Accepted Manuscript Title: Effect of size and shape on specific heat, melting entropy and enthalpy of nanomaterials Author: Madan Singh Sekhants’o Lara Spirit Tlali PII: DOI: Reference:
S1658-3655(16)30079-6 http://dx.doi.org/doi:10.1016/j.jtusci.2016.09.011 JTUSCI 339
To appear in: Received date: Revised date: Accepted date:
19-7-2016 23-9-2016 25-9-2016
Please cite this article as: M. Singh, S. Lara, S. Tlali, Effect of size and shape on specific heat, melting entropy and enthalpy of nanomaterials, Journal of Taibah University for Science (2016), http://dx.doi.org/10.1016/j.jtusci.2016.09.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effect of size and shape on specific heat, melting entropy and enthalpy of nanomaterials
Madan Singh*
[email protected], Sekhants’o Lara, Spirit Tlali
ip t
Department of Physics and Electronics, National University of Lesotho, P.O. Roma 180, Lesotho
us
cr
Tel.: 266 57044674
Abstract
an
A simple theory is proposed to study the size and shape dependent Specific heat, melting entropy and enthalpy of nanomaterials. It is shown that the particle size and shape affect the Specific heat, melting entropy and enthalpy of nanomaterials. The model is applied to Ag,
M
Cu, In, Se, Au and Al nanomaterials in spherical, nanowire and nanofilms shapes. It is found that the specific heat increases with the decrease in particle size. Whereas, melting entropy
ed
and enthalpy decrease as the particle size decreases. Our theoretical predictions agree well
formulation developed.
Ac ce
Keywords
pt
with available experimental and computer simulation results, which support the validity of
Nanomaterials; Specific heat; Melting entropy; Surface properties; Thermodynamic properties
1. Introduction
It has been reported that nanomaterials exhibit interesting physical and chemical properties significantly different as compared to corresponding bulk materials [1-5]. Due to the enormous surface area to volume ratio of nanomaterials, the energy associated with the atoms of these nanomaterials will be different as compared to conventional bulk materials, leading to size dependent thermodynamic properties of nanomaterials [6, 7]. The cohesive energy also known as the heat of sublimation is an important physical quantity to account for the
Page 1 of 27
strength of metallic bonds, as it is the energy required to divide the metallic crystal into individual atoms. Experimental and theoretical studies of cohesive energy of W, Ag, Co, Al and Cu nanoparticles were carried out by many researchers [8-10]. Modelling the size and shape dependent cohesive energy of nanoparticles and its applications in the heterogeneous systems has been calculated theoretically by Li [11] and it is reported that the cohesive energy of the free nanoparticles usually decreases as its size decreases. Considering the
ip t
effects of particles size, lattice and surface packing factors and coordination numbers of the lattice, Shandiz et al. [12] calculated the melting entropy and enthalpy of metallic
cr
nanoparticles. A theoretical study of modelling the melting enthalpy of nanomaterials is
us
debated by defining the conventional shape factor α [13].
The melting temperature of nanosolids (such as nanoparticles, nanowires and nanofilms) has been predicted based on size dependent cohesive energy [14], it is shown that melting
an
temperature of nanomaterials decreases with decrease in particle size. Researchers have calculated the root mean square amplitude model, the size dependent Debye temperature
M
model and size dependent thermal conductivity model [15-16] by considering Lindemann’s criterion and Mott’s equation. It is stated that Debye temperature decreases for nanomaterials as size decreases. Effects of particle size and thermodynamic energy, based on surface
ed
thermodynamics and the atomic bond energy, was reported to calculate the mechanical properties like surface tension and Young’s modulus of nanocrystals [17-18]. The cohesive
pt
energy is the basic thermodynamic property to predict melting temperature, melting enthalpy, melting entropy and specific heat of nanomaterials. Scholars have proposed different models
Ac ce
namely Latent heat model, liquid drop model and surface area difference model [19-21] to predict cohesive energy of nanomaterials. Recently, using the concept of cohesive energy changes with the atomic coordination environment, Qi [22] presented a theory, based on the bond energy model to highlights the thermodynamics for the nanoparticles, nanowires, and nanofilms. The size and coherence dependent cohesive energy, melting temperature, melting enthalpy, vacancy formation energy and vacancy concentration of nanowires and nanofilms are reported [23]. It is found that the variation direction of the thermodynamic properties is determined by the coherent interface and the quantity of variation depends upon the crystal size. Shandiz et al. [12] developed a model for melting entropy and enthalpy of metallic nanoparticles which is based on the effect of packing factors, coordination numbers of lattice and crystalline planes. Thus, it appears that there exist some attempts to study size dependent
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thermodynamical properties. Moreover, thermodynamical properties also depend on the shape, so, it may be valuable to present a model, which incorporates the effects of shape also. In this contribution, we present a surface free energy model, free from any adjustable parameters, which depends upon the size and shape with respect to cohesive energy of nanomaterials. Agreeing the relation between melting temperature and cohesive energy, the
ip t
expressions for size and shape dependent specific heat, melting entropy and enthalpy are obtained. The theoretical predictions of these expressions are applied to Ag, Cu, In, Se, Au
cr
and Al nanomaterials in spherical, nanowire and nanofilms shapes. 1. Methodology
us
The total cohesive energy is defined as the energy due to the contributions of the interior atoms and the surface atoms of the nanomaterial, which is expressed as [14]
an
1 ETotal = E0 (n − N ) + E0 N 2 (1)
M
Where n is the total number of atoms in the nanosolid and N is the number of surface atoms. Therefore, (n − N) is the total number of interiors atoms in the nanomaterial. E0 is the
Eq. (1) may be written as:
ed
cohesive energy of the bulk material per atom. To determine the cohesive energy per mole,
pt
N 1 AETotal / n = AE0 1 − + AE0 N n 2n (2)
Ac ce
Where, A is the Avogadro’s number. Here, AETotal / n is the cohesive energy per mole of the nanomaterial En and AE0 is the cohesive energy per mole of the corresponding bulk material ( Eb ) . On substituting in Eq. (2), one can get N En = Eb 1 − 2n
It is reported [24-25] that the cohesive energy is the linear relation to the melting temperature, we can therefore write the relation for melting temperature of nanomaterials as: N Tn = Tb 1 − 2n (3)
Page 3 of 27
Where, Tb is the melting temperature of bulk material. The Lindemann’s melting criterion states that a crystal melts when the root mean square displacement of atoms exceeds a certain fraction of the interatomic distance in the crystal, which is valid for small particles. Using this theory, the relationship between the melting temperature and Debye temperature of the bulk material can be given as [25] 1/2
Where, M is the molecular mass and V is the volume per atom.
us
Similarly, for the nanomaterials, the expression is 1/2
T ∝ n 2/3 MV
(5)
an
θ Dn
ip t
(4)
cr
θ Db
T ∝ b 2/3 MV
Eq. (3) and Eq. (5) give the following correlation:
(6)
ed
M
2 θ Dn T = n 2 θ Db Tb
On the basis of Debye’s theory [26], a relationship is obtained between specific heat at
1
θ Db 2
Ac ce
C pb ∝
pt
constant pressure and Debye temperature of bulk material [27] that is (7)
Similarly, for the nanomaterials, the expression is C pn ∝
1
θ Dn 2
(8)
From Eqs. (6), (7) and (8), we get
C pn C pb
=
Tb Tn
(9)
On substituting the value of Eq. (3) into Eq. (9), we obtained
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N C pn = C pb 1 − 2n
−1
(10)
This is the relation of specific heat for nanomaterials and bulk materials at different shapes and sizes.
ip t
Now we are deriving the equations for melting entropy and melting enthalpy. The size dependency of nanomaterials’ melting entropy can be calculated by size dependency of their
cr
melting points [28]. The melting entropy for matelic crystals is largely vibrational in nature and electronic entropy is insignificant. The relation between melting entropy and melting
3R Tn ln 2 Tb
(11)
an
Smn = S mb +
us
temperature is derived by considering the vibrational entropy [28] as given below
Where, R is the gas constant; Smn and Smb are the melting entropy of nanomaterials and bulk
M
materials.
S mn = S mb +
3R N ln 1 − 2 2n
ed
From Eq. (3) and (11), we get
(12)
H mb = Tb Smb
pt
The melting enthalpy and melting entropy for bulk materials follow the relation as given (13)
Ac ce
Assuming this relationship is still valid in nanomaterials, we can write
H mn = Tn Smn
(14)
Substituting the values of Eq. (3) and Eq. (12) into Eq. (14) and rearranging, we get 3RTb N N H mn = H mb + ln(1 − ) 1 − 2 2n 2n
(15)
For the values of N/2n, the method has already been reported in the literature [14]. The value of N/2n depends upon the shape and size of the nanomaterials. The value of N/2n is 2d/D for spherical nanosolids, where d is the diameter of an atom and D is the diameter of the spherical nanosolids. For nanowire and nanofilm the values of N/2n are 4d/3l and 2d/3h respectively. Where, l is the diameter of nanowire and h is the width of the nanofilm.
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1. Results and Discussion
The input parameters [15, 29-31] required for the present work are given in Table 1. We
ip t
derived Eqs. (10), (12) and (15) to calculate the size and shape dependence of Specific heat, melting entropy and enthalpy of nanomaterials. Size dependence of specific heat of Ag
cr
nanoparticle is plotted in Fig. 1, along with experimental data [32]. Rupp et al. [32] experimentally measured the specific heat at constant pressure, considering the particles with
us
surface atoms and inner atoms as we discussed in our theory. It is seen that specific heat increases with decreasing the size of nanocrystal indicating that specific heat capacity varies
an
inversely with particle size. The cause of increased specific heat at small sizes is due to the large atomic thermal vibration energies of surface atoms. It is also shown by Sun [33] that the vibrational amplitude of surface atoms is larger for nanosolids than that of bulk materials
M
resulting larger vibrational energy of surface atoms. It is found that the results obtained from Eq. (10) are in good consistent with the experimental values [32].
A good agreement
ed
between theory and experimental results encouraged the authors to extent the model to study specific heat of Cu, In, Se for different shapes and sizes for which experimental values are lacking. Fig. (2) Presents the specific heat of Cu nanosolids in spherical shape calculated by
pt
Eq. (10) as shown by solid lines. For comparison purpose we also drawn the results found
Ac ce
theoretically by Zhu et al. [27]. Our results are very close to the results obtained by Zhu et al. [27] as shown in Fig. 2. We have extended the model to study different shapes, namely spherical, nanowire and nanofilm. Figs. 3-4 compare the model prediction of In and Se nanosolids in different shapes (nanosphere, nanowire and nanofilm). Our predictions are consistent as experimentally measured in Ag nanosolid. It is found that C pn increases with decrease in size for all three shapes with similar trend as shown in Figs. 3-4. Eq. (12) is used to calculate the size dependence melting entropy of Ag, Cu and In spherical nanosolids. The computed values of melting entropy of Ag, Cu and In nanosolids in spherical shape are shown in Figs 5-7, along with the available experimental data [30, 33-35]. As it is revealed from Figs. 5-7, the melting entropy goes down with the decrease in particle size. Melting entropy decreases sharply with the small reduction in particle size. It is realized that our results are in good agreement with the available experimental data. The size and shape
Page 6 of 27
dependent melting entropy of Se, Au and Al nanosolid using Eq. (12) is shown in Figs. 8-10 for different shapes such as nanospherical, nanowire and nanofilm. It is observed that melting entropy decreases with reduction in particle size. Moreover, the reduction in melting entropy is increasing from nanofilm to nanowire and nanosphere. It is evident from Eq. (12), that the melting entropy is dependent upon N/2n. When, we relate N/2n [14] for nanofilms to nanowire to nanosphere, the ratio becomes 1:2:3. Therefore, the melting entropy difference
ip t
for nanofilm to nanowire to nanosphere increases with the same particle size.
The size dependent melting entropy for of Al nanoparticle is shown in Fig. 10, along with the
cr
available experimental facts as reported by Eckert et al.[36] for spherical shape. Eckert et al.
us
[36] performed the experiment in an oxygen atmosphere for Al nanosolid, in his experiment the interaction among Al nanosolids is avoided due to the oxide film on the surface of the particle. It is reported in Fig. 10 that our results agree well with the experimental results for
an
spherical shape.
We used Eq. (15) to calculate the size dependence of melting enthalpy of Ag, Cu, In and Se
M
nanosolids. Figs. 11-14 show the comparison between the model prediction and the experimental records available for these nanosolids in different shapes. It is reported that
ed
melting enthalpy decreases by decrease of particle size. Melting enthalpy for Cu nanosolid in spherical shape is shown in Fig. 12 along with the molecular simulation results obtained by Delogu [35], which support the results obtained in our present work. There is a good
pt
harmony between our theory and Delogu [35] findings. Fig. 13 shows the variation of melting enthalpy of In in spherical shape in terms of their size along with experimental data, it is
Ac ce
shown that Eq. (15) gives fair agreement for the melting enthalpy of In nanosolid, which demonstrates the suitability of the model presented here. The decreasing of melting entropy and enthalpy is because of the surface contribution associated to the large surface area to volume ratio and breaking bonds. The reason is that the atoms at the free surface experience a different background than do the atoms in the bulk of a material. These atoms have excess energy associated with the surface atoms, which is realized as surface free energy [38]. In case of bulk materials, surface free energy is neglected due to the association of few layers of the atoms near the surface. Therefore, the ratio of volume occupied by the surface atoms and the total volume of the bulk material is remarkably small. Whereas, for the nanosphers, nanowires and nanofilms, the surface to volume ratio becomes substantial. Consequently, the surface free energy of nanomaterials increases. Hence, the thermodynamical properties of materials changes at nanoscales.
Page 7 of 27
Page 8 of 27
ed
pt
Ac ce us
an
M
cr
ip t
Page 9 of 27
ed
pt
Ac ce us
an
M
cr
ip t
Page 10 of 27
ed
pt
Ac ce us
an
M
cr
ip t
ip t cr us an M ed pt
1. Conclusions
In conclusion, a model for size and shape dependent specific heat has been introduced. It is
Ac ce
found that the specific heat of nanosolids increases with decrease of the particle size in spherical, nanowire and nanofilm shapes. Moreover, the model is applied to find the shape and size dependence of melting entropy and enthalpy of nanosolids. It has been shown that the prediction of present model for specific heat, melting entropy and enthalpy has good agreement with the molecular dynamics results and available experimental data, signifying that our present model at nanoscale can be applied for a wide range of surface caused phenomena. The present model may has the potential application to calculate the thermodynamical properties of nanomaterials.
Acknowledgements The authors are thankful to the referee for valuable comments which have been useful in revising the manuscript.
Page 11 of 27
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ip t
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cr
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an
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Ac ce
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ip t
Physica E 41 (2009) 359.
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cr
coarse-grained polycrystalline states in element selenium, Phys. Rev. B 54 (1996) 6058.
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[33] C.Q. Sun, Prog. Size dependence of nanostructures: Impact of bond order deficiency, Solid State Chem. 35 (2007) 1.
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[34] W.H. Luo, W.Y. Hu, S.F. Xiao, Size Effect on the thermodynamic properties of silver nanoparticles, J. phys. Chem. C 112 (2008) 2359.
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nanocrystalline aluminium powders, Nanostruct. Mater. 2 (1993) 407. [37] M. Zhang, M. Yu, F. Efremov, E.A. Schiettekatte, A.T. Olsan, S.L. Kwan, T. Lai, J.E. Wisleder, L.H. Greene, Allen, Size-dependent melting point depression of nanostructures:
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Page 14 of 27
30
Ag
ip t
28
27
cr
Cpn(J/mole/K)
29
25 10
20
30
40
50
an
0
us
26
D(nm)
Ac ce
pt
ed
M
Fig. 1. Variation of Specific heat of Ag nanomaterial in spherical shape with size. Values calculated in the present study using Eq. (10) are shown with continuous line and the experimental data [32], with points.
Page 15 of 27
ip t cr us an
34
M
Cu
ed
30
pt
28
26
Ac ce
Cpn(J/mole/K)
32
24
0
5
10
15
20
25
D(nm)
Page 16 of 27
Fig. 2. Variation of Specific heat of Cu nanomaterial in spherical shape with size. Values calculated in the present study using Eq. (10) are shown with continuous line and the
ed
M
an
us
cr
ip t
theoretically by Zhu et al. [27], with dotted line.
40
In
pt
38
Cpn(J/mole/K)
Ac ce
36 34
nanosphere
32 30 28
nanowire
nanofilm
26 5
10
15
20
25
D/h/l(nm)
Page 17 of 27
Fig. 3. Variation of Specific heat of In nanomaterial in spherical, nanowire and nanofilm shape with size. Values calculated in the present study using Eq. (10) are shown with
cr
ip t
continuous lines.
35
Se
us
34
an
32
nanosphere 31
28
nanowire
nanofilm
27 5
ed
29
M
30
pt
Cpn(J/mole/K)
33
10
15
20
25
Ac ce
D/l/h(nm)
Fig. 4. Variation of Specific heat of Se nanomaterial in spherical, nanowire and nanofilm shape with size. Values calculated in the present study using Eq. (10) are shown with continuous lines.
Page 18 of 27
10
Ag
ip t
6
4
cr
Smn(J/mole/K)
8
0 5
10
15
20
30
M
D(nm)
25
an
0
us
2
Fig. 5. Variation of melting entropy of Ag nanomaterial in spherical shape with size. Values calculated in the present study using Eq. (12) are shown with continuous line and the
Ac ce
pt
ed
experimental data [34], with points.
Page 19 of 27
10
Cu
ip t
6
cr
Smn(J/mole/K)
8
0
5
10
15
20
25
30
an
D(nm)
us
4
M
Fig. 6. Variation of melting entropy of Cu nanomaterial in spherical shape with size. Values calculated in the present study using Eq. (12) are shown with continuous line and the
Ac ce
pt
ed
experimental data [35], with points.
Page 20 of 27
In
8
ip t
4
cr
Smn(J/mole/K)
6
0 5
10
15
20
25
30
an
0
us
2
M
D(nm)
Fig. 7. Variation of melting entropy of In nanomaterial in spherical shape with size. Values calculated in the present study using Eq. (12) are shown with continuous line and the
Ac ce
pt
ed
experimental data [30], with points.
Page 21 of 27
Se 10
Nanofilm Nanowire NanoSphere
ip t
Smn(J/mole/K)
9
8
us
cr
7
6 4
6
8 D/l/h(nm)
10
12
14
an
2
Fig. 8. Variation of melting entropy of Se nanomaterial in spherical, nanowire and nanofilm
M
shape with size. Values calculated in the present study using Eq. (12) are shown with
10
Au
Ac ce
Nanofilm
pt
ed
continuous lines.
8
Smn(J/mole/K)
Nanowire
Nanosphere
6
4
2
0 0
5
10
15
20
25
30
D/l/h(nm)
Page 22 of 27
Fig. 9. Variation of melting entropy of Au nanomaterial in spherical, nanowire and nanofilm shape with size. Values calculated in the present study using Eq. (12) are shown with
ip t
continuous lines.
Al
cr
12
us
Nanowire 10
Nanosphere
an
Smn(J/mole/K)
Nanofilm
5
10
M
8
15
20
ed
D/l/h (nm)
Fig. 10. Variation of melting entropy of Al nanomaterial in spherical, nanowire and nanofilm
pt
shape with size. Values calculated in the present study using Eq. (12) are shown with
Ac ce
continuous lines and the experimental data [36] for spherical shape with delta.
Page 23 of 27
12
Ag 10
ip t
Hmn(KJ/mole)
8
6
2 5
10
15
20
an
D(nm)
us
cr
4
Fig. 11. Variation of melting enthalpy of Ag nanomaterial in spherical shape with size.
ed
experimental data [30], with points.
M
Values calculated in the present study using Eq. (15) are shown with continuous line and the
pt
14
Cu
Ac ce
12
Hmn(KJ/mole)
10
8
6
4
2 5
10
15 D(nm)
20
25
30
Page 24 of 27
Fig. 12. Variation of melting enthalpy of Cu nanomaterial in spherical shape with size. Values calculated in the present study using Eq. (15) are shown with continuous line and the molecular dynamics simulation results [35], with points.
ip t
3.5
In
3.0
cr us
2.0
1.5
1.0
0.5
0.0 5
10
15
20
M
0
an
Hmn(KJ/mole)
2.5
25
30
ed
D(nm)
Fig. 13. Variation of melting enthalpy of In nanomaterial in spherical shape with size. Values calculated in the present study using Eq. (15) are shown with continuous line and the
Ac ce
pt
experimental data [37], with points.
Page 25 of 27
7
6
nanofilm
5
nanowire
4
ip t
Hmn(KJ/mole)
Se
nanosphere
cr
3
5
10
us
2 15
20
an
D/l/h (nm)
Fig. 14. Variation of melting enthalpy of Se nanomaterial in spherical, nanowire and
M
nanofilm shape with size. Values calculated in the present study using Eq. (15) are shown
ed
with continuous lines.
Ac ce
Nanomaterials d(nm)
pt
Table 1 The input data used in the calculations [15, 29-31] Smb(J/mole/K) Hmb(kJ/mole) Tb(K)
Cpb(J/mole/K)
0.256
9.76
13.26
1357.6
24.47
0.319
9.16
11.30
1234
25.35
0.288
9.34
12.5
1337.58
25.41
Al
0.258
11.46
10.7
933.25
24.20
In
0.329
7.65
3.29
429
26.75
Se
0.230
9.76
6.69
494
26.65
Cu Ag Au
Page 26 of 27
Page 27 of 27
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