- Email: [email protected]
Shape and size dependent thermophysical properties of nanocrystals
Accepted Manuscript
Shape and Size dependent thermophysical properties of nanocrystals M. Goyal , B.R.K. Gupta PII: DOI: Reference:
S0577-9073(17)30708-6 10.1016/j.cjph.2017.12.014 CJPH 416
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
10 June 2017 19 December 2017 20 December 2017
Please cite this article as: M. Goyal , B.R.K. Gupta , Shape and Size dependent thermophysical properties of nanocrystals, Chinese Journal of Physics (2017), doi: 10.1016/j.cjph.2017.12.014
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Highlights: The thermo-physical properties of nanomaterials are studied theoretically.
The physical properties of nanomaterials are studied with different size and shape.
Good consistency is achieved between the present results and the results reported earlier.
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Shape and Size dependent thermophysical properties of nanocrystals M.Goyal & B.R.K.Gupta Department of Physics GLA University, Mathura-281406, (U.P.), India
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*Corresponding Author email: [email protected]
Highlights:
The thermo-physical properties of nanomaterials are studied theoretically.
The physical properties of nanomaterials are studied with different size and shape.
Good consistency is achieved between the present results and the results reported earlier.
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Abstract
A theoretical model based on thermodynamic variables is employed in the present work to study
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the thermophysical properties of nanomaterials of different shapes and sizes. The model proposed by Qi and Wang [19] is applied to determine the cohesive energy of nanomaterial. The
of shape factor temperature
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number of atoms on the surface to the total number of atoms in nanosolid is considered in terms and size of nanocrystal. The variation of cohesive energy , Debye temperature
, Specific heat capacity
, melting
and Energy band gap
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is studied for spherical, regular tetrahedral, regular hexahedral and regular octahedral nanocrystals. The cohesive energy, melting temperature and Debye temperature are found to
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decrease as the grain size is reduced. However, the energy band gap and specific heat capacity are found to increase with decrease of grain size of nanomaterial. The results achieved in the
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present study are compared with the available experimental and also with those calculated from other theoretical models. The consistency between the present calculated results and the results reported earlier confirms the validity of the present model theory to explain the shape and size dependence of thermophysical properties of nanomaterials. Keywords: Nanocrystals; Cohesive energy; melting temperature; Debye temperature; energy band gap; specific heat capacity.
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1 Introduction
The theoretical and experimental studies on electrical, mechanical, optical and thermal properties of nanomaterials have been of great interest world-wide during the last decade [1, 2, 3]. The physical and chemical properties of nanomaterials have been widely investigated because of their
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industrial and scientific applications. The nanomaterials differ from the corresponding bulk form of material mainly due to their small size. The physical parameters like melting temperature, cohesive energy, activation energy, energy band gap and Young’s modulus remains constant for bulk material at normal conditions, however, these parameters are found to change with the reduction in size of material to nanolevel. Cohesive energy of material varies with grain size and
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hence melting temperature gets affected. Melting-point gets reduced as particle size is reduced and thus the thermal properties of nanomaterials show different behavior as compared to its bulk
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form [4-8]. This is probably due to increase in the ratio of surface area to volume in nanomaterials. The surface area of differently shaped nanocrystals is different and hence the physical properties of the nanocrystal vary with shape and size. It is, therefore, pertinent to
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consider both shape and size effect for understanding the properties of materials at the nanolevel. In order to understand the impact of shape and size on nanomaterials, the expression of cohesive
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energy is shown in terms of number of surface atoms and interior atoms in the nanomaterial. In past years, several theoretical simulations and experimental work have been carried out to study
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the thermophysical properties of nanomaterials [9-12]. The bottom up as well as top down approaches are applied to study the change in thermodynamic properties of solid nanomaterials [13-18]. Top down approaches are mostly based on classical thermodynamics and bottom up approaches are based on complicated simulations. In the present study, a simple model based on top down approach is used to analyze the shape and size dependent mechanical, thermal and optical properties of nanostructured crystals. The theory of formulation is presented in section 2 and the results so obtained are discussed in section 3.
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2 Mathematical formulation The term “ ” denotes the shape parameter of nanomaterials and it is defined as follows [19]: ( )
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(1)
Considering the atoms of the nanosolid as spherical with atomic radius the nanosolid,
the total number of atoms and
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⁄
⁄ ⁄
is diameter of spherical nanosolid and
(2)
is the diameter of an atom of nanosolid [19].
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where
is the radius of
the number of surface atoms in nanosolid, then
⁄
or
and
In accordance with Qi and Wang Model [19], cohesive energy of the solid metallic nanomaterial
where
)
(
)+
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* (
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can be expressed as the sum of bond energies of surface atoms and interior atoms as follows:
is bond energy;
is the number of bonds as each interior atom form bonds with the
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surrounding atoms. In the above expression, ⁄ factor is taken to consider the bond in between two atoms; first term in the bracket corresponds to surface atoms. It is multiplied by
which is
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approximately the number of surface bonds as more than half of the surface atoms bonds remain unsatisfied or dangling bonds. Second term corresponds to interior atoms. Cohesive energy of the solid nanomaterial
is ,therefore, expressed as follows [19]:
*
+ (
)
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where
and
is the cohesive energy of bulk material.
So, the expression of cohesive energy (
of nanomaterial is obtained as follows [19]:
)
(3)
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Melting temperature is related to cohesive energy of a solid [21] as follows: (4)
where k represents Boltzmann constant. In accordance with equations (3) and (4), the melting temperature of nanosolid )
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(
is given by:
Relation between the melting temperature
and Debye temperature
(5)
according to
Lindemann criterion of melting [22, 23] is given by:
⁄
)
⁄
⁄
that
is the molecular mass and
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Where C is constant, .
(6)
the molar volume. Equation (6) thus states
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(
From equations (5) and (6), we get: ⁄
(7)
are Debye temperatures in nanosolid
and its counterpart bulk material
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where
)
PT
(
respectively.
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According to Debye theory [24], the specific heat at constant pressure and
in nanomaterials varies with Debye Temperature as follows:
For bulk:
For nanomaterials : In view of equation (7), the ratio of
to
can be written as:
5
in bulk materials
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(
)
(8)
Arrehenius expression of electrical conductivity in terms of size and temperature is as follows [25]:
where
}
(9)
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{
is pre-exponential parameter,
is the activation energy required by the
particle of diameter D for migration; R is the gas constant at temperature T. Activation energy is the difference between the conduction band energy and Fermi energy , i.e.
.
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Assuming that electrical conductivity is independent of size at melting temperature [26],
Comparing the electrical conductivity of nanomaterial and its bulk form, we get: {
}
}
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Then,
{
(10)
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In most of the semiconductors, Fermi energy level lies in the middle of valence band and conduction band, therefore
is considered to be half of the energy band gap, i.e.
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the difference between the energy band of nano ( and bulk crystal ( ( so in view of equation (10),we get the following expression [27]:
⁄
is expressed as
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(11)
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Using equation (11) and equation (5), the energy band gap in nanomaterials is given by the relation:
where
(
)
and
are energy band gaps in bulk and nanoform.
(12)
In accordance with equation (2),
(13)
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Using above relation in the expressions of cohesive energy temperature
, Debye temperature Energy band gap
, melting , Specific heat capacity
, the equations obtained are given as
under: (
(
)
(
)
(
(14)
( ⁄
) (
)
)
) (
(
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)
)
⁄
) (
)
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(
Since for a spherical nanosolid, the shape parameter temperature
, Debye temperature
) )
(
)
)
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(
(17)
, melting of
(19) (20) (21) (22)
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(
⁄
)
PT
(
(
ED
(
(16)
(18)
and Energy band gap
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spherical nanoparticle can be expressed as:
, Specific heat
, the cohesive energy
(15)
)
(23)
Shape factor for non-spherical nanoparticles can be calculated for polyhedral shaped nanoparticle of edge length
by considering the volume of spherical and non-spherical nanoparticles same
and using the volume relations as follows [20]: For regular tetrahedral shaped nanoparticle of edge length : Volume of regular tetrahedral nanoparticle = Volume of spherical nanoparticle, i.e.,
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√
√ √ R.
or
So, the surface area of regular tetrahedral shaped nanoparticle calculated is 18.725 the shape factor
For
melting temperature
and hence,
in equation (14) to equation (18); cohesive energy
, Debye temperature
, Specific heat
,
and Energy band gap
For regular hexahedral shaped nanosolid of edge length R.
So, the surface area calculated is 15.591
and hence, the shape factor
For
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√
[20]:
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of the regular tetrahedral shaped nanoparticle are obtained.
in equations (14-18); cohesive energy
temperature
, Specific heat
, melting temperature
and Energy band gap
nanosolid are obtained.
, Debye
of the regular hexahedral shaped
√
√
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For regular octahedral shaped nanoparticle [20] of edge length : R
and hence, the shape factor is 1.18.
in equations (14-18); cohesive energy
temperature
, Specific heat
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For
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So, the surface area calculated is 14.857
, melting temperature
and Energy band gap
, Debye
of the regular octahedral shaped
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nanosolid can be obtained.
In accordance to Guisbiers model [28, 29], the melting temperature of nanomaterial as follows:
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related to melting temperature of its bulk form
is
(24)
where D is diameter or length of the nanostructure;
is shape factor and its value depends
on surface energies in solid and liquid phases; melting enthalpy of bulk and also on surface area and volume of the material. As the model depends on large number of parameters, so the approximations are more and accuracy is less. In the present study, the results obtained are also
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compared with Guisbiers model for spherical nanocrystals. The expression of melting temperature from Guisbiers model [28] can be obtained in terms of cohesive energy, Debye temperature, specific heat capacity and Energy band gap using the relations discussed above. Results and Discussion and atomic diameter
as
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The model theory used in the present work requires shape parameter input parameter.The values of atomic diameter
with other known parameters for the bulk
crystal are given in Table 1.The shape parameter
for spherical nanocrystral is found to be 1,
however, if the shape of the nanomaterial is different from the spherical, shape parameter is found greater than one. The shape parameter
for spherical, regular tetrahedral, regular
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hexahedral and regular octahedral nanomaterials is listed in Table 2. We have studied the effect of shape and size on cohesive energy, melting temperature, Debye temperature, energy band gap, and specific heat capacity of nanosolids Ag, Au, W and Sn for the size range from 1nm to 60 nm. The cohesive energy capacity
, melting temperature
and Energy band gap
, Debye temperature
, Specific heat
of the nanomaterials of different shape and size are
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calculated using the equations (14-18) and shown in figures 1-6. In these figures the calculated results are compared with the available experimental data and also with those calculated from
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other theoretical models. Table 1: Input parameters [30-39] Nanocrystals
No
2.
4.
(k)
(k)
(J/g K)
[30-33]
[30]
[30, 38]
[31]
0.3194
-295.9
1235
225
0.235
Au
0.3187
-368
1338
165
0.129
W
0.3108
-824
3695
400
0.132
0.3724
-303
505.1
140
0.227
Ag
AC
3.
(KJ/mol)
[34-39]
CE
1.
d (nm)
PT
S.
Sn
Table 2: Shape parameter Shape of nanoparticle
for different shapes of nanomaterials [20]: Edge length
Shape parameter
Spherical nanosolids
-
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1
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Regular tetrahedral shaped
√
Regular octahedral shaped
√
10
15
1.18
R
√
5
1.24
R
20
25
30
35
-160
45
50
55
60
Hexahedral (present)
-200
Ag
Octahedral (present)
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E cn(KJ/ mole) -------->
-180
40
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Regular hexahedral shaped
0
1.49
√ √ R
-220
Guisbiers model [28]
-240
Spherical(present)
-260
Tetrahedral(present)
-280
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-300
ED
D (nm)--------->
Figure 1 (a): Variation of cohesive energy with shape and size in Ag nanocrystal.
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CE
PT
.
10
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0
5
10
15
20
25
30
35
40
45
50
55
60
-240
Au
-280
Hexahedral (present) Octahedral (present) Guisbiers model [28]
-300
Spherical (present) -320
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Ecn(KJ/mole) ------------>
-260
Tetrahedral (present)
-340 -360 -380
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D (nm) ------------>
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Figure 1(b): Variation of cohesive energy with shape and size in Au nanocrystal.
-300
-500
20
30
40
W
60
Spherical (present) Hexahedral (present) Octahedral (present)
-600
Guisbiers model [28]
CE AC
50
Tetrahedral (present)
PT
Ecn (KJ/mole)--------->
-400
10
ED
0
Exp [33]
-700 -800 -900
D (nm) ------->
Figure 1(c): Variation of cohesive energy with shape and size in W nanocrystal.
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-120 0
10
20
30
40
50
60
70
-160 -180
Sn
-200
Hexahedral (present) Octahedral (present)
-220
Guisbiers model [28]
-240
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Ecn (KJ/mole) ------------->
-140
Spherical (present)
-260
Tetrahedral (present)
-280 -300
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D (nm) --------------->
Figure 1 (d): Variation of cohesive energy with shape and size in Sn nanocrystal.
in spherical, regular tetrahedral, regular hexahedral and
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The variation of cohesive energy
regular octahedral shaped nanomaterials viz. Ag, Au, W and Sn is determined using equation
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(14) and shown in figures 1(a, b, c, d) along with experimental data and theoretical results [28] obtained from other model. It is noted that the shape effect is pronounced only if the size of the
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nanomaterial is less than 30nm.As the size of the nanomaterial increases above 30 nm; the change in cohesive energy is almost negligible. The graphical representation shows that the decreasing behavior of cohesive energy in nanomaterial depends on particle size and shape. The
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results achieved using the present model are quite close to the available experimental results [33]
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for W and also with those values obtained from Guisbiers model [28].
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1300
Ag
1100
Spherical (present)
1000
Tetrahedral (present) Hexahedral (present)
900
Octahedral (present)
800
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TmN (K) ..........>
1200
Guisbiers model [28]
700
Ref. [42]
600 500 400 5
10
15
20
25
30
35
40
45
50
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0
55
60
D (nm) ---------->
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Figure 2(a): Variation of melting temperature with shape and size in Ag nanocrystal.
ED
Au
1100
PT
TmN (K) --------->
1300
Spherical (present) Tetrahedral (present) Hexahedral (present) Octahedral (present) Guisbiers model [28] Exp.[43] Exp [45]
CE
900
AC
700
500 0
5
10
15
20
25
30
35
40
45
50
55
60
D (nm) ------------>
Figure 2(b): Variation of melting temperature with shape and size in Au nanocrystal.
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W
3800 3600
3200 3000
Hexahedral (present)
2800
Octahedral (present)
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TmN (K) -------->
3400
Guisbiers model [28]
2600
Spherical (present)
2400
Tetrahedral (present)
2200 2000 10
20
30
40
50
60
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0
D (nm) ------------->
Figure 2 (c): Variation of melting temperature with shape and size in W nanocrystal.
M
500
ED
400 350
Spherical (present)
Sn
Tetrahedral (present) Hexahedral (present) Octahedral (present)
PT
TmN (K) ------------->
450
Guisbiers model [28]
300
CE
Exp. [40]
AC
250 200
0
10
20
30
40
50
60
70
D (nm) --------------->
Figure 2 (d): Variation of melting temperature with shape and size in Sn nanocrystal.
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The ratio variation in melting temperature
with particle size is calculated
using equation (15) for Ag, Au, W and Sn nanocrystals. It is noted that with decrease in particle size, the melting temperature decreases in nanometals. The melting temperature
for
spherical nanoparticles, regular tetrahedral, regular hexahedral and regular octahedral shaped nanomaterials of different sizes is estimated using equation (15) and shown in figures 2(a, b, c,
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d). The results so obtained are compared with the simulated results for Ag [42], experimental data for Au [43, 45], Sn [40] and also with the results achieved from the Guisbiers model [28]. The figures show that the present calculated values of melting temperature for Ag, Au and Sn are in close agreement with the results reported earlier. It is noted that the melting temperature increases with size of the crystal up to 30 nm, however, if grain size is more than 30 nm, the
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effect is negligible. It is further noted that the melting temperature of spherical nanosolid is found larger comparison to regular tetrahedral shaped nanosolid as particle size increases. Thus, no appreciable change in melting temperature is observed due to size and shape above 30 nm size of the material.
M ED
0.9
Ag
0.8 0.7
Spherical (present) Tetrahedral (present) Hexahedral (present) Octahedral (present) Guisbiers model [28] Ref. [41]
PT
TmN/ TmB or E cn /E o -------->
1
CE
0.6 0.5
AC
0
0.05
0.1
0.15 D-1
0.2
0.25
0.3
0.35
(nm-1)------------>
Figure 3 (a): Cohesive energy variation with reciprocal of grain size in Ag nanocrystal.
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Au
0.9
0.7
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0.8
Spherical (present) Tetrahedral (present) Hexahedral (present) Octahedral (present) Guisbiers model [28] Exp. [45]
0.6
0.5 0
0.05
0.1
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TmN/TmB or Ecn/Eo ------------>
1
0.15
-1
D (nm
0.2
-1)
0.25
0.3
0.35
------------>
Figure 3 (b): Cohesive energy variation with reciprocal of grain size in Au nanocrystal.
ED
M
.
0.9
Hexahedral (present) Octahedral (present) Guisbiers model [28] Spherical (present) Tetrahedral(present)
W
PT
TmN/TmB or Ecn/Eo ---------------->
1
AC
CE
0.8 0.7 0.6 0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
D -1 (nm -1) --------------->
Figure 3 (c): Cohesive energy variation with reciprocal of grain size in W nanocrystal.
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Spherical (present) Tetrahedral(present) Hexahedral (present) Octahedral (present) Guisbiers model [28] Exp.[40]
0.9
Sn 0.8
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TmN/TmB or Ecn/Eo --------------->
1
0.7
0.6
0
0.05
0.1
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0.5 0.15
0.2
0.25
0.3
0.35
D -1 (nm -1) --------------->
ED
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Figure 3 (d): Cohesive energy variation with reciprocal of grain size in Sn nanocrystal.
220
Ag
PT
Debye Temperature θ DN (K) ------>
240
Spherical (present)
200
AC
CE
Tetrahedral (present) Hexahedra (present)l
180
Octahedral (present) Guisbiers model [28]
160
Ref. [44]
140 0
5
10
15
20
25
30
35
40
45
50
55
60
D (nm) ------------->
Figure 4 (a): Variation of Debye temperature with shape and size in Ag nanocrystal.
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Au
160 150 Hexahedral (present) Octahedral (present) Guisbiers model [28] Ref. [44] Spherical (present) Tetrahedral (present)
140 130 120 110 0
5
10
15
20
25
30
35
40
45
50
55
60
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D (nm) --------------->
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Debye Temperature θ DN (K) -------->
170
Figure 4 (b): Variation of Debye temperature with shape and size in Au nanocrystal. It is clear from figures 1 and 2 that cohesive energy as well as melting temperature decreases in
or
and reciprocal of particle size ( or
with the increase in
. Figures 3(a, b, c, d) depict the linear . The results reported in the present
ED
decrease in
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nanomaterials with size. For better understanding, the graphs 3(a, b, c, d) are plotted between
paper are compared with the available experimental [40, 45] and theoretical results [28, 41].
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The variation in Debye temperature with particle size in spherical nanoparticles, regular tetrahedral, regular hexahedral and regular octahedral shaped nanomaterials is determined using
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equation (16) and the results so obtained are shown in figures 4(a-d). It is found that the Debye temperature decreases with reduction in size of differently shaped nanocrystals of Ag, Au, W and
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Sn. The results obtained for Ag, Au, W and Sn are compared with the results of Guisbiers model [28] and also with the simulated results for Ag and Au nanocrystals [44]. It is noted that the calculated values for Ag and Au nanocrystals are close to the available simulated results. The results obtained in the present study somehow follow the same trend as of those calculated from Guisbiers model.
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200
W
190
170 160 Hexahedral (present)
150
Octahedral (present)
140
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θ DN (K) ------------>
180
Guisbiers model [28]
130
Spherical (present)
120
Tetrahedral (present)
110 100 10
20
30
40
50
60
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0
D (nm) --------------->
Figure 4(c): Variation of Debye temperature with shape and size in W nanocrystal.
M
Sn
Spherical (present)
ED
130
Tetrahedral( present) Hexahedral (present)
PT
120
Octahedral (present)
110
CE
θ DN (K)-------------->
140
Guisbiers model [28]
100
AC
0
10
20
30 40 D (nm)------------->
50
60
70
Figure 4(d): Variation of Debye temperature with shape and size in Sn nanocrystal.
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1.5
Ag Spherical (present) Tetrahedral (present) Hexahedral (present)
1.3
Octahedral (present)
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CpN/ CpB --------------->
1.4
Guisbiers model [28]
1.2
1.1
1 10
20
30
40
50
60
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0
D (nm) -------------->
ED
1.3
Au Hexahedral (present) Octahedral (present)
PT
C pN / CpB -------------->
1.5 1.4
with shape and size in Ag nanocrystal.
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Figure 5(a): Variation of
Guisbiers model [28] Spherical (present)
1.2
CE
Tetrahedral (present)
AC
1.1 1 0
10
20
30
40
50
60
D (nm) ------------>
Figure 5 (b): Variation of
with shape and size in Au nanocrystal .
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W
1.4
Hexahedral (present) Octahedral (present) Guisbeirs model [28] Spherical (present) Tetrahedral (present)
1.3 1.2 1.1 1 10
20
30
40
50
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0
CR IP T
CpN/C pB ---------------->
1.5
60
D (nm) ---------------->
1.4
Hexahedral (present) Octahedral (present) Guisbiers model [28]
PT
CpN /CpB -------------->
1.5
Sn
ED
1.7 1.6
with shape and size in W nanocrystal.
M
Figure 5 (c): Variation of
Spherical (present)
1.3
Tetrahedral (present)
CE
1.2
AC
1.1 1 0
10
20
30
40
50
60
D (nm) ------------->
Figure 5(d): Variation of
with shape and size in Sn nanocrystal.
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1.6
Ag Hexahedral (present)
1.4
Octahedral (present) 1.3
Guisbiers model [28] Spherical (present)
1.2
CR IP T
E gN/EgB ---------->
1.5
Tetrahedral (present)
1.1 1 0.9 0
5
10
15
20
25
30
35
Figure 6 (a): Variation of
40
45
50
55
60
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D (nm)------------->
with shape and size in Ag nanocrystal.
Figure 5(a-d) and figure 6(a-d) show the ratio of variation of specific heat capacity in nano and bulk material for Ag, Au, W and Sn nanocrystals
M
and energy bandgap
with particle size. The results are calculated using equation (17) and (18) and compared with the
ED
values obtained from Guisbiers model [28]. The results shown in figures 5 and 6 explain the variation of specific heat capacity and energy bandgap for spherical nanoparticles, regular
PT
tetrahedral shaped, regular hexahedral and regular octahedral structured nanosolids. The graphs indicate that the specific heat capacity and energy band gap in nanostructured crystals increases with decrease in grain size. Figures 5 and 6 depict the same trend of variation of specific heat
CE
capacity and energy band gap with size, however, in small size region, the results are somewhat deviated as compared to other theoretical model [28]. It is thus stated here that the values of
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energy band gap and specific heat capacity are more in nanomaterials as compared to their bulk forms while the melting temperature, Debye temperature and cohesive energy show decreasing behavior with decreasing particle size. A good consistency between the present results and the available experimental or theoretical results achieved from other models supports the validity of the model.
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1.5
Au Hexahedral (present) Octahedral (present)
1.3
Guisbiers model [28] Spherical (present)
1.2
Tetrahedral (present)
1.1 1 0.9 0
5
10
15
20
25
30
35
40
M
1.3
55
60
W Hexahedral (present) Octahedral (present) Guisbiers model [28]
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E gN/ E gB ---------->
1.4
50
with shape and size in Au nanocrystal.
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1.5
45
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D (nm) --------------->
Figure 6 (b): Variation of
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EgN/EgB ------------>
1.4
Spherical (present)
1.2
Tetrahedral (present)
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1.1
AC
1
0.9 0
10
20
30
40
50
60
70
D (nm)------------->
Figure 6 (c): Variation of
with shape and size in W nanocrystal. .
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1.8
Sn Spherical (present) Tetrahedral (present)
1.4
Hexahedral (present) Octahedral (present) Guisbiers model [28]
1.2
1
0.8 0
10
20
30
40
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EgN/EgB ------------->
1.6
50
60
70
with shape and size in Sn nanocrystal.
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Figure 6 (d): Variation of
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D (nm) ------------->
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4 Conclusion:
On the basis of overall descriptions, it is emphasized that the present model is capable of
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explaining the variation of melting temperature, cohesive energy, Debye temperature, specific heat capacity and energy band gap with the shape and size of nanomaterials. However, such variations are not appreciable in those nanomaterials whose sizes are greater than 30 nm. Both
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the size and shape are found to influence the thermodynamic properties of nanostructured materials. From the overall analysis it is stated that the effect of shape becomes less effective if
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the size of nanocrystal is increased above 30 nm and almost the same values of different thermalmechanical-optical parameters are obtained if the size is above 30 nm. Finally, it can be concluded that the present model is able to analyze and explain the thermodynamic and optical properties of nanomaterials of different shape and size satisfactorily and may thus be useful for those researchers who are exploring the studies on electronic, thermal and optical properties of nanomaterials.
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Acknowledgement: The authors are very much thankful to referees for their valuable comments.
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PT
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AC
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ED
[36] A. Safaei, M.A. Shandiz, Size-dependent thermal stability and the smallest nanocrystal,
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low-dimensional systems, Phys. Rev. A, 66 (2002) 013208. [38] S. N. Dickson, J.G.Mullen, Sn-119 Mossbauer study of dilute tin-lead alloys, Phys. Rev. B,
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metallic nanocrystals, J. Mater. Research, 16 (2001), No. 11, 3304. [40] G.L.Allen, R.A. Bayles, W.W. Gile, W.A. Jesser, Small particle melting of Pure metals, Thin Solid Films, 144 (1986) 297. [41] S. F.Xiao, W.Y. Hu ,J.Y. Yang, Melting Behaviors of Nanocrystalline Ag , J Phys. Chem B,109 (2005) 20339-20342. [42] S. F. Xiao, W.Y. Hu, J.Y. Yang, Melting Temperature: From Nanocrystalline to Amorphous Phase, J. Chem. Phys., 125 (2006) 184504.
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[43] Ph. Buffat, J.P. Borel, Size effect on the melting temperature of gold particles, Phys. Rev. A, 13 (1976) 2287-2298. [44] K. Sadaiyandi, Size dependent Debye temperature and mean square displacements of nanocrystalline Au, Ag and Al, Mater. Chem. Phys., 115 (2009)703-706. [45] J.R. Sambles, An electron microscope study of evaporating gold particles: The Kelvin
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28
Shape and Size dependent thermophysical properties of nanocrystals M. Goyal , B.R.K. Gupta PII: DOI: Reference:
S0577-9073(17)30708-6 10.1016/j.cjph.2017.12.014 CJPH 416
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
10 June 2017 19 December 2017 20 December 2017
Please cite this article as: M. Goyal , B.R.K. Gupta , Shape and Size dependent thermophysical properties of nanocrystals, Chinese Journal of Physics (2017), doi: 10.1016/j.cjph.2017.12.014
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Highlights: The thermo-physical properties of nanomaterials are studied theoretically.
The physical properties of nanomaterials are studied with different size and shape.
Good consistency is achieved between the present results and the results reported earlier.
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CE
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M
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Shape and Size dependent thermophysical properties of nanocrystals M.Goyal & B.R.K.Gupta Department of Physics GLA University, Mathura-281406, (U.P.), India
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*Corresponding Author email: [email protected]
Highlights:
The thermo-physical properties of nanomaterials are studied theoretically.
The physical properties of nanomaterials are studied with different size and shape.
Good consistency is achieved between the present results and the results reported earlier.
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Abstract
A theoretical model based on thermodynamic variables is employed in the present work to study
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the thermophysical properties of nanomaterials of different shapes and sizes. The model proposed by Qi and Wang [19] is applied to determine the cohesive energy of nanomaterial. The
of shape factor temperature
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number of atoms on the surface to the total number of atoms in nanosolid is considered in terms and size of nanocrystal. The variation of cohesive energy , Debye temperature
, Specific heat capacity
, melting
and Energy band gap
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is studied for spherical, regular tetrahedral, regular hexahedral and regular octahedral nanocrystals. The cohesive energy, melting temperature and Debye temperature are found to
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decrease as the grain size is reduced. However, the energy band gap and specific heat capacity are found to increase with decrease of grain size of nanomaterial. The results achieved in the
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present study are compared with the available experimental and also with those calculated from other theoretical models. The consistency between the present calculated results and the results reported earlier confirms the validity of the present model theory to explain the shape and size dependence of thermophysical properties of nanomaterials. Keywords: Nanocrystals; Cohesive energy; melting temperature; Debye temperature; energy band gap; specific heat capacity.
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1 Introduction
The theoretical and experimental studies on electrical, mechanical, optical and thermal properties of nanomaterials have been of great interest world-wide during the last decade [1, 2, 3]. The physical and chemical properties of nanomaterials have been widely investigated because of their
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industrial and scientific applications. The nanomaterials differ from the corresponding bulk form of material mainly due to their small size. The physical parameters like melting temperature, cohesive energy, activation energy, energy band gap and Young’s modulus remains constant for bulk material at normal conditions, however, these parameters are found to change with the reduction in size of material to nanolevel. Cohesive energy of material varies with grain size and
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hence melting temperature gets affected. Melting-point gets reduced as particle size is reduced and thus the thermal properties of nanomaterials show different behavior as compared to its bulk
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form [4-8]. This is probably due to increase in the ratio of surface area to volume in nanomaterials. The surface area of differently shaped nanocrystals is different and hence the physical properties of the nanocrystal vary with shape and size. It is, therefore, pertinent to
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consider both shape and size effect for understanding the properties of materials at the nanolevel. In order to understand the impact of shape and size on nanomaterials, the expression of cohesive
CE
energy is shown in terms of number of surface atoms and interior atoms in the nanomaterial. In past years, several theoretical simulations and experimental work have been carried out to study
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the thermophysical properties of nanomaterials [9-12]. The bottom up as well as top down approaches are applied to study the change in thermodynamic properties of solid nanomaterials [13-18]. Top down approaches are mostly based on classical thermodynamics and bottom up approaches are based on complicated simulations. In the present study, a simple model based on top down approach is used to analyze the shape and size dependent mechanical, thermal and optical properties of nanostructured crystals. The theory of formulation is presented in section 2 and the results so obtained are discussed in section 3.
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2 Mathematical formulation The term “ ” denotes the shape parameter of nanomaterials and it is defined as follows [19]: ( )
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(1)
Considering the atoms of the nanosolid as spherical with atomic radius the nanosolid,
the total number of atoms and
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⁄
⁄ ⁄
is diameter of spherical nanosolid and
(2)
is the diameter of an atom of nanosolid [19].
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where
is the radius of
the number of surface atoms in nanosolid, then
⁄
or
and
In accordance with Qi and Wang Model [19], cohesive energy of the solid metallic nanomaterial
where
)
(
)+
PT
* (
ED
can be expressed as the sum of bond energies of surface atoms and interior atoms as follows:
is bond energy;
is the number of bonds as each interior atom form bonds with the
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surrounding atoms. In the above expression, ⁄ factor is taken to consider the bond in between two atoms; first term in the bracket corresponds to surface atoms. It is multiplied by
which is
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approximately the number of surface bonds as more than half of the surface atoms bonds remain unsatisfied or dangling bonds. Second term corresponds to interior atoms. Cohesive energy of the solid nanomaterial
is ,therefore, expressed as follows [19]:
*
+ (
)
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where
and
is the cohesive energy of bulk material.
So, the expression of cohesive energy (
of nanomaterial is obtained as follows [19]:
)
(3)
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Melting temperature is related to cohesive energy of a solid [21] as follows: (4)
where k represents Boltzmann constant. In accordance with equations (3) and (4), the melting temperature of nanosolid )
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(
is given by:
Relation between the melting temperature
and Debye temperature
(5)
according to
Lindemann criterion of melting [22, 23] is given by:
⁄
)
⁄
⁄
that
is the molecular mass and
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Where C is constant, .
(6)
the molar volume. Equation (6) thus states
ED
(
From equations (5) and (6), we get: ⁄
(7)
are Debye temperatures in nanosolid
and its counterpart bulk material
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where
)
PT
(
respectively.
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According to Debye theory [24], the specific heat at constant pressure and
in nanomaterials varies with Debye Temperature as follows:
For bulk:
For nanomaterials : In view of equation (7), the ratio of
to
can be written as:
5
in bulk materials
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(
)
(8)
Arrehenius expression of electrical conductivity in terms of size and temperature is as follows [25]:
where
}
(9)
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{
is pre-exponential parameter,
is the activation energy required by the
particle of diameter D for migration; R is the gas constant at temperature T. Activation energy is the difference between the conduction band energy and Fermi energy , i.e.
.
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Assuming that electrical conductivity is independent of size at melting temperature [26],
Comparing the electrical conductivity of nanomaterial and its bulk form, we get: {
}
}
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Then,
{
(10)
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In most of the semiconductors, Fermi energy level lies in the middle of valence band and conduction band, therefore
is considered to be half of the energy band gap, i.e.
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the difference between the energy band of nano ( and bulk crystal ( ( so in view of equation (10),we get the following expression [27]:
⁄
is expressed as
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(11)
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Using equation (11) and equation (5), the energy band gap in nanomaterials is given by the relation:
where
(
)
and
are energy band gaps in bulk and nanoform.
(12)
In accordance with equation (2),
(13)
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Using above relation in the expressions of cohesive energy temperature
, Debye temperature Energy band gap
, melting , Specific heat capacity
, the equations obtained are given as
under: (
(
)
(
)
(
(14)
( ⁄
) (
)
)
) (
(
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)
)
⁄
) (
)
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(
Since for a spherical nanosolid, the shape parameter temperature
, Debye temperature
) )
(
)
)
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(
(17)
, melting of
(19) (20) (21) (22)
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(
⁄
)
PT
(
(
ED
(
(16)
(18)
and Energy band gap
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spherical nanoparticle can be expressed as:
, Specific heat
, the cohesive energy
(15)
)
(23)
Shape factor for non-spherical nanoparticles can be calculated for polyhedral shaped nanoparticle of edge length
by considering the volume of spherical and non-spherical nanoparticles same
and using the volume relations as follows [20]: For regular tetrahedral shaped nanoparticle of edge length : Volume of regular tetrahedral nanoparticle = Volume of spherical nanoparticle, i.e.,
7
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√
√ √ R.
or
So, the surface area of regular tetrahedral shaped nanoparticle calculated is 18.725 the shape factor
For
melting temperature
and hence,
in equation (14) to equation (18); cohesive energy
, Debye temperature
, Specific heat
,
and Energy band gap
For regular hexahedral shaped nanosolid of edge length R.
So, the surface area calculated is 15.591
and hence, the shape factor
For
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√
[20]:
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of the regular tetrahedral shaped nanoparticle are obtained.
in equations (14-18); cohesive energy
temperature
, Specific heat
, melting temperature
and Energy band gap
nanosolid are obtained.
, Debye
of the regular hexahedral shaped
√
√
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For regular octahedral shaped nanoparticle [20] of edge length : R
and hence, the shape factor is 1.18.
in equations (14-18); cohesive energy
temperature
, Specific heat
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For
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So, the surface area calculated is 14.857
, melting temperature
and Energy band gap
, Debye
of the regular octahedral shaped
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nanosolid can be obtained.
In accordance to Guisbiers model [28, 29], the melting temperature of nanomaterial as follows:
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related to melting temperature of its bulk form
is
(24)
where D is diameter or length of the nanostructure;
is shape factor and its value depends
on surface energies in solid and liquid phases; melting enthalpy of bulk and also on surface area and volume of the material. As the model depends on large number of parameters, so the approximations are more and accuracy is less. In the present study, the results obtained are also
8
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compared with Guisbiers model for spherical nanocrystals. The expression of melting temperature from Guisbiers model [28] can be obtained in terms of cohesive energy, Debye temperature, specific heat capacity and Energy band gap using the relations discussed above. Results and Discussion and atomic diameter
as
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The model theory used in the present work requires shape parameter input parameter.The values of atomic diameter
with other known parameters for the bulk
crystal are given in Table 1.The shape parameter
for spherical nanocrystral is found to be 1,
however, if the shape of the nanomaterial is different from the spherical, shape parameter is found greater than one. The shape parameter
for spherical, regular tetrahedral, regular
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hexahedral and regular octahedral nanomaterials is listed in Table 2. We have studied the effect of shape and size on cohesive energy, melting temperature, Debye temperature, energy band gap, and specific heat capacity of nanosolids Ag, Au, W and Sn for the size range from 1nm to 60 nm. The cohesive energy capacity
, melting temperature
and Energy band gap
, Debye temperature
, Specific heat
of the nanomaterials of different shape and size are
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calculated using the equations (14-18) and shown in figures 1-6. In these figures the calculated results are compared with the available experimental data and also with those calculated from
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other theoretical models. Table 1: Input parameters [30-39] Nanocrystals
No
2.
4.
(k)
(k)
(J/g K)
[30-33]
[30]
[30, 38]
[31]
0.3194
-295.9
1235
225
0.235
Au
0.3187
-368
1338
165
0.129
W
0.3108
-824
3695
400
0.132
0.3724
-303
505.1
140
0.227
Ag
AC
3.
(KJ/mol)
[34-39]
CE
1.
d (nm)
PT
S.
Sn
Table 2: Shape parameter Shape of nanoparticle
for different shapes of nanomaterials [20]: Edge length
Shape parameter
Spherical nanosolids
-
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1
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Regular tetrahedral shaped
√
Regular octahedral shaped
√
10
15
1.18
R
√
5
1.24
R
20
25
30
35
-160
45
50
55
60
Hexahedral (present)
-200
Ag
Octahedral (present)
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E cn(KJ/ mole) -------->
-180
40
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Regular hexahedral shaped
0
1.49
√ √ R
-220
Guisbiers model [28]
-240
Spherical(present)
-260
Tetrahedral(present)
-280
M
-300
ED
D (nm)--------->
Figure 1 (a): Variation of cohesive energy with shape and size in Ag nanocrystal.
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CE
PT
.
10
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0
5
10
15
20
25
30
35
40
45
50
55
60
-240
Au
-280
Hexahedral (present) Octahedral (present) Guisbiers model [28]
-300
Spherical (present) -320
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Ecn(KJ/mole) ------------>
-260
Tetrahedral (present)
-340 -360 -380
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D (nm) ------------>
M
Figure 1(b): Variation of cohesive energy with shape and size in Au nanocrystal.
-300
-500
20
30
40
W
60
Spherical (present) Hexahedral (present) Octahedral (present)
-600
Guisbiers model [28]
CE AC
50
Tetrahedral (present)
PT
Ecn (KJ/mole)--------->
-400
10
ED
0
Exp [33]
-700 -800 -900
D (nm) ------->
Figure 1(c): Variation of cohesive energy with shape and size in W nanocrystal.
11
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-120 0
10
20
30
40
50
60
70
-160 -180
Sn
-200
Hexahedral (present) Octahedral (present)
-220
Guisbiers model [28]
-240
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Ecn (KJ/mole) ------------->
-140
Spherical (present)
-260
Tetrahedral (present)
-280 -300
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D (nm) --------------->
Figure 1 (d): Variation of cohesive energy with shape and size in Sn nanocrystal.
in spherical, regular tetrahedral, regular hexahedral and
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The variation of cohesive energy
regular octahedral shaped nanomaterials viz. Ag, Au, W and Sn is determined using equation
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(14) and shown in figures 1(a, b, c, d) along with experimental data and theoretical results [28] obtained from other model. It is noted that the shape effect is pronounced only if the size of the
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nanomaterial is less than 30nm.As the size of the nanomaterial increases above 30 nm; the change in cohesive energy is almost negligible. The graphical representation shows that the decreasing behavior of cohesive energy in nanomaterial depends on particle size and shape. The
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results achieved using the present model are quite close to the available experimental results [33]
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for W and also with those values obtained from Guisbiers model [28].
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1300
Ag
1100
Spherical (present)
1000
Tetrahedral (present) Hexahedral (present)
900
Octahedral (present)
800
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TmN (K) ..........>
1200
Guisbiers model [28]
700
Ref. [42]
600 500 400 5
10
15
20
25
30
35
40
45
50
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0
55
60
D (nm) ---------->
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Figure 2(a): Variation of melting temperature with shape and size in Ag nanocrystal.
ED
Au
1100
PT
TmN (K) --------->
1300
Spherical (present) Tetrahedral (present) Hexahedral (present) Octahedral (present) Guisbiers model [28] Exp.[43] Exp [45]
CE
900
AC
700
500 0
5
10
15
20
25
30
35
40
45
50
55
60
D (nm) ------------>
Figure 2(b): Variation of melting temperature with shape and size in Au nanocrystal.
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W
3800 3600
3200 3000
Hexahedral (present)
2800
Octahedral (present)
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TmN (K) -------->
3400
Guisbiers model [28]
2600
Spherical (present)
2400
Tetrahedral (present)
2200 2000 10
20
30
40
50
60
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0
D (nm) ------------->
Figure 2 (c): Variation of melting temperature with shape and size in W nanocrystal.
M
500
ED
400 350
Spherical (present)
Sn
Tetrahedral (present) Hexahedral (present) Octahedral (present)
PT
TmN (K) ------------->
450
Guisbiers model [28]
300
CE
Exp. [40]
AC
250 200
0
10
20
30
40
50
60
70
D (nm) --------------->
Figure 2 (d): Variation of melting temperature with shape and size in Sn nanocrystal.
14
ACCEPTED MANUSCRIPT
The ratio variation in melting temperature
with particle size is calculated
using equation (15) for Ag, Au, W and Sn nanocrystals. It is noted that with decrease in particle size, the melting temperature decreases in nanometals. The melting temperature
for
spherical nanoparticles, regular tetrahedral, regular hexahedral and regular octahedral shaped nanomaterials of different sizes is estimated using equation (15) and shown in figures 2(a, b, c,
CR IP T
d). The results so obtained are compared with the simulated results for Ag [42], experimental data for Au [43, 45], Sn [40] and also with the results achieved from the Guisbiers model [28]. The figures show that the present calculated values of melting temperature for Ag, Au and Sn are in close agreement with the results reported earlier. It is noted that the melting temperature increases with size of the crystal up to 30 nm, however, if grain size is more than 30 nm, the
AN US
effect is negligible. It is further noted that the melting temperature of spherical nanosolid is found larger comparison to regular tetrahedral shaped nanosolid as particle size increases. Thus, no appreciable change in melting temperature is observed due to size and shape above 30 nm size of the material.
M ED
0.9
Ag
0.8 0.7
Spherical (present) Tetrahedral (present) Hexahedral (present) Octahedral (present) Guisbiers model [28] Ref. [41]
PT
TmN/ TmB or E cn /E o -------->
1
CE
0.6 0.5
AC
0
0.05
0.1
0.15 D-1
0.2
0.25
0.3
0.35
(nm-1)------------>
Figure 3 (a): Cohesive energy variation with reciprocal of grain size in Ag nanocrystal.
15
ACCEPTED MANUSCRIPT
Au
0.9
0.7
CR IP T
0.8
Spherical (present) Tetrahedral (present) Hexahedral (present) Octahedral (present) Guisbiers model [28] Exp. [45]
0.6
0.5 0
0.05
0.1
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TmN/TmB or Ecn/Eo ------------>
1
0.15
-1
D (nm
0.2
-1)
0.25
0.3
0.35
------------>
Figure 3 (b): Cohesive energy variation with reciprocal of grain size in Au nanocrystal.
ED
M
.
0.9
Hexahedral (present) Octahedral (present) Guisbiers model [28] Spherical (present) Tetrahedral(present)
W
PT
TmN/TmB or Ecn/Eo ---------------->
1
AC
CE
0.8 0.7 0.6 0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
D -1 (nm -1) --------------->
Figure 3 (c): Cohesive energy variation with reciprocal of grain size in W nanocrystal.
16
ACCEPTED MANUSCRIPT
Spherical (present) Tetrahedral(present) Hexahedral (present) Octahedral (present) Guisbiers model [28] Exp.[40]
0.9
Sn 0.8
CR IP T
TmN/TmB or Ecn/Eo --------------->
1
0.7
0.6
0
0.05
0.1
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0.5 0.15
0.2
0.25
0.3
0.35
D -1 (nm -1) --------------->
ED
M
Figure 3 (d): Cohesive energy variation with reciprocal of grain size in Sn nanocrystal.
220
Ag
PT
Debye Temperature θ DN (K) ------>
240
Spherical (present)
200
AC
CE
Tetrahedral (present) Hexahedra (present)l
180
Octahedral (present) Guisbiers model [28]
160
Ref. [44]
140 0
5
10
15
20
25
30
35
40
45
50
55
60
D (nm) ------------->
Figure 4 (a): Variation of Debye temperature with shape and size in Ag nanocrystal.
17
ACCEPTED MANUSCRIPT
Au
160 150 Hexahedral (present) Octahedral (present) Guisbiers model [28] Ref. [44] Spherical (present) Tetrahedral (present)
140 130 120 110 0
5
10
15
20
25
30
35
40
45
50
55
60
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D (nm) --------------->
CR IP T
Debye Temperature θ DN (K) -------->
170
Figure 4 (b): Variation of Debye temperature with shape and size in Au nanocrystal. It is clear from figures 1 and 2 that cohesive energy as well as melting temperature decreases in
or
and reciprocal of particle size ( or
with the increase in
. Figures 3(a, b, c, d) depict the linear . The results reported in the present
ED
decrease in
M
nanomaterials with size. For better understanding, the graphs 3(a, b, c, d) are plotted between
paper are compared with the available experimental [40, 45] and theoretical results [28, 41].
PT
The variation in Debye temperature with particle size in spherical nanoparticles, regular tetrahedral, regular hexahedral and regular octahedral shaped nanomaterials is determined using
CE
equation (16) and the results so obtained are shown in figures 4(a-d). It is found that the Debye temperature decreases with reduction in size of differently shaped nanocrystals of Ag, Au, W and
AC
Sn. The results obtained for Ag, Au, W and Sn are compared with the results of Guisbiers model [28] and also with the simulated results for Ag and Au nanocrystals [44]. It is noted that the calculated values for Ag and Au nanocrystals are close to the available simulated results. The results obtained in the present study somehow follow the same trend as of those calculated from Guisbiers model.
18
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200
W
190
170 160 Hexahedral (present)
150
Octahedral (present)
140
CR IP T
θ DN (K) ------------>
180
Guisbiers model [28]
130
Spherical (present)
120
Tetrahedral (present)
110 100 10
20
30
40
50
60
AN US
0
D (nm) --------------->
Figure 4(c): Variation of Debye temperature with shape and size in W nanocrystal.
M
Sn
Spherical (present)
ED
130
Tetrahedral( present) Hexahedral (present)
PT
120
Octahedral (present)
110
CE
θ DN (K)-------------->
140
Guisbiers model [28]
100
AC
0
10
20
30 40 D (nm)------------->
50
60
70
Figure 4(d): Variation of Debye temperature with shape and size in Sn nanocrystal.
19
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1.5
Ag Spherical (present) Tetrahedral (present) Hexahedral (present)
1.3
Octahedral (present)
CR IP T
CpN/ CpB --------------->
1.4
Guisbiers model [28]
1.2
1.1
1 10
20
30
40
50
60
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0
D (nm) -------------->
ED
1.3
Au Hexahedral (present) Octahedral (present)
PT
C pN / CpB -------------->
1.5 1.4
with shape and size in Ag nanocrystal.
M
Figure 5(a): Variation of
Guisbiers model [28] Spherical (present)
1.2
CE
Tetrahedral (present)
AC
1.1 1 0
10
20
30
40
50
60
D (nm) ------------>
Figure 5 (b): Variation of
with shape and size in Au nanocrystal .
20
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W
1.4
Hexahedral (present) Octahedral (present) Guisbeirs model [28] Spherical (present) Tetrahedral (present)
1.3 1.2 1.1 1 10
20
30
40
50
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0
CR IP T
CpN/C pB ---------------->
1.5
60
D (nm) ---------------->
1.4
Hexahedral (present) Octahedral (present) Guisbiers model [28]
PT
CpN /CpB -------------->
1.5
Sn
ED
1.7 1.6
with shape and size in W nanocrystal.
M
Figure 5 (c): Variation of
Spherical (present)
1.3
Tetrahedral (present)
CE
1.2
AC
1.1 1 0
10
20
30
40
50
60
D (nm) ------------->
Figure 5(d): Variation of
with shape and size in Sn nanocrystal.
21
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1.6
Ag Hexahedral (present)
1.4
Octahedral (present) 1.3
Guisbiers model [28] Spherical (present)
1.2
CR IP T
E gN/EgB ---------->
1.5
Tetrahedral (present)
1.1 1 0.9 0
5
10
15
20
25
30
35
Figure 6 (a): Variation of
40
45
50
55
60
AN US
D (nm)------------->
with shape and size in Ag nanocrystal.
Figure 5(a-d) and figure 6(a-d) show the ratio of variation of specific heat capacity in nano and bulk material for Ag, Au, W and Sn nanocrystals
M
and energy bandgap
with particle size. The results are calculated using equation (17) and (18) and compared with the
ED
values obtained from Guisbiers model [28]. The results shown in figures 5 and 6 explain the variation of specific heat capacity and energy bandgap for spherical nanoparticles, regular
PT
tetrahedral shaped, regular hexahedral and regular octahedral structured nanosolids. The graphs indicate that the specific heat capacity and energy band gap in nanostructured crystals increases with decrease in grain size. Figures 5 and 6 depict the same trend of variation of specific heat
CE
capacity and energy band gap with size, however, in small size region, the results are somewhat deviated as compared to other theoretical model [28]. It is thus stated here that the values of
AC
energy band gap and specific heat capacity are more in nanomaterials as compared to their bulk forms while the melting temperature, Debye temperature and cohesive energy show decreasing behavior with decreasing particle size. A good consistency between the present results and the available experimental or theoretical results achieved from other models supports the validity of the model.
22
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1.5
Au Hexahedral (present) Octahedral (present)
1.3
Guisbiers model [28] Spherical (present)
1.2
Tetrahedral (present)
1.1 1 0.9 0
5
10
15
20
25
30
35
40
M
1.3
55
60
W Hexahedral (present) Octahedral (present) Guisbiers model [28]
PT
E gN/ E gB ---------->
1.4
50
with shape and size in Au nanocrystal.
ED
1.5
45
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D (nm) --------------->
Figure 6 (b): Variation of
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EgN/EgB ------------>
1.4
Spherical (present)
1.2
Tetrahedral (present)
CE
1.1
AC
1
0.9 0
10
20
30
40
50
60
70
D (nm)------------->
Figure 6 (c): Variation of
with shape and size in W nanocrystal. .
23
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1.8
Sn Spherical (present) Tetrahedral (present)
1.4
Hexahedral (present) Octahedral (present) Guisbiers model [28]
1.2
1
0.8 0
10
20
30
40
CR IP T
EgN/EgB ------------->
1.6
50
60
70
with shape and size in Sn nanocrystal.
M
Figure 6 (d): Variation of
AN US
D (nm) ------------->
ED
4 Conclusion:
On the basis of overall descriptions, it is emphasized that the present model is capable of
PT
explaining the variation of melting temperature, cohesive energy, Debye temperature, specific heat capacity and energy band gap with the shape and size of nanomaterials. However, such variations are not appreciable in those nanomaterials whose sizes are greater than 30 nm. Both
CE
the size and shape are found to influence the thermodynamic properties of nanostructured materials. From the overall analysis it is stated that the effect of shape becomes less effective if
AC
the size of nanocrystal is increased above 30 nm and almost the same values of different thermalmechanical-optical parameters are obtained if the size is above 30 nm. Finally, it can be concluded that the present model is able to analyze and explain the thermodynamic and optical properties of nanomaterials of different shape and size satisfactorily and may thus be useful for those researchers who are exploring the studies on electronic, thermal and optical properties of nanomaterials.
24
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Acknowledgement: The authors are very much thankful to referees for their valuable comments.
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PT
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AC
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