- Email: [email protected]
Nucleation theory without Maxwell Demons
Nucleation Theory without Maxwell Demons~ J O S E P H L. K A T Z Department of Chemical Engineering and Institute of Colloid and Surface Science, Clarkson College of Technology, Potsdam, New York 13676 AND H. W I E D E R S I C H Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439 Received December 17, 1975; accepted November 5, 1976 The equations for steady-state nucleation are derived from the rates of growth and decay of clusters with emphasis on a clear distinction between thermodynamic quantities and inherently kinetic quantities. It is shown that the emission rates of molecules from embryos can be related to the equilibrium size distribution of clusters in a saturated vapor. It is therefore not necessary to invoke the existence of an embryo size distribution constrained be in equilibrium with a supersaturated vapor. The driving force for nucleation is shown to be a kinetic quantity called the condensation rate ratio, i.e., the ratio of the rates of acquisition of molecules by clusters in the supersaturated vapor to that in a saturated vapor at the same temperature, and not a thermodynamic quantity known as the supersaturation, i.e., the ratio of the actual pressure to the equilibrium vapor pressure. INTRODUCTION Homogeneous nucleation t h e o r y is usually derived b y considering the rate of evaporation and condensation of molecules from clusters of such molecules. B y a series of kinetic arguments, the rate of nucleation is related to a constrained equilibrium distribution of clusters, i.e., the concentrations of the various sizes of clusters t h a t would exist if one were able to constrain a s u p e r s a t u r a t e d vapor to be in t h e r m o d y n a m i c equilibrium (1). However, a s u p e r s a t u r a t e d v a p o r is n o t in t h e r m o d y namic equilibrium. To justify using the constrained distribution, a Maxwell demon (2) is sometimes invoked. This demon is imagined to take all clusters larger t h a n some given size (any size much larger t h a n the critical size will 1Work supported by the U. S. Energy Research and Development Administration. The work of J. L. Katz supported by USERDA Contract E(11-1)-2183, "Nucleation of Voids."
suffice) and break them into individual molecules t h a t are then available to form clusters, again b y r a n d o m agglomeration. Although words such as "hupothetieal p o p u l a t i o n " and "reference s t a t e , " which acknowledge the nonreality of this distribution, are frequently used, this distribution plays a central role in the theory. T h e r m o d y n a m i c methods are used to calculate the constrained distribution of cluster sizes despite its inherent instability. I n a recent p a p e r on defect cluster nucleation one of us (H.W.) has shown (3) how to construct the constrained distribution from the t h e r m o d y namic equilibrium distribution of embryos in the s a t u r a t e d system, i.e., from a physically realizable embryo distribution. I n the present paper, we show t h a t the artifice of invoking a constrained equilibrium distribution is completely unnecessary. W e show how to convert the problem of predicting rates of nucleation from one t h a t requires 351
C o p y r i g h t ~ 1977 by A c a d e m i c Press, Inc. All r i g h t s of reproduction in a n y form reserved.
Journal of Colloid and Interface £cience, Vol. 61, No. 2, S e p t e m b e r 1977 I S S N 0021-9797
352
KATZ AND WIEDERSICH
knowledge of the distribution of cluster sizes in a supersaturated and not physically realizable state to one requiring the cluster size distribution for a true equilibrium state, the saturated vapor? (Equilibrium cluster size distributions are experimentally accessible in contrast to constrained distributions.) Kinetics and thermodynamics are clearly separated in the present treatment. We begin with a brief review of the usual version of nucleation theory to state clearly the notation and key ideas so that the innovation we have made becomes apparent. We then present the new version and show that while the final equation for the rate of nucleation has the same form as previously, the true driving force for nucleation is the condensation rate ratio fl/3, and not the supersaturation P/P j PREVIOUS VERSION OF NUCLEATION THEORY
In the usual versions of nucleation theory, it is assumed that clusters of molecules, called embryos, increase and decrease in size by acquiring or losing single molecules.4 The net rate at which clusters of size less than x + ½ become clusters of a size greater than x + ½ is
unit time; s(x) is the surface area of a cluster of x molecules; f(x, t) is the concentration of such clusters at time t; and 7(x) is the rate at which molecules evaporate per unit area from a cluster of size x. ]-For dilute vapors, fl can be accurately approximated by its ideal gas value, i.e., fl = aP/(27rmkT)~, where ~ is the condensation coefficient, P is the partial pressure of the condensing species, m is the mass of a molecule, k is Boltzmann's constant, and T is the absolute temperature.3 Note that the assumption has been made that 3'(x) depends on the size of the cluster but not oil the time history of the process nor on the density of the vapor. This is a major assumption of nucleation theory, although it usually is not explicitly stated. Nucleation theory then proceeds to obtain 7(x) in terms of "known" quantities by considering what happens to the system described by Eq. F1-] if the vapor is constrained to be at equilibrium. At equilibrium, J(x) is identically zero; solving Eq. [-11 for 7(x) one thus obtains for all x,
"r(x + 1) = ~(x)n(z)/
Fs(~ + a)n(x + ~)-], E2]
J ( x 27 1 l) = ~ s ( x ) f ( x , t)
-~(x+
1)s(x+ 1 ) / ( ~ + 1, t), Ill where instead of f(x,t) we have written n(x)
where/~ is the number of molecules that arrive and condense on a cluster per unit area per The arguments presented here are equally valid for any equilibrium state. Thus, any undersaturated state at the same temperature can be used as the reference. 3 There has been great controversy about the concentrations of clusters of various sizes (4). Since the same kind of arguments would hold for calculations of the distribution of clusters in a saturated vapor, this controversy is not directly the subject of this paper. Nevertheless, by eliminating the concept of supersaturation from calculations on the dependence of the cluster concentrations on thermodynamic quantities such as surface tension, and on rotation, translation, etc., this paper may be of some help in clarifying this controversy. Furthermore, equilibrium cluster concentrations can, in principle, be obtained from experiments. 4 Nucleation theory has been extended (11) to allow for the condensation and evaporation of dimers, trimers, and larger clusters from the embryos.
to indicate the concentrations of clusters of size x that exist in the hypothetical system, the one constrained to be in equilibrium although supersaturated. Substituting Eq. F2-] into Eq. F1-], it becomes
y(~ + ½, t) = ~s(z)~(~)[f(x, t)/n(~)
- f ( x + 1, t)/n(x + 1)-]. [-3-] Equation [-3-] cannot be solved directly since, even if the n(x) distribution were available, little is known about f(x, t). However, if one divides Eq. [-33 by ~s(x)n(x) and sums to a sufficiently large size Xl (the choice of value for x, is discussed later), the successive terms on the right-hand side will cancel. This procedure, devised by Becker and DSring (5), eliminates the f(x, t) distribution except at the limits at
Journal of Colloid and Interface Science, Vol. 61, No. 2, September 1977
353
NUCLEATION WITHOUT MAXWELL DEMONS
small and large sizes, with the result •,
J(x q- ½, t)
f(1, t)
f(xt + 1, l)
Z
one obtains
[43
Thermodynamic models for n(x) predict that n(x) increases without bound for large x. Because f(x, t), the concentration of embryos of size x, is necessarily finite, the limit of f(x, t)/n(x) goes to zero at large x. At steady state, if only a single nucleation path is available, ~ Y is not only time independent but also size independent and can be factored out of the summation. Doing so, one obtains the steady-state nucleation rate
J(x + ½, 0
](x, t)
js(~:)n.(x) ~°(x)
.,
J(x +
½,
t)
N E W V E R S I O N OF N U C L E A T I O N T H E O R Y
Let us reexamine Fq. [-1]. We need an expression for -/(x + 1), the rate at which molecules evaporate from a cluster. It is not necessary to postulate the existence of a supersaturated vapor constrained to be in equilibrium (by a Maxwell demon?) to be able to calculate "y(x + 1). We need only consider a salurated vapor? A saturated vapor is in equilibrium. Thus, the net rate of nucleation, J(x), will be equal to zero for all sizes. Denoting the concentrations of clusters of size x in a salurated vapor as n~(x), and the rate of condensation on a cluster in a saturated vapor as ¢/,, one obtains from Eq. [1]
v(x + 1) = ~.s(x)n.(x)/ [s(x + 1)n.(x + 173. E6] Note that we have retained the usual assumption that the evaporation rate -y(x) does not depend on the density of the vapor• Substituting Eq. [-6-] into Eq. [17 and rearranging, 5 This is n o t true if multiple p a t h s are available. Such can be the case in multicomponent nucleation or if the cluster shapes are taken into account. Reiss (12) has shown how to convert binary nucleation to a pseudo-one component (and therefore single path) system and Hirschfelder (13) has extended this m e t h o d to multicomponent systems.
E7]
f(1, t)
f(xl + 1, t)
n.(x~ + 1)(~/~e)x~+~" [8]
xl
x=l
.o(x + i)(~/~.)
The device of summing, as used to obtain Eq. [4], will not cancel successive terms since the last term includes a ~/¢~,. However, if we multiply both sides of Eq. [7-] by (~/¢~e)-x, then the cancellation of terms does occur, i.e.,
-
J = . f ( 1 ) [ ~ / . d ) V E Fs(~).(x)-I -~. I-5-]
f(x + L 0
As before, this equation is solved for a steady state3 Factoring J out of the summation of Fq. [8-] and using the argument that for sufficiently large x, f(x)/n,(x) must be of order unity, i.e., few embryos can have grown to large sizes during the time interval elapsed since establishment of the supersaturated state with/3/~, > 1, one finds Iim f(x)/[no (x) (fl/fl~) ~] = O, and obtains
J(x,) = f d ) ~ e / " e ( 1 ) ] / xl
Z [s(x)..(x)(~/~o?-]-'. [9]
x=l
This equation is similar to the previous versions of the equation for the rate of nucleation, e.g., Eq. [-5-]. In fact, the equations can be made identical if one assumes that the constrained equilibrium distribution is related to the cluster distribution in a saturated vapor by the equation
~(~) = ~o(~) (~/~e?.
UO]
6 A true steady-state nucleation process can be obtained in a real situation, e.g., in a diffusion cloud chamber (14) in which the supersaturated m o n o m e r concentration is maintained by m o n o m e r diffusion into the nucleation zone, and large clusters are removed from the zone by falling to the b o t t o m of the chamber•
Journal o] Colloid and Interface Science, Vol. 61, No. 2, September 1977
354
KATZ AND WIEDERSICH
'~ 10 8
WATER, 0°C
-
~3 l0 4
I0°
d =0
b ~. I ° 8
J: 22 cm- s-
£
10-12
,o-'°| o
Aj/J
,
~o
,
,~o
i I,BO I, ~00 I,I~20 140 ;11
eo
~60
, _2OO
~SO
NUMBER OF MOLECULES IN EMBRYO
Fzo. 1. Steady-state embryo size distributions for water at 0°C with a condensation rate ratio B/fie (assumed to be equal to the supersaturation P/P~) equal to 4. T h e p a r a m e t e r AY/Y gives the fractional deviation of the rate Y(xz) from the steady-state n u cleation rate for xz ~ ~ for a n u m b e r of cut-off sizes x~. T h e critical size ~ is 89 molecules. For comparison, the constrained equilibrium distribution is shown as the curve marked Y = 0.
Applying the same methods which are used to obtain the distribution of clusters n(x) in a supersaturated vapor (4) to a saturated vapor, one obtains ne(x) =
NC exp[---as(x)/kr-],
Ell]
where N is a normalization factor, C is a controversial correction which can be size dependent and large, a and a is the surface face energy per unit area. For ideal gas=s, and a pressure-independent condensation coefficient a, the condensation ratio 3/~, reduces to the supersaturation P/P,. Under such conditions Eqs. 1-10-] and [-117 can be combined to obtain {he usual version of the constrained distribution
To evaluate Eq. [-9], one can convert the summation to an integral, write the integrand as an exponential, expand the argument of the exponential in a Taylor series about the size 4 at which the argument's first derivative equals zero, and neglect terms higher than the quadratic term. 7 The quantity 4 is conventionally known as the critical size (though some definitions do not include the s(x) term). However, it has no fundamental significance other than that the largest contributions to the integral occur around this size. A more fundamental definition of critical size is that size at which the probability of a molecule leaving a cluster is equal to the probability of a molecule becoming attached to the cluster. At x > 4, the probability of a molecule becoming attached to the cluster. At x > 4, the probability of growing (adding a molecule) is larger than the probability of decaying and vice versa for x < 4. (All these definitions yield essentially the same value for 4.) The dependence of the solution obtained for the steady-state rate of nucleation on the cutoff size x~ will now be examined. From Eq. F9-] we obtain the steady-state rates J(m) for various xz. The corresponding embryo distributions can then be calculated from the recursion formula
i(-+
= .ol.
oI{_f ( x )
-
1, 1-13-1
which follows from Eq. ~-7~. A number of such steady-state distributions were calculated using water as a sample material (see Fig. 1). The n(x) = NC exp I-x In P/P. - os(x)/kr-]. [-12-1 distributions are almost identical, up to the I t is thus evident that a direct derivation of various cut-off sizes x~, to the distribution the constrained distribution n(x) from thermo- obtained for x~---~oo. The reason for the indynamic arguments is unnecessary and, more- sensitivity of the distributions to the value of over, requires certain restrictive assumptions. x~ is, of course, that the probability of fluctuaFurthermore, there is no need to assign any tions in size decreases rapidly with an increase physical significance to n(x). One should view in the size difference 2xx of the fluctuation, the condensation rate ratio fl/flo as the approThis h a s been done by K a t z (15) for a generalized priate driving force and use it rather than the nucleation equation which can include a n y weakly sizesupersaturation P/P, (to which fl/fle reduces dependent terms. T h e approximations involved have for ideal gases and a pressure-independent been examined in detail by Cohen (16) and are quite accurate. condensation coefficient). Journal of Colloid and Interface Science, Vol. 61, No. 2, September 1977
NUCLEATION WITHOUT MAXWELL DEMONS
because it is the product of the probabilities of losing (or gaining) zXx single molecules. Therefore, the effect of the absence of clusters with x > Xl does not travel for upstream. Also shown in Fig. i are the fractional changes in nucleation r a t e A J / J =~ [ J (xz) -- j ( , e ) ] / J ( o o ) caused by various finite cut-offs, xt. As can be seen, the location of xt, as long as it is large enough, does not significantly affect the calculated value of the rate of nucleation J(xz). The new view of nucleation presented in this paper becomes especially valuable when applied to more complicated nucleation problems. For example, consider the nucleation of " m a t t e r " in the presence of its "antimatter" such as the nucleation of voids in the simultaneous presence of vacancies and interstitials, as occurs in fast-breeder reactors where materials are exposed to energetic displacementproducing irradiation. The additional reactions with antimatter are included into the forward and backward reaction rates, e.g., the acquisition rate of interstitials is added to the rate of decay of void embryos by vacancy emission (6). In earlier papers, special constrained distributions or "effective" free energies of embryos were constructed (7). However, when attempts were made to extend this treatment to nucleation of voids in the presence of both interstitials and gas atoms, the use of binary nucleation theory (8) led to inconsistent results (9). The inconsistency was circumvented by a new way of constructing the constrained distribution (10), but the realization that these distributions were, in essence, not actually being used (3) is what led to this paper. CONCLUSIONS
We have shown that the constrained equilibrium distribution usually used in nucleation theory is an unnecessary artifice which results in the confusion of kinetic and thermodynamic effects. Nucleation theory is derived by considering the rates at which clusters grow and decay. As is the case in previous versions of nucleation theory, the rates of growth are related to the kinetic properties of the supersaturated vapor and the condensation coefficient. We have shown that the rates of decay
355
are related to the concentrations of clusters of various sizes that are present in an equilibrium vapor rather than to the hypothetical constrained distribution in the supersaturated vapor. As a result, the appropriate driving force for nucleation was shown to be the condensation rate ratio, 3/3~. REFERENCES 1. REISS, H., Zeitsch. ]~lectrochemie 56, 459 (1952). FEDER, J., RUSSELL, K. C., LOTttE, J., AND POUND, O. M., Adv. Phys. 15, 111 (1966). 2. McDoNALD, J. E., Amer. J. Phys. 31, 35 (1963). 3. WIEDERSlCIL H., in "The Physics of IrradiationProduced Voids" (R. S. Nelson, Ed.), AERER7934, p. 147, 1975. 4. LOTIt~, J., AND POUND, G. M., J. Chem. Phys. 36, 2080 (1962). REISS, H., ANDKATZ,f. L., dr. Chem. Phys. 46, 2496 (1967). LOTI~E, J., AND POUND, G. M., J. Chem. Phys. 48, 1849 (1968). A~RA~A~, F. F., ANDPOUND, G. M., J. Chem. Phys. 48, 732 (1968). REISS, H., KATZ,J. L., ANDCOHEn, E. R., f . Chem. Phys. 48, 553 (1968). REISS, H., J. Slat. Phys. 2, 83 (1970). BLANDER, M., AND KATZ, J. L., J. Stat. Phys. 4, 55 (1972). Kii~uem, R., J. Star. Phys. 3, 331 (1971). BuRTosr, ~. J., Acts Met. 21, 1225 (1973). NISI~IOI~A,K., ANDPOUND, G. M., Acts Met. 22, 1015 (1974). REISS, H., Advances in Colloid and Interface Science, Vol. 7, p. 1, 1977; KIxuCI¢I, R., ibid., p. 67; NISHIOKA, K., AND POUND, G. M., ibid., p. 205. 5. BECKJgl~, R., AND D6RING, W., Ann. Phys. 24, 719 (1935). 6. WIEDERSICt{,~I., ANDKATZ, J. L., in "Nucleation" (A. C. Zettlemoyer, Ed.), Voh 3, Advances in Colloid and Interface Science. Academic Press, New York, to appear. 7. K.~TZ, J. L., A?,~ WIEDERSICtI,H., ft. Chem. Phys. 55, 1414 (1971). RUSSELL, K. C., Acts Met. 19, 753 (1971). 8. LolL B. T. M., Acts Met. 22, 1305 (1972). 9. RUSSEL, K. C., Scripts Met. 6, 209 (1972). 10. WIEDERSICtt,H., KATZ, ~. L., AND BURTON, jr. f., J. Nud. Mater. 51, 287 (1974). 11. KATZ, J. L., SALTZBURG,H., A~D REISS, H., Y. Colloid Interface Sci. 21, 560 (1966), KATZ, J. L., AND BLANDER, M., J. Colloid Interface Sci. 42, 496 (1973). 12. REISS, H., J. Chem. Phys. 18, 840 (1950). 13. HI~SCtlFELDEIq J. 0., J. Chem. Phys. 61, 2960 (1974). 14. KATz, J. L., J. Chem. Phys. 52, 4733 (1970). KATZ, J. L., ScoPPA II, C. J., KU~AR, N. G., AND MIRABEL, P., dr. Chem. Phys. 62, 448 (1975). 15. KATZ, J. L., dr. Star. Phys. 2, 137 (1970). 16. COHESr,E. R., J. Star. Phys. 2, 147 (1970).
Journal of Colloid and Interface Science, Vol. 61, No. 2, September 1977
suffice) and break them into individual molecules t h a t are then available to form clusters, again b y r a n d o m agglomeration. Although words such as "hupothetieal p o p u l a t i o n " and "reference s t a t e , " which acknowledge the nonreality of this distribution, are frequently used, this distribution plays a central role in the theory. T h e r m o d y n a m i c methods are used to calculate the constrained distribution of cluster sizes despite its inherent instability. I n a recent p a p e r on defect cluster nucleation one of us (H.W.) has shown (3) how to construct the constrained distribution from the t h e r m o d y namic equilibrium distribution of embryos in the s a t u r a t e d system, i.e., from a physically realizable embryo distribution. I n the present paper, we show t h a t the artifice of invoking a constrained equilibrium distribution is completely unnecessary. W e show how to convert the problem of predicting rates of nucleation from one t h a t requires 351
C o p y r i g h t ~ 1977 by A c a d e m i c Press, Inc. All r i g h t s of reproduction in a n y form reserved.
Journal of Colloid and Interface £cience, Vol. 61, No. 2, S e p t e m b e r 1977 I S S N 0021-9797
352
KATZ AND WIEDERSICH
knowledge of the distribution of cluster sizes in a supersaturated and not physically realizable state to one requiring the cluster size distribution for a true equilibrium state, the saturated vapor? (Equilibrium cluster size distributions are experimentally accessible in contrast to constrained distributions.) Kinetics and thermodynamics are clearly separated in the present treatment. We begin with a brief review of the usual version of nucleation theory to state clearly the notation and key ideas so that the innovation we have made becomes apparent. We then present the new version and show that while the final equation for the rate of nucleation has the same form as previously, the true driving force for nucleation is the condensation rate ratio fl/3, and not the supersaturation P/P j PREVIOUS VERSION OF NUCLEATION THEORY
In the usual versions of nucleation theory, it is assumed that clusters of molecules, called embryos, increase and decrease in size by acquiring or losing single molecules.4 The net rate at which clusters of size less than x + ½ become clusters of a size greater than x + ½ is
unit time; s(x) is the surface area of a cluster of x molecules; f(x, t) is the concentration of such clusters at time t; and 7(x) is the rate at which molecules evaporate per unit area from a cluster of size x. ]-For dilute vapors, fl can be accurately approximated by its ideal gas value, i.e., fl = aP/(27rmkT)~, where ~ is the condensation coefficient, P is the partial pressure of the condensing species, m is the mass of a molecule, k is Boltzmann's constant, and T is the absolute temperature.3 Note that the assumption has been made that 3'(x) depends on the size of the cluster but not oil the time history of the process nor on the density of the vapor. This is a major assumption of nucleation theory, although it usually is not explicitly stated. Nucleation theory then proceeds to obtain 7(x) in terms of "known" quantities by considering what happens to the system described by Eq. F1-] if the vapor is constrained to be at equilibrium. At equilibrium, J(x) is identically zero; solving Eq. [-11 for 7(x) one thus obtains for all x,
"r(x + 1) = ~(x)n(z)/
Fs(~ + a)n(x + ~)-], E2]
J ( x 27 1 l) = ~ s ( x ) f ( x , t)
-~(x+
1)s(x+ 1 ) / ( ~ + 1, t), Ill where instead of f(x,t) we have written n(x)
where/~ is the number of molecules that arrive and condense on a cluster per unit area per The arguments presented here are equally valid for any equilibrium state. Thus, any undersaturated state at the same temperature can be used as the reference. 3 There has been great controversy about the concentrations of clusters of various sizes (4). Since the same kind of arguments would hold for calculations of the distribution of clusters in a saturated vapor, this controversy is not directly the subject of this paper. Nevertheless, by eliminating the concept of supersaturation from calculations on the dependence of the cluster concentrations on thermodynamic quantities such as surface tension, and on rotation, translation, etc., this paper may be of some help in clarifying this controversy. Furthermore, equilibrium cluster concentrations can, in principle, be obtained from experiments. 4 Nucleation theory has been extended (11) to allow for the condensation and evaporation of dimers, trimers, and larger clusters from the embryos.
to indicate the concentrations of clusters of size x that exist in the hypothetical system, the one constrained to be in equilibrium although supersaturated. Substituting Eq. F2-] into Eq. F1-], it becomes
y(~ + ½, t) = ~s(z)~(~)[f(x, t)/n(~)
- f ( x + 1, t)/n(x + 1)-]. [-3-] Equation [-3-] cannot be solved directly since, even if the n(x) distribution were available, little is known about f(x, t). However, if one divides Eq. [-33 by ~s(x)n(x) and sums to a sufficiently large size Xl (the choice of value for x, is discussed later), the successive terms on the right-hand side will cancel. This procedure, devised by Becker and DSring (5), eliminates the f(x, t) distribution except at the limits at
Journal of Colloid and Interface Science, Vol. 61, No. 2, September 1977
353
NUCLEATION WITHOUT MAXWELL DEMONS
small and large sizes, with the result •,
J(x q- ½, t)
f(1, t)
f(xt + 1, l)
Z
one obtains
[43
Thermodynamic models for n(x) predict that n(x) increases without bound for large x. Because f(x, t), the concentration of embryos of size x, is necessarily finite, the limit of f(x, t)/n(x) goes to zero at large x. At steady state, if only a single nucleation path is available, ~ Y is not only time independent but also size independent and can be factored out of the summation. Doing so, one obtains the steady-state nucleation rate
J(x + ½, 0
](x, t)
js(~:)n.(x) ~°(x)
.,
J(x +
½,
t)
N E W V E R S I O N OF N U C L E A T I O N T H E O R Y
Let us reexamine Fq. [-1]. We need an expression for -/(x + 1), the rate at which molecules evaporate from a cluster. It is not necessary to postulate the existence of a supersaturated vapor constrained to be in equilibrium (by a Maxwell demon?) to be able to calculate "y(x + 1). We need only consider a salurated vapor? A saturated vapor is in equilibrium. Thus, the net rate of nucleation, J(x), will be equal to zero for all sizes. Denoting the concentrations of clusters of size x in a salurated vapor as n~(x), and the rate of condensation on a cluster in a saturated vapor as ¢/,, one obtains from Eq. [1]
v(x + 1) = ~.s(x)n.(x)/ [s(x + 1)n.(x + 173. E6] Note that we have retained the usual assumption that the evaporation rate -y(x) does not depend on the density of the vapor• Substituting Eq. [-6-] into Eq. [17 and rearranging, 5 This is n o t true if multiple p a t h s are available. Such can be the case in multicomponent nucleation or if the cluster shapes are taken into account. Reiss (12) has shown how to convert binary nucleation to a pseudo-one component (and therefore single path) system and Hirschfelder (13) has extended this m e t h o d to multicomponent systems.
E7]
f(1, t)
f(xl + 1, t)
n.(x~ + 1)(~/~e)x~+~" [8]
xl
x=l
.o(x + i)(~/~.)
The device of summing, as used to obtain Eq. [4], will not cancel successive terms since the last term includes a ~/¢~,. However, if we multiply both sides of Eq. [7-] by (~/¢~e)-x, then the cancellation of terms does occur, i.e.,
-
J = . f ( 1 ) [ ~ / . d ) V E Fs(~).(x)-I -~. I-5-]
f(x + L 0
As before, this equation is solved for a steady state3 Factoring J out of the summation of Fq. [8-] and using the argument that for sufficiently large x, f(x)/n,(x) must be of order unity, i.e., few embryos can have grown to large sizes during the time interval elapsed since establishment of the supersaturated state with/3/~, > 1, one finds Iim f(x)/[no (x) (fl/fl~) ~] = O, and obtains
J(x,) = f d ) ~ e / " e ( 1 ) ] / xl
Z [s(x)..(x)(~/~o?-]-'. [9]
x=l
This equation is similar to the previous versions of the equation for the rate of nucleation, e.g., Eq. [-5-]. In fact, the equations can be made identical if one assumes that the constrained equilibrium distribution is related to the cluster distribution in a saturated vapor by the equation
~(~) = ~o(~) (~/~e?.
UO]
6 A true steady-state nucleation process can be obtained in a real situation, e.g., in a diffusion cloud chamber (14) in which the supersaturated m o n o m e r concentration is maintained by m o n o m e r diffusion into the nucleation zone, and large clusters are removed from the zone by falling to the b o t t o m of the chamber•
Journal o] Colloid and Interface Science, Vol. 61, No. 2, September 1977
354
KATZ AND WIEDERSICH
'~ 10 8
WATER, 0°C
-
~3 l0 4
I0°
d =0
b ~. I ° 8
J: 22 cm- s-
£
10-12
,o-'°| o
Aj/J
,
~o
,
,~o
i I,BO I, ~00 I,I~20 140 ;11
eo
~60
, _2OO
~SO
NUMBER OF MOLECULES IN EMBRYO
Fzo. 1. Steady-state embryo size distributions for water at 0°C with a condensation rate ratio B/fie (assumed to be equal to the supersaturation P/P~) equal to 4. T h e p a r a m e t e r AY/Y gives the fractional deviation of the rate Y(xz) from the steady-state n u cleation rate for xz ~ ~ for a n u m b e r of cut-off sizes x~. T h e critical size ~ is 89 molecules. For comparison, the constrained equilibrium distribution is shown as the curve marked Y = 0.
Applying the same methods which are used to obtain the distribution of clusters n(x) in a supersaturated vapor (4) to a saturated vapor, one obtains ne(x) =
NC exp[---as(x)/kr-],
Ell]
where N is a normalization factor, C is a controversial correction which can be size dependent and large, a and a is the surface face energy per unit area. For ideal gas=s, and a pressure-independent condensation coefficient a, the condensation ratio 3/~, reduces to the supersaturation P/P,. Under such conditions Eqs. 1-10-] and [-117 can be combined to obtain {he usual version of the constrained distribution
To evaluate Eq. [-9], one can convert the summation to an integral, write the integrand as an exponential, expand the argument of the exponential in a Taylor series about the size 4 at which the argument's first derivative equals zero, and neglect terms higher than the quadratic term. 7 The quantity 4 is conventionally known as the critical size (though some definitions do not include the s(x) term). However, it has no fundamental significance other than that the largest contributions to the integral occur around this size. A more fundamental definition of critical size is that size at which the probability of a molecule leaving a cluster is equal to the probability of a molecule becoming attached to the cluster. At x > 4, the probability of a molecule becoming attached to the cluster. At x > 4, the probability of growing (adding a molecule) is larger than the probability of decaying and vice versa for x < 4. (All these definitions yield essentially the same value for 4.) The dependence of the solution obtained for the steady-state rate of nucleation on the cutoff size x~ will now be examined. From Eq. F9-] we obtain the steady-state rates J(m) for various xz. The corresponding embryo distributions can then be calculated from the recursion formula
i(-+
= .ol.
oI{_f ( x )
-
1, 1-13-1
which follows from Eq. ~-7~. A number of such steady-state distributions were calculated using water as a sample material (see Fig. 1). The n(x) = NC exp I-x In P/P. - os(x)/kr-]. [-12-1 distributions are almost identical, up to the I t is thus evident that a direct derivation of various cut-off sizes x~, to the distribution the constrained distribution n(x) from thermo- obtained for x~---~oo. The reason for the indynamic arguments is unnecessary and, more- sensitivity of the distributions to the value of over, requires certain restrictive assumptions. x~ is, of course, that the probability of fluctuaFurthermore, there is no need to assign any tions in size decreases rapidly with an increase physical significance to n(x). One should view in the size difference 2xx of the fluctuation, the condensation rate ratio fl/flo as the approThis h a s been done by K a t z (15) for a generalized priate driving force and use it rather than the nucleation equation which can include a n y weakly sizesupersaturation P/P, (to which fl/fle reduces dependent terms. T h e approximations involved have for ideal gases and a pressure-independent been examined in detail by Cohen (16) and are quite accurate. condensation coefficient). Journal of Colloid and Interface Science, Vol. 61, No. 2, September 1977
NUCLEATION WITHOUT MAXWELL DEMONS
because it is the product of the probabilities of losing (or gaining) zXx single molecules. Therefore, the effect of the absence of clusters with x > Xl does not travel for upstream. Also shown in Fig. i are the fractional changes in nucleation r a t e A J / J =~ [ J (xz) -- j ( , e ) ] / J ( o o ) caused by various finite cut-offs, xt. As can be seen, the location of xt, as long as it is large enough, does not significantly affect the calculated value of the rate of nucleation J(xz). The new view of nucleation presented in this paper becomes especially valuable when applied to more complicated nucleation problems. For example, consider the nucleation of " m a t t e r " in the presence of its "antimatter" such as the nucleation of voids in the simultaneous presence of vacancies and interstitials, as occurs in fast-breeder reactors where materials are exposed to energetic displacementproducing irradiation. The additional reactions with antimatter are included into the forward and backward reaction rates, e.g., the acquisition rate of interstitials is added to the rate of decay of void embryos by vacancy emission (6). In earlier papers, special constrained distributions or "effective" free energies of embryos were constructed (7). However, when attempts were made to extend this treatment to nucleation of voids in the presence of both interstitials and gas atoms, the use of binary nucleation theory (8) led to inconsistent results (9). The inconsistency was circumvented by a new way of constructing the constrained distribution (10), but the realization that these distributions were, in essence, not actually being used (3) is what led to this paper. CONCLUSIONS
We have shown that the constrained equilibrium distribution usually used in nucleation theory is an unnecessary artifice which results in the confusion of kinetic and thermodynamic effects. Nucleation theory is derived by considering the rates at which clusters grow and decay. As is the case in previous versions of nucleation theory, the rates of growth are related to the kinetic properties of the supersaturated vapor and the condensation coefficient. We have shown that the rates of decay
355
are related to the concentrations of clusters of various sizes that are present in an equilibrium vapor rather than to the hypothetical constrained distribution in the supersaturated vapor. As a result, the appropriate driving force for nucleation was shown to be the condensation rate ratio, 3/3~. REFERENCES 1. REISS, H., Zeitsch. ]~lectrochemie 56, 459 (1952). FEDER, J., RUSSELL, K. C., LOTttE, J., AND POUND, O. M., Adv. Phys. 15, 111 (1966). 2. McDoNALD, J. E., Amer. J. Phys. 31, 35 (1963). 3. WIEDERSlCIL H., in "The Physics of IrradiationProduced Voids" (R. S. Nelson, Ed.), AERER7934, p. 147, 1975. 4. LOTIt~, J., AND POUND, G. M., J. Chem. Phys. 36, 2080 (1962). REISS, H., ANDKATZ,f. L., dr. Chem. Phys. 46, 2496 (1967). LOTI~E, J., AND POUND, G. M., J. Chem. Phys. 48, 1849 (1968). A~RA~A~, F. F., ANDPOUND, G. M., J. Chem. Phys. 48, 732 (1968). REISS, H., KATZ,J. L., ANDCOHEn, E. R., f . Chem. Phys. 48, 553 (1968). REISS, H., J. Slat. Phys. 2, 83 (1970). BLANDER, M., AND KATZ, J. L., J. Stat. Phys. 4, 55 (1972). Kii~uem, R., J. Star. Phys. 3, 331 (1971). BuRTosr, ~. J., Acts Met. 21, 1225 (1973). NISI~IOI~A,K., ANDPOUND, G. M., Acts Met. 22, 1015 (1974). REISS, H., Advances in Colloid and Interface Science, Vol. 7, p. 1, 1977; KIxuCI¢I, R., ibid., p. 67; NISHIOKA, K., AND POUND, G. M., ibid., p. 205. 5. BECKJgl~, R., AND D6RING, W., Ann. Phys. 24, 719 (1935). 6. WIEDERSICt{,~I., ANDKATZ, J. L., in "Nucleation" (A. C. Zettlemoyer, Ed.), Voh 3, Advances in Colloid and Interface Science. Academic Press, New York, to appear. 7. K.~TZ, J. L., A?,~ WIEDERSICtI,H., ft. Chem. Phys. 55, 1414 (1971). RUSSELL, K. C., Acts Met. 19, 753 (1971). 8. LolL B. T. M., Acts Met. 22, 1305 (1972). 9. RUSSEL, K. C., Scripts Met. 6, 209 (1972). 10. WIEDERSICtt,H., KATZ, ~. L., AND BURTON, jr. f., J. Nud. Mater. 51, 287 (1974). 11. KATZ, J. L., SALTZBURG,H., A~D REISS, H., Y. Colloid Interface Sci. 21, 560 (1966), KATZ, J. L., AND BLANDER, M., J. Colloid Interface Sci. 42, 496 (1973). 12. REISS, H., J. Chem. Phys. 18, 840 (1950). 13. HI~SCtlFELDEIq J. 0., J. Chem. Phys. 61, 2960 (1974). 14. KATz, J. L., J. Chem. Phys. 52, 4733 (1970). KATZ, J. L., ScoPPA II, C. J., KU~AR, N. G., AND MIRABEL, P., dr. Chem. Phys. 62, 448 (1975). 15. KATZ, J. L., dr. Star. Phys. 2, 137 (1970). 16. COHESr,E. R., J. Star. Phys. 2, 147 (1970).
Journal of Colloid and Interface Science, Vol. 61, No. 2, September 1977