Gibbs Free Energy and Size-Temperature Phase Diagram of ...

Gibbs Free Energy and Size Temperature Phase Diagram
of Hafnium Nanoparticles
Shiyun Xiong, † Weihong Qi,* ,†,‡ Baiyun Huang, § Mingpu Wang, †,‡ and Lanying Wei †
† School of Materials Science and Engineering, Central South University, Changsha, 410083, P.R. China
‡ Key Laboratory of Non-ferrous Materials Science and Engineering, Ministry of Education, Changsha, 410083, P.R. China
§ State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, P.R. China
ABSTRACT: The Gibbs free energy of HCP, FCC, and BCC structures is
calculated, and the size temperature phase diagram is obtained for hafnium
nanoparticles. It is found that FCC, HCP, and BCC are small-size, low-temperature,
and high-temperature stable phases, respectively. The temperature-induced
structure transition is caused by the relative magnitude of lattice vibration
(characterized by the Debye temperature) for different structures, while the sizeinduced
structure transition originates from the different molar volumes. The
observed HCP to BCC and HCP to FCC structure transitions are consistent with
our model predictions. More importantly, we predict that there exists a new
structure transition from FCC to BCC in the size and temperature ranges
3.6 14.6 nm and 1766 1801 K for spherical nanoparticles (R = 1) and
1586 1742 K and 4 14.8 nm for tetrahedral ones (R = 1.49), which has not
been reported in the literature.
1. INTRODUCTION
Hafnium (Hf) and its compounds have attracted an increasing
amount of research in recent decades due to their unique
properties and important applications in modern science and
technology. 1 4 Because of its high cross-section for neutron
absorption and high corrosion resistance, Hf is primarily used in
the control and safety mechanisms of nuclear reactors. 1,2 Hf
oxide based compounds can also be used to replace silicon oxide
as high-k dielectrics in the production of integrated circuits. 3 Due
to their importance in fundamental low-dimensional physics as
well as the potential applications in modern technology, Hf
nanomaterials have been widely studied in recent years. For
instance, Hf oxide nanofilms can be used in energy-efficient
windows because of their high transparency in the visible region
and high reflection property in the near-infrared region. 4 Due to
its good chemical, thermal, and mechanical stability, Hf oxide
nanofilm is also used as CO sensing devices under severe
conditions. 5 The HfB2 nanoparticle of sub-10 nm is a promising
material for single-electron transistors for its high conductivity. 6
However, a serial study shows that the properties (melting
temperature, 7,9 specific heat capacity, 10,11 cohesive energy, 12,13
structure transition temperature, 14 etc.) of nanomaterials are
always size and shape dependent. In addition, the particle size is
usually regarded as an additional thermodynamic variable at
nanoscale, which directly affects the equilibrium properties. 15
The structure of bulk Hf can change from HCP to BCC with
the temperature rising to 2015 K. 16 Nevertheless, Seelam et al.
found that Hf can change from HCP to FCC structure when the
crystalline size reaches several nanometers in the high-energy
milling experiment. 17 These two phenomena lead to a question:
Received: January 5, 2011
Revised: April 15, 2011
Published: May 10, 2011
ARTICLE
pubs.acs.org/JPCC
what structure will exist when both increasing the temperature
and decreasing the grain size of Hf solids? The structure
transition always leads to the change of corresponding properties.
For example, the elastic module of pure Ti, Zr, and Hf in the
HCP structure is larger than that of the BCC structure. 18 Besides,
the structure transition of elements may induce the structure
transition of their compounds. Therefore, the study of size and
shape effects on the structure transition of Hf nanoparticles may
deepen theories of phase transition and extend to possible industrial
applications.
At specified thermodynamic conditions, the metastable phase
may transform into the stable one at minimal Gibbs free energy
(GFE). Therefore, to study the structure transition of Hf
nanoparticles, the GFE of HCP, FCC, and BCC structures
should be calculated. Jiang et al. 14 regarded the GFE difference
of two phases as the sum of the temperature-dependent bulk
term, size-dependent surface free energy term, and size-dependent
elastic energy term induced by pressure. Barnard et al. 19
developed a model accounting for the phase stability of arbitrary
nanoparticles as a function of size and shape by considering the
surface, edge, and corner contributions to GFE. In our previous
work, 20 the Debye model of the Helmholtz free energy (HFE)
for bulk materials has been generalized to calculate the GFE of
nanoparticles by considering the surface effects. In this work, the
method is used to study the structure transformations of Hf
nanoparticles with variation of temperature and particle size.
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The Journal of Physical Chemistry C ARTICLE
2. MODEL
For bulk materials, the Helmholtz free energy Fb in the Debye
model is defined as 21
Fb ¼ Eb þ 3RT lnð1 e Θ=T Þ RTBðΘ=TÞ ð1Þ
in which Eb, R, and Θ are the bulk cohesive energy, gas constant,
and Debye temperature, respectively. B(Θ/T)=3(T/Θ) 3R 0 Θ/T -
((x 3 )/(e x
1))dx is the Debye function. To calculate the HFE of
nanoparticles, one has to consider the surface effect of nanoparticles
for their large surface to volume ratio affects their thermodynamic
properties greatly. The relationship of vibrational
amplitude (x) and frequency (ω) between surface atoms (denoted
with subscript s) and bulk atoms (denoted with subscript b) for
metal elements is expressed as 8 xs/xb =1.43,ωs/ωb = 0.404.
Regarding the lattice vibration as a spring oscillator, the energy of
one atom is directly proportional to the square of its amplitude.
Then the vibrational lattice energy of a surface atom can be written
s
as ED(V,T) =(xs/xb) 2 ED(V,T) =6kBTB(Θ/T). In terms of the
view of statistic mechanics, when the vibrational frequency of an
atom changes from ωb to ωs, the vibrational entropy changes
ΔSD =3kBln(ωb/ωs)=2.71kb. Combining this with the relation
n/N =4Rd/D (n represents the number of surface atoms, N the
number of total atoms, and R the shape factor; 13 d and D are the
diameter of the atoms and nanoparticles, respectively), the mole
HFE of nanoparticles Fp can be rewritten as (see details in ref 20)
Fp ¼ Ep þ 3RT ln 1 e Θ=T
RTBðΘ=TÞ
þ 4Rd
RTð3BðΘ=TÞ 2:71Þ ð2Þ
D
where Ep = Eb(1 3Rd/D) represents the size- and shapedependent
cohesive energy of the nanoparticles. 13 The Debye
temperature of nanoparticles (related with corresponding bulk
value Θb) can be expressed as 20 Θ = Θb(1 3Rd/D). Compared
with the expression of bulk HFE, there is an additional size- and
shape-dependent term for nanoparticles, which is caused by the
different vibrational states of surface atoms. 20 With the obtained
HFE, the GFE of nanoparticles can be easily obtained as follows
G ¼ F þ PVm
ð3Þ
where P and Vm represent the pressure and molar volume,
respectively. For bulk material, if there is no additional pressure,
P equals one atmospheric pressure which can be ignored. However,
for nanomaterial, the pressure varies with particle size in
terms of the Laplace Young equation P= 4f/D, where the
minus symbol means the pressure is from core to shell. 17 The
surface stress of the solid has adopted the value at the isoentropic
temperature Tk with ∂gm(T)/∂T = 0 in the expression 20,22 f =
[((7)/(2(Tmb/T þ 6)))(D0dSvibHm/(kRVm)) 1/2 ] 1/2 ,whereSm,
Hm, andkare the bulk values of the melting entropy, melting
enthalpy and compressibility, respectively, D0 =3d for nanoparticles.
gm(T) = 7Hm(Tbm T)T/[Tbm(Tbm þ 6T)] is the
temperature-dependent solid liquid GFE difference. According
to the above discussion, the surface stress can be obtained as
f = 0.63(d 2 SmHm/(kRVm)) 1/2 . Considering the melting behavior
at high temperature, there is a liquid phase besides the three
structures, and the liquidus temperature is determined by 13
Tpm ¼ Tbmð1 3Rd=DÞ ð4Þ
Figure 1. Temperature-dependent Gibbs free energy of bulk Hf with
HCP, BCC, and FCC structures. The bulk cohesive energy Eb hcp = 947
kJ/mol, Eb fcc = 940 kJ/mol, and Eb bcc = 930 kJ/mol are taken from ref
23 with calculation of first principles, which was performed at 0 K using
the projector augmented-wave method within the generalized gradient
approximation; the bulk Debye temperature Θb hcp = 221 K, Θb fcc = 194 K,
Θb bcc = 154 K are taken from ref 24; they are calculated by first-principles
pseudopotential plane-wave code based on density functional theory
and the conjugate gradients algorithm.
where T bm is the melting temperature of the corresponding bulk
materials. Combining with eqs 2 and 3, the GFE of nanoparticles
can be calculated.
3. RESULTS AND DISCUSSION
In order to test the efficacy of the model, the GFE of bulk Hf
with HCP, BCC, and FCC structures are calculated and the HCP
to BCC transformation temperature is predicted. Figure 1 plots
the Gibbs free energy as a function of temperature for bulk Hf
with HCP, BCC, and FCC structures in terms of eq 1. The GFE
of the three structures decreases with the increase of temperature.
Among the three structures, the relative magnitude of the
entropy S (absolute value of the slope for GFE changes with
temperature) satisfies S bcc > S fcc > S hcp. According to eq 1, there
are two variable parameters for different structures, i.e., the
cohesive energy and Debye temperature, where the cohesive
energy determines the GFE intercept at 0 K and has no effect on
the entropy. Thus, the entropy is only determined by the Debye
temperature of different structures. This can be also examined
from the expression of entropy 20 (there is only vibrational entropy
for elementary metals), S =4kB[B(Θ/T) (3/4) ln(1
e Θ/T )], where Θ is the only parameter in calculation of
the entropy. Therefore, the structure transition (temperature
induced) is determined by lattice vibration (characterized by
Debye temperature). As shown in Figure 1, the most stable
structure is HCP below 1955 K and changes to BCC when the
temperature is higher than 1955 K, which is very close to the
experimental transition results 2015 K. 16 Therefore, the present
model is reliable in describing the structure transition of Hf.
Figure 2 shows the GFE difference (ΔG) of Hf nanoparticles
between two structures varying with particle size and temperature.
The GFE difference, between BCC and HCP (BCC-HCP)
or between FCC and HCP (FCC-HCP), decreases with the
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The Journal of Physical Chemistry C ARTICLE
Figure 2. GFE difference between (a) HCP and BCC, (b) HCP and
FCC, and (c) BCC and FCC as a function of temperature and particle
size (spherical shape). The planes represent ΔG = 0, and the curves in
the X Y plane denote the contour lines of ΔG. Vm hcp =13.4 10 6 m 3 , 17
Vm fcc = 14.3 10 6 m 3 , 17 and Vm bcc = 13.5 10 6 m 325 are experimental
values. As with the Debye temperature, all values for the atomic diameter
and Yang’s module are from ref 24 with the first-principle calculation
d hcp = 3.49 Å, d fcc = 3.56 Å, d bcc = 3.51 Å; B hcp = 1.22 Mbar, B fcc = 1.123
Mbar, B bcc = 1.176 Mbar (the compressibility of materials k is calculated
from Yang’s module with the relation k =1/B); S m = 10.175 J/
(mol 3 K), 26 Hm = 25.5 kJ/mol. 26
increase of temperature for a specified size and decreases with
the decrease of particle size at fixed temperature. The change
of ΔG with D in small sizes is more distinct than that in large
sizes. However, for the GFE difference between FCC and BCC
Figure 3. Variation of the Gibbs free energy of spherical Hf nanoparticles
(8 and 40 nm) with temperature for HCP, FCC, and BCC
structures. GFE curves for particles of 8 nm refer to the left-bottom
coordinates, while particles of 40 nm correspond to the right-top
coordinates.
(FCC BCC), ΔG nearly remains unchanged with the variation
of particle size with the absence of a change in temperature. This
can be observed more clearly from the contour lines, where all
contour lines for FCC BCC are nearly parallel with the size
axis, indicating that the variation of GFE with particle size for
both FCC and BCC structures is the same. The planes in the
figures denote that the GFE difference is zero, and the intersection
lines indicate the critical size of structure transformation as a
function of temperature.
Figure 3 represents the GFE as a function of temperature for
HCP, FCC, and BCC structures with specified sizes. Similar to
bulk, the GFE decreases with the increase of temperature and the
entropy for the three structures satisfies Sbcc > Sfcc > Shcp. The
temperature-dependent structure transition in nanometers is
determined by lattice vibration (characterized by the Debye
temperature), which is the same as the bulk materials. When
T < 1903 K, the most stable phase is the HCP structure for particles
of 40 nm, which can be concluded from the relative magnitude
GFE of the three structures. When T = 1903 K, the relationship
of GFE among the three structures satisfies Gbcc= Ghcp

The Journal of Physical Chemistry C ARTICLE
Figure 4. Size-dependent Gibbs free energy of HCP, FCC, and BCC
structures at 1000 and 2300 K (spherical nanoparticles). GFE curves at
2300 K refer to the left-bottom coordinates, while they correspond to
the right-top coordinates for GFE curves at 1000 K.
high-temperature stable phase, FCC is the small-size stable
phase, and HCP is the low-temperature stable phase.
The variation of the GFE with particle size at 1000 and 2300 K
for HCP, FCC, and BCC structures is shown in Figure 4. Note
that GFE curves at 2300 K refer to the left-bottom coordinates,
while GFE curves at 1000 K correspond to the right-top
coordinates. The GFE of the BCC structure holds the largest
in all size ranges at 1000 K, showing that the BCC structure is the
most unstable phase at this temperature. According to GFE
curves at 1000 K, the stable phase is the FCC structure when D <
5.1 nm and it turns to the HCP structure when D > 5.1 nm; thus,
D = 5.1 nm is the critical size for the possible HCP to FCC
structure transition. Contrary to the case at 1000 K, the GFE of
the BCC structure is the lowest at 2300 K in all size ranges,
indicating that it is the most stable structure at 2300 K and the
structure transition will not take place among the three structures.
From section 2, we know that the effect of the Debye
temperature on the GFE is fixed at a specified temperature, so the
structure transition is induced by the size-dependent internal
pressure term 4fVm/D. The calculated values of the surface
tension for the three structures are nearly the same, and the only
difference in 4fVm/D is their different molar volumes (the
molar volume of HCP, FCC, and BCC is different). Therefore, at
a specified temperature, the structure transformation is only
induced by the different molar volumes of the three structures.
From Figures 1, 3, and 4 it is found that the crossing points
for structure transitions are very shallow (the GFE difference
between the two structures is small around the crossing points),
indicating that the kinetics may dominate the structure transitions.
In general, the energy fluctuation of the system is very large
around the critical point, 27 which may enlarge the temperature
range of the structure transition. For example, the HCP to BCC
structure transition of particles (40 nm) in Figure 3 should take
place at 1903 K; however, because of the small GFE difference
between HCP and BCC around 1903 K, the energy of fluctuation
may be larger than the energy barrier of the HCP to BCC
structure transition. Therefore, this transition may take place
before 1903 K. On the contrary, the transition may also take place
after 1903 K due to energy fluctuation. As a result, the HCP and
Figure 5. Phase diagram of Hf nanoparticles (R = 1 and 1.49) with the
variation of particle size and temperature. The solid triangles denote the
HCP to BCC transition temperature at the bulk state, 16 and the values
denoted by solid squares are the experimental critical size of the HCP to
FCC transition. 17 The symbols of solid and open circles are experimental
values for FCC and HCP structures, respectively. 28 (Note that
the experiments are finished at room temperature; however, the local
temperature will increase (no more than) 200 °C during the high-energy
milling process. 29,30 The experimental values are assumed that there are
100 and 200 °C increases for the local temperatures, respectively.)
BCC structure may coexist in a certain temperature range around
the structure transition temperature, and the transition temperature
predicted by the present model can be regarded as the most
possible phase transition temperature.
The present model is for the phase transitions of nanoparticles;
however, we have not found available experimental data to
test our predictions. Fortunately, the HCP to FCC transition of
Hf nanocrystalline has been reported at room temperature, 17 and
the transition is believed to be caused by interface effects. The
interface effects on nanocrystallines originate from the variation
of bonds at interface, while the surface effects on nanoparticles is
also from the variation of bonds at the surface (characterized by
the present model). As a comparison, we insert the experimental
values of HCP to FCC transition of Hf nanocrystalline into
Figure 5. However, the critical size of the HCP to FCC transformation
for nanocrystallines is smaller than that of nanoparticles
at the same temperature. This is because the number of dangling
bonds for nanocrystalline is less than that of free nanoparticles of
the same size. Although the abnormal structure transition from
HCP to FCC for Hf nanocrystalline was reported, 17 the real
nature of this transformation is questionable. 28 It is believed that
the new FCC phase is stabilized by interstitial impurities introduced
during milling. To validate the idea, two different milling
conditions were introduced in ref 28: one is in regular conditions
and the other in the ultra-high-purity environment. It is found
that there existed a HCP to FCC phase transformation under
regular conditions, while no such transformations were noted
during milling under ultra-high-purity conditions. However, the
crystallite size reached a minimum of 9 nm during ultra-highpurity
milling, whereas it is 3 nm when milled under regular
conditions. Thus, it cannot be concluded that the transformation
from HCP to FCC is not a true allotropic transformation because
the crystallite size in ultra-high-purity milling may not reach the
critical size for the transformation.
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The Journal of Physical Chemistry C ARTICLE
Figure 5 shows the size temperature phase diagram of Hf
nanoparticles with different shapes (R = 1 for spherical shapes,
and R = 1.49 for tetrahedral ones) in terms of the GFE minimum.
As a comparison, available experimental results are also inserted.
Including the liquid phase (the liquidus lines are estimated by
eq 4), there are four single-phase regions and two triple points for
specified shapes. For spherical shape, the two triple points are
3.6 nm, 1766 K for liquid, BCC, and FCC structures and 14.6 nm,
1801 K for HCP, BCC, and FCC structures. In addition, for the
tetrahedral shape, the two triple points become 4 nm, 1586 K and
14.8 nm, 1742 K, respectively. Apparently, the three structures
can transfer into each other with the variation of temperature and
particle size. For instance, spherical particles (R = 1) with a
diameter of 10 nm will change from HCP to FCC first when the
temperature increases to ∼1500 K and then change to the BCC
structure at ∼1750 K before melting. With the increase of shape
factor, the structure transition temperature will decrease at a
specified particle size. In addition, the critical sizes will increase
at specified temperatures. However, the shape effects on the
temperature and size of the structure transition are not distinct.
From Figure 5 it is found that besides the HCP to BCC and HCP
to FCC transitions, there exists an unreported structure transformation
from FCC to BCC in the temperature and size regions
1766 1801 K, 3.6 14.6 nm for R = 1 and 1586 1742 K,
4 14.8 nm for R = 1.49. Since the transition has not been reported
yet, further experiments should be performed to test the present
prediction. The transition temperature from HCP to BCC (or
FCC to BCC) decreases slightly with the decrease of particle size.
Nevertheless, the transition temperature of HCP to FCC decreases
quickly with a little decrease of particle size. Clearly, FCC is the
small-size stable structure, while BCC and HCP are high-temperature
and low-temperature stable structures, respectively. Since
HCP may change to BCC or FCC when increasing temperature
before melting, it will keep solid state in all size ranges.
’ CONCLUSIONS
On the basis of the generalized Debye model, the Gibbs free
energies of HCP, FCC, and BCC structures of Hf nanoparticles
are calculated and the size-temperature phase diagram is obtained
with energy minimization principle. It is found that FCC, HCP, and
BCC are small-size, low-temperature, and high-temperature stable
phases, respectively. The three structures can transform to each
other at certain temperatures and sizes. The temperature-induced
structure transition is caused by the relative magnitude of the lattice
vibration (characterized by the Debye temperature), while the sizeinduced
structure transition originates from the different molar
volumes. Besides HCP to BCC and HCP to FCC structure
transitions, we predict a new structure transition from FCC to
BCC in the size and temperature ranges of 3.6 14.6 nm,
1766 1801 K for R = 1 and 1586 1742 K, 4 14.8 nm for R =
1.49, which will be tested by further experiments.
’ AUTHOR INFORMATION
Corresponding Author
*Phone: þ86-152-7491-1370. E-mail: qiwh216@mail.csu.edu.cn.
’ ACKNOWLEDGMENT
This work was supported by the Program for New Century
Excellent Talents in University (No. NCET-08-0574), Yuyin
Program for Young Talents of Central South University, China
Postdoctoral Science Foundation (No. 200801344), Hunan
Provincial Natural Science Foundation of China (No. 09JJ3106),
and Aid program for Science and Technology Innovative Research
Team in Higher Educational Institutions of Hunan Province.
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