PLEASE SCROLL DOWN FOR ARTICLE

Transcription

1 This article was downloaded by:[mei, Q. S.] On: 14 February 2008 Access Details: [subscription number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Philosophical Magazine Letters Publication details, including instructions for authors and subscription information: Melting of metals: role of concentration and migration of vacancies at surfaces Q. S. Mei a ; K. Lu a a Shenyang National Laboratory for Materials Science, Institute of Metal Research, Shenyang, China Online Publication Date: 01 March 2008 To cite this Article: Mei, Q. S. and Lu, K. (2008) 'Melting of metals: role of concentration and migration of vacancies at surfaces', Philosophical Magazine Letters, 88:3, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Philosophical Magazine Letters, Vol. 88, No. 3, March 2008, Melting of metals: role of concentration and migration of vacancies at surfaces Q. S. MEI* and K. LU Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang , China (Received 24 September 2007; in final form 4 December 2007) The mechanism of melting is investigated by considering the role of surfaces with regard to the concentration and migration of vacancies. For many metals, it is found that while the vacancy concentration in the bulk is 0.37% at the equilibrium melting point (T 0 ), the vacancy concentration at the free surface is as high as 10%, i.e. similar to that in the bulk at the superheating limit. Melting is believed to be associated with a lattice instability induced at a critical vacancy concentration of 10%, both at the surface and within the crystal lattice. The abrupt increase in vacancy concentration from 0.37 to 10% on melting at T 0 can be explained as a result of melting of the surface. The surface pre-melting behaviour of metals is quantitatively interpreted by considering the vacancy migration there. 1. Introduction Melting has always been an interesting topic for scientists, partially on account of the uncertainty concerning the common structural features of different crystals at the melting temperature [1]. Various theories have been proposed to interpret the melting process and the intrinsic instability limit of crystal lattice, such as the Lindemann criterion [2], the Born criterion [3] and the homogenous nucleation catastrophe of liquids [4]. However, the role of surfaces is clearly indicated by the fact that superheating of real solids is normally impossible due to surface melting, i.e. melting initiates at the surface of solids below the equilibrium melting point (T 0 ) as a precursor of bulk melting [5]. Therefore, two different modes should be considered in the melting mechanism: (1) equilibrium melting of real solids initiated at the surface and (2) melting of surface-free ideal solids (i.e. without crystallographic defects, such as surfaces and interfaces, dislocations and grain boundaries, etc.) at the superheating limit. Jin et al. [6] showed that both the Lindemann and Born criteria serve as the underlying mechanism for both melting at the surface and within the lattice. It is believed that the basic questions on melting relate to an understanding of why and how solids melt and what determines the melting temperature [7]. *Corresponding author. qsmei@imr.ac.cn Philosophical Magazine Letters ISSN print/issn online ß 2008 Taylor & Francis DOI: /

3 204 Q. S. Mei and K. Lu Go recki [8, 9] found that, for many metals, melting occurs at a vacancy concentration of 0.37% at T 0 and that this concentration then increases rapidly to 10% at the cost of latent heat. The volume change, as well as the electrical resistivity increase, that occur on melting can be quantitatively interpreted as a result of additional vacancy formation, and the latent heat of melting can be quantitatively related to the vacancy formation energy [8]. The most convincing argument for this model is that the pressure dependence of the melting temperature (Clausius Clapeyron relationship) can be quantitatively obtained by considering the pressure dependence of the vacancy formation energy [9]. Although the vacancy model is more convincing than other models in that it interprets quantitatively almost all the important properties associated with melting [1], it cannot explain the critical problem of why the vacancy concentration increases from 0.37 to 10% on melting and how a small vacancy concentration of 0.37% can affect the stability of the whole crystal lattice [1]. To understand the melting mechanism, several critical problems need to be clarified: (1) Is there a critical vacancy concentration for both equilibrium melting of solids with free surfaces and superheated melting of surface-free ideal solids? (2) Why and how is melting (lattice instability) induced at a critical vacancy concentration? (3) How is surface pre-melting related to the vacancy concentration and vacancy migration at the surface? In the present paper, we try to clarify these problems. It was indicated that melting is a lattice instability induced at a critical vacancy concentration of 10%, both at the surface and in the (bulk) lattice. The increase of vacancy concentration from 0.37 to 10% on melting is explained in terms of surface melting and the surface pre-melting temperature of metals can be quantitatively predicted by considering vacancy migration at the surface. 2. Vacancy concentration at the surface At a given temperature, a certain number of vacancies are formed in a crystal to reduce the total free-energy. The thermal equilibrium concentration of vacancies in the (bulk) lattice (C l v ) is given by [8]: C l v ¼ exp S v k El v ð1þ kt where S v is the entropy change caused by vacancy formation, k is the Boltzmann constant, E l v is the vacancy formation energy in the bulk of the crystal and T is the absolute temperature in degrees K. According to [8] and references therein, equation (1) can be approximated to: C l v ¼ exp 4:1 El v ð2þ kt

4 Vacancy concentration and migration at surfaces during melting 205 Table 1. List of values for E l v used in the calculations (from [10] for Na and from [8] for others). Vacancy formation Metals energy E l v (ev) Cu 1.17 Ag 1.01 Al 0.79 Pt 1.70 Au 1.01 Ni 1.40 Pb 0.53 K 0.31 Na 0.35 Cs 0.26 Rb 0.27 W 3.15 Ta 2.90 Mo 2.24 Nb 2.04 Zr 1.75 Ti 1.55 Mg 0.81 Cd 0.44 Zn 0.50 Similarly, one can determine the equilibrium concentration of vacancies at the crystal surface (C s v ) by: C s v ¼ exp 4:1 Es v ð3þ kt where E s v refers to the vacancy formation energy at the surface. The vacancy formation energy is related to the bonding energy of atoms [8]. For fcc and hcp metals, atoms in the volume have 12 nearest neighbours, while surface atoms have only eight [8]. Therefore, it is reasonable to assume that: E s v ¼ 2 3 El v ð4þ Substituting equation (4) into equation (3), one gets: C s v ¼ exp 4:1 2El v ð5þ 3kT Values of E l v for Al can be obtained by experimental measurements. As listed in table 1, with values of E l v one can calculate the temperature dependence of the vacancy concentration both in the bulk and at the surface for Al via equation (2) and (5), respectively. As shown in figure 1, C s v is significantly higher than Cl v at a given temperature. Interestingly, at T 0, although C l v is very small (0.33% for Al),

5 206 Q. S. Mei and K. Lu s C m Concentration * C m * C v T s 900 T Temperature (K) Figure 1. Variations in C l v, Cs v, Cl m and Cs m with temperature for Al. Cl v is the vacancy concentration in the lattice; C s v is the vacancy concentration at the surface; C l m is the concentration of migratable atoms in the lattice; C s m is the concentration of migratable atoms at the surface; C v is the critical vacancy concentration for melting; C m is the critical concentration of migratable atoms for lattice instability. See text for details. s C v s T m l C m l C v C s v is very high (8.6% for Al). Actually, for a number of metals, it has been found that [8]: E l v ¼ const ¼ 8: ev=k, and C l v T ðt 0Þ¼0:37% ð6þ 0 where C l v ðt 0Þ is the lattice vacancy concentration at T 0. Substituting equation (6) into equation (5), one finds that for many metals: C s v ðt 0Þ10% ð7þ where C s v ðt 0Þ is the surface vacancy concentration at T 0. This indicates that doubts that a vacancy concentration of 0.37% in the bulk lattice is too small to induce a lattice instability in a crystal may be answered if one considers the vacancy concentration at the crystal surface rather than that in the bulk lattice, because the surface vacancy concentration (which is about 10% at T 0 ) may be high enough to induce a lattice instability at the surface. We define T 0.1 as the temperature at which C s v reaches the critical value of C v ¼ 10%, i.e. 2E l v T 0:1 ¼ ¼ El v 3k 4:1 ln C v 9:6k ð8þ Using the available data for E l v for a number of metals, as listed in table 1, T 0.1 can be determined from equation (8). Plotting T 0.1 as a function of T 0, as shown in figure 2, one can see that all data points lie well along the line of T 0.1 ¼ T 0 (note that originally we assumed the relationship of equation (4) to be valid for fcc and

6 Vacancy concentration and migration at surfaces during melting 207 T 0.1 (K) fcc bcc hcp T 0.1 =T 0 Cu Ag Au 1000 Mg Pb Al Zn K Rb Na Cs Cd Ni Pt Ti 2000 Zr T 0 (K) Figure 2. Relationship between the temperature at which the vacancy concentration at the surface reaches the critical value of 10% (T 0.1 ) and the equilibrium melting point (T 0 ) for a variety of metals as indicated. The grey region is the 10% error band. Nb Ta Mo 3000 W hcp metals, but it seems to be valid also for bcc metals). Considering the inaccuracy in determining values for E l v, the consistency of T 0.1 with T 0 is fairly good and deviations on data may originate from this inaccuracy. 3. Critical vacancy concentration for melting Interestingly, in figure 1, one can also see that the lattice vacancy concentration C l v ðt m s Þ at the superheating limit (F J temperature, T m s [11]) is 7%, similar to Cs v ðt 0Þ if one considers the temperature effect on vacancy formation energy at high temperatures. It is, therefore, believed that the isentropic catastrophe is that the crystal becomes unstable against spontaneous lattice collapse on account of the high vacancy concentration [12]. This further suggests that 10% may be a critical vacancy concentration for melting, both for heterogeneous melting induced by surfaces and homogenous melting within the lattice. This is also consistent with a previous model that melting is driven by incorporation of lattice vacancies at a critical concentration of 7.7% [13]. We now tackle the problem as to why and how the vacancy concentration increases from 0.37 to 10% on melting at T 0.AtT 0, the crystal surface melts as the surface vacancy concentration reaches the critical value of C v ¼ 10%. Once surface melting occurs, a solid/liquid interface is formed. Since the vacancy formation energy

7 208 Q. S. Mei and K. Lu is low at the solid/liquid interface (assumed to be similar to E s v ), additional vacancies will be formed at the solid/liquid interface until the whole crystal is transformed into a liquid at C v ¼ 10%. Therefore, the problem as to why and how the vacancy concentration increases from 0.37 to 10% on melting may be answered if one considers the role of the vacancy concentration at the surface. As discussed above, the solid/liquid interface, formed by surface melting of a bulk solid, is instable at T 0 owing to the formation of additional vacancies. Note that, under certain conditions, the solid/liquid interface may be stable, e.g. for surface pre-melting below T 0 and melting of small crystals with special geometry and interfacial structures [14, 15]. According to our analysis, the solid/liquid interface is stable below T 0 because the vacancy concentration does not reach the critical value of 10%. Moreover, many recent studies indicate that for nanoparticles embedded epitaxially in a matrix, the solid/liquid interface may be stable even above T 0 [5]. In this case, particles with epitaxial particle/interfaces are supposed to have a higher vacancy formation energy than bulk solids with a free surface. For ideal crystals without a surface, melting does not happen at T 0,asC l v is only 0.37%, and additional vacancies have to be produced by increasing the temperature. When the vacancy concentration reaches the critical value of C v at the superheating limit, the crystal lattice can no longer maintain such a high vacancy concentration and collapses in the form of homogenous melting. Actually, one can summarize the two types of melting processes as follows: an increase in vacancy concentration can be induced either (1) structurally (by reducing the vacancy formation energy by formation of a surface and liquid/solid interface), leading to the heterogeneous melting of real crystals with surfaces, or (2) thermally (by increasing the temperature), leading to the homogenous melting of surface-free bulk crystals. Interestingly, recent computer simulations [16] have suggested that the concentration of defective atoms in regions close to the surface at T 0 is approximately the same as in the surface-free bulk at the limit of superheating. This agrees well with our calculations that the vacancy concentration at the surface at T 0 is similar to that within the lattice at the superheating limit. It is also necessary to discuss why a vacancy concentration of 10% leads to lattice instability. For a vacancy concentration of 10%, the crystal can be simply viewed as an aggregation of clusters, each of which has approximately 10 nearest atoms around a vacancy, as illustrated in figure 3a. However, such a configuration of clusters, as in figure 3a, is kinetically unstable because it cannot resist the local disordering caused by squashing/annihilating of a vacancy by neighbouring atoms when the mobility of atoms is very high, as it is at high temperatures shown in figure 3b. This scenario was first suggested by the classical simulation experiments of Fukusima and Ookawa [17], that vacancies become more diffuse as the temperature is raised and that the squashing of single vacancies occurs near the melting point, leading to the formation of locally disorderd low-density regions. The local disordering as in figure 3b will eventually lead to the collapse of the whole crystal lattice. Recently, Wang et al. [18] reconsidered this problem by quantitative analysis and found that local disordering around a single vacancy can occur by squashing of the vacancy by neighbouring atoms when the concentration of migratable atoms (atoms that can migrate to a vacancy) reaches a critical value of C m at the superheating limit. For example, C m is 17% for fcc or hcp metals. This is nearly 2C v, which indicates that on average there

8 Vacancy concentration and migration at surfaces during melting 209 Figure 3. (a) Schematic illustration of a crystal lattice with a vacancy concentration of 10%. The configuration of (a) is highly unstable and local disordering will be induced by squashing of a single vacancy by neighbouring atoms due to the high mobility of atoms as indicated in (b), which will lead to the collapse of the whole crystal lattice. are at least two atoms that can migrate simultaneously to each single vacancy, leading to the squashing of vacancies and lattice disordering. 4. Vacancy migration at the surface Similar to the analysis of Wang et al. [18], the concentration of migratable atoms at the surface (C s m ) can be estimated from: C s m ¼ exp 4:1 Es m ð9þ kt where E s m is the vacancy migration energy at the surface, assumed to be 2/3 of that within the lattice. As shown in figure 1, using data for vacancy migration energy from [10], C s m is found to be 34% at T 0 for Al obviously large enough to induce squashing of vacancies. Interestingly, from figure 1, one can see that at T s 820 K, C s m C m 17% while Cs v 4%. This means that the atomic migration is so high as to induce a local disordering at T s 820 K, but the surface disorder layer is not a real liquid until C s v increases to 10% at T 0. Such behaviour corresponds to the surface pre-melting (disordering) of Al. The surface pre-melting temperature of Al obtained by such an analysis (T s 820 K) is in good agreement with experimental observations [19 21]. As listed in table 2, using data of vacancy migration energy from [10], the surface pre-melting temperatures of some other metals can also be determined according to equation (9) by assuming C s m ¼ C m ¼ 17%. The calculated results are in good agreement with experimental observations and computer simulations. Our calculation showed that the surface pre-melting phenomenon can

9 210 Q. S. Mei and K. Lu Table 2. Comparison of the calculated surface pre-melting temperature (T s ) with those determined experimentally and via computer simulations. Metals T 0 (K) Calculated T s (K) T s (K) by experiments or simulations Al > 800 [19, 20], 815 [21] Cu [22] Au < 1000 [23], 770 [24] Ni [25] Pb [26] be quantitatively explained by considering atomic migration at the surface. Note that in previous studies, the Laudau model was also successfully applied to interpret surface pre-melting of bulk solids and small particles [27, 28]. One can now answer the question as to why and how solids melt and what determines the melting temperature: formation of a massive number of vacancies with a large migrability leads to lattice instability by annihilation of vacancies, which may be induced by a free surface (equilibrium melting) or high temperature (superheated melting). The temperature at which the vacancy concentration at surfaces (equilibrium melting) or within the bulk lattice (superheated melting) reaches the critical value of 10% determines the melting temperature. 5. Conclusions We investigated the melting mechanism by considering the role of the surface in vacancy concentration and migration. For a number of metals, we found that, while the vacancy concentration in the bulk lattice is only 0.37% at T 0, the vacancy concentration at a free surface can be as high as 10%, similar to that in the bulk at the superheating limit. Melting is believed to be a lattice instability induced at a critical vacancy concentration of 10%, both at the surface and within the bulk lattice. An increase in the vacancy concentration from 0.37 to 10% on melting is induced by melting of the surface. Quantitatively analysis showed that the surface pre-melting behaviour of metals can be attributed to enhanced vacancy migration at the surface. The atomic mechanism of melting is closely related to the interplay of vacancy formation and migration at high temperatures. Acknowledgements This work was supported by the National Nature Science Foundation of China (NSFC) under grants No and , and the Ministry of Science and Technology (MOST) of China under grant No. 2005CB Q.S.M. thanks Dr. L. W. Wang for valuable discussions.

10 Vacancy concentration and migration at surfaces during melting 211 References [1] R. W. Cahn, Nature (1978). [2] F. A. Lindemann, Z. Phys (1910). [3] M. Born, J. Chem. Phys (1939). [4] K. Lu and Y. Li, Phys. Rev. Lett (1998). [5] Q. S. Mei and K. Lu, Prog. Mater. Sci , (2007), for review. [6] Z. H. Jin, P. Gumbsch, K. Lu, et al., Phys. Rev. Lett (2001). [7] R. W. Cahn, Nature (2001). [8] T. Go recki, Z. Metallkd (1974). [9] T. Go recki, Z. Metallkd (1976). [10] R.W. Cahn and P. Haasen (editors), in Physical Metallurgy (North-Holland, Amsterdam, 1996), vol. 2, p [11] H. J. Fetch and W. L. Johnson, Nature (1988). [12] R. W. Cahn, Nature (1988). [13] H. J. Fetch, Nature (1992). [14] U. Dahmen, S. Hage` ge, F. Faudot, et al., Phil. Mag (2004). [15] J. Chang, T. Sakai and H. Saka, Phil. Mag. Lett (2005). [16] F. Delogu, Phys. Rev. B (2006). [17] E. Fukushima and A. Ookawa, J. Phys. Soc. Jpn (1955). [18] L. W. Wang, L. Zhang and K. Lu, Phil. Mag. Lett (2005); L. W. Wang, Q. Wang and K. Q. Lu, Phil. Mag. Lett (2007). [19] M. Polcˇik, L. Wilde and J. Haase, Phys. Rev. Lett (1997). [20] L. Pedemonte, G. Bracco, A. Robin, et al., Surf. Sci (2003). [21] A. W. D. van der Gon, R. J. Smith, J. M. Gay, et al., Surf. Sci (1990). [22] R. N. Barnett and U. Landman, Phys. Rev. B (1991). [23] F. Ercolessi, S. Iarlori, O. Tomagnini, et al., Surf. Sci. 251/ (1991). [24] A. Hoss, M. Nold, P. von Blanckenhagen, et al., Phys. Rev. B (1992). [25] E. T. Chen, R. N. Barnett and U. Landman, Phys. Rev. B (1990). [26] J. W. M. Frenken and J. F. van der Veen, Phys. Rev. Lett (1985). [27] R. Lipowsky, Phys. Rev. Lett (1982). [28] J. Chang and E. Johnson, Phil. Mag (2005).