Tåõíè÷åñêèå íàóêè/ 3. Îòðàñëåâîå ìàøèíîñòðîåíèå

Banshidhar Choudhary1, Nickolay Zosimovych2

1Sharda University (Greater Noida, India, UP)

2State University of Telecommunications (Kyiv, Ukraine)

CONTACTING PARTICLES OF HCM WITH A MONODISPERSE COMPONENT

 

In these article has been conducted a simulation of the structure of heterogenous condensed mixtures containing one or two monodisperse particles in a wide range of their volume concentrations. It was determined the main statistical characteristics describing the internal structure of HCM. It has been shown that the method of viscous suspension allows simulating a appearance of regular structures when the volume concentration of the particles is close to the maximum possible corresponding close packing.

Key words: Heterogeneous condensed mixtures (HCM), viscous suspension, monodisperse particles, correlation, concentration, coordination number, combustion, cluster, distribution, simulation.

 

Introduction. The problem of random spatial arrangement of solid spheres (or discs in the plane), with a given diameter distribution is not only applied but also the fundamental significance, as occurs in many areas of science [1-4].

The analytical solution of this problem even in the simplest case is missing, so the primary method of research is computer modeling. However, and in this case there is no common method of placing in space particles with diameters of random given distribution [5, 6]. Direct solution of this problem by sorting even for the small number of particles occupies an unacceptable time and very often does not lead to the final result. The main difficulty consists in the fact that the allocation of particles should not overlap.

The greatest difficulties arise when particles are placed in space with wide spectrum of sizes. Methods mentioned above frequently lead to that of the size distribution of particles which is different from the distribution of the initial particles.

The structure of heterogeneous condensed mixtures. The suggested by S.A. Rashkovsky method of viscous suspension [7] may be applied to simulation of a structure having broad class of composite materials which are filled with solid particles having almost spherical form. The author has made a mathematical simulation of metal containing and metal-free HCM structures by the method of viscous suspension.

In general, the system is characterized by volume concentrations of components equal to the ratio of the total volume of particles of a given type to the volume of the entire system. Let us introduce volume concentration of oxidizer particles  and aluminum particles  [8]. We also chose solid and impenetratable walls as a boundary condition to make computations.

Calculations by the method viscous suspension were carried out for a wide class of HCM containing several types of dispersed components, each having its own function in the distribution according to size.  However, a system with particles of the same diameter already has all the properties inherent in real HCM. What's more such a simplification allows distracting from the complex analysis connected with the distribution of particle sizes [8].

HCM with a monodisperse component. Let us consider first a HCM structure, comprising monodisperse particles of the same type.

Such a system corresponds to the metal-free HCM, containing a binding substance and a dispersed oxidizer, such as AP. In this case, the simulation results describe the distribution of AP particles in the volume of HCM.

At the same time, the system under consideration allows to simulate the distribution of the particles of aluminum in metal containing HCM, if aluminum particles are much smaller than particles of oxidizer.  As we know [9, 10], when the system consists of two kinds of particles which sizes differ substantially, large particles can be placed first in the volume, and then in the space between them - fine particles. The typical distance between the large particles will be substantially larger than the diameter of fine particles. It means that we can be distracted from the real structure of the system and treat the distribution of fine particles between large ones as if fine particles were distributed in the free space with an efficient volume concentration  

                                                                                            (1)

In this case, the simulation results will describe the distribution of aluminum particles in the space between the AP particles (1).

Thus, in this study the volume concentration of particles  will be understood as  or  depending from the simulated system. Accordingly all linear dimensions will be referred either to the diameter of the AP particles   or to the diameter of aluminum particles

Many of the properties of HCM, especially acoustic and electromagnetic ones, are determined by the arrangement of the particles of dispersed components. The arrangement of the particles in HCM determines scattering and quenching of acoustic and electromagnetic waves in a material, just as the arrangement of atoms in a solid body or liquid determines the X-ray scattering [11].

The pair correlation function is the most important characteristic of the system, which describes arrangement of the particles.

Fig. 1. The pair correlation function of a system of identical hard spheres, obtained by the method of viscous suspension  

 

Fig. 1 represents the pair correlation function of a system of identical hard spheres, obtained by the method of viscous suspension for different volume concentrations of particles.

Calculations demonstrate that for volume concentrations of particles  the pair correlation function has two maxima if  If  the function  has a shaped characteristic describing contact particles. The amplitude of thisshaped characteristic is determined by the probability of particles' crossing each other when initially filling the space. This probability id determined as  because at low  initially crossing each other particles build further pairs of contact particles. Both maxima at the distance  become apparent due to the presence in the system of contact particles.

Fig. 1 shows that if particles are randomly distributed in the volume because correlations decay very fast and when the correlation function is close to unity. With the increase of  the correlation length increases as well. In such a case change of the correlation function has the character of damped oscillations which fact indicates appearance and growth of regular structures. When  the correlation length reaches 10. This means that when  there already exist quite large regular structures in HCM, in which location of the particles close to the close packing (this is supported by studies conducted by S.A. Rashkovsky, Moscow Institute of Heat Technology, Russia Federation) [12]. Thus, at high volume concentrations of particles of dispersed components HCM forms regular structures of sufficient extent within which the arrangement of the particles can not be considered accidental. However, this is only true for monodisperse particles. For polydisperse particles with a wide range occurrence of regular structures is observed at much higher volume concentrations in comparison to monodisperse ones or they do not occur at all.

It should be noted that the pair correlation function obtained by method of viscous suspension is different from the pair correlation function of "solid atoms" obtained by method of molecular dynamics [11, 13, 14], and the presence of features when and associated with a finite probability of particle contacts in HCM. If we truncate the contact particles that will correspond a uniform distribution of particles in the void space (i.e., the centers of particles in the space between the particles themselves), the standard pair correlation function coincides with the pair correlation function for "solid atoms".

 Another important characteristic of HCM is the coordination number of particles - the number of contacts with neighboring particles characterizing reactivity of HCM, and the ability of the aluminum particles to agglomerate. It has been noticed that when there is a large number of particles which do not contact with any of the particles. In this case, the maximum occurs in 1-2 contacts. And on the contrary, if  the system has very little free particles which do not contact with other. In such a case the maximum occurs in 5 contacts, and the system has a significant number of particles simultaneously in contact with other 8-9 particles.

Contacting particles in HCM form clusters [14]. When a heat wave passes through HCM aluminum particles in clusters fuse to form a quite long structures. These structures play an important role during the combustion process. First, they have higher thermal conductivity and during a combustion process in HCM play a role of "thermal bridges" transferring heat from the surface layers of HCM deep into phase. Second, these clusters represent "germs" of agglomerates that are formed during combustion of metallized HCM and change power characteristics of solid fuel installations [15, 16]. All that points to the fact that clusters of contacting particles are important elements of the HCM structure determining the combustion process [17].

The function of distribution of clusters by weight is a statistical characteristic of the cluster system. For the HCM which is considered here and contains particles of the same size it makes sense to talk about  mass fraction of clusters formed  by an exact number of  particles:   where  is a number of clusters containing an exact number of particles, is a total number of particles in the HCM. Obviously, here  is a relative mass of cluster (cluster mass as respects to the mass of one original particle).

Fig. 2. Distribution of clusters according to their mass in the system of similar hard spheres for different volume concentrations of particles

 

Fig. 2 shows the distribution function of clusters of contacting particles by weight (for the volume concentration of particles 0.2) in HCM for relatively low weight of clusters (less than 30). With the increase of the volume concentration  the fraction of clusters with low weight decreases, but the number of clusters with a large (and very large) weight increases.

We simulated distribution function of clusters of contact particles by weight for  within the whole weight range. It is evident that the system has very large clusters containing more than 5,000 original particles. We analyzed the size of clusters occurring at different volume concentrations  The clusters occurring in the “sample” HCM at different values were of utmost interest. For each of these clusters the maximum size or the largest distance between the centers of two particles in the cluster is determined.

The simulation results show that with the increase of volume concentration of particles increases the maximum size of the largest cluster. With concentrations of particles in HCM less than critical (depending on the size of the "sample"), the system has only isolated clusters with dimensions substantially smaller than the calculated "sample." With the increased concentration of particles the cluster size and a probability of occurring of a cluster that pervades the entire "sample" (percolation cluster) including a large mass of original particles [18] increase (on average) as well. At concentrations of aluminium  in a system with a probability close to unity, there occurs a percolation cluster, whose weight varies significantly from a calculation to a calculation.  With the increase of dimensions of the projected «sample»   slightly increases. Extrapolation of the obtained data to the infinite "sample" HCM shows that the percolation limit of the system of identical spherical particles is the concentration  These data agree well  with the results of [19, 20] which investigated concentration limits of the combustion wave spreading in heterogeneous systems.  Combustion in HCM considered in [19, 20] took place due to the heat transfer through the chain of contact particles and also due to a chemical reaction of contact particles, and stopped when these chains broke. Thus, the presence of a percolation cluster was a necessary condition for stable combustion in the considered HCM which made it possible to establish .

When approaching the percolation limit at low concentrations there are significant fluctuations in the size of the largest cluster, and the smaller "sample" is, the larger these fluctuations are. Existence of a threshold concentration (percolation limit) leads to a phenomenon similar to the phase transition of the second order [21]: in passing through the percolation limit there is an abrupt change in the properties of HCM which is defined by contact particles, such as thermal conductivity, electrical conductivity, etc. In this case, the volume concentration of the particles plays the role of an order parameter.  In metallized HCM when  due to sticking together of the particles in the percolation cluster during a combustion process there may appear a frame made of metal particles that will remain after combustion of the HCM sample and the HCM will retain its original shape.

Fig. 3. The number of original particles in the clusters for different effective concentrations of particles in HCM (line - the upper limit

 

Let's consider the results of the simulation of a system with monodisperse particles with respect to the distribution of aluminum in metallic HCM. Fig. 3 shows numbers of original particles of aluminum in different clusters in different effective concentrations  Here we used data for the following samples 20õ20õ20 and 30õ30õ30.

It is evident that for every value  maximum number of particles combined in one cluster does not exceed a certain value. The line

                                                                                 (2)

limits above all calculated values ​​and can be regarded as a relation (in a statistical sense) of the maximum number of primary particles in the cluster under the given effective concentration of aluminum in HCM. The estimation shows that the relation (2) limits the weight of the maximum cluster with a probability close to one. 

         The formula (2) shows that the weight if the maximum cluster of aluminum particles in HCM grows exponentially with concentration

There have been recent publications which establish the fractal nature of energy release during combustion of HCM [22]. Obviously, this is possible only provided there are fractal structures in the original HCM. Analysis of the results of a mathematical modeling in respect to monodisperse particles showed that any fractal structures (in the strict sense of the word) are absent in the original HCM. But it has been found that clusters of contact particles of aluminum in an original HCM form fractal structures.

The analysis shows that the clusters are by an average stretched in one direction, the calculated points are grouped near the following power-law relation

                                                                                          (3)

which is characteristic for fractal structures where  is a factor which may be considered as the "fractal dimension" of  the cluster; is a constant multiplier.  for identical spherical particles dispersed in a free space.  

It should be noted that a similar relation was obtained for carbon black aggregates appearing during combustion of hydrocarbon fuels [23], and the fractal dimension of the aggregates was 1.74 units. This may indicate a single mechanism of structure formation of aggregates (clusters) of particles, regardless of their nature.

The published data show [23], that clusters of identical contact particles make perforated surface structures. This agrees with the results published in the paper [24] which describes structures formed of aluminum particles which were found in a metallized HCM.

Since clusters extend along an axis, then they can be regarded as linear and heat conductive elements connecting HCM burning surface with its deep layers, and therefore actively participating in the combustion process.

It must be noted that the relation (3) is valid both for particles of ammonium perchlorate (AP) and for aluminum particles in a metallic HCM, when distribution of aluminum particles is not dependent on size and distribution of AP particles (i.e. when AP particles and aluminum may be independently placed in the volume of the HCM).

A comparison of the relations (2) and (3) shows that the maximum size of the largest cluster of contact aluminum particles grows exponentially with the increase in the effective concentration of aluminum in the HCM:

This relation allows us to estimate a range of applicability of the present method of independent distribution of AP particles and particles of finely divided aluminum in modeling the structure of a HCM. Obviously, such a method is reasonable only when the characteristic distance between AP particles is substantially larger than the maximum cluster of aluminum particles:  where the characteristic distance between AP particles and the proportionality factor is

Òhus, independent positioning of AP and aluminum particles in the volume of a HCM is possible in modeling its structure only when

                                                                 (4)

From this relation, for example, it follows that if  and cluster structures will not depend from sizes of AP particles if and if and the same if

If (4) is not satisfied, the average sizes and weights of clusters, even for ultrafine original particles of aluminum will be dependent upon sizes of AP particles.

Fig. 4. Dependence of the average coordination number of the cluster on the number of original particles contained in it

 

Îne of the important characteristics of a cluster is an average coordination number (the average number of contacts per one particle in a cluster) which determines the strength of the cluster and, therefore, the strength of the carcass layer on the combustion surface, and the tendency of the cluster to form a single drop after its melting. If (4) is not satisfied, the average sizes and weights of clusters, even for ultrafine original particles of aluminum  will be dependent upon sizes of AP particles.

Average coordination numbers of clusters obtained as a result of modeling the structure of HCM in terms of number of particles in the cluster are marked in Fig. 4.  As it turned out, the relation decomposes into a discrete number of branches asymptotically approaching which is also a branch of the referred relation. The relation is regular within each branch. Distribution of average coordination numbers of clusters in HCM becomes apparent in the fact that they may belong to different branches of the relation

An analysis has shown that the coordination number of the cluster is related to the number of particles in the cluster by a simple relation

                                                                                                (5)

where is a structural factor determining the type of a cluster. Each branch in the relation has a correspondent value and a cluster type. These relations calculated for different are represented in Fig. 4.

An analysis of clusters belonging to different branches of a relation shows that clusters of different types differ in the number of cyclic structures (loops). The structure factor is related to the number of loops in the cluster by the relation  

 

CONCLUSIONS

1.     There has been proposed an analog model and a new algorithm for random distribution of hard spheres in space (hard disks on the plane), which is applicable to any distribution of particle diameters and volume concentrations up to the maximum possible corresponding to close packing of the particles. The proposed method is called the method of viscous suspension.

2.     There has been conducted a simulation of the structure of heterogenous condensed mixtures containing one or two monodisperse particles in a wide range of their volume concentrations. We determined the main statistical characteristics describing the internal structure of HCM. It has been shown that the method of viscous suspension allows simulating a appearance of regular structures when the volume concentration of the particles is close to the maximum possible corresponding close packing.

3.     It has been demonstrated that the contact particles make in HCM extensive clusters, whose dimensions increase with the increase of volume concentration of the particles. If the volume concentration of particles in the system is greater than 0.15 ... 0.17, the system with probability one will have a percolation cluster that would penetrate through the entire system, and would be capable of changing its thermo-physical and ballistic properties.

4.       We demonstrated that clusters of aluminum contact particles in metal containing HCM have fractal structure, which fractal dimensions change from 2.0 for small concentrations of aluminum particles in HCM to 1.8 for relatively high concentrations.

5.       We have studied the structure of clusters of contact particles. It has been demonstrated that the average coordination number of clusters containing the same number of particles can only take discrete values, which are determined by the presence cyclic elements in the structure.

 

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