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7Journal of Siberian Federal University. Chemistry 2 (2015 8) 181-189
УДК 544.431.7
Simulation of Development of the Solid State Chain Reaction
Marina V. Ananyeva and Alexander V. Kalenskii*
Kemerovo State University 6 Krasnaya Str., Kemerovo, 650043, Russia
Received 12.03.2015, received in revised form 21.04.2015, accepted 09.06.2015
In this work simulation of the kinetics of the explosive decomposition process in silver azide crystals was done. The non local behavior of the chain propagation stage was taken into account. The finite sizes of the sample and the laser radius, the energy transfer through the lattice, which causes the reagents' redistribution, were also taken into account. Optimal area of the crystal for the reaction to develop is the centre of the radiation zone.
Keywords: simulation, difference scheme, model of the chain reaction, explosive decomposition, energy transfer, energetic materials, silver azide.
DOI: 10.17516/1998-2836-2015-8-2-181-189.
Моделирование развития твердофазной цепной реакции
М.В. Ананьева, А.В. Каленский
Кемеровский государственный университет Россия, 650043, Кемерово, ул. Красная, 6
Работа посвящена моделированию кинетики процесса взрывного разложения кристаллов азида серебра при нелокальном характере стадии развития цепи. Моделирование процесса учитывает конечные размеры образца, радиус инициирующего импульса, перенос энергии химической реакции по кристаллической решетке, который приводит к перераспределению концентраций реагентов в кристалле. Оптимальной областью для развития твердофазной цепной реакции является центр облучаемой области.
© Siberian Federal University. All rights reserved Corresponding author E-mail address: [email protected]
*
Ключевые слова: математическое моделирование, разностная схема, модель цепной реакции, взрывное разложение, перенос энергии, энергетические материалы, азид серебра.
1. Introduction
The main purpose of the simulation of the physical and chemical processes is the simultaneous examination of the chemical transformations and the physical processes, among these the main are the diffusion of the reagents and energy transfer [1, 2]. Determination of the mechanisms of the initiation, propagation and development of the energetic materials' explosive decomposition has a wide applied meaning, as long as the inadvertent occurrence of the equipment based on the explosives causes significant material damage [3, 4]. The importance oo the study lies in the necessity of development of the optical detonators [5] based on the initiating and high [6, 7] explosives. There are two ways to describe the explosive decomposition process - thermal t8] and chain model [9]. In terms of the thermal model it is supposed that the material decomposea accord[ng to the one-sSage reaction [10], the constant of this stage has the Atthenius depeortence on the temperature. Io terms os the chain model the self accelerated mode is due to the reagent multiplication [11.. At lot o° tint experimental data on the silver azide explosivedecomposition, in[tiated by the NdYag laser, were explained in terms of the chain model. The aim of this work is tr simulate tOe kinetics of generation, ptepagatirn of the reaction taking into account non-local be3avior of the propogation stage and the finite radius ([3 ihe inieSating pulse, phenomenological model of the process that was proposed [12].
2. Explosive decpmposition model of the silver azide
According to the phenomenological mo3et of tho silvet azide's explosive decomposition thee; energy of the chain reaction might transCet along the crystal leSticeand causes the e-h pairs' generation. The intensity of the reagents' generation is maximal in immediate vicinity oO the reaotion area and then decreases exponentially [13] :
2h-^-> N2 + -^-> 3N2 + 2h + (h + e) ■ exp(-|4r| / r0),
where k1 - constant of the decomposition on N^comptex, k - constant of N6-complex generation. Propagation stage is ihe interaction o3 two radicals Л",0 (A), loocalized an neighbor points of lattice. The constant of the reaction of N6 tormation was estimated taking into occounr Coulomb repulsion Debye screening and tunneling, and its value was equal to k2 =e O.e-lO"11 cm"5«"1 [14]. N6-complex decomposes with molecular nitrogen formation (N2); the sSage constent is k = 3407 s"1 [15]. Reitased energy is accumulated by the electron and vibrational degrees oC fteedom of the nitcogen molecule. In terms of the model of the dipole interactien witii tOe electron system of the crystal the constants of deactivation were estimated foe the casee of e-h pairs' formation (ke~ 109 s"") and energy transfer to a band hole (kv~ 10"12 cm3s-1) [15]. Deactivation of the nitrogen molecules causes thc active particles generation -propagation stage takea place not only io the reaction area, but also outside ft with probability ~exp(-x/r0). r0 wat calculaeed uoing the experimental daea 50 ± U0 (tm [16]. Intensity of the generation stage depended on the illumination inside tiie sample [17], and wis not laken into account in the context of the work.
Simulation of the process of the energy transfer was made taking into account the real geometry of the sample - in cylindrical coordinate system using the difference scheme [18]. For the calculations
Fig. 1. Layout view of the reaction area in a cylindrical coordinates to estimate the efficiency of the reagents' generation in cell n due to the reaction in cell m
the sample was divided into kn cells with the same cross dimensions dx (Fig. 1). The reaction in cell n influences the speed of genehation of the aceive particles (chain carriers) in all cells of the crystal, mathematically this minht be descrihed as an action of the functional on the function of eoncentration. In this work the functional was taken in form of square matrix k„k„. The elements of the matrix, which relate the intensity of the chain carriers' generation in cell n with the speed of the chemical reaction io cell m (Snm), wehe calculaeed by using the following procedure. Tlie foUowing conventional signs were used: Yn - number of holes in cell n, A - concentration or N6 complex, Sn - erea (if cell n. Let us estimate how the reactton speed in cell n influence the reaction speed in cerl m. The distance between cells n and m might be calculated as:
r = Jr2 + r2 — 2 r r cos rn
mn V m •> n m n t
rn = dr\n — rm = dr\m —
Speed of chain carriers' generation in any cell n because of the reaction of N6 decomposition in cell m might Ice calcnlattd using the following exptession:
Snm Tm Xnm, (2)
where x„m - speed of the chain carriers' generation in cell n due to the reaction in cell m, Tm - normalizing factor.
While the distance from the reaction area increases the energy decreases inversely as the square of the distance, and it decreases in e times while the distance is equal to r0 because of the absorption.
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The entire cylindrical segment with radius rn contains Aj-^xdy particles, so the number of particles generated in cell m:
r
_ fmn
A e ro
Ymm = - dxrnd9 , (3)
2%r0rnm ,
where Ymn is quantity oli tlie chain carriers, generated ln cell m pee unit time due to the reaction in cell n. For the layet containing cell m:
xnm = 2\dY,dty
nm J mo t , 0
(4)
rmn
J e ~ Y Snm = 2Tm-^-dq .
=0 2nr0rmn
While n=m value of S„m becomes infinite, what; is why the piecewise function was used: ff/iOX »ic:[l,w-l]u[w + l,£J
[M^), n = m
According lo the; model the; total value of the chain carriers, generated in cell n because of the reaction in cell m, equal to Ym„, i.u one act of transformation generates one e-h pair. Coefficient Tm was estimated using the eollowing normalizntitn:
T --r--. (5)
m rmn
IrjpU
(Con^^eiita^^ion of the chain carrieos generated in cell m:
K
3 J dB nm d9
_ Bnm _ 0_
pnm _ ~ - " " . (6)
fm mr-ar
Calculation of matmix Smn was done in a spatial prognam, calculation ot its components preceded the solution of the equatioe set, describmg the kinetics of the decompositkrn process. Snm is square (knxkn) matrix. Fig. 2 and 3 ehow the lection on the matrix for the different values of n (n = 100 and n2 = 25). Maxima value ot the y-coofdmafd correspsads to iae cf ll, alonf which the tection was made. Ct is obvious that the devgonal elements have the maximal values, ae long as these elementm cortespond to tha peobabilify of e-h peirr' generation in the same celt where the reaction bakes ptace.
Asymmatoy of the elements of matrix S°m ts explained fy the fact that therecombination of the reagents in tlt Ife;^sllslэo]flsooll oX ihe surface it fastet than in the crysXal bulk. IX the consideting cell is
0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005
°0 50 100 150 200 250 300
n
Fig. 2. Probability of the e-h pair generation in cell n because of the reaction in cell m = 25
0.035 0.03 0.025 0.02 0.015 0.01 0.005
°0 50 100 150 200 250 300
n
Fig. 3. Probability of the e-h pair generation in cell n because of the reaction in cell m = 100
in the area next to the surface (distance between the cell and the surface is smaller than r0) reflected energy does not absorbed significantly by the layer of the sample, so the energy of the secondary wave is bigger than the same value in case of the cells , which are far from the surface. This is the case of the cells with numbers 1: (nrl) or (n,+l): n. While moving to the centre of the crystal the curve becomes symmetrical. Maximal values among the elements of each line, as it was mentioned above, correspond to the probability of the generation in the same cell where the reaction takes place. The maximal values of the element ,v„ differ for the different lines (Fig. 3). Dependence of ,v„ on the number of the cell has several extremes. The maximal value corresponds to the crystal centre, because in that case ,v„
- 185 -
is calculated as a sum of probabilities of generation from all cells of the crystal and at the same time always contains the part, which considering the maximal contribution of the centre cell.
Ordinary differential system describing the process of the reaction with non-local development stage (1-6) was solved using the Runge-Kutta method of 1-5 order with time varied pitch. During the calculation of the kinetics the relative error does not exceed 10-12. Results of the simulation of the processes of the chain reaction initiation and development, initiated by the laser pulse with diameter 600 ^m, presented on Fig. 4 and Fig. 5. Constants of the elementary stages, used to simulate the process, were estimated in works [14,15,19]. After the termination of the laser action concentration of reagent A
xli"
i / * f \ - 6« IIS — 90 ns ---- 120 ns ...... 150 ns
/ ! X "i ./■ \ ■■
: f :j :> if/ 3 J t : 1 „ v r ;...........................\
i r......... k
100« -500 0 500
L, mem
Fig. 4 Calculated distributions of A complexes in the crystal in 60, 90, 120, 150 ns after the impulse termination
A A ; ; ! \ ! \ .......;.....• I......\..i......\L A l \ ! \
: U M \f : I i ¥ /........A........A........A...... u..............V V ï i A /«
: ; ; i : i i......i. • > \ \ \ i ; 1 1 ■ • • I \ ■ ' 1 / \ L,\ ' .i f.......v. i j ; i r i ■r \ 1 1 i \ \ \
: I I i ': i X i s ■ ¡•1................... ! \ J v / y \ t : \___Z......! 1 < ; ; 1 V v i A î i := ,'1 : i S
; ! • i : i : ............................ i i j ) \ : / \ : i y V * i 400 ns 500 m
// h. \ i \ * ______; ./ \ ---- 600 ns 700 ns
1000 -500 0 500
L, mem
Fig. 5. Calculated distributions of A complexes in the crystal in 400, 500, 600, 700 ns after the impulse termination
(N6 complex) increases (Fig. 4), while the concentration of the chain carriers decreases (electron-hole pairs) - induction period of the reaction. A similar situation occurs for the homogeneous variant of the chain reaction [20]. Then the reagents' concentrations begin to increase outside the initiated zone at a distance about 100 ^m. Estimation of the diffusion shift Dt (D = 0.25 cm2/s diffusion coefficient [21], t - time of calculation) gives the value ~ 1 ^m. Hence, the main reason of the reaction initiation outside the irradiated zone is not a diffusion, but generation of the reagents due to the non-local character of the branching stage.
When the induction period is over the concentrations of the holes and complex A begin to grow in step both in the irradiated area and in the area, which was not initially acted by the laser pulse. Development of the chain reaction causes decomposition of the anion sublattice and degeneration of the chain reaction. But by this time concentrations of the reagents outside the radiated zone increase and become more than their critical values, and so two reaction fronts are formed. These fronts move towards the crystal edges (Fig. 5).
3. Resume
Non local behavior of the chain propagation stage causes significant active particles' redistribution. The concentrations of the active particles, generated by means of this way, decrease exponentially outside the reaction area. The active particles are generated symmetrically on all sides outside the reaction area. Optimal area of the crystal for the reaction to develop is the centre of the radiation zone.
This work was supported by Russian Foundation for Basic Research for the financial support (grant №14-03-00534 A) and Ministry of Education and Science of the Russian Federation (governmentalproject № 2014/64).
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