The dynamics of condensation and vaporization

ARCHIEF

Forrest R. Gilmore Research Engineer

Office of Naval Research

Contract N6onr-Z4420 (NR-062-059)

THE DYNAMICS

OF.

CONDENSATION AND VAPORIZATION

Prepared by: Approved by:

California Institute of Technology Pasadena1 California Lab. v. Scheepsbouwkwich Technische Hogeschool

0

_DeIft Milton S Plesset Official Investigator

The condensation process in supersaturated vapors is analyzed with the help of thermodynamics and cinetic theory. The simplifying approximations which make the problem tractable are carefully

ex-amined. esults are obtained for the steady rate of codensation which constitute a theoretical improvement over Becker and

rins

values. The uncertainties in the values of the free energy for very small droplets, which re only partially removed by Tolmans theory on the variation of surface tension with curvature, limit this improve-ment in the condensation theory.

Nonsteady condensation situations are also treated in detail by means of numerical and graphical iuterations. It is found that the condensation in such situations generally approaches a steady rate in

time periods of the order of I to lOu microseconds. such tire

peri-ods are negligible in many physical ituations1 but may be si3lxificant

in the case of condensation shocks In wind tunnels.

A similar analysis is carried out for the boiling of superheated li4uids and the rupture of tiuida under tension. Approximations re -duce the analysis of these processes to iathematicai relations very

similar to those for the condensation process. Nonsteady solutions are obtained by numerical integration. t is found that the solutions

ordinarily approach a steady rate of bubble formation in less than a microsecond.

TLBLE CF CONTENTS

Fart

Title ?age

Introduction 1

II Theory of Condensation (Droplet Formation) 2

2. 1 Preliminary Considerations 2

2. 2 Choice of Variables 6

2. 3 Kinetic Fundamentals 8

2.4 The General Càndensatlou EquatIon 11

2. 5 The t,Anearized Equation and the Electrical

Network Analogy 13

2. 6 Evaluation of Droplet

free Energies

15 2. 7 Incompleteness of the MacroscOicallYDeriVed

Free Energy 24

2. 8 The Condensation Equations in Terms of Droplet

Vapor Pressures 28

2.9 The Equilibrium Solution of the Condensation

Equations 32

2.10 The Steady-State Solution 34

11 T1zrie-ependeflt SolutIons of the Condensation

Equations 43

2.12 The Effects of Admxture8 and Impurities 55 Ill Theory of Vmporization (Bubble Formation) 57

1 PrelIminary Considerations 57

3.2 The 1asic

quat1ons for Boiling (Positive

Pressures

3. 3 The Equilibrium Solution of the Vaporization

Equations 62

3.4 'ihe Steady-State Solution 62

3. 5 Time-Dependent Solutions of the Vaporization

Equations 63

3. 6 The Equations for Cavitation or Liquid Rupture

(Negative Pressures) 64

3.7 The Effect of Impurities on Vaporization 66

IV Results and Discussion 67

4. 1 Comparison of Free-Energy Values Given by Macroscopic Formulas with Reed's Numerical

Values 67

4. 2 Steady-State Rates of Condensation 68

4. 3 Computational Method for the Nonsteady

Condensation Process

4.4 Results for the Nonsteady Condensation rocese 4.5 Results for the Nonsteady Vaporization Process

71

V References 77

VI Appendices 79

Thermal Gradients Inside of Droplets 79

Velocity Distributfons of Particles UndergoxLg

Elastic and Inelastic CollisIons 80

Equations for the Solution of the Nonisothermal

Droplet-Growth Problem by a Monte Carlo Method 82 87

VIII Figures 88

I. INTRODUCTION

When a vapor is condensed to a liquidç or when a liquid is vapor.-Ized, the process frequently does not follow the simple ucheme.of a single mass of liquid and a single mass of vapor, one of which stead-il-j grows at the expense of the other. Instead, droplets or bubbles,

-which are embryos of one phaøe dleperaed in the other, are common, ly formed. The8e embryos greatly complicate the procesS, not only from a mechanical standpoint, but also from a thermodynamic one

since the sall embryos have "surface" energies and entropies com-parable with their 'voIume" energies and entropies. It is essentially these surface effects which permit the existence of metactable states

such as supersaturated vapors and superheated liquids.

An adequate theory of the dynamica of this phase transition would contribute to many fields of science and technology. It should

be applicable, for example, to the condensation of droplets in atmos-pheric clouds and iz Wilson cloud chambers, to condensation shocks in superoric wird tunnels, to cavitation in pumps and on marine pro-pallors, and to many types of boiling problems.

An attempt at such a theory, based on thermodynamics and gas kinetic a, was made by Volmer and Weber, Z _{and Improved by}

Faras,

Kaiachew and transki,4 and finally by becker and D8ring5 6

and Zeldovich. A somewhat different development, based an the theory of absolute reaction rates, aas recently been given by Turnbull

and Fisher.9' 10 All of these investigators make a number of

approxi-mations and simplifying aseumptions, the validity of some of which ie very difficult to determine. Moreover, some of their results

condensation shock. and liquid tensile strengths. (It should be empha-sized, however, that in this field the experimental mea8urelfleflts frequently do not agree with each other.)

The aim of the present work has been to remove some of the ap-proximations made by previous investigators and to estimate the ac-curacy of the remaining ones. In the more generalized treatment

pre-se*ted here, the ecker-Ddring theory becomes a special case. IL THEORY CF CONDENSATION (DROPLET FORMATION)

Although the theory of condensation and the theory of

vaporiza-tion to be developed in this paper are quite similar, the significance of the approximations introduced is much more obvious in the

conden-sation case; therefore, condenconden-sation will be studied first.

Z. I. Preliminary Considerations

When a pure vapor 18 brought from an undersaturated or satu-rated state to a sufficiently supersaturated state (by adiabatic

expan-sion, for example), the vapor condenses into liquid droplets. These droplets have presumably been formed by a chain process which starts with the association of two molecules to form a two-molecule aggregate; this aggregate or "droplet" capturea a third molecule, then a fourth, and so on. The mean growth rate of a given droplet will be decreased

or perhaps even made negative by the frequent escape (evaporation) of

moleciles. The occasional

consolidation of two polymolecular

drop-lets, or the converse fission process. may affect over-all condensa-tion significantly. Moreover, as first shown by Frenkel,11' IZ vapors

-3-appreciable numbers of droplets consisting of from two to about ten molecile8; these naturally serve as nuclei for the growth of larger droplets.

t- supersaturated vapor is, by detinitione thermodynamically

un-stable with respect to spherical masses of liquid large enough so that their surface curvature may be neglected; these very

large 'droplets"

will grov at the expense o the vapor molecules. Smaller droplets, however, are lees stable than large droplets, essentially because of their greater proportion of loosely-bound 'aurface rnolecules corn-pared to the tore tightly bound interior molecules.

Thus, as smaller

and smaller droplets are considered a site will be

found which is in

approximate equilibrium with vapor mLecu1es of a given dejree of supersaturation. This

ise is called the 'critical sie;' analytical

expressions for it will be derived later, but generally it corresponds to droplets containing on the order of 10 to ZOO molecules, for

super-saturation ratios of physical interest. The equilibrium between the vapor molecules and the critical droplet is an unstable eui1ibr1urn, because a critical droplet which happens to lose a molecule becomes Less stable and thus will (on the average) rapidly lose more molecules until it is completely vaporized, while, conversely, if it gains one moiecule it will crow at an accelerating rate. The growth of any par-ticular ub-crit1cal droplet up to critical size is therefore an improba-ble process occurring through fluctuations from the average behavior of sub-critical droplets. An individual droplet grows or Shrinks in finite jumps by a stochastic process, and there exists a certain small

)robability that it will attain or exceed critical size. Most

vapor molecules; a significant number of droplets of critical _{size may'} be farmed from these molecules even though the probability for any

given molecule to initiate the growth be very small. Droplets which

grow beyond critical size are Increasingly likely to _{continue to grow.} and a droplet which has reached about _{twice the critical size is} virtu-ally certain to attain macroscopic size.

A question which arises before the condensation process can be treated mathematically is whether the vapor-droplet mixture can be regarded as "homogeneous," or whether significant density and tem-perature gradients will occur in the _{vapor immediately surrounding} each growing droplet. _{In a typical case of water vapor at 00 C and} four times the saturation pressure, a spherical volume of radius equal to the mean free path of the vapor molecules contains about mole-culee, and similarly large values _{are obtained for other vapors and} conditions of Interest. _{Thus any depletion of the vapor in the} neigh-borhood of a droplet growing _{up to 10 molecules, say, would be} equal-ized in a time short compared to the droplet growth time, and also any heat conducted away from the droplet by escaping molecules will be spread ove- a region containing many more molecules in a similarly short time. _{It follows that homogeneity In the vapor can be safely} as-sumed until the droplets grow to many times

_{critical size.}

_Beyond thia point, density and temperature gradients in the vapor may have to be considered; but, on the other hand, for such "large" droplets the actual Irregular molecule-by-molecule growth process can be

the molecular diameter is estimated_{from the volume occupied} by a molecule in the liquid state, It is easily

_{shon in general that the}

number discussed above equals (fl/LOS) (p1 /p) , where _{and Py}

-5-accurately approximated by a continuous steady growth, _{so 'that the}

problem _{is still mathematically tractable. Thua, it is convenient to}

separate the complete condensation procese into _{an early "molecular} domain" and a later _{rnacroscopic domain1' with the dividing point}

taken at droplets containing about 1O4 or molecules. Fortunately, in the transition region all of the approximations _{pertinent to either} domain should be valid, so that the exact choice _{of the dividing point}

is immaterial.

-In treating the condensation process, one might try to follow the growth history of particular droplets, or alternatively, _{one might} ex-amine the change with time of the populations of the, _{various size-groups.} These alternatives correspond to the use of '1grangian" or "Eulerián" coordinates, respectively. In the molecular domain, Lagraugian

co-ordinates yield complicated stochastic equations, since the growth of a molecular-sized droplet is a very erratic process, while Eulerian coordinates yield considerably simpler equations, corresponding to

a "generalized diffusion" process. Hence, Eulerian coordinates will be used in the present work (except in Appendix 3; see section . Z).

In the macroscopic domain, on the other hand, It _{proves simpler} to use Lagrangian coordinates, since the growth of & single macro-scopic droplet can be considered to be steady and continuous. The growth of macroscopic droplets baa been studied by Hazen13 _nd

3arrett and Germain'4, who found that theoretical relations based on macroscopic diffusion and heat conduction give results ingood agree-rnent with experiment. _{This macroscopic domain wiU not be 'treated} in the present paper.

2. 2. Choice of Variables

To treat the condensation process mathematically1 let us char-acterize any given aggregate or droplet by the following three independ-ent parameters: a, the number of molecules it contains (with a I

representing the monomolecular vapor); E, the internal energy of the droplet; and 7, the kinetic energy of motion of the center of mass cf the droplets (E + = total energy). Sy thus limiting the character-igatlon to these three parameters, we tacitLy ignore any variations in size and shape between droplets having the same mass arid energy, and any non-equilibrium distribution of the energy among the various parts of a droplet (i. e.

'internal temperature gradients).

The first ne-glect appears reasonable because droplets containing a dozen or more molecules are virtually certain to be nearly spherical, to minimize the

surface free-energy, while droplets containing fewer molecules can prably be approximated by a moan size and shape with an error not

greater than the other errors which rise when extremely si-nail drop

lets are treated. As will appear later, a considerable error in the

treatment of droplets containing only a iew molecules may affect the calculated condensation process only slightly. The neglect of internal temperature gradients is made plausible by arguments given in Appen-dix. 1, from both a molecular and a macroscopic thermal-conductivity point of view. even with these approximations, the general problem

gives rise to differential equations in four independent variables: n,

E, Vi

, and t

(the time).

The variable ii, can be eliminated by assuming that all droplets of a given size, n, have velocities (and kinetic eriergie) distributed according to Maxwell's law for a certain temperature, T, which wilt

-7-be called the "kinetic-energy temperature," and which is independent of n and E but may depend on time. This assumption appears fairly reasonable, aince any assemblage of moving particles that undergo either completely elastic or completely inelastic (capture) colliion tends toward a Maxweillan velocity distribution (with the usual mass dependence), as shown in detail in Appendix 2, and condensation pro-cesses of interest usually take place over time periods long enough to permit each droplet to undergo many collisions.

The independent variables are thus reduced to three. The dif-ferential eqiations which arise, however, are still too complicated to be solved analytically or even by a conventional numerical method in a reasonable amount of time. The moat feasible method appears to be the "Monte Carlo method" of actually tracing the histories of a large number of typical droplets. An outline of such a procedure for the

con-densation problem is presented in Appendix 3. !1umerical results from such a procedure can be obtained only through extremely lengthy manual computations, or extensive use of automatic computing machines, nei-ther of which has been poesible in the present work. Such computations are planned for the near future.

In order to bring the problem into more tractable form, two fur-ther approximations will be made. First, the droplets will be treated as.if at any instant all the droplets of a given size, n, have the same free energy, which may be a function of n and t. This neglect of the free-energy dispersion is equivalent to the neglect of the internal-energy dispersion, since, under the approximations made, the one quantity is a monotonic function of the other. Each free-energy value,

size n. The second approximation which will be made is that Tn T

for all values of n. This assumption undoubtedly introduces some error, since the growing droplets in general will have higher temper-atures than the surrounding vapor, due to release of the latent heat on

condensation. n cases where the vapor Is mixed with a nou-conderAs-Ing gas, the error should be smaller, since the gas molecules will help carry away this heat. A discussion of the error introduced by this approximation will be given in a later section.

ictually, much of the theory presented below can be carried through for arbitrary T(t). without assuming T = T. The difficulty Is then to find an approximation to T better than T. A successful method for doing this ha net yet been discovered. Even if the mean temperature for all the droplets of a given size, a, could be calcu-lated, use of this value for the effective in the condensation equa-tions may introduce as great an error as use of T, since in general only a very small fraction of the submicroscopic droplets grow to observable size, and these are likely to be those which initially were much colder than the mean droplet temperature.

2. 3. inetic Fundamentals

Let N(t) (a = 1, 2, 3,

. . .) be the number of n-molecule

drop-lets per unit volume, in a mixture of vapor molecules and dropdrop-lets of various sizes, all having velocity distributions corresponding to the kinetic-energy temperature T(t). Then the rate of collision of droplets containing i molecules with droplets containing j molecules is

2 (2kT/m)1 D f( I +

)/J]l/z

N1 N per unit volume, where k is

oltzmann's constant, m is the mass per molecule, and is the mean collision diameter. 12 If the droplets were erfectlf'apberical

and had tho ae density,

_L' as the liquid in bulk thenD would

ë-d/3

p.1/3 .1/3,

qual (3r/41rp,

. _. 3y nu1tiplyng these quantatieB by a

'correction factor1 ne can take as the exact rate of capture.col-lision (not cc-unting colcapture.col-lisions which do not result In the two particles stickin together):

Rate of i _{+ j}

capture collisions =KT..

)%t,

''ere tL.e

antt

. is giver by

(1)

% ?r '2)

nd is costat g0r

any particular liquid, if the vriation of liquid den

ity w.tb ternp rtue

pressure is neglected. The factor

is

intohiccd

ct oz1y to allow for coliisions which do not result in

cap-tur, but also tc correct for any noa-sphericity of the droplets or any of th dropiet density £ron

:en i I and j , the quantity bcomee the conventional

cco:imodtiom cceficient, a, for individual molecules striking a flat liquid uthLce. _{'he be5t experim!ntai evideace indicates that for}

clean liqiid surfices at temperatures not greatly different from room

teperture,

u is probab.y very close to unity.

When

_j

is small, .

can be considered to contain a geometrical factor sligt1y greateT than unity due to non-sphericity of the aggre-gates ed densities less than the liquid in bulk, and a factor omntwhat loss than unity because of collisions which do not result in capture.

'or very sra1i and sinple "droplets, '

such as aggregates of 2 or 3 non&tonic noiccu!e, the latter factor may be several orders of -nagnituzie smai1e than nity, hecaase of the ina'iiity of the limited

number off degrees of freedom of the aggregate tc absorb the collision ererg'i rapidly. iowever, for poiyatorrtic molecules it is believed that

is near unity even for values of j as smaU as 3 or 4.

Similar considerations apply to when both i and j are greater than I. it v'iLl be shown below that only a fraction of the condensation is '1ue to terms containing (i, J I), so that the approximation

I should be satisfactorr in this case, in the absence of more

er.-act lzwledge.

qat1cn (1) can be written in shorter form by trcduc the

quantity

= ;. (

so that

!ate of I + j capture coliiion

_I'T

/\/, At.. (4;

esides the capture process, droplets may also udoro the

con-vese vaoviaticn or fission process. The two opposing procs,

for any particular i and j, can be treated like the opposing processes in a reversible chemical reactior., so that the ratio between t!ie two rates equals the equilibrium constant. If the mixture oi molecules and droplets can be treated like a mixture of perfect gases, the euikibrium constant depends only on the temperature and can be expressed in terms of the standard free-energy change 'o

-F° _'O

y a well - known

thermodynarthc formula:

'

The term ufree energy and symbol F wilt be used throughout for the tGibba free energy," frequently also called the "thermodynamic

potentiaL' The word "potential" and symbol 0 will be reserved for the

kat.e of capture

Rate of

fision

it follows from Eqs. (4) and (5) that

R.te cf i + i fission ex-p

(_0

C)

((a)

where is the standard free energy of an (i + j) droplet at

tempera-ture T and at the standard concentration of one droplet per unit voiume (ccrresionding to the units of N), and F? and are similarly defined

for i and j drc.plets, respectively.

2. 4. The General Condensation Eauation

N, the number (per unit volurne of droplets containing n mole.-cules, will ircreae with time because of captue collisions of N1

drop-lit

ithN . droplets (1 = 1, 2,

.., n -

1) and because of splitting of

larger droplets, into and N. (i = 1, 2,

...,

co), while it will

be decreased by the converse processes of splitting of N droplets and capture of other droplets by the N droplets. Evaluating the rates of these four processes by use of Eqs. (4) and (6). and also remembering that N is the number per unit volume and hence wilt change if the vol-ume V of the mixture changes, one obtains the basic differential

equa-tion for condensation:

d/V7,

KT

-11-- Jquilib. c.onst.

exp

(

I.-,co-/.o

_i

-/ N

e x p (°

fO)?

_ELk

iij

V

dt'7'

where the factor 1/2 appears because the first summation counts each collision twice (since

.

N. N. _..

N N., etc.).

The second

summna-tion counts the n, n collisions twice, which is correct, because each such collision removes two u-molecule droplets.

Equation (7) can be simplified by introducing the quantities

Yc') N7,

V and

Qt)= ex-p

so that it becomes dY,

d

(v

, -

)

- Q1

Y)}.

in a particular condensation situation, the vapor temperature, T, and therefore the temperature-dependent quantities may be known functions of time, either because the temperature is directly controlled by some sort of cooling device, or because the cooling is produced by adiabatic ezpansion with a known volume change and just the earlier stages of the condensation are considered, before so much

condensa-tion has occurred that the latent heat released affects the vapor temper-ature. (During later stages, when the latent heat released is important,

T and _{will depend in a complicated way on the dependent variables} or Y; however, this corresponds to the macroscopic dcmain whre, as already mentioned in section 2.1, other simplifying approzimations can be made.) Thus Eq.(lO) with n

_{1, 2, 3...., yIelds a set of}

non-linear differential equations with variable coefficint which in

principle can be solved numerically, for given initial values of Y, to any desired degree of accuracy. In practice, however, this procedure may be very lengthy.

2. 5. Th Linearized Equation and the Electrical Network Analov

If Eq. (7) is transformed to the following variables:

c')='

;

(il)

C,(z

Vt

exi.7E

- (n/i) 1Q,7()

-

7Va1/-

eo

t('cçzc)_

cø,7...,Ø,, cLt

A4/

d.(C,Ø)

;-e '"

,c.

-13-(13a) /

_()=L'K1,"

TL'\r;

(1 3b)

and the terms for j = 1 and I = a - I in the first suimation and for i = 1 in the second summation are written out separately, the equation be-comes

(I

(øø--)

4,. 4

(14a)

The extra factor o two is inserted into the definition of to allow for the fact that when n = 2 there is only one term for the values i I and

I n- I in the first summation of Eq. (7). Thus a factor of 1/2 in the

-

Ø)/g

term in Eq. (14a) is avoided, although such a factor

must then be included in other places where R1 appears in the equations. The terms in brackets in Eq. (14a) represent contribuUons due to collisions of two polymolecular aggregates, or the converse fission pro-cesses. These are usually inIreueat compared to monomolecular col-lisions, so that to a first approximation they can be neglected, giving simply

c)

_R,,

The accuracy of this approximation can be estimated a posteriori by

using the 0's calculated from Eq. (is) to determine the size of the

neg-lected terms. The results of such calculations will be discussed in a later section.

During the early stages of condensation, the fraction of vapor molecules which condense into liquid droplets is usuaLly quite small, so that the total number of monomolecular vapor molecules, VN1. re-mains virtually constant. Then, if the vapor volume and temperature are known functions of time1 C and R defined by Eqs. (12) and (13)

are known functions of time, and therefore the set of Eqs. (15), with

U = 2, 3, .

..

become a a set of first-order linear differential

equa-tionS with variable coefficients.

Equations (15) are (oirna1ly equivalent to the equations for the electrical network shown in Fig. 1, where the Ø's are voltages.

-15-Rn's resistances, and Cu'S capacitances. Thi8 equivalence makes possible the solution of the linearized condensation equations by means of an electric analogue computer. In general, the circuit, parameters

and C will vary with time, which greatly complicates the conputer design; however, in certain simple cases described below, they are conatant. Another difficulty in the use of an analogue computer is the

necessity of measuring output voltages of the order of times the

input voltage. Probably this difficulty could be eliminated by introduc-ing voltage amplifiers at intervals along the network, together with feed-back circuits so that the later stages would properly influence the

earlier stages.

Because of these difficulties, however, the use of an

electric analogue computer has not been attempted.

One useful by-product of the electric analogy is the application of electrical concepts and terminology to the condensation problem.

This makes obvious many relations which otherwise would require lengthy argument to demonstrate.

2.6. Evaluation of Droplet Tree Energies

Before continuing the discussion of the coudensatic equation, it Is helpful to determine the magnitude of the quantities R and C, which in turn depend upon the droplet free energies.

The free energy of a small liquid droplet, n excess of the free energy if the same amount of matter in the "bulk liquid," is apprOxi mately equal to the ordinary surface tension, s. times the surface area, so that

where is the free energy per unit mass of a large mass of liquid under a pressure equal to its vapor pressure. The above approxirna-tion is poor when n is small &nd worthless when11 = 1. However, an exact expression for can be found from the consideration that the free energy of the liquid equals the free energy of vapor at the satura-tion density, while, if the vapor is assumed to be a perfect gas, its free energy varies according to k T times the logarithm of the density.*

Hence

A/,

m°

_{T/oq(_N5)}

w1ere N_sat. is the number of molecule. per unit volume in a saturated vapor at the temperature T. Equations (16) and (17) can be combined to give

(

6?fl/ rc-

ir/()

(18)

A more exact expression for the free energy of a small droplet can be obtained with the help of Tolman's

theory6

on the variation of surface tension with surface curvature. In carrying out the analysis. the utmost care must be exercised not to neglect terms of the same

17 order of magnitude a5 the surface-tension correction.

The total Internal energy of a system consisting of a single liquid droplet surrounded by vapor in equilibrium can be regarded, following

Gibbs. 18 as made up of three parts: the energy of the liquid calcaiated

(17)

*All _{of the logarithm. used In the text are natural logarithms, to}

the base e 3 2.118...

-17-as if it were in a homogeneous "bulk" state and occupied the volume 4/3 irr3, where r. is the radium taken to some arbitrary point within the surface layer;the energy of the vapor calculated as If itwere

homogeneous and occupied the remainder of the volume; and

acorrec-tion term, E5, needed to make up the remainder of the total energy.

The value of the "eurfece energy," E5,: depends upon the exact choice of r, which is somewhat arbitrary becauee of the finite thicknees of the surface layer. It Is desirable, however, to choose r as the radi-us of the well-defined 'surface of tension" introduced by Gibbs, which

can be considered as the surface at which the surface tension acts, so that other relations derived by Gibbs can be directly applied.

The total energy of the equilibrium liquid-vapor system can thus be written

t+'-,'-

4(V-4ir)&EV *E

-,

(g)

where Is the density the liquid would have in bulk at the tempera-ture and pressure actually existing inside the droplet; EL is the liquid energy per unit mass under the same conditions; and

are the -.

corresponding quantities for the vapor, and V ie the total volume of the system. Similar expressions hold for the total entropy, and the. total mase,

40+?r'i9

(V7r,)&

The total free energy (or thermodynamic potential) of the system te

1-

c-T5)-'-E$-r5#pv,

since Is the pressure actually exerted on the walls of the system.

or this one-component system, Gibbs' Eq. (502)18 can be written, in

the pre8ent notatiDn

ET554Ti/t'crfM,

,

(13)

where F is written for the common free energy (or thermodynamic po-tential) per unit mass of the liquid and vapor phsee In equilibrium. I. e..

FTET5,/,.

(24)

By virtue of Eqs. (23), (24) and (21), Eq. (22) can be written

iç0t

_3-

7TJ3 (t

cL)

(v-4r,A7FM5

-t-= -t

q

The basic eurfacé-tentOn relation (Gibbs. Eq. (500)1 B) l

PL 1'v

eo

_'V

(21)

(25)

(26)

so that Eq. (25) becomes simply

-19-The analysis of the condensation problem in earlier sections treated the vapor-droplet mixture like a mixture of perfect gases, as-suming the vapor density Is BmSU. U the equivalent assumption is

made here, the vapor free energy will be independent of the presence ot the droplet, end hence WILL equal F Mv. The remainder of the total

free energy given by Eq. (27) will be the free energy of the n-molecule droplet

= '7,77

F *

-f

7e

(28)

since the mass of the droplet equals n in. In Eq. (28), F is the free energy per unit mass of the bulk liquidat a temperature T and pressure

Since, in general.15 r external

dF=-5dt+dp/,,

-, (29)

F can be evaluated in terms of the free energy, F, of the bulk liquid at temperature T and pressure equal to ite ordinary vapor pressure.

sat (T):

_# _PeLt

Foj

d

r- L ° e a-

-L.

'

* to

In Eq. (31), is the external pressure exerted on the droplet by the vapor or saturation ratios of Interest, this Is not more than severaL

gas approximabon _{With this approichnation Eq. (28) can be wi4tten}

When tble expresslonli combined with Eq. (17). one obtains finally

'The surface tension, , for a surface of radius r must now be.

evaluated. Atcording to Tolman's theOry,'6

ac.r

i#.-i-t. _I i,.

e_i

., . .

£

A 3...4

where 6 is the distance between the Glbbà surface ot tension and the surface which would make the "surfacà mass" vanish, i.

e.,

/

47r/J

it, (35)

For all Liquida wbtch have been Investigated,19 8. Is somewhat lees than.. the distance beeex& a molecule and its nearest neighbors, and thus Is considerably less than 1 for droplets large enough so that ihe concept of surface tension Is applicable. It is convenient, then, to expand Eq. (34) In pOwers of 6/r:. S d/oqo

r,_1

,.2

53 4 ₍₃₆₎ dit. A2 L A' 3 3. A3

3. ,4#

J 0

emQfl

q

a

e#,,c,,

4

21-If this expression is integrated between r and co. under the assumption that 6 is a constant independent of r, one obtains

S.

= exp1-e

a

3e63

59 1/1

= i_

-;?- t-.. . .j (37)

where Is the ordinary surface tension for a flat surface. It is convenient to rearrange Eq. (35) to read

1 -I

f3/?7i'7 _)f 6

A - 4rr,qJ(J

When this value for r is substituted in Eq. (32) it gives

-'7 =

2'67n

v7[i

+ * e 53 -; iT]

-nTfoq

A. N5

The only published attemptsZO _{to apply Tolxna&s theory to}

con-dertsation phenomena tacitly neglect terms containing 6/r in Eq. (39), while using Eq. (34) or (37) to evaluate a. Such a procedure is incon sistent and in general will lead to error; however, because the first order terms in 6/r fortuitously cancel in Eq. (39), the results thus ob. tamed are first order approximations. A higher order approximation, obtained from Eqs. (37) and (39) is

N

-nip T /a9-:;+

/v5jr..

If a diienionles# qusatity E Is dfiecI:

vrrsrna1l o a maxivam at a po{zit wuich will be cl1ed the tlrttjc

)

gjze,t'

.and then dcreaesard finally becomes iegatve for -lar

(i

-

5,73)

which depen.s oLy upon the liquid zder conideratiàn, Eq. (3) .d:b

transformed to give

(40)

41)

or, inverting th

series,

n=r70

(i-where ,7co=

-T/oqj =0

3,t

3 7Tm C-J

3 (46) 5_/9

TIoqL)3

is the value would have if 6 = 0. 3y combining Eqs. (43) and.

(44), one finds that

(45)

-

f637-

) (47)

y use of £cs. (36) and (40), it may be ehown that this result is equiva-lent to

(F)mox(n=nc)771c

,

(4)

where r and a- are the radius and surface teàsion at the critical sizc.

C C

Suce the free-energy difference has a stationary value at the critical

.sie,- a droplet of this size will be it (unstable) equilibrium with the vapor. quation (48) agrees with Gibbs' E. (560), lB derived by him

(z )+ran5

for the equllibriurn case; one has, therefore, a check on the more gen-eral 'qs. (33) and (43), which have not, to the writer's knowledge, ap-peared in the literature In any form.

2.7. Incompleteness of the Macroscopically-DeTiVOd

Free £uery

In the analysis of the condensation process, the vapor-droplet mixture was treated like a mixture of perfect gases. If droplet. of size n behave like molecules of a perfect gas, their free energy should contain a term of the form kT log

N. It as precisely this term which

permits an underaatiirated or saturated vapor to contain in equilibrium a few polymolacula? droplets, even though the latter have a higher

"intrinic

free energy.

oth the approximate Eq. (18) and the more exact Eq. (43) are seen to lack such a term. This deficiency is related to the fact that if either of these expressions for the free energy (setting N1 = 1, for the standard state) is substituted Into Eq. (11), the ratio 02/01 will be found

to depend upon the units of volume chosen, which is inconsistent with Eqs. (14) and (15) when n 1 or 2.

A plausible way to remedy these deficiencies is to assume that, in treating the droplet thermodyna'ricallY to obtain Eq. (43), the ther-mal motion of the droplet as a whole was,

in effect, disregarded. The

additional free energy the droplet will have due to this motion can be obtained approximately from the translational and rotational partition

functions12, Z. for

a rigid sphere of mass ma and moment of inertia

=

(irmn-.kT)

-25-(Zn)rt

(50)

If the quantity - k T

[Iog(Z)5

+

log(Z)rot]

is added to Eq.

(43), itbecomeè

,,p

4Tb1'-The terms In and from Eq. (43) have been omitted in Zq. (51).

since they are probably smaller than the inaccuracies in the eiation due to the assumption that 6 is independent of r. These terms were originally calculated merely to show that their numerical coefficients are small.

The quantity which actually appears In the condensation equations is the standard free energy difference, F

- nF, corresponding to the

_atandard concentration of one particle per unit volume, i. e., N 1

and N1 1. This quantity was introduced because, in the perfect gas

approximation. it depends only upon n and T, and unlike the actual free energy difference, is independent of droplet and vapor concentration. It follows fom Eq. (51) that

tT.

= A(v

3enheea)4

flhC1l"'5c3t

196v2

₍₅₎

- 4Iocl r?- Ioq

where, for convenience, X designates the dimensionless surface-ten-slon factor:

c',

XT-

_,'

_I

If this expression is substituted Into Eq. (11). the resulting Ø are inde-pendent of the dimensional units used.

An alternative method of introducin the k T log N term, which appears nearly as plausible, is to assume that Eq. (43) gives the free energy for a droplet confined to a volume of order mn/pL. U this vol-ume is then expanded to 1/N, and the droplet behaves like a perfect gas molecule, the resulting isothermal change in the free energy is kT log N + kT Log (mn/p,). If this expression Is added to Eq. (43),

the standard free energy difference becomes

ECompare, _{for example, Frenkel,} 12 334, Ec. (16a) with hi

unnumbered equation for Mg on p. 381.

(53)

/09 '7

_-

),

(54)

instead of Eq. (52).

Equations (52) and (54) are certainly, not equivalent, although, sirprisingly enough, the two expressions are approximately equal for many cases of physical interest. A third, still different expression for the standard free energy difference will be obtained in section 2. 8. Equivalent discrepancieB appear in the condensation relations derived by previous uivestigators, depending upon their different approaches.

7-These differences arise basically from the attempt to extrapolate mac-roscopic relations to. the molecular domain. In the determination of droplet free energies as a function of n, macroscopic measurements enable evaluation of the 'bulk" term proportional to n the surface-tension term proportional to and probably, from Tolman's theory, the next order correction term, proportional to hi'3 Terms cf lower

order, however, such as constant terms or terms prortionat to log n,

may be completely masked and not measurable by any direct macro-scopic means Such lower order terms can make a considerable dif-ference in condensation processes.

At least two approaChe8 tow&rd evaluating such terms are pos-sible. On the one hand, the condensation theory might be worked out for various assumed values of the free energy, and the results com-pared with experiment. This procedure is followed in the present paper. On the other hand, the free energy of very small droplets might be cal-culated directly from the principles of statistical mechanics and quan-tum mechanics. Carrying out the latter procedure is a formidable task; a beginning baa recently been made by Reed, 23 who calculated the free energy for nitrogen aggregates containing 2,

3, 4, 5, 6 and 8 rnolecule,

assuming a Lennard-Jones interaction potential between two molecules and making other approximations. In Figs. 2 an 3, r4eed's values of (F° - nF)/kT are compared with values calculated from Eqs. (52). and

(54), and also with values from an alternate equation derived in section 2.8. The comparison will be discussed in detail later. In any case, these free energy values for a single substance over a limited range of n are hardly sufficient to determine which of the alternate expressions is, most accurate in general. Alt of the expressions, however, are of

the following form:

A ,i /oqA

A'oqn- faq 8 (i,>i)

(55)

where A and B are quantities independent of n. A isa poaltive or nega-tive quantity of order unity; Bt is a large posinega-tive quantity of order

zo

_{czn. When accurate values of the free energy}

_{for n 2, 3,}

are tnown, terrs proportional to

-2/3 etc., might be added

to Eq. (55) to obtain better agreement; however, It Is probably easier to work directly with the numerical free energies for the first few valu of n, and use Eq. (55) for larger values of n.

The theoretical condensation relations developed in this paper will be derived first In terms of arbitrary free-energy functions, and then speclalised to the functional form of Eq. (55). Specific values of A and B will be assumed only for the numerical calculations. Thus, the theory remains readily adaptable to any more accurate free-energy

values which future investigators may obtain.

2. 8. The Condensation Euations In Terms of Droplet Vapor Pressures

Becker and DBring5 developed their theory of condensation in terms of the vapor pressures. of small droplets inateád of their free en-ergies, and used the Gibbs-Thomson formula or the variation of vapor pressure with droplet radius. Basically, the two approaches are e-quivalent, if both are carried out without approximation. It 16 natural, 1owever, to make different types of approximations in the two cases, so that the usual results are only approdmately equivalent. U only a rough answer is required, the Becker-Dkr1ng approach is in some

wrc simpler and followa physical intuition more closely. If a better approximation Is desired, however, tne development of the theory in terms of free-energy differences ha two important advantages. In

the first place, the nonlinear terms in the condensation equation, representing consolidation or fisIon of polymoleculár droplets, can be written dow immediately in terms of the free energies, but could be obtained only by a very involved argument in terms of the droplet vapor pressures. Secondly, the free eflergy of an n-molecule aggre-gate or droplet is a definite well-defined quantity (even though Its numerical value may not be known precisely). The vapor pressure of an n-molecule droplet, on the other hand, might be defined in several different ways; for example, in terms of equilibrium betweena and (a - 1) molecule droplets, or between a and (n + 1) molecule droplets,

and with or without a factor [n/(n j to allow for the difference in

surface areas, etc.

For small n, these definitions are not equivalent,

and the calculated rates of condensation will depend somewhat upon

which vapor pressure is assumed to be given by the extrapolated Gibbs-Thomson Cormul. Significant errors may also result from the mLed use of two different definitions of vapor pressure.

The vapor-pressure definition appropriate to the Becker-Daring theory can be readily derived. i3ecker and Daring expressed therate

of growth of (a - 1) molecule droplets to n-molecule droplets as

- 1)2i'3pN _{(in the present notation), where p is the}

pressure of the monomolecular vapor. Since p = N1kT, this expres-.slon is seen to agree with Eq. (4) when a 1, _j

= a - 1, and

2/3

F;-nr,0

of n-rnolecile droplets to (n - 1), they used Kk T_1

/3 P N

n n

2/3

factor

(n - li/nj

from the cle finition

Equation (58b) differs by the

1.- 23

of? ,iP

used by Reed.

By taking the products of Eq. (58a) for consecutive values of n, one finds, on simplification,

-p7 0 0

II I

pS)

N"

L'e ( Sat. (59) 60)

where is the 'vapor pressures' of an n-molecule droplet. By corn-parison with Eq. (6), settIng i = 1, = - 1, and _1,n-1 = (n

-again, it is seen that

e/

1ir°..r1r,

(56)

T(

exp

T

If ? denotes the ordinary vapor pressure over a flat surface,

so that

(7)

I / #7-I

/c°

F6

r°

)

(58a = dt

)ex(

(58b)

-I

Ioq -#-

fioq

fl -t

-31-Becker- and Dr.ing further assumed that, even for maU values bEn, Pn/Pa,can be approximated by the Gibbs-Thomson formula:15

ea-n7

1oq _

Equation (61) was first deduced, by W. Thomson from mechanical con-aiderations, and later derived more rigorously by Gibb3, 18 using a free-energy argument. When values of r from Eq. (39), o from Eq. (37), 6/r from Eq. (42) and X from q. (53)'are substituted into Eq. (61). one obtains

neglecting higher order terms'. Substitution of values for log (P; from Eq. (62) into Eq. (60) yields

o

r°

--T

(e.-Y3j

364;..2/j

Th.ts, Eq.. (63) becomes, to the same order o1.sproxiTnation.

io(4)

O6i' ),

(62)

)ioq "a #f"oqn

(63)

The summations in Eq. (64) can be expressed with the help of a series for the

ieann Zeta-function (Jahnke nd

Ende,2 p. 269):

/

(n½e6n/.3/3*e.9IC)

iT

-+ (fl-I )'°?

N5 fIG? r'

Eq. (65) 18 seen to be of the form of Eq. (55), with

and

5 N

exp

f.(i.3i3 _e.e9/E)J.

(67)

In their treatment, Becker and DBring simply approximated the summation in Eq. (63) (without the E terms) by the corresponding inte-gral between the limits I and n, and obtained resilts equivalent to

6 = 0,

A 2/3, B N_sat exp(X).

2.9 . The Equilibrium Solution of the Condeaatlon Iqationa

We return now to consideration of the condensation procea and its electrical analogue. If the resistances and capacitances of the e-lectrical network shown in Fig. 1 do not vary with tirn, a possible splution for the network is obtained simply by giving all the voltages the same constant value. Since = 1 from Eq. (Ii), the above condltiou becomes

0 - /

F,

(for all n.

This solution evidently corresponds to the case when there is no current flow in the electrical analogy, or no net growth of droplets in the con-densation situation, i. e., an equilibrium state ha been attained, hold-ing the resistances and capacitances uIed corresponds to holdhold-ing the

tenperature ar.d vc'iue of the vapor

fid. accordin

to s. (12) and

(13).

quation (3) ts readily seen to be a eIutian not only of the 1in

erized Eq. (I3, hut

Iso o the exact nonkiearied

qs. (14a) and

(1-b). This result i to be expected. because it is equivalent to the well-known cheica1 principle that the equilibrium of an over-all

reaction is independent of the assumed intermediate 5ta;es.

The numter of droplets of a given ie in the

eqilibriurrk

situ-ation is found froni p. (11) aild (63) to be

0

NN,exp(

_T

_I

if f'-ee energy values Iron (53) are subst1tted into q. (69)9 one obtairs

NsAh[A(2/3e1/3)J

, (70)

where s is the saturation ratio:

-33-(69)

When s 19 Lq. 70) shows that the series

_

N, which expresses

the total number of droplets of all sizes per unit volume, behaves like the geometric series

s, converging for

undersaturated vapors

< 1) and diverging for supersaturated vaora (s > 1). For saturated vapors (s 1), the series behaves like exP(_n2/'3) arid hence

con-verges. it follows that the equilibrium state is pbysicaUy realizable for undersaturated and saturated vapors, but not for supersaturated vapors. In the realizable cases the values of fall off rapidly with

N,

increasing a, and generally become negligible for a 10.

::quations (69) and (70) for the equilibrium droplet distribution

do not contain the s, and hence ar independent of the uncertainties in numerical value of these quantities. They do, however, contain the free energy difference as an exponent, and thus the calcuiated Values of equilibrium N can differ by factors of 10 to 1000 depending upon

which free energy expression is used, 1. e. upon the values of and

3 assumed in £.q. (70).

2. 10. The steady-State Solution

If P._n and C are held constant, a time-indeendent solution of the_a

!inearized Eq. (15) may be sought by equating the left hand side to zero. This procedure yields

-,

-øn.

Rn-, - (for all n) (72)

or in other wor-s,

Pr,

= con,tôr,t

JV

(73)

The constant in Eq. (73) is denoted by J V because it can be shown that S is then the steady rate at which droplets grow from any size a toa + 1.

This result follows from the fact that CØ = VN. according to Eqs.

(11) and (12); thus the left hand side of Eq. (15) equals VdN/dt (since V Is constant In the steady state), and the terms on the right hand side must then represent "droplet-growth currents" multiplied by V. If

Eq. (73) is writtcn In the form

=JVR

(74)

-35-firt and last will cancel, yielding

-=Jv1.

(75)

'Solving for 3, and using Ø 1, one obtains

Equation (76) hoLds for any value of n. By Eq. (11), Ø

exp LI(F0 -

nF)/kTJ.

As n becomes large, Eq. (55) shows

that this expression behaves like N s. If

is bounded, which must

be 'true in any physical situation, then urn 0 = 0 for supersaturated

vapors (s 1).

Thus, by letting n'co, Eq. (76) becomes for

super-saturated vapors,

Li I

(76)

(77)

The eummatlon Lu the denominator of Eq. (77) behaves Like

and hence converges. For an undersaturated or saturated vapor, both numerator and denominator of Eq. (76) become large for large ii, but -their ratio Is of order N .

If tim N = 0, since, physically, a

drop-fl _n-'cn fl

let of infinite size cannot be found in any finite volume, then it can be

shown that 3 : o for undersatürated or saturated vapors, as

antici-pated.

Equation (77) 'can also be obtained readily from the electric

cir-cuit analogy. If the circuit of Fig. 1 is in a steady state, the charges on the capacitances remain constant, so that the capacitances can be

omitted without changing the physical situation, leaving just a voltage

0 - 0

1 across the series resistances R .

The electric current

1 U

flawing is evidently 1/ R, assuming that the summation converges.

n1

The electric current differs from the "droplet current" by the factor V, so that Eq. (77) is again obtained. (It might be noted that for the

steady state, since V constant, one can set V 1 without loss of generality.) This simplified electrical analogy was discovered by Becker and Daring.

The value of V R needed to calculate J can be found with the

n1

help of Eq. (55):

v'[Rn

K6T'1'4/

r8

i3T

_n'e A i1 ex1[A(n2/3enY3)J/. (78) "-'I,,'

For large values of n, _n 1 (see section 2. 3), and therefore p1 by Eq. (3). Equation (78) would be simplified if

=

n'1

_{even for small values of n; this corresponds to}

= (1 13)_1 (1 A curve for this expression is

shown in Fig. 4; it increases from 0.25 at n 2 to approach 1 ásymp-totically as n approaches infinity. This is not an unreason&ble

as-sumption for a. _ri' when more accurate information is lacking. Ifon the other hand, it is believed that actually u1 = 1, the approximation

l9 n 2/3 is equivalent to neglecting the radiva of the vapor

mole-culea compared to the droplet radius, and neglecting the thermal motion of the droplets.

-

3?-VR

KT'NIBL

/

1e8

(A -

f)lonJ].

(7Q)

The terms in this summation have a maximum for a certain value of n

which will be designated by

n.

and which can be found by equating the

derivative with respect to n of the exponent in Eq. (79) to zero:

-Lo*-4(fl-6)A-f)O

(80)

"*

fl0[/-36?

*(A-f),'3*...]

(81) where (s/o4.J co

leA

3 (82)

The quantity n differs from the 'critjcal size," n defined previously.

in that n is defined to maximize a quantity which has a&edditlonal

term (A -

log n compared to the corresponding quantity for n. In

the Becker-DBring approximation. A 2/3 and thus n* ₌

n. which

result can also be obtained by comparing Eqs. (45) and (81).

l3ecauae of their exponential character, the terms in 1q. (78) or (79) decrease rapidly on either side of n, so that only the terms near

the maximum need to be considered to obtain a reasonably accurate value of 1. Moreover, when Is not extremely small, the initial term ZE/(1 1N1) and the next few terms are negligible compared to later terms, so that the calculated value of S is practically indepetdent

2' etc. ,,Thus, whether theee coefficient, fur

n S are closer to unity or to lO ordinaY1y make a little difference.

however, If they should be as srnll as 10' the reduction In J would be appreciable for high saturation ratios when n is of the order of 10 or 20.

U a simpler expré.sioii for. 3 Ia desired, the ttnati n 1nq.

(79) my *e approxuzzated by the corresponding integral, with a* error which is small when n is reasoitably large Denote by 1(n) the tetins

in tie summation, he.,

It is convenient to introduce a new variable. x. by the equatIon

so that x 0 when n. nd,, and f(n) = maximum. In term$ of thia variable,

U )q. (80) Is multiplied by 3n2x add subtracted from the exponent of

Eq. (85), thelatter reduces to

i' f-'

(3x.#3xx9),oq*Ar,zXxe.

-6n(3A -

e)i09 (/1k]

where

by virtue of Eqs. (81) and (8Z). Hence

00 00

2

=

-39-*3A

/(f#)_(3A-)X]cLx

(88)

The integrand on the right-hand side of Eq. (88) haa a value of unity at x = 0, while it has a value less than 0.01 when lx 1/2, assuming the

usual situation that n log a ) 10.

Thus, with an error usually less

than 1%, Eq. (88) can be written

00

f(n)dn =

3n*(n)fex fXxe-nx,

x - A .]dx

frfex(A--n

ioq)x+

-Ye ezpI_( A

)'Jcx.

(89)

The exponent In Eq. (89) was choSen SO

as to make the x term in the

brackets vanish. The contr1button to the integral from the x,

terms are zero 1y symmetry. Utlie

x4 x6,

..

terms are

neglected, only the exponential remains under the integral sgne The

(87)

so that'

f(n )dh

=

=

The resulting value of

S is

to a good approximation.

The expression for 3 then becomes1 Ifthe Z8/( 1N1) term In Eq. (79) is neglected1'

'

v

'/2

(K UT eN)

43r,

I-

Ari"

-A-eJ

-'

37T

_J

(n)As

(91)

EquatIon (91) corresponds to the simplifying assumption that

2/3

_{in the region n- n. If the different aesumption that}

lismade, eothat

the

the value obtained for S sill be slightly larger. Such an aanmp

tion corrsponde to multiplying the integrand of Eq (89) by a factor

(1 +

.1/3) 2(1

+ n_1)

/2

= (1 +

4/3)Z

+ x

z_l/3

x

(92)

(K5TfV3AT)4Jn

/oq"

-n:<-A-e-e41

initead ci the vsIe given b?

q. (91)

It ii ot ijterest to compare Eqa (91) and (93) with the result obtained by Becker aud Dfr1*.5 They obtMned in the present

(94)

were Interested only 4o a vsrjt rough answer. Becker and

Diug.droppeda tact r equal to é/5. which gen*aUy hós a

zuagni-tude of 10 to 1000. in situstto*s of physical Interest. Aside from this ctor Eq. (4} La a good approximation to Eq. (91) when A = 2/3.

&

Equation(94) differs from Eq. (92)

.1 -z

by an idditional factor, (1 + n

'

} , because Becker end D8ring

neglected the finite radius of the vapor molecules In computing capture

óollisjozts

The inmber of droplets of a given else, in the steady-state

The atesdy-state solution has been derived ftbove for the liflear-med Eq. (iS).

The effct of the nonli ear termà inthe more

(14a) will be to increase J slightly, since both linear and nonlinear terms tend to make the approach the equilibrium value I (see

section 2.9), and this result.e in a "current flow" toward tarer values

of n. On the baia of a numerical ca1culaton. discussed later, the

contribution of the nonlinear term. to 3 t. estimated to b oily a

few

per

nt in most practicalcasea.

Thus far, the steady state has been discussed as a poSsible

mathematical solution to the condensation equations, withont inquiring as to whether such a state is physically

iealisable,

If a .upersaturated vapor were maintained at constant voinme and temperature, and all droplet. which grew very large were removed and replaced by an equal mass of -vapor molecules, it is evident that the system, in time, would approach cb a. steady state In the electrical analogy, this corre-eponds to supplying a constant potential at

and grounding a ps.ilar

Ø (ii large). Because the decrease rapidly beyond the critical n.

-43-Such a physical situation, involving constant removal of droplets and addition of vapor, seldom, if ever, occurs in practice. Previous ixwestlgators, however, have used the, steady-state value of .1 to determine the occurrence or nonoccurrence of'condensation in many types of unsteady physical situations. Because of the exponential factor in Eqs. (91) and (92). a small variation in temperature or

satu-ration ratio generally causes a large variatipn in S. so that in many practical situations very rough approximations

in S are permissible.

It Is not obvious, however, that typical unsteady condensation situ-ations can be even roughly approximated by a steady state, and the arguments of previous investigators on this point have been vague and rather unconvincing. This question will be discussed in the next sec-tion.

Z. 11. Time-Dependent Solutions of the Condensation Equations

In a general condensation situation, the volume V 8hould be a known function of time, and the temperature T either known directly

(as when the system is In a thermal bath) or determined by adiabatic expansion laws, radiation laws. etc. or determined by these laws plus terms due to latent heat released, if this effect is significant. In the first two cases, the condensation Eq. (15) is linear with variable coefficients& in the last case it Is nonlinear. In all of these cases, the

electric circuit analogue involves resistances and capacitances which

vary with time. k general, the

only recourse appears to be to solve

the differential equations numerically, and for this purpose Eq. (10) La probably better suited than Eq. (15).

There exists, however, a particular nonsteady condensation situ-ation which is considerably simpler to treat than the moat general

is suddenly brought to a super saturated state (by adiabatic expansion or otherwise), so rapidly that negligible growth of droplets occurs during the tranaltion period, and thereafter the volume and tempera-ture of the vapor are held constant while the droplet-growth process

occurs. The quantities C andR then remain constant during the

period of significant condensation, and the nonsteady character of the Process is reflected only In the specification of nonequillbriu.m Initial values of the dependent parameters, Ø(t).

The assumption will be made that, before the vapor Is cooled to supersaturation, it is first held In a particular undereatuxated or saturated state Long enough for the corresponding equilibrium distri-bution of droplets to be obtained. (Since N Is generalip negligible under these conditions when n, 7 10, the time required is only that nèceesary.to establish equilibrium among the droplets having n 10,

-which is usually less than 10" sec.) The number of droplets in this equilibrium state is given by Eq. (69) or (70).

If, at

t = 0, the volume

and temperature of the vapor are changed so suddenly that the total number -of droplets, V of each size, is conserved, the new number of dropleti per unit volume will be given by

N (0)

1ex -(;)

(97)

n

.45..

where a bar is used to distinguish the para ters of the old under-saturated or under-saturated state (V. N1, T. etc.) from the corresponding parameters of the new supersaturated state (V, N1, T, etc.). The

initial values of.Øn can then be obtained with the help of Eqs. (11) and

(55):

, tn-_f

/

0 0 O

J=(yJ

eXPI (9&a)

( v8)Lo.)

ei2[ (A -A)n

(AE-ë)n/,(,ii)

(98b)

Like N0(0), the values of Ø(0) decrease rapidly with increasing n, and

generally become negligible for n 'lO. For the significant cases wbe.

n. 10, Eq. (98b) may be only a rough approximation to the exact Eq. (98a); however, it wifl be shown below that a considerable error in the initial values, Ø(0). usually produces only a small chaige in the calcu-lated' condensation process.

During the subsequent condensation process, since V and 'I' are held constant, the parameters C end R remain constant as long as -the total amount of condensation is so ma1l that there is negligible

depletion of the vapor molecules, i. e. negligible decrease in N1. Eventually, in a physical condensation situation, the vapor depletion will become significant, and also the latent heat released on conden-sation will become large enough to affect the temperature; however, it will, be shown that by this time virtually all of the droplets have grown to a sir.c sufficiently large so that their further growth can be calculated by means of simple macroscopic diffusion and

condiction relations.

In treating the condensation1 it is convenient to introdice a di-mensionlese time parameter:

'r=KTN,t

and rewrite Eq. (.15) to read

dø

dv

_a,,.Ø,-(a#617)Ø*bØ ,.(nt

17 fl-I

where the and R have been combined into the quantities

= T '/eA. ç, _, (ri >. / (101)

-' -,

=

[KT'A

_R,71]

,,,, t

expk

.

I

4'(,, )exp [A

n- /1

(P7-IY-eet n

(n >,

-I.

(r08r0\

exp(

T

I

(99)1

(100)

In the Becker-DJring type of approximation. p1 and the

energies are givec by Eq. (63). so that the above expressions simplify

to

=fle/3,(fl>I),

niex

(fl.>i),

(iO4)

tSee1 for inetancC, Caralaw and Jaeger,25 pp. 83-86.

t=ec'exp[4A (- ee)j.

(104b)

Because of the rather complicated variation of b with n (even inihe 'simplest case when 0), it has not been possible to find a general analytic solution for the infinite set of Eqs. (100), with

n

L 3.. 4

..

, or even a particular solution which satisfies the

necessary condition I. One must therefore turn to analytic ap-proxirnationa or numerical methods. A possible approximation is the neglect of eli 0's

beyond a certain Ø. If the a in

is chosen to be considerably larger than ten, one may expect that there will be a cer-tain period of izne during which this approximation is valid, since only a. negligible number of droplets containing more than ten molecules is' present initially. The length of this period may be esthnated by noting the time during which Ø, the last term not.dropped. is negligible. If

ia chosen o be more than twice the critical size, will always be

negligible compared to the 0's near critical size so that the entire be

-haior of droplets near the critical size can be obtained. The

con-tinuing rate of growth of droplets beyond the critical size, however, can be obtained only by increasing n so that it always exceeds the droplet sizó' of Interest.

With tale appraimation. Eqs. (100) 'become a finite set of first-order linear differential equations with constant coefficients, which can be o1ved by standard rnettiods.* The 'steady-state solution is first' sbtracted off to remove the nonhomogeneous condition on the re-mainder of the solution is then of the forrn c1 e - where the C1t5

possible to derive a recurrence r*lation for the secular 4etermlnanta of auccessle orders which considerably simplifies the determination

of tbe'e. When the

's were compute for a typical case having

50, with.tbe help of a punched-card computing.machine..an even more .eerious diffifulty arose. Lt waS found that the summations ex-when n must be of the order of 100 to cover the critical region, this procedure th.volves a vast amoint of coznpxtation. It has been, fomd

pressing, the desixed 0's contained many large terms which nearly '.,cancefled to give totals several orders of magnitude smaller. In

some cases, eight to ten signiftcant figures had to be c&rriedin.inter-muedint calc4attoni to obtain results correct to. 50%.

urthér

in-vestigatton BhOweCl that this difficulty arises because of the large van-ationin magnItude of with .n which is Inherent in the condensatIon

theOry. Calculation of the 0's to a given degree of accuracy i much

more labor ious when this' approximate analytical method is used than when the numerial. method described below is used. Therefore, ftr-ther detatli of tbi8 analytical method will be omitted.

Another type of analytic approximation reeulte from aUowin the. variable. u, originally defined as dlcrete, to take on continuous values. This proáeduxe was followed by Zeldovich7 in treating the simple steady. state situation. if a(n), b(n) and Ø(n. t) are any functions of the con-tinuous' variable n(2 a ) wilch take on the values a. and Ø(-c)

defined by Ecjs. (100). (101) and '(ii), 'respectively, when ñ is an integer. and if Ø(n,v) aLso possesses partial derivatives of aU orders with

Inorder to reduce Eq. (106) to

tractable form. it ia nece8ary to

maie some a.eumption about the magnitude of the higher derivatave In sevçràl direct nui,%erjcál integration$ of q. (100) which have been carried out (see below), It has been found that, through most of tue

range of a and v..

the quantities Ø behave approximately Like a

de-ea3inggeometriC$eTi with.a ratio between

i/z and 1/4.

it follows t1at Ø(n t) is roughly

proportional to e, and therefore

all the deriva-tives .wtth respect to a are approximately of the same magnitude.

-quation (106) can then b approximated by

b)

error of thaorder

of 10%.

Zcp.ation (107), an It stands, represents no significant sizuplifl-cation of the original Eq (100), becauae the functional form of b(n) is too complicated to permit iin&ng an analytic

otution. The first step

1, a2Ø

[a.(n)4- b(i)J

fa(n)b(n)Jf'

. .

(106)

become a

It is seen fi'omn the dimensional form of Eq. (108) that, anycharactertetic condensation time. j which happens to be determined mainJy b the growth processes In the neighborhood of the critical sise should be proo

portional to

n1.

Moreo'Qer. a solution to q. (108) with the boundary

conditions 0(1, 'v) 1, Ø(ti, 0) 0 can be obtained immediately by com

parison with the eqiivaletit heat-conduction

problem;25 itt

According to Eq. (109), O(n,.t) at the critical: sie:w1li réh a

v4tue

of 0 5. which 18 approximately Its eteady-ststw value, at a time

r= 1

1 This result as likely to be a poor approxiifl*tion because

of the boundary condition applied to Eq. (108) at the point a I where the neglected terms are large. An equation similar to Eq4 (109) has

'recently been obtaie.ed

In order to obtain a reasonably accurate picture of the variation the continuous vaçiable a by a discrete variab1e which can beiat be

done by use of Eq. (100) rather than Eq (107). The only regxon wuere Eq (107) appears to be uiefui is in the region near the

critical sIze

where R hae a rnaxixium, so that R

Then, a(n)2b(n) by