  We investigate the energy dynamics of non-crystallized (melted) ions,
confined in a Paul trap. The non-periodic Coulomb interaction experienced by
melted ions forms a medium for non-conservative energy transfer from the
radio-frequency (rf) field to the ions, a process known as rf heating. We study
rf heating by analyzing numerical simulations of non-crystallized ion motion in
Paul trap potentials, in which the energy of the ions' secular motion changes
at discrete intervals, corresponding to ion-ion collisions. The analysis of
these collisions is used as a basis to derive a simplified model of rf heating
energy dynamics, from which we conclude that the rf heating rate is
predominantly dependent on the rf field strength. We confirm the predictability
of the model experimentally: Two trapped $^{40}$Ca$^{+}$ ions are
deterministically driven to melt, and their fluorescence rate is used to infer
the ions' energy. From simulation and experimental results, we generalize which
experimental parameters are required for efficient recrystallization of melted
trapped ions.
