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January 17, 2017
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Modern Physics Letters B
1750011 (13 pages)
c World Scientific Publishing Company
DOI: 10.1142/S0217984917500117
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Specific heat and thermal conductivity of nanomaterials
Sandhya Bhatt∗ , Raghuvesh Kumar and Munish Kumar†
Department of Physics,
G.B. Pant University of Agriculture and Technology,
Pantnagar 263145, India
∗[email protected]
†munish [email protected]
Received 1 September 2016
Revised 26 November 2016
Accepted 5 December 2016
Published 14 January 2017
A model is proposed to study the size and shape effects on specific heat and thermal
conductivity of nanomaterials. The formulation developed for specific heat is based on
the basic concept of cohesive energy and melting temperature. The specific heat of Ag
and Au nanoparticles is reported and the effect of size and shape has been studied. We
observed that specific heat increases with the reduction of particle size having maximum
shape effect for spherical nanoparticle. To provide a more critical test, we extended our
model to study the thermal conductivity and used it for the study of Si, diamond, Cu,
Ni, Ar, ZrO2 , BaTiO3 and SrTiO3 nanomaterials. A significant reduction is found in the
thermal conductivity for nanomaterials by decreasing the size. The model predictions
are consistent with the available experimental and simulation results. This demonstrates
the suitability of the model proposed in this paper.
Keywords: Specific heat; thermal conductivity; size and shape.
1. Introduction
Nanomaterials have been of great research interest in recent years due to their different unique properties. Their size- and shape-dependent properties can be tuned by
using different synthesis techniques. To synthesize or design them, a good knowledge
of their thermodynamic properties like melting temperature, specific heat, thermal
conductivity etc. is required.1 Potential applications for thermoelectric devices in
different areas and solid state managements have made the engineering of thermoelectric materials an active area of research.2,3 Advancements in materials and processing techniques have paved the way to produce scalable, cost effective materials
† Corresponding
author.
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S. Bhatt, R. Kumar & M. Kumar
with desirable properties for an effective thermoelectric device. Because of modern
synthesis and characterization techniques, particularly for nanoscale materials, a
new era of complex thermoelectric materials is approaching. Some advances in this
field, highlighting the strategies used to improve the thermo power and reduce the
thermal conductivity have been critically discussed by Snyder and Toberer.3 The
thermodynamic properties of nanomaterials differ from their bulk counterparts. For
example, the melting temperature and thermal conductivity drop with the reduction in size while specific heat increases.4–6
The reason for these changes corresponds to increased relative surface area and
grain boundaries, which may be explained in terms of defects and impurities. For
bulk material, specific heat is a function of temperature. Moreover, for nanomaterials, the specific heat depends on temperature as well as on the size. Experimental
data7 on specific heat of Cu and Pd reveal that the specific heat is about 10%
and 40% higher as compared to their bulk value. The properties like melting temperature, specific heat and entropy have been investigated for Ag nanoparticles
by calculating its Gibbs free energy.8 Size dependence of the molar heat capacity
of nanogold has been studied by using molecular dynamic theory based on tight
binding potential9 which shows enhancement in specific heat capacity at nanoscale.
It is well known that manufacturing and processing of a material require the
knowledge of its thermal properties. Thermal conduction in nanomaterials is affected by the temperature, size and shape. A molecular dynamic simulation has
been performed to understand the thermal conduction in nanocrystalline silicon.10
The thermal conductivity of Si nanowires was measured11 in the temperature range
20–320 K. Nanocrystalline silicon in thin layer form is a useful material especially in making thin film solar cells, micro-fabricated sensors and transistors.12–14
Various experiments have been performed on these silicon layers with different
sizes.15–17 A phonon transport analysis has been performed4 on ultra thin silicon
layers of grain sizes 20 nm and 100 nm at temperature 20–300 K. Studies of ultra nanocrystalline diamond films (3–5 nm) based on experiment and simulation
work resulted in a large decrement of thermal conductivity at room temperature.18
Grain boundaries have ingenious properties of controlling thermal transport in a
material which is the main reason for such large reduction in thermal conductivity
values at nanoregime. Size dependence of thermal conductivity of copper nanofilm
has been studied using a one-parameter model.19 Nath and Chopra20 measured
the size-dependent thermal conductivity of copper thin film by using an electricalthermal transport analogy. Wang21 prepared nanocrystalline nickel by pulsed current electro-deposition technique and measured its thermal conductivity. A significant change was observed in the thermal conductivity values for samples having
grain sizes below 100 nm. Thermal conductivity of Argon and Silicon thin films was
predicted by using Boltzmann transport equation based on the approximation of relaxation time.22 Grain size dependence in nanocrystalline yttria-stabilized zirconia
(∼ 40 nm) and temperature-dependent studies of its thermal conductivity have been
illustrated in terms of Kapitza thermal resistance.23,24 Grain size effect on thermal
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Specific heat and thermal conductivity
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conductivity of BaTiO3 and SrTiO3 thin film was studied by incorporating the scattering effects of spectral phonons at grain boundaries.25,26 Thus, it seems that a lot
of experimental and simulation studies have been performed to investigate specific
heat and thermal conductivity but no adequate theory is available in the literature.
In this paper, we therefore developed a thermodynamic formulation based on bond
energy model to study the size and shape dependence of specific heat and thermal
conductivity of nanomaterials.
2. Theoretical Formulation
The specific heat (C) may be defined as
C=
dE
dT
(1)
or
E = C(T − T0 ),
(2)
where E is the cohesive energy and T0 is the reference temperature. In terms of the
melting temperature (Tm ), Eq. (2) may be written as
En = Cn (Tmn − T0 )
(3)
Eb = Cb (Tmb − T0 ).
(4)
and
The subscripts n and b refer to nano and bulk material respectively. Using Eqs. (3)
and (4), we get the following relation
En (Tmb − T0 )
Cn
=
.
Cb
Eb (Tmn − T0 )
(5)
By considering surface effects, cohesive energy and melting temperature of
nanosolids have been defined by Qi,27 which reads as follows:
N
En = Eb 1 −
(6)
2n
and
N
Tmn = Tmb 1 −
.
2n
(7)
Here, N and n are the number of surface atoms and total atoms of a nanosolid,
respectively. N/2n depends on shape of the material and is equal to 2d/D, 4d/3L
and 2d/3h with d the diameter of atom and D (diameter), L (length) and h (height)
for spherical nanosolid, nanowire and nanofilm, respectively. Using Eqs. (5)–(7), we
get the following relation for specific heat
−1
Cn
N
N
Tmb
= 1−
1−
.
(8)
Cb
2n
2n Tmb − T0
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S. Bhatt, R. Kumar & M. Kumar
In Eq. (8), N/2n depends on size and shape and thus it can be used to study the
size dependence of specific heat for nanomaterials of different shapes.
A different model accounting for the particle size- and shape-dependent melting
temperature of metallic nanoparticles has also been proposed by Qi and Wang.28 In
this model, the particle shape is considered by introducing a shape factor. According
to this model, the relation of cohesive energy and melting temperature for spherical
nanoparticles reads as follows (using the notations of this paper)
3d
(9)
En = Eb 1 −
D
and
3d
Tmn = Tmb 1 −
.
D
(10)
Thus, in this model, the term N/2n which is 2d/D in Eqs. (6) and (7) for spherical
nanoparticles is replaced by 3d/D. Now, using Eqs. (5), (9) and (10), we get the
following relation for specific heat
−1
Cn
3d
Tmb
3d
1−
= 1−
.
(11)
Cb
D
D Tmb − T0
Equation (11) can also be used to study the size- and shape-dependent specific heat
of nanomaterials.
Now, we proceed to obtain the expression for lattice thermal conductivity. Using
kinetic theory of solids, we can write29
Kb =
1
C b vb lb ,
3
(12)
where Kb is the lattice thermal conductivity, Cb is the specific heat capacity, vb is
the average phonon velocity and lb is the mean free path. Thus, we can write for
nanomaterials
1
Kn = Cn vn ln .
(13)
3
Combination of Eqs. (12) and (13) gives the following relation
Kn
Cn vn ln
=
.
Kb
C b vb lb
(14)
Liang and Li30 discussed that the mean free path and melting temperature are
related as follows:
ln
Tmn
=
.
lb
Tmb
(15)
Combining Eqs. (7) and (15), we get
ln
=
lb
1−
N
.
2n
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(16)
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Specific heat and thermal conductivity
vn and vb are related as follows31
vn
=
vb
N
1−
2n
12
.
(17)
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Making use of Eqs. (8), (16), (17) and (14), we obtained the following relation
5 −1
Kn
N 2
N
Tmb
= 1−
1−
.
(18)
Kb
2n
2n Tmb − T0
Nan and Birringer32 proposed the relation for effective thermal conductivity of
a nanosolid based on Kapitza resistance using effective medium approach, which
reads as
Kn
K=
.
(19)
k Kn
1 + 2RD
Here, Kn is the intra-granular thermal conductivity of the nanomaterial, Rk is
Kapitza thermal resistance and D is the grain size. Yang et al.24 modified Eq. (19)
by assuming that sharing of grain boundary region takes place between two grains
as follows:
Kn
.
(20)
K=
1 + RkDKn
The reason for including these effects may be the reduction in mean free path for
phonons because of the increased phonon scattering effects in intragranular with
decreasing grain size. Thus, the effective thermal conductivity of a nanosolid in
terms of Kapitza resistance can be written using Eqs. (18) and (20)
5 −1
Tmb
N
N 2
1 − 2n
Kb 1 − 2n
Tmb −T0
(21)
K=
5 −1 .
Tmb
N 2
N
1 + RkDKb 1 − 2n
1 − 2n
Tmb −T0
Here, N/2n has the values as discussed above for different shapes, i.e. spherical,
nanowire and nanofilm.
3. Results and Discussion
The properties of solids relate to the bonding energy between atoms, which is characterized by cohesive energy. Different models, to determine the cohesive energy,
have been proposed. The bond-length-strength model,33 used to predict different
properties, shows that bond energy of relax bond increases. In the liquid drop
model,34 which has been applied to explain the size-dependent melting temperature and structural transition of nanoparticles, cohesive energy has been represented
by the volume- and surface-dependent terms. Using the Lindemann criteria, Shi35
proposed a model for the size-dependent amplitudes of the atomic vibrations of
nanosolids. Jiang et al.36,37 generalized the model to predict the cohesive energy of
nanosolids. It is pertinent to mention here that the properties of nanoparticles arise
basically from their surface effect. To account for the surface effect, size, shape and
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Table 1.
Input data used in this paper (Refs. 29, 32, 39 and 40).
Nanomaterial
d (nm)
Tmb (K)
Kb (W/mK)
Ag
Au
Cu
Si
Ni
Ar
Diamond
ZrO2
BaTiO3
SrTiO3
0.304
0.274
0.27
0.234
0.25
0.376
0.154
0.195
0.2272
0.2236
1235
1338
1358
1687
1728
83.81
3800
2973
1898
2350
—
—
401
148
91
1.2∗
1270
2.2
2.5∗∗
7.9∗∗
∗ For
Ar, the value of Kb is at 20 K. All the other values are at room temperature.
BaTiO3 and SrTiO3 thin films, Kb values are obtained by fitting the experimental
∗∗ For
data.
relaxation must be considered. Based on the core-shell structure, bond energy model
has been developed.27 In this model, it is assumed that the cohesive energy of a
solid consists of contributions from both surface and interior atoms. A shape factor
is proposed to describe the shape effect.28 The application of bond energy model to
predict size- and shape-dependent thermodynamic properties including its assumptions has been discussed by Qi.38 This work can be regarded as the extension of the
bond energy model, basically proposed by Qi.27 We have developed formulations
to study the size- and shape-dependent specific heat and thermal conductivity.
Equations (6) and (7) give Eq. (8) for specific heat and Eqs. (9) and (10) give
Eq. (11). In this paper, we used both these equations to study the effect of size and
shape on specific heat for Ag and Au nanoparticles. We selected these nanosolids
because of the fact that experimental data are available so that the suitability of our
model may be judged. The input data29,32,39,40 required are listed in Table 1. The
size-dependent specific heat predictions of Eqs. (8) and (11) for spherical Ag and
Au nanoparticles are reported in Figs. 1 and 2. The calculated values at room temperature are in good agreement with the available computer simulation results.8,9
Both the equations are found to give similar trends of variation. Moreover, the
results obtained by Eq. (8) are found to give better agreement with the available
simulation data for small particle range. It is observed that for the grain size less
than 10 nm, Cn /Cb increases which indicates that specific heat is inversely related
to the grain size. Moreover, the effect is very low for larger size. This enhancement
at nanoscale may be due to the presence of surface atoms and high value of their
atomic thermal vibration energy. Luo et al.8 remarked this discrepancy between
bulk and nanomaterials in terms of surface free energy. To generalize our model, we
extended it to study the effect of shape on these nanomaterials. We used Eq. (8)
for different shapes, i.e. spherical, nanowire and nanofilm. The results are reported
in Figs. 3 and 4. It is observed that Cn /Cb is highest for spherical nanoparticles
and is lowest for nanofilm. This demonstrates that shape effect is the maximum for
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Specific heat and thermal conductivity
Fig. 1. Size dependence of Cn /Cb for Ag (spherical). — and — represent computed values using
Eqs. (8) and (11), respectively. The symbol • represents simulation results.8
Fig. 2. Size dependence of Cn /Cb for Au (spherical). — and - - - represent computed values using
Eqs. (8) and (11), respectively. The symbol • represents simulation results.9
Fig. 3.
Effect of shape on Cn /Cb for Ag nanoparticle computed using Eq. (8).
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Fig. 4.
Effect of shape on Cn /Cb for Au nanoparticle computed using Eq. (8).
Fig. 5. Size effect on thermal conductivity of Si (spherical) using Eq. (21). Open circles and solid
circles refer to the computer simulation results.5,10
spherical and minimum for nanofilm. Thus, in addition to the size, specific heat
of nanomaterials depends on shape also. These investigations for shape effects are
reported in the absence of experimental data which may be helpful for those engaged
in experimental research.
To demonstrate a more critical test of the theory formulated in this paper, we
used Eq. (8) and obtained the formulation to study the size dependence of thermal
conductivity in the form of Eq. (21). The effect of size on thermal conductivity for
nanocrystalline silicon (spherical, nanowire and nanofilm), diamond (thin film), Cu
(thin film), Ni (spherical), Ar (thin film), ZrO2 (spherical), BaTiO3 and SrTiO3 thin
films has been studied using Eq. (21). These materials have been selected because
of the fact that the required40–43 simulation and experimental data are available
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Specific heat and thermal conductivity
Fig. 6. Size effect on thermal conductivity of Si (nanowire) using Eq. (21). Solid circles refer to
the experimental data.11
Fig. 7. Size effect on thermal conductivity of Si (nanofilm) using Eq. (21). Solid circle4 and open
circle4 and triangles17 refer to the experimental data.
Fig. 8. Size effect on thermal conductivity of diamond (nanofilm) using Eq. (21). Open circles
and solid circles refer to the computer simulation results.5,18
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Fig. 9. Size effect on thermal conductivity of Cu (nanofilm) using Eq. (21). Solid circles refer to
the experimental data.20
Fig. 10. Size effect on thermal conductivity of Ni (spherical) using Eq. (21). Solid circles refer to
the experimental data.21
Fig. 11. Size effect on thermal conductivity of Ar (nanofilm) using Eq. (21) at 20 K. Solid circles
refer to the experimental data.41
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Specific heat and thermal conductivity
Fig. 12. Size effect on thermal conductivity of ZrO2 (spherical) using Eq. (21) at 280 K. Solid
circles refer to the experimental data.42
Fig. 13. Size effect on thermal conductivity of BaTiO3 (nanofilm) using Eq. (21). Solid circles
refer to the experimental data.25
Fig. 14. Size effect on thermal conductivity of SrTiO3 (nanofilm) using Eq. (21). Solid circles
and triangles refer to the experimental26 and theoretical data.43
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so that a comparison can be presented. For Si, we performed computational work
at 500 K so that the comparison with the available data can be made. The corresponding input parameter of bulk thermal conductivity is 76 W/mK.39 Rk values
are obtained from nonlinear least square fitting of the corresponding experimental
data. The results are reported in Figs. 5–14, which are in good agreement with the
available experimental and simulation data. These results illustrate that thermal
conductivity decreases very rapidly with the drop in size of nanomaterial.
4. Conclusion
A theoretical model has been formulated for specific heat and thermal conductivity
of nanomaterials. These properties show reverse trend with the reduction of particle
size. The specific heat values are somewhat higher than the corresponding bulk
values while the thermal conductivity values are considerably lower. Therefore, we
conclude that specific heat has weak grain size dependence and thermal conductivity
shows strong grain size dependence. The model predictions agree well with the
available experimental and simulation results. The model predictions of this work
may be used in future for thermal studies on nanomaterials.
Acknowledgments
The authors are thankful to both the referees for their valuable comments, which
have been found very useful in revising the paper. One of the authors (Sandhya
Bhatt) is also thankful to Department of Science and Technology, New Delhi, India
for the financial support in the form of INSPIRE fellowship.
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