Keywords: Nano Particles; Nano Crystalline; Mathematical Modeling; Nano Catalysts; Critical Grain Size
Usually, the nanoparticles show catalytic activity in a very narrow size range [46]. Unique nanoparticles’ properties naturally occur if the particle size does not exceed a certain value [7]. Obviously, this is also due to surface phenomena. However, not all of the effects can be explained by in surface area increasing. For example if the atoms located on the faces are catalytically active, the rate of the catalyzed reaction will be increased by larger particles [8, 9]. The main question remains on how to determine the upper and sometimes the lower limits of the sizes in which nanoproperties appears. These borders could be determined by experiments with the detection of the size effect. Such experiment should be carried out on materials of the same chemical composition but different dispersion in grain size. The important condition of this experiment is equal size of the particles or grains of material investigated. Unfortunately, at the present time it is not possible to carry out such experiments [10].
We believe that these limits can be calculated using the tools of mathematical statistics on the basis of experimental data obtained on materials with significant dispersion of particle sizes. Submitted paper describes the principles of such approach.
In this vein, one of the most important characteristics of nanoparticles is their dimensions. In terms of chemical properties of nanoparticles the most important characteristics aren’t their linear dimensions, weight or volume but their surface area.
The quantitative statistical characteristics used to reflect the average relative values of the active surface of nanomaterials are the rms diameter and polydispersity index of the nanoparticles [12]. The question has arisen: from which dimension of the particles some special properties inheriting namely nanostructures have appeared and at what dimensions they have disappeared? Which features are necessary for a material to be considered a nanomaterial?
For investigation of influence of qualitative changing of properties of nanocomposites it is purposefully to turn the most evident manifestations peculiarities inherent exceptionally to nanoparticles. To such teases can be attributed effects shown by drawing palladium nanocatalysts of lowtemperature oxidation of CO by oxygen. Really exhaustive removal of CO by its catalytic oxidation by oxygen of air at room conditions don’t note neither for too small (solutions, solid solutions) nor too large (massive samples) objects of similar chemical compositions.
In composition of investigated highactive nanocatalysts drawing on polymeric bearers as inorganic and organic nature the active components has presented as particles of complex composition with dimensions 101000 nm [14] containing platinum metals and their compounds. More clearly there particles can be examine on the microphotographs of samples with carbon fibrous materials as bearers (Figure 1). There is a rather high level of the dispersion of the size of the particles.
Let us proposed that at display of high catalytic activity of composites special effects connected with nanodimensions absent. Then in rate of catalytic heterogeneous reactions the role of the summary area of surface of all particles which are accessible to substrates will be very important. In such case it is possible to expect that catalytic activity will be increase smoothly with increasing of total area of surface. Even if this connection isn’t linear never the less it must not be spasmodic on some segment of curve. Proceed from supposition about spherical of particles their volume is proportional to cube of radius and the area of surface  to quadrat of ball diameter. Proceeding from ball with radius d*R d3 balls with less radius R will be obtain it is easily to understand that the total area of surface with decreasing of radius in “d” times will increased at the same times.
Comparison of catalytic activity of samples with palladium on the fiber obtained by carbonization and following activation of MtilonM [16] has shown that there is sharp increasing of activity for nanoparticles with dimensions 3001000 nm (Table 1). The same qualitative changing of catalytic activity was observed for samples on the base of carbonized hydrocellulose with dimensions of particle 40100 nm (area of active surface 300 m^{2}/g).
These data have witnessed in benefit of hypothesis about presence of brightly expressed effect of increasing of catalytic activity at increasing of content of nanoparticles of definite radiuses. Apparently due to the marked sizes of particles with
In the following regime, bold italics are attributed to data on the base of which calculations were carried out.
Average squared diameter, nm 
Specific activity*10^{5}, mole/l*s*g 
Increasing of total surface in times 
Increasing of specific activity in calculation on the unit of area of surface in times 
Bearer – activated carbon fibrous materials from MtilonM, active surface area 2700 m^{2}/g 

2000 
0.2 
1 
1 
1500 
1.4 
1.3 
5 
1200 
1.12 
1.7 
3 
1000 
15.6 
2 
39 
800 
22.1 
2.5 
44 
720 
21.1 
2.8 
38 
601 
24 
3.3 
36 
511 
27.5 
3.9 
35 
380 
15.1 
5.3 
14 
222 
6.4 
9 
4 
104 
2.9 
19.2 
0.8 
62 
0.8 
32.3 
0.1 
47 
0.2 
42.6 
0 
24 
0 
83.3 
0 
12 
0 
166.7 
0 
Bearer –carbon fibrous materials from hydrocellulose, active surface area 300 m^{2}/g 

200 
0 


120 
0.11 
1 
1 
111 
0.09 
1.1 
0.8 
98 
0.84 
1.2 
6.2 
85 
1.4 
1.4 
9 
77 
1.12 
1.6 
6.5 
64 
1.16 
1.9 
5.6 
54 
1.21 
2.2 
5 
44 
1.21 
2.7 
4 
36 
0.41 
3.3 
1.1 
28 
0.27 
4.3 
0.6 
19 
0.12 
6.3 
0.2 
10 
0 
12 
0 
Less than 10 
0 
24 
0 
On the base of obtained data it is shown that effect of spasmodic increasing activity for composites with definite average dimensions of particles can’t be explained by increasing of the total surface of contact of particles with substrate. Thus it is determined that there are effects caused by nanodimensions of composite’s particles.
dimension of particles the nanoproperties have appeared and at what minimal dimension of them they have ended? With others words – how to determine those limits at which particles can be named as nanoparticles and containing them composites – nanocomposites? It is clearly that for each system these values will be different. named "model of balls coloring". Lower the description of this model is cited.
Hypothetical case: It is necessary to paint by different colors the surface of balls which in reality are ellipsoids stretching in one direction, but in first approaching we shall suppose them as balls. Color in this case will be corresponded to definite experimental sample possessed by measured macroproperties – in our case it is correspondently nanocatalysts and their catalytic activity. Ball were in some containers. In each container there are enough many balls – about 10 milliards and by this reason the statistical regularities can be used. The common volume (or mass) of all balls for all containers is similar (naturally, the poured volume of balls will be different, but this fact isn’t important). Balls from each container must be painted in the same color but from different containers – in different colors.
However not all balls must be colored. It is decided no paint too small and too large balls. There are known different and limited amounts of dyes for each color. Expenditure of dye for the unit of surface of balls is the same. It is obvious that these amount from the point of view of chemist will corresponded to observed and measurable macroproperty – in our case to catalytic activity determined as rate of oxidation reaction of CO in the presence of given concrete sample of catalyst.
Required to determine: It is necessary to select such values of minimal and maximal radius of balls that dye will spend optimally.
Measured and available data: Before carrying out of calculations from each container not less than 40 balls have been selected randomly and their radiuses were measured. In result for each container two characteristics were obtained: average squared radius of ball which was calculated on the base of experimental results by formula:
$$\overline{R}=\sqrt{\frac{{\displaystyle \sum _{i=1}^{n}{{\displaystyle R}}_{i}^{2}}}{n}}$$ Measured and available data: Before carrying out of calculations from each container not less than 40 balls have been selected randomly and their radiuses were measured. In result for each container two characteristics were obtained: average squared radius of ball which was calculated on the base of experimental results by formula:
$$L=\frac{{{\displaystyle \sum _{i=1}^{n}\left(\overline{R}{R}_{i}\right)}}^{2}}{n{\overline{R}}^{2}}$$ Quantity of dye (in given case it has corresponded to rate of reaction in the presence of given sample) was known before hand (in given case this quantity was obtained in results of experiments on the base of measuring of catalytic activity that is by decreasing of CO concentration on the outlet).
As example data obtained on the base of real measures by rate determination (V) of catalytic reaction for activated carbon fibers on the base of MtilonM with drawing nanoparticles of palladium are presented below (Table 2). For giving clearness to the mathematical problem values "nm" were substituted for "mm" and V – for volume in m3.
Also the function of balls distribution by their dimensions is necessary for carrying out of abovementioned calculation. We have supposed that this distribution is normal – Gaussian distribution.
For each container values of the total area of surface of balls were calculated. For each iball:
$$Si=4\text{}\pi \text{}Ri2$$ If radiuses of balls have changed on small value dR, than for determination of the total area of balls surface it is necessary to calculate some determined integral:
$$s={\displaystyle {\int}_{m}^{M}4\pi f({R}^{2}})dR$$ Function f (R2 ) is constructed by analytically on the base of data about type of distribution of balls by dimensions – Gaussian distribution; values of their average dimension and coefficient of polydispersity.
In result we have elaborated analytical method of determination of the total area of surface of all painted balls from each container in dependence on given values of "m" and "M" (Figure 2).
Determination of average squared deviation (error): After calculation of analytically common surface of dying balls for each container dependence of calculated and factual amount of dye was constructed. Deviations were determined by method of differentiation (for example, by method of least squares). Given values of "m" and "M" it is possible to find values of error.
Color of painted balls 
Average squared radius, mm 
L, % 
Volume, m^{3} 
Red 
125 
144 
1.12 
Orange 
263 
90 
1.28 
Yellow 
224 
98 
1.92 
Green 
336 
60 
2.56 
Blue 
301 
71 
3.28 
Dark blue 
222 
95 
4.48 
Violet 
336 
57 
4.8 
Brown 
349 
49 
9.84 
White 
268 
38 
22 
Black 
6 
81 
0.08 
Then formula of dependence value of deviation (confidential interval) at fixed probability for given values "m" and "M" was determined. This formula will be final formula for our model.
Determination of values of minimal and maximal radiuses: Now there is formula by which it is possible to find values of the confidential interval X or average squared deviations. By given values of "m" and "M" in this formula sought for values of boundaries of interval of balls dimensions will be supposed those for which confidential interval is minimum that is there is coincidence of calculated values with experimental data (Figure 3).
These values can be obtained by following method. At given step for "m" and “M" X can be calculated. Geometrically this will be the surface where "m" and "M" will be correspondently abscissa and ordinate, and X  applicate. Minimum of this surface is determined and then values of "m" and "M" can be calculated
Using and development of model: For concrete nanosystems (in our case – nanocatalysts) values of "m" and "M" have allowed to give nanostructures the quantitative determination and determine real borders inside of which the special properties have displayed.
At the same time the proposed model don’t take into account some factors which can be very considerable. Properties of nanostructures in particular catalytic activity can be depended on some parameters (in our case – from area of surface) not linearly: form of nanoparticles can differed from spherical or ellipsoidal shape; structure and chemical properties can differ for particles of different dimensions and also in limits of one particle. If proposed approach will be fruitful than this model will be needed in some précises and development.
Mathematical model was proposed which has allowed determining limits of nanoparticles dimensions in borders of which there have displayed some special properties inherited to only nanostructures. The model has allowed obtaining decisions, that is to find values of maximum and minimum radiuses of particles in particular for catalytically active nanocomposites and also can be expended for exposure of others special properties inherited only to nanosystems.
Described method allows finding the answer to one of two questions that’s consist the main scope of investigations of the nanocrystalline state: is there some critical grain (particle) size below which the characteristic properties of nanocrystals become observable, and above which the material behaves as a bulk one.
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