Size and shape dependent melting temperature of

Materials Chemistry and Physics 88 (2004) 280–284

Size and shape dependent melting temperature of metallic nanoparticles W.H. Qi∗ , M.P. Wang School of Materials Science and Engineering, Central South University, Changsha 410083, China Received 12 December 2003; received in revised form 27 March 2004; accepted 8 April 2004

Abstract A new model accounting for the particle size and shape dependent melting temperature of metallic nanoparticles is proposed in this paper, where the particle shape is considered by introducing a shape factor. It is shown that the particle shape can affect the melting temperature of nanoparticles, and the particle shape effect on the melting temperature become larger with decreasing of the particle size. The present calculation results on the melting temperature of Sn, Pb, In and Bi nanoparticles are well consistent with the corresponding experimental values and better than these given by liquid drop model. © 2004 Elsevier B.V. All rights reserved. Keywords: Melting temperature; Nanoparticles; Shape effect

1. Introduction Crystal melting is a very complicated process. Since Takagi first reported the size dependent melting temperature of small particles by means of transmission electron microscope [1], researchers have paid more attention to this basic but still unclear phenomenon [2–5]. Now it is found that the melting temperature of metallic [2,3], organic [4] and semiconductors [5] nanoparticles decreases with decreasing their particle size. In other words, their melting temperature is lower than the corresponding bulk materials. It is known that the melting temperature depression results from the high surface-to-volume ratio, and the surface substantially affects the interior “bulk” properties of these materials. A lot of models try to explain the size dependent melting temperature [6,7], such as liquid drop model [6] and Jiang’s model [7], etc. However, one common characteristic of these models is that the nanoparticles are regarded as ideal spheres. Since the melting temperature depression results from the large surface-to-volume ratio, the surface areas of nanoparticles in different shape will be different even in ∗

Corresponding author. Fax: +86 731 8876692. E-mail address: [email protected] (W.H. Qi).

0254-0584/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2004.04.026

the identical volume, and the area difference is large especially in small particle size. Therefore, it is needed to take the particle shape into consideration when developed models for the melting temperature of nanoparticles. In this contribution, we will introduce a shape factor to account for the particle shape difference, and develop the model for the size and shape dependent cohesive energy of nanoparticles. According to the relation between melting temperature and cohesive energy, the expression for the size and shape dependent melting temperature of nanoparticles will be developed, and the theoretical predictions of this expression for the melting temperature of Sn, Pb, In and Bi nanoaparticles will be compared with the available experimental values and the theoretical results given by liquid drop model.

2. Model To account for the particle shape difference, we have introduced a new parameter [8], i.e., the shape factor α, which is defined by the following equation α=

S S

(1)

W.H. Qi, M.P. Wang / Materials Chemistry and Physics 88 (2004) 280–284

where S is the surface area of the spherical nanoparticle and S = 4πR2 (R is its radius). S is the surface area of the nanoparticle in any shape, whose volume is the same as spherical nanoparticle. It should be mentioned that this definition of the shape factor is dimensionless, and which is easier to be introduced in the theoretical models of spherical nanoparticles as a modified parameter to generalize these models. According to Eq. (1), the surface area of a nanoparticle in any shape can be written as S = α4πR2

(2)

If the atoms of the nanoparticle are regarded as ideal spheres, then the contribution to the particle surface area of each surface atom is πr2 (r is the atomic radius). The number of the surface atoms N is the ratio of the particle surface area to πr2 , which is simplified as N = 4α

R2 r2

(3)

The volume of the nanoparticle V is the same as the spherical nanoparticle, which equals to (4/3)πR3 . Then the number of the total atoms of the nanoparticle is the ratio of the particle volume to the atomic volume ((4/3)πr 3 ), which leads to n=

R3 r3

(4)

If the surface atoms denote the atoms in the first layer of the surface of the nanoparticle, then the number of the interior atoms is n − N. It is known that cohesive energy is an important parameter to estimate the metallic bond, which equals to the energy that can divide the metal into isolated atoms by destroying all metallic bond. For simplicity, we can also regard the metallic bond as the interactions between different atoms. In other words, the metallic bonds of each atom equal to the sum of the interaction energies between the atom and the other atoms (in most cases, only the nearest interactions are considered). Each interior atom forms bonds with its surrounding atoms, and we denote the number of its bonds as β. It is reported that the distance between the surface atoms and the nearest interior atoms is larger than the distance between interior atoms [9]. Therefore, less than half of the volume of each surface atom in the lattice, which means more than half of the bonds of the surface atom are dangling bonds, then we approximately regard the number of the bonds of a surface atom as (1/4)β. The cohesive energy of metallic nanoparticle is the sum of the bond energies of all the atoms. Considering Eq. (4), the cohesive energy of metallic crystal in any shape (Ep ) can be written as Ep =

3 1 1 R2 R R2 β4α 2 + β Ebond − 4α 2 4 r r3 r2

(5)

where Ebond is bond energy. The value 1/2 results from the fact that the each bond belongs to two atoms. For simplicity,

281

we can rewrite Eq. (5) as r 1 (6) Ep = nβEbond 1 − 6α 2 D where D is the size of the crystal and D = 2R. For bulk solids, D 6α r, Eq. (6) is reduced to E0 = (1/2)nβEbond , where E0 is the cohesive energy of solids. We can rewritten Eq. (6) as r Ep = E0 1 − 6α (7) D It should be mentioned that the size D is the diameter of the spherical crystal. For a non-spherical crystal, its size is defined as the diameter of the spherical crystal which has the identical volume with the non-spherical crystal. If D denotes the diameter of the metallic nanoparticles, Eq. (7) can be used to predict size and shape dependent cohesive energy of metallic nanoparticles. In Eq. (7), we have used Ep and E0 . The most difference between Ep and E0 is that Ep is taken the surface effect on the cohesive energy into consideration. For bulk crystal, Ep equals E0 ; for metallic nanoparticles, the surface effect cannot be neglected, Ep and E0 are different. Therefore, Ep can also describe the cohesive energy of nanoparticles, but E0 cannot. According to these discussions, Eq. (7) can be regarded as a more general relation for the cohesive energy of crystals. Rose et al. [10–13] proposed a universal model for solids from the binding theory of solid. Combining their theory with Debye model, they theoretically derived the well-known empirical relation of the melting temperature and the cohesive energy for pure metals Tmb =

0.032 E0 kB

(8)

where Tmb is the melting temperature of bulk pure metals, and kb the Boltzmann’s constant. Similar to the cohesive energy, the melting temperature is also a parameter to describe the strength of metallic bond. Therefore, Eq. (8) can be regarded as the mathematical conversion of both parameters. We replace the cohesive energy of solids E0 by the more general form Ep , then r 0.032 Tm = E0 1 − 6α (9) kB D The main difference between Eqs. (8) and (9) is that Eq. (9) has taken the crystal size and shape (surface effect) into account. For convenience, we denote the size and shape dependent melting temperature as Tm . According to Eq. (8), we can rewrite Eq. (9) as r Tm = Tmb 1 − 6α (10) D Eq. (10) is the more general relation for the size and shape dependent melting temperature of crystals. The relation between Tm and Tmb are similar to the relation between Ep and E0 , i.e., for bulk crystal, Tm and Tmb are the same, but Tm can also describe the melting temperature of nanoparticles. Here

282


we mainly deal with the melting temperature of nanoparticles, therefore, we assume that Tm is the melting temperature of nanoparticles and Tmb denotes the melting temperature of the corresponding bulk materials.

3. Results and discussion The main difference between Eq. (10) and other expressions [6,7] for the size dependent melting temperature is that the particle shape is considered in Eq. (8), where the particle shape is well described by the shape factor α. To calculate the melting temperature of metallic nanoparticles by Eq. (8), it is needed to determine the shape factor for different particle shapes. According to the definition of shape factor, we have calculated the shape factor of different particle shapes, and which is listed in Table 1. For the particle shape is different from each other, the shape factor is only approximate description of the particle shape difference. In most experiments, the nanoparticles are close to regular polyhedral shape [14], so it is needed for us to discuss the shape factor of regular polyhedral nanoparticles in more details. The simplest polyhedral particle is regular tetrahedral nanoparticles, and its shape factor equals 1.49. The shape factors of other regular polyhedral particles are smaller than 1.49 according its definition (Eq. (1)), i.e., the value of 1.49 is the up limitation for regular polyhedral nanoparticles. Furthermore, the down limitation of the shape factor of the regular polyhedral nanoparticles is 1, i.e., the shape factor of the spherical nanoparticles. In the present calculation, we will give the calculation results of the melting temperatures of metallic nanoparticles at the two limitations. It is shown in Eq. (10) that the melting temperature is the function of the particle size and the shape factor. Therefore, we can discuss the melting temperature variation with the shape factor in a specific particle size, and can also discuss the melting temperature variation with the particle size in a specific particle shape. The input values in our calculation are listed in Table 2. The variation tendency of the relative melting temperature with respect to the shape factor calculated by Eq. (10) is shown in Figs. 1 and 2, where the melting temperature of 5, 10 and 20 nm Sn and Pb nanoparticles are calculated. It is shown that the melting temperature of Sn and Pb nanoparticles decreases with increasing of the shape factor. According to the definition of the shape factor, the surface area increases with

Table 2 The input values of present model and liquid drop model Elements

Atomic radius [15] (nm)

δ [6] (nm)

Melting temperature of bulk materials [16] (K)

Sn Pb In Bi

0.140 0.175 0.162 0.174

2.2784 1.7957 2.6500 2.1273

505.1 600.6 429.8 544.5

Fig. 1. Variation of the relative melting temperature of Sn nanoparticles as a function of shape factor. The solid lines are the results calculated from Eq. (10).

increasing of the shape factor in a specific particle size. Therefore, the surface effect on the melting temperature of nanoparticles may be strengthened in large shape factor, which leads to the decreasing of the melting temperature in wide range. Furthermore, it is found that the particle shape have larger effect on small particles than on large particles. For example, the relative melting temperature variation of Sn nanoparticles in 5 nm is 0.15, and is 0.02 in 30 nm, which suggests that the particle shape should be taken into consideration when studied the melting properties of nanoparticles in small size. For In and Bi nanoparticles, the variations of their melting temperature with respect to the shape factor are similar to these of Sn and Pb nanoparticles, which are not plotted. From Figs. 3–6, the theoretical results on the melting temperature of Sn, Pb, In and Bi nanoparticles calculated by present model and liquid drop model are presented, and the

Table 1 The calculated shape factor for different particle shapes Particle shape Spherical Regular tetrahedral Regular hexahedral Regular octahedral Disk-like Regular quadrangular

Shape factor (α) 1 1.49 1.24 1.18 >1.15 >1.24

Fig. 2. Variation of the relative melting temperature of Pb nanoparticles as a function of shape factor. The solid lines are the results calculated from Eq. (10).


Fig. 3. Variation of the melting temperature as the function of the inverse diameter of Sn nanoparticle. The symbols () denote the experimental values [17].

Fig. 4. Variation of the melting temperature as the function of the inverse diameter of Pb nanoparticle. The symbols () denote the experimental values [18].

available experimental results [17–20] are also shown. In liquid drop model, the relation for the size dependent melting temperature is Tm = Tmb (1 − δ/D), where values of the parameter δ for different particles are listed in Table 2 [6]. It is shown that the melting temperature of these nanoparticles decreases with decreasing of the particle size, the variation tendency of our results and these of liquid drop model is consistent with the experimental values. It is also shown that the results given by our present model are more close

Fig. 5. Variation of the melting temperature as the function of the inverse diameter of In nanoparticle. The symbols () denote the experimental values [19].

283

Fig. 6. Variation of the melting temperature as the function of the inverse diameter of Bi nanoparticle. The symbols () denote the experimental values [20].

to the experimental values, and these given by liquid drop model are lower than the experimental values. In other words, our model is better than the liquid drop model in predicting the melting temperature of nanoparticles. For Sn, Pb and Bi nanoparticles, the experimental results are close to our theoretical results of α = 1 when the particle size D is larger than about 10 nm, and in the middle of our theoretical results between α = 1 and 1.49 when the particle size is smaller than about 10 nm, which suggests that the shape of the nanoparticles may be in spherical when the particle size is large, and in polyhedral shape when the particle size is small. For In nanoparticles, the experimental values lie in the middle of the two curves α = 1 and 1.49, which may suggest that the shape of the nanoparticles may be in polyhedral. The shape of the nanoparticles is mainly determined by the preparation methods [21], i.e., the spherical nanoparticles may be prepared by the chemical methods, and the non-spherical nanoparticles may be prepared by physical method. The present theoretical predictions on the particle shape effect on the melting temperature of metallic nanoparticles may be tested by further experiments. The melting temperature depression of nanoparticles is apparent only when the particle size is smaller than 100 nm. If the particle size is larger than 100 nm, the melting temperature of the particles approximately equals to the corresponding bulk materials, in other words, the melting temperature of nanoparticles is independent of the particle size. This fact is supported by experimental results [17–20] and can be explained by the present model. If the particle size is large enough (larger than 100 nm), the percentage of the surface atoms is fairly small. According to the present model, the melting temperature variation results from the effect of surface atoms and the effect of small percentage of surface atoms on the melting temperature can be neglected. In Eq. (10), if the particle size is fairly large, we have 6αr/D 1 and Tm ≈ Tmb . It should be mentioned that the shape factor in the present work only approximately describes the shape difference between the spherical nanoparticles and the polyhedral nanoparticles. The “approximately” is stressed here due to the

284


fact that some different polyhedral nanoparticles may have the identical shape factor, which results from the fact that the shape factor is defined by the surface area. However, the present calculation result shows that the present definition of shape factor is enough for predicting the shape dependent melting temperature of nanoparticles. Furthermore, the shape factor is a new parameter to characterize the nanoparticles, and it can be experimentally determined by measuring the particle shape, which is the subject of further experiments. It is reported that the atomic radius of metallic nanoparticles contracts with decreasing their particle size [22,23], which means that the atomic radius will change a little if the particle size fairly small. However, for most metal particle, the ratio of contraction is less than 1% [23] and which is ignored in our model. The expression for the size dependent melting temperature of nanoparticles in present work is derived from their size dependent cohesive energy. It should be mentioned that the present model for the size dependent cohesive energy is only for the free surface nanoparticles, i.e., the matrix has no effect on the surface of the nanoparticle. If the nanoparticles are embedded in a matrix with high melting point, and the surface of the nanoparticles have coherent or semi-coherent interface with the matrix, the matrix may affect the cohesive properties of nanoparticle. The melting temperature of the nanoparticles may increase with decreasing of the particle size (superheating) [24]. In our further work, we will generalize the present model to explain this special phenomenon.

particle size. For the melting temperature is a very important quantity, we are confident that the model developed in this paper may have potential application in the research of the temperature-related phenomena of nanoparticles.

Acknowledgement One of the authors (W.H. Qi) thanks Professor Q. Jiang for sharing important literature.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

4. Conclusions We have studied the size and shape dependent melting temperature of nanoparticles by our new model, where the shape of the nanoparticles is considered by introducing a new parameter, i.e., the shape factor. It is shown that the present calculated results of the melting temperature of Sn, Pb, In and Bi nanoparticles are well consistent with the experimental values and better than these given by liquid drop model. Furthermore, it is found that the particle shape can affect the melting temperature of nanoparticles, and this effect on the melting temperature become larger with decreasing of the

[17] [18] [19] [20] [21]

[22] [23] [24]

M. Takagi, J. Phys. Soc. Jpn. 9 (1954) 359. T. Ohashi, K. Kuroda, H. Saka, Philos. Mag. B 65 (1992) 1041. K. Sasaki, H. Saka, Philos. Mag. A 63 (1991) 1207. Q. Jiang, H.X. Shi, M. Zhao, J. Chem. Phys. 111 (1999) 2176. A.N. Goldstein, C.M. Ether, A.P. Alivisatos, Science 256 (1992) 1425. K.K. Nanda, S.N. Sahu, S.N. Behera, Phys. Rev. A 66 (2002) 13208. Q. Jiang, S. Zhang, M. Zhao, Mater. Chem. Phys. 82 (2003) 225. W.H. Qi, M.P. Wang, G.Y. Xu, J. Mater. Sci. Lett. 22 (2003) 1333. C. Solliard, M. Flueli, Surf. Sci. 156 (1985) 487. J.H. Rose, J. Ferrante, J.R. Smith, Phys. Rev. Lett. 47 (1981) 675. J.H. Rose, J. Ferrante, J.R. Smith, Phys. Rev. B 25 (1982) 1419. J.H. Rose, J. Ferrante, J.R. Smith, Phys. Rev. B 28 (1983) 1935. J. Ferrante, J.H. Rose, J.R. Smith, Appl. Phys. Lett. 44 (1984) 53. H.K. Kim, S.H. Huh, J.W. Park, J.W. Jeong, G.H. Lee, Chem. Phys. Lett. 354 (2002) 165. E.A. Brands, Smithells Metals Reference Book, 6th ed., Butterworths, London, 1983, pp. 4–5. F. Seitz, D. Turnbull, Solid State Physics, vol. 16, Academic Press, 1964, p. 326. S.L. Lai, J.Y. Guo, V. Petrova, et al., Phys. Rev. Lett. 77 (1996) 99. T.B. David, Y. Lereah, G. Deutscher, R. Kofman, P. Cheyssac, Philo. Mag. A 71 (1995) 1135. V.P. Skripov, V.P. Koverda, V.N. Skokov, Phys. Status Solidi A 66 (1981) 109. G.L. Allen, R.A. Bayles, W.W. Gile, W.A. Jesser, Thin Solid Films 144 (1986) 297. A.S. Edelstein, R.C. Cammarata, Nanomaterials: Synthesis, Properties and Applications, Institute of Physics Publishing, Bristol and Philadelphia, 1998, p. 47. R. Lamber, S. Wetjen, I. Jaeger, Phys. Rev. B 51 (1995) 10968. W.H. Qi, M.P. Wang, Y.C. Su, J. Mater. Sci. Lett. 21 (2002) 877. H. Saka, Y. Nishikawa, T. Imura, Philo. Mag. A 57 (1988) 895.