Melting and undercooling of bismuth nanocrystals by solvothermal synthesis

Article abstract of DOI:10.1016/j.physb.2009.07.158

The nanocrystals of bismuth with nanowire and sphere in shape were synthesized by solvothermal process, and it was found that the amount of nanowires would be reduced by the proper choice of the reaction solvents. The crystal structure of the as-prepared

Full text of DOI:10.1016/j.physb.2009.07.158

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Physica B 404 (2009) 4045–4050
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Melting and undercooling of bismuth nanocrystals by solvothermal synthesis
Ã
C.D. Zou, Y.L. Gao , B. Yang, Q.J. Zhai
Shanghai Key Laboratory of Modern Metallurgy and Materials Processing, Shanghai University, Yanchang Road 149, Zhabei District, Shanghai 200072, PR China
a r t i c l e i n f o
a b s t r a c t
Article history:
The nanocrystals of bismuth with nanowire and sphere in shape were synthesized by solvothermal
process, and it was found that the amount of nanowires would be reduced by the proper choice of the
reaction solvents. The crystal structure of the as-prepared nanocrystals was investigated with X-ray
diffraction (XRD). The morphology of the nanocrystals was observed with the field emission scanning
electron microscope (FE-SEM). Moreover, the melting and solidification processes of the bulk bismuth
and as-prepared nanocrystals were comparatively studied with differential scanning calorimetry (DSC).
During the solidification process, the nanocrystals showed a larger undercooling, the value of which was
about 95 and 31 1C higher than that of the bulk bismuth.
Received 8 September 2008
Received in revised form
14 July 2009
Accepted 15 July 2009
Keywords:
Bismuth
Undercooling
Nanocrystals
Nanowires
& 2009 Elsevier B.V. All rights reserved.
Solvothermal synthesis
1. Introduction
experimentally. In 1909, Pawlow derived the melting temperature
of small particles as a function of surface energy and particle size,
which predicted a depression of melting temperature for small
particles with respect to the bulk value [23]. The size-dependent
melting of bismuth nanocrystals was firstly studied with electron
diffraction (ED) method on in-situ transmission electron micro-
scope (TEM) by Takagi in 1954, which showed an experimental
verification of Pawlow’s prediction for the first time [24]. After
Takagi, TEM has been widely used to study the size-dependent
melting of small particles. Reflection high energy electron
diffraction (RHEED) was also employed to probe the size-
dependent melting of bismuth crystallites [25]. However, the
electron beam has spurious influence on the melting behavior of
small particles [16], and the diffraction technique becomes
increasingly inaccurate due to line broadening as the particle size
decreases [8]. Moreover, TEM cannot measure the heat associated
with the melting process, because it was only limited to structural
measurement. Hence, the ideal experimental technique for
measuring melting of small particles is calorimetry [8]. Olson et
al. [22] studied the size-dependent melting behavior of bismuth
nanoparticles evaporated on a silicon nitride substrate by using
nanocalorimetry. The melting behavior of bismuth nanoparticles
embedded in Al and Zn matrix was studied by means of DSC
[26,27]. These results clearly showed that the melting behavior
depended not only on the particle size but also on the structure
and excess enthalpy of the particle/matrix interface [26].
Nanostructures have attracted extensive interests in recent
years due to their potential use as interconnects and nanoscale
electronic, optoelectronic, and sensing devices [1]. Bismuth (Bi),
as a semimetal with a very small band gap, has attracted more
attention recently as an interesting material for electronics due to
its low effective mass, high anisotropic electronic behavior, low
conduction-band effective mass, high electron mobility, and
potential to induce a semimetal–semiconductor transition with
decreasing crystallite size [2]. There have been some reports on
the shape and size control of bismuth nanocrystals. Wang et al. [1]
have reported a solution-phase process to prepare bismuth
nanocrystals with the shape of wire and sphere by reducing
sodium bismuthate with ethylene glycol in the presence of
polyvinylpyrrolidone (PVP). Bismuth nanocubes, triangular nano-
plates and nanospheres can be obtained in a polyol process by
adjusting the molar ratio of PVP and bismuth [3]. By adopting high
magnetic field, the shape of bismuth nanocrystals can be
controlled from sphere without magnetic field to wire under high
magnetic field [4]. Yet little information about their thermal
stability during the heating and solidification processes was
available in the aforementioned researches and thus further
studies are still necessary to be conducted.
The melting properties of pure metallic small particles such as
Sn [5–8], Ag [9], Cu [10], Al [11–13], In [7,14], Pb [7,15–17] Au
[18,19], Bi [20–22] et al. have been studied both theoretically and
The solidification behavior of small particles has also attracted
interests due to its different process in comparison with the
corresponding bulk precursor. Turnbull [28] studied the formation
of crystal nuclei in liquid metals and pointed out that, in
large continuous masses, nucleation was almost catalyzed by
Ã
Corresponding author. Tel./fax: +86 2156332144.
E-mail address: ylgao@shu.edu.cn (Y.L. Gao).
0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.physb.2009.07.158

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extraneous interfaces, but for very small droplets, the probability
of a catalytic inclusion presence was so little that their minimum
nucleation frequencies were reproducible and therefore a con-
sistent set of values formed. In other words, the undercooling of
very small particles will be more easily obtained attributing to its
homogeneous nucleation. In Takagi’s study of liquid–solid transi-
tion of thin metal films by ED, the undercooling of bismuth
nanocrystals was 124 1C [24]. The solidification behavior of
nanoscaled bismuth particles in an Al matrix was studied by
Goswami by using DSC [27], and found that the solidification of
the embedded bismuth took place in a wide temperature range.
The bismuth nanoparticles with a few tenths of nanometer in size
embedded in germinate glass showed a large undercooling of
182 1C in value [29]. The solidification of the bismuth nanoparti-
cles with the diameter of 22 nm embedded in Al matrix exhibited
significant undercooling, with the value of 165 1C, and the degree
of undercooling increased with decreasing particle size [26].
However, in the above mentioned research on Bi nanoparticles or
nanocrystals, the samples were prepared directly in the measure-
ment equipments or embedded in matrix. It is more important to
consider the real word synthesis and characterization, especially
the thermal stability, of the nanoparticles or nanocrystals for its
application.
ticles and nanowires, the crystal structures of as-prepared
products could be measured by XRD. Fig. 1 shows the XRD
patterns of the as-prepared products. The XRD patterns could be
readily indexed to the rhombohedral phase of bismuth (JCPDS 05-
0519). All the four products synthesized in different cases have the
same phase, and the oxidation trace cannot be detected in the
XRD patterns.
The morphology of the samples was examined by FE-SEM.
Fig. 2 shows the typical SEM images of bismuth prepared in
ethylene glycol and the mixed solvents of ethylene glycol and
absolute ethanol at different volume ratios. The product
synthesized in EG is marked as Bi1 sample. The products
synthesized in the mixed solvents at different volume ratios of
EG/AE ¼ 3:1, EG/AE ¼ 1:1, and EG/AE ¼ 1:3 are marked as Bi2, Bi3,
and Bi4 samples respectively. A typical SEM image of bismuth
nanocrystals synthesized in ethylene glycol (Bi1) is shown in
Fig. 2(a), revealing the product contains a mixture of nanowires
and some sub-micro and nano-sized particles. The length of the
nanowires can reach several micrometers and the diameter of the
nanowires is less than 100 nm. Most of the sub-micro particles are
smaller than 500 nm. The typical SEM images of the products
synthesized in the mixed solvents, i.e. sample Bi2, Bi3 and Bi4, are
shown in Fig. 2(b)–(d), respectively. By adding AE into the solvent,
the morphology of the products will be controllable, which is
clearly shown in Fig. 2(b)–(d). For example, with increasing of AE
in the solvent, both the amount and length of the nanowires in the
products are reduced. For Bi3 and Bi4 samples, the sub-micro
particles are hardly to be seen, and only small amount of
nanowires exist. Obviously, the products of Bi generated by
solvothermal synthesis are the mixture of infinitesimal particles
and nanowires, yet it offers the clear evidence to obtain fully
nanoparticles of Bi by proper process control. And this,
undoubtedly, supplies a promising feasible approach to prepare
pure Bi nanoparticles.
In this paper, the solvothermal process was employed to
prepare bismuth nanocrystals, and the mixed solvents of absolute
ethanol (AE) and ethylene glycol (EG) with different mixture
ratios were used in different cases. The focus of the present work
is, therefore, to study the melting and solidification processes of
the obtained bismuth nanocrystals.
2. Experimental procedure
In a typical process, 0.5 g analytical grade sodium bismuthate
(NaBiO3 ꢀ 2H2O) and 50 ml ethylene glycol (EG) were put into a
beaker with the capacity of 150 ml at room temperature. The
solution was then mechanically stirred intensively for 30 min.
Then the resultant solution was transferred into a Teflon-lined
autoclave of 45 ml capacity, and maintained at 200 1C for 24 h,
followed by being naturally cooled to room temperature. In
different cases, the mixed solvents of absolute ethanol (AE) and
ethylene glycol (EG) with different mixing ratios, i.e. EG/AE ¼ 3:1,
EG/AE ¼ 1:1, EG/AE ¼ 1:3, were chosen to control the morphology
of the products. After the reaction being completed, the obtained
products were centrifugally separated at 4000 rpm for 20 min and
rinsed for several times with large amount of absolute ethanol to
remove all impurities, and then finally dried in vacuum at 60 1C
for 4 h.
In the whole synthesis process, ethylene glycol served as both
solvent and reducing agent [30]. The chemical reaction can be
formulated as
HOCH2–CH2OH-CH3CHO+H2O
(1)
(a)
The products were characterized by means of X-ray diffraction
from 201 to 801, using Cu K radiation (
l ¼ 0.154056 nm) on a
(b)
(c)
a
Rigaku D/Max-2200 X-ray diffractometer. The FE-SEM (JSM-
6700F) was used in order to observe the morphology of the as-
prepared nanocrystals. The melting temperatures of both the
nanocrystals and the bulk bismuth were measured by using a
SDT-Q600 type DSC. The nanocrystals were dispersed in ethanol
by means of ultrasonic vibration, and then dropped into the Al2O3
crucible of the DSC. The sample was heated at a heating rate of
10 1C/min, from room temperature to 350 1C, and then cooled
down to room temperature at the same rate, under the protection
of high-purity nitrogen.
(d)
20
30
40
50
60
70
80
2 Theta (degree)
3. Results and discussion
Fig. 1. XRD patterns of the products synthesized in mixed solvents of EG and AE at
different volume ratios: (a) EG, Bi1; (b) EG/AE ¼ 3:1, Bi2; (c) EG/AE ¼ 1:1, Bi3; (d)
EG/AE ¼ 1:3, Bi4.
On the basis of the above solvothermal process, pure bismuth
nanocrystals could be obtained. Despite the mixture of nanopar-

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Fig. 2. SEM images of Bi nanocrystals synthesized in mixed solvents of EG and AE at different volume ratios: (a) EG, Bi1; (b) EG/AE ¼ 3:1, Bi2; (c) EG/AE ¼ 1:1, Bi3; (d) EG/
AE ¼ 1:3, Bi4.
5CH3CHO+2NaBiO3-2CH3COONa+3CH3COOH+2Bi+H2O
(2)
The melting temperature of bismuth thin film with 5 nm mean
thickness was about 23 1C lower than the bulk bismuth. The size-
dependent melting temperature of bismuth nanoparticles was
also studied by the newly developed means of nanocalorimetry by
Allen [22], the results showed that the melting temperature of
bismuth nanoparticles with 3 nm radius was about 67 1C less than
the bulk bismuth. It is deemed that the relatively small melting
temperature depression derives from the relative larger particle
size and the coupled influence of nanowires in comparison with
the results obtained by other researchers. More detailed analysis
of this difference will be summarized later in this paper.
The formation of bismuth nanowires and nanoparticles is
based on the Ostwald ripening process [31]. Bismuth nanoparti-
cles were formed in the solution according to Eqs. (1) and (2).
Some of these nanoparticles were able to grow into large crystals,
serving as the seeds to continuously grow into nanowires or
nanospheres.
To evaluate the particle size dependence of melting tempera-
ture or undercooling, the determination of particle size is of most
important. Generally, the particle size distribution can be
estimated by image analysis [32–34]. Fig. 3(a) shows the sketch
of the estimation of the particle size distribution, and Fig. 3(b)
shows the typical corresponding particle size histogram of Bi3
sample. By using this image analysis method, the mean size of the
sample can be determined as listed in Table 1.
The heating and cooling DSC curves of the bulk bismuth and
as-synthesized nanocrystals are shown in Fig. 4. Fig. 4(a) and (f)
show the heating and cooling DSC curves of the bulk bismuth,
which indicate the onset melting temperature TM and onset
nucleation temperature TN are 269.93 and 239.48 1C, respectively.
The heating and cooling DSC curves of bismuth nanocrystals
synthesized in EG are shown in Fig. 4(b) and (g), indicating that
the onset melting and onset cooling temperature of these
nanocrystals are 266.45 and 167.34 1C. Fig. 4(c)–(e) show the
heating DSC curves of the samples synthesized in the mixed
solvents of EG and AE at volume ratios: (c) EG/AE ¼ 3:1; (d) EG/
AE ¼ 1:1; (e) EG/AE ¼ 1:3; respectively; the corresponding
cooling DSC curves are shown in Fig. 4(h)–(j), respectively. The
results obtained from the DSC curves are tabulated in Table 1.
From Table 1, it can be found that the onset melting
temperature of bismuth nanocrystals is about 3.5 1C lower than
the bulk bismuth, showing an obvious melting temperature
depression. The melting temperature depression of bismuth
nanocrystals was firstly observed by ED method by Takagi [24].
There are number of different theories on how the melting of
small particles proceeds, and most of which generally predict
melting temperatures in the following form [22]:
ꢀ
ꢁ
ꢂ ꢃ
2ðTmbulk þ 273:15Þ
a
r
TmðrÞ ¼ Tmbulk
ꢁ
ð3Þ
D
Hmbulkrs
where Tm(r) is the size-dependent melting temperature of the
nano-particles, Tmbulk is the melting temperature of the correspond-
ing bulk precursor,
precursor, s is the density of the solid phase of the bulk precursor,
D
Hmbulk is the latent heat of melting for the bulk
r
and r is the radius. The parameter a depends on the melting model
for the selected system.
For bismuth nanocrystals, the homogeneous melting model
(HMM) [18] is usually applied for predicting the size-dependent
melting temperature [22]. The HMM model is based on the
assumption that solid and liquid particles of the same mass are in
equilibrium with their common vapor, and the particles melts
completely when the temperature reaches their melting tem-
perature, without premelting. The parameter a in this model is
determined by the following equation [18]:
ꢄ
ꢅ
2=3
rs
rl
a
¼
ssv
ꢁ
slv
ð4Þ

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12
16
14
12
10
8
6
4
2
0
B
B
Gauss fit of Bi3_B
Gauss fit of Bi1_B
10
8
6
Dav = 118.0 38.4
Dav = 85.1 22.9
4
2
0
-2
0
100 200 300 400 500 600
Particle diameter (nm)
0
100 200 300 400 500 600
Particle diameter (nm)
16
14
12
10
8
6
4
2
0
B
12
10
8
B
Gauss fit of Bi4_B
Gauss fit of Bi2_B
6
Dav = 94.9 20.4
Dav = 144.3 34.3
4
2
0
-2
-2
0
100 200 300 400 500 600
Particle diameter (nm)
0
100 200 300 400 500 600
Particle diameter (nm)
Fig. 3. The particle size histogram of samples (a) Bi1; (b) Bi2; (c) Bi3; (d) Bi4. The lines were fitted by Gaussian model.
Table 1
The results obtained from the DSC curves.
Sample
Bulk Bi
Bi1
Bi2
Bi3
Bi4
Mean particle size D (nm)
N
85.1722.9
266.45
169.74
96.71
94.9720.4
267.34
171.93
95.41
118.0738.4
266.75
170.73
96.02
144.3734.3
266.72
174.78
91.94
Onset melting temperature TM (1C)
Onset nucleation temperature TN (1C)
269.93
239.48
30.66
40.56
44.17
Undercooling
Latent heat of melting
Latent heat of nucleation
D
T ¼ TMꢁTN (1C)
HM (J/g)
HN (J/g)
25.56
26.30
23.40
29.55
D
15.46
29.95
24.99
37.38
D
where rl is the density of the liquid phase of bulk precursor, ssv
and slv are the solid–vapor interface and liquid–vapor interface
energy respectively.
Table 2 lists the data used for calculating the size-dependent
melting temperatures of bismuth nanocrystals based on Eqs. (3)
and (4). The melting temperature and latent heat of melting for
the bulk bismuth were determined by means of DSC
measurements, as given in Table 1.
melting temperature depression can be attributed to the increase
in free energy of the nanocrystals caused by size effect.
The experimental undercooling
to the following equation [35]:
DT can be obtained according
DT ¼ TM ꢁ TN
ð6Þ
Here, TM and TN are both determined on DSC measurements.
On the basis of Eq. (6) combined with the DSC measurement,
Substituting the data from Table 2 into Eqs. (3) and (4), the
size-dependent melting temperature of bismuth can be obtained
by the following equation:
the undercooling
nanocrystals can be obtained, as listed in Table 1. The under-
cooling T of the bulk bismuth was only 30.66 1C. However, the
DT of the bulk bismuth and the as-synthesized
D
undercooling of the nanocrystals, e.g., 96.71 1C for the nanocrys-
tals synthesized in EG, was much larger than that of the bulk
bismuth. In contrast, the undercooling of bismuth thin film with
mean thickness of 5 nm was 147 1C, which was ever demonstrated
by Takagi [24] by means of in-situ transmission electron
microscope (TEM), implying that smaller particle size is helpful
to increase the undercooling.
According to Turnbull’s theory, two steps of solidification of
pure metals should be taken into account: nucleation of crystals
and their subsequent growth [28]. The rate of solid formation is
mainly determined by the rate of nucleation because the latter
step occurs so rapidly. Since bulk metals are expected to contain
248:5
TmðrÞ ¼ Tmbulk
ꢁ
ð5Þ
r
Fig.
5 shows the comparison between the theoretical
calculations for the size-dependent melting temperature and the
experimental results determined by means of DSC for bismuth
nanocrystals. It is interesting to note that the experimental values
are well in agreement with the calculated ones. According to the
calculated results, the melting temperature of the bismuth
nanocrystals should decrease remarkably when the particle
radius decreases below 75 nm. The dominating factor for the

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TNP=239.27˚C
TN=239.48˚C
TM=269.93˚C
ΔHM=40.56J/g
ΔHN=44.17J/g
TMP=273.73˚C
200
225
250
275
300
MP=270.21˚C
300
325
325
325
325
325
150
200
250
300
TNP=160.34˚C
TN=169.74˚C
T
ΔH
N=15.46J/g
TM=266.45˚C
ΔHM=25.56J/g
225
250
275
200
150
175
200
125
TNP=167.34˚C
TN=171.93˚C
=29.95J/g
ΔHN
TMP=270.14˚C
TM=267.34˚C
ΔHM=26.30J/g
200
225
250
275
300
125
150
175
200
TNP=162.73˚C
TN=170.73˚C
ΔHN=24.99J/g
T
MP=270.65˚C
TM=266.75˚C
ΔHM=23.40J/g
200
225
250
275
T
300
125
125
150
175
200
TNP=166.90˚C
TN=174.78˚C
ΔHN=37.38J/g
MP=270.57˚C
TM=266.72˚C
ΔHM=29.55J/g
200
Exo up
225
250
275
300
150
175
200
Temperature (˚C)
Exo up
Temperature (˚C)
Fig. 4. The heating and cooling DSC curves of the bulk bismuth and as-synthesized bismuth nanocrystals (a)–(e): heating DSC curves; (a) bulk bismuth; (b) Bi1; (c) Bi2, (d)
Bi3 and (e) Bi4; (f)–(j) the corresponding cooling DSC curves.
nucleation will occur during the bismuth solidification process.
Table 2
In this work, the bismuth nanocrystals show larger undercooling
during its cooling process of the DSC measurement. For Bi1
sample, the undercooling is 96.71 1C with the ratio of DT/TM 0.179,
Data used in calculating bismuth aimed at Eqs. (1) and (2) [37].
Tmbulk
D
Hmbulk
rs
rl
ssv
slv
indicating the homogeneous nucleation occurs during its solidi-
fication process.
269.93 1C
40.56 J/g
9.7 g/cm3
10.07 g/cm3
550 mJ/m2
378 mJ/m2
In general, undercooling is attributed to the presence of a
nucleation barrier during solidification process that results from
the competition between the decrease in the volume free energy
upon solidification and the increase in the free energy associated
with the existence of a solid–liquid interface [25]. Basing on the
classical homogeneous nucleation theory, Sheng et al. [26]
obtained the relationship between undercooling of small particles
and its size, which can be written as the following equation:
275
250
225
200
Tmbulk
rD
Hmbulk
3aDs
r
Tbulk
þ
ð7Þ
Experimental data in the present study
Calculated melting temperature for
nanoparticles based on Eq. (5)
DT
¼
D
ðrÞ
where
radius of r,
D
T(r) ¼ TMꢁTN is the undercooling of nanocrystals with the
Melting temperature for bulk precursor
D
Tbulk is the undercooling of ‘‘bulk’’ precursor (r-N),
a is the fraction of the liquid/matrix interface that is replaced by
the solid/matrix interface after solid nucleation with the value
between 0 and 1, in this work, a ¼ 1 was used because the
bismuth nanocrystals were assumed to be free in the DSC pan.
r is
1
defined as the average density of solid and liquid phase, i.e.
r ¼
2
0
50
100
150
200
250
(rs+
r
l), and lv. Then, the undercooling of bismuth
Ds ¼ ssvꢁs
Particle size (nm)
nanocrystals as a function of crystal size can be obtained easily by
the equation
Fig. 5. Theoretical calculated and experimental results for the melting tempera-
ture versus the particle size.
698:94
DT
¼
D
Tbulk
þ
ð8Þ
ðrÞ
r
sufficiently many accidental inclusions of external origin, effective
as nuclei of crystallization, the undercooling is unusually observed
[24]. For bismuth, when homogeneous nucleation occurs during
the solidification process, the undercooling will be reached at
Fig.
6
shows the calculated undercooling of bismuth
function of crystal size, the solid circle
nanocrystals as
a
90 1C [28], with the ratio of
DT/TM being 0.165. In other words, if
represents the experimental results estimated by DSC, the solid
line and dotted line represent the calculated results by using
the ratio of T/TM is larger than 0.165, the homogeneous
D

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The solidification undercooling of the bismuth nanocrystals
Exp.
140
120
100
80
obtained by DSC measurement showed that the value of the
undercooling of bismuth nanocrystals was much higher than that
of bulk bismuth at a cooling rate of 10 1C/min. Besides, the
obtained experimental size-dependent undercooling was in
accordance with that of the theoretically estimated results.
Eq. (8) Tmbulk = 90°C
Eq. (8) Tmbulk = 31°C
Acknowledgments
60
This work was supported by the National Natural Science
Foundation of China (Grant no. 50571057), AM Foundation (Grant
no. 08520740500), Shanghai Rising-Star Program (Grant no.
06QA14020) and Shanghai Municipal Education Commission
Program (Grant no. 2006AZ002).
40
20
80
90
100
110
120
130
140
150
Particle size (nm)
Fig. 6. Undercooling of bismuth nanocrystals as
a function of crystal size
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D
Tbulk ¼ 90 1C [28], the maximum undercooling of bismuth
corresponding to
a temperature at which the homogenous
nucleation becomes appreciable, and 31 1C, determined by DSC
in this work, respectively. From Fig. 6, it is found that the
experiment results were much larger than the calculated ones
when the value of 31 1C was used for the calculation of Eq. (8).
However, in this work, the real bulk bismuth was used for the
comparative study of the solidification process of bulk bismuth
and bismuth nanocrystals. In Lu’s model, the undercooling of
‘‘bulk’’ materials was the undercooling of microparticles [26],
which will be much larger than the real bulk materials due to the
size effect on the undercooling of molten metals [36]. Hence, the
value of
D
Tbulk should be determined by the maximum
undercooling of bulk bismuth, the value of which was 90 1C,
given by Turnbull [28]. If the maximum undercooling of bulk
bismuth was used, it would be found that the experimental
results were perfectly agreed with the calculated ones.
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The bismuth nanocrystals have been obtained by solvothermal
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solvents, the morphology of the bismuth nanocrystals reveals the
tendency to change from wire to sphere and also the size decrease
of the sphere will also be obtained, indicating the promising
feasibility to prepare fully nanoparticles of bismuth by this
technique.
Due to the existence of nanowire and the large size of the
crystals, the melting temperature depression of the bismuth
nanocrystals was only about 3.5 1C, in agreement with that
calculated by homogeneous melting model.
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