  Whereas entropy can induce phase behavior that is as rich as seen in
energetic systems, microphase separation remains a very rare phenomenon in
entropic systems. In this paper, we present a density functional approach to
study the possibility of entropy-driven microphase separation in diblock
copolymers. Our model system consists of copolymers composed of freely-jointed
slender hard rods. The two types of monomeric segments have comparable lengths,
but a significantly different diameter, the latter difference providing the
driving force for the phase separation. At the same time these systems can also
exhibit liquid crystalline phases. We treat this system in the appropriate
generalization of the Onsager approximation to chain-like particles. Using a
linear stability (bifurcation) analysis, we analytically determine the onset of
the microseparated and the nematic phases for long chains. We find that for
very long chains the microseparated phase always preempts the nematic. In the
limit of infinitely long chains, the correlations within the chain become
Gaussian and the approach becomes exact. This allows us to define a Gaussian
limit in which the theory strongly simplifies and the competition between
microphase separation and liquid crystal formation can be studied essentially
analytically. Our main results are phase diagrams as a function of the
effective diameter difference, the segment composition and the length ratio of
the segments. We also determine the amplitude of the positional order as a
function of position along the chain at the onset of the microphase separation
instability. Finally, we give suggestions as to how this type of
entropy-induced microphase separation could be observed experimentally.
