Aside from yield point change structural element

Aside from yield point change, structural fdps refinement can also affect the melting temperature. The latter is determined experimentally for micro- and nanodisperse materials, but not as definitively as the yield point. For example, aggregate and powder metallic nanomaterials may have different melting temperatures, whereas metallic nanopowders of pure metals in an inert-gas blanket are characterized by a decrease in the melting point with a decrease in nanoparticle sizes from 20 to 1–2nm [10]. The change in the melting point is, apparently, connected to a reduction of the surface tension coefficient, since the probability of isolated atoms and molecules dislodging from the crystallite increases due to thermal motion. Additionally, the experimental dependence of the melting temperature on the surface tension coefficient is known to be linear for various substances [11]. This experimental behavior indicates that interfacial interaction in crystallites does not have a strong effect on the surface tension forces and the melting process for an average crystallite size of about 100µm. Consequently, the surface energy of crystallites can be regarded as a separate component of the internal energy of the material. The influence of surface tension forces in crystallites on the yield point of the material has not yet been investigated. However, the very experimental procedure for measuring the surface tension forces based on the zero-creep mechanism [12] proves that such an influence exists. Therefore, if the dependence of the surface energy on the crystallite size is known, it is possible to predict the yield point variations while the structure of the metal is being refined. This is a prerequisite for constructing a model that would allow predicting yield point values from the bulk density of the surface tension energy.
An estimate for the relationship between the surface tension coefficient of the particles and their size The Gibbs–Tolman–König–Buff equation [13,14] for a spherical particle has the following form: where σ is the surface tension coefficient; R is the radius of a particle; δ is the Tolman constant equal (in order of magnitude) to the surface layer thickness of a particle. In the general case, this equation is unsolvable in the explicit form, since the constant δ depends on the radius. To solve it, the condition R ≫ δ is accepted, which allows to eliminate the summands of the order . Integrating the Eq. (1) in this case leads to the following well-known formula [15]: where σ0 is the surface tension coefficient for a flat surface. Taking into account the condition R ≫ δ, it is possible to obtain another asymptotic expression for the surface tension coefficient [15]:
The solutions (2) and (3) are typically regarded as the closest to the exact one. It is these solutions that are commonly used for comparisons to the experimental data. However, there is a general solution of Eq. (1), assuming that δ does not depend on R. It can be represented in an analytical form [12]: where are the roots of a cubic equation
Fig. 1 shows the comparison of the exact solution (4) to the most frequently used approximations (2) and (3). Using the known linear relationship between the melting temperature of the material and the surface tension coefficient of the particles, and the solutions (2)–(4), we constructed a model dependence of the melting temperature on the particle size. We obtained the following expressions for the temperatures:
Fig. 2 shows these dependences for pure gold (experimental points and calculated curves). Comparing the plots shows that the exact solution (5) is good for describing the experimental data in the particle size range from 3 to 20nm. The approximate dependences, obtained by the Formulae (6) and (7), produce a substantial error only in the range from 1 to 5nm. Therefore, the exact solution (5) allows to satisfactorily describe the changes in the melting point. This result makes it possible to apply the obtained dependence of the surface tension coefficient on the particle sizes to our further calculations with complete confidence.