  Configurationally disordered crystalline boron carbide, B$_{4}$C, is studied
using first-principles calculations. We investigate both dilute and high
concentrations of carbon-boron substitutional defects. For the latter purpose,
we suggest a superatom's picture of the complex structure and combine it with a
special quasi-random structure approach for disorder. In this way, we model a
random distribution of high concentrations of the identified low-energy
defects: 1) Bipolar defects and 2) Rotation of icosahedral carbon among the
three polar-up sites. Additionally, the substitutional disorder of the
icosahedral carbon at all six polar sites, as previously discussed in the
literature, is also considered. Two configurational phase transitions from the
ordered to the disordered configurations are predicted to take place upon
increasing temperature using a mean-field approximation for the entropy. The
first transition, at 870 K, induces substitutional disorder of the icosahedral
carbon atoms among the three polar-up sites, meanwhile the second transition,
at 2325 K, reveals the random substitution of the icosahedral carbon atoms at
all six polar sites coexisting with bipolar defects. Already the first
transition removes the monoclinic distortion existing in the ordered ground
state configuration and restore the rhombohedral system (R3m). The restoration
of inversion symmetry yielding the full rhombohedral symmetry (R-3m) on
average, corresponding to what is reported in the literature, is achieved after
the second transition. The electronic density of states, obtained from the
disordered phases indicates a sensitivity of band gap to the degree of
configurational disorder in B$_{4}$C.
