1. INTRODUCTION:
Understanding how materials behave at tiny length scales is crucial for developing future nanotechnology. The advances in nanomaterials modeling coupled with new characterization tools are the key to study new properties and capabilities to design devices with improved performance1. This study of size effects on material properties has attracted enormous attention due to their scientific and industrial importance2,3. Nanomaterials have different properties from the bulk due to their high surface area over volume ratio and possible appearance of quantum effects at the nanoscale4,5. Cohesive energy is an important physical quantity to account for the strength of metallic bonds, which equals to the energy to divide the metallic crystal into individual atoms. The cohesive energy, in other words, is the heat of sublimation, which can be determined by experiments. Qi and Wang6 reported the size dependent cohesive energy of Cu nanoparticles (NPs). Experimental study of size dependence of cohesive energy for Mo and W was carried out by Kim et al.7, Xiao et al.8 and further verified by Qi et al.9,10 and Guisbiers.11 Another specific property is the melting point of metallic NPs which depends on size10. Several theoretical models have successfully been applied to understand the size dependency of melting temperature of NPs and a great review on this subject has been presented by Nanda12.
Recently, the study of size dependence melting temperature of Ag NPs has been carried out by Tang et al.13 Among these properties of NPs, the Debye temperature of NPs, has received considerable attention, since it is an essential physical quantity to characterize many material properties, such as the thermal vibration of atoms and phase transitions14. Roy and Chakravorti15 have discussed the details of the experimental data of Debye temperature for Ag-NPs. The size dependent Debye Temperature of Au, Ag and Al NPs are studied by Sadaiyandi16. Thus a good understanding of behavior of nanomaterials is important for deepening one’s knowledge which can help us to make best use of the advantages for further applications and bypass the disadvantages of nanomaterials. Recently, effect of size on cohesive energy, melting temperature and Debye temperature of some nanomaterials has been explored by Kumar and Kumar17. The purpose of present paper is to study size-dependence of these thermodynamic properties for Ag-NPs based on the size-dependent amplitude of the atomic thermal vibrations in terms of the Lindemann’s criterion18. Since all the parameters in our model are only the well known bulk parameters, our work can predict the size-dependent thermodynamic properties of any kind of nanostructures.
2. ANALYSIS:
The melting temperature 
has been determined
in terms of size dependent amplitude of atomic thermal vibration of NPs based
on Lindemann’s criterion on melting. The 
function of
metallic and organic nanocrystals is described by the following expression18,20
… (1)
where 
is the bulk melting
temperature of a material, 
and
denotes a diameter
of the particle and critical diameter at which all atoms of the particle are
located on its surface respectively and 
is defined as the
ratio of the mean square displacement (msd) of atoms on the surface and that in
the interior of crystals. For free standing NPs,
, where 
is the diameter of
atom, 
and vibrational
entropy 
can be expressed as19
……….. (2)
where 
is the ideal gas
constant. For metallic NPs,
,
with 
as bulk melting
entropy. Guisbiers11 reported that the cohesive energy is
responsible for the atomic structure, thermal stability, atomic diffusion,
crystal growth and many other properties and is related to the melting
temperature as:
(3)
where 
is the bulk
cohesive energy of a material.
The Debye temperature 
is a measure of the
vibrational response of the material and therefore intimately connected with
vibrational entropy and Lindmann’s model of melting is adequate and gives
supporting evidence for size dependence of 16. Liang
and Li21 also reported the relationship between 
and as
... (4)
where 
is the bulk Debye
temperature of a material
3. RESULTS AND DISCUSSION:
Kumar and Kumar17 studied the size dependence of cohesive energy, melting temperature and Debye temperature of different nanomaterials using Qi’s model6. This encouraged the authors to compare the model by present model based on Lindemann’s criterian. The values of cohesive energy, melting and Debye temperatures have been calculated using Eqs. (1-4), for free standing, spherical Ag-NPs with input parameters16,22,23 as
|
Table- Values of input parameters16,22,23 for Ag-NPs. |
||||
|
|
|
|
|
|
|
-295.9 |
1234 |
215 |
9.16 |
0.2880 |
Figure 1: Size dependence of melting temperature of Ag nanoparticles. Dashed and dot-dashed lines indicate theoretical results of Nanda et al.24 using liquid drop model12 and Kumar and Kumar17 using Qi model9. Symbols are for experimental values13 and solid line indicate our result. Horizontal dotted line is for bulk melting temperature of Ag.
Figure 1 presents the size dependent melting temperature of
Ag-NPs. As discussed above, the melting temperature goes down with the decrease
of particle size. It is clear from Figure 1 that the size effect on melting
temperature is more and more obvious with the decrease in size. The melting
curves can be divided into two parts, sizes greater than 
and sizes less than
. Melting temperature
changes gently with the variation of size and the curves are nearly horizontal
for 
. However, on the
contrary, the size effect is very distinct in the range of 
. Melting
temperature decreases sharply with further small reduction of particle size.
Our results are in good agreement with experimental data13 measured
using high resolution transmission electron microscopy than other melting
hypotheses reported by Qi6, Kumar and Kumar and Kumar17
and Nanda et al.24. The size dependence of cohesive energy of Ag-NPs
is shown in Figure 2. It is seen that the cohesive energy of Ag-NPs decreases
on increasing the particle size. We compared our results with the available
experimental data8 as well as theoretical results17,24.
It is observed that the trend of variation is same. Moreover, our results agree
well with experimental data8.
Figure 2:Size dependence of cohesive energy of Ag nanoparticles. Dashed and dot-dashed lines indicate theoretical results of Nanda et al.24 using liquid drop model12 and Kumar and Kumar17 using Qi model9. Symbols are for experimental values8 and solid line indicate our result. Horizontal dotted line is for bulk cohesive energy of Ag.
Figure 3: Size dependence of Debye temperature of Ag nanoparticles. Dashed and dot-dashed lines indicate theoretical results of Nanda et al.24 using liquid drop model12 and Kumar and Kumar17 using Qi model9. Symbols are for experimental values15 and solid line indicate our result. Horizontal dotted line is for bulk Debye temperature of Ag. .
A good agreement between theory and experiment encouraged the
authors to extend the model for the study of size dependence of Debye
temperature. The computed values of 
for
Ag-NPs are displayed in Figure 3 along with the experimental data15
and theoretical results17,24. There is good agreement in the present
work with experimental results15 than theoretical results17,24.
This demonstrates the suitability of the model extended for.
4. CONCLUSION:
In conclusion, a simple model using Lindemann’s criterian is used to account for the size dependence of the cohesive energy, melting temperature and Debye temperature of Ag-NPs. It is predicted that, these thermodynamic properties of Ag-NPs agrees well with available experimental results than other theoretical models that suggests the validity of our results. It is finally developed that Lindemann’s theory can successfully be used to study the other size dependent thermodynamic properties of nanostructures.
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Received on 05.10.2014 Modified on 20.10.2014
Accepted on 27.10.2014 © AJRC All right reserved
Asian J. Research Chem. 7(12): December, 2014; Page1013-1015