  This paper concerns the analytic structure of the self-consistent field
theory (SCFT) energy functional of multicomponent block copolymer systems which
contain more than two chemically distinct blocks. The SCFT has enjoyed
considered success and wide usage in investigation of the complex phase
behavior of block copolymers. It is well-known that the physical solutions of
the SCFT equations are saddle points, however, the analytic structure of the
SCFT energy functional has received little attention over the years. A recent
work by Fredrickson and collaborators [see the monograph by Fredrickson, The
Equilibrium Theory of Inhomogeneous Polymers, (2006), pp. 203-209] has analysed
the mathematical structure of the field energy functional for polymeric
systems, and clarified the index-1 saddle point nature of the problem produced
by the incompressibility constraint. In this paper, our goals are to draw
further attention to multicomponent block copolymers utilizing the
Hubbard-Stratonovich transformation used by Fredrickson and co-workers. We
first show that the saddle point character of the SCFT energy functional of
multicomponent block copolymer systems may be high index, not only produced by
the incompressibility constraint, but also by the Flory-Huggins interaction
parameters. Our analysis will be beneficial to many theoretical studies, such
as the nucleation theory of ordered phases, the mesoscopic dynamics. As an
application, we then utilize the discovery to develop the gradient-based
iterative schemes to solve the SCFT equations, and illustrate its performance
through several numerical experiments taking ABC star triblock copolymers as an
example.
