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Gas phase nucleation: Steady state rates for two cluster models
SURFACE
SCIENCE 26 (1971) 1-13 0 North-Holland
GAS PHASE STEADY
STATE
Publishing Co.
NUCLEATION
:
RATES FOR TWO CLUSTER
MODELS
J. J. BURTON Henry Krumb School of Mines, Columbia University, New York, New York 10027, U.S.A.
Received 27 November
1970; revised manuscript received 19 January 1971
Exact thermodynamic properties of small clusters were computed from a model of argon. These thermodynamic properties were used to calculate steady nucleation rates. Steady state rates were also calculated using the drop model to approximate the properties of small clusters; the parameters required in the drop model were determined from the same model of argon used in the exact calculations. The exact model nucleation rates were lower than the drop model results by as much as 1016. The exact calculations suggest that the critical cluster size is nearly temperature independent, while it depends strongly on temperature according to the drop model. It is concluded that the predictions of the classical drop model are both qualitatively and quantitatively incorrect. Inclusion of the Lothe-Pound correction for cluster rotation and translation in the exact results causes the exact model nucleation rates to be nearly equal to the predictions of classical theory (neglecting rotation and translation); this may explain the fact that nucleation rates in many simple systems are correctly predicted by the classical theory.
1. Introduction Steady state gas phase nucleation rate have been studied by a number of investigators1s2). In calculating nucleation rates, it is necessary to assume a model to describe the thermodynamic properties of small clusters. The conventional cluster model is the “drop model”. This model is based on the early work of Thomson3) and HelmholtzJ) on the equilibrium vapor pressure of curved surfaces. The free energy due to a curved surface has more recently been examined rather rigorously by Tolman 5). These thermodynamic investigations are strictly applicable only to drops having large radii. The classical drop model has been questioned on a number of grounds. It neglects the rotational and translational free energies of the clusters’j). There is some uncertainty as to whether this error is significantT_ll). It also apparently neglects the fact that a finite body of N atoms has but 3N-6 vibrational degrees of freedome). This problem has also been the source of some discussionlz-15). There are other important difficulties with the drop model which are often overlooked. Hirth16) has pointed out that the drop model should not be
J. J. BURTON
2
applied to calculating thermodynamic properties of very small clusters; rather exact microscopic calculations should be made for use in nucleation rate computations. Benson17) showed energy of a small cluster is considerably
in 1951 that the excess potential smaller than that of a large cluster.
In particular, the excess potential energy per unit area is roughly 15% lower for clusters of 13 argon atoms than it is for a planar surface. Orianils) examined the energy of water clusters and suggested the drop model may err by 25%. Burton investigated the thermodynamic properties of clusters of 13 to 87 atomsrgpzZ). He found that the excess entropy per atom is not a monotonic function of cluster size ZO-~ZZ),and that the free energy function was quite unlike the predictions of the drop model”O). Waltonas) and Halperne”) have shown that detailed atomic considerations are very important in studying nucleation from the gas phase on surfaces. The above considerations suggest that it is important to calculate nucleation rates from exact properties of small clusters. We adopt the following view in this paper: In general, it is not possible to calculate from first principles the nucleation rate for any material. However, a formalism has been developed to calculate the steady state nucleation rate, subject to a number of assumptions, if the thermodynamic parameters are known for small clusters”j). This formalism has been widely applied to calculations of nucleation ratesI,“), assuming that the internal free energy of the small cluster can be correctly described by the drop model. In section 2, we review briefly the formalism. Recent work 19-22) has shown that the classical drop model for the internal free energy of a small solid cluster is incorrect, at least for a simplified model of solid argon clusters. The free energy problem is developed in section 3. There is no information now available about the exact free energy of small clusters of liquid. Therefore, we compute the nucleation rate for nucleation of solid argon from its vapor in section 4. This work is not, strictly speaking, a critical calculation of nucleation rates for comparison with experiment. Most experimental work is done on complicated molecules such as water or hydrocarbons and, generally, the liquid rather than solid is nucleated from the vapor. The work reported here is basically a bootstrap calculation in which we ask “what are the rates of nucleation from the vapor of a simple solid when the cluster free energy rates are not is calculated in two different ways”. While the calculated applicable exactly to a real material, they give a measure of the error introduced by using the drop model in calculating nucleation rates of real materials. 2. General nucleation theory Many
people have contributed
to the development
of the general
theory
GAS PHASE
of steady state nucleation.
NUCLEATION
3
This work has been reviewed by Hirthr),
and MacDonald 25). We follow here MacDonald’s nucleation in a dilute gas. A dilute gas consists primarily of monomers.
treatment
Feders),
of steady-state
Occasionally
a monomer
captures another monomer to form a dimer. This dimer may either capture another monomer to form a trimer or a monomer evaporates leaving the original monomer. The trimer may grow or shrink, and so on. For small clusters, the evaporation step is much more likely than the capture step. However, for large enough clusters, if the vapor pressure is sufficiently high, the capture process will be more probable than the evaporation process and the cluster will grow. Thus, nucleation depends on the production of large clusters through random fluctuations in the mostly monomeric vapor. This fluctuation is readily described mathematically. Let C, be the rate at which a cluster of II atoms captures a single atom. Let Z?,,be the rate at which a single atom escapes from a cluster of n atoms. We assume than C,, and E, do not depend on time. Let N, represent the concentration of clusters of IZ atoms in the gas. Then
dN,ldt = (G-1%1 + -%+,N,+I)- (GN, + JWJ. (This equation must, of course, be slightly modified current, Z,, between clusters of n and n+ I atoms as Z,(t) = GN,
- E,+,N,+,
.
for IZ= 1.) Define
(1) the
(2)
Then dN,/dt
(t) -I,(t).
= I,-r
(3)
Eq. (3) represents an infinite set of coupled linear equations. They have been solved numerically on computers 26-28). The equations can also be solved exactly in certain cases. The most interesting solution is the unbalanced steady state solution. This solution of (3) corresponds to formation of very large clusters (visible droplets) at a time independent rate. Let In(t) = I (4) be a constant for all IZand t. Here, Zis the unbalanced steady state nucleation rate. Define B,, to be the equilibrium Boltzmann distribution of n atom clusters at the temperature and pressure of the system. Then it can be readily shown25) from (2) and (4) that m
z-1 =
c
1
c,B, .
n=l
(5)
J.J.
4
Eq. (5) allows us to calculate
BURTON
I, the steady state nucleation
rate, if C,, and B,,
are known. Usually, in theoretical computationsz5), it is assumed capture rate C, is given by the ideal gas collision rate C,, = An*p(2nmkT)4,
(6)
where A is a geometrical constant such that An+ is the surface other terms have their co~l~entional meanings. B,,, the concentration of clusters of n atoms, is given by B, = tj (n) enilik?‘, where fl is the chemical potential function, s(n) is given by q(n)
and 4(0) is the partition
qint(n)9
where qt(/z) is the translationsi, qc(n) the rotational, internal partition function. Eq. (7) may be rewritten as B,, = e-
f
area and all
03
of the vapor
= 41(n) 4,(n)
that the
(8) and
cXi,,(~?) is the
enpihT PaI
=B,e
-AF(nfikf
(9b)
,
where Ffn) AF(n)
= - l
WW
= F(B) - nf -t (t - ~1)icT In(p,jp),
(fObI
andj’is free energy per atom in the solid at temperature T, p is the pressure, and pL’is the equilibrium vapor pressure. The central problem of nucleation theory has been the evaluation of F(n).
From (8) we may write Fjtz) = F,:I*fiz) -+ F;(n)
-I- F,(n).
(t1)
Here we write F’ rather than F to call attention to the fact that Fibt(n), the internal free energy is based on only 3n-6 degrees of freedom. F,(n) and Ft(n), the rotational and translational free energies are generally neglected (set equal to zero) in classical nucleation theory. The correct treatment of of considerable co~~troversy~-ll~, In the F,(rr) and F,(n) h as b een the source next section, we also neglect the rotational and translational free energies; their importance will be considered in the discussion at the end of the paper.
3. Cluster free energies fn classical nucleation theory, it is generally obtained from the drop modeIr~ ~85) according AF(n)
assumed that AF(n) may be to which
= A~z’o - nl
(12)
GAS PHASE NUCLEATfON
5
where A was explained in eq. (6) and CFis the surface tension (surface energy per unit area) and is independent of n. Physically, this expression
free can
be regarded as arising from the following two steps in forming the cluster. (1) The n atoms are transferred one at a time from the gas at pressure p to the infinite body. This step lowers the free energy by nkT ln(p/p,). (2) n atoms are transferred from the bulk to the cluster of n atoms. This raises the free energy by the surface free energy of the cluster, An’c. Eq. (12) is not correct for a very small cluster. The remainder of this section is devoted to the correct evaluation of AF(n) in eq. (lob) for small clusters of argon atoms. Though most nucleation experiments deal with nucleation of the liquid from the vapor, we study in this paper the nucleation of the solid. This is done because we are able to calculate the properties of small solid clusters but know nothing about liquid clusters. The properties of small clusters of argon atoms have been studied previouslylQ-22). No other materials have been examined exactly, though the general conclusions about small clusters may be applicable to other materialssi. 22). The discussion here is based on argon. The atoms were assumed to interact via a pair-wise additive Lennard-Jones potential of the form
where r is ZuckeraQ) Many-body have been properties we neglect
the interatomic distance. The constants used were those given by for argon. This potential is not an exact representation of argon. interactions may be significant so-a2) and other pair potentials proposedaa,Qz). Therefore, strictly speaking, we are studying the of a model and not the properties of argon. In our calculations, entirely anharmonic effects and lattice thermal expansion.
Properties of small clusters containing, 13, 19, 43, 55, 79, and 87 argon atoms have been calculated. These sizes are a central atom plus first nearest neighbors, plus first and second, etc. The atoms were originally arranged in an fee configuration and then allowed to relax to a potential energy minimum. The (3~2x 3~) force constant matrix was diagonaljzed on an IBM 360/91 computer to obtain the frequencies of the 3n-6 normal vibrational modes. Some results on the potential energy2*), configurations2Qva5), and entropiesZ0-22) of the clusters have been reported. The interesting part of AF(n), eq. (lob) may be written as is the internal contribution AF,,, (II) = Fi’“, (n) - nf = BE(n)
+ AEZ+(n)
(14a) - TAS(n),
(14b)
6
J. J. BURTON
where AE is the potential energy difference between the cluster and the bulk, AEz.p. is the zero-point energy and AS is the entropy. In writing (14b), it is important to remember that F,‘,,t(n) is based on 3n- 6 degrees of freedom while cf has 3n degrees. Therefore, AEZ,P, and AS have rather strange degrees of freedom. This does not introduce any difficulty if we are careful as (14) is still formally correct; however this has caused considerable discussion 6, rs+15) - the replacement correction controversy. AE(n)/n, AE,,,,(n)/n, and AS(,)/, are in table 1. No configurational contributions to S are included in AS. AFin,(n)/n was calculated from the data in table 1 and is plotted in fig. 1 TABLE
The
exact
excess
1
potential energy, AE, zero-point energy, A/Lt,., per atom of small clusters of II argon atoms AE/n ( X 10 t4 erg/atom)
n
13 19 43 55 79 87
and
~ 0.476 ~ 0.417 m-o.319 ~ 0.247 ~ 0.200 PO.214
I
8.0
I
AS,
AS/PI (k) atom
AEz.,./n (x 10~~r3erg/atom)
8.607 8.242 6.819 5.845 5.451 5.523
entropy,
0.18 0.41 0.69 0.49 0.43 0.57
1
0 ~-
40”
K
A--
60”
K
q -
80”
K
7.0 s L *T o
-
6.0
”
0
25
50
75
100
n
Fig. 1. The excess internal free energy per atom, AFf,,t(n)/n, relative to the bulk material, of small clusters of argon atoms at 40, 60 and SOCK. The heavy curves show the predictions of the drop model. The data points are the results of exact computations. Straight lines have been drawn connecting the data points.
GAS PHASE
NUCLEATION
I
at 40, 60 and 80°K. The excess internal free energy per atom according to the drop model, An-*,, is plotted in fig. 1 for comparison purposes. The value of the surface tension used in the drop model was 0 = 40.5 - 0.1 T erg/cm2 .
(15)
This is the average of the values for the (100) and (I 11) surfaces determined by Allens6) for infinite planar surfaces of solid argon; Allen used the same potential to describe the argon atom interactions as used in this work. Allen also neglected anharmonic contributions to the vibrational free energy. We use here his values of 0 at the lattice parameter appropriate to O”K, that is without thermal expansion. Thus the value of 0 in (15) computed for bulk argon is based on the same model used to compute the free energy of small clusters. Properly, one should not use such a simple way of estimating a; one should consider the details of the shape of the crystal, However, our averaging is justified by the fact that ~(100) and o(lll), as computed by Allen, differ at the melting temperature of argon by less than 0.5 erg/cm2 and are 3 erg/cm2 less than the value of 0 for the (110) surface. Thus the error in using (15) is probably significantly smaller than the uncertainty in most experimentally determined surface entropies. The exact excess internal free energy of the small clusters is clearly unlike that predicted by the drop model. It is greater than expected for the larger clusters and has a rather different dependence on cluster size. The exact excess free energy per atom can be described approximately by two straight lines, of different slopes, which intersect at 55 atoms per cluster.
4. Nucleation
rates
Nucleation rates were calculated for both the classical drop model and the exact treatment of the clusters. We examine first the classical results as these will be useful for understanding the significance of the exact results. 4.1. DROP MODEL Conventionally, the nucleation theorist and experimenter attempt to find the pressure, p*(T) at which the nucleation rate at temperature T is equal to 1 cluster per cubic centimeter per second. This pressure is known as the critical pressure. The ratio, p*(T)/p,(T), where p,(T) is the equilibrium vapor pressure at temperature T, is known at the critical supersaturation S*(T). Nucleation rates were calculated from (5). S*(T) is contained in table 2 for the nucleation of argon at temperatures between 40 and 80°K. (The melting temperature of argon is 84 “K.) In calculating S*(T) according to the drop model, we used the data of Allen and De Wette36) on the
8
J. J, BURTON
surface free energy of solid argon for c in (I), fig. 1 and eq. (I 5). S*(T) in table 2 are given relative to the equilibrium vapor of solid argon37). Solid argon is used throughout this paper as the reference state. TABLE
2
Nucleation of argon according to the drop model; T is the temperature; S*(T) is the critical supersaturation ratio; N*(T) is the number of atoms in the critical cluster In S*(T)
N*(T)
2.968 3.831 5.105 7.110 10.54
48 37 27 19 12
80 70 60 50 40
AF(n) is plotted in fig. 2 at the critical supersaturation at T= 80°K for the drop model. As can be seen, AF(n) has a maximum at N* =48 atoms. This maximum value of AF occurs at the critical cluster size. The critical drop size for the drop model is included in table 2. The critical drop size increases with increasing temperature. In eq. (5), from which the nucleation rates were calculated, the contributions of a cluster of n atoms to f -’ is of the form
8
-
s-__
10-l
I’ ‘\
1
f:‘.I_; \
4
I
\
z ?J
F? h
-z
\
10-S
\
I
X 2
)
\
I
10-7
50
1 u’
-4
0
2
rl 6
\ \
I
;;
1
\
I
0
lo-3
\
I
100
150
200
n (atoms)
Fig. 2. The free energy of formation of, A.F(n), a cluster of n argon atoms from the vapor at the critical supersaturation for T= 80°K according to the drop model (left hand scale, solid curve) and the contribution of a given cluster size, C&,, to the inverse nucleation rate (right hand scale, dashed curve).
GAS PHASE
K is a constant
9
NUCLEATION
which does not depend on n. Clearly, the largest contributions
to 1-l occur for the largest positive values of AF(n). (C,B,)-’ is plotted in fig. 2, and is essentially zero for most values of n; only the clusters quite close in size to the critical drop affect significantly the nucleation rate. 4.2. EXACT CLUSTER MODEL Nucleation rates and critical supersaturations were calculated using the thermodynamic data on small clusters presented in the previous section. Values of AFi.,(n)/n, eq. (14), were calculated for only a limited number of clusters between 13 and 87 atoms using the data in table 1, fig. 1. They were then interpolated linearly. This linear interpolation is justified by the fact that the excess free energy per atom is nearly a linear function of the number of atoms in the cluster, fig. 1. The nucleation rates were then calculated by summing (5) for n between 13 and 87. As shown in fig. 2 and discussed above, it is sufficient to sum (5) over this rather limited range provided that the critical size is well within the range. The error in the nucleation rate due to summing over this limited range is less than 0.001%. Critical supersaturation ratios, S*(T), are in table 3 for the exact cluster TABLE
3
Nucleation of argon according to the exact model of small clusters; Tis the temperature, S*(T) is the critical supersaturation ratio; N*(T) is the number of atoms in the critical cluster; I’ is the approximate nucleation rate, according to the exact model, at the critical supersaturation for the drop model In S*(T)
80 70
3.631 4.593
60 50 40
5.901 7.173 10.66
N* 0-l 33 31 29 26 23
I’ (x-l
cm-3) 10~16 10~13 10~12 10-g 10-l
model. The critical pressure for nucleation is much higher at higher temperatures for the exact model than for the drop model, table 2. This occurs because the exact excess free energy of a small cluster is much larger than is assumed in the drop model, fig. 1. The difference between the exact and drop models is seen quite clearly by comparison of the nucleation rates according to the two models at the same pressure. Accordingly, I’ was calculated from the exact model at the critical pressure for the drop model. I’ is also in table 3. Nucleation at higher temperatures according to the exact model is much slower than according to the drop model; it becomes
10
J. .J.BURTON
slower at very low temperatures. This occurs because the drop model underestimates the excess internal free energy of large clusters and overestimates that of small clusters; the critical cluster size according to the drop model is quite small at low temperatures. AF(n) is plotted in fig. 3 for the exact model at the critical pressures for temperatures between 80°K. The critical cluster sizes for the exact model are in table 3.
40 and
I-
& I
/
0'
’
..
'60" '\ \ '\ '\ '\ 'k. \40" \
n
Fig. 3. The energy of formation of a cluster of n argon atoms from the vapor at the critical supersaturation for various temperatures according to the exact cluster model.
The drop model
predicts
that the critical
n*= ~
cluster
size should
be given by
8A303
(17)
27 k3T3 [In (p/p,)13 ’
which depends
significantly
on temperature
and pressure.
According
to the
exact data, fig. 1, the excess free energy of small clusters can be represented by AFi,,(n)/n
= B - Cn ,
with B and C having one value for y1< 55 and another implies a critical cluster size given by
(18)
for n 2 55. Eq. (18)
This depends much less on T and p than eq. (17). The difference in the temperature dependences of the critical cluster size is seen clearly in tables 2 and 3. The exact free energies for forming the nuclei in fig. 3 show two maxima at 80°K. The second maximum corresponds to the large n branch of (19). It does not appreciably affect the nucleation rates calculated here but may be quite important at higher temperatures or lower pressure.
GAS PHASE
11
NUCLEATION
5. Discussion There are two principal results in this work. 1) Nucleation, according to exact calculations, is much slower than predicted by the drop model at temperatures near the melting temperature. (Most experiments are conducted fairly close to the melting temperature.) 2) The critical cluster size does not depend strongly on temperature and pressure. We wish to explore the relation of these results to experiments. The calculations in this paper were based on argon. Nucleation experiments are generally carried out on H,O, hydrocarbons, and similar materials. We must ask whether our results are general or are specific to our model of argon. Therefore, we have calculated the internal free energy per atom, FLnt(n)/n, for two other simple models - fee and hcp clusters with nearest neighbor interactions only. Fint(n)/n for these models is shown in fig. 4. Fig. 4 also
s L
- 12.0 -
$ 0 ”
-
13.0 -
LLE -14.01 0
1
I
25
50
I 75
100
" Fig. 4. The internal free energy per atom of fee and hcp clusters with nearest neighbor interactions only at 80°K. Straight lines intersecting at 55 are drawn approximately through the points.
shows that the internal free energy per atom can be represented approximately by two straight lines intersecting near 55, as was found in fig. 3. This linearity of F,‘,,(n)/n gave rise to the constancy of the critical cluster size, eq. (19). Therefore, we conclude that this effect does not depend on the nature of the interactions of the atoms. It would also be found in clusters of molecules provided that the molecule-molecule interactions could be represented by a purely radial pair interaction and provided that there is no strong interaction between inter- and intra-molecular forces. Our result that the classical nucleation theory predicts much too high nucleation rates near the melting temperature may explain some experimental
12
J. J. BURTON
results on hydrocarbons. In general, the drop model gives good results for the nucleation rate of hydrocarbons 3s). However, translational and rotational contributions
to the partition
function
drop model. Lothe and Pound6) terms rotational and translational
are omitted
entirely
in the classical
pointed out that q,(n) and q,(n) contribute free energies
GObI to the free energy of the critical cluster; here n* is the critical cluster size, f is its moment of inertia, and B, is the monomer concentration. There has been considerable controversy6-11) as to whether (20b) should contain a correction for the symmetry operations of the cluster. This question is quite unsettled and we use (20) and do enter into this controversy at this time. We have evaluated (20) for the critical cluster and find for argon F, + F, - 29kT,
(21)
for T between 60 and 80°K. This implies that the critical cluster concentraneglecting translation and rotation. tion is IO I3 larger than obtained Hence the nucleation rate is increased by 1013. Examination of the X’ values in table 3, the nucleation rate according to the exact calculations at the critical pressure according to the drop model, indicates that inclusion of translation and rotation in the exact calculations raises the nucleation rates back to approximate agreement with the predictions of the drop model. This may explain why the drop model, despite all of its failings, succeeds in giving reasonable numbers for nucleation ratess8).
This work was supported exclusively Henry Krumb School of Mines.
by Columbia
Universily
and
References 1) 2) 3) 4) 5)
J. P. Nirth and G. M. Pound, Progr. Mater. Sci. 11 (1963). J. Feder, K. C. Russell, J. Lothe and G. M. Pound, Advan. Phys. 15 (1966) 111. W. Thomson, Proc. Roy. Sot. Edinburgh 7 (1870) 63. R. V. Helmholtz, Wiedemanns Ann. 27 (1886) 508. R. C. Tolman, J. Chem. Phys. 17 (1949) 333.
the
GAS PHASE
6) 7) 8) 9) 10)
NUCLEATION
13
J. Lothe and G. M. Pound, 5. Chem. Phys. 36 (1962) 2030.
F. F. Abraham and G. M. Pound, J. Chem. Phys. 48 (1968) 732. H. Reiss, J. L. Katz and E. R. Cohen, J. Chem. Phys. 48 (1968) 5553. A. G. Bashkirov, Phys. Letters 28A (1968) 23. R. Kikuchi, J. Statistical Phys. 1 (1969) 3.51, 11) K. Nishloka and G. M. Pound, Am. J. Phys. 38 1211 (1970). 12) W. J. Dunning, in: Proc. Case Institute ofTechnology Symp. on Nucleation, Cleveland, Ohio, 1965. 13) J. Lothe and G. M. Pound, J. Chem. Phys. 45 (1966) 630. 14) J. Lothe and G. M. Pound, Phys. Rev. 182 (1969) 339. 15) F. F. Abraham and J. Canosa, J. Chem. Phys. 50 (1969) 1303, 16) J. Hirth, Ann. N.Y. Acad. Sci. 101 (1963) 805. 17) G. C. Benson and R. Shuttleworth, J. Chem. Phys. 19 (1951) 130. 18) R. A. Oriani and B. E. Sundquist, J. Chem. Phys. 38 (1963) 2082. 19) J. J. Burton, Chem. Phys. Letters 3 (1969) 594. 20) J. J. Burton, J. Chem. Phys. 52 (1970) 345. 21) J. J. Burton, Chem. Phys. Letters 7 (1970) 567. 22) J. J. Burton, J. Chem. Phys., in press. 23) D. Walton, J. Chem. Phys. 37 (1962) 2182. 24) V. Halpern, Brit. J. Appl. Phys. 18 (1967) 163. 25) J. E. MacDonald, Am. J. Phys. 31 (1963) 31. 26) W. G. Courtney, J. Chem. Phys. 36 (1962) 2009. 27) W. G. Courtney, J. Chem. Phys. 36 (1962) 2018. 28) F. F. Abraham, J. Chem. Phys. 51 (1969) 1632. 29) 1. J. Zucker, Nuovo Cimento 458 (1968) 177. 30) A. E. Sherwood and J. M. Prausnitz, J. Chem. Phys. 41 (1964) 429. 31) L. Jansen, Phys. Rev. 135 (1964) A1292. 32) J. J. Burton, Phys. Rev. 182 (1969) 885. 33) J. H. Dymond and B. J. Alder, J. Chem. Phys. 51 (1969) 301. 34) J. J. Burton, Chem. Phys. Letters 5 (1970) 312. 35) J. J. Burton, Nature 229 (1971) 335. 36) R. E. Allen and F. W. de Wette, J. Chem. Phys. 51 (1969) 4820. 37) P. Flubacher, A. J. Leadbetter and J. A. Morrison, Proc. Phys. Sot. (London) 78 (1961) 1449. 38) J. L. Katz, J. Chem. Phys. 52 (1970) 4733.
SCIENCE 26 (1971) 1-13 0 North-Holland
GAS PHASE STEADY
STATE
Publishing Co.
NUCLEATION
:
RATES FOR TWO CLUSTER
MODELS
J. J. BURTON Henry Krumb School of Mines, Columbia University, New York, New York 10027, U.S.A.
Received 27 November
1970; revised manuscript received 19 January 1971
Exact thermodynamic properties of small clusters were computed from a model of argon. These thermodynamic properties were used to calculate steady nucleation rates. Steady state rates were also calculated using the drop model to approximate the properties of small clusters; the parameters required in the drop model were determined from the same model of argon used in the exact calculations. The exact model nucleation rates were lower than the drop model results by as much as 1016. The exact calculations suggest that the critical cluster size is nearly temperature independent, while it depends strongly on temperature according to the drop model. It is concluded that the predictions of the classical drop model are both qualitatively and quantitatively incorrect. Inclusion of the Lothe-Pound correction for cluster rotation and translation in the exact results causes the exact model nucleation rates to be nearly equal to the predictions of classical theory (neglecting rotation and translation); this may explain the fact that nucleation rates in many simple systems are correctly predicted by the classical theory.
1. Introduction Steady state gas phase nucleation rate have been studied by a number of investigators1s2). In calculating nucleation rates, it is necessary to assume a model to describe the thermodynamic properties of small clusters. The conventional cluster model is the “drop model”. This model is based on the early work of Thomson3) and HelmholtzJ) on the equilibrium vapor pressure of curved surfaces. The free energy due to a curved surface has more recently been examined rather rigorously by Tolman 5). These thermodynamic investigations are strictly applicable only to drops having large radii. The classical drop model has been questioned on a number of grounds. It neglects the rotational and translational free energies of the clusters’j). There is some uncertainty as to whether this error is significantT_ll). It also apparently neglects the fact that a finite body of N atoms has but 3N-6 vibrational degrees of freedome). This problem has also been the source of some discussionlz-15). There are other important difficulties with the drop model which are often overlooked. Hirth16) has pointed out that the drop model should not be
J. J. BURTON
2
applied to calculating thermodynamic properties of very small clusters; rather exact microscopic calculations should be made for use in nucleation rate computations. Benson17) showed energy of a small cluster is considerably
in 1951 that the excess potential smaller than that of a large cluster.
In particular, the excess potential energy per unit area is roughly 15% lower for clusters of 13 argon atoms than it is for a planar surface. Orianils) examined the energy of water clusters and suggested the drop model may err by 25%. Burton investigated the thermodynamic properties of clusters of 13 to 87 atomsrgpzZ). He found that the excess entropy per atom is not a monotonic function of cluster size ZO-~ZZ),and that the free energy function was quite unlike the predictions of the drop model”O). Waltonas) and Halperne”) have shown that detailed atomic considerations are very important in studying nucleation from the gas phase on surfaces. The above considerations suggest that it is important to calculate nucleation rates from exact properties of small clusters. We adopt the following view in this paper: In general, it is not possible to calculate from first principles the nucleation rate for any material. However, a formalism has been developed to calculate the steady state nucleation rate, subject to a number of assumptions, if the thermodynamic parameters are known for small clusters”j). This formalism has been widely applied to calculations of nucleation ratesI,“), assuming that the internal free energy of the small cluster can be correctly described by the drop model. In section 2, we review briefly the formalism. Recent work 19-22) has shown that the classical drop model for the internal free energy of a small solid cluster is incorrect, at least for a simplified model of solid argon clusters. The free energy problem is developed in section 3. There is no information now available about the exact free energy of small clusters of liquid. Therefore, we compute the nucleation rate for nucleation of solid argon from its vapor in section 4. This work is not, strictly speaking, a critical calculation of nucleation rates for comparison with experiment. Most experimental work is done on complicated molecules such as water or hydrocarbons and, generally, the liquid rather than solid is nucleated from the vapor. The work reported here is basically a bootstrap calculation in which we ask “what are the rates of nucleation from the vapor of a simple solid when the cluster free energy rates are not is calculated in two different ways”. While the calculated applicable exactly to a real material, they give a measure of the error introduced by using the drop model in calculating nucleation rates of real materials. 2. General nucleation theory Many
people have contributed
to the development
of the general
theory
GAS PHASE
of steady state nucleation.
NUCLEATION
3
This work has been reviewed by Hirthr),
and MacDonald 25). We follow here MacDonald’s nucleation in a dilute gas. A dilute gas consists primarily of monomers.
treatment
Feders),
of steady-state
Occasionally
a monomer
captures another monomer to form a dimer. This dimer may either capture another monomer to form a trimer or a monomer evaporates leaving the original monomer. The trimer may grow or shrink, and so on. For small clusters, the evaporation step is much more likely than the capture step. However, for large enough clusters, if the vapor pressure is sufficiently high, the capture process will be more probable than the evaporation process and the cluster will grow. Thus, nucleation depends on the production of large clusters through random fluctuations in the mostly monomeric vapor. This fluctuation is readily described mathematically. Let C, be the rate at which a cluster of II atoms captures a single atom. Let Z?,,be the rate at which a single atom escapes from a cluster of n atoms. We assume than C,, and E, do not depend on time. Let N, represent the concentration of clusters of IZ atoms in the gas. Then
dN,ldt = (G-1%1 + -%+,N,+I)- (GN, + JWJ. (This equation must, of course, be slightly modified current, Z,, between clusters of n and n+ I atoms as Z,(t) = GN,
- E,+,N,+,
.
for IZ= 1.) Define
(1) the
(2)
Then dN,/dt
(t) -I,(t).
= I,-r
(3)
Eq. (3) represents an infinite set of coupled linear equations. They have been solved numerically on computers 26-28). The equations can also be solved exactly in certain cases. The most interesting solution is the unbalanced steady state solution. This solution of (3) corresponds to formation of very large clusters (visible droplets) at a time independent rate. Let In(t) = I (4) be a constant for all IZand t. Here, Zis the unbalanced steady state nucleation rate. Define B,, to be the equilibrium Boltzmann distribution of n atom clusters at the temperature and pressure of the system. Then it can be readily shown25) from (2) and (4) that m
z-1 =
c
1
c,B, .
n=l
(5)
J.J.
4
Eq. (5) allows us to calculate
BURTON
I, the steady state nucleation
rate, if C,, and B,,
are known. Usually, in theoretical computationsz5), it is assumed capture rate C, is given by the ideal gas collision rate C,, = An*p(2nmkT)4,
(6)
where A is a geometrical constant such that An+ is the surface other terms have their co~l~entional meanings. B,,, the concentration of clusters of n atoms, is given by B, = tj (n) enilik?‘, where fl is the chemical potential function, s(n) is given by q(n)
and 4(0) is the partition
qint(n)9
where qt(/z) is the translationsi, qc(n) the rotational, internal partition function. Eq. (7) may be rewritten as B,, = e-
f
area and all
03
of the vapor
= 41(n) 4,(n)
that the
(8) and
cXi,,(~?) is the
enpihT PaI
=B,e
-AF(nfikf
(9b)
,
where Ffn) AF(n)
= - l
WW
= F(B) - nf -t (t - ~1)icT In(p,jp),
(fObI
andj’is free energy per atom in the solid at temperature T, p is the pressure, and pL’is the equilibrium vapor pressure. The central problem of nucleation theory has been the evaluation of F(n).
From (8) we may write Fjtz) = F,:I*fiz) -+ F;(n)
-I- F,(n).
(t1)
Here we write F’ rather than F to call attention to the fact that Fibt(n), the internal free energy is based on only 3n-6 degrees of freedom. F,(n) and Ft(n), the rotational and translational free energies are generally neglected (set equal to zero) in classical nucleation theory. The correct treatment of of considerable co~~troversy~-ll~, In the F,(rr) and F,(n) h as b een the source next section, we also neglect the rotational and translational free energies; their importance will be considered in the discussion at the end of the paper.
3. Cluster free energies fn classical nucleation theory, it is generally obtained from the drop modeIr~ ~85) according AF(n)
assumed that AF(n) may be to which
= A~z’o - nl
(12)
GAS PHASE NUCLEATfON
5
where A was explained in eq. (6) and CFis the surface tension (surface energy per unit area) and is independent of n. Physically, this expression
free can
be regarded as arising from the following two steps in forming the cluster. (1) The n atoms are transferred one at a time from the gas at pressure p to the infinite body. This step lowers the free energy by nkT ln(p/p,). (2) n atoms are transferred from the bulk to the cluster of n atoms. This raises the free energy by the surface free energy of the cluster, An’c. Eq. (12) is not correct for a very small cluster. The remainder of this section is devoted to the correct evaluation of AF(n) in eq. (lob) for small clusters of argon atoms. Though most nucleation experiments deal with nucleation of the liquid from the vapor, we study in this paper the nucleation of the solid. This is done because we are able to calculate the properties of small solid clusters but know nothing about liquid clusters. The properties of small clusters of argon atoms have been studied previouslylQ-22). No other materials have been examined exactly, though the general conclusions about small clusters may be applicable to other materialssi. 22). The discussion here is based on argon. The atoms were assumed to interact via a pair-wise additive Lennard-Jones potential of the form
where r is ZuckeraQ) Many-body have been properties we neglect
the interatomic distance. The constants used were those given by for argon. This potential is not an exact representation of argon. interactions may be significant so-a2) and other pair potentials proposedaa,Qz). Therefore, strictly speaking, we are studying the of a model and not the properties of argon. In our calculations, entirely anharmonic effects and lattice thermal expansion.
Properties of small clusters containing, 13, 19, 43, 55, 79, and 87 argon atoms have been calculated. These sizes are a central atom plus first nearest neighbors, plus first and second, etc. The atoms were originally arranged in an fee configuration and then allowed to relax to a potential energy minimum. The (3~2x 3~) force constant matrix was diagonaljzed on an IBM 360/91 computer to obtain the frequencies of the 3n-6 normal vibrational modes. Some results on the potential energy2*), configurations2Qva5), and entropiesZ0-22) of the clusters have been reported. The interesting part of AF(n), eq. (lob) may be written as is the internal contribution AF,,, (II) = Fi’“, (n) - nf = BE(n)
+ AEZ+(n)
(14a) - TAS(n),
(14b)
6
J. J. BURTON
where AE is the potential energy difference between the cluster and the bulk, AEz.p. is the zero-point energy and AS is the entropy. In writing (14b), it is important to remember that F,‘,,t(n) is based on 3n- 6 degrees of freedom while cf has 3n degrees. Therefore, AEZ,P, and AS have rather strange degrees of freedom. This does not introduce any difficulty if we are careful as (14) is still formally correct; however this has caused considerable discussion 6, rs+15) - the replacement correction controversy. AE(n)/n, AE,,,,(n)/n, and AS(,)/, are in table 1. No configurational contributions to S are included in AS. AFin,(n)/n was calculated from the data in table 1 and is plotted in fig. 1 TABLE
The
exact
excess
1
potential energy, AE, zero-point energy, A/Lt,., per atom of small clusters of II argon atoms AE/n ( X 10 t4 erg/atom)
n
13 19 43 55 79 87
and
~ 0.476 ~ 0.417 m-o.319 ~ 0.247 ~ 0.200 PO.214
I
8.0
I
AS,
AS/PI (k) atom
AEz.,./n (x 10~~r3erg/atom)
8.607 8.242 6.819 5.845 5.451 5.523
entropy,
0.18 0.41 0.69 0.49 0.43 0.57
1
0 ~-
40”
K
A--
60”
K
q -
80”
K
7.0 s L *T o
-
6.0
”
0
25
50
75
100
n
Fig. 1. The excess internal free energy per atom, AFf,,t(n)/n, relative to the bulk material, of small clusters of argon atoms at 40, 60 and SOCK. The heavy curves show the predictions of the drop model. The data points are the results of exact computations. Straight lines have been drawn connecting the data points.
GAS PHASE
NUCLEATION
I
at 40, 60 and 80°K. The excess internal free energy per atom according to the drop model, An-*,, is plotted in fig. 1 for comparison purposes. The value of the surface tension used in the drop model was 0 = 40.5 - 0.1 T erg/cm2 .
(15)
This is the average of the values for the (100) and (I 11) surfaces determined by Allens6) for infinite planar surfaces of solid argon; Allen used the same potential to describe the argon atom interactions as used in this work. Allen also neglected anharmonic contributions to the vibrational free energy. We use here his values of 0 at the lattice parameter appropriate to O”K, that is without thermal expansion. Thus the value of 0 in (15) computed for bulk argon is based on the same model used to compute the free energy of small clusters. Properly, one should not use such a simple way of estimating a; one should consider the details of the shape of the crystal, However, our averaging is justified by the fact that ~(100) and o(lll), as computed by Allen, differ at the melting temperature of argon by less than 0.5 erg/cm2 and are 3 erg/cm2 less than the value of 0 for the (110) surface. Thus the error in using (15) is probably significantly smaller than the uncertainty in most experimentally determined surface entropies. The exact excess internal free energy of the small clusters is clearly unlike that predicted by the drop model. It is greater than expected for the larger clusters and has a rather different dependence on cluster size. The exact excess free energy per atom can be described approximately by two straight lines, of different slopes, which intersect at 55 atoms per cluster.
4. Nucleation
rates
Nucleation rates were calculated for both the classical drop model and the exact treatment of the clusters. We examine first the classical results as these will be useful for understanding the significance of the exact results. 4.1. DROP MODEL Conventionally, the nucleation theorist and experimenter attempt to find the pressure, p*(T) at which the nucleation rate at temperature T is equal to 1 cluster per cubic centimeter per second. This pressure is known as the critical pressure. The ratio, p*(T)/p,(T), where p,(T) is the equilibrium vapor pressure at temperature T, is known at the critical supersaturation S*(T). Nucleation rates were calculated from (5). S*(T) is contained in table 2 for the nucleation of argon at temperatures between 40 and 80°K. (The melting temperature of argon is 84 “K.) In calculating S*(T) according to the drop model, we used the data of Allen and De Wette36) on the
8
J. J, BURTON
surface free energy of solid argon for c in (I), fig. 1 and eq. (I 5). S*(T) in table 2 are given relative to the equilibrium vapor of solid argon37). Solid argon is used throughout this paper as the reference state. TABLE
2
Nucleation of argon according to the drop model; T is the temperature; S*(T) is the critical supersaturation ratio; N*(T) is the number of atoms in the critical cluster In S*(T)
N*(T)
2.968 3.831 5.105 7.110 10.54
48 37 27 19 12
80 70 60 50 40
AF(n) is plotted in fig. 2 at the critical supersaturation at T= 80°K for the drop model. As can be seen, AF(n) has a maximum at N* =48 atoms. This maximum value of AF occurs at the critical cluster size. The critical drop size for the drop model is included in table 2. The critical drop size increases with increasing temperature. In eq. (5), from which the nucleation rates were calculated, the contributions of a cluster of n atoms to f -’ is of the form
8
-
s-__
10-l
I’ ‘\
1
f:‘.I_; \
4
I
\
z ?J
F? h
-z
\
10-S
\
I
X 2
)
\
I
10-7
50
1 u’
-4
0
2
rl 6
\ \
I
;;
1
\
I
0
lo-3
\
I
100
150
200
n (atoms)
Fig. 2. The free energy of formation of, A.F(n), a cluster of n argon atoms from the vapor at the critical supersaturation for T= 80°K according to the drop model (left hand scale, solid curve) and the contribution of a given cluster size, C&,, to the inverse nucleation rate (right hand scale, dashed curve).
GAS PHASE
K is a constant
9
NUCLEATION
which does not depend on n. Clearly, the largest contributions
to 1-l occur for the largest positive values of AF(n). (C,B,)-’ is plotted in fig. 2, and is essentially zero for most values of n; only the clusters quite close in size to the critical drop affect significantly the nucleation rate. 4.2. EXACT CLUSTER MODEL Nucleation rates and critical supersaturations were calculated using the thermodynamic data on small clusters presented in the previous section. Values of AFi.,(n)/n, eq. (14), were calculated for only a limited number of clusters between 13 and 87 atoms using the data in table 1, fig. 1. They were then interpolated linearly. This linear interpolation is justified by the fact that the excess free energy per atom is nearly a linear function of the number of atoms in the cluster, fig. 1. The nucleation rates were then calculated by summing (5) for n between 13 and 87. As shown in fig. 2 and discussed above, it is sufficient to sum (5) over this rather limited range provided that the critical size is well within the range. The error in the nucleation rate due to summing over this limited range is less than 0.001%. Critical supersaturation ratios, S*(T), are in table 3 for the exact cluster TABLE
3
Nucleation of argon according to the exact model of small clusters; Tis the temperature, S*(T) is the critical supersaturation ratio; N*(T) is the number of atoms in the critical cluster; I’ is the approximate nucleation rate, according to the exact model, at the critical supersaturation for the drop model In S*(T)
80 70
3.631 4.593
60 50 40
5.901 7.173 10.66
N* 0-l 33 31 29 26 23
I’ (x-l
cm-3) 10~16 10~13 10~12 10-g 10-l
model. The critical pressure for nucleation is much higher at higher temperatures for the exact model than for the drop model, table 2. This occurs because the exact excess free energy of a small cluster is much larger than is assumed in the drop model, fig. 1. The difference between the exact and drop models is seen quite clearly by comparison of the nucleation rates according to the two models at the same pressure. Accordingly, I’ was calculated from the exact model at the critical pressure for the drop model. I’ is also in table 3. Nucleation at higher temperatures according to the exact model is much slower than according to the drop model; it becomes
10
J. .J.BURTON
slower at very low temperatures. This occurs because the drop model underestimates the excess internal free energy of large clusters and overestimates that of small clusters; the critical cluster size according to the drop model is quite small at low temperatures. AF(n) is plotted in fig. 3 for the exact model at the critical pressures for temperatures between 80°K. The critical cluster sizes for the exact model are in table 3.
40 and
I-
& I
/
0'
’
..
'60" '\ \ '\ '\ '\ 'k. \40" \
n
Fig. 3. The energy of formation of a cluster of n argon atoms from the vapor at the critical supersaturation for various temperatures according to the exact cluster model.
The drop model
predicts
that the critical
n*= ~
cluster
size should
be given by
8A303
(17)
27 k3T3 [In (p/p,)13 ’
which depends
significantly
on temperature
and pressure.
According
to the
exact data, fig. 1, the excess free energy of small clusters can be represented by AFi,,(n)/n
= B - Cn ,
with B and C having one value for y1< 55 and another implies a critical cluster size given by
(18)
for n 2 55. Eq. (18)
This depends much less on T and p than eq. (17). The difference in the temperature dependences of the critical cluster size is seen clearly in tables 2 and 3. The exact free energies for forming the nuclei in fig. 3 show two maxima at 80°K. The second maximum corresponds to the large n branch of (19). It does not appreciably affect the nucleation rates calculated here but may be quite important at higher temperatures or lower pressure.
GAS PHASE
11
NUCLEATION
5. Discussion There are two principal results in this work. 1) Nucleation, according to exact calculations, is much slower than predicted by the drop model at temperatures near the melting temperature. (Most experiments are conducted fairly close to the melting temperature.) 2) The critical cluster size does not depend strongly on temperature and pressure. We wish to explore the relation of these results to experiments. The calculations in this paper were based on argon. Nucleation experiments are generally carried out on H,O, hydrocarbons, and similar materials. We must ask whether our results are general or are specific to our model of argon. Therefore, we have calculated the internal free energy per atom, FLnt(n)/n, for two other simple models - fee and hcp clusters with nearest neighbor interactions only. Fint(n)/n for these models is shown in fig. 4. Fig. 4 also
s L
- 12.0 -
$ 0 ”
-
13.0 -
LLE -14.01 0
1
I
25
50
I 75
100
" Fig. 4. The internal free energy per atom of fee and hcp clusters with nearest neighbor interactions only at 80°K. Straight lines intersecting at 55 are drawn approximately through the points.
shows that the internal free energy per atom can be represented approximately by two straight lines intersecting near 55, as was found in fig. 3. This linearity of F,‘,,(n)/n gave rise to the constancy of the critical cluster size, eq. (19). Therefore, we conclude that this effect does not depend on the nature of the interactions of the atoms. It would also be found in clusters of molecules provided that the molecule-molecule interactions could be represented by a purely radial pair interaction and provided that there is no strong interaction between inter- and intra-molecular forces. Our result that the classical nucleation theory predicts much too high nucleation rates near the melting temperature may explain some experimental
12
J. J. BURTON
results on hydrocarbons. In general, the drop model gives good results for the nucleation rate of hydrocarbons 3s). However, translational and rotational contributions
to the partition
function
drop model. Lothe and Pound6) terms rotational and translational
are omitted
entirely
in the classical
pointed out that q,(n) and q,(n) contribute free energies
GObI to the free energy of the critical cluster; here n* is the critical cluster size, f is its moment of inertia, and B, is the monomer concentration. There has been considerable controversy6-11) as to whether (20b) should contain a correction for the symmetry operations of the cluster. This question is quite unsettled and we use (20) and do enter into this controversy at this time. We have evaluated (20) for the critical cluster and find for argon F, + F, - 29kT,
(21)
for T between 60 and 80°K. This implies that the critical cluster concentraneglecting translation and rotation. tion is IO I3 larger than obtained Hence the nucleation rate is increased by 1013. Examination of the X’ values in table 3, the nucleation rate according to the exact calculations at the critical pressure according to the drop model, indicates that inclusion of translation and rotation in the exact calculations raises the nucleation rates back to approximate agreement with the predictions of the drop model. This may explain why the drop model, despite all of its failings, succeeds in giving reasonable numbers for nucleation ratess8).
This work was supported exclusively Henry Krumb School of Mines.
by Columbia
Universily
and
References 1) 2) 3) 4) 5)
J. P. Nirth and G. M. Pound, Progr. Mater. Sci. 11 (1963). J. Feder, K. C. Russell, J. Lothe and G. M. Pound, Advan. Phys. 15 (1966) 111. W. Thomson, Proc. Roy. Sot. Edinburgh 7 (1870) 63. R. V. Helmholtz, Wiedemanns Ann. 27 (1886) 508. R. C. Tolman, J. Chem. Phys. 17 (1949) 333.
the
GAS PHASE
6) 7) 8) 9) 10)
NUCLEATION
13
J. Lothe and G. M. Pound, 5. Chem. Phys. 36 (1962) 2030.
F. F. Abraham and G. M. Pound, J. Chem. Phys. 48 (1968) 732. H. Reiss, J. L. Katz and E. R. Cohen, J. Chem. Phys. 48 (1968) 5553. A. G. Bashkirov, Phys. Letters 28A (1968) 23. R. Kikuchi, J. Statistical Phys. 1 (1969) 3.51, 11) K. Nishloka and G. M. Pound, Am. J. Phys. 38 1211 (1970). 12) W. J. Dunning, in: Proc. Case Institute ofTechnology Symp. on Nucleation, Cleveland, Ohio, 1965. 13) J. Lothe and G. M. Pound, J. Chem. Phys. 45 (1966) 630. 14) J. Lothe and G. M. Pound, Phys. Rev. 182 (1969) 339. 15) F. F. Abraham and J. Canosa, J. Chem. Phys. 50 (1969) 1303, 16) J. Hirth, Ann. N.Y. Acad. Sci. 101 (1963) 805. 17) G. C. Benson and R. Shuttleworth, J. Chem. Phys. 19 (1951) 130. 18) R. A. Oriani and B. E. Sundquist, J. Chem. Phys. 38 (1963) 2082. 19) J. J. Burton, Chem. Phys. Letters 3 (1969) 594. 20) J. J. Burton, J. Chem. Phys. 52 (1970) 345. 21) J. J. Burton, Chem. Phys. Letters 7 (1970) 567. 22) J. J. Burton, J. Chem. Phys., in press. 23) D. Walton, J. Chem. Phys. 37 (1962) 2182. 24) V. Halpern, Brit. J. Appl. Phys. 18 (1967) 163. 25) J. E. MacDonald, Am. J. Phys. 31 (1963) 31. 26) W. G. Courtney, J. Chem. Phys. 36 (1962) 2009. 27) W. G. Courtney, J. Chem. Phys. 36 (1962) 2018. 28) F. F. Abraham, J. Chem. Phys. 51 (1969) 1632. 29) 1. J. Zucker, Nuovo Cimento 458 (1968) 177. 30) A. E. Sherwood and J. M. Prausnitz, J. Chem. Phys. 41 (1964) 429. 31) L. Jansen, Phys. Rev. 135 (1964) A1292. 32) J. J. Burton, Phys. Rev. 182 (1969) 885. 33) J. H. Dymond and B. J. Alder, J. Chem. Phys. 51 (1969) 301. 34) J. J. Burton, Chem. Phys. Letters 5 (1970) 312. 35) J. J. Burton, Nature 229 (1971) 335. 36) R. E. Allen and F. W. de Wette, J. Chem. Phys. 51 (1969) 4820. 37) P. Flubacher, A. J. Leadbetter and J. A. Morrison, Proc. Phys. Sot. (London) 78 (1961) 1449. 38) J. L. Katz, J. Chem. Phys. 52 (1970) 4733.