A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation

Dislocation-assisted phase separation processes in binary systems subjected to irradiation effect are studied analytically and numerically. Irradiation is described by athermal atomic mixing in the form of ballistic flux with spatially correlated stochastic contribution. While studying the dynamics...

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Veröffentlicht in:	Condensed Matter Physics
Datum:	2015
Hauptverfasser:	Kharchenko, D.O., Schokotova, O.M., Bashtova, A.I., Lysenko, I.O.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2015
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spelling	Kharchenko, D.O. Schokotova, O.M. Bashtova, A.I. Lysenko, I.O. 2019-06-14T11:09:05Z 2019-06-14T11:09:05Z 2015 A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation / D.O. Kharchenko, O.M. Schokotova, A.I. Bashtova, I.O. Lysenko // Condensed Matter Physics. — 2015. — Т. 18, № 2. — С. 23003: 1–22. — Бібліогр.: 62 назв. — англ. 1607-324X DOI:10.5488/CMP.18.23003 arXiv:1506.03962 PACS: 05.40.Ca, 64.75 Op, 05.70 Ln https://nasplib.isofts.kiev.ua/handle/123456789/153592 Dislocation-assisted phase separation processes in binary systems subjected to irradiation effect are studied analytically and numerically. Irradiation is described by athermal atomic mixing in the form of ballistic flux with spatially correlated stochastic contribution. While studying the dynamics of domain size growth we have shown that the dislocation mechanism of phase decomposition delays the ordering processes. It is found that spatial correlations of the ballistic flux noise cause segregation of dislocation cores in the vicinity of interfaces effectively decreasing the interface width. A competition between regular and stochastic components of the ballistic flux is discussed. Проведено дослiдження процесiв фазового розшарування за дислокацiйним механiзмом в бiнарних системах, пiдданих дiї опромiнення. Опромiнення описується атермiчним перемiшуванням атомiв, за рахунок уведення балiстичного потоку, що має просторово-скорельовану стохастичну складову. При вивченнi динамiки росту доменiв показано, що дислокацiйний механiзм уповiльнює процес упорядкування. Встановлено, що просторовi кореляцiї шуму балiстичного потоку стимулюють сегрегацiю ядер дислокацiй в околi мiжфазних границь, ефективно зменшуючи ширину мiжфазного шару. Розглянуто конкуренцiю мiж регулярною та стохастичною компонентами балiстичного потоку. Fruitful discussions with Dr. V.O. Kharchenko are gratefully acknowledged. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation Дослiдження процесiв фазового розшарування за наявностi дислокацiй в бiнарних системах, пiдданих опромiненню Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation
spellingShingle	A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation Kharchenko, D.O. Schokotova, O.M. Bashtova, A.I. Lysenko, I.O.
title_short	A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation
title_full	A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation
title_fullStr	A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation
title_full_unstemmed	A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation
title_sort	study of phase separation processes in presence of dislocations in binary systems subjected to irradiation
author	Kharchenko, D.O. Schokotova, O.M. Bashtova, A.I. Lysenko, I.O.
author_facet	Kharchenko, D.O. Schokotova, O.M. Bashtova, A.I. Lysenko, I.O.
publishDate	2015
language	English
container_title	Condensed Matter Physics
publisher	Інститут фізики конденсованих систем НАН України
format	Article
title_alt	Дослiдження процесiв фазового розшарування за наявностi дислокацiй в бiнарних системах, пiдданих опромiненню
description	Dislocation-assisted phase separation processes in binary systems subjected to irradiation effect are studied analytically and numerically. Irradiation is described by athermal atomic mixing in the form of ballistic flux with spatially correlated stochastic contribution. While studying the dynamics of domain size growth we have shown that the dislocation mechanism of phase decomposition delays the ordering processes. It is found that spatial correlations of the ballistic flux noise cause segregation of dislocation cores in the vicinity of interfaces effectively decreasing the interface width. A competition between regular and stochastic components of the ballistic flux is discussed. Проведено дослiдження процесiв фазового розшарування за дислокацiйним механiзмом в бiнарних системах, пiдданих дiї опромiнення. Опромiнення описується атермiчним перемiшуванням атомiв, за рахунок уведення балiстичного потоку, що має просторово-скорельовану стохастичну складову. При вивченнi динамiки росту доменiв показано, що дислокацiйний механiзм уповiльнює процес упорядкування. Встановлено, що просторовi кореляцiї шуму балiстичного потоку стимулюють сегрегацiю ядер дислокацiй в околi мiжфазних границь, ефективно зменшуючи ширину мiжфазного шару. Розглянуто конкуренцiю мiж регулярною та стохастичною компонентами балiстичного потоку.
issn	1607-324X
url	https://nasplib.isofts.kiev.ua/handle/123456789/153592
citation_txt	A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation / D.O. Kharchenko, O.M. Schokotova, A.I. Bashtova, I.O. Lysenko // Condensed Matter Physics. — 2015. — Т. 18, № 2. — С. 23003: 1–22. — Бібліогр.: 62 назв. — англ.
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first_indexed	2025-11-25T20:40:33Z
last_indexed	2025-11-25T20:40:33Z
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fulltext	Condensed Matter Physics, 2015, Vol. 18, No 2, 23003: 1–22 DOI: 10.5488/CMP.18.23003 http://www.icmp.lviv.ua/journal A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation D.O. Kharchenko, O.M. Schokotova, A.I. Bashtova, I.O. Lysenko Institute of Applied Physics of the National Academy of Sciences of Ukraine, 58 Petropavlivska St., 40000 Sumy, Ukraine Received November 20, 2014, in final form March 10, 2015 Dislocation-assisted phase separation processes in binary systems subjected to irradiation effect are studied analytically and numerically. Irradiation is described by athermal atomic mixing in the form of ballistic flux with spatially correlated stochastic contribution. While studying the dynamics of domain size growth we have shown that the dislocation mechanism of phase decomposition delays the ordering processes. It is found that spatial correlations of the ballistic flux noise cause segregation of dislocation cores in the vicinity of interfaces effectively decreasing the interface width. A competition between regular and stochastic components of the ballistic flux is discussed. Key words: phase decomposition, particle irradiation, noise PACS: 05.40.Ca, 64.75 Op, 05.70 Ln 1. Introduction A study of nonequilibrium phenomena observed in materials under sustained particle or laser irra- diation attains an increasing interest in modern theoretical physics, condensed matter physics, material science, and metallurgy. Particle or laser irradiation causes the production of structural disorder with generation of a large amount of point defects. These defects can organize into defects of higher dimension and stimulate the occurrence of nonequilibrium phenomena. In recent decades, numerous experimental data have shown that alloys under sustained irradiation can be considered as nonequilibrium systems manifesting phase transitions, phase separation, pattern formation with rearrangement of point defects in bulk and on a surface (see for example, [1–10]). First observations of ordering/disordering processes in irradiated alloys were discussed six decades ago (see reference [11]). It was shown that nonequilibrium conditions in such systems are caused by interactions of high energy particles with atoms of a target (pure material, alloys). From practical viewpoint, a study of these phenomena remains an urgent problem to predict the be- havior of construction materials. A study of phase stability in various solids and metallic alloys under sustained irradiation received a long-standing attention due to its intrinsic interest and its relevance in technological problems such as: improvement of mechanical properties, radiation resistance, radiation damage, etc. Mechanical stability of construction materials is governed by rearrangement of the defects produced by irradiation and their segregation on phase interfaces and grain boundaries. Perturbation of the atomic configuration by irradiation causes the alternation of the phase stability [12–15]. Therefore, in order to predict the behavior of irradiated materials at different loading, one should know the physical mechanisms leading to self-organization of the defect structure that causes microstructure transforma- tions. It is known that phase transformations in alloys subjected to particle irradiation can be quite differ- ent from that observed in the absence of irradiation [16]. Experimental observations of phase separation © D.O. Kharchenko, O.M. Schokotova, A.I. Bashtova, I.O. Lysenko, 2015 23003-1 http://dx.doi.org/10.5488/CMP.18.23003 http://www.icmp.lviv.ua/journal D.O. Kharchenko et al. (spinodal decomposition) in binary alloys (Ni-Cu, Ni-Cu) have shown that the electron irradiation can increase the solute mobility. It causes phase decomposition at temperatures at which diffusivities un- der thermal conditions are too small to provide this effect (see, for example, references [17, 18]). The same results were obtained for alloys Au-Ni, Cu-Ni, Fe-Mo with different compositions [19–21]. Irradia- tion damage can lead to precipitate dissolution and stagnation precipitate ordering [22–24]. It was found that phase separation in irradiated systems can occur at temperatures above a coherent spinodal. This effect can be described by point defects production with an increase of their mobility to dislocations in such a way that the misfit dislocations move with the composition field relieving strains. The corre- sponding model of a mobile dislocation density field coupled with the composition field was proposed in references [25–27]. It was found that phase separation is possible above the coherent spinodal due to the motion of dislocations with a decrease of misfit strains. Phase decomposition and patterning sustained by the dislocation field dynamics coupled with the composition field in binary systems under sustained irradiation were studied in reference [28]. In this work, the irradiation effect was considered as an ad- ditional contribution to the free energy according to the model proposed in references [29–32]. In this model, the irradiation induced atomic mixing was described by a ballistic (athermal) flux responsible for the production of a structural disorder. In reference [28], the authors have shown that stable patterns characterized by time independent amplitude and wave-length emerge due to misfit dislocations. These linear defects are capable of reducing the coherency strain emergent at an atomic size mismatch. In the proposed model, the authors consider a deterministic case, where irradiation increases the free energy of the system. A problem related to nonequilibrium effects induced by fluctuations of the point defect concentration and local temperature in cascades was not solved for the system with mobile dislocations. In this work, we extend the above model of phase separation with a dislocation mechanism in binary systems subjected to irradiation taking into consideration stochastic conditions. The goal of the paper is to study the role of the above fluctuation effects in the prototype model of binary systems undergoing phase separation assisted by mobile dislocations. We take into account the stochastic component of the ballistic flux proposed in reference [33]. Such a stochastic model was exploited to study phase decomposition processes [34–36] and patterning [37] in irradiated systems. The corresponding stochastic contribution takes care of local fluctuations in the composition field due to stochasticity of point defects concentration and temperature. We consider spatial correlations of these fluctuations and study the effect of spatial correlations onto phase decomposition processes. By taking into account the difference in time scales for composition and dislocation density fields we, initially, consider the simplest case of one slow mode (composition field). This allows us to perform the mean-field analysis for the slow mode and study the effect of the dislocation mechanism strength onto phase decomposition processes. The dynamics of the coupled, simultaneous time evolved composition and dislocation fields is studied numerically. Here, we discuss the statistical properties of phase separation and the domain size growth law. We show that the growth of domain sizes is delayed by dislocations participating in phase decomposition processes under sustained irradiation. This effect was predicted theoretically in unirradiated systems (see references [38, 39]). Considering the segregation of dislocations in the vicinity of interfaces, we discuss a competition between regular and stochastic components of ballistic flux fluctuations. The work is organized as follows. In section 2 we present the stochastic model of a binary system with ordinary thermal fluctuations representing the internal noise and ballistic flux fluctuations playing the role of external noise. In section 3, we study a reduced model, where dislocation density is excluded according to an adiabatic elimination procedure. In section 4, we numerically consider the dynamics of the phase decomposition. We conclude in section 5. 2. Model Considering a binary alloy A-B, one can exploit the Bragg-Williams theory, where the correspond- ing free energy density is fBW(cA) = Z wocAcB/2 + T ci lnci . Here, i = {A,B}, ci = Ni /N , N = NA + NBis the total number of particles, Z is a coordination number, T is the temperature measured in ener- getic units, an ordering energy wo = (2w{A,B} − w{A,A} − w{B ,B}) is defined through pair interaction en-ergies w{·,·}. After expanding fBW around the critical concentration c = 1/2, we arrive at Landau-like potential f (ψ) ' −Aψ2/2+Bψ4/4 with A = T /[c(1− c)]− Z wo and B = [Z wo−1/(1− c)3 +1/c3]/3; the 23003-2 Phase separation processes in binary systems subjected to irradiation quantity ψ measures the deviation from the critical concentration, i.e., ψ ≡ (c − c). Taking into account the inhomogeneity of the alloy and assuming that ψ(r) varies slowly on the scale of lattice parameter a0, i.e., ψ(r+a0) ' ψ(r)+a0 · ∇ψ(r), one can take into account the gradient energy term to the free en- ergy in the form r 2 0 (∇ψ)2/2, where r0 is the interaction radius determining the interface width betweentwo phases enriched by atoms A and atoms B. Following the Krivoglaz-Clapp-Moss expression, one has r 2 0 = dv(k)/dk2, where v(k) is the Fourier transform of the atomic interaction energy. Adopting the Cahn- Hilliard approach, the dimensionless free energy functional assumes the Ginzburg-Landau form [40, 41] F0 = ∫ dV [ − A 2 ψ 2 + B 4 ψ 4 + r 2 0 2 (∇ψ)2 ]. The case A < 0 corresponds to temperatures above the chemical spinodal. An additional contribution to the free energyF0 is given by a lattice mismatch in the form of elasticenergyFe = (ν2E/2) ∫ dVψ2 [28]. Here, E is the Young modulus, ν relates to the lattice parameter change with respect to the composition (Vegard’s law), i.e., a = a0(1+νψ) [42, 43]. The elastic contribution shifts the corresponding coherent spinodal: A = ν2E . Following reference [28], we take into account the dislocation-assisted mechanism for spinodal de- composition by introducing dislocation-dislocation interactions in the formFd = ∫ dV [C 2 \|b\|2 + 1 2E (∇2$)2 ]. Here, the constantC relates to a core energy of dislocations. The elastic strain energy of the system is gov- erned by the Airy stress function $ satisfying the equation ∇$ = E(∇x by −∇y bx ), where bx and by arethe corresponding components of the continuous dislocation density field in a two dimensional problem. This term accounts for the nonlocal elastic interaction between dislocations. To describe the coupling between the composition field ψ and strain field of dislocations, we use the results of the work reference [28] and introduce a relevant contribution to the free energy in the form Fc = ν ∫ dVψ∇2$. This two-dimensional model was previously used to study the melting [44–46], dislocation patterning [47] and phase separation in the misfitting binary thin films [48]. By combining all the above contributions, the total free energy of the actual system reads Ftot = F0 +Fe +Fd +Fc. Therefore, the dynamics of the conserved fields ψ, bx and by is described by thefollowing set of deterministic equations with diffusive dynamics ∂tψ= M∇2 δFtot δψ , (2.1) ∂t bx = ( Mg∇2 x +Mc∇2 y ) δFtot δbx , (2.2) ∂t by = ( Mc∇2 x +Mg∇2 y ) δFtot δby . (2.3) Here,M is the solute mobility,Mg andMc denote the mobility for glide and climb, respectively1.The effect of irradiation leads to an increase in the total free energy due to ballistic mixing of atoms in cascades. One of the models allowing one to describe these processes was proposed in reference [29]. It is based on the introduction of the spatial coupling term relevant to ballistic exchanges under irradia- tion conditions. The related Langevin dynamics with the additive external noise mimicking a stochastic ballistic mixing was studied in reference [32]. It should be noted that this approach does not properly take into account the fluctuations of the solute by a stochastic motion of the defects in cascades. As far as these fluctuations occur in a correlated medium (crystals), the corresponding spatial correlations of fluc- tuations should be considered. The other concept of a ballistic mixing describing the above mentioned fluctuations was proposed in reference [12–14, 50]. It was shown that a ballistic mixing is stochastic in nature since the knocked atoms move at random at the distance R. According to this approach, the bal- listic mixing can be described by introduction of the ballistic diffusion flux with a fluctuating ballistic diffusion coefficient. These fluctuations are induced by irradiation (fluctuations in both concentration of defects and local temperature in cascades). In reference [33] it was shown that such an approach leads to a multiplicative noise Langevin dynamics, where spatially correlated external fluctuations promote the solute flux opposite to the ordinary diffusion flux. The phase decomposition of binary systems under the above assumptions was studied in references [34, 35], while patterning processes in one component crystalline systems under the irradiation effect were discussed in references [36, 37]. 1This set of equations belongs to the models with conserved dynamics according to the classification suggested by Galperin and Hohenberg in reference [49]. 23003-3 D.O. Kharchenko et al. In this paper we exploit the model of a stochastic ballistic flux according to discussions provided in references [12, 14, 33–35, 37, 50]. We assume that the force mixing induced by ballistic jumps occurs with relocation distances b ≡ 〈R〉 = ∫ Rw(R)dR , where R is distributed according to the known distribution w(R). Such ballistic jumps can be considered as a non-thermal diffusion process with a “diffusion coeffi- cient” D0 b . The corresponding ballistic flux is Jb =−D0 b∇ψ [12]. Following reference [33], we assume thatsuch a diffusion occurs in the fluctuating environment. Indeed, collision processes of an energetic parti- cle with an atom result in local fluctuations in the temperature and a number of point defects (Frenkel pairs). It allows one to introduce fluctuations of the ballistic flux assuming D0 b → D0 b(r, t )+ζ(r, t ), where ζ(r, t ) is the random source. Therefore, the quantity Jb has regular (deterministic) and stochastic contri-butions, i.e., Jb = Jdetb + Jstochb . The regular part, Jdetb , is characterized by the quantity Db = φσr b2 defined through a frequency of atomic jumps φσr , where φ and σr are irradiation flux and replacement cross-section, respectively. The corresponding stochastic part, Jstochb , relates to fluctuations in atomic relocation distances. It is characterized by a dispersion 〈(δR)2〉. Therefore, for the ballistic flux, one can write Jb =− [Db +ζ(r, t )]∇ψ. (2.4) Here, we assume that realizations ζ(r, t ) are independent in time but correlated in space. Statistical prop- erties of the external noise ζ(r, t ) are as follows: 〈ζ(r, t )〉 = 0, 〈ζ(r, t )ζ(r′, t )〉 =σ2DbC (r− r′)δ(t − t ′). Here C (r− r′) = ( p 2πrc)−2 exp {−(r− r′)2/2r 2c } is the spatial correlation function with the correlation radius rc; σ2 is an external noise intensity describing a dispersion of the quantity D0 b . The quantity Db in thecorrelator 〈ζ(r, t )ζ(r′, t )〉means that external fluctuations are possible only at nonzero irradiation flux. In such a case, the right-hand side of equation (2.1) can be written as a sum of thermally sustained diffusion flux J =−M∇δF [ψ]/δψ and the ballistic flux Jb .To proceed, we act onto equation (2.2) by the operator ∇y and act onto equation (2.3) by ∇x . Addingthese two equations, we arrive at one equation for the density field φ ≡ ∇4$2. In our consideration, we take into account that the solute mobilityM can depend on the fieldψ asM = M0M̃(ψ). Next, let us move to dimensionless quantities: ψ′ = p B/Aψ, $′ = √ B/Er 4 0$, α = p E/Aν, r′ = √ A/r 2 0 r, t ′ = (M0 A2/r 2 0 )t , D ′ b = Db/M0 A, b′ = √ EBr 2 0 /A3b, M ′c,g = Mc,gr 2 0 E/M0 A2, Mc = Mg, e = AC /Er 2 0 , m ≡ Mc,gr 2 0 E/M0 A2. Considering a general case, let us put M̃(ψ′) = 1−ψ′2. Using the above renormalizations and dropping the primes, we arrive at a system of two equations ∂tψ=∇· M̃(ψ) [ ∂2 ψψΩ(ψ)∇ψ−∇3ψ ] +αφ+∇·ζ(r, t )∇ψ+∇· √ M(ψ)ξ(r, t ), ∂tφ=−m ( φ+α∇2ψ−e∇2φ ) , (2.5) where ∂2 ψψΩ(ψ) = ∂2 ψψ f (ψ)+ Db M̃(ψ) , f (ψ) =−1−α2 2 ψ2 + ψ4 4 . (2.6) The last term in the equation for ψ represents an internal multiplicative noise. It is characterized by 〈ξ〉 = 0 and 〈ξ(r, t )ξ(r, t )〉 = θδ(r−r′; t−t ′), where θ is the parametermeasuring the internal noise intensity proportional to a bath temperature. We assume that no spatio-temporal correlations between fluctuation sources are possible. It should be noted that time scales of the evolution of composition and dislocation density fields de- scribed by the quantitym ∝ Mc,g/M0 can be different. Atm = 0, we get a system with immobile disloca- tions. Limit m →∞ corresponds to extremely mobile dislocations. It means that m ∈ [0,∞) depends on the properties of the studied material and can be considered as a free parameter of the model. A detailed study of the systems characterized by different values ofm was reported in reference [51]. Next, follow- ing reference [51], we consider the system with mobile dislocations by takingm Ê 1. In the simplest case of extremely mobile dislocations (m À 1), one can adiabatically eliminate the fast field considering the behavior of the slow one. In our further study, we discuss statistical properties of the system according to subordination principle. To make a general analysis, we study the behavior of the system with the above two fields by taking into account the above time scales difference. 2As far as φ is defined in terms of gradients of bx and by , we can monitor the strain energy reduction at segregation of disloca-tions at interfaces. 23003-4 Phase separation processes in binary systems subjected to irradiation 3. Subordination principle and mean-field results 3.1. Stability of the reduced system Let us consider the simplest case when mobile dislocations instantaneously adjust the evolving com- position field. To this end, we put m À 1. This allows us to exclude the fast variable φ by assuming m−1∂tφ ' 0. Hence, using the Fourier representation for the Fourier components φk and ψk , we ob- tain the relation φk ' αk2 1+ek2ψk from the second equation of the system (2.5). In the case ek2 ¿ 1, we can expand the denominator up to the first order and obtain an approximation φk 'αk2(1+ ek2)ψk , or φ ' −α∇2(1− e∇2)ψ. Substituting this expression into the first equation of the system (2.5), we get one equation for the slow mode in the form ∂tψ=∇· M̃(ψ)∇µ̃(ψ)+∇· [ ζ(r, t )∇ψ+ √ M(ψ)ξ(r, t ) ] , (3.1) where the notation ∇µ̃(ψ) ≡ ∇µef(ψ)− α2 M̃(ψ) ∇(1− e∇2)ψ is introduced for convenience; µef(ψ) plays the role of the effective chemical potential [∇µef = ∂2 ψψΩ(ψ)∇ψ−∇3ψ]. The obtained equation (3.1) is the main equation for the reduced system analysis. According to the structure of equation (3.1), one should have in mind that for the field ψ we get conserved dynamics, i.e., ∫ ψ(r, t )dr =ψ0, where ψ0 stands forthe initial concentration difference; ψ0 = const according to the mass conservation law.In statistical analysis we study only observable (averaged) quantities. By averaging equation (3.1) one gets noise correlators which can be calculated using the Novikov’s theorem [52] (the correspond- ing averatging procedures are shown in appendix A). The thermal flux (internal) noise correlator reads: ∇ · 〈 √ M̃ξ〉 = −(θ/2)∇ · 〈∇∂ψM̃〉. Calculations for the external noise correlator give: 〈ζ∇ψ〉 = σ2 [ C (0)∇3〈ψ〉+C ′′(0)∇〈ψ〉], where we have to note that C (r− r′) acquires its maximal value at r = r′, which implies that ∇C (r− r′) ∣∣ r=r′ = 0; ∇2 C (r− r′) ∣∣ r=r′ ≡ C ′′(0) < 0 (see references [33, 34, 53, 54] for de- tails). Therefore, after averaging we get ∂t 〈ψ〉 =∇· 〈M̃∇µ̃〉− θ 2 ∇·〈∇∂ψM̃〉+σ2 [ C ′′(0)∇2〈ψ〉+C (0)∇4〈ψ〉] . (3.2) Let us study the stability of the homogeneous state ψ= 0. As far as we consider the system with con- served dynamics, the corresponding stability analysis can be done studying the dynamics of the structure function S(k, t ) as the Fourier transform of the two point correlation function 〈ψ(r, t )ψ(r′, t )〉. Linearizing the system in the vicinity of the state ψ= 0, in the continuous and thermodynamic limit, we arrive at the dynamical equation for the structure function in the form (see appendix B for details) dS(k, t ) dt =−2k2ω(k)S(k, t )+2θk2 − 2k2 (2π)2 ∫ dqS(q, t )+ 2k2Dbσ 2 (2π)2 ∫ dqC (\|k−q\|)S(q, t ), (3.3) with the dispersion relation ω(k) = εef+βefk2. (3.4) Here, εef is the effective control parameter playing the role of an effective temperature counted from thecritical value and βef is the inhomogeneity parameter defined as εef =α2 −1+θ+Db [ 1+σ2C ′′(0) ] , βef = 1−α2e −Dbσ 2C (0). (3.5) It follows that the ballistic diffusion (its regular component) increases the effective temperature of the system, whereas correlation effects governed by the term C ′′(0) < 0 decrease its value. At the same time, ballistic diffusion is capable of decreasing the interface width between two phases [last term in βef inequation (3.5)]. From equation (3.4), one finds that the critical wave-number that bounds the unstable modes is de- fined as kc = √ 1−α2 −θ−Db [ 1+σ2C ′′(0) ] 1−α2e −Dbσ2C (0) . (3.6) 23003-5 D.O. Kharchenko et al. 0.0 0.1 0.2 0.3 0.4 0.5 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 S k ti m e 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 D b =0.15 σ 2 =0.1 D b =0.8 σ 2 =0.1 D b =0.15 σ 2 =0.8 S k k m (a) (b) Figure 1. (Color online) The structure function for the reduced system: (a) the dynamics of S(k, t ) at Db = 0.15, σ2 = 0.1; (b) the dependence S(k) at a fixed time interval (t = 10). Other parameters are: α = 0.5, θ = 0.1, e = 0.2, rc = 1. The most unstable mode is described by the wave-number km = kc/ p 2. For the actual set of the sys- tem parameters at α < 1, one gets a decreasing dependence kc(α). Therefore, at small α, the dislocation mechanism promotes a decrease in the wave-number of unstable modes. With an increase in α, spatial modulations of the composition field are characterized by long-wave modes. At the same time, spatial correlations of the external noise ζ increase the wave-number of unstable modes due to C ′′(0) < 0. Typical dynamics of the structure function S(k, t ) are shown in figure 1 (a), km relates to the positionof the peak in the dependence S(k) [see figure 1 (b)]. From figure 1 (a) one can see that during the system evolution, the peak of S(k, t ) related to the wave-number km moves toward k = 0 and its height increases. Therefore, the corresponding spatial instability promotes the ordering processes with the formation of domains of phases enriched by atoms A or B. The effect of the system parameters onto S(k) is shown in figure 1 (b). Here, one can find that an increase in the ballistic mixing coefficient Db promotes theformation of the structural disorder characterized by realization of long-wave perturbations and small maximal value of the structure function. The stochastic contribution of the ballistic mixing flux acts in an opposite manner stimulating the ordering processes. At elevated values of external noise intensity σ2, the corresponding spatial structures are characterized by lower domain sizes enriched by the atoms of one sort. This effect is caused by spatial correlations of the external fluctuations. The linear stability analysis is valid only on a short time scale. At large time limit (t →∞), one can use the mean-field approximation based on the analysis of the solution of the Fokker-Planck equation for the probability density of the composition field. 3.2. Mean-field approximation To analytically study the statistical properties of phase separation at t →∞ one needs to analyze a stationary probability density Ps([ψ]). The behavior of the system can be described analytically within the framework of the mean-field approach derived for the systems with conserved dynamics [34, 35, 55– 58]. In the Wiess mean-field approximation, one can use the mean field (molecular field) η ≡ 〈ψ〉 as an order parameter for phase transitions and phase decomposition. In such a case, one uses the transforma- tion procedure, which allows us to introduce the order parameter in d -dimensional space as follows: ∆ψ≡→ 2d(〈ψ〉−ψ). (3.7) The mean-field value η is self-consistently defined according to the definition of the mean 〈ψ〉 through the stationary distribution function Ps. In the mean-field theory, the stationary distribution is a function 23003-6 Phase separation processes in binary systems subjected to irradiation of ψ and η. A procedure to obtain the corresponding distribution as a solution of the corresponding Fokker-Planck equation is shown in appendix C. In order to define the transition and critical points at phase separation, we use the procedure pro- posed in references [55, 56, 59]. According to this approach in a deterministic case with Db = 0, one has a model ∂tψ = ∇ ·M∇δF/δψ, where the restriction ψ0 = ∫ V drψ(r, t ) is taken into account, ψ0 is fixedby the initial conditions. For such a system, the transition point is ΘT(ψ0): at Θ > ΘT(ψ0), the homoge- neous state ψ0 is stable; at Θ <ΘT(ψ0), the system separates into two bulk phases, ψ1 and ψ2, fulfilling 〈ψ〉 = ψ0. The transition from a homogeneous state to two-phase state is critical for ψ0 = 0 only, i.e. ΘT(0) = Θc is the critical point. The corresponding steady state solutions are given as solutions of theequation ∇M∇δF/δψ = 0. If no flux condition is applied, then the bounded solution is δF/δψ = h, where h is a constant effective field (in equilibrium systems h is a chemical potential). In the homoge- neous case, the value h depends on the initial conditions ψ0. Above the transition point, the steady stateis not globally homogeneous. Here, the system separates into two bulk phases with the values ψ1 and ψ2. In the case of the symmetric form of the free energy functional where two phases withψ1 =−ψ2 arerealized, we get h = 0 [55]. Hence, if the field h becomes trivial, then the transition point can be defined. By using this procedure, one finds that in the homogeneous case the mean-field is the same every- where and equals the initial value, i.e. η=ψ0. Hence, solving the self-consistency equation η= 1∫ −1 ψPs(ψ,η,h)dψ (3.8) at the fixed mean-field value, we obtain the constant effective field h. Below the thresholdΘT, the systemdecomposes into two equivalent phases with 〈ψ1〉 =−〈ψ2〉, and h should be the same for these two phases and should be zero. Hence, below the threshold only 〈ψ〉 should be defined as a solution of the self- consistency equation with Ps(ψ,η,0). In the actual case, we are interested in phase decomposition phenomena induced by the irradiation effect. Therefore, we define the transition and critical points only for the parameters relevant to irradi- ation, namely Db , σ2 by fixing ψ0. The corresponding dependencies of the effective field h versus Dband the external noise intensity are shown in figure 2. From figure 2 (a) it is seen that the field h takes nonzero values above the transition point DbT. According to the definition of h as a chemical potential, one can say that at fixed ψ0, the quantity h is compensated and phase separation occurs inside the do- main of the parameters where h = 0. From the dependencies h(Db) it follows that the ordered state with the initial concentration ψ0 can be found only before DbT. It is seen that with an increase in the strengthof the dislocation mechanism described by α, the transition point DbT decreases. Considering the depen-dence h(σ2) [see figure 2 (b)], it follows that with an increase in α, the phase separation processes can be 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.01 0.02 0.03 0.04 h D b σ 2 =0.01 α=0.5 r c =2 σ 2 =0.01 α=0.2 r c =2 σ 2 =0.5 α=0.2 r c =2 D bT 1E-3 0.01 0.1 1 1E-10 1E-8 1E-6 1E-4 0.01 σ 2 T0 σ 2 T2 σ 2 T1 h σ 2 α=0.5 D b =0.155 r c =1 α=0.52 D b =0.155 r c =1 α=0.5 D b =0.1 r c =1 α=0.52 D b =0.155 r c =2 (a) (b) Figure 2. (Color online) Constant field h versus Db and σ2 [panels (a) and (b), respectively] at ψ0 = 0.05 and different sets of the system parameters. 23003-7 D.O. Kharchenko et al. 0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.1 0.2 0.3 0.4 0.5 η D b σ 2 =0.1 α=0.5 e=0.2 θ=0.1 σ 2 =0.1 α=0.5 e=0 θ=0.1 σ 2 =0.5 α=0.5 e=0.2 θ=0.1 σ 2 =0.1 α=0.5 e=0.2 θ=0.12 σ 2 =0.1 α=0.52 e=0.2 θ=0.1 1E-3 0.01 0.1 0.02 0.04 0.06 0.08 0.10 σ c σ c2 η σ 2 D b =0.197 α=0.5 r c =2 e=0.2 θ=0.1 D b =0.197 α=0.5 r c =2 e=0 θ=0.1 D b =0.197 α=0.5 r c =1 e=0.2 θ=0.1 D b =0.155 α=0.52 r c =2 e=0.2 θ=0.1 D b =0.155 α=0.5 r c =2 e=0.2 θ=0.12 σ c1 (a) (b) Figure 3. (Color online) Themean-field value η versus ballistic diffusion coefficientDb and noise intensity σ2 [panels (a) and panel (b), respectively]. realized inside the noise intensity interval (σ2T1,σ2T2). Next, let us discuss the mean-field η behavior varying the system parameters. Here, we solve the self-consistency equation at h = 0. According to figure 3 (a), the mean-field decreases with an increase in the coefficient Db . It behaves critically in the vicinity of the value Db = Dbc, when nontrivial values 〈ψ1〉 = −〈ψ2〉 appear. Here, we arrive at the conclusion that the irradiation leads to homogenization ofthe composition field distribution. By increasing the noise intensity, one finds that phase decomposition is realized at lower values ofDb compared to the case of small σ2. An increase in the intensity of internal fluctuations θ suppresses the phase separation at largeDb . The same effect can be found when the inten-sity of the feedback between dislocation density and composition field increases. This result follows even from the analysis of the deterministic system, where the elastic field changes the critical point position. A more interesting situation is observed by varying the external noise intensity [see figure 3 (b)]. Here, one finds that the external noise leads to the emergence of a disordered state (η= 0) at σ2 >σ2c . In otherwords, external fluctuations of large intensity lead to a statistical disorder. On the other hand, at special choice of the system parameters related to ballistic flux properties, a reentrant behavior of themean-field η is observed. Here, phase decomposition is realized in a window of the noise intensity σ2 ∈ [σ2c1,σ2c2]. This phenomenon is caused by the competition between regular and stochastic (correlated) parts of the ballistic flux. Atσ2 <σ2c1, the most essential contribution is given by the regular componentDb leading tohomogenization of the alloy. Inside the interval σ2 ∈ [σ2c1,σ2c2], the correlation effects dominate and lead to a decrease in the effective temperature of the system. At large σ2, external fluctuations destroy the ordered state. Therefore, the correlated ballistic flux is capable of inducing phase separation processes of initially homogeneous alloys. According to dependencies h(σ2), one can conclude that the dislocation mechanism sustains the above reentrance. The corresponding phase diagram illustrating the formation of ordered and disordered phases is shown in figure 4. It is seen that a reentrant phase separation is realized in a narrow interval for σ2 at elevated Db . At small Db , one gets the standard scenario of phase decomposition, where fluctuationssuppress the ordering processes. Comparing different curves, it follows that an increase in the intensity of internal fluctuations θ shrinks the interval for the system parameters bounding the domain of the ordered phase (cf. dash and dash-dot-dot curves). The domain of reentrant ordering extends with an in- crease in the external noise correlation radius rc (cf. dash and dot lines). At an elevated rc, the size of thedomain for the ordered phase grows at small Db . An increase in α shrinks the domain of the orderedphase and extends the domain of the reentrant decomposition (cf. dot and dash-dot curves). According to the obtained results, it follows that phase separation processes can be controlled by the main system parameters (temperature and elastic properties of the alloy) and statistical properties of ir- radiation effect (regular and stochastic contributions in the ballistic flux). Moreover, correlated stochastic contribution of this flux is capable of inducing reentrant phase separation processes. 23003-8 Phase separation processes in binary systems subjected to irradiation 0.12 0.14 0.16 0.18 0.20 0.22 0.0 0.1 0.2 0.3 0.4 0.5 α=0.5 e=0 r c =1 θ=0.1 α=0.5 e=0.2 r c =1 θ=0.1 α=0.5 e=0.2 r c =2 θ=0.1 α=0.5 e=0.2 r c =1 θ=0.12 α=0.52 e=0.2 r c =2 θ=0.1 σ 2 D b order disorder Figure 4. (Color online) The mean-field phase diagram at different sets of the system parameters. Let us study a strong coupling limit, neglecting the spatial interactions term. Assumingψ= η, one gets the stationary distribution in the form Ps(ψ,η) = δ(ψ−η). To obtain an equation for the effective field h, we integrate equation (C.27) and find h = M(η) [ ∂η f (η)+ Db M(η) η ] − θ 2 ∂ηM(η)−2dDbσ 2(C0 −C1)η. (3.9) At h = 0, one has solutions for two bulk phases η± =±1 2 √ 4−2α2 −2 √ α4 +4(Db +θ)−8dDbσ2(C0 −C1). (3.10) The corresponding transition line can be obtained directly from equation (3.10) at η = ψ0, where ψ0 isthe initial value for the composition field. Critical values for the system parameters can be obtained from the condition η = 0. It is interesting to note that ballistic flux parameters lead to renormalization of the effective temperature: Θ counting off the critical one Θc = 1. Indeed, according to the definition of the free energy density f (ψ) for an unirradiated system, the quantity Θ is reduced to α2. In an irradiated system, Θ is reduced to α2+Db +θ−2dDbσ 2(C0−C1). Hence, the regular component of the ballistic flux increases the effective temperature in the same manner as the internal noise does. On the other hand, the external fluctuations reduce this temperature due to their spatial correlations. From equation (3.10) it follows that an increase in Db , θ, and α causes a decrease in the order parameter. The external noiseis capable of extending the interval for Db where the mean-field takes up nonzero values.It is known that the mean-field results are mostly qualitative. To validate the mean-field results, we will use a simulation procedure. In the next sectionwe discuss the behavior of our system considering the dynamics of both quantitiesψ andφ and numerically illustrate a possibility of reentrant phase separation processes. 4. The effect of dislocation density field dynamics 4.1. Stability analysis Considering the system with two fields ψ and φ, let us start with stability analysis. Averaging the system (2.5) over noises, we get dynamical equations for average fields in the form ∂t 〈ψ〉 =∇· 〈M∇µef〉− θ 2 ∇· 〈∇∂ψM〉+α〈φ〉+σ2 [ C ′′(0)∇2〈ψ〉+C (0)∇4〈ψ〉] , 1 m ∂t 〈φ〉 =−〈φ〉−α∇2〈ψ〉+e∇2〈φ〉 . (4.1) 23003-9 D.O. Kharchenko et al. 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 λ(k)<0 σ 22 22 D b r c =0.5 r c =1.5 r c =3 λ(k)>0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 k c D b σ 2 (a) (b) Figure 5. Stability diagram (a) and critical wave-number dependence on Db and σ2 (b) atm = 1, α= 0.5, rc = 0.5. Next, let us consider the stability of the state (ψ= 0,φ= 0) using Lyapunov’s analysis for fluctuations of both ψ and φ. A linearization of the governing equations in the Fourier space yields( d〈ψ〉 dt d〈φ〉 dt ) = ( k2w(k) α mαk2 −m(1+ek2) )(〈ψ〉 〈φ〉 ) , (4.2) where w(k) =α2 −1+θ+Db [ 1+σ2C ′′(0) ]+ [ 1−Dbσ 2C (0) ] k2. (4.3) The corresponding Lyapunov exponent takes the form 2λ= [ k2w(k)−m ( 1+ek2)]± √[ k2w(k)+m ( 1+ek2 )]2 +4m [ 1+k2w(k)+ ( e +α2 ) k2 ] . (4.4) According to the analysis of the Lyapunov exponent (4.4), we can find critical values for Db and σ2, bounding the domain of unstable modes. The corresponding stability diagram is shown in figure 5 (a). It is seen that spatial instability characterized by λ(k) > 0 is possible only inside the window for the ballistic diffusion coefficient. At the same time, a growth in Db shrinks the interval for σ2 where λ(k) > 0. From the stability diagram, one finds that at smallDb , the external noise can sustain a spatial instability even atlarge intensities of fluctuations. An increase in the correlation radius rc of these fluctuations enlarges theinstability domain. Therefore, strongly correlated external fluctuations are capable of inducing spatial instability at short time scales. As figure 5 (b) shows, the critical wave-number bounding wave-number of unstable modes decreases with Db and σ2. Therefore, at a large ballistic mixing intensity and the intensity of external fluctuations, long-wave spatial instabilities should emerge over the whole system. Let us consider the behavior of the structure function S(k, t ). To obtain a dynamical equation for S(k, t ) in the vicinity of the point (ψ = 0,φ = 0), we exploit the approach previously described by con- sidering the system of two equations (2.5). Moving to the discrete representation and multiplying every equation from the system (2.5) by ψ, we arrive at the system of two equations dS(k, t ) dt =−2k2w(k)S(k, t )+αG(k, t )+2θk2 − 2k2 (2π)2 ∫ dqS(q, t )+ 2k2Dbσ 2 (2π)2 ∫ dqC (\|k−q\|)S(q, t ), dG(k, t ) dt =−2m [( 1+ek2)G(k, t )−αk2S(k, t ) ] , (4.5) whereG(k, t ) ≡ 〈ψk(t )φ−k(t )〉 = 〈ψ−k(t )φk(t )〉, and w(k) is given by equation (4.3). 23003-10 Phase separation processes in binary systems subjected to irradiation 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 S k ti m e 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.8 0.9 k c α D b =0.15 σ 2 =0.1 D b =0.50 σ 2 =0.1 D b =0.15 σ 2 =0.5 S k D b =0.15 α=0.5 D b =0.15 α=0.1 D b =0.50 α=0.5 (a) (b) Figure 6. (Color online) The structure function behavior for the original system: a) the dynamics of S(k, t ) at Db = 0.15, σ2 = 0.1; b) the dependence S(k) at t = 10 and a different set of the system parameters. Other parameters are: θ = 0.1, α= 0.5, e = 0.2, rc = 1.0,m = 1.0. The dynamics of S(k, t ) is shown in figure 6 (a). Comparing graphs for S(k, t ) related to actual and reduced models [cf. figures 6 (a), 1 (a)], one finds that the peak of the structure function is larger in the actual two-component dynamical model. From the obtained dependencies for the structure function S(k) shown in figure 6 (b) it is seen that an increase in the regular component of the ballistic flux essentially decreases the structure function; it shifts the peak position toward small wave-numbers. Considering the effect of the dislocation density mechanism strength, one finds that with an increase in α, the wave- number of unstable modes decreases [see the insertion in figure 6 (b)]. At the same time, the height of the peak of the structure function decreases at a short time scale. Therefore, the dislocation mechanism is capable of delaying the ordering processes. Herein below, we will show that this effect can be observed by the dynamics of the average domain size. 4.2. Numerical results To qualitatively describe the system behavior, we numerically solve the system (2.5). In simulation procedure, the Heun method was used. The system was studied on the lattice with square symmetry of the linear size L = 128` with periodic boundary conditions and the mesh size ` = 0.5; ∆t = 0.005 is the time step. We take 〈ψ(0,0)〉 = 〈φ(0,0)〉 = 0, 〈(δψ(0,0))2〉 = 〈(δφ(0,0))2〉 = 0.01 as initial conditions. The obtained results are statistically independent of different realizations of noise terms ζ, ξ. Typical evolution of both the composition fieldψ and dislocation density φ is shown in figure 7. Here, the regions of high values of both fields ψ and φ are represented by white, whereas the black areas relate to lower values of the corresponding field. The coupling between dislocation density and compo- sition field is well observed. A coordinated motion of dislocations and phase boundaries was previously observed at atomic scale using phase field models (see references [28, 60]) It is seen that the dislocation field takes up large values in the vicinity of the phase boundaries; inside the decomposed phases, the dislocation density is around zero. The corresponding oscillations of ψ near the interfaces indicate that the strain energy is reduced due to the atomic size mismatch [28]. Therefore, misfit dislocations segregate on the boundaries. In figure 8, we plot the oscillating structure of dislocation density field φ corresponding to the distribution of the concentration field. It is seen that in the vicinity of the boundary, φ changes the sign; inside the phases, φ' 0. To make a quantitative analysis of phase separation, let us study the dependencies of dispersions of the fields ψ and φ defined as Jψ ≡ 〈(δψ)2〉 and Jφ ≡ 〈(δφ)2〉, where δψ=ψ−〈ψ〉, δφ= φ−〈φ〉. At 〈ψ〉 = 〈φ〉 = 0, these quantities are reduced to the second statistical moments playing the role of effective order parameters at phase decomposition processes (due to conservation laws forψ and φ). The quantity Jψ(t ) is proportional to the area below the structure function Sk (t ), i.e., Jψ(t ) =∑ k Sk (t ). Therefore, the growth 23003-11 D.O. Kharchenko et al. -1 -0.5 0 0.5 1 -0.6 -0.3 0 0.3 0.6 ! t=10 t=100 t=1000 t=10000 Figure 7. Evolution of both the composition field ψ (top panels) and dislocation field (bottom panels) at Db = 0.15, σ2 = 0.5, θ = 0.1, α= 0.5, e = 0.2, rc = 1.0,m = 1. in Jψ (and the related growth of Jφ) corresponds to the phase decomposition process. Both moments Jψand Jφ grow toward nonzero stationary values. The related stationary values J st{ψ,φ} = J{ψ,φ}(t →∞) can be used to define two phases separated during the the long time evolution of the system. If dislocations are excluded from the description, then one can use only Jψ to monitor the formation of two phases. Inour case, the dynamics of phase decomposition can be described by an additional order parameter Jφmanifesting segregation of dislocations in the vicinity of the interface. From naive consideration, one can expect that at large values of both Jψ and Jφ, one gets two well decomposed phases with an increaseddislocation density at the interface. At small Jψ and Jφ, one gets a mixed state where no large deviationsin the composition field are observed. On the other hand, it means that dislocations are distributed inside the phases. In figure 9, we plot the dependencies of the order parameters at different values of the external noise intensity at other fixed system parameters. It is seen that an increase in the noise intensity σ2 results in small values of both Jψ and Jφ. Atσ2 = 0 (see solid lines in figure 9), the order parameter Jψ grows towardstationary value in a monotonous manner. This means an increase in the area under the corresponding structure function S(k, t ) and the formation of well decomposed phases enriched by atoms of different sorts (see right-hand panel illustrating the distribution of the composition field ψ). The order parameter Jφ initially grows meaning the formation of two separated phases with an increasing dislocation densityin the vicinity of the interface. At the next stage, a decaying dependence of Jφ is observed. This means -1 -0.5 0 0.5 1 -0.3 -0.15 0 0.15 0.3 ! Figure 8. (Color online) Snapshots of both concentration and dislocation fields at t = 10000 shown in 3D- view, illustrating the oscillating behavior of dislocation density field near the boundaries of two phases ψ=+1 and ψ=−1. Other parameters are: Db = 0.15, σ2 = 0.1, θ = 0.01, α= 0.5, e = 0.2, rc = 1.0,m = 10. 23003-12 Phase separation processes in binary systems subjected to irradiation 0.0 0.2 0.4 0.6 0.8 10 100 1000 0.000 0.005 0.010 0.015 2 =0 2 =0.01 2 =0.5 J ! 2 =0.5 2 =0.01 -1-0.500.51 -0.6-0.300.30.6 J " time ! " 2 =0.0 Figure 9. (Color online) The dynamics of both order parameters Jψ and Jφ at different values of the ex- ternal noise intensity σ2, and the corresponding snapshots of both concentration and dislocation density fields at t = 4000. Other parameters are: Db = 0.15, θ = 0.01, α= 0.5, e = 0.2, rc = 2.0,m = 10. an agglomeration of the domains belonging to one phase resulting in annihilations of dislocations with opposite signs. At the late stage, the dislocation density goes to its stationary value together with Jψ. Ata small noise intensity σ2 (see dashed curves in figure 9), one observes the same dynamics of both order parameters, where Jψ and Jφ take up low values. Here, the external noise sustains the formation of anordered state characterized by separated phases. This effect is caused by correlation properties of the external noise. The deterministic part of the ballistic flux acts in the manner opposite to the stochastic contribution. By increasing the noise intensity σ2 (in the domain of a disordered phase according to the mean-field analysis), one gets a transition toward disordered state. Here the order parameter Jψ attainsa very small stationary value (see dotted curves in figure 9). The formation of a disordered state is well accompanied by time independence of the quantity Jφ. It fastly attains a small stationary value and fluc-tuates around it. Here, there are no phases enriched by atoms A or B (see the right-hand panel in figure 9), dislocations are distributed over the whole system. Therefore, the effect of fluctuations characterized by large values of σ2 becomes larger than the correlation effects, leading to the formation of a totally disor- dered state. Let us consider in detail the effect of the dislocation density field onto the dynamics of the phase de- composition. An evolution of both order parameters Jψ and Jφ at different values of the coupling constant α is shown in figure 10 (a). Here, one can see that both Jψ and Jφ increase with α (the order parameter Jφ has the corresponding peak at transition to the coarsening regime). Let us consider the behavior ofthe stationary order parameter J stψ = Jψ(t →∞) [see the insertion in the upper panel in figure 10 (a)]. It rapidly increases at small α and slowly grows with α. From the obtained results it follows that a strong coupling between the composition field and the dislocation density field urges the formation of the or- dered state due to redistribution of dislocations over the whole system, as well as their motion to the in- terfaces. Therefore, phase separation is well sustained by a dislocation field. It is interesting to compare the dynamics of the average domain size at different α. According to discussions provided in references [38, 39] it is known that dislocationmechanism is capable of changing the dynamical exponent z, describ- ing the domain size growth law 〈R〉 ∝ t z . To analyze the dependence 〈R(t )〉, we calculate the averaged value 〈k(t )〉 ∝ 1/〈R(t )〉 according to the standard definition 〈k(t )〉 = ∫ kS(k, t )dk/ ∫ S(k)dk. In the stan- dard theory of phase decomposition, the corresponding Lifshitz-Slyozov approach gives z = 1/3 [61]. The 23003-13 D.O. Kharchenko et al. 0.1 0.2 0.3 0.4 0.5 10 100 1000 1E-6 1E-4 1E-3 0.01 0.1 0.2 0.3 0.4 0.5 0.45 0.46 0.47 0.48 J s t ! J !"#$#% !"#$%# !"#$&# J ' time 100 1000 2 4 6 8 !"#"$ !"#$ !"#% z=0.2 z=0.33 z=0.3 <R> time (a) (b) Figure 10. (Color online) (a) Dynamics of both order parameters Jψ and Jφ at different α and Db = 0.12, σ2 = 0.01. (b) Evolution of the average domain size at different α and Db = 0.1, σ2 = 0.5. Other parame- ters are: θ = 0.01, e = 0.2, rc = 2.0,m = 10. same value for z is observed when phase separation is sustained by vacancy mechanism. If dislocation mechanism of phase decomposition plays the major role, then the dynamical exponent takes up lower values z ' 1/6 [39]. In our case, we can control the strength of the dislocation mechanism varying param- eter α. According to the results in figure 10 (b), one obtains z ' 0.33 at α→ 0, as Lifshitz-Slyozov theory predicts. This result was obtained for the system subjected to a stochastic ballistic flux with another form of the function M̃(ψ) (see reference [34]). It was shown that an increase in the external noise intensity σ2 results in disordering processes. Comparing the curves related to α= 0.1 and α= 0.5, it follows that the dislocation mechanism delays the dynamics of the domain sizes growth. In our case, at α= 0.5, we get a lower value for the dynamical exponent. Finally, let us consider the dynamics of the average dislocation density 〈φH 〉 in the vicinity of the in- terfaces and the coherence (interface) width 〈Lφ〉, where this density decreases toward zero value insidethe separated phases. We calculate the quantity 〈φH 〉 as the mean height of φ(r) profile averaged over the whole system. The coherence width 〈Lφ〉 is calculated as the width of the interface on the half-heightof φ(r) profile averaged over the system. In figure 11 (a), solid and dashed lines denote one-dimensional profiles for the composition and dislocation density fields, respectively. The dynamics of both 〈φH 〉 and 〈Lφ〉 is shown in figure 11 (b). Comparing the data related to different sets of Db and σ2, one finds that the dislocation density attains a stationary value during the decomposition process. Considering the dy- namics of 〈φH 〉 at different external noise intensities, it follows that the growth in σ2 increases the values of the dislocation density. Therefore, the external noise is capable of inducing a phase decomposition ac- companied by segregation of dislocations at interfaces. On the other hand, with an increase in Db , thequantity 〈φH 〉 takes up lower values. Here, dislocations are distributed over the whole system due to homogenization of the system produced by a regular component of the ballistic flux. The competition be- tween regular and stochastic components of the ballistic flux can be observed by studying the dynamics of the averaged interface width. From the bottom panel in figure 11 (b), one finds that during the sys- tem evolution, the quantity 〈Lφ〉 attains a maximum. This maximum relates to the stage of the domainsgrowth. A coalescence regime (large domains absorb small ones) is accompanied here by an increase 23003-14 Phase separation processes in binary systems subjected to irradiation -0.2 -0.1 0.0 0.1 0.2 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.5 0.0 0.5 1.0 r/L ! <! H > 100 1000 10000 0.025 0.030 0.035 0.0 0.1 0.2 0.3 0.4 0.5 < L > /L time Db=0,12 ! 2 =0,10 Db=0,12 ! 2 =0,15 Db=0,20 ! 2 =0,15 < H > (a) (b) Figure 11. (Color online) (a) Illustration of typical one-dimensional profiles for solute concentration ψ and the field φ taken at the time t = 400 (Db = 0.12, σ2 = 0.1). (b) Evolution of both the average maxi- mum dislocation density field 〈φH 〉 and a coherence length 〈Lφ〉 at different values of Db and σ2. Other parameters are: α= 0.5, e = 0.2, rc = 1,m = 1, θ = 0.1. in the interface width. A decrease in 〈Lφ〉 corresponds to a coarsening regime. At large time intervals, 〈Lφ〉 attains a stationary value. Comparing the corresponding stationary values at differentDb , it followsthat a regular component of the ballistic flux produces a disordering accompanied by extension of the interface width (cf. the curves marked by circles and triangles). Comparing the curves with different σ2, one finds that the noise is capable of playing a constructive role leading to a decrease in the interface width. Localization of dislocations at interfaces can be stimulated by a correlation effect of the external fluctuations [see the curve with triangles in the upper panel in figure 11 (b)]. It should be noted that the above effect is possible only inside the domain of the ordered phase formation. Therefore, the mentioned above constructive role of external fluctuations is caused by their spatial correlations. 5. Conclusions We have studied phase separation processes driven by the dislocation evolution mechanism in a bi- nary system subjected to sustained irradiation. We describe the irradiation effect by introducing a ballis- tic flux having stochastic properties. We assumed that these fluctuations are spatially correlated. By taking into account different time scales of evolution of both composition and dislocation density fields, we have initially considered a reduced model using the adiabatic elimination procedure. In this case, the dynamics of the composition field playing the role of a slow mode is studied. By making use of the linear stability analysis, we have shown the constructive role of ballistic flux fluctuations. These fluc- tuations induce spatial instability at a short time scale. At a large time scale, we have used the mean-field approach allowing us to describe the properties of phase separation. Corresponding mean-field phase di- agrams illustrating the possibility of phase decomposition are calculated. It was found that ballistic flux components reduced to the intensity of atomic mixing (ballistic diffusion coefficient) and the intensity of the corresponding fluctuations are capable of controling reentrant phase separation processes. It was shown that a reentrant character of phase separation is governed by spatial correlations of the external noise and mobile dislocations. Considering the dynamics of both studied fields by means of computer simulations, we have found that the formation of domains enriched by atoms of different sort is accompanied by an increase of the dislocation density field in the vicinity of the interface. Inside such domains, dislocation density takes up zero values and all dislocations characterized by different signs in two phases segregate on interfaces. The average length of the interface decreases at the formation of the ordered state. In the disordered state, all dislocations are distributed over the whole system. It was shown that spatially correlated ex- ternal fluctuations act in the manner opposite to the regular component of the ballistic flux and induce the ordered state formation. Studying the effect of the dislocation density field onto phase decomposition 23003-15 D.O. Kharchenko et al. we have considered the dynamics of the average domain size at different strengths of the dislocation mechanisms. It was shown that the universal dynamics of the average domain size delays due to a re- distribution of dislocations. We have found that at small contribution of the dislocation density field, the corresponding universal dynamics is described by the standard Lifshitz-Slyozov law with dynamical ex- ponent z = 0.33. With an increase in the dislocation mechanism strength, this exponent takes up lower values and the domains of new phases are characterized by smaller linear sizes. Acknowledgements Fruitful discussions with Dr. V.O. Kharchenko are gratefully acknowledged. Appendix A Let us represent the system on a regular d -dimension lattice. Within the framework of the standard formalism of a discrete representation, the system can be divided onto N d cells of the linear size L = `N , where ` is the a mesh size. Then, the partial differential equation (3.1) is reduced to a set of usual differential equations written for every cell i on a grid in the form dψi dt =∇Li j M̃ j∇Rj l µ̃l +∇Li j √ M̃ jξ j (t )+∇Li j∇Rj kψkζk (t ), (A.1) where the index i labels cells, i = 1, . . . , N d ; the discrete left-hand and right-hand operators are: ∇Li j = 1 ` (δi , j −δi−1, j ), ∇Ri j = 1 ` (δi+1, j −δi , j ), ∇Li j =−∇Rj i , ∇Li j∇Rj l =∆i l = 1 `2 (δi ,l+1 −2δi ,l +δi ,l−1). (A.2) Discrete correlators of stochastic sources are of the form: 〈ξi (t )ξ j (t )〉 = `−2θδi jδ(t − t ′), 〈ζi (t )ζ j (t )〉 = Dbσ 2Ci− jδ(t − t ′), (A.3) where Ci− j is the discrete representation of the spatial correlation function C (r) which in the limit of zero correlation length becomes δi j /`d . For the two-dimensional problem considered below, the quantity C\|i− j \| can be computed as a discrete version of the Fourier transform of C (r− r′) written in the form [53] C (k) = exp {−(r 2c /2)[sin2(kx /2)+ sin2(ky /2)] }. Noise correlators can be calculated according to the recipes shown in references [34, 53, 54, 58]. Next, we consider the case d = 2. To calculate the internal noise correlator 〈pMξ〉, we use the Novikov theorem which can be written in the form 〈√ M̃iξi (t ) 〉 =∑ j t∫ 0 〈 ξi (t )ξ j (t ′) 〉〈 δ √ M̃i δξ j (t ′) 〉 dt ′ = θ `2 t∫ 0 〈 δ √ M̃i δξi (t ′) ∣∣∣∣∣ t=t ′ 〉 dt ′. (A.4) Using a relation δ √ M̃i δξi (t ′) ∣∣∣∣∣ t=t ′ = d √ M̃i dψi (t ) δψi (t ) δξi (t ′) ∣∣∣∣∣ t=t ′ and a formal solution of the Langevin equation (A.1), one can write δψi (t ) δξi (t ′) ∣∣∣∣ t=t ′ =∇Li j √ M̃ j . Substituting it into equation (A.4), we get 〈√ M̃iξi (t ) 〉 = θ `2 ∇Li j 〈 d √ M̃ j dψ j √ M̃ j 〉 = θ 2`2 ∇Li j 〈 dM̃ j dψ j 〉 . (A.5) 23003-16 Phase separation processes in binary systems subjected to irradiation Using the relation between left-hand and right-hand discrete gradient operators and moving to the con- tinuum limit, we get ∇· 〈√ M̃ξ 〉 =−θ 2 ∇·〈∇∂ψM̃ 〉 . (A.6) To calculate the external noise correlator 〈ψζ〉, we again use the Novikov theorem written as follows: 〈ψ j ζ j (t ′)〉 =∑ k t∫ 0 〈ζ j (t )ζk (t ′)〉 〈 δψ j (t ) δζ j (t ′) 〉 dt ′ = Dbσ 2 ∑ k C j−k 〈 δψ j (t ) δζk (t ′) ∣∣∣∣ t=t ′ 〉 . (A.7) According to the formal solution of the Langevin equation, the corresponding derivative with respect to ζ takes the form 〈δψ j /δζk \|t=t ′〉 = ∆ j k 〈ψk〉. Next, substituting it into equation (A.7) and using a discreterepresentation of the Laplacian, we find the sum over the index k allowing us to write 〈ψ j ζ j 〉 = Dbσ 2 `2 [ C1〈ψ j+1〉+C−1〈ψ j−1〉−2C0〈ψ j 〉 ] . (A.8) Acting by Laplacian operator onto this construction, we finally obtain in continuum limit ∆〈ψζ〉 = Dbσ 2 [ C (0)∇4〈ψ〉+C ′′(0)∆〈ψ〉] . (A.9) Appendix B To obtain a dynamical equation for the structure function, we need to construct a dynamical equa- tion for the two-point correlation function 〈ψiψ j 〉. In our computation procedure, we use the proceduredescribed in reference [62]. Multiplying the linearized Langevin equation (A.1) written for ψ j onto ψiand doing the same procedure for the linearized Langevin equation written for ψi , we add these twoequations. This procedure allows us to obtain a dynamical equation for 〈ψiψ j 〉 written in the form d〈ψiψ j 〉 dt = ∆i k [( Db −1+α2)〈ψ jψk〉− ( 1−α2e ) ∆kl 〈ψlψ j 〉+〈ψ jψkζk〉 ] + 〈ψi∇Lj kξk〉−2〈ψi∇Lj kψkξk〉 + ∆ j k [( Db −1+α2)〈ψiψk〉− ( 1−α2e ) ∆kl 〈ψlψi 〉+〈ψiψkζk〉 ] + 〈ψ j∇Li kξk〉−2〈ψ j∇Li kψkξk〉, (B.10) where the sum runs over the repeating indexes. Using the Novikov theorem with recipe shown in Ap- pendix A, one can calculate the following correlators: 〈ψi∇Lj kξk〉 =− θ `2∆i j , 〈ψi∇Lj kψkξk〉 =− θ 2`2 ( ∆ j k〈ψkψ j 〉+∆i k〈ψkψ j 〉 ) , 〈ψ jψkζk〉 = Dbσ 2 ( ∆kl Ck−l 〈ψlψ j 〉+∆ j l Ck−l 〈ψlψk〉 ) . (B.11) Introducing the structure function S(k, t ) ≡ 〈ψk(t )ψ−k(t )〉 in the discrete space Sν(t ) = (N`)−2〈ψν(t )ψ−ν(t )〉 with Fourier components ψν(t ) = `2 ∑ m e−irm kνψm(t ), ψm(t ) = (N`)−2 ∑ ν eirm kνψν(t ) one can define the derivative dSν dt = (`/N )2eikν(r j −rm ) d〈ψmψ j 〉 dt . In the following computations, one needs to exploit definitions:( ` N )2 eikν(r j −ri )∆i s〈ψsψ j 〉 = D̂νSν(t ), (B.12) 23003-17 D.O. Kharchenko et al. ∆i s〈ψsψ j 〉 = (N`)−4eikνr j eikµri D̂µ〈ψµψν〉, (B.13) 2 ( ` N )2 eikν(r j −ri )C\|i− j \|〈ψiψ j 〉 = (N`)−2 ∑ µ CµSµ−ν(t ), (B.14) where D̂ν = `−2 (∑ \|i \|=1 cos(kνri )−1 ) can be understood as the Fourier transform of the discrete Lapla- cian. Using the above definitions and equation (B.10) with correlators (B.11), we arrive at the equation for the quantity Sν(t ): dSν(t ) dt = 2D̂ν [ α2 −1+θ+Db +Dbσ 2∆0mCm − D̂ν ( 1−Dbσ 2C1 )] Sν(t ) − 2θD̂ν+2D̂ν(N`)−2 ∑ µ Sµ(t )−2Dbσ 2(N`)−2D̂ν ∑ µ Cν−µSµ(t ). (B.15) In continuous and thermodynamical limit (N →∞, `→ 0), we obtain a dynamical equation for the struc- ture function in the form dS(k, t ) dt = −2k2 { α2 −1+θ+Db(1+σ2C ′′(0))+k2 [ 1−α2e −Dbσ 2C (0) ]} S(k, t ) + 2θk2 − 2k2 (2π)2 ∫ dqS(q, t )+ 2k2Dbσ 2 (2π)2 ∫ dqC (\|k−q\|)S(q, t ). (B.16) Appendix C To obtain the probability density distribution P ([ψ], t ) in d -dimensional space, let us start with defi- nitions: P ([ψ], t ) = 〈 N d∏ i=1 ρi (t ) 〉 ≡ 〈 ρ(t ) 〉 , ρi (t ) = δ(ψi (t )−ψi )IC , (C.17) where . . .IC and 〈. . .〉 are the averages over initial conditions and fluctuations, respectively. To obtain the corresponding Fokker-Planck equation, we use a standard technique and exploit the stochastic Liouville equation for the distribution ρ(t ) in the form ∂tρ =− ∂ ∂ψi (ψ̇iρ). (C.18) Inserting the expression for the time derivative from equation (A.1) and averaging over the noise, we get ∂t P =− ∂ ∂ψi ( ∇Li j M j∇Rj l µ̃l ) P − ∂ ∂ψi ( 〈∇Li j √ M jξ j (t )ρ〉+〈∆i jψ j ζ j (t )ρ〉 ) . (C.19) Correlators in the second term can be calculated by means of the Novikov theorem, that at ` = 1 gives [52] 〈 ∇Li j √ M j (t )ξ j (t )ρ 〉 = θ t∫ 0 dt ′δ j kδ(t − t ′) 〈 δ∇Li j √ M j (t )ρ δξk (t ′) 〉 , 〈∆i jψ j ζ j (t )ρ〉 = Dbσ 2 t∫ 0 dt ′C\| j−k\|δ(t − t ′) 〈 δ∆i jψ jρ δζk (t ′) 〉 . (C.20) Introducing notations gi j = {(∇L)i j √ M j ,∆i jψ j }, λ= {ξ,ζ}, for the last multiplier, one has δgi jρ(t ) δλk (t ′) =−∑ l gi j ∂ ∂ψk δψl (t ) δλk (t ′) ∣∣∣ t=t ′ ρ. (C.21) Using a formal solution of the Langevin equation, the response functions take up the form δψl (t ) δξk (t ′) ∣∣∣∣ t=t ′ =∇Llk √ Mk , δψl (t ) δζk (t ′) ∣∣∣∣ t=t ′ =∆lkψk . (C.22) 23003-18 Phase separation processes in binary systems subjected to irradiation After some algebra, we obtain the Fokker-Planck equation for the total distribution P in the discrete space ∂t P =− ∂ ∂ψi ( ∇Li j M j∇Rj l µ̃l ) P −θ ∂ ∂ψi ∇Li j √ M j ∂ ∂ψ j ∇Rj i √ Mi P +Dbσ 2 ∂ ∂ψi ∆i jψ j ∂ ∂ψl C\| j−k\|∆klψl P , (C.23) where the relations between left-hand and right-hand gradient operators are used. To proceed, let us obtain an evolution equation for the single-site probability distribution Pi (t ) = ∫ [ ∏ k,i dψk ] P . After integration one has ∂Pi (t ) ∂t = ∂ ∂ψi ∆i j 〈M j 〉Pi (t ), (C.24) where M j = M j [ − ∂ f ∂ψ j − α2 +Db M j ψ j + 1 2d ( 1− α2e M j ) ∆ j rψr ] − θ √ M j ∂ ∂ψ j √ M j +Dbσ 2ψ j ∂ ∂ψn ∆mnC\| j−n\|ψn . (C.25) In the stationary case with no flux, one arrives at the equation ∆i j 〈M j 〉Ps (ψi ) = 0, (C.26) where Ps is a stationary distribution function. By taking i = j , dropping the subscripts, we arrive at the mean-field stationary equation [55] −hPs = { M [ −∂ψ f − α2 +Db M ψ+ ( 1− α2e M ) (η−ψ) ] − θ 2 ∂ψM +2dDbσ 2ψ [ C1η ∂ ∂ψ −C0 ∂ ∂ψ ψ ]} Ps . (C.27) Here we have used the mean-field approximation, allowing us to write ∆i jψ j ≡ ( ∑ nn(i ) ψnn(i ) −2dψi ) → 2d(〈ψ〉−ψ), η≡ 〈ψ〉, (C.28) where nn(i ) denotes the nearest neighbors of a given site. The mean-field value η and the integration constant h should be defined self-consistently. Equation (C.27) has the solution in the form Ps(ψ,η,h) = N exp  ψ∫ dψ′Θ(ψ′,η,h) Ξ(ψ′;η)  , (C.29) where Θ(ψ,η,h) =−M∂ψ f − ( α2 +Db ) ψ+ ( M −α2e ) (η−ψ)− θ 2 ∂ψM −2dDbσ 2C0ψ+h, Ξ(ψ;η) = θM +2dDbσ 2ψ(C0ψ−C1η) (C.30) the normalization constant is a function of the mean-field η and the constant h: N ≡ N (η,h) =  1∫ −1 dψexp  ψ∫ dψ′Θ(ψ′,η,h) Ξ(ψ′;η) −1 . (C.31) 23003-19 D.O. Kharchenko et al. References 1. Schulson E.M., J. Nucl. Mater., 1979, 83, 239; doi:10.1016/0022-3115(79)90610-X. 2. 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(Eds.), Springer, Berlin, 2000, pp. 247-256; doi:10.1007/3-540-45396-2_23. 23003-21 http://dx.doi.org/10.1103/RevModPhys.49.435 http://dx.doi.org/10.1016/S0022-3115(98)00712-0 http://dx.doi.org/10.1063/1.1922578 http://dx.doi.org/10.1103/PhysRevE.60.3597 http://dx.doi.org/10.1209/epl/i1998-00217-9 http://dx.doi.org/10.1103/PhysRevLett.87.020601 http://dx.doi.org/10.1016/j.physa.2008.05.041 http://dx.doi.org/10.1103/PhysRevB.75.064107 http://dx.doi.org/10.1016/0022-3697(61)90054-3 http://dx.doi.org/10.1007/3-540-45396-2_23 D.O. Kharchenko et al. Дослiдження процесiв фазового розшарування за наявностi дислокацiй в бiнарних системах, пiдданих опромiненню Д.О. Харченко, О.М.Щокотова, А.I. Баштова. I.О. Лисенко Iнститут прикладної фiзики НАН України, вул. Петропавлiвська 58, 40000 Суми, Україна Проведено дослiдження процесiв фазового розшарування за дислокацiйним механiзмом в бiнарних си- стемах, пiдданих дiї опромiнення. Опромiнення описується атермiчним перемiшуванням атомiв, за ра- хунок уведення балiстичного потоку, що має просторово-скорельовану стохастичну складову. При ви- вченнi динамiки росту доменiв показано,що дислокацiйниймеханiзм уповiльнює процес упорядкування. Встановлено, що просторовi кореляцiї шуму балiстичного потоку стимулюють сегрегацiю ядер дислока- цiй в околi мiжфазних границь, ефективно зменшуючи ширину мiжфазного шару. Розглянуто конкурен- цiю мiж регулярною та стохастичною компонентами балiстичного потоку. Ключовi слова: фазове розшарування, опромiнення,шум 23003-22 Introduction Model Subordination principle and mean-field results Stability of the reduced system Mean-field approximation The effect of dislocation density field dynamics Stability analysis Numerical results Conclusions

A study of phase separation processes in presence of dislocations in binary systems subjected to irradiation

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