Mathematical Model of Nanostructure Formation in Binary Alloys at Electron Beam Treatment

The paper presents a thermoconcentration and capillary mathematical model, describing the formation of 10 to 100 nm structures in the surface layers of binary alloys irradiated by low-energy, high-current electron beams of submillisecond duration. The model is studied by the example of “ferrum - carbon” and “titanium – carbon” systems. It comprises Navier-Stokes equation, thermoconductivity and diffusion equations, as well as surface kinematic and dynamic boundary conditions. The effect of electron beam on material is specified as various temperature and concentration gradients. A dispersion equation for thermocapillary waves in nanowavelength range is developed and analyzed with thin layer approximation. The critical wavelength, leading to this instability, is revealed. It is found out that its values are 17.39 nm for Fe-С and 69.7 nm for Ti-C at the depth of penetration ~ 10-5 m. Wavelengths are compared to the dimensions of crystallization cells and structures, which are formed in them. The paper shows that the model provides a rational explanation of registered regularities.