Self-Consistent Field Theory and Its Applications by M. W. Matsen

4 1 Self-consistent field theory and its applications
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Linear Homopolymer
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Random Copolymer
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Diblock Copolymer
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Comb Homopolymer
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Star Homopolymer
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Ring Homopolymer
Figure 1.1: Selection of polymer architectures. Molecules consisting of a single monomer type
(i.e., A) are referred to as homopolymer, and those with two or more types (i.e., A and B) are
called copolymers.
tremendous molecular weight, the atomic details play a relative small part in their overall behavior,
leading to universal properties among the vast array of different polymer types. For
example, the characteristic size of a high-molecular-weight polymer in a homogeneous environment
scales with the degree of polymerization (i.e., the total number of monomers) to an
exponent that is independent of the monomer type. The chemical details of the monomer only
affect the proportionality constant. A further advantage of modeling polymers is that their
configurations tend to be very open, resulting in a huge degree of interdigitation among the
polymers, such that any given molecule is typically in contact with hundreds of others. This
has a damping effect on the molecular correlations, which in turn causes mean-field techniques
to become highly effective, something that is unfortunately not true of small-molecule
systems.
With the advantages favoring mean-field theory, it becomes possible to provide accurate
predictions for the equilibrium behavior of polymeric systems. However, polymers are
renowned for their slow dynamics and can remain out of equilibrium for long periods of time.
In fact, this is effectively a rule for the solid state, where both glassy and semi-crystalline polymers
become forever trapped in non-equilibrium configurations. This restricts the application
of statistical mechanics to the melt (or liquid) state, where the dynamics can be reasonably
fast. Even though there are relatively few applications of polymers in their melt state, this is
the phase in which materials are processed and so a thorough understanding of equilibrium
melts is paramount.
This Chapter provides a basic introduction to SCFT for modeling polymeric melts; for further
reading, see Whitmore and Vavasour (1995), Schmid (1998), Fredrickson et al. (2002),
and Matsen (2002a). SCFT is a theory with a remarkable track-record for versatility and
reliability, undoubtedly because of its prudent choice of well-grounded assumptions and ap-