Cluster definitions in nucleation theory

Pergamon

J Aerosol Scl, Vol 25, No 4, pp. 683-688, 1994 Copynght © 1994 Ehewer Saence Lid Pnnted m Great Bntmn. All nghts reserved 0021-8502/94 $700+000

0021-8502(94)E0011-L

CLUSTER

DEFINITIONS

IN NUCLEATION

THEORY*

J. C. BARRETT Department of Nuclear Science and Technology, Royal Naval College, Greenwich, London SE10 9NN, U K.

(First received 23 July 1993; and in final form 20 January 1994) Al~raet--Most theories of nucleation reqmre the evaluation of the partition funcuon q, of a cluster of i interacting vapour molecules anywhere within a container of volume V. Computer simulations and empirical models (such as the classical liquid droplet model~ give the partmon function for a cluster stationary in the laboratory frame, q~,b. Previous ways of relating these two quantities are reviewed and shown to omit some arrangements that should be counted. A new approach, applied to both cubic and spherical clusters of non-interacting molecules leads to the relationship q,_ i3(V/vc) qllab(where v¢ is the cluster volume). The resulting nucleation rate is enhanced by a factor typically between 107 and 109 compared to the classical rate. 1. I N T R O D U C T I O N

When particle-free air containing water vapour is cooled, pure liquid droplets can form, a process known as homogeneous nucleation. This process is important whenever vapour-gas mixtures are cooled very rapidly, as would be the case in certain conceivable (although extremely unlikely) loss of coolant accidents involving a pressurised water reactor. The initial number of particles formed may significantly influence the transport of the aerosol through the containment and its possible subsequent release to the environment. However, nucleation remains a poorly understood process and theories predict vastly different rates of droplet formation from those observed experimentally (see Oxtoby (1992) for a review of the current theoretical and experimental position). A rigorous theory of the behaviour of non-ideal gases has been given by Mayer (Mayer and Mayer, 1940). However, most theories of nucleation adopt a different approach, relating the nucleation rate to the partition function (or a related quantity) of a single cluster of interacting vapour molecules. These two approaches are reviewed in section 2. The single cluster partition function may be obtained from simple models or from computer simulation. In general, however, both these approaches give the partition function for a cluster stationary in a container (independent of the container volume V) whereas what is required is the partition function for a cluster anywhere within the container (which is proportional to V). In section 3 the various ways of relating these two quantities are reviewed and their inadequacies pointed out. A new method is then described which counts arrangements omitted by other methods, leading to an enhancement of the nucleation rate. Numerical values for this enhancement compared to the classical rate are presented in section 4 and the conclusions of this study are summarised in section 5. 2. S T A T I S T I C A L

MECHANICS

OF IMPERFECT

GASES

The starting point of Mayor's theory of imperfect gas behaviour is the Canonical partition function for N interacting molecules of mass m in a container of volume V at a temperature T,

QN=Nv f. . . f exp[-U(rx,r2 . . . . .

rs)/ksT]drx .....

drs,

(1)

where A = (h2/2~mks T) 1/2 is the de Broglie wavelength and U(rl ,r2 . . . . . rs) is the interac* Paper presented at the 7th Annual Conference of the U.K. Aerosol Society (Bristol, 1993). 683

684

J C. BARRETT

tion potential between the N molecules. Mayer showed that, for a pairwise additwe interaction, U = ~ , . j ~b(tr,-rjl), QN can be written as, I-[ (vb')N'

QN= ~N~ , l N,!

(2)

subject to ~ iN, = N. The cluster integrals b, are given by,

b,=~

...

~l-ILk

dr1 . . . . . dr,.

(3)

The sums and products in equation (3) are over all connected graphs of z points, see Mayer and Mayer (1940) for more details. The f:k are given by fjk=exp(--(k(]b--rkl)/ ks T ) - 1 and so vanish for finite range forces when the separation of molecules j and k exceeds the range of the interaction. Generally, the integrals in equation (3) must be evaluated numerically and in principle this can be done by fixing molecule 1 (say) and allowing the remaining ( i - 1) molecules to move throughout the container. For finite range interactions, any positions where a molecule moves out of range of the interaction do not contribute since some of thefj~ then vanish. Finally, the integration over rl gives a factor V. In practice, however, the evaluation of b, for all but the smallest values of i is very complicated due to the large number of terms in the sum over connected arrangements and the multi-dimensional integrals over 1/". A different approach is usually adopted in statistical mechanical treatments of nucleation (Kuhrt, 1952; Reiss et al., 1968; Lee et al., 1973). A supersaturated vapour is assumed to consist of groups or 'clusters' of interacting molecules. The molecules in a cluster do not interact with any molecule external to it and the interaction potential for a cluster containing i molecules is written U~(rl,r2 . . . . . r,). The partition function for the whole vapour can then be written in the form of equation (2) but with (Vb,) replaced by the cluster partition function q, given by q,=~

lffe ...

x p [ - U,(r 1,r2 . . . . . r,)/kB T ] d r l . . . . . dr,.

(4)

Once the q~ are known, the equilibrium number of/-clusters, can be found from the law of mass action

n,=(N/ql)'q,.

(5)

Equation (4) is apparently much easier to evaluate than equation (3) since it involves only a single term. However, the integrand in equation (4) does not vanish when molecules become detached from the cluster. Such arrangements should not be included when evaluating q~ since they are already counted in terms involving q~ for j < i. To evaluate the integral, it is therefore usual to assume that the molecules lie within a small (spherical) volume v¢=~nr~, chosen to be large enough that all arrangements with a large negative interaction energy are included but small enough that spurious arrangements are excluded. The partition function for such a cluster stationary in the laboratory frame is q~"b=i!A3' , c ' ' "

c exp[-- U,(rl,r2,.. ., r,)/kBT]dr~,.

., dr,.

(6)

If the interaction potential U, is known, q~,b can be calculated by Monte-Carlo simulatmn. It is also the quantity given by simple models of clusters such as the classical liquid droplet model. For a 'perfect gas cluster' of/non-interacting molecules in volume v~, q~,b= v,/i!A3,.

3 R E L A T I O N S H I P BETWEEN q~ab AND q, To obtain q~ from q~,b it is necessary to move the cluster throughout the container volume, without counting contributions that have already been included. Several ways have been proposed of doing this. Abraham (1968) fixed a molecule (molecule 1, say) at the centre

Cluster definitions in nucleation theory

685

of the cluster and then allowed this molecule to move throughout V. Since the density at position r within the stationary cluster is given by m,

o(r)=i!A-ri-ql,b

o...

o exp[-

Ui(r, r2. . . . . ri)/kBrldr2 . . . . . dry.

(7)

Abraham's qi can be written

q~=0(0.) Vq~ab.

(8)

ml

For a uniform density cluster, qi = q]*b V/vc. However, this approach does not count all arrangements of the molecules which do not have a molecule at the centre, nor are all arrangements with molecules other than that labelled 1 at the centre counted (some such arrangements are counted, since it is often possible to arrange more than one sphere of radius rc around the molecules). Reiss et al. (1968) proposed a different approach. They argued that a cluster should be defined as a sphere centred on the centre of mass of the molecules, which is then allowed to move throughout the container. Since different positions of the centre of mass necessarily correspond to different arrangements of molecules, they argue that this procedure avoids double counting of arrangements. For a perfect gas cluster, it is possible to show that the relation between qi and q~ab with this cluster definition is (Lee et al., 1973) qt ~ a (i) i 3/2

Vq~ ab,

(9)

where a(i) is a weakly increasing function of i, with asymptotic value (250/97~) 1/2 . However, there are some arrangements counted in q]ab which are not included in equation (9). One such arrangement is illustrated in Fig. 1. The centre of mass of the molecules shown (which lie within a sphere of radius re centred on P) is at P', but the sphere of radius ro centred on P' does not contain all these molecules. Therefore, this arrangement is never counted in the Reiss treatment. A treatment which includes all arrangements for a 'perfect gas cluster' in which all molecules lie within a cube of side re is quite straightforward. The faces of the cube are taken to be perpendicular to the coordinate axes. A molecule is chosen to have the minimum x value and can move through a distance L (the x-dimension of the container). The remaining ( i - 1) molecules must lie within a distance re of this molecule, so the integrations over the x coordinates gives a factor Lr~~- 1). The integrations over the Y and z coordinates yield the same factor. Finally, there are i ways of choosing the 'end' molecule in each dimension (since the same molecule may have both the minimum x and y values, say, in an arrangement) so the final expression for the partition function can be written,

i3 L3 r3.- 1)= i3 ._Vqlab' qi =

i!A3i

(10)

vo

where V = L 3 and vc-- ro3 for a cubic cluster. It should be noted that in evaluating the integrations over molecular positions, the molecules are treated as being distinguishable; the factor i! before the integrals in equation (4) accounts for their actual indistinguishability.

Fig. 1. An arrangement of molecules within the solid circle, centre P, with centre of mass at P'. The dashed circle centred on Iy does not contain all the molecules in the cluster. AS 25:4-G

686

J C BARRETT

For a spherical perfect gas cluster, the treatment is somewhat more complicated. The first problem is to define a unique sphere enclosing the i molecules, since in general there will be infinitely many spheres around each arrangement. This can be done by specifying that the sphere of interest is the one with the minimum z coordinate, for example. This sphere must touch at least one of the i molecules in the cluster and the cases where it touches one, two and three molecules are treated in the Appendix. Cases where more than three molecules touch the sphere contribute negligibly to the volume integrations m equation (4). The total volume integration is the sum of these three contributions so the relationship between q, and q~,b can be written: q,=ig(i) V q ~ "b

(11)

Vc

where 1 3~2 g(i)= 1 +3(t-- ) + - ~ - (i-- 1)(i- 2).

(12)

For values o f / o f interest in nucleation (i> 10), 9(i)~-0.9i z. 4. THE NUCLEATION RATE From equations (5) and (11), the equilibrium number of i clusters can be written: n,=--,g(i)q~ "b - - = P_~_co(i) N q l "b - , vc \ql / ~pve L \ql } J

(13)

where the final form follows from V / N = m/(Spve), the volume per molecule in the vapour, and vc/i = m/pt, the volume per molecule in the cluster, taken to be that in the bulk liquid (P,e is the equilibrium vapour density and p< is the liquid density). Given a model form for q~,b, the nucleation rate can be found from the equilibrium distribution in the usual way. For the classical liquid droplet model, the term in square brackets in equation (13) gives rise to the classical nucleation rate. The/-dependence of g(i) in equation (13) will cause a slight change in the critical cluster size compared to the classical value, but this effect is small (changing the nucleation rate by less than 10%) and to a very good approximation the nucleation rate derived from equation (13) is equal to the classical rate enhanced by a factor f=9(i*)p
(14)

where i*=(32n/3)(m/pe)Z(a/kTlnS) 3 is the classical critical size (a being the bulk liquid surface tension). Since values i* are typically between 10 and 100,fmay vary from 105 to 101° or more. Figure 2 shows the variation o f f with temperature for both water and nonane. As the temperature increases, the saturation decreases to keep the classical nucleation rate constant. However, the equilibrium vapour density Pv, increases rapidly with increasing temperature so the net effect is a decrease o f f with increasing T. 5 CONCLUSIONS A rigorous theory of the statistical mechanics of imperfect gases revolves the evaluation of the duster integrals b~ (equation (3)) which only include contributions from positions where molecules are attached to a duster. Due to the difficulty in evaluating the b,, the usual procedure in nucleation theory involves evaluating the partition function of a cluster defined in a small spherical volume. All distinct arrangements of molecules in this volume as it moves throughout the container must be considered to give the cluster partition function q~. This procedure leads to an enhancement of the nucleation rate by a factor proportional to the ratio of the liquid density to the density of the supersaturated vapour. The factor also depends on the critical cluster size, i*. An approach suggested by Abraham (1968) gives a factor proportional to (i*)- 1, whereas that proposed by Reiss et al. (1968) gives a factor proportional to (i.)1/2. However, both these approaches omit some arrangements and

Cluster definitions in nucleation theory

687

109

108

10 7

......... 230

L. 240

. . . . . . . .

I ......... 250

I ......... 260

i ........ 270

i . 280

. . . . . . . .

i

.........

290

i ....... 300

310

T (K) Fig. 2 Enhancement of classical nucleation rate,f, vs temperature for water (solid lines) and nonane (dashed lines). The upper line of each pair gives values o f f for a classical nucleation rate of 1 cm- 3 s- ~ and the lower line is for a rate of 104 cm- 3 s- L

a treatment that includes all possible arrangements of non-interacting molecules in a cube or a sphere gives a factor proportional to (i*)2. The resulting enhancement of the nucleation rate is typically between 107 and 109 . The approach adopted here is only valid for dusters of non-interacting molecules (or, more generally, ones in which the interaction is independent of molecular position). This is also true of the treatment by Reiss et al. (1968). An important extension would be to treat position-dependent molecular interactions, which are likely to have a significant effect on the results. Experimental nucleation rates tend to be higher than classical theory predictions at low temperatures and lower at high temperatures. The theory presented here does not improve agreement between theory and experiment, except for some substances at low temperatures. Nevertheless, it is likely to form an important element in a more accurate theory. REFERENCES Abraham, E. F. (1968) J. appl. Phys. 39, 3287, 4791. Kuhrt, F. (1952) Ze~tschriftffir Phys. 131, 185. Lee, J. K., Barker, J. A. and Abraham, F. F. (1973) J. Chem. Phys. 58, 3166. Mayer, J. E. and Mayer, M. G. (1940) Statistical Mechanics. Wiley, New York. Oxtoby, D. W. (1992) J. Phys.: Condens. Matter 4, 7627. Reins, H., Katz, J. L. and Cohen, E. R. (1968) J. Chem. Phys. 48, 5553.

APPENDIX In this Appendix, the volume integrals over possible positions of the molecules on a spherical shell gath radius rc and volume v~ are evaluated. The centre of the shell is at (X, Y, Z), where Z has its minimum possible value. The general Idea is to transform from the coordinates of the molecules touching the sphere to those of the sphere's centre and the coordinates relative to the centre. A complication is that only certain positions of the molecules on the bounding sphere should be included to prevent overcounting.

O) Sphere touching one molecule The coordinates of this molecule, (xl, Yl, zl), are related to (X, Y, Z) by xi = X , Yl = Y and zl = Z + r c . Transforming from an integratmn over x t, Yt, z~ to one over X, Y, Z and allowing the sphere's centre to move throughout the container volume gives a factor V. The integrals over the remaining if-l) molecules, which all must he wRhin the shell, give a factor v~(~- 1). Finally, there are i ways of choosing the molecule that the sphere touches so the total contribution from these arrangements is iVv(~'- 1).

688

J C BARRETT

(n) Sphere touching two molecules The centre of the sphere with minimum Z passing through two molecules must lie on the verUcal plane (Le parallel to the z axis) passing through the two molecules These molecules are labelled "1" and "2" and their azimuthal angles 41 and 42 relative to the sphere's centre are related by 4 ~= 7r+ ~b2 The Cartesian coordinates of the two molecules can therefore be written

xl=X +r~slnOlcos~pl, vl= Y+r~smOlSm~pl, zl=r, cos01 x2=X-resln02cos41, ~,'2= Y-rcsmO2sm4~, ,zz=rccosO2 The integrations over the position coordinates of molecules 1 and 2 become

ff

drxdr2 =

f

fr3[sln201cos02

+ sin01 sln02[cos01 +cos02)+cosOlsln202]dX

dYdZ

dO1 d02 d41, (A1)

where the terms in square brackets arise from the Jacobian for the transformation from the six variables (xa, y~, za, x2, Y2, z2) to (X, Y, Z, 01,4~, 02) Integratmn over X, Y, Z gives a factor V as before, but some care must be taken in deciding the limits for the integrations over the polar angles of 1 and 2 Molecule 1 must he in the hemisphere above the sphere's centre, so 0t varies from 0 to ~t/2 (and 4~ from 0 to 2~) Molecule 2 must lie above the diameter through molecule 1 (otherwise the sphere can be rotated to a position with a lower Z value just resting on molecule 1, and such arrangements have already been included in (l) above). Thus 02 varies from 0~ to ~ - 0 : The Integrations on the right-hand side of equation (All therefore give rise to a factor 4~r~V There are i(l-1) distinct ways of choosing molecules 1 and 2 from the ~ molecules in the cluster (interchanging 1 and 2 gives a distinct arrangement because the first molecule is always above the second) so the contnbutmn from these arrangements is 4~r~ Vi(I- 1). Finally, integration over the positions of the remaining t - 2 molecules which must lie within the sphere gives a factor vU 2 so the total contribution is 3 V t 0 - l/vU

(uI) Sphere touching three molecules The first step is to express the coordinates of the three molecules m terms of(X, Y, Z) and their polar coordinates relative to the sphere's centre:

j=

xs=X +rcslnOscosdpj, yj= Y+rcslnOjsln4s, zj=r~cos0j 1, 2, 3. The Jacoblan for the transformation from (xl, A'l, zl, x2, Y2, z2, ~c3,Y3, zs) to (X, Y, Z, 01,41,02,42, 03,

43) IS

Ir~ sin01 sin02 sin03 [sin01 cos02 sin03 sin (41 - 43) + cos01 sin02 SlI]03Sln(43 - 42 ) -~ sin01 sin02 cos03 sin(42 - 41 )][

-[r3sInOlsmO2smO3(r'l

A r 2) r'3[,

where ^ denotes the vector product and r'j is the position vector of molecule j relative to the sphere's centre. The integral over the Jacoblan Is most easily evaluated in terms of 0x2, the angle between r'1 and r 2, and 03, the angle between r'3 and (r'~ ^ r'2) In this coordinate system, the integral becomes

ff

dr,dr2dr3 =

f

fr~lsinO,2cosO3s,nOlsmOxzsInOaldXdYdZdO, d0, 2d03d4~d42d43

(A2)

Evaluating the integrals over all possible angles gives 8~ 4 However, this includes arrangements which do not have the minimum value for Z. Excluding these arrangements gives a value of 1/8 of the total One way of seeing this is as follows: consider a plane through the centre of the sphere and two of the molecules (those labelled "l" and "2", say). Molecule 3 must be above this plane if the sphere's centre is to have the minimum possible Z (otherwise it would be possible to rotate the sphere about the line through molecules 1 and 2 to a position with a lower value of Z). For every position r~ of molecule 3 that satisfies this condition there is a corresponding positron where it is violated (at - r~) so only one half of all positions of molecule 3 should be counted The same argument apphes to molecule 1 (relauve to the plane through molecules 2 and 3) and to molecule 2. The net reductmn is therefore a factor of 1/23 and the required value of the integrals m equation (A2) is ~4r6 V=97~2/16v2 V All arrangements of the three molecules lying on the sphere have been included In this treatment. There are l ( i - 1 ) 0 - 2 ) / 3 ! distinguishable choices of these molecules from the i molecules in the cluster IntegraUon over the positions of the remaining z - 3 molecules gives a factor v~-~, so the total contribution from these arrangements is 3n2/32 VIO- 1 ) 0 - 2 ) v U 1