Modelling of Physical Kinetics of Relaxation in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys
In comparison with classical Lifshitz—Slyozov—Wagner’s theory and, where
 it is possible, with available experimental data, the physical kinetics of evolution
 of a microstructure of f.c.c.-Nі—Fe alloys is simulated by means of the
 Onsager-type equations of microdiffusion an...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2012
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| Zitieren: | Modelling of Physical Kinetics of Relaxation
 in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys / I.V. Vernyhora, S.M. Bokoch, V.A. Tatarenko // Наносистеми, наноматеріали, нанотехнології: Зб. наук. пр. — К.: РВВ ІМФ, 2012. — Т. 10, № 4. — С. 897-916. — Бібліогр.: 37 назв. — анг. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860212503648141312 |
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| author | Vernyhora, I.V. Bokoch, S.M. Tatarenko, V.A. |
| author_facet | Vernyhora, I.V. Bokoch, S.M. Tatarenko, V.A. |
| citation_txt | Modelling of Physical Kinetics of Relaxation
 in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys / I.V. Vernyhora, S.M. Bokoch, V.A. Tatarenko // Наносистеми, наноматеріали, нанотехнології: Зб. наук. пр. — К.: РВВ ІМФ, 2012. — Т. 10, № 4. — С. 897-916. — Бібліогр.: 37 назв. — анг. |
| collection | DSpace DC |
| container_title | Наносистеми, наноматеріали, нанотехнології |
| description | In comparison with classical Lifshitz—Slyozov—Wagner’s theory and, where
it is possible, with available experimental data, the physical kinetics of evolution
of a microstructure of f.c.c.-Nі—Fe alloys is simulated by means of the
Onsager-type equations of microdiffusion and with the account of the effects
caused by magnetic interatomic interactions and elastic interactions of inclusions
of phases. Magnetism proper in f.c.c.-Nі—Fe alloys essentially influences
a tendency to atomic ordering and development of their microstructure;
magnetic interactions promote stabilisation of a precipitated phase and
dilate an interval of two-phase coexistence. In Elinvar alloys, elastic interactions
essentially change morphology of an intermixture of formed phases of
the superstructural L10 and L12 types (or of structural A1 type), giving the
anisotropic character to the shape of inclusions of phases as well as to their
relative spatial arrangement. Meanwhile, for an intermixture of phases of
structural A1 type and superstructural L12 type in Permalloys, the anisotropic
effects of such interactions are inappreciable.
У порівнянні з класичною теорією Ліфшиця—Сльозова—Ваґнера і, де можливо, з наявними експериментальними даними змодельовано фізичнукінетику еволюції мікроструктури стопів ГЦК-Ni—Fe за допомогою рівнань мікродифузії Онсаґерового типу та з урахуванням ефектів, спричинених магнетними міжатомовими взаємодіями та пружніми взаємодіями
вкраплень фаз. Властивий стопам ГЦК-Ni—Fe магнетизм істотно впливає
на тенденцію до атомового впорядкування та розвиток їхньої мікроструктури: магнетні взаємодії сприяють стабілізації фази, що виділяється, та
розширюють інтервал співіснування фаз парами. Пружні ж взаємодії істотно змінюють морфологію суміші утворених фаз надструктурного типу
L10 і L12 (або структурного типу A1) в елінварних стопах, надаючи анізотропного характеру як формі вкраплень фаз, так і їхньому взаємному розташуванню; але для суміші фаз структурного типу A1 та надструктурного
типу L12 в пермалоях анізотропні ефекти таких взаємодій є незначними.
В сравнении с классической теорией Лифшица—Слёзова—Вагнера и, где
возможно, с имеющимися экспериментальными данными смоделирована
физическая кинетика эволюции микроструктуры сплавов ГЦК-Ni—Fe с
помощью уравнений микродиффузии онсагеровского типа и с учётом эффектов, вызванных магнитными межатомными взаимодействиями и
упругими взаимодействиями включений фаз. Присущий сплавам ГЦК-
Ni—Fe магнетизм существенным образом влияет на тенденцию к атомному упорядочению и развитие их микроструктуры: магнитные взаимодействия способствуют стабилизации фазы, которая выделяется, и расширяют интервал попарного сосуществования фаз. Упругие же взаимодействия существенным образом изменяют морфологию смеси образованных
фаз сверхструктурного типа L10 и L12 (или структурного типа A1) в элинварных сплавах, придавая анизотропный характер, как форме включений фаз, так и их взаимному расположению, но для смеси фаз структурного типа A1 и сверхструктурного типа L12 в пермаллоях анизотропные
эффекты таких взаимодействий являются незначительными.
|
| first_indexed | 2025-12-07T18:14:49Z |
| format | Article |
| fulltext |
897
PACS numbers: 61.50.Ah, 61.72.Bb,64.60.De,64.60.qe,64.75.Jk,75.30.Hx, 81.30.-t
Modelling of Physical Kinetics of Relaxation
in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys
I. V. Vernyhora1,2, S. M. Bokoch3,4, and V. A. Tatarenko1
1G. V. Kurdyumov Institute for Metal Physics, N.A.S. of Ukraine,
Department of Solid State Theory,
36 Academician Vernadsky Blvd.,
UA-03680 Kyyiv-142, Ukraine
2Institute for Applied Physics, N.A.S. of Ukraine,
Department of Modelling of Radiation Effects
and Microstructure Transformations in Construction Materials,
58 Petropavlivska Str.,
UA-40030 Sumy, Ukraine
3Institute for Advanced Materials Science and Innovative Technologies,
Department of Materials Design and Technology,
15 Sauletekio Ave.,
LT-10224 Vilnius, Lithuania
4NIK Electronics,
34 Lesi Ukrayinky Blvd.,
UA-01601 Kyyiv, Ukraine
In comparison with classical Lifshitz—Slyozov—Wagner’s theory and, where
it is possible, with available experimental data, the physical kinetics of evo-
lution of a microstructure of f.c.c.-Nі—Fe alloys is simulated by means of the
Onsager-type equations of microdiffusion and with the account of the effects
caused by magnetic interatomic interactions and elastic interactions of inclu-
sions of phases. Magnetism proper in f.c.c.-Nі—Fe alloys essentially influ-
ences a tendency to atomic ordering and development of their microstruc-
ture; magnetic interactions promote stabilisation of a precipitated phase and
dilate an interval of two-phase coexistence. In Elinvar alloys, elastic interac-
tions essentially change morphology of an intermixture of formed phases of
the superstructural L10 and L12 types (or of structural A1 type), giving the
anisotropic character to the shape of inclusions of phases as well as to their
relative spatial arrangement. Meanwhile, for an intermixture of phases of
structural A1 type and superstructural L12 type in Permalloys, the aniso-
tropic effects of such interactions are inappreciable.
У порівнянні з класичною теорією Ліфшиця—Сльозова—Ваґнера і, де мо-
жливо, з наявними експериментальними даними змодельовано фізичну
Наносистеми, наноматеріали, нанотехнології
Nanosystems, Nanomaterials, Nanotechnologies
2012, т. 10, № 4, сс. 897—916
© 2012 ІМФ (Інститут металофізики
ім. Г. В. Курдюмова НАН України)
Надруковано в Україні.
Фотокопіювання дозволено
тільки відповідно до ліцензії
898 I. V. VERNYHORA, S. M. BOKOCH, and V. A. TATARENKO
кінетику еволюції мікроструктури стопів ГЦК-Ni—Fe за допомогою рів-
нань мікродифузії Онсаґерового типу та з урахуванням ефектів, спричи-
нених магнетними міжатомовими взаємодіями та пружніми взаємодіями
вкраплень фаз. Властивий стопам ГЦК-Ni—Fe магнетизм істотно впливає
на тенденцію до атомового впорядкування та розвиток їхньої мікрострук-
тури: магнетні взаємодії сприяють стабілізації фази, що виділяється, та
розширюють інтервал співіснування фаз парами. Пружні ж взаємодії іс-
тотно змінюють морфологію суміші утворених фаз надструктурного типу
L10 і L12 (або структурного типу A1) в елінварних стопах, надаючи анізот-
ропного характеру як формі вкраплень фаз, так і їхньому взаємному роз-
ташуванню; але для суміші фаз структурного типу A1 та надструктурного
типу L12 в пермалоях анізотропні ефекти таких взаємодій є незначними.
В сравнении с классической теорией Лифшица—Слёзова—Вагнера и, где
возможно, с имеющимися экспериментальными данными смоделирована
физическая кинетика эволюции микроструктуры сплавов ГЦК-Ni—Fe с
помощью уравнений микродиффузии онсагеровского типа и с учётом эф-
фектов, вызванных магнитными межатомными взаимодействиями и
упругими взаимодействиями включений фаз. Присущий сплавам ГЦК-
Ni—Fe магнетизм существенным образом влияет на тенденцию к атомно-
му упорядочению и развитие их микроструктуры: магнитные взаимодей-
ствия способствуют стабилизации фазы, которая выделяется, и расши-
ряют интервал попарного сосуществования фаз. Упругие же взаимодей-
ствия существенным образом изменяют морфологию смеси образованных
фаз сверхструктурного типа L10 и L12 (или структурного типа A1) в элин-
варных сплавах, придавая анизотропный характер, как форме включе-
ний фаз, так и их взаимному расположению, но для смеси фаз структур-
ного типа A1 и сверхструктурного типа L12 в пермаллоях анизотропные
эффекты таких взаимодействий являются незначительными.
Key words: Ni—Fe alloys, phase transformation, physical kinetics, coarsening.
(Received 19 September, 2012)
1. INTRODUCTION
Ni—Fe alloys are well known as widely used materials in up-to-date in-
dustrial and technological applications due to their inimitable physical
properties (low thermal expansion, unique elastic properties, high
permeability, and low coercive force), the majority of which is formed
due to the coexistence and significant mutual influence of magnetic
and spatial atomic orders. From the experimentally observed phase di-
agram (that is ‘metastable’ in fact; Fig. 1), one can notice that the tem-
perature decrease leads to the following sequential phase transfor-
mations: the 2
nd-order paramagnetic—ferromagnetic phase transition
and the 1
st-order (dis)order—order phase transformations. Depending
on the Fe (Ni) concentration (cFe (cNi)) and the external conditions (tem-
perature (T), pressure (p), magnetic field (B), etc.), the latter results in
MODELLING OF PHYSICAL KINETICS OF RELAXATION IN F.C.C.-Ni—Fe ALLOYS 899
the formation of the ordered alloys with substitutional f.c.c. L12-type
(super)structures (which are experimentally observed for Ni3Fe stoi-
chiometry and theoretically proposed for NiFe3) or f.c.c. L10-type ones
(e.g., for NiFe) from the disordered solid solutions of f.c.c.-A1 type
(characterized by the short-range atomic order only) [1, 2]. The inter-
play between the magnetic and structural orders becomes apparent
when studying the ordering processes in these alloys. Notably, the
magnetic nature of the alloy components promotes the atomic ordering
and the formation of the ordered (super)structures mentioned above.
Moreover, in the magnetic state of an alloy, the (Kurnakov) ordering
temperatures, TK(cFe), are enhanced (for example, see Refs. [3—6]). In
turn, as was shown previously [5—8], the Curie temperature, TC(cFe), of
the ordered alloys also increases, comparing to the disordered ones (in-
cluding the alloys with a short-range atomic order), e.g., for NiFe-type
Elinvar and Ni3Fe-type Permalloy alloys, the excess is about 100—200
K and 100 K, respectively. Therefore, the theoretical investigation of
magnetic and atomic order effects in these alloys becomes an im-
portant part of deeper understanding a variety of physical phenomena.
In spite of number of works (see, e.g., Refs. [9, 10] and references
Fig. 1. Experimentally obtained ‘metastable’ phase diagram of Ni—Fe alloys
(according to Ref. [2]). The symbol ‘?’ denotes the unidentified authentically
structural and/or magnetic states of the alloys at issue. γ-Fe, α-Fe and δ-Fe
are the f.c.c., low- and high-temperature b.c.c. lattice-based modifications of
an iron. L is the Ni—Fe liquid solution. The equilibrium crystal structures of
the three stoichiometric ordered phases at T = 0 K, L12-Ni3Fe (Permalloy),
L10-NiFe (Elinvar) and L12-NiFe3 (Invar), are also shown (right-to-left).
900 I. V. VERNYHORA, S. M. BOKOCH, and V. A. TATARENKO
therein) devoted to the study of the thermodynamic behaviour of f.c.c.-
Ni—Fe alloys, only a few of them address the investigation of the order—
disorder phase transformation kinetics. For example, in [11], the au-
thors studied all stages (nucleation, growth, and further coarsening)
inherent to the 1
st-order transformation in Ni3Fe Permalloy alloy by
means of experimental measurements (SEM technique and x-ray dif-
fraction). They showed that the degree of order in these alloys is sensi-
tive to the annealing temperature and decreases with temperature in-
creasing; the ordering kinetics, notably, the domain structure and the
impurity effects, was investigated too. Nevertheless, the role of mag-
netism was analysed neither experimentally nor theoretically. It is also
necessary to mention the lack of kinetics investigation for NiFe Elinvar,
where, besides the magnetic effects, the elastic effects can perform the
crucial role in the microstructure formation. Due to the slowness of the
diffusion processes, the experimental observation of ordered structures
in these alloys is rather difficult and, thus, the theoretical modelling
becomes a useful tool in the investigation of the ordering processes.
In a given work, we study the effects of magnetism and elasticity
(induced by the size mismatch between the constituent atoms) on the
formation of the ordered structures in Permalloy and Elinvar alloys. In
Section 2, we formulate a model used to calculate the kinetics of order—
disorder transformation. The obtained results are presented and dis-
cussed in Sec. 3 and are followed by the overall conclusion in Sec. 4.
2. MODEL
On the microscopic level, the physical kinetics of phase transformations
can be described, using the Onsager-type microscopic diffusion equa-
tions. Within the scope of the concentration waves (and in particular,
the static concentration waves–SCW) representation, such an approach
was firstly proposed by Khachaturyan [12, 13]. The morphology of a
two-phase alloy is described by a single-site probability function, p(r,t),
which is the probability of finding a solute atom β (e.g., Fe in Ni3Fe and
NiFe alloys) at a lattice site r and at an instant time t. The diffusional
relaxation of a binary α—β alloy is described by the equation [12, 13]:
(1 )( , )
( )
( , )
B
c cdp t F
L
dt k T p t
β β
′
− δΔ′≈ −
′δ
r
r
r r
r
, (1)
where cβ is the atomic fraction of solute (β) atoms, kB is the Boltzmann
constant, T is the absolute temperature; L(r—r′) is the Onsager kinetic
coefficients related to the substitutional atoms’ mobilities by means of
appropriate diffusion mechanism between the sites r and r′; ΔF is the
configuration-dependent part of the total Helmholtz free energy (per
site) including three contributions: ‘electrochemical’, magnetic and
MODELLING OF PHYSICAL KINETICS OF RELAXATION IN F.C.C.-Ni—Fe ALLOYS 901
elastic ones, i.e. ΔF = ΔFchem + ΔFmagn + ΔEelast. The sum is carried out over
all N sites of the Bravais lattice. The kinetic equation (1) approximates
the evolution rate by the first non-vanishing term of its expansion
with respect to the thermodynamic driving force, δΔF/δp(r,t) (small
driving force). Within the long-wave approximation, Eq. (1) trans-
forms into the conventional Cahn—Hilliard equation [14].
Within the scope of the self-consistent-field approximation, the con-
figuration-dependent part of free energy of binary substitutional alloys
is defined as follows [12, 13]:
2
magn
1
( ) ( ) { ( ) ln ( ) [1 ( )] ln[1 ( )]}
2
; (2)
B
BZ
F w p k T p p p p
N
T S
∈
Δ ≅ + + − − −
− Δ
k r
k k r r r r
here, ( )w k = Σrw(r)exp(−ik⋅r) is the Fourier transform of total ‘mixing’
energies, {w(r)}, and ( )p k = Σrp(r)exp(−ik⋅r) is the Fourier transform of
probability function, p(r). The summation over r is carried out over all
the Bravais lattice sites; the summation over k is over all points of quasi-
continuum within the 1
st
Brillouin zone (BZ) of such lattice permitted by
the periodic boundary conditions. Assuming the interatomic interactions
within the four coordination shells of f.c.c. lattice, ( )w k is defined as
1
2
3
4
( ) 4 (cos cos cos cos cos cos )
2 (cos2 cos2 cos2 )
8 (cos2 cos cos cos2 cos cos cos2 cos cos )
4 (cos2 cos2 cos2 cos2 cos2 cos2 ) ...; (3)
w w h k h l k l
w h l k
w h k l k h l l h k
w h k h l k l
≅ π π + π π + π π +
+ π + π + π +
+ π π π + π π π + π π π +
+ π π + π π + π π +
k
here, w1, w2, w3, w4, … are the values of the effective interchange (‘mix-
ing’) energies of substitutional atoms, w(r) = wprm(r) + wmagn(r) = (wchem(r) +
+ Vsi
ββ
(r)) + wmagn(r), for the 1
st, 2
nd, 3
rd, 4
th, … neighbouring coordination
shell, respectively; Vsi
ββ
(r)) is the strain-induced contribution (due to the
atomic-size mismatch) in the ‘mixing’ energy; (h,k,l) are continuous di-
mensionless coordinates of the wave vector defined as k = (kx,ky,kz) =
= (2π/a)(h,k,l) (a is the equilibrium lattice parameter of f.c.c. lattice).
Equation (2) has a simplest form of free energy, which is used to mod-
el the kinetics of the system. However, we should mention that, in order
to calculate the miscibility gap and the free energy, a more complicated
expression should be used (see Appendix). The ‘paramagnetic’ energies,
{wprm(r)}, are calculated, considering the polynomial approximation pro-
posed in Refs. [5, 9]. Besides, for a magnetic alloy, one should also con-
sider spin-dynamics equations, i.e. it is necessary to take into account
the nonzero derivatives of relative magnetizations, {dσα(r,t)/dt}. How-
ever, as, for considered Ni—Fe alloys, the spin-rotation relaxation rate is
much higher than the rates of atomic microdiffusion jumps, these terms
are neglected in the presented model. Therefore, the magnetic contribu-
902 I. V. VERNYHORA, S. M. BOKOCH, and V. A. TATARENKO
tion to the overall system evolution consists of adding the magnetic en-
ergies (wmagn(r) = J
FeFe(r)sFe
2σFe
2
+ J
NiNi(r)sNi
2σNi
2
− 2JNiFe(r)sFesNiσFeσNi within
the intracrystalline ‘molecular field’ approximation that is the mean
self-consistent-field one) to the total ‘mixing’ energies of constituents
described with the spin numbers, {sα}, ‘coupled’ by means of the ‘ex-
change integrals’, {J
αβ
(r)}. The equilibrium values of {σα} are found by
minimization of the total free energy (see Refs. [5, 9, 15, 16] and Ap-
pendix), accounting a magnetic contribution to the entropy too.
The elastic energy, which arises from the mismatch between the lat-
tice parameters of the matrix and the precipitates of a new phase, can
be calculated within the scope of the Khachaturyan—Shatalov micro-
scopic elasticity theory of structurally inhomogeneous systems [12,
13, 17, 18]. Following Khachaturyan and colleagues [14, 19—23], the
strain-induced energy generated by arbitrary structure inhomogeneity
can be presented in terms of the concentration or long-range order pa-
rameter fields. Assuming the Végard’s law and the equivalency of elas-
tic moduli tensors, {λijkl}, for host-crystal and precipitate phases, the
morphology-dependent part of the strain-induced energy can be writ-
ten in the reciprocal-space representation form as follows:
2
0
elast
( ) ( )
2 BZ
v
E B C
N ∈
Δ ≈
k
n k , (4)
where n = k/|k| is a unit vector along the k direction, ( )C k is the Fouri-
er transform of the inhomogeneous composition field, c(r). The func-
tion ( )B n = ε0
2(λiill − niλijmmΩjk(n)λklm′m′nl) contains all information on the
elastic properties of the system and crystallography of the phase pre-
cipitation, where v0
−2|k|
−2||Ωij(n)|| is the Green function matrix asymptot-
ics (at k → 0), which is inverse to the tensor |k|2||Ωij(n)||
−1
= |k|2||λikljnknl||;
ε0 = (aγ′ − aγ)/{aγ(cβ
(γ′)
− cβ
(γ)
)} is the concentration coefficient of stress-free
lattice-transformation dilatation caused by the changes in composition
of a binary alloy (v0 is the atomic volume); cβ
(γ)
(cβ
(γ′)
) and aγ (aγ') are the
concentration of the solute and lattice parameter of f.c.c. disordered-
matrix (ordered-precipitate) phase, respectively. At k = 0, the function
( )B n has a singularity, since its limit at k → 0 depends on the k vector
direction. This singularity results in a long-range asymptotic behav-
iour of the strain-induced interaction in a real space [24].
3. RESULTS AND DISCUSSION
The microstructure evolution is modelled on a 2D square lattice con-
sisting of 1024×1024 unit cells. The nucleation stage is neglected as we
limit ourselves to the study of the coarsening stage. Therefore, at the
beginning of the simulation, from 200 to 800 small spherical precipi-
tates of the new phase are manually embedded into the homogeneous
MODELLING OF PHYSICAL KINETICS OF RELAXATION IN F.C.C.-Ni—Fe ALLOYS 903
matrix. The concentration of these precipitates is chosen, according to
the equilibrium concentration of the ordered phase. The effects of
magnetic and elastic contributions on the microstructure evolution are
studied by comparison of simulations with and without the respective
terms (wmagn, ΔEelast) in the total ‘mixing’ energies of atoms and the free
energy. The kinetic equation (1) is numerically solved in the reciprocal
space by means of the explicit Euler method. The corresponding atomic
distributions are recovered, using the backward Fourier transform. In
the course of simulations, the periodic boundary conditions are ap-
plied, and the time step is set to Δt = 10
−5.
According to Fig. 1, the temperature T = 650 K is chosen in order to
study the system in two-phase coexistence regions. Such a choice is con-
firmed by the thermodynamic calculations and the free energy plot
(Fig. 2). In Figure 2, the two-phase regions (with cFe ∈ [0.1; 0.185] for
L12 + A1 mixture and cFe ∈ [0.385; 0.425] for L12 + L10 one, respectively)
are defined by the common tangent construction. From such a plot, one
can notice that taking into account magnetic interactions leads to de-
crease of the total free energy and broadening of the phase boundaries,
which testifies that magnetism stipulates the ordering processes and the
stabilization of the ordered phases in the studied alloys. The investiga-
tion of kinetics should give further confirmation of such a hypothesis.
Simulations are performed for some representative compositions
within the specified two-phase regions. In particular, cFe = 0.14 and
0.41 are chosen to study Permalloy and Elinvar alloys, respectively.
The interaction parameters calculated at such thermodynamic condi-
ΔF/N, eV
Fig. 2. The concentration dependence of the configuration part of the total
free energy at T = 650 K.
904 I. V. VERNYHORA, S. M. BOKOCH, and V. A. TATARENKO
tions are listed in Table 1. The values for 1
st
and 2
nd
coordination shells
are extracted from ( )w Γk and ( )Xw k , using the set of linear equations:
1 2 3 4
1 2 3 4
( ) 12 ( ) 6 ( ) 24 ( ) 12 ( ),
( ) 4 ( ) 6 ( ) 8 ( ) 12 ( ),X
w w w w w
w w w w w
Γ ≅ + + +
≅ − + − +
k r r r r
k r r r r
(5)
and considering the ‘strain-induced’ interactions right up to 4
th
shell
(w(r3) = 0.3 meV, w(r4) = 0.7 meV; see details in [5, 9]); the ‘paramagnet-
ic’ and magnetic interactions are considered as short-range ones, i.e.
within the nearest (for magnetic) and next-nearest (for ‘paramagnet-
ic’) neighbourhood only.
3.1. The Effect of Magnetism on the Microstructure
In Figure 3, the results of simulation are presented for both alloys:
Ni0.86Fe0.14 Permalloy (Fig. 3, a) and Ni0.59Fe0.41 Elinvar (Fig. 3, b).
In order to clarify the role of magnetism in the ordered-structure
formation in Permalloy alloys, two kinds of simulations are performed:
with neglecting magnetic contribution, i.e. wmagn ≡ 0, and with consider-
ing it via the appropriate calculation of wmagn ≠ 0. In both cases, the ini-
tial configuration of an alloy is consisted of disordered matrix with
cFe = 0.1 and a set of small L12-ordered nuclei with cFe = 0.185. The time
evolution of a nonconserved system that separates into phases of differ-
ent concentration is studied and shown in Fig. 3, a at reduced time
t*
= {0, 20, 40} (measured in units of Monte Carlo steps per site [MCS/s]).
Neglecting the magnetic nature of an alloy (wmagn ≡ 0; Fig. 3, a, left col-
umn) results in complete disappearance of the nuclei of ordered phase,
which were initially embedded into the disordered matrix. This behav-
iour agrees with the calculated configuration-dependent free energy
curves (Fig. 2) and testifies that, at such temperature and at the condi-
tion of wmagn ≡ 0, the chosen alloy does not undergo an ordering reaction.
In turn, taking into account the magnetic effects, one obtains an op-
posite result. During the evolution, the growth and coarsening stages
TABLE 1. The interaction parameters (‘mixing’ energies) [meV] calculated at
Т = 650 K, considering that w(r3) = 0.3 meV and w(r4) = 0.7 meV [5, 9]. In the
second column, the symbols ‘+’ and ‘—’ denote the presence and absence of
magnetism.
cFe magnetism σNi σFe ( )w Γk ( )Xw k w(r1) w(r2)
0.14
— — — 591.125 −351 58.283 −20.645
+ 0.767 0.806 854.048 −438.667 80.195 −20.648
0.41
— — — 313.255 −229.5 33.322 −17.035
+ 0.88 0.634 515.024 −296.778 50.138 −17.038
MODELLING OF PHYSICAL KINETICS OF RELAXATION IN F.C.C.-Ni—Fe ALLOYS 905
appear sequentially, and at late stages, the large L12-ordered precipi-
tates continue to coarsen the smaller ones until the stationary state is
attained.
The morphology of the obtained structure allows to clearly distin-
guish both A1-type disordered phase and L12-type ordered one (marked
a b
c d
e f
Fig. 3. The microstructure evolution of Ni0.86Fe0.14 Permalloy (a) and
Ni0.59Fe0.41 Elinvar (b) alloys at reduced time t
*
= {0, 20, 40} with (right col-
umns) and without (left columns) taking into account the magnetic (a) and
elastic (b) effects; (c), (d)–respective concentration profiles and (e), (f)–
atomic arrangements corresponding to medium section at t* = 40.
906 I. V. VERNYHORA, S. M. BOKOCH, and V. A. TATARENKO
with black and white colours in Fig. 3, a (right column), respectively).
The latter forms the precipitates of nearly spherical shape. At late
stages of evolution, the system is characterised by a set of the ordered
particles, which differ from each other by a displacement vector
((011)а/2, (101)а/2 or (110)а/2) and are separated by the antiphase
(APB) and interphase (IPB) boundaries.
We should note that, for larger Fe content (closer to the right limit
of the two-phase coexistence interval, e.g., cFe = 0.17), the ordered
phase grows till the size of large domains, and the main evolution
mechanisms in this case will be as follow: 1) the growth of large do-
mains at the expense of the APB movement, 2) the merging of two do-
mains of the same type, and 3) the dissolution of the intermediate do-
mains (see Ref. [25] and references therein).
In Figure 3, c, the local concentration profile for the longitudinal
median section of final structure at t
*
= 40 for simulated Ni0.86Fe0.14
Permalloy is presented. Let us note that the local concentration at each
lattice site is calculated by means of averaging over the 1
st
and 2
nd
co-
ordination shells [14, 19—22, 25—27]. As can be seen, in the disordered
phase, the local concentration approaches the equilibrium concentra-
tion of the disordered phase determined by the phase diagram (Fig. 2),
i.e. cFe ≈ 0.1. In turn, the ordered phase is characterised by the local con-
centration cFe ≈ 0.176 and tends to the equilibrium concentration of the
L12-type phase, i.e. 0.185.
The example of atomic configuration for simulated Ni0.86Fe0.14 Perm-
alloy at late stage of evolution (t
*
= 40) is presented in Fig. 3, e. The black
(white) colour marks the lattice sites, where the Fe (Ni) atoms can be
found with the highest probability; grey colour marks the sites, where
the probabilities to find Fe (Ni) atoms are identical, i.e. the disordered
phase. Inside the ordered precipitates, the atoms alternate in the man-
ner inherent to the 2D projection of the L12-type superstructure.
3.2. The Effect of Elastic Contribution on the Microstructure
The influence of the elastic energy on the microstructure of the Elinvar
alloys is studied by means of two kinds of simulations: neglecting the
elastic term (Eq. (4)), i.e. ΔEelast ≡ 0, and considering it by the explicit cal-
culation of ΔEelast ≠ 0. In both cases, the initial configuration of the alloy
consisted of both L12-type ordered matrix with cFe = 0.385 and a set of
small L10-type ordered nuclei with cFe = 0.425. The time evolution of a
nonconserved system at issue at reduced times t
*
= {0, 20, 40} is shown in
Fig. 3, b. Let us mention that, for Elinvar alloys, it is important to con-
sider the elastic contribution, ΔEelast, to the free-energy functional due to
the tetragonality of L10-type phase and more pronounced crystal-lattice
mismatch between the two adjacent phases (L12 and L10) than in Permal-
loys (L12 and A1). Owing to the small difference in the tetragonal-phase
MODELLING OF PHYSICAL KINETICS OF RELAXATION IN F.C.C.-Ni—Fe ALLOYS 907
parameters, a and b, [28, 29], in the presented simulations, this feature
is neglected, and the following lattice parameters are used [1, 29]:
aA1 ≈ 0.35338 nm, aL12 ≈ 0.356667 nm, aL10 ≈ 0.358229 nm.
One should mention that, for Elinvar alloys, the role of magnetic
contribution is also tested, and the obtained results resemble the out-
comes reported above for the Permalloy alloys. Neglecting the mag-
netism results in dissolution of the embedded nuclei. Thus, further,
the magnetic contribution will be implied by default and, as seen from
Fig. 3, b, the microstructure formation is a result of the growth and
coarsening stages. When elastic energy is neglected (Fig. 3, b, left col-
umn), the obtained microstructure contains precipitates of nearly
spherical shape, which are isotropically distributed over the system.
However, taking into account the elastic energy (in particular, due to
the elastic interactions between the L10-type precipitates) results in both
change of precipitates shape to the plate-like (or rectangular) one and the
anisotropy along (100)* direction in their distribution. Let us note that
the appearance of the anisotropy in the system with L10-type order has
been found previously (see Refs. [19, 20, 22, 25] and references therein),
but as shown, it appears along the (110)*-type direction, and the precipi-
tates form a certain type of pattern, for instance, tweed, twin, chess-
board-like patterns, etc. We assume that such behaviour is a consequence
of the tetragonality of L10-type phase, and, as we have neglected this fea-
ture in our simulation, we do not expect the identical peculiarities.
Among the specific features of the microstructure generated, con-
sidering the elastic effects (ΔEelast), one can note the formation of
‘chains’ of precipitates, which appear due to inability of particles with
different orientation and translation variants to merge (coalesce). (In
general, in 3D case, the precipitates of L10-type phase have three orien-
tation variants and three translation variants, i.e. in total, six variants;
in 2D case, one has three different variants). For visual presentation of
the microstructure, which is formed during the evolution, the local
concentration profile for the longitudinal median section and the ex-
ample of atomic configuration of final structure at t
*
= 40 for simulated
Ni0.59Fe0.41 Elinvar are shown in Fig. 3, d and f, respectively. In the lat-
ter figure, one can clearly see the precipitates of L10-type phase of three
different types (the atoms are altered in a different manner) and the
L12-type phase matrix. The black (white) colour marks the lattice sites,
where the Fe (Ni) atoms can be found with the highest probability. As
seen in Figure 3, d, the local concentration profile smears out approach-
ing the IPB owing to the change of ordered structure; the APB between
the precipitates are not found as a result of wetting by L12 phase.
3.3. Analysis of the Microstructure Evolution
The shape of precipitates can be estimated by the aspect ratio (AR) of
908 I. V. VERNYHORA, S. M. BOKOCH, and V. A. TATARENKO
their sides in 0x and 0y directions following the rules: if AR < 1.5, pre-
cipitates have a shape close to circle (spherical) or square (cubical); if
1.5 < AR < 2.5, particles possess a rectangle shape; if 2.5 < AR < 3.5 or
AR > 3.5, precipitates tend to have a plate-like shape [30]. One can see
in Fig. 4, a that such an analysis adequately reflects the simulated
shapes of precipitates: for Ni0.86Fe0.14 Permalloy and Ni0.59Fe0.41 Elinvar
without elastic contribution, the AR tends to unity that confirms a
spherical shape of precipitates; in turn, considering the elastic contri-
bution for Ni0.59Fe0.41 Elinvar, the AR is evidently > 1.5 and < 2.5, con-
firming a rectangular shape.
In Figure 4, b, the calculated volume fraction of a new phase is shown
for all simulated alloys. When magnetic interactions are neglected, the
dissolution of all particles of the L12-type ordered phase results in a fast
decrease (to 0) of a volume fraction, and therefore, further analysis of
this case will be omitted. During the growth regime, the volume frac-
tion of Ni0.86Fe0.14 Permalloy (with magnetic effects) grows from 0.004
to 0.5 and reaches its equilibrium value already at t
*
≈ 1 that indicates
a b
R3
c
Fig. 4. The time evolution of the aspect ratio, AR (a), volume fraction (b) and
cube of the average precipitate radius (c) for Ni0.86Fe0.14 and Ni0.59Fe0.41 alloys.
MODELLING OF PHYSICAL KINETICS OF RELAXATION IN F.C.C.-Ni—Fe ALLOYS 909
the start of the coarsening regime. Similarly, the volume fraction of
Ni0.59Fe0.41 Elinvars indicates the start of coarsening regime already at
t*
≈ 1 and reaches 0.617 and 0.622, when elastic effects are neglected
and taken into account, respectively. Such a difference can be attribut-
ed to the different number of precipitates contained in the system at
different time intervals; for example, at last time step (t
*
≈ 40), the
studied alloy has 22 and 26 L10-type particles, respectively.
Following the Lifshitz—Slyozov and Wagner’s theory (LSW, [31,
32]), at late stages of evolution, the cube of the average particle radius
obeys a power law:
∝3
R Kt , (6)
where R is an average radius of a precipitate, t is a time of rise, K is
the coarsening rate. In Figure 4, c, such behaviour is noticeable for all
studied alloys. A slope of each curve determines the coarsening rate,
K. One can also visually distinguish the growth and coarsening stages;
at the growth stage, the radius of particles grows very fast and, there-
fore, the slope of the curves is steepest. From the simulated data, one
can estimate the average size of precipitate, and using the linear ap-
proximation, one can estimate the coarsening rate as a slope of the re-
spective linear curve, Kt + const ≅ R3. For Ni0.86Fe0.14 Permalloy, one
finds R ≈ 8.6 nm (at t
*
≈ 40) and K ≈ 2309.52 (in arbitrary units). The
same analysis can be applied to the outcomes for Ni0.59Fe0.41 Elinvar;
the effect of the elastic contribution can be seen in a slowing down the
coarsening reaction and a respective decay of the R3(t) curve.
The fluctuating character of the <R>3(t) dependences for Elinvar al-
loy is explained by a small amount of precipitates at the late stages of
evolution; the averaging over a number of different system realiza-
tions (initial precipitates’ amount and distribution) can possibly
smooth the resulting curve. (In particular, the results in Fig. 4 are ob-
tained after averaging over different precipitates’ distributions with
their initial number equal to 200 and 800.)
The classical LSW theory also predicts a specific time-independent
form of a precipitates’ size distribution (see the thick grey curve on
Fig. 5, a, b), but as seen for all simulated alloys, the size distribution
curves appreciably deviate. For Ni0.86Fe0.14 Permalloy (Fig. 5, a), one can
deduce the asymptotic form of the curves for the precipitation of parti-
cles of some characteristic size, namely, the features inherent to the
classical LSW theory. But, owing to the considerable volume fraction of
the precipitated phase, the form of the curve deviates from the classical
one, namely, one can notice the broadening, the more symmetrical shape
of a distribution, and a shift of the maximum to the region of sizes,
which are smaller than the average precipitates’ radius, <R>. At late
stage (t
* = 40), the distribution widens towards the range of big-particle
910 I. V. VERNYHORA, S. M. BOKOCH, and V. A. TATARENKO
sizes that confirms the increase of larger particles comparing to the pre-
vious evolution stages; the distribution has the ‘tails’ that signifies the
presence of the precipitates (domains) with a size much greater than the
average R. In case of Ni0.59Fe0.41 Elinvar, the size distribution also devi-
ates from its classical form (Fig. 5, b), and one can see that taking into
account the elastic effects changes its form. In particular, when
ΔEelast ≠ 0, the distribution broadens, and the maximum shifts to the
range of smaller precipitate sizes. Similar to the Permalloy, the Elinvar
distribution curves allow one to distinguish a significant amount of pre-
cipitates with a size much greater than the average R.
The deviation of the distribution curves from the classical LSW re-
lation due to the considerable (not null) volume fraction has been re-
ported previously and generated a number of investigations concerned
a b
|k|
|k|
c d
Fig. 5. The time evolution of the size distribution function, h(ρ) (ρ = R/R), (a,
b) and the structure factor (c, d) for Ni0.86Fe0.14 and Ni0.59Fe0.41, respectively.
MODELLING OF PHYSICAL KINETICS OF RELAXATION IN F.C.C.-Ni—Fe ALLOYS 911
with its improvement and modification (see, for example, [33—35]).
For instance, according to the modified LSW theory (LSEM–Lifshitz—
Slyozov encounter modified one [35]), the widening of the distribution
takes place at the expense of the removal of two small particles and the
addition of one big particle instead (see also Ref. [27]).
Another quantity, which allows drawing a conclusion about the or-
der and the scale of the obtained structure, is the structure factor,
which is a Fourier transform of a pair-correlation function:
( )2
Fe
1
( , ) ( , ) ( , )
iS t e c t c t c
N
− ⋅
′
′ ′= + − k r
r r
k r r r , (7)
where c(r) is a local concentration at the site r at the instant of time t,
cFe is a nominal concentration of the alloy, k is a wave vector within the
1st
Brillouin zone of the reciprocal space. The sum and average (...) is
taken over all N crystal-lattice sites. Further, it is more convenient to
use the normalized spherically averaged structure factor:
( )2 2
Fe
( , ) ( , )
( , )
( , ) ( )
S k t S k t
s k t
S t N c c
= =
−
k
k r
. (8)
/2 /2
/2 /2
( , )
( , )
1
k k k k
k k k k
S t
S k t
−Δ < ≤ +Δ
−Δ < ≤ +Δ
=
k
k
k
is a spherically averaged structure factor,
where k = |k| is a length of the wave vector k. In the denominator, there is
a number of points lying in the Δk-thickness vicinity of the wave vector
k. The time evolutions of the calculated structure factors for Ni0.86Fe0.14
and Ni0.59Fe0.41 are shown in Fig. 5, c, d, respectively. As can be seen
from both graphs, the width of structure factor reduces with time that
evidences the formation of sharp interface between two phases (L12 + A1
and L12 + L10, respectively); the maximum height increases and shifts to
lower k values confirming thereby the coarsening reactions and the in-
crease of the precipitates’ size [27, 30, 36, 37]. The elastic contribution
(Fig. 5, d) does not influence the location of the maximum but slightly
increases its height and width. The broadening of the maximum and the
respective smearing of IPB is confirmed by the local concentration pro-
file (Fig. 3, d), which is diffuse close to IPB, and by the appearance of
‘chains’ of precipitates situated close to each other. In this case, the IPB
can be very narrow (up to 2 interatomic distances) and diffuse. For
Ni0.59Fe0.41 with ΔEelast ≡ 0, the number of such IPB is smaller.
Let us mention that the structure factor can be evaluated experi-
mentally due to its straightforward connection to the coherent scatter-
ing intensity. Thus, the conclusions drawn from the computer simula-
912 I. V. VERNYHORA, S. M. BOKOCH, and V. A. TATARENKO
tions can be confirmed by the investigation of the real alloy samples.
4. CONCLUSION
In a given article, the physical kinetics of the atomic orders in Permalloy
and Elinvar alloys is investigated, using the Onsager-type microdiffu-
sion equation. A special attention is paid to the effects generated by the
magnetic and elastic interactions in these alloys. As revealed, the mag-
netic nature substantially influences the atomic ordering in this system
and, in fact, promotes to stabilize the ordered phase (L12 in Permalloys
and L10 in Elinvars) and increases the width of the two-phase coexist-
ence region. In turn, the elastic interactions between the precipitates
significantly change the morphology of the Elinvar alloys, promoting
the anisotropy in the form and the arrangement of precipitates. For
Permalloy alloys, the elastic effects turn out to be negligible.
The calculated statistical characteristics confirm the effects of
magnetic and elastic interactions on the kinetics and structure proper-
ties. The simulated microstructure for the high volume fraction of L12
phase is in a good agreement with the available experimental data [11].
One should also note that, due to the slow diffusion processes in f.c.c.-
Ni—Fe alloys and low temperatures of the order—disorder phase transi-
tion, there are only a few experimental data about the microstructure
in these nonconserved systems. The latter makes the presented study
useful for a general consideration of magnetically and atomically or-
dered systems.
ACKNOWLEDGEMENTS
The authors express their appreciation to Dr. H. M. Zapolsky and Prof.
D. Ledue (GPM, UMR 6634 CNRS, Université de Rouen, France) for
very constructive discussion of the results obtained in the course of a
given work. We are grateful to Prof. B. Schönfeld (ETH, Switzerland)
and Prof. S. Hata (Kyushu University, Japan) for stimulating discus-
sions on atomic ordering phenomena in metallic alloys. I.V.V.
acknowledges the UMR 6634 (France) for partial financial support of a
given work. S.M.B. would like to thank also the Institute for Advanced
Materials Science and Innovative Technologies (Lithuania) for partial
financial support of a given work.
APPENDIX A
The free energy and the phase diagram can be calculated, using the
SCW representation [12, 13], where the occupation probabilities {p(r)}
are defined as follow:
MODELLING OF PHYSICAL KINETICS OF RELAXATION IN F.C.C.-Ni—Fe ALLOYS 913
** *
31 2 22 2
( ) [ ]
4
ii ip c e e e
π ⋅π ⋅ π ⋅
β
η= + + + a ra r a r
r (A1)
for L12-type superstructure and
*
32
( )
2
i
p c e
π ⋅
β
η= + a r
r (A2)
for L10-type superstructure, respectively. η is the long-range order pa-
rameter, which varies from 0 in the disordered state to 1 in the com-
pletely ordered state (at the corresponding stoichiometry); the vectors
a1
*, a2
*, a3
*
are the unit reciprocal-lattice vectors of f.c.c. crystal lattice
along [100], [010] and [001] directions, respectively; |a1
*| = |a2
*| = |a3
*| =
= 1/a. Substituting Eqs. (A1), (A2) into Eq. (2), the expression for the
‘paramagnetic’ free energy of the L12-type structure can be written as:
2
2
prm prm
1 3
( ) ( )
2 2 4
3 ln 3 1 ln 1
4 4 4 4
,
4 3 3 3 3
ln 1 ln 1
4 4 4 4
X
B
F
c w w
N
c c c c
k T
c c c c
β
β β β β
β β β β
Δ η ≅ + +
η η η η − − + − + − + +
+
+ + η + η + − − η − − η
0 k
(A3)
where cβ is the solute concentration.
The magnetic free energy can be defined within the scope of the ‘mo-
lecular field’ approximation [5, 7, 9, 10, 13], and the resulting config-
uration-dependent part of the total free energy will have a form:
( )
2 2 2 2 2 2 2
prm
2 2 2 2 2
prm
1
( ) ( ) ( ) (1 ) 2 ( ) (1 )
2
3
( ) ( ) ( ) 2 ( )
16
3 3 3
ln 1 ln 1
4 4 4
4
X X X X
B
F
c w J s c J s c J s s c c
N
w J s J s J s s
c c c c
k T
αα ββ αβ
β β β β α α β α β α β β β
αα ββ αβ
α α β β α β α β
β β β
Δ ≅ + σ + σ − + σ σ − +
+ η + σ + σ − σ σ +
+ η + η + − − η −
+
0 0 0 0
k k k k
, ,
, ,
3
4
(A4)
3 ln 3 1 ln 1
4 4 4 4
(2 1) ( ) ( )
ln sh ln sh ( ) ( ( ))
2 2
B s
c c c c
s y y
k T c y B y
s s β
β
β β β β
β β β α β β α
β β β α β β α
β β
− η +
−
η η η η + − − + − + − +
+ σ σ
− − − σ σ
, ,
, ,
(2 1) ( ) ( )
(1 ) ln sh ln sh ( ) ( ( ))
2 2
s
s y y
c y B y
s s α
α α α β α α β
β α α β α α β
α α
+
+ σ σ + − − − σ σ
for L12-type phase (for Ni3Fe alloys or for NiFe3 alloys after a trivial
914 I. V. VERNYHORA, S. M. BOKOCH, and V. A. TATARENKO
change of indexes: β ↔ α);
( )
2 2 2 2 2 2 2
prm
2 2 2 2 2
prm
1
( ) ( ) ( ) (1 ) 2 ( ) (1 )
2
1
( ) ( ) ( ) 2 ( )
4
X X X X
F
N
c w J s c J s c J s s c c
w J s J s J s s
ββ αα αβ
β β β β α α β α β α β β β
αα ββ αβ
α α β β α β α β
Δ ≅
≅ + σ + σ − + σ σ − +
+ η + σ + σ − σ σ +
0 0 0 0
k k k k
1 1 1 1
ln 1 ln 1
2 2 2 2
2
ln 1 ln 1
2 2 2 2
B
c c c c
k T
c c c c
β β β β
β β β β
+ η + η + − − η − − η +
+ −
η η η η + − − + − + − +
, ,
, ,
, ,
,
(2 1) ( ) ( )
ln sh ln sh ( ) ( ( ))
2 2
(2 1) ( ) ( )
(1 ) ln sh ln sh ( ) (
2 2
B s
s
s y y
k T c y B y
s s
s y y
c y B y
s s
β
α
β β β α β β α
β β β α β β α
β β
α α α β α α β
β α α β α
α α
+ σ σ − − − σ σ +
+ σ σ
+ − − − σ
,
( ))
(A5)
α β
σ
for L10-type phase (NiFe alloys). prm
( )w k is the k-th Fourier component
of the ‘paramagnetic’ mixing energies {wprm(r − r′)}; ( )Jαβ k
is the k-th
Fourier component of the magnetic ‘exchange’ interactions ‘integrals’
{Jαβ
(r − r′)}; σα (σβ) is the magnetic long-range order parameter (i.e. the
reduced magnetization of α (β) constituents); sα (sβ) is the value of the
total spin number of an atom of α (β) kind; BJ(ζ) is the conventional
Brillouin function defined as follows:
( )mol
1 1 1 1
( ) 1 cth 1 cth ,
2 2 2 2
,
J
B
B B
B
J J J J
J H J g
k T k T
α α α α
α α α α
α
α α
α β αβ β
ζ ≡ + + ζ − ζ
μ
ζ ≡ ≅ − Γ σ
where Jα = sα + lα is the total angular moment of an atom, which consists
of both the spin number (sα) and the orbital momentum number (lα). We
assume that, for transition metals, Jα ≅ sα.
α
αβ ββ
≅ − μ Γ σmol B
H g is the
Weiss intracrystalline ‘molecular field’ (MSCF) with coefficients {Γαβ}.
The equilibrium order parameters can be determined by solving the
set of transcendental equations obtained after minimizing the free en-
ergy.
For L12-type superstructure, one obtains:
MODELLING OF PHYSICAL KINETICS OF RELAXATION IN F.C.C.-Ni—Fe ALLOYS 915
2 2
prm
2 2
2 2
2 2
3
1
( ) ( )4 4
ln ,
3 ( ) 2 ( )
1
4 4
( )(1 ) ( ) (1 )
1
3
(1 ) ( )
16
X X
B X X
s
B X
c c
w J s
k T J s J s sc c
J c s J c c s s
B
c k T J s
α
ααβ β
α α
ββ αβ
β β α β α β
β β
αα αβ
β α α β β α β β
α αα
β α
η − − − η + σ +η =
η + σ − σ σ − + + η
− σ + − σ +
σ = −
− + η
k k
k k
0 0
k
2 2
2 2
,
( )
( ) ( ) (1 )
1
; (A6)3
( ) ( )
16
X
s
B X X
J s s
J c s J c c s s
B
c k T J s J s s
β
αβ
α α β β
ββ αβ
β β β β β α β α
β ββ αβ
β β β α β α
σ − σ
σ + − σ + σ = − + η σ − σ
k
0 0
k k
and for L10-type superstructure:
2 2
prm
2 2
2 2
2
2
1
( ) ( )2 2
ln ,
( ) 2 ( )
1
2 2
( )(1 ) ( ) (1 )
1
(1 ) ( )
4
X X
B X X
s
B X
c c
w J s
k T J s J s sc c
J c s J c c s s
B
c k T J s
α
ααβ β
α α
ββ αβ
β β α β α β
β β
αα αβ
β α α β β α β β
α αα
β α α
η η − − − + σ +η =
η η + σ − σ σ − + +
− σ + − σ +
σ = − η− + σ −
k k
k k
0 0
k
2 2
2
2
,
( )
( ) ( ) (1 )
1
. (A7)
( ) ( )
4
X
s
B X X
J s s
J c s J c c s s
B
c k T J s J s s
β
αβ
α β β
ββ αβ
β β β β β α β α
β ββ αβ
β β β α β α
σ
σ + − σ + σ = − η + σ − σ
k
0 0
k k
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|
| id | nasplib_isofts_kiev_ua-123456789-75893 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1816-5230 |
| language | English |
| last_indexed | 2025-12-07T18:14:49Z |
| publishDate | 2012 |
| publisher | Інститут металофізики ім. Г.В. Курдюмова НАН України |
| record_format | dspace |
| spelling | Vernyhora, I.V. Bokoch, S.M. Tatarenko, V.A. 2015-02-05T17:29:13Z 2015-02-05T17:29:13Z 2012 Modelling of Physical Kinetics of Relaxation
 in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys / I.V. Vernyhora, S.M. Bokoch, V.A. Tatarenko // Наносистеми, наноматеріали, нанотехнології: Зб. наук. пр. — К.: РВВ ІМФ, 2012. — Т. 10, № 4. — С. 897-916. — Бібліогр.: 37 назв. — анг. 1816-5230 PACSnumbers:61.50.Ah,61.72.Bb,64.60.De,64.60.qe,64.75.Jk,75.30.Hx,81.30.-t https://nasplib.isofts.kiev.ua/handle/123456789/75893 In comparison with classical Lifshitz—Slyozov—Wagner’s theory and, where
 it is possible, with available experimental data, the physical kinetics of evolution
 of a microstructure of f.c.c.-Nі—Fe alloys is simulated by means of the
 Onsager-type equations of microdiffusion and with the account of the effects
 caused by magnetic interatomic interactions and elastic interactions of inclusions
 of phases. Magnetism proper in f.c.c.-Nі—Fe alloys essentially influences
 a tendency to atomic ordering and development of their microstructure;
 magnetic interactions promote stabilisation of a precipitated phase and
 dilate an interval of two-phase coexistence. In Elinvar alloys, elastic interactions
 essentially change morphology of an intermixture of formed phases of
 the superstructural L10 and L12 types (or of structural A1 type), giving the
 anisotropic character to the shape of inclusions of phases as well as to their
 relative spatial arrangement. Meanwhile, for an intermixture of phases of
 structural A1 type and superstructural L12 type in Permalloys, the anisotropic
 effects of such interactions are inappreciable. У порівнянні з класичною теорією Ліфшиця—Сльозова—Ваґнера і, де можливо, з наявними експериментальними даними змодельовано фізичнукінетику еволюції мікроструктури стопів ГЦК-Ni—Fe за допомогою рівнань мікродифузії Онсаґерового типу та з урахуванням ефектів, спричинених магнетними міжатомовими взаємодіями та пружніми взаємодіями
 вкраплень фаз. Властивий стопам ГЦК-Ni—Fe магнетизм істотно впливає
 на тенденцію до атомового впорядкування та розвиток їхньої мікроструктури: магнетні взаємодії сприяють стабілізації фази, що виділяється, та
 розширюють інтервал співіснування фаз парами. Пружні ж взаємодії істотно змінюють морфологію суміші утворених фаз надструктурного типу
 L10 і L12 (або структурного типу A1) в елінварних стопах, надаючи анізотропного характеру як формі вкраплень фаз, так і їхньому взаємному розташуванню; але для суміші фаз структурного типу A1 та надструктурного
 типу L12 в пермалоях анізотропні ефекти таких взаємодій є незначними. В сравнении с классической теорией Лифшица—Слёзова—Вагнера и, где
 возможно, с имеющимися экспериментальными данными смоделирована
 физическая кинетика эволюции микроструктуры сплавов ГЦК-Ni—Fe с
 помощью уравнений микродиффузии онсагеровского типа и с учётом эффектов, вызванных магнитными межатомными взаимодействиями и
 упругими взаимодействиями включений фаз. Присущий сплавам ГЦК-
 Ni—Fe магнетизм существенным образом влияет на тенденцию к атомному упорядочению и развитие их микроструктуры: магнитные взаимодействия способствуют стабилизации фазы, которая выделяется, и расширяют интервал попарного сосуществования фаз. Упругие же взаимодействия существенным образом изменяют морфологию смеси образованных
 фаз сверхструктурного типа L10 и L12 (или структурного типа A1) в элинварных сплавах, придавая анизотропный характер, как форме включений фаз, так и их взаимному расположению, но для смеси фаз структурного типа A1 и сверхструктурного типа L12 в пермаллоях анизотропные
 эффекты таких взаимодействий являются незначительными. en Інститут металофізики ім. Г.В. Курдюмова НАН України Наносистеми, наноматеріали, нанотехнології Modelling of Physical Kinetics of Relaxation in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys Article published earlier |
| spellingShingle | Modelling of Physical Kinetics of Relaxation in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys Vernyhora, I.V. Bokoch, S.M. Tatarenko, V.A. |
| title | Modelling of Physical Kinetics of Relaxation in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys |
| title_full | Modelling of Physical Kinetics of Relaxation in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys |
| title_fullStr | Modelling of Physical Kinetics of Relaxation in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys |
| title_full_unstemmed | Modelling of Physical Kinetics of Relaxation in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys |
| title_short | Modelling of Physical Kinetics of Relaxation in Macroscopically Homogeneous F.C.C.-Ni—Fe Alloys |
| title_sort | modelling of physical kinetics of relaxation in macroscopically homogeneous f.c.c.-ni—fe alloys |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/75893 |
| work_keys_str_mv | AT vernyhoraiv modellingofphysicalkineticsofrelaxationinmacroscopicallyhomogeneousfccnifealloys AT bokochsm modellingofphysicalkineticsofrelaxationinmacroscopicallyhomogeneousfccnifealloys AT tatarenkova modellingofphysicalkineticsofrelaxationinmacroscopicallyhomogeneousfccnifealloys |