Theory of the liquid Drop Model HOWARD R E I S S CENTRAL RESEARCH A N D DEVELOPMENT DEPARTMENT, C E L A N E S E C O R P . OF A M E R I C A , S U M M I T , N . J .
I n this paper a brief review of t h e classical theory of gas phase nucleation i n its most modern form is presented and criticized. Subsequently, a brief resume of a recent, more logical theory based on statistical mechanics is given.
I
F A vapor, free of dust and condensation nuclei, is enclosed in
an expansion chamber, when the expansion is performed the vapor does not condense when it becomes saturated. Usually, condensation does not occur until the vapor pressure, p , exceeds the saturation pressure, p,, by a marked amount. It is, apparently, an experimental fact that for a given temperature, T , condensation occurs reproducibly when the supersaturation ratio, p / p , , reaches some definite value-the so-called critical supersaturation ratio. The existence of this critical limit of metaetability has provided an intriguing target for quantitative theoretical considerations, and it is here that one is confronted with the problem of irreversible condensation, sometimes called nucleation. Cnlike reversible condensation the problem of irreversible condensation has not been investigated extensively with tbe tools of statistical mechanics. A passing acquaintance nith the most general treatments of reversible condensation, of which perhaps the 1Iayer cluster theory ( 2 7 ) is the most famous, is sufficient to convey the idea that the irreversible counterpart must be very difficult. Therefore, workers in the field have been content to experiment with semiphenomenologica1 models, designed to provide numerical results when used in certain quasi-thermodynamic arguments. Prominent among these has been the liquid drop model employed by Becker and Doring ( 1 ) and Frenkel (8). With it, the first fragments of the new phaae formed are regarded as welldefined spherical liquid drops having the thermodynamic features of macroscopic drops. The concept becomes extremely vague ( 2 3 ) when it is applied to fragments of nuclear dimensions containing as few as 20 molecules. I n view of this fact it is desirable to develop a more logical theory of nucleation. The natural vehicle for this purpose must still be statistical mechanics. Ideally, the dynamic analog of the Mayer cluster theoryis required. However, this would be fraught with the same if not more formidable calculational difficulties as the original theory. Workers in this field should be content with a more middle of the road approach in which a truly statistical theory is developed but where some rigor is sacrificed in the interest of calculational expediency. I n some recent work soon to appear ( $ I ) , the author presents a theory of this kind.
layer between two phases. This procedure is followed in order to present the classical theory in its natural form.)
F = Fo
+ 4.rrr*u
(1)
Here, FOis the Gibbs free energy which all of the material in the drop would have if it were present a t the same temperature, but a t the pressure outside of the drop. T is the radius, and u, the surface tension of the drop. It is possible to write Equation 1 in the alternative form
where z is the number of molecules in the drop, pe is the molecular chemical potential which the material in the drop would have as part of a bulk phase a t the pressure outside of the drop, and uo is the volume per molecule measured in the bulk liquid. Consider a very large sample of vapor, initially homogeneous, and occupying the fixed volume, V. If p, is the molecular chemical potential in the vapor, then the isothermal reversible work, W ( z ) ,required for the formation of a drop of size, 2, in the midst of this vapor, is
m(z)=
(Pa
- pg)z +
(3)
$xZ'3
This is the change in Helmholtz free energy accompanying the formation of the drop. When the vapor is undersaturated, p, < pLeand W ( z )increases monotonously with 2, but i$-hen the vapor is supersaturated, pQ > perand the first term on the right of Equation 3 is negative. In this case W ( x )passes through a maximum a t z = x* (Figure 1). For drops of size, x > z* dTP dx
