nucleation symposium
HOMOGENEOUS
nucleation from the vapor phase
24
INDUSTRIAL AND ENGINEERING C H E M I S T R '
RONALD P. ANDRES
In the second paper from the Nucleation Symposium, Dr. Andres examines in detail those aspects o f nucleation which are concerned with the important subject o f condensation from the vapor in rapid expansions and other systems where homogeneous nucleation is to be expected. The classical theories of nucleation are compared with data from cloud chamber, nozzle, and molecular beam experiments. The theories assume that condensation takes place via random growth of minute clusters of vapor molecules gas is cooled in the absence of nucleating When catalysts, it will sustain a substantial supera
saturation before embryos of condensed phase appear. Once nuclei of liquid or solid are present, however, the gas does not remain supersaturated but rapidly relaxes toward equilibrium. These facts imply the existence, in a pure vapor, of an appreciable activation barrier which must be overcome before phase transformation can proceed. The nucleation process in such a gas takes place via the random growth of minute condensed aggregates or clusters of the vapor phase molecules. To understand its rate, something about the properties of these microscopic species must be known. This paper, therefore, reviews the classical theories of nucleation with regard to the assumptions made concerning clusters. Predictions of the theory are then compared with nucleation data obtained in cloud chambers, nozzle flows, and molecular beams. Nucleation Mechanism in a Gas
The random process by which embryos of condensed phase are formed can be treated as a series of chemical reactions between a set of cluster species, A,, where the subscript i denotes the number of single molecules in the cluster. The central problem in such a formalism is, of course, the assignment of mean rate constants or transition probabilities to the various reactions. The clusters are all present initially in very small numbers, and, usually to a good approximation, reactions between two clusters can be neglected compared with reactions between clusters and single molecules. The nucleation mechanism, therefore, is that of an infinitely long chain reaction, Ai f Ai f M e A 2 M Ai A2 A3 Ai -43 A4
+ +
+
The origin of the activation barrier is most easily
seen in terms of the thermodynamic properties of a macroscopic embryo. Following Gibbs (2ir), the equilibrium work required to form a spherical embryo of i molecules is ( I I ) :
+
A% = y47rr,2 i(poond - PI) (1) poond(T,Pi) = chemical potential of a molecule of the condensed phase p l ( T , P I ) = chemical potential of a vapor molecule y = surface free energy per unit area of the embryo 113
= radius of the embryo u = molecular volume of the bulk condensed phase This expression exhibits a maximum at a certain critical size given by the Gibbs-Kelvin equation
PI
2yv
p w
rN
kTln-=-
(2)
Pi = pressure of vapor molecules P, = vapor pressure of bulk condensed phase and the work of formation of this critical embryo is : AON =
16a7'u2
3 [kT In (P1/Pm>l2
(3)
If for the moment Equation 1 is assumed to hold for even the smallest clusters, rN defines a critical species, called the nucleus, in metastable equilibrium with the vapor at a given supersaturation. Clusters smaller than this are unstable a.nd will tend to evaporate, while larger clusters are progressively more stable and will tend to grow. Thus, AQN/kT represents an activation barrier which must be surmounted for condensation to proceed. I n terms of our linear chain mechanism this means that for a given supersaturation tbe probability of a forward reaction is less than that of a backward reaction until the nucleus forms. After the nucleus forms, however, the probability of a condensation reaction is greater than that of evaporation, and the small embryos tend to grow to macroscopic size. VOL. 5 7
NO. 1 0 OCTOBER 1 9 6 5
25
RONALD P. ANDRES
In the second paper from the Nucleation Symposium, Dr. Andres examines in detail those aspects of nucleation which are concerned with the important subject of condensation from the vapor in rapid expansions and other systems where homogeneous nucleation is to be expected. The classical theories of nucleation are compared with data from cloud chamber, nozzle, and molecular beam experiments. The theories assume that condensation takes place via random growth of minute clusters of vapor molecules hen a gas is cooled in the absence of nucleating
W catalysts, it will sustain a substantial supersaturation before embryos of condensed phase appear.
Once nuclei of liquid or solid are present, however, the gas does not remain supersaturated but rapidly relaxes toward equilibrium. These facts imply the existence, in a pure vapor, of an appreciable activation barrier which must be overcome before phase transformation can proceed. The nucleation process in such a gas takes place via the random growth of minute condensed aggregates or clusters of the vapor phase molecules. To understand its rate, something about the properties of these microscopic species must be known. This paper, therefore, reviews the classical theories of nucleation with regard to the assumptions made concerning clusters. Predictions of the theory are then compared with nucleation data obtained in cloud chambers, nozzle flows, and molecular beams. Nueleation Mechanism in a Gas
The random process by which embryos of condensed phase are formed can be treated as a series of chemical reactions between a set of cluster species, A,, where the subscript i denotes the number of single molecules in the cluster. The central problem in such a formalism is, of course, the assignment of mean rate constants or transition probabilities to the various reactions. The clusters are all present initially in very small numbers, and, usually to a good approximation, reactions between two clusters can be neglected compared with reactions between clusters and single molecules. The nucleation mechanism, therefore, is that of an infinitely long chain reaction, Ai Ai M e A n M Ai Az e Aa Ai Aa e A4
+ +
+ + . . . . .
+
The origin of the activation barrier is most easily
seen in terms of the thermodynamic properties of a macroscopic embryo. Following Gibbs (21),the equilibrium work required to form a spherical embryo of i molecules is ( I I) :
+
~ 4 n : i(Peond - I ~ S (1) chemical potential of a molecule of the condensed phase r l ( T ,P,)= chemical potential of a vapor molecule y = surface free energy per unit area of the embryo r, = 111 = radius of the embryo AClS
=
had( T, Pi) =
4s3)
(
v = molecular volume of the bulk condewed phase
This expression exhibits a maximum at a certain critical size given by the Gibbs-Kelvin equation
Pi = pressure of vapor molecules P_ = vapor pressure of bulk condensed phase and the work of formation of this critical embryo is: (3)
If for the moment Equation 1 is assumed to hold for even the smallest clusters, rN defines a critical species, called the nucleuq, in metastable equilibrium with the vapor at a given supersaturation. Clusters smaller than this are unstable and will tend to evaporate, while larger clusters are progressively more stable and w i l l tend to grow. Thus, ACIN/kT represents an activation barrier which must be surmounted for condensation to proceed. I n terms of our linear chain mechanism this means that for a given supersaturation the probability of a forward reaction is less than that of a backward reaction until the nucleus forms. After the nucleus forms, however, the probability of a condensation reaction is greater than that of evaporation, and the small embryos tend to grow to macrmopic size. VOL 57
NO. 1 0 O C T O B E R 1 9 6 5
25
A very similar equation suggested by Courtney (73) can be obtained from Equation 6 by approximating the configurational integral by the contribution of (i - 1) molecules of the bulk condensed phase times a correction for the surface free energy of a spherical embryo of i molecules. In this case x is replaced by P,/kT. Recently additional more detailed approximations have been proposed. Kuhrt (34) and Dunning (73, working from an equation such as Equation 5, separated from Q, easily calculated contributions due to the translational and rotational motion of the cluster as a whole. The remainder of the partition function was then approximated by the contribution of (i - 2) molecules of the bulk condensed phase times the usual macroscopic surface correction. Lothe and Pound (36) and Hirth (29), on the other hand, argued that, rather than eliminating the complete bulk contribution of two molecules, correction should be made for only six vibrational modes characteristic of the bulk phase. In essence this is equivalent to multiplying the result of Kuhrt by b2AH"m'RTwhere AHv.n is the molar enthalpy of vaporization. In all these treatments the configurational integral has been approximated by terms characteristic of the bulk condensed phase times a correction based on the
physical clusters important in homogeneous nucleation can be determined for the largest embryos by means of Equation 1 and can be rigorously estimated for the dimer by means of the calculations of Stogryn and Hirschfelder. In between these extremes, an extrapolation formula based on the thermodynamic properties of the bulk phase must be assumed for want of a better approximation. A comparison between the various thermodynamic extrapolations discussed is shown in Table I for the case of argon at 85' K. Insofar as the assumptions used in obtaining Equation 7 are valid for these small clusters, it appears that both the expressions of Courtney and of Kuhrt and Dunning, which are almost the same as the "classical" equilibrium constant of Volmer and Weber, are better than they have any right to be in this region. The result of Lothe and Pound and Hirth, on the other hand, predicts q's that are appreciably tm high. Oriani and Sundquist (40) have pointed out that this latter formula also predicts the unphysical result that clusters of up to eight molecules are more stable than the monomer for a typical water supersaturation observed by Volmer and Flood (.a). Kinetics of Nucleation
The rate at which single molecules react with a cluster species is given by the usual gas kinetic expresion multiplied by a sticking probability a
wt+,., = probability per unit time and concentra-
tion of A, of a transition from AI to A,+,
m
= mass of a single molecule
For large it,a becomes equal to the accommodation coefficient a. for condensation on the bulk condensed phase. For want of a better approximation, this value is also used for the small clusters. The formation of a dimer, however, requires a triple collision, and the transition probability of this reaction will be much smaller than Equation 9. Similarly for the very smallest Figure 7. The gect of ttmpnahrrc on the cqdibriwn conrtant for clusters, a is probablysmaller than acbecause theseclusthe famation of dimer embryos in the UaPOr P h e ' . . terSare unable to absorb the energy of condensation. ..nrawollF1~&?,,?,:; -The backward reaction probabilities are found from a surface free energy of a macroscopic spherical embryo. consideration of microscopic revenibility to be: K i r k w d (33) and Buff (72) have shown that as long as the radius of the cluster is much larger than the range W i , i h = w
