Surface-Area-Difference model for melting temperature of. metallic nanocrystals embedded in a matrix

Transcription

1 Solid State Phenomena Vols (7) pp Online: (7) Trans Tech Publications, Switzerland doi:1.48/ Surface-Area-Difference model for melting temperature of metallic nanocrystals embedded in a matrix Wei-Hong Qi 1,, a, ingpu Wang, Zhou Li and Wangyu Hu 1 School of aterials Science and Engineering, Jiangsu University, Zhenjiang Jiangsu 11, China School of aterials Science and Engineering, Central South University, Changsha Hunan 418, China Department of Applied Physics, Hunan University, Changsha Hunan 418, China a qwh@ujs.edu.cn Keywords: Surface-Area-Difference (SAD) model; melting temperature; superheating; nanocrystals Abstract. The cohesive energy is the energy to divide the crystal into isolated atoms, and the direct result of cohesive energy is to create new surface. The increased surface energy should equal the cohesive energy of the crystal, which results from the surface area difference between the total atoms and the crystal. This is the basic concept of Surface-Area-Difference (SAD) model. The SAD model has been extended to account for the melting temperature of metallic nanocrystals with non-free surface (embedded in a matrix) in the present work. It is shown if the melting temperature of the matrix must be much higher than that of the bulk value of the nanocrystals, and the nanocrystals has coherent or semi-coherent interface with the matrix, the nanocrystals may be superheated. The present results are supported by the available experimental values. 1. Introduction It is known that the melting temperature of nanocrystals in free surface decreases with decreasing of the particle size [1-6], which is explained by different thermodynamic models [7,8]. However, if the nanocrystals are embedded in a matrix, things will be different. Experimental results shows that the melting temperature of nanocrystals embedded in a matrix may be higher than that of the corresponding bulk materials [7,9]. For instance, the melting temperature of In nanocrystals embedded in an Al matrix increases with decreasing of the particle size [9]. The melting temperature of metals is a parameter to estimate the metallic bonds, while the cohesive energy is also a parameter to estimate the metallic bonds. There must be a relation between these parameters. Rose et al [1,11] shows that the All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (ID: , Pennsylvania State University, University Park, USA-7/1/15,1:19:4)

2 118 Nanoscience and Technology cohesive energy and the melting temperature of pure metals have linear relation. Therefore, we can get the size dependent melting temperature of nanocrystals based on their size dependent cohesive energy. Since the cohesive energy can be regarded as the energy to divide the nanocrystals into isolated atoms, which results in increasing the total surface energy, and the energy difference between the surface energy of all the isolated atoms of a nanocrystal and the surface energy of the nanocrystal should be the cohesive energy of the nanocrystal. This is the basic concept of the Surface-Area-Difference (SAD) model [1], which has been introduced in a review article by Professor Sun [1]. However, only the basic concept has been introduced in previous work, and the model is only applied for predicting the cohesive energy of spherical metallic nanocrystals with free surface. In this paper, the SAD model will be generalized to account for the cohesive energy of nanocrystals embedded in a matrix, and further to explain the superheating of nanocrystals embedded in a matrix.. odel For the nanocrystals embedded in a matrix, the previous relation on cohesive energy of nanocrystals cannot be used directly, because that relation is mainly for the nanocrystals with free surface. However, we can generalize SAD model to explain the cohesive energy of nanocrystals in a matrix by considering the difference between the nanocrystals with free surface and non-free surface. It is known that the surface energy is from the bond energy of the dangling bonds of the surface atoms, and the surface energy of the non-free surface atoms of the nanocrystals decreases, which caused by forming new bonds with the atoms of the matrix. Similar to liquid drop model [7], the effective surface energy of non-free surface should equals to the surface energy of free surface of nanocrystals subtracting to the surface energy of the matrix. Based on these considerations, we can generalize the previous SAD model to predict the cohesive energy of nanocrystals in a matrix. Furthermore, the relation for the melting temperature of nanocrystals in a matrix can also be obtained by using the conversion relation of melting temperature and cohesive energy, which can be used to explain the superheating of nanocrystals. By considering the fact that each surface atom of a nanocrystal contributes half its surface area to the total surface area of the nanocrystal, the effective surface area of the nanocrystal is 8π R area is p 8πR, where 1, where R is its radius. We assume that the non-free surface p. Then the free surface area is ( ) 1 p 8πR. If the surface areas per unit area of the nanocrystals and the matrix are. and. respectively, the effective surface energy per unit area of non-free surface is. The total surface energy. of the nanocrystal can be written as. ( 1 p) 8πR + p 8πR ( ) =.

3 Solid State Phenomena Vols i.e. = 8πR p 8πR (1) For the nanocrystals with free surface, we have p =, and if the nanocrystal has coherent interface with the surrounding matrix, we have p = 1. We exert energy E n to make the nanocrystal divide into n isolated atoms, and the surface energy of n atoms is n 4πr. The atomic radius r can be calculated by the lattice parameters (a and c) r = 1 ( ( 8π )) a 1 ( ( 16π )) a a c ( 16π ) 1 ( ) bcc fcc hcp () According to SAD model, E n is the cohesive energy of the nanocrystals with n atoms, which equals to the surface energy of n atoms plus to the surface energy of nanocrystals, i.e. E n ( 8πR p 8πR ) = n 4πr () The volume of the nanocrystal is 4π R, and the volume of the atom is 4π r, then the number of the atoms n is the ratio between 4π R and 4π r, which leads to R n = (4) r Considering equations () and (4), the cohesive energy per atom (E) of the nanocrystal embedded in a matrix can be written as r p E = E 1 1 (5) R where E = 4πr is the cohesive energy per atom of bulk materials. If p =, equation (5) reduces to the expression of the cohesive energy of nanocrystals with free surface. If p > and p >, we have E > E, which means that the cohesive energy of nanocrystals may increase with decreasing the crystal size. Rose et al proposed a universal model for solids from the binding theory of solid [1,11]. Combining their theory with Debye model, they theoretically obtained the well-known empirical relation of the melting temperature and the cohesive energy for pure metals

4 1184 Nanoscience and Technology T mb. = E k B (6) where T mb is the melting temperature of pure bulk metals, and k b the Boltzmann s constant. As mentioned above, the melting temperature is also a parameter to describe the strength of metallic bond. Therefore, equation (6) can be regarded as the mathematical conversion of melting temperature and cohesive energy. The cohesive energy of nanocrystals embedded in a matrix is given by equation (5),and then their melting temperature can be obtained by replacing E (in equation (6)) with E (equation (5)). The size dependent melting temperature can be written as r p T = 1 1 mp Tmb (7) R where Tmp is the melting temperature of nanocrystals in a matrix. Apparently, T mp depends on the particle size and the matrix-induced term. If the surface of the nanocrystals is free or nearly free, i.e. p =, equation (7) reduces to the relation for the size dependent melting temperature of nanocrystals in free surface, and which decreases with decreasing of the particle size. If the nanocrystals are embedded in a matrix, the melting temperature may decrease ( p < ) or increase ( p < ) with decreasing of the particle size, where the condition p < can account for the reason of the superheating of nanocrystals. It is obvious that equation (7) is the more generalized relation accounting for the size dependent melting temperature of nanocrystals.. Results and discussion To confirm the efficiency of our present model, the melting temperature of In nanocrystals in in Al matrix have been calculated. For the In nanocrystal embedded in the matrix, p equals to 1, where it is assumed that all the surface of the nanocrystals is non-free surface. In our calculation, the surface energies per unit area of In and Al are 1 mj/m and 6 mj/m respectively [14]. The melting temperature of bulk In is 49.8 K [15], and the atomic radius is.175 nm [16]. The present calculation results are shown in Fig.1.As a comparison, the results of liquid drop model and the available experimental values are also presented in Fig. 1 It is found that the melting temperature of In nanocrystals in a Al matrix increases with decreasing of the particle size. Compared with the results of liquid drop model [7], our present theoretical predictions are more close to the experimental values. Furthermore, our present model is correct in predicting the size dependent superheating of nanocrystals even in several nanometers. From equation (7), it is shown that the melting temperature of nanocrystals may

5 Solid State Phenomena Vols increase or decrease with decreasing of the particle size, which depends on the surface energies per unit area of the nanocrystals and the matrix. If the surface energy of nanocrystals is larger than that of the matrix, the melting temperature of nanocrystals decreases with decreasing of the particle size. If the surface energy is smaller than that of the matrix, the melting temperature of nanocrystals increases with decreasing of the particle size, which means that the nanocrystals may be superheated. It can be confirmed that the superheating of nanocrystals is induced by the matrix. By equation (7), we can get a qualitatively criterion for superheating, i.e., if >, the nanocrystals embedded in a matrix may be superheated, where and denote the surface energy of bulk materials. As mentioned above, surface energy is linear to the cohesive energy, and the cohesive energy is linear to the meting point, therefore, the surface energy is linear to the melting point. Apparently, we can choose the melting temperature as the criterion for superheating. In other words, if the melting temperature of the matrix is higher than that of the bulk value of the nanocrystals, the superheating phenomena of nanocrystal may appear; Furthermore, if the melting temperature difference between the bulk value of the nanocrystals and the matrix is fairly large, the superheating is more obvious. Take In nanocrystals embedded in an Al matrix for instance, we known that the melting temperatures of bulk Al (9.5K) and bulk In are 9.5K and 49.8K [15], so the superheating of In nanocrystals exists. However, if In nanocrystals embedded in a Se matrix (the melting temperature of Se is 494K), the superheating is difficult to measure because their melting temperatures are close. This criterion can be used to predict the superheating temperature of the other nanocrystals embedded in matrixes. The cohesive energy of nanocrystals may increase or decrease with the crystal size, which depend on the bonds of surface atoms of nanocrystals. If the surface atoms have large dangling bonds, the cohesive energy of nanocrystals decreases with decreasing the crystal size. However, if the surface atoms have formed bonds with the matrix and the strength of the bonds are higher than that of the corresponding bulk materials of nanocrystals, the cohesive energy increases with decreasing the crystal size. The variation of cohesive energy may lead to the variation of other thermodynamic properties, such as the melting temperature. In this point, the SAD model can be used to describe other size and coherence dependence thermodynamic properties of nanocrystals. It is reported that the lattice parameter of metallic nanocrystal contract with decreasing their particle size [17,18], which is correct for the nanocrystals in free surface. However, the lattice parameters of the nanocrystals embedded in a matrix have not been reported in literatures, which will be studied in our further work. In this paper, the size effect on the lattice parameters of the nanocrystals embedded in a matrix is not considered. It is also reported that the nanocrystals may be in spherical or polyhedral shapes [19,], where we assumed that all nanocrystals are spherical and the shape difference has not been taken into consideration presently.

6 1186 Nanoscience and Technology 4. Conclusion The Surface-Area-Difference model is generalized in this paper to account for the size dependent superheating of nanocrystals embedded in a matrix. It is found that the superheating mainly results from the surface energy difference per unit area between the nanocrystals and the matrix. Only if the surface energy per unit area of nanocrystal is smaller than that of the matrix, the melting temperature of nanocrystals can increase with decreasing of the particle size (superheating). Furthermore, It is shown presently that the conditions for the existence of the superheating are: 1) the melting temperature of the matrix must be much higher than that of the bulk value of the nanocrystal; ) the nanocrystals has coherent or semi-coherent interface with the matrix. The present theoretical predictions of In nanocrystals agree with the corresponding experimental results. Acknowledgement: This work was supported by National Natural Science Foundation of China (No. 5411). References [1].. J.Takagi, Phys. Soc. Japan 9(1954)59. []..Hasegawa, K. Hoshino,.Watabe, J. Phys. F 1(198) 619. []. F.G.Shi, J. ater. Res. 9(1994)17. [4]. A.N.Goldstein, C..Ether, A.P.Alivisatos, Science 56(199)145. [5]. C.L. Jackson, G.B.cKenna. Chem. ater. 8(1996) 18. [6]. S.L.Lai, J.Y. Guo, V. Petrova,et al. Phys. Rev. Lett. 77(1996)99. [7]. K.K.Nanda, S.N.Sahu and S.N.Behera, Phys.Rev. A 66()18. [8]. Q.Jiang, N.Aye and F.G.Shi, Appl.Phys. A 6(1997)67 [9]. H.Saka, Y.Nishikawa and T.Imura, Phil.ag. A 57(1988)895 [1]. J.H. Rose, J. Ferrante and J.R. Smith, Phys. Rev. Lett. 47(1981)675 [11]. J.H. Rose, J. Ferrante and J.R. Smith, Phys. Rev. B 5(198)1419. [1]. W.H. Qi,.P. Wang, J. ater. Sci. Lett. 1()174. [1]. C.Q.Sun, Physics Reports (invited), 5 (to be published). [14]. A.R. iedema, Z.etallkd 69(1978)87 [15]. C.Kittel, Introduction to Solid State Physics, 7th edition, New York :Wiley, [16]. C.S.Barrett, T.B. assalski,, Structure of etals,rd revised Ed., Pergamon Press,198, p.66. [17]. R. Lamber, S. Wetjen, I. Jaeger., Phys. Rev. B 51(1995)1968. [18]. W.H. Qi,.P. Wang, Y.C. Su., J. ater. Sci. Lett. 1()877. [19]. S.Link, C. Burd, B. Nikoobakht, et al. J.Phys.Chem.B 14()61 []. A.V.Simakin, V.V.Voronov, G.A.Shafeev et al. Chem.Phys.Lett. 48(1)18.

7 Solid State Phenomena Vols T mp (K) R (nm) Fig.1 Comparison of experimental size-dependent superheating of In nanocrystals embedded in Al matrix with different models. The solid line is the present results calculated by equation (7) with p = 1, the dash dot line is the results of liquid drop model [7], and the symbols denote the experimental values are the experimental results [9]

8 Nanoscience and Technology 1.48/ Surface-Area-Difference odel for elting Temperature of etallic Nanocrystals Embedded in a atrix 1.48/