  We study the thermodynamical properties of crystals of trapped ions which are
laser cooled to two different temperatures in two separate regions. We show
that these properties strongly depend on the structure of the ion crystal. Such
structure can be changed by varying the trap parameters and undergoes a series
of phase transitions from linear to zig-zag or helicoidal configurations. Thus,
we show that these systems are ideal candidates to observe and control the
transition from anomalous to normal heat transport. All structures behave as
`heat superconductors', with a thermal conductivity increasing linearly with
system size and a vanishing thermal gradient inside the system. However,
zig-zag and helicoidal crystals turn out to be hyper sensitive to disorder
having a linear temperature profile and a length independent conductivity.
Interestingly, disordered 2D ion crystals are heat insulators. Sensitivity to
disorder is much smaller in the 1D case.
