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315

NUCLEATION THEORY: IS REPLACEMENT FREE ENERGY NEEDED?

ROBERT H. DOREMUS Materials Engineering Department, New York, USA

Rensselaer Polytechnic Institute,

Troy,

INTRODUCTION The classical theory of nucleation of liquid from its vapor as developed by Volmer and Weber [1] agrees well with experimental data on rain formation of various liquids in a cloud chamber [2] and with more recent results in the diffusion cloud chamber [3-5]. Nevertheless a number of authors have suggested that the "classical" theory needs modification with a factor sometimes called the "replacement free energy", and that the "capillarity approximation" on which the classical derivations are based is in error [6-8]. The size of this correction factor has varied from a factor of 1017 in the nucleation rate to a negligible correction. In this paper Volmer's equation for the rate of nucleation is derived from fluctuation theory [9,10], the result of Gibbs [11] for the reversible work to form a critical nucleus, and the rate of collusion of gas molecules with a surface essentially following Volmer's original derivation. The capillarity approximation is not used. An objection of Reiss [8] to this use of fluctuation theory is considered. The chemical potential of small drops is next considered, and it is shown that the "capillarity approximation" can be derived from thermodynamic equations. The corrections that have been suggested for the Volmer equation are based on questionable assumptions of equilibrium between clusters and the surrounding vapor. DERIVATION OF EQUATIONS From fluctuation

theory at

constant

temperature

the probability P of a

fluctuation to form one thermodynamic state from another is

given by [9,10]:

P = e

(1)

where w is the reversible work required to provide the change in state, k is Boltzmann's constant and T the temperature. If the probability to form an embryo of liquid of radius r from a supersaturated vapor containing N vapor molecules (all the same) per unit volume is Nr/N, where Nr is the number of these embryos per unit volume, then NrIN

=

-w/kT e(2

(2)

The rate of nucleation I can be considered as the rate Z at which molecules strike the surface of a critical nucleus times the number Nr* of these critical embryos per unit volume times their surface area Ar*: I = Z A * Nr* r r Z is

the

collision

frequency of molecules with a surface

Z = P(2TrnkT)-I/2

(3) from kinetic

theory:

(4)

316 where m is the mass of the colliding molecule and P the gas pressure. The reversible work w* to form a spherical critical nucleus, that is, one that is in (unstable) equilibrium with the surroundings, was found by Gibbs [11] to be 3

w* = 167yy /3(AP) 2 where y is the surface energy of the liquid and AP between the inside and the outside of the critical saturation P/Po, where Po is the equilibrium vapor of the liquid at the temperature of the nucleation AP by: AP = -kTTZn P/P v

(5) is the pressure difference nucleus. The superpressure of a flat surface experiment, is related to

(6)

0

where v is the volume of a m