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

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