  A two-phase model, where the plasma expansion is an isothermal one when laser
irradiates and a following adiabatic one after laser ends, has been proposed to
predict the maximum energy of the proton beams induced in the ultra-intense
laser-foil interactions. The hot-electron recirculation in the ultra-intense
laser-solid interactions has been accounted in and described by the
time-dependent hot-electron density continuously in this model. The dilution
effect of electron density as electrons recirculate and spread laterally has
been considered. With our model, the scaling laws of maximum ion energy have
been achieved and the dependence of the scaling coefficients on laser
intensity, pulse duration and target thickness have been obtained. Some
interesting results have been predicted: the adiabatic expansion is an
important process of the ion acceleration and cannot be neglected; the whole
acceleration time is about 10-20 times of laser pulse duration; the larger the
laser intensity, the more sensitive the maximum ion energy to the change of
focus radius, and so on.
