CC BY
8References
1. Alotaibi B., Elleithy K. A passive fingerprint technique to detect fake access points // IEEE Wireless Telecommunications Symposium. 2015. pp. 1-8.
2. Agarwal M., Biswas S., Nandi S. Detection of De-authentication Denial of Service attack in 802.11 networks // 2013 Annual IEEE India Conference (INDICON), Mumbai, 2013. pp. 1-6. doi: 10.1109/INDCON.2013.6726015.
3. Weiner E. S., Simpson J. A. Oxford English Dictionary - Oxford University Press, 2nd ed, 2004 -the word "timeliness"
4. Voevodin V. A., Burenok D. S., Chernyaev V. S. On the application of a simulation model for assessing the probability of server failure in the conditions of DDoS attacks // Collection of reports of the All-Russian conference with the participation of Radio-electronic devices and systems for infocommuni-cation technologies REDS-2021, Moscow, 2021, 0204 June. [Published in Russian].
5. RFC 7540. Hypertext Transfer Protocol Version 2 (HTTP/2). 2015.
6. RFC 2818. HTTP Over TLS. 2000.
7. Kunda D., Chihana S., Muwanei S. Web Server Performance of Apache and Nginx: A Systematic Literature Review // Computer Engineering and Intelligent Systems. 2017. Vol. 8. No. 2. pp. 43-52.
8. Apache HTTP Server Documentation // Apache HTTP Server Project [Available online]. -URL : https ://httpd.apache.org/docs/current/
9. Nginx documentation // Nginx [Available online]. - URL: https://nginx.org/en/docs/
10. Oktrifianto R., Adhipta D., Najib W. Page Load Time Speed Increase on Disease Outbreak Investigation Information System Website // International Journal of Information Technology and Electrical Engineering. 2019. Vol. 2. No. 4. pp. 114-119. doi: 10.22146/ijitee.46599.
11. Verma J. P., Abdel-Salam G. Testing Statistical Assumptions in Research - Wiley, 1st edition, 2019. - 224 p. doi: 10.1002/9781119528388.
12. Voevodin V. A, Burenok D. S. Result of experimental study on detecting Wi-Fi access points // The scientific heritage. 2021. Vol 1. No 73. pp. 32 -44.
13. Arkadov G. V., Getman A. F., Rodionov A. N. Probabilistic Safety Assessment for Optimum Nuclear Power Plant Life Management (PLiM). - Wood-head Publishing Series in Energy, 2012. - 368 p.
UDC 621.454.3
MODELING AND OPTIMIZATION OF THE STRUCTURE OF A HIGHLY FILLED POLYMER COMPOSITE MATERIAL IN THE PROCESS OF MIXING COMPONENTS
Kryvanos A.K., Ilyushchanka A.Ph., Piatsiushyk Y.Y., Buloichyk V.M.
1State Research and Production Powder Metallurgy Association, Minsk, Belarus
Abstract
One of the approaches to modeling the structure formation of a highly filled polymer composite material (HFPCM) by mixing its components represented by liquid and solid phases is considered. To develop a model of the mixing process, one of the heuristic algorithms was used, i.e. the "metal annealing" method. The model is formalized with the condition of averaging the particle sizes inside each fraction, as well as the morphology of their surface. As a representative element of the model, a unit cell in the form of hexogonally densely packed particles around one introduced into the composition of the composite material (in small quantities, up to 5%) was adopted. The voids in the cell are filled with liquid polymer. The developed model is based on the objective function, which involves obtaining a uniform distribution of components with the required packing density and filling the voids with the liquid phase while minimizing the number of mixing iterations.
The results of the study, obtained during modeling, can be used in the choice of technological equipment and the determination of its operation modes to obtain the specified characteristics of HFPCM, as well as in predicting the possibility of obtaining the required properties in the process of mixing its components.
Keywords: highly filled polymer composite materials, structure formation, modeling, mixing of powder components in a polymer binder
Introduction
Highly filled polymer composite materials (HFPCMs) are a mixture of uniformly distributed pol-ydispersed particles of a solid phase ranging in size from 40 to 200 ^m (depending on the composition of HFPCMs, some components may be nanosized particles) in an amount of 75-85 wt.% in a liquid-phase polymer media [1]. Due to the selection of fractions of
solid-phase components, their quantitative ratio, and surface morphology, their most dense packing is achieved. The most appropriate number of mixing cycles and correctly chosen technological modes of mixing equipment ensure uniform distribution of particles of each component in the polymer material, filling of all voids with this polymer and cladding of the particles surface of solid-phase components.
In most cases, the methods and modes of mixing the components of the HFPCM are determined empirically for each composition and type of technological equipment [2], taking into account the analysis of a large number of experiments involving the selection of a suitable sequence for adding components to the overall composition, its mixing time, rotation speed of the working mixer body, temperature, vacuum in the operating chamber of the mixer and other characteristics of the technological process. As is well known, this approach is quite long-time, since it involves a periodic stop of the manufacturing process of the composite material at a certain stage for sampling and conducting appropriate measurements. It should be noted that the data obtained during the measurements are not always changed accordingly when scaling the manufacturing process of the HFPCM.
In turn, the manufacture of the HFPCM in the required volumes for measurements due to the high price of raw materials and the need for subsequent disposal of the experimental sample is a rather expensive research method. One of the approaches that minimize the number of experiments and, consequently, reduce the cost of the desired result, is the modeling of the process under study. In our case, it is the process of mixing components and production of the HFPCM with desired properties. As a rule, the first stage of modeling is the formalization of the objective function. Its quality of the implementation largely determines the performance of the subsequent stages of the model construction of the studied process.
1. The construction of a unit cell model of HFPCM
In the issue under consideration, particles of solid-phase components of four types are subject to mixing: coarse and fine fractions of calcium carbonate (CaCOs) of MTD-2 grade
(GOST 17498-72), bakelite powder and antioxidant, which is used as a diamond-containing charge mixture obtained by the method of detonation synthesis of a mixture of trinitrotoluene (C6H2CHs(NO2)3) and cy-clotrimethylenetrinitroamine (CH2)3№(NO2)3) in a proportion of 50:50 [3]. The qualitative composition of the diamond-containing charge mixture and its properties are described in reference [4].
To formalize the objective function of the mixing process and calculate the appropriate values of the amount of each component and the characteristics of their fractional composition, at the first stage, we will construct a unit cell model formed during the packing of solid phase particles. The basis for the development of this model is the approach described in reference [5], where a unit cell with a coordination number of 12 is formed during the hexagonal dense packing of particles of solid phase components. The most acceptable description of these particles to simplify the calculations would be their representation in the form of a sphere. In the center of this unit cell is a particle of bakelite powder, to which 12 particles of calcium carbonate are adjoined. Its view is shown in Figure 1.
Calcium carbonate particles
Bakelite powder particle
Figurel. The model of unit cell packing of the HFPCM solid phase
To obtain the most dense particle packing, it is necessary that the linear sizes of the fractions of calcium carbonate and bakelite powder would be relatively equal, and the mass ratio, according to the values of their density, would be approximately 24:1. For any other components of the HFPCM solid phase, making a system-forming matrix, and the chosen approach for constructing a unit cell (1+12), the following dependences of the mass ratios were established in reference [6]
m„pr (1)
m - •
where mr- mass of the component whose particle (bakelite powder) is located in the center of the unit cell;
mo - the mass of the component whose particles (calcium carbonate) are adjoined to the central one;
Pr; Po - densities of the considered solid-phase components (1 and 12), respectively;
N - coordination number inherent in the formed unit cell.
For other mass ratios of the components, the number of particles of each of them in the unit cell can be determined through the following dependence
n0_ mopr (2)
nr mrpO
where no and nr- the number of particles in the unit cell of the o and r components, respectively.
Taking into account expression (1), the fractional composition of bakelite powder and calcium carbonate with a particle size of 180 ^m was chosen. At the same time, in order to obtain the particle packing density in
the considered unit cell at a level of 0.84 - 0.86 (corresponding to a certain packing density ^), the calcium carbonate particles are chosen by two fractions. When determining the appropriate particle size ratio, the approaches described in reference [7] were used, which were refined using the capabilities of the KOMPAS-3D software product and expressed as follows
Di ~ (0,2^0,225)-Di,. (3)
where Di - fine particle diameter;
Db.- coarse particle diameter.
According to the established dependence, the median values of calcium carbonate particles for coarse and fine fractions were 180 and 40 ^m, respectively.
The developed model of the cell, consisting of 13 particles, forms 13 octahedral and 26 tetrahedral pores (voids) [7], in which particles of a fine fraction of calcium carbonate and other components of the HFPCM solid phase are distributed. Options for such a distribution are shown in Figure 2.
Figure2. Options for filling the voids with particles of a fine fraction of calcium carbonate (a)
tetrahedral pore; b) - particles in the octahedral pore)
a particle in a
It is assumed that the components of the solid phase with a particle size of less than 10 ^m (including those having a nanosize) together with liquid-phase components will fill voids formed between the filler particles. For such a unit cell, according to the chosen
fractional composition, its basic characteristics are calculated and the appropriate stoichiometric composition is determined. The calculation procedure is described in reference [6], and their results for solidphase components are given in the table.
Table
Characteristics ^ of the solid-phase components of the HFPCM forming the unit cell
Component name Particle radius, m Volume of one particle of substance, m3 Density, kg/m3 Mass of one particle, kg The number of particles in the unit cell, units Mass of particles in the unit cell, kg Particle surface area, m2 The amount of component in the composition of the HFPCM, wt.%
Calcium 910-5 3,05-10-12 8,27-10-9 12 0,9910-7 1,2210-6 72,6
carbonate (CaCOs) 210-5 3,35-10-14 2710 9,0810-11 143 1,3 10-8 7,2-10-7 11,04
Bakelite powder 910-5 3,05-10-12 1360 4,1510-9 1 4,15 10-9 1,0210-7 3
Diamond-
containing charge 210-8 3,3510-23 3850 1,2910-19 9,026 109 1,1610-9 4,5 10-5 0,9
mixture
The liquid phase in the considered HFPCM is 15 wt. %. The components of the liquid phase fill the voids formed between the particles of the solid phase and cladding their surface. The volume of the liquid phase of the unit cell takes on the value
VrV.+V^ (4)
where Vt - volume of the liquid phase of the unit cell of the HFPCM;
Vc - volume of the liquid phase of the HFPCM necessary to fill the voids in the unit cell;
V p - volume of the liquid phase of the HFPCM
necessary for cladding the particle surface of the unit cell.
In accordance with the established stoichiometry, each unit cell (providing their uniform mixing) accounts for 3.6 10-8 kg or 3.8 10-11 m3 of a liquid-phase polymer binder. At the same time, the calculation of the liquid phase values for the chosen unit cell, due to the absence of clearly defined external boundaries, can be carried out only with a sufficiently large inaccuracy. With this in mind, to determine the characteristics of all the components that make up the unit cell, a repre-
sentative element in the form of a hexagonal prism repeating in a hexagonal dense packing is chosen with a height
H = 2 •
and base area
!
• D
_ 3^3 2
Sb — ~ D2
(5)
(6)
Sb - base area of the representative element
(hexoprism).
A view of the unit cell model with the mutual arrangement of the particles of bakelite powder and the coarse fraction of the filler, as well as its representative element with the mutual arrangement of the particles of bakelite powder, coarse and fine fractions of the filler are shown in Figure 3.
where Hc - height of the representative element (hexoprism);
Figure 3. Unit cell model for hexagonal dense packing ofparticles with a coordination number of 12 and its
representative element
The representative element has one particle of bakelite powder in the center of the figure and 18 segments (types A, B and C) equal in volume to 5 particles of a large fraction of calcium carbonate. A fine fraction of calcium carbonate is located in 8 tetrahedral pores (1 particle in one pore) and 6 octahedral pores (9 particles in one pore). Using the capabilities of the KOMPAS-3D software product, the ratio of the linear dimensions of the representative element and the chosen unit cell was determined, which was 6/13.
Then, based on expressions (5) and (6), the unit cell volume will be
Vi = 6,5^2D
b,
(7)
and the volume of the components of the solid phase in it will be
Vi = 6,5V2Db3f-,
(8)
where Vc' - volume of the i unit cell;
V} - solid phase volume of the i unit cell; - particle packing density in the i unit cell.
In terms of the established ratio, the volume of voids in the unit cell, which had to be filled with a polymer binder, is 2.2610-11 m3. The remaining polymer material (1.5410-11 m3) during mixing should be evenly distributed over the surface of the particles (5.93-10-5 m2) of the solid phase of the unit cell. The specific value of the distribution of the polymer binder to the size of the particle surface of the HFPCM solid phase will be 2.6 10-4 l/m2. This is equivalent to a coating of 260 nm.
Further on, to simplify the calculations, we will consider not all components of the solid phase, but only particles of bakelite powder and two fractions
(coarse and fine) of calcium carbonate. We denote the particles of bakelite powder and a coarse fraction of calcium carbonate as m= 1 and n2 = 12, respectively. As was shown during the construction of the unit cell, the voids between these particles are filled with particles of a fine fraction of calcium carbonate in an amount of n3. In real terms, n3 varies in the range from 39 (provided that at least one particle of a fine fraction of calcium carbonate gets into each formed void) up to 143. The volumes of these particles are denoted by Vi, V2 and V3, respectively, and the volume of the formed voids is Vo. Then the volume of the entire HFPCM (V) mixture will take the value
Vc - Vo + Vi + V2 + V3 + Vl. (9)
The constructed model of the unit cell is the basis for the formalization of the process of mixing components and obtaining a HFPCM mixture.
2. Formalization of the mixing process of the HFPCM.
The process of mixture formation can be represented as a random process of filling an elementary cell with particles of three types and liquid polymer material, where the following tasks are solved:
To obtain a uniform distribution of all solid-phase components in the HFPCM volume and creating relatively the same composition;
To reach the densest packing of particles of the solid phase due to their rational distribution in the HFPCM volume;
To ensure the most complete filling of voids formed in the particle packing of solid-phase components with a liquid polymer;
To clad the entire surface of the particles of the solid phase with liquid-phase components.
This process will be reproduced using the "metal annealing" algorithm [8].
We assume that the volume (v:) of an individual particle of the first type, which is part of the unit cell, with an average diameter d\, will be
Vr
nd3
6 (10) the volume of particles of the second (v2) and third (v3) type, included in the unit cell, with average diameters d2 and d3, will be
nd 2 nd3 .. ,
v9 =-h v =-respectively.
v2 6 v3 6 F J (11)
And the total volume of particles in the unit cell, according to the specific example shown in table 1, will be
V1 + V2 + V3 _n(d3 + 12d2 + 143d3).
6 (12) Provided that in the considered task di=d2, we simplify expression (12) by designating the particle diameter of the first and second type di,2
V1 + V2 + V3 = ^(d 3,2 + 11d I) .
6 (13)
The volume of polymer material vi in the unit cell, according to expressions (4) and (9) and under the condition vo^0, as well as in our example Db = d:A will be:
vi = 2,36d32 -
3 143nd3
6
-V,
(14)
Considering the distribution of particles in the initial material to be uniform, we estimate the probability that a particle of bakelite powder will get into the cell at the beginning of this process
Pi =-
Vi
nd 3
Vi + V2 + V3 13%(d 1,2 + iidi) (15)
For the probability of particles of coarse and fine fractions of calcium carbonate entering the unit cell, we can write down, respectively
P2 =
nd2
13n(d 3,2 + iid 3)
P3 =
nd3
13n(d 3,2 + iid 3)
(16)
From the normalization condition, the probability of the formation of voids within the chosen unit cell will take the value
po -1-Pi - P2 - P3.
(17)
The process of compaction of a cell is the search for such a combination of particles to be chosen, in which the volume of voids (vo) is the smallest, that is v0 ^ 0. This event corresponds to the smallest volume of the cell itself, that is Vc ^ v^1" . In accordance with expression (9), the voids in the unit cell are filled
with the liquid-phase HFPCM component. In this case, the part of the polymer material remaining after filling the voids is intended for cladding the surface of the particles of solid-phase HFPCM components. The volume of this part of the liquid phase spent on the complete cladding of all particles will be proportional to the surface area of the particles of each type, their quantity, as well as the thickness of the cladding layer and will be
Vp - ¿(«iSi+«2S2 + «3S3) = hsc,
where sc - total surface area of the particles of solid-phase components of the unit cell;
si , s2 , s3 - particle surface area of the corresponding type (taken from the table);
h - thickness of the cladding layer. The cladding process is a search for such a combination of the distribution of the fuel binder, where the area of the cladding layer (s p) will correspond to the surface area of the particles of the solid-phase components of the unit cell, i.e. sp ^ sc
Thus, the cyclic step-by-step (i=1,2,...) procedure for obtaining the required cell consists in:
the formation of the densest packing of particles of each type with the volumes and quantities m, n2 and n3 chosen for them;
the maximum filling of the space between the particles of the solid phase with the liquid-phase HFPCM component, minimizing the volume of voids v0 ^ 0 (or minimizing the volume of the cell itself, that is
Vc ^ vr);
cladding the surface of each of the particles with the liquid-phase HFPCM component, the area of which for the considered cell will be sc .
The required cell, representing the final solution of the task, will be characterized by a vector that has coordinates
Xopt -X[ni,«2,n3,V0 ^0,Sp ^Sc]. (19)
The solution to the optimization task will lie in the search for particular solutions that satisfy the abovementioned conditions for obtaining the required cell, and their subsequent coordination to choose the most appropriate approach to prepare the required mixture of the HFPCM.
3. Development of an algorithm for constructing a particular solution
The procedure for forming the contents of a unit cell is represented by the following steps.
We denote the number of particles of the first, second, and third type, respectively, randomly chosen into the unit cell during mixing at the i step. Such a cell will be characterized by a vector having coordinates
X' -X'[ni,«2,«3,v0]. Then it can be written
vi - ni ^nd3, v2 - «2 ^nd2, v3 - «3 ^ndf. (20) 6 6 6
To obtain the value of the void volume, we use the relation
Vo = vC - (vi + V2 + v3 ). (21)
Using a random number sensor, we choose a number q uniformly distributed over the interval [0;1].
If qe [0,pi], then we make a decision on getting particles of the first type into the unit cell.
If qe [ p 1, p 2 ], then we make a decision on getting particles of the second type into the unit cell.
If qe[p2,p3], then we make a decision on getting particles of the third type into the unit cell.
If qe [p3, 1], then we make a decision that not a single particle gets into the unit cell (that is, the presence of a void in this cycle of the procedure).
The abovementioned calculations are repeated N = ni +n2+ m number of times corresponding to the number of all particles that should be part of the future "ideal" cell. As a result, a unit cell will be formed at the first step of the iterative procedure, characterized
by the vector X1 = X 1[n11, n21, n31, v^], and its
volume will take the value
VC = vc = 6,5>/2D3, . (22)
According to the given scheme, unit cells with a random structure are formed at each i step of the iterative procedure. That is, the so-called particular i solution is being framed, which, in accordance with the metal annealing algorithm, should be further improved (the cell will become denser) while it is mixed.
The quality of the resulted cells (the efficiency of each particular solution X,) will be characterized by the proximity of the vector X, to the vector
Xopt = X[«1,«2,«3,vcmin]. The value is calculated for this
Et = [(«1 -«1 )2r,3 + («2 -«2)2r3 + («3 -«3)2r3], (23)
and the difference is determined at each subsequent (i+1) step
AEf+1, t = E+1 - Et. (24)
The algorithm for framing a particular solution is shown in Fig. 4.
Based on the considered algorithm for framing a particular solution, an algorithm is synthesized that implements the "annealing" procedure and the search for the best solution. Such a solution will be when the unit cell contains the smallest volume of voids and at the same time the surface of all particles was clad with the liquidphase component of the HFPCM.
The subject matter of this procedure includes the following actions: if AEi+u<0, then the new solution is better than the previous one and is remembered. Otherwise, before discarding it and proceeding to the next iteration, the probability of maintaining the resulting "bad" solution is estimated. This probability depends on the so-called "annealing temperature". An analogue of the "annealing temperature" is the volume of the
unit cell 1/+1, which decreases from iteration to iteration. It is proposed to adopt the law of changing the volume of the mixture (i.e., the law of compaction of the unit cell) as follows:
vC = Vco[1 + — ], i = 1,2,...
i
Here ae [1; 2; 3] is a tuning factor, regulating the annealing rate; Vco is a limit value of the unit cell volume.
Figure 4. Algorithm development of a particular solution (25)
At the end of each i iteration, the volume vci is adjusted (in accordance with the expression
vC - vco[i + -i]) and the probabilities of choosing
1
particles of each type are recalculated according to the
a
following
formulas
,+!_ Vi +1_ v2
Pi -— ,P2 - —
Vc Vc
p3 - — v c
In accordance with the normalization requirements po = 1 - pi - p2 - P3 •
The process of forming the contents of the unit cell is repeated, i.e. (i+1) iteration is performed.
The calculations continue until the value AEi+\j with a predetermined accuracy approaches zero, i.e. AEi+i,i * 0.
The number of iterations (mixing cycles) i, under which the above condition is fulfilled, characterizes the required mixing time of bakelite powder and two fractions of calcium carbonate. A comparison of the values resulted during the modeling with the measured values of the mixing results on a specific mixer will make it possible to form a suitability scale of the modeling results to the operating modes of technological equipment.
4. Verification of mixing model of the HFPCM
In order to assess the functional completeness, accuracy, and reliability of the simulation results using the proposed optimization method, a model was developed that describes the mixing of the HFPCM and its verification was carried out. The model is based on the optimization task X opt = X[«i, «2, «3, Vcmin] of finding the conditions for obtaining the minimum unit cell volume
for a given number of particles of each of the solid-state components of the HFPCM.
Modeling was carried out for a volume of a mixture of 2.375 l, which is mixed in a SP-15 planetary mixer (Russian Federation) with two mixers and a diameter of a mixing bowl of 335 mm. Similarity criteria for the chosen mixing equipment (Reynolds, Frood, Euler, and Weber criteria) were calculated in accordance with the approaches described in Reference [9] for dispersed systems with a particle diameter of 180 ^m. Based on the calculation results, the following modes for mixing were determined: blade rotation speed - 20 rpm; the temperature of the mixed mixture - (T) 303-308 K; pressure in the bowl - (P) 0.075 MPa.
The resulted simulation values showed that for given mixing modes, the smallest unit cell volume (8,29-10 -11 m3) will correspond to 20,5 - 21 minutes of mixing. The dynamics of a decrease in the unit cell volume during mixing is shown in Fig. 5 a.
Verification of the model was carried out by comparing the results obtained from the simulation results with the data obtained in the study of samples taken during mixing the components of the experimental composition of the HFPCM. Sampling was carried out every two minutes of mixing. As a comparable parameter, the density of the mixed composition was measured as the reciprocal of its volume, formed by a set of unit cells (simulated parameter). During the mixing of the experimental composition, it was found that the maximum density (1922 kg / m3) was achieved from 21 to 22 minutes of mixing. The dynamics of changes in the density of the mixed composition is shown in Fig. 5 b.
£
DC JJ
ISl
<D Q)
<U £
o >
13
o
'c Z>
8 10 12 14 16 18 20 22 24
Time of mixing (min)
Figure 5. Verification results of the mixing model: a) dynamics of obtaining the minimum unit cell volume; b)
the dynamics of changes in the density of the mixture
A comparison of the diagram characterizing the dynamics of the change in the unit cell volume obtained during the simulation and the diagram reflecting the nature of the change in the density of the mixed composition makes it possible to conclude that the model matches the real mixing conditions.
Conclusions
The approach of modeling the process of mixing the HFPCM components, based on the use of one of the heuristic methods, i.e. "metal annealing", is considered. The process of a uniform distribution of all components in the HFPCM is formalized. When formalizing the mixing process, the probabilities of obtaining the densest packing of particles of solid-phase components, cladding of their surface with polymer material were taken into account, and the possibility of minimizing voids by filling them with a liquid-phase component of the HFPCM was analyzed.
The proposed modeling method avoids local errors in the study of the formation process of the densest packing of particles of solid-phase components during their mixing, filling the formed voids with a liquidphase polymer material and cladding the surface of the mixed particles with it. The presence of such a model makes the mixing process more predictable, providing the HFPCM manufacturer with the required technological information. This makes it possible to improve the technological process by obtaining appropriate modes at the stage of its development and thereby obtain the methodological basis for the formation of a quality management system in production.
References
1. Bondaletova, L.I. Polymer composite materials (Part 1): textbook / L.I. Bondaletova, V.G. Bon-daletov. Tomsk: Publishing House of Tomsk Polytechnic University, 2013. P. 118.
2. Kerber, M.L. Polymer composite materials: structure, properties, and technology: textbook / M.L.
Kerber [et al.]; under the editorship of A.A. Berlin. St. Petersburg: Profession, 2008. P. 560.
3. Kryvanos, A.K. Technology for modifying an energy-saturated composite material with ultradis-persed diamonds / A.K. Kryvanos // Powder Metallurgy: Rep. Inter. Collection of scientific papers / under the editorship of A.Ph. Ilyushchanka [et al.]. Minsk: NAS of Belarus, 2018. Issue 41. P. 194-199.
4. Vityaz, P.A. Detonation synthesis nanodia-monds: production and application / P.A. Vityaz [et al.]; under the general editorship of P.A. Vityaz. Minsk: Belarus. Belaruskaya Navuka, 2013. P. 381.
5. Ilyushchanka, A.Ph. Formalization of the process and development of an algorithm for solid-phase mixing of components of a heterogeneous composite material / A.Ph. Ilyushchanka, V.M. Buloichyk, A.K.Kryvanos, Y.Y.Piatsiushyk // Vestsi of NAS of Belarus. Ser. of physics and technical sciences. 2018. Vol. 63. No. 3. P. 263-270.
6. Kryvanos, A.K. The choice of criteria for assessing the quality of mixing the components of an energy-saturated heterogeneous composite material / A.K. Kryvanos, A.Ph. Ilyushchanka, Y.Y. Pi-atsiushyk, A.A. Prokharau // Powder Metallurgy: Rep. Inter. Collection of scientific papers / under the editorship of: A.Ph. Ilyushchanka [et al.]. Minsk: NAS of Belarus, 2019. Issue 42. P. 146-153.
7. Bogdan, T.V. Description of the crystal structures of metals in terms of ball packing and laying [Electronic resource] / T.V. Bogdan. Access mode:
http ://www. chem. msu. su/rus/cryst/cryschem/li-ter/bogdan/bogdan05.pdf. Access date: 23.06.2019.
8. Kirkpatrick, S. Optimization by simulated annealing / S.Kirkpatrick, C. D.Gelatt Jr., M. P. Vecchi // Science. 1983. Vol. 220, N 4598. P. 671-680.
https://doi.org/10.1126/science.220.4598.671
9. Strenk, F. Stirring and devices with stirrers. F.Strenk / Translated from Polish. Edited by Schuplyak I.A. - L., "Chemistry", 1975. P. 384.
