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Prediction of spinodal wavelength in continuously cooled metallic liquid.

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Ann. Phys. (Berlin) 18, No. 1, 4 ? 12 (2009) / DOI 10.1002/andp.200810330
Prediction of spinodal wavelength in continuously cooled
metallic liquid
Christine Borchers1,? , Jan Schroers1,2 , and Ralf Busch3
1
2
3
Institut fu?r Materialphysik, Universita?t Go?ttingen, 37077 Go?ttingen, Germany
Mechanical Engineering, Yale University, P.O. Box 208284, New Haven CT 06520-8284, USA
Lehrstuhl fu?r Metallische Werkstoffe, Universita?t des Saarlands, Campus, Geb. C6.3, 66123 Saarbru?cken,
Germany
Received 29 July 2008, accepted 10 October 2008 by B. Kramer
Published online 21 January 2009
Key words Spinodal decomposition, bulk metallic glass, transmission electron microscopy.
PACS 61.25.Mv; 68.37.Lp
The spinodal decomposition of a deeply undercooled metallic liquid Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 between
800 and 700 K is analysed in the framework of the theory of Cahn and Hilliard for continuous cooling, and
the wavelength with maximum amplification is predicted, using as input parameters thermodynamic values gained in experiments. Electron microscopical studies show the microstructure of glass forming alloys
Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 . The as-cast material exhibits a two phase mixture of amorphous regions
with different compositions. Evidence for spinodal decomposition is given, and the computed maximumamplitude wavelength corresponds well with the one found in the experiments.
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
Spinodal decomposition occurs when a mixture of two or more materials unmix into distinct regions with
different material concentrations. This decomposition differs from classic nucleation and growth in that
phase separation occurs throughout the material, and not just at nucleation sites. The concomitant growth
of a composition modulation in an initially homogeneous mixture implies uphill diffusion or a negative
diffusion coefficient. This is the case, when the free energy as a function of concentration has a negative
curvature [1]. The so-called gradient energy coefficient, which replaces the interfacial energy in the case
of spinodal decomposition, ensures that large concentration gradients are levelled out. The reaction can
be treated purely as a diffusional problem and many of the characteristics of the decomposition can be
described by an approximate analytic solution to the diffusion equation. The only prerequisite is that the
values for the effective diffusion coefficient, or, in lieu thereof, the free energy of mixing, and of the
gradient energy coefficient have to be known.
In recent years, multicomponent alloy systems have been found with an excellent resistance against crystallization that form bulk metallic glasses. Examples are the La-Al-Ni [2] the Zr-Al-Ni-Cu [3] and the ZrTi-Cu-Ni-Be [4] alloy systems of which the latter one has a critical cooling rate as low as 1 K/s [5]. Amorphous ingots typically up to 10 ? 20 mm in the smallest dimension can be produced. Unmixing phenomena
in metallic glasses have been reported since long, see e.g. [6?12]. Numerous investigations have been performed that correlate the microstructure of crystallized metallic glass forming alloys with the nucleation
and growth behaviour, see e.g. [13?15]. In contrast, transmission electron microscopy (TEM), [16?18],
small angle neutron scattering (SANS), [19?22], and atom probe field ion microscopy (AP/FIM) [9,23,24]
?
Corresponding author
E-mail: chris@ump.gwdg.de, Phone: +49 551 395584, Fax: +49 551 395012
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Ann. Phys. (Berlin) 18, No. 1 (2009)
5
studies were performed that reveal a periodic spatial arrangement of regions of different composition which
was taken as an indicator for a spinodal-type chemical decomposition in the liquid phase. Further evidence
for spinodal decomposition in the liquid is given by the theoretical study by Assadi and Schroers [25], who
find that observed nucleation densities in crystallization of, e.g. Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 , can be well
understood as the result of preceeding spinodal decomposition in the undercooled liquid. Since the initial
homogeneous state is near a deep eutectic [9, 11, 14], i.e. a liquid mixture at such composition that the
melting point is as low as possible, and all components crystallize simultaneously into crystalline phases
with compositions more or less far from the initial one, it is consensus that decomposition in the liquid
state eases subsequent crystallization, since the composition of at least one of the liquid phases resulting
from decomposition is nearer to the composition of crystalline phases than the homogeneous liquid.
Spinodal decomposition has been analysed numerous times by computer simulations [26, 27], and theoretical calculations [28?32]. Further, experimental results gained by TEM, AP/FIM and SANS have been
analysed to derive thermodynamic values subsequently (summarised in Wagner and Kampmann [33]), but
to our knowledge, spinodal decomposition has up to date not been predicted in detail a priori. In this work,
this decomposition is analysed in the framework of the theory of Cahn and Hilliard for continuous cooling,
and the wavelength with maximum amplification is computed. The microstructure of the as-prepared alloy
is studied. Evidence for spinodal decomposition in the as-prepared material is given. The free energy of
mixing is known from experiments [9, 23], and the gradient energy coefficient is computed with the aid of
a pair interaction model [34]. It is the aim of this work to demonstrate that it is possible to predict spinodal
decomposition of alloys in a straightforward manner, as soon as crucial thermodynamic parameters are
known.
2 Calculation of spinodal wavelength
Cahn and Hilliard [1] published the theory of spinodal decomposition: Deviations from homogeneous
composition are premised, expanding the composition field into Fourier components, which in the onedimensional form reads as follows:
c(x, t) = co + A exp(??x + Rt)
(1)
with c the local concentration, co the initial concentration, ? = 2?/? the wave vector, A the amplitude of
initial fluctuations, and R the time factor. When the gradient energy of the diffusion process is taken into
account in absence of strain, the time factor R is negative for large ?, levelling out short wavelengths:
R(?) = ?De? ?
2
1 + 2?
?2f
?c2
?1
?
2
(2)
where De? is an effective interdiffusion coefficient, ? the gradient energy coefficient, and f the free energy
density. This expression for R(?) is valid when no stress contributions have to be included, which, as a first
approximation, is true for liquids. Concentration fluctuations with large wavelengths are expected to grow,
whereas fluctuations with short wavelengths are expected to vanish, and there is a wavelength ?max with
maximum amplification. Note that the interdiffusion coefficient De? merely affects the height of the peak,
but not its position in ?. However, when spinodal decomposition does not occur in isothermal surroundings
but during continuous cooling, the temperature dependence of the free energy density f has to be taken into
account. This has been done by Huston, Cahn and Hilliard [35]. The temperature dependence is now given
by an explicit consideration of the entropy, and they give the following expression for the wave vector ?max
corresponding to the wavelength ?max that is most strongly amplified. For the case, that the cooling rate is
slow enough for spinodal decomposition to be completed before the final temperature is reached, this can
now be given as follows:
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C. Borchers et al.: Prediction of spinodal wavelength
2
?max
=
=
kB ?T E ? 2 s
3E 2kB ? ?c2
?T ? 2 s
6? ?c2
(3)
where ?T is the temperature range in which the decomposition occurs, E is the activation energy for
atomic mobility, s the entropy density, and kB the Boltzmann constant. The gradient energy coefficient ?
can be given by ? = a2 , where is an interaction energy and a an interaction distance. The interaction
energy can in turn be given in the framework of the regular solution model: f = ?c2 ? T ?s, assuming a
parabolic course of the internal energy around the initial concentration [34, 36].
We will now demonstrate that all necessary values are known for the prediction of spinodal decomposition of Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 liquid during continuous cooling. Busch and coworkers [9, 23]
studied Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 bulk metallic glasses in the as-prepared state. They found that the undercooled melt decomposes during cooling in the liquid state and forms a two phase mixture of amorphous
regions, where the one is Be-rich and Zr-poor, while the other one is Zr-rich and Be-poor. The numbers of
Ti, Ni, and Cu atoms do not vary in the amorphous regions. The length scale of this composition fluctuation
is of the order of tens of nanometers. In atom probe measurements, they could establish a concentration
jump of about �% Be and � Zr. They observed a heat release of ?f ? 150 J/g-atom between 800
and 700 K, which they ascribed to unmixing in the undercooled liquid during the cooling process, which
leads to ?T ? 100 K [9, 23].
The glass transition temperature of Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 is at around 620 K [5], so the unmixing is actually taking place in the undercooled liquid regime. The entropy of a liquid is given by the
entropy of mixing, a vibrational component, and a structural component. The total entropy was determined
as 16 J/(g-atom K) [37]. Calculating the entropy of mixing for Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 with the aid
of s = ?NA kB (cZr ln cZr + cTi ln cTi + cCu ln cCu + cNi ln cNi + cBe ln cBe ) yield s ? 12 J/g-atom K
(NA : Avogadro constant). Since the vibrational as well as the structural entropy contributions can be assumed to be somewhat smaller than the entropy of mixing [38?40], this result seems reasonable. As the
latter contributions are to the first approximation independent of the local concentration [41], the second
derivative of the entropy can be given as follows:
1
1
?2s
? 56 J/g-atom K
(4)
= ?NA kB
+
?c2
cBe
cZr
when, as found in experiments [9, 23], only Be and Zr are participating in the unmixing process.
The interaction energy is given in the framework of the regular solution model as ?f = ?c2 ? T ?s.
Since to the first approximation there is no structural change in unmixing undercooled liquids, ?s is the
change in configurational entropy density, given by ?NA kB (?cBe ln ?cBe + ?cZr ln ?cZr ) ? 3.6 J/gatom. With ?cBe ? 0.15, ?cZr ? 0.05 and T ? 750 K, the interaction energy can easily be calculated
as ? 125 kJ/g-atom. For the interaction distance a we choose the hopping distance of Be atoms in
the undercooled liquid Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 , given by Tang et al. [42] as 0.45 nm. Whereas for
diffusion in the bulk metallic glass hopping of individual Be atoms was established [42], atomic motion
in the undercooled liquid requires collective motion of about 10 atoms [42, 43]. Incidentally, the volume
of ten atoms (Zr: 4, Ti, Cu, Ni: one each, Be: 3) is 1.25 � 10?28 m3 , the side length of the corresponding
cube being 0.5 nm. This, too, confirms that 0.45 nm is a reasonable interaction distance for undercooled
2
Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 liquid. These values (?T = 100 K, ? = 3.4 � 10?9 J/m, ??c2s = 7.5 �
106 J/m3 K), put into Eq. (3), yield a wavelength with maximum amplification: ?max ? 30 ? 35 nm.
It should be stressed that the thermodynamic input values from this prediction are gained entirely from
experimental results, and not from computed databases.
To estimate the maximum amplitude is difficult: Since the diffusion coefficient De? goes exponentially
into the amplification factor, see Eq. (1), an exact evaluation seems to deliver rather fortuitous results. In
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Berlin) 18, No. 1 (2009)
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Fig. 1 TEM micrograph of crystalline inclusions in the amorphous matrix. The amorphous matrix shows no contrast, except for
large-scale dark streak near the inclusions. This is due to thickness
augmentation of the amorphous bulk near the inclusions as an effect
of ion-milling.
contrast, the prediction of the wavelength ?max is quite robust: all values except the interaction distance a
enter as square roots into the calculation of ?max , a entering in linear form.
3 Comparison with experiments
Amorphous Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 samples were prepared by arc-melting the constituents with a
purity ranging from 99.5 at% to 99.995 at% in a titanium gettered argon atmosphere. The maximum cooling
rates were 10 K/s. We performed transmission electron microscopy (TEM) on a Philips EM 420 ST. The
electron microscopical techniques used were conventional bright-field TEM, and hollow cone dark field
(HCDF) imaging, the latter being a chemically sensitive technique [44]. Samples were prepared by cutting
discs of 3 mm diameter and 0.5 mm thickness from the bulk sample by spark erosion, followed by dimpling
and subsequent ion milling. During the milling process, the samples were cooled with liquid nitrogen to
avoid heating of the samples.
Figure 1 shows a bright-field TEM micrograph of an as-prepared sample. Crystalline inclusions can be
seen in the amorphous matrix. The inclusions are facetted with a diameter of about 500 to 1000 nm. They
have a surrounding shell of further crystalline material. The volume fraction of the crystalline inclusions
can be estimated to be well below 1% from SEM micrographs (not shown). It is known that in Al94 Ti5 B1
alloys, Al3 Ti particles form during solidification with protective shells of TiB2 that keep the Aluminide
from further growth [45]. It is likely that a similar mechanism is active here, i.e. the inclusions form
in the melt, but a protective shell of another composition prevents them from growing. The presence of
similar crystalline phases in as-quenched glass have been reported for Zr52 Ti5 Cu18 Ni15 Al10 [18] and for
Cu50 Zr50 [46]. It should be stressed that these inclusions do not seem to trigger further crystallization, and
have no influence on a possible spinodal unmixing at temperatures around 750 K. In the context of this
work, they are included to demonstrate that thickness variations within the sample can be seen in brightfield micrographs: The darker and brighter regions in the amorphous matrix reflect thicker and thinner
regions in the sample. It is highly probable that in the vicinity of the crystalline inclusions, the sample
remains a bit thicker upon ion milling.
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C. Borchers et al.: Prediction of spinodal wavelength
Fig. 2 HCDF micrograph of a region without inclusion. The wavy
contrast is due to concentrational fluctuations.
Figure 2 shows a HCDF image of a region without inclusions. A wavy contrast not seen in Fig. 1 is
apparent, with dark or bright structures of several 100 nm length, being about 40 nm apart. Since the HCDF
technique is chemically sensitive and suppresses strain contrast [44], the observation can be interpreted as
compositional fluctuations with a regular spatial arrangement. Both micrographs were recorded from the
same sample in the same TEM session, so that it can be excluded that the differences seen between Fig. 1
and Fig. 2 is due to different samples and/or some preparation artefact, such as thickness variations on the
50?100 nm scale. Since the total examined volume is no bigger than about 1 ?m2 � 20 nm, it is also highly
improbable that the micrographs were recorded from regions with different thermal history.
4 Discussion
Since the negative enthalpy of mixing is one empirical rule for glass formation, a repulsive interaction between constituents, necessary for spinodal decomposition, is not expected in glass forming alloys.
This issue was discussed in detail by Abe and co-workers [47]: In multi-component systems, homogeneous solutions can decompose into solutions with different compositions even if the mixing enthalpy
is all negative, when one of the binary mixtures is significantly more negative compared to the others [47]. With calculations using the sub-regular solution model, they found phase separation in numerous ternary glass-forming alloys, in particular in the ternary ZrTiCu system [47], where the glass-forming
region and the two-phase region overlap. From this point of view, it is not surprising to find unmixing in
Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 , containing Zr, Ti, and Cu.
As stated above, Busch and coworkers [9,23] found a recalescence effect during cooling of Zr41.2 Ti13.8
Cu12.5 Ni10 Be22.5 from the melt. This is a glowing in a cooling metal caused by the release of heat of transformation, which is obviously an unmixing process. Whether this process is of spinodal type, or nucleationand-growth type, is determined by the local curvature of the free energy as a function of the concentration.
A convex curvature, prerequisite for spinodal decomposition, is encountered when it is appropriate to describe the result of decomposition as one single phase with two (meta-)stable compositions, lowering the
free energy as compared to the mixed state. Since as a first approximation no structural change is encountered, it seems reasonable to choose one amorphous phase with a local maximum between two minima, as
sketched in Fig. 3. In the past, unmixing in amorphous alloys has been described thermodynamically as
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Berlin) 18, No. 1 (2009)
9
Fig. 3 Schematic Gibbs free energy diagram of the undercooled
liquid state with respect to two crystalline phases.
the existence of two distinct amorphous phases [7,8]. But, as for one crystalline phase exhibiting unmixing
tendency like CuCo [34], one should chose one amorphous phase with a convex maximum between two
concave minima.
We are aware of the fact that Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 is not a binary mixture, but nonetheless,
we use the theory of decomposition of binary mixtures. Since according to experiments [9, 23] mainly the
species Be and Zr are involved in the decomposition process observed during continuous cooling, it seems
legitimate to use a theory for binary mixtures. Also the wavy structures in the micrograph Fig. 2 suggest
that the theory can locally be used in its one-dimensional form. The predicted wavelength and the observed
one are in satisfactory agreement, confirming the procedure chosen in this work. According to AP/FIM
studies [48], the species Be and Ti are involved in spinodal decomposition of Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 ,
but this study is only partly comparable, since they subjected the samples to an isothermal heat treatment
and not to continuous cooling. As was elegantly shown by Zhao and Notis [49] for a ternary Cu77 Ni15 Sn8
(wt.%) alloy, a multitude of different phases with and without spinodal decomposition and/or ordering can
be observed in this one alloy, depending on the respective heat treatment. All the more can such a behaviour
be expected for a quinternary alloy such as Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 . Therefore, a comparison with
results obtained by other groups is only of limited value.
There is some controversy over the question whether spinodal decomposition is encountered in
Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 at all. Martin and co-workers [50] investigated an alloy with nearly the same
composition and found no spinodal decomposition in the as-quenched state whatsoever. They prepared
their samples by the melt-spinning technique, which ensures high quenching rates on the order of 105 K/s.
In contrast, the samples reported of in this work were quenched at rates up to 10 K/s. Therefore, it is
not surprising to find decomposition upon cooling in the slowly cooled material, and none in the rapidly
quenched. On the other hand, newest results of Way, Wadhwa & Busch [51], who investigated the influence
of shear rate and temperature on the viscosity and fragility of the Zr41 Ti14 Cu12 Ni10 Be23 liquid, found effects that can most plausibly be explained by a phase separation in the undercooled liquid upon cooling.
This also corroborates the decomposition stated in this work.
Isothermal ageing of Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 bulk metallic glasses at temperatures between 621 K
and 661 K were studied with SANS [19, 20, 22]. In contrast to Martin et al. [50], it was found that at
these annealing temperatures a spinodal-type decomposition takes place in the undercooled liquid prior
to crystallization, and the wavelength of the decomposition grows with increasing annealing temperature.
Furthermore, wavelengths significantly smaller than the ? 40 nm found in the as-quenched alloys in this
work were observed, namely in the nm range. This shows that spinodal decomposition during continuous
cooling has a quite different quality than isothermal decomposition. The reason for this is that the properties
of the decomposition depends upon cooling rate instead of holding temperature, and the distribution of
wavelengths around the one receiving maximum amplification is less sharply peaked [35]. In studying
meteorite material, it is possible to reconstruct the thermal history by evaluating the microstructure [52,53].
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C. Borchers et al.: Prediction of spinodal wavelength
In order to estimate cooling rates of meteorite material, Weinbruch and Mu?ller [52] performed extensive
modelling of spinodal growth of so-called exsolution lamellae of pigeonite/diopside, two silicates. Their
qualitative results are, however, applicable to metallic liquids as well. They computed continuous timetemperature-transformation diagrams for cooling rates covering four orders of magnitude. The main result
is the following: At the beginning of the cooling, the peak wavelength grows, but after a short time, which
is itself depending on the cooling rate, the peak wavelength will stop growing. Interestingly, they find a
strong growth of the peak wavelength when the cooling rate falls. On the other hand, Huston, Cahn and
Hilliard [35] state that for slow cooling rates, a change of the rate itself does not affect the peak wavelength
strongly any more. This is the case, when spinodal decomposition is finished during cooling, which seems
to be true here, as the observed heat release is completed well before the glass temperature Tg is reached.
It is interesting to note that in crystallization experiments, the length scale for crystallization is strongly
dependent on the crystallization temperature [14, 15], and the observed length scales at 700 K and 600 K
correspond nicely to the spinodal length scales reported in this work and in literature (cited above) for
the respective temperatures. Additionally, Be-rich ZrBe2 and Zr-rich Zr2 Cu are the phases observed in the
primary crystallization process of Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 [16, 50, 54]. This seems to confirm that
crystallization is indeed related to decomposition into Be-rich and Be-poor areas in the undercooled liquid.
The decomposition process is always overshadowed by nanocrystallization in one of the decomposed
amorphous phases, after an incubation time to that decreases with increasing temperature. When the quench
rate is sufficiently high, crystallization can be suppressed during cooling to Tg . Hu and Mathot [55] have
shown that in the case of polymer blends, liquid-liquid demixing can be driven solely by the componentselective crystallizability, i.e., that demixing in the melt facilitates subsequent crystallization. This is consistent with the finding of Assadi and Schroers [25], that only a spinodal decomposition in the melt can
account for the number densities of crystallites found in experiments. It may be necessary, though, to develop new thermodynamic models to account for the resulting driving forces. One possible approach has
been sketched by Lohwongwatana et al. [56]. They suggest that a homogeneous bulk metallic glass forming liquid can lower its overall viscosity by phase separating into two distinct liquids. These, in turn, have a
higher crystallization probability than the homogeneous melt. As a reason for this, they specify that internal
stresses lead to a loss of viscosity [56]. Greer [57] analysed stress effects on interdiffusion in amorphous
materials. He stated that diffusional asymmetry can lead to significant stress. There is certainly a diffusional
asymmetry in the interdiffusion of Zr and Be in undercooled Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 . At 750 K, the
diffusion coefficient of Zr can be estimated to be on the order of 10?19 m2 /s [58], whereas that of Be
can be estimated to be on the order of 10?16 m2 /s [42]. Guerdane and Teichler [59] examined the structure of an amorphous, massive-metallic-glass forming NiZrAl alloy with molecular dynamics simulations.
Two of their results are of interest here: First, they observe medium range order associated with clustering tendencies of Al, and second, they observe atomic-level Mises shear stress accompanied with density
fluctuations, the latter also accompanying the respective atomic species. The latter result of Guerdane and
Teichler [59] may lead to the assumption that these fluctuations are, when the critical temperature is passed
during the quench, initially amplified. However, in accordance with Weinbruch and Mu?ller [52], very soon
larger wavelengths receive preferential amplification, leading eventually to the observed fluctuation lengths
of about 30 nm. Another aspect of the diffusional asymmetry between Be and Zr is demonstrated by Mao
et al. [30]: Such asymmetry triggers a dramatic decrease in connectivity of the respective decomposed regions, leading to isolated structures that are more or less aligned in locally parallel arrays, as can be found
in Fig. 2 of this work. The wavevector subjected to maximum amplification is, in contrast, not affected.
These last considerations show that the thermodynamics used in this work may be somewhat simplifying, since, as always, real circumstances are a bit more complicated. On the other hand, it is known that
chemical contributions to the free energy of a non-crystalline system are mostly the greatest contributions
by far (see, e.g., Spaepen [60]), and the same holds for the configurational contributions to the entropy.
Therefore, the approach chosen in this work can be expected to yield results not too far from what is
reasonable, a statement that is corroborated by the experimental results presented in this work.
c 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Berlin) 18, No. 1 (2009)
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5 Conclusions
The wavelength of spinodal decomposition of Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 was predicted using the CahnHilliard theory modified for continuous cooling. The necessary thermodynamic values were derived from
experiments. The predicted wavelength is 30 ? 35 nm, which is in satisfactory agreement with our observations gained by electron microscopy, where 30 ? 40 nm are observed in Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5
after continuous cooling. The applied method to predict spinodal wavelengths expected after continuous
cooling with low cooling rates seems to be appropriate.
Acknowledgements This work was supported by the National Aeronautics and Space Administration (Grant No.
NCC8-119), which is gratefully acknowledged.
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