Nonlinear Diffusion Kinetics In Nanometer-scale Multilayers

Due to the nanoscale modulation structure and amounts of interfaces in nanometer-scale multilayers, free energy of the multilayers is consistent with that of the nonuniform system and the diffusion coefficients of atoms depend strongly on their composition. The nonlinearity of diffusion in the nanometer-scale multilayers is manifested in the change of the range of the validity of the classical diffusion laws and in peculiar concentration distribution and its time evolution. At present, diffusion in nanometer-scale multilayers concentrates on experimental research. In the theoretical study, classical diffusion laws were used to investigate diffusion in the multilayers, however, nonuniformity of nanometer-scale multilayers was ignored. In order to study the nonlinear diffusion in nanometer-scale multilayers and further to perfect the traditional diffusion theory, nonlinear substitutional and interstitial diffusion in the nanometer-scale multilayers have been studied by the theory analysis and the experimental measurement.The correlation of the nonlinear kinetic discrete model for substitutional diffusion in a nonuniform system proposed by Martin and the classical diffusion laws has been established. In the regular solution approximation, the nonlinear kinetic discrete model can be transformed into the presences of Fick's first law, Fick's second law and Cahn-Hilliard equation. The substitutional diffusion of AB binary nanometer-scale multilayers was calculated using the nonlinear kinetic discrete model. The diffusion coefficients of atoms which strongly dependent on their local concentration were given and the diffusion asymmetry coefficient m' and the ordering energy V were introduced to characterize the nonlinear substitutional diffusion with variation of (V_Bb-V_aa), the ordering energy V, diffusion temperature and diffusion direction in the AB binary nanometer-scale multilayers. With decreasing the diffusion asymmetry coefficient m', the interfaces of nanometer-scale multilayers changed from broadening to sharpening during diffusion. When the ordering energy V decreased, the changes in concentration profile and interface structure were same for the multilayers with the same diffusion asymmetry coefficient m', but the diffusion time decreased correspondently. According to the calculated decay of intensity of low-angle x-ray diffraction peaks of Mo/V multilayers as a function of diffusion time, it is found that Fick's laws, Cahn-Hilliard equation and the nonlinear kinetic discrete model are all valid for description of diffusion for larger modulation wavelength of multilayers, but the errors of the two formers increase gradually with decreasing the modulation wavelength, only the nonlinear kinetic model is valid for description of nanoscale diffusion in nanometer-scale multilayers.A nonlinear kinetic discrete model for interstitial diffusion in a nonuniform system is presented, based on Hillert's sublattice theory and Martin's nonlinear kinetic discrete model for substitutional diffusion. The concentration-dependent correlation coefficientγand the relaxation timeÏ„are introduced to characterize the nonlinear interstitial diffusion of interstitial atoms. With increasing the concentration-dependent correlation coefficientγfrom 2.14 to 10.7 and 21.4, interface in the interstitial solid solution changed from broadening to sharpening and shifted towards the surface of solid solution with thickness of interface decreasing during diffusion. The effect of relaxation timeÏ„on the diffusion rate of solid solution has been discussed for three different matrix sublattices in the solid solution. The diffusion of interstitial carbon in the nanometer-scale multilayers of MC/M (M=3d transition metal) and TiC/Ti nanometer-scale multilayers were calculated using the model. The diffusion of carbon in the nanometer-scale multilayers was dependent on the pair interaction energies of carbon V_CC and modulation ratio r. With V_CC decreasing from -0.02 to -0.10 eV, interfaces in the multilayers changed from broadening to sharpening and shifted towards side of MC sublayer with thickness of interface decreasing. Diffusion process of carbon in the multilayers was divided into three stages in terms of modulation ratio r. Thickness evolution of titanium carbide in the TiC/Ti nanometer-scale multilayers during diffusion was relative to the modulation ratio r. The ratio of diffusion time in the stages for the TiC/Ti multilayers with same thickness ratio was found to be in square proportional to the ratio of the modulation wavelengths. The calculative diffusion process of carbon in the TiC/Ti multilayers was in agreement with the reported experimental result.An interstitial diffusion model for foreign interstitial atoms in nanometer-scale multilayers is proposed based on Pasturel's hydrogenation model in bilayer films and the chemical potential of interstitial atoms in the interstitial solid solution by sublattice theory, where Fick's second law was applied in the sublayers and chemical potential of interstitial atoms in the interfaces of the multilayers. Hydrogenation process of nanometer-scale multilayers has been calculated by the model. The effects of composition and modulation structure of multilayers, hydrogenation atmosphere and diffusion coefficient of hydrogen atoms on the hydrogenation process of the nanometer-scale multilayers have been studied. The continuity of the chemical potential for interstitial atoms in the interfaces of multilayers is the most important characteristic of the interstitial diffusion model for foreign interstitial atoms in nanometer-scale multilayers. The affinity between hydrogen and composition element of multilayers results in the uphill diffusion of hydrogen at the interfaces of multilayers and determines the concentration evolution of hydrogen in the multilayers. Fe/Ti nanometer-scale multilayers of modulation wavelength A=10-200 nm with alternating Fe and Ti sublayers thickness ratio r=1:1 were deposited by direct current magnetron sputtering in the duplex chamber chamber magnetron sputtering apparatus, the in suit hydrogenation of Fe/Ti nanometer-scale multilayers was carried out under NH3 atmosphere at the temperatures of 423 and 463 K for 0-300 min. The modulation structure, phase state and concentration profile of as-deposited and hydrided Fe/Ti nanometer-scale multilayers were studied by Small/Wide Angle X-ray Diffraction (SA/WAXRD), Rutherford Backscattering Spectroscopy (RBS), Transmission Electron Microscopy (TEM) and Secondary Ion Mass Spectrometry (SIMS). After hydrogenation, the hydrided Fe/Ti nanometer-scale multilayers kept the modulation structure as the as-deposited multilayers, with concentration of Fe, Ti and H distributed periodically in the multilayers. There was no solid state reaction at the interfaces of multilayers and no intermetallic formed after hydrogenation.α-Ti changed intoδ-TiH_2-ε andβ-TiH_x at the hydrogenation temperature of 423 K and intoδ-TiH₂ at higher temperature of 463 K, withα-Fe remaining. Concentration of hydrogen in the multilayers increased with increasing hydrogenation time, and decreased with decreasing modulation wavelength of the multilayers. The concentration profile of hydrogen calculated by the interstitial diffusion model for foreign interstitial atoms in nanometer-scale multilayers was consistent with the experimental results, verifying the validity of the interstitial diffusion model for foreign interstitial atoms.