Final distribution density of material fragment sizes at slow fragmentation process

The process of slow material fragmentation is studied when the diffusion approximation is applicable. Final distribution density of fragment sizes is calculated in the case of scale-homogeneity of subdivision mechanism.

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Date:	2012
Main Authors:	Brodskii, R.Ye., Virchenko, Yu.P.
Format:	Article
Language:	English
Published:	Single Crystal Institute 2012
Series:	Вопросы атомной науки и техники
Subjects:	Section D. Theory of Irreversible Processes
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/107097
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Cite this:	Final distribution density of material fragment sizes at slow fragmentation process / R.Ye. Brodskii, Yu.P. Virchenko // Вопросы атомной науки и техники. — 2012. — № 1. — С. 206-208. — Бібліогр.: 3 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	nasplib_isofts_kiev_ua-123456789-1070972025-02-23T18:03:51Z Final distribution density of material fragment sizes at slow fragmentation process Финальная плотность распределения размеров фрагментов материала в условиях медленности фрагментации Фiнальна густина розподiлу розмiрiв фрагментiв матерiалу в умовах повiльностi фрагментацiї Brodskii, R.Ye. Virchenko, Yu.P. Section D. Theory of Irreversible Processes The process of slow material fragmentation is studied when the diffusion approximation is applicable. Final distribution density of fragment sizes is calculated in the case of scale-homogeneity of subdivision mechanism. Рассмотрен процесс медленной фрагментации материала в условиях, когда применимо диффузионное приближение. Вычислена финальная плотность распределения размеров фрагментов в случае масштабной однородности механизма дробления. Розглянуто процес повільної фрагментації матеріалу в умовах, коли можна застосувати дiфузiйне наближення. Обчислена фінальна густина розподілу розмiрiв фрагментiв у випадку масштабної однорiдностi механiзму дроблення. 2012 Article Final distribution density of material fragment sizes at slow fragmentation process / R.Ye. Brodskii, Yu.P. Virchenko // Вопросы атомной науки и техники. — 2012. — № 1. — С. 206-208. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 05.20.Dd https://nasplib.isofts.kiev.ua/handle/123456789/107097 en Вопросы атомной науки и техники application/pdf Single Crystal Institute
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Section D. Theory of Irreversible Processes Section D. Theory of Irreversible Processes
spellingShingle	Section D. Theory of Irreversible Processes Section D. Theory of Irreversible Processes Brodskii, R.Ye. Virchenko, Yu.P. Final distribution density of material fragment sizes at slow fragmentation process Вопросы атомной науки и техники
description	The process of slow material fragmentation is studied when the diffusion approximation is applicable. Final distribution density of fragment sizes is calculated in the case of scale-homogeneity of subdivision mechanism.
format	Article
author	Brodskii, R.Ye. Virchenko, Yu.P.
author_facet	Brodskii, R.Ye. Virchenko, Yu.P.
author_sort	Brodskii, R.Ye.
title	Final distribution density of material fragment sizes at slow fragmentation process
title_short	Final distribution density of material fragment sizes at slow fragmentation process
title_full	Final distribution density of material fragment sizes at slow fragmentation process
title_fullStr	Final distribution density of material fragment sizes at slow fragmentation process
title_full_unstemmed	Final distribution density of material fragment sizes at slow fragmentation process
title_sort	final distribution density of material fragment sizes at slow fragmentation process
publisher	Single Crystal Institute
publishDate	2012
topic_facet	Section D. Theory of Irreversible Processes
url	https://nasplib.isofts.kiev.ua/handle/123456789/107097
citation_txt	Final distribution density of material fragment sizes at slow fragmentation process / R.Ye. Brodskii, Yu.P. Virchenko // Вопросы атомной науки и техники. — 2012. — № 1. — С. 206-208. — Бібліогр.: 3 назв. — англ.
series	Вопросы атомной науки и техники
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fulltext	FINAL DISTRIBUTION DENSITY OF MATERIAL FRAGMENT SIZES AT SLOW FRAGMENTATION PROCESS R.Ye. Brodskii 1 and Yu.P. Virchenko 2∗ 1Single Crystal Institute, 61000, Kharkov, Ukraine 2Belgorod State University, 308015, Belgorod, Russia (Received November 2, 2011) The process of slow material fragmentation is studied when the diffusion approximation is applicable. Final distrib- ution density of fragment sizes is calculated in the case of scale-homogeneity of subdivision mechanism. PACS: 05.20.Dd 1. INTRODUCTION A lot of works are dedicated to theoretical researching of material fragmentation process from the statistical physics viewpoint (see, for example, [1–3]). Impor- tance of such investigations is stipulated by its prac- tical necessity. The complexity of the study of such physical process on the basis of statistical physics is connected with serious difficulties if microscopic rep- resentations are used. In such conditions, main inves- tigation approach to study fragmentation dynamics consists in the construction of the most general prob- abilistic fragmentation models which are phenomeno- logically reasonable . After that it is necessary to study the most general properties of such models. 2. PROBLEM FORMULATION Let N(r, t) is the number of fragments which have sizes being not larger than r at the time moment t and g(r, t) = ∂N(r, t)/∂r is the corresponding distri- bution density. We use further the following kinetic equation that describes the temporal evolution of the density ġ(r, t) = ∞∫ r K(r, r′; t)g(r′, t)dr′ − μ(r, t)g(r, t) . (1) It has been formulated in [3]. Here K(r, r′; t) is a non- negative function. From the physical point of view, it is the average number of fragments which are formed during the small time interval (t, t+dt), μ(r, t) is the average number of fragments having the size r which are disintegrated during the same temporal interval. It is supposed that the conservation of total volume of all fragments takes place that is formulated in the form ∫ ∞ 0 g(r, t)r3dr = const [2]. It is assumed that the energy expended on the breaking of intermolec- ular bonds during the time interval dt, which is pro- portional to the value ∫ ∞ 0 g(r, t)r2dr of total surface area of all fragments [2], is also constant. At the slow- ness condition of fragmentation process, the diffusion equation is applicable [3]. Using it and also using conditions of volume conservation and the constancy of the expending energy intensity, the equation (1) is transformed to the following [3] ġ(r, t) = γ(r, t)g(r, t) + 2 3 ∂ ∂r [rγ(r, t)g(r, t)]+ + 1 6 ∂2 ∂r2 [r2γ(r, t)g(r, t)] , where γ(r, s) > 0 is the intensity of the fragment for- mation. At the assumption that the fragment subdivision is steady-state stable at all scales after sufficiently long evolution when γ(r, t) ∼ c(r)γ(t), we have ∂g ∂s = c(r)g(r, s) + 2 3 ∂ ∂r [rc(r)g(r, s)]+ + 1 6 ∂2 ∂r2 [r2c(r)g(r, s)] , (2) where we introduce the effective time scale with ele- mentary interval ds = γ(t)dt. Taking into account that g(r, s) is concentrated near the point r = 0, g(r, s) ∼ N(s)δ(r) (N(s) is the total number of frag- ments) when the time evolution is sufficiently large, it is important to investigate the function g(r, s) by such a way that, firstly, to find its structure near this point. In this situation there are two qualita- tively different cases connected with the behavior of the function c(r): the scale invariant (Kolmogorov) case when c(0) > 0 and the scale homogeneous case when c(r) ∝ rβ , β > 0. In this work we calculate the final distribution density in the last case. In such ∗Corresponding author E-mail address: virch@bsu.edu.ru 206 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1. Series: Nuclear Physics Investigations (57), p. 206-208. a situation, the equation (2) is transformed to the following ∂g ∂s = rβg(r, s) + 2 3 ∂ ∂r [rβ+1g(r, s)]+ + 1 6 ∂2 ∂r2 [r2+βg(r, s)] , (3) where the transformation to dimensionless variable r ⇒ r/r∗, r∗ = c −1/β 0 is done in the density g(r, s). It is necessary to solve the equation (3) on the semi-axis (0,∞) taking into account the boundary conditions rg(r, s) → 0 at r → 0 and r4g(r, s) → 0 at r → ∞ which follow from the integrability of den- sity g(r, s) near the zero point and the finiteness of integral ∫ ∞ 0 r3g(r, s)dr < ∞. 3. FINAL DISTRIBUTION DENSITY For solving the equation (3), we make the following transformations. At first, we introduce a new func- tion h(r, s) = rβ+3g(r, s) and after that we pass to a new independent variable x = r−β/2. In a result, the equation (3) in terms of new values takes the form ∂h ∂s = β2 24 ∂2h ∂x2 + β 6x [(1 + β/2)/2 − 1] ∂h ∂x . (4) Here, we essentially simplify in comparison with the work [3] the procedure of the solution building and the calculations connected with it (in particular, it permits to remove the inaccuracy that takes place in the cited work). The initial boundary problem on the axis x ∈ (0,∞) for the equation obtained is necessary to solve at the conditions xβ+2h(x, s) → 0 at x → 0 and xβ−1h(x, s) → 0 at x → ∞. Further, in this work we analyze only the case β = 2, when the equation (4) has the most simple form ∂h ∂s = 1 6 ∂2h ∂x2 . (5) The initial boundary problem on the semi-axis is solved on the basis of the Laplace transformation on the time s. Its solution can be represented by the formula h(x, s) = ∞∫ 0 G(x, x′; s)h(x′, 0)dx′ , (6) where the Green function G(x, x′; s) has the following form G(x, x′; s) = √ 3 2πs × × [ exp ( −3(x − x′)2 2s ) − exp ( −3(x + x′)2 2s )] . (7) From (6) one may obtain the following formula for the function g(r, s) g(r, s) = = ∞∫ 0 g(r′, 0) · [ r∗ r′ ]2 · [ r′ r ]5 · G (r∗ r , r∗ r′ ; s ) d ( r′ r∗ ) . The expression g(r, 0) = δ(r − r0) corresponds to very important special case, when the fragmentation process starts from one fragment having the size r0. In this case g(r, s) = [ r2 ∗r 3 0 r5 ] · G ( r∗ r , r∗ r0 ; s ) . To obtain the asymptotic expression for this distrib- ution density at s → ∞, i.e., in probabilistic termi- nology, the final density, it is sufficient to take into account that, after the long evolution, the density g(r, s) should be concentrated in the region of small values r � r0. Assuming also that it takes place by such a way that r2∗ � rr0 (the asymptotic formula obtained in this case is nonuniform on r at r → 0, i.e. it is not right at very small r), one may find that g(r, s) = C(s) r6 exp ( −α(s) r2 ) , (8) where α(s) = 3r2 ∗/2s, C(s) = 6r4 ∗r 2 0(3/2πs3)1/2. Then the average fragment number N(s), corre- spondingly normalized to unit probability distribu- tion density f(r, s) = g(r, s)/N(s) and the average fragment size 〈r̃〉 are determined by formulas N(s) = 3 8 √ π C(s) α5/2(s) , f(r, s) = 8 3 √ π α5/2(s) r6 exp ( −α(s) r2 ) , 〈r̃〉 = 3 4 √ π α1/2(s) . The relative squared variation of the fragment size is constant, since Dr̃ = 2 3 ( 1 − 8 3 √ π ) α(s) . 4. CONCLUSIONS The research made in the present work permits essen- tially simplify the calculation of the final distribution density of fragment sizes when the scale homogeneous mechanism of fragment subdivision takes place. We find general equation (4) defining the final distribu- tion density at any homogeneity parameter β which is reduced to the Schrödinger equation with the imag- inary time and with the potential being proportional to ∼ x−2. We illustrate the calculation of the fi- nal distribution density explicitly on the case when β = 2. It is remarkable that the distribution density in the case under consideration is decreased by the power way when the size r tends to infinity ∼ r−6. Besides, it is very important that the values N(s), 〈r̃〉 vary essentially different from those which take place in the Kolmogorov case, N(s) ∼ s, 〈r̃〉 ∼ s−1/2. 207 References 1. A.N. Kolmogorov. On logarithmically normal distribution law of particle sizes at division process // DAN SSSR. 1941, v. 31, p. 99-101. 2. R.Z. Sagdeev, A.V. Tur, V.V. Yanovskii. Forma- tion and universal properties of the size distribu- tion in the fragmentation theory // DAN SSSR. 1987, v. 294, p. 1105-1110. 3. R.Ye. Brodskii, Yu.P. Virchenko. Final probabil- ity distribution of random sizes at the self-similar division mechanism // Scientific Bulletin of Bel- gorod State University. Mathematics & Physics. 2008, v. 13 (53), p. 23-32. �� !�� " � �# �� "# ��"� � � �� "��"� � �� $�� " � � � �� " � ��"�"� �� "� � �� !�� # �� "� ��" �� "�� %&'

Final distribution density of material fragment sizes at slow fragmentation process

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