Kinetic theory of heterogeneous nucleation on charged particles

A. L. Itkin

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

Abstract A new kinetic theory of heterogeneous nucleation is proposed as a generalization of the unimolecular condensation theory (UCT) put forward by the author and Dr.Kolesnichenko [1]. In the present paper the main provisions of the theory are shortly formulated in order to demonstrate why and how heterogeneous condensation on the positive ions differs from condensation on electrons.

1 Introduction

In the present report we describe a new theory of heterogeneous condensation on charged particles - ions and electrons. Since 1899 when Wilson observed this phenomenon it has been well-known that the rate of water vapor condensation is dramatically accelerated in the presence of free electrons, however so far there has been no adequate theory of such processes. In the existing attempts to describe it based on the classical nucleation theory (CNT) all restrictions of CNT are presented.

The new theory proposed is a generalization of UCT put forward by the author and Dr.Kolesnichenko [1]. Below the main provisions of the theory are shortly formulated (a more detailed description see in [2] and the difference between heterogeneous condensation on positive ions and electrons is explained.

2 Formulation of the problem

Below we investigate a mixture of gas molecules, free ions and electrons and both neutral and ionic clusters containing only one positive or negative ion and a few neutral molecules. Each cluster and molecule in addition to the total number of molecules j are characterized by their total internal vibrational energy E_{j}(k) where k = 0 … G_{j} is the quantum number. The translational and rotational temperatures of the mixture are equal except the translational temperature of the electronic gas. We consider the kinetic equations describing simultaneously condensation and relaxation processes as well as recombination and ionization of molecules and clusters. However, the cluster ions having two or more charges were left aside. In addition we take into account only formation and decay of clusters gaining or losing a single molecule or ion.

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

Based on the results obtained in [1] let the mixture of clusters be characterized by a set of numerical densities x_{j}^{ν}(k) where superscript ν = 0 denotes the homogeneous clusters, ν = e - the heterogeneous ones including an additional electron (electronic clusters) and ν = i - the heterogeneous clusters with an additional positive ion (ionic clusters), G_{j}^{ν} denotes the top - most energetic level of the corresponding cluster [1]. In particular, x_{1}^{i}(k) is a population of the vibrational states of free ions, x_{1}^{e}(0) ≡ x_{1}^{e} - a concentration of free electrons; by definition x_{1}^{e}(k) ≡ 0 if k ≠ 0. The kinetic equations describing the evolution of the mixture are the following

\frac{dx_{j}^{ν}(k)}{dt} = J_{j}^{ν}(k) + I_{j}^{ν}(k),  j ≥ 2  (2.1)

In (2.1) the first term describes processes of collisional relaxation of the cluster vibrational energy, the second one - variation of the cluster concentration due to condensation and evaporation. In detail they can be written as follows

J_{j}^{ν}(k) = ∑ J_{j}^{νμ}(k) ≡ ∑_{μ} ∑_{l,i,m} J_{j}^{νμ}(l, i | k, m)  (2.2)
I_{j}^{μ}(k) = I_{j−1}^{μ0}(k) − I_{j}^{ν0}(k) + I_{j}^{0ν}(k)(1 − δ_{ν0}) − ∑_{μ=e,i} I_{j}^{νμ}(k)δ_{0ν}

I_{j−1}^{νμ}(k) = ∑_{l,i} I_{j−1}^{νμ}(l, i | k)(1 + δ_{j1}δ_{li})  (2.3)

J_{j}^{νμ}(l, i | k, m) = R_{j}^{νμ}(l, i | k, m)x̄_{j}^{ν}(l)x̄_{1}^{μ}(i) − R_{j}^{νμ}(k, m | l, i)x̄_{j}^{ν}(k)x̄_{1}^{μ}(m)

I_{j−1}^{ν0}(l, i | k) = C_{j−1}^{ν0}(l, i | k)x̄_{j−1}^{ν}(l)x̄_{1}^{0}(i) − E_{j}^{ν0}(k | i, l)x̄_{j}^{ν}(k), ν = 0, e, i,

I_{j}^{0μ}(l, i | k) = C_{j}^{0μ}(l, i | k)x̄_{j}(l)x̄_{1}^{μ}(i) − E_{j+1}^{0μ}(k | i, l)x̄_{j+1}^{μ}(k), μ = i, e

Equations for monomers and free ions and electrons can be written in the form

\frac{dx_{1}^{0}(k)}{dt} = − ∑_{j} ∑_{ν} Ī_{j}^{ν0}(k) − ∑_{j} ∑_{ν} J̄_{j}^{ν0}(k)  (2.4)

\frac{dx_{1}^{μ}(k)}{dt} = − ∑_{j} Ī_{j}^{0μ}(k) − ∑_{j} ∑_{ν} J̄_{j}^{νμ}(k),  μ = e, i
J̄_{j}^{νμ}(k) = ∑_{l,i,m} J_{j}^{νμ}(l, k | i, m)(1 + δ_{lk}δ_{j1}δ_{νμ})
Ī_{j−1}^{νμ}(k) = ∑_{l,i} I_{j−1}^{νμ}(l, k | i)(1 + δ_{j1}δ_{lk})

Here t is the time; C_{j−1}^{00}(l, i | k) is the rate of formation of the homogeneous j-mer with energy E_{j}(k) from the homogeneous (j − 1) mer with energy E_{j−1}(l) by the monomer attachment with energy E_{1}(i); E_{j}^{00}(k, | l, i) is the rate of the reverse process; R_{j}^{00}(l, i | k, m) is the rate of relaxation of the homogeneous j - mer from the state with energy E_{j}(l) into the state with energy E_{j}(k) together with the relaxation of a monomer from the state with energy E_{1}(i) into the state with energy E_{1}(m); x̄_{j}^{ν}(k) = x_{j}^{ν}(k)/s_{j}^{ν}(k), s_{j}^{ν}(k) - a statistical weight of the k-th vibrational level of the respective cluster, C_{j}^{ν}, E_{j}^{ν}, R_{j}^{ν0}, ν = i, e - the respective constants for ionic and electronic clusters describing processes of their interaction with monomers, R_{j}^{νμ}, ν, μ = i, e - the relaxation constant describing the interaction between the heterogeneous clusters and ions or electrons.

The populations x_{j}^{ν}(k) are to be normalized by the condition

∑_{j=1}^{N} j ∑_{ν} ∑_{k=0}^{G_{j}} x_{j}^{ν}(k) ≡ ∑_{j=1}^{N} j ∑_{ν} x_{j}^{ν} = n ≡ N^{0}/V,  (2.5)

∑_{j=1}^{N+1} ∑_{k=0}^{G_{j}} x_{j}^{i}(k) ≡ ω ≡ N^{i}/V,  ∑_{j=0}^{N+1} ∑_{k=0}^{G_{j}} x_{j}^{e}(k) ≡ ω ≡ N^{i}/V

where V is the volume of the system, N^{ν}, ν = 0, i, e is the total number of molecules, positive and negative charged particles in the system, respectively. Further, for the sake of simplicity we assume N^{i} = N^{e},

3 Equilibrium solutions

It is obvious the equilibrium populations are connected with the equilibrium concentrations x_{j}^{ν} = ∑_{i} x_{j}^{ν}(i) by the conditions x_{je}^{ν}(k) = x_{je}^{ν}s_{j}^{ν}(k)Z_{j}^{ν}(k). Here Z_{j}^{ν}(k) = exp(−E_{j}^{ν}(k)/k_{B}T) /Q_{j}^{ν}, ν = 0, e, i and Q_{j}^{ν} are the Boltzmann factors of level k and the vibrational partition functions of the homogeneous and heterogeneous clusters.

To determine equilibrium concentrations x_{je}^{ν} we use the method [3] together with the simplest model in which the charged particle is assumed to be located in the center of the cluster, all clusters are suggested to have a shape close to the sphere, the influence of the electric field of the ion on the intermolecular vibration frequencies and the gas polarization is assumed to be neglegible but the contribution of the spontaneous polarization of the cluster molecules is taken into account. As shown in [2] with allowance for the above assumptions one can get

x_{je}^{0} = Ā_{j}(T) exp(−bj^{2/3}), x_{je}^{ν} = x_{1e}^{ν}Ā_{j} exp(−bj^{2/3})K_{j}^{ν}, j ≥ 2 (3.1)

x_{1e}^{ν} = ω [1 + ∑_{j=2}^{N} Ā_{j}(T) exp(−bj^{2/3})K_{j}^{ν} ]^{−1} , ν = e, i

Here b ≡ (4πσ/k_{B}T)η^{2}, η = (3m_{1}/4πρ_{l})^{1/3} , K_{j}^{ν} = λ_{1}^{3} [j^{−1} + (m_{1}/m_{ν})]^{3/2} × exp (−ΔΨ_{j}^{ν}/k_{B}T ) , m_{ν}, ν = 1, i, e are the masses of a single molecule and positive and negative ions, respectively, ρ_{l} is the liquid density , σ is the surface tension, ΔΨ_{j}^{ν} is the barrier height, λ_{1} is the De-Broil wave length. For Ā_{j} and ΔΨ_{j}^{ν}, ν = i, e the complete representation is given in [2].

As shown in [2] x_{1e}^{0} coincides with the thermodynamic equilibrium density determined by the Clapeirone - Clausius equation if n > n_{1e}. If n < n_{1e} one can get x_{1e}^{0} solving the equation x_{1e}^{0}[1 + ωK_{j}^{i}(1 + x_{1e}K_{j}^{i})^{−1} +ωK_{j}^{e}(1+ x_{1e}K_{j}^{e})^{−1}] ≈ n. In particular, if n ≫ ω, then x_{1e}^{0} = n (it means that in the unsaturated vapor without ions there are almost no clusters). If ω ≫ n, then x_{1e}^{0} ≈ (n/ω)(K_{j}^{i} + K_{j}^{e})^{−1}. This equation is valid at n/[ω(K_{j}^{i} + K_{j}^{e})] < n_{1e}. Note, that for many substances (for example, water vapor, nitrogen etc.) x_{1e}K_{J}^{ν} ≪ 1, ν = e, i, so that x_{1e}^{0} = n[1 + ω(K_{j}^{i} + K_{j}^{e})]^{−1}.

Thus, there is a situation when the vapor being in fact in the equilibrium state without ions and electrons becomes supersaturated in their presence. Actually, let the vapor firstly be consisted only of neutral monomers and n < x_{1e,s}. Then it is followed that x_{1e}^{0} ≈ n or supersaturation S = n/x_{1e}^{0} < 1. An addition of ions or electrons to the vapor leads to x_{1e} decrease so that S becomes more than a unity.

4 Nonequilibrium solutions

In the present work we have modified method [1] to solve the kinetic equations (2.1), (2.4). The main idea of UCT used to solve (2.1) is to subdivide all x_{j}^{ν} variables into fast and slow ones, i.e. into the variables for which the specific rates of relaxation are quite different. Then one could study a behaviour of the whole system in the time scale corresponding to the slow variable evolution because the fast variables in this scale depend only upon the slow variables (a steady-state regime). The method of finding these steady-state solutions proposed in [2] consists of three steps.

Step 1. The initial step is connected with a special variable exchange. We substitute in (2.1) the concentrations x_{j}^{ν} for x_{j}^{ν}(0). The respective equations for x_{j}^{ν} can be obtained from (2.1) by summing the equations for x_{j}^{ν}(k) over all k. Further it is managed to prove that all populations x_{j}^{ν}(k), j = 2...N, k = 1... are the fast variables with respect to the slow concentrations x_{j}^{ν}.

Step 2. Since all variables are subdivided into the slow and the fast ones we can use methods of the asymptotic integration for singular perturbated systems. Doing that once again we have used the idea of [1] to introduce size r such that at j > r the rate of condensation is much more than the rate of relaxation for the top-most levels but for j < r the opposite relation takes place. The problem to express r for the charged clusters through the gas properties as well as the problem of the rate constant determination for the above processes are discussed in [4].

Expressing all the fast variables x_{j}^{ν}(k), k = 0... through the low concentrations and substituting the obtained relationships into the equations for x_{j}^{ν} we get the closed system with respect to x_{j}^{ν} [2]. In case of the fast relaxation of monomers at j > r the derived equations for neutral clusters coincide with the quasichemical model in contrast to all other situations (j ≤ r, the slow monomer relaxation) where these equations differ significantly from the generally used ones.

Step 3. To solve the steady-state equations for x_{j}^{ν} we used the special method proposed in [1]. It allows to find an analytical expression of the cluster concentrations through T and S for all j excluding very large j. Once again this method is based upon the nonlinear variable substitution x_{j}^{ν} → f_{j}^{ν} = x̄_{j}^{ν}/x̄_{j−1}^{ν}S_{0}, j > r_{ν}; f_{j}^{ν} = x̄_{j}^{ν}/S_{0}^{j}, j < r_{ν}, so that in new variables all initially coupled equations become independent. To manage this one should assume the following: 1) For large clusters ∑_{l,i} _{j}(k | i, l) ∼ j^{ν}, ν ≤ 1; 2) the equilibrium concentrations x_{je}^{ν} satisfy relationship x_{je}^{ν}/x_{1e}^{ν} ≪ 1; 3) at j > r variables f_{j} smoothly depend upon j. As shown in [2] for the model values of the rate constants found in [4] all these assumptions could be proved to be valid.

The final reduced equations for the only low variables of the system S^{ν} are as follows

\frac{dS}{dt} = − K_{1}^{i+}β^{i}(1)S(S_{i} − 1) − K_{1}^{e+}β^{e}(1)S(S_{e} − 1)  (4.1)
− ∑_{ν=0,i,e} \frac{2K^{ν+}(q_{ν} + 2)β^{ν}(q_{ν} + 1)S^{q_{ν}+2}ϕ^{ν}(q_{ν})ϕ^{ν}(q_{ν} + 1)b}{9q_{ν}^{4/3}}

\frac{dS_{i}}{dt} = K_{1}^{i+}S(1 − S_{i}), q_{ν} = max(r_{ν}, h_{ν}) K^{ν}(j) ≡ K_{j}^{ν}
\frac{dS_{e}}{dt} = −\frac{K_{2}^{e+}SK_{1}^{e+}x_{1e}^{e}/x_{2e}^{e}}{K_{2}^{e+}S + K_{1}^{e+}x_{1e}^{e}/x_{2e}^{e}} S(1 − S_{e})

Here S_{ν} = x_{1}^{ν}/x_{1e}^{ν} - three supersaturations corresponding to neutral, ionic input2.hp

Figure 1. The dependences of h_{i} (the dashed line), h_{e} (the dash-dot line) and j_{∗} (the solid line) on lnS at T = const.

and electronic clusters (compare it with homogeneous condensation where the only S exists!), β_{j}^{ν} = x_{j}^{ν}/x_{je}^{ν}, ϕ^{ν}(q_{ν}) = S exp(−2b/3h_{ν}^{1/3}), ν = e, i, ϕ^{0} ≡ 1, h_{0} ≈ j_{∗} - the classical critical size, K_{j}^{ν+} = ∑_{l,i} C_{j−1}^{ν0}(l, i | G_{j}^{ν})Z_{j−1}^{ν}(l)Z_{1}^{0}(i), K_{j}^{ν−} = ∑_{l,i} E_{j}^{ν0}(G_{j}^{ν} | i, l)Z_{j}^{ν}(G_{j}^{ν}). ^{1}

Solving (4.1) three supersaturations can be found and using these values and T one can get all concentrations of the clusters and populations of their vibrational levels based on the analytical expressions given in [2]. The only uncertain parameter remained in (4.1) is h_{ν}, ν = e, i, which is the root of the equation with respect to j

\frac{1}{S_{0}} exp (\frac{2b}{3j^{1/3}} + \frac{1}{k_{B}T} \frac{∂ΔΨ_{j}^{ν}}{∂j} ) = 1  (4.2)

As shown in [2] under the above assumptions we manage to rewrite (4.2) in the form

1 − \frac{δ_{1}}{j} [1 − \frac{δ_{2}}{j^{1/3}} (1 − \frac{δ_{3}}{j} )] = (\frac{j}{j_{∗}} )^{1/3}  (4.3)

δ_{1} = \frac{22πBe^{2}}{3ηk_{B}T} , δ_{2} = \frac{24η̄}{11η}, δ_{3} = \frac{8BeP_{0}}{η̄η^{2}P_{∞}^{2}} , B = \frac{1}{16π} (\frac{1}{ϵ_{v}} − \frac{1}{ϵ_{v}} )

Here e is the elementary charge, ϵ_{v} and ϵ_{l} are the dielectric penetrations of the vapor and liquid phases, respectively, P_{0} is the spontaneous polarization, P_{∞} = pρ_{l}, where p is the dipole momentum of a single molecule, η̄ is the displacement of the phase splitting surface (for water vapor η̄ ≈) 1A.

The form of solution of (4.3) essentially depends on the sign of the term eP_{0}. For instance, for water clusters P_{0} < 0 therefore for positive ionic clusters δ_{3} < 0. In such a case the qualitative analysis of (4.3) gives that at S about 1 h_{i} ≈ j_{∗} and h_{i} decreases like j_{∗} with S increase (see Fig.1, curve 1). For negative electronic water clusters δ_{3} > 0 and the dependence of h_{e} upon S takes quite an unusual form (curve 2 in Fig.1). Indeed, the nucleation rate strongly depends upon q_{ν} = max(r_{ν}, h_{ν} (see (4.1)). The typical value of r_{ν} for water vapor is about 6 [2], therefore the nucleation rate at small j is determined by h_{ν}. As at S < 1 we have h_{0} = h_{i} = j_{∗} = ∞ the rate of nucleation on neutral molecules and positive ions is negligible because β^{ν}(j_{∗})S^{j_{∗}} → 0 at j_{∗} → ∞. However, for the electronic clusters at S < 1 there exists the second root of (4.3) h_{e} = h_{2e} > 0 which is rather small that means the term β^{e}(q_{e})S^{q_{e}} is large enough. Thus, even in the unsaturated water vapor heterogeneous nucleation on negatively charged particles can be carried out but in any case condensation elapses immediately after the moment when the vapor became supersaturated. The typical value of supersaturation in the Wilson point in such a case is about 1.1 though in the absence of electrons it is about 5 − 6. This situation was observed by Wilson in his famous experiments with the chamber as early as at the end of the last century.

1Generally speaking h_{0} is equal to the ratio of some complex terms created by the constants presented in (2.1) but this ratio suddenly has turned out about j_{∗}. However, in our case we should extend the j_{∗} definition supposing j_{∗} = ∞ at S < 1.

Bibliography

1. Itkin, A.L., Kolesnichenko, E.G. and Gorbachev, Y.E. (1991). Preprints of Ioffe Phys.Tech.Inst.Acad.Sci. USSR, Quasy - stationary state method in the theory of condensation. 1488, Asymptotical methods in the theory of condensation (unimolecular model). 1490, The analyses of the nucleation kinetics experimental data from the point of unimolecular condensation theory view. 1523.

2. Itkin, A.L., and Kolesnichenko, E.G. (1995). Nonempirical theory of heterogeneous condensation. Soviet Journal of Chemical Physics. 12, No.1, 121–131, No.2, 215-239.

3. Bogdanov, A.V., ..., Itkin, A.L., Kolesnichenko, E.G....(1992). Analytical investigation of equilibrium and steady state solutions for a quasichemical condensation model. Soviet Journal of Chemical Physics. 9 No.5, 1113–1129.

4. Itkin, A.L., and Kolesnichenko, E.G. (1993). Calculation of dissociation rate constants for vibrational-excited clusters and an estimation of the main parameter of unimolecular condensation theory. Soviet Journal of Chemical Physics. 10, No.11, 2403–2418.

8 Kinetic theory of heterogeneous nucleation

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