ARTICLE pubs.acs.org/JPCC
Melting Enthalpy and Entropy of Freestanding Metallic Nanoparticles Based on Cohesive Energy and Average Coordination Number Hamed Omid,† Hamid Delavari H.,‡ and Hamid R. Madaah Hosseini*,† †
Department of Materials Science and Engineering and ‡Institute for Nanoscience and Nanotechnology, Sharif University of Technology, 14588-9694, Iran ABSTRACT: An analytical model is proposed to study the effect of particle size on melting enthalpy and entropy of metallic nanoparticles (NPs). The Mott's and Regel's equations for melting entropy in the combination of core average coordination number (CAC) and surface average coordination number (SAC) of freestanding NPs are considered. Clusters of icosahedral (IC), body centered cubic (BCC), and body centered tetragonal (BCT) structure without any vacancies and defects are modeled. Using the variable coordination number made this model to be in good agreement with experimental and molecular dynamic (MD) results of different crystal structures. The model predicts melting entropy and enthalpy of freestanding NP free of any adjustable parameter or physical constant.
1. INTRODUCTION Thermodynamic parameters are requisite factors to study the material behaviors in various conditions. It is important to know these special thermodynamic properties for nanocrystals to design new materials. Several evidences show that the physical and chemical properties of nanoparticles (NPs) differ from their bulk counterparts, such as melting temperature (Tm),1,2 Curie temperature (Tc),3 cohesive energy (EC),4 melting enthalpy (Hm), and entropy (Sm).5,6 The micro (nano)-electronics industry has a great interest in the thermodynamics of nanomaterials, where transistors and metal interconnectors are in nanoscale. One particular phenomenon of interest is the size-dependent melting point of NPs, which drops with decreasing particle size. Many theoretical models have been developed by researchers to explain and calculate the melting temperature of NPs.2,712 To fathom melting transition behavior in nanostructures, a clear understanding of enthalpy and entropy of melting is vital. Nevertheless, there is some limited research on melting enthalpy and entropy.5,6,1316 Lai et al.17 developed a model based on excluded volume (δV), which is the volume of the liquid surface layer and the solid core region, to describe the heat of fusion for small Sn particles. They defined δV in terms of the critical thickness, t0, of the liquid layer. In this model, t0 has been considered as the fitting parameter. However, Lereah et al.18 and Kofman et al.19 showed that the thickness of the critical liquid layer is size-dependent. Jiang et al.5,20 developed a model for the melting entropy of metallic and organic nanocrystals with a combination of Mott's21 and Regel's and Glazov’s22 equation for the melting entropy and a model for the size-dependent melting point.23 It is worthy to mention that their model can only support NPs above the 4 nm in diameter, i.e., below this size the results are unreliable based on r 2011 American Chemical Society
the third law of classical thermodynamics, which predicts that the entropy is permanently positive. Guisbiers and Buchaillot24 generated a model for the melting enthalpy of nanomaterials according to classical thermodynamics. In this model, the Gibbs free energy of a nanostructure has been expressed as a sum of the bulk free energy with a term considering the effect of the surface at the nanoscale. The melting enthalpy of NPs has been estimated by the size-dependent melting temperature of NPs, which was similar to the Lu et al. relationship.25 Their approach was not possible to deduce the size effect on the melting entropy of NPs; by other means, they found that the melting entropy of NPs is equal to the bulk melting entropy. The reason is due to the mismatch of the nano to bulk ratio of melting enthalpy and melting temperature (ΔHmn/ΔHmb = Tmn/Tmb), notably for less than 10 nm NPs.26 Safaei and Shandiz15,16 developed a model based on the cohesive energy and packing factor of surface (PS) and lattice (PL) to calculate melting entropy and enthalpy of nanoparticles. The q factor (the ratio of the surface to core coordination number), which has been implemented in this model, gains constant values for different crystal structures. By contrast,27 it has been shown that the CAC and SAC are variable. Therefore, the q is particle-size-dependent and increases with decreasing particle size. In addition, the surface of particles actually consists of several plates with various Miller indices; however, in this model the surface of particles are covered by the most closed packed plane. Furthermore, according to the third law of classical thermodynamics, the entropy cannot gain the negative value. Accordingly, all of the above-mentioned theories are unable to Received: May 3, 2011 Revised: July 9, 2011 Published: August 04, 2011 17310
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predict a reasonable amount of entropy for NPs when the size of NP decreases to a critical point (typically below 2 nm). Therefore, developing a comprehensive model to calculate the melting entropy and enthalpy of metallic NPs with different crystal structures is very important. The aim of the present work is to use the average coordination number to predict the melting entropy and enthalpy dependent on the size of clusters. It is shown that the proposed model could predict the melting entropy and enthalpy of freestanding NPs as a function of their sizes. The results are compared with the experimental results, MD simulations, and other theoretical models reported by others to show the capability of the model.
2. THEORETICAL CALCULATIONS For a cluster that consists of N atoms, the cohesive energy (EPc ) can be written as follows:27 Ci Cσ N σ P b Ec ¼ Ec 1 μ ð1Þ Ci 3 Ni Ebc
is the cohesive energy of bulk; Ci and Cσ are the where, coordination number of interior (core) and surface atoms, respectively; μ is the shape factor; and Nσ is the number of atoms on the surface of NP. In addition, the relationship between the cohesive energy and the melting point of NP can be written as follows:8,28 Tmp Ep ¼ Tmb Eb
Spm
¼
Sbm
3R Tp ln mb þ 2 Tm
! ð6Þ
where, R is the universal gas constant, Spm and Sbm are the melting entropies of NP and bulk, respectively. Notice that in this model the ratio of liquid to crystal ultrasound propagation for bulk and nanosized material has been considered to be independent of particle size. The melting entropy of NP can be calculated by a combination of eq 3 and eq 6 as follows: 3R Ci Cσ N σ p b ln 1 μ ð7Þ Sm ¼ Sm þ 2 Ci 3 Ni In addition, the melting enthalpies of NP and bulk can be expressed as the product of their corresponding melting entropy and melting point as follows: Hmp ¼ Spm 3 Tmp
ð8Þ
Table 1. Melting Temperature, Enthalpy, and Entropy of Bulk Metalsa element
ð2Þ a
With the combination of eq 1 and eq 2, the melting temperature of NP can be calculated as follows: Tmp Ci Cσ N σ ¼ 1μ Tmb Ci 3 Ni
and Regel’s22 equation by
structure
Tmb [K]
Hmb [kJ mol1]
Smb [J mol1 K1]
Cu
FCC
1357.8
13.01
9.58
Ag
FCC
1234
11.3
9.16
V In
BCC BCT
2175 429.9
17.57 3.28
8.08 7.63
The melting entropy has been calculated by eq 9.
ð3Þ
Here, we consider perfect clusters of IC and BCC structures. These clusters can be built up by locating atoms at various shells around a central atom. A shell is constituted by the atoms that are all at the same distance from center. The number of crust, ν, defines the order of the cluster which is formed by the atoms at various shells to form a perfect cluster. On the basis of our previous work,27 the CAC (Ci) and the SAC (Cσ) of a particle can be calculated by Ci ¼
2ψðυ 1Þ Nðυ 1Þ
ð4Þ
Cσ ¼
2ψσ ψ þ σc Nσ Nσ
ð5Þ
where ψ(υ 1) and N(υ 1) are the total number of bonds between the nearest atoms and the total number of atoms at the lower crust order (ν 1), respectively. Nσ, ψσ, and ψσc are the total number of atoms on the surface of NP, total number of bonds between surface atoms of NP, and the number of bonds between surface atoms and the core of NP, respectively. For more details, the reader is invited to refer to the original work of Delavari H. et al.27 As mentioned previously, Jiang et al. 5,20 expressed the melting entropy of metallic nanocrystals based on Mott's21
Figure 1. Relationship between melting entropy (A) and enthalpy (B) of Cu NPs with their diameter. 17311
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Figure 2. Relationship between melting entropy (A) and enthalpy (B) of Ag NPs with their diameter.
Hmb ¼ Sbm 3 Tmb
ð9Þ
where Hpm and Hbm are the melting enthalpies of NP and bulk, respectively. By combining eqs 79, the melting enthalpy of NP can be calculated as follows: Hmp ¼
1μ
C i C σ Nσ 3R b C i C σ Nσ b T þ ln 1 μ H m 2 m C i 3 Ni 3 C i 3 Ni
ð10Þ In this work, the proposed models have been compared with other models, experimental data, and MD simulation results. It should be noticed that models of Shandiz et al.15 with two different values of q (1/2 and 1/4) and Jiang and Shi5 have been considered for entropy. Models of Jiang et al.6 and Guisbers and Buchaillot24 have been used for enthalpy.
3. RESULTS AND DISCUSSION To calculate the melting entropy and enthalpy of nanoparticles based on eq 7 and 10, respectively, the melting temperature, enthalpy, and entropy of bulk have been summarized in Table 1. In Figure 1, three models are compared to the MD results of the melting enthalpy and entropy of copper (Cu) NPs.13 It can be observed that all models predict decreasing entropy and enthalpy with decreasing particle size. These phenomena can be
Figure 3. Relationship between melting entropy (A) and enthalpy (B) of V NPs with their diameter.
explained with a decreasing cohesive energy as particle size decreases. However, all other models underestimate the melting enthalpy value of Cu NPs; our model is in good agreement with the MD results (Figure 1A). Despite the fact that the Guisbiers and Buchaillot24 approach is not able to consider the size effect on the melting entropy, it gives a good estimation of the melting enthalpy of Cu NPs (Figure 1B). Our model overestimates the nanoscaled melting enthalpy of copper. It is worthy to mention that the melting enthalpy and entropy converge to zero when the particle size decreases. With considering one nanometer NP in diameter, the Jiang et al.5,6 and Attarian Shandiz et al.15 models predict a negative value: 13.8 J/mol 3 K (for q = 1/4) and 60 J/mol 3 K, respectively. Figure 2 shows the comparison of theoretical models with MD results of melting entropy and enthalpy of silver NPs.29 The model of Safaei and Shandiz16 has been drawn for different values of q. However, according to Shandiz et al.15 values equal to 1/2 and 1/4 are more appropriate for particles having a diameter higher and lower than 10 nm, respectively. It is clear that our model is in good agreement with the MD results of melting entropy and enthalpy of silver NPs in comparison with those of other models. In addition, all other models underestimate the melting entropy and enthalpy of Ag NPs. In Figure 3, our model has been compared with MD results of vanadium (V), Safaei and Shandiz's model,16 Jiang et al.'s model,5,6 and Guisbiers' model.24 As mentioned previously, the q 17312
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’ AUTHOR INFORMATION Corresponding Author
*Tel/Fax: +98-21-66005717. E-mail:
[email protected].
’ REFERENCES
Figure 4. Relationship between melting entropy (A) and enthalpy (B) of In NPs with their diameter.
value in the Safaei and Shandiz models equal to 0.5 and 0.25, respectively, is more appropriate for particles having a diameter higher and lower than 10 nm. It can be observed that their models for q = 0.5 have favorable agreement in small size. Nevertheless, our model is in good agreement with MD results of V NPs. Figure 4 shows a comparison between theoretical models and experimental data of melting entropy and enthalpy for indium (In) NPs with body center tetragonal (BCT) structures, which have been considered as body center cubic (BCC).14 This figure shows that it is rather difficult to obtain accurate experimental data in nanoscale sizes. All of the models show that the melting entropy and enthalpy of In decreases with decreasing particle size. It can be observed that all models converge on each other when the particle size increases. In addition, for small particles (below 3 nm) the Shandiz et al. model15 (for q = 1/2) converges on our model.
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4. CONCLUSIONS A simplified model has been proposed that is based on the cohesive energy of atoms to predict the melting enthalpy and entropy of nanoclusters with their size and crystal structures. SAC and CAC, in combination with Mott's and Regel’s equation for melting entropy, were considered in the model. This model, as an improved model of Shandiz and Safaei, has been considered to predict melting entropy and enthalpy of freestanding NPs. Results show that the model is in good agreement with experimental and molecular dynamic (MD) results. 17313
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