size and temperature effect on thermal expansion coefficient and lattice parameter of nanomaterials

September 19, 2013 11:35 WSPC/147-MPLB S0217984913501807 1–11

Modern Physics Letters BVol. 27, No. 25 (2013) 1350180 (11 pages)c© World Scientific Publishing Company

DOI: 10.1142/S0217984913501807

SIZE AND TEMPERATURE EFFECT ON THERMAL EXPANSION

COEFFICIENT AND LATTICE PARAMETER OF

NANOMATERIALS

RAGHUVESH KUMAR, GEETA SHARMA and MUNISH KUMAR∗

Department of Physics, G.B. Pant University of Agriculture and Technology,Pantnagar 263145, India∗munish [email protected]

Received 11 June 2013Revised 22 July 2013Accepted 22 July 2013

Published 19 September 2013

A simple theoretical model is developed to study the effect of size and temperature on thecoefficient of thermal expansion and lattice parameter of nanomaterials. We have studiedthe size dependence of thermal expansion coefficient of Pb, Ag and Zn in different shapeviz. spherical, nanowire and nanofilm. A good agreement between theory and availableexperimental data confirmed the model predictions. We have used these results to studythe temperature dependence of lattice parameter for different size and also includedthe results of bulk materials. The temperature dependence of lattice parameter of Znnanowire and Ag nanowire are found to present a good agreement with the experimentaldata. We have also computed the temperature and size dependence of lattice parameterof Se and Pb for different shape viz. spherical, nanowire and nanofilm. The results arediscussed in the light of recent research on nanomaterials.

Keywords: Nanomaterials; thermal expansion; equation of state.

1. Introduction

Nanomaterials are a new class of materials with properties vastly different and of-

ten superior to those of the conventional course-grained materials. Applications of

these materials are being actively explored. Because of the high surface to volume

ratio, their properties and structure stability display many differences as compared

with the bulk materials. Due to the small size, nanomaterials consist of two compo-

nents with comparable volume fractions viz. a crystalline component comprising all

atoms located in the lattice of the crystallites (grains) and an interfacial component

formed by all atoms situated in the interfaces (grain boundaries). Various physical

∗Corresponding author.

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http://dx.doi.org/10.1142/S0217984913501807

mailto:[email protected]


R. Kumar, G. Sharma & M. Kumar

properties such as hardness, melting temperature, sintering ability and electronic

structure may be dependent upon particle size. The thermal expansion is the funda-

mental property of nanomaterials, which directly relate to the application. However,

few efforts have been paid for the thermal expansion. The experimental measure-

ments show that the thermal expansion coefficients of nanocrystalline materials are

larger than those of their polycrystalline counterparts. Lu and Sui1 reported that

the average linear thermal expansion coefficient of nanocrystalline Ni-P increases

with the decrease of grain size. Zhang and Mitchell2 studied nanocrystalline Se and

found the similar trend of variation. Yang et al.3 studied the size dependence of

volume thermal expansion coefficient, αv(D), of Se and Pb nanocrystals based on

size dependent root mean square amplitude model. It has been discussed that model

can predict αv (D = 40 nm) value accurately in comparison with the experimental

results, while a big divergency between them is found for the αv (D = 16 nm).

Moreover, they concluded that αv(D) increases with decreasing D. Xu et al.4 stud-

ied the thermal expansion of as-prepared and annealed silver nanowires embedded

in anodic Lumina membrane with different diameters up to 800 K. For both the

as-prepared and annealed samples, the coefficients of thermal expansion have “V ”

shape change as the diameters increases and minimum values of the coefficient of

thermal expansion do not correspond to the same diameters of nanowires. Zhao

and Jiang5 extended the use of classical thermodynamics to nanoscale and studied

the size effect on thermal properties of low dimensional materials. Two different

approaches to study the thermal expansion and compression of nanosystems have

been unified.6–8 The unified theory have been used to study the effect of temper-

ature (at constant pressure), the effect of pressure (at constant temperature) as

well as combined effect of pressure and temperature on nanomaterials. The model

predictions have been found to present a good agreement with the experimental

data. Moreover, the effect of size and shape is very important for nanomaterials,

which have been incorporated in the present work to investigate the coefficient of

thermal expansion and lattice parameter. The theoretical formulation is described

in Sec. 2, results and discussion in Sec. 3.

2. Theoretical Formulation

In general, materials expand when heated and contract when cooled for a given

temperature. The change in dimensions is usually linear related to the change in

temperature. The coefficient of volume thermal expansion α is defined as

α =1

V

(∂V

∂T

)P

, (1)

where V is the volume and T is the temperature. Prakash9 determined α of single-

walled carbon nanotube using molecular dynamics simulations. During these stud-

ies, Prakash9 used the temperature dependence of α as given below

α = a+ bT + cT 2 . (2)

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Size and Temperature Effect on Thermal Expansion Coefficient of Nanomaterials

Here a, b and c are constants. It should be pointed out that Eq. (2) is not consistent

with the initial boundary conditions, viz. α = α0 at T = T0. In order to satisfy this

condition, we write Eq. (2) as follows:

α = a+ b(T − T0) + c(T − T0)2 (3)

or

α = α0 + α′0(T − T0) + α′′

0(T − T0)2 , (4)

where α′ and α′′ are the first and second order derivatives of α with T and 0 refers

to initial condition. It is good approximation in many materials to express α′0 and

α′′0 in terms of α0 as follows:10

α′0 = α2

0δT , (5)

α′′0 = α3

0δ2T , (6)

where δT is called the Anderson–Gruneisen parameter. Thus, Eq. (4) may be rewrit-

ten as follows:

α = α0 + α20δT (T − T0) + α3

0δ2T (T − T0)

2 . (7)

Equation (7) is true provided that α depends on T quadratically. Moreover, con-

sidering the temperature dependence of α given by Eq. (7) neglects the higher

order terms, which may introduce the error on temperature dependence of α. It

is therefore necessary to consider the complete form of Eq. (7), which we write as

follows:

α = α0 + α20δT (T − T0) + α3

0δ2T (T − T0)

2 + · · ·∞ (8)

orα

α0= [1− α0δT (T − T0)]

−1 . (9)

Using the definition of α, the integration of Eq. (9) gives

V

V0= [1− α0δT (T − T0)]

−1/δT , (10)

where V is the volume, T is the temperature and o refers to their initial value.

It should be mentioned that Eqs. (9) and (10) were also derived by Kumar and

Upadhyay11 using thermodynamic approach, which have been used to study the

thermal properties of bulk materials.12,13

If a is the lattice parameter, then we can rewrite Eq. (10) as follows:

a

a0= [1− α0δT (T − T0)]

−1/3δT . (11)

In Eq. (11) α0 is the initial value of the coefficient of volume thermal expansion of

nanomaterials, which we write αn for simplicity. Here αn is the parameter, which

depends on size and shape of nanomaterials and may be written as follows:14

αn = αb

(1− N

2n

)−1

, (12)

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where αb is the coefficient of volume thermal expansion of bulk material. N is

the number of surface atoms and n is the total number of atoms of nanosolid. The

surface atoms refer to the first layer of nanosolid. Now the task is to calculate N/2n

to find αn. Actually, N/2n depends on the size and shape of the nanomaterials.

The method to find N/2n for different shape of nanomaterials has already been

discussed by Qi.15 According to this model, N/2n is 2d/D for spherical nanosolid

with d as the diameter of atom and D the diameter of spherical nanosolid. For

nanowire, N/2n is 4d/3L, where L is the diameter of nanowire. For nanofilm, N/2n

is 2d/3h, where h is the height (size) of nanofilm. Thus, we can write Eq. (11) for

different type (shape) of nanosystems.

For spherical nanosolid:

a

a0=

[1− αbδT

(1− 2d

D

)−1

(T − T0)

]−1/3δT

. (13)

For nanowire:

a

a0=

[1− αbδT

(1− 4d

3L

)−1

(T − T0)

]−1/3δT

. (14)

For nanofilm:

a

a0=

[1− αbδT

(1− 2d

3h

)−1

(T − T0)

]−1/3δT

. (15)

Equations (13)–(15) give the size and temperature dependent of different type

(shape) of nanomaterials. We make use of these relations in the present paper.

3. Results and Discussion

We have thus developed a simple theoretical model to study the effect of size and

temperature on lattice parameter for different type of nanomaterials viz. spherical

nano solid, nanowire and nanofilm. To show the real connection with the nanosys-

tems, we make use of these relations to study the size and temperature dependence

of lattice parameter of Se, Pb and Ag. The input data required16,17 are given in

Table 1. The present formulation needs the values of size dependent of αn as input

data. We have therefore used Eq. (12) to study the size dependence of αn. The re-

sults obtained are reported in Figs. 1–3. The size dependence of αn of Se (spherical)

Table 1. Input parameters used in present work.16,17

Nanomaterials Atomic size [d (in nm)] αb (10−5 K−1)

Se 0.437 9.45

Pb 0.350 8.70

Ag 0.288 1.50

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Fig. 1. Size dependence of thermal expansion coefficient of Se using Eq. (12). Experimentaldata18 have been shown by • for spherical nanosolid.

Fig. 2. Size dependence of thermal expansion coefficient of Pb using Eq. (12). Experimentaldata19 has been shown by • for spherical nanosolid.

computed using Eq. (12) is reported in Fig. 1 alongwith the available experimental

data18 for the sake of comparison. It is found that αn increases with decreasing

size. Our computed results are in good agreement with the available experimental

data.18 This demonstrates the suitability of the formulation used. We have also

computed the size dependence of αn of Se nanowire and Se nanofilm. The similar

trends of variation are found. To test the validity of the model we have repeated

our studies for the Pb and Ag as spherical, nanowire and nanofilm. The results ob-

tained are reported in Figs. 2 and 3. For Pb (spherical) the available experimental

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Fig. 3. Size dependence of thermal expansion coefficient of Ag using Eq. (12).

data19 is shown in the Fig. 2, which agrees well with our computed result. This

further confirms the validity of the model used. It should be mentioned that the ex-

perimental data are not available for other systems viz. Pb nanowire and nanofilms

and for Ag. We are reporting our model predictions in the absence of experimental

data. These predictions may be of current interest to the researchers engaged in

the experimental studies.

Now, we proceed to compute the temperature and size dependence of lattice pa-

rameter. This needs the values of δT , which are not yet available for nanomaterials.

Moreover, Kumar and Kumar20 discussed that δT ≈ B′0 is a good approximation,

which is equal to four. We have used this approximation in the present paper. We

have selected Zn nanowire (40 nm) and Ag nanowire (55 nm) because of the fact

that experimental data21 are available in these systems so that a comparison can

be made. We have used Eq. (14) and the results obtained are reported in Figs. 4

and 5. For Zn nanowire, there is a very good agreement up to 600 K. Above this

temperature our results seems to be slightly higher than the experimental data.

Moreover, in the case of Ag nanowire a good agreement is obtained up to 900 K

with our results slightly low above this temperature. We found that overall agree-

ment is good. An important aspect of the present formulation is that it also includes

the effect of size in addition to the effect of temperature. We have used Eq. (13) to

Eq. (15) to compute the temperature dependence of lattice parameter for different

size of Se and Pb for different cases viz. spherical, nanowire and nanofilm as well

as for bulk. In Eq. (12), when n is very large viz. αn ≈ αb, the nanomaterial ap-

proaches to the bulk value. We have thus computed the temperature dependence

of lattice parameter of bulk materials also and reported in corresponding figures

for the sake of comparison. The results obtained are reported in Figs. 6 to 11. It

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300 350 400 450 500 550 600 650 7001.0000

1.0005

1.0010

1.0015

1.0020

1.0025

a/a 0

T(K)

Fig. 4. Temperature dependence of a/a0 of Zn nanowire (40 nm) using Eq. (14). Experimentaldata21 are shown by •.

300 400 500 600 700 800 900 10001.00000

1.00005

1.00010

1.00015

1.00020

1.00025

1.00030

1.00035

a/a 0

T(K)

Fig. 5. Temperature dependence of a/a0 of Ag nanowire (55 nm) using Eq. (14). Experimentaldata21 are shown by •.

is observed that lattice parameter increases with temperature. The value is higher

as compared with the bulk and it increases with decreasing the size. The lattice

parameters for some nanomaterials have also been measured by means of XRD

technique, and they have been found to be slightly increased with respect to the

equilibrium lattice parameters of bulk materials.22 Thus, our model predictions are

in good agreement with the experimental observations. It has been discussed5,23

that the cohesive energy increases (the absolute value decreases) with a decrease

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300 400 500 600 700 800 900 10001.000

1.005

1.010

1.015

1.020

1.025

1.030

10nm25nm50nmBulk

a/a 0

T(K)

Fig. 6. (Color online) Temperature dependence of lattice parameter of Se (Spherical) usingEq. (13).

300 400 500 600 700 800 900 10001.000

1.005

1.010

1.015

1.020

1.025

1.030

10nm25nm50nm

Bulk

a/a 0

T(K)

Fig. 7. (Color online) Temperature dependence of lattice parameter of Se (Nanowire) usingEq. (14).

in size. This demonstrates the instability of nanocrystals in comparison with the

corresponding bulk crystals. This trend is expected since the surface to volume ra-

tio increases with decreasing size, while the surface atoms have lower coordinates

and thus higher energetic state, and consequently cohesive energy as a mean value

of all atoms increases. Thus lattice parameter increases by decreasing the size and

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300 400 500 600 700 800 900 10001.000

1.005

1.010

1.015

1.020

1.025

1.030

10nm

25nm50nm

Bulk

a/a 0

T(K)

Fig. 8. (Color online) Temperature dependence of lattice parameter of Se (Nanofilm) usingEq. (15).

300 400 500 600 700 800 900 10001.000

1.005

1.010

1.015

1.020

1.025

1.030

10nm

20nm

30nmBulk

a/a 0

T(K)

Fig. 9. (Color online) Temperature dependence of lattice parameter of Pb (Spherical) usingEq. (13).

is greater than its bulk value. The effect of size decreases as we go from spherical

to nanowire and nanofilm. It is also observed that the similar trends of variation

are available for Pb. It should be mentioned that such experimental data are not

available. Our results may help the researchers engaged in the experimental studies

of nanomaterials.

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300 400 500 600 700 800 900 10001.000

1.005

1.010

1.015

1.020

1.025a/

a 0

T(K)

10nm20nm

30nm

Bulk

Fig. 10. (Color online) Temperature dependence of lattice parameter of Pb (Nanowire) usingEq. (14).

300 400 500 600 700 800 900 10001.000

1.005

1.010

1.015

1.020

1.025

a/a 0

T(K)

10nm20nm30nmBulk

Fig. 11. (Color online) Temperature dependence of lattice parameter of Pb (Nanofilm) usingEq. (15).

Acknowledgment

We are thankful to both the referees for their valuable comments, which have been

used in the revised manuscript.

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size and temperature effect on thermal expansion coefficient and lattice parameter of nanomaterials

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