay. For the rane of expermenta condtons t yeds 3 8 for pure water, 8 16 for pure n-propano ths estmate s obtaned usn the macroscopc contact ane 19.1, mcroscopc zero contact ane eads to nfnte vaues and, for exampe, =1 eads to vaues 10, and 11 for the bnary cases wth the dervatves taken wth respect to both as-phase actvtes. As a further check of our theorem, and aso the correctness of dervatves of the knetc pre-factor, we aso have enerated nuceaton probabty data wth the cassca eroeneous nuceaton theory, apped the eroeneous nuceaton theorem to ths data, and checked that the resutn numbers of moecues n the crtca custer are equa to the numbers of moecues ven by the cassca theory. Somewhat surprsny, the knetc contrbuton to the frst nuceaton theorem s not n the rane of 1 as n the oeneous one and two-component cases. 13,17,18 Ths can be understood, for smpcty n a one-component system, by comparn Eq. 61 to the oeneous counterpart K = c 1 Z, 74 where c 1 s the number of monomers n the nuceatn vapor. The monomer concentraton s proportona to the as-phase actvty, c 1 p A, the coson rate between crtca custers and monomers s proportona to the monomer concentraton and the ace area of the custer, p 4r A r, and the Zedovch factor s nversey proportona to the area of the custer 9 Z 1/r. In a these cases the proportonaty constants do not depend on the as-phase actvty A. Thus the dependence of the knetc factor on the radus of the crtca custer cances out, K A, and the contrbuton of the pre-factor equas K =c 1 Z /A n the 1/S-verson of the cassca theory, 30 n whch case the contrbuton s 1. In the eroeneous aan for smpcty one-component case Eq. 6 ves, c s ads A, Eqs. 63 and 66 yed Rav= sn c s ads sn A,, where cos =X m/ 1 mx+x and X=R p /r, and the Zedovch factor aso depends on the crtca custer radus as 31 Z 1/r mx 4mX m 3X 1 mx+x 3/ Thus the knetc pre-factor depends on the as-phase actvty A n a compcated way K A /r sn 4, mx 4mX m 3X 1 mx+x 3/ wth X=R p /r, and r dependn on the as phase actvty accordn to the Kevn Eq. 5. The nonspherca eometry of the custer, and the fact that the contact ne between the substrate and the custer pays the same roe n the coson rate as the ace area of the custer n the oeneous case, resut n the dependency of K on r not cancen out, Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see

11 Heteroeneous mutcomponent nuceaton J. Chem. Phys. 16, FIG. 4. Coor onne Numbers of moecues s the crtca custer obtaned from the expermenta data usn the frst nuceaton theorem. Sod nes wth crces refer to water moecues, sod nes wth squares to n-propano. Cassca theory predctons for water are ndcated by dashed nes, and for n-propano by dot-dashed nes. The seven sub-pots show data for dfferent as-phase actvty fractons X marked beow each pot. The bottom x-axes show the as-phase actvty of n-propano, whe the top x-axes ve the as-phase actvty of water. and thus the contrbuton of the pre-factor dffers from the oeneous case. Test cacuatons show that the contrbuton of the pre-factor s senstve to the crtca custer sze, as we as to the contact ane: the vaues obtaned usn the cassca theory for very are custers are not appcabe to the expermenta resuts wth much smaer custer szes. Subtracton of the cassca vaues for the knetc contrbuton from the resuts ven by the data anayss woud ead to Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see

12 Vehkamaek et a. J. Chem. Phys. 16, neatve numbers when the experments ndcate the custers to be sma. Thus, n F. 4 the effect of the knetc pre-factor has not been taken nto account. A more sophstcated mode for the knetcs s needed to estmate the roe of the knetc pre-factor n the eroeneous nuceaton theorem. For pure water the crtca custers contan ess than 17 moecues, for pure n-propano the arest custers have around 300 moecues. Ths dfference n the crtca custer szes of pure components foows drecty from the ceary steeper sope of nuceaton probabty for pure n-propano compared to that of pure water n F.. For as-phase actvty ratos X =0.18, 0.5, and custers contan more water than n-propano, for X =0.541 haf of the custer moecues are water, haf n-propano, and for X =0.819 n-propano ceary domnates. The arest bnary custers contan ess than 90 moecues. Our anayss demonstrates that the expermentay obtaned eroeneous nuceaton probabty does not chane too steepy as a functon of the as-phase actvty, thus aown a meannfu data anayss. X. CONCLUSIONS The formaton of new sma custers on seed partces has earer been studed n we-defned experments. 4 Ths paper shows how these experments can be utzed to dentfy the sze of newy formed moecuar custers usn the frst eroeneous nuceaton theorem. The custers studed typcay contan ony moecues. The crtca custer szes obtaned from the experments usn the eroeneous nuceaton theorem are dramatcay smaer than the predctons by the cassca nuceaton theory. Ths resut s reated to the fact that the sopes of the nuceaton probabty curves ven by the cassca nuceaton theory 14 are eneray found to be consderaby steeper compared to the expermenta data. 4 The dscrepancy between the cassca theory and experments ceary ndcates that the nuceatn custers are too sma to be quanttatvey descrbed usn the macroscopc Fetcher theory. 14 The custers are on the vere of a fu quantum mechanca descrpton to be computatonay feasbe. The eroeneous nuceaton theorems provde drect expermenta access to nanocuster propertes. In the future, when eroeneous nuceaton experments w be conducted at varous temperatures, the second nuceaton theorem provdes the means to anayze the eneretcs of the custers, and ad the deveopment of accurate modes for moecuar nteractons between the nuceatn moecues, and between the custer and the underyn aces. The appcaton of eroeneous nuceaton theorems eads to an mproved understandn of nanocuster formaton. ACKNOWLEDGMENTS The assstance of Ka Ruusuvuor s ratefuy acknoweded. Ths work was supported by the Academy of Fnand and the Austran Scence Foundaton, Project No. P N0. We thank Professor G. P. Resch for hs vauabe hep wth the determnaton of the expermenta seed partce sze dstrbuton. 1 P. Ham, R. P. Turco, C. S. Kan, O. B. Toon, and R. C. Whtten, J. Aeroso Sc. 13, H. Korhonen, K. Lehtnen, and M. Kumaa, Atmos. Chem. Phys. 4, R. J. Charson, S. E. Schwartz, J. M. Haes, R. D. Cess, J. A. Coakey, J. E. Hansen, and D. J. Hofmann, Scence 55, P. A. Stott, S. F. B. Tett, G. S. Jones, M. R. Aen, J. F. B. Mtche, and G. J. Jenkns, Scence 90, V. Ramanathan, P. J. Crutzen, J. T. Keh, and D. Rosenfed, Scence 94, S. Menon, A. D. De Geno, D. Koch, and G. Tseouds, J. Atmos. Sc. 59, M. Kumaa, K. E. J. Lehtnen, and A. Laaksonen, Atmos. Chem. Phys. 6, D. Kashchev, Nuceaton: Basc Theory wth Appcatons Butterworth- Henemann, Oxford, S. Toschev, Crysta Growth: An Introducton North-Hoand, Amsterdam, D. W. Oxtoby and D. Kashchev, J. Chem. Phys. 100, H. Ress, Methods of Thermodynamcs Dover, New York, F. F. Abraham, Advances n Theoretca Chemstry Academc, New York, I. J. Ford, J. Chem. Phys. 105, N. Fetcher, J. Chem. Phys. 9, T. Youn, Phos. Trans. R. Soc. London 95, A. Määttänen, H. Vehkamäk, A. Laur, S. Merkao, J. Kauhanen, H. Savjärv, and M. Kumaa, J. Geophys. Res. 110, E I. J. Ford, Phys. Rev. E 56, H. Vehkamäk and I. J. Ford, J. Chem. Phys. 113, M. Lazards, M. Kumaa, and B. Z. Gorbunov, J. Aeroso Sc. 3, H. R. Pruppacher and J. D. Kett, Mcrophyscs of Couds and Precptaton Kuwer, Norwe, Massachusetts, A. Määttänen, H. Vehkamäk, A. Laur, I. Napar, and M. Kumaa, J. Chem. Phys. to be pubshed. A. Inada, Ph.D. thess, Kobe Unversty, Japan M. Kumaa, A. Laur, H. Vehkamäk, A. Laaksonen, D. Petersen, and P. E. Waner, J. Phys. Chem. B 105, P. Waner, D. Kaer, A. Vrtaa, A. Laur, M. Kumaa, and A. Laaksonen, Phys. Rev. E 67, G. P. Resch, J. M. Mäkeä, and J. Necd, Aeroso Sc. Techno. 7, R. Strey and Y. Vsanen, J. Chem. Phys. 99, R. Strey, Y. Vsanen, and P. E. Waner, J. Chem. Phys. 103, Matab Reference Gude, Verson R14 Servce Pack 3 The MathWorks, Inc., MA, H. Vehkamäk, Cassca Nuceaton Theory n Mutcomponent Systems Sprner, Bern, Hedeber, H. Ress, W. K. Kee, and J. I. Katz, Phys. Rev. Lett. 78, H. Vehkamäk, A. Määttänen, A. Laur, I. Napar, and M. Kumaa, Atmos. Chem. Phys. 7, Downoaded 07 May 007 to Redstrbuton subject to AIP cense or copyrht, see