- Email: [email protected]
Melting and superheating of low-dimensional materials
Current Opinion in Solid State and Materials Science 5 (2001) 39–44
Melting and superheating of low-dimensional materials K. Lu*, Z.H. Jin State Key Laboratory for Rapidly Solidified Non-equilibrium Alloys, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110015, China
Abstract The state-of-art of melting and superheating for low-dimensional materials is reviewed. Irregular size dependence of melting kinetics was found for ultrafine particles. Substantial melting point elevation (superheating) was observed in both nanoparticles and thin films with an epitaxial confinement. These findings might significantly advance our understanding of the nature of melting of solids providing the relevant theoretical and computer simulation investigations are stimulated. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Superheating; Melting; Low-dimensional materials; Nanoparticles; Thin films
1. Introduction Melting of solids is a common phenomenon in nature and is normally initiated at solid surface or interfaces [1]. As low-dimensional materials (such as nanoparticles, nanowires, thin films, and nanophase materials) possess large areas of surfaces or interfaces, their melting kinetics deviate much from that for conventional bulk solids. For example, the melting points (T m ) of free-standing nanometer-sized metal particles are remarkably depressed relative to the equilibrium T m for bulk materials [2,3]. Meanwhile, it is also observed that when the nanoparticles are properly coated by (or embedded in) a high-T m metal, the melting point can be elevated above the equilibrium T m for bulk solids [4–10]. The dramatic T m variations for various low-dimensional materials have stimulated increasing interest in recent years. Investigations on melting and superheating of low-dimensional materials, on the one hand, provide a unique opportunity for advancing our understanding of the nature of melting, and on the other hand, are crucial for the technological applications of this new materials family with novel properties. Because of the significant role of the free surface on melting, there have been extensive studies on the surface melting (see a recent review article [*11]). A typical example is the melting of a two-dimensional (2D) sample which may represent the outermost layer of any real substance. To clarify how the liquid layer consumes the *Corresponding author. Tel.: 186-24-2384-3531; fax: 186-24-23998660. E-mail address: [email protected] (K. Lu).
adjacent solid still needs further efforts, which will help to understand the melting of low-dimensional materials such as nanotubes [12] as well as other materials to be discussed in this short review. In this paper, we restrict our attention to the most recent progress on melting and superheating of low-dimensional materials, and especially with emphasis on nano-granular structures such as embedded nanoparticles and confined thin films that mainly constitute metallic elements. Several key issues on melting and superheating will be discussed in terms of experimental measurements, theoretical analyses, as well as computer simulations.
2. Size-dependence of melting point for nanoparticles It has been well known that free-standing nanometersized particles may melt below the equilibrium bulk T m due to the extremely high surface / volume ratio. The depression of melting point (DT m ) is found to be proportional to 1 /D (D the particle size) even when D is as small as a few nanometers. Such a particle size dependence of melting point can be well interpreted by the classical thermodynamic arguments. However, when the particle size is reduced to be of tens or hundreds or thousands of atoms, very different melting behavior appears. The melting point of the particles (or clusters) depends sensitively on their size, as observed in Na particles by Schmidt et al. [**13,14]. Highly irregular patterns were detected for the variation of melting point and latent heat of fusion with the particle size. Changing the size of the Na clusters by a few atoms can change their
1359-0286 / 01 / $ – see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S1359-0286( 00 )00027-9
40
K. Lu, Z.H. Jin / Current Opinion in Solid State and Materials Science 5 (2001) 39 – 44
melting temperature by tens of percent. Local maxima in melting point and latent heat of fusion seem to exist at several specific cluster sizes with ‘magic numbers’ of atoms. Though the effects of geometric shells and electronic shells were considered to be responsible for this irregular variation in melting point, this phenomenon has not yet been fully understood theoretically and needs further in-depth investigations. When nanoparticles are embedded randomly (in crystallographic orientations) in a high-T m matrix (i.e. a nanogranular structure), a depression of melting point is also observed, analogous to that for free-standing nanoparticles. In this case, the T m depression is found to depend upon not only the particle size but also the contact angle between the solid matrix and the liquid nucleus, which is determined by the nature of the particle / matrix interface. In other words, beside the size-effect, the particle / matrix interface structure is also a vital parameter controlling the T m depression. Recent experimental measurements of several kinds of metal nanoparticles embedded in Al indicated that both the melting point and the latent heat of fusion for the nanoparticles decreases linearly with an increase of 1 /D [15]. The contact angle between the liquid nucleus and the solid matrix is determined by the nature of the two elements (particle and matrix), and it increases with a reduction of the heat of mixing between two solids.
3. Effect of interfacial structure on superheating The melting kinetics of the embedded nanoparticles depends significantly upon the conditions of the particle / matrix interfacial structure. When a low-energy coherent or semi-coherent interface is formed between the nanoparticle and the matrix, melting of the nanoparticle can be effectively suppressed at elevated temperatures and the melting point is higher than the equilibrium bulk T m , i.e. superheating of the nanoparticle occurs. This phenomenon has been observed in a number of metallic systems, such as Pb /Al [5–7,9], In /Al [5,16,17], Ag / Ni [18], etc. In all of these cases, a common feature is that a specific shape of the nanoparticle is formed and every particle / matrix interface is semi-coherent with a cubic–cubic orientation relationship. The elevated melting point is believed to result from an effective suppression of the heterogeneous nucleation of melting at the epitaxial particle / matrix interfaces. When the epitaxy is broken at the interface, no superheating will be obtained [1], as experimentally demonstrated in the Pb /Al [9] and In /Al samples (see Fig. 1). Two kinds of nanogranular In /Al samples were prepared via different approaches: rapidly quenching from the In /Al melt to form In nanoparticles embedded in Al with a specific shape and semicoherent In /Al interfaces; ball-milling of the mixture of In and Al powders to form nanogranular In in Al with
Fig. 1. Variation of melting point with the particle size for In nanoparticles embedded in an Al matrix in two kinds of In /Al nanogranular samples prepared by means of melt-spinning and ball-milling. The remarkable different melting point variations in these two cases might be attributed to the In /Al interfacial structures. For the melt-spun sample a semi-coherent In /Al interface was formed as evidenced by the HRTEM image of the In particle and the electron diffraction pattern, and for the ball-milled sample, the In /Al interfaces are random (see inserted HRTEM image).
K. Lu, Z.H. Jin / Current Opinion in Solid State and Materials Science 5 (2001) 39 – 44
random interfaces. The melting points of the In nanoparticles (with comparable particle sizes) in these two samples are completely different. Superheating is observed in the melt-quenched sample but melting point depression in the ball-milled one. In the first case, the degree of superheating is also strongly dependent upon the particle size. The smaller the particle, the higher the superheating, as clearly seen in Fig. 1. Molecular dynamics (MD) simulation of the melting process in confined nanoparticles has been performed to illustrate the effect of interface structure on superheating [*19]. Comparison of the melting behavior for Pb clusters (of 201 and 249 atoms, respectively) coated by Al with and without epitaxial interfaces, simulated by using the Sutton–Chen type many-body potential, indicated quite different melting pictures. The coated cluster with a semicoherent Pb /Al interface is superheated up to 750 K (the equilibrium T m for the Sutton–Chen bulk Pb is about 610610 K), while for the coated Pb cluster without semicoherent interfaces, premelting dominates and the melting point is merely 500 K. This observation coincides with the experimental evidence.
41
Fig. 2. Cross-sectional snapshot views (after energy minimizations) of the MD simulation of an Ag 3055 cluster coated by Ni at different temperatures (1322 K (a), 1335 K (b), 1333 K (c), and 1328 K (d)) around the melting point to illustrate the nucleated melting [20].
4. Nucleated melting in superheated crystals Obviously, nucleation of melt is the most important process for superheating of solids. In order to identify the melt nucleation in superheated nanoparticles, a MD simulation on an Ag cluster (3055 atoms) coated by Ni was performed. Experimental measurements indicated that the Ag nanoparticles embedded in Ni with a cubic–cubic epitaxial Ag / Ni interfaces can be substantially superheated (up to 40 K for an average particle size of about 30 nm) owing to the relatively smaller lattice mismatch (|16% compared with 23% for Pb /Al) [18]. MD results demonstrated that the coated Ag 3055 cluster can be superheated, and the melting process of the superheated Ag cluster is initiated at the defective interfacial region and then propagates inwards, as shown in Fig. 2 [20]. This observation suggests that heterogeneous nucleation at the interface dominates the melting of the superheated nanoparticles even a low-energy semicoherent interface is constructed. If the lattice mismatch across the interface vanishes, as demonstrated in MD simulations of a model thin-film crystal fully confined by Al matrix, the melting point is much elevated and the melting is initiated homogeneously from the interior of the confined thin films [21]. The degree of superheating in this perfectly-confined crystal agrees well with the theoretical prediction based on the kinetic stability limit determined by homogeneous nucleation events [*22]. The results clearly show that the confinement (the nature of the interfaces) is the key factor controlling the melt nucleation behavior, which, in turn, determines the extent of superheating for the confined crystals.
It should be pointed out that superheating of confined crystals might be due to two different kinetic mechanisms. One is the kinetic nucleation of melting and the other is kinetic melt front growth. Both mechanisms, with their thermodynamic basis, differ from that for the kinetic superheating due to sluggish melting kinetics under fast heating rates. The fast heating technique is an effective approach to obtain kinetic superheating for surfaces as well as bulk solids. While the effect of confinement on melt nucleation has been demonstratively studied by means of MD simulations, the (interface) confinement effect on the growth process of the melt is still unclear, though that seems to be equally important for understanding the melting of solids. This effect is believed to be more significant for the melting of confined nanowires and thin films. Studies on this aspect (experimental, theoretical, and simulation) are highly needed.
5. Superheating of thin films Analogous to that for nanoparticles, experimental observations [23] and thermodynamic analysis [24] indicate that the melting point of free-standing 2D thin films is depressed relative to the bulk T m . While there are plenty of observations of superheating in confined nanoparticles, superheating of 2D thin films is rarely seen. Those necessary conditions for superheating of particles (effective suppression of melt nucleation by the epitaxial interfaces)
42
K. Lu, Z.H. Jin / Current Opinion in Solid State and Materials Science 5 (2001) 39 – 44
are not practically feasible for 2D thin films. Even if a thin film could be sandwiched by two high-T m films with epitaxial interfaces, heterogeneous nucleation of melt at various defects in the thin film (such as grain boundaries) and at the defective interfaces would not be effectively suppressed. Therefore, 2D thin films are usually regarded to be hardly capable of being superheated. However, a very recent experimental exploration showed, for the first time, that confined thin films can be substantially superheated [**25]. In the experiment, Pb thin films (of about 20 nm thick) were sandwiched by Al layers (about 40 nm thick). In the multilayer thin film sample, Pb films are polycrystalline fragments with a grain size (plan-view) of about 50|70 nm. For a small fraction of Pb layers, a cubic–cubic orientation relationship between Pb and Al was detected from electron diffraction patterns. An in situ X-ray diffraction analysis of the Pb /Al thin film sample heated to elevated temperatures clearly indicated that the confined Pb thin films refuse to melt up to 3348C, which is about 68C higher than the equilibrium bulk T m for Pb. Repeated measurements demonstrated that a metastable superheating of the confined thin film varies in a range of 3|108C. Such a superheating phenomenon in confined thin films was reasonably attributed to suppression of growth of the molten droplet by the epitaxial Al / Pb /Al confinement, instead of suppression of melt nucleation (as in superheated nanoparticles) which may not be preventable due to various kinds of defects in the polycrystalline films and at the Pb /Al interfaces. A simple model was developed based on thermodynamic analysis of the interfacial energy conditions for the growth of the Pb liquid droplet in the confined 2D thin film, as schematically shown in Fig. 3. The degree of superheating induced by suppression of melt
growth is DT m 5 (2gsl T m cos 2u ) /DLv (p / 2 2 u ) (where D is the film thickness, Lv is the latent heat per unit volume, u is the wetting angle which depends upon the Pb /Al solid interfacial energy and gsl is the solid / liquid interfacial energy). For the 20 nm thick Pb confined films, the calculated DT m is about 2|98C depending upon the value of u, which agrees well with the experimental observations. Based on this model it is anticipated that with a reduction of the film thickness, or a decrease of the epitaxial interface energy, the extent of superheating would be increased. This has been verified by our latest measurement [26]. The finding of superheating in 2D thin films provides an important clue that superheating might be achieved alternatively by suppression of melt growth without suppression of nucleation. This is challenging our traditional understanding of the superheating phenomenon. At the same time, superheating of thin films is extremely crucial for further development and application of novel thin film materials.
6. Melting point limit It is of extreme interest to ask if there is any upper (temperature) limit for the existence of a superheated crystal. There are several models dealing with the question. Here we may put them into the following three categories. 1. By analogy to the argument of Kauzmann on the supercooled liquid-to-glass transition, an isentropic melting point (T s ) can be defined at which the entropy of a solid undergoing superheating is equal to that of the equilibrium liquid [27]. The superheated solid
Fig. 3. A schematic illustration of interfacial conditions for the growth of a Pb liquid droplet confined by Al layers. The solid / liquid interface for Pb at the equilibrium melting point is stable when an epitaxial Pb /Al interface is formed (a), while it is unstable when a high-energy Pb /Al interface exists (b) [**25].
K. Lu, Z.H. Jin / Current Opinion in Solid State and Materials Science 5 (2001) 39 – 44
cannot exist at any temperature above T s and it must undergo the melting transition below that critical temperature. It is predicted theoretically that T s could be located in the range 1.3–2.0 T m [27,28]. In order to reach this temperature, the system must be heated via an ultrafast heating rate to suppress any nucleation or growth of liquid phase which may lead to complete melting otherwise. Hence, in this sense such a catastrophic limit will be directly related to the dynamic origin of melting transitions, as in the case for supercooled liquid-to-glass transitions. 2. Similar limits exist when one puts the volume and / or rigidity into consideration. It is found that the rigidity limit is lower than both the isotropic and the isochoric limit. The rigidity melting criterion was originally developed by Born and later was found to be closely related to possible superheating (instead of normal melting) of a crystal [29]. It can be regarded as belonging to the same type of lattice instability models as proposed by Lindemann at the beginning of this century. 3. On the basis of the homogeneous nucleation theory for melting, a kinetic limit for superheated crystals was proposed [*22]. According to both the classical nucleation theory [*22] and a density functional argument [30], a superheating of about 10% can be obtained with respect to the equilibrium T m by neglecting the additional 5 to 10% of the superheating due to the elastic strain effects originating from the density difference between the liquid and the solid. This is in good agreement with the MD simulation result of the superheating in a surface-free bulk single crystal [31]. It is interesting to note that superheating by 20% of T m is in numerical accordance with the prediction based on the rigidity model (22% of T m ), that underlies some correlation between the lattice instability model and the kinetic theory. To clarify the argument we again benefit from the atomistic scale information provided by MD simulations. Our recent simulations [32,33] showed that loss of either crystallinity or, equivalently, lattice rigidity (from calculation of the temperature dependence of shear modulus), not globally but locally, acts as the key factor to induce melting for a homogeneous crystal undergoing superheating. Those thermally destabilized atoms, which satisfy Lindemann’s criterion, will also satisfy Born’s criterion, and they appear in a cooperative manner as the temperature is elevated. Melting occurs as those destabilized liquid-like clusters start to accumulate and percolate throughout the crystal. Thus, according to this picture, the instability model of either Lindemann type or Born type are basically correlated to each other and serves as the nucleation mechanism at the atomic scale. And based on such a scenario, a unified view of the melting origin can be achieved, which should be of value to narrow the different
43
opinions in regard to the prediction of the melting point limit [**32]. Other evidence for our opinion exists. Firstly, it is demonstrated by MD simulation that, the effective elastic modulus of the (310) symmetric tilt grain boundary under shear softens significantly and goes to zero at about 98% of the bulk melting point, at which point the boundary will start to melt [34]. Secondly, a very recent detection of the so-called ‘non-thermal melting’ by ultrafast X-ray diffraction from a Ge single crystal surface under heating by an ultrashort-pulse laser indicated that there exists a large difference in the melting speed between the crystal surface and the internal crystal which is undergoing transient superheating [**35]. The result indicated that homogeneous nucleation of melt should occur for superheated single crystals. Though detailed information is still lacking from the experimental side, MD simulations should be used extensively to understand the important problems concerning the structure and properties of superheated crystals and transient liquid phase growth from the solid lattice. The results also encourage us to make further possible experimental verifications of the theoretical prediction based on MD simulations.
7. Summary The work reviewed here has shown that low-dimensional materials exhibit very different melting behavior from that of conventional bulk solids. With a reduction of size into the nanometer regime, irregular variation of the melting kinetics with size appears for ultrafine particles. For the embedded nanoparticles, the melting behavior is controlled not only by the particle size but also by the particle / matrix interfaces. Melting point elevation has been observed in confined nanoparticles and in confined thin films as well, in which the epitaxial interface plays a key role. Although theoretical analysis and computer simulation investigations have deepened, to some extent, the understanding of these observations, further development along this direction is still highly needed. From a practical point of view, experimental studies of the melting process of thin films and nanowires that are finding more and more technological applications in modern industries, especially to explore effective approaches to elevate their thermal stability against melting, seem to be of great importance.
Acknowledgements The authors would like to acknowledge the financial support by the National Science Foundation of China (Grants no. 59931030, 59841004, and 59801011) and Max-Planck Society of Germany.
44
K. Lu, Z.H. Jin / Current Opinion in Solid State and Materials Science 5 (2001) 39 – 44
References Papers of particular interest, published within the annual period of review, have been highlighted as: * of special interest; ** of outstanding interest. [1] Cahn RW. Melting and the surface. Nature 1986;323:668–9. [2] Buffat Ph, Borel JP. Size effect on the melting temperature of gold particles. Phys Rev A 1976;13:2287–98. [3] Jiang Q, Shi HX, Zhao M. Melting thermodynamics of organic nanocrystals. J Chem Phys 1999;11:2176–80. [4] Daeges J, Gleiter H, Perepezko JH. Superheating of metal crystals. Phys Lett A 1986;119:79–82. [5] Zhang DL, Cantor B. Melting behavior of In and Pb particles embedded in an Al matrix. Acta Metall Mater 1991;39:1591–602. [6] Grabak L, Bohr J. Superheating and supercooling of lead precipitates in aluminum. Phys Rev Lett 1990;64:934–7. [7] Andersen HH, Johnson E. Structure, morphology and melting hysteresis of ion-implanted nanocrystals. Nucl Inst Meth Phys Res B 1995;106:480–91. [8] Chattopadhyay K, Goswami R. Melting and superheating of metals and alloys. Prog Mater Sci 1997;42:287–300. [9] Sheng HW, Ren G, Peng LM, Hu ZQ, Lu K. Superheating and melting-point depression of Pb nanoparticles embedded in Al matrices. Philos Mag Lett 1996;73:179–86. [10] Allen GL, Gile WW, Jesser WA. The melting temperature of microcrystals embedded in a matrix. Acta Metall 1980;28:1695– 701. [*11] Dash JG. History of the search for continuous melting. Rev Modern Phys 1999;71:1737–43. [12] Kwon YK, Tomanek D. Orientational melting in carbon nanotube ropes. Phys Rev Lett 2000;84:1483–6. [**13] Schmidt M, Kusche R, von Issendorff B, Haberland H. Irregular variation in the melting point of size-selected atomic clusters. Nature 1998;393:238–40. [14] Schmidt M, Kusche R, Kronmueller W, von Issendorff B, Haberland H. Experimental determination of the melting point and heat capacity for a free cluster of 139 sodium atoms. Phys Rev Lett 1997;79:99–102. [15] Sheng HW, Lu K, Ma E. Melting and freezing behavior of embedded nanoparticles in ball-milled Al-10 wt.% M (M5In, Sn, Bi, Cd, Pb) mixtures. Acta Mater 1998;46:5195–202.
[16] Saka H, Nishikawa Y, Imura T. Melting temperature of In particles embedded in an Al matrix. Philos Mag A 1988;57:895–906. [17] Sheng HW, Ren G, Peng LM, Hu ZQ, Lu K. Epitaxial dependence of the melting behavior of In nanoparticles embedded in Al matrices. J Mater Res 1997;12:119–23. [18] Zhong J, Zhang LH, Jin ZH, Sui ML, Lu K. Superheating of Ag nanoparticles embedded in Ni matrix (to be published). [*19] Jin ZH, Sheng HW, Lu K. Melting of Pb clusters without free surfaces. Phys Rev B 1999;60:141–9. [20] Xu FT, Jin ZH, Lu K. Superheating and melting behaviors of an Ag3055 cluster with a Ni coating. Sci China E 2000; in press. [21] Jin ZH, Lu K. To what extent can a crystal be superheated? Nanostruct Mater 1999;12:369–72. [*22] Lu K, Li Y. Homogeneous nucleation catastrophe as a kinetic stability limit for superheated crystal. Phys Rev Lett 1998;80:4474– 7. [23] Willens RH, Kornblit A, Testardi LR, Nakahara S. Melting of Pd as it approaches a two-dimensional solid. Phys Rev B 1982;25:290–6. [24] Jiang Q, Zhang Z, Li JC. Superheating of nanocrystals embedded in matrix. Chem Phys Lett 2000;332:549–52. [**25] Zhang L, Jin ZH, Zhang LH, Sui ML, Lu K. Superheating of confined Pb thin films. Phys Rev Lett 2000;85:1484–7. [26] Zhang L, Jin ZH, Lu K. Superheating and melting kinetics of Pb thin films confined by Al. To be published. [27] Fecht HJ, Johnson WL. Entropy and enthalpy catastrophe as a stability limit for crystalline materials. Nature 1988;334:50–1. [28] Lele S, Ramachandra P, Dubey KS. Entropy catastrophe and superheating of crystals. Nature 1988;336:567–8. [29] Tallon JL. A hierarchy for catastrophes as a succession of stability limits for crystalline state. Nature 1989;342:658–60. [30] Iwamatsu M. Homogeneous nucleation for superheated crystal. J Phys Cond Matter 1999;11:L1–5. [31] Jin ZH, Lu K. Melting of surface-free bulk single crystals. Philos Mag Lett 1998;78:29–35. [**32] Jin ZH, Lu K, Gumbsch P, Ma E. Melting mechanisms at the limit of superheating. Phys Rev Lett to be published. [33] Jin ZH, Lu K. Melting as a homogeneously nucleated process within crystals undergoing superheating. Z Metallkd 2000;91:275–9. [34] Broughton JQ, Gilmer GH. Grain-boundary shearing as a test for interface melting. Model Simul Mater Sci Eng 1998;6:87–97. [**35] Siders CW, Cavalleri A, Sokolowski-Tinten K, Toth C, Guo T, Kammler M, Horn von Hoegen M, Wilson KR, von der Linde D, Barty CPJ. Detection of nonthermal melting by ultrafast X-ray diffraction. Science 1999;286:1340–2.
Melting and superheating of low-dimensional materials K. Lu*, Z.H. Jin State Key Laboratory for Rapidly Solidified Non-equilibrium Alloys, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110015, China
Abstract The state-of-art of melting and superheating for low-dimensional materials is reviewed. Irregular size dependence of melting kinetics was found for ultrafine particles. Substantial melting point elevation (superheating) was observed in both nanoparticles and thin films with an epitaxial confinement. These findings might significantly advance our understanding of the nature of melting of solids providing the relevant theoretical and computer simulation investigations are stimulated. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Superheating; Melting; Low-dimensional materials; Nanoparticles; Thin films
1. Introduction Melting of solids is a common phenomenon in nature and is normally initiated at solid surface or interfaces [1]. As low-dimensional materials (such as nanoparticles, nanowires, thin films, and nanophase materials) possess large areas of surfaces or interfaces, their melting kinetics deviate much from that for conventional bulk solids. For example, the melting points (T m ) of free-standing nanometer-sized metal particles are remarkably depressed relative to the equilibrium T m for bulk materials [2,3]. Meanwhile, it is also observed that when the nanoparticles are properly coated by (or embedded in) a high-T m metal, the melting point can be elevated above the equilibrium T m for bulk solids [4–10]. The dramatic T m variations for various low-dimensional materials have stimulated increasing interest in recent years. Investigations on melting and superheating of low-dimensional materials, on the one hand, provide a unique opportunity for advancing our understanding of the nature of melting, and on the other hand, are crucial for the technological applications of this new materials family with novel properties. Because of the significant role of the free surface on melting, there have been extensive studies on the surface melting (see a recent review article [*11]). A typical example is the melting of a two-dimensional (2D) sample which may represent the outermost layer of any real substance. To clarify how the liquid layer consumes the *Corresponding author. Tel.: 186-24-2384-3531; fax: 186-24-23998660. E-mail address: [email protected] (K. Lu).
adjacent solid still needs further efforts, which will help to understand the melting of low-dimensional materials such as nanotubes [12] as well as other materials to be discussed in this short review. In this paper, we restrict our attention to the most recent progress on melting and superheating of low-dimensional materials, and especially with emphasis on nano-granular structures such as embedded nanoparticles and confined thin films that mainly constitute metallic elements. Several key issues on melting and superheating will be discussed in terms of experimental measurements, theoretical analyses, as well as computer simulations.
2. Size-dependence of melting point for nanoparticles It has been well known that free-standing nanometersized particles may melt below the equilibrium bulk T m due to the extremely high surface / volume ratio. The depression of melting point (DT m ) is found to be proportional to 1 /D (D the particle size) even when D is as small as a few nanometers. Such a particle size dependence of melting point can be well interpreted by the classical thermodynamic arguments. However, when the particle size is reduced to be of tens or hundreds or thousands of atoms, very different melting behavior appears. The melting point of the particles (or clusters) depends sensitively on their size, as observed in Na particles by Schmidt et al. [**13,14]. Highly irregular patterns were detected for the variation of melting point and latent heat of fusion with the particle size. Changing the size of the Na clusters by a few atoms can change their
1359-0286 / 01 / $ – see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S1359-0286( 00 )00027-9
40
K. Lu, Z.H. Jin / Current Opinion in Solid State and Materials Science 5 (2001) 39 – 44
melting temperature by tens of percent. Local maxima in melting point and latent heat of fusion seem to exist at several specific cluster sizes with ‘magic numbers’ of atoms. Though the effects of geometric shells and electronic shells were considered to be responsible for this irregular variation in melting point, this phenomenon has not yet been fully understood theoretically and needs further in-depth investigations. When nanoparticles are embedded randomly (in crystallographic orientations) in a high-T m matrix (i.e. a nanogranular structure), a depression of melting point is also observed, analogous to that for free-standing nanoparticles. In this case, the T m depression is found to depend upon not only the particle size but also the contact angle between the solid matrix and the liquid nucleus, which is determined by the nature of the particle / matrix interface. In other words, beside the size-effect, the particle / matrix interface structure is also a vital parameter controlling the T m depression. Recent experimental measurements of several kinds of metal nanoparticles embedded in Al indicated that both the melting point and the latent heat of fusion for the nanoparticles decreases linearly with an increase of 1 /D [15]. The contact angle between the liquid nucleus and the solid matrix is determined by the nature of the two elements (particle and matrix), and it increases with a reduction of the heat of mixing between two solids.
3. Effect of interfacial structure on superheating The melting kinetics of the embedded nanoparticles depends significantly upon the conditions of the particle / matrix interfacial structure. When a low-energy coherent or semi-coherent interface is formed between the nanoparticle and the matrix, melting of the nanoparticle can be effectively suppressed at elevated temperatures and the melting point is higher than the equilibrium bulk T m , i.e. superheating of the nanoparticle occurs. This phenomenon has been observed in a number of metallic systems, such as Pb /Al [5–7,9], In /Al [5,16,17], Ag / Ni [18], etc. In all of these cases, a common feature is that a specific shape of the nanoparticle is formed and every particle / matrix interface is semi-coherent with a cubic–cubic orientation relationship. The elevated melting point is believed to result from an effective suppression of the heterogeneous nucleation of melting at the epitaxial particle / matrix interfaces. When the epitaxy is broken at the interface, no superheating will be obtained [1], as experimentally demonstrated in the Pb /Al [9] and In /Al samples (see Fig. 1). Two kinds of nanogranular In /Al samples were prepared via different approaches: rapidly quenching from the In /Al melt to form In nanoparticles embedded in Al with a specific shape and semicoherent In /Al interfaces; ball-milling of the mixture of In and Al powders to form nanogranular In in Al with
Fig. 1. Variation of melting point with the particle size for In nanoparticles embedded in an Al matrix in two kinds of In /Al nanogranular samples prepared by means of melt-spinning and ball-milling. The remarkable different melting point variations in these two cases might be attributed to the In /Al interfacial structures. For the melt-spun sample a semi-coherent In /Al interface was formed as evidenced by the HRTEM image of the In particle and the electron diffraction pattern, and for the ball-milled sample, the In /Al interfaces are random (see inserted HRTEM image).
K. Lu, Z.H. Jin / Current Opinion in Solid State and Materials Science 5 (2001) 39 – 44
random interfaces. The melting points of the In nanoparticles (with comparable particle sizes) in these two samples are completely different. Superheating is observed in the melt-quenched sample but melting point depression in the ball-milled one. In the first case, the degree of superheating is also strongly dependent upon the particle size. The smaller the particle, the higher the superheating, as clearly seen in Fig. 1. Molecular dynamics (MD) simulation of the melting process in confined nanoparticles has been performed to illustrate the effect of interface structure on superheating [*19]. Comparison of the melting behavior for Pb clusters (of 201 and 249 atoms, respectively) coated by Al with and without epitaxial interfaces, simulated by using the Sutton–Chen type many-body potential, indicated quite different melting pictures. The coated cluster with a semicoherent Pb /Al interface is superheated up to 750 K (the equilibrium T m for the Sutton–Chen bulk Pb is about 610610 K), while for the coated Pb cluster without semicoherent interfaces, premelting dominates and the melting point is merely 500 K. This observation coincides with the experimental evidence.
41
Fig. 2. Cross-sectional snapshot views (after energy minimizations) of the MD simulation of an Ag 3055 cluster coated by Ni at different temperatures (1322 K (a), 1335 K (b), 1333 K (c), and 1328 K (d)) around the melting point to illustrate the nucleated melting [20].
4. Nucleated melting in superheated crystals Obviously, nucleation of melt is the most important process for superheating of solids. In order to identify the melt nucleation in superheated nanoparticles, a MD simulation on an Ag cluster (3055 atoms) coated by Ni was performed. Experimental measurements indicated that the Ag nanoparticles embedded in Ni with a cubic–cubic epitaxial Ag / Ni interfaces can be substantially superheated (up to 40 K for an average particle size of about 30 nm) owing to the relatively smaller lattice mismatch (|16% compared with 23% for Pb /Al) [18]. MD results demonstrated that the coated Ag 3055 cluster can be superheated, and the melting process of the superheated Ag cluster is initiated at the defective interfacial region and then propagates inwards, as shown in Fig. 2 [20]. This observation suggests that heterogeneous nucleation at the interface dominates the melting of the superheated nanoparticles even a low-energy semicoherent interface is constructed. If the lattice mismatch across the interface vanishes, as demonstrated in MD simulations of a model thin-film crystal fully confined by Al matrix, the melting point is much elevated and the melting is initiated homogeneously from the interior of the confined thin films [21]. The degree of superheating in this perfectly-confined crystal agrees well with the theoretical prediction based on the kinetic stability limit determined by homogeneous nucleation events [*22]. The results clearly show that the confinement (the nature of the interfaces) is the key factor controlling the melt nucleation behavior, which, in turn, determines the extent of superheating for the confined crystals.
It should be pointed out that superheating of confined crystals might be due to two different kinetic mechanisms. One is the kinetic nucleation of melting and the other is kinetic melt front growth. Both mechanisms, with their thermodynamic basis, differ from that for the kinetic superheating due to sluggish melting kinetics under fast heating rates. The fast heating technique is an effective approach to obtain kinetic superheating for surfaces as well as bulk solids. While the effect of confinement on melt nucleation has been demonstratively studied by means of MD simulations, the (interface) confinement effect on the growth process of the melt is still unclear, though that seems to be equally important for understanding the melting of solids. This effect is believed to be more significant for the melting of confined nanowires and thin films. Studies on this aspect (experimental, theoretical, and simulation) are highly needed.
5. Superheating of thin films Analogous to that for nanoparticles, experimental observations [23] and thermodynamic analysis [24] indicate that the melting point of free-standing 2D thin films is depressed relative to the bulk T m . While there are plenty of observations of superheating in confined nanoparticles, superheating of 2D thin films is rarely seen. Those necessary conditions for superheating of particles (effective suppression of melt nucleation by the epitaxial interfaces)
42
K. Lu, Z.H. Jin / Current Opinion in Solid State and Materials Science 5 (2001) 39 – 44
are not practically feasible for 2D thin films. Even if a thin film could be sandwiched by two high-T m films with epitaxial interfaces, heterogeneous nucleation of melt at various defects in the thin film (such as grain boundaries) and at the defective interfaces would not be effectively suppressed. Therefore, 2D thin films are usually regarded to be hardly capable of being superheated. However, a very recent experimental exploration showed, for the first time, that confined thin films can be substantially superheated [**25]. In the experiment, Pb thin films (of about 20 nm thick) were sandwiched by Al layers (about 40 nm thick). In the multilayer thin film sample, Pb films are polycrystalline fragments with a grain size (plan-view) of about 50|70 nm. For a small fraction of Pb layers, a cubic–cubic orientation relationship between Pb and Al was detected from electron diffraction patterns. An in situ X-ray diffraction analysis of the Pb /Al thin film sample heated to elevated temperatures clearly indicated that the confined Pb thin films refuse to melt up to 3348C, which is about 68C higher than the equilibrium bulk T m for Pb. Repeated measurements demonstrated that a metastable superheating of the confined thin film varies in a range of 3|108C. Such a superheating phenomenon in confined thin films was reasonably attributed to suppression of growth of the molten droplet by the epitaxial Al / Pb /Al confinement, instead of suppression of melt nucleation (as in superheated nanoparticles) which may not be preventable due to various kinds of defects in the polycrystalline films and at the Pb /Al interfaces. A simple model was developed based on thermodynamic analysis of the interfacial energy conditions for the growth of the Pb liquid droplet in the confined 2D thin film, as schematically shown in Fig. 3. The degree of superheating induced by suppression of melt
growth is DT m 5 (2gsl T m cos 2u ) /DLv (p / 2 2 u ) (where D is the film thickness, Lv is the latent heat per unit volume, u is the wetting angle which depends upon the Pb /Al solid interfacial energy and gsl is the solid / liquid interfacial energy). For the 20 nm thick Pb confined films, the calculated DT m is about 2|98C depending upon the value of u, which agrees well with the experimental observations. Based on this model it is anticipated that with a reduction of the film thickness, or a decrease of the epitaxial interface energy, the extent of superheating would be increased. This has been verified by our latest measurement [26]. The finding of superheating in 2D thin films provides an important clue that superheating might be achieved alternatively by suppression of melt growth without suppression of nucleation. This is challenging our traditional understanding of the superheating phenomenon. At the same time, superheating of thin films is extremely crucial for further development and application of novel thin film materials.
6. Melting point limit It is of extreme interest to ask if there is any upper (temperature) limit for the existence of a superheated crystal. There are several models dealing with the question. Here we may put them into the following three categories. 1. By analogy to the argument of Kauzmann on the supercooled liquid-to-glass transition, an isentropic melting point (T s ) can be defined at which the entropy of a solid undergoing superheating is equal to that of the equilibrium liquid [27]. The superheated solid
Fig. 3. A schematic illustration of interfacial conditions for the growth of a Pb liquid droplet confined by Al layers. The solid / liquid interface for Pb at the equilibrium melting point is stable when an epitaxial Pb /Al interface is formed (a), while it is unstable when a high-energy Pb /Al interface exists (b) [**25].
K. Lu, Z.H. Jin / Current Opinion in Solid State and Materials Science 5 (2001) 39 – 44
cannot exist at any temperature above T s and it must undergo the melting transition below that critical temperature. It is predicted theoretically that T s could be located in the range 1.3–2.0 T m [27,28]. In order to reach this temperature, the system must be heated via an ultrafast heating rate to suppress any nucleation or growth of liquid phase which may lead to complete melting otherwise. Hence, in this sense such a catastrophic limit will be directly related to the dynamic origin of melting transitions, as in the case for supercooled liquid-to-glass transitions. 2. Similar limits exist when one puts the volume and / or rigidity into consideration. It is found that the rigidity limit is lower than both the isotropic and the isochoric limit. The rigidity melting criterion was originally developed by Born and later was found to be closely related to possible superheating (instead of normal melting) of a crystal [29]. It can be regarded as belonging to the same type of lattice instability models as proposed by Lindemann at the beginning of this century. 3. On the basis of the homogeneous nucleation theory for melting, a kinetic limit for superheated crystals was proposed [*22]. According to both the classical nucleation theory [*22] and a density functional argument [30], a superheating of about 10% can be obtained with respect to the equilibrium T m by neglecting the additional 5 to 10% of the superheating due to the elastic strain effects originating from the density difference between the liquid and the solid. This is in good agreement with the MD simulation result of the superheating in a surface-free bulk single crystal [31]. It is interesting to note that superheating by 20% of T m is in numerical accordance with the prediction based on the rigidity model (22% of T m ), that underlies some correlation between the lattice instability model and the kinetic theory. To clarify the argument we again benefit from the atomistic scale information provided by MD simulations. Our recent simulations [32,33] showed that loss of either crystallinity or, equivalently, lattice rigidity (from calculation of the temperature dependence of shear modulus), not globally but locally, acts as the key factor to induce melting for a homogeneous crystal undergoing superheating. Those thermally destabilized atoms, which satisfy Lindemann’s criterion, will also satisfy Born’s criterion, and they appear in a cooperative manner as the temperature is elevated. Melting occurs as those destabilized liquid-like clusters start to accumulate and percolate throughout the crystal. Thus, according to this picture, the instability model of either Lindemann type or Born type are basically correlated to each other and serves as the nucleation mechanism at the atomic scale. And based on such a scenario, a unified view of the melting origin can be achieved, which should be of value to narrow the different
43
opinions in regard to the prediction of the melting point limit [**32]. Other evidence for our opinion exists. Firstly, it is demonstrated by MD simulation that, the effective elastic modulus of the (310) symmetric tilt grain boundary under shear softens significantly and goes to zero at about 98% of the bulk melting point, at which point the boundary will start to melt [34]. Secondly, a very recent detection of the so-called ‘non-thermal melting’ by ultrafast X-ray diffraction from a Ge single crystal surface under heating by an ultrashort-pulse laser indicated that there exists a large difference in the melting speed between the crystal surface and the internal crystal which is undergoing transient superheating [**35]. The result indicated that homogeneous nucleation of melt should occur for superheated single crystals. Though detailed information is still lacking from the experimental side, MD simulations should be used extensively to understand the important problems concerning the structure and properties of superheated crystals and transient liquid phase growth from the solid lattice. The results also encourage us to make further possible experimental verifications of the theoretical prediction based on MD simulations.
7. Summary The work reviewed here has shown that low-dimensional materials exhibit very different melting behavior from that of conventional bulk solids. With a reduction of size into the nanometer regime, irregular variation of the melting kinetics with size appears for ultrafine particles. For the embedded nanoparticles, the melting behavior is controlled not only by the particle size but also by the particle / matrix interfaces. Melting point elevation has been observed in confined nanoparticles and in confined thin films as well, in which the epitaxial interface plays a key role. Although theoretical analysis and computer simulation investigations have deepened, to some extent, the understanding of these observations, further development along this direction is still highly needed. From a practical point of view, experimental studies of the melting process of thin films and nanowires that are finding more and more technological applications in modern industries, especially to explore effective approaches to elevate their thermal stability against melting, seem to be of great importance.
Acknowledgements The authors would like to acknowledge the financial support by the National Science Foundation of China (Grants no. 59931030, 59841004, and 59801011) and Max-Planck Society of Germany.
44
K. Lu, Z.H. Jin / Current Opinion in Solid State and Materials Science 5 (2001) 39 – 44
References Papers of particular interest, published within the annual period of review, have been highlighted as: * of special interest; ** of outstanding interest. [1] Cahn RW. Melting and the surface. Nature 1986;323:668–9. [2] Buffat Ph, Borel JP. Size effect on the melting temperature of gold particles. Phys Rev A 1976;13:2287–98. [3] Jiang Q, Shi HX, Zhao M. Melting thermodynamics of organic nanocrystals. J Chem Phys 1999;11:2176–80. [4] Daeges J, Gleiter H, Perepezko JH. Superheating of metal crystals. Phys Lett A 1986;119:79–82. [5] Zhang DL, Cantor B. Melting behavior of In and Pb particles embedded in an Al matrix. Acta Metall Mater 1991;39:1591–602. [6] Grabak L, Bohr J. Superheating and supercooling of lead precipitates in aluminum. Phys Rev Lett 1990;64:934–7. [7] Andersen HH, Johnson E. Structure, morphology and melting hysteresis of ion-implanted nanocrystals. Nucl Inst Meth Phys Res B 1995;106:480–91. [8] Chattopadhyay K, Goswami R. Melting and superheating of metals and alloys. Prog Mater Sci 1997;42:287–300. [9] Sheng HW, Ren G, Peng LM, Hu ZQ, Lu K. Superheating and melting-point depression of Pb nanoparticles embedded in Al matrices. Philos Mag Lett 1996;73:179–86. [10] Allen GL, Gile WW, Jesser WA. The melting temperature of microcrystals embedded in a matrix. Acta Metall 1980;28:1695– 701. [*11] Dash JG. History of the search for continuous melting. Rev Modern Phys 1999;71:1737–43. [12] Kwon YK, Tomanek D. Orientational melting in carbon nanotube ropes. Phys Rev Lett 2000;84:1483–6. [**13] Schmidt M, Kusche R, von Issendorff B, Haberland H. Irregular variation in the melting point of size-selected atomic clusters. Nature 1998;393:238–40. [14] Schmidt M, Kusche R, Kronmueller W, von Issendorff B, Haberland H. Experimental determination of the melting point and heat capacity for a free cluster of 139 sodium atoms. Phys Rev Lett 1997;79:99–102. [15] Sheng HW, Lu K, Ma E. Melting and freezing behavior of embedded nanoparticles in ball-milled Al-10 wt.% M (M5In, Sn, Bi, Cd, Pb) mixtures. Acta Mater 1998;46:5195–202.
[16] Saka H, Nishikawa Y, Imura T. Melting temperature of In particles embedded in an Al matrix. Philos Mag A 1988;57:895–906. [17] Sheng HW, Ren G, Peng LM, Hu ZQ, Lu K. Epitaxial dependence of the melting behavior of In nanoparticles embedded in Al matrices. J Mater Res 1997;12:119–23. [18] Zhong J, Zhang LH, Jin ZH, Sui ML, Lu K. Superheating of Ag nanoparticles embedded in Ni matrix (to be published). [*19] Jin ZH, Sheng HW, Lu K. Melting of Pb clusters without free surfaces. Phys Rev B 1999;60:141–9. [20] Xu FT, Jin ZH, Lu K. Superheating and melting behaviors of an Ag3055 cluster with a Ni coating. Sci China E 2000; in press. [21] Jin ZH, Lu K. To what extent can a crystal be superheated? Nanostruct Mater 1999;12:369–72. [*22] Lu K, Li Y. Homogeneous nucleation catastrophe as a kinetic stability limit for superheated crystal. Phys Rev Lett 1998;80:4474– 7. [23] Willens RH, Kornblit A, Testardi LR, Nakahara S. Melting of Pd as it approaches a two-dimensional solid. Phys Rev B 1982;25:290–6. [24] Jiang Q, Zhang Z, Li JC. Superheating of nanocrystals embedded in matrix. Chem Phys Lett 2000;332:549–52. [**25] Zhang L, Jin ZH, Zhang LH, Sui ML, Lu K. Superheating of confined Pb thin films. Phys Rev Lett 2000;85:1484–7. [26] Zhang L, Jin ZH, Lu K. Superheating and melting kinetics of Pb thin films confined by Al. To be published. [27] Fecht HJ, Johnson WL. Entropy and enthalpy catastrophe as a stability limit for crystalline materials. Nature 1988;334:50–1. [28] Lele S, Ramachandra P, Dubey KS. Entropy catastrophe and superheating of crystals. Nature 1988;336:567–8. [29] Tallon JL. A hierarchy for catastrophes as a succession of stability limits for crystalline state. Nature 1989;342:658–60. [30] Iwamatsu M. Homogeneous nucleation for superheated crystal. J Phys Cond Matter 1999;11:L1–5. [31] Jin ZH, Lu K. Melting of surface-free bulk single crystals. Philos Mag Lett 1998;78:29–35. [**32] Jin ZH, Lu K, Gumbsch P, Ma E. Melting mechanisms at the limit of superheating. Phys Rev Lett to be published. [33] Jin ZH, Lu K. Melting as a homogeneously nucleated process within crystals undergoing superheating. Z Metallkd 2000;91:275–9. [34] Broughton JQ, Gilmer GH. Grain-boundary shearing as a test for interface melting. Model Simul Mater Sci Eng 1998;6:87–97. [**35] Siders CW, Cavalleri A, Sokolowski-Tinten K, Toth C, Guo T, Kammler M, Horn von Hoegen M, Wilson KR, von der Linde D, Barty CPJ. Detection of nonthermal melting by ultrafast X-ray diffraction. Science 1999;286:1340–2.