Size-dependent melting of matrix-embedded Pb-nanocrystals
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Size-dependent melting of matrix-embedded Pb-nanocrystals 1
H. Ehrhardt,1 J. Weissmüller1,2 and G. Wilde1 Institut für Nanotechnologie , Forschungszentrum Karlsruhe, Karlsruhe, Germany 2 Technische Physik, Universität des Saarlandes, Saarbrücken, Germany
Abstract We report calorimetric data for the size-dependence of the melting temperature as well as the enthalpy and entropy of melting for nanoscale Pb particles in an Al matrix, prepared by high energy ball-milling. The results are discussed with respect to various models for the melting of small confined systems. We can rule out models based on a temperature-independent Gibbs excess free energy of the particle-matrix interface, and a model based on an increased meansquare displacement of the interfacial layer. The best agreement to the data is provided by modeling the interface as an inert layer of finite thickness, which does not participate in the phase transition. Introduction It is well known that a decrease of the size D of a particle leads to a shift of the melting temperature, TM, [1-4], both for free and for matrix-isolated particles [5-7]; a recent review is given in Ref. [8]. There is as yet no single generally accepted model for the size dependence, and the individual models yield different predictions for the variations of measurable quantities, such as the temperature and enthalpy of melting, on the particle size. Since Pb has a conveniently low bulk melting point and is insoluble in Al, size-effects on the melting of Pb nanoparticles in Al has been repeatedly studied. Al-Pb nanocomposites have been prepared by melt spinning [9] and by high energy ball milling [10], the former technique leading to an increase of TM with decreasing size, whereas the latter results in a decrease with decreasing size. More recently, a melting point increase was also reported for Pb in Al-Pb multilayers [11]. Besides TM, calorimetry measures the heat of melting, and provides estimates for the entropy of melting. The proposed models make different predictions on the size-dependence of these quantities, and the purpose of our paper is to assess the models by comparing their predictions to experimental data. The data were obtained by a calorimetric investigation of the melting of Pb particles, with sizes ranging from 2 to 100 nm, embedded in an Al matrix by means of high energy ball milling. Theory We denote by g, h and s the free energy, enthalpy, and entropy, respectively, of the Pb particles per volume of Pb (e.g., total enthalpy over total volume); the subscript ‘0’ refers to the SURSHUWLHVRIWKHEXONPDWHULDOLQWKHDEVHQFHRILQWHUIDFHVDQGZHGHQRWHE\
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hM denotes the volumetric latent heat of melting, liquid minus solid). Formally, it is always possible to decompose the actual values of g, h, and s into the values for the bulk material, g0, h0, and s0, and the respective Gibbs excess quantity per area, A, of interface, {G}, {H}, and {S}. For instance,
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