Published in : Physics, Chemistry and Applications of Nanostructures, Eds. V.E. Borisenko, S.V. Gaponenko, V.S. Gurin Singapore, World Scientific, 2005, pp. 219-222.
MAХIMUM OF INFORMATION ENTROPY AND SIZE DISTRIBUTION FUNCTION OF NANOCLUSTERS S.P. FISENKO A.V. Luikov Heat&Mass Transfer Institute, National Academy of Sciences of Belarus, 15,P.Brovka St., Minsk, Belarus, E-mail:
[email protected] The new form of size distribution function of small nanoclusters is obtained by means of the principle of the maximum of the information entropy. Surprisingly, for saturated and supersaturated vapor this function is not monotonic one. It was shown that normalization constant of this distribution function depends on supersaturation in the system. Some new results of calculation of nucleation rate of graphite like carbon clusters are presented. The transition to barrierless nucleation of carbon clusters is predicted for relatively small supersaturations.
1
Introduction
The explicit form of steady-state size distribution function f(g) of clusters of g molecules, formed at a vapor or a plasma, is actual and important problem for many applications. Currently, there is a sharp contradiction between Frenkel’s estimation of this distribution function and modern theoretical [1] and experimental results. In particular, such contradiction exists for small carbon clusters [2]. The aim of this work is to clarify this situation using modern results of nonequilibrium statistical mechanics. We regard as that any cluster has at least two molecules. For simplicity of mathematical treatment we consider that the number of molecules g is the continuous variable. The supersaturation S is the important parameter of "mother " phase: mechanisms: S n 1 (T ) / n s
where ns(T) is the numerical density of the saturated vapor at temperature T, and n1 is the actual density of monomers of a vapor. S is the measure of nonequilibrium situation in gas phase relatively of phase transition. For derivation of this nonequilibrium distribution function we use the principle of the maximum of the Boltzmann - Gibbs information entropy Si. This principle is widely used at nonequilibrium statistical mechanics [3]; the information entropy Si is defined as following
Si k f (g ) ln(f (g ))dg ,
Fisenko_nanomeeting_2005.doc submitted to World Scientific 12/18/2014 : 2:46:41 PM 1/4
where k is Boltzmann’s constant. The nonequilibrium size distribution function, obtained by such method, gives usually good description although it may "camouflage" some details. We accept, based on results of the nucleation theory [1,4], that the ensemble of clusters is well enough characterized by the average dimensionless free energy of cluster formation , where (g ) F(g ) gF1 ,
(1)
where -1 = kT, T is the temperature of vapor, F(g) is the free energy of cluster from g molecules, F1 the free energy per one vapor molecule. Here the averaging procedure of any function A(g) means averaging with the size distribution function f(g)
A f (g )A(g)dg .
(2)
Following [5], we put forward the natural condition that f(0)=0. Therefore we have additionally to accept that the ensemble of clusters is characterized by the average value of . From the condition of conserving normalization of the size distribution function, the conservation values of , the value of , and the maximum of information entropy Si, we have the explicit form of size distribution function: f (g ) Cg exp (g )
(3)
Normalization constant C and are the Lagrange multipliers. For convergence of normalization integral parameter has to be greater that one. It worth to note that for binary nucleation =1 exactly [4], therefore below we will use the same value of . 2
Size distribution function of graphite like nanoclusters
We apply the size distribution function (3) with =1 for description of graphite like carbon nanoclusters, which have a broad industrial applications. For calculation of size distribution of graphite like carbon nanoclusters we use the capillary approximation, which was successfully exploited in the nucleation theory [1] for description of nanoclusters. The free energy of cluster formation (g), which includes the contributions of volumetric part and interfacial one, is equal to: (g ) g ln S 4R 2
Fisenko_nanomeeting_2005.doc submitted to World Scientific 12/18/2014 : 2:46:41 PM 2/4
where R is the cluster radius, is the surface tension. For 2000C the density of saturated carbon vapor ns is about 2 1016atoms/m3; for graphite the surface tension is equal to 96mJ/m2 [6]. The calculated size distribution functions for different supersaturations of carbon are displayed in Figure 1. Interestingly that for equilibrium vapor (S=1), the maximum of cluster distribution function is for three atoms that corresponded well with the results of [2]. We see that for higher supersaturation the amplitude of maximum decreases, but the width of peak significantly increases. Probability to find dimers and trimers is higher for unsaturated carbon vapor.
Probability
0.18 0.16
S=0.9
0.14
S=1
0.12
S=1.05
0.10 0.08 0.06 0.04 0.02 0.00 0
4
8
12
16
20
24
28
32
36
40
The number of carbon atoms Figure 1. Size distribution function of clusters in carbon vapor at 2000K
3
Nucleation rate of graphite like clusters
Developed results permit to calculate nucleation rate of graphite like carbon clusters. To emphasize hat we use the distribution function (3) as the boundary condition for small g for modernized version of the kinetic equation. The steady state kinetic equation looks as following:
g f (g )L11 g ln f (g ) ln(g ) (g ) g I 0
The nucleation rate I is
Fisenko_nanomeeting_2005.doc submitted to World Scientific 12/18/2014 : 2:46:41 PM 3/4
I Cg * L11 exp (g*) /
(4)
where g* is the critical size of clusters, L11 is the flux of vapor atoms on the critical cluster, =0.5gg(g*). From consideration above it follows that the normalization constant C in the expressions (3) and (4) depends on supersaturation. Indeed, curves, which are shown in Figure 1, give clear understanding that classical estimation of normalization constant Cn n1 is very crude [7]. For example for unsaturated vapor and using expression (3), we have now estimation Cn 0.5n1. For supersaturated vapor, easily to show that Cn
