Open Journal of Applied Sciences, 2012, 2, 163-167
doi:10.4236/ojapps.2012.23023 Published Online September 2012 (
Phase Behavior of Sphere-Forming Triblock
Copolymers in Films
Hong-Ge Tan1*, Qing-Gong Song1, Xin-Huan Yang2, Ya-Jing Deng1
1College of Science, Civil Aviation University of China, Tianjin, China
2College of Science, Hebei University of Technology, Tianjin, China
Email: *
Received July 8, 2012; revised August 5, 2012; accepted August 16, 2012
The self-assembly of sphere-forming triblock copolymers confined between two thin homogeneous surfaces is investi-
gated based on mean-field dynamic density functional theory. The morphologies deviating from the bulk sphere-form-
ing phase are revealed, including cylinders oriented perpendicular to the surface, cylinders oriented parallel to the sur-
face, perforated lamellae and lamellae by varying film thickness and surface field strength. The phase diagram of sur-
face reconstruction is also constructed. By comparing the present phase diagram with the other relevant phase diagram
for the cylinder-forming triblock copolymer film, the difference between the sphere-forming and the cylinder-forming
triblock copolymer thin film is discussed.
Keywords: Triblock Copolymer; Mean-Field Dynamic Density Functional Theory; Thin Film
1. Introduction
Block copolymer molecules are composed of two or
more chemically distinct blocks of monomer units. Since
the blocks are joined covalently, they can form micro-
scopic phase separation and are thus forced to form rich
ordered microstructures. Those ordered microstructures
could be useful in nanotechnology application. For bulk
block copolymers, these microdomains have been ob-
tained by controlling the molecules weight, molecules
architecture and interaction parameter between different
segments (blocks). When a block copolymer melt is con-
fined in the film, two additional factors have to be con-
sidered. One is the surface energy, the other is the film
thickness. In order to obtain controllable patterns on a
nanometer scale, films of block copolymers have re-
ceived considerable attention. The self-assembly of dib-
lock copolymers confined in films has been studied ex-
tensively both theoretically and experimentally [1-4].
Under confinement, diblock copolymers can not only
change the orientation, but also alternate between differ-
ent morphologies deviating from the bulk structure.
In contrast to confined diblock copolymers, phase be-
havior of confined triblock copolymers is more compli-
cated [5-9]. Stcoker et al. studied surface reconstruction
of the lamellar morphology in triblock copolymers by the
experiment [5]. Rehse et al. studied the microstructure
near the surface in lamellar ABC triblock copolymers
and observed the distorted lamellar [6]. Knoll et al. in-
vestigated the thin film of the cylinder-forming triblock
copolymers and compared the experimental and theo-
retical result [7]. Horvat et al. simulated the phase be-
havior of the cylinder-forming triblock copolymer film
[8]. They obtained the morphologies deviating from the
bulk and phase diagram of the surface reconstruction by
changing the film thickness and surface field. So far, the
sphere-forming triblock copolymer film has received
much less attention and most of the work in this aspect
focused on the packing and packing transition of the
spherical domain in the film [9]. A fundamental under-
standing of morphology reconstruction due to the con-
finement and the surface field is lacking. In this paper,
we employ a mean-field dynamic density functional the-
ory (DDFT) to study confined triblock copolymers with
spherical bulk phase. This model has been used exten-
sively in many aspects. Huinink et al. employed this
model to study asymmetric diblock copolymers with cy-
lindrical bulk phase confined in the films [3]. Horvat et
al. used this model to study the asymmetric triblock co-
polymer film with cylinder-forming bulk phase [8]. The
paper is arranged as follows: in Section 2 we simply in-
troduce the method and parameters. In Section 3, we fo-
cus on the morphologies of sphere-forming triblock co-
polymer films by tuning the surface field and film thick-
ness. Also, the corresponding discussion is given in this
section. Finally, the conclusions are presented in Section
*Corresponding author.
Copyright © 2012 SciRes. OJAppS
2. Method
For our simulation, we used the standard MESODYN
code [10]. The MESODYN is based on the mean-field
DDFT approach developed by Fraaije et al. [11-16]. In
MESODYN, the polymer melt is modeled as a com-
pressible system, consisting of Gaussian chain molecules
in a mean-field environment. In this paper, we consider a
compressible melt of n ABA triblock copolymers in a
volume V at a temperature T. The total degree of polym-
erization of the triblock copolymer is N and the degree of
polymerization of block I is fI N, where fI is the fraction
of segments on each chain belonging to type I. Thus, the
free energy F used in the MESODYN has the form
 
ln d
FkT uF
 
is the single-chain partition function for ideal
Gaussian chains in an external field uІ [12],
is the
density of the copolymer bead І and nonideal term
includes bead-bead and bead-surface interac-
tion as well as Helfand penalty function for compressi-
bility [15]. The details of nonideal term
can be
found in [3] and [14]. DDFT is used to describe the tem-
poral evolution of the system. Thus, Langevin equation is
used to describe the time evolution of the density.
is intrinsic chemical potential field
of a bead of type I, MІ is the mobility parameter of a bead
of type I.
is Gaussian noise which satisfies the fluc-
tuation-dissipation theorem[16]. In the neighborhood of
hard objects, rigid wall boundary conditions are used.
These conditions are implemented by using 0
where n is the normal point toward the wall. The same
boundary conditions are used for the noise
. Utilizing
Equation (2) and boundary conditions, these fluxes will
drive the system to a steady state, which corresponds to
the minima of free energy.
In the paper, we model the polymer film as a collec-
tion of Gaussian chain A8B4A8 with a total length N = 20.
The simulations are completed on a cubic grid of dimen-
sions X × Y × Z = 32 × 32 × H + 1 with a mesh size h (h
= 1 nm). One surface is positioned at z = 0 and the other
surface is positioned at H + 1 because of periodicity of
boundary condition. Thus, film thickness is H. The in-
teraction between different blocks is characterized by the
interaction parameter AB
in units of kJ/mol and the
interaction between blocks and interfaces is characterized
by corresponding interaction parameters AM
An effective interface-copolymer interaction parameter is
equal to M
, which characterizes the
strength of the surface field. In our case, T = 413 K, and
kJ/mol. Thus, surface field strength
. The mobility parameters were assumed to
be equal (MA = MB = M). The dimensionless parameters
in MESODYN program are chosen as: the grid parameter
with a the Gaussian bond length,
which is optimal ratio for DDFT approach [17]; the time
20.5Mt hkT
 , which is optimal value [3];
the noise scaling parameter 3100hv with v the
bead volume , which is the best numerical performance
for pure diffusive system [12,17]; and compressibility
parameter 8
, which allows small fluctuation.
Thus, the phase behavior of the ABA triblock copolymer
film is determined by A-B bead interaction parameter
, surface field strength
and the film thickness H.
In this paper, the unit of
is kJ/mol and we neglect it
from now on. As in [3] and [8], we follow the temporal
evolution in the system and stop it when both the free
energy and the order parameter do not change signifi-
cantly. The average simulation time is 5000 dimension-
less time steps and the stability of some nonperfect struc-
tures is checked by continuing simulations till 20,000 or
more time steps.
3. Results and Discussion
In order to guarantee that the system is sphere-forming
phase in the bulk, the simulation is done with MESODYN
in a cubic box with 32 × 32 × 32 grids and periodic
boundary condition. We find that B-rich domains of
spherical shape form in an A-rich matrix in the range 7.4
8.4 in the bulk. Figure 1 shows the isodensity
profile of the B-beads for AB
= 8, which is fixed
throughout this paper for discussion. The averaged dis-
tance between subsequent layers of spheres along z axis
is estimated as c0 = 6 - 7 grid units by MESODYN.
Now, triblock copolymers with spherical phase are put
into the thin film. As mentioned above, the phase behav-
ior of the confined ABA triblock copolymer film is con-
trolled by two parameter
and H. Considering the
effect of the surface field, we fix film thickness with H =
12, which is compatible with natural bulk domain space
Figure 1. Bulk isodensity profile (ρBν =0.2) for A8B4A8
triblock copolymer melts at εAB = 8 kJ/mol. In the following
gures, we use the same unit for ε and will neglect it.
Copyright © 2012 SciRes. OJAppS
H.-G. TAN ET AL. 165
By varying the surface field strength from 4 to 25, we
obtain the surface structures deviating from the bulk
structure, as shown in Figure 2. For εM 4, component B
is weakly attracted to the surface and spheres touching
the surface are formed (see Figures 2 and 3).
Huinink et al. [3] and Wang et al. [4] have found that
neutral wall has weak preference for the short block be-
cause of purely entropic effect. We think that the surface
can be regarded as neutral in the vicinity of εM = 4, be-
cause the surface attracts B component weakly in the
vicinity of εM = 4. The fact that the neutral wall is not at
εM = 0 has also been observed in [3] and [8]. With the
increase of the surface field strength, component B is
weakly repelled by the surface and then spheres (S) not
connected to surface are formed at εM 5 (see Figures 2
and 3). Furthermore, the spheres (S) are somewhat com-
pressed at εM = 5, compared with those at εM = 4. Besides,
the array of spheres is hexagonal at εM 4 and εM 5.
Continuing to increase the surface field strength, we find
that cylinders oriented parallel to the surface (C||) appear.
Cylinders are compressed with the increase of the surface
field and then convert to the perforated lamellae (PL) at
εM 20. Finally, the perforated lamellae (PL) convert to
the lamellae (L) under very large surface field. The ap-
pearance of these surface reconstructions mentioned in
turn is because that surface-field additivity leads to the
decrease of the averaged mean curvature.
Considering the effect of the confinement, we choose
surface field strength at εM = 4, at which the surface can
be regarded as neutral. By varying the film thickness
from 6 to 18, we obtain the confinement-induced struc-
tures, as shown in Figure 4. We find the bulk sphere-
forming phase is kept for some film thicknesses which
are approximately compatible with natural bulk domain
space such as H = 6, 7, 11, 12, 13, 14, 17 and 18. One
layer of spheres is formed for H = 6 and 7, two layers of
spheres are formed for H = 11 - 13 and three layers of
spheres are formed for H = 17 and 18. When the film
thickness is very incompatible with natural bulk domain
space such as H = 8, 9, 10, 15 and 16, the bulk
sphere-forming phase is frustrated. Moreover, cylinders
oriented perpendicular to the surface (C) and touching
the surface are formed. Here, we define the length of
perpendicular cylinder is more than or equal to the length
in H = 8 nm, which is an artificial definition (we choose
length in H = 8 as criterion of perpendicular cylinder).
Comparing Figures 2 and 4, we find that surface field
only induces the cylinders oriented parallel to the surface
(C||), perforated lamellae (PL) and lamellae (L) in
sphere-forming triblock copolymer thin film. Cylinders
oriented perpendicular to the surface (C) are induced by
incommensurability between the thin film thickness and
the bulk period of the triblock copolymers.
Figure 2. Three-dimensional isodensity profile (ρBν = 0.2)
for A8B4A8 triblock copolymer film with H = 12 at εM = 4, 5,
7, 19, 20, and 30. Solid triangle () represents the parallel
cylinder; solid () and hollow () boxes represent the la-
mella and the perforated lamella respectively, while solid
sphere () represents the spherical phase.
εM = 4 εM = 5
Figure 3. Isodensity profile of the component B where the
mask is added at εM = 4 and 5 as a side view of those in Fig-
ure 2.
Figure 4. Three-dimensional isodensity profile (ρBν = 0.2)
for A8B4A8 triblock copolymer film at εM = 4 with H = 6, 8,
9, 10, 11, 15, 16 and 18. The solid inverse triangle ()
represents the perpendicular cylinder. The definitions of
other symbols are the same as those in Figure 2.
In order to study the cooperating effect of the con-
finement and the surface field on structures, we simulate
the phase behavior of triblock copolymers at different
surface fields (4 εM 40) under different film thickness
(6 H 14) and draw the phase diagram of surface re-
construction as shown in Figure 5.
From the phase diagram, we can obtain more informa-
tion on surface reconstructions. First, spheres (S) do not
appear in case of H = 8 and 9 at all the surface fields.
Furthermore, the range of perpendicular cylinders (C)
exist is wider for H = 8 and 9 than that for other film
Copyright © 2012 SciRes. OJAppS
Figure 5. Phase diagram of surface reconstructions, i.e. film
thickness H versus surface field strength εM. The half hol-
low box indicates that perforated lamella and lamella coex-
ist in the surface layer. The half hollow triangle indicates
that spheres and parallel cylinders coexist in the surface
layer. The definitions of other symbols are the same as those
in Figure 2 and Figure 4.
thicknesses. These phenomena might relate to the in-
commensurability of sphere layer period of c0 with the
film thickness. Second, perforated lamellae (PL) do not
appear in thin film not only at H = 9 but also at H = 10.
As a result, the pathway from parallel cylinders to lamel-
lae at H = 9 and H = 10 is different from that at other
film thicknesses. The intermediate state from parallel
cylinders to lamellae is undulated lamellae (connected
parallel cylinders) for H = 9 and 10, while the intermedi-
ate stable state is perforated lamellae (PL) for other film
thicknesses. Third, the critical surface field required to
induce a surface reconstruction with nonspherical phase
(C||, PL and L) increases with the film thickness which
are the integer multiple of a nature layer thickness by
comparing H = 6 (c0) and H = 12 (2c0). Lamellae (L),
perforated lamellae (PL) and parallel cylinders (C||) are
observed at εM 15, εM 10 and εM 6, respectively, for
the thickness of H = 6, while the phases are observed
correspondingly at εM 27, εM 20 and εM 7 for the
thickness of H = 12. It indicates that the strength of sur-
face field needed to form these microstructures in thin
film is smaller than that in thick film because of the in-
terference effect of two surface field strengths which is in
agreement with the conclusion obtained by DDFT in thin
film of cylindrical phase [8]. Fourth, the surface fields εM,
where the lamella (L) is formed at H = 8, is stronger than
that for other thicker films. In general, the strength of
surface field needed to form lamellae in thin film is
smaller than that in the thick film. However, we find that
one lamella is formed for H = 8 and the width of the la-
mella is very wider for H = 8 than for other thicker films,
which leads to the chain stretching and the decrease of
conformation entropy for H = 8. Thus, the surface field
needs to be strong enough to compensate for unfavored
entropic decrease. Finally, there are some coexistence of
two phases, for example, L and PL, and C|| and S as well.
By coexistence of two phases, we can learn how struc-
ture evolves from one phase to another phase.
The phase behavior of sphere-forming tirblock co-
polymer slit is analogous to that of cylinder-forming
triblock copolymer slit studied by Horvat et al. [8]. First,
the surface reconstructions in sphere-forming triblock
copolymer slit are found in cylinder-forming triblock
copolymer slit except for spherical phase. It has been
proved theoretically that sphere-to-cylinder transition
will occur under an external field (confinement, shear,
electric field) and hence the transitions due to confine-
ment in cylinder-forming systems can be seen as a subset
of the transitions observed in sphere-forming systems.
Second, Cylinders oriented perpendicular to the surface
(C) are observed at the certain thickness which is in-
compatible with natural bulk domain space for both sys-
tems. However, there is an important difference in both
systems. For cylinder-forming systems in slit [8], the
stable surface reconstructions like C||, PL and L are found
to be nested into each other to respond to the incom-
mensurability. Apparently, the situation for spheres is
different. The difference between sphere-forming and
cylinder-forming systems may be due to the enhanced
flexibility of the sphere-forming system to respond to
incommensurability between the film thickness and the
domain distance.
4. Conclusion
In summary, the effects of the film thickness and the
surface field strength on the phase behavior of the
sphere-forming copolymer film are investigated. The
surface field can induce spheres with hexagonal packing,
parallel cylinders, perforated lamellae and lamellae near
the surfaces. Cylinders oriented perpendicular to the sur-
face is induced by incommensurability between the thin
film thickness and the bulk period of the triblock co-
polymers. When incommensurability cooperates with the
surface field, the phase behavior of sphere-forming tri-
block copolymer film becomes more complex. Phase
diagram of surface reconstruction is constructed and
various morphologies are predicted. Comparing the pre-
sent phase diagram with other relevant phase diagram for
the cylinder-forming triblock copolymer film, we find
that sphere-forming system in a slit is more flexible to
respond to the incommensurability.
5. Acknowledgements
We appreciate the financial support from the Fundamen-
tal Research Funds for the Central Universities (ZXH
2009D010) and Scientific Research Foundation of Civil
Aviation University of China (No. 05qd05s).
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Copyright © 2012 SciRes. OJAppS
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