Fe-Ni invar alloy: an example of fractal decomposition?
The important experimental results of irradiated and unirradiated Ni-Fe alloys and some approaches to explain the various contradictory behaviours are discussed. We outline the known results of the scattering experiments and suggest possible directions for theoretical investigations to explain the d...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-804042015-04-18T03:01:28Z Fe-Ni invar alloy: an example of fractal decomposition? Abromeit, C. Ananthakrishna, G. Физика радиационных повреждений и явлений в твердых телах The important experimental results of irradiated and unirradiated Ni-Fe alloys and some approaches to explain the various contradictory behaviours are discussed. We outline the known results of the scattering experiments and suggest possible directions for theoretical investigations to explain the decomposition kinetics of the FeNi Invar alloy. Обговорюються важливі експериментальні результати опромінених та не опромінених NiFe-сплавів, а також деякі апроксимації для пояснення різних суперечливих характеристик. Ми підкреслюємо відомі результати, отримані в експериментах з розсіянням, та пропонуємо можливі напрямки для теоретичних досліджень, щоб пояснити кінетику розкладу FeNi ИНВАР сплава. Обсуждаются важные экспериментальные результаты облученных и необлученных Ni-Fe-сплавов, а также некоторые аппроксимации для объяснения различных противоречивых характеристик. Мы подчеркиваем известные результаты, полученные в экспериментах с рассеянием, и предлагаем возможные направления для теоретических исследований, чтобы объяснить кинетики разложения FeNi ИНВАР сплава. 2005 Article Fe-Ni invar alloy: an example of fractal decomposition? / C. Abromeit, G. Ananthakrishna // Вопросы атомной науки и техники. — 2005. — № 3. — С. 32-37. — Бібліогр.: 33 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/80404 539.438 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Физика радиационных повреждений и явлений в твердых телах Физика радиационных повреждений и явлений в твердых телах |
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Физика радиационных повреждений и явлений в твердых телах Физика радиационных повреждений и явлений в твердых телах Abromeit, C. Ananthakrishna, G. Fe-Ni invar alloy: an example of fractal decomposition? Вопросы атомной науки и техники |
| description |
The important experimental results of irradiated and unirradiated Ni-Fe alloys and some approaches to explain the various contradictory behaviours are discussed. We outline the known results of the scattering experiments and suggest possible directions for theoretical investigations to explain the decomposition kinetics of the FeNi Invar alloy. |
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Article |
| author |
Abromeit, C. Ananthakrishna, G. |
| author_facet |
Abromeit, C. Ananthakrishna, G. |
| author_sort |
Abromeit, C. |
| title |
Fe-Ni invar alloy: an example of fractal decomposition? |
| title_short |
Fe-Ni invar alloy: an example of fractal decomposition? |
| title_full |
Fe-Ni invar alloy: an example of fractal decomposition? |
| title_fullStr |
Fe-Ni invar alloy: an example of fractal decomposition? |
| title_full_unstemmed |
Fe-Ni invar alloy: an example of fractal decomposition? |
| title_sort |
fe-ni invar alloy: an example of fractal decomposition? |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2005 |
| topic_facet |
Физика радиационных повреждений и явлений в твердых телах |
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http://dspace.nbuv.gov.ua/handle/123456789/80404 |
| citation_txt |
Fe-Ni invar alloy: an example of fractal decomposition? / C. Abromeit, G. Ananthakrishna // Вопросы атомной науки и техники. — 2005. — № 3. — С. 32-37. — Бібліогр.: 33 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT abromeitc feniinvaralloyanexampleoffractaldecomposition AT ananthakrishnag feniinvaralloyanexampleoffractaldecomposition |
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2025-07-06T04:21:43Z |
| last_indexed |
2025-07-06T04:21:43Z |
| _version_ |
1836869960281882624 |
| fulltext |
УДК 539.438
Fe-Ni INVAR ALLOY: AN EXAMPLE OF FRACTAL
DECOMPOSITION?
C. Abromeit1 and G. Ananthakrishna2
1Hahn-Meitner Institut GmbH, Glienicker Str. 100, D-14109 Berlin, Germany;
2Materials Research Centre, Indian Institute of Science, Bangalore 560012, India
The important experimental results of irradiated and unirradiated Ni-Fe alloys and some approaches to explain
the various contradictory behaviours are discussed. We outline the known results of the scattering experiments and
suggest possible directions for theoretical investigations to explain the decomposition kinetics of the FeNi Invar al
loy.
INTRODUCTION
The purpose of this report is to examine possible
theoretical approaches to some of the unexplained ex
perimental results on decomposition kinetic of the FeNi
Invar alloy. The subject has received considerable atten
tion. There exists in the literature a critical review of the
experimental status on various properties related to the
decomposition kinetics of FeNi alloys at various com
positions as also at the compositions of the Invar alloy
[ Russell and Garner 1988, and Russell and Garner
1989]. On the basis of several measured properties,
these authors suggest that there is a miscibility gap at
the Invar alloy composition which is very narrow. Most
recent experiments of Wiedenmann et al. [1989, 1990,
1992, 1994] appear to be the most reliable evidence for
the decomposition of the alloy even though one should
expect to see the signature of the decomposition in some
direct experiments which one does not (TEM, FIM etc. )
. Even so, we outline the known results of the scattering
experiments and suggest possible directions for theoret
ical investigations.
CRITICAL DISCUSSION
OF THE EXPERIMENTAL RESULTS
ABOUT A FE-NI DECOMPOSITION
In this section we briefly summarize experimental
results on Fe Ni which has been critically reviewed by
Russell and Garner [1988]. Some relevant conclusions
of this study will be recalled below.
The information on the decomposition tendency of
Invar alloys comes from two sources. These alloys ex
hibit a near zero thermal expansion coefficient over a
wide temperature range. For this reason they have been
used in reactors as well. Under these conditions they ex
hibit remarkable resistance to dimensional changes
arising due to void swelling under neutron as well as
heavy ion irradiation. Therefore, some information
about these alloys is available from such studies. Addi
tional information regarding certain properties can also
be obtained from metallic meteorites, which have the
approximate composition of Fe-35 at.%Ni. In the critic
al review by Russell and Garner [1988], the authors ex
amine evidence relevant to high temperature miscibility
gap from variety of properties such as magnetic proper
ties, thermal expansion coefficient, lattice parameter,
electrical resistivity, interdiffusion coefficient and micro
structural tools. Some of the conclusions they arrive at
are important for the discussion attempted here.
Invar alloys exhibit low thermal expansion coeffi
cient below Curie temperature. This gradual transition
gives the volume contraction which approximately can
cels out the volume expansion due to thermal expan
sion. Asano [1969] and Crangle and Hallan [1963] had
attributed this gradual transition from ferromagnetic to
paramagnetic phase to the development of microscopic
compositional inhomogeneities. Further it is known that
the Curie temperature of FeNi alloys increases with Ni
contents. Due to this, lower Ni regions transforms at
lower temperature compared to the regions of higher Ni
contents. Asano suggest domain size of 60 atoms.
In the context of studies of thermal expansion Mor
ita et al [1984] using splat quenching conclude that the
results are consistent with decomposition into high and
low Ni regions.
Lattice parameter measurements also indicate anom
alous behaviour in the composition region of 25 to 45 %
of Ni. Kacho and Asano [1969] find broad diffraction
peaks which are indicative of decomposition into re
gions of different lattice parameter. The mean lattice
parameter is considered as the mean of paramagnetic
higher density lower Ni content regions with ferromag
netic lower density higher Ni content. Elastic constant
C L=C11C 12C44 / 2 also shows anomalous be
haviour up to 200 K above the Curie temperature Tc ac
cording Hausch and Warlimont [1973], while one
should have expected this anomalous behaviour for tem
peratures below Tc. They proposed that the alloy con
tained very small precipitates of Fe3Ni and FeNi. Jago
and Rossiter [1982] also found some extra diffraction
spots.
Kondorsky and Sedov [1960] found anomalously
large residual resistivity for alloys in 30…50%Ni which
could be caused by some kind of fine scale heterogen
eities. Kachi and Asano [1969] on the basis of Moss
bauer spectra measurements came to the conclusion that
this alloy consists of ferromagnetic and antiferromag
netic domains. The electron scattering was attributed to
the scattering from the interface. Thermoelectric poten
tial of paramagnetic FeNi measurements carried out by
Tanji et al. [1978-1979] also show a pronounced low
temperature hump around the same composition where
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other properties also show anomalies. They attribute this
anomaly to fluctuations involving large number of small
Ni rich and Fe rich clusters. They note that the anomaly
persist up to 1073 K.
There is some additional evidence from Mossbauer
spectroscopy suggesting the possibility of an instability
in this composition regime [Becker 1990].
Even though mixing enthalpies of FeNi alloys are
negative indicating ordering rather than phase separa
tion (Kubaschewski et al. [1977]), there is a pronounced
hump around 25% Ni. This might be considered as evid
ence for phase separation in these alloys. On the basis of
some of these studies Tanji and coworkers [1978-1979],
suggest a coherent miscibility gap and coherent spinodal
in the free energy. They also estimate the peak of coher
ent miscibility gap at 1100 K. Russell and Garner
[1989] suggest that the coherent miscibility gap and the
coherent spinodal region span a narrow composition
range and these are depressed well below the chemical
miscibility gap.
Additional important evidence for the existence of
spinodal decomposition comes from the measurement of
interdiffusion coefficient D. The measurement of inter
diffusion coefficient D for Fe-Ni alloys for the entire
range 0…100 %Ni at temperatures 1123…1373 K has
been carried out by Nakagawa et al. [1979]. The results
show a deep minima in D in the Invar region suggesting
the proximity to the spinodal.
It is also known that a disordered Fe3Ni phase with
fcc structure and an ordered FeNi is found in meteorites
of these compositions. The application of energetic
particle irradiation (electrons, neutrons and ions) in or
der to accelerate the diffusion has clearly shown a phase
separation in NiFe alloy of various compositions
[Chamberod et al. 1979, Garner et al. 1986]. Very long
wave lengths were observed under neutron irradiations
[Garner et al. 1986], which could be explained if the
dominant wavelength λmax= 2p/qmax in a spinodal decom
position is shifted due to irradiation-induced mixing
[Abromeit &Krishan 1986, Abromeit & Martin 1999].
However, under proton irradiation such a wavelength
were not observed. Instead a Porod law
dσ q /d~ q−4 was found, which would be in
agreement with the nucleation and growth in a system
with radiation enhanced diffusion [Wiedenmann 1990].
On the basis of several of these results Russell and
Garner [1988] postulate the existence of miscibility gap
and the spinodal boundary which is narrow in the com
position range having the peak around 1200 K.
The above discussion suggests that there is sufficient
evidence that the Invar alloy is undergoing decomposi
tion and the structure of the decomposition changes with
increasing annealing time.
SUMMARY OF RESULTS OF SANS
EXPERIMENTS
In this section we briefly recapitulate the main res
ults of the experiments by Wiedenmann et al.
[1989,1990,1992,1997] and the interpretation by the au
thors. Some effort has been made by these authors to
provide a phenomenological basis for the experiment
ally measured decomposition kinetics of Ni-Fe alloys in
the composition interval 0.26 < x <0.45. The main res
ults to be considered here are taken from the Small
Angle Neutron Scattering (SANS) differential cross-
sections dσ/dΩ obtained from the Fe-34 at. %Ni alloy
after subtraction of the magnetic scattering contribution.
They are [Li et al. 1997] (Fig. 1 and Fig. 2 are examples
taken from Li et al. 1997):
1) The scattering intensities show no pronounced
maximum in the investigated interval of the scattering
vector interval ξmin= 0.03 nm-¹ < q < 1 nm-¹ = ξmax.
2) The scattering intensity follows a potential law q-α
with 1.1 < α < 4.0.
3) The lower value ξmin and the exponent α increases
with the aging time t.
4) Both parameters can be scaled according to ho
mologous time given by th= t exp(2.3eV/kB(1/898 -1/T).
5) The value of α(th) approaches 3 - 4, and ξmin(th) =
α(th) b with b = 0.2.
Fig. 1: The excess scattering intensity for x=0.34 after
annealing at 675°C following power laws (solid line)
with exponents increasing with increasing time (accord
ing Fig 6 of Li et al. 1997)
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Fig. 2. The exponents α(th) as a function of a homolo
gous annealing time th for x=0.34 (open symbols) and
x=0.30 (full symbols). Different symbols characterize
different time-temperature combinations (according Fig
7 of Li et al. 1997)
It is stated by these authors that these results can be
explained only with the assumption of a fractal geo
metry of the decomposed material, with a maximum
concentration fluctuation of Δc = 0.5…4 at%.
However, these results should be compared with the
observations of three other experimental investigations
performed on similar material. A Small Angle X-ray
Scattering (SAXS) experiment by Simon et al. [Simon
1992] could not confirm a decomposition after a long
time annealing, which may be due to a low sensitivity of
SAXS for this special alloy system [Wiedenmann,
1994]. A first FIM-AP investigation of 34 at. %Ni alloy
also failed as a martensitic transformation in the FIM tip
was occurring [Li 1992]. However, using a highly soph
isticated data analysis (repeated smoothing procedure
RSP) Rьsing et al. [Rьsing 1998] could show in a Fe-37
at.%Ni Invar alloy that at 898 K a concentration fluctu
ation develops with increasing amplitude for increasing
annealing time and a size larger than 2 nm. Decomposi
tion could not be detected by Transmission Electron Mi
croscopy (TEM) using the same samples that were used
in SANS experiments [Wanderka, 1992]. It must be
stated here that the scattering power of Fe and Ni are
nearly the same for X-rays, electrons and neutrons. This
might add to the complication.
CRITICAL DISCUSSION OF THE SANS
RESULTS
As for the quantitative aspects, we examine some of
the results of Wiedenmann and coworkers critically to
see what results of the experiments should be taken seri
ously. We present below some discussion on points 2 -
5 of the last section.
α< 2: The low exponent value α is only seen for two
scattering cross sections [Li 1992, Li 1997]. The scatter
ing data corresponding to the lowest value of α ~ 1.08
has not been shown in these papers. In addition, the
scattering vector interval is limited to ξmin= 0.1 nm-¹ < q
< 1 nm-¹ = ξmax and shows significant error bars. A fit by
a slightly curved function e.g., a broad distribution of
frequencies would also be possible. Therefore, there is
no compelling reason to explain values α < 2. Further, if
the basic mechanism is triggered by decomposition of
the alloy, values less than 2 are improbable since this
would amount to the restriction of the concentration
fluctuations (profiles) to two dimensions. However,
scattering is not anisotropic. Instead, it is probable that
the scattering could be from a surface fractal. But an
analysis along similar lines would show that α > 2.
2 < α < 3: This range appears to be the most reliable
region in the estimation of the value of α(t). The interval
of the scattering vector for the scaling regime is ξmin =
0.03 nm-¹ < q <1 nm-¹ = ξmax. Any theoretical attempt
should explain the value of α(t) and ξmin(t).
3 < α < 4: Only data in the scattering vector interval
ξmin = 0.03 nm-¹< q < 0.1 nm-¹= ξmax is shown. They
show a possible maximum at ξmin for α ≈ 3. The value of
α ≈ 4 is shown in the same way as the thermal data in
their figure but this actually corresponds to proton irra
diated results and does not fit in the homologous time.
Since this dependence corresponds to proton irradiated
situation, it also increases the pace of the kinetics by in
creasing the effective diffusion in the sample. This
might perhaps explain the higher value of α = 4.
The scaling according to an homologous time should
be taken as a general guideline but the activation energy
2.3 eV should not be taken too seriously.
It must be pointed out that the above emerging pic
ture is the sum total of the evidence gathered on the
basis of various properties and must be taken with some
caution. A careful reading of the literature will show
that there is considerable scope for interpreting these
results differently and this can lead to conflicting evid
ences.
CLASSICAL APPROACHES
FOR THE DECOMPOSITION OF ALLOYS
The main features of the scattering intensity profiles
will have to examined critically in view of the fact that
these results are not consistent with the principal feature
of the decomposition of other metallic alloys. Before
doing this we will recapitulate some basic facts about
decomposition of alloys.
The conventional approach to modelling an alloy un
dergoing decomposition is based either on nucleation
and growth process described by the LSW theory or by
a spinodal decomposition [Gunton 1983]. The LSW the
ory predicts a well-defined size distribution f(R) of reg
ularly shaped clusters of radii R having sharp interfaces
[Lifshitz and Slyozov 1961]. The corresponding scatter
ing cross section for monodisperse distribution of equal
size is
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dσ qR
d
==V R Δρ3j qR
qR
2
.
This gives for qR << 1 a Guinnier law dσ(qR)/dΩ ~
exp(-q(3R/5)/3), and for large qR >> 1 a Porod law
dσ(qR)/dΩ)~ q-4. However, a specially chosen size dis
tribution f(R) can in principal produce a power law de
pendence of dσ(q)/dΩ in the experimentally observed q
regime by providing appropriate weight factor in the
following way
dσ qR
d
==∫ f R dσ qR
d dR .
Such a distribution should be highly asymmetric
with its peak at very small radii R/<R> ~ 0.1 having a
very long tail for large particle sizes. However, such a
distribution cannot result from the known nucleation
and growth theories [Lifshitz and Slyozov 1961]. This
result is not changed essentially, if a distance distribu
tion G(D,R) of interacting particles of radii R and dis
tance D is taken into account.
It is pertinent to note that the scattering intensity of a
spinodal decomposing system normally exhibits a max
imum in dσ/dΩ at some qmax with approximate power
laws q² for q < qmax and q-4 for q >> qmax. In the conven
tional experiments on decomposition, a power law with
some exponent does not extend over more than an order
in q.
From the above facts, the above authors conclude
that both the LSW and the spinodal decomposition fail.
The SANS data were therefore interpreted in terms of a
fractal-like decomposition [Wiedenmann 1994].
However, the kinetics could not be explained by the au
thors. We will comment on this aspect later.
DIRECTIONS OF POSSIBLE
THEORETICAL APPROACH
From the discussion presented in the previous sec
tion, we believe that any theoretical approach could take
the following points with reasonable level of confid
ence.
a) The discussion in the previous section suggests
that there is sufficient evidence that the Invar alloy is
undergoing decomposition and there is evidence for the
existence of a miscibility gap which is narrow centered
around the Invar alloy composition.
b) The structure of the decomposition changes with
increasing annealing time. As for the quantitative as
pects, we need to explain the change in the exponent as
a function of annealing time with the exponent value α
in the range 2 - 3.
c) It may be important to consider the magnetic as
pects the alloy for the decomposition kinetics.
d) Lastly, it may be crucial to offer some explana
tion of the scaling structure at the early annealing times.
It must be stated that the above experimental results
are intriguing in several respects. First, the scattering in
tensity shows no conceivable maxima which is normally
expected of a decomposing system. Second, this is
coupled with the existence of a scaling regime, even
though the interval is only one to two orders, and which
is rather unusual since there have been no such reports
for alloys undergoing decomposition. It is pertinent to
note here that scaling regimes are seen in conventional
spinodal decomposition from the intermediate time
scales when nonlinearities govern the time develop
ment. In this regime normally dynamic scaling is resor
ted to, where in the typical length scale (which corres
ponds to the mean length of the A rich regime) consti
tutes the key to casting the structure factor in the scaled
form. However, for the present case, this length scale
cannot be fixed due to the absence of the peak in the
structure factor. Even so, it may be still worthwhile to
represent the structure factor for various times in this
fashion by resorting to intelligent guess. In addition, in
the present case, the scaling regime if anything is larger
for smaller times, unlike the spinodal.
So the primary objective of any theory is to give a
reasonable explanation for the existence of scattering in
tensity with only scaling property, (i.e. without a max
ima) for all values of times. Due to this above unusual
scaling feature, it may be worthwhile to split the prob
lem into three parts each of which should be considered
in parallel. They are given according to the time evolu
tion of the decomposition process.
i) Starting from sudden deep quenches of solution
annealed (1513 K , 48 h), the system already contains
extended heterogeneities. The SANS scattering intensit
ies for small q < 3 nm-№ are different for different
samples and were attributed to grain boundaries.
However, they are not considered to influence the de
composition kinetics during the subsequent annealing
[Wiedenmann 1994]. In fact, the existence of initial in
homogeneities may have a bearing on further decom
position.
ii) To consider the time development of the decom
position kinetics by considering the magnetic order
parameter coupled appropriately with the concentration
variable.
iii) It is possible that some additional relaxational
kinetics may be involved. One way of including this is
to carry out a Monte Carlo simulation.
Below we comment on each of the aspect based on
known facts in the literature and some simple calcula
tions that we have carried out.
EFFECT OF LARGE SCALE
HETEROGENEITIES
In the following we attempt to include the effect of
heterogeneities into the early stages of decomposition
by assuming the Cahn-Hilliard's theory. Consider the
early stages of spinodal decomposition where the local
concentration deviation
δc(r,t) = c(r,t) - <c> with the average concentration
<c> is given by
Δc r , t =∫ d 3q exp iqr exp Rq t .
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Here R(q²)=-Mf′′q²- MKq4 is the Cahn-Hilliard amp
lification factor, which is time independent. Due to the
large heterogeneities we assume that that different re
gions of the sample is undergoing decomposition with
slightly different dominant wavelength qmax. To include
the effect qmax, we introduce a distribution of qmax given
by f(qmax) into the above equation. Then the average
structure factor takes the form
¿ S q >=∫dqmax S q , qmax f qmax .
The corresponding average correlation function
<G(r)> is the Fourier transform of <S(qmax >. The res
ults have to be compared with the spatial particle-
particle correlation function for a mass fractal structure
of dimension D [Teixera 1986]
G r ∣fractal=D /4π ξ max
D r D−3 exp −ξ min t .
The three dimensional Fourier transform of this
function gives the desired power law in the range ξmax <
q < ξmin given by
S q =∫G r exp iqr d 3 r=4π∫r 2 [sinqr /qr ] q /ξ max
−D dr .
For D=2 the particle-particle correlation function
G(r)│fractal and G(r) obtained for the early stages of
spinodal decomposition has vary similar form if a broad
distribution of qmax is considered so that the G(r) is
smeared out. This result may be improved if the higher
order correction term according to Cook and Langer
[Gunton 1983] is taken into account. Thus, it appears
that spinodal decomposition in the early stages can be
described by power laws for q > qmax.
COUPLED MAGNETIC
AND COMPOSITIONAL ORDER
PARAMETERS
From the discussion in section 2, it is clear that mag
netic fluctuations are present during the decomposition.
Thus a natural approach would be to consider a coupled
Ginzburg-Landau type description for the two order
parameter variables [Gunton 1983]. Let m(r,t) represent
the magnetic order parameter. Then the spinodal decom
position of the system should be governed by the fol
lowing equations.
dc r ,t
dt
=−Γ c Δ δF c , m
δc
;
dm r , t
dt
=−Γ m Δ δF c , m
δm
.
Here Δ=∇ 2 , Γ is the mobility and F(c,m) is the
free energy density whose form has to be chosen appro
priately for the present problem. The input for this
choice has to be obtained from the literature on the mag
netic properties of the Fe Ni system. In the above equa
tion the relaxation of the nonconserved magnetic order
parameter is much faster than the conserved order para
meter. Thus, the competition between the two relaxing
variables could give rise to much more complicated
time development than in one variable. In the present
case, since the miscibility gap is very narrow, particular
attention must be made to include this feature in the free
energy expression.
FAILURE OF DLA LIKE MODELS
A different approach could be based on a diffusion-
limited cluster-cluster aggregation (DLA). Computer
simulations have shown that irreversible aggregation
processes can lead to ramified fractal aggregates
[Meakin 1983, Kolb et al. 1983]. However, the minim
um fractal dimension D for the Euclidian dimension d=3
obtained by this process is always significantly larger
than 2 ( D=2.4) which would not explain the observed
value α=2. No time dependence of the fractal dimension
with increasing cluster size is predicted.
In our opinion the formation of diffusion limited ag
gregates (DLA) like fractal structure is suspect for vari
ety of reasons. First, diffusion limited aggregates are
formed only in the limit of zero concentration, ie., a
highly fluctuating situation where the aggregation pro
ceeds with single monomers aggregating in an irrevers
ible manner. This clearly is not consistent with the phys
ical picture of an alloy decomposition where neither the
zero concentration (ie., single particles being present in
the neighbourhood of the aggregating point) is satisfied,
nor the particles aggregate in an irreversible way.
Second, the fractal dimension of such a cluster is D=1.7
in Euclidean dimension d=2 and D=2.4 in d=3. Further,
the fractal dimension remains constant as a function of
time. As can be seen these points are against the known
facts in the present case. However, if there is possibility
of the formation of concentration fluctuations with a
ramified geometry, such a situation cannot be ruled out.
Granting the formation of DLA like structure, fur
ther evolution of these initial structures cannot be de
scribed in the framework of the theories given above.
The structures are determined by the overall growth and
additional atomic rearrangements on local scale due to
thermal relaxation. They contain a dissolution and re-
aggregation of atoms and/or atom diffusion into regions
of higher, local concentrations but smaller concentration
gradients. There are several examples of a thermal re
laxation reported in the literature [References in Irisawa
et al. 1985, Toyoki 1985]. They predict the evolution of
a ramified structure into a more compact form.
MONTE CARLO SIMULATION
In parallel to the analytical approaches, it is neces
sary to undertake a Monte Carlo simulation to get fur
ther insight into various mechanisms leading to ramified
fractal structure and also to understand increase in
fractal dimension as a function of time. Furthermore, in
order to describe the transition from an spinodally cre
ated ramified structure into a more compact cluster-like
structure, an appropriate description of the geometrical
atomic arrangements is necessary. In the presence of
small scale inhomogeneities arising from the magnetic
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aspects of the alloy, different types of relaxation effects
could be envisaged. It is worthwhile to investigate the
effect of such relaxations on the kinetics
[Ananthakrishna 1996]. From the experimental side,
only the structure factor S(q) in the limited scattering
vector interval ξmin< q < ξmax is known. The required
quantity is therefore the spatial density-density correla
tion function G(ΔR)= <ρ(ΔR+R)ρ(R)> with the main
contribution in the length scale ξmin
-¹> l > ξmax
-¹ because
S(q) is the Fourier transform of G(ΔR). It has to be
checked whether it is possible to use the computer simu
lation results to construct the structure factor S(q), but
also to model the initial and the final state of the struc
ture.
COARSENING REGIME
In the final state of the decomposition, the alloy con
sists of well-shaped clusters with sharp interfaces
between the matrix phase and the precipitated phase.
These clusters are quite large and compact. Their contri
bution to the SANS scattering intensity are well de
scribed by a Porod law dσ(q)/dΩ~ q-4. This late stage is
obtainable under irradiation. The proton irradiation ac
celerates the relaxation into a more compact form by
direct ballistic re-dissolution inside the ramified struc
ture, but also be an enhanced growth and coarsening due
to enhanced-diffusion.
DECOMPOSITION UNDER ENERGETIC
PARTICLE IRRADIATION
An interesting test of this model is provided by the
ion- and neutron irradiation results. The ion-irradiation
already destroys the initial state of the decomposition in
the spinodal region leading to long-wave fluctuations.
Such process can be understood assuming a underlying
thermal spinodal decomposition, however with a shift of
a length scale due to ballistic diffusion under irradiation
(Abromeit and Krishan 1986, Abromeit and Martin
1999). No thermal relaxation occurs in such structure.
The fluctuation is conserved also after long times,
which is in accordance with the experimental findings.
REFERENCES
1.Papers in `On growth and form', eds. H. Stanley and
N. Ostrowsky. Martinus Nijhoff, Hague. The Nether
lands. 1986.
2.C. Abromeit and K. Krishan //Acta Metall. 1986,
v. 34, p. 1515.
3.C. Abromeit and G. Martin //J. Nucl. Mater. 1999, v.
271&272, p. 251.
4.G. Ananthakrishna and S. J. Noronha //Physica A.
1996, v. 224, p. 412.
5.E. Becker //Thesis, University Duisburg. 1990.
6.G. Crangle and G.C. Hallam //J. Proc. Rov. Soc. Ja
pan. 1963, v. A 272, p. 119.
7.Chamberod, J. Langier, and J.M. Penisson //J. Mag.
Mat. 1979, v. 10, p. 139.
8.F.A. Garner, H.R. Brager, R.A. Dodd and T. Laur
itzen //J. Nucl. Inst. Meth. 1986, v. B 16, p. 224.
9.J.D. Gunton, M. San Miguel and P.S. Sahni in `Phase
transitions and critical phenomena, Eds. C. Domb and
J.L. Lebowitz (Academic, London, 1983),
v. 8.
10.G. Hausch and H. Warlimont //Acta Metall. 1973,
v. 21, p. 401.
11.S. Kachi and H. Asano //J. Phys. Soc. Japan. 1969,
v. 27, p. 536.
12.W. Klein //Phys. Rev. Lett. 1990, v. 65, p. 1462.
13.K. Krishan and C. Abromeit //J. Phys. F. 1984, v.
14, p. 1103.
14.M. Kolb, R. Jullien and R. Botet //Phys. Rev. Lett.
1983, v. 51, p. 1119.
15.E.I. Kondorsky and V.L.Sedov //J. Appl. Phys. 1960,
v. 31, p. 331S.
16.T. Irisawa, M. Uwaha and Y. Saito //Europhys. Lett.
1995, v. 30, p. 139.
17.R.A. Jago and P.L. Rossiter //Phys. Stat. Sol. (a).
1982, v. 72, p. 497.
18.Q. Li //Thesis, TU Berlin. 1992, D83.
19.Q. Li, A. Wiedenmann and H. Wollenberger
//J. Materials Researc. 1997, v. 12, p. 83.
20.I.M. Lifshitz and V.U. Slyozov //J. Phys. Chem.
Solids. 1961, v. 19, p.35.
21.P. Meakin //Phys. Rev. Lett. 1983, v. 51, p. 1123.
22.J. Rьsing, V. Naundorf, N. Wanderka and H.
Wollenberger //Ultramicroscopy. 1998, v. 73, p. 267.
23.K.C. Russell and F.A. Garner //Metallurgical Trans
action. 1988.
24.J.-P. Simon, O. Lyon, F. Fandot, L. Boulanger and
O. Dimitrov //Acta metall. 1992, v. 40, p. 2693.
25.Y. Tanji, H. MOriya and Y. Nakagawa //J. Phys.
Soc. Japan. 1978, v. 45, p. 1244.
26.Y. Tanji, Y. Nakagawa, Y. Saito, K. Nishimura, and
K. Nakatsuka //Phys. Stat. Sol. (a). 1979, v. 56, p. 513.
27.H. Toyoki and K. Honda //Phys. Lett. 1985, v. 111,
p. 367.
28.T. Viscek `Fractal growth phenomena' //World Sci
entific. London, 1988.
29.N. Wanderka //Unpublished, 1992.
30.Wiedenmann, Q. Li, W. Wagner and W. Petry
//Physica B. 1992, v. 180 & 181, p. 793.
31.Wiedenmann, W. Wagner and Wollenberger in
`Physical Metallurgy of Controlled Expansion Invar-
type alloys, eds. K.C. Russell and D.F. Smith (The Min
erals, Metals & Materials Society, Warrendale, USA,
1990) p. 47.
32.Wiedenmann, W. Wagner and Wollenberger
//Scripta Metall. 1989, v. 23, p. 603.
33.H. Wright, R. Muralidhar and D. Ramakrishna
//Phys. Rev. A. 1992, v. 46, p. 5072.
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Fe-Ni ИНВАР СПЛАВ КАК ПРИМЕР ФРАКЦИОННОГО РАСПАДА
С. Абромайт и Ж. Анантакришна
Обсуждаются важные экспериментальные результаты облученных и необлученных Ni-Fe-сплавов, а так
же некоторые аппроксимации для объяснения различных противоречивых характеристик. Мы подчеркиваем
известные результаты, полученные в экспериментах с рассеянием, и предлагаем возможные направления
для теоретических исследований, чтобы объяснить кинетики разложения FeNi ИНВАР сплава.
Fe-Ni ІНВАР СПЛАВ ЯК ПРИКЛАД ФРАКЦІЙНОГО РОЗПАДУ
С. Абромайт та Ж. Анантакришна
Обговорюються важливі експериментальні результати опромінених та не опромінених NiFe-сплавів, а та
кож деякі апроксимації для пояснення різних суперечливих характеристик. Ми підкреслюємо відомі ре
зультати, отримані в експериментах з розсіянням, та пропонуємо можливі напрямки для теоретичних дослі
джень, щоб пояснити кінетику розкладу FeNi ИНВАР сплава.
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