The three-dimensional equilibrium crystal shape of Pb: Recent results of theory and experiment

Abstract

The three-dimensional equilibrium crystal shape (ECS) is constructed from a set of 35 orientation-dependent surface energies of fcc Pb which are calculated by density functional theory in the local-density approximation and distributed over the [110] and [001] zones of the stereographic triangle. Surface relaxation has a pronounced influence on the equilibrium shape. The (111), (100), (110), (211), (221), (411), (665), (15,1,1), (410) and (320) facets are present after relaxation of all considered surfaces, while only the low-index facets (111), (100) and (110) exist for the unrelaxed ECS. The result for the relaxed Pb crystal state is in support of the experimental ECS of Pb at 320-350 K. On the other hand, approximating the surface energies of vicinal surfaces by assuming a linear relationship between the Pb(111) first-principles surface energy and the number of broken bonds of surface atoms leads to a trivial ECS that shows only (111) and (100) facets, with a sixfold symmetric (111) facet instead of the correct threefold symmetry. It is concluded that the broken bond rule in this simple linear form is not a suitable approximation for obtaining the proper three-dimensional ECS and correct step formation energies.PACS 05.70.Np; 61.50.Jr; 68.35.Md; 71.15.Mb IntroductionThe surface free energy is a fundamental property of solids and liquids. For crystalline solids it is generally anisotropic, i.e., it depends on the orientation of the surface. In this case the surface free energy is particularly difficult to measure, especially for structurally and chemically well-defined surfaces [1][2][3][4]. Hence it is not surprising that many theoretical attempts have been undertaken to generate hopefully trustworthy values of this quantity. Some were empirical, starting from surface free energies of liquids (e.g., metals) and transposing them to the solid state [5,6]. Other semi-empirical studies have been summarized by Galanakis et al. [7]. More recently, full potential first principles calculations have produced surface energies of many crystalline materials [7][8][9][10][11][12][13][14][15][16][17][18] mostly for low-index orientations. Although there can be considerable scatter in theoretical values for a single material and orientation [16,19], it has been suggested by u Fax: +49 2461 61 2950, E-mail: h.bonzel@fz-juelich.de some authors that the lack of experimental surface energy data for well-defined orientations may be replaced by ab initio calculations [15,17,20].Theoretical attempts of describing the more or less complete anisotropy of the surface free energy of crystals by invoking attractive pairwise atomic interaction potentials have been numerous, beginning in 1931 with the work by Stranski et al. [21][22][23][24][25][26] followed by others [27][28][29][30]. A different phenomenological description of the anisotropy and to some extent of the temperature dependence of the surface free energy was based on the terrace-step-kink model of a crystalline surface [31] assuming the existence of steps of monoat...

Metallic nanocrystals synthesized in solution: a brief review of crystal shape theory and crystallographic characterization

“…Attempting to perform molecular dynamics calculations using DFT will presently require the usage of very thin slabs, leading to errors in the surface energy. Conversely, the Wulff construction can be used to extract surface energies at finite temperature from experimental measurements of crystal shapes . Figure shows the differences between the surface energies extracted from experiments at finite temperatures and those computed using DFT at 0 K.…”

Formation of Facets for an Effective Model of Crystal Growth

“…Namely, for the few initial values of k = 1, 2, ..., k c the size c k N of the k-th monolayer at the moment of its creation is strictly smaller than the size of the underlying (k − 1)-th layer. Thus, the picture is qualitatively the same as that of the lead crystal -there is an extensive physical literature on the latter subject, for instance see Figure 2 in [5] and discussions in [21,6] However, for k > k c this is not the case any more, and the size c k N is exactly the same as the size of the underlying (k − 1)-th layer at the creation moment.…”

The Microstructure in an Al–Ti Alloy Melt: The Wulff Cluster Model from a Partial Structure Factor

“…All the calculations in this paper are performed by the Vienna ab initio simulation package (VASP) [32,33], based on the density functional theory (DFT) [34,35]. The exchange correlation functional was described through the generalized gradient approximation (GGA) with the Perdew, Burke, and Ernzerhof (PBE) parameter, which has already been proved to give a reasonable description of metal systems [36,37]. The ionic cores were represented by projector augmented wave (PAW) potentials [38,39].…”