Formation of glass-metal metamaterials via reactive diffusion: A model

Diffusion of hydrogen in metal-doped glasses leads to the reduction of metals and to the growth of metallic nanoparticles in the glass body that allows the formation of metamaterials. The nanoparticles grow due to the supersaturation of the glass matrix by neutral metals, whose solubility in glasses is low compared to initial concentration of metal ions. In some cases, these metallic nanoparticles are self-arranging to quasi-periodic layered structure. A theoretical analysis of the reactive hydrogen diffusion accompanied by the interdiffusion of protons, metallic ions and neutral metals allowed us to study the temporal evolution of the average size of the metallic nanoparticles and their spatial distribution. The developed model of the formation of metallic nanoparticles defines range of parameters providing the formation of layered structures of metallic inclusions in silver and copper doped glasses. The layered structure arises at relatively low supersaturation of the diffusion zone by a neutral metal as the result of the competition of the enrichment of the glass by neutral metal atoms via reducing of metal ions by diffusing hydrogen and the depletion of the glass by the metal atoms caused by their diffusion to the nanoparticles. The results of numerical calculations are compared with the data of optical spectroscopy of the glass-metal metamaterials containing silver and copper nanoparticles.