Monte-carlo modeling for unstable particle ensembles with thermal fluctuations
In this presentation we study binary alloy systems subjected to particle irradiation and thermal noise influence. We discuss two competing mechanisms of the system evolution: dynamics driven by irradiation and stochastic influences bringing the system toward thermal equilibrium. Using a phase-fie...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2009 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2009
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Monte-carlo modeling for unstable particle ensembles with thermal fluctuations / D.O. Kharchenko // Вопросы атомной науки и техники. — 2009. — № 4. — С. 52-60. — Бібліогр.: 18 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859790008776392704 |
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| author | Kharchenko, D.O. |
| author_facet | Kharchenko, D.O. |
| citation_txt | Monte-carlo modeling for unstable particle ensembles with thermal fluctuations / D.O. Kharchenko // Вопросы атомной науки и техники. — 2009. — № 4. — С. 52-60. — Бібліогр.: 18 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | In this presentation we study binary alloy systems subjected to particle irradiation and
thermal noise influence. We discuss two competing mechanisms of the system evolution:
dynamics driven by irradiation and stochastic influences bringing the system toward thermal
equilibrium. Using a phase-field formalism and Monte-Carlo modeling we consider selforganization
processes in binary alloy systems with field/concentration dependent
mobility/diffusion coefficient. Generalization of the phase separation scenario and patterning
in such systems are presented.
Досліджуються системи бінарних сплавів, підданих опроміненню часток і впливу
теплових шумів. Обговорюються два конкуруючих механізми еволюції системи: динаміка,
обумовлена опроміненням і стохастичні впливи, що переводять систему до теплової
рівноваги. З використанням формалізму теорії фазового поля і статистичного
моделювання розглядаються процеси самоорганізації у системах бінарних сплавів з
рухомістю, яка залежить від поля концентрації. Запропоноване узагальнення сценаріїв
фазового розшарування і структуроутворення у таких системах.
Исследуются системы бинарных сплавов, подверженных облучению частиц и
воздействию тепловых шумов. Обсуждаются два конкурирующих механизма эволюции
системы: динамика, вызванная облучением, и стохастические воздействия, переводящие
систему к тепловому равновесию. С использованием формализма теории фазового поля и
статистического моделирования рассматриваются процессы самоорганизации в системах
бинарных сплавов с подвижностью, зависимой от поля концентрации. Предложено
обобщение сценария фазового расслоения и структурообразования в таких системах.
|
| first_indexed | 2025-12-02T11:26:43Z |
| format | Article |
| fulltext |
MONTE-CARLO MODELING FOR UNSTABLE PARTICLE
ENSEMBLES WITH THERMAL FLUCTUATIONS
D.O. Kharchenko
Institute of Applied Physics, National Academy of Science of Ukraine,
Sumy, Ukraine
E-mail: d.kharchenko@ipfcentr.sumy.ua
In this presentation we study binary alloy systems subjected to particle irradiation and
thermal noise influence. We discuss two competing mechanisms of the system evolution:
dynamics driven by irradiation and stochastic influences bringing the system toward thermal
equilibrium. Using a phase-field formalism and Monte-Carlo modeling we consider self-
organization processes in binary alloy systems with field/concentration dependent
mobility/diffusion coefficient. Generalization of the phase separation scenario and patterning
in such systems are presented.
1. INTRODUCTION
It is well known that the sustained
irradiation of materials produces a disorder
which can be structural (isolated point defects,
traps, dislocation loops, clusters of vacancies
and interstitials). Such point defects in alloys
are considered as unstable particles which can
exist for some fixed lifetime and can lead to
self-organizational processes: if its density is
large, then the interaction processes start to
play a crucial role leading to a new phase
appearance caused by collective effects. If the
above disorder is continuous then a net flux of
defects can be induced that results in driving
the material into stationary non-equilibrium
states. For irradiation processes that occur at
finite temperature a thermally activated
dynamics leads to annealing such non-
equilibrium disorder. Therefore, the question
which is of fundamental interest is to study
competing dynamics caused by regular and
irregular forces leading to microstructural
change in such materials. It is known that
dynamical systems often reach some steady
state (quasi-steady state in the case of alloys
under irradiation), and one appealing approach
is to develop an effective theoretical and
thermodynamical framework to address the
stability of these steady states.
Usually, to investigate such non-
equilibrium phenomena one can exploit multi-
scale modeling: molecular dynamics (MD)
methods or kinetic Metropolis procedures that
allow one to find statistical information that
can be transferred to the next hierarchical level
of calculation, namely kinetic Monte-Carlo
(KMC). The principle idea lies in the fact that
on atomic length (~10-10 m) and time scales
(~10-13 s) the system dynamics can be captured
by MD. Unfortunately, the number of atoms in
MD simulations is still small (~109 atoms), the
time interval for predictions of MD
simulations is too short (~10-8 s) to describe
microstructure transformations on a atomic
length scales and mesoscopic time scales [1].
The problem can be partially solved by KMC.
The principle idea of the original KMC can be
stated as follows: given a system and its phase
space, a distribution of configuration at initial
(starting) time, and a set of transition rates
between configurations, one can generate
temporal trajectories of the system in its phase
space. Such trajectories should be produced by
correct statistical weight. Both the average
evolution of the system and its fluctuations
around this average can be obtained from a
large set of temporal trajectories. From the
mathematical viewpoint the problem is to
solve the numerically master equation of the
Chapmen-Kolmogorov’s kind. This method
has some limitations because one needs to
calculate all possible macroscopic transition
rates in one time step.
Progress that alleviates this limitation has
been made recently by the introduction of the
phase field theory adapted to description of
crystals [2]. Using this method one can
consider a local atomic density field in which
atomic vibrations have been integrated out up
52
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2009. №4-1.
Серия: Физика радиационных повреждений и радиационное материаловедение (94), с. 52-60.
to diffusive time scales. Therefore, dissipative
dynamics is governed by the temporal
evolution such local density field.
Deterministic dynamics can be generalized
introducing stochastic sources into the
corresponding evolution equations. As usual,
fluctuating sources represent statistical
information about the influence of microscopic
processes onto mesoscopic ones. Considering
alloys under irradiation, such fluctuating
sources can be divided by its physical
character: thermal fluctuations obeying the
fluctuation-dissipation relation and external
noise addressed to local dynamics driven by
irradiation caused by energetic particles, some
transformation of defects, chemical reactions,
etc. Such KMC simulations allow one to
investigate the role of fluctuating sources on
microstructural transformations of alloys under
irradiation, and to predict the system behavior
under fluctuating sources influence. Despite
some useful advantages of this method, one
needs to highlight that due to the diffusive
time scales considered this method does not
contain a mechanism to simulate elastic
interactions to study deformation properties of
alloys.
Returning back to the main question noted
above, one needs to say that in the case of
alloys under particle irradiation, in order to
describe the non-equilibrium steady states and
the corresponding system behavior one needs
to construct an associated non-equilibrium
potential/functional to find a stationary
distribution. If such a functional can be found
exactly, then one can describe local physical
fields such as concentration deviation, local
magnetization, polarization in real space, and
thus investigate their spatial profiles.
Considering the related problem of irradiation
of alloys, this can be shown to be a
consequence of patterning which in turn is a
result of the competing dynamics between two
of the above noted dynamics [3]. Formation of
patterns of anti-structure defects induced by
irradiation, was studied in [4]. It was shown
that under irradiation stable and unstable
configurations of anti-structure defects can be
observed. The authors explain that at a fixed
irradiation dose an increase in the temperature
leads to a transition from unstable into stable
ordered configurations. Some attempts to
study the influence of fluctuation terms on
microstructure transformations were recently
reported [5,6]. It was shown that in binary
alloys under irradiation a macroscopic phase
separation manifested as a patterning can be
observed (see for example [7-9]). In these
works the authors studied systems when the
dynamical governing equation is identical to
that describing a binary alloy undergoing a
quasi-chemical reaction A↔B with phase
separation processes. The term related to such
a quasi-chemical reaction (local force) was
assumed to be linear (Debye relaxation
processes) and to describe ballistic mixing or
birth-death processes for unstable particle
ensembles. It was shown that by controlling
the irradiation intensity one arrives at different
regimes of the system’s behavior: patterning,
macroscopic phase separation, and solid
solution.
In this paper we discuss a similar model
reduced to the quasichemical reactions in
alloys A↔B that occur with the help of
additional product (Schlögl model for
fluctuations in moving fronts between two
phases: A+2X↔3X, X↔B). Such a model can
be obtained directly from the dynamical
approach if the possible stationary states are
well known. It is possible then to describe
clustering and dissociation effects caused by
irradiation. In addition we introduce into the
model a flux of a local atomic density and a
fluctuating source obeying a fluctuation-
dissipation relation. Moreover, we assume that
mobility, related to the diffusion coefficient, is
a field-dependent function that leads to the fact
that our fluctuating force is also field-
dependent function. We will show that under
such noise influence the system can undergo
phase separation, patterning with different
kinds of structures: bubbles/porous, spinodal
decomposition, and strip patterns with
dislocations. Our analytical investigation is
compared with computer simulations and
verified with well known results of phase
modeling of alloys under irradiation.
2. GENERALIZED MODEL
We study the problem using a Cahn-
Hillard-type description of fronts, allowing us
to simulate the walls of the labyrinthine
patterns. The equation describing the temporal
evolution of the concentration field x=x(r,t) is
53
composed of two terms, one for mixing caused
by irradiation (relocation of atoms), and
another one for thermal diffusion. Formally,
such a reaction-diffusion model can be
described by the generalized continuity
equation of the form
Jxfxt ⋅∇−=∂ )( , (1)
where f(x) stands for the local dynamics, and J
is the flux for transport phenomena.
Considering the non-Fickian diffusion, we
exploit a gradient of interaction potential U(r)
( , where the
spherically symmetric interaction potential
u(r) between particles separated by a distance
|r| is introduced). The corresponding force
given by the gradient of U(r) governs the
transport phenomena. In the case of the small
interaction radius compared to the diffusion
length the concentration field x will not vary
significantly within the interaction radius. It
allows us to approximate the integral by
κx+βΔx, where
')'()'()( rrrrr dxuU ∫ −−=
;
2
1)(||
2
1
;0)(;)(
22
crdu
dudu
κβ
κ
≈=
==
∫
∫∫
rrr
rrrrr
rc is a correlation radius. Therefore, the
obtained flux allows one to describe phase
separation processes with mutual (lateral)
interactions; the combined model with local
dynamics can be used to consider the spatial
patterns induced by the fluctuations of the
bath.
Generally, the local dynamics is defined by
a force assumed to be of the form
, where the local potential is
assumed as .
Here ε and μ are control parameters related to
rates of quasi-chemical reactions in the
system. For the diffusion flux one use the
definition:
dxxdVxf /)()( =
4/3/2/)( 432 xxxxV ++−= με
xFxDJ δδ /)( ∇−= , where D(x) is a
field-dependent diffusion coefficient, and the
free energy functional is of the from
, related to the
lateral interactions between particles. For the
diffusion coefficient we will use an
approximate formula describing a bell shaped
form of D versus the atomic density field, i.e.
. Such the approximation is
widely used in the study of phase separation
dynamics in a large class of physical systems.
The parameter α is usually reduced to the ratio
between bulk and surface diffusion
coefficients (α≈1-D
∫ ∇+−= ]2/)(2/[ 22 xxdrF βκ
12)1()( −+= xxD α
b/Ds).
Considering the system under real
conditions, one needs to introduce fluctuating
source related to the problem under
consideration. Formally, such fluctuations can
be included in an ad hoc form. Using
variational principles one can rewrite Eq.(1) in
the form , where Λ is a
Lyapunov’s functional related to the right hand
side of the Eq. (1):
xxDxt δδ /)]([ 1 Λ−=∂ −
.
Next we introduce Gaussian fluctuations ζ into
such an equation in order to satisfy the
fluctuation-dissipation relation
∫ ∇+−=Λ )]/)(()()()([ xFxDxDxDxfxdr δδδδ
)'()'()]([)',';(),;( 21 ttrrxDtrxtrx −−= − δδσζζ ,
σ2 is the noise intensity reduced to the
temperature of the effective bath; 0),;( =trxζ .
As a result we arrive at a stochastic evolution
equation for the mass density field in the form
),,(
)(
1 trx
xxDt
x ζ
δ
δ
+
Λ
−=
∂
∂ (2)
treated in the Stratonovich sense. This
equation can be used to provide KMC
simulations of dynamical regimes of the
system. Statistical properties of the system
states can be found from the stationary picture
that is described by the corresponding
stationary distribution, following from the
stationary solution of the Fokker-Planck
equation. Performing the standard calculations
one arrives at the distribution functional in the
form
.)(ln
2
][][
),/][exp(][
2
2
∫−Λ=
−∝
xDdrxxU
xUxP
ef
ef
σ
σ
(3)
It follows, that the stationary distribution
functional P[x] or the effective functional
Uef[x] are obtained exactly: the form of initial
functional Λ[x] is supposed to be known, the
second term in Uef[x] can be calculated if
needed. Let us note, if we assume that Λ[x]
plays a role of an effective free energy
functional, then rewriting the integral in
Eq. (4) as ∫ −= 1)](ln[
2
1 xDdrSef , the expression
54
for Uef[x] can be transformed into the
thermodynamic relation between free energy,
internal energy and entropy functionals:
Uef[x]=Λ[x]+Sef[x]. Therefore, according to
such a relation the noise intensity σ2 reduces to
an effective temperature of the bath, whereas
Sef[x] plays the role of an effective entropy.
Such a situation is well known in stochastic
systems theory. It appears when the
multiplicative fluctuations corresponded to the
internal noise. The later one results in the
entropy change that yields entropy driven
phase transitions [10-13]. In this paper we will
not discuss the above phase transitions, but we
will consider the ability of the noise to sustain
or induce formation of spatial structures.
3. RESULTS
3.1. Internal noise influence on the phase
separation scenario
Firstly, let us consider the generalized
approach based on the Cahn-Hillalrd-type
theory allowing us to describe a phase
separation scenario of a microstructure
transformation. Here we assume that no quasi-
chemical reactions are possible, i.e. f(x)=0, the
free energy functional F[x] is assumed in the
Ginzburg-Landau form, i.e.
∫ ∇++−= ]2/)(4/2/[ 242 xxxdrF βε .
Therefore, an evolution of the concentration
field x=x(r,t) is governed by the Langevin
equation of the form
),()(][)( txD
x
xFxDxt rξ
δ
δ
∇+⎟
⎠
⎞
⎜
⎝
⎛ ∇⋅∇=∂ , (4)
where )'()'()','(),( 2 tttt −−= δδσξξ rrrr ,
and 0),( =trξ . Statistical properties of such
a system can be described by the probability
density functional P[x], that can be exactly
found as a solution of the corresponding
Fokker-Planck equation. It was shown that
such a functional is of the form [14]
.)(ln
2
][][
),/][exp(][
2
2
∫+=
−∝
xDdrxFxU
xUxP
ef
ef
σ
σ
(5)
Fig. 1. Typical spatial patterns as a solution of the
Langevin equation on a regular 2-dimensional lattice of
N=L2, L=120: a - spinodal decomposition; b - nucleation.
Other parameters are: β=4, ε=1, α=0.5, σ2=0.2
a
b
In such kinds of stochastic models, the
scenario of the phase separation depends on
the initial conditions: at 0)0,( =rx the
system evolves by a spinodal decomposition
scenario (see Fig. 1,a), whereas at 0)0,( ≠rx
a nucleation process is realized (Fig. 1,b). In
further investigation herein, we will study only
the spinodal decomposition.
Next, let us consider the early stages of the
system’s evolution. To this end we calculate
the structure function )()()( txtxtS kkk −= , in
the vicinity of the mixed state x=0, where
rr kr
k dexx id −− ∫= )()2( π . In the framework of
a linear stability analysis we obtain the
55
evolution equation for the structure function in
the form
qqk
k
dtSkktS
kktS
dt
d
d ∫−++−
−=
)(
)2(
)()
()(
2
1
2
2222
22
π
σσασε
β
(6)
from which it follows that the fluctuating
source does not lead to instability of the
mixed/disordered state. From exponential
solutions of Eq. (6) one can see that only
modes with Ddkk c /)(2 2ασε −=< are
unstable and grow at early stages of evolution.
With an increase in α or σ2 the size of the
unstable domain modes k<kc decreases. Modes
with k>kc remain stable during the linear
regime. One needs to stress that the unstable
modes cannot be realized at the condition
ε<ασ2. As it follows, the domain growth
should be different for additive and
multiplicative nois.
In Fig.2 we present solutions of the
evolution equation (6) at different values of
the parameter α. It can be seen that an increase
in α leads to a shift of the peak position toward
smaller values of k. The peak of S(k) is less
pronounced in the multiplicative noise case
than in the case of the additive noise. It
follows that, if the multiplicative noise is
considered, then the dynamics is slowed. A
decrease in the peak height means that the
interface is more diffuse in the case of
multiplicative noise (see insertions in Fig.2).
We compare analytical results with computer
simulations at the same time t in the two-
dimensional lattice. In the insertions of Fig. 2
typical patterns and images of spherically
averaged structure functions are shown. It is
seen that in the multiplicative noise case the
pattern has a more diffuse interface and the
resonance ring in S(k)-dependence is less
pronounced than for the additive noise.
Fig. 2. Evolution of the structure function at
an early stage t=10 at β=4, ε=1, σ2=0.3.
Different values of the parameter α are used to
compare the influence of additive α=0 and
multiplicative α=0.9 noises (solid and dashed
lines, respectively). Insertion shows typical
patterns and corresponding images of
spherically averaged structure functions at the
same time obtained from numerical solutions
of Eq.(4) at x3
a
b
Fig. 3. Power law for domain size growth: a - log-log plot of the evolution of R(t) at different
values of the parameter α (insertion shows universal behavior of the function R(t) at large times,
indicated in the rectangle); b - dependence of the power law exponent z versus parameter α.
Other parameters are ε=1.0, β=4.0, σ2=0.2
56
At late stages of the system’s evolution one
can estimate in what manner such thermal
fluctuations can modify the domain size
growth law. In order to obtain the linear
domain size growth law R(t) we use following
relations:
∫
∫
== −
max
max
0
01
),(
),(
)(;)()( k
k
dktkS
kdktkS
tktktR , (7)
where in this calculation we have used the
spherically averaged structure function S(k,t)
[10]. The power law behavior of the function
is verified at different values of the
parameter α, where the domain growth
exponent depends on α, i.e. z=z(α) (see Fig. 3).
It is seen that in the case of additive noise
(α=0) the exponent z≈1/3, whereas at α=1.0 we
obtain z≈1/4. Therefore, with an increase in α,
a crossover of dynamical regimes is observed.
Our results are in good correspondence with
deterministic and stochastic approaches which
indicate that an increase in the parameter α
delays the dynamics [15-17]. Comparing our
results with results related to phase separation
regimes in alloys under irradiation, one can
conclude that with an α increase the crossover
from strong- to weak-segregation regimes can
be realized [7].
zttR ∝)(
3.2. Patterning scenario under irradiation
and fluctuating source influence
In this subsection we discuss the influence
of internal noise on the pattern formation
scenario in systems under irradiation. To this
end we assume that there is a local dynamics
caused by the irradiation, described by the
force f(x). Moreover, we assume that the
thermal diffusional processes are possible. The
latter is described by non-Fickian diffusion
with interaction potential U(r). Introducing the
corresponding fluctuating source into the
evolution equation for the concentration field,
the probability density functional takes the
form of Eq. (3). At first let us investigate the
structure of the effective potential considering
a homogeneous (zero-dimensional) system
where the mass density field does not depend
on the spatial coordinate. To find the
homogeneous solutions we need to compute
the extrema positions of the function Uef(x)
when the noise intensity is changed, and
calculate the corresponding phase diagram,
illustrating the change in the number of
extrema of the function Uef(x).
Fig. 4. Bifurcation diagram for noise induced
transitions (a change number of extrema of the
function Uef(x)) at α=0.2, ε=0.2, μ=-0.5. Solid
lines define stable states, the dashed line
responds to the unstable solution. The form of th
Fig. 5. An averaged structure function at different
values of the noise intensity at t=4000 (the model of
lateral interactions): triangles correspond to
σ2=0.25; circles correspond to σ2=σ0
2; and stars
relate to σ2=1.5. Other parameters are: ε=0.2,
μ=0.5, κ=1.0, β=1.0, and α=0.2
cor e
effective potential is shown in insertion
57
In our investigations we assume that our
effective concentration field can take values in
the interval [-1,1] to describe two possible
dense phases, whereas for the mixed one it
corresponds to x=0. To consider a case
let us assume values of both ε and μ
to locate a minimum U
]1,1[−∈x
ef(x_) at x<0, a
minimum Uef(x+) we locate at x>0. An
appropriate choice is ε=0.2, μ=-0.5, α>0. The
corresponding dependencies are
shown in Fig. 4. To understand
transformations of the system states let us use
the noise induced transitions formalism [18].
As it follows from naive considerations, the
bimodal stationary distribution
becomes unimodal
with an increase in the noise intensity. In the
case under consideration here, the transition
occurs in the following manner. In the
noiseless case, a form of the effective potential
U
)( 2σ±x
)/)(exp()( 2σxUxP efst −∝
ef is topologically identical to a form of the
initial potential V(x). With an increase in the
noise intensity a minimum of Uef located at x_
tends to zero, at the effective
potential has a double degenerated point,
x
αεσσ /22 == s
0=x_=0. Therefore, the values define a
spinodal curve. At the point x
2
sσ
2
0
22 σσσ <<s 0
relates to a minimum, whereas x_ defines a
maximum position of the function Uef. These
two minima differ in depth. At one
has U
2
0
2 σσ =
ef(0)=Uef(x+), therefore, defines a
coexistence line (binodal). With a further
increase in the noise intensity we get
U
2
0σ
ef(0)<Uef(x+). At
one gets another spinodal. At the
effective potential has one well only.
Therefore, in such a noise induced transition
we have a shift of the potential extreme,
transformation of the global minimum into a
local one, loss of its stability and, finally, a
change in the number of extrema of the
function U
/4)( 2122 μεασσ +== −
c
22
cσσ >
ef. To analytically study a possibility
of patterning in the system under
consideration, one needs to solve a variational
problem 0/ =xUef δδ . Indeed, the stationary
structures x(r) should correspond to extrema
positions of the effective functional Uef[x].
Solutions of the corresponding variational
problem allows one to find the stable
structures that are formed in the vicinity of the
positions’ of local minima of the effective
potential Uef(x). The corresponding stationary
profiles of the most probable concentration
field x(r) obtained, have one period only. This
result is verified by a numerical solution of the
Langevin equation (2) where the spherically
averaged structure function has only one peak
(see Fig. 5).
Fig.6. One dimensional concentration profiles (top) and two-dimensional stationary patterns
at σ2=0.2 (a), σ2= σ0
2 (b) and σ2=2 (c). Other parameters are: α=0.2, ε=0.2,
μ=-0.5, β=1.0, κ=1
a b c
58
The corresponding simulations of spatial
patterns in a two-dimensional lattice allows
one to find that at small noise intensities
the system is organized in patterns of
the type “vacancy clusters” with very small
values of the concentration field (see Fig. 6, a).
At when two minima of the effective
potential U
22
sσσ <
2
0
2 σσ =
ef(x) located at x=0 and x=x+ are
equivalent in depth, a condition of phase
separation is realized (Fig.6, b). Here the
average concentration x(r,t) is the constant
value (conserved dynamics), but there is no
domain size growth law, as is realized in phase
separation processes (see Sec.3.1). Here, the
linear size of the domain R saturates and does
not grow in time, it decreases with as
. At large noise intensities
the system is organized into strip
phases with liner defects of the dislocation
type (Fig. 6, c). It is interesting to note that at
noise intensities
fluctuations destroy the stable ordered patterns
and stable structures can not be formed.
Patterns cannot be formed if irradiation effects
are not considered.
2σ
01.06.122 )()( ±−∝ σσR
22
cσσ >
ακσσσ /4+ 2222
sT =>
4. CONCLUSIONS
We have discussed the possibility of phase
separation processes and pattern formation in
stochastic systems such as binary alloys under
irradiation conditions. We have examined the
phase separation scenario of the system with
internal multiplicative noise related to the
field-dependent mobility. Analysis was
performed for early and late stages of the
evolution by computer simulations. We have
generalized the well known results of phase
separation theory. Comparing the noise
induced transition picture and pattern
formation processes, it was shown that the
system follows the entropy driven mechanism
by analogy with entropy driven phase
transition theory. Our study shows that at a
small noise intensity the system manifests a
nucleation regime, at fixed values of the noise
strength a spinodal decomposition is realized,
and at large noise the system exhibits strip
patterns with liner defects. Strip structures
exist in the fixed interval of the noise intensity
– large fluctuations destroy patterns.
The obtained results can be applied to study
patterns in adsorption/desorption processes in
metal deposition of a monolayer of molecules
and in processes of microstructure
transformations of materials subject to
intensive irradiation.
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59
СТАТИСТИЧЕСКОЕ МОДЕЛИРОВАНИЕ АНСАМБЛЕЙ
НЕСТАБИЛЬНЫХ ЧАСТИЦ С ТЕПЛОВЫМИ ФЛУКТУАЦИЯМИ
Д.О. Харченко
Исследуются системы бинарных сплавов, подверженных облучению частиц и
воздействию тепловых шумов. Обсуждаются два конкурирующих механизма эволюции
системы: динамика, вызванная облучением, и стохастические воздействия, переводящие
систему к тепловому равновесию. С использованием формализма теории фазового поля и
статистического моделирования рассматриваются процессы самоорганизации в системах
бинарных сплавов с подвижностью, зависимой от поля концентрации. Предложено
обобщение сценария фазового расслоения и структурообразования в таких системах.
СТАТИСТИЧНЕ МОДЕЛЮВАННЯ АНСАМБЛІВ НЕСТАБІЛЬНИХ
ЧАСТОК З ТЕПЛОВИМИ ФЛУКТУАЦІЯМИ
Д.О. Харченко
Досліджуються системи бінарних сплавів, підданих опроміненню часток і впливу
теплових шумів. Обговорюються два конкуруючих механізми еволюції системи: динаміка,
обумовлена опроміненням і стохастичні впливи, що переводять систему до теплової
рівноваги. З використанням формалізму теорії фазового поля і статистичного
моделювання розглядаються процеси самоорганізації у системах бінарних сплавів з
рухомістю, яка залежить від поля концентрації. Запропоноване узагальнення сценаріїв
фазового розшарування і структуроутворення у таких системах.
60
СТАТИСТИЧЕСКОЕ МОДЕЛИРОВАНИЕ АНСАМБЛЕЙ НЕСТАБИЛЬНЫХ ЧАСТИЦ С ТЕПЛОВЫМИ ФЛУКТУАЦИЯМИ
|
| id | nasplib_isofts_kiev_ua-123456789-96336 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-02T11:26:43Z |
| publishDate | 2009 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kharchenko, D.O. 2016-03-15T09:46:53Z 2016-03-15T09:46:53Z 2009 Monte-carlo modeling for unstable particle ensembles with thermal fluctuations / D.O. Kharchenko // Вопросы атомной науки и техники. — 2009. — № 4. — С. 52-60. — Бібліогр.: 18 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/96336 In this presentation we study binary alloy systems subjected to particle irradiation and thermal noise influence. We discuss two competing mechanisms of the system evolution: dynamics driven by irradiation and stochastic influences bringing the system toward thermal equilibrium. Using a phase-field formalism and Monte-Carlo modeling we consider selforganization processes in binary alloy systems with field/concentration dependent mobility/diffusion coefficient. Generalization of the phase separation scenario and patterning in such systems are presented. Досліджуються системи бінарних сплавів, підданих опроміненню часток і впливу теплових шумів. Обговорюються два конкуруючих механізми еволюції системи: динаміка, обумовлена опроміненням і стохастичні впливи, що переводять систему до теплової рівноваги. З використанням формалізму теорії фазового поля і статистичного моделювання розглядаються процеси самоорганізації у системах бінарних сплавів з рухомістю, яка залежить від поля концентрації. Запропоноване узагальнення сценаріїв фазового розшарування і структуроутворення у таких системах. Исследуются системы бинарных сплавов, подверженных облучению частиц и воздействию тепловых шумов. Обсуждаются два конкурирующих механизма эволюции системы: динамика, вызванная облучением, и стохастические воздействия, переводящие систему к тепловому равновесию. С использованием формализма теории фазового поля и статистического моделирования рассматриваются процессы самоорганизации в системах бинарных сплавов с подвижностью, зависимой от поля концентрации. Предложено обобщение сценария фазового расслоения и структурообразования в таких системах. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Monte-carlo modeling for unstable particle ensembles with thermal fluctuations Статистичне моделювання ансамблів нестабільних часток з тепловими флуктуаціями Статистическое моделирование ансамблей нестабильных частиц с тепловыми флуктуациями Article published earlier |
| spellingShingle | Monte-carlo modeling for unstable particle ensembles with thermal fluctuations Kharchenko, D.O. |
| title | Monte-carlo modeling for unstable particle ensembles with thermal fluctuations |
| title_alt | Статистичне моделювання ансамблів нестабільних часток з тепловими флуктуаціями Статистическое моделирование ансамблей нестабильных частиц с тепловыми флуктуациями |
| title_full | Monte-carlo modeling for unstable particle ensembles with thermal fluctuations |
| title_fullStr | Monte-carlo modeling for unstable particle ensembles with thermal fluctuations |
| title_full_unstemmed | Monte-carlo modeling for unstable particle ensembles with thermal fluctuations |
| title_short | Monte-carlo modeling for unstable particle ensembles with thermal fluctuations |
| title_sort | monte-carlo modeling for unstable particle ensembles with thermal fluctuations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/96336 |
| work_keys_str_mv | AT kharchenkodo montecarlomodelingforunstableparticleensembleswiththermalfluctuations AT kharchenkodo statističnemodelûvannâansamblívnestabílʹnihčastokzteplovimifluktuacíâmi AT kharchenkodo statističeskoemodelirovanieansambleinestabilʹnyhčasticsteplovymifluktuaciâmi |