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Experimental Test of the Volmer Theory of Heterogeneous Nucleation S. Twomey Citation: The Journal of Chemical Physics 30, 941 (1959); doi: 10.1063/1.1730131 View online: http://dx.doi.org/10.1063/1.1730131 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/30/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Experimental test of quasilinear theory Phys. Fluids B 3, 2747 (1991); 10.1063/1.859911 Line tension effects in heterogeneous nucleation theory J. Chem. Phys. 75, 2441 (1981); 10.1063/1.442309 On the application of cluster growth rates to heterogeneous nucleation theory J. Appl. Phys. 44, 2902 (1973); 10.1063/1.1662672 Experimental test of classical nucleation theory in a liquidliquid miscibility gap system J. Chem. Phys. 58, 896 (1973); 10.1063/1.1679343 An Experimental Test of Nucleation Theory: Formation of Rare Gas Crystals J. Vac. Sci. Technol. 6, 468 (1969); 10.1116/1.1315657 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 150.135.239.97 On: Thu, 18 Dec 2014 02:30:24

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Page 1: Experimental Test of the Volmer Theory of Heterogeneous Nucleation

Experimental Test of the Volmer Theory of Heterogeneous NucleationS. Twomey Citation: The Journal of Chemical Physics 30, 941 (1959); doi: 10.1063/1.1730131 View online: http://dx.doi.org/10.1063/1.1730131 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/30/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Experimental test of quasilinear theory Phys. Fluids B 3, 2747 (1991); 10.1063/1.859911 Line tension effects in heterogeneous nucleation theory J. Chem. Phys. 75, 2441 (1981); 10.1063/1.442309 On the application of cluster growth rates to heterogeneous nucleation theory J. Appl. Phys. 44, 2902 (1973); 10.1063/1.1662672 Experimental test of classical nucleation theory in a liquidliquid miscibility gap system J. Chem. Phys. 58, 896 (1973); 10.1063/1.1679343 An Experimental Test of Nucleation Theory: Formation of Rare Gas Crystals J. Vac. Sci. Technol. 6, 468 (1969); 10.1116/1.1315657

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Page 2: Experimental Test of the Volmer Theory of Heterogeneous Nucleation

CONJUGATION AND POLAR EFFECTS IN BUTADIENE 941

We may summarize our investigation as follows. The nonorthogonal valence bond representation has proved helpful in giving a qualitative picture of the importance of ionic contributions to the wave function of the 11"

electrons of butadiene. These ionic contributions seem to enter primarily as a means of introducing correla­tions between the motion of electrons in different double bonds. Care must be taken if any quantitative interpretation is to be given to the valence structures, particularly in the case of conjugation. The valence bond approach is not particularly helpful in studying bond lengths because it seems to break the problem into one of several phenomena of comparable magnitude,

THE JOURNAL OF CHEMICAL PHYSICS

rather than providing a viewpoint in which one effect is dominant. In addition, its nonorthogonal functions do not even correspond to complete separation of the effects which the method selects.

ACKNOWLEDGMENTS

The author would like to express his great debt to the late Professor William Moffitt, for his guidance and aid during the course of the work which forms the basis of this investigation. He would also like to thank Professors R. S. Mulliken and M. J. S. Dewar for the interest they have shown and for their very helpful and stimulating suggestions.

VOLUME 30, NUMBER 4 APRIL, 1959

Experimental Test of the Volmer Theory of Heterogeneous Nucleation S. TWOMEY

C.S.I.R.O., Radiophysics Laboratory, Sydney, Australia

(Received November 10, 1958)

The prediction of the Volmer theory that the critical supersaturation for nucleation from the vapor phase on a fiat surface depends upon (1- COS</>2) (2+ cos</>>' , where </> is the contact angle of the condensate on the surface, has been tested experimentally. Critical supersaturations for visible fogging on test surfaces coated with various transparent plastics were found to agree within observational error with the theoretical values computed for the observed contact angles.

INTRODUCTION

I T is well known that a high supersaturation is neces­sary before homogeneous condensation is initiated

in a pure vapor, whereas condensation commences upon foreign particles or "nuclei" at much more modest degrees of supersaturation.

If the nucleus is a droplet of the liquid phase, then condensation will proceed if the free energy of the vapor is sufficient to provide for the increasing surface energy of the growing droplet. The growth of droplets in this way is determined by the Gibbs-Thomson equation

log (Pr/ Pee,) = (2aM / pRT). (l/r), (1)

which relates the vapor pressure p, at which a liquid surface of radius r is in equilibrium, with the vapor pressure Poo of a plane surface under the same condi­tions. In this equation a, M, and p are the surface ten­sion, molecular weight, and density of the liquid, R is the universal gas constant, and T the temperature (OK).

If the nucleus is a solid perfectly wetted by the con­densed liquid, no change in surface free energy is in­volved when the solid-vapor interface is replaced by solid-liquid and liquid-vapor interfaces. From the point of view of nucleation, therefore, an insoluble but perfectly wettable particle is equivalent to a droplet of

the same radius. Hence condensation will take place on either liquid droplets or perfectly wettable insoluble nuclei providing the supersaturation exceeds a certain critical value. If supersaturation is defined as (p- Poo)/ Poo, then from Eq. (1), the critical supersaturation for a droplet or wettable particle of radius r is given by

Sr= exp[(2aM/pRT) ·(l/r)J-1. (2)

If the nucleus is soluble, allowance must be made for the lowering of the vapor pressure, in accordance with Raoult's law. Thus for a soluble nucleus, the critical supersaturation will be somewhat less than that given by Eq. (2). The fundamental Gibbs-Thomson equation has been universally accepted as rigorously true, but direct experimental verification has been obtained only recently.l

If the nucleating particle is not perfectly wettable, then replacing the solid-vapor interface with solid­liquid and liquid-vapor interfaces alters the surface free energy. Such nuclei should not therefore be taken as equivalent to liquid nuclei. The problem of heter­ogeneous nucleation was examined in detail by Volmer,2

IV. K. La Mer and R. Gruen, Trans. Faraday Soc. 48, 410 (1952).

2 M. Volmer, Kinetik deT Pllasenbildung. (Th. Steinkopff, Dresden and Leipzig, 1939).

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Page 3: Experimental Test of the Volmer Theory of Heterogeneous Nucleation

942 S. TWOMEY

who showed that the free energy change A involved in the formation of a spherical-cap embryo of the critical radius rc on a plane surface, with which the condensed phase possessed a contact angle cf>, was given by

A =t'll"r/cr(2- coscf>-2 sin2cf> coscf». (3)

The critical radius r c can be derived directly from the Gibbs-Thomson Eq. (1). It is then readily deduced that the critical supersaturation S 9 for production of 1t critical embryos per cm2 per sec is given by

[loge (1 +S9) J2 = (4'll"cr3M2/3pW2k3TS)

.[(1- coscf»2(2+ coscf»/(logeA- logen)]. (4)

In Eq. (4), A is a numerical constant of order 1026

cm-2 secI, N is Avogadro's number and k Boltzmann's constant. The remaining symbols have the same sig­nificance as in Eqs. (1) and (2).

The Volmer theory, therefore, predicts that a sur­face exhibiting a finite contact angle for the condensing liquid shall require a definite supersaturation for the initiation of condensation, even if the radius of curva­ture is infinite. If the Gibbs-Thomson relationship alone determined the question, a flat surface (r= 00) would promote condensation as soon as saturation was reached. Many workers appear to doubt the cogency of the Volmer theory in this respect. It has been claimed that adsorption eliminates the first stage of condensation, i.e., that a particle acquires an adsorbed film which makes it equivalent, for condensation, to a droplet of the same radius, about which condensation will proceed providing the critical supersaturation S" given by the Gibbs-Thomson theory, is exceeded.

The very existence of adsorbed layers at vapor pres­sures below saturation would seem to require that the adsorbate be quite different, from a thermodynamic viewpoint, to the bulk liquid. A full discussion of the physical properties of adsorbed layers has been given by Brunauer,3 where several important differences are indicated. Further support is given to this argument in several papers by Bangham and co-workers, who examined the relationship between adsorption and condensation. Of particular relevance is an experiment by Bangham and Razouk,4 in which it was found that layers of adsorbed methyl alcohol in the macropores of wood charcoal did not act as nuclei for condensation in an environment highly supersaturated with methyl alcohol vapor. Recent experiments in this laboratory6

showed that the critical supersaturation for water con­densation on an aerosol of silver iodide, a substance which strongly adsorbs water from the vapor,6 agreed within experimental error with that given by the Volmer theory, whereas application of Gibbs-Thomson

3 S. Brunauer, Adsorption of Gases and Vapors (Princeton University Press, Princeton, New Jersey, 1942), Pt. I, Chap. XII.

4 D. H. Bangharn and R. I. Razouk, Trans. Faraday Soc. 33, 1463, (1952).

5 N. H. Fletcher, J. Meteorol. (to be published). 6 S. Birstein, J. Meteorol. 12, 324 (1954).

relationship alone gave a much lower value than was actually observed.

The question of the validity of the Volmer theory is important, not only to the study of adsorption phe­nomena, but also to problems of condensation and ice nucleation in atmospheric physics. 7 The theory can be given an exacting test if the critical supersaturation for nucleation on a flat surface is measured for a wide range of contact angle. If a relationship similar to that given by Eq. (4) is obtained, the result provides ex­perimental verification of the Volmer theory; if, on the other hand, no such relationship is found, it must be concluded that adsorption nullifies the effect of con­tact angle, so that the theory is invalidated.

EXPERIMENTAL

Supersaturated conditions can be produced by iso­thermal diffusion of the vapors of two suitably chosen miscible components between surfaces of mixtures differing in composition. A cloud chamber in which water and HCI vapors diffused between water surface and the surface of an aqueous solution of HCI has been in use in this laboratory, and will be discussed in greater detail elsewhere; it is found that the super­saturation produced is a function of Hel concentration, ranging from 0.3% for Hel concentration 1 g per 100 g solution to 100% for Hel concentration 30 g per 100 g solution. These supersaturations were computed from published vapor pressure data for aqueous Hel, and were tested experimentally by comparison with adia­batic expansions.

To examine the dependence of critical supersatura­tion on contact angle, a number of clean Pyrex beakers were partially filled with Hel solutions of different concentrations, covered with a wetted porcelain plate and the concentration increased in small steps until visible fogging of the walls was observed. This pro­cedure was then repeated using sets of beakers coated with different transparent materials (such as methyl methacrylate, cellulose nitrate, polyvinyl formal, and polystyrene) by application of solutions of these com­pounds in suitable volatile solvents. Thus a minimum concentration for visible fogging was determined for each surface. Prior to each run, a sessile drop was placed on the wall of each beaker and photographed to determine the contact angle. In all cases the results were not accepted unless the contact angles were con­stant within the observational error (about ±3°) for all beakers of the set.

The results obtained have been plotted in Fig. 1. It is apparent that there was, as the Volmer theory re­quires, a definite relationship between critical super­saturation and contact angle. To compare the results quantitatively with the Volmer theory, it is necessary to make an estimate of the rate of nucleation which corresponded to visible fogging of the test surfaces. Fortunately, this estimate need not be at all precise,

7 N. H. Fletcher, J. Chern. Phys. 29,572 (1958).

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Page 4: Experimental Test of the Volmer Theory of Heterogeneous Nucleation

T EST 0 F VOL MER THE 0 R Y 0 F H E T E R 0 G ENE 0 USN U C LEA T ION 943

FIG. 1. Observed de­pendence of supersatu­ration on contact angle.

200

,l!\ .

2

1100 20· 40° ~ 8 100· CONTACT ANGLE

as the variation of 58 with n according to Eq. (4) is very slow; an increase of 108 in nucleation. rate, for example, requires an increase in supersaturatlOn of less than one-third. Visible fogging of the test surfaces was found to represent droplet concentrations in the ap­proximate range 10L I07 per cm2• The surface was ob­served for some five minutes in each experiment, hence the nucleation rate n may be taken as being between 103 and 107 per cm2 per sec. In Fig. 1 curves of super­saturation versus contact angle have been plotted, using Eq. (4), for n= 1 cm-2 sec-I and n= 1?8 cm-2

sec-I, respectively. It is apparent that the expenmental

results are in excellent agreement with the predictions of the Volmer theory.

The presence of hydrochloric acid in the condensate is not likely to affect the results, as this compound has an almost negligible effect on surface tension, even in concentrated solutions. 8 The surfaces used were not attacked by hydrochloric acid vapor.

CONCLUSIONS

The main conclusions to be drawn are as follows: (i) The Volmer theory is correct in predicting a de­

pendence of critical supersaturation on contact angle in heterogeneous nucleation.

(ii) Adsorbed layers are sufficiently different to the bulk liquid to render them inoperative as nuclei for condensation.

(iii) The particles which act as condensation centers in natural clouds (in which usually 5 < 1 %) must not only exceed the critical radius given by the Gibbs­Thomson formula, but must also exhibit very small contact angles «6°) to the condensed phase (water). Few insoluble particles are likely to satisfy such a stringent requirement, so that the nuclei active in natural cloud condensation are likely to consist largely of soluble particles or mixed particles containing a proportion of soluble material.

8 P. Volkmann, Ann. Physik u. Chern. 17, 353 (1882).

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