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Langmuir 1992,8, 1868-1877
Nucleation in Condensation Don H. Rasmussen,* Ming-Tsai Liang, Etop Esen, and Mary R. Appleby Department of Chemical Engineering, Clarkson University, Potsdam, New York 13699 Received September 10, 1990. In Final Form: April 13, 1992
Nucleation in condensation is defined to occur by local fluctuations within a supersaturated system. These density fluctuations are sufficient for local volume elements to increase from the supersaturated parent state density to the density at the spinodal boundary. If io monomers are required to form a nucleus, then, for 6io monomers in a fluctuation, spinodal decomposition to (6 - l)iovapor atoms of parent density and io atoms of a single liquidlike cluster may occur when 6iou,= (6 - l)iovp+ iou,. 6 is a multiplier determined by conservation of matter, and vi is the appropriate specific volume. If the io atoms in the liquid cluster are in unstable equilibrium with the parent vapor, nucleation will have occurred. A capillarity approximation which depends on the density difference across the interface (Le., a MacleodSugden capillarity) and the classical Kelvin equation provide a measure of the size of fluctuation, 6io monomers, which must occur to yield a nucleus. Fluctuation theory provides a measure for the rate of occurrence of density fluctuations containing 6io monomers from the parent phase density to the spinode boundary. Spinodal decomposition and completion time theory provides a measure of the fraction of fluctuations which survive long enough to decompose to a droplet within a supersaturated parent phase during an experimentalobservation. The nucleation rate is determined by application of all three concepts, and this theory is supported by the best experimental data available on nucleation during condensation.
I. Introduction Nucleation is defined here as the occurrence of a density, concentration, or order fluctuation in a small volume of a supersaturated system which, as a temporarily isolated subsystem, decomposes by spinodal decomposition into two phases in local equilibrium. The nucleation process is complete when the subsystem opens to the supersaturated parent phase at constant pressure and the nucleus which has formed is able to grow. The fluctuation theory of Landau and Lifshitzl describes the number of fluctuations per unit time per unit volume of parent phase. The process of spinodal decomposition is described by the theory of Cahn and Hilliard.2 The fraction of the fluctuations which decompose and ripen to a single nucleus during the lifetime of the fluctuation is described by the completion time theory of Langer and S ~ h w a r t z .Nu~ cleation is the product of all three processes and is complete when isolation of the subsystem and cluster is removed from the rest of the supersaturated parent phase and the cluster is stable and capable of growth. This theory is a "once over the barrier" theory. There is no need for a permanent population of subcritical unstable clusters through which diffusion of nuclei occurs. Once a fluctuation has yielded a cluster, it is either stable or unstable. It will evaporate or grow depending upon its size. There is no consideration of growth by monomer addition in determining the kinetics of nucleation. While any one fluctuation may achieve cluster growth along the classical pathway as opposed to an Ostwald ripening of a finer cluster distribution which started as spinodal decomposition, no consideration of the classical critical complex is necessary to arrive at a kinetic expression for the nucleation rate. The classicalpathway is very unlikely for a subsystem which reaches the spinodal boundary. The volume to undergo density fluctuation must be large enough for the resulting new phase to be stable within the supersaturated parent phase. The size of the required density fluctuation is determined by the combination of the Kelvin relation applied to the original supersaturated
parent phase and conservation of mass for the fluctuation from the spinodal boundary to the cluster and residual supersaturated parent phase. For application to real systems, an equation of state4or solution model and a suitable theory of capillarity5 are required. This information is available for describing nucleation during condensation of simple vapors, nucleation of bubbles in superheated simple liquids, and nucleation of a new liquid phase within a supersaturated regular solution. The development of the energetics and kinetics of the nucleation process is given in what follows. The kinetics are then compared with experimental results from the literature. Real gases5 undergo pressure-volume fluctuations in the form of pressure or density waves. The fluctuations to be described here in terms of small subsystems or enclosed pistons can be considered to represent volume elements which are compressed during the onset of a pressure wave, held compressed for an amount of time determined by the wavelength and velocity of the pressure wave, and then reexpanded to the parent phase stateduring rarefaction. In three-dimensional systems, nodes of high pressure may occur with large amplitude fluctuations in density and/or long lifetimes. For such localized density fluctuations, the idealized description to follow may be reasonable. 11. A Cluster May Exist in Either Stable or Unstable Equilibrium Consider the vapor-liquid system presented in Figure 1.
In (A), this system consists of a piston which is pinned to maintain the enclosed volume constant. The chamber is filled with a vapor which is just at equilibrium with bulk liquid, though no liquid is present in the chamber. Assume that there is no gravity in the environment or vessel, the walls are nonwetting, and the pressure in the environment is adjustable. The following process takes place. In (B),a hypodermic needle is inserted into the volume element and a drop of radius R is formed on the end of ~~
(1) Landau, L. D.; Lifshitz, E. M. Statistical Physics;Pergamon Press:
London, 1958; pp 344-408. (2) Cahn, J. W.; Hilliard, J. E. J. Chem. Phys. 1959,31, 688-699. (3) Langer, J. S.; Schwartz, A. J. Phys. Reu. A 1980,21, 948.
~
(4) Modell, M.; Reid, R. C. Thermodynamics and Its Applications, 2nd ed.; Prentice Hall: Englewocd Cliffs, NJ, 1983; p 239. ( 5 ) Reid, R. C.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1966; pp 372-388.
Q743-7463/92/2408-1868$03.00/0 0 1992 American Chemical Society
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7.
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D.
,
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Figure 1. Figure for thought problem no. 1.
Figure 2. Figure for thought problem no. 2.
the needle and the needle is quickly removed, suspending the drop in the center of the volume element. Because this drop has a curved interface and a higher equilibrium vapor pressure than the pressure at bulk phase equilibrium, the drop will begin to evaporate. This evaporation will increase the pressure in the vapor space, and depending upon the size of the vapor space and the initial drop radius, either the drop will disappear completely or evaporate to a smaller radius, re, where stable phase equilibrium will be achieved between the drop and the vapor phase. Any necessary heat transfer is permitted to take place with the reservoir to keep the system isothermal. Equilibrium is achieved for a drop containing io monomers at a higher vapor phase pressure, P,,, than the bulk phase equilibrium pressure,Pe,and the system equilibrium is completely stable. In (C),the exterior environment is adjusted in pressure to equal the pressure within the enclosed volume, PS,,= P,,e.This pressure adjustment in no way affects the stable equilibrium which exists within the volume element which contains the drop and supersaturated vapor. Since this pressure adjustment in the exterior space does not affect either the drop or the vapor, the constancy of Gibbs free energy and equality of chemical potential which exists at stable global equilibrium is unaffected. If the droplet were to evaporate one monomer, the increase in pressure within the gas phase would be greater than the increase in vapor pressure due to the change in the surface curvature and the free energy of the system would increase. If one more monomer were to condense, the decrease in the gas phase pressure would be greater than the decrease in the liquid vapor pressure due to curvature and the free energy of the system would increase. Finally, in (D), the pin which held the piston in place is removed. The removal of the pin is the removal of one constraint from the system (constant volume) and the establishment of a new constraint (constant pressure). This action does not change the temperature, pressure, or potentials of either the vapor phase or the droplet phase, and these two phases remain in local equilibrium with each other. However, the nature of the equilibrium has changeddramaticallyasthetwophasesarenowinunstable equilibrium. If an atom or monomer were to condense onto the droplet, the growth of the droplet would decrease its vapor pressure, and in a constant vapor phase pressure environment, the probability of further growth would be enhanced. On the other hand, if the droplet were to evaporate an atom initially, its radius of curvature would decrease and by the Kelvin relation its vapor pressure would increase above the constant vapor phase pressure. This would move the piston so as to enlarge the enclosed vapor space, and the probability of further evaporation would be enhanced. Stable global equilibrium changed to unstable local equilibrium by the change of constraints on the system. It was not necessary to make any changes in the temperature, pressure, or potentials of either of the phases within
the system. Therefore, it was not necessary to move from the constraints of stable thermodynamic phase equilibrium, AP = 0, AT = 0, AG = 0, and Ap = 0, to describe unstable phase equilibrium. It was also not necessary to incorporate a work of formation of the droplet in describing its phase equilibrium. The curvature of the interface increased the Gibbs free energy per atom in the drop over that of bulk liquid the same amount as the increase in gas pressure increased the Gibbs free energy per atom of the gas phase over that of the gas in equilibrium with bulk liquid. The equilibrium of a global system is defined by the constraints on that system. Phases within the global system, however, always obey the Constraints on phase equilibrium, and if the global system Constraints constrain the phases to be in equilibrium, then phase equilibrium also exists. However, when the constraints on the global system change, local phase equilibrium still exists but may become unstable. The response to changes in global system constraints determineswhether or not nucleation has taken place.
111. Nuclei Are in Unstable Equilibrium with the Parent Phase Consider the following variation on the above thought problem for the vapor-liquid system presented in Figure 2.
In (A), this system consists of a chamber which has a diathermic wall immersed in a reservoir at temperature TR and a frictionless piston pinned to maintain the enclosed volume constant with the vapor compressed to the same pressure which was present in step C of the first thought problem. Assume that there is no gravity in the environment or vessel, that the vessel walls are nonwetting and that no liquid exists initially. The following process takes place. In (B), a second smaller piston is placed within the large piston and it is closed and pinned around a volume element which contains 6iomonomers at the same specific volume as that of the supersaturated parent phase, V,,. The magnitude of io is determined by the local equilibrium to be established between a nucleus and the supersaturated vapor as existed in the original thought problem. The Kelvin equation applied to the original vapor phase supersaturation and a capillarity model for the interface will determine the numerical value of io. 6 is a multiplier determined from a lever rule application to an equation of state such that the volume change to occur in part C, AV, turns out to equal 6io( V ,- V,) where V, is the specific volume of the vapor at the spinodal limit to vapor phase stability. In (C), the large piston is unpinned and a further fluctuation in the total volume of the system, AV (the magnitude of which was determined in (B)above), takes place by a slight movement of the large piston and this piston is repinned at the newer smaller volume. The change in volume is infinitesimal in comparison to the total volume of the whole system so that the supersatu-
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1870 Langmuir, Vol. 8, No. 7, 1992
ration does not change appreciably. The work done on the system, P,AV, is one measure of the work required to create the fluctuation. Finally, in (D), the pin which holds the small subsystem piston in place is removed and the small subsystemvolume is allowed to fluctuate. A fluctuation which changes the small subsystem volume by A Vwill bring the parent system back to the parent phase supersaturation, and the subsystem will be compressed to ita spinodal limit. This is not a probable process, and only a small fraction of all fluctuations will so concentrate the vapor in the subsystem. Following such a fluctuation, the pin is replaced in the piston for the subsystem and the subsystem volume is thus fixed at the spinodal limit for the subsystem vapor. Once the spinodal boundary is achieved within the subsystem, there is no barrier to the spontaneousformation of a drop of liquid and drop formation is considered to occur by spinodal decomposition and Ostwald ripening, though a single drop could also form by cluster growth by monomer addition along the classical pathway. This description of nucleation does not eliminate the possibility of the classical process generating the drop. It is a matter of probability as to which part process occurs and a matter of semantics as to which process one might prefer in the description of the mechanism. Enough time elapses during the spinodal decomposition process for ripening to leave but one cluster centered in the small volume element or for one droplet of maximum size to have grown by the classical process of growth by monomer addition. If the classical process has occurred, the cluster has grown larger than the classical critical nucleus and has achieved equilibrium with the vapor in the subsystem. This cluster size is the Ostwald cluster size for the subsystem. The fraction of spinodal fluctuations which actually achieve a single liquid drop surrounded by supersaturated parent phase vapor in time t is given by the completion time theory. The kinetics will be discussed later. For now, it is sufficient that a single equilibrium drop in the subsystem at the supersaturated parent vapor pressure is achieved. As in the original thought problem, the drop is in complete equilibrium with the vapor in the small subsystem as long as the pin is retained in the piston which separates the subsystem from the supersaturated parent phase. (Actually, all droplets formed from subsystems of different sizes will be internally in equilibrium with the vapor in their respective subsystems, but drops of different sizes will require different final equilibrium vapor pressures, and only drops equal to or larger than the Ostwald cluster will become nuclei to the external supersaturated parent phase.) The nucleation event will only be completed when a window is opened between the subsystem and the parent system, and only if this change in constraint leads to growth of the dorplet. A Maxwell's demon could open a window between the subsystem and the parent system as easily as pulling the pin on the piston. The result will be the same. The constraint of constant volume will change to a constraint of constant pressure. Drops larger than io will grow, and drops smaller than io will evaporate. There is no significance to the classical critical size whatsoever in this treatment of nucleation. Nucleation in condensation occurs by density fluctuations within a supersaturated system which decompose by spinodal decomposition or classical cluster growth into liquid drops and a remnant parent phase. For 6io monomers in a density fluctuation, a spinodal decomposition to (6 - l ) i o vapor atoms of parent density and io atoms of a liquidlike cluster is possible when iouB= (6 l)ioup + iOul. If the io atoms in the liquidlike cluster are in unstable equilibrium with the parent vapor as determined by a common tangent construct on a free energy
versus molar volume diagram for a system containing io monomers, nucleation will have occurred. The above description of nucleation is in violation of the conditions specified by Gibbs6 in his determination of the work of formation of a fluid of different phase within any homogeneous fluid. Gibbs required that the outer of the two phases separated by a spherical interface be infiiite in extent and at constant pressure. He also noted,6'When the interior mass and the surface of discontinuity are formed entirely of substances which are components of the surroundingmass, the equilibrium is always unstable... Thus, the equilibrium of a drop of water in an atmosphere of vapor is unstable ..." (emphasis added). Note that, for the constant volume subsystems above, a droplet of water would be completely stable and in equilibrium, and therefore, the limitations that Gibbs placed on his definition of an equilibrium interface as existing only between bulk phases with infinite radii of curvature are not correct when considering the nucleation of a new phase within a supersaturated parent phase, especially within a constant volume system. With the help of a cubic equation of state for bulk systems, we7 developed a modified equation of state for small systems in which two phases are separated by a spherically curved interface. The modified equation of state requires that the effect of the interface be accounted for in the determination of the properties of the interior phase. The interior of the drop to form in a small system can only interact with the surrounding parent phase through the spherical interface. Equilibrium in the small system can be established when the enhanced vapor pressure of the drop is equal to the increased pressure in the surrounding vapor. Since a fluid interface does not contain a structure capable of supporting a pressure gradient, the pressure inside the droplet will be achieved by a balance of expansion and contraction along the bulk fluid isotherm combined with the positive Laplacian pressure from the interface. The total internal pressure will be the same as the enhanced vapor phase pressure outside the equilibrium Ostwald drop-again, because of the inability of the interface to support a pressure gradient. This equilibrium pressure will depend on the interface curvature and will be given by the classical Laplace and Kelvin equations. The critical complex in the classical theory of nucleation is generally treated as a new phase particle, but according to this analysis, the classical critical complex (asubsystem compressed to the spinodal boundary) does not satisfy the necessary conditions of thermodynamic equilibrium with the supersaturated parent phase. The nucleus in the current model does and is, in fact, larger in size than the classical critical complex. The nucleus or Ostwald cluster fulfills the necessary conditions of either stable or unstable equilibrium in the supersaturated vapor; Le., AT = 0, AP = 0, AG = 0, and A p = 0. For the small subsystem when isolated from the rest of the parent phase, the Ostwald cluster is at a minimum in the Gibbs free energy with respect to the size of the condensed cluster, not at a maximum as for the classical critical cluster at constant pressure. The above thought problems indicate that stable and unstable equilibria both exist at the same location on the molar change in the Gibbs free energy surface but differ in the nature of the global system constraints. That is, there can be no difference in the Gibbs free energy (6) Gibbs, J. W. The Collected Works of Josiah Willard Gibbs. Vol. I. Thermodynamics; Longmans Green & Co.: New York, 1931; pp 219274. (7) Rasmussen, D. H.; &en,
E.;Appleby, M. R. Atmospheric Aerosols and Nucleation. In Lecture Notes in Physics 309; Wagner, P. E., Vali, G., Eds.; Springer Verlag: Berlin, 1988, pp 438-441.
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Nucleation in Condensation 3.6 L
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Volume (cc/mole) Figure 3. Helmholtz and Gibbs potentials as a function of volume for the Peng-Robinson EOS (T= 0.9Tc).
between the nucleus and the supersaturated parent phase from which it forms. This position on the change in the Gibbs free energy surface, the product of the size of the cluster and the molar Gibbs free energy change, is not at the maximum or saddle point where the classical critical nucleus exists but at a minimum in AG versus i, where AG = 0, for a constant volume subsystem or at the point of crossover of the change in free energy from positive to negative with respect to size, where AG = 0, for a constant pressure parent phase. Phase equilibrium exists on the change in the free energy surface where AG = 0. The constraints on the system determine whether the equilibrium is stable or unstable.
IV. Energetics of Cluster Formation A Peng-Robinson fluid4 will be used to illustrate the theory though similar results have been obtained for a van der Waals fluid. Any ternary equation of state (EOS) would give similarresults. The fluid modeled by the PengRobinson EOS is like nonane. Figure 3 is a plot of the Helmholtzpotential and the Gibbs potential for the PengRobinson fluid at a temperature of 0.9Tc. The reference state is the ideal gas at 1 atm of pressure. The PengRobinson equation of state is4 RT QT (1) = - (u(u + b) + b(u - b ) ) where p is the pressure, R is the gas constant, T is the absolute temperature, u is the volume, aTis a temperaturedependent constant, and b is a constant under all conditions. UT and b depend on the molecular structure of the fluid. The Helmholtz potential of the Peng-Robinson fluid is given by4
E
u + b(1- 2 1 / 2 9 ( 2 ) A = R T h ( F ) +-ln( QT u - b) 2(2b)’/2 u + b(l + 2’12) The Gibbs potential for the Peng-Robinson fluid is
G = A + p~ = A - u(dA/du) (3) The common tangent to the Helmholtz potential defines the equilibrium gas and liquid specific volumes for the fluid at this temperature. The slope of the common tangent is equal to the negative of the equilibrium pressure, and the common tangent extrapolates to a zero volume intrinsic chemical potential which is equal to the partial molar Gibbs free energy in the equilibrium fluids. Note from Figure 3 that the Gibbs free energy is the same for the two equilibrium fluids and that the maximum and minimum in the Gibbs free energy correspond to the spinodal points of the Helmholtz free energy. This can be
2.6
Molar V o l u m e
(cc/mole)
Figure 4. Helmholtzand Gibbs potentials for the modified PengRobinson EOS (T= 0.9Tc,i = 430, V, = 1.6 X 10-3m3/m0l.
observed by equating the derivative of eq 3 with respect to volume to zero. Figure 4 is a plot of the Helmholtz potential and the Gibbs potential for a modified Peng-Robinson EOS for a cluster containing 430monomers at a parent phase molar volume of 1600 cm3/mol, at a temperature of 0.9Tc. The reference state is the ideal gas at 1 atm of pressure. The modified equation of state (MEOS)for fluctuations within a supersaturated parent Peng-Robinson fluid which contains i monomers is given by
where p’ and u’ are the pressure and molar volume in the cluster which contains i monomers, R is the gas constant, T is the absolute temperature, and UT and b are the constants of the Peng-Robinson EOS. d(us/i)/du‘, the Laplacian contribution to the pressure within the enclosed phase, is evaluated from the derivative of the product of the surface area separating the fluctuated volume from the parent volume, s,and the surface free energy per unit area, u, as determined from the Macleod-Sugden5 correlation. Gibbs’ assumption is equivalent to making u constant so that only the geometric derivative of s with respect to u’ remains. The Macleod-Sugden correlation5for the liquid-vapor interface is &4 = [PI(P’ - P”) (5) where [PIis the parachor, a molecular structure dependent parameter relating the surface tension to the difference in fluid phase densities, p’ = l/u’ and p” = 1/u”. The Helmholtz potential of the modified EOS is given by
where i is the number of monomers in the second phase cluster and u’ is the specific volume of this phase. The common tangent to the modified Helmholtz potential is tangent at the parent phase specific volume and at the equilibrium cluster specific volume and extrapolates to a zero volume intrinsic chemical potential which is equal to the partial molar Gibbs free energy at the cluster specific volume and the parent phase specific volume. The modified Gibbs free energy of the fluctuated region containing i monomers is G(i,u’) = A(i,u’) + p’u’ = A(i,u’) = u’(d A(i,u’)/du’) (7) According to Figure 4,the criteria for phase equilibrium, AG = 0, A p = 0, AT = 0, and AF’ = 0, exist for the
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1872 Langmuir, Vol. 8, No. 7,1992
430-monomer cluster at a specific volume of VI= 2.45 X 10"' m3/mol and the supersaturated parent phase at a specific volume of Vp = 1.6 X m3/mol. The equilibrium pressure for the bulk fluids is 13.5 atm, the equilibrium specific volume of the gas Vv,e= 1.982 X 10-3 m3/ mol, and the equilibrium specific volume of the liquid Vlte = 1.92 X m3/mol. The equilibrium for the cluster and supersaturated parent phase is stable or unstable depending on the constraints of the system whereas the two bulk phases are in stable thermodynamic equilibrium. In a constant parent phase pressure environment, growth of the cluster will reduce the system free energy and lower the chemical potential of the cluster below that of the parent phase. On the other hand, if the cluster were to fluctuate to a smaller size, the system free energy would be increased and both the chemical potential and the vapor pressure of the cluster would be increased over the chemical potential and pressure in the supersaturated vapor. The increase in the system free energy does not outweigh the increase in the vapor pressure of the cluster. The change in free energy with cluster size around the Ostwald size is given by AG(r) = n dp' = v' dp' = nRT In (p,/p&
3. P o
w 0
Figure 5. Free energy surface that fluctuations must traverse or spinodally decompose to reach the nucleus at 245 cm3/mol and AC = 0.
(8)
where Pr is the vapor pressure of a cluster of radius r and po is the vapor pressure of the Ostwald cluster of radius r,, and v' is the volume of the cluster. For radii smaller than r,, AG(r) is positive and Pris larger than p,, the vapor pressure of the Ostwald cluster which is equal to the parent phase pressure. Nucleation occurs not at the classical critical complex size but at the larger Ostwald cluster size. The Ostwald nuclei are at the same thermodynamic pressure, -(dA/dV),I =-(dA/dV)a, as the supersaturated parent phase. When one creates a curved interface at the equilibrium bulk phase pressure, there will always exist a pressure differential attributable to the elastic tension of the interface as given by the classical Laplace equation. However, when the parent phase is compressed to a specific volume such that the supersaturation is sufficientto restore equilibrium with the curved interface separating the two phases, the pressure differential across the interface disappears. Only clusters with spherical morphology are possible because only one radius of interface curvature is permissible for unstable equilibrium between the fluid nucleus and the fluid supersaturated parent phase. All other cluster radii will still experience a Laplacian pressure in the supersaturated vapor, though now both positive andmegative Laplacian pressures are possible. For any real supersaturation, all clusters which have a higher free energy, chemical potential, or vapor pressure than the Ostwald cluster are unstable to evaporation. That is, all clusters smaller than r, have a higher probability of disappearing by evaporation than of growing by condensation, and this includes the classical critical complex. V. Change in Free Energy Surface For the modified equation of state, the state variables are pressure, temperature, volume, and cluster size. For a system at constant temperature and given parent state pressure, the independent variables can be cluster specific volume and cluster size. The change in the Gibbs potential for cluster formation asa function of cluster specificvolume and cluster size is presented in Figure 5. The potential surface has been calculated for the modified Peng-Robinson EOS at a parent phase molar volume of 1600 cm3/ mol and at a temperature of 0.9Tc. Note that the parent state is a low flat valley on the left of this figure (all cluster sizes have the same specific volume in the parent state) and that the classical saddle point can be observed on the
Figure 6. Change in free energy per atom on cluster formation. Note the lack of a saddle point versus cluster size.
right side of this figure as the maximum in the modified Gibbs potential of the liquidlike cluster valley. The Ostwald cluster contains 430 monomers, and it lies in the trough where AG equals zero. The classical pathway from the parent state to the Ostwald cluster is around the back of the free energy surface and through the saddle point. The model of nucleation advocated here consists of the fluctuation of a group of ai, atoms or monomers within a small subsystem from the parent phase to the spinodal boundary. This model has an activation energy which is related to the height of the inflection in the ridge of free energy per atom at specific volumes intermediate between those of the parent supersaturated vapor and the liquid state. The magnitude of the fluctuation in free energy per atom remains almost constant; see Figure 6. The rate of nucleation will be proportional to the number of atoms or monomers in the fluctuation and the magnitude of the fluctuation in free energy per atom to reach the spinodal limit. VI. Kinetics of Nucleation If the spinodal decomposition and ripening process within the small subsystemis fast compared to the lifetime of the fluctuation, the kinetics of the nucleation process
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will be described by the rate of occurrence of suitable large fluctuations in subsystems. The fluctuation theory of Landau and Lifshitzl provides a first approximation for the rate of fluctuation of 6io monomers to the spinodal boundary. Since these spinodal processes are diffusion controlled and the spinodal boundary is a series of inner critical points, the process of decomposition will have a characteristic completion time which is temperature dependent. The completion time, 7 , and the fraction of fluctuations which achieve single droplet formation in time t have been treated by Langer and S c h ~ a r t zThe . ~ scaling of this completion time and the scaling of the correlation length on which it is based have been treated by Langer and Turski.8 A combination of these treatments of phase transformation kinetics yields our theory of nucleation. The model of nucleation presented here is similar to that advocated by Lovettsas the first passage over a barrier. It is also similar in some respects to the kinetic theory of nucleation advanced by Ruckenstein.loJ1 The fluctuation which is sufficient to generate an equilibrium Ostwald cluster in an isolated subsystem is sufficient to generate a nucleus in the supersaturated parent phase. Such a fluctuation may be considered to be very improbable because it has a higher total free energy barrier than that of the classical critical complex in the classical theory (note the difference between the maximum in the Gibbs free energy depending on whether or not the fluctuation is by monomer addition (passage through the saddle point on the right-hand side of Figure 5 ) or by density fluctuation (passage from the vapor phase on the left over/ through the hill at intermediate specific volumes to the droplet on the right-hand side of Figure 5). The barrier to fluctuation to the spinodal boundary obviously disappears as the initial supersaturation approaches the mechanical spinodal. Therefore, this mechanism of nucleation is expected to dominate at high supersaturation. Furthermore, the classical mechanism of nucleation requires that constant pressure exists at all times during the clustering process and this constant pressure requirement demands substantial uphill diffusional flows to exist if the cluster growth is rapid as these flows must run counter to the higher vapor pressure of the clusters smaller than the Ostwald cluster; this pertains even to the classical critical cluster. These flows may not be attainable without additional local fluctuations in supersaturation of the parent vapor. The activation energy is determined by the Ostwald cluster size, io, and the magnitude of the modified equation of state fluctuation in free energy per atom for the constant volume densification from the supersaturated parent state to the spinodal boundary, Up,.Following the approach of Landau and Lifshitz,’ the probability of formation of the Ostwald cluster by density fluctuation of a group of ai,, monomers becomes
J = Q exp(-6ioAAp,/kZ‘) exp(-t/T)
(9)
where the normalizing coefficient, Q, is a parameter describing the total number of fluctuations per unit time per volume of supersaturated parent phase. Q will be discussed later. The first exponential term describes the fraction of the fluctuations which are large enough in size to yield a stable nucleus on complete spinodal decomposition and ripening. The second exponential describes the fraction of these fluctuations which decompose to (8) Langer, J. S.;Turski, L. A. Phys. Reu. A 1973,8, 3230-3243. (9) Lovett, R. J . Chem. Phys. 1984,81, 6191. (10) Ruckenstein, E.;Nowakowski, B. J . Colloid Interface Sei. 1990, 137,583-592. (11) Narsimhan, G.;Ruckenstein, E. J. Colloid Interface Sci. 1989, 128, 549-565; Erratum. J . Colloid Interface Sci. 1989, 132, 289.
stable nuclei in time t . iois the totalnumber of monomers in the stable Ostwald cluster. 6 is a multiplier found by applying conservation of mass and the modified equation of state to the spinodal decomposition process within the subsystem. AAw is the necessary fluctuation in the Helmholtz free energy per atom to locally compress 6io atoms from the supersaturated parent state to the local critical complex condition from which spinodal decomposition into an Ostwald cluster and supersaturated parent vapor can spontaneously occur. The magnitude of AA,, is given by where V,is the molar volume and P,is the pressure at the spinodal boundary and Ppis the pressure in the parent phase. By using the modified equation of state, the influence of the generation of a new interface as the cluster forms is accountedfor in the calculation. 7 is the relaxation time for the local spinodal decomposition process. This process has been modeled by Langer and Turski8 and Langer and Schwartz3for spinodal decomposition near the critical point, and their scaling law will be used in the following. There is reason to believe that this scaling behavior occurs along the line of inner critical points, i.e., along the spinodal boundary, since the instability in the vapor phase is similar. First, the free energy barrier to reach the spinodal boundary with a large enough fluctuation is proportional to the inverse cube of the logarithm of the supersaturation. This results from the choice of the equilibrium nature of the Ostwald cluster and the application of the Kelvin equation to this size cluster. The “free energy barrier to the fluctuation”, 6ioAAps,scales in temperature as given in eq 11. a is a proportionality constant, T, is the reduced
temperature, and S is the supersaturation, P/P,.Exactly the same scaling behavior is obtained with the van der Waals equation of state combined with the Macleod-Sugden5 correlation. Only the numerical constant, a, is affected by the differences in the equations of state. Numerical results for a Peng-Robinson fluid with parameters for nonane are plotted versus the scaling law expectation inFigure 7. The barrier height, 6ioAAp8,was evaluated for a critical cluster size of 100 atoms or monomers over the temperature range from 0.95Tcto 0.4Tc. The proportionality constant depends on the equation of state and surface parameters for a specific material. To apply the proposed theory to the temperature-supersaturation dependence of the nucleation rate, the rest of the terms in the nucleation rate expression, eq 9, must be evaluated. a, the preexponential factor, is a measure of the number of density fluctuations per unit time per unit volume. Langer and Turski8 developed a hydrodynamic model of the process where the fluctuations decompose spinodally without a free energy barrier to fluctuation size. They estimate the statistical prefactor to be Q = Q,(l- T,)3.57where Q, = This prefactor is similar to the estimates of the prefactor in classical nucleation theory except for the temperature dependence. The second exponential term in eq 9 is the probability that any one fluctuation will last for a sufficient time, t , to decay to a single droplet, or if the fluctuation is spatially large, that it will last long enough for the droplets which form spinodally to grow by Ostwald ripening to the critical size. The relaxation time for the spinodal decomposition process is 7 . The method of experimental measurement limits t to some definite value, and therefore only a fraction of all possible large fluctuations actually yield nuclei. The
Rasmussen et al.
1874 Langmuir, Vol. 8, No. 7, 1992
ratio of the relaxation time for the decay of fluctuations to yield nuclei from the spinodal decomposition and ripening process to the experimental observation time, t, is given by Langer and Schwartz3 (eq 3.15, ref 3) as
7/t = Dx?/ 24C2 (12) where D is diffusivity, is the correlation length, and x,, a measure of the original supersaturation, is given by (eq 2.18, ref 3) 4[~[~/kT,]'/~ (13) where u is the surface tension, k is Boltzmann's constant, and T, is the critical temperature. Langer and Turski8 (eqs 7.3 and 7.4, ref 8) note the following scaling behavior for [ and u:
100 lo-'
c
C
h
x,
= [,(l- T,)-# u = ~ ~ ( T,)2J 1 (14) where toand uo are constants. The diffusivity, D, is assumed to scale in temperature along the line of inner critical points inversely to the scaling of the correlation length. Combining these scaling laws
10-7 10-7 10-6
SI-^] x
exp[-8(1- TJ21 (16) Classical nucleation theory, on the other hand, predicts that the logarithm of the nucleation rate should be a linear function of the inverse square of the logarithm of the supersaturation ratio as given by
J = A, e x p [ - ( 1 6 ? r / 3 ) ( ~ ~ ~ / ( ~ ~ ~ ) ( l(17) n Insertion of the Eotvos relation yields12 J=A,exp[-(?)(~) ke 3 l-T, 3 (18)
(7 IS')) -^]
where A, is the classical preexponential factor, ke is the Eotvos constant, and S is the supersaturation.
VII. Fit to Nucleation Data The two supersaturation dependencies are compared in Figure 8. The data from Katz et al.,13J4Adams et a l . , l 5 and Wagner and Strey16 for the condensation of nonane at 238 K cover the widest range of rates and supersaturation available. Each investigator determined the nucleation rate over a range of supersaturation at 238 K, and the total data cover a range of nucleation rates from (12) Rasmueeen, D. H.; Babu, S. V. Chem. Phys. Lett. 1984,108,449452. (13) Katz, J. L.; Hung, C.-H.;Krasnopoler, M. Atmospheric Aerosols and Nucleation. In Lecture Notes in Physics 309; Wagner, P. E., Vali, G., E&.; Springer-Verlag: Berlin, 1988; pp 356-359. (14) Katz, J. L. J. Chem. Phys. 1970, 52, 4733. (15)Adams, G. W.; Schmitt, J. L.; Zalabsky, R. A. J . Chem. Phys. 1984,82, 5074. (16) Wagner, P. E.; Strey, R. J . Phys. Chem. 1984,80, 5266.
10-5 10-4
10-3 10-2
/ Tr ' Figure 7. Gi,AA, versus (1 - Tr)5/T;L. (1
10.75 2'50
10-1
100
101
-Tr)5
5
9.00
-
7.25
5.50
7 ~
J = Q,(l-T,)3.57 ex,[ -a( q ) ( l n
10-4
10-6
v
7/t = /3-'(1- TJ3" (15) where is a constant related to the intrinsic diffusivity, [D], surface tension, u,, correlation length, to,and critical temperature, T,. The scaling exponent 'v theoretically must equal or exceed 2/d where d is the dimensionality of the system even though experimentally 'v has been noted to equal 0.62 for condensation of C02.8 Combining eqs 9-15 gives the following expression for the nucleation rate of a Peng-Robinson fluid by localized spinodal decomposition of density fluctuations into Ostwald nuclei and supersaturated parent phase vapor:
10-2
Y
\ 10-3
3.75
0
- 2.00 0.25 -1.50 -3.25
.
-5.00' " " " " " 1 0.10 0.14 0.18 0.22 0.26 0.30 0.34 0.38 0.42 0.46 0.50 "
"
(A) In(S)-3
"
(8)ln(S)-2
Figure 8. Data (238 K) from Katz et al.,13J4Adams et al.,16and Wagner and Strey16versus (In and (In S)-2.
to lo9. A linear regression of the data approximately against both (In and (In S)-2gives excellent fits to the data. Figure 8 demonstrates that the data fit both functionalities with excellent correlation, the (In SI4 functionality has a Pearson's correlation of 0.99, and the (In functionality has a Pearson's correlation of 0.985. The data deviate from the (In correlation more than from correlation, with a systematic curvature, low the (In at both ends, high in the middle. The middle set of data is noted to average about 1 K colder than the external data seta, but this should have lowered this set relative to the outer seta instead of raised it. Thus, the curvature and the deviation may be just part of the variation in and between the experimental data seta. The temperature dependence of the slope of the log J versus (In S)(-3Or -2) is a better test of the theory. Katz,13 on analyzing the complete data set on nonane, has already noted "One sees that classical theory results in deviations from experimentthat range from lo*'to almost 105."Figure 9 presents the overall fit to the fluctuation treatment from eq 16 as log J vs (log S)-3 in the following form: J = 1026.5(1- T,)3.57
x
exp[ -54.12( ?)(log
S)-'] exp[-5.76(1- T,)-2] (19)
The data of Katz et al.13J4for rates between lo4 and 10 nuclei/(cm3/s) are for temperatures from right to left of 315,299,285,273,268,258,248,238, and 233 K. The data of Adams et al.15 for rates between 2 X lo2and lo5 are for temperatures from right to left of 266,257,248,237,228, and 217 K. The data of Wagner and Streyls are for rates between lo6 and lo9at temperatures of 238,219, and 203 K. The theory does an excellentjob of fitting the composite
Langmuir, Vol. 8, No. 7, 1992 1875
Nucleation in Condensation .
I
.
I
.
I
.
8
.
I
.
!
.
I
.
13 11
9 7
1 -1 -3 -5
0.0
0.3
0.6
0.9
1.2
1 .5
1 .8
2.1
2.4
2.7
3.0
0.0
0.5
Figure 9. All data from Katz et al.,13J4 Adams et al.,lSand Wagner and Streyle versus (In S)-3.
data at all temperatures though there is again some variation between the fit to each experimental data set which may be due, presumably, to measurement errors and data reduction. It appears that the preexponential factor, Q,, remains constant over the entire range of the data at 1028.5. Katz13 also noted that the total pressure affects the nucleation rate of nonane in the diffusion cloud chamber, and this could not be explained by classical nucleation theory because the supersaturation and the bulk planar surface tension of the capillarity approximation are not functions of total gas pressure. These results are explainable with the present model because of the effect of noncondensable compression of the nonbulklike liquid clusters. The gas phase density of the supersaturated parent phase does not change significantly on the addition of a bath gas, but if this bath gas is noncondensable, then compression of the cluster will increase the density difference across the interface and the barrier to nucleation as affected by a higher surface tension for the Ostwald cluster will increase the required size of the Ostwald cluster for the same supersaturation of the parent vapor. This increase in the Ostwald cluster size will decrease the nucleation rate at constant supersaturation. Three sets of data on the nucleation of water droplets from the vapor are available which cover the range of nucleation rates from 1to 1Olo nuclei/(cm3/s); see Miller et al.,17Anderson et al.,18and Wagner and Strey.lg These sets of data are presented in Figure 10 along with the theory prediction according to the relationship
[
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
log(s)-3
Iog (s)-3
J = 10'8.5(l- Tr)3.57 exp -0.82 ((1 ;F5)(lOg
1.0
V ]x
The data sets overall fit the form of the present model though there is considerable discrepancy within any one set. Further consideration of the equation for the nucleation rate will be required to determine if the numerical values are consistent with the known values for the parachor and equation of state. There are additional experimental data sets which are inconsistent with the temperature dependence of the nucleation rate predicted by the classical theory of (17) Miller, R. C.; Anderson, R. J.; Kassner, J. L., Jr.; Hagen, D. E. J. Chem. Phys. 1983, 78,3204-3211. (18) Andereon, R. J.; Miller, R. C.; Kassner, J. L., Jr.; Hagen, D. E. J. Atmos. Sci. 1980,37, 2508-2520. (19) PWagner, P. E.; Strey, R. J . Phys. Chem. 1981, 85, 2694-2698.
Figure 10. Nucleation rate of water as predicted by eq 20 and
measured by Miller et al.,17Anderson et al.,18 and Wagner and Strey.19
nucleation. Peters and Paikert2OPz1have determined the temperature dependence of the supersaturation at which anucleation rate of 108is observed in shock tube expansions of water, methanol, ethanol, and propanol. These data cover a larger temperature range than that of the above data sets on condensation of water in cloud chambers and indicate that a higher supersaturation can be obtained at high temperatures than that predicted by the classical theory. In the case of methanol, Peters and Paikert attempt to explain this effect as being due to liberation of the heat of association to dimers, trimers, and tetramers in the expanding vapor. This extra heat liberation would result in a higher final temperature of expansion than predicted by the transport equations used to describe the expansion. Since this final temperature is not measured but calculated, this correction to the data is only qualitative. The anomalous temperature dependence of the supersaturation as experimentally determined may be explained by the present nucleation model and the Binder and Stauffer22argument that completion times are longer near critical points. In addition the correlation length increases and the effective diffusion ~ o e f f i c i e n t ~ ~ decreases as temperature rises toward the critical point. Since the above model of nucleation incorporates completion time analysis, eq 16 has been rewritten in the following form to compare with Peter and Paikert's data for water, methanol, ethanol, and propanol:
P = lop,
-a((l- TrI5/T3
[log (J/Q,(l - T1)3.57) + o(1-
1ii3
(21)
where Q, is the preexponential factor and a and /3 are constants which are material dependent. Figure 11 compares the results of eq 21 with Peters and Paikert's data for water with the coefficients 52, = 1018.5,a = 0.82, and j3= 2.2, as determined from the fit to the composite set of data in Figure 10. The theory is also consistent with Peters and Paikert's22 results for methanol, ethanol, and propanol as demonstrated in Figures 12-14. The coefficients for these material are as follows: for methanol, Q, = 1020,a = 0.196, and j3 = 2.65; for ethanol, Q, = 1020, a = 0.355,and /3 = 2.52; and for propanol, Q, = 1020,a = 1.2, and @ = 2.5. The fit for methanol is poor at low temperatures but excellent at high temperatures. Even the fit to the anomalously (20) Peters, F.; Paikert, B. Exp. Fluids 1989, 7,521-530. (21) Peters, F.; Paikert, B. J . Chem. Phys. 1989,91, 5672-5678. (22) Binder, K.; Stauffer, D. Ado. Phys. 1976,25, 343. (23) Kang, K.; Redner, S. Phys. Rev. A 1985, 32, 435-447.
Rasmussen et al.
1876 Langmuir, Vol. 8, No. 7,1992 lo-'
10-4
......................
........................ -
10-6
................................
R .............
.............Q?
...................................................
// ......................................................
........, 6..........
+ I "
I
-
0.25 0.27 0.29 0.31 0.33 0.35 0.37 0.39 0.41 0.43 0.45
0.35 0.37 0.39 0.41 0.43 0.45 0.47 0.49 0.51 0.53 0.55
REDUCE0 TEMPERATURE
REDUCED TEMPERATURE
Figure 11. Reduced pressure at which nucleation is observed Figure 14. Reduced pressure at which nucleation is observed in shock tube expansionsof water (data of Peters and Paikert20*21). in shock tube expansions of propanol (data of Peters and Paikert20lz1). 2
lo-' K
3
t2 E a B 10-4
!lo-'
a
w u10-4
W
n K
u 3
........................ fl...........-p.:....................
....................
~
n 3
K
B E
2
10-6
10-5 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58
10-6
REDUCED TEMPERATURE
Figure 12. Reduced pressure at which nucleation is observed in shock tube expansions of methanol (data of Peters and Paikert20,21). 1c-3
............,c.s?............*:................A 4 ..................................................... .&o: ........... ....... ...... .........
w
....................
ad
/
2 1 0-4 d v)
................................... @V..........y.:........./...... ....................... ........................... .... .&Q?......qdy ... .........................
5an.
,c
2 1 0-5
-
d
1
%
2
13-6
I
.
I
'
"
"
"
"
'
"
"
"
"
'
0.15 0.19 0.23 0.27 0.31 0.35 0.39 0.43 0.47 0.51 0.55
0.35 0.37 0.39 0.41 0.43 0.45 0.47 0.49 0.51 0.53 0.55 REDUCED TEMPERATURE
Figure 13. Reduced pressure at which nucleation is observed in shock tube expansions of ethanol (data of Peters and Paikert20.21). high supersaturation a t high temperature is excellent. The poor fit at low temperatures may be real or it may be due to an inaccurate vapor pressure function. The results were obtained here with the Harlacher equation and parameters given in the Appendix to the paper by Reid, Prausnitz, and S h e r w ~ o d .The ~ anomalously high supersaturation at high temperatures is theoretically caused by the onset of completion time limits to achieving nucleation as the supersaturation approaches the spinodal boundary. An excellent test of the new theory would include measurement of the supersaturation to constant rate of nucleation in the high temperature regime where each rate of nucleation will undergo dispersion in the supersaturation a t a different reduced temperature. Lower nucleation rates will require higher reduced temperatures for the observation of the onset of completion time limits to the experimentally measured nucleation rate.
REDUCED TEMPERATURE
Figure 15. Reduced pressure at which nucleation is observed in shock tube expansions of argon (data of Wegener et al.24925).
A final data set which is inconsistent with classical nucleation theory is that of Wegener et al.24*25 for the temperature dependence of the supersaturation to constant nucleation rate for argon condensation. The composite data of Wegener et al. are presented in Figure 15 along with the theoretical curve for 0, = 1020,a = 5.33, and 0 = 3.03. In all of the above cases, determining the exact correspondence between the fluctuation theory of nucleation and the data will require theoretical evaluation of the constants of the nucleation rate expression, Q,, a,and 0, from the material parameters which determine them.
VIII. Discussion The foregoing analysis replaces the state of equilibrium used in classical nucleation theory and in classical capillarity with a state of equilibrium between the nucleus or Ostwald cluster and the supersaturated parent phase in a small constant volume system. As a result, the critical complex of classical theory at the maximum in the free energy versus size is not the nucleus because it is unstable to evaporation. The nucleus or Ostwald cluster, which is located at the point of equality of the Gibbs potential with the supersaturated vapor, is in unstable condensationevaporation equilibrium in a constant pressure system. The barrier to nucleation is related to achieving extremum conditions at the spinode for a region containing a multiple of io monomers. This complex can spinodally decompose to the Ostwald cluster and parent vapor in stable or unstable equilibrium. If the system were of the size of the fluctuation which would just create the Ost(24) Wegener, P. P.; Mirabel, P. Natuurissenschaften 1987, 74,111-
119. (25) Wu, B.J. C.; Wegener, P. P.; Stein, G. D. J. Chem. Phys. 1978, 69,1776-1777.
Nucleation in Condensation
Langmuir, Vol. 8, No. 7,1992 1877
wald cluster, then this cluster would be absolutely stable and in equilibrium with the vapor in the constant volume system. However, in real systems the Ostwald cluster is created in a larger volume; therefore, the phase equilibrium is unstable because the global system will act like the constant pressure piston in the hypothetical system of thought problem 2. The use of a Macleod-Sugden capillarity permits the development of a theory of nucleation which is internally consistent and consistent with experimental results. The theory will require refinement in terms of material constant evaluation, but the fit to Figures 10-15 is a substantial improvement in describing nucleation data over that possible with classical nucleation theory. This theory was developed on a variation of the Gibbsian approach. The droplet was considered to be a phase which incorporates the effects of the interface which separates the droplet from the surrounding parent phase. The properties of both phases were used to describe the state of the interface, and the "bulk" of the droplet phase was given freedom to expand andlor contract in order to minimize the total system free energy. The classical capillarity approximation does not allow such independent bulk phase behavior for the droplet liquid because an equilibrium liquid can only exist in contact with a planar interface and an equilibrium bulk vapor at a constant pressure within both the liquid and the vapor. This model of nucleation is based on the assumption of independent fluctuations in density or supersaturation. The lifetime of any one short-range fluctuation must be sufficient to permit spinodal decomposition to an Ostwald cluster. Binder26has modified the Cahn-Hilliard theory to incorporate such random thermal fluctuations but limited his analysis to decomposition within the spinodd. Here, the short-range fluctuation must decay along other pathways at a rate slower than that of decay by spinodal decomposition and Ostwald ripening for the process of nucleation to be observed. At high supersaturation a t high temperatures, the independence of the fluctuations is questionable because of the increase in the coherence length and the onset of completion time limitations to the ripening processes. It may, under these circumstances, be possible to have fluctuations large in extent which yield many new phase particles from a single fluctuation. Such phenomena will destroy the stochastic character of the nucleation process and merge homogeneous nucleation with nonlinear spinodal decomposition. The nucleation rate calculated from this treatment is a rate of appearance of supercritical fluctuations, not a rate of appearance of new phase particles. At low supersaturations and nucleation rates, the number of particles should equal the number of supercritical fluctuations. The modifications to the energetics of nucleation which have been used in this analysis should also be derivable from van der Waals capillarityz7where the density of the droplet is considered a continuous function of the radius (26) Binder, K. J. Chem. Phys. 1983, 79,6387. (27) Rowlinson, J. S.; Widom, B . Molecular Theory Clarendon Press: Oxford, 1982.
of
Capillarity;
starting from bulk liquid in the core of the droplet and decreasing through the interface to the density of the surrounding vapor. However, it is not readily apparent how the interior droplet density can be permitted to vary to achieve equilibrium for droplets in small closed constant volume systems when the chemical potential is defined in terms of a reference state for bulk liquid. The bulk liquid density is the equilibrium density only for an infinite radius of curvature for the droplet. Once this modification of the boundary conditions are made to the van der Waals treatment, the results presented here should be derivable in a more rigorous fashion. This treatment of nucleation permits both homogeneous and heterogenous nucleation to occur, as does classical theory. In addition, heterogeneous nucleation will be expected to be affected by external fields and by heterogenousfluctuational phenomena. Dynamic effects which can reduce the free energy of formation of the unstable extremum complex or concentrate the decay of an initial fluctuation which was large in extent but small in degree can now also affect the rate of nucleation. Fluctuations which can be stabilized to longer times such as fluctuations in the vicinity of damping surfaces, etc., may also catalyze nucleation. It is not necessary that all heterogeneous effects be expressible in terms of a factor which is a geometric function of the Ostwald cluster size and shape. Of course, all colligative effects on the Ostwald nucleus will still catalyze or depress the rate of nucleation. These concepts applied to nucleation during condensation are also applicable to evaporation or boiling nucleation, liquid-liquid phase separation, and nucleation of crystals from the melt. The Gibbsian capillarity approximation with its restriction to bulk liquid behavior has significantly limited the quantitative application of classical nucleation theory to these transformations. The Macleod-Sugden approximation simply insists that the cluster be treated as an independent fluid or a material with the same interactions as in the parent phase from which it forms and that the interfacial energy be a function of the difference in density across the interface. With this capillarity approximation, nucleation merges with spinodal decomposition at the spinodal boundary since the nucleation process is a localized spinodal decomposition within a fluctuation which results in a heterophase Ostwald cluster. The stability of the cluster depends on its size and on the parent phase supersaturation. At the spinodal boundary, the free energy barrier to the formation of the smallest possible Ostwald cluster disappears, and inside the spinodal boundary there is no barrier to nucleation. Acknowledgment. We thank Professor John Schmitt for furnishing the raw data of Adams et al., Professor Joseph Katz for furnishing the data for Katz et al., Professor Franz Peters for furnishing reprints and data of Peters and Paikert, and Professor Dim0 Kashiev for helpful discussions concerning the theory. This work was sponsored in part by a grant from the Army Research Office (DAAG29-85-K-0196) and in part by a grant from the National Science Foundation (DMR-8605556). M.R.A. thanks NASA for Training Grant 33-011-800.
