CHINESE JOURNAL OF PHYSICS
VOL. 50, NO. 2
April 2012
Two-Dimensional Time-Resolved X-ray Diffraction Study of Liquid/solid
Fraction in Fe-C Binary System with an Electrostatic Levitator Furnace
Mitsuharu Yonemura,1, ∗ Junpei Okada,2 Yasuhiro Watanabe,3
Takehiko Ishikawa,2 Susumu Nanao,2 Takahisa Shobu,4
Ayumi Shiro,4 Tomoyuki Fujishiro,4 and Hidenori Toyokawa5
1
Sumitomo Metal Industries, Ltd., Fuso-cho 1-8, Amagasaki, 660-0891, Japan
2
Japan Aerospace Exploration Agency,
Sengen2-1-1, Tsukuba, 305-8505, Japan
3
The University of Tokyo, Komaba4-6-1, Meguro-ku, 153-8505, Japan
4
Japan Atomic Energy Agency, Kouto1-1-1, Sayo-cho, 679-5148, Japan
5
Japan Synchrotron Radiation Research Institute,
Kouto1-1-1, Sayo-cho, 679-5148, Japan
(Received July 29, 2011)
Liquid state provides a function of matter transport or reaction field and plays an important role in manufacturing processes such as refining, forging or welding. However, experimental procedures are significantly difficult for an observation of solidification process
of iron and iron-based alloys in order to identify rapid changing phenomena subjected to
fast temperature evolution. Therefore, in order to study the solidification in iron and ironbased alloys, we considered a combination of high energy X-ray diffraction measurements
and an electrostatic levitation method (ESL). ESL allows us to eliminate the confounding
high-temperature-environment setup problems because it can levitate the specimen without a vessel in a high vacuum, and so ESL is suited to measure X-ray scattering of high
temperatures or undercooled melts. In order to analyze the liquid/solid fraction, the solidification of melted spherical specimens was measured at a time resolution of 0.1 seconds
during rapid cooling using the two-dimensional time-resolved X-ray diffraction. The mole
fraction of solid phases was analyzed as a function of solidification time at each temperature
and experimentally-studied the solidification phenomena for several seconds.
PACS numbers: 61.05.cp; 78.47.jd; 81.05.Bx
I. INTRODUCTION
Liquid state provides a function of matter transport and reaction field which plays
an important role in manufacturing processes such as refining, forging and welding. It is
important for the industrials or materials science society to crystallographically-comprehend
the nature of solidification phenomena. In contrast, the simulation techniques for the
microstructure formation such as the phase-field method [1, 2], which is a mathematical
model for solving interfacial problems, have been developed in a field of model simulation
and property predictions. In order to correlate the mathematical theory and practice, the
∗
Electronic address: [email protected]
http://PSROC.phys.ntu.edu.tw/cjp
243
c 2012 THE PHYSICAL SOCIETY
⃝
OF THE REPUBLIC OF CHINA
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experimental kinetics that synchronizes computing time and real time is important for
the microstructure formation. However, experimental setups are significantly difficult for
an observation of solidification process of iron or iron-based alloys because of their high
melting points, a high evaporation pressures and very fast time-dependent phenomenon.
Some groups have studied the solidification of the pure iron or the iron-based alloys
using X-ray probes. Babu et al. have investigated the primary weld solidification in the
Fe-C-Al-Mn steel weld using the time-resolved X-ray diffraction technique [3]. Their remarkable results were devoted to the heat affected zone of welding. Volkman et al. have
investigated solidification behavior of undercooled Fe-Cr-Ni melts of different compositions
using electromagnetic levitation with respect to the competitive formation of δ-Fe and γ-Fe
phases [4]. The tendency of metastable δ-Fe phase formation in a wide composition range
is confirmed. Recently, Mizuno et al. have performed the time-resolved X-ray diffraction
on solidification behavior of undercooled liquid of binary iron-based alloys [5]. The difference of nonequilibrium phase of Fe-C alloys and Fe-B alloys under solidification using the
gas levitation method was analyzed. Furthermore, Yasuda et al. have observed the dendrite growth with a real space in-situ visualization technique using X-ray imaging [6]. The
dendrite arm fragmentation at δ/γ boundaries was reported in Fe-C binary system alloys.
Authors also have observed solidification and phase transformation processes of weld metals by the in-situ X-ray diffraction technique of a directional solidification, which results
in the dendrite growth during rapid cooling [7, 8]. This technique is significantly effective
for an observation of a weld metal solidification and an observation of early solidification
or nucleation. However, it was very challenging to quantify the phase fraction during rapid
solidification since the integral intensities have some influences of the optical system such
as a reflection mode fixed the incident angle and the crystal preferred orientation due to
the texture. That is why it is significantly difficult to evaluate the integral intensities.
Therefore, experimental environments and conditions were questionable.
Our interest is the details of the growth process of solid phase in a melted iron on
the base of Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation [9–11]. We focus on the
observation of the homogeneous nucleation at an isothermal kinetics. Then, the high temperature and evaporation pressure of iron reduced the specimen size rapidly. Further, the
dendrite growth from internal wall in a crucible influences the integral intensity. Furthermore, X-ray diffraction intensity is significantly weak for high time resolutions in a rapid
crystal growth. Therefore, to facilitate the X-ray transmission method without the diffraction attenuation by the sample container is essential for the study of isothermal transition
or nucleation kinetics in order to obtain better integral intensities during solidification.
Then, the electrostatic levitator (ESL) [12, 13] provides the capability of undercooling and
solidifying metals and alloys in a contactless high vacuum, and quiescent environment. The
levitation significantly reduces the environmental-induced inhomogeneous nucleation from
internal wall, which could yield a lot of more direct information on a homogeneous nucleation during solidification. Especially, the ESL furnace provides much better conditions
for X-ray transmission method at a containerless. Further, with the availability of the
high energy intense X-ray beams from synchrotron radiation facilities and the high sensitive X-ray photon counting pixel detector, it is now possible to directly observe in-situ
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phase transformation and microstructural formation simultaneously in the pure iron and
the iron-based alloys in real-time as a function of cooling times. In the present study, we
analyzed the isothermal transition of solid phase fraction in melts using the two-dimensional
time-resolved X-ray diffraction (2D-TRXRD) system with the ESL. Our target is to obtain
isothermal transformation curves of γ-Fe as a solid phase in the melts experimentally. We
also found phase fractions of γ-Fe during solidification which leads to our experimental
confirmation of the classical nucleation in the solidification theory.
II. EXPERIMENTAL PROCEDURE
II-1. Electrostatic Levitation Furnace
The measurements were made using an electrostatic levitator that consisted of a
chamber with the base vacuum of a ∼ 10−6 Pa before the start of the processing procedure.
The chamber housed a specimen charged by electronic emission and levitated between electrodes via a feedback loop (the two disk electrodes are for the vertical position control and
the four spherical electrodes are for the horizontal control). The positioning control relied
on two sets of orthogonally arranged He–Ne lasers and the associated position detectors.
The three-dimensional specimen position information was fed to a computer that produces
position (x, y, and z) control voltages to high voltage amplifiers so that a prefixed specimen
position can be maintained. The specimen was heated and melted using the focused radiation of three 100 W semiconductor laser beams emitting at a wavelength of 808 nm. This
configuration provided temperature homogeneity, specimen position stability, and helped
to control specimen rotation. Then, the temperature of specimen was measured by a radiation thermometer to obtain cooling curves. The radiance temperature was measured with a
single-color pyrometer (0.90 µm, 120 Hz acquisition rate) covering a 550 ∼ 2500 K interval.
The specimen was observed by a color camera offered a view of both the electrodes and
the specimen. This helped to monitor the specimen position in the horizontal plane and to
align the heating laser beams to minimize photoninduced horizontal specimen movement
and specimen rotation. The features of electrostatic levitation are (i) the suppression of
impurities from outside by a high vacuum and a containerless, (ii) the convenient absorption correction by a rotation of spherical droplet, (iii) the decrease of X-ray absorption
and transmit at high energy by levitated specimens of a 1 ∼ 2mm size, (iv) an installation
available by a compact instrument size and (v) no blind for incidence and diffracted beams.
II-2. Specimen preparation
The pure iron has the melting point at 1538 ◦ C and the evaporation pressure at 1447
◦ C. The specimen evaporates remarkably and the size of specimen reduces by melting. The
liquidus is able to be decreased by an addition of carbon due to the dual phase diagram of
Fe-C. Therefore, the Fe-2mass%C, that passes an eutectic point and has a wide temperature
range of solid-liquid phase equilibrium, was prepared for the candidate material. The ingots
were produced by arc melting of 99.999 mass% electrolytic irons and 99.7 mass% carbon
powders in argon. The specimen with a spherical shape of a diameter of approximately 2
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mm was prepared by arc melting of tips cut from ingot on a copper cooling plate. The
evaporation rate during melting was confirmed to decrease remarkably at the preliminary
test. The specimen component was conclusively expected Fe-1.7mass%C through some
arc-melting processes as we will discuss later.
II-3. Two-dimensional time-resolved X-ray diffraction system
In this study, the solidification of iron-based alloys was dynamically observed by collaboration with an intense X-ray of an undulator beamline, the ESL technique and an X-ray
photon counting pixel detector of PILATUS-100K [14, 15]. The 2D-TRXRD system is a
unique method, which does not only enable a direct observation of the homogeneous nucleation, but can also help to identify the nature of the observed reactions at the boundaries
of phase fields from the real time evolution of the intensities of a particular phase. The
X-ray integral intensity is related to the volume fraction of phases, although the quality of
the diffraction is inferior compared to powder diffraction for textures. Occasional effects
of textures are mostly diminished by the inherent rotation of the levitated polycrystalline
specimen.
Fig. 1 shows the experimental set-up that performed at the BL22XU beamline of
SPring-8 that is the third generation synchrotron radiation facilities in Japan. We tried
the two-dimensional time-resolved X-ray diffraction system (2D-TRXRD) along with our
previous work. However, a transmission mode was applied as the X-ray diffraction system
for a quantification of integral intensities. In the BL22XU, the high intense and highenergy X-rays can be obtained by using an undulator and double crystal monochromator
with {111}Si cooled with liquid nitrogen. The X-ray energy was accurately confirmed 69.5
KeV. Then, the penetration probability of 2 mm length in the iron was estimated at 32
%. The focused monochromatic beam was passed through 0.1 x 0.1 mm slit and irradiated
to spherical specimens that was levitated and melted by the ESL furnace. The diffraction
patterns were collected with a uniquely-sensitive two-dimensional pixel detector PILATUS
that was developed by SPring-8 in collaboration with Paul Scherrer Institute (PSI) in
Switzerland [15]. The PILATUS was mounted on an arm of the four-axis diffractometer
at a distance of 212.6 mm behind a specimen. This 2θ range was optimized to contain six
diffraction peaks from the FCC phase (γ-Fe). That is, the qx of reciprocal space coordinate
(= 4π sin θ/λ) become 0 ∼ 10 /nm. Then, the solidification was dynamically observed at
a time resolution of 0.1 seconds as functions of time and temperature at various cooling
rates.
III. RESULTS AND DISCUSSIONS
III-1. Cooling curves in the ESL furnace
Fig. 2 shows the cooling curves and the equilibrium phase diagram of Fe-C binary
system. The phase transformation sequences were calculated from equilibrium thermodynamic relationships using the THERMOCALC software version R with the SSOL4 solutions
database which is a sophisticated important thermochemical database for many non-ideal
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FIG. 1: Experimental set-up that performed at the BL22XU beamline of SPring-8. The focused
monochromatic beam of 69.5 KeV was passed through 0.1 × 0.1 mm slit and irradiated to spherical
specimens that was levitated and melted by ESL furnace. The diffraction patterns collected with
the uniquely-sensitive two-dimensional pixel detector.
multicomponent solution phases. First, the recalescence is measured after supercooling
by a homogeneous nucleation. The recalescence means that the sudden temperature raise
due to the release of the latent heat of fusion of an supercooled specimen upon solidification. In pure metals, the temperature of solidification after a recalescence maintains a
constant until solidification finish. In contrast, in case of alloys, the specimen temperature
increases steeply until the solidus temperature by the recalescence and decreases rapidly
until the temperature of eutectic point. Since alloys have the temperature range of solidliquid phase equilibrium between the solidus and the liquidus, the solidus of present alloys
is non-equilibrium in the phase rule. The dual system alloys are considered to become an
equilibrium by reaching the temperature of eutectic point with a solid fraction gradient.
Furthermore, in passing the temperature of eutectic point, cooling curves vibrate until passing the precipitation line of Fe3 C carbide. That is, the vibration of cooling curve means the
precipitation and the dissolution of Fe3 C carbides along the precipitation line of Fe3 C carbides comparing with an equilibrium phase diagram. Next, a recalescence is also observed
at 750 ◦ C though the cooling rate is low. It means the uniform solid transformation from
an austenite (γ-Fe) phase to a ferrite (α-Fe) phase along with a homogeneous nucleation
at high temperature. Moreover, we confirmed that the microstructure before melting in
the ESL furnace has many cracks and coarse-grained texture but the microstructure after
solidification in the ESL furnace shows extremely fine and uniformly texture. That is, a
homogeneous nucleation and an uniform transformation are suggested.
The inflection points such as recalescence points or some shoulders were related to
the crystallization of γ-Fe phase, the precipitation of Fe3 C carbides and the solid phase
transformation from γ-Fe phase to α-Fe phase exhibiting at 1200 ◦ C, 1050 ◦ C, 750 ◦ C. Furthermore, the shoulder in the vicinity of 1150 ◦ C corresponds to the temperature of eutectic
point. Consequently, the element of carbon was estimated to be 1.7 mass%. Therefore, the
evaporation of 0.3 mass%C is caused by a formation of CO2 gas during the specimen preparation and experiment. True temperature was estimated with the corresponding between
inflection points of cooling curve and transformation points of equilibrium phase diagram.
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FIG. 2: Cooling curves (a) and equilibrium phase diagram of Fe-C binary system (b). The inflection
points such as recalescence points or some shoulders were related to the crystallization of γ-Fe phase,
the precipitation of Fe3 C carbides and the solid phase transformation from γ-Fe phase to α-Fe phase
exhibiting at 1200 ◦ C, 1050 ◦ C, 750 ◦ C. Further, a shoulder in the vicinity of 1150 ◦ C corresponds
to the temperature of the eutectic point.
Furthermore, the 2D-TRXRD patterns were observed at several cooling rates for the isothermal transition of solid phase fraction. The cooling rates, which are 53 ◦ C/s, 40 ◦ C/s, 26
◦ C/s and 13 ◦ C/s at 1300 ◦ C, were controlled by some heating laser powers.
III-2. 2D-TRXRD patterns during cooling
Fig. 3 shows the example of the 2D-TRXRD patterns that were observed with a
cooling rate of 53 ◦ C/s at 1300 ◦ C and at a time resolution of 0.1 seconds. The bright
regions in the 2D-TRXRD patterns correspond to the high intensity of Bragg reflections of
the crystalline phases. The origin of reciprocal lattice lies on the left hand side parallel to xaxis. The right hand side corresponds to the high q value. In Fe-1.7mass%C, the diffraction
pattern of liquid halo pattern is first observed at approximately 1300 ◦ C as shown in Fig. 3
(a). Next, the spotty γ-Fe phase appears in the halo pattern at 1220 ◦ C as shown in
Fig. 3 (b). With a decrease of temperature, the broaden halo pattern transforms to sharp
diffraction patterns. It means the crystallization of γ-Fe phase. The diffraction of γ-Fe phase
broadens with the residual halo pattern along Laue rings at 1100 ◦ C with a temperature
drop as shown in Fig. 3 (c). After increasing to the temperature of the solidus by the
recalescence, the liquid phase remains at non-equilibrium until the temperature of eutectic
point. At 1000◦ C, the diffraction pattern of Fe3 C carbides appears as shown in Fig. 3 (d).
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At 725◦ C, the diffraction patterns of α-Fe phases appear while the diffraction intensity of
γ-Fe phase decreases as shown in Fig. 3 (e), suggesting the solid-state transformation from
γ-Fe phase to α-Fe phase. Then, a somewhat finer grain size increases the number of Laue
spots. A still finer grain size produces the smooth, continuous Laue ring. That is, the α-Fe
phase continues to increase in amount as the solid-state transformations proceed. Then, the
diffraction patterns of α-Fe and Fe3 C phases mainly coexist as the solid-transformation to
the pearlitic microstructure which consists of the α-Fe phase and the Fe3 C carbide during
cooling is well known. Consequently, it is considered that the formation of continuous
Laue-rings is referred to the fine microstructure due to the formation of the dual phase
microstructure of α-Fe phase and Fe3 C carbide in a phase transformation.
FIG. 3: 2D-TRXRD patterns that were observed a cooling rate of 53 ◦ C/s at 1300 ◦ C and at a
time resolution of 0.1 seconds. The bright regions in 2D-TRXRD patterns correspond to the high
intensity of Bragg reflections of crystalline phases.
In order to quantify integral intensities, 2D-TRXRD patterns were converted to onedimensional patterns. First, the q-value on a detector plane was mapped using the standard
silicon powder of NIST674b. Then the pixel coordinates were converted to the q-value of
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reciprocal space coordinates by the q-map. Afterwards, one dimensional X-ray intensity
profiles were produced by an integration along Laue rings and corrected by the background,
absorption, polarization and normalization. Fig. 4 shows the one-dimensional diffraction
intensity mapping as a function of time during cooling. The change of diffraction profiles
is obviously corresponded to that of phase transformation. The diffractions of solid γ-Fe
phase appear in the broaden liquid-Fe pattern. The liquid remains until the precipitation
of Fe3 C carbide and the solid phase transformation from γ-Fe phase to α-Fe phase is obviously observed. Moreover, the displacement of diffraction patterns was confirmed with a
temperature drop. Particularly, the lattice spacing of γ-Fe phase increases with a temperature drop. Though carbon atoms would diffuse from γ-Fe phase to liquid phase, the carbon
atoms in γ-Fe phase, which are interstitial solute atoms, are arranged at the center of octahedron constructed by iron atoms. Then the interaction, which is the repulsive energy,
between carbon atoms induces the lattice strain but the formation free energy decreases.
Furthermore, the unit cell of γ-Fe phase is enhanced because the repulsive energy increases
with the crystal growth of γ-Fe phase.
FIG. 4: One-dimensional integrated diffraction intensity mapping under cooling. The change of
diffraction profiles is obviously corresponded to that of phase transformation.
The integral intensity of γ-Fe phases was analyzed for the normalized volume fraction
of solid/liquid phases. First, the sharp {111}γ diffraction peaks at each temperature were
separated from liquid phase by Lorentz function in order to evaluate the integral intensity
of {111}γ . Then, it was assumed that the atomic scattering factor of solid is approximately
equivalent to that of liquid since γ-Fe phase crystallizes in liquid-Fe phase. The influence
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of a small amount of carbon atoms was disregarded. Second, the integral intensity ratio of
{111}γ (Ir ) was expressed by dividing the integral intensity of {111}γ by the total integral
intensity of observed profile. Finally, the volume fraction of solid phases was normalized by
dividing Ir at each temperature by the saturated Ir at the temperature of eutectic point.
The curves of volume fractions during solidification were compared at any temperature.
Here, it was impossible to normalize the volume fraction at the cooling rate of 13 ◦ C/s
since the Ir saturates without reaching the temperature of eutectic point. Fig. 5 shows the
volume fraction of solid phase during solidification at the cooling rate of 26 ◦ C/s, 40 ◦ C/s
and 53 ◦ C/s. The solid phase fraction increases as a function of cooling time. The volume
fraction falls into disorder greatly over the temperature of the eutectic point. The influence
of precipitation of Fe3 C carbides is expected. Further, the lower cooling rate is marginally
delayed the crystallization of solid phases. However, the volume fraction includes both
temperature and time factors.
FIG. 5: Volume fraction of solid phase during solidification at the cooling rate of 26 ◦ C/s, 40 ◦ C/s
and 53 ◦ C/s. The solid phase fraction increases as a function of cooling time. The volume fraction
falls into disorder greatly over the temperature of eutectic point.
III-3. Isothermal transformation of solid phase fraction
Variations of solid phase fraction along cooling curve includes the both information
of time and temperature. In order to obtain the isothermal transition of solid phase fractions, the distribution of normalized volume fractions of {111}γ was mapped by time and
temperature like well-known Time-Temperature-Transformation (T-T-T) diagrams due to
analysis of volume fractions along cooling curves. Then, as the difference of degree of supercooling involved a gap in the recalescence, the degree of supercooling was meaningly
conformed. The counter curves were approximated by measured points that are indicated
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by plus marks as shown in Fig. 6 (a). Then, it is possible to obtain the isothermal transition
of volume fraction at any temperature. Fig. 6 (b) shows the isothermal transition of volume
fraction at 1474 K. The curve of volume fraction with a cooling time were approximated
by the isothermal kinetics JMAK equation. Then the n-value in the JMAK equation was
expected 4 in order to indicate the nucleation behavior of surface growth controlling [16].
Consequently, the volume fraction was saturated at 4.9 seconds. This isothermal crystal
growth interval is reasonable with the expected value. The activation energy was estimated
200 kJ/mol with fitting by the JMAK equation. The diffusion of iron atoms and carbon
atoms is dominant for Fe-C binary alloy system. The activation energy of self-diffusion
of iron atoms and the diffusion of carbon atoms in the γ-Fe phases are 291 kJ/mol and
155 kJ/mol at 1801 K, respectively [17]. Therefore, the crystallization of solid phase γ-Fe
is related to both of the self-diffusion of iron atoms and the diffusion of carbon atoms in
γ-Fe phase. It is considered that the γ-Fe crystallizes with the diffusion of carbon atoms
to the melts and the supersaturated carbon in the melts produces Fe3 C carbides. Next,
the Isothermal transition of volume fraction was compared at any temperature as shown in
Fig. 7. The crystal growth rate decreased with a decrease of temperature and the saturated
time was marginally delayed. This behaviors are correspond to a nose of γ-Fe phase in the
T-T-T diagram. Thus, it is possible to obtain isothermal transitions of volume fractions
at any temperature. It is expected that the diffusion of carbon atoms are enhanced with
a temperature drop and follows the formation of Fe3 C carbides as the activation energy
decreases with a temperature drop.
FIG. 6: Counter map of normalized integral intensities (a) and isothermal transition of volume
fraction at 1474 K (b). The variation of volume fraction with cooling time were approximated by
the isothermal kinetics JMAK equation.
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FIG. 7: Isothermal transitions of volume fraction at any temperature. The crystal growth rate
decreased with a decrease of temperature and saturated time is marginally delayed.
Next, the mole fraction of solid phases was estimated corresponding with an equilibrium phase fraction since the volume fraction is normalized by a saturated volume fraction.
Fig. 8 shows the result of equilibrium phase fraction calculation. The carbon component
that was estimated by behavior of cooling curves was 1.7 mass% as discussed previously.
Then, the mole fraction of γ-Fe phase shows 100 % at analyzed points. Therefore, in this
study, it was concluded that the normalized volume fraction was equivalent to the mole
fraction in the γ-Fe single phase area.
Finally, in the discussion about the Full Width of Half Maximum (FWHM) in the
vicinity of the melting point, it is necessary to consider the influence of Thermal Diffuse
Scattering (TDS). The incoherent part of TDS becomes a background, and the coherent
part becomes halo-patterns of liquid phase. It is considered that the former becomes the
uniform background and has been deleted by the background removal. The latter has been
separated by the Lorentz function. It is guessed it might be extremely weak since the
second halo-pattern as it guesses from the intensity of the first halo-pattern. Moreover, it is
guessed that the thermal vibration might influence the FWHM from the thermal expansion
of about 1400◦ C by about 1.5%. The influence of TDS is not remarkable.
IV. SUMMARY
The system of in-situ high-energy X-rays of a synchrotron source with an X-ray
photon counting pixel detector combining with the containerless electrostatic levitation
technique has been developed for a homogeneous nucleation and was applied to quantify
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FIG. 8: Equilibrium phase fraction. The carbon component that was estimated by cooling curve
was 1.7 mass%. The mole fraction of γ-Fe phase shows 100% at analyzed points.
the solid phase fraction during solidification.
(1) Two-dimensional time-resolved X-ray diffraction for a homogeneous nucleation was developed with the electrostatic levitator and the X-ray photon counting pixel detector.
The homogeneous nucleation and the crystal growth were observed at various cooling
rates by melting and solidifying the levitated spherical Fe-1.7wt%C specimens.
(2) The isothermal transitions of volume fraction were analyzed by the two-dimensional
time-resolved X-ray diffraction. The isothermal transition of volume fraction at different temperature levels was obtained. It was confirmed that the volume fraction
of γ-Fe phase estimated by integral intensities was equivalent to the mole fraction of
γ-Fe phase. The crystal growth was related with both diffusion of iron atoms and
carbon atoms on the base of Johnson-Mehl-Avrami-Kolmogorov theory.
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Acknowledgements
This work was performed under the Common-Use Facility Program of Japan Atomic
Energy Agency (JAEA).
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