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.
References
1.
Strauss D.J. A model for clustering. Buiometrica, 1975, V.63,
p.467-475.
2.
Stoyan, D., Kendall, W.S., Mecke, J. Stohastic Geometry and its
Applications. Akademie-Verlag Berlin, 1987, 345 p.
3.
Ripley, B.D. Spatial Statistics. J. Wiley&Songs, New
York/Chichester. 1981. 252 p.
4.
Jodrey W.S., Tory E.M. Computer simulation of close random packing of
equal spheres. Physical Review A, 1985, V.32, ¹4, P.2347-2351.
5.
Jackson, T.L., Buckmaster, J. Heterogeneous
Propellant Combustion. AIIAA Journal, 2002, Vol. 40, pp. 1122-1130.
6.
Rashkovsky S.A. Simulation of composite explosives statistical
structure. In: Proceeding of Eleventh symposium on Chemical Problems, Connected
with the Stability of Explosives, Bastad, Sweden, 1998, P.17-18.
7.
Rashkovskii S.A. Structure of heterogeneous condensed mixtures. Combustion,
Explosion and Shock Waves, 1999, pp. 523-531.
8.
Nickolay Zosimovych, Banshidhar Choudhary. The Structures of
Heterogeneous Condensed Mixtures. PARIPEX
- Indian Journal of Research, Vol. 3, Issue: 4, pp. 135-141, May, 2013.
9.
Ji-Guang Li, Takayasu Ikegami, Jong-Heun Lee, Toshiyuki Mori, Yosiuki
Yajima. Co-precipitation synthesis and sintering of yttrium aluminum garnet
(YAG) powders: The effect of precipitant. Journal of the European Ceramic
Society, #20, pp. 2395-2405, 2000.
10. Sullivan K.T., Piekiel N.W., Wu C., Choudhury
S., Kelly S.T., Hufnagel T.C., Fezzaa K., Zachariah M.R. Reactive sintering: An
important component in the combustion of nanocomposite thermites. Combustion
and Flame, #159, pp. 2-15, 2012.
11. Croxton, Clive A. Liquid State Physics. A
Statistical Mechanical Introduction. Cambridge University Press, 421 pp., 1974.
12. Rashkovsky S.A. Simulation of composite explosives
statistical structure. In: Proceeding of Eleventh symposium on Chemical
Problems, Connected with the Stability of Explosives, Bastad, Sweden, 1998,
P.17-18.
13. Balescu Radu. Equilibrum and Nonequilibrum
Statistical Mechanics, Wiley, New York, 1975.
14. Allergini P., Grigolini P., West B.J.
“Preface”, Chaos, Solutions&Fractals, 2007.
15. Robert A. Beddini, Ted A. Roberts. Effects of
Turbulence on Stationary and Non-Stationary Process in C-Systems, University of
Illinois, Urbana, Fianal Technical Report ¹AAE 87-1 UILU ENG870501, 1989.
16. Garry A. Flando, Sean R. Fischbach, and
Joseph Majdani. Nonlinear rocket motor stability prediction: Limit amplitude,
triggering, and mean pressure shift. Physics of Fluids, 19, 094101, 2007.
17. Ãðèãîðüåâ Â.Ã., Êóöåíîãèé Ê.Ã., Çàðêî Â.Å. Ìîäåëü
àãëîìåðàöèè àëþìèíèÿ ïðè ãîðåíèè ñìåñåâûõ êîìïîçèöèé. ÔÃÂ, Ò. 17, ¹ 4, Ñ. 9-17,
1981.
18. Yupu Zang, Xu Xu, He Liu, Yujuan Zhai, Ye
Sun, Hangi Zhang, Aimin Yu and Yighna Wang. Matrix solid-Phase Dispersion
Extraction of Sulfonamides from Blood. Journal of Chromatographic Science, #50,
pp. 131-136, 2012.
19. Ôðîëîâ Þ.Â., Ïèâêèíà À.Í., Íèêîëüñêèé Á.Å. Âëèÿíèå
ïðîñòðàíñòâåííîé ñòðóêòóðû ðåàêöèîííîé ñðåäû íà òåïëîâûäåëåíèå ïðè îáðàçîâàíèè
àëþìèíèäîâ íèêåëÿ è öèðêîíèÿ. Ôèçèêà ãîðåíèÿ è âçðûâà, Ò. 28, ¹5, Ñ.95-100,
1988.
20.
Ôðîëîâ Þ.Â., Ïèâêèíà À.Í., Íèêîëüñêèé Á.Å. Êîíöåíòðàöèîííûå ïðåäåëû
ðàñïðîñòðàíåíèÿ âîëíû ãîðåíèÿ â ãåòåðîãåííûõ ñèñòåìàõ. Ãîðåíèå ãåòåðîãåííûõ è
ãàçîâûõ ñèñòåì, IX Âñåñîþçíûé ñèìïîçèóì ïî ãîðåíèþ
è âçðûâó. – ×åðíîãîëîâêà, Ñ.17-21, 1989.
21.
Adams Nicholas J. Dynamic Trees: A Hierarchial
Probabilistic Approach to Image Modelling. Edinburgh Research Archive, PhD
Dissertation, 2001.
22.
Ôðîëîâ Þ.Â., Ïèâêèíà À.Í. Ôðàêòàëüíàÿ
ñòðóêòóðà è îñîáåííîñòè ïðîöåññîâ ýíåðãîâûäåëåíèÿ (ãîðåíèÿ) â ãåòåðîãåííûõ
êîíäåíñèðîâàííûõ ñèñòåìàõ. Ôèçèêà ãîðåíèÿ è âçðûâà, Ò. 33, ¹5, Ñ.3-19,
1997.
23.
Koylu U.O., McEnally C.S.,
Rosner D.E., Pfefferle L.D. Simultaneous Measurements of Soot Volume Fraction
and Particle Size/Microstructure in Flames Using a Thermophoretic Sampling
Technique. Combustion and Flame, V.110, P. 494-507, 1997.
24.
Sambamurthi T.K., Price
E.W., Sigman R.K. Aluminium agglomeration in solid-propellant combustion //
AIAA Journal. V. 22, ¹8, p. 1132-1138, 1984.