Fig. 4 Phase diagram for cubic 64 64 64 lattice. Free energy increases in
the direction of brighter color intensity. (eu,ev) = (0.05,0.05), (u,%
% v) = (0.5,0.0).
L = 1. For s = 60 we use L = 1.05 to correct for size-dependent free energy.
The system evolves 10 000 cycles and we record configurations with the
least energy. Typical morphologies are shown.
Fig. 3 (a1–a3) TEM images of a confined block copolymer. (a4) STEM
image. (b1–b3) Three-dimensional reconstructed images of nanoparticles
with PS and PI phases shown in blue and green, respectively. (a1 and a2) are
adapted from ref. 81 and (b3) is adapted from ref. 60. (c1–c4) Isosurfaces of
the numerical solution of coupled Cahn–Hilliard eqn (4) and (6) with the PS
and PI phases shown in blue and green, respectively. Simulation parameters:
(c1) s = 40, a = 0.0, tv = 1, F = 0.2551; (c2) s = 60, a = 0.0, tv = 1, F = 0.2787;
(c3) s = 60, a = 0.01, tv = 100, F = 0.2699; (c4) s = 60, a = 0.01, tv = 1,
F = 0.2780.
value of tv, which leads to a slower evolution of the microphase
separation in comparison to the speed at which the macrophase
separation takes place.
Phase diagram of confined diblock copolymers
In this section we present a systematic survey of the parameter
space and the corresponding phase diagrams, along with configuration energies for diﬀerent morphologies.
3.2.1 Phase diagram for small cubic lattices. Small systems
are important to consider because frustration eﬀects are more
noticeable in small confining enclosures. Fig. 4 shows a phase
diagram for a cubic 64 64 64 lattice, which can be considered
small enough, as strong frustration eﬀects are present in this
system size. Three realizations were used to construct this phase
diagram, each one starting from a diﬀerent initial condition,
namely, tennis, ring and layer configuration. These morphologies
are shown in Fig. 3(c2–c4). The reason behind choosing these
three initial morphologies is that they seem to appear more often
when we probe the system using different parameter sets. There
is some freedom to choose the initial conditions of the phase
diagram because the model eqn (4) and (6) guarantee that the
system will eventually settle down in its preferred state. To illustrate
this approach let us consider an example. When we initialize the
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simulation using the layer morphology as initial conditions, we
consider that this is a minimizer candidate of the system because
this morphology persists even when we initialize the simulation
from, say, multipods.
According to the survey, the most common minimizers in this
phase diagram are stacked layers (left hand side in Fig. 4) and
multipods or k-noids (right hand side). Multipods are structures
in which a component with v o 0 and having several holes,
surrounds another component with v 4 0 which has several pods
that pass through the holes of the first component. Rings can
be regarded as a 2-noid morphology. Tennis ball and onion also
appear in some regions of the phase diagram. Around the boundary
between the two regions of the phase diagram, transition from one
morphology to another one occurs gradually.
There are some unusual morphologies like bent ring (see the
top central area in Fig. 4). This deformation might be caused by
the mismatches between the characteristic length of the candidate
morphology and the system size. To avoid these mismatches, we
need to find suitable system sizes for each candidate morphology.
It is possible to evaluate the energy values of candidate morphologies
by controlling the system size at each point on the phase diagram,
i.e., finding the least energy point with respect to the system size.
This in principle could allow us to remove deformations from the
typical shape of morphologies. The cell size for this system size in
most cases is L = 1. In the case of bent ring, we assessed several
values of the cell size about L = 1. Interestingly, bent morphologies
persist even when lower values of free energy were achieved. We used
L = 1.05 in other cases to assess the size-dependent free energy,
thence we make sure that data on the phase diagram are consistent.
3.2.2 Phase diagram for medium cubic lattices. Fig. 5
shows a phase diagram for the cubic 128 128 128 lattice.
Similarly as in the previous section, three realizations were
used to construct this phase diagram, each one starting from a
diﬀerent initial condition, namely multipod, onion and stacked
layer configurations. These morphologies seem to appear more
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