  Phase separation has been observed by several different experiments and it is
believed to be closely related with the physics of cuprates but its exactly
role is not yet well known. We propose that the onset of pseudogap phenomenon
or the upper pseudogap temperature $T^*$ has its origin in a spontaneous phase
separation transition at the temperature $T_{ps}=T^*$. In order to perform
quantitative calculations, we use a Cahn-Hilliard (CH) differential equation
originally proposed to the studies of alloys and on a spinodal decomposition
mechanism. Solving numerically the CH equation it is possible to follow the
time evolution of a coarse-grained order parameter which satisfies a
Ginzburg-Landau free-energy functional commonly used to model superconductors.
In this approach, we follow the process of charge segregation into two main
equilibrium hole density branches and the energy gap normally attributed to the
upper pseudogap arises as the free-energy potential barrier between these two
equilibrium densities below $T_{ps}$. This simulation provides quantitative
results %on the hole doping and temperature %dependence of the degree of the
charge inhomogeneity in agreement with %some experiments and the simulations
reproduce the observed stripe and granular pattern of segregation. Furthermore,
with a Bogoliubov-deGennes (BdG) local superconducting critical temperature
calculation for the lower pseudogap or the onset of local superconductivity, it
yields novel interpretation of several non-conventional measurements on
cuprates.
