The Rate Constants of the Unimolecular Decay and Condensation Coefficients of

Neutral Clusters and Cluster Ions

A. L. Itkin

International Institute for Problems of the Asteroid Hazard, Moscow, Russia

Abstract A new approach to calculate the rate constant of the unimolecular decay of a cluster having developed the RRKM theory for such a case is presented. Based on the result obtained and the equilibrium cluster distribution function [1] we have got the rate constant for the reverse process - the growth of a cluster gaining a single molecule and the value of the cluster condensation coefficient depending upon j and T. In the limit of large j one should get the extreme value of α_{c} which doesn’t depend on j. It turns out that, for instance, for water vapor this extreme value considerably differs from α_{c} = 1 as usually assumed and strongly depends on T. In such a way the rate constants of formation and decay for cluster ions can be obtained.

1 Introduction

A recent progress in the theoretical description of such processes as homogeneous nucleation in nozzles, jets and thermodiffusion chambers [2], photoinduced nucleation [3] as well as heterogeneous nucleation on charged particles (see the other report of the author) has been connected with a new common microscopic (nonempirical) approach put forward by the author and Dr.E. Kolesnichenko. This method known as the unimolecular condensation theory (UCT) allows to overcome many principal problems of the existing nucleation theories.

The analytical methods proposed in [2] to solve the kinetic equations describing coupled processes of condensation and relaxation of the cluster internal degrees of freedom give the possibility to calculate simply enough the real multidimensional flows with such processes [4]. However, in the final expressions of UCT there are some constants and parameters which should be specified in an explicit form outside from the nucleation theory. First of all as such parameters one should consider the cluster dissociation and relaxation rate constants as well as r - the main parameter of UCT.

Acknoledgment: We acknowledge support from grant No. 286 of the Russian Foundation for Basic Research program

In the present report we suggest a new approach to calculate the rate constant of the unimolecular decay of the cluster having developed the RRKM theory for such a case. Based on the result obtained while using the microscopic reversibility law and the equilibrium cluster concentrations [1] we have also got the rate constant for the reverse process - the growth of the cluster gaining a single molecule. The ratio of this constant to a free molecular flux on the cluster surface gives the value of a cluster condensation coefficient α_{c} depending upon j and T. In the limit of large j one should get the extreme value of α_{c}.

The explicit expression for the main parameter of UCT - R - follows from these results. The physical meaning of this parameter is rather simple because r is the number of molecules contained in the cluster for which the rate of its dissociation from the top-most level M_{r} is of the same order as the vibrational relaxation rate at the same level. For clusters with j > r the rate of relaxation of intra-cluster vibrations is much greater than the rate of the cluster dissociation but at j < r there is an opposite situation. The details of this model are discussed in [2].

2 Cluster dissociation rate constants

As shown in [2] at j > r kinetic equations of UCT describing evolution of the cluster concentrations exactly correspond to the well-known quasichemical condensation model and the term describing the j–cluster decay can be written in the form K_{j}^{−}n_{j} where K_{j}^{−} = ∑_{k,l,i}^{M} E_{j}(k | i, l)s_{j}(k)z_{j}(k), n_{j} is concentration of j-clusters in the system, E_{j}(k | i, l) is the rate of dissociation of j-mer with the energy E_{j}(k) into (j − 1)-mer with the energy E_{j−1}(l) and a monomer with the energy E_{1}(i), M_{j} is the topmost level of j–cluster, s_{j}(k)z_{j}(k) is the Boltzmann factor of the k–level (see below). In the present paper we discuss some paths of calculation Ē_{j}(k) ≡ ∑_{l,i} E_{j}(k | i, l)s_{j}(k)z_{j}(k) by means of the quantum variant of RRKM [5].

First of all a physical model of the cluster should be more precisely specified. The total number of intermolecular vibrations l_{j} in the cluster is l_{j} = 6j − 6. ^{1} We subdivide all the intermolecular vibrations proceeding in the cluster into two groups - translational and librational ones with three vibrations of each type per monomer (l_{j}/2j ≃ 3) considering them in a harmonic approximation. Moreover, let us suppose that all vibrations of the determined type are of the same frequency equal to the geometric mean frequency (g.m.f.) of normal vibrations of the corresponding modes ( ν_{t} and ν_{l}, respectively). Based on the analysis of vibrations inside a harmonic lattice of N oscillators [5], it can be shown that g.m.f. of intermolecular vibrations in a cluster doesn’t depend on N and \sqrt{2} times less than the maximum frequency of the normal vibrations.

1For linear monomer one has l_{j} = 5j − 6 and for monoatomic gas l_{j} = 3j − 6. As the common results could be simply generalized for these situations they are not discussed here.

The general expression of RRKM for the dissociation rate constant Ē_{j}(k) is introduced in the form

Ē_{j}(k) = Ls_{j}(k)\frac{W (E_{j}^{+} − E_{a})}{hρ(E)} \frac{Q_{j}^{+}}{Q_{j}} ,  (2.1)

Here ρ_{j}(E) = dW_{j}(E)/dE - the density of energetic levels, h - the Plank constant, E_{a} - an activation energy, Q_{j} - a partition function of the cluster, L - the number of reaction’s channels, superscript ”+” denotes an activated complex, s_{j}(k) is the degree of degeneracy of level k. For the system of s identical oscillators consisted of k quanta (k = E/hν), W (E) = C_{k}^{k+s}, s_{j}(k) = C_{k}^{k+s−1}, where C_{k}^{k+s} = (k+s)!/k!s! and in our case s = 3j − 3. Below for the sake of simplicity the frequencies ν_{t} and ν_{l} are assumed to be unmultiplied. Thus, the total energy of the intermolecular vibrations can be written as follows: E = hν_{t}k_{t} + hν_{l}k_{l}, where variables k_{t} and k_{l} for the specified value of E are uniquely determined.

Note, that (2.1) is obtained under the assumption of a random fluctuative redistribution of energy over the cluster’s degrees of freedom that means anharmonicity was indirectly accounted for.

Now let us bring the final representation of the cluster dissociation rate constant omitting all intermediate results for the sake of brevity (the details of our method are described in [5]).

E_{j}(k) = \frac{8hν_{l}}{k_{B}T} Z_{1,2,r}^{+} (\frac{η}{d_{1}} )^{2} ϕ(k_{t}, k_{l}, s)j^{2/3}  (2.2)

ϕ(k_{t}, k_{l}, s) = \frac{(k_{t} − D_{t} + s)!(k_{l} + s)!}{(k_{t} − D_{t})!k_{l}!s!s!} (\frac{k_{t} + s}{sν_{l}} + \frac{k_{l} + s}{sν_{t}})^{−1}

Here D_{t} is defined by relationship E_{a} = hν_{t}D_{t} because the dissociation of a monomer is carried out by the energy supplement into translational vibrations, Z_{1,2,r}^{+} - the partition function of free rotations of the activated complex around the axes being perpendicular to the reaction coordinate. For Z_{1,2,r}^{+} we take representation Z_{1,2,r}^{+} = 2μ_{j}(r^{+})^{2}k_{B}T π/h^{2}, where μ_{j} - the reduced mass of fragments (μ_{j} ≈ m_{1}), r^{+} is about twice the bond length between the dissociating monomer and the cluster of the other j − 1 molecules.

To derive (2.2) we supposed that as the monomer tears itself off the cluster then its two degrees of freedom corresponding to librational vibrations in the activated complex turn into free rotations (the ”loosened” complex model). As far as the number of reaction channels L is concerned we take into account that the most capable for dissociation are molecules located on the surface of the cluster. For large clusters having a spherical shape the number of surface molecules is 4πη^{2}j^{2/3}/(πd_{1}^{2}/4) = 16(η/d_{1})^{2}j^{2/3}, where d_{1} is the effective diameter of a monomer in the cluster.

3 The top-most level M_{j}

It is obvious that for the used model of the harmonic oscillators there is an infinite number of energetic levels in contrast to the real anharmonic oscillators for which a number of levels is limited by the dissociation barrier. In order to take this effect into account we need to introduce the important conception of the top-most level.

In principal, in any j-cluster at large k the dissociation rate is always higher than the relaxation one. Fortunately these levels don’t bring a contribution to K_{j}^{−}. Indeed, by definition z_{j}(k) = Q_{j}^{−1} exp (−hνk/k_{B}T) , Q_{j} = ∑_{k_{l},k_{t}} s_{j}(k_{t})s_{j}(k_{l}) exp (−hνk/k_{B}T), where νk ≡ ν_{l}k_{l} + ν_{t}k_{t}. As it can be easily shown s_{j}(k)z_{j}(k) is almost δ–function with a maximum in the point k = M determined by relationship d[s_{j}(k)z_{j}(k)]/dk |_{k=M}= 0. Thus, the considerable contribution in K_{j}^{−} brings only those levels which are close to M and more high levels could be omitted. In our case of unmultiplied modes one can get

M_{i} = \frac{s − 1}{α_{i}}, α_{i} = exp (\frac{hν_{i}}{k_{B}T} ) − 1, i = t, l  (3.1)

For large clusters ratio M_{j}/s doesn’t depend on j. Thus, for the cluster with the internal energy corresponding to M_{j}-level the mean energy per oscillator is constant and depends only uponT. In view of the above mentioned one can rewrite K_{j}^{−} in the form K_{j}^{−} = E_{j}(M). At s → ∞

ϕ(M_{t}, M_{l}, s) → ν_{t} (1 + \frac{1}{α_{t}} ) exp (−\frac{E_{a}}{k_{B}T} ) ,  (3.2)

Thus, all terms in (2.1) are presented in the explicit form and the final expression for K_{j}^{−} can be found

K_{j}^{−} = \frac{8hν_{l}}{k_{B}T} Z_{1,2,r}^{+} (\frac{η}{d_{1}} )^{2} ϕ(M_{t}, M_{l}, s)j^{2/3} ≈  (3.3)
≈ W_{0}(T)j^{2/3}  at large j
W_{0}(T) ≡ \frac{8hν_{l}}{k_{B}T} Z_{1,2,r}^{+} (\frac{η}{d_{1}} )^{2} ν_{t}(1 + 1/α_{t}) exp(−E_{a}/k_{B}T)

This formula has been preliminary obtained in [5]. The similar expression was derived by Klots in [6]. It is seen that for large clusters the dissociation rate constant is the linear function of their surface as it is commonly used.

4 The activation energy

The activation energy E_{a} presents in (3.2) as still an uncertain parameter. In principal, this energy should depend upon the cluster size j. In the zero approximation let E_{a} be considered in the form E_{a} = E_{b}y_{j}, where E_{b} is the average dissociation energy per bond, y_{j} is the average number of bonds per monomer which could be capable to leave the cluster with other molecules in the cluster. ^{2} Unfortunately, at present there is no available experimental and especially theoretical data on y_{j} variation under j growth. Therefore, for simulating such an effect let us use the simplest approximation ^{3} y_{j} = l_{1} − [l_{2}m/(j + m − 2)], where l_{1}, l_{2} and m for the specified substance are constants. We choose l_{1}, l_{2} and m by the indirect way, namely: we combined above approximation with (3.2) in the frame of UCT in order to calculate the condensation process in nozzles, jets, aerodynamic tubes, thermodiffusion chambers etc [2] and compared the results of our prediction with the experimental data. It turns out that the best agreement can be reach at l_{1} = 3, l_{2} = 1 (it corresponds to the circular structure of dimer). The variation of m has a weak influence on the final results on condensation kinetics however this parameter allows to change y_{j} variation rate. In our calculations we assume that the cluster containing more than 150 molecules is a macro one so the difference between y_{∞} and y_{1}00 should not exceed 5%. Hence, m=16. In the limiting cases j = 2 j → ∞ this approximation gives the right values of y_{j}. Note, that for water molecules E_{b} is of the order of 5–6 kkal/mol so that E_{b}/k_{B}T ≈ 2500/T ≈ 8 at T = 300K. Therefore, K_{j}^{−} considerably decreases under j growth. That is why for large clusters the relaxation rate from M–level is much greater than the dissociation one in contrast to small clusters for which the opposite relationship takes place.

For the Van-der-Waals clusters y_{j} = 1 at j = 2. For the large cluster supposing its molecules located in the junction of a cubic lattice one can find that y_{j} ≈ 5 − −6. Thus, l_{1} = 5 − −6, l_{2} = 4 − −5.

5 The condensation coefficient

Using the above expression for K_{j}^{−} it is possible to derive the explicit representation for condensation coefficient α_{c}. To do that one should input1.hp

2For instance, it is known [7] that small water clusters can exist in two different configurations: 1) like the circular one in which each molecule is bonded with its neighbours by two hydrogenous bonds (the most energy profitable configuration for clusters with j > 2); 2) a linear structure in which the neighbouring molecules are connected by the only bond. In contrast to that for large clusters containing over than 100 molecules a molecule located at the cluster surface (the most capable to dissociate) can have up to the maximum number of bonds which for water clusters is equal to 4 [8].

3Generally speaking, the energy of the hydrogenous bond per molecule increases with j growth [7].

Figure 1. Condensation coefficient for macro clusters α_{c∞} vs. T for water vapor (1 — E_{b} = 5kkal/mol, 2 —- E_{b} = 6kkal/mol).

Figure 2. Parameters r vs. T along the phase equilibrium curve for water vapor ( 1,2 — E_{b} = 5kkal/mol; 3,4 — E_{b} = 6kkal/mol, 1,3 — m = 30, 2,4 —- m = 10).

express the constant K_{j}^{+} of the reverse process - condensation through K_{j}^{−} based upon the microscopic reversibility law K_{j}^{+} = K_{j}^{−}n_{j+1,e}/n_{je}n_{1e}, where n_{je} is the cluster equilibrium distribution of sizes. As shown in [1] n_{j+1,e}/n_{je} ≈ exp (−2b/3j^{1/3}) , b ≡ (4πσ/k_{B}T)η^{2}, η ≡ (3m_{1}/4πρ_{l})^{1/3}, where m_{1} is a molecular mass, ρ_{l} is the liquid density , σ is the surface tension and n_{1e} is defined by the Clapeyrone–Clausius equation. Because the generally used expression K_{j}^{+} = α_{c}Kj^{2/3} where K = (k_{B}T/2πm_{1})^{1/2}η^{2}, it is possible to find α_{c}(j, T)

α_{c}(j, T) = \frac{8hν_{l}}{k_{B}T} Z_{1,2,r}^{+} (\frac{η}{d_{1}} )^{2} ϕ(M_{t}, M_{l}, s) exp (−\frac{2b}{3j^{1/3}} ) (Kn_{1e})^{−1} (5.1)

At large j (5.1) transforms into the form

α_{c} = \frac{8hν_{l}}{k_{B}T} Z_{1,2,r}^{+} (\frac{η}{d_{1}} )^{2} (\frac{ν_{t}}{Kn_{1e}} ) (1 + \frac{1}{α_{t}} ) exp (−\frac{E_{b}l_{1}}{k_{B}T} )  (5.2)

Thus, α_{c∞} ≡ α_{c} |_{j→∞} doesn’t depend on j. In Fig.1 the dependence of α_{c∞} vs. T is presented for water vapor. It is seen α_{c∞} significantly depends on E_{b} but unfortunately, the quantum chemistry data on E_{b} for water are ranged from 4 kkal/mol to 6 kkal/mol.

The present approach allows to determine the main parameter of UCT - r if we introduce some relationship defining the rate of vibrational relaxation for the top-most level M_{j}. For harmonic vibrations the familiar expression could be taken: R̄_{j}(M) = Ms_{j}(M)(P_{M,M_{1}}+P_{M,M+1})2\sqrt{π}η^{2}j^{2/3} × \sqrt{2k_{B}T/m_{1}} ≈ Ms_{j}(M)2\sqrt{π}η^{2}j^{2/3}\sqrt{2k_{B}T/m_{1}}

Solving equation R̄_{j}(M)n_{1} − Ē_{j}(M) |_{j=r}= 0 we can find the number r. Thus, r = r(T, P) where P = n_{1}k_{B}T is the gas pressure. In Fig. 2 the results of r calculation along the phase equilibrium curve are presented for water vapor.

In this way the rate constants of formation and decay for the cluster ions can be obtained. In such a case the dissociation energy also depends upon the sign and the value of the ion charge.

Bibliography

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8 The rate constants and condensation coefficients of clusters

app.37, Taganskaya 44, Moscow, 109147, Russia, e-mail: itkin@momipao.msk.su