  We inquire about the possible coexistence of macroscopic and microstructured
phases in random Q-block copolymers built of incompatible monomer types A and B
with equal average concentrations. In our microscopic model, one block
comprises M identical monomers. The block-type sequence distribution is
Markovian and characterized by the correlation \lambda. Upon increasing the
incompatibility \chi\ (by decreasing temperature) in the disordered state, the
known ordered phases form: for \lambda\ > \lambda_c, two coexisting macroscopic
A- and B-rich phases, for \lambda\ < \lambda_c, a microstructured (lamellar)
phase with wave number k(\lambda). In addition, we find a fourth region in the
\lambda-\chi\ plane where these three phases coexist, with different,
non-Markovian sequence distributions (fractionation). Fractionation is revealed
by our analytically derived multiphase free energy, which explicitly accounts
for the exchange of individual sequences between the coexisting phases. The
three-phase region is reached, either, from the macroscopic phases, via a third
lamellar phase that is rich in alternating sequences, or, starting from the
lamellar state, via two additional homogeneous, homopolymer-enriched phases.
These incipient phases emerge with zero volume fraction. The four regions of
the phase diagram meet in a multicritical point (\lambda_c, \chi_c), at which
A-B segregation vanishes. The analytical method, which for the lamellar phase
assumes weak segregation, thus proves reliable particularly in the vicinity of
(\lambda_c, \chi_c). For random triblock copolymers, Q=3, we find the character
of this point and the critical exponents to change substantially with the
number M of monomers per block. The results for Q=3 in the continuous-chain
limit M -> \infty are compared to numerical self-consistent field theory
(SCFT), which is accurate at larger segregation.

Soft Condensed Matter;Statistical Mechanics