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DOI: 10.1142/S021798491002495X
September 28, 2010 10:21 WSPC/147-MPLB S021798491002495X
Modern Physics Letters B, Vol. 24, No. 26 (2010) 2647–2657c© World Scientific Publishing Company
NANOMATERIALS UNDER HIGH TEMPERATURE
AND HIGH PRESSURE
R. KUMAR, UMA D. SHARMA and MUNISH KUMAR∗
Department of Physics, G.B. Pant University of Agriculture and Technology,
Pantnagar-263145, India∗munish [email protected]
Received 4 January 2010Revised 15 February 2010
Two different approaches to study thermal expansion and compression of nanosystemsare unified, which have been treated quite independently by earlier workers. We providethe simple theoretical analysis, which demonstrates that these two approaches may beunified into a single theory, viz. one can be derived from other. It is concluded that thereis a single theory in the place of two different approaches. To show the real connectionwith the nanomaterials, we study the effect of temperature (at constant pressure), theeffect of pressure (at constant temperature) as well as the combined effect of pressureand temperature. We have considered different nanomaterials viz. carbon nanotube,AlN, Ni, 80Ni–20Fe, Fe–Cu, MgO, CeO2, CuO and TiO2. The results obtained arecompared with the available experimental data. A good agreement between theory andexperiment demonstrates the validity of the present approach.
Keywords: Nanomaterials; high pressure; equation of state; thermal expansion.
1. Introduction
Nanomaterials differ significantly from the bulk materials by virtue of their ex-
tremely small size. Iijima1 discovered a new form of carbon which is now popularly
known as carbon nanotube (CNT). They belong to a class of naturally occur-
ring form of carbon called fullerenes. Their unique mechanical, electrical, thermal
and electronic properties make them suitable for developing several multi func-
tional applications. Some useful thermal properties are thermal conductivity, coef-
ficient of thermal expansion and heat capacity. Following the discovery of fullerenes
such as C60, CNT, perfect cylinders formed of rolled graphite monolayer, are now
being considered as important building blocks of nanotechnology.2,3 Thermal prop-
erties of CNTs are of great importance, since the electrical characteristic of com-
plex nanotubes-based circuitry will change significantly due to differential contrac-
tion imposing internal stress.4,5 Another important aspect of thermal expansion
∗Corresponding author.
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2648 R. Kumar, U. D. Sharma & M. Kumar
of nanotubes is the effect on the mechanical properties of nanotubes-based high-
performance composites.6 Kwon et al.7 performed molecular dynamics simulations
to study shape changes of carbon fullerenes and nanotubes with increasing temper-
ature. These investigators found that high temperatures due to anharmonicity in
the vibrational modes cause an overall expansion. Prakash8 determined the coeffi-
cient of thermal expansion of single-wall CNTs using molecular dynamics simulation
under isobaric condition (constant pressure).
Due to the possibilities of substantially different behavior compared to the bulk,
the studies of nanocrystalline materials under high pressures are of considerable
current interest. The equation of state (EOS) is one of the most fundamental prop-
erties of a solid since it reveals how the volume of a sample changes under applied
pressure. Even though the individual lattice parameters of a solid may decrease
or increase under pressure, its volume must always decrease. High pressure Ra-
man studies on single-wall CNT bundles upto 25.9 GPa have been performed.9,10
These investigators9,10 used Murnaghan EOS for the analysis of the experimental
data. Poloni et al.11 reported a detailed experimental and theoretical study of the
Rb6C60 and Cs6C60 systems under pressure. X-ray diffraction and X-ray absorp-
tion experiments were coupled with ab initio calculations in order to understand
the mechanisms taking place during the compression of these intercalated systems.
These investigators11 also used Murnaghan EOS for the analysis of the experimen-
tal data. Thus, Murnaghan EOS seems to have a central role during high pressure
studies (at constant temperature) for nanomaterials.9–11 A review of the literature
of high pressure behavior of nanomaterials shows that the Birch–Murnaghan (BM)
third-order EOS has been widely used.12–21 During these studies,12–21 the pres-
sure derivative of bulk modulus, B′
0, has been fixed at 4. It seems that there exists
some methodology7,8 to investigate thermal expansion (at constant pressure), and
to study compression (at constant temperature) either by Murnaghan EOS9–11 or
BMEOS.9–11 These approaches have been used quite independently. It is therefore
legitimate and may be useful to demonstrate that these approaches are not different
from each other and may be unified as a single theory for nanomaterials, which is
the purpose of the present paper.
2. Theory
The BM third-order EOS reads as follows22:
P =3
2B0
[(
V0
V
)7/3
−
(
V
V0
)5/3 ]
×
[
1 +3
4(B′
0− 4)
{(
V
V0
)2/3
− 1
}]
, (1)
where P is the pressure, B bulk modulus and V/V0 is the relative change in volume.
“0” refers to their value at reference condition. It should be mentioned that by
taking B′
0 = 4, Eq. (1) takes the following form:
P =3
2B0
[(
V0
V
)7/3
−
(
V0
V
)5/3 ]
. (2)
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Nanomaterials Under High Temperature and High Pressure 2649
Equation (2) is known as BM second-order EOS. For bulk materials, dense oxides,
such as Al2O3, FeO, stishovite and MgSiO3 perovskite have values B′
0 = 4. Con-
sequently, the BM second-order EOS is often a close approximation to the BM
third-order EOS and in past,22 it has been used to estimate high compression data
when appropriate values of B′
0were not known. The simpler form of EOS is the
Murnaghan EOS22,23 which reads as follows:
P = −
B0
B′
0
[
1−
(
V
V0
)
−B′
0]
. (3)
In the next section, we shall discuss that BM second-order EOS (Eq. (2)) and
Murnaghan EOS (Eq. (3)) are very much similar for the compression range of the
order of 0.8 < V/V0 < 1. Actually, this is the compression range generally used for
nanomaterials.
In general, materials expand when heated and contract when cooled for a given
temperature. The changes in dimensions are usually linearly related to the change
in the temperature. The coefficient of thermal expansion α is the proportionality
constant and it is unique for each material. It is defined as
α =1
V
(
dV
dT
)
P
, (4)
where V is the volume, and T the temperature. Prakash8 determined α of single-wall
CNTs using molecular dynamics simulation. The author8 used the temperature-
dependence of α as given below:
α = a+ bT + cT 2 . (5)
Here, a, b and c are constants. It should be pointed out that Eq. (5) is not consistent
with the initial boundary condition, viz. α = α0 at T = T0. In order to satisfy this
condition, we write Eq. (5) as follows:
α = a+ b(T − T0) + c(T − T0)2 , (6)
or
α = α0 + α′
0(T − T0) + α′′
0 (T − T0)2 , (7)
where α′ and α′′ are the first and second-order derivatives of α with T . It is possible
to define α′
0 and α′′
0 in terms of α0, which reads as follows24:
α′
0 = α2
0δT , (8)
α′′
0 = α3
0δ2
T , (9)
where δT is called the Anderson–Gruneisen parameter. Thus, Eq. (7) may be rewrit-
ten as follows:
α = α0 + α2
0δT (T − T0) + α3
0δT2(T − T0)
2 . (10)
Equation (10) is true provided that α depends on T quadratically. Moreover, consid-
ering the temperature-dependence of α given by Eq. (10) neglects the higher-order
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September 28, 2010 10:21 WSPC/147-MPLB S021798491002495X
2650 R. Kumar, U. D. Sharma & M. Kumar
terms, which may introduce the error on the temperature-dependence of α. It
is therefore necessary to consider the complete form of Eq. (10), which reads as
follows:
α = α0 + α2
0δT (T − T0) + α3
0δ2T (T − T0)
2 + · · ·+∞ , (11)
orα
α0
= [1− α0δT (T − T0)]−1 . (12)
Using the definition of α, the integration of Eq. (12) gives the following relation:
V
V0
= [1− α0δT (T − T0)]−1/δT . (13)
The thermal pressure PTh is defined as22
PTh = α0B0(T − T0) , (14)
where B is the isothermal bulk modulus. Thus, Eq. (13) reads as follows:
V
V0
=
[
1−δTB0
PTh
]
−1/δT
(15)
or
PTh =B0
δT
[
1−
(
V
V0
)
−δT ]
. (16)
When applied, pressure P is not equal to zero, Eq. (16) takes the following form:
PTh − P =B0
δT
[
1−
(
V
V0
)
−δT ]
, (17)
when PTh = 0, Eq. (17) may be rewritten as follows:
P = −
B0
δT
[
1−
(
V
V0
)
−δT ]
. (18)
Using the well-known approximation,22 δT = B′
0, where B′
0is the first-order pres-
sure derivative of B, Eq. (18) becomes
P = −
B0
B′
0
[
1−
(
V
V0
)
−B′
0]
(19)
or
V
V0
=
[
1 +
(
B′
0
B0
)
P
]
−1/B′
0
(20)
or
V
V0
= exp
[
−
1
B′
0
ln
{
1 +
(
B′
0
B0
)
P
}]
. (21)
Equation (21) is the same relation as given by Murnaghan.23 It was also derived by
Murnaghan in a slightly different way, viz. assuming that bulk modulus depends
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Nanomaterials Under High Temperature and High Pressure 2651
Table 1. Values of input parameters.
Materials B0 (GPa) Reference
Bulk material
NaF 46.5 25
NaCl 24.0 25
NaBr 19.9 25
NaI 15.1 25
Nanomaterial
Carbon nanotube (individual) 230 27
AlN (hexagonal wurtzite phase) 321 19
AlN (cubic rocksalt phase) 359 19
Ni 185 13
Fe–Cu 129 12
MgO 178 16
CeO2 (cubic fluorite phase) 328 15
CeO2 (orthorhombic phase) 326 15
CuO 81 17
TiO2 243 18
linearly on pressure, which reads as
B = B0 +B′
0(P − P0) . (22)
3. Results and Discussion
For the high pressure behavior of nanomaterials, BM third-order EOS (Eq. (1))
has been used widely.12–21 During these studies, B′
0has been taken to be 4. In
the present paper, we discuss that BM third-order EOS (Eq. (1)), which reduces
to Eq. (2) when B′
0= 4, is similar to the Murnaghan EOS (Eq. (3)) for the com-
pression range 0.80 < V/V0 < 1. To demonstrate this, we have considered four
bulk materials, viz. NaF, NaCl, NaBr and NaI. The input parameters25 required
are given in Table 1 and B′
0has been taken to be 4. The results obtained by both
these equations are reported in Fig. 1, which are the same. Thus, it may be con-
cluded that BM third-order EOS is same as Murnaghan EOS for B′
0 = 4 for the
compression range 0.80 < V/V0 < 1. It has also been discussed in the earlier lit-
erature that Murnaghan EOS is the best EOS for a particular pressure range.26
Equation (5) has been used successfully7,8 to study the properties of nanosys-
tems under the effect of temperature (at constant pressure). On the other hand,
Murnaghan EOS (Eq. (21)) as well as BM third-order EOS (Eq. (1)) has been
used to study the high pressure behavior (at constant temperature). Thus, these
two formulations have been treated quite independently. In the present paper, we
presented Eq. (21) from the theory of thermal expansivity7,8 used for nanomaterial
(Eq. (5)). We are first to report that these formulations are related to each other,
viz. one can be derived from other, which provides the unification of these models.
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2652 R. Kumar, U. D. Sharma & M. Kumar
0.80
0.85
0.90
0.95
1.00
0 20 40 60 80 100 120 140 160 180
Pressure (kbar)
V/V
0(Eq. (2))
(Eq. (3))
NaI
NaF
NaC
lNaB
r
Fig. 1. Pressure-dependence of V/V0 using Eqs. (2) and (3).
The unification presented in the present paper may be of interest to the researchers
engaged in the study of nanomaterials at high temperature and high pressure.
We used this unified theory to study the effect of pressure (at constant tem-
perature), the effect of temperature (at constant pressure) as well as the combined
effect of pressure and temperature. We have considered different types of nano-
systems. We used Eq. (21) to compute the values of V/V0 at different pressures of
CNTs (individual tube). The results are reported in Fig. 2 along with the avail-
able experimental data.27 It is found that Eq. (21) gives the results, which are
in good agreement with the experimental data.27 The compression of nanostruc-
tured Fe–Cu (14 nm) material prepared by mechanical milling has been investi-
gated in situ high pressure X-ray diffraction using synchrotron radiation by Jiang
et al.12 We included these studies for comparison purposes. We used Eq. (21) to
compute the compression behavior of Fe-Cu. The results obtained are reported
in Fig. 2 along with the experimental data.12 A good agreement between theory
and experiment12 is obtained. To investigate the effect of particle size on the com-
pressibility of MgO, Rekhi et al.16 performed an X-ray diffraction study on MgO
with particle size 100 nm. We included these studies in the present work. We have
computed the compression behavior of MgO (100 nm) using Eq. (21). The results
are reported in Fig. 2 along with the experimental data.16 A good agreement is
obtained. The EOS of nanocrystalline CuO (24 nm) has been studied by Wang
et al.17 up to 16 GPa using high energy synchrotron radiation and Raman spectro-
scopic techniques. We include these results for comparison purposes in the present
work. We have computed the compression behavior of nano CuO using Eq. (21).
The results thus obtained are reported in Fig. 2 along with experimental data.17
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Nanomaterials Under High Temperature and High Pressure 2653
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.02
0 10 20 30 40
Fe-Cu
CuO
MgO
TiO2
CNT
Pressure (GPa)
V/V
0
Fig. 2. Pressure-dependence of V/V0 of nanomaterials. “—” represents calculated results(Eq. (21)). Experimental data12,16–18,27 are represented by N for Fe–Cu, • for MgO, ◦ for CuO,? for TiO2 and × for CNT.
There is good agreement between theory and experiment. Compression behavior of
nanocrystalline anatase TiO2 (40 nm) has been studied by Swamy et al.18 using
synchrotron X-ray diffraction. We have computed the compression behavior of nano
anatase TiO2 using Eq. (21). The results obtained are reported in Fig. 2 along with
the experimental data.18 There is a good agreement between theory and experi-
ment. Figure 2 demonstrates that CNT is least compressible and CuO the most
compressible. The EOS of other materials lie somewhere between these two cases.
The simplicity and applicability of the present formulation encouraged the au-
thors to extend the formulation for some other nanomaterials. We have selected
AlN which is a very important material as a bulk and nanomaterial. The in situ
X-ray diffraction study of AlN nanocrystals under hydrostatic conditions has been
performed19 to the pressure of 36.9 GPa, using an energy dispersive synchrotron-
radiation technique in a diamond anvil cell. Hexagonal AlN nanocrystals have a
particle size of 10 nm on average and display an apparent volumetric expansion
as compared with bulk AlN polycrystal. During compression to 14.5 GPa, AlN
nanocrystals start to transform to a rock salt phase. We have used Eq. (21) to
study the EOS of AlN in the hexagonal wurtzite phase and cubic rock salt phase.
The results obtained are reported in Fig. 3 along with the experimental data.19
A good agreement between theory and experiment demonstrates the validity of
Eq. (21) for both phases of AlN. The X-ray diffraction study of nanosized CeO2
was carried out to the pressure of 38.6 GPa by Wang et al.15 using an energy dis-
persive synchrotron radiation in a diamond anvil cell. At a pressure 23.3 GPa, nano
CeO2 (15 nm) starts to transform to an orthorhombic structure. This pressure is
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2654 R. Kumar, U. D. Sharma & M. Kumar
20
40
60
80
100
120
140
160
180
0 5 10 15 20 25 30 35 40
Cubic rocksalt phase
Hexagonal wurtzite phase
CeO2
AlN
Pressure (GPa)
V(A
3) Calculated (Eq. (21))
Experimental19
Experimental15
Cubic fluorite phase
Orthorhombic phase
Fig. 3. Pressure-dependence of volume.
significantly lower than the transition pressure of 31 GPa for phase transition in the
bulk CeO2. We included these studies in the present work and compute the com-
pression behavior of nano CeO2 in both phases using Eq. (21). The results obtained
are reported in Fig. 3 along with the experimental data.15 A good agreement be-
tween the theory and experiment supports the validity of the formulation developed
in the present work. The values of pressure-dependence of bulk modulus obtained
from the corresponding relation (Eq. (22)) are plotted in Fig. 4 for CNT (individual
tube) and Fe–Cu (14 nm). The experimental data on the pressure-dependence of
bulk modulus are not available in the literature. We are therefore reporting the
results in the absence of experimental data.
We have computed the temperature-dependence of V/V0 (constant pressure)
using Eq. (13) for (20 nm) Ni and 80Ni–20Fe (15 nm). The results are reported in
Fig. 5. It seems that V/V0 depends linearly on temperature for these nanomaterials.
It should be mentioned that linear behavior have also been obtained experimentally
by Lu et al.28 for nanocrystalline Ni–P alloy in the temperature range 300 K to
400 K. Thus our results are consistent with the experimental observations of Lu
et al.28 To understand the size effect on the bulk modulus and to look for possible
new high pressure phases, nanocrystalline nickel (20 nm, Ni) was studied under
high pressure by Chen et al.13 X-ray diffraction data, using a synchrotron source,
was collected under non-hydrostatic and quasi-hydrostatic conditions upto 55 GPa.
The bulk modulus was found to be different during the two processes. No phase
transition was observed for either non-hydrostatic or quasi-hydrostatic compression
for the pressure range of these experiments. We included these results for the com-
parison purposes in the present paper. We used Eq. (17) to study the compression
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Nanomaterials Under High Temperature and High Pressure 2655
0 1 2 3 4 5 6 7 8
150
200
250
300
CNT (Individual)B
ulk
mo
du
lus (
GP
a)
Pressure (GPa)
Fe-Cu (14 nm)
Fig. 4. Pressure-dependence of bulk modulus using Eq. (22).
300 350 400 450 500
1.000
1.001
1.002
1.003
1.004
1.005
1.006
1.007
V/V
0
T(K)
80 Ni-20 Fe(15nm)
Ni(20 nm)
Fig. 5. Temperature-dependence of V/V0 using Eq. (13).
behavior of (20 nm) Ni at different temperatures, viz. 300 K, 400 K and 500 K.
This needs the values of α0 as input, which is readily available.29 The results are
reported in Fig. 6. The experimental data13 are available at 300 K, which have been
included for the comparison purposes. There is a good agreement between theory
and experiment. In addition to this, we report the compression behavior at higher
temperatures, viz. 400 K and 500 K. There is a shift of the isotherm but it is small.
This shows a small effect of temperature on compression for this temperature range.
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2656 R. Kumar, U. D. Sharma & M. Kumar
0.82
0.84
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
10 20 30 40 50 60
Pressure (GPa)
V/V
0
Calculated (Eq. (17))
Experimental13
(300 K)
300 K
400 K
500 K
Fig. 6. Pressure-dependence of V/V0 of Ni (20 nm) at different temperatures.
4. Conclusion
Thermal expansion (constant pressure) of nanomaterials has been studied7,8 using
linear behavior of the thermal expansion coefficient on the temperature (Eq. (5)).
Murnaghan EOS has also been used for some materials to study compression be-
havior (constant temperature).9–12 Birch–Murnaghan EOS with B′
0= 4 is widely
used EOS12–21 for the compression (constant temperature) for nano materials. In
the present work, we present unification of all these ideas. The Murnaghan EOS
is presented from the linear expansion of the thermal expansion coefficient. It is
also pointed out that the Birch–Murnaghan EOS with B′
0 = 4 is equivalent to the
Murnaghan EOS. This provides a set of equations to be used. We demonstrated the
application of this unified approach to study the effect of temperature (constant
pressure), the effect of pressure (constant temperature) as well as the combined
effect of pressure and temperature. We have considered different nanosystems for
this purpose. A good agreement between theory and available experimental data
demonstrates the validity of the present approach. Due to the simplicity and appli-
cability, the present approach may be of interest to the researchers engaged in the
study of nanomaterials under high temperature and high pressure.
Acknowledgment
The authors are thankful to the referee for his valuable comments, which have been
used in the revised manuscript.
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Nanomaterials Under High Temperature and High Pressure 2657
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