Theoretical calculation of dense random packings of equal and non-equal sized hard spheres applications to amorphous metallic alloys

Abstract

A model describing the structure of amorphous metallic alloys is proposed using a packing of non-equal-sized hard spheres. Models containing up to 1000 spheres were generated by computation. Hypotheses guiding the calculation are chosen in order to obtain a model as compact as possible and to take into account affinity between metal and metalloids: therefore two small spheres are not allowed to be in hard contact. This method of calculation is first tested by building up heaps of equal-sized spheres. Both the radial distribution function and the interference function are calculated for each model and the variations of these curves with the number of spheres are studied. In the case of two sizes of spheres heaps calculations are devoted to the study of the so-called ‘shoulder interference function’ which characterizes many amorphous alloys such as NiP, PdSi, etc. It is shown that the existence of such a shoulder depends on two parameters: the ratio of sphere diameters and the relative concentration of small and large spheres. These results are in good agreement with some recent experimental observations. This shoulder is well-marked when the diameter ratio reaches 10% and when the concentration in small spheres lies in the range 10–15%. In such a structure (e.g. amorphous NiP and PdSi alloys) it is observed that small spheres are always surrounded by nine large spheres such as P or Si atoms in crystalline Ni 3P or Pd 3Si. Large spheres, i.e. metallic atoms, form distorted icosahedra, the distortion of which varies from one to the other. The short-range order has a five-fold symmetry which is not compatible with any long-range order.