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Description of the Oxygen Order evolution and its relation to the Superconducting Transition in $La_2CuO_{4+y}$

The segregation of oxygen in the high critical temperature cuprate superconductor $La_2CuO_{4+y}$ has been systematically studied along the years. In a recent set of experiments, Poccia et al related, for the first time, time ordering ($t$) of oxygen interstitials with the corresponding superconducting transition temperature $T_c(t)$. We develop a phenomenological description of the time ordering forming pattern domains and show how it may affect the superconducting interaction. The superconducting self-consistent calculations in a system with electronic granular structure of varying hole doping yields also different local d-wave amplitudes. These amplitudes are of the order of magnitude of scanning tunneling microscopy measurements and they vanish at $T^*(t)> T_c(t)$. Then, calculations with Josephson coupling among the isolated charge domains reveal that the superconducting interaction is likely to be scaled by the local free energy and capture the details of $T_c(t)$. The accurately reproduction of these apparently disconnected phenomena establishes routes to the important physical mechanisms involved in the connection between sample production and on the origin of the superconductivity of cuprates.