J. Phys. Chem. 1994,98, 6408-6412
6408
The Factor 1/S in the Classical Theory of Nucleation Cheryl Li Weakliem and Howard Reiss' Department of Chemistry, University of California, Las Angeles, California 90024- 1569 Received: October 5, 1993; In Final Form: March 29, 1994'
It is shown that there is no fundamental justification for the factor 1 / S frequently used to adjust the classical theory of nucleation, and that one can arrive at an arbitrary factor (including unity), depending upon how the translational degrees of freedom of the nucleus are separated (within the model) from the remaining ones. The most correct and unambiguous result (within the model) can only be obtained by means of molecular theory. In addition, it is demonstrated that (again within the model) the use of the full machinery of chemical-like detailed balance is unnecessary for the purpose of deriving cluster evaporation coefficients. Introduction In the classicaltheory of vapor-phase nucleation, the application of the principleof detailed balance leads to the following expression for the equilibrium concentration of clusters
Comparison of this equation with eq 4 shows that it differs from that equation by having an additional factor of C I in the denominator of the term on the left. In contrast, making the same substitution in eq 3 leads to
where a = 47r+-) 3v
213
and where u andv are the surface tension and volume per molecule for the bulk liquid while S = p/po is the supersaturation of the vapor for which po is the saturation pressure at the temperature T, in question, andpis the actual pressure. C1is theconcentration of single molecules in the supersaturated vapor while k is the Boltzmann constant. Several authors, arguing from different vantage point~,l-~ but ultimately from the same basic reason, have suggested that eq 1 should be replaced by Cn= Coexp(-
1 kT [(-kTIn S)n + an2/']1
(3)
where CO = po/kT is the concentration of single molecules (essentially all of the molecules) in a vapor at equilibrium with the liquid, i.e. in the saturated vapor. To pass from eq 1 to eq 3, it is necessary to divide C i by C I / C O = p ~ / p o= S (pl, the partial pressure of single molecules, is essentially indistinguishable from p ) . The effect of this division by S is to also require that the nucleation rate derived, using C',, in the application of detailed balance, also be divided by S. A very simple argument (also based ultimately on the same reasons advanced by various authors) for the need for this factor 1/S is the following. Clusters containing n molecules are in equilibrium with single molecules. Since the vapors in question are usually nearly ideal, we may use concentrations in place of thermodynamic activities so that, using the law of mass action, it is possible to formulate an equilibrium constant expression, namely (4) where Kn is the equilibrium constant. Using S = p/po = kTCI/ PO, in eq 1, allows it to be arranged into Abstract published in Advance ACS Abstracts. May 15, 1994.
0022-3654/94/2098-6408%04.50/0
which, in form, agrees precisely with eq 4! Since eq 3 can be derived from eq 1 through multiplication of the right side by 1/S, this factor appears to receive justification. The problem however remains that eq 1 has a highly modelistic origin and its roots are not well based in fundamentals. Multiplication by a factor of 1/S then represents a 'band-aid" process intended to bring, by fiat, an originally nonfundamental relation into conformance with the law of mass action. The wellknown problem of the "replacement free en erg^"^.^ is another face of the same coin, as are several other problems that involve the classical theory. The theory is still useful in many applications, as long as its limitations are understood. At the same time, "band-aid" repairs (which have attracted more popularity than they deserve) can be hazardous, especiallybecause one can generate almost any desired result by choosing the proper band-aid. Because there still seems to be confusion and controversy over this issue, the whole matter of 1/Scan still use clarification that puts it into proper perspective with regard to molecular theory. We attempt this further clarification in the present paper, developing the discussion in such a manner as to illuminate both the issue of the replacement free energy and the significance of the early commentary of Courtney' that first stimulated the consideration of 1/S. Courtney's Argument
In this section we show that Courtney's argument can lead to almost any preexponential factor in analogs of eqs 1 and 3, including C1 and Co as special cases. Courtney employed a thermodynamic "cycle" in arriving at his result, eq 3. Although cycles have a long and respectable history in thermodynamics, they are not always easy to understand, and it was left to Gibbs to introduce the direct analytical approach into the derivation of most thermodynamic relations. The analytical approach is usually straightforward and easy to understand. In the following we use it to rederive Courtney's result in order to expose his assumptions in a simple manner. 0 1994 American Chemical Society
1/S Factor in the Classical Theory of Nucleation
The Journal of Physical Chemistry, Vol. 98, No. 25, 1994 6409
Courtney1 pointed out that Frenkel? in deriving theequilibrium distribution of clusters, wrote for the chemical potential of a cluster containing n molecules, the expression p,, = pi
+ k T In Nn
(7)
where N is the total number of molecules in the vapor and pi is a chemical potential in a standard state. p,, has a usual form for the chemical potential of a component of an ideal gas, but in the form shown in eq 7, it is necessary for pi to depend on both the temperature and pressure of the gas, Le., p i = p i (T,p). If one wishes to use a p i that depends on temperature alone, it is known (as any elementary text on physical chemistry will show) that eq 7 should be replaced by
quantity appearing in eq 9. However, since that quantity represents the free energy of a drop at rest in the laboratory coordinate system, Frenkel’s pi, at least superficially, contains no contribution from translational degree of freedom. We use the term ‘superficially” because, in fact, eq 9 does contain some translational free energy! To see this, consider that the drop is delineated by a Gibbs dividing surface at rest in the laboratory frame. However, its center of mass can still fluctuate? and this corresponds to translational motion. For a large drop the fluctuation is vanishingly small, but for a small drop it may be appreciable. At face value, it therefore appears that Frenkel meant his pi to represent internal free energy, thus implying the following relation.
+
np,
Then eq 12 shows that where p(n) is the partial pressure of the cluster in the gas. Unfortunately, Frenkel wrote for pi the expression pi
= np, + an2I3
pi
(9)
where 1.11 is the chemical potential of the bulk liquid. pi in eq 9 is therefore a free energy of a condensed system (i.e., of a liquid) and is therefore relatively insensitiveto pressure. Thus, Courtney assumed that Frenkel was choosing the standard state chemical potential appearing in eq 8, i.e., a p i that was a function of temperature alone, and that some correction to eq 7 had to be introduced in order to achieve thermodynamic consistency. However, since Frenkel is now deceased, he cannot tell us what he really intended in adopting eq 9. Like Courtney, we are forced to make assumptions about what he meant. Before proceeding further, we note that eq 8 should really be written as p,, = pi ( T )
+ k T In(p(“)/p)
(10)
where p represents unit pressure, e.g., 1 atm. To gain a further perspective on eq 10, we note that the standard statistical mechanical formula for the chemical potential of a component of an ideal gas is
pi
= -kT In q,,
(13)
should be written as
(7‘) = k T In
A> + np, + an2I3 kT
Like Frenkel, Courtney also did not consider translation, but he did attempt to adjust Frenkel’s expression toward a form of the type of eq 8. In this connection we note that it would be equally valid to use PO in place of p in eq 11 since whatever quantity is used actually cancels out of the expression. In that case eq 11 would be replaced by
*”
p,, = kTln--
kTlnq,,+ k T l n P (n)
(15)
Po
kT
but p,, in eq 15 would have exactly the same value as in eq 11. Now k T ln(A)g,/kT) in eq 15 and k T ln(A>/kT) in eq 11 are the chemical potentialsof ideal gases consistingof structureless molecules at pressures PO and p , respectively. In other words, these quantities represent translational free energy. To follow Frenkel, therefore, they should be omitted in the specification of p,. If the omission occurs in eq 15 and eq 13 is adhered to, one obtains Courtney’s modification of Frenkel’s p,,. If the omission occursin eq 11, we obtain still another modification. We examine, first, the consequences of omitting it in eq 15, Le., in following Courtney’s program. For the individual molecules of the vapor, the correct expression for the chemical potential is p1 = p ; ( T ) + k T l n - P1
= k T l n -kT
kTlnq,,+ kTlne
P
B
(11)
where
where A,, is the thermal de Broglie wavelength of the cluster and q, is its internal partition function. Comparison of eq 11 with eq 10 shows that p:(T)
A> = kTln-kT
-
(17) Chemical equilibrium in the vapor then requires
k T In q,, Pn
-
Now Frenkel’s expression, eq 7, does restrict what Frenkel could have meant by choosing p,, p: when N,, N. Under thisconditionthere arenoother “molecules”in thesystem besides clusters of size n. Then, since the chemical potential p,, is the free energy per ‘molecule”, and the term ‘molecule” is being used to denote a cluster of size n, it must be the free energy of a cluster. Thus pi is the free energy of a cluster when there are no clusters in the vapor save those of size n. Since the cluster is modeled as a liquid drop, the free energy per cluster, not unreasonably, is the
= W1
or pi
IP + k T l nPo(n)P = pi + k T l n XPo
where X,,= N n / N is the mole fraction of the cluster species.
6410 The Journal of Physical Chemistry, Vol. 98, No. 25, 1994
Rearrangement of this equation yields
where we have used the fact that p1 is essentially p. Equilibrium between the saturated vapor and the bulk liquid gives rise to the relation pI = p ;
+ k T In Po T P
or p ; = p l - k T I nPo~
P and substitution of this expression, together with eq 9, into eq 20 gives
1
1 c,, = exp - -[(-kT c‘po P kT
In S)n + anY3]}
where S = p / p ~ is the supersaturation. Now sincepolp = Co/Cl, eq 22 may be written as
1
1 C,, = Coexp - -[(-kT
+
In S)n an2/’]) (23) kT This is eq 3, the result obtained by Courtney and used by others, and demonstrates that Courtney’s cycle is equivalent to both adding and subtracting In POin the correct formula for p,,, Le., in eq 11, and then ignoring the translational contribution in the resulting expression, eq 15, in accordance with Frenkel’s apparent intent. If we had omitted the translational term in eq 11, rather than in eq 15, and had followed the same steps as in passing from eq 16 to eq 23, we would have obtained C,, =
clexp1- [(-kT klT
In S)n + an2/’]}
(24)
where Cl is the concentration of single molecules in a vapor at the same temperature but at unit pressure (1 atm). However, which result is the correct equation in which the omission should be made is academic since both results, eqs 23 and 24, are wrong and, worse yet, the repeated canceling pressure could have been any pressure and we could have therefore arrived at any preexponential factor! Courtney chose to use POrather than p because he reasoned that Frenkel, in using eq 9 for the specification of pn0, was choosing a standard state in which the drop was subjected to the saturation pressure of the liquid. However, in eq 1 1 or eq 15 thep orpo, respectively, do nor refer to the pressure of the surrounding vapor but to the partial pressure of a vapor made up only of clusters of size n, so there is no compelling reason to choose either p or po. The only proper way to proceed is to not omit the translational term and to use eq 14. Indeed, ifthesamesteps had beencarriedout,usingeq 14as thedefinition of p:, the result would have been
1 k:.
1 C,, = -exp - -[(-kT
4
In S)n
+ an2/’]] cl
(25)
The quantity Ai3 is much larger than either COor so that C,, and the corresponding rate of nucleation far exceed the classical value, as Lothe and Pound4 discovered. The possession of distributions involved in equations such as eqs 1,3, and 25 is imperative, within the confines of the classical theory of nucleation, in order to deduce cluster evaporation coefficientsfrom the more easily derivedcondensationcoefficients. However,as we show in the Appendix, as long as only the classical
Weakliem and Reiss
theory is involved, the evaporation coefficients can be obtained with satisfactory accuracy in an extremely simple manner that never invokes chemical-like detailed balance and correspondingly has no requirement for the above mentioned distributions. On the other hand, somewhat paradoxically, the distributions must be used to prove that they are not necessary.
The Replacement Free Energy In eq 25 the translational free energy of the cluster appears to be given by-kTln(V/Ai), but Lothe and Pound4 realized that some translational contributions were also present in the bulk liquid to which the chemical potential p~corresponded, and that these effects should properly be included in the exponent in eq 25 so that the overall translational contribution would be the difference between -kT ln(V/Ai) and the original free energy in the bulk liquid. This difference they referred to as the “replacement free energy”. However, they considered the effects in the bulk liquid as associated with vibrational degrees of freedom that were converted to translational motion, i.e., to the translation of the center of mass, in the cluster. Actually, Lothe and Pound also considered that the cluster could “rotate” and that the corresponding rotational degrees of freedom were originally torsional vibrations in the bulk liquid. Thus rotation as well as translation gave rise to a replacement free energy, and both of the replacement free energies contrived to increase C i in eq 1 or C, in eq 3 by a factor of the order of loL7.As a result, the predicted rate of nucleation was also increased by a factor of this magnitude. The largest part of this factor originated in the rotational replacement free energy. For a liquid cluster it is almost meaningless to consider rigid rotation of the cluster as a whole; Le., Euler axes that “rotate” with thecluster cannot beconveniently lodged in it. Furthermore, in the evaluation of the classical liquid partition function, integration over the momenta (implicitly including angular momenta) is always performed exactly so that pl already contains most of the rotational effects. Therefore, it has become increasingly clear that a rotational replacement free energy is unwarranted. However, sometranslational contributions must be taken into account. Reiss, Katz, and CohenS showed that these effects could be summarized by the fluctuation of the center of mass of the cluster and concluded that, by eliminating the rotational replacement and including the center of mass fluctuation, the factor of 1017could be reduced to only 104-105. They based their estimate on an approximate evaluation of the fluctuation of the center of mass. Later authors, in various contexts,7.* arrived at even smaller values for the replacement factor, until now it appears to lie in the range 10-1O2.* The latest estimate is probably valid, and the somewhat larger factor suggested by Reiss, Katz, and Cohen is apparently rooted in the approximate nature of their calculation of the fluctuation. To state the problem in a slightly different manner, part of the large result obtained by Lothe and Pound4 comes from the uncritical separation, represented by eq 14, of translational and internal degrees of freedom. As we have indicated, because of the fluctuation of the center of mass of the drop,’ the terms np1 + an2I3contain some translational free energy. Yet the full translational free energy of the cluster has already been accounted for by the use of VA;’ (we have set V - 1.Ocm3),the translational partition function, in the theory leading to eq 25. Thus eq 14 counts some of the translational free energy more than once, and eq 25 cannot be corrected by simply dividing by the translational partition function (see Reiss et al. (ref 5)). Translation free energy is a negativequantity so that if the redundant contribution is eliminated from n N l + a$/’, and therefore also from p,,, the work of cluster formation, in curly brackets in eq 25, will be
The Journal of Physical Chemistry, Vol. 98, No. 25, 1994 6411
1/S Factor in the Classical Theory of Nucleation increased. This reduces thevalueof C,andcounteracts theeffect of a small Ai.
and substitution of this relation into eq- A8 gives -
Appendix Simple Estimate of the Evaporation Coefficient. Chemicallike detailed balance is used in the Frenkel-Zeldovich theory of nucleation. In that theory, thenet rateat whichclusterscontaining n molecules become clusters of n + 1 molecules is expressed as Jn
= 0 SJn - ~
(AI)
n + Sn+l l f&l
where S, is the surface area of a cluster (drop) of n molecules and fn is the possibly time-dependent number of clusters of size n. fl and ?,,+I are condensation and evaporation coefficients, respectively, defined as the rates at which molecules condense or evaporate on or from unit area of surface of a cluster (drop) containing n molecules. Sticking coefficients (for condensation) are usually assumed to be unity, but in any event, they are not crucial to the present argument. Furthermore, both fl and ?,,+I are approximatedas the correspondingquantities for a flat surface. For such a surface in contact with an ideal gas,
~=-E--M
(-42)
T
where p is the pressure of the gas and m is the molecular mass. For later use, we note that the vapor pressure pn of a liquid drop containing n molecules is prescribed by the Gibbs-Thomson relation? namely Zm/r,,kT
Pn=Poe
Using eqs A3 and A4 and a little algebra, we find, from eq A10,
Using eqs A3 and A4 again, together with the expansion
and some additional algebra, we find
or
Substitution of this result into eq AI 1 then yields
(A3)
in which po is the equilibrium vapor pressure of bulk liquid (flat surface), u is the liquid-vapor interfacial tension (also for a flat surface), u is the volume per molecule in the liquid, and r, is the radius of the drop given by
since, even for n = 10, the exponent 1/9n2 is only 1/900. Next we examine the ratio S,/S,+l in eq AS. Since
we find
-
If the system is constrained into chemical-like equilibrium, Jn may be set to zero and fn Nn where N, is the equilibrium distribution of clusters. Equation A1 is then reduced to
Yn+l = 0 SPnIsn+, Nn+1
-=("r"= Sn S,,
1
n+l
(1 +
2/3
n)
2 5 = 1 -+...
3n
9n2
(A51
In the usual theory (see Frenkel, ref 6)
Substitution of eqs A2, A15, and A17 into eq A5 yields
N, = N , e+'JkT
(A61
where N1is the number of single molecules in the system and W, is the reversible work of formation of a drop of size n prescribed by
W,=
-(k T In E)n + 4ua(
$)213
n2I3
(A7) If we simply replace n
wherep is the pressure of the supersaturated vapor and all of the other quantities have been defined previously. We proceed to study the ratio Nn/Nn+lthat appears in eq A5. Using eqs A6 and A7, we find
3 = exp(- 4ua k~ (q,)Iis 3vn (1 - ( 1 + !-)l/l)l (A8) Nn+1 P Expanding ( 1 we obtain
+1
Ti)*/3
in powers of 1 out to the quadratic terms,
+ 1 by n in this equation, we obtain
If we compare this result with the condensation coefficient specified by eq A2, its physical meaning is immediately clear. Whereas the condensation rate is proportional to the pressure p of the surroundingvapor (sticking and accommodationcoefficients assumed to be unity), the evaporation rate is proportional to the vapor pressurep,of the drop. Indeed the proportionality constants are identical (equal to ( 2 ~ m k 2 7 4 ~ )If. a flat surface were involved, pn would reduce to p, and eq A19 could have been deduced from the simplest consideration of vapor-liquid equi-
6412 The Journal of Physical Chemistry, Vol. 98, No. 25, 1994
librium. As long as we are concerned with the classical theory, the evaporation coefficient is always given simply in terms of the vapor pressurep, and the rather ponderousmachinery ofchemicallike detailed balance and the equilibrium distribution of clusters need not be involved. In considering this, however, one should not lose sight of the fact that the various approximations, eqs A9, A12, A15, A17, and A18, all depend on n not being too small. Fortunately, in most cases, n in the neighborhood of the nucleus is fairly large, e.g., n > 20, so that the approximations are not too bad. The simple result is a consequence of the very fact that the classical theory omits consideration of the translational degrees of freedom of the cluster. In eq A l , for example, a cluster of size n can be produced by evaporation of a molecule from one of size n 1 at a position that is not related to the location where a cluster of size n 1 is produced by condensation of a molecule on one of size n. This uncertainty of position is a reflection of the translational entropy of the cluster, not dealt with in the
+
+
Weakliem and Reiss classical theory. As a result, the classical theory yields a probability of nucleus formation that is not proportional to the volume of the vapor in which it is occurring, whereas, obviously, it should be proportional to that volume.
Acknowledgment. Research supported by the National Science Foundation under Grant No. CHE 90-22215. References and Notes Courtney, W. G. J . Chem. Phys. 1966, 35,2249. Blander, M.; Katz, J. L. J. Stat. Phys. 1972, 4, 55. Oxtoby, D.W. J. Phys.: Condens. Matter 1992,4, 7627. Lothe, J.; Pound, G. M. J. Chem. Phys. 1962.36, 2080. ( 5 ) Reiss, H.; Katz, J. L.; Cohen, E. R. J . Chem. Phys. 1968,48,5553. (6) Frenkel, A. Kineric Theory ojliquids; Oxford University Press: New York, 1946; J. Chem. Phys. 1939, 7 , 324. (7) Abraham, F. F.; Lee, J. K.; Barker, J. A. J. Chem. Phys. 1974,60, (1) (2) (3) (4)
246.
( 8 ) Talanquer, V.; Oxtoby, D. W. J . Chem. Phys. 1994, April 1, 100. (9) Lewis,G. N.; Randall, M. In Thermodynamics;Pitzer, K. S.,Brewer, L., Eds.; McGraw-Hill: New York, 1961; Chapter, 29.
