NON-EQUILIBRIUM CLUSTER PROPERTIES AND NON-ISOTHERMAL
NUCLEATION
R. HALONEN, E. ZAPADINSKY and H. VEHKAMÄKI
Department of Physics, University of Helsinki, P.O.Box 64, FIN-00014 University of Helsinki,
Finland
Keywords: NUCLEATION, CONSTRAINT EQUILIBRIUM, LENNARD-JONES, CLUSTER.
INTRODUCTION
Almost all theories of the vapour to liquid nucleation are based on well-known kinetic approach.
The kinetic part of the theories comprise the solution of death-birth equations with two boundary
conditions giving two different steady state solutions, which can be related to each other. The
balanced steady state solution is identified with equilibrium cluster distribution and this is crucial
assumption sometimes called constraint equilibrium hypothesis (CEH). The validity of CEH was
questioned many times (Lushnikov and Kulmala, 1998; Barrett, 2002; Ford and Harris, 2004;
Bartell, 2009). The citation list is not full, and, perhaps, even the authors of the kinetic approach
were in doubts equalising the balanced steady state to conditions for equilibrium. However it was
the only way to advance in nucleation theory at that time. Nowadays the computer simulations
can help in studying the validity of CEH. It has been reported about the break of CEH (Zhuo,
2005) in MD simulations of argon nucleation. In our opinion it is not possible to deduce from those
simulations about the brake of CEH. They rather demonstrate that evaporation and condensation
rates are different in balanced and unbalanced steady states. That is not surprising since the
conditions of the simulations were at the vapour densities close to the liquid ones, so the process
can hardly be called vapour to liquid nucleation. In the present study we carefully examine the
assumptions involved in the basis of the nucleation theories. Particularly we study the conditions
of the kinetic scheme applicability and limits when CEH can be used. The Lennard-Jones vapour
is used as an example.
RESULTS AND DISCUSSION
Suppose we have ensemble of n-clusters where their energy E is distributed according to normalized
function
1 dNn (E)
ϕ(E) =
,
(1)
Nn dE
where Nn (E) is the number of n-clusters with energy less than E, Nn is the total number of nclusters. We section the distribution function in equal narrow energy interval ∆E and label them
with i index, so the number of cluster with energy Ei equals to
∆Nn,i = Nn ϕ(Ei )∆E.
(2)
The clusters evolution with time t is described by death-birth kinetic equations
∂∆Nn,i
∂t
= αn+1,→i Nn+1 + βn−1,→s Nn−1 − (αn,i→ + βn,→i )∆Nn,i
−
X
(γn,i→k ∆Nn,i − γn,k→i ∆Nn,k ),
k
(3)
where β is the monomer condensation rate, α is the evaporation rate, γn,i→k is a rate of transition
for the n-cluster from energy Ei to Ek induced by collision with carrier gas molecules, in the indexes
of evaporation and condensation rates the sharp end of the arrow points out the final energy value
of transition and the blunt end of the arrow shows the initial one. If one of them is not specified
it means that the quantity is integral and evaporation/condensation occurs to or from all possible
energy values. Knowing all the coefficients in Eq. (3) provides the possibility to calculate nucleation
rate.
To obtain the condensation rates, the collision cross section (CCS) between the cluster and the
monomer is geometrically estimated by a simple brute-force Monte Carlo method. We consider
each atom of a cluster as a sphere with radius r, and the cluster is association of such spheres.
The random trajectories of colliding monomers are produced by generating two random points on a
sphere with a radius larger than rref serving as a reference surface area measure. The line between
these two points is the axis of a cylinder with radius r. The corresponding hits are counted if the
cylinder has an intersection with the reference sphere or the cluster. The possible intersection can
be calculated by basic vector calculus: if the distance from a cluster point to the line is less than r
the monomer hits the cluster. Several thousands of random trajectories are generated in order to
get good statistics. The results are presented in Figure 1(b).
10
0
400
(a)
(b)
Collision cross section (Å 2)
-1
Evaporation rate (ps )
350
10 -1
10
-2
n = 10
n = 25
10 -3
-2.5
-2
-1.5
-1
-0.5
0
Total energy per molecule (ǫ)
300
250
200
150
100
50
r = 2.5 Å
r = 2.15 Å (liquid drop)
5
10
15
20
Cluster size
25
30
Figure 1: (a) Evaporation rate as a function of the total cluster energy per molecule. (b) Collision
cross section as a function of cluster size obtained with the brute-force Monte Carlo method (solid
line) for r = 2.5 Å. The dashed line corresponds the collision cross section of spherical cluster with
bulk liquid density of argon at 50 K.
The evaporation rates are obtained though (MD) simulations. The examples of the results are
presented in Figure 1(a). The transition rates γn,i→k are calculated assuming that that the carrier
gas atoms elastically collide with the cluster atoms. The calculations of CCS have free parameter
r. To define it we consider the system at formally infinitely high concentrations of the carrier
gas. According to general principle of Statistical Mechanics the carrier gas serving as a thermostat
brings all the cluster to equilibrium in this case. Therefore the balanced steady state solution of
Eqs. (3) must satisfy Boltzmann distribution. In other words, the detailed balance in the following
form must hold
∞
∞
Z
Nn
Z
βn (E)ϕn (E)dE = Nn+1
−∞
αn+1 (E)ϕn+1 (E)dE,
−∞
(4)
where ϕn (E) has the energy distribution corresponding to the Boltzmann one. As a matter of fact,
to satisfy Eq. (4) radius r appears to have reasonable value for each cluster size. Provided it is
done we have everything for solving Eqs. (3) at arbitrary conditions.
The massive data of evaporation rates in wide ranges of energies and cluster sizes obtained in
this study for the solution of the kinetic equations has also self-worth. It can be used to analyze
the validity of the existing theories for the cluster evaporation. To our knowledge this is a first
systematic study of such kind. However, the main goal is to check the validity of CEH and compare
the results to non-isothermal nucleation theory. The study is in progress and the results are to be
presented at the Conference.
ACKNOWLEDGEMENTS
This work was supported by the European Research Council under grant 692891-DAMOCLES. We
thank the CSC-IT Center for Science in Espoo, Finland, for computational resources.
REFERENCES
A. A. Lushnikov and M. Kulmala (1998). Dimers in nucleating vapors. Phys. Rev. E, 58, 3157.
J. Barrett (2002). The significance of cluster lifetime in nucleation theory. J. Chem. Phys., 116,
8856.
I. J. Ford and S. A. Harris, (2004). Molecular cluster decay viewed as escape from a potential of
mean force. J. Chem. Phys., 120, 4428.
L. Bartell (2009). Failure of the constraint equilibrium hypothesis. J. Chem. Phys., 131, 174505.
L. Zhuo (2005). A Critical Reexamination of the fundamentals of the classical nucleation theory.
PhD thesis. Yale University.