Phase Behavior of Sphere-Forming Triblock Copolymers in Films

doi:10.4236/ojapps.2012.23023

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Open Journal of Applied Sciences, 2012, 2, 163-167

doi:10.4236/ojapps.2012.23023 Published Online September 2012 (http://www.SciRP.org/journal/ojapps)

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: *[email protected]

Received July 8, 2012; revised August 5, 2012; accepted August 16, 2012

ABSTRACT

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.

H.-G. TAN ET AL.

164

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



 



nid

ln d

FkT uF





 



Vrrr



(1)

where



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





nid



includes bead-bead and bead-surface interac-

tion as well as Helfand penalty function for compressi-

bility [15]. The details of nonideal term





nid



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.

III







 (2)

where





 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





n,

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



and



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

1.1543dah



 with a the Gaussian bond length,

which is optimal ratio for DDFT approach [17]; the time

step





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

≤ AB



≤ 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

(c0).

Figure 1. Bulk isodensity profile (ρBν =0.2) for A8B4A8

triblock copolymer melts at εAB = 8 kJ/mol. In the following

ﬁgures, we use the same unit for ε and will neglect it.

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

H.-G. TAN ET AL.

166

εM

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).

H.-G. TAN ET AL.

167

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