A kinetic model for the influence of external noise, such as fluctuations of the point-defects’ production
rate and inhomogeneity of irradiated f.c.c. crystal, on the formation of modulated defect-distribution structure
is considered. Defect-production rate and density of sinks for point defects are simulated as independent
uniform and stationary stochastic fields with certain parameters. The interaction between vacancies is
taken into account. Such stochastic fields can induce a spatial point-defects’ distribution, which is a stationary
uniform stochastic field. Its mean value and correlation functions are estimated, and restricting
conditions are determined when this stochastic field becomes unstable because of interaction between defect-
density fluctuations and a stochastic field with a spatially-periodic mean value is formed. A formula
for evaluating its spatial period is analysed. This geometrical parameter of such a dissipative structure is
determined also by kinetic characteristics.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3543

Oliinyk, O.V.

Park, Y.B.

Selyshchev, P.O.

Tatarenko, V.A.

Electronic Sumy State University Institutional Repository

PROCEEDINGS OF THE INTERNATIONAL CONFERENCE NANOMATERIALS: APPLICATIONS AND PROPERTIES Vol. 1 No 4, 04RES09(4pp) (2012)   2304-1862/2012/1(4)04RES09(4) 04RES09-1  2012 Sumy State University Influence of Fluctuations of the Point-Defects’ Generation Rate and Inhomogeneity of Irra-diated F.C.C. Crystal on the Temperature Dependence of the Dissipative-Structure Period in a Spatial Distribution of Radiation Vacancies  O.V. Oliinyk1,*, V.A. Tatarenko1,†, P.O. Selyshchev2, Y.B. Park3  1 G. V. Kurdyumov Institute for Metal Physics, Department of Solid State Theory, 36 Academician Vernadsky Blvd., UA-03680 Kyyiv-142, Ukraine 2 University of Pretoria, Department of Physics,Private Bag X20, 0028 Hatfield, South Africa 3 Department of Materials Science and Metallurgical Engineering, Nano Materials Research Center,  Sunchon National University, Sunchon 540-742, Korea  (Received 22 June 2012; revised manuscript received 02 July 2012; published online14 August 2012)  A kinetic model for the influence of external noise, such as fluctuations of the point-defects’ production rate and inhomogeneity of irradiated f.c.c. crystal, on the formation of modulated defect-distribution struc-ture is considered. Defect-production rate and density of sinks for point defects are simulated as independ-ent uniform and stationary stochastic fields with certain parameters. The interaction between vacancies is taken into account. Such stochastic fields can induce a spatial point-defects’ distribution, which is a sta-tionary uniform stochastic field. Its mean value and correlation functions are estimated, and restricting conditions are determined when this stochastic field becomes unstable because of interaction between de-fect-density fluctuations and a stochastic field with a spatially-periodic mean value is formed. A formula for evaluating its spatial period is analysed. This geometrical parameter of such a dissipative structure is determined also by kinetic characteristics.  Keywords: Vacancies, Dissipative structure, ’Electrochemical’ interaction, Strain-induced interaction.   PACS numbers: 61.50.Lt, 61.72.Bb, 61.72.jd, 61.72.Qq                                                             * neutrino@ukr.net  † tatar@imp.kiev.ua  1. INTRODUCTION  The main causes for the stochastic behaviour of ra-diation-defects’ density are external ones. Firstly, these are fluctuations of the rate of defects’ generation. Sec-ondly, it is the random distribution of various imperfec-tions of a crystal lattice, which serve as sinks for the point defects. Because of these, the fluctuations of the density of radiation point defects are essentially non-equilibrium. Unlike the thermodynamic fluctuations, they do not decrease in inverse proportion to a system size and can reach the noticeable values [1]. The role of random disturbances becomes especially important for systems with non-linear feedbacks between their ele-ments, and for processes, which have a threshold char-acter and bifurcation points [2]. In a given work, the analysis of the formation of the spatially-periodic structure of the radiation point-defects’ distribution that occurs due to non-linear in-terdefect interaction, which was carried out in Refs. [3-5] and extended to irradiated solids with fluctuating parameters [2], is used to predict and consider the pos-sible modulated structure in a spatial distribution of radiation defects formed in irradiated f.c.c. crystal. To describe the probabilistic nature of a point-defects’ dis-tribution, it is suggested to be a stochastic field, and for its description a stochastic differential equation is used. If the rate of production of defects and density of their sinks are modelled by the relevant random uniform stationary fields, the random distribution of point de-fects can also be homogeneous and stationary. Howev-er, under certain conditions of irradiation, it becomes unstable due to the interaction of fluctuations of a de-fects’ density through the strain-induced (‘elastic’) fields and ‘electrochemical’ interaction between defects. Because of this, a stochastic field with a spatially-periodic behaviour of its average value is formed.  2. THE BASIC EQUATIONS   We consider an f.c.c. crystal in which due to an ex-ternal irradiation the vacancies and self-interstitial atoms are randomly formed. Right after [2], the rate of generation of defects, K(r, t), is considered to be a ran-dom uniform and stationary function of spatial co-ordinates (r) and time (t), respectively. Its average, 3 3 10( , ) ( 0.5 10 5 10 s )K t Kr  and variance are the constants and supposed to be predetermined.  Cor-relation functions of this field depend only on a differ-ence of respective arguments, i.e. ( , ) ( , ) ( , )K t K t t f tr r r r , and their Fourier-transform components, i.e. the spectral densities, G(k, t), are also supposed to be given functions. The defects mi-grate and are absorbed by sinks (for instance, by dislo-cations, dislocation loops, etc.) with the density d  d(r). As, due to the ‘fast’ migration of intrinsic interstitial atoms during the relaxation, their concen-tration in a bulk is rapidly decreasing, a residual con-centration of self-interstitial atoms is comparatively small, atom-to-atom distances are large, and a total contribution of their interactions between themselves and with vacancies is much more weaker than contri-bution of interaction between vacancies. Then, neglect-ing both the former and the recombination of these point defects, the evolution of the density of ‘slow’ va- O.V. OLIINYK, V.A. TATARENKO, P.O. SELYSHCHEV, Y.B. PARK PROC. NAP 1, 04RES09 (2012)   04RES09-2 cancies (v), n(r, t), will be considered hereinafter and described by the following stochastic equation [2, 6]:   ( , )( , ) ( ) ( , ) ( ( , ))n tK t n t ttrrr r r j r  (1)  ( r  / r is the Hamilton differentiation operator ‘na-bla’; (... ...) symbolizes the scalar product). Here (r)  zdD d(r) is the uniform random field, which de-scribes the probability of an absorption of point defect by sinks [2], with (r)   0, the factor zd  1 for a va-cancy, and the spectral density – G (k),   ( , ) ( , )( , ) ( , )Bn t tt D n tk Trr F rj r r  (2)  is a vacancy-flux density vector [2, 5, 6], where D  D0exp( Em/{kBT}) is the vacancy-diffusion coeffi-cient (diffusivity), Em is the migration-activation ener-gy, T is the temperature of an irradiated specimen, kB is the Boltzmann constant, ( , ) ( ) ( , )vvt W n t drF r r r r r  is the force of interac-tion between the vacancy located at the point r with all other vacancies with co-ordinates {r }, Wvv(r  r ) is the pairwise-interaction energy of vacancies at the points r and r . Separating in (1) the deterministic and fluctuation-noise components of n(r, t), (r), and K(r, t), we have a following set of equations [2, 6]:  0 0( , )( ) ( , ) ( ) ( , ) ( , )n tK n t n t D n ttrr r r r r   ( , ) ( , ) ( )vvBDn t n t W dk Tr rr r r r r   ( , ) ( , ) ( ) ,vvBDn t n t W dk Tr rr r r r r  (3)  0( , )( , ) ( , ) ( ) ( , ) ( , )n tK t n t n t D n ttrr r r r r   ( , ) ( , ) ( )vvBDn t n t W dk Tr rr r r r r   ( , ) ( , ) ( )vvBDn t n t W dk Tr rr r r r r   ( ) ( , ) ( ) ( , )n t n tr r r r   ( , ) ( , ) ( , ) ( , ) ( ) ,vvBDn t n t n t n t W dk Tr rr r r r r r r (4)  where ( )r r  is the Laplace operator, and  0( , ) ( , ) ( , ), ( , ) ( ) ,n t n t n t tr r r r r   0( , ) ( , )K t K t Kr r . 3. THE RANDOM UNIFORM STATIONARY FIELD OF DEFECTS’ DENSITY  Amongst the solutions of Eq. (1) or the set of Eqs. (3) and (4), there can be a solution, which is a random uniform stationary field – n0(r, t). The average density of vacancies for it, 0 0( , )n t nr  is constant in space and in time, and the correlation function of a defects’ density depends only on a difference of arguments [2]. Then, for the average value n0, we have [2]   0 0 0 0 0r r( ) ( , ) .K n n t  (5)  Neglecting the fluctuations of a product of stochas-tic functions, we solve Eq. (4) and construct the correla-tion functions as follows [2]:   0 00( )( ) ( , ) ,( ; )Gn t nnkr rk (6)   ( )20 1 0 2 0 20( )( , ) ( , )( ; )iG e dn t n t nnk r rk kr rk  1 2 11 2 1 2( ) ( )( ) 2 ( ) ( )1 2 2( , ) ,t t ti t td e d e d G ek r r k k kk k (7)  where 2 00 0( )( ) ( ; ) 1vvBn Wn k Dk Tkk k ,    ( ) ( )exp( )vv vvW W i dk r k r r . (8)  The equations (5), (6) form a closed system for n0(r, t) and together with expression (7) completely de-scribe a random homogeneous stationary field of the density of defects within the correlation approximation. For the determination of stability conditions for a random uniform stationary field, let us consider the evolution of a small perturbation of a probability distri-bution of the density of vacancies, at which the pertur-bation of average value is ( , )n tr  (fexp{ }t ik r  with a damping factor f  f(k)), and a perturbation of the fluctuation part is ( , )n tr [2]. Due to the anisotropy of an f.c.c. crystal, the interaction-energy Fourier com-ponent, ( )vvW k , reaches a minimum value for a partic-ular crystallographic direction. Therefore, with a change of parameters, for instance, the temperature of an irradiated specimen (and/or its content), one of the modes will become unstable with a wave vector k = kc having the same direction. Among other factors, it al-lows to restrict the analysis of stability to one-dimensional perturbations by directing Ox axis along kc. Substituting 0( , ) ( , )n x t n n x t into Eq. (3) and 0( , ) ( , )n x t n n x t in Eq. (4), we obtain the equations of the evolution of small perturbations of the average value and fluctuations of a homogeneous stationary random distribution of point defects [2, 6]   2 200 2 2( , )( , ) ( ) ( . ) ( . ) ( , ) ( )vvBDnn x tn x t x n x t D n x t n x t W x x dxt k Tx x   INFLUENCE OF FLUCTUATIONS OF THE POINT-DEFECTS’… PROC. NAP 1, 04RES09 (2012)   04RES09-3  0 0( , ) ( , ) ( , ) ( , ) ,vvBDn x t n x t n x t n x t W x x dxk T x x (9)   20 2( . )( , ) ( , )n x tn x t D n x tt x    202( , ) ( ) ( ) ( , )vvBDnn x t W x x dx x n x tk T x    0 0( ) ( )( , ) ( , ) ( , ) ( , ) .vv vvBD W x x W x xn x t n x t dx n x t n x t dxk T x x x (10)  The set of Eqs. (9), (10) has variable coefficients and is not closed, as it contains ( ) ( , )x n x t  and 0( , ) ( , )n x t n x t  [2]. But, as in Eq. (10), the variable coefficients are only at ( , )n x t  we consider the rele-vant terms as a inhomogeneity and find ( , )n x t , which is now a functional of ( , )n x t . Then, we substitute it in Eq. (9) and by averaging obtain the following expres-sion for a damping decrement of above-mentioned per-turbations [2]:   0 1 0 2ff( ) ( , ) ( , )( ) ( ) 1 1( ( ) ( )) ( ) ( )B BG n DS n DSd Sk T k Tk k k k kk k kk k k k k, (11)  where   2 0 0 01 2 3 1 2 1 2 3 1 2 f 32( , ) ( , ) ( , )exp{ ( )( ) ( ( ) ( )) },( )BDS d d d d S S Gk Tk k k k k k k k k k     1( , ) (( ) { ( ) ( )}),vv vvS W Wk k k k k k k k     2( , ) ( { ( ) ( ) ( )}).vv vvS W Wk k k k k k k k k    In the derivation of expression (11), the statistical independence of ( )r  and ( , )K tr  was supposed, i.e. their mutual correlation function and the relevant spectral density were taken to be equal to zero. Evi-dently, the damping decrement f is a function of k. If the energy of thermal motion of vacancies considerably exceeds the energy of their force interaction, f  0 for any mode. Nevertheless, with decreasing temperature for some k  kc (where c( ) 0vvW k ), the damping dec-rement passing through zero becomes positive (as am-plification factor), the instability in relation to the transition to a probability distribution with an average, the period of spatial change of which being equal to 2 /|kc| [2]. We consider the conditions for a spatially-periodic (inhomogeneous) distribution of the defects’ density to arise in their stochastic formation. To obtain further results a form of a spectral density should be rendered concrete. For the Gauss temporal spectrum 2( , ) exp( )G t tk , there will be an unstable mode and the homogeneous stationary random field of a defects’ density will not be realized. Let us consider the case when fluctuations of a rate of defect-production in time and in space are statistically independent and small so that the period of a structure of a defects’ density which arises is close to a deterministic one and much more than a radius of their correlation, rcor, and the time of their correlation, cor, is much less of all characteristic times of a problem. Then for the component in (11) caused by fluctuations we have [2]   21 2 cor2( ) ( , ) ( , ).( )( ( ) ( ))2( )BG S SDS dk Tk k k k kkk k k k (12)  In the stochastic description, it is not possible to re-duce parameters 0n , vvW0 , T to the one parameter—0 / ( )vvBnW k T0 , where { }max lim ( )vv vvW W0k 0nk . We will search for conditions of the development of instabil-ity by changing K0 ( f depends on K0 only through ), and the temperature and coefficient of a variation of the rate of a creation of displacements 2 0/K K  will be considered as fixed [2].  Correspondingly, the radius of correlation and the time of correlation may be evaluated as follows [7, 8]:  cor 2( , ) ( , ) ( )( , )K t K t drK tr r r rr, cor 2( , ) ( , ) ( )( , )K t K t t d tK tr rr.  A condition of the development of instability ob-tained without taking into account the fluctuations is the following:   2 0( ; ) 1 ( ) 0k D Uk k , (13)  O.V. OLIINYK, V.A. TATARENKO, P.O. SELYSHCHEV, Y.B. PARK PROC. NAP 1, 04RES09 (2012)   04RES09-4  ( ) ( ) /vv vvU W W0k k . The properties of an even function (k; ) are investigated in detail in [3, 5]. If   1, then (k; )  0. At  = 1 at a point k = km, there is a extre-mum. km| monotonously increases with increasing , and the value of (km; ) increases and becomes equal to zero at  = cr and k = kcr [2, 6]. Because of this the integrand in (12), decreasing with an increase of k as  k2 is localised within the interval ( km|, km|) pro-vided that km|  kcr|. The damping decrement f will become zero at   cr, because the numerator of an integrand in (11) is non-negative within the interval ( km|, km|), and S  0 [2]. It is also clear from this consideration that in a region of stability (k; )  0 for all k. The value of kcr is that, for the large-scale inhomo-geneity of sinks’ distribution when k0 is small, ( ) lim ( )vv vv vvcrW W Wnk 0k k , where k  0 along the direction ncr  kcr/|kcr|. If a distance between the de-fects largely exceeds the host-lattice period, the Fourier components, ( )vvW k , of the interaction energies for vacancies can be written as a power series:  2el.chem( ) ( ) ( ) ...vv vv vv vv vvW V w knk k k n( ) (14)  vvV k( )  is the Fourier component of strain-induced v–v-interaction energies. Within the small finite region near k  0, vvV k( )  may be represented as follows [9, 10]: 2( ) ( ) ( ) .vv vv vv vvV A B k Qk n n  Here, the well-known first term is based on the long-wave-limit ap-proximation [9, 10]; the second term is a correction to this approximation [10, 11], and the third term is a gauge, which eliminates a strain-induced self-action of the vacancies [9, 12]. The coefficient Bvv(n) was derived in revised form in Ref. [11]. el.chem( )vv k  is the Fourier component of energies for direct ‘electrochemical’ v–v-interactions. Within both the long-wavelength approx-imation and the cohesive-energy-estimation approach developed in Refs. [13–17], 2el.chem el.chem( ) ( )vv vv vvkk 0  [17], ( ) ( )vv vv vvBn n  is the expansion coefficient in Eq. (13) ( kn k ), el.chem( ) ( )vv vv vvw A Qn n 0 . As shown in Ref. [17], the coefficient Bvv(n) and hence, ( )vv n  have the minimum values along the di-rection (100) for f.c.c. crystal, when k  0. The Fourier component of v–v-interaction energies along the direc-tion (100) within the small finite region near k  0 may be represented as follows:  2( ) ( 1 (100) )vv vvW W k0k B ,  where (100) (100) /vv vv vvW0B . In a numerator of an integrand in (12), we will be restricted to the first term of an expansion, and in, (k; ) (13),  to the second term because it determines km [2]. Going to (12) from an integration to the summation and taking into account 2cor cor,0G K r0  [2], then substituting (12) into (11) and keeping the terms of the same order of smallness, we find [2]  5/4 1/40c 0c cr d1( ) (100)( ) 2vvvvBK Wb TT k TB ,  2 2 1/4 5/4c cr d11 ( ) (100)4k k b T B ,  where 2 2cor cor 0( ) ( ) /b T r D T K K  [2].  REFERENCES  1. V. Herstemke and R. Lefebvre, Noise-Induced Transitions (Moscow: Mir: 1987) (in Russian). 2. P. A. Selishchev, Metallofiz. Noveishie Tekhnol. 21 No9, 11 (1999) (in Russian) 3. V.J. Sugakov, Preprint ITF-84-70, ITPh AS Ukr.SSR, Ki-ev, 1984 (in Russian). 4.  P. A. Selishchev, V. I. Sugakov, Radiat. Eff. and Defects in Solids 133, 231 (1995). 5. V.J. Sugakov, Fundamentals of Synergetics, (Kyyiv: Ober-egy: 2001) (in Ukraine). 6. P. A. Selishchev, Self-Organization in Radiation Physics, (Kiev: Aspekt-Poligraf: 2004) (in Russian). 7. P.A. Selishchev, D.V. Shchur, Bulletin of Kyiv University. Ser. Phys-Math. Science 2, 505 (1999) (in Ukraine). 8. I. A. Kvasnikov, Thermodynamics and Statistical Physics. Theory of Non-Equilibrium Systems (Moscow: MGU: 1987) (in Russian). 9. A.G. Khachaturyan, Theory of Structural Transformations in Solids (Mineola, NY: Dover Publications, Inc.: 2008). 10. A.A. Katsnelson, A.I. Olemskoi, Microscopic Theory of Nonhomogeneous Structures (Moscow: Mir Publ.: 1990). 11. O.V. Oliinyk, V.A. Tatarenko, S.M. Bokoch, Metallofiz. Noveishie Tekhnol. 33 No10, 1383 (2011) (in Ukrainian); Erratum, ibidem, 34 No1, 137 (2012). 12. V.A. Tatarenko, C.L. Tsinman, Physics of Real Crystals, 244 (Kiev: Naukova Dumka: 1992) (in Russian). 13. V.G. Weizer and L.A. Girifalco, Phys. Rev. 120 No3, 837 (1960). 14. R. Yamamoto and M. Doyama, J. Phys. F: Metal Phys. 3 No8, 1524 (1973). 15. W.A. Harrison, Pseudopotentials in the Theory of Metals (New York: W.A. Benjamin, Inc.: 1966). 16. W.A. Harrison, Solid State Theory (New York: McGraw-Hill: 1970). 17. O.V. Oliinyk, V.A. Tatarenko, J. Nano- Electron. Phys. 4 No1, 01023 (2012).  

Influence of Fluctuations of the Point-Defects’ Generation Rate and Inhomogeneity of Irradiated F.C.C. Crystal on the Temperature Dependence of the Dissipative-Structure Period in a Spatial Distribution of Radiation Vacancies

Influence of Fluctuations of the Point-Defects’ Generation Rate and Inhomogeneity of Irradiated F.C.C. Crystal on the Temperature Dependence of the Dissipative-Structure Period in a Spatial Distribution of Radiation Vacancies

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