nanomaterials under high temperature and high pressure

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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http://dx.doi.org/10.1142/S021798491002495X


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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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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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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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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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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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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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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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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