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On the validity of homogeneous nucleation theory
ON
THE
VALIDITY
OF HOMOGENEOUS J.
J.
NUCLEATION
THEORY*
BURTON7
Exect thermodymunio properties of small clusters of 3-87 atoms have been computed for 8 simple model system. The di&rences between the exect results end the classic81 o8pillerity model 8re examined. The exact thermodvrmmic nronerties 8re used to oelcuiete the homoneneous vaoor I Ioh8se nucleation rates, Jexsct, and the results 8re compared with the predictions of cl&sic81 nucleation theory, Jeans. It is found that Jexrct/Jcisas, varies from lo-* to 10’ and depends strongly on temperature end supersaturcltion. The predicted values ofJexsct 8re compared with existing experiment81 results involving a wide range of temperatures end materials. The agreement between the theory and experiments is quite good. I
VALIDITE
I
DE
I
LA
THEORIE
DE
LA
GERMINATION
HOMOGENE
Les+grandeurs thermodynamiques de petits agglomhets de 3 8 87 atomes ont Bte celcul~s de feyon exacts dans un modele simple. L’auteur etudie les dii%rences existent entre ces result&s et le modele clessique bese sur 18 capillarit& Ces gmndeurs thermodyxmmiques ex8ctes sont utilis&s pour oalculer les vitesses de germination en phase vapeur homogene, J eXact, et les r&rultats sont cornperes aux veleurs prevues par 18 theorie classique de 18 germination, Je~ass. L’auteur trouve que Jexs&ciass varie de 1O-L 8. 10’ et depend fortement de la temperature et de la surseturation. Les veleurs calculees Jclact sont comparees aux resultats experimenteux existants, dans un large On constste un t&s bon 8coord entre domaine de temperetures et pour une gmnde serie de materiaux. 18 theorie et les experienoes. ZUR
GULTIGKEIT
DER
THEORIE
DER
HOMOGENEN
KEIMBILDUNG
Fur em einfeches Modellsystem wurden exekte thermodyn8misohe Eigensoheften kleiner Agglomerate von 3 bis 87 Atomen berechnet. Die Unterschiede zwischen den exekten Ergebnissen und dem klsesischen Kapilllr-Model1 werden untersuoht. Mit Hilfe der exakten thermodynamischen Eigensoheftan wird die homogene Demf-Keimbildungsr8te (J exact) berechnet und mit den Vorhersagen der klessischen Keimbildungstheorie (Jcias.) vergliohen. Es zeigt sioh. dell J Bxset/JcisI. zwischen 10-s und 10’ veriiert und stark von Temperatur und Ubers&ttigung abhiingt. Die vorhergesegten Jexset- Werte werden mit vorliegenden experimentellen Ergebnissen fiir einen grol3en Tempereturbereioh und verschiedene Materislien verglichen. Die Ubereinstimmung zwischen Theorie und Experimenten ist befriedigend. 1.
INTRODUCTION
Homogeneous nucleation theory attempts to describe the kinetics ofa first order phase transition such as condensation or crystallization in a pure perfect system. Though most theoretical and experimental studies are done on condensation of simple gases, the theory has been of great interest to metallurgists. This interest arises from the fact that many solid state transformations are of the nucleation and growth variety and nucleation theory is used to describe them; however condensation remains the only transformation for which the theory may be quantitatively tested. The basic features of homogeneous vapor phase nucleation theory were developed in the 1920s by Volmer and Webert’) and Farkas.t2) Further progress occurred in the 1930s and is associated with Volmer and Flood(3) and Becker and Doering. At this point, many investigators felt that homogeneous nucleation was well understood; experimental results involving waterf3) agreed reasonably well with the theoretical predictions. This early theory is now known as classical nucleation theory and its development has been reviewed.(5’ * Received Deoember18, 1972; revised February 16, 1973. t Henry Krumb School of Mines, Columbia University, New York, New York 19927, U.S.A. ACTA METALLURGICA,
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1973
In 1955 Band(s) questioned some of the basic assumptions of classical nucleation theory in a statistical mechanical treatment of cluster thermodynamics. In particular, he pointed out that classical nucleation theory overlooked the translational partition fraction of the clusters. Band’s contributions were not widely recognized and in 1962 Lothe and Pound(‘) revived the statistical mechanical approach to cluster theory. Lothe and Pound discussed rotational and translational contributions to the thermodynamic properties of small clusters. They concluded that experimental nucleation rates might be about 1015 larger than predicted by classical nucleation theory. These contributions, which are referred to as Lothe and Pound’s factors, which are the subject of considerable controversy.@-14) Experimental nucleation rate studies in some systems such as water and etl~anol~16J6)apparently support the predictions of classical nucleation theory. However, in other systems, such as chloroform and freon,“6’ the experimenters reported that nucleation rate agreed with Lothe and Pound’s predictions. This dichotomy, some experiments agreeing with the classical theory and some with the Lothe-Pound theory, led to discussion of the distinction in surface structure between classical materials and LothePound materials.d7)
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All nucleation theorists, until 1970, used the capillarity approximation in some form to describe thermodynamic properties of small clusters. Band@) and Lothe and Pound(‘) only added important correction terms to the capillarity approximation, but did not introduce corrections for the fact(ls) that the capillarity approximation fails to correctly describe the energy of very small clusters. Lothe and Poundus) suggested that this fundamental problem of the errors in the capillarity approximation might account for the discrepancies between nucleation theory and experiment. Accordingly, a number of workers began to study the thermodynamic properties of simple model systems.‘20) In this paper we discuss our results from a study on a very simple model system-atoms on face-centered cubic (f.c.c.) lattice sites with nearest neighbor harmonic forces only. In particular, we examine with great care various contributions to exact corrections to the homogeneous nucleation rate. The thermod_ynamic properties of the clusters are reported elsewhere.(21) As classical theory provides simple analytic expressions for nucleation rates, we find it relevant to ask and answer for one model system the question: “If one uses only classical nucleation theory to predict the outcome of an experiment, how wrong will the prediction be?” We will compare our predicted corrections to classical nucleation rates with the result of previous experimental investigations. The agreement is quite good even though our calculations are based on simple crystalline clusters while most nucleation rates are determined by properties of small clusters which are probably liquid-like. 2. STATISTICAL
THERMODYNAMICS
In this section, we derive, using statistical mechanics, the important parameters which are required to calculate homogeneous gas phase nucleation rates. This derivation is fairly standard and similar to previously published derivations;‘7u’mz1) we include it here as it is required in order to understand the subsequent discussion. The homogeneous vapor phase nucleation rate is given by(22) J=
=
-1 51 ( i=lCi?Zd1
(14
zci.ni*
W
where ci is the rate of capture of atoms by clusters of
VOL.
21, 1973
size i; n, is the equilibrium Boltzman concentration of clusters i ; Z is the Zeldovich non-equilibrium correction factor which is approximately 0.1,(S) and i* is the critical cluster size at which nni.is a minimum and which dominates the sum in equation (la). According to classical nucleation theory(5) ni = nr exp [-AF(i)/kT]
(2)
where AF(i), the excess free energy of a cluster of size i, is given by the capillarity approximation AF(i) = Ai2J3a -
ikT lnplp,
(3)
where A is a numerical factor such that AP is the surface area, o is the surface tension, and p/p0 is the supersaturation. We will now derive an appropriate expression for AF(i) from statisticalmechanicsforuseinequation (2). Our intention is to obtain an exact expression for the free energy to compare with the predictions of the capillarity approximation (equation 3). According to statistical mechanics, the equilibrium concentration of clusters of size i in the vapor phase is given by’s) ni = Zi exp [ip/kT]
(4)
where Zi is the partition function of a cluster of size i in a box of unit volume and p is the chemical potential of the monomer. When the motions of the cluster can be separated into translation of the whole cluster, rotation of the whole cluster, and internal motion of the atoms with the cluster, then the partition function can be written as(20*21) 4 =
2 Ztrans(i, X)Z,,,(G x = conflg
XV&,(&
X)
(5)
z tram
is the translational partition function for a cluster of i atoms in a box of unit volume Ztrans(i, X) =
r?y
where m is t’he mass of a single atom. rotational partition function
w Zrot is the
(6b) where o(X) is the symmetry number of a cluster and I3 is the product of three principal moments of inertia of the cluster. Zint is the internal partition function. For a solid-like cluster with harmonic forces Z,,,(X,
i) = exp [- U(X, i)/kT] 3i--8
XII j=l 1 -
e-hvJ2kT exp [-hv,/kT]
(W
BURTON:
VALIDITY
OF HOMOGENEOUS
where U(X, i) is the potential energy of the cluster and vj are the normal mode vibrational frequencies. The sum over X in equation (6) means that each distinguishable configuration of i atoms is counted exactly once. The translational partition function, equation 6a, is independent of the cluster configuration; the rotational partition function depends on configuration primarily through the symmetry number, a(X). As the summation over all configurations required in equation (5) is difficult, it is convenient to write
NUCLEATION
THEORY
1227
F osailg(i) as the logarithms of the appropriate partition functions in equation (8a). Using equations (9b and lob), equation (4) may be rewritten as Fto,,,(i) + FfJi)
+
(i) Fconiio
-
iF, -
+ Fio,Ji)
ikT In:
11
kT
1
. (11)
If we write ni in the form used in classical nucleat,ion theory n, = nl esp [-AF(i)/kT]
(4)
then from equation (II) where the superscript zeros refer now to the most probable configuration (i.e. minimum free energy or maximum partition function) of a cluster of size i. We can write equation (7) in a more revealing fashion (3a)
where zconng(i)is
the
configurational
partition
function
(8b) Comparing now equations (lb, 4 and 8a), wesee that the configurational partition function tells us how much configurations other than the most probable enhance the concentration of small clusters and hence the nucleation rate. (In introducing the configuration partition function, we are explicitly considering the possible contribution of clusters of irregular shape to the nucleation rate.) As explained above, our intention is to obtain a correct expression for AF(i) for use in equation (2). If the gas is nearly ideal, then ,u = kTInp.
(94
AF(i) = {J%,,,(i) + {e,,(i)
+ F&,(i) + kT ln nJ -
iP,}+
Fconfig(i) -
ikT In ?- . PO (12)
In the nest section we shall compare AF(i) as computed from equation (12) for the model system with A F( i) as computed from the capillarity approximation, equation (3), for the same model system. The terms in the first brackets in equation (12) constitute the Lothe-Pound(‘) correction to the free energy of a small cluster. The second bracketed quantity, the excess free energy of the most probable configuration of a small cluster relative to the bulk, has not been previously discussed; we shall compare this term directly with the predictions of the capillarity approximation. The F conilg(i) term in equation (12) will be examined briefly; it is not of great numerical significance in the temperature range of interest. The last term in equation (12) is the supersaturation, which appears in both the capillarity approximation, equation (3), and the statistical mechanical treatment. 3.
COMPARISON OF EXACT THERMODYNAMIC PROPERTIES AND THE CAPILLARITY APPROXIMATION
Remembering that the free energy of the bulk phase, F,, must equal the chemcial potential of the vapor phase when the two are in equilibrium (at pressure po) we write
Initially we will compare the excess internal free energy of the most probable cluster, the second term in brackets in equation (12)
kT In y + F,. PO As the free energy is related to the partition function
wit,h the predictions of the capillarity approximation at the equilibrium pressure, p = p,, in equation (3),
,u =
by F=
-kTInZ
(104
or equivalently Z = exp [- F/kT] it is appropriate to define P:,,(i),
(lob) FFJi),
F:&(i) and
AFkt(i) = Y&(i) -
AF,,,(i)
iF,
(13)
= AC%.
(14)
In the usual way we will separate the free energy into potential energy, zero-point energy and entropy. AF = AU,,,
+ AU,.,.
We now examine AU,,,
-
TAti’,,,.
and AS,,,.
(15)
The errors in
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as predicted by the ctlpillarity approximation, AU,.,., are small compared to the errors in AUpot@l) and affect nucleation r8tes by less than 8 factor of 100; therefore we will not discuss AU,.,. The simplest way to represent AU,,, is in terms of the bond deficit per atom. An infinite f.c.c. solid has 6 bonds per 8tom. A two atom cluster has only 1 bond and hence a bond deficit of ll(2 x 6 - 1) bonds or 5.5 bonds/atom. The bond deficit per atom for small f.c.c. clusters is shown in Fig. 1. Exact results and capillarity 8pproximation results are both shoun. The smooth curve through the circles is sn analytic function which fits the exact data. The computations of the two curves in Fig. 1 (and in
VOL.
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2
I
-I,1 i f
2
3
E vi-
2
‘0
-
EXACT
---
CAPILLAR ITY
25
50 n
75
0 ? 55 -2
E
-3
f-
z
-4
100
FIU. 2. Internal vibrational entropy per atom, Si,,t, and excess entropy relative to an infinite solid, ALSint, as a function of cluster size. I
0 t ;; S $
4
a2
0
25
50
75
100
n FIG. 1. Bond deficit per atom for small clusters as a function of aluster size. The pm&&ions of the oapillarity approxlmatlon are shown, as are exact results foqminimum free energy configurations.
succeeding figures) are discussed in detail elsewhere.(21) Comparison of the two curves in Fig. 1 is informative. For large cluster sizes the capillarity approximation underestimates the bond deficit. This occurs because the capillerity approximation assumes that all of the atoms lie on infinite bulk surfaces and have coordination numbers 8ppropriat.e to such surfaces; it neglects entirely the fact that there are many edges and corners on a small cluster. Edge and corner atoms have lower coordination numbers than regular surface atoms; hence the bond deficit is increased. At small cluster sizes, the exact and capillarity results are in good agreement. This occurs because the cspillarity approximation overestimates the number of surface atoms for small clusters; for instance, according to the capillerity approximation, 8 10 atom cluster has 20 surface 8toms.fN) ASiDt is plotted in Fig. 2. The capillarity approximation is fairly good for Isrger clusters but fails badly for smaller clusters. The error for smaller clusters arises from the fact that 8 small cluster has only 3i - 6
vibration81 modes; 6 degrees of freedom are used in rotation and translation. The excess free energy of a small cluster is plotted in Fig. 3. Here we 8ssume we are at the equilibrium vapor pressure, p = p,. The data are plotted at 8 temper8ture of 6O’K for potential parameters which give a critic81 temperature of 108’K; this is 8 reduced temperature of T/T, = 0.55 where T, is the critical temperature. Note first that excess internal free energy of 8 small cluster at this temperature is substantially higher than predicted by the capillerity approximation; this occurs primclrily becsuse the capillarity approximation underestimates the bond deficit, Fig. 1. When the Lothe-Pound correction factor, the first term in brackets in equation (12), is 20
-INTERNAL ---T+ R+I ---CAPlLLARlTY 0
0
20
40
60
00
too
n FIG. 3. Excess free energy of small alusters relative to an infinite solid at a reduaed temperature of T/T, = 0.66. The ourve marked “Internal” considers only the internal degrees of freedom of the most probable cluster. The curve marked “T + R + I” considers all degrees of freedom of the most probable oluster. The capillarity approximation result is also shown.
BURTOIi:
VALIDITY
OF
HOMOGENEOUS
NUCLEATIOS
THEORY
1229
TABLE 1. Corrections to the equilibrium concentration of clusters nj due to various errors in the capillarity approximation Error
Correction
10-10_ IO-‘0
AU,ot
10”
0
20
40
60
80
T (OK) FIG. 4. Configurational free energy of a 49 atom cluster as a function of temperature. (The critical temperature 1s 108%) The curves marked “0,” “1,” and “2” are improvmg approximetions to Pconflr. The cross hatched region is a guessed range of reasonable extrapolations to include all configurations.
included in the curve labelled “T + R + I” the free energy is reduced. At this particular temperclture the intern81 free energy error virtuslly cancels the LothePound correction factor; this is, as we will see, accidental. In Fig. 4, we have plotted the coni&urational free energy for a 49 atom cluster 8s 8 function of temperature. The deteils of this contribution 8re discussed elsewhere.(ms21) The curves labelled “O”, “1” snd “2” are successive approximations to Fcontig.(4s) The crosshatched region is a range of possible extrapolations of F coniig to include 811 possible cotiguretions. Comparison of Figs. 3 and 4 indicates that Fconiig lowers the free energy by rv5 per cent. 4.
CORRECTIONS NUCLEATION
TO
CLASSICAL RATES
ASint
10-1 -
F config
10’ -
Lothe-Pound (RotationTranslation)
10’4 -
10’6
Combined
10-Z -
1019
10-Z 10’
Comments Large clusters, depends on temperature. Small clusters, independent of temperature. Independent of temperature and cluster size. Depends on temperat,ure. Depends weakly on temperature. Depends on cluster size and on temperature.
For small cluster sizes the capillarity approximation is fairly good, Teble 1. The energy correction will depend somewhat on the system studied. However, the qualitative features of the energy correction are independent of the system. The capillerity approximlation neglects edges and corners for 8ny system and hence always underestimates the bond deficit for large cluster sizes. For small cluster sizes, the cepillarity approximation always overestimates the number of surface atoms and this error tends to cancel the neglect of edges and corners. B. The internal entropy correction The intern81 entropy correction, Teble 1, will depend on the system studied, but, as it is small, the variation between systems will not be serious. C. The conjgwational correction
We are now ready to examine the effects of all the different terms discussed above on the nucleation rate. We will give the numerical corrections and discuss qualitatively whether these correctVions are model dependent. Initirtlly we will examine the effects of the corrections on the equilibrium concentrstion of 8 particular size cluster. Then we will examine nucleation rates. The discussion is separated into these two steps as t,he critical cluster size calculated exactly is different from that predicted by the capillarity approximation.
The rotation-transletion correction increases the cluster concentretion by m1016 and is based on fundemental statistical mechanic81 considerations.(‘) It is essentially independent of the system.
A. The potential energy correction
E. Free energy
For large critical cluster sizes the cepillarity approximrttion seriously underestimates the bond deficit. Hence the potential energy correction greatly lowers the cluster concentration. This correction demnds stronelv on temnerature. Table 1.
Above we have examined various contributions to errors in the free energy of small clusters. In this section we examine the free energy itself. In Fig. 5, we plot the total excess free energy for three c8ses. Exact results 8re comuared with the
L
VY
L
The configurational correction increases the cluster concentration by 10 to 10,000. Any small cluster can assume many configurations like an amoeba. This configurational term will occur for any system. D. The Lothe-Pound
(rot&ion-trandution) correction
L
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The capillarity approximation gives somewhat smaller critical cluster sizes than predicted by the exact theorY. The principal result of this work, the correction to the classical nucleation rate, Jeract/Jclass, as a function of supersaturation and reduced temperature, is in Table 2. We do not give results here at low supersaturations as the critical cluster sizes are significently
EXACT CAPILLARITY
TABLE 2. Corrections to the classical nucleation rate, .JeXsEl/ Jelass, as a function of supersaturation. p/pe. and reduood temperature, T/T,, where T, 1s the critical temperature 72 \ 0.35 0.39 0.43 0.47
Fm. 5. Excess free energy of small olusters 8s a funotion of reduoed pressure p/pi, (p,, is the equilibrium vapor pressure), and reduoed temperature T/To (T, is the critic81 temperature of the vapor). Exact results are oomp8red with the predictions of the capillarity approximetion.
predictions of the capillsrity approximation. Note that the free energy corrections to the cspillarity approximation depend on both temperature 8nd supersaturation. F. Nucleation
rates
In Fig. 6 we show the dependence of the critical cluster size* on both supersrtturation and temperature.
.50
\’ \
P/P,
= 16
are compared with the predictions larity epproximetion.
102 10’ 10’ 106
10’ 105 :;:
10s 10’ 10’ 10’
lerger than the biggest clusters which we have studied, 87 atoms. (Larger clusters were not studied because of difficulties associated with finding the vibrational frequencies for equation (60) ; one must diagormlize 3n x 3n matrices to obtain the normal mode vibrational frequencies.) Nucleation rates at lower aupersatur8tions might be very crudely estimated by extrapolation. The results in Table 2 8re exact results for our model system. As pointed out above, the contributions are not strongly dependent on the to JexactlJo,ass system. We expect that the numbers in the table would not change by more than ~10~ when other systems are studied. The difference between classic81 nucleation theory and our exact calculations may be seen in 8 different w8y. Figure 7 shows the supersaturation (p/p,Jexsot at which the nucleation rate at 8 given reduced temperature, T/T,, is equal to that predicted by classical The results in this figure 8re theory at (P/z4J)c*ae.s*cal. exact only for our model system. However, as we exis not strongly system dependent, pect that JexactlJclass we expect that Fig. 7 is fairly general and applicable to many experimental systems. * * The reasons for this are explained in Section 4. A-D. Essentially they are that the major souroes of the errors in the oapillerity approximation 8re a geometrioel error in the potential energy and the neglect of rotation and translation. Neither of these errors is model dependent. Note that this presumed generality of our result is analogous to the oorresponding states assumption whioh is widely spplied in the study of vapors. Corresponding stat+8 is known to apply reasonably well to critical point phenomena, which oan also be described in terms of cluster propertiesus As some oluster
1
FIG. 6. Critical cluster size, a*, 8s a function of reduced pressure, p/pa. and reduced temperature, T/To. Exaat results
10-z 10’ 10’ 106
of the oapil-
l The critical oluster size is that cluster size for whioh the i.e. the least probable excess free energy is 8 msximum; cluster. See Refs. (6) and I**‘.
properties can be calculated from critical indioes, critioal point properties have been recently used by Kisng(*s’ to 08lcUl8m nucleation rates. Comparison of the sosling parameters introduced in Kiang’s work with those whioh appear in ol8ssioal nucleation theory,” indicates that corresponding states is applicable to the correction to classical nucleation theory, Jexsct/JcrarsieB1, except very near the oritieal point. Henoe, as claimed, Jeract/Jclaasiral, caloulated here for our simple model s.ystem should apply to a wide range of materials.
BURTON:
VALIDITY
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NUCLEATION
THEORY
TABLE 3. Experiment81 and our theoretical corrections to cl8ssioal gas phase nucleation r&es, J/Jels.sic~l. p/p0 is the supersaturation. T/T, is the reduced temperature
200
IO0
Meterial (Reference)
PIP0
50
H,O’“” H,O’**’ C,H,OH’rs’ CH,OH’**’
E :: w $
1231
20
0.41 0.45 0.47 0.47
5.5 ;.z 212
J/Jcl*8llIC*l,
Experimental
Theoretical, this work
10-p 100 100 100
.? IO
5’
5
10
50
20
100
200
500
‘P’Po’CLASSICAL
Fm. 7. The supersaturation (P/P,),,.,t 8t which the nucleation r8t.e at 8 given reduced temperature, T/Tc (T, is the vapor phase critical temperature), is equal to the nucleation rate predicted by classical nucleation theory at (P/;D,,)~~.~~~~~I. The figure is used 8s follows: We w8nt to know, for instance, what superseturetion is required to obmin the nucleation rete at T/T, = 0.49 prediated by classical theory at (P/P,,) = 50. We follow the (P/Ps)C~~~~i~~l = 50 line verticelly until it interseots with the T/Z’, = 0.49 curve 8t (P/Po)exaet= 17. This is then the predicted supersatur8tion required to obtain the desired nucleation rate. 5.
COMPARISON
WITH
EXPERIMENT
In this section we compare our calculated nucleation rates with some previous experiments. We have made no attempt to search all of the nucleation literature and compare our t,heoretical work with all experiments. This exhaustive comparison is not warranted until thermodynamic properties of clusters of more than 87 atoms have been calculated so that conclusive theoretical statements can be made even at low supersaturations. Two types of experimental results are often reported. Either the observed nucleation rate at some pressure and temperature is measured directly and reported and also compared with the predictions of the capillarity approximation, or the pressure and temperature at which some phenomenon such as
condensation occurs in a particular experiment is measured and compared with the pressure and temperature predicted by the capillarity approximation. In Table 3 we compare results from cloud chamber experiments at low supersaturations with our predictions. The experiments were carried out at temperature T, or reduced temperature T/T, relative to the critical temperature, and supersaturation p/p,,. The experimenters reported either the numerical value of the nucleation rate J and the correction to classical theory J/Jclaasioal or reported that the experimental nucleation rates agreed quantitatively with the predictions of classical theory. For each reduced temperature and supersaturation we oan find the predicted value of J/Jclwsic.l from data such as in Table 2. Unfortunately, these experiments were all carried out at lower supersaturations than those for which we can now make statements. However, extrapolation of our data such as in Table 2 to lower supersttturations suggests upper limit,s for J/Jclaasloal which are included in Table 3. The agreement between experiments and our predictions is satisfactory. In Table 4 we give the supersaturations (P/PrJLmIlL?ntslat which condensation was observed to occur in wind tunnel and diffusion chamber experiments for various materials at reduced temperature T/T,. The experimenters also reported the superat which classical nucleation saturations (z&Jo~aeSica~
TABLE 4. Classical, experimental, 8nd our predictions for the supersaturation, p/p,. at which condensation occurs for 8 given reduoed temperature, T/T, Material (Reference) H,O’u NH,‘=’ CeH,“6’ CHCl,“a’ CCl,F”” C,H,OH”” C,H,,‘**’ C,H,,‘C&HI,‘-’ C8Ht0’*@’
(P/PO)throretical. this work 0.40 0.49 0.37 0.37 0.36 0.46 0.45 0.46 0.44 0.44
15 7.5 390 170 440 11 11 11 ::
11 12 4.5 *1 110 * 20 40 * 15 100 * 30 61t2 10.5 * 2 10 f 1 16 f 2 13 & 2
13 6.5 100 65 115 a.7 9 8.7 ::
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theory predicts condensation. We then used Fig. 7 to find the corresponding supersaturations at which we predict condensation should occur; these are also in the table. The agreement is very good with two exceptions-NHs and CHCl,. In both of these oases we expect significantly higher supersaturations than found experimentally. In the case of ammonia, the investigators reported major difficulties with water contamination,d6) and chloroform is very difficult to produce with great purity. Hence it is possible that heterogeneous nucleation was actually observed at a lower supersaturation than required for homogeneous nucleation. 6.
CONCLUSIONS
We have calculated exactly the homogeneous vapor phase nucleation rate, Jexact, for a simple material and have compared this result with the predictions of the classical capillarity approximation, Jo,_, for the same material. The ratio Jexact/Jolaescan vary from 10-s to 10’ and depends strongly on temperature and supersaturation. Generally at low temperatures and low supersaturations Jexeot corresponds fairly closely while at high temperatures and large superto Jalarrs saturations the differences are large. We have compared our predicted corrections to the classical homogeneous vapor phase nucleation rates with the corrections found in a wide range of experiments. The agreement between theory and experiment is generally quite good. We believe therefore that homogeneous nucleation rates can be satisfactorily computed from first principles when all corrections to classical nucleation theory, including the Lothe-Pound rotation and translation factors,“) are considered. Accordingly, we feel that the long standing controversy about the necessity of rotation-translation corrections(8-14) is resolved, and that the apparent discrepancies between experiments and the Lothe-Pound theory(‘) reflected the need for consideration of corrections besides rotation-translation ; these further corrections were in fact suggested by Lothe and Pound.(19) The results of a wide range of experiments are satisfactorily accounted for in an exact theory.
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ACKNOWLEDGEMENTS
The author wishes to express his appreciation to F. F. Abraham, G. M. Pound and F. H. Stillinger for their invaluable suggestions and criticisms of this work; to C. Briant and J. K. Tien for comments on the manuscript ; and to Columbia University and the Henry Krumb School of Mines for supporting the work. REFERENCES Chem. 119, 227 1. N. VOLMER and A. WEBEIL. 2. Phys. (1920). 2. L. FARKAS, 2. Phys. Chem. 136,236 (1927). 3. N. VOLMERand H. FLOOD, 2. Phys. Chem. 17OA, 273 (1934). 4. R. BECKERand W. DOERINO,Ann. Phya. 94, 719 (1936). 5. J. FEDER, K. C. RUSSELL, J. LOTHE and G. M. POUND, Adv. Phys. 15, 111 (1966). van Nostrand (1966). 0. W. BAND, Quantum Stat&ice. and G. M. POOND, J. Chum. Phy8. 96, 2080 7. J&IEE \----I.
(1967). 8. H. REISS and J. L. KATZ, J. Chem. Phyu. 46.2496 9. F. F. ABRAHAM and G. M. POUND, J. C&m. Phya. 49, 732 (1068). 10. J. LOTHE and G. M. POUND, J. Chem. Phys. 49, 1849
11968~ \____,.
11. H. REISS, J. L. KATZ and E. R. COHEN,J. Cbm. Phy8. 49,6563 (1968). 12. K. NISHIOKA and G. M. POUND, Am. J. Phya. 99, 1211 (1970). (1972). 13. B. V. DERJA~UIN,J. Coil. Int. Sci. 99,617 Phys. 4, 66 (1972). 14. M. BLANDEX end J. L. KATZ, J. Stat. 16. H. L. JAEOER, E. J. WILSON, P. G. HILL and K. C. RUSSELL,J. Chem. Phya. 61, 5380 (1969). 16. D. B. DAWSON, E. J. WILSON, P. G. HILL and K. C. RUSSELL,J. Chem. Phye. 51,638s(1060). Appl. Phys. Lett. 18,208 (1968). 17. F. F. ABRAHAM, J. Chem. Phya. 19, 18. G. C. BENSONand R. SHUTTLEWORTH, 130 (1961). 19. J. LOTHEand G. M. POUND, Nucleation, p. 109. Ed. A. C. ZETIZEMOYER. D&her (1969). 20. J. J. BWTON, Chem. Phya. Lett. 17, 199 (1972) contains Feferencesto many of the recent papers in this field. J. J. BURTON,J.C.S. Foradcly II, 69,640 (1973). ::. J. E. MACDONALD,Am. J. Phya. 81. 31 (1963). 23: K. L. MURTY and J. E. DORN, J. Phys. Chem. SOlide 99, 767 (1972). 24. F. F. ABRAHAMand J. V. DAVE, J. Chem. Phy8. 66, 1687 (1971). M. E. FISHER, Phyeics 3, 255 (1967). ;:: C. S. KIANQ, D. STAUFFER,G. H. WALKER, 0. P. PURI, J. D. WISE and E. RI. PATTERSON,J. Atmos. Sci. 98, 1222 (1971). 27. L. B. ALLEN and J. L. KASSSER, J. Coil. Int. Sci. 89, 81 (1960). 28. J. L. KATZ and B. J. OSTERMIER,J. Chem. Phya. 47, 478 (1967). 29. J. L. KATZ, J. Chem. Phys. 52, 4733 (1970).
THE
VALIDITY
OF HOMOGENEOUS J.
J.
NUCLEATION
THEORY*
BURTON7
Exect thermodymunio properties of small clusters of 3-87 atoms have been computed for 8 simple model system. The di&rences between the exect results end the classic81 o8pillerity model 8re examined. The exact thermodvrmmic nronerties 8re used to oelcuiete the homoneneous vaoor I Ioh8se nucleation rates, Jexsct, and the results 8re compared with the predictions of cl&sic81 nucleation theory, Jeans. It is found that Jexrct/Jcisas, varies from lo-* to 10’ and depends strongly on temperature end supersaturcltion. The predicted values ofJexsct 8re compared with existing experiment81 results involving a wide range of temperatures end materials. The agreement between the theory and experiments is quite good. I
VALIDITE
I
DE
I
LA
THEORIE
DE
LA
GERMINATION
HOMOGENE
Les+grandeurs thermodynamiques de petits agglomhets de 3 8 87 atomes ont Bte celcul~s de feyon exacts dans un modele simple. L’auteur etudie les dii%rences existent entre ces result&s et le modele clessique bese sur 18 capillarit& Ces gmndeurs thermodyxmmiques ex8ctes sont utilis&s pour oalculer les vitesses de germination en phase vapeur homogene, J eXact, et les r&rultats sont cornperes aux veleurs prevues par 18 theorie classique de 18 germination, Je~ass. L’auteur trouve que Jexs&ciass varie de 1O-L 8. 10’ et depend fortement de la temperature et de la surseturation. Les veleurs calculees Jclact sont comparees aux resultats experimenteux existants, dans un large On constste un t&s bon 8coord entre domaine de temperetures et pour une gmnde serie de materiaux. 18 theorie et les experienoes. ZUR
GULTIGKEIT
DER
THEORIE
DER
HOMOGENEN
KEIMBILDUNG
Fur em einfeches Modellsystem wurden exekte thermodyn8misohe Eigensoheften kleiner Agglomerate von 3 bis 87 Atomen berechnet. Die Unterschiede zwischen den exekten Ergebnissen und dem klsesischen Kapilllr-Model1 werden untersuoht. Mit Hilfe der exakten thermodynamischen Eigensoheftan wird die homogene Demf-Keimbildungsr8te (J exact) berechnet und mit den Vorhersagen der klessischen Keimbildungstheorie (Jcias.) vergliohen. Es zeigt sioh. dell J Bxset/JcisI. zwischen 10-s und 10’ veriiert und stark von Temperatur und Ubers&ttigung abhiingt. Die vorhergesegten Jexset- Werte werden mit vorliegenden experimentellen Ergebnissen fiir einen grol3en Tempereturbereioh und verschiedene Materislien verglichen. Die Ubereinstimmung zwischen Theorie und Experimenten ist befriedigend. 1.
INTRODUCTION
Homogeneous nucleation theory attempts to describe the kinetics ofa first order phase transition such as condensation or crystallization in a pure perfect system. Though most theoretical and experimental studies are done on condensation of simple gases, the theory has been of great interest to metallurgists. This interest arises from the fact that many solid state transformations are of the nucleation and growth variety and nucleation theory is used to describe them; however condensation remains the only transformation for which the theory may be quantitatively tested. The basic features of homogeneous vapor phase nucleation theory were developed in the 1920s by Volmer and Webert’) and Farkas.t2) Further progress occurred in the 1930s and is associated with Volmer and Flood(3) and Becker and Doering. At this point, many investigators felt that homogeneous nucleation was well understood; experimental results involving waterf3) agreed reasonably well with the theoretical predictions. This early theory is now known as classical nucleation theory and its development has been reviewed.(5’ * Received Deoember18, 1972; revised February 16, 1973. t Henry Krumb School of Mines, Columbia University, New York, New York 19927, U.S.A. ACTA METALLURGICA,
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In 1955 Band(s) questioned some of the basic assumptions of classical nucleation theory in a statistical mechanical treatment of cluster thermodynamics. In particular, he pointed out that classical nucleation theory overlooked the translational partition fraction of the clusters. Band’s contributions were not widely recognized and in 1962 Lothe and Pound(‘) revived the statistical mechanical approach to cluster theory. Lothe and Pound discussed rotational and translational contributions to the thermodynamic properties of small clusters. They concluded that experimental nucleation rates might be about 1015 larger than predicted by classical nucleation theory. These contributions, which are referred to as Lothe and Pound’s factors, which are the subject of considerable controversy.@-14) Experimental nucleation rate studies in some systems such as water and etl~anol~16J6)apparently support the predictions of classical nucleation theory. However, in other systems, such as chloroform and freon,“6’ the experimenters reported that nucleation rate agreed with Lothe and Pound’s predictions. This dichotomy, some experiments agreeing with the classical theory and some with the Lothe-Pound theory, led to discussion of the distinction in surface structure between classical materials and LothePound materials.d7)
1226
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All nucleation theorists, until 1970, used the capillarity approximation in some form to describe thermodynamic properties of small clusters. Band@) and Lothe and Pound(‘) only added important correction terms to the capillarity approximation, but did not introduce corrections for the fact(ls) that the capillarity approximation fails to correctly describe the energy of very small clusters. Lothe and Poundus) suggested that this fundamental problem of the errors in the capillarity approximation might account for the discrepancies between nucleation theory and experiment. Accordingly, a number of workers began to study the thermodynamic properties of simple model systems.‘20) In this paper we discuss our results from a study on a very simple model system-atoms on face-centered cubic (f.c.c.) lattice sites with nearest neighbor harmonic forces only. In particular, we examine with great care various contributions to exact corrections to the homogeneous nucleation rate. The thermod_ynamic properties of the clusters are reported elsewhere.(21) As classical theory provides simple analytic expressions for nucleation rates, we find it relevant to ask and answer for one model system the question: “If one uses only classical nucleation theory to predict the outcome of an experiment, how wrong will the prediction be?” We will compare our predicted corrections to classical nucleation rates with the result of previous experimental investigations. The agreement is quite good even though our calculations are based on simple crystalline clusters while most nucleation rates are determined by properties of small clusters which are probably liquid-like. 2. STATISTICAL
THERMODYNAMICS
In this section, we derive, using statistical mechanics, the important parameters which are required to calculate homogeneous gas phase nucleation rates. This derivation is fairly standard and similar to previously published derivations;‘7u’mz1) we include it here as it is required in order to understand the subsequent discussion. The homogeneous vapor phase nucleation rate is given by(22) J=
=
-1 51 ( i=lCi?Zd1
(14
zci.ni*
W
where ci is the rate of capture of atoms by clusters of
VOL.
21, 1973
size i; n, is the equilibrium Boltzman concentration of clusters i ; Z is the Zeldovich non-equilibrium correction factor which is approximately 0.1,(S) and i* is the critical cluster size at which nni.is a minimum and which dominates the sum in equation (la). According to classical nucleation theory(5) ni = nr exp [-AF(i)/kT]
(2)
where AF(i), the excess free energy of a cluster of size i, is given by the capillarity approximation AF(i) = Ai2J3a -
ikT lnplp,
(3)
where A is a numerical factor such that AP is the surface area, o is the surface tension, and p/p0 is the supersaturation. We will now derive an appropriate expression for AF(i) from statisticalmechanicsforuseinequation (2). Our intention is to obtain an exact expression for the free energy to compare with the predictions of the capillarity approximation (equation 3). According to statistical mechanics, the equilibrium concentration of clusters of size i in the vapor phase is given by’s) ni = Zi exp [ip/kT]
(4)
where Zi is the partition function of a cluster of size i in a box of unit volume and p is the chemical potential of the monomer. When the motions of the cluster can be separated into translation of the whole cluster, rotation of the whole cluster, and internal motion of the atoms with the cluster, then the partition function can be written as(20*21) 4 =
2 Ztrans(i, X)Z,,,(G x = conflg
XV&,(&
X)
(5)
z tram
is the translational partition function for a cluster of i atoms in a box of unit volume Ztrans(i, X) =
r?y
where m is t’he mass of a single atom. rotational partition function
w Zrot is the
(6b) where o(X) is the symmetry number of a cluster and I3 is the product of three principal moments of inertia of the cluster. Zint is the internal partition function. For a solid-like cluster with harmonic forces Z,,,(X,
i) = exp [- U(X, i)/kT] 3i--8
XII j=l 1 -
e-hvJ2kT exp [-hv,/kT]
(W
BURTON:
VALIDITY
OF HOMOGENEOUS
where U(X, i) is the potential energy of the cluster and vj are the normal mode vibrational frequencies. The sum over X in equation (6) means that each distinguishable configuration of i atoms is counted exactly once. The translational partition function, equation 6a, is independent of the cluster configuration; the rotational partition function depends on configuration primarily through the symmetry number, a(X). As the summation over all configurations required in equation (5) is difficult, it is convenient to write
NUCLEATION
THEORY
1227
F osailg(i) as the logarithms of the appropriate partition functions in equation (8a). Using equations (9b and lob), equation (4) may be rewritten as Fto,,,(i) + FfJi)
+
(i) Fconiio
-
iF, -
+ Fio,Ji)
ikT In:
11
kT
1
. (11)
If we write ni in the form used in classical nucleat,ion theory n, = nl esp [-AF(i)/kT]
(4)
then from equation (II) where the superscript zeros refer now to the most probable configuration (i.e. minimum free energy or maximum partition function) of a cluster of size i. We can write equation (7) in a more revealing fashion (3a)
where zconng(i)is
the
configurational
partition
function
(8b) Comparing now equations (lb, 4 and 8a), wesee that the configurational partition function tells us how much configurations other than the most probable enhance the concentration of small clusters and hence the nucleation rate. (In introducing the configuration partition function, we are explicitly considering the possible contribution of clusters of irregular shape to the nucleation rate.) As explained above, our intention is to obtain a correct expression for AF(i) for use in equation (2). If the gas is nearly ideal, then ,u = kTInp.
(94
AF(i) = {J%,,,(i) + {e,,(i)
+ F&,(i) + kT ln nJ -
iP,}+
Fconfig(i) -
ikT In ?- . PO (12)
In the nest section we shall compare AF(i) as computed from equation (12) for the model system with A F( i) as computed from the capillarity approximation, equation (3), for the same model system. The terms in the first brackets in equation (12) constitute the Lothe-Pound(‘) correction to the free energy of a small cluster. The second bracketed quantity, the excess free energy of the most probable configuration of a small cluster relative to the bulk, has not been previously discussed; we shall compare this term directly with the predictions of the capillarity approximation. The F conilg(i) term in equation (12) will be examined briefly; it is not of great numerical significance in the temperature range of interest. The last term in equation (12) is the supersaturation, which appears in both the capillarity approximation, equation (3), and the statistical mechanical treatment. 3.
COMPARISON OF EXACT THERMODYNAMIC PROPERTIES AND THE CAPILLARITY APPROXIMATION
Remembering that the free energy of the bulk phase, F,, must equal the chemcial potential of the vapor phase when the two are in equilibrium (at pressure po) we write
Initially we will compare the excess internal free energy of the most probable cluster, the second term in brackets in equation (12)
kT In y + F,. PO As the free energy is related to the partition function
wit,h the predictions of the capillarity approximation at the equilibrium pressure, p = p,, in equation (3),
,u =
by F=
-kTInZ
(104
or equivalently Z = exp [- F/kT] it is appropriate to define P:,,(i),
(lob) FFJi),
F:&(i) and
AFkt(i) = Y&(i) -
AF,,,(i)
iF,
(13)
= AC%.
(14)
In the usual way we will separate the free energy into potential energy, zero-point energy and entropy. AF = AU,,,
+ AU,.,.
We now examine AU,,,
-
TAti’,,,.
and AS,,,.
(15)
The errors in
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as predicted by the ctlpillarity approximation, AU,.,., are small compared to the errors in AUpot@l) and affect nucleation r8tes by less than 8 factor of 100; therefore we will not discuss AU,.,. The simplest way to represent AU,,, is in terms of the bond deficit per atom. An infinite f.c.c. solid has 6 bonds per 8tom. A two atom cluster has only 1 bond and hence a bond deficit of ll(2 x 6 - 1) bonds or 5.5 bonds/atom. The bond deficit per atom for small f.c.c. clusters is shown in Fig. 1. Exact results and capillarity 8pproximation results are both shoun. The smooth curve through the circles is sn analytic function which fits the exact data. The computations of the two curves in Fig. 1 (and in
VOL.
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2
I
-I,1 i f
2
3
E vi-
2
‘0
-
EXACT
---
CAPILLAR ITY
25
50 n
75
0 ? 55 -2
E
-3
f-
z
-4
100
FIU. 2. Internal vibrational entropy per atom, Si,,t, and excess entropy relative to an infinite solid, ALSint, as a function of cluster size. I
0 t ;; S $
4
a2
0
25
50
75
100
n FIG. 1. Bond deficit per atom for small clusters as a function of aluster size. The pm&&ions of the oapillarity approxlmatlon are shown, as are exact results foqminimum free energy configurations.
succeeding figures) are discussed in detail elsewhere.(21) Comparison of the two curves in Fig. 1 is informative. For large cluster sizes the capillarity approximation underestimates the bond deficit. This occurs because the capillerity approximation assumes that all of the atoms lie on infinite bulk surfaces and have coordination numbers 8ppropriat.e to such surfaces; it neglects entirely the fact that there are many edges and corners on a small cluster. Edge and corner atoms have lower coordination numbers than regular surface atoms; hence the bond deficit is increased. At small cluster sizes, the exact and capillarity results are in good agreement. This occurs because the cspillarity approximation overestimates the number of surface atoms for small clusters; for instance, according to the capillerity approximation, 8 10 atom cluster has 20 surface 8toms.fN) ASiDt is plotted in Fig. 2. The capillarity approximation is fairly good for Isrger clusters but fails badly for smaller clusters. The error for smaller clusters arises from the fact that 8 small cluster has only 3i - 6
vibration81 modes; 6 degrees of freedom are used in rotation and translation. The excess free energy of a small cluster is plotted in Fig. 3. Here we 8ssume we are at the equilibrium vapor pressure, p = p,. The data are plotted at 8 temper8ture of 6O’K for potential parameters which give a critic81 temperature of 108’K; this is 8 reduced temperature of T/T, = 0.55 where T, is the critical temperature. Note first that excess internal free energy of 8 small cluster at this temperature is substantially higher than predicted by the capillerity approximation; this occurs primclrily becsuse the capillarity approximation underestimates the bond deficit, Fig. 1. When the Lothe-Pound correction factor, the first term in brackets in equation (12), is 20
-INTERNAL ---T+ R+I ---CAPlLLARlTY 0
0
20
40
60
00
too
n FIG. 3. Excess free energy of small alusters relative to an infinite solid at a reduaed temperature of T/T, = 0.66. The ourve marked “Internal” considers only the internal degrees of freedom of the most probable cluster. The curve marked “T + R + I” considers all degrees of freedom of the most probable oluster. The capillarity approximation result is also shown.
BURTOIi:
VALIDITY
OF
HOMOGENEOUS
NUCLEATIOS
THEORY
1229
TABLE 1. Corrections to the equilibrium concentration of clusters nj due to various errors in the capillarity approximation Error
Correction
10-10_ IO-‘0
AU,ot
10”
0
20
40
60
80
T (OK) FIG. 4. Configurational free energy of a 49 atom cluster as a function of temperature. (The critical temperature 1s 108%) The curves marked “0,” “1,” and “2” are improvmg approximetions to Pconflr. The cross hatched region is a guessed range of reasonable extrapolations to include all configurations.
included in the curve labelled “T + R + I” the free energy is reduced. At this particular temperclture the intern81 free energy error virtuslly cancels the LothePound correction factor; this is, as we will see, accidental. In Fig. 4, we have plotted the coni&urational free energy for a 49 atom cluster 8s 8 function of temperature. The deteils of this contribution 8re discussed elsewhere.(ms21) The curves labelled “O”, “1” snd “2” are successive approximations to Fcontig.(4s) The crosshatched region is a range of possible extrapolations of F coniig to include 811 possible cotiguretions. Comparison of Figs. 3 and 4 indicates that Fconiig lowers the free energy by rv5 per cent. 4.
CORRECTIONS NUCLEATION
TO
CLASSICAL RATES
ASint
10-1 -
F config
10’ -
Lothe-Pound (RotationTranslation)
10’4 -
10’6
Combined
10-Z -
1019
10-Z 10’
Comments Large clusters, depends on temperature. Small clusters, independent of temperature. Independent of temperature and cluster size. Depends on temperat,ure. Depends weakly on temperature. Depends on cluster size and on temperature.
For small cluster sizes the capillarity approximation is fairly good, Teble 1. The energy correction will depend somewhat on the system studied. However, the qualitative features of the energy correction are independent of the system. The capillerity approximlation neglects edges and corners for 8ny system and hence always underestimates the bond deficit for large cluster sizes. For small cluster sizes, the cepillarity approximation always overestimates the number of surface atoms and this error tends to cancel the neglect of edges and corners. B. The internal entropy correction The intern81 entropy correction, Teble 1, will depend on the system studied, but, as it is small, the variation between systems will not be serious. C. The conjgwational correction
We are now ready to examine the effects of all the different terms discussed above on the nucleation rate. We will give the numerical corrections and discuss qualitatively whether these correctVions are model dependent. Initirtlly we will examine the effects of the corrections on the equilibrium concentrstion of 8 particular size cluster. Then we will examine nucleation rates. The discussion is separated into these two steps as t,he critical cluster size calculated exactly is different from that predicted by the capillarity approximation.
The rotation-transletion correction increases the cluster concentretion by m1016 and is based on fundemental statistical mechanic81 considerations.(‘) It is essentially independent of the system.
A. The potential energy correction
E. Free energy
For large critical cluster sizes the cepillarity approximrttion seriously underestimates the bond deficit. Hence the potential energy correction greatly lowers the cluster concentration. This correction demnds stronelv on temnerature. Table 1.
Above we have examined various contributions to errors in the free energy of small clusters. In this section we examine the free energy itself. In Fig. 5, we plot the total excess free energy for three c8ses. Exact results 8re comuared with the
L
VY
L
The configurational correction increases the cluster concentration by 10 to 10,000. Any small cluster can assume many configurations like an amoeba. This configurational term will occur for any system. D. The Lothe-Pound
(rot&ion-trandution) correction
L
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The capillarity approximation gives somewhat smaller critical cluster sizes than predicted by the exact theorY. The principal result of this work, the correction to the classical nucleation rate, Jeract/Jclass, as a function of supersaturation and reduced temperature, is in Table 2. We do not give results here at low supersaturations as the critical cluster sizes are significently
EXACT CAPILLARITY
TABLE 2. Corrections to the classical nucleation rate, .JeXsEl/ Jelass, as a function of supersaturation. p/pe. and reduood temperature, T/T,, where T, 1s the critical temperature 72 \ 0.35 0.39 0.43 0.47
Fm. 5. Excess free energy of small olusters 8s a funotion of reduoed pressure p/pi, (p,, is the equilibrium vapor pressure), and reduoed temperature T/To (T, is the critic81 temperature of the vapor). Exact results are oomp8red with the predictions of the capillarity approximetion.
predictions of the capillsrity approximation. Note that the free energy corrections to the cspillarity approximation depend on both temperature 8nd supersaturation. F. Nucleation
rates
In Fig. 6 we show the dependence of the critical cluster size* on both supersrtturation and temperature.
.50
\’ \
P/P,
= 16
are compared with the predictions larity epproximetion.
102 10’ 10’ 106
10’ 105 :;:
10s 10’ 10’ 10’
lerger than the biggest clusters which we have studied, 87 atoms. (Larger clusters were not studied because of difficulties associated with finding the vibrational frequencies for equation (60) ; one must diagormlize 3n x 3n matrices to obtain the normal mode vibrational frequencies.) Nucleation rates at lower aupersatur8tions might be very crudely estimated by extrapolation. The results in Table 2 8re exact results for our model system. As pointed out above, the contributions are not strongly dependent on the to JexactlJo,ass system. We expect that the numbers in the table would not change by more than ~10~ when other systems are studied. The difference between classic81 nucleation theory and our exact calculations may be seen in 8 different w8y. Figure 7 shows the supersaturation (p/p,Jexsot at which the nucleation rate at 8 given reduced temperature, T/T,, is equal to that predicted by classical The results in this figure 8re theory at (P/z4J)c*ae.s*cal. exact only for our model system. However, as we exis not strongly system dependent, pect that JexactlJclass we expect that Fig. 7 is fairly general and applicable to many experimental systems. * * The reasons for this are explained in Section 4. A-D. Essentially they are that the major souroes of the errors in the oapillerity approximation 8re a geometrioel error in the potential energy and the neglect of rotation and translation. Neither of these errors is model dependent. Note that this presumed generality of our result is analogous to the oorresponding states assumption whioh is widely spplied in the study of vapors. Corresponding stat+8 is known to apply reasonably well to critical point phenomena, which oan also be described in terms of cluster propertiesus As some oluster
1
FIG. 6. Critical cluster size, a*, 8s a function of reduced pressure, p/pa. and reduced temperature, T/To. Exaat results
10-z 10’ 10’ 106
of the oapil-
l The critical oluster size is that cluster size for whioh the i.e. the least probable excess free energy is 8 msximum; cluster. See Refs. (6) and I**‘.
properties can be calculated from critical indioes, critioal point properties have been recently used by Kisng(*s’ to 08lcUl8m nucleation rates. Comparison of the sosling parameters introduced in Kiang’s work with those whioh appear in ol8ssioal nucleation theory,” indicates that corresponding states is applicable to the correction to classical nucleation theory, Jexsct/JcrarsieB1, except very near the oritieal point. Henoe, as claimed, Jeract/Jclaasiral, caloulated here for our simple model s.ystem should apply to a wide range of materials.
BURTON:
VALIDITY
OF HOMOGENEOUS
NUCLEATION
THEORY
TABLE 3. Experiment81 and our theoretical corrections to cl8ssioal gas phase nucleation r&es, J/Jels.sic~l. p/p0 is the supersaturation. T/T, is the reduced temperature
200
IO0
Meterial (Reference)
PIP0
50
H,O’“” H,O’**’ C,H,OH’rs’ CH,OH’**’
E :: w $
1231
20
0.41 0.45 0.47 0.47
5.5 ;.z 212
J/Jcl*8llIC*l,
Experimental
Theoretical, this work
10-p 100 100 100
.? IO
5’
5
10
50
20
100
200
500
‘P’Po’CLASSICAL
Fm. 7. The supersaturation (P/P,),,.,t 8t which the nucleation r8t.e at 8 given reduced temperature, T/Tc (T, is the vapor phase critical temperature), is equal to the nucleation rate predicted by classical nucleation theory at (P/;D,,)~~.~~~~~I. The figure is used 8s follows: We w8nt to know, for instance, what superseturetion is required to obmin the nucleation rete at T/T, = 0.49 prediated by classical theory at (P/P,,) = 50. We follow the (P/Ps)C~~~~i~~l = 50 line verticelly until it interseots with the T/Z’, = 0.49 curve 8t (P/Po)exaet= 17. This is then the predicted supersatur8tion required to obtain the desired nucleation rate. 5.
COMPARISON
WITH
EXPERIMENT
In this section we compare our calculated nucleation rates with some previous experiments. We have made no attempt to search all of the nucleation literature and compare our t,heoretical work with all experiments. This exhaustive comparison is not warranted until thermodynamic properties of clusters of more than 87 atoms have been calculated so that conclusive theoretical statements can be made even at low supersaturations. Two types of experimental results are often reported. Either the observed nucleation rate at some pressure and temperature is measured directly and reported and also compared with the predictions of the capillarity approximation, or the pressure and temperature at which some phenomenon such as
condensation occurs in a particular experiment is measured and compared with the pressure and temperature predicted by the capillarity approximation. In Table 3 we compare results from cloud chamber experiments at low supersaturations with our predictions. The experiments were carried out at temperature T, or reduced temperature T/T, relative to the critical temperature, and supersaturation p/p,,. The experimenters reported either the numerical value of the nucleation rate J and the correction to classical theory J/Jclaasioal or reported that the experimental nucleation rates agreed quantitatively with the predictions of classical theory. For each reduced temperature and supersaturation we oan find the predicted value of J/Jclwsic.l from data such as in Table 2. Unfortunately, these experiments were all carried out at lower supersaturations than those for which we can now make statements. However, extrapolation of our data such as in Table 2 to lower supersttturations suggests upper limit,s for J/Jclaasloal which are included in Table 3. The agreement between experiments and our predictions is satisfactory. In Table 4 we give the supersaturations (P/PrJLmIlL?ntslat which condensation was observed to occur in wind tunnel and diffusion chamber experiments for various materials at reduced temperature T/T,. The experimenters also reported the superat which classical nucleation saturations (z&Jo~aeSica~
TABLE 4. Classical, experimental, 8nd our predictions for the supersaturation, p/p,. at which condensation occurs for 8 given reduoed temperature, T/T, Material (Reference) H,O’u NH,‘=’ CeH,“6’ CHCl,“a’ CCl,F”” C,H,OH”” C,H,,‘**’ C,H,,‘C&HI,‘-’ C8Ht0’*@’
(P/PO)throretical. this work 0.40 0.49 0.37 0.37 0.36 0.46 0.45 0.46 0.44 0.44
15 7.5 390 170 440 11 11 11 ::
11 12 4.5 *1 110 * 20 40 * 15 100 * 30 61t2 10.5 * 2 10 f 1 16 f 2 13 & 2
13 6.5 100 65 115 a.7 9 8.7 ::
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theory predicts condensation. We then used Fig. 7 to find the corresponding supersaturations at which we predict condensation should occur; these are also in the table. The agreement is very good with two exceptions-NHs and CHCl,. In both of these oases we expect significantly higher supersaturations than found experimentally. In the case of ammonia, the investigators reported major difficulties with water contamination,d6) and chloroform is very difficult to produce with great purity. Hence it is possible that heterogeneous nucleation was actually observed at a lower supersaturation than required for homogeneous nucleation. 6.
CONCLUSIONS
We have calculated exactly the homogeneous vapor phase nucleation rate, Jexact, for a simple material and have compared this result with the predictions of the classical capillarity approximation, Jo,_, for the same material. The ratio Jexact/Jolaescan vary from 10-s to 10’ and depends strongly on temperature and supersaturation. Generally at low temperatures and low supersaturations Jexeot corresponds fairly closely while at high temperatures and large superto Jalarrs saturations the differences are large. We have compared our predicted corrections to the classical homogeneous vapor phase nucleation rates with the corrections found in a wide range of experiments. The agreement between theory and experiment is generally quite good. We believe therefore that homogeneous nucleation rates can be satisfactorily computed from first principles when all corrections to classical nucleation theory, including the Lothe-Pound rotation and translation factors,“) are considered. Accordingly, we feel that the long standing controversy about the necessity of rotation-translation corrections(8-14) is resolved, and that the apparent discrepancies between experiments and the Lothe-Pound theory(‘) reflected the need for consideration of corrections besides rotation-translation ; these further corrections were in fact suggested by Lothe and Pound.(19) The results of a wide range of experiments are satisfactorily accounted for in an exact theory.
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ACKNOWLEDGEMENTS
The author wishes to express his appreciation to F. F. Abraham, G. M. Pound and F. H. Stillinger for their invaluable suggestions and criticisms of this work; to C. Briant and J. K. Tien for comments on the manuscript ; and to Columbia University and the Henry Krumb School of Mines for supporting the work. REFERENCES Chem. 119, 227 1. N. VOLMER and A. WEBEIL. 2. Phys. (1920). 2. L. FARKAS, 2. Phys. Chem. 136,236 (1927). 3. N. VOLMERand H. FLOOD, 2. Phys. Chem. 17OA, 273 (1934). 4. R. BECKERand W. DOERINO,Ann. Phya. 94, 719 (1936). 5. J. FEDER, K. C. RUSSELL, J. LOTHE and G. M. POUND, Adv. Phys. 15, 111 (1966). van Nostrand (1966). 0. W. BAND, Quantum Stat&ice. and G. M. POOND, J. Chum. Phy8. 96, 2080 7. J&IEE \----I.
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