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carried out in the lipid-water system, however, would lead to a transition from
lamellar to Ia3d, and thus, with an increasing curvature (see Fig. 1.1). This
sequence, also refered to as “anomalous,” is a signature of complexity of the
lipid-water system and illustrates well why attempts at modeling the phase
diagram in these complex fl uids has been very unsuccessful compared to other
systems such as block copolymers.
1.3.1
Self-Consistent Field Theory
Self - consistent fi eld theory (SCFT) is a coarse-grained fi eld - theoretic model
solved within the mean-fi eld approximation (De Gennes 1969; Edwards, 1965).
This method has been successfully applied to many different examples of
polymer systems (Fredrickson et al., 2002; Matsen and Bates, 1996; Matsen and
Schick, 1994). In the case of diblock copolymers, the agreement found between
theory (Matsen and Bates, 1994) and experiments (Khandpur et al., 1995) is
remarkable. This shows that SCFT is a reliable enough framework for describ-
ing the equilibrium self-assembly of complex fl uids that possess structured
lyotropic and thermotropic phases.
A mesoscopic fi eld-based polymer model is often derived from a particle-
based model, which uses positions and momenta of particles as the degree of
freedom while a fi eld-based model utilizes continuous fi elds. Once a particle-
based model is made, there are formally exact methods for transforming it into
afi eld-based model, and by this particle-to-fi eld transformation, a many-
interacting-chain problem is decoupled into several single-chain problems in
the presence of the external fi elds. The result of the transformation is func-
tional integrals with regard to several fi elds. Since these functional integrals
for any nontrivial model do not have an analytic solution in closed form,
numerical methods have to be used (Fredrickson et al., 2002). The main idea
behind SCFT prediction of phase diagrams is that the particular profi le of
segmental densities that minimizes the total Hamiltonian of the system
expressing the total free energy also corresponds to the equilibrium structure
that should occur experimentally.
For the purpose of illustration, we introduce the SCFT formalism in the
context of a blend of AB diblock copolymer and A homopolymer using the
continuous Gaussian chain model. This system has some common features
with lipid-water mixtures since a lipid has the same structure and role as an
AB diblock copolymer, notwithstanding the shorter chain length, presence of
unsaturated bonds, and rigid head group. Similarly, an A homopolymer can be
viewed as a “solvent” for the A blocks of the copolymer, analogous to the role
that water plays in lipid-water systems. A summary of the SCFT equations for
this system is given in an earlier work (Mezzenga et al., 2006).
The results of SCFT on such a model system indicate that all the phases—
lamellar, hexagonal, and bicontinuous—can be recovered by the theory and
found to correspond to energy minima (e.g., equilibrium morphologies).
Unfortunately, however, their locations in the phase diagram do not corre-
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